Local structure-mobility relationships of confined fluids reverse uponsupercooling

Jonathan A. Bollinger,1 Avni Jain,1 James Carmer,1 and Thomas M. Truskett1, a)

McKetta Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712,USA

(Dated: 30 October 2018)

We examine the structural and dynamic properties of confined binary hard-sphere mixtures designed tomimic realizable colloidal thin films. Using computer simulations, governed by either Newtonian or over-damped Langevin dynamics, together with other techniques including a Fokker-Planck equation-basedmethod, we measure the position-dependent and average diffusivities of particles along structurally isotropicand inhomogeneous dimensions of the fluids. At moderate packing fractions, local single-particle diffusivitiesnormal to the direction of confinement are higher in regions of high total packing fraction; however, thesetrends are reversed as the film is supercooled at denser average packings. Auxiliary short-time measure-ments of particle displacements mirror data obtained for experimental supercooled colloidal systems. Wefind that average dynamics can be approximately predicted based on the distribution of available space forparticle insertion across orders of magnitude in diffusivity regardless of the governing microscopic dynamics.

PACS numbers: Valid PACS appear here

Confined fluids exhibit inhomogeneous structural andrelaxation properties, which are general features of ma-terials subjected to position-dependent external fields.Because confined fluids emerge in a diverse array of nat-ural and technological contexts (e.g., water in biologicalmedia, polymer thin films, etc.), considerable attentionhas been directed at understanding how their static anddynamic properties relate to bulk fluid physics observedunder similar conditions. As a result, the static prop-erties of confined fluids, such as local one-body den-sity ρ(z), are now well-understood in terms of physi-cal intuition (e.g., emergence of particle layering nearboundaries to relieve packing frustration1,2) and canbe predicted using microscopic approaches like densityfunctional theory3,4. However, much less is understoodabout what controls the dynamics of inhomogeneous flu-ids, and only recently have efforts broadened to includedeveloping theories5–11 and other tools12–16 for charac-terizing particle dynamics both on a spatially-averagedbasis and as a function of position.

Given the difficulty of applying first principles to un-derstand the dynamics of such systems, progress hasbeen made by virtue of use pragmatic approaches, e.g.,application and testing of semiempirical, quasi-universalscaling laws that relate transport coefficients of inter-est to static properties17–27. To wit, it has been shownthat single-particle diffusivities, relaxation times, andviscosities along structurally-invariant (i.e., isotropic)dimensions of simple confined fluids can be predictedbased on knowledge of how dynamic properties of thebulk fluid relate to static quantities including excess en-tropy sex (relative to the ideal gas) and fractional avail-able space exp{c(1)} (or insertion probability p0), whichcharacterize short-range static correlations and particlepackings, respectively28–37.

In this spirit, one might expect that local particle mo-

a)Electronic mail: truskett@che.utexas.edu

bility in an inhomogeneous fluid should similarly corre-late with position-dependent static properties; in otherwords, the way particles navigate through the inhomo-geneous environment might be encoded in the physics ofmotion observed in a bulk, homogeneous fluid. However,the validity of such a connection has yet to be carefullyand systematically evaluated. Despite providing otherimportant insights, previous investigations directly mea-suring inhomogeneous dynamics have studied a varietyof fluids governed by disparate interactions, externalfields, and conditions, and they have also used differ-ent protocols to characterize the dynamics9,11,13,14,38–40.As a result, even fundamental questions related to con-fined (and more generally inhomogeneous) fluids remainopen: Do local and average correlations between parti-cle mobility and structure universally reflect bulk be-haviors? Do new structure-mobility relations emerge asinhomogeneous fluids are supercooled toward glass tran-sitions? And does the choice of microscopic dynamicsaffect these qualitative trends?

As a step toward addressing these questions, we ex-amine computer simulations of bulk and confined binary

FIG. 1. (color online). Illustration of confined binary mix-ture comprising small (blue) and large (red) particles.

