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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
[http://www.deas.harvard.edu/weitzlab/broedersz.sm.2010.pdf "Measurement of nonlinear rheology of cross-linked biopolymer gels"]<br />
<br />
C. P. Broedersz, K. E. Kasza, L. M. Jawerth, S. Münster, D. A. Weitz, ''Soft Matter'', '''6''', 4120-4127 (2010).<br />
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<br />
== Keywords ==<br />
rheology, gel, biopolymer<br />
<br />
== Overview ==<br />
<br />
[[Image:SM201064120_1.jpg|300px|thumb|right|'''Figure 1.''' Linear rheology of various systems. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
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[[Image:SM201064120_4.jpg|300px|thumb|right|'''Figure 4.''' Comparison between prestress and strain ramp methods. The graphs are the tangent (strain ramp) or differential (prestress) modulus K normalized by the linear modulus <math>G_0</math> plotted against stress <math>\sigma</math>. Symbols represent the prestress method and lines represent the strain ramp protocol at various strain rates. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
<br />
[[Image:SM201064120_A.jpg|300px|thumb|right|'''Figure A.''' An illustration of the system the model represents - a nonlinear Kelvin-Voigt element in series with another dashpot. (created for this entry; not from paper)]]<br />
<br />
[[Image:SM201064120_5.jpg|300px|thumb|right|'''Figure 5.''' (a) Strain as a function of time for prestress pulses as predicted by the model. Strain accumulates in a normal network(red), but not in a crosslinked one (blue). (b) Differential modulus as a function of stress calculated for both the prestress method (symbols) and the strain ramp (lines).]]<br />
<br />
Biopolymer networks, both intracellular and extracellular, exhibit interesting mechanical responses: highly nonlinear, strain stiffening, as well as large, negative normal stresses under shear. The stiffening may prevent large deformations that would be harmful to cells; however, there are situations (invatsion, division) where remodeling of networks is necessary. The two seem to be contradictory, and this complication means that performing traditional rheology will be insufficient.<br />
<br />
The authors of this paper study the nonlinear response of biopolymer gels (primarily F-actin, with some ) using two different methods: a prestress protocol and a strain ramp, to determine which one is more suitable. Using the data collected, they also create a model to represent the system. It was found that for permanent networks, the two protocols are equally suitable, but for transient networks, it is true only at high strain rates. The prestress protocol was insensitive to creep.<br />
<br />
== Results and discussion ==<br />
<br />
The actin system's linear viscoelastic modulus was characterized first and was found to have a very weak frequency dependence (Figure 1). The fibrin network's elastic modulus was independent of frequency.<br />
<br />
The nonlinear response was measured by the two methods. The prestress method measures the response at a specific frequency; the strain ramp uses a fixed rate: <br />
<br />
''Strain ramp protocol''<br />
Both crosslinked and uncrosslinked F-actin networks were found to have measurements that depended on <math>\dot\gamma</math>.<br />
''Prestress protocol''<br />
<br />
Again, both permanent F-actin networks and pure solutions had similar responses - the elastic differential modulus <math>K'</math> increased rapidly with applied prestress but had no time-dependence. In all systems tested, <math>K'</math> leveled off (the rate differed though). The response relaxed back to the initial linear modulus very soon after the stress was removed and both linear and nonlinear properties displayed no hysteresis.<br />
<br />
''Comparison''<br />
The main difference between the two results was that the strain ramp method had a rate dependence (Fig 4) while the prestress method had no hysteresis despite creep in the system.<br />
<br />
== Model ==<br />
<br />
Based on these results, the authors proposed a model for use while studying these systems:<br />
<br />
''Assumptions''<br />
# network repsonds instantaneously to an applied stress<br />
# two components of the strain: reversible (<math>\gamma_e</math>) and network flow (<math>\gamma_f</math>) for a total strain of <math>\gamma = \gamma_e + \gamma_f</math>; this implies that stresses are always equal <math>\sigma=\sigma_e=\sigma_f</math><br />
#single relaxation timescale, so the network is treated like a simple liquid with viscosity <math>\zeta\gg\eta</math><br />
<br />
Reversible deformation can be described as: <math>\frac{d\sigma}{dt}=\left[k(\sigma)+\eta\frac{d}{dt}\right]\frac{d\gamma_e}{dt}</math><br />
<br />
where <math>k(\sigma)</math> is the elasticity and <math>\eta</math> is the viscosity and they are acting in parallel. This is a nonlinear generalization of the Kelvin-Voigt model (dashpot and spring in parallel).<br />
<br />
Stress relaxation is given by: <math>\frac{d\sigma}{dt}=\zeta\frac{d^2\gamma_f}{dt^2}</math><br />
<br />
This describes a Newtonian liquid-like system.<br />
<br />
By the second assumption, the stresses in the above equations can be equated to represent a second dashpot in series with the Kelvin-Voigt system (Figure A).<br />
<br />
In the cases of the protocols tested in this paper, the equations could be further developed. The prestress protocol had a time-independent prestress <math>\sigma_0</math> and an oscillatory stress <math>\delta\sigma(t)</math>. There was also a time-dependent creep response <math>\gamma_0(t)</math> and a small amplitude oscillatory strain <math>\delta\gamma(t)</math>. The paper shows that taking these into account produces a good fit with the experimental data.<br />
<br />
The general nonlinear response is given by the differential equation:<br />
<br />
<math>\left(1+\frac{\eta}{\zeta}\right)\frac{d^2\sigma}{dt^2}+\frac{k(\sigma)}{\zeta}\left(1-(\eta+\zeta)\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d\sigma}{dt}=k(\sigma)\left(1-\eta\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d^2\gamma}{dt^2}+\eta\frac{d^3\gamma}{dt^3}</math><br />
<br />
Figure 5 shows that the calculated response based on these equations. In 5a, we see strain accumulation in the normal networks (red line), but none for crosslinked ones (blue).<br />
<br />
== Conclusions ==<br />
<br />
The prestress and strain ramp methods are both fine for cross-linked networks. For networks that creep, the prestress method is the better choice.<br />
<br />
The model describes experimental data and shows how the differential nonlinear elastic response can be determined by the prestress method when there is creep. This is because the differential and steady stress and strain components are decoupled in the equations.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Microrheology_of_entangled_F-actin_solutions&diff=16117Microrheology of entangled F-actin solutions2010-12-03T21:40:02Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
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== Reference ==<br />
<br />
[http://crocker.seas.upenn.edu/Gardel2003.pdf "Microrheology of Entangled F-Actin Solutions"]<br />
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M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, D. A. Weitz, ''Physical Review Letters'', '''91''', 158302 (2003).<br />
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== Keywords ==<br />
microrheology, biopolymer, semiflexible networks<br />
<br />
== Overview ==<br />
<br />
The dynamics of networks of semiflexible polymers such as filamentous actin (F-actin) are affected by many characteristic length scales and frequency scales. The polymers become entanled at very low volume fractions and the resulting networks have large elastic moduli and long relaxation times compared to flexible polymers. Since the volume fraction is so low, individual fibers are sterically hindered. However, instead of a constant elastic modulus (over various frequencies), bulk rheological measurements show a monotonically increasing <math>G'(\omega)</math> that approaches a plateau asymptotically. Thus, the authors determined the frequency and length scale dependencies in order to understand the system. They used one-particle (1P) and two-particle (2P) microrheology to accomplish this.<br />
<br />
== Results and discussion ==<br />
<br />
[[Image:PRL158302_1.jpg|300px|thumb|right|'''Figure 1.''' Comparison of 1P (filled symbols) and 2P (open symbols) MSDs.]]<br />
<br />
[[Image:PRL158302_3.jpg|300px|thumb|right|'''Figure 3.''' Comparison between elastic modulus G' (closed symbols) and loss modulus G" (open symbols) from 1P (squares) and 2P (circles) data. Conventional rheometer data (triangles) is also represented. (a) 1.0mg/ml F-actin, (b) 0.3mg/ml F-actin.]]<br />
<br />
Using the data obtained during experiments, the authors calculated the one-dimensional ensemble averaged mean-squared displacement (1P MSD) and scaled it by the particles' radii for the size-dependent viscous drag. Figure 1 shows that there was little change in this value over time for particles 0.32<math>\mu</math>m and greater, but at 0.23<math>\mu</math>m, the value increases. This was found to be due to the fact that 0.23<math>\mu</math>m particles were traveling through the network whereas the larger particles were trapped.<br />
<br />
The 2P MSD gave information about dynamics at larger length scales than the radius. It represents the one-particle motion from long-wavelength modes. Assuing the material was incompressible, the scaling factor should be 2/radius.<br />
<br />
The 1P and 2P MSDs are very different until about <math>\tau</math>=10s, where they converge (the right edge of Figure 1).<br />
<br />
The generalized Stokes-Einstein relation was used to approximate the bulk elastic modulus G'(<math>\omega</math>) and viscous modulus G"(<math>\omega</math>). Figure 3 shows that these approximations were close to the measured bulk values. 2P microrheology measures a viscoelastic response and indicates that at low frequencies (<0.1rad/s), the elastic modulus dominates. However, at intermediate frequencies (<30rad/s), longitudinal fluctuations of the filaments affect the bulk response.<br />
<br />
Looking at the 1P microrheology with the generalized Stokes-Einstein relation produced information on the origins of the viscoelasticity observed using 2P microrheology. Since 1P microrheology significanly underestimates viscoelasticity in the bulk material, particles are again permeating through the network. 1P viscoelasticity seems to be independent of both frequency and particle size. Thus, the authors suggest that the differences between 1P and 2P microrheology comes from coupling between particles.<br />
<br />
Entanglements also affect the bulk viscoelasticity. They determine the plateau elasticity <math>G_0 ~ \rho k_b T/l_e</math>, which contains terms for the filament density <math>\rho</math>. 1P and 2P microrheology were shown to both effectively measure the low frequency plateau of the modulus due to entanglement, so local heterogeneities had little to no effect on the measurements.<br />
<br />
1P microrheology may be applied to ''in vitro'' or ''in vivo'' measurements because crosslinking proteins reduce the importance of longitudinal fluctuations, which 2P microrheology can account for.<br />
<br />
== Experimental Setup ==<br />
<br />
Actin was polymerized in glass sample chambers and then imaged with CCD cameras.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Highly_Anisotropic_Vorticity_Aligned_Structures_in_a_Shear_Thickening_Attractive_Colloidal_System&diff=16116Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System2010-12-03T21:39:11Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
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== Reference ==<br />
<br />
[http://seas.harvard.edu/weitzlab/osuji.softmatter.2008.pdf "Highly anisotropic vorticity aligned structures in a shear thickening attractive colloidal system"]<br />
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C. O. Osuji, D. A. Weitz, ''Soft Matter'', '''4''', 1388-1392 (2008).<br />
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== Keywords ==<br />
[[colloid]]<br />
<br />
== Overview ==<br />
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[[Image:SM2008_2.gif|300px|thumb|left|'''Figure 2.''' Microstructure under shear (a) cylindrical flocs at <math>\dot\gamma = 6.67 s^{-1}</math>, (b) <math>\dot\gamma = 133 s^{-1}</math>, (c) math>\dot\gamma = 1330 s^{-1}</math>.]]<br />
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[[Image:SM2008_4.gif|300px|thumb|right|'''Figure 4.''' Quenched samples starting at zero shear rate and going up to math>\dot\gamma = 10 s^{-1}</math>. Vorticity is indicated by the white line in the first panel.]]<br />
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[[Image:SM2008_5.gif|300px|thumb|right|'''Figure 5.''' (a) Optical micrographs over time. (b) the FFTs of images in (a).]]<br />
