Forming Binary Mixtures

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Microscopic Theory of Coupled Slow Activated Dynamics in Glass-Forming Binary Mixtures Rui Zhang, and Kenneth S. Schweizer J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10568 • Publication Date (Web): 18 Jan 2018 Downloaded from http://pubs.acs.org on January 19, 2018

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The Journal of Physical Chemistry

submitted to Journal of Physical Chemistry B,

October 2017

revised January, 2018

Microscopic Theory of Coupled Slow Activated Dynamics in GlassForming Binary Mixtures

Rui Zhang1 and Kenneth S. Schweizer1,2,3,*

Department of Materials Science1, Department of Chemistry2, Frederick Seitz Materials Research Laboratory3, University of Illinois, Urbana, IL 61801

*corresponding author: [email protected]

1 ACS Paragon Plus Environment

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ABSTRACT The Elastically Collective Nonlinear Langevin Equation theory for one-component viscous liquids and suspensions is generalized to treat coupled slow activated relaxation and diffusion in glass-forming binary sphere mixtures of any composition, size ratio and inter-particle interactions. A trajectory-level dynamical coupling parameter concept is introduced to construct two coupled dynamic free energy functions for the smaller penetrant and larger matrix particle. A two-step dynamical picture is proposed where the first step process involves matrix-facilitated penetrant hopping quantified in a self-consistent manner based on a temporal coincidence condition. After penetrants dynamically equilibrate, the effectively one-component matrix particle dynamics is controlled by a new dynamic free energy (second-step process). Depending on the time scales associated with the first and second-step processes, as well as the extent of matrix correlated facilitation, distinct physical scenarios are predicted. The theory is implemented for purely hard core interactions, and addresses the glass transition based on variable kinetic criteria, penetrant-matrix coupled activated relaxation, self-diffusion of both species, dynamic fragility and shear elasticity. Testable predictions are made. Motivated by the analytic ultra-local limit idea derived for pure hard sphere fluids, structure-thermodynamicsdynamics relationships are identified. As a case study for molecule-polymer thermal mixtures, the chemically matched fully miscible polystyrene-toluene system is quantitatively studied based on a predictive mapping scheme. The resulting no-adjustable-parameter results for toluene diffusivity and the mixture glass transition temperature are in good agreement with experiment. The theory provides a foundation to treat diverse dynamical problems in glass-forming mixtures, including suspensions of colloids and nanoparticles, polymer-molecule liquids, and polymer nanocomposites.


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I. INTRODUCTION Understanding and predicting slow dynamics in glass-forming materials remains a great challenge. Although there is no widely accepted theory that explains all the key phenomena, theoretical progress has steadily occurred over the past several decades1,2. Compared to onecomponent systems, glass-forming mixtures introduce significantly more complexity. A wide variety of experimental3-8, simulation9-11 and theoretical work (both thermodynamics-based12,13 and kinetics-based14) has been carried out to investigate glassy mixture properties and phenomena. These studies have both fundamental scientific importance and broad practical relevance. However, many questions remain to be addressed, even qualitatively. The central goal of this article is to present a new microscopic theory that aims to predict relaxation, transport and viscoelastic properties for model systems studied via computer simulation and in laboratory colloidal, molecular and polymeric materials. Given glassy dynamics involves high barriers and activated motion in 1-component systems, mixtures raise new questions associated with multiple time and length scales. These include whether both species experience activated dynamics, the degree to which relaxation and mass transport involves in collective re-arrangements of both species (e.g., slaved versus decoupled), new sources of dynamic heterogeneity, and what features control mixture elasticity at different time scales. Chemical complexity enters not only via molecular shape, interactions, and thermodynamic state, but also via mixture concentration, relative sizes of species, and possible specific attractions between unlike molecules or colloids. Even the basic question of whether adding a relatively small molecule to a glass forming liquid speeds up its dynamics (plasticization) or slows it down (anti-plasticization) is a complex issue of high practical