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mixtures of small (sm) and large (lg) hard spheres (HS)approximated by a steeply-repulsive Weeks-Chandler-Andersen (WCA) pair potential41 between particles iand j, adapted for multiple particle diameters: ϕi,j(r) =4ε([σsm/(r + ∆)]48 − [σsm/(r + ∆)]24) + ε for r ≤(21/24σsm −∆) and ϕi,j(r) = 0 for r > (21/24σsm −∆),where ε is the characteristic energy scale; r is the in-terparticle separation; σ denotes particle diameter; and∆ = σsm−(1/2)(σi+σj). The binary mixtures are com-posed of spheres with size ratio σlg/σsm = 1.3, volume-proportional masses mlg/msm = (σlg/σsm)3, and com-position defined by the fraction of small particles xsm =0.75. These parameters mimic colloidal mixtures inves-tigated in recent experiments14,38. Below, we implicitlynon-dimensionalize quantities via appropriate combina-tions of the characteristic lengthscale σsm and energyscale ε = kBT , where kB is Boltzmann’s constant, andT is temperature.

Periodic boundary conditions are applied in all direc-tions for the bulk systems, while for the confined sys-tems (see Fig. 1), particles are situated in slit-poresof size H = 5 between two reflective walls placed atz = ±H/2, with periodic boundary conditions appliedin the x- and y-directions. The wall-particle interac-tions are analogous to the hard-sphere-like interactionsbetween particles, but defined such that the center ofparticle i can access −(H − σi)/2 . z . (H − σi)/2.Spatially averaged packing fractions are given by φavg =(π/6)ρ∗[xsm +σ3

lg(1−xsm)], where ρ∗ = (Nsm +Nlg)/Vis the combined number density of both species and Vis volume. Here, the φavg values for the confined fluidsare defined by the total (surface- rather than center-accessible) slit pore volume. We generate particle tra-jectories governed by either conventional molecular dy-namics (MD) or Brownian dynamics (BD) (i.e., over-damped Langevin ignoring hydrodynamic interactions)using GROMACS 4.5.542 with implementation detailsare provided in the Supplemental Material (SM)43.

To characterize particle motions, we calculate mean-squared displacements (MSDs) and diffusivities in thestructurally isotropic and inhomogeneous directions ofthe bulk and confined systems. Average diffusivitiesDavg in the bulk systems and parallel to the walls inthe confined systems characterize motions in isotropicdirections, and are derived by fitting the long-time be-havior of the MSD of all the particles to the Einsteinrelation 〈∆r2〉 = 2dD∆t. In the bulk (confined) case,〈∆r2〉 is the MSD in the x-, y-, and z-directions (x- andy-directions) over lag-time ∆t and dimensionality d = 3(d = 2).

Diffusivities in the inhomogeneous z-direction of theconfined pores are position-dependent and cannot becalculated via the Einstein relation because particles aresubjected to locally non-cancelling potentials of meanforce44. Particle displacements along the z-coordinateare instead accurately described13 by the 1D Fokker-Planck (FP) equation

∂G

∂t=

∂

∂z

(Dz(z)e

−F (z) ∂

∂z[eF (z)G]

)(1)

-2 -1 0 1 2z

10-2

10-1

100

Dz (z

)/D

∞

10-3

10-2

10-1

Dz(z

)

0.1

0.3

0.5

0.7

φ(z

)

MD

BD

φavg

φavg

φavg

FIG. 2. (color online). Local total packing fractions φ(z)(top) and local diffusivities in the z-direction Dz(z) of smallparticles calculated from MD simulations (middle) and BDsimulations (bottom) of pore size H = 5 and average totalpacking fractions φavg = 0.20, 0.30, 0.35, 0.40, 0.45, 0.48,0.50, 0.51, and 0.52. The BD profiles are normalized bythe infinite dilution diffusivity D∞

σsm. Dz(z) profiles for the

large particles exhibit shapes in line with the small-particleprofiles, as shown in the SM43.

where Dz(z) are position-dependent diffusivities. Here,G(z, t0 + ∆t|z′, t0) is the Markovian propagator char-acterizing temporal single-particle displacements giventhe potential of mean force F (z) = − ln{ρ(z)} + C,where C is an arbitrary constant. To obtain Dz(z)from simulation data, we use a mean-first passage times(MFPT) method15,45,46 applied to the steady-state (i.e.,∂G/∂t = 0) limit of the FP equation, which is known toprovide equivalent information compared to alternativeFP treatments13,16. Additional implementation detailsare discussed elsewhere39,40.

