{'fraction_non_alphanumeric': 0.027335193686800492, 'fraction_numerical': 0.008941698922795331, 'mean_word_length': 5.01697057230547, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
From Hindered to Promoted Settling in Dispersions of Attractive Colloids: Simulation, Modeling, and Application to Macromolecular Characterization

The settling of colloidal particles with short-ranged attractions is investigated via highly resolved immersed boundary simulations. At modest volume fractions, we show that inter-colloid attractions lead to clustering that reduces the hinderance to settling imposed by fluid back flow. For sufficient attraction strength, increasing the particle concentration grows the particle clusters, which further increases the mean settling rate in a physical mode termed promoted settling. The immersed boundary simulations are compared to recent experimental measurements of the settling rate in nanoparticle dispersions for which particles are driven to aggregate by short-ranged depletion attractions. The simulations are able to quantitatively reproduce the experimental results. We show that a simple, empirical model for the settling rate of adhesive hard sphere dispersions can be derived from a combination of the experimental and computational data as well as analytical results valid in certain asymptotic limits of the concentration and attraction strength. This model naturally extends the Richardson-Zaki formalism used to describe hindered settling of hard, repulsive spheres. Experimental measurements of the collective diffusion coefficient in concentrated solutions of globular proteins are used to illustrate inference of effective interaction parameters for sticky, globular macromolecules using this new empirical model. Finally, application of the simulation methods and empirical model to other colloidal systems are discussed.

For sufficient attraction strength, increasing the particle concentration grows the particle clusters, which further increases the mean settling rate in a physical mode termed promoted settling. The immersed boundary simulations are compared to recent experimental measurements of the settling rate in nanoparticle dispersions for which particles are driven to aggregate by short-ranged depletion attractions. The simulations are able to quantitatively reproduce the experimental results. We show that a simple, empirical model for the settling rate of adhesive hard sphere dispersions can be derived from a combination of the experimental and computational data as well as analytical results valid in certain asymptotic limits of the concentration and attraction strength. This model naturally extends the Richardson-Zaki formalism used to describe hindered settling of hard, repulsive spheres. Experimental measurements of the collective diffusion coefficient in concentrated solutions of globular proteins are used to illustrate inference of effective interaction parameters for sticky, globular macromolecules using this new empirical model. Finally, application of the simulation methods and empirical model to other colloidal systems are discussed.

I. INTRODUCTION
Settling of dispersed colloidal particles is central to the processing and analysis of a wide range of industrial and scientific materials. Centrifugation is used as both a processing and analytical tool in laboratory and commercial settings [1], and gravity-driven particle motion controls the shelf life of many consumer and food products [2]. In the context of environmental science, sedimentation of suspended particles plays a crucial role in the engineering of pollution remediation strategies [3] as well as natural processes of erosion and deposition [4].
Additionally, there is a linear relationship between the sedimentation coefficient and the collective diffusivity in colloidal dispersions. Thus, important industrial separation processes such as ultrafiltration applied to purification of biologically derived macromolecules, whose design requires accurate models of the collective diffusivity, depend on knowledge of the sedimentation coefficient in concentrated dispersions [5]. In a vast majority of these materials and systems, the colloidal particles are attractive. However, most fundamental studies of settling in colloidal dispersions concern hard or repulsive particles. In the present work, we apply the immersed boundary method to study the settling rate in concentrated dispersions of spherical colloids with short-ranged attractions.
The settling dynamics of concentrated colloidal dispersions are controlled by the hydrodynamic interactions induced as moving particles displace the surrounding fluid. The flows generated by settling particles decay to leading order as the inverse of the distance from the particle. The entrainment of other particles by these disturbance flows is a critical factor in settling dispersions. The combined flow produced by all the settling particles appears to diverge with the physical size (extent) of the dispersion because of the long-ranged nature of these hydrodynamic interactions [6]. However, mass conservation at the system boundaries produces a back-flow driven by a pressure gradient balancing the buoyant weight of the particles. When the back-flow is combined with the induced flows, an intensive sedimentation rate, one that is independent of the system size, results. The interplay of entrainment and back-flow is a subtle one that makes the sedimentation a sensitive function of the microstructure of the colloidal dispersion. In dispersions of attractive colloids, which tend to form clusters, entrainment and back-flow act synergistically so that attractions always accelerate the settling process. The reasons for this are straightforward: clusters in isolation naturally sediment more quickly than individual particles, and the widened spaces between clustering particles lead to a weaker resistance due to the back-flow. Quantifying these effects represents a major challenge in quantitative colloid science. In the present work, we consider a limit in which the Péclet number for the settling particles is vanishingly small.
