Densification and Depression in Glass Transition Temperature in

Sep 11, 2014 - *E-mail: [email protected] 33-2 97 87 45 55. Abstract. Abstract Image. Ellipsometry and X-ray reflectivity were used to cha...
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Densification and Depression in Glass Transition Temperature in Polystyrene Thin Films G. Vignaud,*,† M. S. Chebil,‡ J. K. Bal,§ N. Delorme,‡ T. Beuvier,‡ Y. Grohens,† and A. Gibaud‡ †

Université Bretagne-Sud, EA 4250, LIMATB, F-56100 Lorient, France LUNAM Université, IMMM, Faculté de Sciences, Université du Maine, UMR 6283 CNRS, Le Mans Cedex 9, 72000, France § Centre for Research in Nanoscience and Nanotechnology, University of Calcutta, Technology Campus, Block JD2, Sector III, Saltlake City, Kolkata 700098, India ‡

S Supporting Information *

ABSTRACT: Ellipsometry and X-ray reflectivity were used to characterize the mass density and the glass transition temperature of supported polystyrene (PS) thin films as a function of their thickness. By measuring the critical wave vector (qc) on the plateau of total external reflection, we evidence that PS films get denser in a confined state when the film thickness is below 50 nm. Refractive indices (n) and electron density profiles measurements confirm this statement. The density of a 6 nm (0.4 gyration radius, Rg) thick film is 30% greater than that of a 150 nm (10Rg) film. A depression of 25 °C in glass transition temperature (Tg) was revealed as the film thickness is reduced. In the context of the free volume theory, this result seems to be in apparent contradiction with the fact that thinner films are denser. However, as the thermal expansion of thinner films is found to be greater than the one of thicker films, the increase in free volume is larger for thin films when temperature is raised. Therefore, the free volume reaches a critical value at a lower Tg for thinner films. This critical value corresponds to the onset of large cooperative movements of polymer chains. The link between the densification of ultrathin films and the drop in their Tg is thus reconciled. We finally show that at their respective Tg(h) all films exhibit a critical mass density of about 1.05 g/cm3 whatever their thickness. The thickness dependent thermal expansion related to the free volume is consequently a key factor to understand the drop in the Tg of ultrathin films.

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INTRODUCTION The confinement of polymer in thin films provides scientists with an extraordinary opportunity to study how their physical properties may be altered in comparison to the ones of their bulk counterparts. This is a very important issue as polymer thin films have many applications in biomedical,1 packaging,2 and flexible photovoltaic3 industries, just to cite a few. Although industrial processes using polymer in thin films are well controlled, the effects of confinement remain intensively debated. All the discussion started 20 years ago, with the pioneering work of Keddie et al., who evidenced that the depression in the Tg of polystyrene (PS) films on native silicon could be dropped by more than 25 °C for films less than 10 nm thick.4 Anomalous behavior like dewetting, processes much faster than suggested by bulk viscoelasticity5,6 and unexpected instabilities of these films7,8 were later reported. Even though most of the experimental works, supported simulations9,10 and different theoretical approaches11−14 converge proving that these deviations exist, a clear self-consistent theory explaining why this is the case, is not available at present.15 This suggests that so far all the relevant parameters have not yet been taken into account. Among these an essential physical quantity which © 2014 American Chemical Society

can be thickness dependent is the mass density of the polymer. By reducing the thickness of a film, the conformations, local packing, and therefore entanglements of the polymer chains are perturbed.9,10,16−19 The presence of a free surface and of an impenetrable substrate is believed to yield an inhomogeneous distribution of density in thin films. McCoy and Wyss20 proposed establishing a correlation between the average density of films and the shift in Tg. So far, whether the density of a polymer in a confined state should be different from that of the bulk still remains controversial. A recent publication by Mondal et al. 21 shows that the average electron density of polyacrylamide (PAM) films becomes superior for thinner films with a maximum increase of about 12% compared to their bulk counterpart. The same trend was observed in thin films of poly(methyl methacrylate) (PMMA)22−24 and of PS.25 Nevertheless, this is not generally the case as reported in Reiter’s Xray analysis of thin PS films.25−27 Tsui et al.28 show that the depression of Tg could be interpreted by a decrease of the mass Received: April 28, 2014 Revised: September 10, 2014 Published: September 11, 2014 11599

