Theory of the Spatial Transfer of Interface-Nucleated Changes of Dynamical Constraints and Its Consequences in Glass-Forming Films
AAPS/123-QED
Theory of the Spatial Transfer of Interface-Nucleated Changes of DynamicalConstraints and Its Consequences in Glass-Forming Films
Anh D. Phan and Kenneth S. Schweizer Department of Physics, University of Illinois, 1110 West Green St, Urbana, Illinois 61801, USA Department of Materials Science and Chemistry,Frederick Seitz Materials Research Lab, University of Illinois at Urbana-Champaign ∗ (Dated: January 7, 2019)We formulate a new theory for how caging constraints in glass-forming liquids at a surface orinterface are modified and then spatially transferred, in a layer-by-layer bootstrapped manner, intothe film interior in the context of the dynamic free energy concept of the Nonlinear LangevinEquation (NLE) theory approach. The dynamic free energy at any mean location (cage center)involves contributions from two adjacent layers where confining forces are not the same. At themost fundamental level of the theory, the caging component of the dynamic free energy variesessentially exponentially with distance from the interface, saturating deep enough into the film witha correlation length of modest size and weak sensitivity to thermodynamic state. This imparts aroughly exponential spatial variation of all the key features of the dynamic free energy required tocompute gradients of dynamical quantities including the localization length, jump distance, cagebarrier, collective elastic barrier and alpha relaxation time. The spatial gradients are entirely ofdynamical, not structural nor thermodynamic, origin. The theory is implemented for the hardsphere fluid and diverse interfaces which can be a vapor, a rough pinned particle solid, a vibrating(softened) pinned particle solid, or a smooth hard wall. Their basic description at the level of thespatially-heterogeneous dynamic free energy is identical, with the crucial difference arising from thefirst layer where dynamical constraints can be weaken, softened, or hardly changed depending on thespecific interface. Numerical calculations establish the spatial dependence and fluid volume fractionsensitivity of the key dynamical property gradients for five different model interfaces. Comparison ofthe theoretical predictions for the dynamic localization length and glassy modulus with simulationsand experiments for systems with a vapor interface reveal good agreement. The present advancesets the stage for using the Elastically Collective NLE theory to make quantitative predictions forthe alpha relaxation time gradient, decoupling phenomena, T g gradient, and many film-averagedproperties of both model and experimental (colloids, molecules, polymers) systems with diverseinterfaces and chemical makeup. I. INTRODUCTION
Activated dynamics, mechanical properties and vitri-fication in thin films of glass-forming liquids of diversechemical nature (atoms, colloids, molecules, polymers)with highly varied boundary conditions is a problem ofgreat intrinsic scientific interest, which additionally may(or may not) shed light on the physics of the bulk glasstransition [1–5]. Thin films are also important in manymaterials applications [6–9]. Despite intense experimen-tal, simulation and theoretical effort over the past twodecades [2, 5, 10–18], the key physical mechanisms un-derlying the observed phenomena remain not very wellunderstood. We believe that this reflects the complexityof activated relaxation in bulk liquids [10] in concert withthe formidable complications of geometric confinement,interfaces and spatial inhomogeneity.A particularly rich aspect of thin films is the qual-itatively varied impact of boundary conditions. Freestanding thin films with two vapor interfaces, or semi-infinite thick films with one vapor interface/surface, arethe simplest realizations of confined systems. Extensive ∗ Electronic address: [email protected] experimental [1–3, 19–28] and simulation [2, 5, 14, 29–32] efforts suggest a spatially inhomogeneous large speedup of structural relaxation with mobile layers extend-ing rather deep into the film with correspondingly largefilm-averaged reductions of the glass transition temper-ature, T g . In contrast, experiments and simulationsfind that near a solid substrate the dynamics is verynon-universal – it can modestly speed up, slow downdrastically, or hardly change at all relative to the bulk[2, 5, 13, 14, 24, 25, 33–36]. The origin of such complex-ity often seems puzzling. A confining surface or substratecan be topographically smooth or rough, can promoteliquid adsorption or not, and can have a mechanical stiff-ness varying from infinitely rigid (pinned particles) to asoft surface [37–39] to even liquid substrates [40] thatare thermodynamically hard but dynamically fluid. Itappears all these features are important, often qualita-tively, for determining the glassy dynamics of real worldfilms.Recently, a quantitative force-level statistical mechani-cal approach for structural (alpha) relaxation in isotropiccolloidal, molecular and polymer bulk liquids, the ”Elas-tically Collective Nonlinear Langevin Equation” (EC-NLE) theory [41–46], has been developed and general-ized to treat glassy dynamics in free-standing films [47–49]. Structural relaxation is described as a coupled ac- a r X i v : . [ c ond - m a t . s o f t ] J a n tivated process involving a large amplitude cage-scaleparticle hopping event that is facilitated by a small am-plitude longer-range collective elastic deformation of thesurrounding liquid. Quantitative tractability for molecu-lar and polymeric liquids is achieved based on an a priorimapping of chemical complexity to a thermodynamic-state-dependent effective hard sphere fluid [42, 45]. Thetheory for free-standing films predicts strongly acceler-ated and spatially inhomogeneous relaxation for purelydynamical reasons.Most recently, Phan and Schweizer [50] formulated animproved technical treatment of the collective elasticityaspect in free-standing thin films and semi-infinite thickfilms with vapor interfaces, and addressed qualitativelynew questions. For example, the mobile layer length scaleis predicted to grow strongly with cooling, and corre-lates nearly linearly with the dynamic barrier deducedfrom the bulk alpha time. A new type of spatially in-homogeneous ”decoupling” was predicted, an effect firstdiscovered by Simmons and coworkers using computersimulation in the weakly supercooled regime [57]. Specif-ically, this type of decoupling corresponds to a remark-able effective factorization of the total barrier into itsbulk temperature-dependent value multiplied by a func-tion that depends only on location in the film. Quan-titative no-fit-parameter comparisons of the theory forfree standing films with experiment and simulation for T g shifts of polystyrene and polycarbonate are in rea-sonable accord with the theory, and testable predictionswere made [19, 51–54]However, major puzzles remain even for films with va-por interfaces. Conceptual ones include precisely howmobility changes are nucleated at an interface or sur-face, and how they are ”propagated” or transferred deepinto the film. How such questions can be theoreticallyaddressed for films with solid interfaces is open. Cru-cial motivations for the present article are puzzles suchas the long standing simulation finding that the relax-ation time gradient for free standing and solid substratefilms appears to have (to leading order) a ”double ex-ponential” form [13, 14, 31, 55–58]. This behavior im-plies the effective barrier varies roughly in an exponentialmanner with distance from an interface. However, theassociated length scale only modestly grows with cool-ing, and appears to already saturate in the lightly su-percooled regime probed in simulation [55–58]. Such be-havior is in apparent disagreement with entropy crisis orthermodynamic-based theories of glassy dynamics whichargue the relevant length scale should continue to grow allthe way down to the laboratory vitrification temperature[10, 16, 56]. A seemingly related behavior revealed bysimulation is a spatial dependence of the ”decoupling ex-ponent” that varies roughly exponentially with distancefrom the interface, and the strong correlation of this be-havior with gradients of the activation barrier [57].The present paper reports the first and most criticaladvance required to generically address the above issueswithin the ECNLE theoretical framework. Specifically, we formulate a new treatment of how local dynamicalconstraints, quantified via a cage scale ”dynamic free en-ergy”, are modified at an interface, and how they aretransferred into the film. The ideas are applied to studythe spatial dependence of the particle localization lengthand glassy elastic modulus, and also to establish how alldynamic free energy properties that determine the totalactivation barrier and alpha time gradient are modified.This sets the stage for future efforts that will employECNLE theory to quantitatively predict the alpha relax-ation time gradient and other properties for films withdiverse boundary conditions.The remainder of the article is as follows. We brieflyreview in Section II the key elements of the existing EC-NLE theory of bulk liquids and vapor interface films.Section III presents our new formulation of how cagescale dynamical constraints are modified for various softand hard interfaces. Five different hard, soft and vaporinterfacial models are considered. Application to treatthe dynamic localization length and glassy modulus infilms is the subject of Section IV, and quantitative no-fit-parameter comparisons are made with experiment andsimulation. Section V establishes how all other featuresof the dynamic free energy in films are modified. The pa-per concludes with a discussion in Section VI. The Ap-pendix compares predictions for the localization lengthobtained from two different formulations of the new the-oretical idea. II. BACKGROUND: ECNLE THEORY OF BULKLIQUIDS AND FREE-STANDING THIN FILMS
For context, we briefly review the present state of EC-NLE theory for bulk liquids [41–46] and free standingthin films [47–50] in the simplest context of sphericalparticle liquids; all details are in prior papers. In thisarticle, we will implement the new ideas for the founda-tional hard sphere system.
