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# Role of isostaticity and load-bearing microstructure in the elasticity of yielded colloidal gels

Edited by William R. Schowalter, Princeton University, Princeton, NJ, and approved August 30, 2012 (received for review April 21, 2012)

## Abstract

We report a simple correlation between microstructure and strain-dependent elasticity in colloidal gels by visualizing the evolution of cluster structure in high strain-rate flows. We control the initial gel microstructure by inducing different levels of isotropic depletion attraction between particles suspended in refractive index matched solvents. Contrary to previous ideas from mode coupling and micromechanical treatments, our studies show that bond breakage occurs mainly due to the erosion of rigid clusters that persist far beyond the yield strain. This rigidity contributes to gel elasticity even when the sample is fully fluidized; the origin of the elasticity is the slow Brownian relaxation of rigid, hydrodynamically interacting clusters. We find a power-law scaling of the elastic modulus with the stress-bearing volume fraction that is valid over a range of volume fractions and gelation conditions. These results provide a conceptual framework to quantitatively connect the flow-induced microstructure of soft materials to their nonlinear rheology.

Colloidal gels form sample-spanning networks (1) and are used to generate solid-like properties in a broad range of materials such as direct-write inks (2), nanoemulsions (3), tissue scaffolds (4), and membranes (5). When gels undergo large deformations, their network ruptures into clusters via a complex process. The ability to connect structural changes to suspension rheology is critical in understanding the mechanism of yielding. The network of clusters that makes up colloidal gels arises due to percolation, dynamic arrest, and phase separation, where the volume fraction and pair potential play an important part in their structure and rheology (6⇓–8). Interparticle bonds rupture under a sufficiently large stress; the result is a flow-induced fluidization transition accompanied by the formation of voids and aggregates that continuously break and reform along the principal axes of flow (9, 10). A complex two-step yielding process has been observed in gels at intermediate volume fractions (0.05 ≤ *ϕ* ≤ 0.30) (11, 12). Within this regime, a small number of bonds are broken in gels undergoing steady shear yielding (9, 10). Methods that track the evolution of ensemble-averaged structure, such as mode coupling theory (6) and light scattering, lack sensitivity to these subpopulations. On the other hand, micromechanical treatments directly model the contributions of local microstructure to the macroscopic elasticity of the material (13⇓–15). However, experiments to connect the yield stress to different interparticle potentials, volume fractions, and particle sizes show little agreement. Presently, these theories can only provide estimates of colloidal rheology under specific conditions.

We demonstrate that a simple, general correlation between microstructure and strain-dependent rheology exists for colloidal depletion gels undergoing large deformations at high shear rates. Our experiments harness confocal laser scanning microscopy (CLSM) to capture the three-dimensional (3D) morphology of colloidal gels. Nonadsorbing polystyrene and photopolymer are used to induce depletion attraction of various strengths (Table 1 and Fig. S1). Quiescent gels exhibit structures (Fig. 1 *A* and *B*) and rheology (Fig. 1*C*) that are typical of colloidal gels (7, 11); specifically, there is a prominent peak in the radial distribution function *g*(*r*) at the first coordination shell, and the structure factor *S*(*q*) is elevated at low scattering vectors *q*. The rheological response shows a crossover between the elastic and viscous moduli at the yield strain. (Complete rheological data can be found in Fig. S2.) We generate gels with a variety of initial microstructures by changing the depletion potential, the colloidal volume fraction, the gelation time, and the degree of preshearing (16) (Fig. S3).

Previous attempts to visualize yielding have been complicated by shear banding (17), concurrence of rupture and reaggregation in steady flow, and difficulties in resolving particle positions at large deformation rates (9, 10, 18, 19). We resolve these issues (particularly shear banding; Fig. S4) by using unidirectional, high-rate step strains within a shear cell mounted to the CLSM (Fig. 1*D*). Incorporation of ultraviolet-initiated photopolymer to the solvents allows structures to be rapidly locked in place for imaging. We confirm that the immobilized sample represents the postdeformation structure by comparing the arrest time of the photopolymerization to the characteristic stress relaxation time of the gel. The difference between them is negligible (Fig. S5), indicating that photopolymerized gels accurately reflect the structure after cessation of flow.