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polymer film and the angle of incidence. The refractive index of the polymer film was modeled using a transparent Sellmeier dispersion relation:

density of a mobile layer. They estimated that this decrease should reach 5% for a depression of 20 °C corresponding to a thickness of about 30 Å. Other studies using twin neutron reflectivity29 do not support such conclusions and conversely show that films do not even exhibit any changes in mass density when thickness changes. Several studies also reveal that the density profile normal to the surface exhibits some fluctuations in thinner films.23,29,30 In this study, X-ray reflectivity (XRR) and ellipsometry are combined to probe the evolution of the mass density of thin PS films of a thickness range from 6 to 150 nm (from approximately 0.4Rg to 10Rg). The combination of these two complementary techniques is compulsory as XRR provides a reliable and precise value for the thickness that is used as an input for fitting the ellipsometric data in order to decouple the contributions of refractive index and film thickness. We have measured the Tg of the same thin films by ellipsometry. We raise a question of whether the determined density is compatible with the evolution of Tg as a function of the thickness. In response to this question, we propose an interpretation which is supported by free volume theory that emphasizes the importance of the link between the thermal expansion of a thin film and the critical free volume concept.

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n2(λ) = 1 + B

λ2 λ − λ02 2

(1)

where λ is the wavelength of the incident light, B is a dimensionless parameter that determines the shape of the refractive index in the visible range, and λ0 is the resonance wavelength for which the refractive index diverges. λ0, which is far from the wavelength range of our measurements, it was fixed from 288 to 1700 nm. The calculated optical constants were determined by simultaneously fitting the data obtained at three different incident angles (65°, 70°, and 75°). It should be noted that the Sellmeier relation is not consistent with Kramers−Kroning relation and therefore parameters B and λ0 do not have any physical meaning. Any physical interpretations must be directly derived from the refractive index. A spectroscopic ellipsometer equipped with a hot stage (LinKAM TMS600) was also used to detect thermal variations (30−145 °C) and transitions in PS samples. For each sample, the measurements were made at every 5 °C interval from 30 to 150 °C and before each measurement the sample was held at a fixed temperature for 5 min to equilibrate. The rate of heating was 1°/min between two measurements. The thickness h(T) and the refractive index n(T) of the different films were determined from the fitting of data at each temperature. While the film is being heated, the thickness h(t) increases and the density ρ(T) decreases. In accordance with the universal principle of mass conservation and assuming that thermal dilation is negligible parallel to the sample surface, we have checked that the surface density ρh remains constant with temperature (see Supporting Information, Figure S1). Nevertheless, Tg was evaluated from h(T) vs T plot where the intercept position of two different slopes corresponding glassy and rubbery states of PS films determines Tg . X-ray Reflectivity Measurements. XRR measurements were carried out on a versatile X-ray reflectometer to investigate the position of qc, the thickness of PS films, and the electron density profile (EDP) in a direction normal to the sample surface for all the films. The reflectometer (Empyrean Panalytical) was equipped with a Cu source (sealed tube) followed by a W/C mirror to select and enhance the Cu Kα radiation (λ = 1.54 Å). All measurements were carried out in θ−θ geometry for which the sample was kept fixed during the measurements. The intensity was measured with a Pixel 3D detector using a fixed aperture of three channels (0.165°) in the 2θ direction. Under such conditions, the wave vector qz is given by (4π/ λ)sin θ with a resolution of 0.0014 Å−1. The diffuse scattering arising from the PS films is extremely weak in comparison of the specular signal and did not show any sign of correlated roughness (see Supporting Information, Figure S2 left) unlike the results of MüllerBuschbaum et al.31,32 measured on thin PS films. Except at very high qz where the diffuse scattering can be modeled by a constant background, its contribution is negligible in most of the qz range measured in the XRR curves. For this reason, XRR curves were analyzed without taking in account the diffuse scattering except for a constant background to adjust its contribution at qz > 0.4 Å−1 (see Supporting Information, Figure S2 right). The EDP was accessed from a least-square fit to the observed XRR data using the matrix technique.30 Best fits were obtained by modeling the EDP of the PS film with three layers. For each layer, the free fitted parameters are the electron density (critical wave vector), the thickness and the interfacial root-mean square roughness. The EDP is then calculated from these parameters by modeling the interface between the layers by an error function with a width which depends of the interfacial roughness. This assumes that the interfacial roughness can be described by an error function that yields a Gaussian function in the derivative of the EDP.

