Effect of Interfaces on the Glass Transition of ... - ACS Publications

Jun 9, 2015 - Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States. ABSTRACT: We investigate the glass transition ...
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Effect of Interfaces on the Glass Transition of Supported and Freestanding Polymer Thin Films Ronald P. White, Christopher C. Price, and Jane E. G. Lipson* Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, United States ABSTRACT: We investigate the glass transition behavior in polymer thin films using a model equation-of-state approach, which involves molecular parameters whose values are determined from fits to bulk information only (pressure− volume−temperature and surface tension data). Following an earlier proof-ofconcept application to freestanding polystyrene (PS) films, here we both extend the study to poly(methyl methacrylate) (PMMA) films and generalize the model so that it is applicable for either freestanding or supported films. In the case of the freestanding PMMA film, model predictions for the Tg suppression (relative to bulk) as a function of film thickness are in very good agreement with the corresponding experimental data, reflecting the fact that freestanding PMMA films are evidently less perturbed by the presence of free surfaces than those made of PS. We then turn to the case of PMMA films supported on a silica substrate by accounting for possible polymer−substrate interactions, such that when these are switched off the same model maps smoothly back to the case of a PMMA freestanding film. We then probe the origin of the interaction required such that the model can capture the experimentally observed Tg enhancement for supported PMMA films while also accounting for the relative lack of impact on the Tg behavior of supported PS films relative to their freestanding counterparts. Finally, we make connections between related experimental and simulation studies and our own results for the differences between supported PMMA and PS films.

1. INTRODUCTION The study of the effect of confinement on glassy polymeric systems has continued to draw strong research interest over the past two decades.1−5 Polymer thin films have been a particularly popular system to study. Beyond important material and engineering applications, they show a diverse range of behavior, depending both on the polymer’s characteristic properties and on the nature of the confinement. Films of interest include both freestanding films (two free surfaces) and films supported on a substrate. These supported films have one free surface and one surface in contact with a substrate (e.g. silica, gold, a substrate coating, a polymeric underlayer, or other). The effect that confinement has on a film’s glass transition temperature (Tg) has been of particularly strong interest and much remains to be understood. Although many experimental investigations have shown that there can be a strong and varied change in the film Tg relative to the corresponding bulk value, other investigations have shown little or no change in Tg, and adding to this is evidence that such effects can depend on cooling rate.6−9 In general, experimental results show that for freestanding films (e.g., refs 10−12), as the film gets thinner (e.g., less than 100 nm) the film Tg is suppressed (lowered) compared to the corresponding bulk Tg value. For supported films, the behavior can be quite diverse. In these cases, the film Tg can either be enhanced or suppressed relative to the bulk, depending on the polymer, the substrate, and also on the possible existence of an additional underlayer.13−16 To give one example, the Tg of polystyrene (PS) supported on a silica substrate decreases © XXXX American Chemical Society

relative to bulk, whereas that of supported poly(methyl methacrylate) (PMMA) increases relative to bulk. A variety of experimental methods have been used to study these systems, a subset of which fall into the category of probing film thickness as a function of temperature. For a fixed amount of sample, the temperature dependence of the thickness is directly related to the thermal expansion coefficient. In the bulk, this quantity undergoes a discontinuous change on cooling as the sample goes from melt to glass at Tg. Similarly for films, it is the change from meltlike (rubbery) values to glasslike values that is associated with the glass transition. Methods that have been applied to polymer films include ellipsometry,11 Brillouin light scattering (BLS),10 fluorescence labeling methods,17 and X-ray reflectivity.18 In addition to probing the behavior experimentally, there have also been modeling investigations, including both coarse-grained and atomistic simulation,19−24 and a number of theoretical approaches.25−38 The model in the present work is an equation-of-state approach (i.e., one that is built from a thermodynamic point of view). It shares some similarities with other treatments, for example that of Truskett and Ganesan,37 which is also a thermodynamic model for confined systems, where both fluid−fluid and fluid-interface interactions are considered in the total system energy. In addition, there is the model of McCoy and Curro38 where inhomogeneities in the Received: March 10, 2015 Revised: May 22, 2015

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DOI: 10.1021/acs.macromol.5b00510 Macromolecules XXXX, XXX, XXX−XXX