In this limit, the rate of particle diffusion far exceeds the rate of sedimentation and the microstructure of the colloidal dispersion is dictated by thermodynamic equilibrium.
Of the limited studies on the sedimentation of attractive suspensions, the most important result, a dilute-limit approximation for the mean settling rate U of a suspension of spherical particles, accurate to first order in the particle volume fraction φ, was derived by Batchelor more than fifty years ago [7,8]: where U 0 is the settling speed of a single isolated particle and B * 2 = B 2 /B HS 2 , where B 2 is the second virial coefficient of the particles and B HS 2 is the second virial coefficient of a reference hard sphere dispersion. For dilute hard spheres, entrainment and back-flow combine to reduce the relative settling rate of particles by an amount of 6.55 φ. Attractive interactions cause B * 2 to drop from unity (hard spheres) to negative values. This has the effect of increasing the settling speed relative to hard spheres. For sufficiently strong attractions (B * 2 ≤ −0.861), Batchelor's theory predicts that a dispersion can settle faster than an isolated particle. Moncho-Jordá et al. used a series of stochastic rotation dynamics (SRD) simulations to test Batchelor's dilute limit prediction as well as to study sedimentation in concentrated attractive dispersions. They observed that particles with attractive interactions sediment more quickly than hard spheres at all volume fractions and that, within a specific range of volume fractions, sufficiently strong attractions produce a settling rate that is larger than even the isolated particle value. For dispersions that exhibited such promoted settling, the sedimentation rate exhibited a non-monotonic dependence on particle volume fraction. At low volume fractions, the rate increases above the single particle value. Beyond a critical volume fraction, the settling rate decreased as expected of conventional hindered settling. These simulations and the resulting data possess a number of physical and quantitative issues, including: being performed at finite but small Péclet and Reynolds numbers, being limited in the number of particles represented in the periodic simulation box, and having made no systematic finite system size corrections to the measured sedimentation rates.
Inspired by these simulations, Lattuada et al. conducted an experimental study designed to corroborate the simulation results [10]. They found a qualitative agreement between the experiments and simulations. However, they observed that the SRD simulations systematically under-predict the settling rate of attractive suspensions measured experimentally and also under-predict the range of volume fractions over which promoted settling is observed.
These studies are the first steps towards understanding the settling of concentrated solutions of attractive particles, but much remains unexplored. For example, a predictive microstructural model for the settling rate would be very useful in applications involving attractive colloidal dispersions.
In the present work we use immersed boundary simulations in the limits of zero Péclet and Reynolds numbers to systematically investigate the effect of attraction strength and particle volume fraction on the mean settling rate. We show that these simulations, which account properly for finite system size effects reproduce experimental measurements quantitatively.
Furthermore, we develop an empirical model for the settling rate based on a measure of microstructure in attractive dispersions that quantitatively matches experimental data and our own simulation data at all the volume fractions studied.

II. SIMULATION METHODOLOGIES
We compute the settling rate of attractive colloidal dispersions using an immersed boundary method referred to colloquially as the composite bead [11] or rigid multi-blob [12] approach. The details of the computational method are described elsewhere [11]. Here we give a brief review. In this approach, the surface of each colloidal particle is approximated by a collection of beads that interact hydrodynamically. Each bead generates a regularized Stokeslet flow and is entrained by flows induced by other particles in a reciprocal fashion [13,14]. In our formulation, the Rotne-Prager-Yamakawa tensor [15] that linearly relates the force exerted on one bead to the entrained velocity of another bead is used to represent these flows. Additionally, the beads tessellating the surface of each particle are constrained to move as a rigid body. A set of Lagrange multipliers, forces that ensure the constraints are satisfied, are introduced and a system of linear equations is solved to determine the rigid body motions of the colloidal particles and the Lagrange multipliers in response to any set of imposed external forces. In the present work, the spectral Ewald method [16] is used to evaluate the product of the Rotne-Prager-Yamakawa tensor for all the N bead , at different volume fractions, φ. We find that N bead = 2562 is sufficient to exactly reproduce the published data of Zick and Homsy [18]. The polyhedra used in the calculations are shown above the associated points. The black dashed line denotes 5% relative error.
beads with an arbitrary set of forces acting on the beads. The linear equations governing the rigid body motions and the Lagrange multipliers are solved using the GMRES methodology with constraint preconditioning [17]. In summary, this approach can be used to evaluate the transport properties of macromolecular and colloidal solutions of arbitrarily shaped particles in O(N) time where N is the total number of beads used to discretize the surface of all the colloids.
