Substrate Curvature Resulting from the Capillary Forces of a Liquid Drop

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Entry by Emily Redston, AP 226, Spring 2012

Reference

Substrate curvature resulting from the capillary forces of a liquid drop by F. Spaepen. J. Mech. Phys. Solids 44, 675 – 681 (1996)

Keywords

surface tension, interface stress, Young's equation, curvature, thermodynamics, interfaces, contact angle

Introduction

We typically characterize the surface of solids using two thermodynamic quantities:

  • 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
  • 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>

In this paper, Spaepen illustrates the difference between these two quantities by considering a hemispherical liquid drop on a solid surface.

Geometry of the Droplet

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.

Capcur dgrm.png
Fig. 1. Diagram of the droplet on the substrate, indicating the relevant interfaces and quantities

Equilibrium Shape of the Droplet

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:

<math> \Delta F = A_{sl}(\gamma_{sl} - \gamma_{sv}) + A_{lv} \gamma_{lv}\ [1]</math>

Taking the geometry of Fig. 1 into account, we can also write this as:

<math> \Delta F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv}\ [2]</math>


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:

<math> F = 2Rsin\theta(\gamma_{sl} - \gamma_{sv}) + 2\theta R\gamma_{lv} + \lambda R^2(\theta - sin\theta cos\theta)\ [3]</math>

Minimization gives the conditions:

<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>


<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>


Equating <math>\lambda R</math> found from both equations and simplifying the trigonometric functions gives:

<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>

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,


<math>cos\theta = - {\gamma_{sl} - \gamma_{sv} \over \gamma_{lv}}\ [6]</math>

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.

Capcur fbal.png
Fig. 2. Vector diagram showing the horizontal balance of the interfacial tensions that yields the wetting angle <math>\theta</math>


Solving for the Lagrange multiplier in equilibrium:

<math> \lambda = -{\gamma_{lv} \over R}\ [7]</math>

This is the pressure difference between the liquid and vapor across the curved interface.

Curvature of the Substrate

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.

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.

Capcur pdiff.png
Fig. 3. Portion of the liquid-vapor interface, indicating the relevant quantities for Laplace's calculation of the pressure difference between liquid and vapor


The component of the force normal to the surface is balanced by the components of <math>f_{lv}</math> in that direction:

<math>2Rd\theta \Delta p = 2f_{lv}d\theta\ [8]</math>

For the liquid <math>f_{lv} = \gamma_{lv}</math>, so we can write:

<math>\Delta p = {\gamma_{lv} \over R}\ [9]</math>

A free body diagram of the vertical forces on the substrate is shown in Fig. 4.

Capcur frbdy.png
Fig. 4. Free body diagram of the substrate with the vertical capillary forces


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:

<math>\epsilon_0(x) = {6Fx \over Et^2}({x \over L} - 1)\ [10]</math>

where E is the Young's modulus of the substrate. The total elongation of the top fiber is

<math>\Delta L = \int_0^L \epsilon_0(x)dx = -{FL^2 \over Et^2}\ [11]</math>

This translates into an average curvature of

<math>\kappa_1 = {2\Delta L \over tL} = -{2FL \over t^3E} = -{4 \gamma_{lv} R sin^2\theta \over t^3E}\ [12]</math>


The free body diagram for the horizontal forces on the substrate is shown in Fig. 5.

Capcur frbdy2.png
Fig. 5. Free body diagram of the substrate with the horizontal capillary forces


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.

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)

<math>\kappa_2 = {6(f_{sv} - f_{sl} - \gamma_{sl} cos\theta) \over t^2E}\ [13]</math>

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.

Conclusion

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.

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.