To formulate a general correlation between rheology and microstructure, we look to the physics of granular materials for inspiration. In 1864, Maxwell determined that a particular configuration of nodes interacting with central, pairwise forces can only be isostatic and rigid if the number of constraints is equal to the degrees of freedom (20). This definition motivates a criterion for rigidity in colloidal gels, because gels support stress just as macroscopic structures are designed to withstand loads. For example, triangular and tetrahedral arrangements of trusses are used in architecture because of their structural stability. Analogous colloidal structures manifest themselves as Bernal spirals in gels (21, 22) and as tetrahedra in dodecagonal quasicrystals (23). Distinguishing between particles in rigid clusters and those connected by soft bonds allows us to parameterize micromechanical models proposed for colloidal systems (24⇓–26). Similar ideas have been used in fiber networks (27) and granular materials (28) to quantify the contribution of rigidity to the mechanics of 3D networks. Frictionless spheres in granular matter, for example, jam at the isostatic contact number of six (29). These treatments suggest that the contact number may be used as a metric of isostatic connectivity (30). In order to examine their implications in colloidal gels, we identify stress-bearing colloids as particles with *z* greater or equal to the isostatic contact number, *z*_{iso}, of spheres interacting through central forces (i.e., *z*≥*z*_{iso}, where *z*_{iso} = 6).

## Results and Discussion

We observe a striking transformation in the microstructure as the applied step strain, γ, is increased. At lower applied strains (0.5 ≤ γ ≤ 5), no changes are visually resolved (Fig. 1 *E*–*H*). As the applied strain increases, there is a progression from a dense network at γ = 10 (Fig. 1*I*) to a disordered, fluid-like suspension by γ = 20 (Fig. 1*J*). Locally dense clusters form between γ = 30 to 60 (Fig. 1 *K*–*N*). We provide two contrasting examples (γ = 5, 60) of yielding in Movies S1 and S2. Over the range of our CLSM observations, *G*^{′}(γ) decreases by almost two orders of magnitude. Despite this large decrease, the *G*^{′} at γ = 60 is much greater than that of a suspension of noninteracting particles, for which *G*^{′} is not measurable when γ > 15. Inspection of image volumes reveals a significant number of large clusters, even in the large-strain amplitude regime. The potential role of this type of cluster has previously been discussed in the context of thixotropic systems, for which the existence of large intact aggregates at high shear rates was inferred from measurements of the hydrodynamic stresses that were substantially larger than that of hard sphere suspensions (31, 32). What properties of these residual clusters are correlated with the nonlinear elastic stress?

To address this question, we acquire particle coordinates in 3D and plot the contact number distribution as a function of strain (Fig. 2*A*, *Inset*). Particles are considered to be in adhesive contact if they are within a distance less than that of the first minimum in *g*(*r*) (Fig. 1*A*). The breakdown of the network structure in Fig. 1 *E*–*N* is accompanied by a decrease in the mean contact number, , by a factor of two as γ increases from 0.5 to 60 (Fig. 2*A*). The modest decrease in is insufficient to explain the observed changes in *G*^{′}, as we address later. This failure suggests that theories based on the averaged ensemble of all particles in the system cannot fully describe rheological changes in yielding. Instead, it appears that certain subpopulations of particles are responsible for supporting elasticity in the ruptured gel.

We search for candidate subpopulations based on contact number by examining the difference between the contact number distribution for each γ and for the quiescent gel (Fig. 2*B*). The results show a decrease in the fraction of particles with six neighbors (*z* = 6) and a corresponding increase in the number of particles with *z* = 2. These observations are markedly different from proposed models. For example, within micromechanical theories, soft bonds are postulated to be important in yielding (9, 12). Other experiments suggest shear-induced densification occurs as flow progresses (9, 10). Neither possibility explains our observations of high strain-rate yielding, because Fig. 2*B* suggests that the principal change in microstructure is caused by the gradual erosion of particles with large *z*. Thus, it appears that high contact number particles are key determinants of the nonlinear *G*^{′}(γ) of ruptured gels. We hypothesize that these particles are localized in rigid clusters that are load-bearing. As strain-induced yielding proceeds, a reduction in the number of these clusters causes a drop in the *G*^{′}(γ) of the gel.

