# Effect of the Nature of the Solid Substrate on Spatially Heterogeneous
Activated Dynamics in Glass Forming Supported Films

Anh D. Phan<EMAIL_ADDRESS>Faculty of Materials Science and
Engineering, Phenikaa University, Hanoi 12116, Vietnam  Phenikaa Institute for
Advanced Study, Phenikaa University, Hanoi 12116, Vietnam  Kenneth S.
Schweizer<EMAIL_ADDRESS>Departments of Materials Science, University
of Illinois, Urbana, IL 61801, USA  Departments of Chemistry, University of
Illinois, Urbana, IL 61801, USA  Departments of Chemical $\&$ Biomolecular
Engineering, University of Illinois, Urbana, IL 61801, USA  Materials Research
Laboratory, University of Illinois, Urbana, IL 61801, USA

###### Abstract

We extend the force-level ECNLE theory to treat the spatial gradients of the
alpha relaxation time and glass transition temperature, and the corresponding
film-averaged quantities, to the geometrically asymmetric case of finite
thickness supported films with variable fluid - substrate coupling. The latter
typically nonuniversally slows down motion near the solid-liquid interface as
modeled via modification of the surface dynamic free energy caging constraints
which are spatially transferred into the film, and which compete with the
accelerated relaxation gradient induced by the vapor interface. Quantitative
applications to the foundational hard sphere fluid and a polymer melt are
presented. The strength of the effective fluid-substrate coupling has very
large consequences on the dynamical gradients and film-averaged quantities in
a film thickness and thermodynamic state dependent manner. The interference of
the dynamical gradients of opposite nature emanating from the vapor and solid
interfaces is determined, including the conditions for the disappearance of a
bulk-like region in the film center. The relative importance of surface-
induced modification of local caging versus the generic truncation of the long
range collective elastic component of the activation barrier is studied. The
conditions for the accuracy and failure of a simple superposition
approximation for dynamical gradients in thin films is also determined. The
emergence of near substrate dead layers, large gradient effects on film-
averaged response functions, and a weak non-monotonic evolution of dynamic
gradients in thick and cold films, are briefly discussed. The connection of
our theoretical results to simulations and experiments is briefly discussed,
as is extension to treat more complex glass-forming systems under
nanoconfinement.

##  I Introduction

Understanding the spatially heterogeneous activated dynamics and kinetic
vitrification of diverse glass-forming liquids near interfaces and under thin
film confinement remains a frontier scientific challenge 1 ; 2 ; 3 ; 4 ; 5 ; 6
; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 ;
23 ; 24 . Vapor and highly variable solid interfaces can induce spatial
gradients of structural relaxation that can slow down or speed up relaxation,
resulting in major shifts of the local and film-averaged glass transition
temperature out to large length scales well beyond those that characterize
changes of thermodynamics or structure near the interface 1 ; 2 ; 3 ; 4 ; 5 ;
6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22
; 23 ; 24 . For the two-interface thin film confinement case, the surfaces can
be symmetric and composed of either solids of highly variable chemistry,
stiffness and topography corresponding to capped films, or two vapor
interfaces corresponding to free standing films. Asymmetric films with one
vapor and one solid surface are even more complex, with competing spatial
gradients of dynamics of opposite nature emanating from the different
interfaces. Many theoretical models have been proposed for various aspects of
these systems (for reviews see refs. 1 ; 10 ; 17 ), which typically are built
on very different physical ideas for the origin of glassy dynamics in the
bulk. Simulations have been extensively applied to study film dynamics, with
many important findings 1 ; 2 ; 8 ; 10 ; 12 ; 13 ; 15 ; 16 ; 17 ; 24 .
However, computational limitations restrict simulations from approaching the
ultra-long time scales probed experimentally.

Recently, a microscopic, particle level, homogeneous bulk dynamical theory has
been created and widely applied to atomic, colloidal, molecular, and polymeric
glass forming fluids – the Elastically Collective Nonlinear Langevin Equation
(ECNLE) theory 25 ; 26 ; 27 ; 28 ; 29 . It is built on the view that the alpha
relaxation process is of a coupled local-nonlocal nature, where large
amplitude activated hopping on the cage scale is strongly coupled with longer
range collective elastic distortions of all particles outside the cage. Based
on an a priori coarse-grained mapping or complexity reduction strategy that
retains key aspects of chemical specificity 25 ; 26 ; 27 ; 28 ; 29 , the
theory has been successfully applied to understand the temperature dependence
of the alpha relaxation time of diverse families of equilibrium liquids over a
range up to 14 orders of magnitude 26 ; 27 ; 28 . Both the high temperature
Arrhenius regime and the lower temperature strongly non-Arrhenius regime have
been addressed in a unified and predictive manner. The central concept is the
idea of a dynamic free energy that quantifies cage scale localization and the
local activation barrier, and also the jump distance and emergent dynamic
elastic modulus in sufficiently cold liquids that underlies the collective
elastic component of the activation event.

The ECNLE theory has been extended to glass forming liquids near surfaces in
thick films of both a soft (vapor) 1 ; 2 ; 3 ; 5 ; 6 ; 7 ; 30 ; 31 ; 32  and
hard (microscopically corrugated) nature 3 ; 4 ; 32 . The consequences of
confinement in free standing films with two vapor interfaces has been
extensively worked out. Novel predictions include 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ;
30 ; 31 ; 32  (i) factorization of the film location and temperature (or
density) dependences of the dynamic activation barrier, (ii) strong spatial
gradients of the alpha time and vitrification temperature of a double and
single exponential form, respectively, (iii) a crossover to an inverse in
distance from the interface power law decay of these quantities due to cutoff
of the elastic field at the surface, (iv) power law decoupling of the alpha
time in films from its bulk analog with apparent exponents that depend on
location in the film, and (v) nonadditive gradient interference effects that
emerge in thin enough films. Essentially all these predictions have been
verified by computer simulations 1 ; 2 ; 7  for free standing thick and thin
films. Experimental evidence has also been obtained for the double exponential
form of the dynamical gradient close to the vapor interface of a thick film 33
; 34 , and the results are also relevant to polymer nanocomposites 35 .
Predictions have been made for both microscopically rough surfaces and smooth
hard walls 4 .

The goal of this article is to build on these recent advances to address new
aspects motivated by both their fundamental scientific interest and
experimental relevance. The first new theme is to extend the theory to
asymmetric films composed of one vapor interface and one solid surface. A
second theme is to propose and study an effective model for the fluid-solid
interface that tunes the surface dynamic free energy to mimic the highly
variable and nonuniversal manner that a substrate can change the alpha
relaxation process at a surface. Both aspects are studied for the foundational
hard sphere fluid and a polymer melt over a wide range of film thicknesses,
and most importantly for how the magnitude of the nonuniversal substrate-fluid
coupling affects the dynamical gradients. How the opposite sign gradients
associated with a solid and vapor surface interfere in thin films is also
studied, including nonadditive collective effects beyond a linear
superposition ansatz. The new model and numerical studies are broadly
motivated by the experimental ability to widely tune the fluid-solid interface
via topography (corrugation, smooth walls), fluid-surface potentials, and/or
mechanical stiffness. Realization of the latter includes substrates composed
of soft crosslinked elastomers [36], grafted polymer chains 10 ; 17 ; 37 , and
a viscoelastic liquid [38]. We suggest the new results are testable in
carefully designed simulation studies which construct substrate models that
specify the alpha time at the solid interface, as recently done by Simmons et.
al 7 ; 24  to achieve an effectively “neutral” solid interface. Our analysis
of this generalized substrate model employs the existing ECNLE theory ideas in
the bulk, near interfaces, and under thin film confinement.

The article is organized as follows. In section II we first briefly review the
well documented bulk and film ECNLE theories, and then present its
generalization to variable substrates modeled in an implicit manner. A few
model calculations are presented, along with the mapping from the reference
hard sphere fluid to thermal liquids. Our core new results for the variable
substrate thick and thin supported films are presented in sections III and IV,
respectively. The article concludes in section V with a summary, discussion,
and future outlook. All technical and conceptual details of ECNLE theory in
the bulk, near surfaces, and in thin films have been documented in great
detail in prior publications 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 30 ; 31 ; 32 , and
for economy of expression are not repeated.

##  II Theory: Background and Extension to Supported Films

###  II.1 Model and dynamic free energy

We first consider the foundational hard sphere (diameter, $d$) fluid
characterized by a number density, $\rho$, and corresponding volume or packing
fraction, $\Phi=\pi d^{3}/6$, which are taken to be the same in the bulk and
under film conditions. The basic theoretical quantity is the dynamic free
energy as a function of the displacement of a tagged particle from its initial
position, $r$, which is given by the sum of ideal and caging contributions
which in the isotropic bulk fluid is 25 ; 26 ; 27 ; 28 ; 29

$\displaystyle F_{dyn}^{bulk}(r)$ $\displaystyle=$ $\displaystyle
F_{ideal}(r)+F_{caging}^{bulk}(r),$ (1) $\displaystyle F_{ideal}(r)$
$\displaystyle=$ $\displaystyle-3k_{B}T\ln\left(\frac{r}{d}\right),$ (2)
$\displaystyle\frac{F_{caging}^{bulk}(r)}{k_{B}T}$ $\displaystyle=$
$\displaystyle\rho\int\frac{d\mathbf{q}}{(2\pi)^{3}}\frac{S(q)C^{2}(q)}{1+S^{-1}(q)}$
(3) $\displaystyle\times$
$\displaystyle\exp\left[-\frac{q^{2}r^{2}}{6}\left(1+S^{-1}(q)\right)\right].$

Here, $k_{B}$ is Boltzmann’s constant, $T$ is temperature, $q$ is the
wavevector, $C(q)=\rho^{-1}\left[1-1/S(q)\right]$ is the direct correlation
function in Fourier space, and $S(q)$ is the corresponding static structure
factor computed using the Ornstein-Zernike (OZ) integral equation approach
with the Percus-Yevick (PY) closure 39 . Within the nonlinear Langevin
equation (NLE) stochastic evolution equation for a tagged particle trajectory,
the quantity $-dF_{dyn}(r)/dr$ is the systematic, displacement-dependent,
effective force on a moving tagged particle due to the surrounding particles
25 ; 26 ; 27 ; 28 ; 29 .

