Abstract
Colloidal gels formed by arrested phase separation are found widely in agriculture, biotechnology, and advanced manufacturing; yet, the emergence of elasticity and the nature of the arrested state in these abundant materials remains unresolved. Here, the quantitative agreement between integrated experimental, computational, and graph theoretic approaches are used to understand the arrested state and the origins of the gel elastic response. The microstructural source of elasticity is identified by the lbalanced graph partition of the gels into minimally interconnected clusters that act as rigid, load bearing units. The number density of clustercluster connections grows with increasing attraction, and explains the emergence of elasticity in the network through the classic CauchyBorn theory. Clusters are amorphous and isostatic. The internal cluster concentration maps onto the known attractive glass line of sticky colloids at low attraction strengths and extends it to higher strengths and lower particle volume fractions.
Introduction
In soft solids composed of colloidal suspensions, emulsions, foams, and biomaterials, elasticity is governed by the spatial distribution and interactions among amorphous mesoscale structures^{1}. Identifying and understanding the behavior of these fundamental building blocks are the underlying challenges for developing structureproperty relations that are essential to controlling and tailoring the mechanics of such materials. Among soft solids, an important and ubiquitous class are colloidal gels, in which attractive interactions between suspended colloidal particles drive a thermodynamic instability that promotes aggregation, arresting in a space spanning network structure possessing unique mechanical and transport properties^{2,3}. Commonly, phase separation is induced by the addition of nonadsorbing polymer to a suspension of repulsive colloids by the wellknown depletion interaction^{4,5,6,7}. Depletion gels are often found in industrial processes and products where fine solids are dispersed in polymer solutions, including agrochemicals, consumer care products, and pharmaceuticals, and have frequently served as model experimental systems^{8,9}.
In applications, the rheology of a gel is its principal material property of interest, including its elasticity^{10} and yielding^{11}. At low volume fractions and strong interaction energies between particles, colloidal gels are effectively modeled as fractal flocs formed through diffusioncontrolled aggregation processes, which grow together to form a percolating microstructure^{12}. The flocs are the principal load bearing units of the gel and theories connecting the floc architecture to the gel modulus remain a stateoftheart description^{13,14}. Yet, there exists no definitive microstructural theory for the elasticity of colloidal gels at higher volume fractions and lower strengths of interaction. What are the fundamental structural units imparting elasticity to the network, and what physical principles govern their formation?
Linear elasticity in depletion gels has been postulated to result from the spatial organization of particles into locally dense clusters, each cluster acting as a rigid, mechanical unit that propagates the elastic deformation^{10,15,16,17}. For instance, the work of Zaccone, Wu, and Del Gado^{16} theorized that the role of clustercluster contacts were central to gel elasticity. The authors showed that one model of cluster microstructure, based on contact number distributions for hard spheres and the CauchyBorn theory for the affine elastic response of amorphous solids^{18}, can fit experimental measurements of the shear modulus. Similar models with different approaches to enumerating clusters and cluster contacts based on mode coupling theory (MCT) were also explored by Ramakrishnan and coworkers^{10}. Signatures of clustering are evident in light scattering^{10} and confocal microscopy^{19}, which probe longrange fluctuations in colloid number density, and active microrheology, which examines the elastic deformation in response to a local perturbation^{20}.
In the present work, we combine experimental, computational, and graphtheoretic approaches to systematically identify clusters in depletion gels and show that they indeed constitute the fundamental structural units that lead to the gel’s elastic response. For the same interparticle interactions, the structure and elastic modulus of the experimental and simulated gels agree quantitatively. Our experiments and simulations show that the gel elastic modulus is a convex function of the interparticle attraction. However, this property cannot be accounted for by any macroscopic structural change or the change in interparticle bond stiffness. To resolve this important puzzle, we use a graph theoretic approach to identify the fundamental elastic units of the gels, a set of clusters, and to measure the density of clustercluster contacts. The physical size of the clusters is independent of the strength of the depletion attraction, but their number density and the number of clustercluster contacts grows with increased attractive strength. These changes, when combined with the CauchyBorn theory, yield a prediction of the elastic modulus with the correct convexity and in quantitative agreement with experimental measurements and calculations from simulations. Additionally, we find that the internal cluster structure becomes less dense with increasing attraction. The volume fraction of colloids within the clusters falls on and then extends the attractive glass line (AGL) of colloids determined via MCT, suggesting that the arrest of the gel forming process is due to glassy physics within the fundamental elastic units. Colloidal gels formed via arrested phase separation should be viewed as dense packings of locally glassy grains. There is a striking similarity to polycrystalline solids: elasticity and, potentially, plasticity emerges from the few weak bonds between clusters^{21}.
