^{1,a)}, Jade M. Noble

^{1}and Laura J. Kaufman

^{1,b)}

### Abstract

Experimental studies that follow behavior of single probes embedded in heterogeneous systems are increasingly common. The presence of probes may perturb the system, and such perturbations may or may not affect interpretation of host behavior from the probe observables typically measured. In this study, the manifestations of potential probe-induced changes to host dynamics in supercooled liquids are investigated via molecular dynamics simulations. It is found that probe dynamics do not necessarily mirror host dynamics as they exist either in the probe-free or probe-bearing systems. In particular, for a binary supercooled liquid, we find that smooth probes larger than the host particles induce increased translational diffusion in the host system; however, the diffusion is anisotropic and enhances caging of the probe, suppressing probe translational diffusion. This in turn may lead experiments that follow probe diffusion to suggest Stokes-Einstein behavior of the system even while both the probe-free and probe-bearing systems exhibit deviations from that behavior.

The work was supported by National Science Foundation (NSF) CHE 0744322.

INTRODUCTION

SIMULATION DETAILS

RESULTS AND DISCUSSION

Structure and dynamics of the system

Dynamics of the probe

Temperature dependence of probe dynamics

Probe dynamics relative to system dynamics

Stokes-Einstein behavior

CONCLUSION

### Key Topics

- Diffusion
- 60.0
- Anisotropy
- 11.0
- Molecular dynamics
- 6.0
- Particle distribution functions
- 6.0
- Viscosity
- 6.0

## Figures

(a) Radial distribution function (rdf) of A particles, g_{AA}(*r**), in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan) at *T** = 0.48. Rdfs are nearly identical and thus nearly completely overlap. (b) Rdfs of A particles relative to the probe g_{Ap}(*r**) for *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan) at *T** = 0.48. Rdfs are shifted by *σ* _{p} /2 to highlight peak positions relative to the probe. The inset shows the unshifted g_{Ap}(*r**) functions. Data is averaged over five simulations.

(a) Radial distribution function (rdf) of A particles, g_{AA}(*r**), in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan) at *T** = 0.48. Rdfs are nearly identical and thus nearly completely overlap. (b) Rdfs of A particles relative to the probe g_{Ap}(*r**) for *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan) at *T** = 0.48. Rdfs are shifted by *σ* _{p} /2 to highlight peak positions relative to the probe. The inset shows the unshifted g_{Ap}(*r**) functions. Data is averaged over five simulations.

(a) MSDs for A particles in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Each MSD presented is the average of individual MSDs from five simulations. Inset shows a region of the MSDs in the linear, diffusive regime. (b) *F* _{ s }(*q*,*t**)s for A particles in systems with probes from *σ* _{p} = 1 – 5 at *q* = 7.25, the peak of the static structure factor. Each *F* _{ s }(*q*,*t**) presented is the average of individual *F* _{ s }(*q*,*t**)s from five simulations. Color scheme is the same as in (a). (c) Average translational diffusion constant, *D* _{A(p)}, (left axis, black circles) and average alpha-relaxation times, *τ* _{ α },_{A(p)}, (right axis, red squares) of large A particles in systems with σ_{p} = 1 – 5. Averages are obtained from the five simulations used to construct (a) and (b) and error bars are standard deviations over those five data sets.

(a) MSDs for A particles in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Each MSD presented is the average of individual MSDs from five simulations. Inset shows a region of the MSDs in the linear, diffusive regime. (b) *F* _{ s }(*q*,*t**)s for A particles in systems with probes from *σ* _{p} = 1 – 5 at *q* = 7.25, the peak of the static structure factor. Each *F* _{ s }(*q*,*t**) presented is the average of individual *F* _{ s }(*q*,*t**)s from five simulations. Color scheme is the same as in (a). (c) Average translational diffusion constant, *D* _{A(p)}, (left axis, black circles) and average alpha-relaxation times, *τ* _{ α },_{A(p)}, (right axis, red squares) of large A particles in systems with σ_{p} = 1 – 5. Averages are obtained from the five simulations used to construct (a) and (b) and error bars are standard deviations over those five data sets.

(a) MSDs for probe particles in the same systems shown in Figs. 1 and 2, with *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Additionally, the MSD for the A particles (black dotted line) is shown. Each MSD presented is the average of individual MSDs from five simulations. Inset shows a region of the MSDs in the linear, diffusive regime. (b) Ratio of diffusion constant of the A particles for the system with probe of given size, *σ* _{p}, to the diffusion constant of the probe multiplied by the probe size as a function of temperature (*D* _{A(p)}/(*D* _{p}**σ* _{p})) for *σ* _{p} = 1 (black circles), *σ* _{p} = 2 (red squares), *σ* _{p} = 3 (green diamonds), *σ* _{p} = 4 (blue triangles), *σ* _{p} = 5 (cyan downward triangles). Positive deviations from 1 reflect probe motion that is decreased relative to motion of the A particles. Points are averages over five simulations and error bars are standard deviations.