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[[Image:SM2008_6.gif|300px|thumb|right|'''Figure 6.''' Microstructure after shear thickening in parallel plate geometry with gap sizes, (a) <math>d = 25 \mu m</math>, (b) <math>d = 50 \mu m</math>, (c) <math>d = 100 \mu m</math>, (d) <math>d = 250 \mu m</math>.]]<br />
<br />
Soft materials and complex fluids often form structures in response to flow around them. In Brownian systems with hard spheres, distortion will occur when the timescale for flow is less than for the particles' diffusion, as described by the Péclet number (represents flow force over thermal/diffusive). In a system where flow dominates, the particles separate in one direction and aggregate in another, forming strings in dilute solutions. Shear thickening happens in more concentrated solutions, but usually not in colloidal systems with attractive interactions (they tend to form flocculated gels).<br />
<br />
The authors studied steady state flow behavior of dilute, simple hydrocarbon dispersions of carbon black particles and were actually able to observe shear thickening under certain conditions (above a critical flow rate <math>\dot \gamma_c\approx 10^2-10^3 s^{-1}</math>. They also found that the shear modulus of the gels had a power law dependence on the pre-shear stress in the system, and deforming the thickened gells at lower shear rates created ordered vorticity aligned aggregates that broke down into small isotropic clusters given enough time (~300s).<br />
<br />
== Results and discussion ==<br />
<br />
Samples under steady shear displayed thixotropy, meaning they are normally thick and highly viscous but flow when stressed. The curves also indicate a composition-dependent shear thickening transition, mentioned previously. Optically observing the systems showed that at low shear stress, the system contained large pieces of broken gel that broke into subsequently smaller pieces when shear was increased. The system's viscosity began to increase at <math>\dot \gamma\approx 10^0-10^1 s^{-1}</math> and clusters of particles aggregated along the vorticity axis (Figure 2a). This was considered to be the steady state response of the system.. They then found that as the shear rate increased, the structures went from being cylindrical to isotropic clusters that become more and more dense until the aggregates break and the transition to shear thickening flow happens.<br />
<br />
The system was found to have large negative normal stresses at high shear rates (no changes seen at low rates). If the shear was stopped suddenly (quenching), the effect persisted for long times unless shear flow was applied again, in which case it went back to the original state quickly. This relaxation of internal shear stresses was determined to have a power law dependence <math>\sigma_i ~ t^{-0.1}</math>.<br />
<br />
Deforming quenched shear thickened gels quickly resulted in highly anisotropic vorticity aligned structures that were much more defined than the steady state response the authors looked at first (Figure 4). The aspect ratios of the cylinders were greater than the macroscopic portions of the shear cell as well. They studied this phenomenon using Fourier transformations of the images to monitor the development of alignment in the system (Figure 5) - it turned out to be rapid and peaked at 20s with <math>\gamma=200</math>.<br />
<br />
The authors observed that the width of the cylindrical structures was slightly larger than and proportional to the gap but periodicity doesn't change much, meaning the flocs become less dense as they become thicker.<br />
<br />
Structure dissolution occurred first at the outside edge of the rheometer geometry both when using a parallel plate (highest shear at edge) as well as with an angled cone (constant shear across plate), so the authors suggest that confinement is important for stabilization of the flocs. (Figure 6)<br />
<br />
The frequency and strain dependence of the elastic modulus of the flocs was different from the shear thickened gel, but both showed strong elastic responses. Yield strain was lower for structures than for gel and its presence indicates that the rolling of the flocs is not sufficient to account for the displacement and deformation that comes from the applied shear.<br />
<br />
The observations made by the authors of this paper are similar to an elastic instability associated with what is seen in semi-dilute non-Brownian nanotubes that are undergoing clustering due to flow. Also, these results provide insight as to the effect of confinement and composition on the mechanics of such structures.<br />
<br />
== Experimental Setup ==<br />
<br />
Tetradecane dispersions of 8% 0.5 <math>\mu</math>m carbon black particles were used. Experiments were done using rheometers, sometimes combined with an optical observation component.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Smooth_Cascade_of_Wrinkles_at_the_Edge_of_a_Floating_Elastic_Film&diff=16115Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film2010-12-03T21:38:33Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
[[Image:PRL038302_1.jpg|300px|thumb|right|'''Figure 1.''' (a) Image of the wrinkled PS sheet, (b) geometry of the system, (c) wavelength of wrinkles as a function of thickness, which is fit with <math>q_0=(\frac{\rho g}{B})^{1/4}</math> .]]<br />
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== Reference ==<br />
<br />
[http://arxiv.org/abs/0908.4358 "Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film"]<br />
<br />
J. Huang, B. Davidovitch, C. D. Santangelo, T. P. Russell, N. Menon, ''Physical Review Letters'', '''105''', 038302 (2010).<br />
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<br />
== Keywords ==<br />
[[thin films]]<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL038302_2.jpg|300px|thumb|right|'''Figure 2.''' (a) Wave number as a function of distance from the edge, which is almost independent of thickness. (b) Wave number vs. distance scaled with capillary length. (c) same as (a), scaled with distance.]]<br />
<br />
[[Image:PRL038302_4.jpg|300px|thumb|right|'''Figure 4.''' (a) Image of cascade. (b) Histogram of separation between crests (scaled with <math>q_0d/(2\pi)</math> at x distances from the edge.]]<br />
<br />
At the edge of a pattern, the symmetry usually breaks due to the boundary's tendency to be flat. The authors of this paper studied wrinkling using this phenomenon by looking at a thin sheet that has a pattern due to elastic instability and proposed a mechanism by which the pattern and the flat boundary can both exist.<br />
<br />
When a thin rectangular sheet floating on a liquid surface is compressed from two sides along the same axis, it forms the pattern in Figure 1 - large wrinkles of wavelength <math>\lambda \ll</math> the width of the sheet and much smaller wrinkles near the boundary. The wave amplitude is expected to decrease to minimize the surface energy of the interface and the wave number should increase to maintain inextensibility. This model has a point where the cost of bending offsets the gain in surface energy. The paper explores parameters that affect the wave cascade, the amplification of wavelength, and the length of the cascade.<br />
<br />
== Results and discussion ==<br />
<br />
The authors propose that 2 principles determine the pattern we see:<br />
# a thin sheet is basically inextensible, so the wavelength and amplitude are proportional<br />
# the wavelength is a compromise between the bending energy (favors long wavelengths) and gravitational energy (favors small ones).<br />
They demonstrate that the scaling of the wrinkles in the bulk goes as <math>q_0=(\frac{\rho g}{B})^{1/4}</math> where <math>q_0</math> is the wave number, B is the bending modulus, <math>\rho</math> is the fluid density. (see Figure 1c for the graph)<br />
<br />
Figure 2 shows that the increase in wave number at the edge happens at approximately the same distance for all film thicknesses (with systematic deviations). This penetration distance is around <math>1.8 \pm 0.2mm</math>. They used energy calculations, estimating the cost of a wave number at the edge as well as the effect of breaking symmetry in the wave pattern. They found that persistence length <math>l_p \approx</math> between <math>l_c</math>, the capillary length, and <math>1/q_e</math> at various wave numbers between the edge and the large bulk wrinkles.<br />
<br />
A current model for such elastic cascades was proposed by Pomeau and Rica, which explained the smaller wrinkles near the edge using a branching hierarchy. However, the system observed in the paper behaved differently than this would predict, particularly that the smooth amplitude reduction reflects a finite number of Fourier components being mixed as you approach the edge. The Pomeau-Rica case would predict sharp ridges and folds rather than a smooth cascade. Figure 4 shows the observed cascade and also a histogram of the separation between wave crests. If the branching theory were correct, there would be peaks at the lower end of the separation axis for each distance from the edge as the two waves become one.<br />
<br />
Gravity, bending, and capillarity forces are all important in this system but they all scale differently. The authors suggest that they can be used to tune the system independently and allow for further studies.<br />
<br />
== Experimental Setup ==<br />
<br />
Polystyrene (PS) sheets of dimensions 3x2cm with thicknesses between 50-400nm were prepared by spin coating onto glass substrates, then transferred to a dish of DI water. The sheet floated because of the hydrophobic nature of PS.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Microrheology_Probes_Length_Scale_Dependent_Rheology&diff=16114Microrheology Probes Length Scale Dependent Rheology2010-12-03T21:37:46Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
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== Reference ==<br />
<br />
[http://crocker.seas.upenn.edu/Liu2006.pdf "Microrheology probes length scale dependent rheology"]<br />
<br />
M. L. Gardel, K. Kroy, E. Frey, B. D. Hoffman, J. C. Crocker, A. R. Bausch, D. A. Weitz, ''Physical Review Letters'', '''96''', 118104 (2006).<br />
<br />
<br />
== Keywords ==<br />
[[rheology]]<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL118104_1.jpg|300px|thumb|right|'''Figure 1.''' Comparison of 1P and 2P MSDs with lengths (a) 0.5<math>\mu</math>m, (b) 2<math>\mu</math>m, (c) 5<math>\mu</math>m, (d) 17<math>\mu</math>m. The open boxes are 1P values and the filled ones are for 2P.]]<br />
<br />
[[Image:PRL118104_2.jpg|300px|thumb|right|'''Figure 2.''' Comparison of 1P and 2P <math>G'</math>, <math>G''</math> against the frequency, <math>\omega</math> for lengths (a) 0.5<math>\mu</math>m, (b) 2<math>\mu</math>m, (c) 5<math>\mu</math>m, (d) 17<math>\mu</math>m. The open boxes are <math>G'</math> and the filled ones are for <math>G''</math>.]]<br />
<br />
The researchers looked at the mechanical response of a semiflexible polymer (F-actin in this case) at different length scales using microrheology. Semiflexible polymers become entangled at low concentrations, and they are sterically hindered at the entanglement length <math>l_e</math>, which is related to the distance between polymers and the persistence length. At intermediate frequencies, there is a transition where the mechanical response is dependent on the filament length, <math>L</math>, which is not predicted by theory. The researchers identified fluctuations over <math>L</math> as a relaxation mechanism between 0.1-30rad/s and used 2-particle (2P) microrheology to look at lengths >5<math>\mu</math>m and 1-particle microrheology for lengths ~<math>l_e</math>. 2P microrheology showed increased viscoelastic relaxation for intermediate frequencies that scaled as <math>L^2</math><br />
<br />
== Results and discussion ==<br />
<br />
For varied L values, the researchers demonstrated that longer filaments show a transition in particle motion where before a certain point, mean squared displacement (MSD) changes with respect to <math>\tau</math> (raised to a factor) but then switches to little time evolution. Figure 1 shows the MSD values for various particle sizes. For <math>L > 2 \mu m</math>, the 1P MSD is more constrained, but behavior is similar. The switching time between regimes was similar across lengths. The 2P displacement correlation tensor was scaled to the 2P MSD, <math>a</math>. When <math>L</math> was about <math>a</math>, 2P and 1P MSD values were similar. However, the 2P MSD changed in both slope and magnitude as <math>L</math> changed, unlike the 1P MSD.<br />
<br />
The generalized Stokes-Einstein relation was used to compare 1P to 2P MSDs. For small <math>L</math>, they mostly matched, but longer filaments show differences. (see Figure 2, showing <math>G'</math>, <math>G''</math> against the frequency, <math>\omega</math>) A transition from sloping up to a plateau is once again seen for the 1P microrheology over <math>L</math>, whereas 2P microrheology is dependent on <math>L</math>. They converge to similar values at low frequency.<br />
<br />
