View source for Measurement of nonlinear rheology of cross-linked biopolymer gels

Entry by [[Helen Wu]], AP225 Fall 2010

== Reference ==

"Measurement of nonlinear rheology of cross-linked biopolymer gels"

C. P. Broedersz, K. E. Kasza, L. M. Jawerth, S. Münster, D. A. Weitz, ''Soft Matter'', '''6''', 4120-4127 (2010).


== Keywords ==
rheology, gel, biopolymer

== Overview ==

[[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.]]

[[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.]]

[[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)]]

[[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).]]

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.

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.

== Results and discussion ==

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.

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: 

''Strain ramp protocol''
Both crosslinked and uncrosslinked F-actin networks were found to have measurements that depended on <math>\dot\gamma</math>.
''Prestress protocol''

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.

''Comparison''
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.

== Model ==

Based on these results, the authors proposed a model for use while studying these systems:

''Assumptions''
# network repsonds instantaneously to an applied stress
# 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>
#single relaxation timescale, so the network is treated like a simple liquid with viscosity <math>\zeta\gg\eta</math>

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>

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).

Stress relaxation is given by: <math>\frac{d\sigma}{dt}=\zeta\frac{d^2\gamma_f}{dt^2}</math>

This describes a Newtonian liquid-like system.

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).

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.

The general nonlinear response is given by the differential equation:

<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>

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).

== Conclusions ==

The prestress and strain ramp methods are both fine for cross-linked networks. For networks that creep, the prestress method is the better choice.

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.