When we apply the rigidity criterion (configurations with *z*≥*z*_{iso} = 6 are considered rigid) to yielded gels, renderings illustrate differences between the rigid microstructure of gels sheared at small (γ = 0.5) and large strain (γ = 60). Rigid clusters at γ = 0.5 span the image volume (Fig. 3*C*), whereas at γ = 60, the clusters form only a small subset (Fig. 3*D*) of the original volume (Fig. 3*A* and *B*). In addition, we find that the volume fraction of rigidly bonded particles, *ϕ*_{rigid}, is a strongly decreasing function of γ (Fig. 3*E*). Between γ = 0.5 and γ = 60, *ϕ*_{rigid} drops from 0.08 ± 0.01 to 0.008 ± 0.006. Correspondingly, the mean number of particles in rigid clusters, , decreases by more than a factor of seven. (Note that the gel volume fraction, *ϕ*_{gel} = 0.20, is constant for these cases.)

Simulation of the clusters identified from the CLSM images supports that particles with *z*≥*z*_{iso} behave rigidly. We perform Brownian dynamics simulations (HOOMD-blue) on these clusters by initializing particles from the experimental configurations and using a harmonic spring approximation to connect those in the same cluster (33). The configurations are allowed to thermalize (Movie S3), and the relative rigidity of the clusters is characterized by the average angular mobility parameter, (Fig. 3*F*), and Fourier shape-matching parameter (34) (Fig. S6). The value of drops approximately fivefold from *z*≥1 to *z*≥9, indicating that the rotation of particles is increasingly constrained with large *z*. Clusters from *z*≥4 to *z*≥6 experience the largest decrease in . The simulations support the idea that rigidity in the experimental system can be correlated to the subpopulation of high contact number particles, allowing the effects of rigidity on gel rheology to be studied.

A mechanism for the elasticity of ruptured gels in terms of the mean size and volume fraction of the high contact number clusters can now be hypothesized. Within yielded gels, there exists a distribution of slowly diffusing clusters comprised of particles with large *z*. The low contact number particles relax from their nonequilibrium, shear-induced configurations rapidly, and are incapable of supporting elastic stress. These floppy modes with *z* ≤ *z*_{iso} need not be considered in the rheological model for elasticity; the elastic rheology is then a function of the size and number of interacting rigid clusters. To test this idea, we plot the strain-dependent elastic modulus normalized by the cluster volume and thermal energy, , as a function of *ϕ*_{rigid}(γ) (defined for particles with *z*≥6) (Fig. 4). The ordinate is based on the classical scaling for hard sphere rheology (35), with the volume of a hard sphere, *V*_{p} ∼ (2*a*)^{3}, replaced by the mean volume of a rigid cluster, . Fig. 4 is generated by plotting nondimensional variables suggested by the isostaticity approach. These variables obtained from the CLSM microstructural results have no adjustable parameters. The high quality of the correlation suggests that the nonlinear *G*^{′}(γ) is described by the manner in which shear reduces the size and volume fraction of load-bearing clusters. Remnant clusters contribute a Brownian stress that scales as a power law of *ϕ*_{rigid}(γ) with an exponent of 2.7 ± 0.4. This scaling is generated because of the slow relaxation of rigid clusters, mediated by hydrodynamic interactions between clusters. We obtain a similar power-law correlation with the Brownian stress, *G*_{B}(*t*,γ), which we measure during stress relaxation after a nonlinear step strain deformation (31, 36) (Fig. S7). These experiments provide additional support for the hypothesis that structural rigidity is the principal determinant of the stress response of gels ruptured through different types of nonlinear deformation. Because the large amplitude oscillatory deformation of *G*^{′}(γ) and the stress relaxation after step strain, *G*_{B}(*t*,γ), differ in the relative contributions of hydrodynamic and Brownian components of the stress, there is scope for rigidity to contribute to these rheological measures in different ways. These differences have potential implications for rheological properties, such as in thixotropic systems (31).