From their data, the researchers say that the 1P microrheology probes bending fluctuations of single filaments at various frequencies, where single-filament dynamics dominate until they become entangled at <math>l_e</math>. The plateau comes from the entanglement. Based on the data, the authors suggest that 1P microrheology may be useful for measurements of cross-linked networks of semiflexible filaments.<br />
<br />
In contrast, the 2P results scale as <math>\tau_m </math>~<math> L^2</math>, like in diffusion, and indicate density fluctuations along the filament (expected because transverse thermal fluctuations result in varying quantities of material present at a specific point). 2P microrheology probes longer lengths than 1P.<br />
<br />
The authors conclude that mechanical response changes as length scales in the system vary and that rheology can be used to learn about network geometry and filament properties.<br />
<br />
<br />
== Experimental Setup ==<br />
<br />
G-actin mixed with polystyrene particles and then polymerized. <math>L</math> was varied using gelsolin. Particle motions were recorded with a fast camera.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Bubble_formation_via_multidrop_impacts&diff=16113Bubble formation via multidrop impacts2010-12-03T21:35:48Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
[http://www.chms.ucdavis.edu/research/web/ristenpart/bick_POF_2010.pdf "Bubble formation via multidrop impacts"]<br />
<br />
A. G. Bick, W. D. Ristenpart, E. A. van Nierop, H. A. Stone, ''Physics of Fluids'', '''22''', 042105 (2010).<br />
<br />
[[Image:PhF042105_1.jpg|300px|thumb|right|'''Figure 1.''' (a) experimental setup used in paper - syringe pump suspended over liquid. (b) sequence of drop images. the crater can be seen in the center panels and a bubble is in the rightmost one.]]<br />
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== Keywords ==<br />
bubbles<br />
<br />
== Overview ==<br />
<br />
[[Image:PhF042105_6.jpg|300px|thumb|right|'''Figure 6.''' (a) histograms showing <math>\delta t</math> for impacts that didn't form a bubble, and (b) ones that did.]]<br />
<br />
[[Image:PhF042105_2.jpg|300px|thumb|left|'''Figure 2.''' (a) phase diagram of the bubble formation regimes. (b) phase diagram comparing regular and multidrop bubble formation regions, Froude vs Weber numbers.]]<br />
<br />
Foams consist of many gas bubbles trapped in a continuous solid or liquid phase. One common scenario where a foam arises is when 2 liquids are mixed, which can be an undesirable state for certain procedures.<br />
<br />
Foams have been studied for a long time, but their generation is not well characterized, especially in situations with less control (liquid being poured into another liquid). This group of researches demonstrated that bubbles could be formed by the impact of smaller and faster drops than previously used (Figure 2, Franz label) if the timing between 2 successive drops is correct. They call this using a "multidrop impact" to create a bubble. Their results suggest mechanical reasons for lack of bubble formation when liquids are mixed.<br />
<br />
== Results and discussion ==<br />
<br />
The geometry of the surface as the drop falls was seen to be a crater (Figure 1b). When nothing else impacts the area afterwards, the surface recovers its initial flat state. Occasionally (~2% of the time), bubbles can be generated by vortices created by the single drop. When a second drop does hit the surface, a bubble can be formed. Qualitatively, a similar result was obtained both in cases where a surfactant was added and in carbonated liquids (they used beer).<br />
<br />
Part of the analysis was based on the Froude number (comparision between the inertial and gravitational effects) and the Weber number (comparision between inertial to surface tension effects). Figure 2 shows that multidrop bubbles formed at similar Froude numbers as regular bubbles, but the Weber numbers were an order of magnitude lower (higher surface tension effect).<br />
<br />
Analysis of crater depths showed that there is a critical depth which the crater must exceed in order for bubbles to form, and that depth was about the same as the capillary length <math>\ell_c=\sqrt{\gamma / \rho y} \approx 2mm</math>. <5% of craters fulfilled this criterion. Drop size and order did not seem to affect probability of bubble generation.<br />
<br />
Figure 6 in the paper contains a histogram indicating that the critical time between drops is <math>\delta t \le 5-20ms</math>.<br />
<br />
The authors propose that capillary forces and inertia govern the crater dynamics, and that this determines that the critical time interval is <math>t_c ~ (\frac{\rho h^3}{\gamma})^{1/2}</math>.<br />
<br />
== Description of Experimental Setup ==<br />
<br />
Drops were formed using a syringe pump suspended over a pool of liquid, usually distilled water, shown in Figure 1a.<br />
<br />
A high speed camera was used to observe the drops.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Curvature_condensation_and_bifurcation_in_an_elastic_shell&diff=16112Curvature condensation and bifurcation in an elastic shell2010-12-03T21:33:48Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
[http://www.seas.harvard.edu/softmat/downloads/2007-01.pdf "Curvature condensation and bifurcation in an elastic shell"]<br />
<br />
M. Das, A. Vaziri, A. Kudrolli, L. Mahadevan, ''Physical Review Letters'', '''98''', 014301 (2007).<br />
<br />
<br />
== Keywords ==<br />
elastic shell, defects<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL014301_2.jpg|300px|thumb|left|'''Figure 2.''' Figure of the system being studied. (a-d) The deformation of the sheet along the axis of symmetry for the cylinder. (e-h) Colormap representation of the curvature going from blue (low) to red (high).]]<br />
<br />
Elastic systems often have a hierarchy in structural scales, but the process by which defects form is unknown. The researchers study a thin mylar sheet that has been bent into a half-cylindrical elastic shell, then indented the sheet at one edge along the axis of symmetry. The sheet was reconstructed using laser aided tomography.<br />
<br />
== Results and discussion ==<br />
<br />
[[Image:PRL014301_3.jpg|300px|thumb|right|'''Figure 3.''' (a) The transition between global and local deformation modes is indicated. Filled and open circles are the condensate before and after twinning, respectively. (b) Looking at the Gauss curvature along the axis of symmetry shows a transition at a secondary maximum in curvature.]]<br />
<br />
[[Image:PRL014301_4.jpg|300px|thumb|right|'''Figure 4.''' (a) The location of the condensate as a function of the indentation (normalized) for different thicknesses. Dotted & dashed lines represent saturation. (b) Location of defect near a bifurcation for various thicknesses.]]<br />
<br />
They observed that for small indentations, the deformation was strong in the immediate area of the edge, but the curvature decayed monotonically away from the edge. Past a critical threshold, another maximum in curvature appears on the symmetry axis that looks like a parabolic defect (a "curvature condensate"). Indenting even more results in the defect bifurcating, and farther beyond that point, the shell folds inwards rather than continuing to move defects. There is no dependence on material parameters (the system only has a dimensionless parameter <math>R/t</math> (R=radius, t=thickness) because of the Young's modulus and thickness).<br />
<br />
The authors used numerical methods to study the system, minimizing the elastic energy of the shell with an energy density that had components of energy from in-plane deformations and from out-of-plane deformations. Figure 2 shows the system as observed experimentally and the simulation results.<br />
<br />
They studied the transition between 2 modes of deformation - global and local - and determined that there is a threshold at which it happens, indicated by an inflection point in the graph of the Gauss curvature <math>\kappa_G(0,y)</math> as a function of the scaled indentation <math>\delta/R</math>. This only happens in 2D systems.<br />
<br />
It was also shown that the location where the condensation happens scales with the indentation size until a saturation point (Figure 4a), and that saturation value increases with <math>t</math>. The indentation size necessary for bifurcation, or twinning, was found to go as <math>t^{1/2}</math>. Above the indentation size <math>\delta_{bif}</math> where this happens, the condensates move from the axis of symmetry in a symmetrical manner (Figure 4b).<br />
<br />
The curvature condensate was localized along a crescent shape and the width and radius of curvature were similar to results from a previous paper about the size of the defect core when a sheet is bent into a cone shape.<br />
<br />
The Donnell-Föppl-von Kármán equations for large deflections of thin, flat plates were successfully applied to this system to represent the deformations.<br />
<br />
The paper gives a general, qualitative overview of curvature condensates, but more work still needs to be done to obtain a quantitative result.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Curvature_condensation_and_bifurcation_in_an_elastic_shell&diff=16111Curvature condensation and bifurcation in an elastic shell2010-12-03T21:28:06Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Curvature condensation and bifurcation in an elastic shell"<br />
<br />
M. Das, A. Vaziri, A. Kudrolli, L. Mahadevan, ''Physical Review Letters'', '''98''', 014301 (2007).<br />
<br />
<br />
== Keywords ==<br />
elastic shell, defects<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL014301_2.jpg|300px|thumb|left|'''Figure 2.''' Figure of the system being studied. (a-d) The deformation of the sheet along the axis of symmetry for the cylinder. (e-h) Colormap representation of the curvature going from blue (low) to red (high).]]<br />
<br />
Elastic systems often have a hierarchy in structural scales, but the process by which defects form is unknown. The researchers study a thin mylar sheet that has been bent into a half-cylindrical elastic shell, then indented the sheet at one edge along the axis of symmetry. The sheet was reconstructed using laser aided tomography.<br />
<br />
== Results and discussion ==<br />
<br />
[[Image:PRL014301_3.jpg|300px|thumb|right|'''Figure 3.''' (a) The transition between global and local deformation modes is indicated. Filled and open circles are the condensate before and after twinning, respectively. (b) Looking at the Gauss curvature along the axis of symmetry shows a transition at a secondary maximum in curvature.]]<br />
<br />
[[Image:PRL014301_4.jpg|300px|thumb|right|'''Figure 4.''' (a) The location of the condensate as a function of the indentation (normalized) for different thicknesses. Dotted & dashed lines represent saturation. (b) Location of defect near a bifurcation for various thicknesses.]]<br />
<br />
They observed that for small indentations, the deformation was strong in the immediate area of the edge, but the curvature decayed monotonically away from the edge. Past a critical threshold, another maximum in curvature appears on the symmetry axis that looks like a parabolic defect (a "curvature condensate"). Indenting even more results in the defect bifurcating, and farther beyond that point, the shell folds inwards rather than continuing to move defects. There is no dependence on material parameters (the system only has a dimensionless parameter <math>R/t</math> (R=radius, t=thickness) because of the Young's modulus and thickness).<br />
<br />
The authors used numerical methods to study the system, minimizing the elastic energy of the shell with an energy density that had components of energy from in-plane deformations and from out-of-plane deformations. Figure 2 shows the system as observed experimentally and the simulation results.<br />
<br />
They studied the transition between 2 modes of deformation - global and local - and determined that there is a threshold at which it happens, indicated by an inflection point in the graph of the Gauss curvature <math>\kappa_G(0,y)</math> as a function of the scaled indentation <math>\delta/R</math>. This only happens in 2D systems.<br />
<br />
It was also shown that the location where the condensation happens scales with the indentation size until a saturation point (Figure 4a), and that saturation value increases with <math>t</math>. The indentation size necessary for bifurcation, or twinning, was found to go as <math>t^{1/2}</math>. Above the indentation size <math>\delta_{bif}</math> where this happens, the condensates move from the axis of symmetry in a symmetrical manner (Figure 4b).<br />
<br />
The curvature condensate was localized along a crescent shape and the width and radius of curvature were similar to results from a previous paper about the size of the defect core when a sheet is bent into a cone shape.<br />
<br />
The Donnell-Föppl-von Kármán equations for large deflections of thin, flat plates were successfully applied to this system to represent the deformations.<br />
<br />