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separate dependencies on h and L2. Also note that the total outside area of the supported film sample is ∼2L2 (because h ≪ L); this is the sum of the areas of the “upper” and “lower” faces (each face having an area of L2). For the supported film, the upper face is a free surface and the lower face is the interface with the substrate, whereas for the freestanding film, both faces are a free surface. The model film sample contains N chain molecules. Each molecule is comprised of r segments, and each segment occupies a single lattice site of volume, v. Nearest-neighbor segments interact with a nonbonded interaction energy, ε. To account for fluid compressibility, not all of the lattice sites will be occupied. Nvac denotes the number of vacant sites. Thus, the total number of lattice sites is Nr + Nvac, and the total volume is V = hL2 = v(Nr + Nvac). The volume fractions of occupied (ϕ) and vacant sites (ϕvac) are ϕ = Nr/(Nr + Nvac) and ϕvac = Nvac/ (Nr + Nvac). We begin with a formulation of the film’s model internal energy. First consider the simplest van der Waals type mean field energy expression for a bulk sample, which is given by Ebulk = (1/2)rNϕzcε. This expression is appropriate for the polymer segment−segment interactions in the interior of the film; however, correction terms need to be added to account for the fact that the sample is of finite thickness (i.e., a thin film). The polymer segments near the free surface lack the interactions that would have otherwise extended into the bulk. These missing interactions need to be subtracted. Furthermore, the polymer segments near the substrate have polymer segment− substrate interactions, for which an accounting is required. We make the assumption that only the interactions for segments that are located on an outermost lattice layer need to be adjusted; recall that because h ≪ L essentially all of the outer sites are located on one of the two large faces (either the upper or lower face) each of area L2. The number of sites in a row along one of the long sides is L/v1/3, therefore the number of sites in a lattice layer is (L/v1/3)2. This is the total number of outer sites for a single face (either an upper or lower face). Not all of these sites will be occupied, so in order to estimate the number of segments occupying the outer sites for a single face (i.e., the number of segments in an outer lattice layer), we continue with our mean field view and assume that the density of the entire thin film sample is uniform, even in an outer layer. Therefore, the number of segments in a single outer lattice layer can be approximated as [(L/v1/3)2] × [(Nrv)/(hL2)]. In this picture, this is simply the number of sites in a single lattice layer (first factor) multiplied by the probability (ϕ) that a site is occupied (second factor). We now consider the correction term to the energy that accounts for the effect of the free surface. Noting that (1/2) ϕzcε = (1/2)((Nrv)/(hL2))zcε is the interaction energy assigned to any one segment in the bulk, we assume that we can account for the effect of a free surface by removing some fraction (f) of these interactions for the case of a surface segment, i.e., we subtract (f/2)((Nrv)/(hL2))zcε for each surface segment. Multiplying this by the result for the number of segments in a lattice layer ([(L/v1/3)2] × [(Nrv)/(hL2)]), the total interaction energy lost because of one outer face of free surface is thus [(L/v1/3)2] × [(Nrv)/(hL2)] × [(f/ 2)((Nrv)/(hL2))zcε]. In words, it is the total number of segments in one lattice layer multiplied by the energy lost per segment. In the case of a supported film, this will be the energy adjustment accounting for the upper outer face of free surface, whereas for a freestanding film, this energy adjustment (which

density profile caused by confining surfaces/interfaces are represented as an equivalent simplifying change in the average density. We also note that recent applications by Xu and Freed39 of a thermodynamic model for bulk systems have shown a strong connection of thermodynamic properties to material fragility, noteworthy because some recent experimental studies40 have also connected fragility with Tg shifts in confined systems. In a recent publication,41 we developed an equation-of-state model that was applied to the case of a freestanding film. The model was parametrized on the basis of bulk properties only (pressure−volume−temperature (PVT) and surface tension data), and there were no adjustable parameters upon application to the film. The freestanding film model balanced the entropic drive to expand (entropy increase with increasing volume) against the attractive energy lost between the polymer segments because of the existence of two free surfaces. The balance depends on the relative amount of surface (i.e., the film thickness); it therefore affects the volumetric behavior of the film and thus the intersection point for the melt and glass, viz., the glass transition temperature Tg. Parameterized to bulk PS, the model predictions for the change (suppression) in Tg as a function of film thickness were in good agreement with experimental data for freestanding PS films.10 In this work, we expand the application of the model considerably such that it can also handle the case of supported films reducing to the case of freestanding films in the limit of zero substrate interaction. We demonstrate that depending on the nature of the substrate interactions the model is capable of predicting both Tg suppression and Tg enhancement, and we apply it to study both PMMA and PS films supported on a silica substrate. In addition, we also apply the original freestanding film model to make predictions about unsupported PMMA films, which are then contrasted with our earlier results for PS films. In all cases, comparisons are made with experimental results. The remainder of the paper is as follows: In section 2, we describe the theoretical background for the film model and provide a general derivation to cover both the case of a supported film as well as that of a freestanding film. Section 3 comprises two parts: In the first part, new results are presented for the freestanding PMMA film, which are compared both with experimental data and with our (analogous) previous results for PS. In the second part, results are shown for supported films of both PMMA and PS and are analyzed in terms of what the model predictions suggest regarding the strength of energetic interactions between polymer and substrate. This section also includes further analysis, connecting our results with both experimental and simulation studies from the literature. A summary and concluding remarks are presented in section 4.

2. THEORY AND IMPLEMENTATION The model we describe here is such that as the interactions with the substrate go to zero the model maps from the supported film over to the freestanding film. Therefore, the derivations below will be developed in such a way that application to either freestanding or supported films remains possible throughout. We begin with a compressible lattice model for chain molecules. The lattice coordination number is zc. Although the volume, V, of a bulk sample is characterized by the product L3, the volume of the thin film analogue is described by V = hL2: the product of the film thickness (h) and the area, L2, such that h ≪ L. Note that we are introducing B

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therefore assumed to be constant. We therefore absorb the entire product, fsubϕsubzcεsub, into a single characterization parameter, ωsub = fsubϕsubzcεsub. In principle, ωsub could be fit to experimental polymer−substrate interfacial tension data; in the absence of such data, it can simply be treated as adjustable and used for exploring qualitative trends. Another option is to fit it to supported film data for Tg as a function of thickness. (Here, it is important to note that fitting Tg vs film thickness is not done in the case of the freestanding film because the freestanding model requires only pure component bulk data for parametrization.) More details on quantifying ωsub will be provided in section 3. In examining eq 1, it is useful to note the volume dependence for a sample of constant area. The second term, accounting for the polymer−polymer interaction loss at the free surface and substrate interfaces, is proportional to 1/V2, i.e., proportional to (1/hL2)2. The first term (the van der Waals like bulk term) is proportional to 1/V, and the third term (associated with interactions gained at the substrate) is also proportional to 1/V because ϕsub is taken to be roughly constant. Having obtained an expression for the internal energy, we move now to an expression for the model entropy. Here, we employ a well-known mean field result for lattice chains,42 wherein the number of ways, W, to arrange N chain molecules of r segments on a lattice with Nvac vacancies is given by