A. Bulk Liquids
Consider a one-component liquid of spherical parti-cles (diameter, d ) of packing fraction Φ. The fun-damental theoretical quantity is an angularly-averagedparticle displacement-dependent ”dynamic free energy”, F dyn ( r ) = F ideal ( r ) + F caging ( r ) , the derivative of whichis the effective force on a moving particle in a stochasticnonlinear Langevin equation (NLE) [59]: F dyn ( r ) k B T = − r − ρ (cid:90) d q (2 π ) S ( q ) C ( q )1 + S − ( q ) × exp (cid:20) − q r (cid:0) S − ( q ) (cid:1)(cid:21) = F ideal ( r ) k B T + F caging ( r ) k B T , (1)where β = ( k B T ) − , k B is the Boltzmann’s constant, T is temperature, ρ is number density, r is the displacementof a particle from its initial position, S ( q ) is the struc-ture factor, q is wavevector, and C ( q ) = ρ − (cid:2) − S − ( q ) (cid:3) is the direct correlation function. The leading term inEq.(1) is an ideal entropy-like contribution that favorsthe fluid state, and the second term is due to interpar-ticle forces which favors cage localization. The latteris determined from knowledge of fluid density and pairliquid structure. As the density (or temperature) ex-ceeds (goes below) a critical value, a local barrier F B in F dyn ( r ) emerges (at Φ ≈ .
43 for hard spheres [59] basedon Percus-Yevick theory [60] input) signaling transientlocalization. Figure 1 shows an example dynamic freeenergy, its ideal and caging components, and defines keylength and energy scales including the localization length, r L , barrier location, r B , jump distance, ∆ r = r B − r L ,and local cage barrier, F B . B r B F d y n (r) / k B T r/d r L r F r ee ene r g y ( k B T ) r/d ideal caging FIG. 1: (Color online) Dynamic free energy as a function ofreduced particle displacement for a hard sphere fluid of pack-ing fraction Φ = 0 .
58 ; important length and energy scales aredefined. The inset shows the corresponding ideal and cagingcomponents of F dyn ( r ). For hard sphere fluids with barriers beyond a few k B T ,much insight has been gained based on the so-calledultra-local analytic analysis [61]. The crucial result isthat, to leading order, all aspects of the dynamic free en-ergy enter via a universal function multiplied by a single”coupling constant”, λ [61]: F caging ( r ) = λ (Φ) .f cage ( r/d ) , λ (Φ) ∝ Φ g (Φ) , (2)where g ( d ) is the contact value of g ( r ). The first equal-ity in Eq.(2) is a factorization-like property which im-plies the functional form of the caging dynamic free en-ergy (and corresponding force, ( − ∂F caging ( r ) /∂r ) is, toleading order, not dependent on thermodynamic state.The local structure and packing fraction enter solely ina multiplicative manner via a coupling constant, λ . Thisis a striking prediction of NLE theory that holds whenbarriers are relatively high and motion is strongly ac-tivated. Detailed analysis shows the coupling constant can be physically interpreted as proportional to an effec-tive mean square caging force experienced by a taggedparticle. It is dominated by nearest neighbor forces forshort range interactions (high q contributions dominatein Eq.(1)). Prior analytic analysis has derived [61]: r L ≈ √ π g ( d ) ∝ λ − ∝ ( βF B ) − ,r B = 1 q c (cid:112) g ( d )) , q c = π/d. (3)The predicted relation d/r L ∝ βF B ∝ λ connects shortand long time dynamics, a hallmark of NLE theory. Thedynamic (relaxed high frequency) shear modulus, G (cid:48) , ispredicted (not assumed) to obey a micro-rheology likerelation [41, 61]: G (cid:48) ≈ k B T πdr L . (4)These connections remain useful for thermal liquids sincethey are a priori mapped to effective hard sphere fluids[42, 45]. In Eq.(2), the coupling constant then becomesa function of temperature, pressure and chemistry. Theconnections also remain useful in thin films. ECNLE elastic barrier r u ( r )≤ r L rF B r L r B F dyn NLE local barrier(a)(b)1234 F dyn ( r , z ) r z z = 0 zzinterface rough interfacevapor FIG. 2: (Color online) (a) Schematic of the fundamental re-laxation event in bulk liquid ECNLE theory involving twocoupled physical processes: (1) local/cage-scale hopping asdescribed by the dynamic free energy, and (2) a nonlo-cal/spatially longer range collective harmonic elastic motionoutside the cage region required to allow the large amplitudelocal rearrangement to occur. Various key length and energyscales are indicated. (b) Cartoon illustration of the layer-like model of the surface nucleated dynamic caging constrainttransfer idea and spatial variation of the dynamic free energy.
In ECNLE theory, large amplitude hopping is stronglycoupled to a long range collective elastic spontaneousfluctuation of all particles outside the cage required tocreate the small amount of extra space to accommodatea hop; conceptual elements are sketched in Fig. 2a. Theradially-symmetric solution for the required elastic dis-placement field decays as an inverse square power law[41, 62]: u ( r ) = ∆ r eff r cage r , r ≥ r cage ∆ r eff ≈ r / r cage ≤ r L . (5)The amplitude is set by a small mean cage expansionlength, ∆ r eff , which follows from assuming each particlein the cage independently hops in a random directionby ∆ r . The elastic barrier is determined by summingover all harmonic displacements outside the cage regionthereby yielding [41] F elastic = ρ K (cid:90) ∞ r cage dr πr u ( r ) g ( r ) ≈ K Φ∆ r eff (cid:16) r cage d (cid:17) , (6)where r is relative to the cage center, and K = 3 k B T /r L is the curvature of the dynamic free energy at its mini-mum. The sum of the coupled (and in general temper-ature and density dependent) local and elastic collectivebarriers determine the total barrier for the alpha process, F total = F B + F elastic . A generic measure of the structuralor alpha relaxation time follows from a Kramers calcula-tion of the mean first passage time for barrier crossing.For barriers in excess of a few k B T one has [41, 59]: τ α τ s = 1 + 2 π ( k B T /d ) √ K K B exp F B + F elastic k B T , (7)where K B is the absolute magnitude of the barrier cur-vature. The alpha time is expressed in units of a ”shorttime/length scale” relaxation process (cage-renormalizedEnskog theory), τ s , the explicit formula for which is givenelsewhere [41, 42] . Physically, it captures the alphaprocess in the absence of strong caging defined by theparameter regime where no barrier is predicted (e.g.,Φ < .
43 for hard spheres). The latter condition cor-responds to being below the nave mode coupling theory(NMCT [59, 63]) ideal dynamic glass transition which inECNLE theory is manifested as a smooth crossover.The theory can be applied to any spherical particleof colloidal fluid, and to molecular and polymeric liq-uids based on an appropriate mapping [42, 45]; here weconsider only the hard sphere fluid. To place our calcu-lations in broader context, we recall how packing frac-tion, reduced temperature and alpha time are related forthe prototypical glass-forming molecular liquid orthoter-phenyl (OTP) [42]: Φ = 0 . , . , . , . , .