Our model for supported films (thickness, $H$) with one vapor and one variable
solid substrate is schematically illustrated in Fig. 1. The dynamic free
energy and particle trajectories in films become a function of location in the
direction ($z$) orthogonal to the interfaces and is treated as homogeneous in
the two transverse directions. The minimalist theory 1 ; 2 ; 3 ; 5 ; 6 ; 7 ;
30 ; 31 ; 32  employed here does not consider any equilibrium variations in
spatial density or alterations of the pair structure induced by an interface
corresponding to the so-called ”neutral confinement” 40 ; 41 ; 42 ; 43 ; 44 ;
45 ; 46  scenario. Hence, all surface and confinement induced changes of
dynamics arise solely from kinetic considerations, not changes of structure or
thermodynamics. For a single vapor or solid surface thick film
($H\rightarrow\infty$), the $z$-dependent dynamic free energy has been
previously formulated in a manner that can be conceptually (not literally)
described in terms of layers labeled by a discrete index $i$ with $z=(i-1)d$.
Modification of the bulk dynamic free energy is ”nucleated” at the interface
and transferred in a layer-by-layer or ”bootstrapped” manner into the film
interior as 1 ; 2 ; 3 ; 5 ; 7

$\displaystyle F_{dyn}^{(i)}(r)$ $\displaystyle=$ $\displaystyle
F_{ideal}(r)+\left(1-\frac{1}{2^{i}}\right)F_{caging}^{bulk}(r)+\frac{F_{caging}^{surface}(r)}{2^{i}}$
(4) $\displaystyle=$ $\displaystyle
F_{ideal}(r)+\frac{1}{2^{i}}\left(F_{caging}^{surface}(r)-F_{caging}^{bulk}(r)\right),$

where $F_{caging}^{surface}(r)$ is the first layer (surface) caging part of
the dynamic free energy which depends strongly on the nature of the interface.
Using $2^{-i}=0.5\exp\left(-\frac{z}{d}\ln{2}\right)$ reveals the underlying
theoretical physical idea that caging constraints vary exponentially as a
function of distance from the interface. Since the film exhibits broken
symmetry, the dynamics is in principle spatially anisotropic. A full treatment
of this aspect is extremely difficult. Moreover, conceptually, the dynamic
local equilibrium idea underlying the construction of the dynamic free energy
1 ; 2 ; 3 ; 5 ; 7  involves a minimum averaging length scale of order the cage
diameter ($\sim 2-3d$). Hence, in films the discrete index $i$ or distance $z$
defines the center of a cage of radius $r_{cage}$ (see Fig. 1) within which
the tagged particle displacement is taken to be isotropic, corresponding to an
angular averaging of motion on the cage scale.

Figure 1: (Color online) Schematic representation of the finite thickness
supported film model and the transfer of caging constraints nucleated at the
surface. For the one-surface thick film, $H\rightarrow\infty$. The cartoon of
the solid interface is relevant for the “maximally rough” model of the
substrate corresponding to $\kappa=1$ where the substrate is a pinned
amorphous solid configuration of the hard sphere fluid 4 . The various
particle colors schematically depict the variation of mobility across the
polymer film.

###  II.2 Influence of a rough solid substrate on dynamical caging

The model reported in Ref. 3 ; 4  to describe vapor and solid surfaces is
schematically shown in Fig.1 where the “maximally rough” substrate consists of
pinned particles identical to those of the mobile liquid. Thus, half of the
cage experienced by a tagged particle at the interface ($z=0$) contains pinned
particles, which are modeled as randomly distributed within a full spherical
cage that includes the other half of mobile particles. This is a tractability-
motivated simplification that assumes the critical effect of the solid
interface is the fraction of neighboring particles of a tagged particle within
a cage are immobilized, rather than their precise spatial arrangement. To
determine the corresponding dynamic free energy, we adopt the idea of ”neutral
confinement” where the pinning process does not affect the liquid pair
correlation function nor induces density layering or any other change of the
liquid thermodynamics. This model means that all effects of interfaces and
confinement are purely dynamical, not tied to any interface-induced changes of
equilibrium structure or thermodynamics.

The solid substrate model has been generalized to an arbitrary fraction
$\alpha$ of pinned particles in the surface cage. The corresponding caging
part of the dynamic free energy is 3 ; 4 :

$\displaystyle\frac{F_{caging}^{pinned}(\alpha,r)}{k_{B}T}$ $\displaystyle=$
$\displaystyle-\int\frac{d\mathbf{q}}{(2\pi)^{3}}\left[\frac{C(q)S_{12}(q)e^{-q^{2}r^{2}/6}}{\rho(1-\alpha)\left[1-\rho(1-\alpha)C(q)\right]}+\frac{\rho(1-\alpha)C(q)^{2}e^{-q^{2}r^{2}\left[2-\rho(1-\alpha)C(q)\right]/6}}{\left[1-\rho(1-\alpha)C(q)\right]\left[2-\rho(1-\alpha)C(q)\right]}\right]_{\alpha=0.5}$
(5) $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{F_{caging}^{bulk}(r)}{k_{B}T}+\frac{1}{2}\frac{F_{caging}^{surface}(r)}{k_{B}T},$

where $S_{12}(q)$ is collective static structure factor for density
fluctuations of the immobilized and mobile particles 46 :

$\displaystyle S_{12}(q)=\frac{\rho^{2}(1-\alpha)\alpha
C(q)}{[1-\rho(1-\alpha)C(q)][1-\rho\alpha C(q)]-\rho^{2}(1-\alpha)\alpha
C^{2}(q)},$ (6)

Equation (5) provides a definition of the dynamic free energy associated with
the solid surface. When $\alpha=0.5$, which is our present (and prior 4 )
focus, one has
$F_{caging}^{surface}=2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)$, and Eq.
(4) becomes

$\displaystyle
F_{dyn}^{(i)}(r)=F_{dyn}^{bulk}(r)+\frac{F_{caging}^{pined}(r)-F_{caging}^{bulk}(r)}{2^{i-1}}$
(7)

or in terms of the continuous variable distance from an interface as:

$\displaystyle
F_{dyn}^{(i)}(r)=F_{dyn}^{bulk}(r)+\frac{F_{caging}^{pined}(r)-F_{caging}^{bulk}(r)}{2^{z/d}}$
(8)

###  II.3 Model for variable substrate-fluid coupling and caging constraints

Within the framework of ECNLE theory, we previously established [4] that a
maximally rough solid substrate has a very large impact on the alpha
relaxation time gradient in a thick film. A second limiting case examined was
to consider a smooth hard wall where all lateral forces between the surface
and fluid are absent. This limit could mimic a solid surface with a very small
degree of corrugation on the length scale of fluid molecules or polymer
segments, or perhaps a high surface tension liquid substrate that appears
”dynamically smooth” on the alpha time scale of the confined fluid. A large
reduction of dynamical constraints is expected since the surface does not
exert any forces on the fluid in 2 of the 3 spatial directions. These
considerations motivated our modeling of this limit by reducing the surface
caging dynamic free energy by a factor of 3, i.e.,
$F_{caging}^{surface}=\left[2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)\right]/3$.
The theoretical predictions for this smooth hard wall model are qualitatively
consistent with simulations 8 ; 15 ; 16 ; 40 ; 47  which compared the dynamics
of model supercooled liquids confined by a single smooth hard wall to its
maximally rough substrate analog.

The focus of the present work is to explore the predictions of ECNLE theory
for films with a continuously tunable dynamical constraint at a hard surface,
with the maximally rough case studied previously 4  as the limiting reference
system. We do this by introducing a single parameter $\kappa$
($0\leq\kappa\leq 1$) to rescale the surface caging dynamic free energy to
mimic different degrees of topographic surface roughness or corrugation, or
more crudely a change of substrate rigidity (e.g., an elastomer substrate or a
surface with soft grafted polymers), or another modification of substrate-
fluid interactions. This is an “effective”, not explicit, model of the many
possible complexities at a fluid-solid interface associated with repulsive
interactions. Fluid-substrate attractive interactions and adsorption are not
explicitly included. However, from a highly qualitative perspective, since a
major consequence of such attractions is to slow down the alpha relaxation
time in the immediate vicinity of the surface, we speculate that they might be
crudely mimicked by varying $\kappa$.

Mathematically, the surface caging dynamic free energy then becomes

$\displaystyle
F_{caging}^{surface}=\kappa\left[2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)\right],$
(9)

where $\kappa=1$ and 1/3 corresponds to the previously studied thick film case
composed of a maximally rough and smooth solid surface, respectively 4 . When
$0\leq\kappa\leq 1/3$, the caging force is even weaker than a smooth hard
wall, which might mimic a lower density substrate, or systems with weaker than
hard core substrate-fluid repulsive forces which could arise from microscopic
mechanisms such as a rubbery surface 17 ; 36 ; 48 ; 49 , or an interface
coated with flexible polymer grafts 10 ; 17 ; 37 , or a viscoelastic fluid
interface 38 . When $\kappa=0$, there is no surface caging force corresponding
to $F_{caging}^{surface}=0$, which models the previously studied vapor
interface where half of the nearest neighbors of a tagged particle at the
surface are missing 2 ; 3 ; 5 ; 6 ; 7 . From Eqs. (4) and (9), a general
expression of the gradient dynamic free energy is then

$\displaystyle\frac{F_{dyn}(r,z)}{k_{B}T}$ $\displaystyle=$
$\displaystyle\frac{F_{dyn}^{bulk}(r)}{k_{B}T}+\frac{\kappa}{2^{z/d}}\frac{F_{caging}^{pinned}(r)}{k_{B}T}-\frac{1+\kappa}{2\times
2^{z/d}}\frac{F_{caging}^{bulk}(r)}{k_{B}T}$ (10) $\displaystyle=$
$\displaystyle-3\ln\left(\frac{r}{d}\right)+\left(1-\frac{1+\kappa}{2}\frac{1}{2^{z/d}}\right)\frac{F_{caging}^{bulk}(r)}{k_{B}T}+\frac{\kappa}{2^{z/d}}\frac{F_{caging}^{pinned}(r)}{k_{B}T}.$

Figure 2: (Color online) The dynamic free energy (in units of the thermal
energy) at the substrate as a function of normalized particle displacement for
a hard sphere fluid with a packing fraction $\Phi=0.57$ for different values
of $\kappa$. Characteristic length and energy scales are indicated.

The dynamic free energy $F_{dyn}(r,z)$ is the crucial quantity. For bulk hard
spheres, when $\Phi\geq 0.43$ a cage barrier is present which signals
transient localization and a crossover to activated relaxation. Figure 2 shows
an example of how the dynamic free energy varies with $\kappa$. Also indicated
in Fig.2 are the 3 key length scales (localization length, $r_{L}$, barrier
position, $r_{B}$, jump distance, $\Delta r=r_{B}-r_{L}$), and the 3 key
energy scales (harmonic curvatures $K_{0}=\frac{3k_{B}T}{r_{L}^{2}}$ and
$K_{B}=\left|\frac{\partial^{2}F_{dyn}(r,z)}{\partial r^{2}}\right|_{r=r_{B}}$
at $r_{L}$ and $r_{B}$, respectively, and cage barrier,
$F_{B}(z)=F_{dyn}(r_{B},z)-F_{dyn}(r_{L},z)$. As $\kappa$ varies from 0 to 1,
the interfacial caging force increases leading to a decrease of the
localization length, increase of the jump distance, and increase of the local
cage barrier.