Results
Gel structure, rheology, and interparticle interactions
The structure, rheology, and interparticle interactions for the depletion gels are summarized in Fig. 1. In experiments, a recently developed model depletion gel is employed, which enables the rheology, microstructure, and particle interactions to be measured in concert for complete determination of the microscopic properties and macroscopic elastic response^{22}. In simulations, a highperformance Brownian dynamics simulation algorithm generates representative depletion gel structures and enables us to compute the elastic modulus of those gels^{23,24}.
Colloidal gels formed by arrested phase separation (Fig. 1a, experiment; 1b, simulation) exhibit no significant change in number density correlation length and only subtle changes in local bonding structure with increasing strength of attraction between the colloids. However, the elastic modulus of the gels measured experimentally and computed in simulations over the same range of attractions varies by more than a factor of 5 (Fig. 1c). Figure 1d depicts the number density fluctuations within spherical volumes of radius r measured from particle tracking applied to confocal microscopy of depletion gels and computed from results of Brownian Dynamics simulations. For large r, all density fluctuations decay as the powerlaw r^{−3}, reflecting a homogeneous structure on large length scales. Below the correlation length ξ (discussed below) shown in Fig. 1e, the number density fluctuations scale with a different powerlaw sensitive to the local structure of the gels. In both simulations and experiments, these differences are plainly evident. Over length scales similar to the particle size, the experimentally measured variations in the number density fluctuations are overwhelmed by statistical noise in the particle tracking algorithm arising from static and dynamic tracking errors.
The correlation length ξ (Fig. 1e) is visible in the experimental micrographs and renderings of the simulations. We calculate it from the peak of the static structure factor computed from both the experiments and the simulations. Across a broad range of interaction strengths, the microstructural correlation length is nearly constant and in close agreement between experiment and simulation. The value ξ ≈ 10a, where a = 0.560 μm is the colloid radius, is commonly observed in experiments with gels formed via arrested phase separation^{10}. It is well understood that a diffusionlimited aggregation process precedes the arrest of the gel in its terminal, elastic state, which freezes in the same correlation length regardless of the strength of the attractive interaction^{9,25,26}. But the key question remains: How are the local number density fluctuations to be understood, and what impact do these have on the elasticity of the colloidal gels?
CauchyBorn theory
In order to understand the emergence of elasticity in the particle network, a model of gel elasticity can be formulated from the CauchyBorn theory, in which the elastic modulus arises from a few weak connections between locally rigidified clusters^{10,15,16}. The CauchyBorn elastic modulus is written as the product of three physical quantities: G′ = 4n_{e}κ 〈r^{2}〉, where n_{e} is the number density of elastically active bonds in the material, κ is the bond stiffness, and \(\left\langle {r^2} \right\rangle = (1/3){\int} {(r_xr_y + r_yr_z + r_xr_z)} P({\mathbf{r}})\mathrm{d}{\mathbf{r}}\) is a squared length scale associated with the elastically connected domains^{18}. Formally, it is an average over the crosssecond moments of the vector r, connecting each cluster’s geometric center to the point at which it is bonded to its neighboring clusters taken over the distribution of clustercluster contacts, P(r).
