Anisotropy of Shear Relaxation in Confined Thin Films of Unentangled

Oct 6, 2015 - We show that the classical Rouse model unsurprisingly fails to predict the thin-film relaxation functions in response to out-of-plane sh...
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Anisotropy of Shear Relaxation in Confined Thin Films of Unentangled Polymer Melts Brendan C. Abberton, Wing Kam Liu, and Sinan Keten* Theoretical and Applied Mechanics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States S Supporting Information *

ABSTRACT: The anisotropic shear relaxation functions of confined thin films of unentangled polymer melts are measured via nonequilibrium step−strain simulations of in-plane and outof-plane shear using the finitely extensible, nonlinear-elastic (FENE) model. We show that the classical Rouse model unsurprisingly fails to predict the thin-film relaxation functions in response to out-of-plane shear, due in part to non-Gaussian conformation statistics in the dimension perpendicular to the sub/superstrate. Using an alternate empirical model for the outof-plane response, we quantify decreases in the plateau modulus GP⊥, relaxation time λ⊥, and viscosity η⊥ and an increase in the logarithmic relaxation rate r⊥ as functions of film thickness, and we discuss these anisotropic changes in stress-relaxation properties in terms of structural/conformation changes on the microscopic level, namely the relative contraction and non-Gaussian quality of polymer conformations in the dimension normal to the substrate and the resulting phenomenon of cooperative relaxation. We then incorporate these into a semiempirical extension to the Rouse model which closely predicts our computational results and which will be useful for further study of polymer thin films.

I. INTRODUCTION Thin polymer films have increasingly numerous applications in sensors, micro/organic electronics, coatings, multilayered materials, nanocomposites, microfluidics, microelectromechanical systems (MEMS), membranes, biotechnology, and medicine. Advances in these applications are calling for increasingly thinner films approaching the characteristic dimensions of polymer chains, giving rise to challenges in describing their thermo-mechanical properties for reliable designs. Computational and experimental research in thin films has indicated that confinement-induced spatial variation in polymer mobility on different length-scales contributes significantly to the macroscopic relaxation times and glass transition behavior. In supported polymer thin films, changes in the glass-transition temperature Tg have been observed in early studies 1,2 and later further explored by a wealth of experimental3−13 and computational studies.14−28 The range and magnitude of confinement effects have been singled out by systematically varying the boundary conditions (free-standing film, supported film, and confined film) and surface/polymer chemistry. Systematic coarse-grained molecular dynamics (CGMD) simulations29,30 have recently aimed to elucidate the effects of interfacial energy and intermolecular cohesive forces on local and average glass transition temperatures in supported PMMA films using atomistically informed potentials.31 These studies revealed that the average Tg in a supported thin film increases linearly with the energy of attraction to the substrate up to a plateau value; that local © XXXX American Chemical Society

deviations from bulk Tg extend tens of nanometers into the film, in agreement with both experimental results32 and theoretical treatments;33,34 and that the effects of confinement and free surfaces are stronger for rubbery films than for glassy films, which is useful to interpret controversial findings on polymers that can show either appreciation or depreciation of Tg depending on confinement conditions. However, studies have shown that the confinement effect extends beyond changes in T g . Changes in polymer conformation, density, relaxation times, chain-end alignment and segregation (leading to increased free volume near the interface), directional dependence of mobility (increases in lateral and total mobility accompanied by a decrease in perpendicular mobility), and entanglement topology have also been observed.14−21,35,36 Moreover, significant changes in the local viscoelastic properties of polymers in thin films and near interfaces have been measured.37−44 It was found using atomicforce-microscopy-based techniques that low-molecular-weight polystyrene shows liquid-like behavior at the surface though glassy in the bulk.41,42 Temperature- and rate-dependent lateral force microscopy measurements in combination with time− temperature superposition were used to determine that the relaxation time at the surface obeys an Arrhenius law in Received: June 3, 2015 Revised: September 15, 2015