EXPERIMENTAL SECTION

Sample Preparation. Polymer thin films having different thickness were prepared on silicon wafers (100) by spin coating from different concentrations of atactic PS solutions (Mw = 136 kg/ mol, I = 1.05, from Polymer Source). Toluene was used as a solvent to dissolve PS. Silicon wafers initially have a very thin layer of silicon oxide (≈15 Å as measured by ellipsometry) on the surface of the crystalline silicon. Prior to spin coating, the native oxide layer of these substrates was removed by hydrofluoric acid (HF). This treatment consisted with the immersion of substrates in a HF acid solution (5% in volume) for 5 min followed by rinsing with Milli-Q water (resistivity 18.2 MΩ cm) and drying the surface under filtered nitrogen. The chemical treatment efficiency on the quality of the substrate surface was controlled by contact angle measurement (∼82° with water). The polar and dispersive components of surface energies extracted from the Owens and Wendt model are 7 and 32 mJ/m2, respectively. This yielded a hydrophobic hydrogen terminated silicon surface (Si−H). The measurements were all taken under ambient conditions (55% of relative humidity). To avoid any contamination, the polymer solutions were directly spin-coated onto Si−H. These solutions were spin-cast at a rotation speed of 2000 rpm during 1 min. Before analysis, these films were annealed at 160 °C during 24 h in vacuum. After switching off the oven, the samples were cooled down naturally until the temperature reached 25 °C under vacuum. Their thickness, measured by XRR within 2 days after annealing, ranges between 6 and 150 nm. Before the measurements, the samples were stored in a climate controlled room (relative humidity 55% and T = 23 °C). Atomic force microscopy (AFM) images also revealed that the films were homogeneous and extremely flat with typical root-mean-square (rms) roughness of the order of 3 ± 1 Å on a large area (100 μm2). Ellipsometry Measurements. Spectroscopic ellipsometry was used to determine the refractive index of all films. The Jobin Yvon− Uvisel instrument was mainly composed of a Xenon source, ranging from long-range infrared to ultraviolet (250-1700 nm), a polarizer, an analyzer, and a monochromator handling the dispersion and the selection of the wavelength to a photomultiplier. Each sample was measured at three incident angles 65°, 70°, and 75°. The measurements were all taken at ambient conditions (55% of relative humidity and an ambient temperature of 20 °C). The refractive index spectrum n(λ) for all the films was fitted using a single layer model in which the film thickness was kept fixed to the value obtained by XRR. The model contained a polymer layer and a silicon substrate. The fitting parameters were the refractive index of the 11600

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RESULTS AND DISCUSSION Ellipsometry Results. Figure 1 shows the evolution of the refractive index (n) versus the thickness (h) of PS thin films at a

To go further in the interpretation of the evolution of n versus h, we consider the Clausius and Mossotti equation. This equation links the refractive index of a substance, n, to its density ρv: αNa (n2 − 1) ρ = 2 3Mε0 v (n + 2)

where Na is Avogadro’s number, M is the molecular weight of the polymer repeat unit, ε0 is the vacuum permittivity, and α is the average polarizability of the polymer repeat unit in the international system units (C m2/V). From this equation, it can be shown that increasing the refractive index is equivalent to increasing the molecular density. To calculate the density from Clausius and Mossotti relationship, it is necessary to know the polarizability α(λ) at 605 nm. The polarizability of an atom or molecule describes the response of the electron cloud to an external field. Frequencydependent or dynamic polarizabilities are needed for electric fields which vary in time, except for frequencies which are much lower than electron orbital frequencies, where static polarizabilities suffice. As the values of α(λ) for PS in thin films are unknown, we have to estimate it from the known physical properties of the bulk according to the following equation:

Figure 1. Calculated refractive index of the PS films as a function of thickness. Open squares, our first set of samples; close squares, our second set of samples; closed circles, Li et al.33 work; and triangle symbols, Ata et al.25 work.

wavelength λ = 605 nm for two sets of measurements on different samples. By using the Sellmeier model it is possible to find out the wavelength and thickness dependency of n. Ellipsometric data were very well fitted with the set of parameters B and λ0 tabulated in Table 1 for some samples.

