http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Redston&feedformat=atomSoft-Matter - User contributions [en]2022-05-21T12:47:52ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24331Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:29:41Z<p>Redston: </p>
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<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
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==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
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==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], [[thermodynamics]], [[interfaces]], [[contact angle]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
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==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
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==Equilibrium Shape of the Droplet==<br />
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Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
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Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
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To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
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Minimization gives the conditions:<br />
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<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
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<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
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Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
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<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
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Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
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<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
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which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
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[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
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Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
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This is the pressure difference between the liquid and vapor across the curved interface.<br />
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==Curvature of the Substrate==<br />
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After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
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The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
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The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
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<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
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For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
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A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
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[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
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The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
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<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
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where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
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This translates into an average curvature of<br />
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<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
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The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
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[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
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When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
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The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
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<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
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Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
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==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on solid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
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This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the resulting substrate curvature in depth. This seems like it would be an important thing to consider for thin solid films, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Contact_angle&diff=24330Contact angle2012-04-21T19:29:24Z<p>Redston: </p>
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<div>''Edited by Pichet Adstamongkonkul, AP225, Fall 2011''<br />
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==Introduction==<br />
<br />
[[image:Contact_angle.png]]<br />
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The contact angle is a quantitative measure of the wettability of the surface, represented by the angle at which a liquid or vapor interface makes with a solid surface.[1] The angle is specific and determined by the interactions across the three interfaces. Typically, the contact angle is illustrated by a drop of liquid on the surface. The shape of the drop is governed by the Young-Laplace equation (contact angle is incorporated as a boundary condition of the equation.) Normally, the contact angle can be measured using the so-called''goniometer''.[2]<br />
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The contact angle is independent of geometry and hence a material property. Recent publications on contact angles on deformable surfaces can be included.<br />
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==Connection to Capillarity/Wettability==<br />
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The figure (from the lecture) illustrated different contact angles corresponding to different wettability of the surface, which depends on the relative hydrophobicity/ hydrophilicity of the surface compared to the liquid. Conventionally, the contact angle is measured as the angle between the solid surface and the liquid drop surface. In other words, the contact angle is the angle between the solid-liquid interfacial (surface) force (denoted as gamma_SL) and the surface tension or the liquid-vapor interfacial force (denoted as gamma_LG). The larger the angle, the more the drop is repelled from the surface, indicating a relatively higher hydrophobicity of the surface, in the case of water drop.<br />
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[[image:surface_tension.png]][[image:contact_angle.jpg]]<br />
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If the liquid strongly attracts to the surface, the drop of the liquid would spread out on the solid surface. On highly hydrophobic surfaces, the contact angle can be as big as <math>-120^o</math>. However, materials with high degree of roughness on the surface can increase the angle up to <math>-150^o</math>; the materials in this group are called [[superhydrophobic surfaces]].[2]<br />
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From the surface tensions at all three interfaces, we can explicitly write the Young equation that the system must satisfy at equilibrium:<br />