61, cor-responds to
T /T g ≈ . , . , . , . , . τ α ∼ τ α = 100 s at T = T g . B. Vapor Interface Films
For films with interfaces every property (thermody-namic, structural, dynamic) is spatially heterogeneousand anisotropic. Treating such complexity theoreticallyis intractable. In the past a minimalist approach wasadopted based on the hypothesis that the most impor-tant effects are purely dynamical with no changes of ther-modynamics or structure in the film [47–50]. This idea isconsistent with recent machine-learning based analysis ofsimulations of free standing films which found the largedynamical changes are not related to any change of struc-tural or static properties [64]. It also is relevant for sim-ulation studies of films performed under so-called ”neu-tral confinement” conditions [13, 14, 55, 56, 65] where thesolid substrate is constructed to have no effect on liquidpacking.Of course, in real thin films there are changes of ther-modynamic properties, the one-body density profile per-pendicular to the interface, and (anisotropic) packingcorrelations near an interface. But these changes are usu-ally small, highly localized near the interface in a denseliquid, depend sensitively on the nature of the interfaceor surface, and are chemically specific. In this article, forboth simplicity and our desire to focus on the purely dy-namical physics, we ignore such complications and adopta step-function density profile in the direction orthogonalto the interface and a density or volume fraction identicalto that in the bulk liquid.How a vapor interface modifies the alpha process inthe prior ECNLE theory involves two coupled effects: (i)local caging as encoded in the dynamic free energy, and(ii) the collective elastic displacement field and associatedbarrier. The cage remains the elementary dynamical unitand is characterized locally by (pre-averaged) isotropicsymmetry. The goal is to predict how it changes as afunction of distance from the film interface, z . A zerothorder approach [47–50] [47-50] for free standing films wasconstructed as follows. For point (i), near the surface(0 ≤ z ≤ r cage where for a sharp interface the center ofparticles of the first layer define z = 0) caging constraintsare softened due to losing nearest neighbors. The fractionof bulk cage particles present at location z follows fromgeometry as [47]: γ ( z ) = 12 − (cid:18) zr cage (cid:19) (cid:20) − (cid:16) r cage z (cid:17) (cid:21) . (8)For z = 0, γ ( z ) = 0 . e − q r / S ( q ) in Eq.(1) for the fraction 1 − γ ( z ) of particlesmissing from the effective cage, while the remaining par-ticles (quantified by the factor γ ( z )) have the same col-lective dynamic Debye-Waller factor as in the bulk. For z = r cage , the full cage is recovered and γ ( z ) = 1. This isa highly local approximation, where surface-induced mo-bility is assumed to not extend into the film beyond thecage radius (it is this approximation that is re-visitedin the present work). The dynamic free energy is thusmodified as [47]: F dyn ( r ) = − k B T ln r + γ ( z ) F caging ( r ) . (9)Near the surface all properties of the dynamic freeenergy behave as a liquid with weaker dynamical con-straints. Importantly, note the multiplicative manner theinterface modifies dynamical constraints where F caging ( r )remains the same as in the bulk. Given Eq.(2), this im-plies, to leading order, a ”double factorization” type ofmathematical structure: F caging ( r ) γ ( z ) .λ (Φ) .f cage ( r/d ).This can have potentially profound consequences. For ex-ample, to leading order a type of ”corresponding states”behavior is expected for the caging constraints since f cage ( r/d ) is universal but a continuum of values of ( z, Φ)in principle exist such that the net amplitude of thecaging dynamic free energy and force (determined by theproduct γ ( z ) .λ (Φ) remains constant. To address point(ii) above, a simple ”cutoff” of the bulk isotropic elasticfield assumption was adopted, formulated in two tech-nically different, but qualitatively the same, manners[41]. Since this article focuses solely on point (i), wedo not elaborate further, except to emphasize that allthe information required to determine the elastic bar-rier (jump distance and dynamic localization length since K = 3 k B T /r L ) also follows from knowledge of the dy-namic free energy in the film. III. NEW FORMULATION OFINTERFACE-INDUCED SPATIALLYINHOMOGENEOUS CAGING CONSTRAINTS
The first and foremost critical issue is: (i) how is thecaging force modified near an interface? Prior work [38–41] for a vapor surface assumed that beyond a radius r cage ∼ . − . d the dynamic free energy recovers itsbulk form. Additional simplifications were invoked torender the theory tractable and/or for internal consis-tency with the bulk formulation. (ii) The liquid-vaporinterface is perfectly sharp. (iii) The ensemble-averagedpair structure, liquid density and thermodynamic prop-erties are unchanged in the film. (iv) The mobility ofall particles in a spherical cage region are the same. As-sumptions (ii) and (iii) can be relaxed at the expenseof technical complexity. Assumption (iv) pre-averagesdynamic heterogeneity inside the cage scale of ∼ d , re-taining the spirit of bulk NLE theory.Here we propose a new general formulation of the dy-namic free energy idea for films that we believe qualita-tively improves the treatment of (i) and (iv). Point (i)is the most fundamental, and we aim to understand howmobility near the surface can affect particles in a layerdirectly above it, and how such a gradient of dynamicalconstraints extends further into the film. For a vapor in-terface where dynamics speeds up, one could view this asa form of ”dynamic facilitation”, albeit of literal broken spatial symmetry origin of different physical origin thanin an isotropic bulk fluid. [10] For a solid surface thatslows down particles near it, the effect would be akin to”anti-facilitation”.We first recall that bulk NLE theory is built on the sin-gle particle (naive) version of ideal mode coupling theory(so-called NMCT [59, 63]) as encoded in a self-consistentnonlinear equation for strict kinetic arrest based on anensemble-averaged localization length. NMCT relatespair structure, forces, thermodynamic state and cagingconstraints. Given the film problem is more complex, wefirst explore two different approaches which are in thesame spirit physically. Both adopt a finer resolution ofspace than a cage diameter to formulate dynamic con-straints, namely a ”layer” which can be interpreted asa region of one particle diameter or cage radius thick-ness; here we adopt the former perspective. See Fig. 2bfor a sketch. The layer picture is a conceptual device toquantify constraints in a spatially discrete manner. Itdoes not require any density gradient perpendicular tothe flat interface.The first approach is in the NMCT framework and onlyaddresses the ideal glass question. The second generalapproach is formulated directly in terms of the dynamicfree energy concept. As shown in the Appendix, for theonly question these two formulations can both address,the gradient r L ( z ), the numerical results are similar. Thesecond approach is the focus of our present and futureefforts. A. NMCT Gaussian Dynamical Formulation
The NMCT self-consistent localization relation for anideal glass in the isotropic bulk is [59]:9 r L = (cid:90) d k (2 π ) | kC ( k ) | ρS ( k ) e − k r L (1+ S − ( k )) / , (10)where (cid:10) r ( t → ∞ ) (cid:11) ≡ r L , and e − k r L / and e − k r L / S ( k ) are the kinetically arrested single and collective dynamicpropagators (Debye-Waller factors), respectively. PerFig. 2b, for a film we change perspective to a finer res-olution of the cage corresponding to a layer-like modelor (in practice) resolving a cage into two halves. Sincein-plane particle localization is taken to be uniform at agiven distance from the interface, the arrested dynami-cal state in layer i (or z = ( i − d in terms of spatialposition) is described by r L,i . We continue to adopt thephysical picture of a cage of diameter 3d surrounding atagged particle which encapsulates particles from threelayers. Focusing on a particle at the cage center, we viewit as experiencing forces from an equal number of parti-cles above and below (if present) it. As our starting pointansatz, a central particle is modeled as experiencing twotypes of dynamical environments in a film depending onits distance from the interface. Within each half of acage, we average over particle mobility, in contrast tobulk NLE theory which averages over all particles in aspherical cage. Now, based on the idea that dynami-cal inhomogeneity is initiated at the interface, the cagingconstraints on a particle in a given layer are constructedin a democratic fashion. This corresponds to a collectiveDebye-Waller factor in Eq. (10) that has two contribu-tions yielding a modified self-consistent NMCT equation:9 r L,i = (cid:90) d k (2 π ) | kC ( k ) | ρS ( k ) e − k r L / × (cid:20) e − k r L,i / S ( k ) + 12 e − k r L,i − / S ( k ) (cid:21) . (11)The first (second) term inside the bracket corresponds tothe dynamic constraints from half of a particle cage ofcenter assigned to layer i ( i − r L, = (cid:90) d k (2 π ) | kC ( k ) | ρS ( k ) e − k r L / (cid:20) e − k r L, / S ( k ) (cid:21) . (12)This is identical to the Mirigian-Schweizer (MS) ap-proach [47–49] for z = 0. For a supported film, we firstconsider the case where the substrate is modeled as aquenched fluid composed of literally pinned particles ofthe same size, density and pair structure as the mobileparticle liquid that defines the film (often called ”neutralconfinement”). Then the first layer localization length isdetermined by Eq.(11) with r L, = 0. The localizationlength of particles in first layer of the mobile liquid isthus: 9 r L, = (cid:90) d k (2 π ) | kC ( k ) | ρS ( k ) e − k r L / × (cid:20) e − k r L, / S ( k ) + 12 (cid:21) . (13)Importantly, an equation identical to that above followsif we employ our recent theory [66] of the bulk pinned-mobile hard sphere system with the fraction of pinnedparticles set to 0.5. The reason is that in both casesa tagged particle experiences one half of its constraintsfrom immobile but otherwise identical particles. This ex-poses a key assumption: in broken symmetry films it isthe number of particles that are mobile versus immobilein a spherically-averaged cage which quantifies (to lead-ing order) the dynamical constraints on a tagged particlein a cage; the precise spatial arrangement is angularlypre-averaged. This essential approximation is what ren-ders the theory tractable, and allows us to think and cal-culate in a manner analogous to prior NLE theory workin bulk and thin films.The full dynamic localization length gradient then fol-lows immediately from the above ideas and Eq.(11). Note that the localization length in layer i follows from knowl-edge of its analog in the underlayer i −
1. Thus, one canpredict the full gradient in a sequential layer-by-layer orbootstrapped manner starting at the surface, resulting ina simple physical picture and easy numerical solution.