A

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Macromolecules temperature with a much lower apparent activation energy compared to the bulk.43,44 Stress relaxation in polymers, which is the source of viscoelasticity, arises from motions that occur over disparate length and time scales, and the dynamics of these motions can change in seemingly contradictory ways in response to confinement. For example, experimental observations of segmental dynamics, i.e., motions on the scale of several monomers, may in some cases show confinement-induced rate increases, while observations of whole-chain movement show marked decreases,45−47 suggesting that the behavior of the chain is not necessarily the direct sum of the contributions of the smaller units that compose it.48,49 The key challenge in predicting the dynamic mechanical performance of a polymer thin film is then to quantify the confinement-induced changes in constitutive behavior on both the segmental (i.e., Rouse-like) and whole-chain (i.e., reptationdominated) levels, as functions of film thickness. This requires investigation of polymer films in both the unentangled and entangled regimes. While an exponential increase in the reptation relaxation time with decreasing film thickness was predicted theoretically by Semenov,49 a corresponding treatment for segmental modes does not exist. We present in the current work the computationally derived stress relaxation behavior of thin films of unentangled polymers confined between weakly interacting sub- and superstrates as a function of film thickness. In this analysis, we seek to answer (1) whether the Rouse model can be modified to treat the stress relaxation of thin films, (2) what changes occur in the relaxation behavior in response to chain confinement, and (3) what molecular mechanisms are responsible for these changes. We model the polymers as freely bending chains of monomers connected by finitely extensible, nonlinear−elastic (FENE) bonds. We calculate the stress relaxation functions in response to in-plane and out-of-plane simple shear deformations via nonequilibrium molecular dynamics step-strain simulations. The film-thickness dependence of the plateau moduli GPi , logarithmic relaxation rates ri, relaxation times λi, and the viscosities ηi are established. We discuss the changes in stress relaxation properties in terms of confinement-induced microstructural changes within the polymer film, namely the gyration tensor profiles in the interphase regions which indicate increased contraction of polymer chains in the dimension normal to the substrate. For the sake of clarity, we define the mathematical terms relevant to the geometry and physics of the problem at hand in Table 1. These terms will be used frequently throughout the following.

Table 1. Mathematical Definitions term

definition

N ρ T h b∥ b⊥ Q0 or Q0 Q∞ or Q∞ S Rg∥

number of bonds per chain bead density temperature film thickness Kuhn length in the plane of the film Kuhn length out of the plane of the film zero-film-thickness result for quantity Q bulk result for quantity Q chain gyration tensor radius of gyration in the plane of the film, Rg∥2 = Sxx + Syy

Rg⊥

radius of gyration out of the plane of the film, Rg⊥2 = Szz

Rg

total radius of gyration, Rg2 = Rg∥2 + Rg⊥2

κ

relative chain contraction out of the plane of the film, κ = 1 − 3Rg⊥2/⟨Rg2⟩∞

G⊥(t) GP⊥ r⊥ λ⊥ GR∥(t) GR⊥(t) λR∥ λR⊥ μ(t) Gκ⊥(t)

measured relaxation function for out-of-plane shear fitted plateau modulus for out-of-plane shear fitted logarithmic relaxation rate for out-of-plane shear fitted relaxation time for out-of-plane shear Rouse relaxation function for in-plane shear Rouse relaxation function for out-of-plane shear Rouse relaxation time in the plane of the film Rouse relaxation time out of the plane of the film secondary relaxation function for out-of-plane shear total relaxation function for out-of-plane shear, Gκ⊥(t) = ⟨GR⊥(t)μ(t)⟩ static Gaussian parameter

f

9g

⊥

2

surrogate gyration profile, 9 g 2 = f α R g 2 with mapping ⊥ ⊥ exponent α

⎡ ⎛ r ⎞2 ⎤ 1 UFENE − WCA = − KR 0 2 ln⎢1 − ⎜ ⎟ ⎥ + 4ϵ ⎢⎣ 2 ⎝ R 0 ⎠ ⎥⎦ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ϵ, r < 21/6σ ⎝r⎠ ⎦ ⎣⎝ r ⎠

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ULJ = 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ , r < 2.5σ ⎝r⎠ ⎦ ⎣⎝ r ⎠

(1)

(2)