For the spherical colloids studied here, the beads are made to sit on the vertices of an icosphere formed by subdividing the faces of a Goldberg polyhedron (see Figure 1). We have shown how such a discretization can be used to replicate the hydrodynamic interactions among a pair of spherical particles in a previous publication [11]. Convergence of the calculation with respect to the number of beads is slow for relative motion between particles because of the effects of hydrodynamic lubrication. However, convergence is fast for collective modes of motion. Figure 1 also depicts the convergence of the sedimentation rate of a cubic array of spherical colloids as a function of the number of beads tessellating the surface. Zick and Homsy performed high accuracy calculations of lattice sedimentation and published their results with three digits of accuracy [18]. We find that 2562 beads per colloid is sufficient to exactly reproduce the published results, and therefore use the solution with 2562 as the reference state for the convergence study. With just 162 beads per colloid, the classic results of Zick and Homsy are recovered to within 5% for the full range of volume fractions studied. Therefore, in the present calculations we use 162 beads per colloid. This level of discretization is chosen to accommodate a balance between the number of colloids required to accurately model the microstructure of an attractive dispersion, the number of realizations of that microstructure needed to compute statistically meaningful averages, and the total computation time of the simulation. It is known that number density fluctuations are large in attractive colloidal dispersions. Additionally, the magnitude of velocity fluctuations among sedimenting colloids can be of the same order of magnitude as the mean sedimentation velocity [19]. Thus, 1000 colloids are modeled and averages of the settling velocity are constructed from 1000 independent snapshots of the dispersion microstructure.
Throughout this work, fluctuations in the settling rate among different configurations are used to quantify a standard error at the 95% confidence level.
The experiments of Lattuada et al. control inter-particle attraction by adding surfactant micelles to solution in order to induce depletion [10]. Therefore, we model the interaction potential, U(r) with a short-ranged attraction described by the Asakura-Oosawa depletion potential [20] plus a hard core repulsion at inter-particle contact, r = 2a: where ε is the strength of the attraction at contact and δ is the range of the attraction. The Asakura-Oosawa potential was validated in these experiments through measurements of the osmotic compressibility in the colloidal dispersion. The range of the depletion potential is chosen to match the experiments: δ/a = 0.028. The depletion potential at contact, ε, is varied and can be chosen so that simulated dispersions match the experimentally measured second virial coefficient, B * 2 . Consistent with experiments, 5% dispersity in particle size is introduced to suppress crystallization in the simulations.
The experimental value of the Péclet number is small, where ∆ρ is the density difference between the immersed colloids and the fluid, g is the gravitational constant, and k B is Boltzmann's constant, so the distribution of particle positions during settling is very close to the equilibrium distribution. Therefore, simple, freely draining Brownian dynamics simulations of single beads are used to generate representative configurations of the particles from which the mean settling rate can be computed. For each combination of φ and ε (or B * 2 ), an initial hard sphere configuration of spherical colloids is allowed to relax for 1000 bare diffusion times (6πηa 3 /k B T ). Then, snapshots of the particle configuration are taken every 10 bare diffusion times until 1000 snapshots have been accumulated. For each snapshot, a composite-bead representation of all the particles is constructed from 162 bead icospheres having randomly assigned orientations. The mean settling rate for each configuration is determined by applying a uniform force to each bead and computing the mean of the resulting rigid body translational velocities of the colloids. The computed mean velocity is sensitive to the periodic boundary conditions in the simulations, and the infinite system size limit U ∞ is found by which is the finite size correction given by Ladd et al. [21], where S(0) is the value of the static structure factor at zero wavenumber, η s is the solvent viscosity, and η (φ) is the high-frequency viscosity, which we compute for each configuration. It should be noted that, although we use the depletion potential to model the particle attractions, with this procedure any sufficiently short-ranged potential will produce equivalent equilibrium structures [22].
The Noro-Frankel rule of corresponding states suggests that samples of these equilibrium structures can be drawn from a mapping onto the Baxter adhesive hard-sphere interaction potential. In section III B we discuss this mapping explicitly. Therefore, the results presented in this manuscript should be generic for any colloidal dispersion aggregating due to shortranged attractions.