relevance. Our starting point is the recently developed Elastically Collective Nonlinear Langevin Equation (ECNLE) theory of 1-component suspensions and liquids15. This theory can accurately predict single particle dynamical properties of colloidal suspensions over the ~6 decades experimentally measureable and of molecular and polymeric liquids over 14 decades of time scale, typically with no adjustable parameters15-18. Here we report a full generalization of ECNLE theory to two-component glass-forming mixtures of spheres. The key new idea is to explicitly account for trajectory-level dynamical correlation between the two species based on the concept of coupled penetrant and matrix dynamic free energies. The latter are constructed using a statistical description of mixture structure and a dimensionless dynamic coupling parameter which self-consistently quantifies the degree of matrix dynamical fluctuation relative to the penetrant jump distance required to achieve an irreversible penetrant hopping event. We very recently reported a theory for the limiting dilute penetrant limit19. Here we focus on the general hard sphere mixture system as the simplest and foundational model for more complex interacting mixtures. With regards to materials science and engineering applications, glassy mixtures are ubiquitous. Polymer-molecule mixtures are central in diverse applications such as membranebased gas separation and water purification20,21, modifying polymer mechanics via plasticizer or anti-plasticizer additives, and capsule-based self-healing polymer composites22,23. Recent fundamental experimental studies of molecular mixtures are of great value for elucidating general principles. For example, Rössler et al. reported a series of comparative experimental studies4-6 that revealed the local dynamics and glass transition trends in polymer-molecule and molecule-molecule mixtures are very similar. They chose two model systems, one in each


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category, with similar large size asymmetry ratios and large glass transition temperature contrast between the two species. The observed glass transition diagrams, two-step relaxation profiles, and many other dynamic properties are qualitatively identical for the two systems. These findings strongly support a key modeling simplification we invoke to treat real mixture materials involving polymers and molecules by mapping them to an effective, properly coarse-grained spherical particle mixture19. The latter is also of direct relevance to many colloid or nanoparticle mixtures of interest in soft matter. The remainder of the article is organized as follows. In Section II we briefly review the established dynamical theory. Our new mixture theory is constructed in Section III. The mixture model and required structural input are briefly discussed in Section IV. Section V presents results for kinetic arrest maps. Section VI studies the cooperative relaxation time and self-diffusion constants. An analytic analysis is performed in Section VII to acquire some intuitive understanding of dynamics-structure-thermodynamics connections. Glassy shear elasticity is studied in Section VIII. The dynamical theory is applied and compared to experiments in Section IX for self-diffusion in a fully miscible polymer-molecule mixture of experimental interest. The article concludes in Section X with a summary, discussion and future opportunities outlook.

II. THEORETICAL BACKGROUND We first briefly review the foundational theoretical elements established in our prior work. Detailed derivations are given elsewhere15,16,24-26. A. Naïve Mode Coupling Theory The starting point for a binary mixture is two standard generalized Langevin equations (GLE) which describe the motion of a tagged particle of each species25,26. The central quantity is



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the time correlation (memory) function of the slowly relaxing part of the total force on a tagged particle which is approximately calculated based on a mode-coupling theory (MCT) approximation of projecting forces onto the bilinear field of collective and tagged particle density fluctuations and using the pair level factorization procedure25,26. The result in Fourier space is r r fα (0) ⋅ fα (t ) =

1 6 β 2π 2

∞

2

0

j , k =1

4 ∫ dqq ∑ Cα j (q)S jk (q, t )Ckα (q)Γ s,α (q, t ),

α = 1 or 2

(1)

where β = 1 / k BT , C( q ) is the 2 × 2 direct correlation function matrix, S ( q, t ) is the collective density fluctuation dynamic structure factor matrix25,26, and Γ s ,α (q, t ) is the single particle density fluctuation dynamic structure factor or propagator of species α . Binary mixture naïve mode coupling theory (NMCT)25,26 invokes additional approximations to obtain two coupled self-consistent equations to describe ideal localization transitions. Briefly, arrested states are modeled as harmonic Einstein solids (localization length of species α is denoted as rloc ,α ) with a long time single particle Debye-Waller factor: Γ s,α (q,t → ∞) = e

2 − q 2 rloc ,α / 6

, α = 1 or 2

(2)