We begin our discussion by considering Fig. 2, wherewe compare local total packing fraction φ(z) and localparticle diffusivities in the z-direction Dz(z) for con-fined systems over a wide range of φavg and governedby either Newtonian or Brownian microscopic dynamics.Here, we use φ(z) because it more economically quan-tifies the local aggregate packing frustration comparedto component density profiles ρ(z), where the latter areprovided in the SM43. Remarkably, we find that whilethe packing structure in the confined pores undergoesan apparent shift from four to five dense particle layersupon increasing φavg, the shapes of the Dz(z) profilesare qualitatively insensitive to this considerable struc-tural rearrangement. Thus, for φavg ≤ 0.40, particlesdiffuse more quickly though densely-packed regions (ex-cept very close to the walls, where particles slow downdue to impenetrability), but for φavg ≥ 0.45, particlesinstead move more slowly through densely-packed re-

3

10-1

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101

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103

∆t

10-2

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101

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103

MS

Dxy

(∆t)

10-3

10-2

10-1

100

101

∆t

10-3

10-2

10-1

100

101

102

MD

BD

φavg

φavg

FIG. 3. (color online). Mean-squared displacements (MSD)per particle in the x- and y-directions versus lag-times ∆tfor small (solid lines) and large (dashed lines) particles fromMD (main) and BD (inset) simulations of pore size H = 5and average total packing fractions φavg = 0.20, 0.30, 0.35,0.40, 0.45, 0.48, 0.50, 0.51, and 0.52. In (b), only small-particle curves are shown for clarity and lag-times have beennormalized by D∞

σsm/σ2

sm. Symbols in (a) denote times cor-responding to the profiles in Fig. 4.

gions.

This gradual reversal from positive to negative lo-cal correlations between packing fraction and mobilitybridges observations based on previous measurementsof particle dynamics in confined pores–measurementsthat seemingly pointed to inconsistent local trends, butwhere comparisons were also complicated by differentprotocols and dynamic regimes. Mittal et al.13 mea-sured local FP-based diffusivities in Newtonian HS sim-ulations at equilibrium conditions (φavg ≤ 0.40) and ob-served positive correlations between local density ρ(z)(or φ(z)) and Dz(z). In contrast, Nugent and co-workers14,38 experimentally measured short-time MSDsalong the z-coordinate as a function of position for su-percooled thin films of pseudo-HS colloids, results whichpointed to negative correlations between local densityand mobility. While the latter results more intuitivelycorrelate to expectations based on bulk HS densitytrends, Mittal et. al. provide a plausible physical basisfor the observed positive correlations. Specifically, theycorrectly note that higher-density regions in such inho-mogeneous HS systems also exhibit the greater fractionof locally available space for inserting additional parti-cle centers, i.e., more locally free volume, which mightcorrelate with dynamics1,47,48.

The results in Fig. 2 clearly demonstrate within asingle framework for measuring dynamics that eitherpositive or negative correlations between density anddiffusivity can be observed in these systems dependingon whether equilibrium or supercooled conditions arebeing studied. By considering Fig. 2 in conjunction

-2 -1 0 1 2z0

10-3

10-2

10-1

100

MS

Dz(∆

t|z 0)

-2 -1 0 1 2z0

10-3

10-2

10-1

100

101

MS

Dxy

(∆t|

z 0)

0.3

0.5

0.7

φ(z

)

0

1

2

3

ρ(z)

∆t

φavg= 0.35 φavg

= 0.52

FIG. 4. (color online). Local total packing fractions φ(z)and individual component densities ρ(z) (top) and MSDs perparticle in the x- and y-directions (middle) and z-direction(bottom) for various lag-times ∆t plotted as a function ofparticle position z0 at ∆t = 0. Apart from φ(z) profiles(line-symbols), results for small and large particles are plot-ted with darker and lighter curves, respectively. Results arecalculated from MD simulations for pore size H = 5, whereleft panels show results for φavg = 0.35 and ∆t = 0.05, 0.1,0.2, 0.5, 2.0, and 10.0 and right panels for φavg = 0.52 and∆t = 0.1, 1.0, 50.0, and 500.0. Lag-times are also plotted inFig. 3 as symbols.

with Fig. 3, where the latter shows particles MSDs par-allel to the confining walls as a function of lag-time ∆tand φavg, we observe that the reversal in local structure-mobility correlations approximately coincides with theemergence of plateaus in the MSDs at φavg & 0.45, asignature of sub-diffusive “particle caging” characteris-tic of supercooling14,49.