###  II.4 Collective elasticity, alpha relaxation time gradient, and mapping
to thermal liquids

The elementary alpha relaxation event in ECNLE theory is of a local-nonlocal
nature where particle hopping over the cage barrier is causally coupled with a
collective elastic distortion of all particles outside the cage. The physical
idea is that local rearrangement requires a small cage expansion which induces
a power law, scale invariant, elastic displacement field. The total barrier
consists of correlated local cage and longer range collective elastic
contributions. The presence of a vapor or solid interface modifies not only
the dynamic free energy, but also the elastic displacement field. Here, we
utilize our latest approach to this complex problem developed in Refs. 2 ; 4 ;
5 ; 6 ; 7 . Diverse novel predictions based on this approach have been
successfully compared with simulations of free standing films, and a supported
film with a solid neutral surface 2 ; 4 ; 5 ; 6 ; 7 . The adopted displacement
field is taken to have the same functional form as in the bulk. It is derived
using a continuum elasticity analysis modified to enforce boundary conditions
whereby the amplitude vanishes at the surface (a no displacement or “zero
strain” boundary condition at the interface). The result is 2 ; 4 ; 5 ; 6 ; 7

$\displaystyle u(r,\theta,z)=A_{s}(\theta,z)r+\frac{B_{s}(\theta,z)}{r^{2}}$
(11)

where $A_{s}(\theta,z)$ and $B_{s}(\theta,z)$ are chosen to enforce the
boundary condition $u(r,\theta,z=0)=0$, and $\theta$ is defined in Figure 1.
At the cage surface where the displacement field is nucleated,

$\displaystyle u(r=r_{cage},\theta,z)$ $\displaystyle=$
$\displaystyle\Delta{r_{eff}}$ (12) $\displaystyle=$
$\displaystyle\frac{3}{r_{cage}^{3}}\left(\frac{\Delta
r^{2}r_{cage}^{2}}{32}-\frac{\Delta r^{3}r_{cage}}{192}+\frac{\Delta
r^{4}}{3072}\right)$ $\displaystyle\approx$
$\displaystyle\frac{3}{32}\frac{\Delta r^{2}}{r_{cage}}.$

where $r_{cage}$ is the cage radius equal to the location of the first minimum
of $g(r)$. The cage expansion amplitude is dependent on position in the film
as previously derived 2 ; 4 ; 5 ; 6 ; 7 :

$\displaystyle A_{s}(\theta,z)$ $\displaystyle=$
$\displaystyle-\frac{\Delta{r_{eff}}r_{cage}^{2}\cos^{3}\theta}{z^{3}-r_{cage}^{3}\cos^{3}\theta},$
$\displaystyle B_{s}(\theta,z)$ $\displaystyle=$
$\displaystyle\frac{\Delta{r_{eff}}(z)r_{cage}^{2}z^{3}}{z^{3}-r_{cage}^{3}\cos^{3}\theta},$
(13)

The corresponding elastic barrier follows by integrating over all film
particles outside the cage (the center of which is at a distance $z$ from the
surface) 2 ; 4 ; 5 ; 6 ; 7 :

$\displaystyle F_{e}=\frac{3\Phi}{\pi
d^{3}}\int_{V_{film}}d\textbf{r}u^{2}(r,\theta,z)K_{0}(r,z).$ (14)

Before proceeding, we briefly review the physical ideas 1 ; 2 ; 3 ; 4 ; 5 ; 6
; 7 ; 30 ; 31 ; 32  that underlie our approximate modeling of the elastic
displacement field and calculation of the elastic barrier for vapor and solid
interfaces. First, we emphasize that using continuum mechanics to construct
the functional form of the elastic displacement field in our molecular-based
theory is a simplification. A continuum perspective is not used to quantify
the amplitude of the displacement field, nor compute the magnitude of the
elastic barrier. Rather, the latter are done using the microscopic dynamic
free energy idea, and the Einstein glass picture of localized particles
outside the cage 29 . Such a ”mixed” microscopic-continuum analysis of the
collective elastic barrier contrasts with phenomenological elastic models in
the literature, e.g., the ”shoving model” 25 ; 26 ; 27 ; 28 ; 29 . Concerning
boundary conditions, we have adopted the vanishing of the elastic displacement
field at the interface (akin to a “zero strain” condition in mechanics) for
both vapor and solid interfaces. This is a natural and simple choice for a
rigid solid surface, or perhaps if there are strong surface-liquid attractions
at the solid interface although we are not explicitly considering this case.

An alternative possible choice for a vapor interface is a “zero stress”
boundary condition. We have not done this, with our motivation as follows. (i)
Since the displacement fields are of low amplitude, a linear elastic scenario
applies where stress and strain are proportional, so naively the adopted zero-
displacement condition might be crudely viewed as in the same spirit as it
zero-stress analog. (ii) If surface tension plays a dominant dynamical role at
a vapor interface, the zero-displacement boundary condition may be suitable
since it minimizes surface area under elastic deformation. (iii) Given the
“mixed” continuum-microscopic nature of our analysis of the elastic barrier, a
priori whether a zero-strain or zero-stress boundary condition is more
appropriate is unclear to us. Of course, none of the above arguments are
rigorous. But we emphasize that use of the zero-displacement boundary
condition for free standing films has led to many striking predictions by
ECNLE theory for the role of collective elasticity in thick and thin films,
which have major support from simulations (Refs. 1 ; 2 ; 3 ; 6 ; 7 ). At the
most fundamental level, one can view the problem as open and requiring more
work, and whether for vapor surfaces the zero-stress boundary condition may be
more appropriate could depend on the questions asked within the ECNLE theory
framework.

Now, by combining all the above results, the alpha or structural relaxation
time gradient is determined using the Kramer’s mean first passage time for
barrier crossing expression 2 ; 4 ; 5 ; 6 ; 7

$\displaystyle\frac{\tau_{\alpha}(z)}{\tau_{s}}=1+\frac{2\pi}{\sqrt{K_{0}(z)K_{B}(z)}}\frac{k_{B}T}{d^{2}}\exp\left(\frac{F_{B}(z)+F_{e}(z)}{k_{B}T}\right),$
(15)

where $\tau_{s}$ is a ”short time/length scale” non-activated dissipative
process taken to be unaffected from interfaces and confinement, and is given
by 2 ; 4 ; 5 ; 6 ; 7 ; 26 :

$\displaystyle\tau_{s}$ $\displaystyle=$
$\displaystyle\tau_{0}g(d)\left[1+\frac{1}{36\Phi}\int_{0}^{\infty}dq\frac{q^{2}\left[S(q)-1\right]^{2}}{S(q)+b(q)}\right],$
$\displaystyle\tau_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{24\rho
d^{2}}\sqrt{\frac{M}{\pi k_{B}T}},$ (16)

where $b^{-1}(q)=1-j_{0}(q)+2j_{2}(q)$, $j_{n}(x)$ is the spherical Bessel
function of order $n$, $M$ is the particle mass, and $\tau_{0}$ is a ”bare”
Boltzmann-like time scale.

To quantitatively apply ECNLE theory for hard sphere fluids to experiments and
simulations on thermal liquids we employ the well-developed coarse-grained
mapping or complex reduction strategy to a thermalized effective hard sphere
model. The physical idea is to equate the dimensionless compressibility
(rigorously equal to the non-dimensionalized amplitude of long wavelength
thermal density fluctuations) predicted by integral equation theory for hard-
sphere fluids, $S_{0}^{HS}$, and the corresponding experimental quantity,
$S_{0}(T)=\rho k_{B}T\kappa_{T}$, deduced from equation of state data for
liquids in equilibrium. A key physical idea for molecules and polymers is that
it is the amplitude of density fluctuations at the level of the number of
elementary rigidly moving interaction sites, $N_{s}$, that enters the mapping.
The resulting mapping relation based on the OZ-PY approach is 2 ; 4 ; 5 ; 6 ;
7 ; 26 ; 27 ; 28 :

$\displaystyle
S_{0}^{HS}=\frac{\left(1-\Phi_{eff}\right)^{4}}{\left(1+2\Phi_{eff}\right)^{2}}\equiv
S_{0,expt}\approx\frac{1}{N_{s}}\left(-A+\frac{B}{T}\right)^{-2}$ (17)

Eq. (17) determines $\Phi_{eff}$ as a temperature and chemistry-specific
effective hard sphere fluid packing fraction. The second approximate equality
has been shown to be highly accurate 26 ; 27 ; 28 , with $A$ and $B$
corresponding to an interaction site level defined entropic packing and
cohesive energy quantities, respectively. As mentioned above, $N_{s}$ is the
number of interaction sites for a rigid molecule or the corresponding number
in a Kuhn segment of a polymer chain. Using this mapping, the mean alpha time
can be predicted with no adjustable parameters. In this article, we present
results both for the literal hard sphere model, and for the widely studied
thermal polystyrene (PS) melt for which prior work 2 ; 4 ; 5 ; 6 ; 7 ; 28  has
shown $d=1.16$ nm (Kuhn segment space filling diameter), A = 0.618, $B=1297$
$K$, and $N_{s}=38.4$.

###  II.5 Confined thin films

We model a finite thickness supported film with a vapor surface at $z=H$ as
illustrated in Fig. 1. The same basic analysis for a single interface thick
film can be employed to describe a vapor layer. We have proposed that the
combined effects of the two interfaces on the dynamic free energy can be
superimposed at the fundamental level of the dynamic free energy 2 .
Importantly, this does not mean that physical quantities predicted by the
theory (e.g., alpha time, $T_{g}$ gradients) obey naive superposition. Given
this idea, Eq. (10) for the dynamic free energy of the tagged particle in the
$i^{th}$ layer can then be written as

$\displaystyle\frac{F_{dyn}(r,z,H)}{k_{B}T}=-3\ln\left(\frac{r}{d}\right)+\left(1-\frac{1+\kappa}{2}\frac{1}{2^{z/d}}\right)\frac{F_{caging}^{bulk}(r)}{k_{B}T}+\frac{\kappa}{2^{z/d}}\frac{F_{caging}^{pinned}(r)}{k_{B}T}-\frac{1}{2\times
2^{(H-z)/d}}\frac{F_{caging}^{bulk}(r)}{k_{B}T}.$ (18)

As before, the local and elastic barriers can be calculated as a function of
position in the film, the overall film thickness, and temperature to predict
the gradient of the alpha relaxation times and glass transition temperatures
of finite-size supported films.

##  III Thick Supported Films

###  III.1 Analytic analysis

Before presenting our numerical results for thick supported films, we first
employ the ”ultra-local” analytic analysis method 50  developed for bulk hard
sphere fluids to construct an initial qualitative understanding of the effect
of $\kappa$ on the most elementary feature in films: the dynamic localization
length. This analysis also provides insights concerning the local cage barrier
46 ; 50 .