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<math>0=\gamma_{SG}-\gamma_{SL}-\gamma_{LG}cos(\theta_c)</math> where <math>\theta_c</math> is the contact angle.[3]<br />
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In the capillary effect, the driving force that causes water to go up the capillary is the net surface tension, balancing between the solid-vapor interfacial tension that pulls in the upward direction , and the solid-liquid interfacial tension that pulls downward.<br />
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This is commonly stated but is incorrect. The Young-Dupre equation resolves all the inbalances in energy - no unresolved force is left to cause a lift. Capillary rise is due to Laplace pressures.<br />
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Examples of surfaces where the contact angles play an important role:<br />
*Lotus leaf: superhydrophobic surface that causes the water droplet to roll over the surface without “wetting” the surface.[4] <br />
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[[image:lotus_leaf.jpg]]<br />
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*Human cornea: an extremely hydrophobic surface and, together with hydrophilic tears, maintain the lachrymal layer.<br />
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*Modified surfaces with enhanced hydrophilicity, via plasma treatment<br />
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This entry needs references to advancing and receding contact angles.<br />
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==References==<br />
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[1] Lecture on Capillarity and Wetting, AP225 Fall 2011<br />
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[2] Wikipedia contributors. “Contact angle.” Wikipedia, The Free Encyclopedia. 27 Nov 2011.<br />
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[3] Robert J. Good. “Contact angle, wetting, and adhesion: a critical review.” Journal of Adhesion Science and Technology. 6.12 (1992): 1269-302.<br />
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[4] Wikipedia contributors. “Lotus Leaf.” Wikipedia, The Free Encyclopedia. 27 Nov 2011.<br />
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==Keyword in references:==<br />
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[[Contact angle associated with thin liquid films in emulsions]]<br />
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[[Dewetting-Induced Membrane Formation by Adhesion of Amphiphile-Laden Interface]]<br />
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[[Surface-Tension-Induced Synthesis of Complex Particles Using Confined Polymeric Fluids]]<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Interfaces&diff=24329Interfaces2012-04-21T19:27:35Z<p>Redston: </p>
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<div>Entry needed.<br />
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== Keyword in references: ==<br />
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[[Surface-Tension-Induced Synthesis of Complex Particles Using Confined Polymeric Fluids]]<br />
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[[Thermodynamics of Solid and Fluid Surfaces]]<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Emily_Redston&diff=24328Emily Redston2012-04-21T19:27:09Z<p>Redston: </p>
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<div>Wiki entries:<br />
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Fall 2011<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
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! width=250 | Topic<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
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|- valign = "left" align = "left"<br />
| ''1 - General Introduction''<br />
| [[Biofilms as complex fluids]]<br />
| [[biofilms]], [[colloids]], [[polymers]], [[gels]], [[viscoelasticity]], [[elasticity]], [[cross-linking]], [[volume fraction]]<br />
|- valign = "left" align = "left"<br />
| ''2 - Surface Forces''<br />
| [[Crystalline monolayer surface of liquid Au–Cu–Si–Ag–Pd: Metallic glass former]]<br />
| [[metallic glasses]], [[crystal structure]], [[surface freezing]], [[liquid alloys]], [[surface crystals]], [[phase transition]], [[eutectics]]<br />
|- valign = "left" align = "left"<br />
| ''3 - Capillarity''<br />
| [[Diffusion through colloidal shells under stress]]<br />
| [[encapsulation]], [[colloids]], [[osmotic pressure]], [[diffusion]], [[emulsification]], [[permeability]]<br />
|- valign = "left" align = "left"<br />
| ''4 - Polymers and Polymer Solutions''<br />
| [[The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures]]<br />
| [[polydisperse]], [[monodisperse]], [[morphology]], [[colloids]], [[gels]], [[phase separation]], [[polymers]], [[non-adsorbing]], [[volume fraction]], [[poroelastic]], [[transient gelation]], [[spinodal decomposition]]<br />
|- valign = "left" align = "left"<br />
| ''5 - Surfactants''<br />
| [[Bacteria Pattern Spontaneously on Periodic Nanostructure Arrays]]<br />
| [[biofilms]], [[self-assembly]], [[nanoposts]], [[Fourier transform]], [[fluorescence microscopy]], [[ordering]], [[periodicity]]<br />
|- valign = "left" align = "left"<br />
| ''6 - Equilibria and Phase Diagrams''<br />
| [[David Turnbull (1915-2007). Pioneer of the kinetics of phase transformations in condensed matter]]<br />
| [[metallic glasses]], [[eutectics]], [[phase transition]], [[liquid undercooling]], [[liquid structure]], [[crystal structure]], [[free-volume model]], [[viscosity]], [[glass-transition temperature]]<br />
|- valign = "left" align = "left"<br />
| ''7 - Charged Interfaces''<br />
| [[Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model]]<br />
|[[beads-on-a-string model]], [[electrostatic interactions]], [[polymers]], [[polymer folding]], [[RNA]], [[triboelectricity]], [[self-assembly]]<br />
|- valign = "left" align = "left"<br />
| ''10 - Foams and Emulsions''<br />
| [[Mechanical Inhibition of Foam Formation via a Rotating Nozzle]]<br />
| [[bubbles]], [[capillarity]], [[confined flow]], [[drops]], [[foams]], [[nozzles]], [[rotational flow]], [[foam-suppression]]<br />
|}<br />
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Spring 2012<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
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|- valign = "left" align = "left"<br />
| [[Five-Fold Symmetry in Liquids]]<br />
| [[liquid structure]], [[five-fold symmetry]], [[x-ray scattering]], [[hard sphere]], [[dense random packing]]<br />
|- valign = "left" align = "left"<br />
| [[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]<br />
| [[omniphobicity]], [[biomimetics]], [[self-healing]], [[surface texture]], [[SLIPS]], [[liquid-repellent surface]]<br />
|- valign = "left" align = "left"<br />
| [[Thermodynamics of Solid and Fluid Surfaces]]<br />
| [[thermodynamics]], [[interfaces]], [[excess free energy]], [[Phase Rule]], [[Gibbs-Duhem]]<br />
|- valign = "left" align = "left"<br />
| [[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]<br />
| [[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], [[thermodynamics]], [[interfaces]], [[contact angle]]<br />
|}</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Young%27s_equation&diff=24327Young's equation2012-04-21T19:26:27Z<p>Redston: </p>
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== Keyword in references: ==<br />
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[[Thermodynamic deviations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations]]<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Thermodynamics&diff=24326Thermodynamics2012-04-21T19:26:08Z<p>Redston: </p>
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<div>==Keyword in references==<br />