is per face) is multiplied by 2 to account for both faces of free surface. Note that none of the characteristic polymer parameters required are free parameters. f will be fit to surface tension data, and the energetic parameter (zcε) will be obtained by fitting to PVT data. Now, we consider the adjustment to the internal energy that is required in order to account for the effect of one surface of a supported polymer film interacting with the substrate. Here, we imagine that polymer segments in the lower surface lattice layer will both lose a fraction of polymer−polymer interactions (ε) and then gain a fraction of polymer−substrate interactions (εsub). We thus imagine this as a two-step process, wherein first we cut out the lower face from the polymer bulk to make a free surface and then bring the substrate material up into contact with this face. As described above, the interaction energy lost for a single polymer segment in the lower lattice layer is ( f/ 2)((Nrv)/(hL2))zcε. Multiplying this by the total number of segments in the lower lattice layer, the total interaction energy lost is [(L/v1/3)2] × [(Nrv)/(hL2)] × [(f/2)((Nrv)/(hL2))zcε]. In exchange, the interaction energy gained for a single polymer segment in the lower lattice layer is (fsub/2)(ϕsub)zcεsub; thus, the total interaction energy gained is [(L/v1/3)2] × [(Nrv)/ (hL2)] × [( fsub/2)(ϕsub)zcεsub]. Here, ϕsub is the packing fraction for the substrate material (the probability that a site is occupied by substrate material), εsub is the polymer−substrate interaction energy, and fsub is the fraction of a polymer segment’s total possible interactions (zc) that coordinate with the substrate. Finally, the above interaction energy adjustments are used to construct the overall expression for the internal energy of the sample. This overall energy expression comprises, first, a bulklike energy term, which is given by (1/2)Nr((Nrv)/(hL2)) zcε; this term assigns bulklike interior polymer−polymer interactions to all of the polymer segments in the sample. Then, the energy adjustments are included as described above to account for the losses and gains in the interaction energy for the segments that reside in the upper and lower faces. We write the expression for the total energy, E, in a general form that is applicable for both the supported film and the freestanding film (the latter applying in the limit of zero substrate interactions). This general expression is as follows. E=

⎛ rNv ⎞2 1 ⎛ rNv ⎞ rN ⎜ 2 ⎟zcε − fv−2/3L2⎜ 2 ⎟ zcε ⎝ hL ⎠ 2 ⎝ hL ⎠ ⎛ rNv ⎞ 1 + fsub v−2/3L2⎜ 2 ⎟ϕsubzcεsub ⎝ hL ⎠ 2

(r − 1)N (rN + Nvac)! ⎛ zc − 1 ⎞ W≈ ⎜ ⎟ N! Nvac! ⎝ rN + Nvac ⎠

(2)

As noted above, the presence of the lattice vacancies makes the film model compressible. Equation 2 is completely analogous to the enumeration in Flory−Huggins theory for the case of an incompressible polymer−solvent mixture; in that case, Nvac is replaced by the number of solvent molecules. An expression for the entropy, S = kB ln W, follows after using Stirling’s approximation and dropping the volume-independent additive constants. ⎡ rN + Nvac ⎤ ⎡ rN + Nvac ⎤ S = NkB ln⎢ + kBNvac ln⎢ ⎥ ⎥ ⎦ ⎣ rN ⎣ Nvac ⎦ ⎤ ⎡ hL2 ⎤ ⎛ hL2 − rNv ⎞ ⎡ hL2 = NkB ln⎢ ⎥ ⎥ + kB ⎜ ⎟ ln⎢ 2 v ⎣ rNv ⎦ ⎠ ⎣ hL − rNv ⎦ ⎝

(1)

As noted above, the first term is the bulk energy term. The second term accounts for the loss of polymer−polymer type interactions. It is the total loss of polymer−polymer interactions for two outer faces because the total loss of polymer−polymer interactions ends up being the same amount whether it is a freestanding film or a supported film. The third term is for the gain in polymer−substrate interactions in the case of a supported film. In the case of a freestanding film, the limit of zero substrate interactions, εsub = 0, applies and the third term would vanish. The third term in eq 1 is defined in terms of the quantities, fsub, ϕsub, and zcεsub, in keeping with the theoretical lattice model construction; however, we will not resolve values for each of them explicitly. ϕsub for the solid substrate is not expected to vary strongly with temperature and pressure (less sensitive dependence compared to the polymer) and is

(3)

The expressions for the energy (eq 1) and entropy (eq 3) are both independent of temperature. The entropy depends only on volume, whether it be by adjusting the thickness h or the area L2. However, because of the free-surface term and the substrate term (for supported films), the model energy expression depends on both thickness and area independently. (Therefore, two of the three variables V, h, L2 must be specified.) Temperature dependence appears because the balance of energy and entropy is defined in the form of the Helmholtz free energy, A = E − TS. Constructing this expression from eqs 1 and 3 gives C

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Macromolecules A=

⎛ rNv ⎞2 1 ⎛ rNv ⎞ rN ⎜ 2 ⎟zcε − fv−2/3L2⎜ 2 ⎟ zcε ⎝ hL ⎠ 2 ⎝ hL ⎠ ⎡ hL2 ⎤ ⎛ rNv ⎞ 1 + v−2/3L2⎜ 2 ⎟ωsub − NkBT ln⎢ ⎥ ⎝ hL ⎠ 2 ⎣ rNv ⎦ ⎤ ⎛ hL2 − rNv ⎞ ⎡ hL2 − kBT ⎜ ⎥ ⎟ ln⎢ 2 v ⎝ ⎠ ⎣ hL − rNv ⎦

γ=

2 1 ⎛ ∂A ⎞ 1 −2/3⎛ rNv ⎞ ⎜ ⎟ ⎜ ⎟ zε fv = − ⎝ hL2 ⎠ c 2 ⎝ ∂L2 ⎠ N , T , hL2 2

(6)

In taking the derivative an increase in L2 corresponds to an increase in the area of each face thus leading to the factor of 1/2.