The ultra-local analytic analysis is based on the approximate dominance of
high wavevector contributions ($q>q_{c}$) to the dynamic force vertex in Eq.
(10) for systems with a sufficiently high cage barrier. Specifically, in the
high wavevector regime, $S(q)\approx 1$ and based on the PY closure for hard
sphere fluids 39 , $C(q)\approx-4\pi d^{3}\cos(qd)(qd)^{-2}$. Using these
simplifications in the condition that defines the minimum of the dynamic free
energy and hence localization length, $\left.\frac{\partial
F_{dyn}(r,z)}{\partial r}\right|_{r=r_{L}}=0$, one can easily derive an
expression for the inverse dynamic localization length:

$\displaystyle\frac{3d}{r_{L}}=\frac{8\Phi
g(d)^{2}r_{L}}{\pi}\left[\int_{q_{c}}^{\infty}dqe^{-q^{2}r_{L}^{2}/3}\left(1-\frac{1+\kappa}{2}\frac{1}{2^{z/d}}\right)+\int_{q_{c}}^{\infty}dqe^{-q^{2}r_{L}^{2}/6}\left(\frac{1}{2}+\frac{1}{2}e^{-q^{2}r_{L}^{2}/6}\right)\frac{\kappa}{2^{z/d}}\right].$
(19)

Since localization lengths must be small compared to the inverse crossover
wavevector for the applicability of the ultra-local analysis 46 ; 50 , one has
$q_{c}r_{L}\ll 1$, and hence the lower limit of the integral of Eq. (19) can
be approximated as zero. Equation (19) then simplifies to:

$\displaystyle\frac{3d}{r_{L}}=\frac{4\sqrt{3}\Phi
g(d)^{2}}{\sqrt{\pi}}\left[\left(1-\frac{1+\kappa}{2}\frac{1}{2^{z/d}}\right)+\left(\frac{\sqrt{2}}{2}+\frac{1}{2}\right)\frac{\kappa}{2^{z/d}}\right].$
(20)

From this, the dynamic localization length follows as

$\displaystyle
r_{L}(z)=\frac{r_{L,bulk}}{1-\cfrac{1-\kappa}{2}\cfrac{1}{2^{z/d}}+\cfrac{\sqrt{2}-1}{2}\cfrac{\kappa}{2^{z/d}}}.$
(21)

Equation (21) predicts that $1/r_{L}(z)$ scales linearly with $\kappa$, i.e.,
stronger substrate caging constraints lead to smaller dynamic localization
lengths, a physically expected trend. For the $\kappa=0$ vapor surface one
has, $r_{L}(z)=\cfrac{r_{L,bulk}}{1-\cfrac{1}{2}\cfrac{1}{2^{z/d}}}$ and
$r_{L}(z)=2r_{L,bulk}$. When $\kappa=1$, the surface is a maximally rough
solid and
$r_{L}(z)=\cfrac{r_{L,bulk}}{1+\cfrac{\sqrt{2}-1}{2}\cfrac{1}{2^{z/d}}}$,
which at the interface ($z=0$), becomes
$r_{L}(z)=\cfrac{2r_{L,bulk}}{1+\sqrt{2}}$. Interestingly, for
$\kappa=1/\sqrt{2}\approx 0.707$, we find $r_{L}(z)=r_{L,bulk}$, and hence the
localization length (and the corresponding localization well harmonic spring
constant $K_{0}(z)$) equal their bulk values unaffected by the solid surface
at all film locations. This striking prediction is discussed further below.
More generally, the alpha time and vitrification temperature gradients require
$K_{0}(z)$, $K_{B}(z)$, $F_{B}(z)$, and $\Delta r(z)$ (see Fig. 2) which are
determined numerically from the theoretically predicted $F_{dyn}(r,z)$ in
Eq.(10).

###  III.2 Local cage and collective elastic barrier

We first illustrate the role of $\kappa$ on dynamical properties for a hard
sphere fluid packing fraction for which the homogeneous bulk fluid alpha time
is approximately 100 ns. This choice is motivated by the fact that solid
surfaces typically strongly increase the relaxation time, and also by the fact
that 100 ns is a typical longest relaxation time probed in standard molecular
dynamics simulations. This timescale is achieved based on bulk ECNLE theory 7
; 25 ; 26  for a packing fraction of 0.5788.

Figure 3a shows the local cage barrier over the entire range of $\kappa$ of
present interest, at four locations near the surface. The cage barrier grows
with $\kappa$ in a weakly nonlinear manner, with a slope that decreases with
distance from the interface. These trends are expected, however the near
intersection of all the curves at $\kappa\approx 0.44$ is surprising. The cage
barrier at the near intersection is close to its bulk value of $7.9k_{B}T$,
suggesting a subtle “cancellation” of surface effects. Further insight to this
and other behaviors in Fig. 3a can be gleaned from our analytic analysis.

First recall that based on using the OZ-PY theory for the fluid structure as
input to the ECNLE theory, a near linear proportionality between $F_{B}$ and
$1/r_{L}$ was previously discovered and analytically understood 46 ; 50 .
Operationally, our numerical results in Fig. 3a are roughly consistent with a
linear relation $F_{B}(z)\sim\kappa$, as expected if the cage barrier scales
as the inverse localization length given in Eq. (21). This $F_{B}-\kappa$
linearity also strongly suggests that the dependence of $r_{B}$ and $r_{L}$ on
$\kappa$ plays a minor role in the determination of $F_{B}$. The slope of the
lines in Fig. 3a significantly decreases deeper into the film, as expected
from Eq. (21) and the linear $F_{B}(z)\sim\kappa$ connection.

The insensitivity of $F_{B}(z)$ to position in the film for $\kappa\approx
0.44$ can be qualitatively explained mathematically using Eq. (10). This
equation can be rewritten as

$\displaystyle
F_{dyn}(r,z)=F_{dyn}^{bulk}(r)+\frac{2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)}{2\times
2^{z/d}}\left[\kappa-\frac{F_{caging}^{bulk}(r)}{2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)}\right].$
(22)

If $F_{dyn}(r,z)$ is nearly independent of position near the solid surface,
then $\kappa$ is

$\displaystyle\kappa=\frac{F_{caging}^{bulk}(r)}{2F_{caging}^{pinned}(r)-F_{caging}^{bulk}(r)}.$
(23)

In the ultra-local analytic limit, we have

$\displaystyle\frac{F_{dyn}^{bulk}(r)}{k_{B}T}$ $\displaystyle=$
$\displaystyle-\frac{12\Phi g^{2}(d)}{\pi
d}\int_{q_{c}}^{\infty}\frac{dq}{q^{2}}e^{-q^{2}r^{2}/3},$ (24)
$\displaystyle\frac{F_{dyn}^{pinned}(r)}{k_{B}T}$ $\displaystyle=$
$\displaystyle-\frac{24\Phi g^{2}(d)}{\pi
d}\int_{q_{c}}^{\infty}\frac{dq}{q^{2}}\left[\frac{e^{-q^{2}r^{2}/6}}{2}+\frac{e^{-q^{2}r^{2}/3}}{4}\right],$

Substituting Eqs. (III.2) and (LABEL:eq:25) into Eq. (23), and using
$q_{c}r\ll 1$, then yields $\kappa=\cfrac{1}{2e^{q_{c}^{2}r^{2}/6}}$. Since
the barrier position $r_{B}$ falls in the particle displacement range of
$0.4d$ to $0.6d$ when $\Phi=0.5-0.61$ (Fig. 2), the inequality $q_{c}r\ll 1$
is roughly applicable (though not overly strong). Based on these arguments,
tuning $\kappa$ to a value just below 0.5 results in $F_{dyn}(r,z)\rightarrow
F_{dyn}^{bulk}(r)$. Our full calculations in Fig. 3a suggest that the dynamic
gradient vanishes for $\kappa\approx 0.44$. Of course, quantitative deviations
between the full numerical calculations and the analytic analysis are expected
since the latter is approximate.

Figure 3: (Color online) (a) The local cage barrier and (b) collective elastic
barrier as a function of $\kappa$ for $\Phi=0.5788$ corresponding to a bulk
alpha time of 100 ns, at different positions in the thick film. The bulk local
cage and bulk elastic barriers are $F_{B}^{bulk}\approx 7.9k_{B}T$ and
$F_{e}^{bulk}\approx 3.87k_{B}T$, respectively.

At a more detailed level, one can ask how the value of $\kappa$ at the
intersection point varies with packing fraction. In prior work 46 , the
characteristic wavelength was approximated as $q_{c}=\cfrac{2\pi}{r_{cage}}$.
Increasing packing fraction weakly decreases the cage radius $r_{cage}$,
thereby slightly increasing $q_{c}$, leading to a weak decrease of $\kappa$ at
the intersection point. But overall, this is a small effect.

The elastic barrier depends not only on the finite-size nature of the films
via the geometric cutoff of the elastic field at an interface, but also on the
nature of the surfaces via the dynamic free energy properties which enter the
elastic barrier (harmonic spring constant, jump distance) in a $z$-dependent
manner 2 ; 4 ; 5 ; 6 ; 7 . To compare with $F_{B}(z)$ in Fig. 3a, Figure 3b
shows the dependence of $F_{e}(z)$ on $\kappa$ near the solid surface. In
contrast to the cage barrier, the elastic barrier does not grow linearly with
$\kappa$ since its magnitude is related to the jump distance and harmonic
curvature, not the inverse localization length. However, interestingly,
$F_{e}(z)$ is roughly independent of $z$ when $\kappa\approx 0.4$. This
finding is qualitatively consistent with a relatively uniform
($z$-independent) local cage barrier in the first few layers near the surface
when $\kappa\approx 0.44$, per Fig. 3a. The differences in this behavior of
the two barriers are attributed to a greater influence of interfacial effects
on the elastic barrier compared to its local barrier analog. Overall, the
different dependences of the local and elastic barriers on $\kappa$ imply the
theory predicts tuning this substrate parameter will modify the relative
importance of local caging and long-range collective elasticity, which will
have multiple dynamical consequences. For example, how interfaces modify
dynamic fragility, and the quantitative aspects of the alpha time and glass
transition temperature spatial gradients.

###  III.3 Alpha time and glass transition temperature gradients

Previously we showed 7  that ECNLE theory predicts the alpha time gradient is
of a double exponential decay form near the interface, which then crosses over
to an inverse power law decay far enough from the interface before attaining
its bulk fluid limit. Moreover, the glass transition temperature displays an
exponential dependence on $z$, also with a weak power law tail. Within the
present formulation of ECNLE theory for symmetry-broken films, these core
predictions are qualitatively identical for all surface boundary conditions,
and have been verified in simulations 1 ; 2 ; 7 ; 24 ; 40 ; 41 ; 42 ; 43 ; 44
; 45 ; 51  and experiment 1 ; 33 ; 34 ; 35 . Hence, in this section on thick
films, we first we focus on the dependence of the alpha time and $T_{g}$ on
$\kappa$ only at selected locations in the thick film.

Figure 4 shows the normalized alpha relaxation time and glass transition
temperature near the solid interface of a thick film. The local $T_{g}$ is
determined based on two dynamic criteria that mimic experimental and
simulation studies, $\tau_{\alpha}(T_{g}(z))=100$ s and 100 ns, respectively.
As expected, increasing $\kappa$ induces more slowing down (Fig. 4a), and in a
manner that varies in a supra-exponential manner with $\kappa$. This behavior
results in the glass transition temperature gradient trends in Fig. 5b, and
generally a nonlinear variation with $\kappa$. Interestingly, we again find an
”invariant” to interface state at $\kappa\sim 0.42$, qualitatively consistent
with results in Fig. 3a and 3b.

In prior works, it was proved that within ECNLE theory the $T_{g}$ gradient
normalized by its bulk value in free-standing films, and also in supported
films with rough or smooth solid surfaces, is nearly unaffected by the
vitrification criterion over a wide range of timescales 4 ; 5 ; 6 . Thus, we
expect such an insensitivity of $T_{g}(z)/T_{g,bulk}$ holds as $\kappa$ is
varied, which is confirmed in Fig. 4. Note that the $T_{g}(z)/T_{g,bulk}$
curves corresponding to $\tau_{\alpha}(T_{g,bulk})=100$ s and 100 ns
determined as a function of $\kappa$ near the solid interface are close. This
suggests that the normalized $T_{g}$ gradients determined from simulations can
be utilized to predict the gradient under typical experimental conditions.

Figure 4: (Color online) (a) Logarithm of the inverse alpha time, and (b)
local glass transition temperature, of polystyrene melts, normalized by their
bulk counterparts, near the substrate surface as a function of $\kappa$. The
solid and dotted curves correspond to calculations based on vitrification
timescale criteria of 100 s and 100 ns, respectively.