The paper gives a general, qualitative overview of curvature condensates, but more work still needs to be done to obtain a quantitative result.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL014301_4.jpg&diff=16110File:PRL014301 4.jpg2010-12-03T21:26:48Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL014301_3.jpg&diff=16109File:PRL014301 3.jpg2010-12-03T21:26:33Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL014301_2.jpg&diff=16108File:PRL014301 2.jpg2010-12-03T21:26:12Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Curvature_condensation_and_bifurcation_in_an_elastic_shell&diff=16107Curvature condensation and bifurcation in an elastic shell2010-12-03T21:24:04Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Curvature condensation and bifurcation in an elastic shell"<br />
<br />
M. Das, A. Vaziri, A. Kudrolli, L. Mahadevan, ''Physical Review Letters'', '''98''', 014301 (2007).<br />
<br />
<br />
== Keywords ==<br />
elastic shell, defects<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL014301_2.jpg|300px|thumb|right|'''Figure 2.''' Figure of the system being studied. (a-d) The deformation of the sheet along the axis of symmetry for the cylinder. (e-h) Colormap representation of the curvature going from blue (low) to red (high).]]<br />
<br />
Elastic systems often have a hierarchy in structural scales, but the process by which defects form is unknown. The researchers study a thin mylar sheet that has been bent into a half-cylindrical elastic shell, then indented the sheet at one edge along the axis of symmetry. The sheet was reconstructed using laser aided tomography.<br />
<br />
== Results and discussion ==<br />
<br />
[[Image:PRL014301_3.jpg|300px|thumb|right|'''Figure 3.''' (a) The transition between global and local deformation modes is indicated. Filled and open circles are the condensate before and after twinning, respectively. (b) Looking at the Gauss curvature along the axis of symmetry shows a transition at a secondary maximum in curvature.]]<br />
<br />
[[Image:PRL014301_4.jpg|300px|thumb|right|'''Figure 4.''' (a) The location of the condensate as a function of the indentation (normalized) for different thicknesses. Dotted & dashed lines represent saturation. (b) Location of defect near a bifurcation for various thicknesses.]]<br />
<br />
They observed that for small indentations, the deformation was strong in the immediate area of the edge, but the curvature decayed monotonically away from the edge. Past a critical threshold, another maximum in curvature appears on the symmetry axis that looks like a parabolic defect (a "curvature condensate"). Indenting even more results in the defect bifurcating, and farther beyond that point, the shell folds inwards rather than continuing to move defects. There is no dependence on material parameters (the system only has a dimensionless parameter <math>R/t</math> (R=radius, t=thickness) because of the Young's modulus and thickness).<br />
<br />
The authors used numerical methods to study the system, minimizing the elastic energy of the shell with an energy density that had components of energy from in-plane deformations and from out-of-plane deformations. Figure 2 shows the system as observed experimentally and the simulation results.<br />
<br />
They studied the transition between 2 modes of deformation - global and local - and determined that there is a threshold at which it happens, indicated by an inflection point in the graph of the Gauss curvature <math>\kappa_G(0,y)</math> as a function of the scaled indentation <math>\delta/R</math>. This only happens in 2D systems.<br />
<br />
It was also shown that the location where the condensation happens scales with the indentation size until a saturation point (Figure 4a), and that saturation value increases with <math>t</math>. The indentation size necessary for bifurcation, or twinning, was found to go as <math>t^{1/2}</math>. Above the indentation size <math>\delta_{bif}</math> where this happens, the condensates move from the axis of symmetry in a symmetrical manner (Figure 4b).<br />
<br />
The curvature condensate was localized along a crescent shape and the width and radius of curvature were similar to results from a previous paper about the size of the defect core when a sheet is bent into a cone shape.<br />
<br />
The Donnell-Föppl-von Kármán equations for large deflections of thin, flat plates were successfully applied to this system to represent the deformations.<br />
<br />
The paper gives a general, qualitative overview of curvature condensates, but more work still needs to be done to obtain a quantitative result.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=16106Helen Wu2010-12-03T20:36:01Z<p>Huayinwu: </p>
<hr />
<div>Definitions:<br />
<br />
<br />
<br />
Weekly wiki entries:<br />
<br />
[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
<br />
[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
<br />
[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
<br />
[[Microrheology Probes Length Scale Dependent Rheology]] - ''October 20, 2010''<br />
<br />
[[Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film]] - ''November 1, 2010''<br />
<br />
[[Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System]] - ''November 3, 2010''<br />
<br />
[[Microrheology of entangled F-actin solutions]] - ''December 2, 2010''<br />
<br />
[[Measurement of nonlinear rheology of cross-linked biopolymer gels]] - ''December 3, 2010''<br />
<br />
[[Curvature condensation and bifurcation in an elastic shell]] - ''December 3, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=16105Helen Wu2010-12-03T20:35:02Z<p>Huayinwu: </p>
<hr />
<div>Definitions:<br />
<br />
<br />
<br />
Weekly wiki entries:<br />
<br />
[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
<br />
[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
<br />
[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
<br />
[[Microrheology Probes Length Scale Dependent Rheology]] - ''October 20, 2010''<br />
<br />
[[Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film]] - ''November 1, 2010''<br />
<br />
[[Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System]] - ''November 3, 2010''<br />
<br />
[[Microrheology of entangled F-actin solutions]] - ''December 2, 2010''<br />
<br />
[[Measurement of nonlinear rheology of cross-linked biopolymer gels]] - ''December 3, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Measurement_of_nonlinear_rheology_of_cross-linked_biopolymer_gels&diff=16104Measurement of nonlinear rheology of cross-linked biopolymer gels2010-12-03T20:33:40Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Measurement of nonlinear rheology of cross-linked biopolymer gels"<br />
<br />
C. P. Broedersz, K. E. Kasza, L. M. Jawerth, S. Münster, D. A. Weitz, ''Soft Matter'', '''6''', 4120-4127 (2010).<br />
<br />
<br />
== Keywords ==<br />
rheology, gel, biopolymer<br />
<br />
== Overview ==<br />
<br />
[[Image:SM201064120_1.jpg|300px|thumb|right|'''Figure 1.''' Linear rheology of various systems. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
<br />
[[Image:SM201064120_4.jpg|300px|thumb|right|'''Figure 4.''' Comparison between prestress and strain ramp methods. The graphs are the tangent (strain ramp) or differential (prestress) modulus K normalized by the linear modulus <math>G_0</math> plotted against stress <math>\sigma</math>. Symbols represent the prestress method and lines represent the strain ramp protocol at various strain rates. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
<br />
[[Image:SM201064120_A.jpg|300px|thumb|right|'''Figure A.''' An illustration of the system the model represents - a nonlinear Kelvin-Voigt element in series with another dashpot. (created for this entry; not from paper)]]<br />
<br />
[[Image:SM201064120_5.jpg|300px|thumb|right|'''Figure 5.''' (a) Strain as a function of time for prestress pulses as predicted by the model. Strain accumulates in a normal network(red), but not in a crosslinked one (blue). (b) Differential modulus as a function of stress calculated for both the prestress method (symbols) and the strain ramp (lines).]]<br />
<br />
Biopolymer networks, both intracellular and extracellular, exhibit interesting mechanical responses: highly nonlinear, strain stiffening, as well as large, negative normal stresses under shear. The stiffening may prevent large deformations that would be harmful to cells; however, there are situations (invatsion, division) where remodeling of networks is necessary. The two seem to be contradictory, and this complication means that performing traditional rheology will be insufficient.<br />
<br />
The authors of this paper study the nonlinear response of biopolymer gels (primarily F-actin, with some ) using two different methods: a prestress protocol and a strain ramp, to determine which one is more suitable. Using the data collected, they also create a model to represent the system. It was found that for permanent networks, the two protocols are equally suitable, but for transient networks, it is true only at high strain rates. The prestress protocol was insensitive to creep.<br />
<br />
== Results and discussion ==<br />
<br />
The actin system's linear viscoelastic modulus was characterized first and was found to have a very weak frequency dependence (Figure 1). The fibrin network's elastic modulus was independent of frequency.<br />
<br />
The nonlinear response was measured by the two methods. The prestress method measures the response at a specific frequency; the strain ramp uses a fixed rate: <br />
<br />
''Strain ramp protocol''<br />
Both crosslinked and uncrosslinked F-actin networks were found to have measurements that depended on <math>\dot\gamma</math>.<br />
''Prestress protocol''<br />
<br />
Again, both permanent F-actin networks and pure solutions had similar responses - the elastic differential modulus <math>K'</math> increased rapidly with applied prestress but had no time-dependence. In all systems tested, <math>K'</math> leveled off (the rate differed though). The response relaxed back to the initial linear modulus very soon after the stress was removed and both linear and nonlinear properties displayed no hysteresis.<br />
<br />
''Comparison''<br />
The main difference between the two results was that the strain ramp method had a rate dependence (Fig 4) while the prestress method had no hysteresis despite creep in the system.<br />
<br />
== Model ==<br />
<br />
Based on these results, the authors proposed a model for use while studying these systems:<br />
<br />
''Assumptions''<br />
# network repsonds instantaneously to an applied stress<br />
# two components of the strain: reversible (<math>\gamma_e</math>) and network flow (<math>\gamma_f</math>) for a total strain of <math>\gamma = \gamma_e + \gamma_f</math>; this implies that stresses are always equal <math>\sigma=\sigma_e=\sigma_f</math><br />
#single relaxation timescale, so the network is treated like a simple liquid with viscosity <math>\zeta\gg\eta</math><br />
<br />
Reversible deformation can be described as: <math>\frac{d\sigma}{dt}=\left[k(\sigma)+\eta\frac{d}{dt}\right]\frac{d\gamma_e}{dt}</math><br />
<br />
where <math>k(\sigma)</math> is the elasticity and <math>\eta</math> is the viscosity and they are acting in parallel. This is a nonlinear generalization of the Kelvin-Voigt model (dashpot and spring in parallel).<br />
<br />
Stress relaxation is given by: <math>\frac{d\sigma}{dt}=\zeta\frac{d^2\gamma_f}{dt^2}</math><br />
<br />
This describes a Newtonian liquid-like system.<br />
<br />
By the second assumption, the stresses in the above equations can be equated to represent a second dashpot in series with the Kelvin-Voigt system (Figure A).<br />
<br />
In the cases of the protocols tested in this paper, the equations could be further developed. The prestress protocol had a time-independent prestress <math>\sigma_0</math> and an oscillatory stress <math>\delta\sigma(t)</math>. There was also a time-dependent creep response <math>\gamma_0(t)</math> and a small amplitude oscillatory strain <math>\delta\gamma(t)</math>. The paper shows that taking these into account produces a good fit with the experimental data.<br />
<br />
The general nonlinear response is given by the differential equation:<br />
<br />
<math>\left(1+\frac{\eta}{\zeta}\right)\frac{d^2\sigma}{dt^2}+\frac{k(\sigma)}{\zeta}\left(1-(\eta+\zeta)\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d\sigma}{dt}=k(\sigma)\left(1-\eta\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d^2\gamma}{dt^2}+\eta\frac{d^3\gamma}{dt^3}</math><br />
<br />
Figure 5 shows that the calculated response based on these equations. In 5a, we see strain accumulation in the normal networks (red line), but none for crosslinked ones (blue).<br />
<br />
== Conclusions ==<br />
<br />
The prestress and strain ramp methods are both fine for cross-linked networks. For networks that creep, the prestress method is the better choice.<br />
<br />