It is natural to expect 〈r^{2}〉 to scale with one of the two length scales evident in a depletion gel: the particle size, a, or the correlation length, ξ. Since neither varies significantly with interparticle interaction, changes in length scale cannot explain the variation of the elastic modulus in gels formed by arrested phase separation. The bond stiffness κ, can be estimated from the interparticle potential through equipartition by computing kT/〈h^{2}〉, where kT is the thermal energy and 〈h^{2}〉 is the average of the squared surface to surface separation of a pair of particles whose interparticle distance satisfies a MaxwellBoltzmann distribution. It may also be inferred in the simulations of arrested gels through explicit calculation of kT/〈h^{2}〉. Both measures are plotted in Fig. 1f. The bond stiffness with respect to the strength of interparticle interaction increases modestly and has the opposite concavity of the elastic modulus, and thus it alone cannot explain how elasticity in the gels develops. What remains from the CauchyBorn theory to explain the change in elasticity is a change in the number density of elastically active bonds.
Visual inspection of gel structures measured experimentally and computed in simulations offers no clear delineation of cluster and intercluster bonds beyond the correlation length ξ associated with the number density fluctuations. Laser tweezer experiments have identified rigidly clustered regions with similar characteristic length scale through mechanical interrogation of the network by an oscillatory driving force^{20}. This form of mechanical interrogation is purely local and identifies only single clusters, and cannot, in a practical way, provide a statistical description of the cluster number density or their interconnectedness. Yet, these two quantities: the number density of rigid clusters, n_{c}, and the average number of clustercluster contacts per cluster, z_{c}, are sufficient to determine the remaining parameter for the CauchyBorn theory: the number density of elastically active bonds, \(n_e = \frac{1}{2}n_{\mathrm{c}}z_{\mathrm{c}}\). An alternative means of identifying rigid clusters and their connectedness is needed. In the present work, we employ a graph theoretic approach^{27,28}.
lbalanced graph partition of gels
To calculate n_{c}, z_{c}, and 〈r^{2}〉, we apply the lbalanced graph partition (Fig. 2a) to represent the rigidly connected clusters comprising the gel in a form that encodes the topology of the interparticle bond network but that is abstracted from the particles’ physical spatial arrangement. An unweighted, undirected graph representing each simulated gel is built by associating every particle with a vertex and every bond between two particles with an edge connecting the corresponding vertices. Decomposition proceeds by cutting edges to divide the graph into clusters containing no more than l particles. The partition is chosen such that there are nearly equal numbers of vertices in each cluster (balanced) while cutting the fewest possible edges. Subgraphs given by the partition represent the most loosely connected set of particle clusters containing l particles. The bound of l can be chosen arbitrarily, and here we use bisection to efficiently find a value l^{*} such that the average radius of gyration, R_{G}, of the clusters identified in the subgraphs is equal to the correlation length of the gel, ξ. The number density of clusters is simply n_{c} = n/l^{*}, (where n is the number density of particles) and is plotted in Fig. 2c. Importantly, the decomposition shows that the number of rigid clusters, and thus n_{c}, is an increasing function of U/kT. With the cluster size and overall number density fixed, the internal density of particles internal to the clusters is a decreasing function of U/kT. The colored clusters identified by the graph partitioning (Fig. 2b) exhibit this property.
The connections between subgraphs determines the average cluster contact number z_{c} and the square length scale 〈r^{2}〉. Figure 2c depicts n_{c}, z_{c}, and 〈r^{2}〉 computed from the graph decomposition. Like the correlation length ξ, 〈r^{2}〉 is not a systematic function of the attraction strength. However, z_{c} is an increasing function, and crucially, the product n_{c}z_{c} is a convex function of U/kT, such that the number of elastically active contacts alone is able to reproduce trends in the elastic modulus measured from experiments and simulations. When these measures of the connectivity among the fundamental elastic subunits are substituted into the CauchyBorn theory (Fig. 1c), the elastic modulus is reproduced quantitatively as well.
We compute a histogram of the number of bonds formed by particles within the individual clusters to verify that the clusters identified by the lbalanced graph decomposition are themselves rigid (Fig. 3). This distribution reveals that the mean number of bonds per particle exceeds six for all cases studied. For particles with central interactions, six bonds are necessary for isostaticity and thus rigidity. The contact number distribution cannot prove definitively that the clusters identified by the decomposition are internally rigid. However, that the mean number of contacts per particle in a cluster is larger than six is consistent with clusters having a rigid core. Specific notions of isostaticity and rigidity of the whole gel structure or parts has been highlighted recently through alternative methods^{29,30,31}.