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[[Thermodynamics of Solid and Fluid Surfaces]]<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Curvature&diff=24325Curvature2012-04-21T19:26:08Z<p>Redston: New page: ==Keyword in references== Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop</p>
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<div>==Keyword in references==<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Interface_stress&diff=24324Interface stress2012-04-21T19:25:51Z<p>Redston: New page: ==Keyword in references== Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop</p>
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<div>==Keyword in references==<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Surface_tension&diff=24323Surface tension2012-04-21T19:25:44Z<p>Redston: </p>
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<div>==Definition==<br />
Surface tension is a property of liquid surfaces caused by cohesion. Cohesion is the physical property resulting from the intermolecular forces attracting like-molecules. The molecules on the surface of a liquid have a greater attraction to like-molecules around them than to unlike-molecules.<br />
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Molecules on the surface of a liquid experience an inward force balanced by the resistance to compression. Another important point in understanding surface tension is the liquid molecules seek the lowest possible surface area. This is the reason that liquids form droplets on hydrophobic surfaces. The interface of lke-molecules has a lower energy than the interface of unlike-molecules, therefore surface molecules seek to have as many like-molecule interfaces as possible resulting in the lowest surface area.<br />
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==Units==<br />
Surface tension (<math>\gamma</math>) has dimensions of force per unit length, <math>\frac{F} {L}</math>.<br />
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==References==<br />
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http://en.wikipedia.org/wiki/Cohesion_%28chemistry%29<br />
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http://en.wikipedia.org/wiki/Surface_tension<br />
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== Keyword in references: ==<br />
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[[Capillary rise between elastic sheets]]<br />
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[[Contact angle associated with thin liquid films in emulsions]]<br />
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[[Controlled Assembly of Jammed Colloidal Shells on Fluid Droplets]]<br />
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[[Controlling the Fiber Diameter during electrospinning]]<br />
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[[Krafft Points, Critical Micelle Concentrations, Surface Tension, and Solubilizing Power of Aqueous Solutions of Fluorinated Surfactants]]<br />
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[[Surface-Tension-Induced Synthesis of Complex Particles Using Confined Polymeric Fluids]]<br />
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[[The Deformation of an Elastic Substrate by a Three-Phase Contact Line E. R. Jerison]]<br />
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[[Thermodynamic deviations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations]]<br />
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[[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Phase_Rule&diff=24322Phase Rule2012-04-21T19:24:38Z<p>Redston: </p>
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<div>Entry by [[Andrew Capulli]]<br />
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'''Definition: Phase Rule''' (Gibbs' Phase Rule)<br />
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The phase rule relates:<br />
*F: The degrees of freedom of the system; see below.<br />
*P: The number of phases that can coexist; any separable material in the system. A phase can be a pure compound (say water for example) or a mixture (solid or aqueous), but the phase must "behave" as a consistent substance. For example, ice and liquid water are two separate phases in a one component system (H20). Similarly, higher component systems may have phases made up of multiple components (ie a phase can be composed of more than one component).<br />
*C: The number of components (that make up the phases)<br />
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'''The Phase Rule States: the degrees of freedom of a system is equal to the number of components minus the number of phases plus two'''... the 2 comes from the extensive variables Temperature and Pressure.<br />
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[[Image:Phase1.jpg]]<br />
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The 'degrees of freedom' of the system (at chemical equilibrium) refer to the number of conditions or variables that can be altered, independent of each other, without effecting the number of phases in the system. Essentially, the degrees of freedom of a system describe the dependency of parameters such as temperature and pressure on each other.<br />
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The Phase Rule describes the number of variables (and equations) that can be used to describe a system (at chemical equilibrium). The number of chemical components (C in the equation above) in addition to the "extensive variables" (temperature and pressure) comprise the 'variables' of a system. The degrees of freedom of a system dictate the number of phases (as described above in the bullet list) that can occur in the system. <br />
<br />
''Note''<br />
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The critical point (on a phase diagram) can only exist at one temperature and pressure for a substance or system and thus the degrees of freedom at any critical point is zero.<br />
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''An Example of the Phase Rule: 1 Component System'' :Take the generic 1 component phase diagram below (from class). So, at A, B, and C (and all points for that matter, we consider the system to have one component, ie C = 1. At A, there are two possible phases at a fixed temperature as shown by the red line drawn through point A; these phases are gas and liquid. Since there is one component and two phases, using the Phase Rule equation, the degrees of freedom of the system at A is one. At B there are three possible phases (gas, liquid, and solid) and consequently the degrees of freedom of the system is 0 (ie B is a critical point). Similarly we can find at C the degrees of freedom to be one because there are two possible phases (fluid or solid) as indicated by the red line drawn through C. <br />
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[[Image:Phase2.jpg]] <br />
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The Wikipedia article "Gibbs Phase Rule" has a number of examples using the phase rule on a phase diagram to determine the degrees of freedom of a system at a given point; it can be found at: http://en.wikipedia.org/wiki/Gibbs%27_phase_rule.<br />
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References:<br />
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[1] http://www.chemicool.com/definition/phase_rule.html<br />
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[2] http://serc.carleton.edu/research_education/equilibria/phaserule.html<br />
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[3] http://en.wikipedia.org/wiki/Gibbs%27_phase_rule<br />
<br />
In Reference:<br />
<br />
[[The Science of Chocolate: interactive activities on phase transitions, emulsification, and nucleation]]<br />
<br />