3. RESULTS AND DISCUSSION 3.1. Freestanding Films. The first step in applying the model to a polymer thin film system is to determine the values of the molecular characterization parameters. These are r (segments per molecule), v (volume per lattice site), zcε (energy per near neighbor interaction, scaled by the coordination number), and f (fraction of interactions lost by a segment at a free surface). The parameters for PS and PMMA are obtained by fitting to their respective bulk physical property data. First, r, v, and zcε are obtained by fitting the equation for the pressure (eq 5) to experimental pressure−volume− temperature (PVT) data.44 Here, eq 5 is applied in the bulk limit where L2/rN → 0; thus, the fourth and fifth terms are zero. The resulting values for r, v, and zcε are then utilized in eq 6, where the one remaining parameter, f, is obtained by fitting the surface tension as a function of temperature at atmospheric pressure to the corresponding experimental data.45 Figure 1 illustrates our characterization of bulk PMMA. The lower panel shows the fit of the model equation of state (eq 5) to experimental PVT data (to determine r, v, and zcε). The

(4)

A is a function of four independent variables: N, T, and two of the three variables h, L2, and V. In eq 4 the first, fourth, and fifth terms remain upon reducing to an N, V, T (bulk) system and become equivalent to the expression for the Helmholtz free energy in the lattice fluid theory of Sanchez and Lacombe.43 For a freestanding film, the third term is zero because in that case ωsub = 0. Using the result for the Helmholtz free energy, we can derive an equation for the pressure by envisioning a scenario in which the area of the sample (2L2) is held fixed while the thickness h varies, for instance, with temperature or external applied pressure. At equilibrium, a change in h at constant N, T, and L2 leads to an amount of work done by the system, −(∂A/∂h) dh = Fnorm dh, where Fnorm is the total force exerted normal to the large faces, i.e., it is the normal force at the free surface and at the substrate interface (supported film) or other free surface (freestanding film). The pressure normal to the large faces is thus −(∂A/∂h)/L2 = Fnorm/L2 and is given by ⎛ 1 ⎞⎛ ∂A ⎞ P = − ⎜ 2 ⎟⎜ ⎟ ⎝ L ⎠⎝ ∂h ⎠ N , T , L2 ⎡ ⎤ ⎛ N ⎞ hL2 = kBTv−1 ln⎢ 2 ⎥ + kBT (1 − r )⎜ 2 ⎟ ⎝ hL ⎠ ⎣ hL − rNv ⎦ 2 ⎛ L2 ⎞⎛ rNv ⎞3 1 ⎛ rNv ⎞ + v−1⎜ 2 ⎟ zcε − 2fv−5/3⎜ ⎟⎜ 2 ⎟ zcε 2 ⎝ hL ⎠ ⎝ rN ⎠⎝ hL ⎠ ⎛ L2 ⎞⎛ rNv ⎞2 1 + v−5/3⎜ ⎟⎜ 2 ⎟ ωsub 2 ⎝ rN ⎠⎝ hL ⎠

(5)

The first three terms describe the pressure when surface and/or substrate effects are negligible. The fourth term accounts for the adjustment to the pressure because of the loss of polymer− polymer interactions experienced by polymer segments at both of the outer film faces (freestanding and supported films). The fifth term is the adjustment to the pressure coming from the gain in polymer interactions when there is a substrate (supported film). Given that the near-neighbor polymer− polymer interaction energy (ε) is intrinsically negative, the fourth term serves to increase the pressure over that of the corresponding bulk sample modeled (with the same parameters) at the same N, T, and V = hL2. The fifth term, however, will serve to reduce the pressure (because εsub and ωsub must also be intrinsically negative.) It is useful to define the interfacial tensions for each type of interface: γ, corresponding to free surface, and, γsub, associated with the substrate interface. In this work, we will only apply the surface tension result for the free surface (γ) and therefore restrict ourselves to the freestanding film model (εsub, ωsub = 0). When the long sides of the film are increased at fixed temperature and fixed volume (fixed hL2, where again, L ≫ h), the surface work done on the system is γ2 dL2. Thus, taking the derivative of the freestanding film Helmholtz free energy with respect to L2, we obtain

Figure 1. Model fitting results. Upper panel shows model fit (curves) to pressure−volume−temperature data for PMMA (points, ref 44). Results are plotted in the form of isobars of P = 0, 20, 40, 60, 80, and 100 MPa (from top to bottom). Lower panel shows model fit (curve) to surface tension data for PMMA (points, ref 45) at atmospheric pressure. D

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Macromolecules model is well able to capture the behavior of bulk PMMA over the experimental temperature range of 385−450 K and pressure range of 0−100 MPa. The lower panel of Figure 1 shows the fit to the PMMA bulk experimental surface tension data (to determine the f parameter). The complete set of characteristic parameters are summarized in Table 1 for both PMMA and PS. Table 1. Polymer Molecular Parametersa polymer

r

v (mL/mol)

zcε (J/mol)

f

PS PMMA

6414.5 6393.4

14.089 12.530

−11975 −12413

0.322 0.280

a

For comparison, the r parameters for PS and PMMA in the table both correspond to polymer molecular weights (M) of 100 000 g/mol. Note that the PVT data for PS (ref 44) is for M = 110 000 g/mol; in that case, r thus scales to a value of 7055.9.

