Percolation model of interfacial effects in polymeric glasses

Transcription

1 Eur. Phys. J. B 72, (2009) DOI: /epjb/e y Regular Article THE EUROPEAN PHYSICAL JOURNAL B Percolation model of interfacial effects in polymeric glasses J.E.G. Lipson 1 and S.T. Milner 2,a 1 Department of Chemistry, Dartmouth College, Hanover NH 03755, USA 2 Department of Chemical Engineering, The Pennsylvania State University, University Park PA 16802, USA Received 15 April 2009 / Received in final form 25 August 2009 Published online 7 October 2009 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2009 Abstract. Recent observations using fluorophore probes of local dynamics in polymer films have provided new insight into the glass transition. Using a percolation model, we predict the local T g in a polymer film, as a function of distance from a substrate or a free surface. Our predictions are in good agreement with the observed elevation of T g near a substrate, whereas the observed reduction of T g near a free surface is too strong to be accounted for by percolation effects. PACS pj Polymers Q- Theory and modeling of the glass transition p Liquid-solid interfaces 1 Introduction The dramatic effect of proximity to an interface on the glass transition in polymers has been the subject of numerous investigations, both experimental and theoretical, in recent years [1,2]. Two cases of particular interest are supported films [3 9], in which there are both air and substrate interfaces, and free-standing films [10 13]. Most experimental investigations report a single glass transition, as a function of total film thickness. We have become interested in the results of a recent series of experiments which yield the local (e.g. for a slice as thin as nm) glass transition in a film, as a function of distance from an interface which may be substrate or air. The availability of such data motivates the need for a theory which can predict both the depth dependence of the glass transition as well as a film-averaged transition temperature. In this paper we progress beyond a percolation model introduced by Long and Lequeux [14], and propose the first predictive theoretical description for the local glass transition as a function of distance from a substrate a level of description sought by but absent from previous approaches [15]. Using our percolation calculations for a face-centred-cubic (FCC) lattice, along with literature data on poly(methyl methacrylate) (PMMA), we predict the depth dependence of the glass transition for a PMMA film supported on a substrate, and insulated from the effects of a free surface by a thick overlayer of PMMA. Our predictions are in reasonable agreement with recent experiments of Torkelson et al. [16]. However, this model does not capture the effect of a free surface on the film glass a smilner@engr.psu.edu transition. We conclude with remarks about the strengths and limitations of this percolation-based approach. Since the initial paper in 1994 by Keddie et al. [3], which reported a significant drop in the glass transition temperature for a supported thin film of polystyrene (PS) relative to the bulk value, there have appeared hundreds of such studies in the literature [17]. Two commonly used experimental techniques in these studies are ellipsometry [10,11] and X-ray reflectivity [12]. Both routes yield results for the average T g of a film as a function of film thickness, and although there is still considerable disagreement in the literature about the details the basic phenomena are clear: free surfaces depress T g by up to tens of degrees within a few tens of nanometers of the surface, whereas strongly interacting substrates serve to raise T g by up to tens of degrees over similar length scales. A large number of studies have also appeared on T g reduction in free-standing films, for which even more dramatic shifts are observed [10,11,13]. For both free-standing and supported films there are still unresolved issues regarding the precise magnitude of the shift versus film thickness, in addition to the effects of changing molecular weight. Some discrepancies may arise from comparison of different experimental probes, which couple to different relaxations on vastly different timescales [2]. The emerging experimental consensus with regard to molecular weight effects appears to be (1) no molecular weight dependence is observed for T g shifts near strongly interacting substrates [4,7], while (2) molecular weight effects are commonly reported for T g shifts in freestanding films [1,2]. In any case, it appears to be widely accepted that in these systems large shifts in the glass transition are observed