The model describes experimental data and shows how the differential nonlinear elastic response can be determined by the prestress method when there is creep. This is because the differential and steady stress and strain components are decoupled in the equations.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM201064120_A.jpg&diff=16103File:SM201064120 A.jpg2010-12-03T20:31:36Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM201064120_5.jpg&diff=16102File:SM201064120 5.jpg2010-12-03T20:31:26Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM201064120_4.jpg&diff=16101File:SM201064120 4.jpg2010-12-03T20:31:16Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM201064120_1.jpg&diff=16100File:SM201064120 1.jpg2010-12-03T20:30:56Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Measurement_of_nonlinear_rheology_of_cross-linked_biopolymer_gels&diff=16099Measurement of nonlinear rheology of cross-linked biopolymer gels2010-12-03T20:27:08Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Measurement of nonlinear rheology of cross-linked biopolymer gels"<br />
<br />
C. P. Broedersz, K. E. Kasza, L. M. Jawerth, S. Münster, D. A. Weitz, ''Soft Matter'', '''6''', 4120-4127 (2010).<br />
<br />
<br />
== Keywords ==<br />
rheology, gel, biopolymer<br />
<br />
== Overview ==<br />
<br />
[[Image:SM201064120_1.jpg|300px|thumb|right|'''Figure 1.''' Linear rheology of various systems. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
<br />
[[Image:SM201064120_4.jpg|300px|thumb|right|'''Figure 4.''' Comparison between prestress and strain ramp methods. The graphs are the tangent (strain ramp) or differential (prestress) modulus K normalized by the linear modulus <math>G_0</math> plotted against stress <math>\sigma</math>. Symbols represent the prestress method and lines represent the strain ramp protocol at various strain rates. (a) NeutrAvidin-crosslinked F-actin. (b) F-actin solution. (c) filamin-crosslinked actin. (d) fibrin.]]<br />
<br />
[[Image:SM201064120_A.jpg|300px|thumb|right|'''Figure A.''' An illustration of the system the model represents - a nonlinear Kelvin-Voigt element in series with another dashpot. (created for this entry; not from paper)]]<br />
<br />
[[Image:SM201064120_5.jpg|300px|thumb|right|'''Figure 5.''' (a) Strain as a function of time for prestress pulses as predicted by the model. Strain accumulates in a normal network(red), but not in a crosslinked one (blue). (b) Differential modulus as a function of stress calculated for both the prestress method (symbols) and the strain ramp (lines).]]<br />
<br />
Biopolymer networks, both intracellular and extracellular, exhibit interesting mechanical responses: highly nonlinear, strain stiffening, as well as large, negative normal stresses under shear. The stiffening may prevent large deformations that would be harmful to cells; however, there are situations (invatsion, division) where remodeling of networks is necessary. The two seem to be contradictory, and this complication means that performing traditional rheology will be insufficient.<br />
<br />
The authors of this paper study the nonlinear response of biopolymer gels (primarily F-actin, with some ) using two different methods: a prestress protocol and a strain ramp, to determine which one is more suitable. Using the data collected, they also create a model to represent the system. It was found that for permanent networks, the two protocols are equally suitable, but for transient networks, it is true only at high strain rates. The prestress protocol was insensitive to creep.<br />
<br />
== Results and discussion ==<br />
<br />
The actin system's linear viscoelastic modulus was characterized first and was found to have a very weak frequency dependence (Figure 1). The fibrin network's elastic modulus was independent of frequency.<br />
<br />
The nonlinear response was measured by the two methods. The prestress method measures the response at a specific frequency; the strain ramp uses a fixed rate: <br />
<br />
''Strain ramp protocol''<br />
Both crosslinked and uncrosslinked F-actin networks were found to have measurements that depended on <math>\dot\gamma</math>.<br />
''Prestress protocol''<br />
<br />
Again, both permanent F-actin networks and pure solutions had similar responses - the elastic differential modulus <math>K'</math> increased rapidly with applied prestress but had no time-dependence. In all systems tested, <math>K'</math> leveled off (the rate differed though). The response relaxed back to the initial linear modulus very soon after the stress was removed and both linear and nonlinear properties displayed no hysteresis.<br />
<br />
''Comparison''<br />
The main difference between the two results was that the strain ramp method had a rate dependence (Fig 4) while the prestress method had no hysteresis despite creep in the system.<br />
<br />
== Model ==<br />
<br />
Based on these results, the authors proposed a model for use while studying these systems:<br />
<br />
''Assumptions''<br />
# network repsonds instantaneously to an applied stress<br />
# two components of the strain: reversible (<math>\gamma_e</math>) and network flow (<math>\gamma_f</math>) for a total strain of <math>\gamma = \gamma_e + \gamma_f</math>; this implies that stresses are always equal <math>\sigma=\sigma_e=\sigma_f</math><br />
#single relaxation timescale, so the network is treated like a simple liquid with viscosity <math>\zeta\gg\eta</math><br />
<br />
Reversible deformation can be described as: <math>\frac{d\sigma}{dt}=\left[k(\sigma)+\eta\frac{d}{dt}\right]\frac{d\gamma_e}{dt}</math><br />
<br />
where <math>k(\sigma)</math> is the elasticity and <math>\eta</math> is the viscosity and they are acting in parallel. This is a nonlinear generalization of the Kelvin-Voigt model (dashpot and spring in parallel).<br />
<br />
Stress relaxation is given by: <math>\frac{d\sigma}{dt}=\zeta\frac{d^2\gamma_f}{dt^2}<br />
<br />
This describes a Newtonian liquid-like system.<br />
<br />
By the second assumption, the stresses in the above equations can be equated to represent a second dashpot in series with the Kelvin-Voigt system (Figure A).<br />
<br />
In the cases of the protocols tested in this paper, the equations could be further developed. The prestress protocol had a time-independent prestress <math>\sigma_0</math> and an oscillatory stress <math>\delta\sigma(t)</math>. There was also a time-dependent creep response <math>\gamma_0(t)</math> and a small amplitude oscillatory strain <math>\delta\gamma(t)</math>. The paper shows that taking these into account produces a good fit with the experimental data.<br />
<br />
The general nonlinear response is given by the differential equation:<br />
<br />
<math>\left(1+\frac{\eta}{\zeta}\right)\frac{d^2\sigma}{dt^2}+\frac{k(\sigma)}{\zeta}\left(1-(\eta+\zeta)\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d\sigma}{dt}=k(\sigma)\left(1-\eta\frac{\frac{d}{dt}k(\sigma)}{k(\sigma)^2}\right)\frac{d^2\gamma}{dt^2}+\eta\frac{d^3\gamma}{dt^3}</math><br />
<br />
Figure 5 shows that the calculated response based on these equations. In 5a, we see strain accumulation in the normal networks (red line), but none for crosslinked ones (blue).<br />
<br />
== Conclusions ==<br />
<br />
The prestress and strain ramp methods are both fine for cross-linked networks. For networks that creep, the prestress method is the better choice.<br />
<br />
The model describes experimental data and shows how the differential nonlinear elastic response can be determined by the prestress method when there is creep. This is because the differential and steady stress and strain components are decoupled in the equations.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Shear_thickening_and_scaling_of_the_elastic_modulus_in_a_fractal_colloidal_system_with_attractive_interactions&diff=16098Shear thickening and scaling of the elastic modulus in a fractal colloidal system with attractive interactions2010-12-03T19:38:49Z<p>Huayinwu: Redirecting to Shear Thickening and Scaling of the Elastic Modulus in a Fractal Colloidal System with Attractive Interactions</p>
<hr />
<div>#REDIRECT [[Shear Thickening and Scaling of the Elastic Modulus in a Fractal Colloidal System with Attractive Interactions]]</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL158302_3.jpg&diff=16094File:PRL158302 3.jpg2010-12-02T17:43:27Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL158302_1.jpg&diff=16093File:PRL158302 1.jpg2010-12-02T17:43:02Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Microrheology_of_entangled_F-actin_solutions&diff=16092Microrheology of entangled F-actin solutions2010-12-02T17:35:59Z<p>Huayinwu: New page: Entry by Helen Wu, AP225 Fall 2010 == Reference == "Microrheology of Entangled F-Actin Solutions" M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, D. A. Weitz, ''Physical...</p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Microrheology of Entangled F-Actin Solutions"<br />
<br />
M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, D. A. Weitz, ''Physical Review Letters'', '''91''', 158302 (2003).<br />
<br />
<br />
== Keywords ==<br />
microrheology, biopolymer, semiflexible networks<br />
<br />
== Overview ==<br />
<br />
The dynamics of networks of semiflexible polymers such as filamentous actin (F-actin) are affected by many characteristic length scales and frequency scales. The polymers become entanled at very low volume fractions and the resulting networks have large elastic moduli and long relaxation times compared to flexible polymers. Since the volume fraction is so low, individual fibers are sterically hindered. However, instead of a constant elastic modulus (over various frequencies), bulk rheological measurements show a monotonically increasing <math>G'(\omega)</math> that approaches a plateau asymptotically. Thus, the authors determined the frequency and length scale dependencies in order to understand the system. They used one-particle (1P) and two-particle (2P) microrheology to accomplish this.<br />
<br />
== Results and discussion ==<br />
<br />
[[Image:PRL158302_1.jpg|300px|thumb|right|'''Figure 1.''' Comparison of 1P (filled symbols) and 2P (open symbols) MSDs.]]<br />
<br />
[[Image:PRL158302_3.jpg|300px|thumb|right|'''Figure 3.''' Comparison between elastic modulus G' (closed symbols) and loss modulus G" (open symbols) from 1P (squares) and 2P (circles) data. Conventional rheometer data (triangles) is also represented. (a) 1.0mg/ml F-actin, (b) 0.3mg/ml F-actin.]]<br />
<br />
Using the data obtained during experiments, the authors calculated the one-dimensional ensemble averaged mean-squared displacement (1P MSD) and scaled it by the particles' radii for the size-dependent viscous drag. Figure 1 shows that there was little change in this value over time for particles 0.32<math>\mu</math>m and greater, but at 0.23<math>\mu</math>m, the value increases. This was found to be due to the fact that 0.23<math>\mu</math>m particles were traveling through the network whereas the larger particles were trapped.<br />
<br />
The 2P MSD gave information about dynamics at larger length scales than the radius. It represents the one-particle motion from long-wavelength modes. Assuing the material was incompressible, the scaling factor should be 2/radius.<br />
<br />
The 1P and 2P MSDs are very different until about <math>\tau</math>=10s, where they converge (the right edge of Figure 1).<br />
<br />
The generalized Stokes-Einstein relation was used to approximate the bulk elastic modulus G'(<math>\omega</math>) and viscous modulus G"(<math>\omega</math>). Figure 3 shows that these approximations were close to the measured bulk values. 2P microrheology measures a viscoelastic response and indicates that at low frequencies (<0.1rad/s), the elastic modulus dominates. However, at intermediate frequencies (<30rad/s), longitudinal fluctuations of the filaments affect the bulk response.<br />
<br />
Looking at the 1P microrheology with the generalized Stokes-Einstein relation produced information on the origins of the viscoelasticity observed using 2P microrheology. Since 1P microrheology significanly underestimates viscoelasticity in the bulk material, particles are again permeating through the network. 1P viscoelasticity seems to be independent of both frequency and particle size. Thus, the authors suggest that the differences between 1P and 2P microrheology comes from coupling between particles.<br />
<br />
Entanglements also affect the bulk viscoelasticity. They determine the plateau elasticity <math>G_0 ~ \rho k_b T/l_e</math>, which contains terms for the filament density <math>\rho</math>. 1P and 2P microrheology were shown to both effectively measure the low frequency plateau of the modulus due to entanglement, so local heterogeneities had little to no effect on the measurements.<br />
<br />
1P microrheology may be applied to ''in vitro'' or ''in vivo'' measurements because crosslinking proteins reduce the importance of longitudinal fluctuations, which 2P microrheology can account for.<br />
<br />
== Experimental Setup ==<br />
<br />
Actin was polymerized in glass sample chambers and then imaged with CCD cameras.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Shear_thickening_and_scaling_of_the_elastic_modulus_in_a_fractal_colloidal_system_with_attractive_interactions&diff=16090Shear thickening and scaling of the elastic modulus in a fractal colloidal system with attractive interactions2010-12-02T00:18:31Z<p>Huayinwu: New page: helen</p>
<hr />