On the basis of these results, PS segments adjacent to a free surface may be expected to lose out on a slightly higher fraction (by about 15%) of slightly weaker (by about 70 J/mol, or 3.7%) neighboring interactions than will PMMA segments in the same position. Using our model characterization for the bulk behavior, we apply eq 5 to predict how PS and PMMA film thickness varies as a function of temperature. To do so, we first determine the amount of material per area (N/L2) that gives a polymer sample of some desired thickness, h°, at a particular reference temperature and pressure; we choose the reference temperature to be 370 K and the pressure to be 0.1 MPA. (P = 0.1 MPa is used throughout all calculations.) This sample area per amount of material (the L2/N value) is then kept constant throughout any subsequent calculations for that h° thick film, e.g., how its thickness, h, varies from h° as T is changed, how the specific volume varies, etc. This mirrors the setup used in ellipsometry measurements. The upper panel of Figure 2 summarizes our predictions for the relationship between specific volume and temperature for freestanding PMMA films of varying thickness (varying h°). As expected, at h° = 10 000 nm the theoretical curve (solid black) matches the experimental bulk melt data. The remaining curves in the set show model predictions for how the melt specific volume changes with temperature for films that diminish in thickness. We see a noticeable, albeit small, shift once the thickness (h°) is down to 70 nm, with films thinner than that exhibiting a significant shift away from the bulk curve. Also shown in the figure are the bulk data for glassy PMMA, connected by a dashed line as a visual aid. In what follows, we assume that the V(T) behavior of the glassy phase remains fixed because of the kinetically arrested nature of this phase. In the bulk, the glass transition is identified as the intersection temperature between the melt and glassy V(T) curves, reflecting the discontinuous change in the coefficient of thermal expansion. As the film becomes thinner, the predicted intersection with the glassy data occurs at lower and lower temperatures, signifying a depression in the glass transition that becomes more marked as the film thickness decreases. Qualitatively, the model predicts identical behavior for PS and PMMA, as illustrated in the lower panel of Figure 2. However, for PS there is a somewhat stronger shift as the film becomes thinner. We ascribe this difference to two factors: First, as noted above, PS segments adjacent to the free surface lose a slightly greater percentage of neighboring interactions than PMMA segments, and this would have an increasing

Figure 2. Model freestanding thin-film specific volume as a function of temperature at constant pressure (P = 0.1 MPa): PMMA shown in upper panel and PS in lower panel. Results in each case are shown for films of four different reference thicknesses, h° = 10, 30, 70, and 10 000 nm (values of h at T = 370 K and P = 0.1 MPa). The model results (shown as solid curves) correspond to the melt/rubbery state. Also shown are the experimental bulk data (points, ref 44); points are shown for both the melt and glassy states (0 MPA; i.e., very close to atmospheric pressure). The dashed curve interpolates and extrapolates the glassy data.

impact as the films became thinner. Second, we also could point to a difference in cohesive energy density (CED) between the two polymers. For example, in a recent application of our locally correlated lattice (LCL) theory, we have found that PMMA has a stronger CED than PS by about 50 J/mol (when characterized and compared under the same temperature and pressure conditions of 425 K/0.1 MPa).46 This would suggest that at the same temperature and pressure it would take more energy to swell a film of PMMA than of PS. Here it is important to clarify that our results are compared to experimental data for Tg shifts that are independent of molecular weight (over a range of molecular weights, M < 500 000 g/mol). For higher PS molecular weights, Pye and Roth47,48 have discovered two Tg’s: an upper Tg, associated with a strong transition and found to be molecular-weightindependent in the range studied (and in good agreement with results in ref 10 for M’s ranging from 115 900 to 347 000 g/ mol), and a lower Tg, much weaker in strength and molecularweight-dependent. These authors correlate the stronger upper transition to the solidification of 80−90% of the film. They further note a connection of this upper Tg to creep complianceE

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Macromolecules derived Tg’s reported in O’Connell et al.,49 which is sensible because the creep studies are a measure of the film’s entire mechanical properties. Our model does not currently incorporate a specific chain-entropy term capable of reflecting the possible packing effects that might be expected when the chain radius of gyration becomes commensurate with the film thickness. We therefore focus on experimental results corresponding to the (evidently more fundamental) molecular-weight-independent Tg. Our prediction of a weaker film effect for PMMA than for PS is most clearly illustrated in Figure 3, which shows model