The above results imply that local packing structureas measured by φ(z) does not generally correlate in anontrivial way to position-dependent diffusive mobility(this is also true of more “microscopic” local static quan-tities like p0(z), as shown in the SM43). In turn, giventhat bulk HS fluids exhibit simple negative correlationsbetween packing fraction and mobility, it is apparentthat local static-dynamic correlations in confined-fluidsystems cannot be naıvely extrapolated (or predicted)from the bulk physics, in agreement with findings formore idealized density-varying HS systems39. Interest-ingly, for the systems examined here, the choice of mi-croscopic dynamics had no qualitative impact on theshapes of the Dz(z) profiles, though recent results40 in-dicate this is not generally true of inhomogeneous fluids.

We next provide data reinforcing the idea that theopposing correlations in Fig. 2 between packing frac-tion and diffusivity at equilibrium versus supercooledconditions may also emerge in real colloidal thin films:in Fig. 4, we compare position- and time-dependentparticles MSDs of the confined films against φ(z) and

4

component ρ(z), results that mirror experimental mea-surements by Nugent and co-workers (see, e.g., Figs.7-8 from38) for supercooled thin films of pseudo-HS. Inparticular, we show results calculated from MD simu-lations for φavg = 0.35 and 0.52, which correspond toequilibrium and supercooled conditions, respectively.

Mirroring the experimental findings, for the super-cooled conditions in Fig. 4, MSDs in the xy-plane areinsensitive with respect to originating position z0 (i.e.,position at lag-time ∆t = 0) for all ∆t, while MSDsin the inhomogeneous z-direction are negatively corre-lated with respect to φ(z) (ρ(z)) for sufficiently short∆t. At longer ∆t, the MSD dependence on z0 disap-pears as particles are no longer generally situated nearz0. At φavg = 0.35, MSDs in the xy-plane also donot vary with z0, but MSDs in the z-direction are in-stead positively correlated with respect to φ(z) (ρ(z))for ∆t ≤ 0.2, though the correlations appear to reverseat longer ∆t ≥ 0.50 before washing out at long lag-times. To our knowledge, no analogous data for realnon-supercooled thin films for has yet been published.

Surprisingly, the MSDs in the z-direction at many∆t approximately reflect the Dz(z) profiles in Fig. 2even though the quantified motions are by necessity sub-diffusive and accrued when a particle is no longer at z0.In turn, only profiles at the very shortest times provideinformation about motions precisely at z0, but theseare also furthest from the diffusive regime. Nonethe-less, given that the results for the supercooled systemin Fig. 3 are consistent with the available experimentaldata, it is plausible that the FP-derived results for diffu-sive motions–including the positive correlations betweenφ(z) and Dz(z) and their reversal at high φavg–can beobserved in real confined colloids.

Given that the previous results undermine any notionof a universal connection between local structure andlocal mobility, it is natural to wonder whether averagediffusivities in directions parallel and perpendicular tothe confining walls reflect bulk fluid physics and can bepredicted based on average static properties. In Fig. 5,we address this by comparing average diffusivities Davg

xy

and Davgz (z) =

∫H/20

Dz(z)ρ(z)dz /∫H/20

ρ(z)dz for thesmall and large particles from the confined pores againstcurves for bulk mixtures. Here, we plot these dynamicquantities against component-specific average insertionprobabilities pavg0 (or available volumes for insertion),which have been shown to provide the most quantita-tively robust connection between bulk diffusivity andDxy in confined slit pores of HS governed by Newtoniandynamics32. Details of pavg0 calculations are provided inthe SM43.

As is evident in Fig. 5, the average diffusivities Dxy

and Davgz (z) of the confined fluids approximately col-

lapse onto the relevant bulk curves over many orders ofmagnitude in pavg0 for systems governed by either New-tonian and Brownian dynamics. Notably, even at highφavg associated with supercooling, Davg

z (z) values onlydiffer from the bulk by factors of 2-3 based on com-ponent pavg0 ; if one instead plots diffusivities against aless “microscopic” static property, e.g., component ρavg,

10-16

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p0avg

10-3

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Dav

g 10-16

10-12

10-8

10-4

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p0avg

10-2

10-1

100

Dav

g /D∞

MD

BD

φavglarge

small

FIG. 5. (color online). Average component diffusivities Davg

in the xy-plane (squares) and z-direction (triangles) versusaverage component insertion probabilities pavg0 for small andlarge particles (filled and unfilled symbols, respectively) cal-culated from MD simulations (main) and BD simulations(inset) of pore size H = 5 and average total packing frac-tions φavg = 0.20, 0.30, 0.35, 0.40, 0.45, 0.48, 0.50, 0.51, and0.52. Average diffusivities for bulk mixtures shown as solidblack lines with ±20% bounds shown as dashed lines.

confined and bulk diffusivities differ by up to an order ofmagnitude. Overall, the data support the idea that, de-spite the difficulty of rationalizing position-dependentdiffusivity behaviors based on bulk physics, the aver-age dynamics of inhomogeneous fluids are nonethelessstrongly encoded with bulk correlations between mobil-ity and available space.