<div>helen</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Curvature_condensation_and_bifurcation_in_an_elastic_shell&diff=16089Curvature condensation and bifurcation in an elastic shell2010-12-02T00:17:37Z<p>Huayinwu: New page: helen</p>
<hr />
<div>helen</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Measurement_of_nonlinear_rheology_of_cross-linked_biopolymer_gels&diff=16088Measurement of nonlinear rheology of cross-linked biopolymer gels2010-12-01T21:18:30Z<p>Huayinwu: New page: helen</p>
<hr />
<div>helen</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=15566Helen Wu2010-11-04T02:46:13Z<p>Huayinwu: </p>
<hr />
<div>Definitions:<br />
<br />
<br />
<br />
Weekly wiki entries:<br />
<br />
[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
<br />
[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
<br />
[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
<br />
[[Microrheology Probes Length Scale Dependent Rheology]] - ''October 20, 2010''<br />
<br />
[[Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film]] - ''November 1, 2010''<br />
<br />
[[Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System]] - ''November 3, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Highly_Anisotropic_Vorticity_Aligned_Structures_in_a_Shear_Thickening_Attractive_Colloidal_System&diff=15565Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System2010-11-04T02:45:37Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Highly anisotropic vorticity aligned structures in a shear thickening attractive colloidal system"<br />
<br />
C. O. Osuji, D. A. Weitz, ''Soft Matter'', '''4''', 1388-1392 (2008).<br />
<br />
<br />
== Keywords ==<br />
[[colloid]]<br />
<br />
== Overview ==<br />
<br />
[[Image:SM2008_2.gif|300px|thumb|left|'''Figure 2.''' Microstructure under shear (a) cylindrical flocs at <math>\dot\gamma = 6.67 s^{-1}</math>, (b) <math>\dot\gamma = 133 s^{-1}</math>, (c) math>\dot\gamma = 1330 s^{-1}</math>.]]<br />
<br />
[[Image:SM2008_4.gif|300px|thumb|right|'''Figure 4.''' Quenched samples starting at zero shear rate and going up to math>\dot\gamma = 10 s^{-1}</math>. Vorticity is indicated by the white line in the first panel.]]<br />
<br />
[[Image:SM2008_5.gif|300px|thumb|right|'''Figure 5.''' (a) Optical micrographs over time. (b) the FFTs of images in (a).]]<br />
<br />
[[Image:SM2008_6.gif|300px|thumb|right|'''Figure 6.''' Microstructure after shear thickening in parallel plate geometry with gap sizes, (a) <math>d = 25 \mu m</math>, (b) <math>d = 50 \mu m</math>, (c) <math>d = 100 \mu m</math>, (d) <math>d = 250 \mu m</math>.]]<br />
<br />
Soft materials and complex fluids often form structures in response to flow around them. In Brownian systems with hard spheres, distortion will occur when the timescale for flow is less than for the particles' diffusion, as described by the Péclet number (represents flow force over thermal/diffusive). In a system where flow dominates, the particles separate in one direction and aggregate in another, forming strings in dilute solutions. Shear thickening happens in more concentrated solutions, but usually not in colloidal systems with attractive interactions (they tend to form flocculated gels).<br />
<br />
The authors studied steady state flow behavior of dilute, simple hydrocarbon dispersions of carbon black particles and were actually able to observe shear thickening under certain conditions (above a critical flow rate <math>\dot \gamma_c\approx 10^2-10^3 s^{-1}</math>. They also found that the shear modulus of the gels had a power law dependence on the pre-shear stress in the system, and deforming the thickened gells at lower shear rates created ordered vorticity aligned aggregates that broke down into small isotropic clusters given enough time (~300s).<br />
<br />
== Results and discussion ==<br />
<br />
Samples under steady shear displayed thixotropy, meaning they are normally thick and highly viscous but flow when stressed. The curves also indicate a composition-dependent shear thickening transition, mentioned previously. Optically observing the systems showed that at low shear stress, the system contained large pieces of broken gel that broke into subsequently smaller pieces when shear was increased. The system's viscosity began to increase at <math>\dot \gamma\approx 10^0-10^1 s^{-1}</math> and clusters of particles aggregated along the vorticity axis (Figure 2a). This was considered to be the steady state response of the system.. They then found that as the shear rate increased, the structures went from being cylindrical to isotropic clusters that become more and more dense until the aggregates break and the transition to shear thickening flow happens.<br />
<br />
The system was found to have large negative normal stresses at high shear rates (no changes seen at low rates). If the shear was stopped suddenly (quenching), the effect persisted for long times unless shear flow was applied again, in which case it went back to the original state quickly. This relaxation of internal shear stresses was determined to have a power law dependence <math>\sigma_i ~ t^{-0.1}</math>.<br />
<br />
Deforming quenched shear thickened gels quickly resulted in highly anisotropic vorticity aligned structures that were much more defined than the steady state response the authors looked at first (Figure 4). The aspect ratios of the cylinders were greater than the macroscopic portions of the shear cell as well. They studied this phenomenon using Fourier transformations of the images to monitor the development of alignment in the system (Figure 5) - it turned out to be rapid and peaked at 20s with <math>\gamma=200</math>.<br />
<br />
The authors observed that the width of the cylindrical structures was slightly larger than and proportional to the gap but periodicity doesn't change much, meaning the flocs become less dense as they become thicker.<br />
<br />
Structure dissolution occurred first at the outside edge of the rheometer geometry both when using a parallel plate (highest shear at edge) as well as with an angled cone (constant shear across plate), so the authors suggest that confinement is important for stabilization of the flocs. (Figure 6)<br />
<br />
The frequency and strain dependence of the elastic modulus of the flocs was different from the shear thickened gel, but both showed strong elastic responses. Yield strain was lower for structures than for gel and its presence indicates that the rolling of the flocs is not sufficient to account for the displacement and deformation that comes from the applied shear.<br />
<br />
The observations made by the authors of this paper are similar to an elastic instability associated with what is seen in semi-dilute non-Brownian nanotubes that are undergoing clustering due to flow. Also, these results provide insight as to the effect of confinement and composition on the mechanics of such structures.<br />
<br />
== Experimental Setup ==<br />
<br />
Tetradecane dispersions of 8% 0.5 <math>\mu</math>m carbon black particles were used. Experiments were done using rheometers, sometimes combined with an optical observation component.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2008_6.gif&diff=15564File:SM2008 6.gif2010-11-04T02:45:09Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2008_5.gif&diff=15563File:SM2008 5.gif2010-11-04T02:44:59Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2008_4.gif&diff=15562File:SM2008 4.gif2010-11-04T02:44:45Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2008_2.gif&diff=15561File:SM2008 2.gif2010-11-04T02:44:25Z<p>Huayinwu: </p>
<hr />
<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Highly_Anisotropic_Vorticity_Aligned_Structures_in_a_Shear_Thickening_Attractive_Colloidal_System&diff=15560Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System2010-11-04T02:42:25Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
<br />
== Reference ==<br />
<br />
"Highly anisotropic vorticity aligned structures in a shear thickening attractive colloidal system"<br />
<br />
C. O. Osuji, D. A. Weitz, ''Soft Matter'', '''4''', 1388-1392 (2008).<br />
<br />
<br />
== Keywords ==<br />
[[colloid]]<br />
<br />
== Overview ==<br />
<br />
[[Image:SM2008_2.jpg|300px|thumb|left|'''Figure 2.''' Microstructure under shear (a) cylindrical flocs at <math>\dot\gamma = 6.67 s^{-1}</math>, (b) <math>\dot\gamma = 133 s^{-1}</math>, (c) math>\dot\gamma = 1330 s^{-1}</math>.]]<br />
<br />
[[Image:SM2008_4.jpg|300px|thumb|right|'''Figure 4.''' Quenched samples starting at zero shear rate and going up to math>\dot\gamma = 10 s^{-1}</math>. Vorticity is indicated by the white line in the first panel.]]<br />
<br />
[[Image:SM2008_5.jpg|300px|thumb|right|'''Figure 5.''' (a) Optical micrographs over time. (b) the FFTs of images in (a).]]<br />
<br />
[[Image:SM2008_6.jpg|300px|thumb|right|'''Figure 6.''' Microstructure after shear thickening in parallel plate geometry with gap sizes, (a) <math>d = 25 \mu m</math>, (b) <math>d = 50 \mu m</math>, (c) <math>d = 100 \mu m</math>, (d) <math>d = 250 \mu m</math>.]]<br />
<br />
Soft materials and complex fluids often form structures in response to flow around them. In Brownian systems with hard spheres, distortion will occur when the timescale for flow is less than for the particles' diffusion, as described by the Péclet number (represents flow force over thermal/diffusive). In a system where flow dominates, the particles separate in one direction and aggregate in another, forming strings in dilute solutions. Shear thickening happens in more concentrated solutions, but usually not in colloidal systems with attractive interactions (they tend to form flocculated gels).<br />
<br />
The authors studied steady state flow behavior of dilute, simple hydrocarbon dispersions of carbon black particles and were actually able to observe shear thickening under certain conditions (above a critical flow rate <math>\dot \gamma_c\approx 10^2-10^3 s^{-1}</math>. They also found that the shear modulus of the gels had a power law dependence on the pre-shear stress in the system, and deforming the thickened gells at lower shear rates created ordered vorticity aligned aggregates that broke down into small isotropic clusters given enough time (~300s).<br />
<br />
== Results and discussion ==<br />
<br />
Samples under steady shear displayed thixotropy, meaning they are normally thick and highly viscous but flow when stressed. The curves also indicate a composition-dependent shear thickening transition, mentioned previously. Optically observing the systems showed that at low shear stress, the system contained large pieces of broken gel that broke into subsequently smaller pieces when shear was increased. The system's viscosity began to increase at <math>\dot \gamma\approx 10^0-10^1 s^{-1}</math> and clusters of particles aggregated along the vorticity axis (Figure 2a). This was considered to be the steady state response of the system.. They then found that as the shear rate increased, the structures went from being cylindrical to isotropic clusters that become more and more dense until the aggregates break and the transition to shear thickening flow happens.<br />
<br />
The system was found to have large negative normal stresses at high shear rates (no changes seen at low rates). If the shear was stopped suddenly (quenching), the effect persisted for long times unless shear flow was applied again, in which case it went back to the original state quickly. This relaxation of internal shear stresses was determined to have a power law dependence <math>\sigma_i ~ t^{-0.1}</math>.<br />
<br />
Deforming quenched shear thickened gels quickly resulted in highly anisotropic vorticity aligned structures that were much more defined than the steady state response the authors looked at first (Figure 4). The aspect ratios of the cylinders were greater than the macroscopic portions of the shear cell as well. They studied this phenomenon using Fourier transformations of the images to monitor the development of alignment in the system (Figure 5) - it turned out to be rapid and peaked at 20s with <math>\gamma=200</math>.<br />
<br />
The authors observed that the width of the cylindrical structures was slightly larger than and proportional to the gap but periodicity doesn't change much, meaning the flocs become less dense as they become thicker.<br />
<br />
Structure dissolution occurred first at the outside edge of the rheometer geometry both when using a parallel plate (highest shear at edge) as well as with an angled cone (constant shear across plate), so the authors suggest that confinement is important for stabilization of the flocs. (Figure 6)<br />
<br />
The frequency and strain dependence of the elastic modulus of the flocs was different from the shear thickened gel, but both showed strong elastic responses. Yield strain was lower for structures than for gel and its presence indicates that the rolling of the flocs is not sufficient to account for the displacement and deformation that comes from the applied shear.<br />
<br />
The observations made by the authors of this paper are similar to an elastic instability associated with what is seen in semi-dilute non-Brownian nanotubes that are undergoing clustering due to flow. Also, these results provide insight as to the effect of confinement and composition on the mechanics of such structures.<br />