counter the free-surface effect operating at the other interface, producing an overall Tg enhancement. However, because of lack of relevant experimental data, characterization of pure component bulk PMMA alone cannot capture a substrate effect. In the absence of direct experimental quantification of the polymer−silica interaction strength, we have thus introduced (section 2) the simplifying combined parameter ωsub = fsubϕsubzcεsub, which represents the energetic contribution to the part of the film’s free energy that accounts for the film− surface interaction. For convenience in assessing trends, we now re-express this interaction in terms of a reference value, ωref, that we might expect to be typical. Specifically, we construct ωref from a representative set of individual values for fsub, ϕsub, and zcεsub. We take a value of fsub = 0.3, noting that the free surface f values in Table 1 are close to this; for ϕsub, we take a value of 0.9, noting that occupied site fractions for lattice models of polymer melts are typically 0.8 or greater and for a solid phase substrate the occupied site fraction should be at the larger end of this spectrum. For zcεsub, we take −12 000 J/mol, noting that this is close to the values in Table 1 and very close to the interaction energy in the PS melt. Using these values, we obtain ωref = −3240 J/mol. We think of ωref loosely, as a PS-like reference value, because it is roughly the amount of energy associated with the segments occupying a plane in a typical molten polymer similar to PS. The working parameter that now factors in the difference between this PS-like reference system and the particular system of interest (e.g., PMMA interacting with a silica surface) is denoted by λ, giving ωsub = λωref. This result for ωsub is what we now use in the equation of state (eq 5, section 2). We therefore reduce the problem of accounting for the film−substrate interaction effect to that of adjusting a single parameter, λ; it accounts for the strength with which the boundary layer segments interact with a substrate. Specific volumes of PMMA films are shown in Figure 4, which illustrates the results of varying λ from 0.00 (upper left; recovering the freestanding film case) to 10.0 (lower right), thus representing an order of magnitude increase in strength of PMMA−silica interaction relative to the reference polymer interaction. For the case of λ = 10.0, the film specific volume is considerably reduced (at a given value of T) relative to what the bulk sample would show. The hypothetical set of Tg−thickness shifts are predicted to be at the successive intersections of the new film V(T) curves with the bulk glassy curve (dashed line), the latter being extrapolated to higher temperatures. As the figure illustrates, the predicted Tg for these (λ = 10) films lie considerably above the bulk Tg. By increasing the strength of the substrate interactions (λ), the model thus predicts that there is a transition from Tg suppression to Tg enhancement. Two other panels are shown in the figure: In the upper right panel is the result when λ = 3.75, which is the value needed in order for the free-surface and substrate effects to cancel exactly, such that for all film thicknesses the bulk results are recovered. Also shown (lower left panel) is what happens when λ = 5.3, viz., that the prediction is qualitatively similar to that using λ = 10.0 but less extreme. This is the value of λ giving film predictions that agree best with experimental results.15 Comparison with experiment is shown in Figure 5, where the model predictions for Tg versus PMMA film thickness for all four values of λ are shown, along with experimental data for both freestanding12 (diamond) and supported15 (circle) PMMA films. As expected, using λ = 0 recovers the freestanding film prediction, which agrees well with experiment, as noted in the previous section. For λ = 3.75, the model

Figure 3. Model freestanding film Tg as a function of film thickness, h°, (curves) and comparison with experimental data (symbols). The upper curve and symbols (circles and triangles) correspond to the PMMA freestanding film, and the lower curve and symbols (circles) correspond to the freestanding PS film. Experimental data for the freestanding PS film are from ref 10 for a molecular weight (M) of 115 900 g/mol, and data for the PMMA film are from ref 12, where the circles correspond to atactic PMMA of M = 509 000 g/mol and the triangles correspond to M = 159 000 g/mol.

results for the Tg shift with film thickness (h°) compared with the experimental data10,12 for the two polymers. The theory does an excellent job of capturing the similarities, which involve an initially modest decrease in Tg from the bulk as the sample is reduced in thickness and then a rapid plunge in the glass transition for films thinner than about 40 nm. The theory shows a slightly better match for the somewhat gentler falloff in the PMMA Tg at 40 nm: it predicts a precipitous drop in Tg for PS at about 30 nm, which is roughly 10 nm thinner than experimental results would indicate. However, the agreement is still excellent, particularly given that the sample differences are so ably captured by simple characterization of just the bulk behavior in both cases. 3.2. Supported Films. A focus on PMMA in this paper was initially motivated by the goal of making a comparison with PS systems to demonstrate that for freestanding films the theory could capture material differences while predicting a qualitatively similar effect. However, another interesting feature of PMMA films is that samples supported on a silicon oxide surface exhibit a notable increase in Tg relative to the bulk. This contrasts with the data for supported PS films, which behave similarly to their freestanding counterparts in that both show Tg suppression. Presumably, there is a significant degree of polymer− substrate interaction in supported PMMA films, which acts to F

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Figure 4. Comparing effect of the strength of substrate interactions (λ) on the specific volume as a function of temperature at P = 0.1 MPa. Cases shown for model supported thin films with λ = 3.75, 5.30, 10.0, and the freestanding thin film (λ = 0). Results are shown for model melt state PMMA films (solid curves) of four different reference thicknesses h° = 10, 30, 70, and 10 000 nm (values of h at T = 370 K and P = 0.1 MPa). Also shown are the experimental bulk data (points, ref 44) for both the melt and glassy states. The dashed curve interpolates and extrapolates the glassy data. (See text for detailed definition of λ.)

The most striking result is the excellent agreement between the model results using λ = 5.3 and experiment. Evidently, working through the energetic implications of our simple one-layer PMMA−substrate interaction, an energetic interaction about five times that of the reference (PS/PS-like interaction) captures to a significant degree the behavior of a PMMA film supported on silica. Having quantified the strength of the PMMA−substrate interaction needed to capture the experimentally observed behavior, the question is now whether that is consistent with the explanation that the interaction originates because ester groups in the repeat units of PMMA form hydrogen bonds with the silanol groups on the silica surface; this has been proposed as the underlying mechanism for Tg enhancement. The rationale here is that the polymer−substrate attraction causes local ordering and compression, which accomplishes some of the work needed for transitioning to the glassy phase and therefore allows the melt to glassify at a higher temperature than that required in the bulk. There are number of ways to interpret and compare the model polymer−substrate energy obtained above. For instance, given that there are 6.4 model segments for each actual PMMA polymeric repeat unit, using λ = 5.3 and ωref = −3240 J/mol translates to a prediction that there is 110 kJ of substraterelated interaction energy for every mole of polymeric repeat units that are situated next to the substrate surface (i.e., 6.4 × 5.3 × 3240 J/mol = 110 kJ/mol). Each PMMA repeat unit contains a carbonyl group that can form hydrogen bonds with