A. Comparison of Simulations and Experiments
There is limited experimental data available for the mean settling rate of colloids as a function of concentration for different well-characterized attraction strengths. Lattuada et al.
provide data at three different attraction strengths, characterized by the reduced second virial coefficients: B * 2 = 1.0, −0.27, −1.08 [10]. These values correspond to hard sphere interactions (no attraction), a modest attraction (ε = 4.84 k B T ), and a strong attraction

Present Work
Ref. [10] Ref. [9] FIG. 2. Normalized settling velocity as a function of particle volume fraction for B * 2 = −1.08 (red), B * 2 = −0.27 (blue), and B * 2 = 1 (black) from the experimental data of Ref. [10] (stars with lines) and computed in the present work (filled circles) and the SRD simulations of Ref. [9] as interpolated in Ref. [10] (open squares). Error bars represent the standard error of the observed measurements.
(ε = 5.18 k B T ) that is near the boundary for liquid-liquid phase separation, respectively.
The experiments were performed with particle volume fractions, φ, as large as φ = 0.20. We compute the mean settling rate for each of the experimental values of B * 2 for 0.01 ≤ φ ≤ 0.40. ulations and the experimental data, and predict a much sharper decay of U with φ than is observed experimentally for the two attractive dispersions studied. The agreement between experiments and our simulations affirms our claim that the proposed simulation methodology is appropriate to perform a systematic study the sedimentation rate as a function of particle volume fraction and attraction strength.
It is a challenge to assess the discrepancy between the SRD simulations and our own. A number of factors may be at play. One central problem is a failure of Moncho-Jordá et al.
to account for finite system size effects in the calculation of the settling rate. As in Equation (3), in periodic geometries transport properties like settling rate and diffusivity are depressed by an amount proportional to aV −1/3 where V is the volume of the simulation cell. For simulations with a fixed number of particles, increasing the volume fraction acts to decrease V , which further depresses the computed values of transport properties. However, to change volume fraction, Moncho-Jordá et al. fix the simulation volume and increase N. So a systematic depression of the settling rate due to finite system size effects alone is not enough to explain their data. A further issue that may explain the discrepancy with their simulations is the finite Péclet number used in that study. The authors estimate that Pe ≈ 2.5 in their dynamic simulations, which means that deviations from equilibrium distribution for the particle configuration are inevitable. The effect of Pe on the mean settling rate of dispersions is also not well studied. However, for hard-spheres in the dilute limit and at large Peclet numbers, Cichocki and Sadlej [23] have shown that a different expression for the sedimentation rate is expected: U/U 0 = 1 − 3.87φ. The settling rate is a sensitive function of the structure, and the settling process modeled dynamically by the SRD simulations may have exhibited changes from the equilibrium structure and even structural anisotropy promoted by the hydrodynamic interactions among the particles. In the present study, the Peclet number is set asymptotically to zero and only the equilibrium structure is used for the calculation of the transport properties.

B. Microstructural Characterization and Corresponding States
As is common for colloids with short-ranged attractions, Lattuada et al. characterized the attraction strength in their dispersion using the Baxter temperature, τ . The Baxter temperature is a parameter that arises in the adhesive hard sphere (AHS) model of attractive particles, in which the inter-particle potential has vanishing width and infinite depth, but is characterized by a single parameter, τ , reflecting the propensity of particles to adhere to one another [24]. This parameter is a measure of the effective strength of attraction in the suspension, with smaller values corresponding to stronger attractions. B * 2 for the AHS model can be related to the Baxtern temperature, τ , as B * 2,AHS = 1−1/ (4τ ). In the approach taken by Lattuada et al., τ is computed by matching the measured B * 2 to the analytical expression for B * 2 of AHS model, so that: Assigning a value to τ in this fashion provides a way of determining the dilute limit thermodynamic and transport properties of dispersions with short-ranged attractions in a fashion that is agnostic to the details of the interaction potential. However, values of τ computed by (4) are valid only in the dilute limit. As the dispersion concentration increases, τ changes even for fixed B * 2 , and as a result more detailed models are required to infer τ for dispersions of arbitrary concentration.