Bulk polymer melts were generated with an initial monomer density of ρ = 0.85σ−3 using a partially self-avoiding randomwalk code and equilibrated using a soft cosine potential,50 followed by a Monte Carlo bond-swap algorithm52 in combination with the potential described by eqs 1-2. They were further equilibrated at temperature T ≈ 1ϵ/kB and pressure P ≈ 0ϵ/σ3 in the isobaric−isothermal (NPT) ensemble using a Langevin thermostat and a Berendsen barostat. Films of different thicknesses h were then generated with an additional wall-avoidance criterion such that their average density was that of the equilibrium zero-averagepressure case for the bulk systems; namely, ρ = 0.8867σ−3. Implicit (i.e., smooth/unstructured) LJ walls with the same potential parameters as the nonbonded monomer interactions were used to bound the film geometries in the z-direction, while the films were periodic in the x- and y-directions. The films were equilibrated using an initial soft-potential push-off followed by a multiple-step NVT procedure involving repeated activation and deactivation of the bond-swap algorithm, which greatly accelerates the equilibration process. The total time

II. METHODOLOGY Bonded monomers interact via the finitely extensible nonlinear elastic (FENE) potential and Weeks−Chandler−Anderson (WCA) potential, which is the 12−6 Lennard-Jones (LJ) potential using a cutoff radius of 21/6 σ,50,51 as in eq 1. Nonbonded monomers interact via the LJ potential using a cutoff radius of 2.5 σ, such that the attractive well is included, as in eq 2. B

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Macromolecules spent with the bond-swap algorithm active was approximately equal to the bulk relaxation time, which is considerably longer than required to properly equilibrate even significantly entangled systems in bulk.53 All of the systems examined in this work consisted of 376 chains with 85 monomers per chain, whereas the entanglement length of the FENE model is approximately Ne ≈ 85,54 so the polymers in our simulations are in the unentangled regime. This approximates the conditions of a flexible, unentangled polymer melt confined between weakly interacting interfaces. Stronger adhesion, covalent bonds, or electrostatic interactions with the interface are likely to produce different or even contradictory changes in stress relaxation properties. Structured interfaces will also produce different results, especially concerning dynamics in the x- and y-directions, but these differences will depend greatly on adhesion strength and the actual structure used. We leave these items for future work, and instead focus on the effects of general confinement on generic polymer systems. Equilibrium molecular dynamics simulations were performed on both the bulk and thin-film systems in the NVT ensemble at T = 1ϵ/kB using a Langevin thermostat. The gyration tensor S and the z-component of the center of mass zmol of each chain were output at intervals of 120τ over a period of 48,000τ. The stress-relaxation dynamics were measured via nonequilibrium molecular dynamics simulations under prescribed deformations. The deformations were applied over an interval of 12τ in order to approximate step-strains, and block averages of the stress tensors were output during relaxation every 1.2τ. Zero time was taken to be the midpoint in the deformation process, and the first two decades of stress data were discarded in order to eliminate the effects of the finite deformation time, similar to the methods of Hou et al. in their study of stress relaxation in bulk polymers.55 This step-strain method allows us to capture the entire relaxation function at once, but is less useful than constant-strain-rate methods in examining nonNewtonian viscosities (unless via the Cox−Merz rule56) and flow-induced changes in structure and optical behavior.57,58 Under assumptions of isotropy and incompressibility, the constitutive relationship for a polymer melt is shown in eq 3, where G(t) is the relaxation modulus, B = F·FT is the left Cauchy-Green deformation tensor (where the isochoric deformation gradient F = ∂x/∂X = x∇0 is the map from the undeformed coordinates X to the deformed coordinates x), and Δσ is the viscous Cauchy overstress, which decays to zero as t → ∞.

Figure 1. Deformations applied and corresponding relaxation functions determined. (a) Simple shear in the xz/yz-planes (out-ofplane shear) determine G⊥(t), while (b) simple shear in the xy-plane (in-plane shear) determines G∥(t).

relaxation functions: (a) G⊥(t), which controls the stress response to out-of-plane shear, and (b) G∥(t), which controls the stress response to in-plane shear. In our simulations we use γ∥ = 2 and γ⊥ between 2 and 10. Larger values of γ⊥ are required for thinner films in order to reduce the need for additional statistical sampling. The applied deformation gradient associated with each deformation is also shown in each panel of Figure 1. The apparent relaxation functions for these deformations are then G⊥(t) = Δσiz(t)/γ⊥ with i ≠ z and G∥(t) = Δσxy(t)/γ∥, where we have included the step-strain assumption. For bulk systems, these relaxation functions will be identical and are expected to agree with the classical result of the Rouse model61 shown in eq 4, where ρ is the monomer density, kB is Boltzmann’s constant, T is the absolute temperature, N + 1 is the polymer chain length, λR is the Rouse time (the longest relaxation time relevant to the Rouse chain), and p = 1, 2, ..., N is the normal mode index. GR (t ) =

