^{1}, Jeffrey R. Errington

^{2}and Thomas M. Truskett

^{1,a)}

### Abstract

Partial pair-correlation functions of colloidalsuspensions with continuous polydispersity can be challenging to characterize from optical microscopy or computer simulation data due to inadequate sampling. As a result, it is common to adopt an effective one-component description of the structure that ignores the differences between particle types. Unfortunately, whether this kind of simplified description preserves or averages out information important for understanding the behavior of the fluid depends on the degree of polydispersity and can be difficult to assess, especially when the corresponding multicomponent description of the pair correlations is unavailable for comparison. Here, we present a computer simulation study that examines the implications of adopting an effective one-component structural description of a polydisperse fluid. The square-well model that we investigate mimics key aspects of the experimental behavior of suspended colloids with short-range, polymer-mediated attractions. To characterize the partial pair-correlation functions and thermodynamic excess entropy of this system, we introduce a Monte Carlo sampling strategy appropriate for fluids with a large number of pseudo-components. The data from our simulations at high particle concentrations, as well as exact theoretical results for dilute systems, show how qualitatively different trends between structural order and particle attractions emerge from the multicomponent and effective one-component treatments, even with systems characterized by moderate polydispersity. We examine consequences of these differences for excess-entropy based scalings of shear viscosity, and we discuss how use of the multicomponent treatment reveals similarities between the corresponding dynamic scaling behaviors of attractive colloids and liquid water that the effective one-component analysis does not capture.

We thank Professor Stuart Rice and Professor C. P. Royall for their helpful input. T. M. T. acknowledges support of the Welch Foundation (F-1696) and the National Sciece Foundation (CBET-1065357). J. R. E. acknowledges financial support of the National Science Foundation (CBET-0828979). M. J. P. acknowledges the support of the Thrust 2000 – Harry P. Whitworth Endowed Graduate Fellowship in Engineering. The Texas Advanced Computing Center (TACC), the University at Buffalo Center for Computational Research, and the Rensselaer Polytechnic Institute Computational Center for Nanotechnology Innovations provided computational resources for this study.

I. INTRODUCTION

II. METHODS

A. Model for the polydisperse fluid

B. Monte Carlo simulations

C. Static pair correlations

D. Structural order metrics

E. Thermodynamic excess entropy

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Entropy
- 20.0
- Colloidal systems
- 16.0
- Monte Carlo methods
- 11.0
- Shear rate dependent viscosity
- 7.0
- Suspensions
- 7.0

## Figures

Effect that the magnitude of the interparticle attraction ε/*k* _{B} *T* has on the static structure of the polydisperse SW fluid described in the text in the dilute (ρσ^{3} → 0) limit. (a) [Main panel] Comparison of the reduced two-body excess entropy, −*s* ^{(2)}/ρσ^{3} *k* _{B} (identical to −*s* ^{ex}/ρσ^{3} *k* _{B} in this limit), computed using Eq. (A3), as well its effective one-component counterpart, , calculated using Eqs. (5) and (A5). [Inset] Comparison of the reduced structural order metric, τ/ρ^{1/3}σ, computed using Eq. (A1), as well its effective one-component counterpart, τ_{eff}/ρ^{1/3}σ, calculated using Eqs. (7) and (A5). Effective one-component quantities spuriously indicate the presence of structural anomalies (attractions apparently weaken structure) for energies ε/*k* _{B} *T* less than the values indicated by the black dots. (b) [Main panel] The effective one-component radial distribution function *g* _{eff}(*r*), computed using Eq. (A5), plotted versus reduced interparticle separation *r*/σ. [Inset] The interparticle potential, , of the effective one-component fluid.

Effect that the magnitude of the interparticle attraction ε/*k* _{B} *T* has on the static structure of the polydisperse SW fluid described in the text in the dilute (ρσ^{3} → 0) limit. (a) [Main panel] Comparison of the reduced two-body excess entropy, −*s* ^{(2)}/ρσ^{3} *k* _{B} (identical to −*s* ^{ex}/ρσ^{3} *k* _{B} in this limit), computed using Eq. (A3), as well its effective one-component counterpart, , calculated using Eqs. (5) and (A5). [Inset] Comparison of the reduced structural order metric, τ/ρ^{1/3}σ, computed using Eq. (A1), as well its effective one-component counterpart, τ_{eff}/ρ^{1/3}σ, calculated using Eqs. (7) and (A5). Effective one-component quantities spuriously indicate the presence of structural anomalies (attractions apparently weaken structure) for energies ε/*k* _{B} *T* less than the values indicated by the black dots. (b) [Main panel] The effective one-component radial distribution function *g* _{eff}(*r*), computed using Eq. (A5), plotted versus reduced interparticle separation *r*/σ. [Inset] The interparticle potential, , of the effective one-component fluid.

Partial radial distribution functions *g* _{30j }(*r*) describing correlations between the and the pseudo-components (*j* ∈ [1, 60]) of the polydisperse SW fluid obtained via MC simulations. Results are shown as a function of reduced interparticle separation *r*/σ and interaction diameter (σ_{30} + σ_{ j })/2. Color (red-to-blue) corresponds to magnitude of *g* _{30j }(*r*) (high-to-low). Data are for a reduced density of ρσ^{3} = 1.05 and interparticle attractions of (top panel) ε/*k* _{B} *T* = 0 (hard-sphere limit) and (bottom panel) ε/*k* _{B} *T* = 2.5.