Figure 5. Model PMMA film Tg as a function of film thickness, h°, (curves) for four different strengths of substrate interactions (λ) and comparison with experimental data (symbols). Cases are shown for model supported thin films with λ = 3.75, 5.30, 10.0, and the freestanding thin film (λ = 0). The case of λ = 5.3 fits well to the experimental data (ref 15, circles) for a PMMA supported film on a silica substrate. The curve for the model freestanding film (λ = 0) is a prediction, requiring bulk molecular parametrization only, and is in good agreement with corresponding experimental data (ref 12, diamonds).

predicts that the film will behave as the bulk, aside from a small decrease in Tg for film thicknesses on the order of a nanometer. G

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to that observed for freestanding PS films. As the figure shows, it is a model value of λ = 1.0 that optimizes agreement with experiment, which is significantly different from the value of 5.3 required to capture the PMMA−silica interaction strength. Indeed, a value of λ = 1.0 suggests a picture in which the PS− substrate interaction strength is similar to that associated with neighboring PS surfaces. Related to this scenario, there have been some experimental DSC studies by Simon and coworkers54,55 carried out on stacked PS films corresponding to various implementations of “polystyrene supported on polystyrene”. The authors note that their data are reasonably consistent with other data on supported PS films, e.g., Tg suppression similar to that reported in ref 13 that would again make them fairly close to the λ = 1 curve for PS-like interactions. Other tests in refs 54 and 55 involved films of PS stacked with polyisobutylene (PIB); though qualitatively similar Tg suppression was observed, it was evidently muted relative to a comparable PS-on-PS film. This suggests that a more sophisticated treatment of the polymer−polymer interface might be interesting to pursue. Adding to the comparison of PS interactions with different kinds of substrate material is the work by Bodiguel and Fretigny56 and Wang and McKenna,57 where a liquid-dewetting method was used to obtain Tg’s of PS films in contact with liquid glycerol. Here, it was observed that the Tg suppression was weaker than that of PS films on silica substrates. Those results, shown in Figure 10 of ref 57, would be reasonably captured through a model curve with a λ value of about 2 (between 1 and 3). One might therefore infer that the PS− glycerol interaction is effectively about twice as strong as the PS−silica-substrate interaction. Other notable PS-containing studies include the work on films of PS star polymers by Green and co-workers.58 Here, compared to the linear polymer (which showed the most Tg suppression), as the number of arms increased or as arms got shorter, systems generally showed less Tg suppression and ultimately, in some cases, Tg enhancement. With respect to qualitative interpretation in the context of our model, this would correspond to a stronger interaction (λ) between the branched polymer and substrate, and this might not be surprising given the expectation (discussed in ref 58) that star polymers should tend to be attracted toward an interface (relative to linear). Returning to a consideration of PS versus PMMA, we can further compare the model silica substrate interaction for these two polymers with some experimental measurements in Johnson et al.59 where competitive binding was investigated between PS and PMMA adsorbed on silica from solution. For instance, the strength of the PMMA−silica interaction (quantified in terms of a model60 parameter, χ, for the excess free energy) was found to be a factor of 4 stronger than the corresponding PS−silica interaction. This relative strength of 4:1 appears to be very consistent with the results presented here because as we noted above the λ value for PMMA is found to scale up the reference PS-like interaction, ωref, by a factor of 5.3. Some interesting comparisons can also be drawn from recent simulations by Lang et al.61 investigating the glassification of nanoscale coarse-grained Kremer−Grest bead−spring model polymers. The systems studied were nanolayers of two different types of polymers, one a low-Tg polymer and the other a highTg polymer. The layers were in contact with each other, stacked in alternating fashion in one coordinate direction, and infinite/

silanol (OH) groups on the silica, so it makes sense to compare the above estimated model energy value (110 kJ per mole of repeat unit) with typical hydrogen bond energies,50 which commonly range from 10 to 40 kJ/mol (and are sometimes stronger). Although the model value is on the same scale, it appears to overestimate the anticipated hydrogen bonding energy. However, taking 6.4 model surface segments per repeat unit assumes that all atom types in PMMA are equally likely to be adjacent to the surface, e.g., this assumes there would be no difference between carbonyl oxygens and backbone carbons in their frequency of being in direct contact with the surface. In fact, favorable energetic interactions may well cause a preferential ordering in order to maximize hydrogen bonding, even at the expense of an entropic penalty. This would increase the probability that a carbonyl oxygen, as opposed to a backbone carbon, would be adjacent to the substrate. An alternative interpretation would be to simply consider the model energy in terms of energy per surface area. From the fit to bulk PVT data, we determined the size of a PMMA lattice site to be 12.530 mL/mol. It follows that there are 13.2 sites per square nanometer and multiplying this by a typical melt occupied-site fraction of 0.85, with λ = 5.30 and ωref = −3240 J/mol, we calculate 193 kJ/(mol nm2). (For scale, 1 nm spans the total length of about 7 polymeric carbon−carbon backbone bonds.) If the silica surface can contain up to almost 5 OH (silanol) groups per square nanometer (see ref 51), then dividing the energy per area value above by 5 gives about 40 kJ/ mol for each hydrogen bond, a number that is clearly much closer to typical values but does assume a maximized coordination with the available silanol groups. We now turn to examine supported PS films. Unlike PMMA, PS does not have groups that can participate in H-bonding with silica; therefore, it is not expected to interact as strongly with a silica substrate. Figure 6 shows how the model results for supported PS films are indeed consistent with this picture. A key feature (compared to PMMA) is that the experimental results13,52,53 for PS supported on silica show that there is Tg suppression upon decreasing thickness, qualitatively analogous