<br />
== Experimental Setup ==<br />
<br />
Tetradecane dispersions of 8% 0.5 <math>\mu</math>m carbon black particles were used. Experiments were done using rheometers, sometimes combined with an optical observation component.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Highly_Anisotropic_Vorticity_Aligned_Structures_in_a_Shear_Thickening_Attractive_Colloidal_System&diff=15557Highly Anisotropic Vorticity Aligned Structures in a Shear Thickening Attractive Colloidal System2010-11-04T01:38:18Z<p>Huayinwu: New page: Helen</p>
<hr />
<div>Helen</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Mitosis:_Taking_the_Measure_of_the_Spindle_Length&diff=15544Mitosis: Taking the Measure of the Spindle Length2010-11-03T21:27:26Z<p>Huayinwu: New page: Helen</p>
<hr />
<div>Helen</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=15512Helen Wu2010-11-02T03:58:54Z<p>Huayinwu: </p>
<hr />
<div>Definitions:<br />
<br />
<br />
<br />
Weekly wiki entries:<br />
<br />
[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
<br />
[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
<br />
[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
<br />
[[Microrheology Probes Length Scale Dependent Rheology]] - ''October 20, 2010''<br />
<br />
[[Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film]] - ''November 1, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Smooth_Cascade_of_Wrinkles_at_the_Edge_of_a_Floating_Elastic_Film&diff=15511Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film2010-11-02T03:58:23Z<p>Huayinwu: </p>
<hr />
<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
[[Image:PRL038302_1.jpg|300px|thumb|right|'''Figure 1.''' (a) Image of the wrinkled PS sheet, (b) geometry of the system, (c) wavelength of wrinkles as a function of thickness, which is fit with <math>q_0=(\frac{\rho g}{B})^{1/4}</math> .]]<br />
<br />
== Reference ==<br />
<br />
"Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film"<br />
<br />
J. Huang, B. Davidovitch, C. D. Santangelo, T. P. Russell, N. Menon, ''Physical Review Letters'', '''105''', 038302 (2010).<br />
<br />
<br />
== Keywords ==<br />
[[thin films]]<br />
<br />
== Overview ==<br />
<br />
[[Image:PRL038302_2.jpg|300px|thumb|right|'''Figure 2.''' (a) Wave number as a function of distance from the edge, which is almost independent of thickness. (b) Wave number vs. distance scaled with capillary length. (c) same as (a), scaled with distance.]]<br />
<br />
[[Image:PRL038302_4.jpg|300px|thumb|right|'''Figure 4.''' (a) Image of cascade. (b) Histogram of separation between crests (scaled with <math>q_0d/(2\pi)</math> at x distances from the edge.]]<br />
<br />
At the edge of a pattern, the symmetry usually breaks due to the boundary's tendency to be flat. The authors of this paper studied wrinkling using this phenomenon by looking at a thin sheet that has a pattern due to elastic instability and proposed a mechanism by which the pattern and the flat boundary can both exist.<br />
<br />
When a thin rectangular sheet floating on a liquid surface is compressed from two sides along the same axis, it forms the pattern in Figure 1 - large wrinkles of wavelength <math>\lambda \ll</math> the width of the sheet and much smaller wrinkles near the boundary. The wave amplitude is expected to decrease to minimize the surface energy of the interface and the wave number should increase to maintain inextensibility. This model has a point where the cost of bending offsets the gain in surface energy. The paper explores parameters that affect the wave cascade, the amplification of wavelength, and the length of the cascade.<br />
<br />
== Results and discussion ==<br />
<br />
The authors propose that 2 principles determine the pattern we see:<br />
# a thin sheet is basically inextensible, so the wavelength and amplitude are proportional<br />
# the wavelength is a compromise between the bending energy (favors long wavelengths) and gravitational energy (favors small ones).<br />
They demonstrate that the scaling of the wrinkles in the bulk goes as <math>q_0=(\frac{\rho g}{B})^{1/4}</math> where <math>q_0</math> is the wave number, B is the bending modulus, <math>\rho</math> is the fluid density. (see Figure 1c for the graph)<br />
<br />
Figure 2 shows that the increase in wave number at the edge happens at approximately the same distance for all film thicknesses (with systematic deviations). This penetration distance is around <math>1.8 \pm 0.2mm</math>. They used energy calculations, estimating the cost of a wave number at the edge as well as the effect of breaking symmetry in the wave pattern. They found that persistence length <math>l_p \approx</math> between <math>l_c</math>, the capillary length, and <math>1/q_e</math> at various wave numbers between the edge and the large bulk wrinkles.<br />
<br />
A current model for such elastic cascades was proposed by Pomeau and Rica, which explained the smaller wrinkles near the edge using a branching hierarchy. However, the system observed in the paper behaved differently than this would predict, particularly that the smooth amplitude reduction reflects a finite number of Fourier components being mixed as you approach the edge. The Pomeau-Rica case would predict sharp ridges and folds rather than a smooth cascade. Figure 4 shows the observed cascade and also a histogram of the separation between wave crests. If the branching theory were correct, there would be peaks at the lower end of the separation axis for each distance from the edge as the two waves become one.<br />
<br />
Gravity, bending, and capillarity forces are all important in this system but they all scale differently. The authors suggest that they can be used to tune the system independently and allow for further studies.<br />
<br />
== Experimental Setup ==<br />
<br />
Polystyrene (PS) sheets of dimensions 3x2cm with thicknesses between 50-400nm were prepared by spin coating onto glass substrates, then transferred to a dish of DI water. The sheet floated because of the hydrophobic nature of PS.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL038302_4.jpg&diff=15510File:PRL038302 4.jpg2010-11-02T03:54:31Z<p>Huayinwu: </p>
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<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Smooth_Cascade_of_Wrinkles_at_the_Edge_of_a_Floating_Elastic_Film&diff=15308Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film2010-10-25T18:17:58Z<p>Huayinwu: New page: Helen Wu</p>
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<div>Helen Wu</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=And_others&diff=15307And others2010-10-25T18:17:47Z<p>Huayinwu: </p>
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<div>[[Main_Page | Home]]<br />
<br />
Papers from outside SEAS, interesting and important. These have already been entered in the wiki. Edits are welcome. Additional papers are welcome.<br />
<br />
209. [[Spontaneous breakdown of superhydrophobicity]], Mauro Sbragaglia, Alisia M. Peters, Christophe Pirat, Bram M. Borkent, Rob G. H. Lammertink, Matthias Wessling, and Detlef Lohse, Phys. Rev. Lett. 99, 156001 (2007).<br />
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210. [[Surface Viscoelasticity of Individual Gram-Negative Bacterial Cells Measured Using Atomic Force Microscopy]] V. Vadillo-Rodriguez, T J. Beveridge, and T R Dutcher Journal of Bacteriology, Vol. 190, No. 12, June 2008, p. 4225-4232<br />
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211. [[Pearl drops]] . Bico, C. Marzolin and D. Quere, Europhys. Lett. 47, 220-226 (1999).<br />
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212. [[Patterned Superhydrophobic Surfaces: Toward a Synthetic Mimic of the Namib Desert Beetle]] L. Zhai, M. C. Berg, F. C. Cebeci, Y. Kim, J. M. Milwid, M. F. Rubner, and R. E. Cohen, Nano Lett. 6, 1213-1217 (2006).<br />
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213. [[Variable-focus liquid lens for miniature cameras]] S. Kuiper, and B.H.W. Hendriks Applied Physics Letters, Volume 85, Number 7, pp.1128-1130 (16 August 2004) <br />
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214. [[Assembly of Colloidal Particles by Evaporation on Surfaces with Patterned Hydrophobicity]], Fengui Fan, et. al., Langmuir, 2004, 20(8)<br />
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215. [[Splashing on elastic membranes: The importance of early-time dynamics]] Rachel E. Pepper, Laurent Courbin, and Howard A. Stone Physics of Fluids 20, 082103 (2008)<br />
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216. [[Continuous Convective Assembling of Fine Particles into Two-Dimensional Arrays on Solid Surfaces]] Antony S. Dimitrov and Kuniaki Nagayama, Langmuir 1996, 12, 1303-1311.<br />
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217. [[Impalement of fakir drops]] M. Reyssat, J. M. Yeomans, and D. Quere, Europhys. Lett. 81, 26006 (2008). <br />
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218. [[Drop Splashing on a Dry Smooth Surface]] Lei Xu, Wendy W. Zhang, and Sidney R. Nagel PRL 94, 184505 (2005) <br />
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219. [[Osmotically driven shape-dependent colloidal separations]] Mason, T.G. Physical Review E 66, 060402(R) 2002<br />
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220. [[Surface-induced droplet fusion in microfluidic devices]] Luis M. Fidalgo, Chris Abell, and Wilheml T.S. Huck Lab Chip, 2007, 7, 984-986<br />
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221. [[From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath]] Y. Couder, E. Fort, C.H. Gautier, and A. Boudaoud PRL 94, 177801 (2005)<br />
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222. [[Mechanistic Principles of Colloidal Crystal Growth by Evaporation-Induced Convective Steering]] Damien D. Brewer, Joshua Allen, Michael R. Miller, Juan M. de Santos, Satish Kumar, David J. Norris, Michael Tsapatsis, and L. E. Scriven Langmuir 2008, 24, 13683-13693 <br />
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223. [[Bursting of soap films. I. An experimental study]] Damien D. Brewer, Joshua Allen, Michael R. Miller, Juan M. de Santos, Satish Kumar, David J. Norris, Michael Tsapatsis, and L. E. Scriven Langmuir 2008, 24, 13683-13693 <br />
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224. [[The hydrodynamics of water strider locomotion]] David L. Hu, Brian Chan and John W. M. Bush, Nature 424, 663 (2003).<br />
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225. [[Optically trapped aqueous droplets for single molecule studies]] J. E. Reiner, A. M. Crawford, R. B. Kishore, Lori S. Goldner, K. Helmerson Applied Physics Letters, 2006, 89, 013904<br />
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226. [[Experimental observations of the squeezing-to-dripping transition in T-shaped microfluidic junctions]] Christopher GF, Noharuddin NN, Taylor JA, Anna SL., Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Sep;78(3 Pt 2):036317.<br />
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292. [[Fabrication of Hierarchical Structures by Wetting Porous Templates with Polymer Microspheres]] Jiun-Tai Chen, Dian Chen, Thomas P. Russell Langmuir 2009 25 (8), 4331-4335<br />
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293. [[Shape-Tunable Polymer Nanofibrillar Structures by Oblique Electron Beam Irradiation]] Tae-il Kim, Changhyun Pang, Kahp Y. Suh Langmuir 2009 25 (16), 8879-8882<br />
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294. [[Highly-Ordered Selective Self-Assembly of a Trimeric Cationic Surfactant on a Mica Surface]] Yanbo Hou, Meiwen Cao, Manli Deng, Yilin Wang Langmuir 2008 24 (19), 10572-10574<br />
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295. [[Creating, Transporting, Cutting, and Merging Liquid Droplets by Electrowetting-Based Actuation for Digital Microfluidic Circuits]] Sung Kwon Cho, Hyejin Moon, and Chang-Jin Kim. Journal of Microelectromechanical Systems 2003, 12, 70-80.<br />
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296. [[Superhydrophobic Aluminum Surfaces by Deposition of Micelles of Fluorinated Block Copolymers]] Simon Desbief, Bruno Grignard, Christophe Detrembleur, Romain Rioboo, Alexandre Vaillant, David Seveno, Michel Voue, Joel De Coninck, Alain M. Jonas, Christine Jerome, Pascal Damman, and Roberto Lazzaroni. Langmuir 2009, DOI: 10.1021/la902565y.<br />
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297. [[Microfluidic control of cell pairing and fusion]] Alison M Skelley, Oktay Kirak, Heikyung Suh, Rudolf Jaenisch, and Joel Voldman. Nature Methods 2009, 6(2), 147-152.<br />
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298. [[A Biodegradable and Biocompatible Gecko-Inspired Tissue Adhesive]] Alborz Mahdavi, Lino Ferreira, Cathryn Sundback, Jason W. Nichol, Edwin P. Chan, David J. D. Carter, Chris J. Bettinger, Siamrut Patanavanich, Loice Chignozha, Eli Ben-Joseph, Alex Galakatos, Howard Pryor, Irina Pomerantseva, Peter T. Masiakos, William Faquin, Andreas Zumbuehl, Seungpyo Hong, Jeffrey Borenstein, Joseph Vacanti, Robert Langer, and Jeffrey M. Karp. PNAS 2008 105 (7), 2307-2312. <br />