Taking the long time limit of eq 1 and enforcing self-consistency yields two coupled localization relations: r r 1 3 β fα (0) ⋅ fα (t → ∞ ) rloc2 ,α = k BT , 2 2 ∞

−2 loc,α

r

2 2 1 4 − q rloc ,α / 6 = dqq e 18π 2 ∫0

2

∑ Cα

j,k =1

j

α = 1 or 2

(q)S jk (q,t → ∞)Ckα (q), α = 1 or 2

(3)

(4)

Equation 4 requires a dynamic closure for S ( q, t → ∞ ) . Details of how this is done and the corresponding mathematical equations are in ref 25. Here we simply note that S ( q, t ) obeys


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d S ( q , t ) = − q 2 H ( q ) S −1 ( q ) S ( q , t ) dt

(5)

where the mobility matrix element H µα (q) = (k BT / ζ s ,α )δ µα and ζ s ,α is the short time friction constant for species α . Study of the long time limit localized state is addressed via the mapping 2 25 or replacement 6k BT ζ µ−,1α t → rloc , which mathematically closes eq 4. ,α

To facilitate our theory development in Section III and better reveal its self-consistent nature, we rewrite eqs 3 and 4 as: ∞

−2 rloc, = α

1 − q2 r 2 / 6 dqq 4 e loc ,α 2 ∫ 18π 0

r r fα (0) ⋅ fα (t → ∞) =

∞

1 6β π 2

2

2

∑ Cα

j,k =1

4 ∫ dqq e 0

j

(q)S (jkNMCT) (q,rloc,1 ,rloc,2 )Ckα (q), α = 1 or 2

2 − q 2 rloc ,α /6

2

∑ Cα

j

(6)

( q ) S (NMCT) ( q, rloc ,1 , rloc ,2 )Ckα ( q ), α = 1 or 2 (7) jk

j , k =1

The explicit algebraic expressions for S ( NMCT) (q, rloc ,1 , rloc ,2 ) are given in ref 25. To briefly summarize, the key quantity of the above theory is the dynamically arrested component of the force-force correlations experienced by a tagged particle which contains contributions from all harmonically localized particles of both species in the mixture. Dynamical constraints are set by the so-called vertex in eq 4 which is quantified by the mixture pair structure. Self-consistency between these arrested force correlations and particle localization enters via a product of Debye-Waller (DW) factors. The “self” component of the latter involves the localization length of the tagged particle of interest, while the “collective” components contain the localization lengths of all other particles (of both species) and the partial collective static structure factors of the binary mixture. In this way the motion of the two species influence each other which at the NMCT level is manifested in the two coupled self-consistent localization length equations.



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Equation 6 has three possible types of solutions corresponding to distinct dynamical states at the NMCT level. The mixture is fluid if only the trivial solution rloc ,1 = ∞, rloc ,2 = ∞ exists. If a double finite localization length solution exists, it is called the double glass state. If only single-finite localization length solution exists, it is a single glass state. If the two components are identical, eq 6 reduces to the one-component NMCT equation24,27 ∞

−2 loc

r

1 − q 2 r 2 / 6 − q 2 r 2 / 6S ( q ) = dq ρq 4C 2 (q)S(q)e loc e loc 2 ∫ 18π 0

(8)

where ρ is the number density. No species subscripts are used for one-component systems.

B. Beyond NMCT: One-Component Fluids The NMCT localization transition is not a true nonergodicity transition, but rather signals a dynamic crossover beyond which particles relax via activated barrier hopping24,27 which for one-component systems we describe by ECNLE theory15-18. We briefly recall the key elements of the latter required to extend this approach to binary mixtures. The fundamental physical idea is that a single particle activated hopping event involves two distinct, but inter-related, local cage and longer range collective elastic barriers. The original nonlinear Langevin equation (NLE) theory24 adequately addresses the local barrier physics. For a one-component fluid (henceforth called pure matrix), the NLE24 in the overdamped limit for the single-particle scalar dynamical displacement from its initial position, r (t ) , is: −ζ s