B. Dynamic Free Energy Formulation
We now consider the problem directly from viewpointof the dynamic free energy. The physical idea for in-troducing sub-cage resolution of dynamical constraintsremains the same as above. Consider a particle at a cagecenter. We again assume dynamic constraints on it arisefrom equal contributions of particles in two adjacent lay-ers. Given we assume packing structure is not changedin the film, the dynamic free energy in layer i is: F ( i ) dyn ( r ) = 12 F bulkdyn ( r ) + 12 F ( i − dyn ( r ) , i ≥ i = 0 is the first layer of the substrate. The ”1/2-1/2” weighting form is the same as in Eq. (11). For thepurpose of analyzing layer i , the constraints from the up-per half of the cage are quantified as in the bulk. This isanother key approximation, but one we believe is consis-tent with the assumed invariance of equilibrium structurein the film. But the particles in the lower half of a cageare affected by the interface in a manner that dependson the nature of, and distance from, the interface. Thus,the idea is again that film perturbations are nucleated inthe first layer, and via modification of the caging part ofthe dynamic free energy are spatially transferred into thefilm. For the first liquid layer one has, F (1) dyn ( r ) = 12 F bulkdyn ( r ) + 12 F (0) dyn ( r )= F ideal ( r ) + 12 F bulkcaging ( r ) + 12 F surfacecaging ( r ) , (15)where the crucial quantity is the ”surface layer cagingdynamic free energy”, the last term above. The dynamicfree energy of the film is constructed by iterating Eq.(15).For the second layer and third layers one has F (2) dyn ( r ) = F ideal ( r ) + 12 F bulkcaging ( r ) + 12 F (1) caging ( r ) , = F ideal ( r ) + (cid:18)
12 + 12 (cid:19) F bulkcaging ( r ) + 12 F surfacecaging ( r ) , (16) F (3) dyn ( r ) = 12 F bulkdyn ( r ) + 12 F (2) dyn ( r )= F ideal ( r ) + (cid:18)
12 + 12 + 12 (cid:19) F bulkcaging ( r )+ 12 F surfacecaging ( r ) , (17)One can obviously write a general expression for the dy-namic free energy in n th layer F ( n ) dyn ( r ) = F ideal ( r ) + (cid:18) − n (cid:19) F bulkcaging ( r ) + F surfacecaging ( r )2 n , = F bulkdyn ( r ) + 2 − n ∆ F caging ( r ) (18)where ∆ F caging ( r ) = F surfacecaging ( r ) − F bulkcaging ( r ) . (19)The physical essence of this approach is effectively ahypothesis of a geometric-like transfer of dynamical con-straint information nucleated at the surface into the film.The amplitude of the change of constraints enter via a dif-ference in caging dynamic free energy (Eq.(19)) which isexpected to be positive (negative) for a pinned solid (va-por) surface. The generic form above implies the dynamicfree energy varies essentially exponentially in space if onemathematically passes from a discrete layer descriptionto a continuous space description:2 − n = e − n ln 2 = e − z/ξ , where z = nd, ξ = d/ ln 2 . (20)Importantly, the corresponding ”decay length” is a uni-versal constant of ∼ . d , but only at the most funda-mental level of the caging dynamic free energy. Of coursethe latter is a theoretical construct that is not directlyobservable, and thus this simplicity does not genericallyapply for various dynamical properties derived from thedynamic free energy and full ECNLE theory. The am-plitude of the change of dynamical constraints in Eq.(19) depends on chemistry, thermodynamic state, andnature of the surface. Moreover, the amplitude and z -dependence of caging constraints effectively factorize.Given the ultra-local analytic understanding of bulk NLEtheory [61] reviewed above, qualitatively one then ex-pects the local barrier and all other key aspects of thedynamic free energy vary roughly exponentially as a func-tion of distance from the interface (as shown numericallybelow). If true, this immediately provides a generic phys-ical mechanism for the simulation observations of a ”dou-ble exponential” form of alpha time gradients [13, 14, 55–58].Recall from the discussion below Eq. (9) of Section IIBthat the fundamental form of the caging part of the dy-namic free energy of the prior ECNLE theory [50] for free-standing films obeyed the ”double factorization” form.Eqs. (18)-(20) continue to obey this general form for thedifference between the caging component of the dynamicsin the bulk and at a location z in the film. This propertyof the theory is expected to have many consequences. Forexample, as shown below, the spatial gradients of dimen-sionless ratios of a dynamic property in the film relativeto in the bulk can often be (to leading order) invariantto temperature, volume fraction and chemistry. More-over, the ”corresponding states” structure mentioned inSection IIB continues to hold to leading order. We note that the existence of the simplicities describedabove rely on several physical ansatzes of the theory:high wavenumber dominance of the caging dynamic freeenergy, no changes of equilibrium properties in the filmrelative to the bulk, and the multiplicative manner thatthe location in the film variable modifies the dynamic freeenergy corresponding to a z -dependence that does not di-rectly depend on thermodynamic state or chemistry.Finally, we comment on two fundamental theoreti-cal aspects of our present formulation. First, Eq.(14)plus Eq.(20) may perhaps be interpreted as a forward-difference approximation of a gradient expansion of thedynamic free energy. This was not our perspective informulating the theory, especially since dynamical spa-tial gradients are often very sharp for supercooled liquidsnear interfaces. Rather, we have chosen to formulate thetheory in a discrete manner that explicitly acknowledgesthe finite size of particles and cages which are the ele-mentary scales of NLE theory and the dynamic free en-ergy concept. A second question is whether there could,or should, be an explicit coupling of layer i with bothlayers i − i + 1, perhaps in the spirit of a 1 − d Ising model. We note that such a formulation would in-troduce much additional technical and conceptual com-plexity since all layers become effective coupled and thedynamics of the entire film would need to be treated self-consistently. This is in contrast to our simpler formula-tion which has a layer-by-layer ”bootstrapping” charac-ter. Moreover, our approach is in the spirit of the ofteninvoked physical notion that dynamic changes at an in-terface ”propagate” or are transferred in a directionalmanner from the interface into the film.