[[Thermodynamics of Solid and Fluid Surfaces]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Interfaces&diff=24321Interfaces2012-04-21T19:24:25Z<p>Redston: </p>
<hr />
<div>Entry needed.<br />
<br />
<br />
<br />
<br />
<br />
== Keyword in references: ==<br />
<br />
[[Surface-Tension-Induced Synthesis of Complex Particles Using Confined Polymeric Fluids]]<br />
<br />
[[Thermodynamics of Solid and Fluid Surfaces]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Gibbs-Duhem&diff=24320Gibbs-Duhem2012-04-21T19:24:04Z<p>Redston: New page: ==Keyword in References== Thermodynamics of Solid and Fluid Surfaces</p>
<hr />
<div>==Keyword in References==<br />
<br />
[[Thermodynamics of Solid and Fluid Surfaces]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Excess_free_energy&diff=24319Excess free energy2012-04-21T19:23:56Z<p>Redston: New page: ==Keyword in references== Thermodynamics of Solid and Fluid Surfaces</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Thermodynamics of Solid and Fluid Surfaces]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Thermodynamics&diff=24318Thermodynamics2012-04-21T19:23:32Z<p>Redston: New page: ==Keyword in references== Thermodynamics of Solid and Fluid Surfaces</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Thermodynamics of Solid and Fluid Surfaces]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Biomimetics&diff=24317Biomimetics2012-04-21T19:22:30Z<p>Redston: </p>
<hr />
<div>Contributed by [[Daniel Daniel]]<br />
<br />
==Introduction==<br />
<br />
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs and the cat's eye reflectors which were the results of studying the mechanism of cat's eyes. <br />
<br />
[[image:biomimetics1.jpg]]<br />
Figure 1. Tiny hooks found on the surface of burs.<br />
<br />
Examples of biomimetic systems can be found in the wikipedia article on bionics. <br />
http://en.wikipedia.org/wiki/Bionics<br />
<br />
==Biomimetics Chemistry==<br />
<br />
From the point of view of chemistry, biological systems are able to synthesize complex chemical compounds efficiently at relatively low temperature (e.g. human body's temperature ~37 degrees celsius), whereas we often requires the use of high temperature, high energy and huge reactors. Biological systems often achieve these through enzymatic reactions and it will interesting to study the way biological systems snynthesize chemical compounds to better optimize the way we do chemistry. <br />
<br />
A good article discussing this by Ronald Breslow in the Journal of Biological Chemistry. <br />
http://www.jbc.org/content/284/3/1337.full<br />
<br />
==Difference between biological systems and artificial systems==<br />
<br />
One main difference between biological and artificial system is that the former is responsive to the environment, exhibit homeostasis and self-repair properties, while the latter is often static, lacks self-regulatory abilities and is relatively unresponsive. The interest in studying biological systems is in part hoping to incorporate their design principles in smart material in the future which can responds to different environments appropriately. Examples include glass windows that can regulate the amount of sunlight entering the room to optimize energy efficiency.<br />
<br />
== Keyword in references: ==<br />
<br />
[[A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal]]<br />
<br />
[[Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Biomimetic Morphogenesis of Calcium Carbonate in Mixed Solutions of Surfactants and Double-Hydrophilic Block Copolymers]]<br />
<br />
[[Pitcher plant inspired non-stick surface]]<br />
<br />
[[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]] (version 2)</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid-repellent_surface&diff=24316Liquid-repellent surface2012-04-21T19:22:12Z<p>Redston: New page: ==Keyword in references== Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=SLIPS&diff=24315SLIPS2012-04-21T19:21:54Z<p>Redston: New page: ==Keyword in references== Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Surface_texture&diff=24314Surface texture2012-04-21T19:21:41Z<p>Redston: New page: ==Keyword in references== Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Self-healing&diff=24313Self-healing2012-04-21T19:21:30Z<p>Redston: New page: ==Keyword in references== Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Hard_sphere&diff=24312Hard sphere2012-04-21T19:20:13Z<p>Redston: </p>
<hr />
<div>Final Project for AP225 Fall 2011, written by Hyerim Hwang<br />
<br />
==[[Definition]]==<br />
Most of matter is made up of atoms and molecules, and then phase of the condensed matter is solid or liquid. That gives us the fact that there is an attractive force between molecules. There is also a repulsive force, which prevents matter from completely collapsing. The short-range interaction is essentially quantum mechanical; the electron orbitals of neighboring molecules begin to interact. Much of the liquids and solids assumes that the repulsion is infinite when the molecules overlap, but there is a certain long-range interaction for larger separation which is known as a hard sphere potential. <br />
In shortly, hard spheres are thought to be model particles in the statistical mechanical theory of fluids and solids which are studied by analytically with simulations. They cannot overlap in space and mimic the strong repulsion that atoms and spherical molecules experience at very close distances. In comparison with soft spheres, internal structure of hard spheres does not change with concentration. There is a change in soft spheres at high concentration due to overlapping and deformation of the surfaces. <br />
[[Image:hard sphere1.png|thumb|400px|left| Figure 1. Interactions of Hard Spheres Among Themselves and With the Wall.]]<br />
<br />
==[[Colloidal Interactions]]==<br />
Entropic interactions are exlained in the context of suspensions of hard spheres. Hard-sphere colloids lack attractive and long-range interactions, which compete with entropic effects to produce ordered phases. It was noted that in mixtures of different size spherical particles an ordered arrangement of large spheres can increase the total entropy of the system by increasing the entropy of the small spheres. The box in the Figure 1 contains a few large spheres and many small spheres. The entropy of a small sphere depends on the number of positions it can occupy in the box. More free volume means more entropy for the small spheres. Since the center of mass of the small sphere cannot penetrate within a/2 of the large sphere surface, a region of "excluded volume" surrounds each large sphere. Figure 1 illustrates the interactions of hard spheres. The small sphere centers are excluded from the shaded blud regions. The red regions correspond to the overlap of excluded volumes which means the increased volume for small spheres. <br />
<br />
==[[References]]==<br />
1. Israelachvili, Jacob N. (2011). Intermolecular and Surface Forces. Academic Press. ISBN 9780123919274.<br />
<br />
2. Jones, Richard A. N. (2002). Soft Condensed Matter. University Press. ISBN 9780198505891.<br />
<br />
3. Eckert, T., Richtering, W., 2008. "Thermodynamic and Hydrodynamic Interaction in Concentrated Microgel Suspensions: Hard or Soft Sphere Behavior?" J. Chem. Phys. 129, 124902. �<br />
<br />
==[[Additional Readings]]==<br />
1. Cloitre, M. (2010). High Solid Dispersions (Advances in Polymer Science). Springer 1st Edition. ISBN 9780521864299.<br />
<br />
2. Fischer, Earl K. (2008) Colloidal Dispersions. Fischer Press. ISBN 9781443729345. <br />
<br />
== Keyword in references: ==<br />
<br />
[[Pickering Emulsions - Particles as Surfactants]]<br />
<br />