∫−∞ ∂G(t∂t−′ t′) (B(t′) − 1) dt′

⎛ 2p2 t ⎞ exp ∑ ⎜− ⎟ ⎝ λR ⎠ p=1 N

(4)

In the Introduction, we implied that the Rouse model of stress relaxation, broadly applicable to unentangled polymers in bulk, requires modification for thin films. It is known that, although Gaussian quality is retained in the directions parallel to the substrate (in x and y),62 the chain conformations do not remain Gaussian in the direction perpendicular to the substrate (in z).63,64 This non-Gaussian quality invalidates the normal mode transformation used in the Rouse model, and as such there is no known analytical solution. However, as a starting point, we consider the intermediate case where conformations are anisotropic, but still Gaussian in each dimension, as we wish the end result to trivially reduce back to the classical Rouse model when asymmetry and non-Gaussian quality are not present. In this case, the geometrical asymmetries present in the system can be incorporated into the Rouse model, in which case GR∥ is still given by eq 4 (replacing λR with λR∥), while GR⊥ is given by eq 5 (see Supporting Information for derivation),

t

Δσ (t ) =

ρkBT N+1

(3)

However, in thin-film systems, spatial symmetry is broken, and the assumption of isotropy is no longer valid. In general, the relaxation modulus should be replaced with a fourth-order tensor function (G(t) → Gijkl(t)) to be doubly contracted with the strain measure; if we relax the incompressibility assumption, this tensor would contain five independent relaxation functions. On the basis of applications of interest in nanocomposites and layered materials where load transfer is accomplished according to shear-lag or tension-shear-chain models,59,60 we consider in this work the stress response to two specific deformation types: (a) simple shear in the xz- (or yz-) plane (out-of-plane shear) by a strain magnitude γ⊥ and (b) simple shear in the xy-plane (in-plane shear) by a strain magnitude γ∥, as illustrated in Figure 1, parts a and b, respectively. Thus, we will measure two C

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Macromolecules where the Kuhn lengths bi and the characteristic times λRi = (N + 1)2bi2/(π2Di) are functions of local chain conformations (and segmental mobility in the case of the latter, due to dependence on the diffusion constants Di). ρkBT ⎛ b 2⎞ ⎜⎜1 + ⊥2 ⎟⎟ 2(N + 1) ⎝ b ⎠ N ⎡ ⎛ ⎞⎤ ∑ exp⎢−p2 t ⎜⎜ 1 + 1 ⎟⎟⎥ ⎢⎣ λR ⊥ ⎠⎥⎦ ⎝ λR p=1

GR ⊥(t ) =

(5)

From eqs 4 and 5, we define the parallel and perpendicular Rouse plateau moduli (GPR∥ and GPR⊥, respectively) as follows. G RP =

ρkBT , N+1

G RP⊥ =

ρkBT ⎛ b 2⎞ ⎜⎜1 + ⊥2 ⎟⎟ 2(N + 1) ⎝ b ⎠

(6)

We discuss in the following sections that this theoretical extension remains insufficient to predict the results we observe, due to the non-Gaussian nature of conformation statistics in the ⊥ direction, and the increased need to account for additional interactions between chains; however, we will also show in the discussion that an empirical extension of the model that incorporates cooperativity and uses surrogate distributions dependent on the degree of non-Gaussian behavior of each chain leads to an accurate reproduction of the observed computational results. Before this, in order to define and quantify the observed relaxation properties of the films overall, we use the empirical expression in eq 7, where GPi is the plateau modulus, ri > 0 is the logarithmic relaxation rate, λi is the relaxation time, and the subscript i is either ⊥ or ∥. This expression approximates the Rouse result for ri → 1/2, the Doi−Edwards reptation result for ri → 0,65 and the reptation result incorporating anisotropic alignment of chain ends for 0 < ri < 1/2.66 ⎛ t ⎞−ri ⎛ t⎞ Gi(t ) = GiP ⎜ ⎟ exp⎜ − ⎟ ⎝ λi ⎠ ⎝ λi ⎠

(7)

The (zero-rate) viscosity ηi, defined as the integral of the relaxation function for t > 0, can be then be calculated as ηi = GPi λiΓ(1 − ri), where Γ is the gamma function. Note that this model exhibits singularities in Gi(t) as t → 0 and in ηi as ri → 1, but it is sufficient to capture the important aspects of the relaxation functions measured here. The loss and storage moduli are discussed in the Supporting Information.