It is important to recognize that the lbalanced graph decomposition is not unique. For example, consider the graph with five vertices connected in a ring. The lbalanced graph decomposition with subgraphs having three or fewer vertices (l = 3) could result in any pair of subgraphs containing three and two vertices selected from the ring by cutting just two edges. We employ a spectral decomposition method that determines the decomposition through kmeans clustering using a stochastically sampled initial condition. Degeneracy of the decomposition should only affect the loose connections between rigid subunits and not the subunits themselves. The effect of degeneracy in the decomposition on the predicted macroscopic elasticity was quantified by determining 95% confidence intervals on the inferred quantities from five replicates of the decomposition process. These uncertainties are smaller than the size of the symbols in Fig. 2c, but are propagated forward to the elastic modulus determined from the CauchyBorn theory in Fig. 1e. During the process of bisection for determining the value of l, we observe that the mean radius of gyration of clusters varies monotonically in the neighborhood of l^{*}, which itself is much smaller than the total number of particles in the gels. Therefore, based on smoothness of the objective function and reproducibility of the decomposition, we conclude that the graph decomposition is stable and insensitive to the precise value of l^{*} when the gel contains a large number of clusters.
While it is ultimately desirable to use the experimental confocal image data to perform an identical cluster analysis, the present experimental system sacrifices some particle tracking resolution (index mismatch) to enable both rheology and optical trapping for direct measurements of the particle interactions by laser tweezers^{22}. This increases both the static and dynamic contributions to the variation in particle positions. Figure 1d clearly shows the small wave vectors and short length scales of these contributions, while both experiment and theory capture the longrange structure of the gel.
Discussion
The model data in Fig. 1c are the quantity: 2n_{c}z_{c}κ〈r^{2}〉, with n_{c}z_{c} and 〈r^{2}〉 determined via graph theoretic methods. The results show that CauchyBorn theory yields a prediction of the elastic modulus with the correct convexity and in quantitative agreement with experimental measurements and calculations from simulations. The graph decomposition, which is agnostic to the absolute spatial arrangement of the colloids, has revealed the underlying elastic structure of these heterogeneous gels. Similar graph theoretic approaches have proven useful for understanding the mechanical properties of other amorphous solids such as granular packings^{32}.
The graph theoretic approach provides further insight into the microstructure of gels formed during arrested phase separation on length scales below the correlation length, ξ. The number density of clusters n_{c} can be used to define an effective cluster volume fraction ϕ_{c} = 4πn_{c}ξ^{3}/3, representing the fraction volume in the gel occupied by the elastic subunits. Through conservation of particle number, the volume fraction of colloids internal to the subunits is ϕ_{g} = ϕ/ϕ_{c}. Because n_{c} is an increasing function of the strength of interparticle attraction, the internal volume fraction of the clusters is a decreasing function of the same. That is, the elastic subunits contain fewer particles with increasing strength of attraction. Figure 4a depicts both ϕ_{c} and ϕ_{g} determined from the graph decomposition. Following the work of Zaccone, Wu, and Del Gado^{16}, an approximation for ϕ_{c} is also calculated by solving the implicit equation G′ = ϕ_{c}z_{c}(ϕ_{c})κ/(5πξ), at each strength of attraction and measured value of G′. This equation is a restated form of the CauchyBorn theory with G′ the modulus determined in the simulations, 〈r^{2}〉 = ξ^{2}/15, and z_{c}(ϕ_{c}) drawn from the hardsphere equation of state^{33,34}, in which the radial distribution function at contact, \(g(2;\phi ) = \frac{{1  \frac{1}{2}\phi }}{{(1  \phi )^3}},\) gives z_{c}(ϕ) = ϕg(2;ϕ). The resultant volume fraction of clusters and their internal volume fraction are also plotted in Fig. 4a. The hardsphere approximation provides a reasonable estimate of the volume fraction of elastic subunits and their internal volume fraction; both differ from the graphtheory result by roughly 10%. The data in Fig. 4a come from the simulated G′ to provide a direct comparison between different models of the cluster architecture: graph decomposition and hard sphere equation of state. A quantitatively equivalent set of cluster volume fractions can be generated using the hardsphere approximation and experimentally measured values of G′, which are indistinguishable from the the simulated values within the experimental uncertainty.