[[Five-Fold Symmetry in Liquids]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Dense_random_packing&diff=24311Dense random packing2012-04-21T19:19:45Z<p>Redston: New page: ==Keyword in references== Five-Fold Symmetry in Liquids</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Five-Fold Symmetry in Liquids]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=X-ray_scattering&diff=24310X-ray scattering2012-04-21T19:19:32Z<p>Redston: New page: ==Keyword in references== Five-Fold Symmetry in Liquids</p>
<hr />
<div>==Keyword in references==<br />
<br />
[[Five-Fold Symmetry in Liquids]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_structure&diff=24309Liquid structure2012-04-21T19:19:11Z<p>Redston: </p>
<hr />
<div>==Keyword in References==<br />
<br />
[[David Turnbull (1915-2007). Pioneer of the kinetics of phase transformations in condensed matter]]<br />
<br />
[[Five-Fold Symmetry in Liquids]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Five-fold_symmetry&diff=24308Five-fold symmetry2012-04-21T19:19:07Z<p>Redston: New page: ==Keyword in references:== Five-Fold Symmetry in Liquids</p>
<hr />
<div>==Keyword in references:==<br />
[[Five-Fold Symmetry in Liquids]]</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Emily_Redston&diff=24307Emily Redston2012-04-21T19:16:48Z<p>Redston: </p>
<hr />
<div>Wiki entries:<br />
<br />
<br />
Fall 2011<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=250 | Topic<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| ''1 - General Introduction''<br />
| [[Biofilms as complex fluids]]<br />
| [[biofilms]], [[colloids]], [[polymers]], [[gels]], [[viscoelasticity]], [[elasticity]], [[cross-linking]], [[volume fraction]]<br />
|- valign = "left" align = "left"<br />
| ''2 - Surface Forces''<br />
| [[Crystalline monolayer surface of liquid Au–Cu–Si–Ag–Pd: Metallic glass former]]<br />
| [[metallic glasses]], [[crystal structure]], [[surface freezing]], [[liquid alloys]], [[surface crystals]], [[phase transition]], [[eutectics]]<br />
|- valign = "left" align = "left"<br />
| ''3 - Capillarity''<br />
| [[Diffusion through colloidal shells under stress]]<br />
| [[encapsulation]], [[colloids]], [[osmotic pressure]], [[diffusion]], [[emulsification]], [[permeability]]<br />
|- valign = "left" align = "left"<br />
| ''4 - Polymers and Polymer Solutions''<br />
| [[The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures]]<br />
| [[polydisperse]], [[monodisperse]], [[morphology]], [[colloids]], [[gels]], [[phase separation]], [[polymers]], [[non-adsorbing]], [[volume fraction]], [[poroelastic]], [[transient gelation]], [[spinodal decomposition]]<br />
|- valign = "left" align = "left"<br />
| ''5 - Surfactants''<br />
| [[Bacteria Pattern Spontaneously on Periodic Nanostructure Arrays]]<br />
| [[biofilms]], [[self-assembly]], [[nanoposts]], [[Fourier transform]], [[fluorescence microscopy]], [[ordering]], [[periodicity]]<br />
|- valign = "left" align = "left"<br />
| ''6 - Equilibria and Phase Diagrams''<br />
| [[David Turnbull (1915-2007). Pioneer of the kinetics of phase transformations in condensed matter]]<br />
| [[metallic glasses]], [[eutectics]], [[phase transition]], [[liquid undercooling]], [[liquid structure]], [[crystal structure]], [[free-volume model]], [[viscosity]], [[glass-transition temperature]]<br />
|- valign = "left" align = "left"<br />
| ''7 - Charged Interfaces''<br />
| [[Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model]]<br />
|[[beads-on-a-string model]], [[electrostatic interactions]], [[polymers]], [[polymer folding]], [[RNA]], [[triboelectricity]], [[self-assembly]]<br />
|- valign = "left" align = "left"<br />
| ''10 - Foams and Emulsions''<br />
| [[Mechanical Inhibition of Foam Formation via a Rotating Nozzle]]<br />
| [[bubbles]], [[capillarity]], [[confined flow]], [[drops]], [[foams]], [[nozzles]], [[rotational flow]], [[foam-suppression]]<br />
|}<br />
<br />
<br />
Spring 2012<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| [[Five-Fold Symmetry in Liquids]]<br />
| [[liquid structure]], [[five-fold symmetry]], [[x-ray scattering]], [[hard sphere]], [[dense random packing]]<br />
|- valign = "left" align = "left"<br />
| [[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]<br />
| [[omniphobicity]], [[biomimetics]], [[self-healing]], [[surface texture]], [[SLIPS]], [[liquid-repellent surface]]<br />
|- valign = "left" align = "left"<br />
| [[Thermodynamics of Solid and Fluid Surfaces]]<br />
| [[thermodynamics]], [[interfaces]], [[excess free energy]], [[Phase Rule]], [[Gibbs-Duhem]]<br />
|- valign = "left" align = "left"<br />
| [[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]<br />
| [[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], [[thermodynamics]]<br />
|}</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24306Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:16:34Z<p>Redston: /* Conclusion */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], thermodynamics<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
<br />
The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on solid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the resulting substrate curvature in depth. This seems like it would be an important thing to consider for thin solid films, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Emily_Redston&diff=24305Emily Redston2012-04-21T19:14:41Z<p>Redston: </p>
<hr />
<div>Wiki entries:<br />
<br />
<br />
Fall 2011<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=250 | Topic<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| ''1 - General Introduction''<br />
| [[Biofilms as complex fluids]]<br />
| [[biofilms]], [[colloids]], [[polymers]], [[gels]], [[viscoelasticity]], [[elasticity]], [[cross-linking]], [[volume fraction]]<br />
|- valign = "left" align = "left"<br />
| ''2 - Surface Forces''<br />
| [[Crystalline monolayer surface of liquid Au–Cu–Si–Ag–Pd: Metallic glass former]]<br />
| [[metallic glasses]], [[crystal structure]], [[surface freezing]], [[liquid alloys]], [[surface crystals]], [[phase transition]], [[eutectics]]<br />
|- valign = "left" align = "left"<br />
| ''3 - Capillarity''<br />
| [[Diffusion through colloidal shells under stress]]<br />
| [[encapsulation]], [[colloids]], [[osmotic pressure]], [[diffusion]], [[emulsification]], [[permeability]]<br />
|- valign = "left" align = "left"<br />
| ''4 - Polymers and Polymer Solutions''<br />
| [[The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures]]<br />
| [[polydisperse]], [[monodisperse]], [[morphology]], [[colloids]], [[gels]], [[phase separation]], [[polymers]], [[non-adsorbing]], [[volume fraction]], [[poroelastic]], [[transient gelation]], [[spinodal decomposition]]<br />
|- valign = "left" align = "left"<br />
| ''5 - Surfactants''<br />
| [[Bacteria Pattern Spontaneously on Periodic Nanostructure Arrays]]<br />
| [[biofilms]], [[self-assembly]], [[nanoposts]], [[Fourier transform]], [[fluorescence microscopy]], [[ordering]], [[periodicity]]<br />
|- valign = "left" align = "left"<br />
| ''6 - Equilibria and Phase Diagrams''<br />
| [[David Turnbull (1915-2007). Pioneer of the kinetics of phase transformations in condensed matter]]<br />
| [[metallic glasses]], [[eutectics]], [[phase transition]], [[liquid undercooling]], [[liquid structure]], [[crystal structure]], [[free-volume model]], [[viscosity]], [[glass-transition temperature]]<br />
|- valign = "left" align = "left"<br />
| ''7 - Charged Interfaces''<br />
| [[Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model]]<br />
|[[beads-on-a-string model]], [[electrostatic interactions]], [[polymers]], [[polymer folding]], [[RNA]], [[triboelectricity]], [[self-assembly]]<br />
|- valign = "left" align = "left"<br />
| ''10 - Foams and Emulsions''<br />