2 134 The European Physical Journal B over length scales which are at least an order of magnitude greater than the size of the cooperatively rearranging region (CRR) associated with the glass transition. Of particular relevance to this paper are results from the Torkelson group that exploit the sensitivity to local density of the fluorescence emission of certain dyes [6,8,9]. Through labeling experiments on polymers including polystyrene (PS), poly(methyl methacrylate) (PMMA) and others, data have been collected for experiments in which different layers of the sample have been labelled. Thus, information from a reporting layer of controlled thickness and distance from the substrate and/or air interface is collected. The configuration of special interest involves labeled slabs of varying thickness that are adjacent to the substrate (surface), and are protected from the effects of the surface (substrate) by an over (under) layer several hundred nm thick [6,16]. In this way one can distinguish between the effects of surface and substrate, thereby allowing for a more discriminating test of model predictions. At the most basic level, these experiments raise the question: how does the influence of the interface on the glass transition extend over tens of nanometers into the polymer? Two basic mechanisms have been proposed. (1) Polymer connectivity propagates the enhanced mobility at a free interface deeper into the sample [18,19]. (Though not discussed in these papers, conceivably polymer connectivity may likewise suppress mobility near a stronglybinding interface). (2) Glassy behavior of a material results when small slowly-relaxing regions percolate through the sample [14]. In this paper, we explore this latter percolation [20] mechanism, to see whether it can account for the range and magnitude of local T g shifts in the vicinity of interfaces. Long and Lequeux (LL) first proposed this approach [14], which we now summarize. Above T g, a sample undergoes equilibrium density fluctuations. The relaxation time τ K of a so-called cooperatively rearranging region (CRR) is a strong function of the local density. Above a threshold density ρ, τ K is slower than the experimental timescale (1 s, say). As a sample cools, the average density increases, and the filling probability p that a CRR has density above ρ increases. At T g, slow CRRs percolate, and the system exhibits macroscopic glassy behavior. The size of a CRR has been estimated variously by different authors [21,22]. Lodge and McLeish [23] simply asserted that in absence of special considerations it should be of order the Kuhn length. Merabia and Long constructed a self-consistent dynamical argument which ultimately results in a similar-size estimate [24]. For present purposes we shall estimate the CRR size as a Kuhn length as in reference [23]. LL were particularly interested in the reduction of average T g in freely-suspended films. They identified T g with the percolation threshold for slow CRRs. They argued that in finite-slab geometries, the percolation threshold would be suppressed relative to a bulk sample, because of the crossover from d =3tod = 2 percolation behavior [25]. Because of the diverging correlation length at the percolation threshold, the effect of an interface would be felt at a distance many times the size of a CRR. 2 Theory and results We move beyond the LL approach by predicting local T g shifts in polymeric films with either a free surface or a substrate adjacent. We propose to relate the local onset of glassy behavior at a depth z in a in a semi-infinite slab adjacent to either a substrate or a free surface, with the probability P conn (z) thatacrratdepthz is part of an infinite percolating cluster of slow regions. Although the percolation threshold in such a system does not depend on z [26], we expect P conn to depend on distance from the substrate and free surface, as we now show. Strongly interacting substrates, to which the polymer strongly adsorbs, may be argued to increase the density and decrease mobility within a few nm. We model this as a layer of CRRs adjacent to the substrate that are always slow. Then, a CRR near the substrate may be part of a percolating slow cluster, simply by the presence of a short path of slow CRRs leading to the substrate. This will enhance glassy behavior near the substrate. In contrast, sites near the free surface will have lower values of P conn relative to sites well away from any interface, because of the absence of percolating paths that would connect the site via the missing upper halfspace of material. To implement this model, we have collected percolation results for bond percolation on an FCC lattice. The simulation volume is a three-dimensional array with n = 128 sites per side. To account for the presence of a strongly interacting substrate, we fill all bonds connected to sites in the first layer; thus, a site which connects to the glassy layer is therefore immediately part of a spanning cluster. The opposite boundary is a free surface. The probability that a site on a given layer z is on a transversely spanning cluster is then determined as a function of bond