C. Specialization to a Specific Interface
The nature of the interface or substrate enters solelyvia the ”surface” component of the caging dynamic freeenergy in Eq. (19). Per Eq(18), the modification ofcaging constraints at the surface always decreases atlarger distances from the interface and bulk behavior isrecovered deep enough into the thick film. We introduce6 models for the ”surface” component of the caging dy-namic free energy that mimic to varying degrees of real-ism specific physical systems of experimental and simu-lation interest, as sketched in Fig. 3. In each case thereis a sharp interface between the liquid (top) and sub-strate (bottom) which are of macroscopic extent. Herewe consider only physical systems where the dynamicalstructure of the substrate is a priori specified, i.e., thesubstrate sets boundary conditions and serves as an ”ex-ternal force field” felt by the liquid. We envision suchmodels as directly relevant to simulations that employpinned particle substrates, a film with a vapor inter-face, and as a simple model for amorphous or crystallinesubstrates (e.g., silica, silicon, gold) that are employedat temperatures far below their melting or glass transi-tion temperature. Of course the latter can interact withthe liquid via attractive interactions, and variable surfacecorrugation or roughness can play a role, surface effectsnot considered here. (a) (b)(c) Φ s Φ ΦΦ Φ s =Φ Φ Φ s =Φ FIG. 3: (Color online) Illustration of different interfaces: (a)macroscopic bilayer with one sharp interface but (in general)different packing fractions in the two thick films, (b) free-standing or vapor interface thick film, and (c) pinned particlerough surface film which may or may not have the same pack-ing fraction as the overlayer fluid. The image on the rightindicates the technical approximation employed to map thereal system to a first layer description based on an isotropicrandom pinned particle system [66].
1. General Bilayer System
This is the most general system considered here andis depicted in Fig. 3a–two macroscopic layers (bilayer)with one sharp interface. The film and substrate are thesame type of system (spheres of equal sizes) but, in gen-eral, can be at different volume fractions. In the first layer the cage center lies at the bilayer interface ( z = 0).The dynamic free energy experienced by a liquid parti-cle in this first layer is given by Eq. (15). This modeldoes not address polymeric bilayers composed of two dis-tinct glass-forming materials [37–39] since in that casethe bottom layer modifies the dynamics of the top layerand vice-versa. The bilayer system can be analyzed withour approach and will be studied in a future publication.
2. Vapor interface
Per Figure 3b, since there are no particles in the vaporlayer, one has:Φ s = 0 , F surfacecaging ( r ) = 0 . (21)
3. Rough pinned substrate
Here the substrate is composed of literally pinned par-ticles identical in every other way to the liquid particles.This is the simplest example of a rough solid substrate.It has been extensively studied in simulations and allowsone to focus entirely on interface-induced changes of liq-uid dynamics. This system is of course anisotropic whichrenders the problem extremely complex. However, asdiscussed in Section III, the dynamic free energy of NLEtheory is formulated at the cage scale based on an ap-proximate angular averaging procedure. We implementthis idea per the schematic of Fig.3c. At the interface,the half of a cage that are pinned particles are modeled asdistributed randomly in a full spherical cage with the mo-bile particles. This zeroth order simplification assumesthe most important consideration is the fraction of neigh-bors in a cage that are pinned versus mobile, and nottheir precise spatial arrangement.Given the above simplification, we employ our previ-ously developed NLE theory for bulk randomly pinnedparticle fluids under neutral confinement conditions [66]for a fraction of pinned particles in a cage equal to α = 0 . F roughpinnedcaging ( α, r ) = − (cid:90) d q (2 π ) (cid:34) C ( q ) S ( q ) e − q r / ρ (1 − α ) [1 − ρ (1 − α ) C ( q )] + ρ (1 − α ) C ( q ) e − q r [2 − ρ (1 − α ) C ( q )] / [1 − ρ (1 − α ) C ( q )] [2 − ρ (1 − α ) C ( q )] (cid:35) α =0 . = 12 F bulkcaging ( r ) + 12 F pinnedsurfacecaging ( r ) , (22)where S ( q ) is the cross collective static structure factorbetween pinned and mobile hard spheres as discussed inRef. [66]. The second equality defines the rough pinned surface dynamic free energy: F pinnedsurfacecaging ( r ) = 2 F roughpinnedcaging (0 . , r ) − F bulkcaging ( r ) . (23)
4. Rough vibrating pinned substrate
A simple variant of model 3) allows the randomlypinned particles to harmonically vibrate via a small pre-scribed localization length r L,s . This model is relevantto recent simulation studies of Simmons and co-workers [29] that examined the influence of substrate mechanicalstiffness or Debye-Waller factor on film dynamics. Theparameter r L,s enters the theory via the first contribu-tion on the right hand side of Eq. (22) which is modifiedby introducing the appropriate collective Debye-Wallerfactor of the vibrating pinned particles: F roughvibratingcaging ( α, r ) = − (cid:90) d q (2 π ) (cid:34) C ( q ) S ( q ) e − q r / e − q r L,s / S ( q ) ρ (1 − α ) [1 − ρ (1 − α ) C ( q )] + ρ (1 − α ) C ( q ) e − q r [2 − ρ (1 − α ) C ( q )] / [1 − ρ (1 − α ) C ( q )] [2 − ρ (1 − α ) C ( q )] (cid:35) α =0 . = 12 F bulkcaging ( r ) + 12 F vibratingsurfacecaging ( r ) . (24)
5. Smooth Rigid Wall
For decades simulations have studied model super-cooled liquids confined by a smooth hard wall (no corru-gation, no attraction) which have no transverse wall-fluidforces. They find the locally anisotropic liquid dynamicsis modified in a qualitatively different manner than forrough particle-based walls – motion speeds up parallel tothe wall and also in an angularly average manner rela-tive to the bulk versus slowing down near rough walls ofpinned particles [13, 58, 67–69]. The hard smooth wallsystem can be viewed as simply a toy model, but it alsomay be crudely relevant to two classes of experimentalsystems as we briefly discuss.Some hard substrates (dense crystalline or amorphoussolids) are composed of atoms (size b ) that are muchsmaller than the size of the molecules or polymer seg-ments that constitute the fluid film, i.e., b (cid:28) d . Thesubstrate-fluid potential energy is thus of a corrugatedform in the transverse direction which implies an os-cillating (about zero) spatial variation of the wall-fluidforces on a length scale small compared to the fluid par-ticles. If true, then for a nonadsorbing atomic substratethe transverse forces could average out to zero (at zerothorder) on the longer length and time scales relevant forthe structural relaxation process of the larger fluid par-ticles – an effectively smooth wall. On the other hand,for short time and length scale dynamics (e.g., transientfluid particle localization near the substrate and ”rattlingdynamics”) this picture will surely be less accurate andmay not apply.Another experimental system perhaps related to thesmooth wall model is a liquid substrate (e.g., glycerol[40]) that is immiscible with the fluid film. It behavesin a thermodynamically hard manner corresponding toexerting repulsive forces on the fluid particles perpendic-ular to the interface. If the liquid substrate is of lowenough viscosity such that its structural relaxation timeis very small compared to the alpha time of the super-cooled fluid, then the substrate particles are effectively ergodic from the perspective of the film particles. WithinNLE theory this suggests an effective in-plane dynamicalinterfacial smoothness could apply on the alpha relax-ation time scale of the film particles, and hence liquidsubstrates might be crudely viewed (to zeroth order) asa ”smooth wall”. Again such a viewpoint may not applyto the shorter time and length scale rattling dynamics ofthe fluid particles.In any case, our interest in this article is not the abovetwo experimental systems, but solely to study the smoothwall surface as a limiting model where all wall-fluid forcesparallel to the interface vanish. Compared to the roughpinned particle system, caging constraints exerted by thesubstrate on fluid particles in two spatial directions areabsent. We crudely mimic this situation in an averagemanner by reducing the caging component of the dy-namic free energy at the interface by a factor of 3. Thus,a smooth wall is modeled by dividing the pinned particledynamic caging free energy by a factor of 3 correspondingto using in Eq.(19): F smoothwallcaging ( r ) = 13 F pinnedsurfacecaging ( r )= 13 (cid:104) F roughpinnedcaging (0 . , r ) − F bulkcaging ( r ) (cid:105) (25)
6. Attractive Rough Walls
We consider a variant of rough substrate models 3)and 4) where there is an attractive interaction betweenthe mobile liquid and the immobilized substrate parti-cles. Treating this fully is difficult given the high degreeof nonuniversality of surface-fluid interactions, substratestructure, and the often presence of an explicit attractiveforce between the substrate and fluid particles. However,prior theoretical and simulation studies have found thata rather generic consequence of such an attraction is fluiddensification near the wall, and typically only in the firstlayer [13, 67, 70]. We consider a model that is a crude0mimic solely of this effect by assigning a packing fractionin the first liquid layer that is higher than in other layerswhere it takes on the bulk value [49]:Φ = λ Φ bulk , λ > j = Φ bulk , j ≥ . (26)Density enhancements are chemistry specific, but can beas large as 10-15 %. As a specific example, for glycerolin contact with a silica surface, a recent computationalstudy found [70] a first layer enhancement of 1.038. Al-ternatively, if the surface is weakly dewetting, the fluiddensity could be reduced, λ <
1. Explicit attractiveforces are not taken into account dynamically, but theywould serve to further slow down the mobile liquid par-ticles near the surface. Treating the latter may requiremodifying the dynamic force vertex using the ”projecteddynamics theory” approach [71].In subsequent sections we present representative nu-merical results for the key dynamical quantities of NLEtheory for models 2), 3), 4), 5) and 6). A full analysisusing ECNLE theory and the treatment of how variousinterfaces modify the collective elastic aspect of the alphaprocess in films is beyond the scope of this initial workand will be addressed in future publications.