Figure 5 shows calculations of the spatial gradient of the inverse alpha time
normalized by its bulk counterpart as a function of distance from the
substrate for a PS thick film at a temperature of $T_{g,bulk}$. The results in
the main frame are presented in a log-linear format. Near the interface
($z\leq 0.5d$), the normalized mobility gradient obeys a double-exponential
spatial form for all surfaces. For $\kappa-0.5$, one sees in the inset of
Fig.5 that $\log_{10}\left(\tau_{\alpha,bulk}/\tau_{\alpha}(z)\right)\approx
0.05-0.15$ over a large range of film locations spanning from $z\sim 0$ to
$8d$. The range of variation of this normalized mobility gradient is
”flattened” when $\kappa=0.4$, consistent with the alpha time being nearly
$z$-independent (Fig. 4a). Far from the interface ($z\geq 10d$), the double-
exponential behavior is replaced by an inverse power law form that was
previously predicted and observed in simulation for a thick free-standing film
7 . The existence of this inverse power law contribution is why a double
exponential form for the alpha time gradient in thick films and its linear
superposition cannot describe numerous dynamic features at the center of thin
films, as discussed in Ref. 2 . Since a solid and vapor interface truncate the
collective elastic field in the same manner within the present theory, the
power-law tail decay is identical in functional form for different interfaces.

Concerning the inset of Fig. 5, there is a striking degree of quantitative
universality of the power law tail. This is understandable since this long-
range tail is determined by the universal cutoff of the elastic field at an
interface mechansim, and that far enough from the interface the dynamic free
energy attains its bulk liquid form. To elaborate, recall that we previously
derived far from an interface that
$\log_{10}\left(\cfrac{\tau_{\alpha,bulk}(T)}{\tau_{\alpha}(T,z)}\right)\approx\log_{10}e\cfrac{F_{e,bulk}}{k_{B}T}\cfrac{r_{cage}}{4z}$
7 . The tail amplitude is not universal with regards to chemistry since it
depends on the elastic barrier in the bulk, the magnitude of which plays an
essential role in determining the dynamic fragility in ECNLE theory 4 ; 5 ; 29
. However, in the present context, it is indeed independent of $\kappa$ which
nucleates a change of the dynamic free energy at the interface that is
transferred into the film interior, but it effect decays exponentially as
stated above. Thus, the magnitude of $\kappa$ does strongly modify both the
local cage and collective elastic barriers close enough to the interface where
the dynamic free energy is spatially heterogeneous, but far enough from the
interface where the power law form of the gradient cleanly emerges due to the
pure cutoff effect the dependence on $\kappa$ vanishes, per the good overlap
of curves region in the inset of Fig. 5.

Figure 5: (Color online) Logarithm of the inverse alpha time normalized by its
bulk value for PS films at $T_{g,bulk}\approx 502.6$ $K$ (or $\Phi=0.5788$)
based on the indicated 100 ns kinetic vitrification criterion, as a function
of distance from the solid surface, with $\kappa=0,0.3,0.5,0.8,$ and 1.0.
Inset: identical results as the main frame but presented in a double-
logarithmic format.

The above findings suggest a simple analytic (though approximate) form for the
alpha time gradient discussed previously 7 :

$\displaystyle\log_{10}\left(\cfrac{\tau_{\alpha,bulk}(T)}{\tau_{\alpha}(T,z)}\right)=A_{\kappa}(T)e^{-z/\xi_{\kappa}(T)}+\frac{B_{\kappa}(T)}{z}.$
(26)

Here, $A_{\kappa}(T)$ is the amplitude at the surface, $\xi_{\kappa}(T)$ is
the penetration or decay length (predicted by ECNLE theory to be nearly
independent of temperature 4 ; 5 ), and $B_{\kappa}(T)$ is the amplitude of
the universal in form power law decay part of the gradient due solely to the
interfacial cut-off of the long-ranged collective elastic field. We shall use
this expression in section IV to study deviations from the naive gradient
superposition approximation.

Concerning the accuracy of Eq. (26) within the ECNLE theory framework, we note
that prior work 7  on thick supported films with one neutral solid surface and
one vapor interface has verified Eq. (26) provides a good representation of
the numerical ECNLE theory predictions. In addition, based on the present
formulation of collective elasticity effects in ECNLE theory for symmetry-
broken films, we analytically know that very far from an interface in thick
films that
$\log_{10}\left(\cfrac{\tau_{\alpha,bulk}(T)}{\tau_{\alpha}(T,z)}\right)\approx\log_{10}e\cfrac{F_{e,bulk}}{k_{B}T}\cfrac{r_{cage}}{4z}$
7 . This result implies that
$B_{\kappa}=\log_{10}e\cfrac{F_{e,bulk}}{k_{B}T}\cfrac{r_{cage}}{4}$ in Eq.
(26) is always positive. Hence, as the bulk behavior is approached far from
the interface, the dynamics is slightly faster than in the bulk. On the other
hand, $A_{\kappa}$ can be positive or negative, and strongly depends on a
nature of the interface; a negative $A_{\kappa}$ is the dominant origin of
slowing down of relaxation near a solid interface. Taking Eq. (26) seriously
would then appear to suggest that a liquid near a solid surface could exhibit
a different glass transition behavior compared to the behavior near the
interface or in the film center. As a speculative comment, such opposite
effects of $A_{\kappa}$ and $B_{\kappa}$ on glassy dynamics could be a
possible origin of the observation of two $T_{g}$’s in the pore confined
liquids studied by McKenna and coworkers 52 ; 53 . More generally, one expects
within ECNLE theory that the suppression of dynamics near solid surfaces
becomes more significant if particles are confined within (spherical-like, or
cylindrical) pores since such a change of geometry amplifies the coupled
dynamical effects of interfacial cut-off of the long-ranged collective elastic
field and modification of the dynamic free energy.

##  IV Confined Thin Films

Within ECNLE theory, confinement modifies the structural relaxation time and
glass transition temperature via two distinct physical effects 2 : (i)
strengthening/weakening of the local caging constraint near interfaces which
is embedded in the surface dynamic free energy and spatially transferred into
the film, and (ii) a reduction of the collective elasticity cost for cage
scale hopping due to the interfacial cutoff of the elastic displacement field.
Here we explore the evolution of these effects for supported films of
variables thicknesses with different interfaces as a function of $\kappa$.

###  IV.1 Glass transition temperature gradients

We first calculate the spatially resolved ratio of the vitrification
temperature $T_{g}(z)/T_{g,bulk}$ as a function of $\kappa$ near the substrate
for various film thicknesses. Since we have previously shown that the $T_{g}$
gradient normalized by its bulk value remains nearly unchanged with
modification of vitrification criterion, we adopt the experimental
vitrification time scale of 100 s. For the PS liquids of present interest, the
latter criterion corresponds to $T_{g,bulk}=428.3$ $K$. The corresponding
$T_{g}(z)$ curves for polymer films are shown in Fig. 6 for $H=10d$ and $30d$
at the near surface locations of $z=1d$ and $z=2d$. Note that the curves
intersect at $\kappa=0.42$, a value close to what was found for $T_{g}(z)$ in
thick supported polymer films (Fig. 4b). This finding implies that near a
solid substrate with $\kappa=0.42$, there is an interfacial region with a
nearly uniform distribution of glass transition temperatures. However, as the
film becomes thinner, $T_{g}(z)$ decreases due to the cutoff of the elastic
barrier at the surfaces. For the polymer film with $H=5d$,
$T_{g}(z)/T_{g,bulk}\sim 0.9$ in the first few interfacial layers for
$\kappa=0.3$ since both the local and elastic barrier are significantly
reduced.

Figure 6: (Color online) The local glass transition temperatures of PS films
of thicknesses $H=5d,10d$, and $30d$, normalized by their respective bulk
values, as a function of $\kappa$. The solid and dotted curves correspond to
calculations at $z=1d$ and $2d$, respectively. The bulk $T_{g}$ equals 428.3
$K$ and corresponds to the typical experimental vitrification time scale of
100 s.

The nearly linear relationship in Fig. 6 between $T_{g}(z)/T_{g,bulk}$ and
$\kappa$ at $z=2d$ can be understood based on Eq. (15) which indicates how
$T_{g}(z)/T_{g,bulk}$ depends on variations of the local and elastic barriers
as a function of $\kappa$. As seen in Fig. 3, both $F_{B}$ and $F_{e}$ are
proportional (to leading order) to $\kappa$ beyond the first two layers from
the solid interface. On the other hand, very close to the surface at $z=0$ or
$1d$, the gradient $T_{g}(z)/T_{g,bulk}$ does not vary linearly with $\kappa$.
The origin of this nonlinearity is attributed to the strongly nonlinear
relationship between $F_{e}/k_{B}T$ and $\kappa$.

Figure 7: (Color online) Log-linear plot of the local glass transition
temperature normalized by its bulk value as a function of location in a
supported PS film for different values of $\kappa$ based on a vitrification
criterion of 100 s with film thicknesses of (a) $5d$, (b) $10d$, and (c)
$30d$. Here, $T_{g,bulk}$ = 428.3 K corresponds to the typical experimental
vitrification time scale of 100 s. For PS, $d\sim 1.1$ nm.

To investigate the dynamic coupling of asymmetric interfacial effects in a
thin film, Figure 7 shows the rich variation of the spatial gradient of
$T_{g}(z)/T_{g,bulk}$ for different values of $\kappa$ at various film
thicknesses. The presence of the vapor layer ($\kappa=0$) speeds up
relaxation, and this acceleration spreads towards the film center. This
mechanism accounts for the lower $T_{g}(z)$ in the film compared to the bulk
$T_{g}$. Increasing $\kappa$ enhances caging at the interface, leading to an
elevation of the local glass transition temperature. The maximally rough solid
substrate ($\kappa=1$) system exhibits very strongly slowed down dynamics near
the interface, and thus $T_{g}(z)>T_{g,bulk}$. When the film is sufficiently
thin ($H\leq 10d$) there is a large separation between the curves in Fig. 8a
which indicates significant interfacial coupling between the two surfaces.
Particularly, at the film center ($z=H/2$), one finds $2^{-H/2d}\geq
2^{-5}\approx 3.1\%$, and the mathematical forms of Eqs. (14) and (18) reveal
that the effects of both surfaces on the local and collective elastic aspects
of the dynamics are important. For $H\geq 20d$, the local dynamics recovers
its bulk behavior at the film center since $2^{-H/2d}\leq 2^{-10}\approx
0.1\%$ . Thus, we conclude that the behavior of $T_{g}(z>H/2)/T_{g,bulk}$ is a
consequence of only the vapor layer, and the interactions of the two surfaces
on the cage scale are dynamically decoupled. Rather, the difference between
$T_{g}(z)$ and $T_{g,bulk}$ arises from the suppression of the elastic barrier
associated with the truncation of the displacement field at the interfaces.