Partial radial distribution functions *g* _{30j }(*r*) describing correlations between the and the pseudo-components (*j* ∈ [1, 60]) of the polydisperse SW fluid obtained via MC simulations. Results are shown as a function of reduced interparticle separation *r*/σ and interaction diameter (σ_{30} + σ_{ j })/2. Color (red-to-blue) corresponds to magnitude of *g* _{30j }(*r*) (high-to-low). Data are for a reduced density of ρσ^{3} = 1.05 and interparticle attractions of (top panel) ε/*k* _{B} *T* = 0 (hard-sphere limit) and (bottom panel) ε/*k* _{B} *T* = 2.5.

Effect that magnitude of the interparticle attraction ε/*k* _{B} *T* has on the static pair correlations of the polydisperse SW fluid at a reduced density of ρσ^{3} = 1.05. Data obtained via MC simulations. (a)–(f) Partial radial distribution functions *g* _{30j }(*r*) between and the pseudo-components for *j* = 10, 20, 30, 40, and 50 (with corresponding characteristic interaction diameters σ_{30j } = 0.90, 0.95, 1.0, 1.05, and 1.10, respectively) and various values of ε/*k* _{B} *T*. (g) Radial distribution function for the effective one-component description of the fluid, *g* _{eff}(*r*), as a function of reduced particle center separation *r*/σ for conditions used to generate the data for panels (a)–(f).

Effect that magnitude of the interparticle attraction ε/*k* _{B} *T* has on the static pair correlations of the polydisperse SW fluid at a reduced density of ρσ^{3} = 1.05. Data obtained via MC simulations. (a)–(f) Partial radial distribution functions *g* _{30j }(*r*) between and the pseudo-components for *j* = 10, 20, 30, 40, and 50 (with corresponding characteristic interaction diameters σ_{30j } = 0.90, 0.95, 1.0, 1.05, and 1.10, respectively) and various values of ε/*k* _{B} *T*. (g) Radial distribution function for the effective one-component description of the fluid, *g* _{eff}(*r*), as a function of reduced particle center separation *r*/σ for conditions used to generate the data for panels (a)–(f).

Effect that magnitude of interparticle attraction ε/*k* _{B} *T* has on different structural order metrics for the polydisperse SW fluid at a reduced density of ρσ^{3} = 1.05. Data obtained via MC simulations. [Main panel] Negative thermodynamic excess entropy, −*s* ^{ex}/*k* _{B} (diamonds), and its two-body approximation based on multicomponent, −*s* ^{(2)}/*k* _{B} (squares), and effective one-component, (circles), descriptions of the system. [Inset] Translational structure metrics based on the effective one-component, τ_{eff} (circles), and multicomponent, τ (squares), descriptions of the system. Curves are guides to the eye.

Effect that magnitude of interparticle attraction ε/*k* _{B} *T* has on different structural order metrics for the polydisperse SW fluid at a reduced density of ρσ^{3} = 1.05. Data obtained via MC simulations. [Main panel] Negative thermodynamic excess entropy, −*s* ^{ex}/*k* _{B} (diamonds), and its two-body approximation based on multicomponent, −*s* ^{(2)}/*k* _{B} (squares), and effective one-component, (circles), descriptions of the system. [Inset] Translational structure metrics based on the effective one-component, τ_{eff} (circles), and multicomponent, τ (squares), descriptions of the system. Curves are guides to the eye.

Reduced shear viscosity ηρ^{−2/3}(*mk* _{B} *T*)^{−1/2} plotted versus various structural order metrics based on static pair correlations for the polydisperse SW fluid: (a) [Main panel] negative two-body excess entropy, −*s* ^{(2)}/*k* _{B} and [Inset] its effective one-component counterpart, . (b) [Main panel] translational structure metric, τ, and [Inset] its effective one-component counterpart, τ_{eff}. Structural data were obtained via the MC simulations of this study, and the dynamic data were extracted from the earlier molecular dynamics investigation of Krekelberg *et al.* ^{45} Open symbols represent data for ε/*k* _{B} *T* > 2 and filled symbols for ε/*k* _{B} *T* < 2. For the latter conditions, shear viscosity anomalously increases with ε/*k* _{B} *T* (see Fig. 2 of Ref. 45)

Reduced shear viscosity ηρ^{−2/3}(*mk* _{B} *T*)^{−1/2} plotted versus various structural order metrics based on static pair correlations for the polydisperse SW fluid: (a) [Main panel] negative two-body excess entropy, −*s* ^{(2)}/*k* _{B} and [Inset] its effective one-component counterpart, . (b) [Main panel] translational structure metric, τ, and [Inset] its effective one-component counterpart, τ_{eff}. Structural data were obtained via the MC simulations of this study, and the dynamic data were extracted from the earlier molecular dynamics investigation of Krekelberg *et al.* ^{45} Open symbols represent data for ε/*k* _{B} *T* > 2 and filled symbols for ε/*k* _{B} *T* < 2. For the latter conditions, shear viscosity anomalously increases with ε/*k* _{B} *T* (see Fig. 2 of Ref. 45)

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