IV. SHORT LENGTH SCALE RESULTS:DYNAMIC LOCALIZATION LENGTH ANDSHEAR MODULUS
We first numerically apply the theory to study themost spatially local questions of the dynamic localiza-tion length for a thick film and the elastic shear modulusand its spatial gradient in a thin film.
A. Dynamic Localization Length: Vapor vs PinnedRough Solid Interfaces
Figure 4 shows the spatial variation of the dynamiclocalization length, r L , normalized to its bulk value fortwo very different values of volume fraction for vaporinterface (main frame) and rough pinned solid interface(inset) thick films. For both systems, this relative de-pendence depends very weakly on volume fraction, re-flecting the near ”factorization” property of the NLEdynamic free energy discussed in sections II and III.As expected, the localization length is larger (smaller)near the vapor (solid) surface. Moreover (see the in-set and figure caption), we find that it decays to thebulk value in an exponential manner with an essentiallyvolume-fraction-independent characteristic length scaleof ∼ .
83 and 1.7 particle diameters for vapor and solidsurfaces, respectively. These results are akin, at ze-roth order, with Eq.(20) that suggests a decay lengthof ∼ d/ln (2) ∼ . d . The shorter penetration length forthe solid surface compared to the ”softer” vapor interface is interesting, especially since the amplitude of the sur-face perturbation (deviation of r L (0) /r L,bulk ) from unitis larger for the vapor film. This reveals a nontrivial dis-crimination by the theory between the surface amplitudeversus penetration depth aspects of soft and hard inter-faces. Moreover, the predicted trend for the localizationlength appears to be in qualitative accord with the exper-imental finding that the spatial range of T g perturbationsnear soft interfaces are greater than for hard interfaces[38, 39]. = 0.55 = 0.61 500 K 200 ps 400K 200 ps 400K 20 ps0 2 4 6 8 10 12e e r L ( z ) / r L , bu l k z (nm)e -0.25 r L ( z ) / r L , bu l k z (nm) FIG. 4: (Color online) Spatial gradient of the dynamic local-ization length (normalized to its bulk value) for vapor in-terface (main frame) and pinned rough solid (inset) thickfilms. Results are shown in terms of an absolute lengthscale relevant to polystyrene ( d ∼ . . .
55 are well described by the exponential fits r L ( z ) /r L,bulk = 1+1 . e − z/ . d and 1 − . e − z/ . d ,respectively. The dotted, dashed and dot-dashed curves aresimulation results of [31] based on using MSD data evaluatedat the time scales and temperatures indicated in the legend ofthe inset. The inset shows the same theoretical results plottedin a natural log-linear format. Our results for the vapor interface are also compared(with no fitting) to the recent free-standing film simula-tions of an atomistic polystyrene liquid model of Zhouand Milner [31]. Although there is some ambiguity asso-ciated with the extraction of a transient dynamic local-ization length via the intermediate time segmental meansquare displacement (MSD) in simulation [31], there isbroad consistency between the data and theory includ-ing the relative insensitivity to thermodynamic state, theexponential decay, and the amplitude of the change atthe surface. Note that the MSD normalized by its bulkvalue is nearly independent of temperature over the range400 K − K . This agrees well with our predicted weakdensity dependence of r L ( z ) /r L,bulk . The normalizedgradient of the simulation also seems to be insensitiveto the analyzed MSD time.Figure A1 shows our corresponding predictions usingthe inhomogeneous film NMCT of section IIIA. One sees1very good agreement with the dynamic free energy basedanalogs for the vapor interface film, but a significantlyshorter range gradient for the pinned solid surface sys-tem. Another important point is the comparsion tothe prior NLE-based theory of MS [47–49]. Figure A.1shows the localization length enhancement is of very sim-ilar magnitude near the surface, but decays much morequickly to the bulk value at a distance of ∼ . − . d ,as expected. B. Smooth Wall and Vibrating Rough ParticleInterfaces
Figure 5 shows representative calculations of the nor-malized localization length gradient for the smooth walland vibrating particle rough wall (for 3 values of sur-face particle localization lengths) models, and contraststhem with the vapor and pinned particle results of theprevious section. Interestingly, the smooth hard wallsystem exhibits enhancements of the localization length,and hence behaves more akin to a vapor interface thana rough pinned particle substrate for this property. Thevibrating particle rough wall systems evolve from sup-pression of the localization length for small vibrationalamplitude ( r Ls = 0 . d ), to weak enhancement for largevibrational amplitude ( r Ls = 0 . d ). We find that all thesystems studied show a good exponential decay profile(see the caption for fit functions), with a characteristiclength scale of ∼ − d . Vapor r = 0.05 r = 0.025 r = 0.01 Smooth Rough r L ( z ) / r L , bu l k z/d F B ( z ) / F B , bu l k z/d FIG. 5: (Color online) Normalized (to the bulk value) lo-calization length gradient at Φ = 0 .
57 for 4 types of in-dicated surfaces: vapor, and vibrated pinned and literallypinned solid treated in two ways to mimic a rough andsmooth hard surface. The vibrating pinned rough substrateresults are shown for the 3 indicated small values of vibra-tional amplitude (surface localization length). Inset: Theanalogous calculations for the normalized local cage bar-rier, F B ( z ) /F B,bulk for smooth and vibrating surfaces with r L,S /d = 0 . , . , .
01, which we find are well fit by theexponential forms 1 − . e − z/ . d , 1 + 1 . e − z/ . d ,1 + 1 . e − z/ . d , and 1 + 1 . e − z/ . d , respectively. C. Elastic Modulus
We employ Eq.(4) with a z -dependent localizationlength to calculate the glassy elastic modulus gradient.Results are shown in the inset of Fig. 6 in the format ofshear modulus at location z divided by its bulk analogfor two volume fractions. The value of the latter does notmatter in a practical sense in the normalized format. Vi-sually, the glassy modulus gradient extends 4-5 particlediameters into the film. The modulus softens at the va-por surface by a factor of ∼
3, while at the pinned roughsurface there is hardening by ∼ E f il m / E bu l k h (nm)rough solid G f il m ( z ) / G bu l k z/dvapor FIG. 6: (Color online) Main frame: Normalized to the bulkfilm-averaged elastic moduli of a free-standing vapor interfacefilm at volume fractions Φ = 0 .