| [[Mechanical Inhibition of Foam Formation via a Rotating Nozzle]]<br />
| [[bubbles]], [[capillarity]], [[confined flow]], [[drops]], [[foams]], [[nozzles]], [[rotational flow]], [[foam-suppression]]<br />
|}<br />
<br />
<br />
Spring 2012<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| [[Five-Fold Symmetry in Liquids]]<br />
| [[liquid structure]], [[five-fold symmetry]], [[x-ray scattering]], [[hard sphere]], [[dense random packing]]<br />
|- valign = "left" align = "left"<br />
| [[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]<br />
| [[omniphobicity]], [[biomimetics]], [[self-healing]], [[surface texture]], [[SLIPS]], [[liquid-repellent surface]]<br />
|- valign = "left" align = "left"<br />
| [[Thermodynamics of Solid and Fluid Surfaces]]<br />
| [[thermodynamics]], [[interfaces]], [[excess free energy]], [[Phase Rule]], [[Gibbs-Duhem]]<br />
|- valign = "left" align = "left"<br />
| [[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]<br />
| [[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], thermodynamics<br />
|}</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24304Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:14:13Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], thermodynamics<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
<br />
The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on solid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the resulting substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24303Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:13:26Z<p>Redston: /* Keywords */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]], thermodynamics<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
<br />
The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on solid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the resulting substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24302Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:11:49Z<p>Redston: /* Conclusion */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
<br />
The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on solid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the resulting substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24301Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:10:18Z<p>Redston: /* Curvature of the Substrate */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure in the droplet exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
<br />
The free body diagram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney equation (which applies exactly for the infinitesimally thin surfaces in which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. The contribution to the curvature is an essential result of the action of interface stresses instead of tensions. The factor in parentheses would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on s olid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24300Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T19:03:14Z<p>Redston: /* Equilibrium Shape of the Droplet */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface with solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume, <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
<br />
Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly,<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
The free body digram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney euqation (which applies exactly for the infinitesimally thin surfacesin which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. THe contribution to the curvature is an essential result of the action of interface stresses instead of tensions. THe factor in parenthese would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on s olid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24299Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T18:57:24Z<p>Redston: /* Geometry of the Droplet */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet on a solid surface surrounded by a vapor phase. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, and <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications like surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
The free body digram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney euqation (which applies exactly for the infinitesimally thin surfacesin which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. THe contribution to the curvature is an essential result of the action of interface stresses instead of tensions. THe factor in parenthese would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on s olid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24298Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T18:41:52Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
The free body digram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney euqation (which applies exactly for the infinitesimally thin surfacesin which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. THe contribution to the curvature is an essential result of the action of interface stresses instead of tensions. THe factor in parenthese would be zero by the Young equation, (6), if the interfacial tensions were used.<br />
<br />
==Conclusion==<br />
<br />
In this paper, Spaepen compared interfacial tension and interface stress by looking at the example of a hemispherical liquid drop on s olid substrate. The equilibrium shape was determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions were calculated: one arising from the hydrostatic pressure of the drop and the other from the imbalance of the interfacial stresses.<br />
<br />
This is a very neat little derivation that fits in nicely with a lot of the things we discussed in class. We talk about Young's equation all the time, but I had never really considered the substrate curvature in depth. From thin films, this seems like it would be an important thing to consider, since I imagine you would be able to see the effect of these stresses.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24297Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T18:35:03Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math></center><br />
<br />
The free body digram for the horizontal forces on the substrate is shown in Fig. 5. <br />
<br />
[[Image:capcur_frbdy2.png|center|]]<br />
<center>Fig. 5. Free body diagram of the substrate with the horizontal capillary forces</center><br />
<br />
<br />
When balancing the forces for the system to one side of a cut AA', the capillary contributions are: the horizontal component of the liquid-vapor interfacial tension, <math>\gamma_{lv} cos\theta</math>, the interface stress <math>f_{sl}</math> from the solid-liquid interface at the top, and the interface stress <math>f_{sv}</math> from the solid-vapor interface at the bottom. The interface stress at the bottom is taken to be the same as that at the top-vapor interface; otherwise the substrate outside the droplet would have a net curvature. <br />
<br />
The curvature under the droplet is constant in this case, and can be obtained directly from the well-known Stoney euqation (which applies exactly for the infinitesimally thin surfacesin which forces on either side of the substrate act)<br />
<br />
<center><math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math></center><br />
<br />
Note that having <math>f_{sv}</math> acting at the bottom is equivalent to having <math>-f_{sv}</math> acting at the top. THe contribution to the curvature is an essential result of the action of interface stresses instead of tensions. THe factor in parenthese would be zero by the Young equation, (), if the interfacial tensions were used.<br />
<br />