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299.[[Measuring the Surface Dynamics of Glassy Polymers]] Z. Fakhraai and J. A. Forrest, Science 2008, 319, 600. <br />
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300. [[A Microfabrication-Based Dynamic Array Cytometer]] Joel Voldman, Martha L. Gray, Mehmet Toner, and Martin A. Schmidt. Analytical Chemistry 2002, 74(16), 3984-3990.<br />
<br />
301. [[Three-Dimensional Interstitial Nanovoid of Nanoparticulate Pt Film Electroplated from Reverse Micelle Solution]] Sejin Park, Sun Young Lee, Hankil Boo, Hyun-Mi Kim, Ki-Bum Kim, Hee Chan Kim, Youn Joo Song, and Taek Dong Chung. Chem. Mater. 2007, 19, 3373-3375<br />
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302. [[Reversible phase transition from vesicles to lamellar network structures triggered by chain melting]] Yuwen Shen, Jingcheng Hao, Heinz Hoffmann and Zhonghua Wu. Soft Matter, 2008, 4, 805–810<br />
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303. [[Ultrasensitive detection of bacteria using core-shell nanoparticles and a NMR-filter system]] Hakho Lee, Tae-Jong Yoon, and Ralph Weissleder. Angewandte Chemie Internation Edition 2009, 48(31), 5657-5660.<br />
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304. [[Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium]] I.B. Burgess, W.E. Shimmell, K. Saravanamuttu, J. Am. Chem. Soc. 129, 4738-4746 (2007).<br />
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305. [[All terrain droplet actuation]] M. Abdelgawad, S.L.S. Freire, H. Yang and A.R. Wheeler Lab on a Chip 8, 672–677 (2008).<br />
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306. [[Electrospun polymer membrane activated with room temperature ionic liquid: Novel polymer electrolytes for lithium batteries]] Gouri Cheruvally, Jae-Kwang Kim, Jae-Won Choi, Jou-Hyeon Ahn, Yong-Jo Shin, James Manuel, Prasanth Raghavan, Ki-Won Kim, Hyo-Jun Ahn, Doo Seong Choi, Choong Eui Song. Journal of Power Sources 172 (2007) 863–869<br />
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307. [[Hydrodynamic metamaterials: Microfabricated arrays to steer, refract and focus streams of biomaterials]] Keith J. Morton, Kevin Loutherback, David W. Inglis, Ophelia K. Tsui, James C. Sturm, Stephen Y. Chou, and Robert H. Austin. Proceedings of the National Academy of Science (2008), 105(21), 7434-7438.<br />
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308. [[Electronic detection of DNA by its intrinsic molecular charge]] Jurgen Fritz, Emily B. Cooper, Suzanne Gaudet, Peter K. Sorger, and Scott R. Manalis, PNAS 99, no. 22, 14142-14146 (2002)<br />
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309. [[Cell-free synthesis of functional proteins using transcription/translation machinery entrapped in silica sol-gel matrix]] Kyeong-Ohn Kim, Seong Yoon Lim, Geun-Hee Hahn, Sahng Ha Lee, Chan Beum Park, and Dong-Myung Kim, Biotechnol Bioeng 102(1), 303-307<br />
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310. [[Interaction Forces between Colloidal Particles in Liquid: Theory and Experiment]] Yuncheng Liang, Nidal Hilal, Paul Langston, and Victor Starov, Advances in Colloid and Interface Science 134-135, 151-166 (2007).<br />
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311. [[Development of functionalized superparamagnetic iron oxide nanoparticles for interaction with human cancer cells]] A. Petri-Finka, M. Chastellaina, L. Juillerat-Jeanneretb, A. Ferraria, H. Hofmann. Biomaterials 26 (2005) 2685–2694<br />
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312. [[On-chip natural assembly of silicon photonic bandgap crystals]] Y.A. Vlasov, X.-Z. Bo, J.C. Sturm, D.J. Norris, Nature 414, 289-293 (2001).<br />
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313. [[Preparation of Hard Mesoporous Silica Spheres]] Q. Huo, J. Feng, F. Schuth, G.D. Stucky, Chemistry of Materials, 9, 14-17 (1997)<br />
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314. [[Surface-directed assembly of cell-laden microgels]] Y. Du, M. Ghodousi, E. Lo, M.K. Vidula, O. Emiroglu and A. Khademhosseini, Biotechnol. Bioeng. 105 (3) 655-662 (2010)<br />
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315. [[Directional water collection on wetted spider silk]] Y. Zheng, H. Bai, Z. Huang, X. Tian, F. Nie, Y. Zhao, J. Zhai & L. Jiang, Nature. 463 (4) 640-643 (2010)<br />
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316. [[Reversible stress softening of actin networks]] O. Chaudhuri, S.H. Parekh & D.A. Fletcher, Nature, 445 295-298 (2007)<br />
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317. [[Supramolecular self-Assembly of lipid derivatives on carbon nanotubes]] C. Richard, F. Balavoine, P. Schultz, T.W. Ebbesen, & C. Mioskowski, Science, 300 775-778 (2003)<br />
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318. [[Water Freezes Differently on Positively and Negatively Charged Surfaces of Pyroelectric Materials]] D. Ehre, E. Lavert, M. Lahav & I. Lubomirsky, Science, 327 672-675 (2010)<br />
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319. [[Aggregation and vesiculation of membrane proteins by curvature-mediated interactions]] B.J. Reynwar, G. Illya, V.A. Harmandaris, M.M. Muller, K. Kremer & M. Deserno, Nature, 447 461-464 (2010)<br />
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320. [[Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film]] J. Huang, B. Davidovitch, C. D. Santangelo, T. P. Russell, & N. Menon, Phys. Rev. Letters 105, 038302 (2010)<br />
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[[Main_Page | Home]]</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=15254Helen Wu2010-10-21T03:35:43Z<p>Huayinwu: </p>
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<div>Definitions:<br />
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Weekly wiki entries:<br />
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[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
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[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
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[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
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[[Microrheology Probes Length Scale Dependent Rheology]] - ''October 20, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Helen_Wu&diff=15253Helen Wu2010-10-21T03:35:11Z<p>Huayinwu: </p>
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<div>Definitions:<br />
<br />
<br />
<br />
Weekly wiki entries:<br />
<br />
[[Cell Migration Driven by Cooperative Substrate Deformation Patterns]] - ''September 13, 2010''<br />
<br />
[[Microbristle in gels: Toward all-polymer reconfigurable hybrid surfaces]] - ''September 20, 2010''<br />
<br />
[[Bubble formation via multidrop impacts]] - ''October 4, 2010''<br />
<br />
[[Microrheology probes length scale dependent rheology]] - ''October 20, 2010''</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL118104_1.jpg&diff=15252File:PRL118104 1.jpg2010-10-21T03:34:19Z<p>Huayinwu: uploaded a new version of "Image:PRL118104 1.jpg"</p>
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<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL118104_2.jpg&diff=15251File:PRL118104 2.jpg2010-10-21T03:33:05Z<p>Huayinwu: </p>
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<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=File:PRL118104_1.jpg&diff=15250File:PRL118104 1.jpg2010-10-21T03:32:50Z<p>Huayinwu: </p>
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<div></div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Microrheology_Probes_Length_Scale_Dependent_Rheology&diff=15249Microrheology Probes Length Scale Dependent Rheology2010-10-21T03:29:18Z<p>Huayinwu: </p>
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<div>Entry by [[Helen Wu]], AP225 Fall 2010<br />
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== Reference ==<br />
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"Microrheology probes length scale dependent rheology"<br />
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M. L. Gardel, K. Kroy, E. Frey, B. D. Hoffman, J. C. Crocker, A. R. Bausch, D. A. Weitz, ''Physical Review Letters'', '''96''', 118104 (2006).<br />
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== Keywords ==<br />
[[rheology]]<br />
<br />
== Overview ==<br />
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[[Image:PRL118104_1.jpg|300px|thumb|right|'''Figure 1.''' Comparison of 1P and 2P MSDs with lengths (a) 0.5<math>\mu</math>m, (b) 2<math>\mu</math>m, (c) 5<math>\mu</math>m, (d) 17<math>\mu</math>m. The open boxes are 1P values and the filled ones are for 2P.]]<br />
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[[Image:PRL118104_2.jpg|300px|thumb|right|'''Figure 2.''' Comparison of 1P and 2P <math>G'</math>, <math>G''</math> against the frequency, <math>\omega</math> for lengths (a) 0.5<math>\mu</math>m, (b) 2<math>\mu</math>m, (c) 5<math>\mu</math>m, (d) 17<math>\mu</math>m. The open boxes are <math>G'</math> and the filled ones are for <math>G''</math>.]]<br />
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The researchers looked at the mechanical response of a semiflexible polymer (F-actin in this case) at different length scales using microrheology. Semiflexible polymers become entangled at low concentrations, and they are sterically hindered at the entanglement length <math>l_e</math>, which is related to the distance between polymers and the persistence length. At intermediate frequencies, there is a transition where the mechanical response is dependent on the filament length, <math>L</math>, which is not predicted by theory. The researchers identified fluctuations over <math>L</math> as a relaxation mechanism between 0.1-30rad/s and used 2-particle (2P) microrheology to look at lengths >5<math>\mu</math>m and 1-particle microrheology for lengths ~<math>l_e</math>. 2P microrheology showed increased viscoelastic relaxation for intermediate frequencies that scaled as <math>L^2</math><br />
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== Results and discussion ==<br />
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For varied L values, the researchers demonstrated that longer filaments show a transition in particle motion where before a certain point, mean squared displacement (MSD) changes with respect to <math>\tau</math> (raised to a factor) but then switches to little time evolution. Figure 1 shows the MSD values for various particle sizes. For <math>L > 2 \mu m</math>, the 1P MSD is more constrained, but behavior is similar. The switching time between regimes was similar across lengths. The 2P displacement correlation tensor was scaled to the 2P MSD, <math>a</math>. When <math>L</math> was about <math>a</math>, 2P and 1P MSD values were similar. However, the 2P MSD changed in both slope and magnitude as <math>L</math> changed, unlike the 1P MSD.<br />
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The generalized Stokes-Einstein relation was used to compare 1P to 2P MSDs. For small <math>L</math>, they mostly matched, but longer filaments show differences. (see Figure 2, showing <math>G'</math>, <math>G''</math> against the frequency, <math>\omega</math>) A transition from sloping up to a plateau is once again seen for the 1P microrheology over <math>L</math>, whereas 2P microrheology is dependent on <math>L</math>. They converge to similar values at low frequency.<br />
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From their data, the researchers say that the 1P microrheology probes bending fluctuations of single filaments at various frequencies, where single-filament dynamics dominate until they become entangled at <math>l_e</math>. The plateau comes from the entanglement. Based on the data, the authors suggest that 1P microrheology may be useful for measurements of cross-linked networks of semiflexible filaments.<br />
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In contrast, the 2P results scale as <math>\tau_m </math>~<math> L^2</math>, like in diffusion, and indicate density fluctuations along the filament (expected because transverse thermal fluctuations result in varying quantities of material present at a specific point). 2P microrheology probes longer lengths than 1P.<br />
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The authors conclude that mechanical response changes as length scales in the system vary and that rheology can be used to learn about network geometry and filament properties.<br />
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== Experimental Setup ==<br />
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G-actin mixed with polystyrene particles and then polymerized. <math>L</math> was varied using gelsolin. Particle motions were recorded with a fast camera.</div>Huayinwuhttp://soft-matter.seas.harvard.edu/index.php?title=Microrheology_Probes_Length_Scale_Dependent_Rheology&diff=15098Microrheology Probes Length Scale Dependent Rheology2010-10-16T20:58:34Z<p>Huayinwu: New page: Helen Wu</p>
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<div>Helen Wu</div>Huayinwu