As is well known, an effective glass transition temperature of a thin polymer
film can be estimated multiple ways 1 ; 6 ; 10 ; 30 ; 31 ; 32 ; 53 ; 54 ; 55 .
A thermodynamic-like approach 1 ; 6 ; 30 ; 31 ; 32 ; 54  is based on a
”democratic” average over the film,
$\left<T_{g}\right>_{H}=\frac{1}{H}\int_{0}^{H}T_{g}(z)dz$. Such pseudo-
thermodynamic calculations can potentially be compared to experimental heat
capacity and ellipsometry measurements. Although dynamics near the solid
interface is significantly slowed down, particularly if $\kappa=1$, motion is
faster near the vapor interface, and can be bulk-like or even faster in the
middle of sufficiently thin films. The results in Fig. 8 reveal
$\left<T_{g}\right>_{H}$ is smaller than its bulk counterpart, and thus in
this film-averaged sense the presence of the surfaces enhances the overall
molecular mobility. This may seem like a surprising result. But we caution it
is not a general result, but rather applies to the present model of a
supported film and the parameters that mimic PS. However, it does appear to be
qualitatively consistent with experimental findings in Ref. 56  which found
the film-averaged glass transition temperature of the supported film is
suppressed compared to its bulk value, despite the significant slowing down of
dynamics close to a solid substrate. The authors of Ref. 56  speculated this
is due to the emergence of a so-called ’dead layer” near the solid surface. In
our theory, the detailed mechanism for the average $T_{g}$ reduction involves
the cutoff of collective elasticity at the interface. The specific nature of
the substrate-liquid coupling does also play an important quantitative role in
the shift of $T_{g}$, including “dead layers” under some conditions (see
Section IV-C), but we find that it is not the primary reason for lowering
$T_{g}$ compared to its bulk value, at least for the model and parameters
studied here. For example, for $H=21.5d$ (roughly $H=25$ nm), we predict
$\left<T_{g}\right>_{H}/T_{g,bulk}\approx 0.966$ for $\kappa=0.5$ and
$T_{g,bulk}\sim\left<T_{g}\right>_{H}\approx 15$ $K$, which, surprisingly and
perhaps accidentally, seems to agree quantitatively with experiments in Ref.
56 . Possible caveats to the generality of this conclusion include the model
assumption that surfaces do not perturb the structure or density in the film,
and the direct effect of strong surface-fluid attractions (adsorbed layers)
that are not considered. When $H\rightarrow\infty$, $\left<T_{g}\right>_{H}$
recovers the bulk $T_{g}$, as it must.

As a side comment, one might think that the results in Fig. 8 that suggest the
mean film $T_{g}$ is always suppressed relative to bulk is not consistent with
some experiments which measure enhancements in thin films. But the latter are
typically materials with significant polymer-surface attractions, which are
not in our model. Nor do we allow for attractive surface induced densification
near the solid interface. Both these effects would lead to a local elevation
of $T_{g}$, and potentially an increase of the film-averaged vitrification
temperature. Moreover, the results in Fig. 8 do seem natural for our model
given the form of the predicted $T_{g}$ gradients in Fig. 7c. Of course, if we
allowed $\kappa$ to be sufficiently larger than unity, no doubt the film-
averaged $T_{g}$ would be predicted to be larger than its bulk value. We do
not consider this parameter regime since, as discussed in Refs. 3 ; 4 , we
view $\kappa=1$ as a natural upper bound of the model sketched in Fig. 1.
Finally, we note that in Fig. 9 discussed below, the predicted film-averaged
$T_{g}$ can be larger than the bulk value for high $\kappa$ systems based on a
specific gradient averaging criterion, even within our film model.

Figure 8: (Color online) Film-averaged glass transition temperatures
(normalized by their bulk value) as a function of film thickness as determined
from $\left<T_{g}(z)\right>_{H}$ (pseudo-thermodynamic approach) for a
vitrification criterion of $\tau_{\alpha}\left(T_{g}(z)\right)=100$ s at
different values of $\kappa$. Inset: same results as main frame but plotted
$T_{g,bulk}/\left<T_{g}\right>_{H}$ versus $d/H$.

An oft-employed empirical expression used to fit experimental and simulation
data for how $T_{g}$ shifts depend on film thickness is based on a naive ”two-
layer” model 1 ; 10 ; 30 ; 40 ; 41 :

$\displaystyle\left<T_{g}(z)\right>_{H}=\frac{T_{g,bulk}}{1+\cfrac{\delta}{H}},$
(27)

where $\delta$ is an adjustable parameter characterizing the interfacial
regime. Of course, our theory does not predict the alpha time or $T_{g}$
gradient is of a step function form corresponding to constant values in two
layers, but rather predicts continuous gradients. Moreover, it has been
previously shown that the typically valid Taylor series expansion of Eq. (27),
$\left<T_{g}(z)\right>_{H}/T_{g,bulk}=1-\delta/H$, must hold as a mathematical
fact for any film for which the dynamic gradient is of finite range 1 . This
latter equation provides a good description for the thickness variation of the
film-averaged glass transition temperatures irrespective of the specific
details of the gradient 1 .

Motivated by Eq. (27), we plot our numerical results for
$T_{g,bulk}/\left<T_{g}(z)\right>_{H}$ as a function of $d/H$ in the inset of
Fig. 8. We find the inverse film-averaged glass transition temperature for
diverse solid surfaces is roughly proportional to $1/H$. This linearity
validates the empirical usefulness of Eq. (27). Note that the slope of the
curves increases with decreasing $\kappa$, implying that as the caging
constraint at the interface is softened, the apparent interfacial layer length
scale ($\delta$) is shortened.

An alternative approach to determining the effective glass transition
temperature of a finite-size film is explicitly based on dynamics 1 ; 10 ; 30
; 31 ; 32 ; 53 ; 54 ; 55 . One can define a dynamic glass transition
temperature as the temperature at which the film-averaged alpha relaxation
time
$\left(\left<\tau_{\alpha}(T,z)\right>_{H}=\frac{1}{H}\int_{0}^{H}\tau_{\alpha}(T,z)dz\right)$
reaches a selected vitrification criterion relevant to dielectric spectroscopy
or other dynamic measurements. Figure 9a shows representative results. We note
that the corresponding $\left<T_{g}(z)\right>_{H}$ based results in Fig. 8
exhibit a more substantial decrease compared to their purely dynamic
counterparts. When $\kappa\leq 0.5$, the predictions based on the pseudo-
thermodynamic and dynamic approaches exhibit qualitative, but not
quantitative, agreement.

On the other hand, for the more rigid substrates of $\kappa=0.8$ and $1$ the
behavior is very different. Specifically, recall that we showed that
$\left<T_{g}(z)\right>_{H}<T_{g,bulk}$ in Fig. 8, but the effective film-
averaged mean value of $T_{g}$ determined by
$\left<\tau_{a}lpha(T_{g},z)\right>_{H}=100$ s in Fig. 9a is greater than the
bulk $T_{g}$ out to a large film thickness of $H=100d$. Such behavior might
seem unphysical. However, the construction of the theory guarantees the bulk
alpha time will be achieved sufficiently far from the interface, and hence the
film averaged $T_{g}$ will equal its bulk value at “large enough” film
thicknesses. Hence, this different behavior for the very rigid substrate cases
is a consequence of two factors: (i) how the mobility gradient is averaged,
and (ii) our assumption in the calculations that the film structure (and hence
alpha time) is fully equilibrated on all length scales. Consideration (i)
relates to the democratic weighting of the predicted surface-induced large
slowing down of the alpha time near the interface versus the more modest (in a
relative, not absolute, sense) vapor-induced speeding up of mobility.
Consideration (ii) is highly relevant for large values of $\kappa$ since the
theory predicts huge increases of the barrier and alpha time near a rigid
rough substrate 4 . However, the latter may not be observed in typical
experimental measurements that do not perform ultra long-time aging in which
case the near surface layers can appear as effectively “dead”, and the alpha
process close to the substrate may not contribute the measured observable from
which $T_{g}$ is determined. For example, if $\Phi=0.58$, we find the alpha
time in the equilibrated first layer for $\kappa=1$ is predicted to be
$10^{7}$ s, and the corresponding average alpha time for a thick film of
$H=100d$ is still very large, of order $\sim 10^{4}$ s, corresponding to a
“glass” based on averaging of the alpha time gradient approach. Thus, we
indeed expect that $H\gg 100d$ is required to recover the bulk $T_{g}$.

Figure 9: (Color online) (a) Film-averaged glass transition temperatures
(normalized by the bulk analog) as a function of thickness determined using
the dynamic criterion $\left<\tau_{\alpha}\left(T_{g},z\right)\right>_{H}=100$
s (dynamic approach for the experimental vitrification timescale criterion
corresponding to $T_{g,bulk}=428.3$ $K$) at different values of $\kappa$. (b)
Analogous results but based on the vitrification criterion
$\left<\log_{10}\left[\tau_{\alpha}\left(T_{g},z\right)\right]\right>\equiv
2$.

Of course, if the practical situation in experiment or simulation is the near
rigid substrate film is not equilibrated, then our calculations for the high
values of $\kappa$ in Fig. 9a are not directly relevant. We have chosen to
present them since they are of conceptual interest as indicating what would be
expected in a fully equilibrated film. Moreover, we again emphasize that in
the $H\rightarrow\infty$ limit, the theory recovers a film-averaged alpha time
and glass transition temperature equal to their bulk values. Explicit
numerical verification based on the dynamic averaging criterion requires
numerical calculations out to ultra-large values of H which are beyond our
ability to computationally treat.

Figure 9b shows results analogous to those of Fig. 9a but for the different
vitrification criterion of
$\left<\log_{10}\left[\tau_{\alpha}\left(T_{g},z\right)\right]\right>\equiv
2$. The motivation for this calculation is that when $\kappa>0.5$, the results
in Fig. 10a are dominated by the contribution of the very long alpha times
near the solid surface. The alternative criterion acknowledges the fact, as
seen from Figure 7c, that the normalized gradient of glass transition
temperatures near the solid surface is roughly of equal but opposite form
(absolute magnitude of shift from the bulk value and spatial range) as that at
the vapor surface. Thus, one expects from the alternative averaging approach
above that the normalized to the bulk $T_{g}$ film-averaged glass transition
temperature will more rapidly approach its limiting value of unity at large
film thicknesses, and its shift from the bulk will decrease and nearly vanish
as $\kappa$ becomes large and approaches unity. These anticipated behaviors
are indeed found in the numerical results in Fig. 9b. Taken together, the
results in Fig. 9 show (as is well known 1 ; 54 ) that in a supported film
with strong competing spatial gradients, prediction of an average $T_{g}$ is a
subtle issue that depends on the criterion for averaging over the spatially
heterogeneous dynamics.

###  IV.2 Alpha relaxation time gradients

Numerical ECNLE theory calculations of confinement effects on the alpha time
gradient are shown in Fig. 10 and will be discussed in detail below. But we
first wish to note that one can attempt to build an analytic description of
the gradients as follows. From simulations and our prior analysis 2 ; 7 ; 24 ,
it is known that the mobility gradient of thin films is sometimes quite well
approximated by a linear superposition model corresponding to combining the
thick-film gradients of the relaxation time from the solid and vapor interface
in a linear manner as 2 ; 7 :

$\displaystyle\log_{10}\left(\cfrac{\tau_{\alpha,bulk}(T)}{\tau_{\alpha}(T,z,H)}\right)=A_{\kappa}(T)e^{-z/\xi_{\kappa}(T)}+\frac{B_{\kappa}(T)}{z}+A_{\kappa=0}(T)e^{-(H-z)/\xi_{\kappa=0}(T)}+\frac{B_{\kappa=0}(T)}{H-z}$
(28)

Here, $A_{\kappa=0}(T)$ is the amplitude at the surface, $\xi_{\kappa=0}(T)$
is the decay length (essentially $T$-independent in ECNLE theory [4,5]), and
$B_{\kappa=0}(T)$ quantifies the amplitude of the long range collective
elastic part of the gradient for $\kappa=0$ which mimics a vapor interface.
Adoption of this equation assumes there are no new finite-size effects in the
thin-film scenario compared to the thick film.