Figure 2. (a) Gyration profiles within the film of thickness h = 33.03 σ, and (b) film-averaged gyrations as functions of film thickness. Solid lines overlapping the ⟨Rg⊥2⟩ and ⟨Rg∥2⟩ data in (b) are fitted analytical functions of the type ⟨Sij⟩(h) = ⟨Sij⟩0 + Δ⟨Sij⟩exp(−h0/h), while the solid line overlapping the ⟨Rg2⟩ data is a linear combination of the two.

III. RESULTS The profiles of the gyration tensor components Sij within the film of thickness h = 33σ are shown in Figure 2(a), where zmol is the position of the molecular center of mass, while the filmaveraged structural metrics are shown as functions of h in Figure 2(b). A clear interphase region of thickness proportional to the bulk radius of gyration ⟨Rg2⟩∞1/2 (the ∞ sub- or superscript denotes the bulk result) can be seen in the profile, wherein the relative contraction of chains in the ⊥ dimension, which we define here as κ = 1−3Rg⊥2/⟨Rg2⟩∞, increases monotonically as the confining interface is approached. The solid lines in Figure 2b are fitted analytical functions of the form ⟨Sij⟩(h) = ⟨Sij⟩0 + Δ⟨Sij⟩ exp(−h0/h), where ⟨Sij⟩0 is the result for a hypothetical film of zero thickness (the zero superscript denotes the zero-thickness result), and Δ⟨Sij⟩ is the

difference between the bulk result and the zero-thickness result. We can approximate the zero-thickness result by observing the behavior of chains at the interface (the far left in Figure 2a). At the interface, we see that the perpendicular component ⟨Rg⊥2⟩ approaches zero, while the trace ⟨Rg2⟩ = ⟨Rg⊥2 + Rg∥2⟩ has approximately the same value as in the bulk (⟨Rg2⟩∞/3 = 7.66σ2, in excellent agreement with the pioneering computational study on bulk polymer by Kremer and Grest50). Thus, we assume ⟨Rg⊥2⟩0 = 0 and ⟨Rg∥2⟩0 = ⟨Rg2⟩∞. Note that the bulk and zero-thickness results for ⟨Rg⊥2⟩ imply ⟨κ⟩ must be between 0 (bulk) and 1 (zero-thickness). D

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Macromolecules G⊥ and G∥ determined from simulations of simple shear deformation in the xz/yz- and xy-planes are shown in Figure 3,

Figure 4. (a) Plateau modulus GP⊥, logarithmic relaxation rate r⊥, relaxation time λ⊥, and viscosity η⊥ as functions of film thickness h taken from the fitted functions in Figure 3a and normalized by their bulk values. Solid lines are separate functions fitted to the property results to guide the eye. The relaxation time and viscosity increase linearly, while the plateau modulus increases as GP⊥/GP∞ ⊥ = exp(−h0/ h), and the logarithmic relaxation rate decays as r = 1 + (r∞ ⊥ − 1) exp[−(h0/h)δ]. (b) Schematic definitions of stress relaxation properties.

Figure 3. Apparent relaxation functions (a) G⊥(t) and (b) G∥(t) determined from simple shear simulations. Solid lines are the results of the analytical model developed here. Relaxation of shear stresses in the xy-plane mimics bulk behavior for each film, while relaxation in the xz/ yz-plane becomes faster as the film thickness decreases. The legend in part b applies to both subfigures.