The power-law correlation based on *G*^{′}(γ) and *ϕ*_{rigid}(γ) holds for all samples in Table 1. Moreover, the correlation based on the properties of the rigid clusters is statistically much better (, *n* = 24) than the analogous correlation of *G*^{′} with the ensemble-averaged contact number (, *n* = 9). Here, is the normalized chi-squared value of the correlation, and *n* is the number of experimental conditions tested. When we expand the analysis to consider correlations for *z*≥0 to *z*≥9, we find highly significant correlations for *z*≥4 to *z*≥6, with values of that are close to 1 (Fig. 4, *Inset*). This rigidity range is also supported by the simulation results (Fig. 3*F*). Interestingly, these bounds match the *z*_{iso} for 3D granular matter, which decreases from six to four as friction between particles increases (29). The lower limit of the rigidity criterion (*z*≥4) could be linked to friction and/or noncentral, tangential interactions in colloids, which has been previously measured in particle chains (14). Thus, we propose using *z* = 4 and *z* = 6 as lower and upper bounds for evaluating the rigidity of colloidal gels.

Identification of the role of isostaticity and rigidity in gel elasticity provides a new method to understand and manipulate the viscoelastic properties of soft matter in high strain-rate flows. The general correlation that we show in Fig. 4 can be applied to predict colloidal structure in large strain and strain-rate flows that are characteristic in the processing of soft matter. We have shown that nonlinear elasticity is a result of the slow relaxation of rigid and hydrodynamically coupled clusters; it would be interesting to investigate the orientational anisotropy of these clusters along the principal axes of the flow (10) as well as the nature of this hydrodynamic coupling. Our work also demonstrates that rheological properties can be quantitatively predicted using the underlying microstructure of stress-bearing clusters, and introduces new possibilities for the design of soft materials in which the shape of such clusters can be modified. For example, gels with Bernal spiral microstructure (21, 22) might potentially display an even more defined rheological response than that found in this work, because of the structural uniformity imposed by their rigid tetrahelical motif.

## Materials and Methods

### Preparation of Colloidal Gels.

A suspension of sterically stabilized poly(methyl methacrylate) (PMMA) dyed with Nile Red dispersed in a refractive index matched solvent was used. We synthesized monodisperse dimethyl-diphenyl siloxane (DPDM) stabilized PMMA spheres of two diameters (37), 2*a* = 861 nm ± 5% and 1.06 μm ± 4%, as well as poly(12-hydroxystearic acid) (PHSA) stabilized PMMA spheres (38) of diameter 2*a* = 1.67 μm ± 4%. Gels were created by adding a combination of nonadsorbing, linear polystyrene of various concentration (*M*_{W} = 900,000 g/mol, *M*_{w}/*M*_{n} ≤ 1.10, *R*_{g} = 41 nm, ) and a UV-triggered photopolymer (39) to particles suspended in two solvents: (*i*) 55% cyclohexyl bromide (CHB):10% decalin:35% dioctyl phthalate (DOP) and (*ii*) pure DOP. Tetrabutylammonium chloride of concentration ranging from 1 to 10 μM (Debye length of κ^{-1} = 69 nm) was added. Direct observation of the motion of the colloidal particles in an electric field device established an upper bound of the zeta potential ≤ 10 mV (40).

### Confocal Microscopy, Shearing Device, and Structural Measurements.

Confocal microscopy images were acquired using a Leica TCS SP2 scanning head attached to a Leica DMIRE 2 inverted microscope and 100×, 1.4 numerical aperture oil immersion objectives. A shear cell with parallel glass plates was used for the visualization of gels before and after the application of a step strain (39). The top plate was attached to a linear stepper motor. We ensured that the parallelism of the plates was within ± 10 μm. The gap distance between the plates was set to 120 μm for 5 ≤ γ ≤ 60, and doubled to 240 μm for 0.5 ≤ γ ≤ 5 to compensate for the very small displacements needed.