∂ d r − Fdyn ( r ) + δ f (t ) = 0 dt ∂r

(9)

where ζ s is the short time friction constant, δ f (t ) is the corresponding white noise random force, and Fdyn ( r ) is the dynamic free energy function the negative gradient of which quantifies a displacement-dependent effective caging force on the tagged particle. Taking eq 8 as describing


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the ideal localization limit of eq 9 in absence of the noise term ∂Fdyn ( r ) / ∂r

rloc

=0

(10)

and invoking a local equilibrium idea that replaces the ensemble-averaged localization length in eq 10 by its instantaneous dynamic variable analog r , an equation relating ∂Fdyn ( r ) / ∂r to the pure matrix structure was heuristically constructed20 yielding: ∞  q2r 2  ρq 2C 2 (q)S(q) r 1 = −3ln   − 2 ∫ dq exp (1+ S −1 (q))  − −1 k BT  σ  2π 0 1+ S (q)  6 

Fdyn (r)

(11)

where σ is the matrix particle diameter. NLE theory was subsequently more microscopically motivated using time-dependent statistical mechanics within a dynamical variable version of dynamic density functional theory27. We employ the heuristic route24 to construct our new dynamical theory for mixtures. Beyond the NMCT crossover, Fdyn ( r ) in eq 11 has a local minimum (transient localization length) and maximum (barrier location), rloc and rB , respectively. The local barrier FB = Fdyn (rB ) − Fdyn (rloc ) and jump distance ∆r = rB − rloc . When FB exceeds a few k BT , it has been derived15,16 that the dynamical constraints entering the dynamic free energy are dominated by the local cage scale wavevectors. Physically, this means NLE theory does not capture longer range collective re-arrangements of particles outside the cage. Including this effect in a microscopic “elastic model” spirit28,29 introduces an additional barrier contribution, Felastic , which quantifies the cost of cage expansion that accommodates the local large amplitude particle rearrangement. ECNLE theory a priori computes Felastic . Key results15,16 are as follows. The required cage expansion (dilation) length scale, ∆reff , is



∆reff =

2 2 rcage ∆r 3 ∆r 4  3  rcage ∆r − +   3 rcage 192 3072   32

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(12)

the harmonic spring constant, K 0 , governing small particle adjustments outside the cage is: K 0 = ∂ 2 Fdyn ( r ) / ∂ 2 r

rloc

(13)

and the elastic barrier is: Felastic = 12 φ

3 rcage

σ3

∆reff2 K 0

(14)

The cage radius in eqs 12 and 14 is rcage = rmin , where rmin is the first minimum of the pair correlation function g (r ) and φ ≡ ρπσ 3 / 6 is the matrix volume fraction. The physical idea of ECNLE theory is the activated alpha relaxation process is of a coupled local-nonlocal character whereby the longer range elastic fluctuation “facilitates” the large amplitude cage scale particle hopping. Hence, using Kramers theory30, the mean barrier hopping time is16,

τ hop =

2τ s e

Felastic / k B T rloc + ∆r

σ

2

∫

r

dre

Fdyn ( r ) / k B T

rloc

∫ dr 'e

− Fdyn ( r ') / k B T

(15)

rloc

where τ s = ζ sσ 2 / k BT is a characteristic short time scale15, 16.

III. SELF-CONSISTENT COOPERATIVE HOPPING THEORY A. Basic Ideas We first discuss basic, motivating points. First, one can always classify the two components into a “slow” species (subscript ‘1’) and “fast” (subscript ‘2’) species by the rule:

τ hop ,1 ≥ τ hop ,2 . For convenience of discussion, we consistently refer to the “slow” species as matrix and the “fast” species as penetrant, regardless of the composition. Thus subscript ‘1’ and ‘m’ are interchangeable, and subscript ‘2’ and ‘p’ are interchangeable. 10 ACS Paragon Plus Environment