α=

Sellmeier parameters B

λ0 [nm]

refractive index

5.6 8.7 34.8 98.3 132.8

2.322 1.96 1.624 1.454 1.434

26.54 31.17 118.1 144.2 146.1

1.83 1.722 1.64 1.595 1.589

(nbulk 2 − 1)3Mε0 (nbulk 2 + 2)5ρvbulk

(3)

Indeed, when the film thickness increases, n and ρv are supposed to approach the values of the bulk material,36 that is, ρbulk = 1.06 g/cm3 and nbulk = 1.58985 at λ = 605 nm.37 This v yields a polarizability α = 1.46 × 10−33 cm3 at λ = 605 nm. This value is then used to obtain the evolution of the density as a function of the film thickness using eq 2 (shown in Figure 4b). Note that the density calculated from the refractive index is an average density which does not take into account a possible film anisotropy. Indeed, the polarizability is a tensor which depends on the anisotropy in the sample. For spherically symmetric charge distribution, it reduces to a single number. Ata et al.25 have shown that for thinner films the refractive index is incident angle dependent, suggesting an out-of-plane anisotropy of the polarizability. Hence, the refractive index should depend on the polarization and on the direction of light propagation. As the Sellmeier model does not incorporate the birefringence, when the incident beam impinges at a fixed incident angle, the refractive index determined from this model is a weighted average of the two distinct refractive indexes n⊥ and n∥ in the perpendicular and parallel directions to surface of the substrate, respectively. Since our refractive index is averaged from three measurements at three distinct angles of incidence, this leads to an average refractive index and an average polarizability independent of the chain orientation. The question is to know whether this average polarizability will change with the films thickness. Quantum chemical calculations25 show that the average polarizability of a PS chain with a density of 1.4 g/cm3 (density of our thinner film) is α = 1.4643 × 10−33 cm3 which is very close to the polarizability in the bulk state α = 1.4607 × 10−33 cm3. Thus, despite the fact that anisotropy present in the thinner films, average polarizability appears constant and hence it is quite reasonable to consider the same average polarizability value irrespective of film thickness. X-ray Reflectivity Results. The mass density was also extracted from XRR measurements on a series of films used in ellipsometric measurements. As shown in Figure 2, XRR curves

Table 1. Examples of Parameters of the Sellmeier Equation Used to Determine the Refractive Index as a Function of the Film Thickness thickness [nm]

(2)

The most striking feature in Figure 1 is the increase of n with the decrease of h. It can be observed that n converges toward 1.589 for thicker films which is consistent with the reported value of bulk polystyrene at 605 nm.29 This behavior is very similar to the one reported by Li et al.33 and Ata et al.25 Before discussing the physics, we need to clarify the methodology used in our data analysis. In ellipsometry, the thickness and refractive index values are strongly correlated in particular for the thinner films.34 Ellipsometry is indeed sensitive to the product of the refractive index and the film thickness (i.e., the optical path length or optical thickness). Hence, for an unknown thickness, the absolute value of the index of refraction may be dubious. This problem is more acute for thinner films typically less than 10 nm thick. Hence, we prefer to fix the thickness to the one determined by XRR to avoid any bias in the calculation of the refractive index reported in Figure 1. The thickness deduced from XRR datasets can be accurately obtained by direct Fourier transform. This has the great advantage of being model independent.35 In addition, the thickness was further cross-checked by AFM yielding a good agreement with the results of XRR. 11601

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where Ai, Zi, and ci are atomic mass, atomic number and the number fraction of element i in the chemical formula of the polymer, respectively. Na is Avogadro’s number. Equation 4 shows that the mass density is proportional to the electron density so that one can conclude that thinner films are denser than thicker ones. It is evident from Figure 2 that, for ultrathin films (h < 30 nm), qc is not directly measurable from the plateau of total external reflection, but rather it is only accessible via a fit to the data. Figure 3 shows a few fits to the XRR data with their corresponding EDPs. It is clear from the EDPs that the average

Figure 2. Total external reflection plateau for PS films on Si showing the location of the critical wave vector qcPS of the PS film (marked by a vertical solid line), the critical qcSi of the silicon (marked by a vertical dashed line). Curves are shifted vertically for clarity. The areas designed by (1), (2), and (3) are described in the text. Also the first derivative of the reflectivity curves is plotted in the inset to carefully define the position of qc.