Remarkably, the internal volume fraction of clusters ϕ_{g} extrapolate from the AGL calculated from MCT, which includes a bilinear coordinate transformation to superimpose the MCT theory predictions onto Monte Carlo simulations^{35,36}. It is important to note that glassiness is conventionally defined in terms of long or diverging relaxation times and not merely the density of a phase. We observe in both experiments and simulations that the gel as a whole is arrested. This highlights the importance of work that tracks the morphology and dynamics of individual clusters over longer time scales than studied here, in order to understand how clusters form, relax, and eventually arrest^{30}. In Fig. 4b, each interparticle attraction strength corresponding to a depletant concentration is represented by the reduced Baxter temperature τ_{B}^{3,9,37,38,39}. The Baxter temperature is related to the reduced second virial coefficient \(B_2^ \ast\) by \(\tau _B = \frac{1}{{4(1  B_2^ \ast )}},\) where \(B_2^ \ast = \frac{{B_2}}{{\frac{2}{3}\pi \sigma _{{\mathrm{eff}}}^3}}\), \(\sigma _{{\mathrm{eff}}}^3\) is the effective hard sphere diameter, and B_{2} is the second virial coefficient calculated from the pair interaction potential U(r) by \(B_2 = 2\pi {\int}_0^\infty {r^2} \left[ {1  e^{  U(r)/kT}} \right]\mathrm{d}r.\) The AsakuraOosawa potential is used to calculate \(B_2^ \ast\) for the experimental system of PMMA in CH and CHB (mass fraction w_{CH} = 0.37) with polystyrene depletant. Based on the principle of corresponding states for shortrange attractive interactions^{40}, the particle volume fraction and reduced Baxter temperature τ_{B} for each experiment is plotted on the phase diagram for adhesive hard spheres (AHS) with open black circles in Fig. 4b. The solid black circles represent the corresponding ϕ_{g} values derived from the cluster model. We also plot τ_{B}, ϕ, and ϕ_{g} obtained from fitting the cluster model to the elastic moduli of depletion gels from octadecyl silica nanoparticles in decalin^{10} (Details of the model fits are provided in Supplementary Note 2). The results are remarkably consistent between depletion gels that differ substantially in size and chemistry, but that otherwise correspond to the same scaled range and strength of the interaction potential. Finally, without a graphtheoretic approach to identify gel clusters, we calculated the local packing fraction of dense particle strands in the network, following the approach of Lu et al.^{9} These values, shown in Fig. 4b, are roughly constant at a volume fraction ϕ ~ 0.57 and agree with their earlier study using PMMA colloids (a = 0.560 μm) in density matched cyclohexyl bromide and decahydronaphthalene with tetrabutylammonium chloride.
The CauchyBorn cluster model of gel rheology accurately describes the elastic modulus of depletion gels based on the bond rigidity, the cluster size, and the cluster density. As the attractive strength increases with depletant concentration, the gel modulus also increases. However, the higher modulus is due mainly to the lower density of particles within clusters, which coincides with the AGL extrapolated into the coexistence region of the phase diagram. These results are supported by a long line of evidence that there is an intimate connection between the gelation of colloids with shortrange attraction and phase separation. For particles with centrosymmetric forces, a thermodynamic driving force in the twophase region drives aggregation, and locally arrested density fluctuations, clusters that are sparsely connected, give the network elastic properties.
This work raises a number of interesting possibilities for engineering the mechanical strength of colloidal gels, similar to controlling crystallite grain size in metals^{41,42}. During the quiescent formation of gels, the constant value of ξ at different attractive strengths results from the diffusionlimited aggregation kinetics. By imposing a flow or external field, the cluster size can be altered and clusters can even be made anisotropic. Thus, one could rely on processing, with a focus on altering the cluster size ξ or the cluster shape as a means of controlling the material stiffness. The CauchyBorn theory has a more primitive tensorial form that can account for such anisotropies. The lbalanced graph decomposition provides the necessary tool to detect anisotropic clusters from which these anisotropies derive. Finally, the graph theory methods introduced here should also be powerful tools for identifying and characterizing mesoscale structures from realspace position data that dictate the mechanics of other networked colloidal materials, like capillary suspensions^{43} and particle networks at interfaces^{44,45,46}.