In the original development of the AHS model, Baxter produced an analytical expression for the Percus-Yevick (PY) approximation to the pair distribution function g(r) of sticky particles [24,25]. The AHS model results in an expression for g(r) that is completely parameterized by the particle size a, φ, and the effective "stickiness" parameter -the Baxter temperature -τ . An analytical expression for the static structure factor, S(q), of an AHS suspension can be directly computed from Baxter's result [26]. Applying the Noro-Frenkel rule of corresponding states, which states that the thermodynamic properties, including dispersion microstructure, of colloids with short-ranged attractions can all be mapped back onto the adhesive hard-sphere model, the stickiness parameter, τ , for a concentrated dispersion of colloids with short-range attractions can be inferred from calculations of either g(r) or S(q) (or any other state function) for that dispersion. For any of these functions, the calculated values can be fit by analytical expressions from the adhesive hard sphere model at the same particle concentration to determine a corresponding value for τ . In the dilute limit, using the osmotic pressure as the state function will recover (4). For more concentrated dispersions, S(q) (or g(r)) is required to determine the particular value of τ that characterizes the thermodynamic state of the dispersion in the corresponding adhesive hard-sphere model. For this reason, throughout this manuscript τ is an a priori unspecified parameter which is inferred from simulation data.  the suspension microstructure in terms of the generalized stickiness parameter. This fitting procedure can be generalized to experimental data with sticky macromolecules of known number density, n as discussed in section III D. In this way, measurements of S(q), allows mapping onto the AHS model without a priori knowledge of either B * 2 . The pair τ and the volume fraction serve as unique descriptors of the attractive suspension microstructure and as we will show, the value of this pair also fixes the mean settling rate.

C. Settling rate as a function of φ and τ
We performed simulations systematically varying B * 2 between −1.48 and 1.00 and φ from 0.01 to 0.40 for a dispersion of spherical colloids interacting via a depletion potential with δ/a = 0.028. Each set of parameters is characterized by the τ value inferred from S(q). Figure 4a is the mean settling rate as a function of volume fraction and inferred Baxter temperature normalized by the single particle value, U/U 0 , where the color of the At a fixed concentration, as τ increases, the settling velocity decreases and approaches the hard sphere limit, τ → ∞. Furthermore, although the shape of U/U 0 is strongly dependent on τ at low φ, at sufficiently large φ, the settling velocity exhibits a qualitative similarity for all τ . With increasing concentration, the influence of the stickiness parameter on the settling rate is diminished. In this regime, particles fill space rather homogeneously, regardless of τ , and the sedimentation rate decays to zero much as with hard spheres. Attractions between particles still increase the sedimentation rate slightly by introducing small heterogeneities in the microstructure that are indicated by an increase in the value of S(q) as q → 0. These void spaces allow back-flow to occur with less resistance, thus slightly increasing the settling rate.

Shown in
The simulation data show a transition in the shape of the settling curves as both τ and φ are increased. For hard spheres, U/U 0 is monotonically decreasing and concave up for all φ. At modest attractions, U/U 0 is still monotonically decreasing, but is initially concave down until an inflection point around φ = 0.2. An inflection point is also observed for strong attractions, τ 0.2, but U/U 0 is no longer monotonic. The settling rate increases until φ ≈ 0.10, where it exhibits a maximum rate of promoted settling. Then, the settling rate decays much as for hard spheres.
The ratio of the sedimentation rate of attractive particle dispersions to a hard sphere dispersion at the same concentration is shown in Figure 4b as a function of φ/τ . φ/τ is the natural parameter that appearing in Batchelor's dilute limit result, when substituting the second virial coefficient with the AHS model [8]. Batchelor's model describes the ratio well for for φ/τ 0.5. For φ/τ 0.5, the data is no longer a function of φ/τ alone and exhibits a sharp downturn with increasing φ/τ .
To develop an empirical model for sedimentation in attractive dispersions at arbitrary volume fractions, we start with the Richardson-Zaki correlation [27], which is a common model for settling in suspensions, This expression is widely used in engineering applications even though it is known to deviate The value of m is determined to be 3.77 by fitting Equation (6)  Inspired by this approach, we notice that further modifying (5) with a denominator 1 + (6.55 − m)φ − 0.89φ/τ would also recover Batchelor's predictions in the dilute limit the literature: squares [28], triangles [29], upside down triangles [30], diamonds [31], solid circles [32], open circles [33].
for attractive dispersions. Because the settling rate bears such qualitative similarity with hard spheres at high concentrations, we proceed with the ansatz that the net effect of interparticle attractions can then be accounted for by simply replacing Batchelor's: −0.89φ/τ with a more general term −0.89f (φ, τ ). This function, f (φ, τ ) must be bounded from above to avoid singularities. It also must scale as φ/τ in the dilute limit and must approach zero in the hard sphere limit. In principle this function serves to interpolate between hard sphere settling rate and the settling rate of aggregated dispersions. The empirical expression we propose for the settling rate is, where a form for f (φ, τ ) remains to be specified.