==Conclusion==</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24296Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T18:14:00Z<p>Redston: /* Curvature of the Substrate */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displacements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostatic. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substrate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces, <math>2\gamma_{lv}sin\theta</math>, spaced a distance <math>L= 2Rsin\theta</math> apart, are balanced by the force from the hydrostatic pressure <math>\Delta p</math>. The substrate curvature resulting from this load was estimated using simple beam bending with a strain that varies linearly though the thickness. This approximation is reasonable if the thickness of the substrate, t, is less than L. There is no curvature to the left and right of the droplet. Under the droplet, the curvature varies, being maximum in the middle and going to zero at the ends. Spaepen focused on calculating an average curvature, since that's what one measures in condensation experiments. Standard balancing of forces and moments gives for the strain in the top surface:<br />
<br />
<center><math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math></center><br />
<br />
where E is the Young's modulus of the substrate. The total elongation of the top fiber is<br />
<br />
<center><math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math></center><br />
<br />
This translates into an average curvature of<br />
<br />
<center><math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}</math></center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24295Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T17:42:51Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displcements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostaticc. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center><br />
<br />
<br />
The vertical components of the capillary forces</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24294Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T17:37:47Z<p>Redston: /* Curvature of the Substrate */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displcements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostaticc. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math></center><br />
<br />
A free body diagram of the vertical forces on the substate is shown in Fig. 4. <br />
<br />
[[Image:capcur_frbdy.png|center|]]<br />
<center> Fig. 4. Free body diagram of the substrate with the vertical capillary forces</center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24293Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T17:34:39Z<p>Redston: /* Curvature of the Substrate */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displcements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostaticc. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center><br />
<br />
The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:<br />
<br />
<center><math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math></center><br />
<br />
For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:<br />
<br />
<center><math>\Delta p = {\gamma_{lv} \over R}\ [9]</math><\center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24292Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T17:29:34Z<p>Redston: /* Equilibrium Shape of the Droplet */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center><br />
<br />
Equation <math>\lambda R</math> found from both equations and simplifyling the trigonometric functions gives:<br />
<br />
<center><math>\theta cos\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta) = sin\theta({\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}} + cos\theta)\ [5]</math></center><br />
<br />
Equation 5 can only be satisfied in two ways: <math>\theta</math> = 0 (complete wetting), which requires <math>\gamma_{sl} + \gamma_{lv} < \gamma_{sv}</math>, or, more interestingly<br />
<br />
<br />
<center><math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math></center><br />
<br />
which is the well-known Young equation for the equilibrium wetting angle <math>\theta</math>. This scalar equation has an equally well-known vector representation, shown in Fig. 2. Although the interfacial tensions are formally represented as vectors in this diagram, it is important to remember that these vectors are not forces. <br />
<br />
[[Image:capcur_fbal.png|center|]]<br />
<center> Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math></center><br />
<br />
Solving for the Lagrange multiplier in equilibrium:<br />
<center><math> \lambda = -{\gamma_{lv} \over R}\ [7]</math></center><br />
<br />
This is the pressure difference between the liquid and vapor across the curved interface.<br />
<br />
==Curvature of the Substrate==<br />
<br />
After establishing the shape of the droplet from the relation between the tensions, Spaepen considered the strains in the substrate from the forces exerted by the droplet and its interfaces. Since the displcements under consideration are elastic, the interfacial stresses are the relevant quantities. <br />
<br />
The stress inside the droplet is hydrostaticc. The pressure exceeds that in the vapor by <math>\Delta p</math>, which is found by the well known Laplace force equilibrium, illustrated in Fig. 3. <br />
<br />
[[Image:capcur_pdiff.png|center|]]<br />
<center> Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor</center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24291Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T16:10:25Z<p>Redston: /* Equilibrium Shape of the Droplet */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center><br />
<br />
Minimization gives the conditions:<br />
<br />
<center><math>{\partial F \over \partial R} = 2sin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta \gamma_{lv} + 2\lambda R(\theta - sin\theta cos\theta) = 0\ [4a]</math></center><br />
<br />
<br />
<center><math>{\partial F \over \partial \theta} = 2Rcos\theta(\gamma_{sl} - \gamma_{sv}) + 2R\gamma_{lv} + \lambda R^2(1 - cos^2\theta + sin^2\theta) = 0\ [4b]</math></center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24290Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T16:02:23Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.<br />
<br />
==Geometry of the Droplet==<br />
Figure 1 shows a droplet surrounded by a vapor phase on a solid surface. The problem is kept two-dimensional for simplicity. Associated with the three types of interfaces are the tension <math>\gamma_{lv}</math>, <math>\gamma_{sv}</math>, and <math>\gamma_{sl}</math>, as well as the stresses <math>f_{lv}</math> (=<math>\gamma_{lv}</math>), <math>f_{sv}</math>, <math>f_{sl}</math> (corresponding to the only applicable strain, <math>\epsilon_{11}</math>, in the direction of the interface). Spaepen also assumes that the three phases consist of the same single element to avoid complications due to surface segregation. <br />
[[Image:capcur_dgrm.png|400px|center|]]<br />
<center>Fig. 1 Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities</center><br />
<br />
==Equilibrium Shape of the Droplet==<br />
<br />
Consistent with <math>\gamma_{lv}</math> being isotropic, the liquid-vapor interface is considered semi-circular and has a radius of curvature R. The angle between the radii to the end points (OA and OB) is taken to be 2<math>\theta</math>. The equilibrium shape of the droplet for a given volume is determined by minimizing the free energy of the system with respect to <math>\theta</math> or R. Placing the droplet on the substrate replaces the solid-vapor interface by solid-liquid interface over an area <math>A_{sl}</math>, also creating an area <math>A_{lv}</math> of liquid-vapor interface. The associated changes in free energy are:<br />