(a) MSDs for probe particles in the same systems shown in Figs. 1 and 2, with *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Additionally, the MSD for the A particles (black dotted line) is shown. Each MSD presented is the average of individual MSDs from five simulations. Inset shows a region of the MSDs in the linear, diffusive regime. (b) Ratio of diffusion constant of the A particles for the system with probe of given size, *σ* _{p}, to the diffusion constant of the probe multiplied by the probe size as a function of temperature (*D* _{A(p)}/(*D* _{p}**σ* _{p})) for *σ* _{p} = 1 (black circles), *σ* _{p} = 2 (red squares), *σ* _{p} = 3 (green diamonds), *σ* _{p} = 4 (blue triangles), *σ* _{p} = 5 (cyan downward triangles). Positive deviations from 1 reflect probe motion that is decreased relative to motion of the A particles. Points are averages over five simulations and error bars are standard deviations.

(a) Temperature dependence of diffusion as expressed via log(*D*) vs log(*T*–*T* _{ c }) for A particles in a *σ* _{p} = 1 system (open black circles, dotted lines), B particles in a *σ* _{p} = 1 system (open black squares, dashed line), and probe particles (*σ* _{p} = 1, filled black circles; *σ* _{p} = 2, red squares; *σ* _{p} = 3, green diamonds; *σ* _{p} = 4 blue triangles; *σ* _{p} = 5, cyan downward triangles). Each point represents the average of diffusion constants over five simulations and error bars are standard deviations. *T* _{ c } = 0.436 is the best-fit *T* _{ c } value and was determined by finding the best-fit *T* _{ c } for each of the data sets represented in the figure and then averaging those values. (b) Temperature dependence of alpha-relaxation as expressed via log(*τ* _{ α }) vs. log(*T* – *T* _{ c }) for systems described in (a). Here *T* _{ c } = 0.447, obtained via the same process described in (a). (c) Values of *γ*, 1/*K*, and -*γ* as determined from fits to *D* ∼ (*T* – *T* _{ c })^{ γ } (black circles; data and fits shown in (a)), *D* ∼ *T* exp(−1/*K*(*T*/*T* _{ o }−1) (red squares; data shown in (a); fits not shown), and *τ* _{ α } ∼ (*T* − *T* _{ c })^{−} ^{ γ } (green diamonds; data and fits shown in (b)), respectively. For fits to the Vogel-Fulcher-Tamman expression (red squares), 1/*K* values are best-fits for *T* _{ o } = 0.314, determined in the same manner T_{c} values are determined.

(a) Temperature dependence of diffusion as expressed via log(*D*) vs log(*T*–*T* _{ c }) for A particles in a *σ* _{p} = 1 system (open black circles, dotted lines), B particles in a *σ* _{p} = 1 system (open black squares, dashed line), and probe particles (*σ* _{p} = 1, filled black circles; *σ* _{p} = 2, red squares; *σ* _{p} = 3, green diamonds; *σ* _{p} = 4 blue triangles; *σ* _{p} = 5, cyan downward triangles). Each point represents the average of diffusion constants over five simulations and error bars are standard deviations. *T* _{ c } = 0.436 is the best-fit *T* _{ c } value and was determined by finding the best-fit *T* _{ c } for each of the data sets represented in the figure and then averaging those values. (b) Temperature dependence of alpha-relaxation as expressed via log(*τ* _{ α }) vs. log(*T* – *T* _{ c }) for systems described in (a). Here *T* _{ c } = 0.447, obtained via the same process described in (a). (c) Values of *γ*, 1/*K*, and -*γ* as determined from fits to *D* ∼ (*T* – *T* _{ c })^{ γ } (black circles; data and fits shown in (a)), *D* ∼ *T* exp(−1/*K*(*T*/*T* _{ o }−1) (red squares; data shown in (a); fits not shown), and *τ* _{ α } ∼ (*T* − *T* _{ c })^{−} ^{ γ } (green diamonds; data and fits shown in (b)), respectively. For fits to the Vogel-Fulcher-Tamman expression (red squares), 1/*K* values are best-fits for *T* _{ o } = 0.314, determined in the same manner T_{c} values are determined.

2D projections of 3D trajectories of large A particles at *T** = 0.48 surrounding a (a) *σ* _{p} = 1 probe (b) *σ* _{p} = 2 probe, and (c) *σ* _{p} = 5 probe all at *t** = 500 with frames separated by d*t** = 20.

2D projections of 3D trajectories of large A particles at *T** = 0.48 surrounding a (a) *σ* _{p} = 1 probe (b) *σ* _{p} = 2 probe, and (c) *σ* _{p} = 5 probe all at *t** = 500 with frames separated by d*t** = 20.