In closing, by characterizing the particle dynamics ofhighly confined binary HS mixtures in both inhomo-geneous and isotropic dimensions, we find that diffu-sive mobility is not universally predicated upon packingstructure according to bulk HS behaviors, as exempli-fied by the reversal from positive to negative correlationsbetween local total packing fraction φ(z) and single-particle diffusivity Davg

z (z) coinciding with the onsetof supercooling. In contrast, average diffusive mobil-ity is strongly encoded by the bulk physics, and can beapproximately predicted via knowledge of the distribu-tion of available space. For the confined fluids studiedhere, results are insensitive to whether Newtonian orBrownian (i.e., overdamped Langevin) microscopic dy-namics govern particle trajectories, though it is an openquestion as to whether similar classes of behavior willemerge in real colloidal thin films treated within theFP formalism. More speculatively, the shapes of theDavg

z (z) profiles (and their qualitative insensitivity toφavg) suggest that there may simply be a “universal”oscillatory signature of local diffusivity that emerges fornon-continuum fluids proximal to confining potentialsregardless of microscopic dynamics or the specific na-ture of any emergent structural inhomogeneity. We are

5

presently investigating this possibility.

ACKNOWLEDGMENTS

We kindly thank Dr. Vincent Shen for providing ther-modynamic data for the bulk and confined systems.This work was supported by the Robert A. Welch Foun-dation (F-1696), and the National Science Foundation(CBET-1403768). We also acknowledge the Texas Ad-vanced Computing Center (TACC) at The Universityof Texas at Austin for providing HPC resources for thisstudy.

APPENDIX A: SIMULATION PROTOCOLS

To generate particle trajectories governed by eithermolecular dynamics (MD) or Brownian dynamics (BD),we simulate systems of Nsm+Nlg = 2400 particles usingGROMACS 4.5.542. MD trajectories are generated byintegrating the Newtonian equations of motion with atime step of 0.001 while fixing temperature with a Nose-Hoover thermostat. BD trajectories are generated viathe overdamped Langevin equation (ignoring hydrody-namic interactions), where the position ri of particle iis propogated with a time-step of 0.01 according to50,51:ri(t + ∆t) = ri(t) + D∞σi

∆tFi(ri(t)) + ξi(t). Here, D∞σi

is the infinite dilution diffusivity, Fi(t) is the net forcedue to interparticle and wall interactions, and ξi(t) isthe stochastic contribution. We set D∞σsm

= 0.001 andD∞σlg

/D∞σsm= σsm/σlg, and in each direction, ξi(t) =

rG(t)√

2D∞σi∆t, where rG(t) is a Gaussian noise with

〈rG(t)〉 = 0 and variance σ2 = 1.To generate bulk and confined packings at high φavg,

we initialize systems at φavg < 0.30 and compressthem to the desired packing fractions via the methodof Lubachevsky and Stillinger52, in which particle di-ameters are grown linearly with time according to thedimensionless growth rate Γ. We execute compressionsvia MD simulations with effective Γ < 1x10−6, which al-lows us to avoid generating partially jammed (i.e., non-equilibrated) structures for all presented φavg. Furtherequilibration and production runs (MD and BD) arethen initialized with the final structures.

APPENDIX B: DENSITY PROFILES ACROSSCONFINED PORES

In the main text, we largely plot local total pack-ing fractions φ(z) to characterize fluid structure. How-ever, a few noteworthy aspects concerning fluid struc-ture are apparent by simultaneously considering indi-vidual component density profiles ρ(z), as shown herein Fig. 1. First, for select conditions (e.g., φavg = 0.40),a given individual component ρ(z) profile can be out ofphase with φ(z). This does not happen frequently, andgiven the characteristics of the mixture (e.g., composi-tion xsm = 0.75, size ratio σlg/σsm = 1.3) studied here,

0.3

0.5

0.7

φ(z

)

-2 -1 0 1 2z0

0.3

0.5

0.7

φ(z)

0

1

2

3

ρ(z)

0

1

2

3

ρ(z)

0

1

2

3

ρ(z)

0

1

2

3

ρ(z)