4 ∞ 3 linearly to theirs (λ∞ ⊥ = 1.59 × 10 τ and η = 110ϵτ/σ ). Note that expanding exp(−h0/h) at infinite film thickness gives 1− h0/h, so the empirical fits for the relaxation modulus and gyration tensor components are similar to the interphasevolume-fraction model Q = Q∞(1−h0/h)+Q0 h0/h, where Q is an observed property or structural metric, except that the unreasonable divergence to ± ∞ as h → 0 is removed.

parts a and b, respectively. There is negligible change in G∥ as the film thickness is varied. However, significant changes occur in G⊥ as the film thickness is decreased. Increasing confinement leads to a decrease in GP⊥ and λ⊥, and an increase in r⊥, which all contribute to reduction in η⊥. These properties are plotted as functions of film thickness h in Figure 4a, along with a schematic of their meanings in Figure 4b. The solid lines in Figure 4b are fitted analytical functions to guide the eye. We find that GP⊥ increases as ∼ GP∞ ⊥ exp(−h0/h) and r⊥ decays as δ ∼1 + (r∞ ⊥ − 1) exp[−(h0/h) ] to their respective bulk values 3 ∞ (GP∞ ⊥ = 0.67ϵ/σ and r⊥ = 0.54), while λ⊥ and η⊥ both increase

IV. DISCUSSION That we see no change in G∥(t) implies a cancellation of opposing effects on its characteristic time. From Figure 2, we E

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Macromolecules see an increase in ⟨Rg∥2⟩ ∼ b∥2, and since λR∥ ∼ b∥2/D∥, the simultaneous increase in lateral mobility (increase in D∥, not shown here) must be strong enough to balance the in-plane stress relaxation dynamics. On the other hand, decreases in ⟨Rg⊥2⟩ ∼ b⊥2 and D⊥ must not balance, leading to a decrease in λR⊥ ∼ b⊥2/D⊥, which, along with a decrease in GPR⊥ ∼ b⊥2/b∥2, causes accelerated relaxation of G⊥(t). Note that these scaling relations come from the Gaussian assumption (true for ∥ but not for ⊥), but they should still hold qualitatively for contracted non-Gaussian chains. Consider the limiting case ⟨Rg⊥2⟩ = 0, wherein simple shear in the xz/yz-plane would induce sliding of platelet-like, noninterpenetrating molecules parallel to each other; these motions would be opposed by minimal stress. If we consider the case where the chains are Gaussian in each dimension but have the same observed gyration tensor profiles, the anisotropic Rouse theory alone would predict much smaller changes in λ⊥ and GP⊥ compared to our observations. That is, if we average eq 5 over the observed conformations using the 2 2 ∞ 2 2 2 2 adjustments λR⊥/λ∞ R⊥ = 3Rg⊥ /⟨Rg ⟩ and b⊥ /b∥ = 2Rg⊥ /Rg∥ , where the gyration terms are the equilibrium measurements, the result (shown in Supporting Information) is still far from Figure 3a and predicts no change in r⊥. Our results for the bulk (h → ∞) systems agree with the classical Rouse theory,61 wherein ri = 1/2 from the integral approximation of the sum in eq 4. Similar results were obtained by Hou et al. for low-molecular-weight samples in a computational study of stress relaxation in bulk polymers.55 However, it is clear from Figures 3 and 4 that the logarithmic relaxation rate nearly doubles within the range of thicknesses studied here. Again, considering the case where the chains are Gaussian but have the same observed gyration tensor profiles, this result cannot be predicted by the anisotropic Rouse model, as the logarithmic relaxation rate of GR⊥(t) depends similarly on the power of the mode number p within the exponential sum. This change in the logarithmic relaxation rate suggests cooperative (i.e., positively coupled) relaxation between neighboring chains. This is a secondary effect of the increasing contraction in the ⊥ dimension within the interphase. The fast relaxation of the “pancaked” chains dynamically releases constraints on neighboring chains, causing them to relax faster than they normally would. A similar constraint-release phenomenon famously occurs in the reptation of entangled chains, in which context the relaxation function can be multiplied by itself to account for the effects of the changing environment.55,67 In our case, a cooperativity gradient exists throughout each film, so we cannot simply square GR⊥(t). The observed stress relaxation properties are replotted against the average relative ⊥-contraction ⟨κ⟩ in Figure 5a. All properties but r⊥ decay to zero as κ increases, while r⊥ increases linearly. This indicates that an appropriate secondary relaxation function to choose should be proportional to GR⊥ raised to the power κ. Thus, we define the relaxation function Gκ⊥(t) = ⟨GR⊥(t)μ(t)⟩, where μ is unitless and approximated empirically in eq 8, and the ensemble average can be approximated by averaging over the gyration tensor profiles in each film. We have binned the conformation profiles into 376 bins per film, which for each film leads to 376 values each of the monomer density ρ, the variance ratio b⊥2 /b∥ 2 = 2Rg ⊥ 2/Rg ∥2 , the perpendicular contraction κ = 1 − 3Rg⊥2/⟨Rg2⟩∞, and both the in-plane and