MSD values as a function of distance from the probe to particle centers (defined at *t** = 0), for (a) *σ* _{p} = 1, (b) *σ* _{p} = 2, and (c) *σ* _{p} = 5. Figure 6(a) also shows the rotated frame relative to the probe, with MSD_{par} = MSD_{x} + MSD_{z} and MSD_{perp} = MSD_{y}. In (a)–(c), solid lines with filled symbols are values at *t** = 50, all multiplied by 5, and dashed lines with open symbols are values at *t** = 500. Black lines and black circles represent total MSD values, red squares and red lines represent MSD_{par} values, and blue lines and symbols represent MSD_{perp} values. (d) Ratio of MSD values parallel and perpendicular to the probe (MSD_{par}/MSD_{perp}) at *t** = 50 (solid lines and filled symbols) and *t** = 500 (dashed lines and open symbols) as a function of distance from the probe for probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), and *σ* _{p} = 5 (cyan). (e) Maximum values of binned MSD_{par}/MSD_{perp} values from (d) as a function of probe size for *t** = 50 (black circles) and *t** = 500 (red squares). Best-fit lines are shown. (f) Various quantities (X) normalized by those quantities for the *σ* _{p} = 1 system: X = Maximum values of binned MSD_{par}/MSD_{perp} values at t* = 50 (filled black circles) and t* = 500 (filled red squares), total MSD values for those systems at *t** = 50 (open black circles) and *t** = 500 (open red squares), and the diffusion constants obtained from the full MSDs (filled blue diamonds). All data in Fig. 6 are attained from an average of two simulations performed at each probe size.

MSD values as a function of distance from the probe to particle centers (defined at *t** = 0), for (a) *σ* _{p} = 1, (b) *σ* _{p} = 2, and (c) *σ* _{p} = 5. Figure 6(a) also shows the rotated frame relative to the probe, with MSD_{par} = MSD_{x} + MSD_{z} and MSD_{perp} = MSD_{y}. In (a)–(c), solid lines with filled symbols are values at *t** = 50, all multiplied by 5, and dashed lines with open symbols are values at *t** = 500. Black lines and black circles represent total MSD values, red squares and red lines represent MSD_{par} values, and blue lines and symbols represent MSD_{perp} values. (d) Ratio of MSD values parallel and perpendicular to the probe (MSD_{par}/MSD_{perp}) at *t** = 50 (solid lines and filled symbols) and *t** = 500 (dashed lines and open symbols) as a function of distance from the probe for probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), and *σ* _{p} = 5 (cyan). (e) Maximum values of binned MSD_{par}/MSD_{perp} values from (d) as a function of probe size for *t** = 50 (black circles) and *t** = 500 (red squares). Best-fit lines are shown. (f) Various quantities (X) normalized by those quantities for the *σ* _{p} = 1 system: X = Maximum values of binned MSD_{par}/MSD_{perp} values at t* = 50 (filled black circles) and t* = 500 (filled red squares), total MSD values for those systems at *t** = 50 (open black circles) and *t** = 500 (open red squares), and the diffusion constants obtained from the full MSDs (filled blue diamonds). All data in Fig. 6 are attained from an average of two simulations performed at each probe size.

(a) Stokes-Einstein behavior as revealed by examining A particles via *D* _{A(p)}**τ* _{ α } _{,A(p)}/*T** in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Stokes-Einstein breakdown is apparent in all systems and is not dependent on probe size. (b) Stokes-Einstein behavior as revealed by examining the diffusion of probe particles and alpha-relaxation of A particles via (*D* _{p}**τ* _{ α } _{ A(p)}**σ* _{p})/*T** in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), and *σ* _{p} = 5 (cyan). Stokes-Einstein breakdown appears suppressed when examining this quantity. In both (a) and (b), values are obtained by averaging *D* and *τ* _{ α } values obtained from five independent simulations and error bars are standard deviations.

(a) Stokes-Einstein behavior as revealed by examining A particles via *D* _{A(p)}**τ* _{ α } _{,A(p)}/*T** in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), *σ* _{p} = 5 (cyan). Stokes-Einstein breakdown is apparent in all systems and is not dependent on probe size. (b) Stokes-Einstein behavior as revealed by examining the diffusion of probe particles and alpha-relaxation of A particles via (*D* _{p}**τ* _{ α } _{ A(p)}**σ* _{p})/*T** in systems with probes of *σ* _{p} = 1 (black), *σ* _{p} = 2 (red), *σ* _{p} = 3 (green), *σ* _{p} = 4 (blue), and *σ* _{p} = 5 (cyan). Stokes-Einstein breakdown appears suppressed when examining this quantity. In both (a) and (b), values are obtained by averaging *D* and *τ* _{ α } values obtained from five independent simulations and error bars are standard deviations.

## Tables

Parameters for systems investigated with σ_{p} = 1 – 5. P* is averaged over five simulations.

Parameters for systems investigated with σ_{p} = 1 – 5. P* is averaged over five simulations.

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