We allowed the gels to rest for a specified waiting time, *t*_{w}, after initial sample loading. Step strains were applied to the quiescent gel sample at a shear rate of 40 s^{-1} (for DPDM-stabilized PMMA gels) and 50 s^{-1} (for PHSA-stabilized PMMA gels) to induce structural changes. For the PHSA-stabilized PMMA gels (case 7 in Table 1), we also applied an oscillatory preshear procedure (γ = 120 at 50 s^{-1} each oscillation for a total of 30 s) prior to equilibration. We chose high shear rates to avoid shear banding, as shown in the case of 0.033 s^{-1} where regions of nonuniform density were observed in the gels (Fig. S4). We also checked for the existence of shear banding in the rheometer using the same shear rates. These combined results suggest the existence of unstable flows consisting of sections of fluid moving at low shear rates. Consequently, for most of the cases listed in Table 1, gels were deformed at the high shear rate of 40 s^{-1}. For gels suspended in pure DOP, rheological instrument constraints limit us to deformations with a shear rate of 5 s^{-1}.

Three independent experiments were performed for each step strain. Gels were photopolymerized immediately after yielding. Image volumes (27.5 × 27.5 × 25 μm^{3}) were taken at three locations away (distances above the coverslip at 10 μm, 35 μm, 60 μm) from sample boundaries. Image analysis was performed on 96% of the total collected volumes, with the rest rejected because of anomalous, extreme structural heterogeneity that is possibly caused by the deficient initial bonding of the gel to the surface of the shear cell. Centroids of particles in the image volumes were identified in 3D space (41). The static error in particle location was previously found to be ± 35 nm in the objective plane and ± 45 nm in the axial direction (42). Using this image processing procedure, we obtain the contact number distribution of each image volume. Particles are considered nearest neighbors if the interparticle separation is less than that of the first minimum in the radial distribution function, *g*(*r*).

The static structure factor, *S*(*q*), was calculated from particle centroids using , where **q** are the wave vectors chosen in Cartesian coordinates, *N* is the total number of particles in the system, and **r**_{i} is the distance vector of particle *i* to the origin. We removed particle centroids near the boundaries to eliminate finite size effects of the image volume (7).

### Rheological Measurements.

Rheological measurements were done using a stress-controlled rheometer (AR-G2, TA Instruments) with a 40-mm parallel steel plate at a gap of 300 μm. The temperature was fixed at 25 °C. A solvent trap was fitted over the geometry to reduce solvent evaporation. The samples were presheared at 200 rad/s for 2 min and allowed to rest for *t*_{w}. For PHSA-stabilized PMMA gels (case 7 in Table 1), the sample was presheared for an additional 30 s at 50 rad/s. To address the inhomogenous shear flow in the parallel disk geometry, we correct our oscillatory rheology and step strain data using the method of Soskey and Winter (43).

The elastic moduli, *G*^{′}, of the gels were measured for strain amplitudes ranging from 0.0001 to 100 at a shear rate of 40 rad/s. The shear rate was reduced to 5 rad/s for samples from case 2 to obtain measurements of *G*^{′} at large γ due to the stress limits on the instrument. We checked that wall slip did not affect our results by attaching sandpaper (roughness length scale of approximately 26 μm) to the top and bottom of the rheometer geometry and by changing the geometry gap to 600 μm (44). We also performed an oscillatory measurement using a 60-mm cone-and-plate geometry. The standard deviation of these measurements for all conditions and all strains (± 40%) is comparable to that of sample-to-sample variability (± 50%).

In order to quantify the degree of gelation arising from the added polystyrene and the photopolymer, we measured the linear oscillatory elastic modulus, , as a function of the concentration of the depletants, *c*_{depl}. We take the onset of gelation to occur at *c*_{depl} = *c*_{depl,gel}, where two distinct power laws intersect. We include corresponding CLSM images for these data points (Fig. S1).