Figure 5. (a) Properties of stress relaxation normalized by bulk values (plateau modulus G⊥P /G⊥P∞, logarithmic relaxation rate r⊥/r⊥∞, ∞ relaxation time λ⊥/λ∞ ⊥ , and viscosity η⊥/η⊥ ) vs relative ⊥-contraction ⟨κ⟩. (b) Anisotropic Rouse model with cooperative relaxation Gκ⊥(t) = ⟨GR⊥(t)μ(t)⟩ eqs 5, 8, and 9. The legend here is the same as in Figure 3b. 2 2 ∞ out-of-plane relaxation times, λR∥/λ∞ R∥ = 3Rg∥ /(2⟨Rg ⟩ ) and ∞ 2 2 ∞ λR⊥/λR⊥ = 3Rg⊥ /⟨Rg ⟩ , respectively.

μ(t ) κ ⎧ ⎡ N ⎛ ⎞⎤⎫ ⎛ ⎪ ⎪ b⊥2 ⎞ 1 1 1 2 ⎢ ⎥ ⎟ ⎜1 + 2 ⎟⎟ ∑ exp −p t ⎜ ≈⎨ ⎜ λ + λ ⎟⎥⎬ ⎢ ⎪ 2(N + 1) ⎜⎝ b ⎠ p=1 R⊥ ⎠⎦⎪ ⎝ R ⎣ ⎩ ⎭ (8)

In addition to this local cooperativity, conformations that are folded across the confining interfaces and others whose tails are segregated toward them can extend significantly into the interior of the film, allowing interaction of the midplane of the F

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Macromolecules film with the soft interphase region,63,64 leading to enhanced relaxation. Moreover, those chains forming loops can relax quickly via de Gennes’ sliding mechanism.68 In either case sliding or nonlocal cooperativitythe non-Gaussian quality exhibited by a chain is a good indicator of whether its relaxation will be sped up. Thus, we can calculate a static measure of Gaussian quality as a function of location within each film and use this to construct surrogate gyration tensor profiles that demonstrate local speed-up and optimally reproduce the computational results. That is, if we make the substitution R g 2 → 9 g 2 = f α R g 2 when calculating the relaxation time, ⊥ ⊥ ⊥ plateau modulus, and relative contraction of each chain used in the ensemble average Gκ⊥(t) = ⟨GR⊥(t)μ(t)⟩, where the static Gaussian parameter f is defined in eq 9, then we find that a mapping exponent of α ≈ 1.556 (see Supporting Information) reproduces our computational results with the most fidelity, as shown in Figure 5(b), where the legend is the same as in Figure 3b. The static Gaussian parameter is calculated for each of the aforementioned 376 conformation bins by integrating a virtual Gaussian distribution with variance equal to the perpendicular gyration tensor component within the bounds of the measured gyration tensor profile (i.e., z− and z+ are the minimum and maximum chain centers of mass of the relevant profile). Thus, smaller values of f(zmol) indicate that the existence of the Gaussian distribution at zmol is less plausible. f (zmol) =

∫z

z+ −

⎡ ⎤ (z′ − zmol)2 ⎥ exp⎢ − ⎢⎣ 2R g 2(zmol) ⎥⎦ 2πR g 2(zmol) ⊥ ⊥

and the incorporation of side functional groups can lead to additional energetic penalties associated with conformation changes, which the Rouse model would underestimate even in the bulk state. In addition, strongly interacting or structured interfaces will produce similar conformation profiles, but may lead to the slowing down of stress relaxation dynamics, causing increases in λ⊥ and λ∥, and decreases in r⊥. In this case, the secondary relaxation function, the static Gaussian factor, or both might be raised to negative exponents.