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Second, we envision the following intuitive two-step process underlying the mixture alpha relaxation: (1) matrix facilitated penetrant hopping, and (2) matrix full relaxation after penetrants diffuse and dynamically equilibrate. The time scale of the first process may be much shorter than the second one corresponding to a type of “decoupled” dynamics, or the second process is faster than the first process and then the penetrant and matrix relaxations are essentially “slaved” in practice. For the latter case, the penetrant and matrix relax almost simultaneously via a cooperative co-hopping process whereby penetrant hopping triggers the irreversible matrix displacement, a mechanism we call “constraint release”. Addressing the firststep penetrant-matrix cooperative process is the major theoretical challenge since it is a “coupled two dynamic parameter” problem. The second process can be readily modeled as a “one dynamic parameter” problem. Third, we recently proposed a novel approach, called self-consistent cooperative hopping theory (SCCHT), to treat correlated matrix-fluctuation-mediated activated transport of dilute penetrants in glass-forming liquids and suspensions19. This work introduced the “dynamical coupling parameter” concept, which is again key in formulating our general mixture theory (schematically illustrated in Figure 1). We refrain from reviewing its details since our full mixture theory follows a more general route to construct the relevant equations, starting from the two coupled NMCT equations (eq 6). However, our general mixture theory exactly reduces to the dilute penetrant SCCHT when the penetrant volume fraction (denoted as f p ) approaches zero. We call the new theory formulated below mixture SCCHT. In the dilute limit, we previously studied19 how the penetrant-matrix size ratio and interactions influence the penetrant diffusivity and the quantitative extent of matrix facilitation. The former (latter) quantity universally decreases (increases) with larger penetrant size and



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stronger penetrant-matrix attraction. The new theory enables us to systematically study the composition dependence, and the rich set of behaviors that emerge.

B. First-Step Dynamics: Matrix Facilitated Penetrant Hopping To describe matrix-facilitated penetrant hopping we propose two coupled, trajectorylevel, NLE equations for the tagged penetrant and matrix dynamic displacement variables:

−ξ s ,2

dr2 (t ) ∂Fdyn ,2 (r2 ; γ ) − + δ f 2 (t ) = 0 dt ∂r2

(16)

dr1 (t ) ∂Fdyn ,1 ( r1 ; γ ) − + δ f1 (t ) = 0 dt ∂r1

(17)

−ξ s ,1

A key simplifying approximation is to introduce a dynamical coupling parameter γ to characterize via a single scalar quantity a matrix-penetrant cooperative “reactive eigenvector”. In a dynamic mean-field at the trajectory level spirit, this parameter characterizes a trajectoryaveraged matrix-penetrant cooperative mode of motion for achieving a large amplitude activated hop of a penetrant particle. It is initially unknown, but as explained below it is uniquely determined via a temporal self-consistency condition. The first step heuristically24 constructs the γ -dependent penetrant and matrix dynamic free energy functions (see Figure 1) to obtain:

∂Fdyn ,1 ( r1 ; γ ) ∂r1

∂Fdyn ,2 ( r2 ; γ ) ∂r2 Here, ⋅

r1 , r2

r r = β f1 (0) ⋅ f1 (t → ∞) r r = β f 2 (0) ⋅ f 2 (t → ∞ )

r1 , r2 ( γ )

r1 ( γ ), r2

r1 − 3k BT / r1

r2 − 3k BT / r2

(18)

(19)

refers to replacing on the right side of eq 7 rloc ,1 by r1 and rloc ,2 by r2 (local

equilibrium idea). The second step introduces mathematically simple and physically sensible relations between r2 (γ ) and r1 (γ ) that enter eqs 18 and 19: 12 ACS Paragon Plus Environment

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r2 (γ ) = rloc,2 + γ (r1 − rloc,1 )

(20)

r1 (γ ) = rloc,1 + (r2 − rloc,2 ) / γ

(21)

Combining eqs 18-21 and performing the integration then yields the penetrant and matrix dynamic free energy functions relevant to the first-step cooperative penetrant hopping process; each function takes the displacement of the corresponding species and γ as basic variables. The so-constructed dynamic free energy functions properly guarantee that Fdyn ,1 ( r1 ; γ ) and Fdyn ,2 ( r2 ; γ ) have rloc ,1 and rloc ,2 as the local minimum location (localization length) of NMCT of

binary mixtures. To solve for γ , we employ Kramers theory30 to calculate the mean penetrant barrier hopping time, τ hop , p ( ∆rp (γ )) , where ∆rp = rB , p − rloc , p is the penetrant jump distance. To obey temporal coincidence, the mean first passage time that the matrix reaches the displacement ∆rm ,c = ∆rp / γ , τ dis ,m ( ∆rm ,c (γ )) (facilitates penetrant hopping) is required to be the same as