Figure 10: (Color online) Logarithm of the gradient of the inverse alpha
relaxation time normalized by its bulk value in supported films of thicknesses
$H=5d,10d,$ and $30d$ plotted versus position in the film normalized by the
film thickness. Results are shown for (a) $\kappa=0$, (b) $\kappa=0.3$ and (c)
$\kappa=1$, at the bulk glass transition of a typical standard MD simulation,
which corresponds to a packing fraction of $\Phi=0.5788$ or a bulk alpha time
of $\sim$ 100 ns. Data points and solid curves are the full thin-film
numerical calculations and their gradient superposition (Eq. (28) analogs,
respectively. The dotted curves correspond to the gradient superpositions
without the power law decaying term. The inset in Figure 10(c) plots the same
data as in the main frame but in a log-linear format.

Figure 10 compares predictions of the approximate Eq. (28) against our full
numerical ECNLE theory results. Within the context of the theory, such a
comparison provides direct insight concerning the influence of non-additive
gradient effects in thin films. This comparison can be done with, or without,
the long-range power law contribution to the gradients in order to further
probe the importance of the explicitly long-range consequence of collective
elasticity on dynamics.

We now discuss the more detailed trends and features in Fig. 10. Consider
first the case of where our supported film model becomes a symmetric free-
standing film as studied in Ref. 2 ; 6 . We find a significant flattening in
the mid-film region (Fig. 10a) for intermediate film thicknesses ($H\approx
20-30d$) compared to the gradient superposition approximation without the
power law decay term. Considering the power-law tail originates solely from
the interfacial truncation of the elastic contribution to the barrier,
including it in the analytic superposition expression is expected to improve
the accuracy of analytic form. For a thinner film of $H=10d$, the power-law
decay contribution plays a minor role due to the strong cutoff of the elastic
field from the two interfaces, and the superposition approximation with and
without a $1/z$ power-law tail in $\tau_{\alpha}(z)$ describe quite well the
numerically computed mobility gradients. However, the superposition
approximation even with the power law tail gradually becomes less accurate in
ultrathin films ($H\sim 5d$) due to strong coupling between the different
dynamics nucleated at the two interfaces associated with the gradient of cage
scale physics. This behavior emerges even though most of the elastic barrier
is largely cut-off at the interfaces since the long-range power-law tail
disappears in ultra-thin films 2 ; 7 .

Figures 10b and 10c contrast our full ECNLE theory calculations, analogous
superposition approximation results, and calculations based on Eq. (28) for
the logarithm of the alpha time as a function of $z/H$ for smaller values of
$\kappa=0.3$ and $0.5$ and different film thicknesses. The same level of
accuracy and limitations of Eq. (28) and the superposition approximation found
for the $\kappa=0$ free-standing film case are again evident. One sees a good
quantitative agreement between numerical ECNLE theory results and the analytic
approximation when $H\geq 10d$. The inverse power-law decay of the relaxation
time gradient is the origin of the gradient flattening in the middle of the
film. An important technical limitation of Eq. (28) is that it cannot
accurately capture glassy dynamics near the interface due to the (artificial)
divergence associated with the contributions $1/z$ and $1/(H-z)$. The alpha
time gradient only obeys a power law decay form far enough from the surface.
In the interfacial region, the analytic formula describing structural
relaxation could be much more complicated.

Given the above analysis, we return to our numerical normalized $T_{g}$
gradient results discussed above. Motivated by prior work in 7 , we explore
whether they can be approximated by the same analytic form as the logarithm of
the normalized alpha relaxation time (Eq. (28)). Thus, we write

$\displaystyle\frac{T_{g}(z,H)}{T_{g,bulk}}$ $\displaystyle=$ $\displaystyle
A_{g,\kappa}e^{-z/\xi_{g,\kappa}}+\frac{B_{g,\kappa}}{z}+A_{g,\kappa=0}e^{-(H-z)/\xi_{g,\kappa=0}}$
(29) $\displaystyle+$ $\displaystyle\frac{B_{g,\kappa=0}}{H-z},$

where
$A_{g,\kappa},B_{g,\kappa},\xi_{g,\kappa},A_{g,\kappa=0},B_{g,\kappa=0},$ and
$\xi_{g,\kappa=0}$ are fit parameters. If we ignore the power decay
contributions in Eq. (29), the film average $T_{g}(H)$ becomes

$\displaystyle\frac{\left<T_{g}(z,H)\right>_{H}}{T_{g,bulk}}$ $\displaystyle=$
$\displaystyle\frac{A_{g,\kappa}\xi_{g,\kappa}}{H}\left(1-e^{-H/\xi_{g,\kappa}}\right)$
(30) $\displaystyle+$
$\displaystyle\frac{A_{g,\kappa=0}\xi_{g,\kappa=0}}{H}\left(1-e^{-H/\xi_{g,\kappa=0}}\right).$

Equation (30) suggests that $\cfrac{\left<T_{g}(z,H)\right>_{H}}{T_{g,bulk}}$
is approximately linearly proportional to $\frac{d}{H}$, particularly when the
film is thick. To test this relationship, we replot in Fig. 11 the theoretical
data in the main frame of Fig. 8 as
$\cfrac{\left<T_{g}(z,H)\right>_{H}}{T_{g,bulk}}$ versus $d/H$. We find a good
linearity applies for $\kappa=0,0.3,$ and 0.5 over a wide range of film
thicknesses. However, the film averaged $T_{g}$ normalized by its bulk value
becomes somewhat nonlinear for $\kappa=0.8$ and 1, particularly when the film
is very thin.

Figure 11: (Color online) Film-averaged glass transition temperatures
normalized by the bulk value of $T_{g,bulk}=428.3$ $K$ (corresponding to the
typical experimental vitrification time scale of 100 s) as a function of $d/H$
determined by the pseudo-thermodynamic approach at different values of
$\kappa$.

###  IV.3 Glass melting and dead layers in very cold bulk films

We briefly consider the situation when the bulk liquid is very cold
corresponding to being at its experimental bulk glass transition (occurs at
$\Phi=0.61$) with an alpha time of 100 s. This situation is particularly
germane to the question of the emergence of “dead layers” near a solid
interface 10 ; 56 ; 57 ; 58 ; 59 ; 60  (as briefly commented on above), and
also to the inverse question of how a vapor interface can lead to melting of a
bulk glass which is germane to the formation of so-called ultra-stable
glasses. For the former case, the slowing down of relaxation at a solid
interface can increase the alpha time beyond what can be measured, and thus in
a limited region of space the material is a solid exhibiting a “dead layer”.
Of course, in this situation the material in that spatial region is very
likely out of equilibrium, so employing equilibrium integral equation theory
for the structural input in ECNLE theory calculations is less justified. Our
results below should be viewed to within this caveat, but we believe they
provide some zeroth order insight to the dead layer question.

Figure 12a shows the logarithm of the alpha time as a function of $\kappa$ at
the PS bulk glass transition temperature at four locations close to the
surface in a thick film ($H=30d$). For sufficiently small values of $\kappa$,
the bulk glass re-fluidizes, while for larger values of $\kappa$ a dead layer
behavior is evident corresponding to an alpha time (far) greater than 100 s.
The dead layer is only present for $\kappa>0.5$, and widens with increasing
$\kappa$, extending to beyond 3 particle diameters when $\kappa>0.8$ for the
example shown.

Figure 12: (Color online) The logarithm of the alpha time (in seconds)
calculated at $\Phi=0.61$ or $436.2$ $K$ (bulk laboratory glass transition
criterion) as a function of $\kappa$ (a) at positions ($z$) near the solid
surface for $H=30d$ and (b) at the solid-liquid interface ($z=0$) for
different finite-size films. The results in both panels include the limiting
$\kappa=0$ case where the ”solid surface” reduces to a vapor interface and
hence the system is a free-standing film.

Figure 12b shows analogous results in the first layer at the solid surface
over a wide range of film thicknesses. Now a dead layer emerges not only by
increasing $\kappa$ at fixed $H$, but also by increasing film thickness at
fixed $\kappa$. This behavior arises from the decoupling of gradients induced
by the vapor and solid surface interfaces. Decreasing film thickness
facilitates dynamic interference between the gradients emanating from the two
interfaces, which reduces the tendency to form a dead layer at the solid
surface given the other interface displays faster than bulk dynamics. A much
deeper theoretical analysis of this problem is possible, but is beyond the
scope of the present article.

###  IV.4 Time and frequency domain correlation functions

The discussion of dead layers in the prior section, and the results in Fig. 9a
that show dramatically enhanced (to high $H$) values of the film-averaged
$T_{g}$ which are likely unobservable due to nonequilibrium effects, are
relevant in a fixed temperature sense. However, many experiments near the
laboratory bulk glass transition temperature do not directly report a
relaxation time, but rather a film-averaged response function which reflects a
wide distribution of relaxation times due to strong dynamical gradients. The
theoretically predicted massive solid substrate enhancement in relaxation
times could lead to a very slow and lower amplitude component of a time domain
relaxation function, or a low frequency wing (or perhaps even a distinct peak)
in a frequency domain measurement. Such features could be unobservable for
practical reasons in typical experiments, and if so, the reported film-
averaged mean relaxation time would miss the large substrate induced slowing
down effect. This complicated issue would occur even in equilibrium.

The above considerations motivate us to briefly consider what our theory
predicts for the film-averaged relaxation function. We ignore any “intrinsic”
dynamic heterogeneity that is present even in the bulk, and focus on how
interfaces and confinement induce a (spatial) distribution of the alpha time.
This corresponds to constructing a film-averaged relaxation function that
averages over exponential relaxation processes with time constants that depend
on location in the film, i.e., the alpha time gradient considered in this
article. Specifically, we define the time and frequency domain film-averaged
relaxation functions:

$\displaystyle
C(t)=\frac{1}{H}\int_{0}^{\infty}dz\exp\left(-\frac{t}{\tau_{\alpha}(z)}\right),$
(31) $\displaystyle
C"(\omega)=\frac{1}{H}\int_{0}^{\infty}dz\frac{\omega\tau_{\alpha}(z)}{1+\left[\omega\tau_{\alpha}(z)\right]^{2}}.$
(32)

Figures 13 and 14 show calculations for the high $\kappa=1$ solid substrate
case, two choices of film thickness ($H=10d$ and $30d$), three temperatures
close to, and straddling, the bulk $T_{g}$ germane to the conditions studied
in the previous section and in Fig. 9. The selected temperatures correspond to
bulk alpha times of $10^{x}$ s with $x=0.441,2.107$ and $4.03$. There are
multiple interesting features.