55 (red points) and Φ = 0 . d = 1 . E film /E bulk = 1 / (cid:0) δ E /h (cid:1) with δ E = 4 . .
55 (red points) and Φ = 0 . The calculations have been done at two hard sphere2volume fractions, with effectively the same results found.The theoretical spatial gradients for hard spheres are nat-urally represented in terms of z/d , where d is the effectiveparticle diameter. Conversion of the x-axis to real unitsallows comparison with experimental data on polystyrenefilms [54]; we use the known value of Kuhn segment di-ameter, d ∼ . ∼ ∼
10 nm thickfilm, and bulk behavior is recovered only for films ap-proaching 100 nm thick.Various experimental and simulation data sets are alsoshown in Fig. 6. Although there is some disagree-ment among experimentalists [36, 53], all simulationsand the large majority of experimental studies find a va-por interface induces a softening of the elastic modulusnear the surface. The simulations of [72] and [73] em-ployed coarse-grained molecular dynamics (CGMD) tocompute the size-dependent Youngs modulus of the poly-mer diglycidyl ether Bisphenol with 3.3-diaminodiphenylsulfone (DGEBA/33DDS) and polymethylmethacrylate(PMMA) free-standing films, respectively. Since our gra-dients of normalized localization length agree with thesimulations [31] (see Fig. 4), and given Eq.(4), we ex-pect good agreement between theory and simulation forthe spatially-dependent Young’s modulus normalized byits bulk value, E ( z ) /E bulk . This expectation is verifiedin Fig. 6.The (rather noisy) experimental data shown is forpolystyrene thin films [74] of thicknesses that range fromfrom 7 nm to 220 nm. The films were deposited on wa-ter to avoid gravitational deformation. Measurements ofstress-strain response until the polymer film breaks in abrittle manner are used to extract Youngs modulus. Theaveraged experimental moduli ratio depend to some ex-tent on sample width. These real world complicationsintroduce some uncertainty in comparing to our theoret-ical calculations based on linear response and two vaporinterfaces to the experimental data. Nevertheless, thereis rough consistency between theory and experiment forthe magnitude of modulus changes and variation withfilm thickness. We note that the bulk value of the modu-lus is experimentally recovered at a smaller film thicknessthan in our calculations, but nearly quantitative agree-ment is found for thicknesses of 20 nm and smaller.Given we find that the theoretical localization lengthis well described by an exponential decay function, it isnot surprising that we find the modulus gradients of Fig.6 also follow an exponential form to a high degree ofaccuracy (not shown). However, as known from prior ex-perimental and simulation studies, other functional formscan also fit well the data. As an example of this point,the dotted curve in the main frame shows that our film-averaged normalized elastic modulus results can also bewell fit using a popular empirical function in the litera-ture, 1 / (1 + δ E /h ). V. JUMP DISTANCE AND LOCAL CAGEBARRIER: VAPOR, PINNED ROUGH ANDSMOOTH SOLID INTERFACES r( z ) / r bu l k z/d = 0.55 = 0.61 r( z ) / r bu l k z/drough solidvapor FIG. 7: (Color online) The particle jump distance normal-ized to its bulk value for vapor interface (main frame) andpinned particle rough surface (inset) thick films as a functionof distance from interface for Φ = 0 .
55 and 0 .
61. The solidcurves correspond to calculations using the new dynamic cageconstraint transfer idea of this article, and the dashed-dotcurves are based on the prior theory [47–49] which did notinclude this effect. We find that the ratio ∆ r ( z ) / ∆ r bulk forthe vapor interface films is well fit by 1 − . e − z/ . d and1 − . e − z/ . d for Φ = 0 .
55 and 0 .
61, respectively, andthe pinned particle rough surface film results are well fit by1 + 1 . e − z/ . d and 1 + 0 . e − z/ . d for Φ = 0 .
55 and0 .
61, respectively.
A crucial additional dynamical property needed toquantify the elastic barrier in ECNLE theory is the ef-fective jump distance of Eqs. (5) and (6). To predictthe alpha time gradient also requires knowledge of thelocal cage barrier gradient, F B ( z ). In this section, weuse the theory to study these two dynamical propertiesin films with vapor, pinned rough, and smooth hard wallsurfaces.The main frames of Fig. 7 and 8 show results for theabove two quantities at two volume fractions for the va-por and pinned solid interface models. Also shown forcomparison are the analogous results for the vapor in-terface based on the simpler MS [47–49] approach. Fora vapor interface, the jump distance (Fig. 7) and localbarrier (Fig. 8) are strongly reduced at the surface, andmore so at lower packing fraction. The gradients visiblydecay on a length scale of ∼ d . We find they are allwell fit by an exponential function (see figure captions)with decay lengths in the range of ∼ − d . The lat-ter depend relatively weakly on property and interface,and almost not at all on volume fraction, trends whichcan be understood from the general nature of the the-ory discussed in section IIIB. Although the direction ofchanges of these properties at the surface are the same3as in the prior approach [47–49], incorporation of longerrange mobility transfer physics leads to a much slowerspatial decay with a different functional form. = 0.55 = 0.610 2 4 6 8 100.00.20.40.60.81.01.2 0 2 4 6 8 100.81.21.62.02.42.8 F B / F B , bu l k z/d F B / F B , bu l k z/drough solidvapor FIG. 8: (Color online) The spatial gradient of the local cagebarrier height normalized by its bulk value for vapor interface(main frame) and pinned particle rough surface (inset) thickfilms as a function of distance from the interface at Φ = 0 . .
61. The solid curves correspond to the new theoryresults and the dashed-dot curves are those of the prior theory[47–49] that ignored the longer range cage constraint transfereffect. We find that the ratio F B ( z ) /F B,bulk of vapor interfacefilms are well fit by 1 − . e − z/ . d and 1 − . e − z/ . d for Φ = 0 .
55 and 0 .
61, respectively, and the pinned particlerough surface results are well fit by 1 + 1 . e − z/ . and1 + 1 . e − z/ . d for Φ = 0 .
55 and 0 .
61, respectively.
The insets of Fig. 7 and 8 show the analogous re-sults for the rough pinned solid surface. The qualitativetrends of the gradients, compared to each other and tothe prior more local approach [47–49], are the same asfound for the vapor surface, although the exponential de-cays lengths are non-trivially larger. On the other hand,the relative enhancement of the two properties at thesurface is a factor of ∼ − . r L ,∆ r , and F B at the surface to their corresponding bulkvalues vary monotonically over the range from ∼ . − ∼ . − .
53 and ∼ . − .
3, respectively; the corre- sponding values for the pinned surface are ∼ . − . ∼ . − .
65 and ∼ . − .
1. One sees that for all prop-erties there is a stronger volume fraction dependence forthe vapor interface system. r( z ) / r bu l k z/d F B ( z ) / F B , bu l k z/d FIG. 9: (Color online) The normalized (by its bulk value)jump distance calculated for pinned particle rough surface(dot-dashed curves) and smooth hard wall surface (solidcurves) thick films as a function of distance from interfacefor Φ = 0 .
55 and 0 .
61. We find that the ratio ∆ r ( z ) / ∆ r bulk for the two smooth hard wall films are, to leading order, bothwell fit by 1 + 0 . e − z/ . d for Φ = 0 .
55. The inset showsthe analogous calculations for the normalized local barriers invapor interface (solid curves) and smooth hard wall (dash-dotcurves) films.