0

1

2

3

ρ(z)

0.3

0.5

0.7

φ(z

)

0.3

0.5

0.7

φ(z

)-2 -1 0 1 2

z0

0.3

0.5

0.7

φ(z

)

0.3

0.5

0.7

φ(z

)

φavg= 0.35

φavg= 0.52

φavg= 0.45

φavg= 0.20

φavg= 0.50 φavg

= 0.51

φavg= 0.48

φavg= 0.40

φavg= 0.30

FIG. 6. (color online). Local total packing fractionsφ(z) (line-symbols) and individual component densities ρ(z)(lines) from simulations of pore size H = 5 at various aver-age total packing fractions φavg, where results for small andlarge particles are plotted with darker and lighter curves,respectively.

this behavior is only observed for small particle pro-files. This underlines the importance of characterizingthe spatial distribution of both particle species in orderto gain a complete picture of packing structure; oth-erwise, one may come to qualitatively incorrect conclu-sions about which regions of a fluid are densely packed.However, for the size ratio studied here, qualitative vari-ations in φ(z) can mostly be derived from knowledgeonly of the large particle ρ(z) profiles (as would likelybe possible for all xsm ≤ 0.75).

As discussed in relation to Figs. 2 and 5 in the maintext, by comparing these ρ(z) profiles against local par-ticle diffusivities Davg

z (z), it is evident that there is noφavg-independent local correlation between componentdensity and diffusive mobility. Likewise, the local avail-able space for particle insertion, as quantified by inser-tion probability p0(z) in these systems, does not univer-sally correlate with mobility because (see Section IV)p0(z) = ρ(z)/ξ, where ξ is the spatially-invariant com-ponent activity47. Of course, this implies that otherposition-dependent measures of structural correlations

6

-2 -1 0 1 2z

10-3

10-2

10-1

Dzlg

(z)

10-3

10-2

10-1

Dzsm

(z)

0.1

0.3

0.5

0.7

φ(z

)φavg

φavg

φavg

FIG. 7. (color online). Local total packing fractions φ(z)(top) and local diffusivities in the z-direction Dz(z) of small(middle) and large (bottom) particles calculated from MDsimulations of pore size H = 5 and average total packingfractions φavg = 0.20, 0.30, 0.35, 0.40, 0.45, 0.48, 0.50, 0.51,and 0.52.

that are positively correlated with p0(z), such as the lo-cal two-body excess entropy s(2)(z), would likewise ex-hibit no consistent correlation with diffusive mobility39.Taken altogether, the results in the supplemental Fig.1 underline that idea that local structure does not con-trol, nor can be used to predict, local diffusive mobilityin any straightforward way.

APPENDIX C: COMPONENT LOCAL DIFFUSIVITIES

In supplemental Fig. 2, we show local particle diffu-sivities Davg

z (z) for both small and large particles gov-erned by MD at the various φavg conditions studied. Asnoted in the main text, it is evident that the Davg

z (z)profiles for the two types of particles have qualitativelysimilar shapes and exhibit the same insenstivity to thestructural rearrangment that occurs upon the onset ofsupercooling at high φavg. Note that there is slight drift(i.e., overestimaton of Davg

z (z)) near the edges of thelarge-particle profiles at higher φavg due to the relativelymeager amount of particle trajectory data obtained forthis species (xlg = 0.25). Davg

z (z) profiles for small andlarge particles governed by overdamped Langevin dy-namics are also qualitatively similar.

APPENDIX D: CALCULATING AVERAGE INSERTIONPROBABILITY

To calculate pavg0 for each component, we note that ifthere is no external field at position z (i.e., ϕext(z) = 0),

the local insertion probability53 for bulk or inhomoge-neous HS is a ratio47 p0(z) = ρ(z)/ξ of the local com-ponent density ρ(z) and the spatially-invariant compo-nent activity ξ = exp(βµ)/λ3, where the latter is de-fined by the component chemical potential µ and thede Broglie wavelength λ. Given that we have com-ponent ρ(z) profiles measured from the MD and BDsimulations, all that is required to obtain p0(z) pro-files are activities ξ for the bulk and confined mix-tures at the various φavg values. These ξ values areobtained via grand canonical transition matrix MonteCarlo (GC-TMMC) simulations54, with implementationdetails presented elsewhere32. It is then straightforward

to calculate pavg0 = H−1∫H/20

p0(z)dz, which for bulkmixtures is simply pavg0 = ρ/ξ.

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