This hypothesized model can be tested by solving (7) for f (φ, τ ), and computing this quantity from known settling rates. Figure 6 shows the values of f (φ, τ ), as a function of φ/τ with the settling rates given by our simulation data and the experimental data of Lattuada et al.. All the data in Figure 6 falls along a single master curve. For small values of φ/τ , f (φ, τ ) is linear in φ/τ with a slope of 1. This is required in order to recover Batchelor's There are many possible choices for this function, but a compact and convenient one is where p is a fitting parameter that is varied along with n to minimize the sum of squared errors between the model prediction for U and the data. The optimal values computed by a nonlinear least squares fit to the entire data set are n = 0.41 and p = 1.51.
The model accurately describes the settling rate across the wide range of φ and τ studied.
The relative error of the model with respect to the data is plotted in Figure 6b. The error is smaller than 10% for all the cases studied. The model does tend to slightly over-predict the settling rate at modest τ and is the least accurate for the strongest attraction studied. This can be understood by recognizing that for the most strongly attractive particles studied, τ ≈  F (q, 0), and its rate of change can be used to compute a wave vector dependent diffusivity, D(q) = −q −2 (d/dt) log(F (q, t)/F (q, 0)) as t → 0. As q → 0, D(q) is just the collective diffusivity D C , and is related to the sedimentation coefficient, U/U 0 , of the macromolecule by the simple relation: Here, D 0 is the diffusion coefficient of the macromolecule at infinite dilution. Therefore  where, From the present work, equation (7) models the sedimentation coefficient, U/U 0 = D C S(0)/D 0 .
To relate experimental data to the two models, the colloidal volume fraction must be defined in terms of the molar concentration c as φ = vc. Then, the experimental system has the properties of the sticky sphere solution at concentration c and parameterized by a molar volume v and stickiness parameter τ , just as in the dilute limit. We envision three different modes of inference with these models: • repeated measurements of S(0) and D C at a particular concentration, c and in a particular solution condition (temperature, ionic strength, pH), denoted p, are used to determine the values of τ (c, p) and v(c, p) for a corresponding AHS dispersion.
• measurements of S(0) and D C are made at different macromolecular concentrations and a particular solution condition, and a nonlinear least squares fit assuming a con-stant molar volume but concentration dependent stickiness parameter is used to find the set of parameters: v(p), τ (c, p).
• S(0) and D C are measured as function of the solution conditions at a particular macromolecule concentrations, and a nonlinear least squares fit of the data to the model is used to determine the parameters, v(p) and τ (p) descriptive of the data across the entire range of concentrations.
Choice of inference scheme depends on the particular macromolecule under study, one could proceed from one scheme to the next should the assumptions of concentration independent molar volume and stickiness prove sound.    mM and (472, 920) mM. Consequently, the electrostatic interactions between the proteins are similar screened in each of these cases. However, the stickiness parameters and molar volumes inferred from the AHS hard sphere models show no correspondence. This same lack of correspondence is evident in the S(0) and U/U 0 data itself. It is known that there are specific ion effects in concentrated electrolyte solutions that sensitively affect the interactions between charged colloids such as proteins [37]. These specific effects alter the structure factor and transport properties of the lysozyme solutions in ways that are difficult to predict with classical DLVO theory [38]. However, inference of the molar volume and stickiness parameter from fitting the AHS model to DLS data is sufficient to characterize solution in a way that yields an accurate description of thermodynamic properties (isothermal compressibility) and transport properties (sedimentation coefficient).

IV. CONCLUSION
We present coarse-grained simulations of the sedimentation of sticky particle suspensions that are shown to be in quantitative agreement with published experimental data. We systematically studied the effect of inter-particle attraction strength and volume fraction on the mean sedimentation rate of the suspensions, observing a smooth transition with increasing attraction strength from hard sphere sedimentation to non-monotonic promoted settling due to particle aggregation. We developed a simple model for the sedimentation rate by modifying a Richardson-Zaki-like expression to match Batchelor's dilute limit prediction and including a function that extrapolates between hard sphere-like and aggregated states. The model quantitatively describes the simulation data up to the percolation phase boundary, and an example application of this model to the study of collective protein diffusion is illustrated. The mapping of sticky particle suspensions onto an effective AHS model is generic, and can be used to infer sedimentation and diffusion behavior across a range concentrations from light scattering measurements made at only a few concentrations, even in suspensions where the particles are not uniform or spherical, provided the attractions are sufficiently short-ranged.