τ hop , p ( ∆rp (γ )) . We then obtain the crucial self-consistency equation of mixture SCCHT:

τ hop, p (∆rp (γ )) = τ dis,m (∆rm,c (γ ))

(22)

Figure 1 schematically illustrates eq 22. We note that an alternative approach to treat coupled activated dynamics in binary mixtures that has been previously investigated is to construct a 2-dimensional (2-d) dynamic free energy surface25. In principle, such an approach can allow a more explicit treatment of correlations between matrix and penetrant particle trajectories and their fluctuations. However, besides the computational intractability of this approach when barriers are very high (per our present interest in thermal glass-formers), it suffers from the conceptual problem of an “integration path dependence” since construction of the 2-d surface cannot be uniquely done from knowledge solely of its partial first derivatives (the effective forces of the NLE approach). 13 ACS Paragon Plus Environment


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In contrast, SCCHT is well defined for any mixture composition, size ratio and inter-particle interactions. We now introduce all other ingredients required to determine the self-consistent solution of γ based on eq 22. The relevant Kramers theory formulas in the ECNLE framework are:

τ hop, p (∆rp (γ )) =

2τ s, p e

Felastic , p ( γ ) / k B T rloc , p + ∆rp ( γ )

d

τ dis,m (∆rm,c (γ )) =

∫

2

2τ s,m e

drp e

rp

Fdyn , p ( rp ,γ ) / k B T

rloc , p

∫

− Fdyn , p ( rp' ,γ ) / k B T

(23)

rloc , p

Felastic , p ( γ ) / k B T rloc ,m + ∆rm ,c

σ2

drp' e

∫

rm

drm e

Fdyn ,m ( rm ) / k B T

rloc ,m

∫

drm' e

− Fdyn ,m ( rm' ) / k B T

(24)

rloc ,m

The short-time scales obey τ s , p = d 2 / Ds , p , τ s ,m = σ 2 / Ds ,m , ζ s , p = k BT / Ds , p , ζ s ,m = k BT / Ds ,m , where d and σ are the penetrant and matrix diameter, respectively. We adopt the “binary collision mean field” (BCMF)-type theory of ref 31 to estimate these short-time quantities: −1

Ds, p

∞   ρp ∞ ρm DE, p 2 2 2 2 = DE, p 1+ dkk h (k) + dkk h (k)  pp pm 2 ∫ 6π 2 (DE ,m + DE, p ) ∫0  12π 0 

Ds,m

∞   ρ p DE ,m ρm ∞ 2 2 2 2 = DE ,m 1 + dkk h (k) + dkk h (k)  mm pm 2 ∫ 6π 2 ( DE,m + DE , p ) ∫0  12π 0 

(25)

−1

(26)

Here, h(k) functions are Fourier transform of h(r)=g(r)-1. The two Enskog diffusivities for Newtonian hard sphere mixture fluids obey the following relations32,33 1/ 2

DE , p

3 k T  =  B  8 πm 

1/ 2

DE ,m

3 k T  =  B  8 π M 

1/ 2   2 ( contact ) 2 ( contact )  2 M   ρ p d g pp + ρm d pm g pm  M + m    

1/ 2  2m   ( contact ) 2 ( contact )   ρmσ 2 g mm + ρ p d pm g pm  M + m    

−1

(27)

−1

where m and M are the mass of a penetrant and matrix particle, respectively. 14 ACS Paragon Plus Environment

(28)

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As discussed previously24, for hard sphere colloidal suspensions where inertia effects are not important and the elementary process is dilute particle Fickian diffusion in a solvent, the analogous Enskog-type diffusivities obey different formulas and inertial mass is irrelevant. Specifically, a Stokes-Einstein relation including a correction factor of 1 / g(σ ) to characterize binary collisions effects is employed: DE(colloid ) = k BT / (3πηsolventσ g(σ )) , where η solvent is the solvent viscosity. For hard sphere colloidal mixtures, an averaged contact value is introduced to yield:

(colloid ) (contact ) DE, = k BT / (3πηsolvent dgave, ) p p

(colloid ) (contact ) DE,m = k BT / (3πηsolventσ gave,m ) ,

and

where

(contact ) ) ) (contact ) ) (contact ) gave, = f p g (contact + (1− f p )g (contact = f p g (contact + (1− f p )gmm and gave,m . Here we focus p pp pm pm

on the Newtonian short time dynamics model, where m = M (d / σ )3 is employed. The relative penetrant to matrix activated dynamics time scales is affected very little by any difference in short time dynamics. The collective elastic barrier accompanying the cooperative penetrant hopping event (Figure 1) is constructed following the same basic method of ref 16. The key is that in mixtures a penetrant (matrix) hop is treated as on average requiring an outward radial motion of amplitude ∆rp / 4 ( ∆rm ,c / 4 )16. Hence, the angularly averaged effective cage dilation distance ∆reff , p

follows as before as16: rcage , p

fp ∆reff , p =

∫

rcage , p

dz ( z + ∆rp / 4 − rcage , p ) z 2 + (1 − f p )

rcage , p −∆rp /4

∫

dz ( z + ∆rm ,c / 4 − rcage, p ) z 2

rcage , p −∆rm ,c / 4 rcage , p

∫

dzz 2

0

=

3 3 rcage ,p

2 2  r  rcage ∆r rcage, p ∆r ∆rp4  rcage, p ∆rm3,c ∆rm4,c   , p ∆rm , c − + − +  f p   + (1 − f p )   192 3072  32 192 3072      32 2 cage , p

2 p

3 p

(29)



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The geometric fact that a tagged penetrant hop at the cage center is always small enough that it does not contribute to cage expansion is used in deriving eq 29. We also treat the particles outside the cage as randomly distributed, a reliable simplification employed in 1-component ECNLE theory16. The factors f p and 1 − f p in eq 29 are consequences of such a random distribution simplification. The calculation of Felastic , p is straightforward following the pure liquid derivation16:

(

3 β Felastic, p = 2π∆reff2 , p rcage, ρ p K0, p + ρm K0,m p

)

rcage, p = f p rpp,min + (1 − f p )rpm,min

(30) (31)

The two spring constants in eq 30 are the well curvatures of the corresponding dynamic free energy functions. The penetrant hopping diffusion constant is simply estimated per Fick’s law as:

Dhop , p = ∆rp2 / 6τ hop , p

(32)

C. Second-Step Dynamics: Matrix Relaxation After Penetrants Equilibrate The effective matrix dynamic free energy function and NLE equation in the equilibrated penetrant limit are documented in detail in ref 25. In the mixture SCCHT framework, this limit exactly describes the second-step relaxation of matrix. We denote this second-step matrix relaxation time as τ 2,m (subscript ‘2’ here does not indicate species type). The mean matrix hopping time is then a sum of relaxation time for the first and second step

τ hop ,m = τ hop , p + τ 2,m

(33)

Below we only present the new theoretical elements beyond ref 25 required to calculate τ 2,m . First, the effective matrix short-time diffusivity is smaller than Ds ,m (eq 26) as a consequence of the medium being more viscous after penetrants equilibrate. We denote the


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resulting “renormalized” matrix short-time diffusivity as D% s ,m . We compute D% s ,m using eq 26. Specifically, DE , p in eq 27 is replaced by D p(2) (‘2’ means this is the effective penetrant

= k BT / (ξs, p + ξhop, p ) , where diffusivity in the second-step dynamics), D(2) p

ξhop, p = k BT / Dhop, p = 6k BT τ hop, p / ∆rp2

(34)

The final result is: ∞ ρ D ρ ∞   2 2 D% s ,m = DE ,m 1 + m 2 ∫ dkk 2 hmm (k ) + 2 p E , m (2) ∫ dkk 2 hpm (k )  6π ( DE ,m + D p ) 0  12π 0 