filling probability p. By transversely spanning we mean that a cluster extends across the entire simulation volume in the x or y directions. We then randomly fill the bonds with probability p, and use the Hoshen-Kopelman algorithm [27] to find any spanning clusters, recording the number of sites in each layer that are part of a spanning cluster. This process is repeated for 100 realizations, and the average P conn (z; p) thereby computed. In Figure 1 we show the percolation results as a series of plots. Each curve represents the probability, P conn (z; p), that a site randomly chosen from a given layer z is part of a spanning cluster, plotted as a function of filling probability p. The thick curve gives the result for bulk percolation on the FCC lattice, P conn = A(p p c ) β ; the percolation threshold (p c =0.119) and exponent β =0.41 are well known [20]. Curves to the left of the bulk result correspond to layers near a substrate, with those nearest to the substrate being most strongly perturbed from bulk behaviour, with elevated values of P conn. Curves to the right of the bulk result correspond to layers near a free surface, with those nearest to the free surface being most strongly affected, with suppressed values of P conn. Although we find

3 J.E.G. Lipson and S.T. Milner: Percolation model of interfacial effects in polymeric glasses 135 Fig. 1. (Color online) Probability P conn of a site (CRR) in a given layer of an FCC lattice being part of a transversely spanning cluster, as a function of bond filling fraction p, for layers near a strongly interacting substrate (curves to left) or near a free surface (curves to right). Dark solid curve is result for sites far from any interface. Results shown are for nearsubstrate layers z =2, 4,..., and near-surface layers z = n 1,n 3,... Horizontal line is P conn =1/Q. reduction of P conn, the effect is quite weak compared to the substrate effect. Note that sites near a substrate have a finite probability for any nonzero p (scaling as p z ) of connecting to the substrate, and thus have no percolation threshold (see Fig. 1). Also, observe that while P conn is suppressed for sites near a free surface, p c is independent of z as expected. We must therefore refine our percolation criterion for identifyingalocalt g. In place of the percolation threshold, we propose that the glass transition be identified with the condition that a randomly chosen site has a finite probability 1/Q (where Q = 12 is the FCC lattice coordination number) of being on a percolating slow cluster. In this way, a randomly chosen site at T g would typically have at least one neighbor on a slow cluster. To predict the z-dependence of the local glass transition, we first find the filling fraction p (z) at which P conn (z; p (z)) = 1/Q. Graphically (see Fig. 1) thisis the intersection of the horizontal line P conn =1/Q with the curve for the layer of interest. The intersection of P conn =1/Q with the curve for bulk percolation yields a value for the filling probability we denote as p b. This value (p b ) is slightly higher than the FCC percolation threshold. At this filling probability the bulk system has sufficiently connected slow regions to be glassy. A filling factor less than p b is needed for those layers which are close to the substrate; this will ultimately translate into alocalt g which is shifted above T g,bulk,asweshallshow below. Conversely, for layers close to the surface a filling factor larger than p b will be required, which leads to a predicted T g less than T g,bulk. However, we anticipate from the relatively weak effect of the free surface on P conn of nearby layers, that the predicted T g shifts will be quite small. The next step is to map from the filling fraction p needed to turn the zth layer glassy, to the temperature Fig. 2. Probability P (ρ) of a CRR having density ρ, at T = T g,bulk (dark curve) and at T = T g,bulk ± 30 K (dashed curves). Shaded regions are densities in excess of ρ ;areaof shaded region gives filling factor p. Inset: shift in glass transition temperature ΔT g(p) as a function of filling factor p. T (p) at which the probability of a CRR being slow is p. We construct this mapping following reference [14]. We take the density fluctuations within a small region of sample to be Gaussian, controlled by the compressibility κ. The regions of interest are individual CRRs, which we assume to have a volume equal to the Kuhn volume v k (cube of the Kuhn length). Thus we write [ ] P (ρ, T ) exp (ρ ρ eq(t )) 2 v k 2Tρ 2 (1) eq (T )κ for the probability distribution of density ρ within the Kuhn volume, in which T is the temperature, ρ eq (T )the equilibrium density, and κ the compressibility. This is shown in Figure 2 for PMMA, with values l K =1.56 nm, T g = 379 K, and κ = /MPa. We have taken a rather high value for κ relative to reported bulk values in the liquid state just above T g (typically around /MPa, which may be rationalized as a rough compensation for the more-than-gaussian likelihood of large fluctuations in a small region. The filling factor p(t ) as a function of temperature is the fraction of the density distribution that exceeds a threshold ρ at which a local CRR becomes slow by some experimental measure. We identify the value of ρ such that at the bulk glass transition, p is equal to the filling factor p b at the bulk glass transition. Hence dρ P (ρ, T )=p(t) (2) ρ and p b = p(t g,bulk). In turn, p b is chosen such that the probability for a site in the bulk to be connected to an infinite cluster is 1/Q, which is our criterion for the bulk glass transition. In Figure 2, ρ is shown by the vertical line, and the integral of equation (2) is indicated by the shaded regions. As the sample is cooled, the density distribution shifts to higher values by thermal contraction, and the value

4 136 The European Physical Journal B Fig. 3. T g shift relative to bulk, versus thickness of reporting layer next to substrate, for PMMA. Points, results of reference [16]; solid curve, present calculation. Dashed curve, predicted local T g value. of p(t ) steadily increases. The inverse of p(t ) is the desired mapping function T (p), which (if we subtract T g,bulk ) determines the shift in glass transition temperature as a function of filling factor, shown in Figure 2 inset. Composing the functions T (p)andp (z) yields our prediction for the local glass transition temperature T g (z). The function T g (z) can then be averaged over some portion of the sample, to compare to experiments on labeled reporting layers of reference [16]. Figure 3 shows theoretical (solid curve) and experimental (points) results for PMMA. The experimental data are for T g measurements from fluoresence-labeled supported films of varying thickness, each of which is insulated from the effects of the free surface by a thick overlayer. Also shown is the predicted local T g (z) (dashed curve). The data (symbols) show a maximum T g shift of about 10 K, for a 15 nm thick layer next to the substrate. As the reporting layer increases in thickness there is a fairly linear decrease in its average T g, with recovery of the bulk value for reporting layers thicker than about 60 nm. By comparison, while our predicted results (solid curve) for the average T g of the reporting layer show a shift of comparable magnitude for the thinnest film, there is a more gradual approach to the bulk value. The unaveraged local T g (dashed curve) shows a dramatic elevation in T g for the first 5 10 nm, but by 40 nm the local shift in T g is predicted to be less than 1 K. 3 Discussion The slower approach to bulk behavior of the predicted average T g is a consequence of the averaging process, which gives equal weight to every slice of the reporting layer. The democratic average f(h) (1/h) h 0 dz f(z) ofa positive function f(z) that decays to zero as z increases, cannot approach zero any faster than O(1/h), regardless of how quickly the local f(z) approaches zero. Note that the experiment also involves an averaging process in its result of a single T g for the entire reporting layer. The dye molecules have been argued to respond to the local density ρ(t ), which in bulk samples shows a break in slope in its temperature dependence at T g.thus the single experimental T g involves treating a superposition of individual lineshapes ρ(t,z)as if it were a single lineshape, and identifying a single apparent break in slope as the marker for T g. If the individual lineshapes are only shifted in temperature, this amounts to a democratic average. If however the break in slope is broader or weaker near the substrate, this would lead to heavier weighting of the signal farther from the substrate, and a more rapid approach of the average T g to the bulk value. It would be useful if experiments were performed in which a the distance from the substrate of a constant-thickness reporting layer were systematically varied. Such experiments could be compared directly with the predicted local T g without averaging, as well as revealing how the local lineshapes ρ(t,z)vary with distance from the substrate. In conclusion: we have presented the first prediction of local glass transition temperatures in a polymer thin film, using the percolation model. Averaging over the local depth-dependent T g (z) to compare to average T g in labeled reporting layers yields results in fair agreement with experiment. Differences between model and experiment lead to questions regarding the averaging process. A more direct comparison to theory could be made with data on T g for a labeled layer of constant thickness, as a function of its distance from substrate. As is evident in Figure 1, the free surface does reduce the number of paths that connect a near-surface site to percolating clusters. 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