We now consider smooth hard walls. The localizationlength calculations of Fig. 5 suggest this system behavesin a manner intermediate between a vapor surface and arough solid surface. Figure 9 shows calculations of thenormalized jump distance (main frame) and local bar-rier (inset), and contrasts the results with the vapor andtwo different solid interface analogs. The latter two sys-tems exhibit a large suppression and enhancement of thejump distance, respectively. The smooth surface showsonly a very weak enhancement of this quantity, but theform of the spatial decay is again exponential and of arange similar to that of the other two systems (see cap-tion). While the normalized local barrier in the insetof Fig. 9 qualitatively behaves as if the smooth surfacewas more like a vapor interface, its suppression is quan-titatively much weaker. Given the smooth surface showsan enhanced jump distance compared to the bulk (whichwould increase the collective elastic barrier in ECNLEtheory) but shows a smaller local cage barrier, how thealpha time mobility gradient will change is subtle andunclear. But since all changes for the smooth surfacerelative to the bulk are rather small, one expects the mo-bility modifications for this system will be modest. In theexperimental polymer film community, such a situationhas been inferred, for example, for polystyrene films sup-ported on substrates such as silicon and silica [1–3]. Thephrase ”neutral substrate” is typically invoked to indi-4cate a hard surface that has little effect on the dynamicsor T g of the film.The inset of Fig. 5 shows results for the local cagebarrier at a representative volume fraction of 0.57 for asmooth hard wall compared to the other systems; in thenormalized barrier format shown the results are nearly in-dependent of packing fraction although small variationsare typically found in the high packing fraction regime(0.55-0.61). Interestingly, the smooth wall system nowshows a suppression of the local barrier, albeit ratherweak. In conjunction with the smooth wall results inFig. 8, this again buttresses the view that the smoothwall model may be relevant to nearly atomically smoothhard surfaces (e.g., silica, silicon) where molecule-surfaceor polymer-surface adhesion is weak—a ”neutral hardsurface”. One also sees that allowing pinned particlesto vibrate modestly reduces the local cage barrier, butthe degree of change relative to the bulk is smaller thanfor the localization length. These trends seem physicallysensible given the barrier is determined by motion on alength scale far beyond a vibrational amplitude. But forall systems, the spatial range of the local barrier gradi-ents are essentially the same, and the same as the otherkey features of the dynamic free energy. Bulk behavior isrecovered in a practical sense at ∼ − = = 1.038 = = 1.038 r L ( z ) / r L , bu l k z/d F B ( z ) / F B , bu l k z/d FIG. 10: (Color online) Main frame: Normalized (to the bulkvalue) dynamic localization length gradient for pinned parti-cle rough surface films as a function of distance from interfaceat Φ = 0 .
55 and 0 .
61 with (dashed and dotted curves) andwithout (solid curves) densification in the first layer. Inset:The analogous calculations for the normalized local cage bar-rier. The first layer density enhancement factor is 1.038. Thebarriers in the bulk are 4.7 and 12.9 in thermal energy units.
Finally, Figure 10 presents one example of how surface-induced densification of the first liquid layer affects thedynamic localization length and local cage barrier. Thechosen value of density enhancement of 3.8 % is moti-vated by a recent computational study of liquid glycerolexposed to a silica surface [66]. Calculations are shownfor two values of volume fraction, 0.55 and 0.61, where the bulk local barrier in thermal energy units is 4.7 and12.9, respectively. The corresponding results if there isno first layer densification are shown for comparison.The main frame of Fig. 10 shows that such a modestdensification results in a major enhancement of particlelocalization near the hard surface. However, the changesrelative to the bulk are almost the same at the two dif-ferent volume fractions studied with and without densifi-cation. Moreover, the length scale for visually recoveringbulk behavior is almost the same for all calculations, ∼ k B T (10.3 k B T ) for the lower (higher) volume frac-tion system. Since the alpha relaxation time scales asthe exponential of the barrier, even only taking into ac-count this change of the local cage barrier would resultin an increase of the alpha time by a factor of ∼
45 or ∼ , VI. DISCUSSION
We have constructed a new particle-level microscopictheory for how dynamic caging constraints at a surface orinterface are modified and spatially transferred in a layer-by-layer manner into the film interior in the context ofthe dynamic free energy concept of the force-based NLEtheory. The basic idea is to reduce the resolution of thecage level description to acknowledge different dynami-cal constraints in the upper and lower halves of a cage.The effective dynamic free energy at any mean location(cage center) then involves contributions from two ad-jacent regions where confining forces are not the same.The z -dependence of the caging component of the dy-namic free energy varies essentially exponentially as afunction of distance from the interface, with a universaldecay length of modest size and weak sensitivity to ther-modynamic state. Such a variation imparts a roughlyexponential variation of all key features of the dynamicfree energy required to treat dynamical gradients of thelocalization length, jump distance, cage barrier, and al-pha time. As an important consequence we expect that,to leading order, a double exponential form of the alphatime spatial gradient is predicted.Diverse systems were considered where the surface wasa vapor, a rough pinned particle solid, a vibrating (soft-ened) pinned particle solid, a smooth hard wall, and asolid substrate which densifies the first layer of the liq-uid. The fundamental manner that they enter the the-ory at the level of the dynamic free energy is the same,with the crucial difference arising solely from the first5layer where the non-universal dynamical constraints canbe weaken, softened, or hardly changed depending on theinterface. However, both the amplitude of the modifica-tion and its quantitative spatial range of penetration intothe film varies with interface type, although the penetra-tion depth is weakly dependent on density or tempera-ture. Numerical calculations for the hard sphere fluidestablished the spatial dependence and volume fractionsensitivity of the changes of key dynamical properties for5 different models. No adjustable comparison of the the-oretical predictions for the dynamic localization lengthand glassy modulus against simulation and experimentfor systems with vapor surface(s) reveal good agreement.Future work will fully integrate the new advance re-ported in this article with all aspects of ECNLE the-ory for films with vapor and solid interface films includ-ing the collective elasticity contribution. This will allowus to make quantitative predictions for quantities suchas the alpha relaxation time gradient, dynamic decou-pling phenomena, T g gradient, and film-averaged prop-erties for both model systems and experimental materialswith diverse interfaces and chemical nature of the build-ing blocks (colloids, molecules, polymers) Key open ques-tions such as the near double exponential variation of thealpha time gradient, how the amplitude of the alpha timechanges at the surface, the length scale of the dynamicgradient, how the apparent decoupling exponent [57] pre-cisely varies with location in a film, and the role of solidsubstrate elasticity or Debye-Waller factor [29] will beaddressed in detail for vapor, pinned particle solid andother interfaces. The impact of the now longer range na-ture of surface-induced changes of dynamics emanatingfrom the interface compared to the prior formulation ofthe ECNLE theory of thin films [47–50] on how importantthe cutoff at the interface of the collective elastic compo-nent of the alpha process is will be re-visited. Finally, thebasic new idea of the present paper is generalizable to dif-ferent confined geometries (spherical droplets, cylindricalpores) and polymer or molecular bilayers. Appendix A: NMCT formulation of DynamicLocalization Lengths in Films
The normalized dynamic localization length gradientsare calculated using the NMCT formulation of Eqs.(3)and (4), and the results are compared to the layer-basedNLE dynamic free energy formulation. Figure A.1 showsrepresentative results, and ones sees the two theoriesmake very similar predictions for the vapor interface, but there are quantitative differences for the pinned particlesurface model. All results can be described by an expo-nential function. Both calculations show an insensitivityof the normalized gradient to the fluid packing fraction.Also shown are the predictions of the prior formulation ofMS [47–49] for a vapor interface which assumed surface-nucleated reduction of the caging constraints is confinedto a distance of only r cage ∼ . d from the interface. Ob-viously including the new physics developed in this workgreatly extends the spatial modification of r L ( z ) /r L,bulk relative to this prior formulation. rL(z)/rL,bulk z / d rL(z)/rL,bulk z / d
FIG. A.1: (Color online)Normalized gradient of the dynamiclocalization length, r L ( z ) /r L,bulk , for vapor interface (mainframe) and rough pinned particle surface (inset) thick filmscalculated using NLE theory with the dynamic free energyconcept without (dashed-dot curves) and with (solid orangeand green curves). the new cage constraint transfer effect.The analogous latter results based on the ideal NMCT for-mulation of Eq. (11) are shown as the solid red and bluecurves.
Acknowledgments
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