D p(2) =

−1

1 −1 s, p

D

+ 6τ hop , p / ∆rp2

(35)

(36)

Second, one needs to incorporate collective elastic physics in the second-step matrix dynamics. The bulk 1-component system elastic barrier analysis16 holds for the second-step matrix dynamics since the equilibrated penetrants are effectively a viscous liquid background. Now the volume fraction of matrix particles becomes φm = (1 − f p )φT , where φT denotes the mixture total packing fraction. The collective elastic barrier then straightforwardly follows the ECNLE theory pure matrix result

Felastic,m / k BT ≈ 12φm ∆reff2 ,m (rcage,m / σ )3 K0,m

(37)

where the cage expansion length ∆reff ,m is a function of the matrix jump distance calculated (2) based on the effective second-step matrix dynamic free energy function Fdyn , m ( rm ) and

(2) rcage,m = f p rpm,min + (1 − f p )rmm,min , and K 0,m is the local curvature of Fdyn , m ( rm ) at r%loc , m (new

matrix localization length). Detailed relevant equations are documented in ref 16. (2) As a technical note, it is possible that no localized state (minimum) exists in Fdyn , m ( rm ) . That



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is, penetrant hopping and equilibration has reduced cage constraints felt by matrix particles below the critical NMCT value required for transient localization. In this case, we set τ 2,m = 0 . In practice, as long as τ 2,m dij gij (r ) = 0, r < dij

(40)

where d ij = ( di + d j ) / 2 . To be consistent with notation in Section III, d1 = σ , d 2 = d , and we define d pm = ( d + σ ) / 2 . These integral equations can be solved analytically35. Both total density 18 ACS Paragon Plus Environment

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and concentration fluctuations enter. As developed in previous one-component thermal liquid ECNLE theory studies17,18, the general sphere mixture model can be applied to study real mixtures either relatively straightforwardly (e.g., for colloidal mixtures) or via a mapping scheme for thermal molecular and polymer liquids to an effective hard sphere system.

V. DYNAMIC GLASS TRANSITION DIAGRAMS The hard sphere mixture properties are controlled by three key parameters: size ratio

d / σ , total packing fraction φT , and volume fraction of smaller penetrant species f p . We present dynamic glass transition or kinetic arrest diagrams for different representative parameters in this section. In addition to their intrinsic interest, these set the stage for understanding dynamical properties discussed later. We consider three different kinetic arrest criteria. The first is the NMCT ideal kinetic arrest phase diagram. Starting from a low φT (no localization solution exists), if a single localization solution (finite rloc ,m but infinite rloc , p ) is first found after increasing φT to a critical value, then this φT represents a fluid-to-single glass transition. A single glass-to-double glass transition is found at a higher φT corresponding to when a double localized solution (finite rloc ,m and rloc , p ) first emerges. Under some conditions, a direct fluid-to-double glass transition is found, corresponding to the direct emergence of a double localization solution.

A. NMCT Ideal Kinetic Arrest NMCT arrest boundaries signal the onset of activated dynamics described by NLE and ECNLE theories. They can alternatively be viewed as the simplest version of the full MCT for an ideal nonergodicity transition where the wavevector-dependent dynamic order parameter is the



arrested collective density fluctuation Debye-Waller factor function. Thus, it is of some interest to compare our NMCT arrest diagrams with their full ideal MCT analogs14. The solid and dashed curves shown in Figure 2 present our NMCT ideal glass transition boundaries for 4 different p-m size ratios. Overall, the major trends are qualitatively consistent with the following aspects of the more complex full MCT analogs reported in the literature14. (i) For large enough d / σ < 1 , a single glass phase is not predicted. As Figure 2 shows, the single glass window is already very small at d / σ =0.5, and disappears for d / σ =0.75. Quantitatively, full MCT calculations find that for d / σ =0.42, a “partially frozen” phase is already absent14. We note in passing, however, that based on different dynamic vitrification criteria (results shown later) the SCCHT predicts that the single glass phase always exists for binary hard sphere mixture with d / σ

Forming Binary Mixtures

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