Methods
Depletion gels
Experiments are performed using a recentlydeveloped model colloidal gel that enables the simultaneous measurement of the structure, rheology, and particle interactions^{22}. The depletion gels consist of poly(methylmethacrylate) latex (PMMA) particles dispersed in a mixture of cyclohexane and hexadecane. With this model system, we measured the bulk shear modulus with a rotational rheometer, the microstructure using confocal microscopy, and the particle interactions with laser tweezers.
Depletion gels were prepared with PMMA particles (2a = 1.11 μm ± 3%) suspended in a mixture of cyclohexane (CH) and cyclohexyl bromide (CHB) with the CH mass fraction w_{CH} = 0.37 at a particle volume fraction ϕ = 0.2. Polystyrene (M_{w} = 900,000 g/mol, c* = 10.8 mg/mL, R_{g} = 32 ± 2 nm) is added at different concentrations (c/c^{*} = 0.35, 0.46, 0.58, 0.69, 0.80, and 0.93) to adjust the strength of the attraction between the particles. The results from the model system are compared with depletion gels reported by Ramakrishnan et al.^{10} (Supplementary Note 2 contains information on the experimental system). The material properties of the two model systems are summarized in Table 1.
Confocal imaging of gel structure
An inverted confocal microscope (Nikon A1Rsi) equipped with a resonant scanner head and a highspeed piezo stage is used to image the colloidal gels. Gel samples are loaded into glass capillaries with 300 μm spacers to match the gap used in the rheometer. The top and bottom of the capillaries are glass coverslips suitable for microscopy (0.17 mm thickness). After loading, the gels sit quiescently for 30 min before imaging. 3D image stacks are taken from the bottom coverslip at a speed of 15 slices per second. The image stacks are 42 μm × 42 μm × 10 μm in dimension. The voxel dimensions are 83 nm × 83 nm × 83 nm.
Confocal image volumes are analyzed using feature finding code based on the particle tracking algorithm of Crocker and Grier^{47}, which uses a Gaussian mask to filter out digital noise and that identifies centroids based on their intensity maxima. The particle tracking is modified from the original twodimensional algorithm to locate the center of a particle within a threedimensional voxel with subvoxel resolution. Errors in centroid location are determined in the xy and the xzplanes using immobilized samples. The error is ±34 nm in the xyplane and ±67 nm in the xzplane. Particle position data are used to calculate several measures of the gel structure, including the particle contact number distribution and the number density fluctuations. Particles are considered to be nearest neighbors if they are closer than the distance that defines the first minimum in the radial distribution function, g(r).
Bulk rheology
Oscillatory rheometry is performed on a stresscontrolled rheometer (ARG2, TA Instruments) at T = 25 °C. Colloidal gels are loaded onto a Peltier plate and a stainless steel parallel plate geometry (d = 6 cm, gap h = 300 μm) is lowered to the gap distance while rotating at ω = 1 rad/s to minimize the formation of bubbles at the sample interface. The geometry minimizes confinement effects and a solvent trap is used to prevent the loss of the volatile solvent. We employ a correction factor for the nonhomogeneous strain rate in the parallel plate geometry^{48}. We verified that gap size effects are negligible for h > 150 μm, which is determined by measuring the straindependent linear elastic modulus for poly(ethylene oxide) standards (Molecular weight M_{w} = 2 × 10^{6} g/mol, 4 wt%) at h = 50, 100, 150, 200, 300, 400, and 500 μm, and for colloidal gel samples at h = 300 and 500 μm. Slip does not occur; a smooth fixture and a fixture with a sandblasted surface (parallel plate geometry d = 6 cm) give identical moduli to within experimental error.