Figure 6. PS supported films. Model film Tg as a function of film thickness, h°, for three different strengths of substrate interactions (λ = 1, 3, and 5; curves) and comparison with experimental data (symbols) for PS films on silica substrates. Data shown are from three sources: circles are from ref 52 with M = 263 000 g/mol, squares are from ref 53 with M = 200 000 g/mol, and triangles are from ref 13 with M = 120 000 g/mol. H

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for film behavior, including Tg as a function of thickness. Our model is able to capture the experimentally observed trends common to the two polymer films and also succeeds in predicting how the two polymers differ, e.g., the lesser perturbation in PMMA film Tg (as a function of film thickness) relative to that observed for PS. Furthermore, the model was generalized such that it is now able to describe both freestanding films (two free surfaces) as well as supported films (one free surface and one surface in contact with a substrate). As such, it now has the ability to describe both Tg enhancement relative to bulk and Tg suppression. A feature of the model is a smooth mapping from the supported film to the freestanding film as the strength of the polymer−substrate interactions (scaled by the coupling variable, λ) is reduced to zero. The model was applied to supported PMMA films on a silica substrate, wherein agreement with experimental results (Tg enhancement as films get thinner) was obtained via adjustment of λ. On reducing λ to 0 while holding all other molecular parameters constant, the model recovered its prediction of the experimental results for PMMA freestanding films (T g suppression). Application to supported PS films revealed that experimental results, which show evidence of little or no interaction between the polymer and a silica substrate, could be accounted for by assuming the polymer−substrate interaction strength to be qualitatively similar to that between two adjacent PS layers. In addition, we analyzed in greater detail the energetic strength indicated by λ that was required to bring the model into agreement with the supported PMMA film results. Strong H-bonding interactions are expected for this system between PMMA and silanol groups on the silica. We evaluated whether the value of λ provided a reasonable approximate prediction of the actual polymer−substrate interactions. The model indicates that the interaction energy with the surface is 193 kJ/(mol nm2), which is a reasonable value considering typical single-Hbond energies along with the likelihood of multiple silanol groups (i.e., multiple H-bonds) per square nanometer of substrate surface. We anticipate that the simple thermodynamic model described here will prove to be useful as additional experimental data on freestanding and supported polymer films continues to accrue, particularly for cases in which greater experimental characterization results become available regarding the interfacial energetics of polymer films next to a nonpolymeric underlying support.

periodic in the other two directions. An analogy can be drawn to the case of supported polymer films if one considers only the behavior of the low-Tg polymer while taking the high-Tg polymer to play the role of a substrate. In this case, the lowTg polymer is like a capped polymer film with substrate above and below. The simulation results showed that as the attractive interactions (interfacial energy) between segments in different nanolayers fell to zero (which is similar to the present model λ going to 0) the Tg behavior approached that of the corresponding freestanding film (same degree of Tg suppression relative to bulk). Furthermore, in the simulations, as the strength of the attractive interactions between the two types of segments increased (similar to increasing the present model λ), the Tg behavior crossed over from Tg suppression to Tg enhancement, exactly as predicted here in our supported film model. Regarding the high-Tg polymer (which had the stronger polymer−polymer interactions in the simulations), the authors61 found a lower influence on its Tg when the interfacial energy changed. The analogy with the results reported here is that we found PMMA with its stronger polymer−polymer interactions ε to be less perturbed by surface than PS (Figure 2). Experimentally, it has been a challenge to resolve and compare the actual absolute density of different films having different average thicknesses (h°). Very recently, Vignaud et al.62 reported experimental (ellipsometry) measurements for a set of supported thin films of PS on a silicon substrate; this study indicated that film density increases with decreasing film thickness (h°). This contradicts the present model results for supported PS, where we find behavior analogous to that of a freestanding film. However, as shown above, our model does predict the same density−thickness trend for supported films where the substrate interactions are strong enough (e.g., PMMA). One point to bear in mind is that in the recent supported-PS study, the most noticeable density increases occurred for films with thickness less than the characteristic dimensions of the polymer, e.g., 6 nm thickness corresponding to 0.4 gyration radius (Rg). We believe it likely that in such cases the polymers may well be transitioning to 2D behavior and plan to address the possible model impact of this in future work. Finally, some recent simulation results by Hudzinskyy et al.21 for supported PS films show an increase in density in regions near the substrate, becoming more noticeable when model substrate interactions were increased (similar to the present results). The same simulation study also showed a density decrease for regions near the free surface, consistent with the present model when the free surface dominates. These results illustrate the trade-off implicit in applying a uniformdensity approximation, such as used in our approach. However, early efforts we have taken in formulating a multilayer model make clear that the increased level of sophistication to be gained in a multilayer model will almost certainly come at the cost of a considerable gain in complexity.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We gratefully acknowledge the financial support provided by the National Science Foundation (DMR-1403757). C.C.P. also greatly benefited from the support of Dartmouth College via the Neukom Institute and a Zabriskie Undergraduate Fellowship.

4. SUMMARY AND CONCLUSIONS In this work, we have used an equation-of-state approach to model the glass transition behavior in thin polymer films. We have applied the model to freestanding PMMA films and quantitatively compared it to our previous results on freestanding PS films and to experimental data for both. Molecular parameters for both systems were characterized using only bulk data (PVT and surface tension data) for each corresponding polymer, and the theory was then applied to make predictions

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