We performed stress relaxation measurements after a nonlinear step strain to determine the Brownian stress of the ruptured gels and to determine the adequacy of the photopolymer arrest time (36). The Brownian stress, *G*_{B}(*t*,γ), was determined as the residual stress immediately after the step strain has been completed, after the hydrodynamic contribution to the stress has ceased (Fig. S7). We measured *G*_{B}(*t*,γ) for a number of cases listed in Table 1 (except for γ = 0.5, 1, 20, 30, 40 in case 1, and γ = 20, 60 in case 7). To test the performance of the photopolymer, a step strain (γ = 50) was applied (case 1 from Table 1) and the stress relaxation measured after the deformation (Fig. S5). After the cessation of the step strain, the stress magnitude remained close to 1.5 Pa for approximately 10 s. This stress plateau is much longer than the photopolymerization time (< 0.8 s). The comparison shows that the photopolymer arrests structure much more rapidly than the diffusion and reorientation of the gels.

### Clustering of Particles Based on Contact Number and Statistical Analysis.

We assigned particles to clusters for which the cutoff distance is the location of the first minimum of *g*(*r*) of the quiescent gel. Linear regression was used to find the best power-law fit for versus *ϕ*_{rigid} for each value of *z*. A chi-squared value was calculated to quantify the agreement between the experimental values and the power-law fit using the relation , where *d* is the number of degrees of freedom, *O*_{k} and *E*_{k} are the observed and expected *y*-axis values, and *n* is the total number of observations. Larger values of is a result of significant disagreement between the observed and expected distributions (45). If the value of is close to 1, there is a significant probability that the measurements follow the expected distribution. We used the same method to obtain a power-law fit of versus *ϕ*_{rigid} (Fig. S7). Variability in the magnitude of the rigid volume fraction arises from sample-to-sample differences, the particular value of the *g*(*r*) selected as the cutoff for nearest neighbors, and the number of nearest neighbors used as the rigidity criterion.

### Brownian Dynamics Simulations and Rigidity Metrics.

Molecular dynamics simulations were conducted using the molecular dynamics package HOOMD-blue (http://codeblue.umich.edu/hoomd-blue). Particle coordinates were taken from experimental data. All particles were modeled with a soft-core repulsion via the Weeks–Chandler–Anderson potential (46), with a core diameter of 1.06 μm. Particles within the distance defined by the first minimum of the *g*(*r*) were bonded with a harmonic approximation of the interaction potential. The spring constant was set at *k* = 4,800, similar to the estimated width of the experimental depletion potential. The equilibrium bond distance was chosen to be the interparticle distance. The simulation was allowed to thermalize for 1 million time steps. The connectivity of the bonded network determined the extent to which particles can locally rearrange and move during the simulations.

Characterization of the rigidity of clusters was done using two metrics: the angular mobility metric and the Fourier shape-matching parameter. Angular mobility for each cluster of particles was measured by calculating the average square displacement of all angles in each cluster as they evolved, and can be quantified using a time-averaged mobility metric, , where θ_{i}(*t*_{0}) and θ_{i}(*t*) are angles between two adjacent bonds in the initial configuration at *t*_{0} and the simulation configuration at time *t*, and *N*_{c} is the number of particles within the cluster. The Fourier parameter was computed from shape descriptors that determine the correlation between the shape of a cluster with the shape at another frame (33). These descriptors can be used to calculate an order parameter based on the goodness of fit. The relative rigidity of a cluster was measured by the time average of this order parameter, , with larger values denoting a more rigid structure. Both parameters were calculated from 20 simulation frames and clusters containing 3 to 100 particles.

## Acknowledgments

We are grateful to R. Ziff and J. Hart for useful discussions on the subject of rigidity percolation and structural stability, and K. Kohlstedt for the static structure factor computations. M.J.S and L.C.H were supported in part by the National Science Foundation, Division of Chemical, Bioengineering, Environmental, and Transport Systems Award CBET 0853648, and the International Fine Particles Research Consortium. S.C.G. and R.S.N. were supported by the Department of Defense/Assistant Secretary of Defense (Research and Engineering) under Award N00244-09-1-0062.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: mjsolo{at}umich.edu.

Author contributions: L.C.H. and M.J.S. designed research; L.C.H., R.S.N., S.C.G., and M.J.S. performed research; L.C.H., R.S.N., S.C.G., and M.J.S. contributed new reagents/analytic tools; L.C.H., R.S.N., S.C.G., and M.J.S. analyzed data; and L.C.H., R.S.N., S.C.G., and M.J.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1206742109/-/DCSupplemental.

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