After sample loading, gels are presheared unidirectionally for 1 min (strain = 37,700, shear rate = 628.3 s^{−1}). Oscillatory strain sweep (fixed angular frequency, ω = 10 rad/s) and frequency sweep measurements (fixed strain amplitude, γ = 0.01) are performed after waiting 30 min. The waiting time is chosen to allow direct comparison between the gel microstructure and the rheological measurements on the gels. The average of three independent measurements of the gel rheology are made for each depletant polymer concentration.
Brownian dynamics simulations
Computer simulations are performed using a recently developed a method for rapid calculation of hydrodynamic interactions (HI) in suspensions of monodisperse spheres^{24}. The positivelysplit Ewald (PSE) algorithm makes the cost of computing Brownian displacements in simulations of colloidal scale particles with HI comparable to the cost of computing deterministic displacements in freely draining simulations. Here, the RotnePragerYamakawa tensor (RPY)^{49} is used to account for the longranged HI with great fidelity. The method relies on a new formulation for Ewald summation of the RPY tensor, which guarantees that the realspace and wavespace contributions to the tensor are independently symmetric and positivedefinite for all possible particle configurations. Brownian displacements are drawn from a superposition of two independent samples: a wavespace (farfield) contribution, computed using techniques from fluctuating hydrodynamics and nonuniform fast Fourier transforms; and a realspace contribution, computed using a Krylov subspace method. The combined computational complexity of drawing these two independent samples scales linearly with the number of particles enabling hydrodynamic simulations with system sizes up to 10^{6} particles.
The shortranged depletion attraction is modeled with an AsakuraOosawa form^{4}:
for particle separations r in the range of 2a < r < 2(a + δ). The width of the attraction is δ = 0.057a and the value of the potential at contact, U, relative to the thermal energy scale, kT, is chosen to match the depletion strength of the polymer concentration c/c^{*} used in the experiments.
Simulations of colloidal dispersions with 216,000 spherical particles of mean radius a and 5% polydispersity are conducted in a cubic simulation box of size L and periodic boundary conditions. We impose a particle volume fraction, ϕ = 0.20, matching that of the experiments. The attractive dispersion is allowed to gel in a solvent of viscosity η for 10^{3} bare diffusion steps τ_{D} = 6πηa^{3}/kT at which point each sample is found to be percolated and the measurements are performed. All results are averaged over 5 independently generated samples for each value of U/kT.
In order to measure the structural correlation length in the simulated gels we compute the static structure factor using all particle positions, \({\mathbf{x}} \in {\Bbb R}^{3N}\):
S(q, t) measures spatial correlations between particle positions over distances proportional to ~2π/q. As the samples are arrested, the S(q, t) curves are unchanging at long times. The location of the peak in S(q, t) at small wave vectors (q → 0) is a measure of the characteristic correlation length of the particles in the gel. For each sample we identify the value of the wavevector at the maximum, q_{max}, using a local fit to a parabola, and use it to compute ξ = 2π/q_{max}.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The simulation method, Brownian Dynamics with hydrodynamic interactions, is made publicly available through a plugin to HOOMDblue, the open source molecular dynamics suite, at the authors website: http://web.mit.edu/swangroup/software.shtml, as well as via email requests. HOOMDblue version 1.3.4 is used in this study.
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Acknowledgements
E.M.F. and M.J.S. acknowledge funding from the International Fine Particles Research Institute. E.M.F. also thanks the National Science Foundation (CBET1235955) and fruitful discussions with A. Lemaître. J.W.S. acknowledges ACSPRF Grant No. 56719DNI9 and an NSF CAREER grant (CBET1554398) for financial support.
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E.M.F., M.J.S. and J.W.S. designed the research; K.A.W. and L.C.H. collected the experimental data; K.A.W., L.C.H., E.M.F., M.J.S. and J.W.S. analyzed data; Z.V. and J.W.S. performed computer simulation and contributed to theory; and E.M.F., J.W.S., K.A.W., L.C.H. and M.J.S. wrote the manuscript.
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Whitaker, K.A., Varga, Z., Hsiao, L.C. et al. Colloidal gel elasticity arises from the packing of locally glassy clusters. Nat Commun 10, 2237 (2019). https://doi.org/10.1038/s4146701910039w
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