<br />
<center><math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math></center><br />
<br />
Taking the geometry of Fig. 1 into account, we can also write this as:<br />
<br />
<center><math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math></center><br />
<br />
<br />
To minimize this free energy at constant volume <math> V = R^2(\theta - sin\theta cos\theta)</math>, a Lagrange multiplier, <math>\lambda</math>, is introduced:<br />
<br />
<center><math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math></center></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24289Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T15:31:52Z<p>Redston: /* Introduction */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar quantity equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24288Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-21T15:30:09Z<p>Redston: /* Introduction */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar equal to the work required to create a unit area of new interface at constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work required to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24254Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-19T16:58:16Z<p>Redston: /* Introduction */</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface (or interface) tension <math>\gamma</math>, which is a scalar equal to the work required to create a unit area of new interface and constant strain in the solid<br />
*surface (or interface) stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work require to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math><br />
<br />
In this paper, Spaepen illustrates the difference between these two quantities</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24253Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-19T16:56:36Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of solids using two thermodynamic quantities:<br />
*surface tension <math>\gamma</math>, which is a scalar equal to the work required to create a unit area of new interface and constant strain in the solid<br />
*surface stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work require to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24252Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-19T16:55:37Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
[[surface tension]], [[interface stress]], [[Young's equation]], [[curvature]]<br />
<br />
==Introduction==<br />
We typically characterize the surface of a solids using two thermodynamic quantities:<br />
*surface tension <math>\gamma</math>, which is a scalar equal to the work required to create a unit area of new interface and constant strain in the solid<br />
*surface stress <math>f_{ij}</math>, which is a 2x2 tensor defined such that the surface work require to strain a unit surface elastically by <math>d {\epsilon}_{ij}</math> is <math>f_{ij} d {\epsilon}_{ij}</math></div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24251Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-19T16:38:58Z<p>Redston: </p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids '''44''', 675 – 681 (1996)<br />
<br />
==Keywords==<br />
<br />
<br />
==Introduction==</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Substrate_Curvature_Resulting_from_the_Capillary_Forces_of_a_Liquid_Drop&diff=24250Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop2012-04-19T16:38:44Z<p>Redston: New page: Entry by Emily Redston, AP 226, Spring 2012 Work in Progress ==Reference== ''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. ...</p>
<hr />
<div>Entry by [[Emily Redston]], AP 226, Spring 2012<br />
<br />
Work in Progress <br />
==Reference==<br />
''Substrate curvature resulting from the capillary forces of a liquid drop'' by F. Spaepen. J. Mech. Phys. Solids'''44''',675 – 681 (1996)<br />
<br />
==Keywords==<br />
<br />
<br />
==Introduction==</div>Redstonhttp://soft-matter.seas.harvard.edu/index.php?title=Emily_Redston&diff=24249Emily Redston2012-04-19T16:22:06Z<p>Redston: </p>
<hr />
<div>Wiki entries:<br />
<br />
<br />
Fall 2011<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=250 | Topic<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| ''1 - General Introduction''<br />
| [[Biofilms as complex fluids]]<br />
| [[biofilms]], [[colloids]], [[polymers]], [[gels]], [[viscoelasticity]], [[elasticity]], [[cross-linking]], [[volume fraction]]<br />
|- valign = "left" align = "left"<br />
| ''2 - Surface Forces''<br />
| [[Crystalline monolayer surface of liquid Au–Cu–Si–Ag–Pd: Metallic glass former]]<br />
| [[metallic glasses]], [[crystal structure]], [[surface freezing]], [[liquid alloys]], [[surface crystals]], [[phase transition]], [[eutectics]]<br />
|- valign = "left" align = "left"<br />
| ''3 - Capillarity''<br />
| [[Diffusion through colloidal shells under stress]]<br />
| [[encapsulation]], [[colloids]], [[osmotic pressure]], [[diffusion]], [[emulsification]], [[permeability]]<br />
|- valign = "left" align = "left"<br />
| ''4 - Polymers and Polymer Solutions''<br />
| [[The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures]]<br />
| [[polydisperse]], [[monodisperse]], [[morphology]], [[colloids]], [[gels]], [[phase separation]], [[polymers]], [[non-adsorbing]], [[volume fraction]], [[poroelastic]], [[transient gelation]], [[spinodal decomposition]]<br />
|- valign = "left" align = "left"<br />
| ''5 - Surfactants''<br />
| [[Bacteria Pattern Spontaneously on Periodic Nanostructure Arrays]]<br />
| [[biofilms]], [[self-assembly]], [[nanoposts]], [[Fourier transform]], [[fluorescence microscopy]], [[ordering]], [[periodicity]]<br />
|- valign = "left" align = "left"<br />
| ''6 - Equilibria and Phase Diagrams''<br />
| [[David Turnbull (1915-2007). Pioneer of the kinetics of phase transformations in condensed matter]]<br />
| [[metallic glasses]], [[eutectics]], [[phase transition]], [[liquid undercooling]], [[liquid structure]], [[crystal structure]], [[free-volume model]], [[viscosity]], [[glass-transition temperature]]<br />
|- valign = "left" align = "left"<br />
| ''7 - Charged Interfaces''<br />
| [[Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model]]<br />
|[[beads-on-a-string model]], [[electrostatic interactions]], [[polymers]], [[polymer folding]], [[RNA]], [[triboelectricity]], [[self-assembly]]<br />
|- valign = "left" align = "left"<br />
| ''10 - Foams and Emulsions''<br />
| [[Mechanical Inhibition of Foam Formation via a Rotating Nozzle]]<br />
| [[bubbles]], [[capillarity]], [[confined flow]], [[drops]], [[foams]], [[nozzles]], [[rotational flow]], [[foam-suppression]]<br />
|}<br />
<br />
<br />
Spring 2012<br />
{| cellspacing = "1" border = "1" style="margin: 0em 0em 1em 0em"<br />
|- valign = "left" align = "left"<br />
! width=300| Weekly entry<br />
! width=400| Keywords<br />
<br />
|- valign = "left" align = "left"<br />
| [[Five-Fold Symmetry in Liquids]]<br />
| [[liquid structure]], [[five-fold symmetry]], [[x-ray scattering]], [[hard sphere]], [[dense random packing]]<br />
|- valign = "left" align = "left"<br />
| [[Bioinspired Self-Repairing Slippery Surfaces with Pressure-Stable Omniphobicity]]<br />
| [[omniphobicity]], [[biomimetics]], [[self-healing]], [[surface texture]], [[SLIPS]], [[liquid-repellent surface]]<br />
|- valign = "left" align = "left"<br />
| [[Thermodynamics of Solid and Fluid Surfaces]]<br />
| [[thermodynamics]], [[interfaces]], [[excess free energy]], [[Phase Rule]], [[Gibbs-Duhem]]<br />
|- valign = "left" align = "left"<br />
| [[Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop]]<br />
|<br />
|}</div>Redston