V. CONCLUSIONS As a first step toward establishing a full picture of the constitutive behavior of confined polymer thin films necessary for describing dynamic mechanical properties in increasingly sophisticated applications, we have reported here the relaxation functions G⊥(t) and G∥(t) of a film of unentangled FENE polymer chains confined between weakly interacting interfaces in response to out-of-plane and in-plane pure shear deformations, respectively. The influence of the confining walls and the breaking of spatial symmetry lead to increasingly anisotropic behavior as the film thickness is reduced, causing no change in G∥ but drastic changes in G⊥, and the classical Rouse model is no longer applicable as a result. Using an alternate empirical model to quantify stress relaxation properties (plateau modulus GP⊥, logarithmic relaxation rate r⊥, relaxation time λ⊥, and viscosity η⊥), we find that r⊥ increases from the classical result of ≈1/2 to nearly 1 as the film thickness is decreased within the range studied here, while GP⊥, λ⊥, and η⊥ all decrease. We explain these changes in terms of correlated microstructural changes in polymer conformations using an extended Rouse model that accounts for conformational anisotropy, nonGaussian statistics, and cooperative relaxation. The chains contracted in the ⊥-direction offer significantly less resistance to out-of-plane shear deformations, leading to fast local relaxations and propagation of dynamic constraint releases toward the center of the film. We hope that further investigations into the mechanics of thin films will address the effects of cross-linking, molecular weight (specifically, differences between entangled and unentangled systems), more complicated interactions between chains and with the confining surfaces, and the offset in temperature from that of the glass transition on stress relaxation properties. Additional consideration should be given to equilibrium structural and dynamic measures (e.g., gyration vs mobility) in order to determine which has stronger control over stress relaxation dynamics in which situations, and exactly why. We also hope that future investigations will address the response of films to uniaxial and biaxial extensions in order to develop a fuller picture of the complete constitutive equations.

dz′

(9)

This result reproduces all three trends observed in the computational results (increased logarithmic relaxation rate, reduced plateau modulus, and reduced relaxation time), with some slight quantitative discrepancies between the predicted and observed properties, which is not surprising, as the cooperativity treatment only accounts for interactions in an approximate way. It is interesting to note that preliminary results of the equilibrium measurement of G⊥ for the film h = 11.30 σ via the computationally expensive Green−Kubo relation69,70 (see Supporting Information) slightly overshoot the computational results (magenta crosses), and thus also the predictions of the cooperative surrogate model. This indicates that large deformation and finite extensibility may play a concerted role in further accelerating stress relaxation. This role could be clarified by examining the evolution of the gyration tensor profiles and local stresses during both the imposition of the step strain and the following relaxation process, although this would require a considerable amount of statistical sampling. Nonetheless, the predictive capability of our model is more than sufficient for constitutive modeling in continuum treatments of thin films and layered nanocomposite materials made with flexible polymers and weakly interacting interfaces. Furthermore, the model reduces to the classical Rouse model when polymer conformations are symmetric and Gaussian, as in this case κ → 0, f → 1, and Rg⊥2 → ⟨Rg2⟩∞/3. However, whether equilibrium conformation statistics are sufficient to predict the dynamic relaxation behavior of a wide range of polymers, or polymer−substrate combinations, remains to be seen. We believe it works here because the polymer chains are flexible and interact most significantly with the confining interfaces through the reflection of their conformations. For inflexible or nonlinear polymers, additional potential energy terms (e.g., angle and dihedral interactions)

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01204. Derivation of anisotropic Rouse model, predictions without cooperative relaxation, and preliminary Green− Kubo results (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(S.K.) E-mail: [email protected].. G

DOI: 10.1021/acs.macromol.5b01204 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Author Contributions

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The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

The Air Force Office of Scientific Research (W.K.L., Grant No. FA9550-14-l-0032), the National Science Foundation (S.K., DMREF Award CMMI-1235480), and the Army Research Office (S.K., Grant No. W911 NF-13-1-0241). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS B.C.A. is grateful to T. Wylie Stroberg, John Moore, and Margarita Rayzberg for their participation in multiple helpful discussions. Resources of the Quest and Hercules computing clusters at NU were utilized in this research.

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ABBREVIATIONS

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REFERENCES

CGMD, coarse-grained molecular dynamics; FENE, finitely extensible nonlinear elastic; LJ, Lennard-Jones; NPT, number− pressure−temperature; NVT, number−volume−temperature; MEMS, microelectromechanical systems

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