Figure 13: (Color online) Film-averaged time dependent relaxation function of
Eq. (31) for a PS film of thickness (a) $10d$ and (b) $30d$ at temperatures of
436.2 $K$, 427.9 $K$, and 419.6 $K$. The corresponding bulk liquid mean alpha
relaxation times expressed as $10^{x}$ s are $x=0.441,2.107$ and $4.03$ with
decreasing temperature. The circles on the x-axis indicate the bulk alpha time
at the corresponding temperature. The vertical dashed lines are a guide-to-
the-eye.

The time domain results in Fig. 13 display a smeared or continuous two regime
form for the thin film ($H=10d$) where gradients from the vapor and solid
surfaces strongly overlap. On the other hand, for the thicker film ($H=30d$),
a well defined, 2-step decay form is predicted. The time constants for a $1/e$
decay of $C(t)$ are close to the bulk alpha time analog for both film
thicknesses. For context, we note that the alpha time close to the solid
surface at $z=d$ and at the vapor surface are both off scale in Fig. 13. For
the three temperatures shown, the film alpha times expressed as $10^{y}$s at
the solid surface ($z=1d$) for $H=10d(30d)$ are given with cooling by
$10^{y}$s with $y$ = 12.2 (13.4), 15 (16.4), and 18.2 (19.8). These timescales
are smaller for the thinner film reflecting the significant impact of the
vapor interface gradient near the solid surface, and the interface-driven
cutoff of the elastic barrier. The corresponding results at the vapor surface
are essentially the same for the two film thicknesses, with the alpha times of
$10^{y}$s and exponents of $y$ = 9.98, 9.94 and 9.87 as temperature decreases.

Figure 14: (Color online) The dielectric loss like function of Eq. (32) at
temperatures of 436.2 $K$, 427.9 $K$, and 419.6 $K$ as a function of frequency
for PS film thicknesses (a) $10d$ and (b) $30d$. The circles indicate the
logarithm of the inverse bulk liquid alpha relaxation time at the
corresponding temperatures. The vertical dashed lines are a guide-to-the-eye.

The analogous frequency domain results are shown in Fig. 14. For each
temperature, the primary peak frequency is nearly the same for the $H=10d$ and
$30d$ films, which in turn is nearly identical to that of the corresponding
bulk systems. For both film thicknesses, there is a fast (high frequency)
component, with a weak peak or shoulder-like feature attributed to the near
vapor interface dynamics and truncation of the elastic displacement field. Its
location in frequency space is close to the alpha relaxation rate at the vapor
surface. At low frequencies, on the scale shown in Fig. 14, the thicker film
does not exhibit any unusual behavior. In contrast, the thinner film exhibits
a broad wing feature extending to ultra-low frequencies that reflects the
enormous slowing down of relaxation at the solid surface. Another feature of
interest is the peak breadth. For a simple exponential decay process, the
response function is of a Lorentzian form with a full width at half maximum
(FWHM), $W$, of roughly unity in terms of number of decades. The loss peak of
the thick film displays a $W\sim 2$ which is essentially invariant to
temperature. In contrast, the thinner film peak is enormously broader, with
values of $W\sim 6-8$ that increase as temperature decreases.

Overall, these preliminary calculations support the scenario discussed in the
first paragraph of this section, and provide a concrete example of the very
large effects of dynamical gradients in confined films on experimentally
observable film-averaged relaxation functions.

###  IV.5 Undershoot in the $T_{g}$ gradient

As discussed in the two sub-sections above, for $\kappa=0.8$ and 1.0 the solid
interface greatly slows down the interfacial relaxation time compared to its
bulk analog, and the alpha time gradient exhibits a steep double exponential
decay form near the interface. When the thickness of the supported film is
sufficiently large, the impact of the vapor layer on the relaxation of the
half of the film near the solid substrate significantly diminishes. The alpha
time gradient then follows an inverse power law decay far enough from the
interface. The local relaxation time is smaller than its bulk counterpart and
progressively increases toward the film center.

Since the dynamics near the two different interfaces in a supported film are
affected in an opposite manner compared to the bulk, and given the long range
tail of the gradients associated with collective elastic effects, it is
conceivable that an undershoot or overshoot in the normalized mobility
gradient in the film might occur under certain conditions. In this section we
present calculations that investigate this point. We caution that the very
existence of a weak undershoot or overshoot is likely to be a subtle and
nonuniversal issue, sensitive to not only the specific model of a supported
film and the thermodynamic state studied, but also the technical theoretical
approximations adopted and their quantitative consequences for quantitatively
predicting the competing gradients at a solid and vapor interface. For
example, precisely how the displacement field is constructed (especially the
choice of boundary conditions at different interfaces, as discussed in section
II), which impacts the quantitative spatial variation of the collective
elastic barrier. Such caveats should be kept in mind when considering the
significance of the calculations presented below in the context of the
validity of ECNLE theory. Indeed, the core ECNLE theory predictions in prior
work and the present article are not tied to any possible existence of subtle
undershoots or overshoots in mobility gradients.

Figure 15 shows calculations of $T_{g}(z)/T_{g,bul}$ for $\kappa=0.8$ and 1.0
at rather large film thicknesses of $H=50d$, $80d$, and $100d$. One sees a
very low amplitude undershoot (less than a 1 $\%$ effect) at $z/H\sim 0.1$ is
predicted. The tiny nature of this effect reinforces the potential concerns
expressed above, although the results may also motivate looking for such a
feature in simulations. Of course, the latter would be a very challenging task
given the small amplitude of this feature. We note that the simulation studies
in thick supported films reported in Refs. 15 ; 16  might be suggestive of a
weak non-monotonic variation of a mobility gradient, but those results are not
definitive for multiple reasons. Another reason for caution concerning the
existence of such an undershoot feature is that we expect that when the
supported film is ”thin enough” the influence of the vapor surface on the
dynamics of the entire film becomes sufficiently large that the subtle
undershoot feature could disappear. This is indeed what the present theory
predicts, as seen from our results for $H=30d$ and thinner supported films in
Figs. 7 and 10, which seems to be consistent with various simulations of
polymer films 13 ; 24 . Future theoretical, simulation and experimental work
is required to more deeply investigate whether such weak undershoots or
overshoots in dynamic gradients truly exist.

Figure 15: (Color online) The normalized $T_{g}$ gradient in supported films
of thicknesses $H=50d,80d,$ and $100d$ plotted versus position in the film
normalized by the film thickness. The results are for (a) $\kappa=0.8$ and (b)
$\kappa=1$. Here, $T_{g,bulk}=428.3$ $K$ corresponding to the typical
experimental vitrification time scale criterion of 100 s.

##  V Summary and Discussion

We have extended ECNLE theory to analyze the gradients of the alpha relaxation
time and glass transition temperature, and the corresponding film-averaged
quantities, to the geometrically asymmetric case of finite thickness supported
films of variable substrate-fluid coupling. The latter involves a complex
surface science problem and can be very material specific. In principle, a
substrate can either slow down or speed up the dynamics at the interface
relative to the bulk behavior due modification of the caging constraints in
the dynamic free energy which nucleates a mobility gradient that is spatially
transferred into the film. When the film is sufficiently thin, the spatially
heterogeneous interfacial relaxation induced by the solid and vapor interfaces
are dynamically coupled at both the local cage and longer-range collective
elasticity levels. Although increasing the film thickness dramatically reduces
the dynamic coupling between the two interfaces with regards to the local
dynamics, sharp interfaces of any kind truncate the elastic barrier and lead
to generic long-range power law tails in dynamic gradients. Near the
interface, the alpha time gradient can be described by a double-exponential
form, with an amplitude that depends on density, temperature, and $\kappa$ but
with an intrinsic decay length scale that is nearly invariant to thermodynamic
state. Overall, the continuous mobility gradient can be qualitatively
visualized as composed of three regions: near the solid surface, near the
vapor surface, and a “flattened” or “bulk-like” region in the film center. The
gradient superposition approximation that includes the power law decay term
associated with collective elasticity can provide a rather accurate
description of the numerical ECNLE theory predictions for the spatial
distribution of relaxation time within the film, particularly in the central
region of the film, and for films not too thin. The emergence of near
substrate dead layers, strong spatial gradient induced broadening of time and
frequency dependent relaxation functions, and the prediction of an unusual but
very weak and subtle non-monotonic evolution of dynamic gradient in
sufficiently thick and cold films, have also been briefly discussed.

We suggest our new results can be tested using simulations that carefully
design in the extent of perturbation of the bulk alpha time at a hard
substrate by tuning, for example, its corrugation, mechanical stiffness, or
introducing appropriate surface-fluid interactions as recently done by Simmons
et al. 7 ; 24 ; 51 . Although direct experimental measurement of such
dynamical gradients in supported films is very difficult, our results may
provide insight to the highly varied behaviors observed for supported polymer
films depending on the nature of the fluid-surface interactions.

Another interesting issue is the question of the level of predictive power of
the model and theory. This mainly involves gaining a priori insight to the
”proper” value of $\kappa$ for specific surface-fluid systems. This might be
achieved based on atomistic computational studies of the very local solid-
fluid interface, and/or calibration of the theory against a specific
experimental measurement sensitive to the near substrate physics that can
allow the deduction of this key parameter, which can then be used to make a
priori predictions for other dynamical questions or properties. The most
obvious route to achieving the latter is to use fine scale simulation to
determine the alpha time of the fluid in the first layer against the substrate
and require the theory reproduces it by choice of a single number, $\kappa$.
Such an “at the surface” local calibration strategy could then endow
predictability of the theory for the rest of the dynamical gradient over a
wide range of temperatures and film thicknesses.

Our model and theory can potentially be further developed to study more
complex systems and issues such as: (i) dynamical gradients of the polymer
alpha relaxation time induced by solid particles or fillers in nanocomposites,
(ii) the effects of a densified near surface layer on the film dynamics, and
(iii) capped films with two solid interfaces of either identical or different
levels of fluid-surface coupling as encoded in the parameter(s) $\kappa$. We
emphasize that the basic theoretical ideas for the dynamic free energy
gradient have been formulated for any geometry. For example, for a non-flat
(curved) solid interface, the pinning fraction of particles at the surface is
not 0.5 corresponding to half of the dynamical cage, but rather will depend on
the radius of curvature of the interface.

Concerning capped films with two solid surfaces, we note that the analytic
formula of Eq. (26) would seem to imply that in a “thick enough” film the tail
of the elastic barrier gradient would result in slightly faster dynamics in
the film center than in the bulk, despite the huge slowing down in the near
surface region. Such a possibility may seem counterintuitive, but much future
work is required to investigate this point within ECNLE theory since we have
not performed full numerical calculations for such systems, and Eq. (26) is
based on an approximate analytic analysis that naively superimposes asymptotic
behaviors. For any such subtle effect, it is unclear if the latter analytic
simplification reflects what ECNLE theory really predicts. Moreover, what is
predicted for small effect by the present formulation of ECNLE theory could be
sensitive to the details of the approximate construction of the elastic
displacement fields and boundary condition choice discussed in section IID.

Finally, if attractive substrate-film interactions are present, this will
induce a short-range density gradient orthogonal to the interface. Such a
consequence of interfacial interactions was previously treated 3  to model the
densified layer(s) and can be extended to the present variable substrate-fluid
coupling model. Explicit treatment of in-plane spatial variation of the alpha
relaxation time, and the explicit consequences of a substrate-fluid attractive
forces, remain open challenges.

###### Acknowledgements.

We acknowledge many enlightening and informative discussion of the issues in
this article with Professor David Simmons.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

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