exhibit two critical wave vector transfer qc on the plateau of total external reflection: one for silicon substrate which is fixed at qcSi = 0.0317 Å−1 and another one at a lower value qcPS which is characteristic of the average electron density of the PS films. According to the value of qz, in comparison with qcPS and qcSi, one can distinguish three regions in the XRR curves (Figure 2): (1) qz < qcPS, total external reflection occurs and the normalized reflected intensity is constant. As XRR measurements start below the critical angle of external reflection, the illuminated area (footprint of the beam) on the sample is larger than the length of the sample along the direction of the incident beam. As a consequence, even though the incident angle is below the critical angle for external reflection, only a part of the incident beam is totally reflected by the sample surface which in turn explains why the reflectivity deviates from unity below the critical angle.38 (2) qcPS < qz < qcSi, the incident beam is split in a reflected and refracted beam into the film, hence producing a drop in the total reflected intensity. In this region, the refracted beam is entirely reflected by the silicon substrate. (3) qz > qcSi, a part of the beam now penetrates into the substrate leading to a strong drop of the total reflected intensity. The location of the dip in the plateau of total reflection gives a qualitative estimation of the electron density of the film. Using the fist derivative of the XRR as shown in the inset of Figure 2, one can obtain an approximate value on the qcPS. Since qc is related to the average electron density of the film by the expression, qc = 0.0375(ρe)1/2, such an observation gives an unbiased estimation of the electron density of the PS films. It means that a model is not necessary for ascribing a density of the film. Results from Figure 2 show that the critical qc and hence ρe of the film increase for thinner films. Notably, mass density (ρv) determined by ellipsometry is related to the electron density (ρe) by this equation:23 ρe ∑ c A i i i ρv = = 3.085ρe ∑ Na i ciZi (4)

Figure 3. Normalized XRR data (symbols) and analyzed curves (solid line) of different PS films of thicknesses: 6, 13, 17, 22, and 35 nm from the top to the bottom. Curves are shifted vertically for clarity. Inset: Corresponding electron density profile.

electron density of thinner films is superior to the one of thicker films. The calculated qc reported in Figure 4a is a weighted average of the three qci associated with each layer (i) of our three layer model according to qc =

h1qc1 + h2qc2 + h3qc3 h1 + h2 + h3

(5)

where hi is the thickness of the layer (i). Figure 4 summarizes all the critical wave vectors qc and the mass densities obtained from XRR and ellipsometry. For films thinner than 50 nm, the higher density is 30% higher than that of the bulk polymer. Possible causes of such high density will be discussed in the next section.

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DISCUSSION This study shows that the mass density of a thin film increases when the film thickness decreases. In view of all the changes that are reported in the physical properties of thin films (h < 100 nm) compared to their bulk counterpart, it is not so striking that differences in the density of bulk and thin films are observed. This raises the first question which is why is the density of polymer confined in the thin film higher than that of the bulk? A possible explanation may emerge from the internal structure of a thin film near the substrate. A three-layer model consisting of a mobile surface layer at the top of the film hydrodynamically coupled to the bottom layer in contact with the substrate with a bulklike layer in between is frequently adopted.39−42 Molecular dynamics simulations have also revealed that density profiles inside the films are heterogeneous, with a clear stratified structure.10,43−45 With increasing strength of attraction, the density near the substrate becomes greater 11602

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Figure 4. (a) Average critical wave vector of differently thick PS films as a function of film thickness obtained from electron density profile and directly from the position of the first dip in the plateau of total external reflection. (b) Average mass density as a function of film thickness obtained from the refractive index (ellipsometry) and critical wave vector (XRR analysis). The dashed line is to guide to the eye.

conformation in thin PS films under confinement. When the film thickness h becomes smaller than approximately 6 times the radius of gyration Rg,bulk, there appears a significant increase of the normalized in-plane chain dimension Rg,∥/Rg,bulk as expected in the case of elongated chain in thin films. Hence an increase of the chain conformation along the direction parallel to the surface of substrate could be inferred. Based on these results, one could expect that flatter chains would favor denser films. This behavior of the density opens of course the following new question: is the reduction in film thickness accompanied by an increase in the density compatible with a depression in the Tg as commonly reported in the literature? It has been shown that the density of a glass system can change irrespective of Tg.53,54 As shown in Figure 5, the glass transition

than that of the bulk. By decreasing the film thickness, the weight of the bulklike layer inside the film decreases in proportion to the film thickness to finally vanish when the thickness approaches the radius of gyration Rg (≈15 nm) of the polymer. Thus, for thinner films, the compact layer at the PS/ substrate interface becomes predominant and a decrease in film thickness leads to an increase of the average total film density. Experimentally this suggests that the interaction between PS and a HF treated silicon surface is sufficiently strong to produce a denser layer near the substrate. This chemical treatment makes the silicon substrate hydrophobic which is attributed to a surface layer of silicon hydride (Si−H) groups. A recent study by Fujii et al.46 shows that a very strong interaction exists between the PS film and the Si−H terminated surface. This statement was obtained by observing that rinsing thick films in toluene yields an very thin irreversibly adsorbed PS layer at the H−Si supporting surface. It should be noted that all our calculated electron density profiles (Figure 3) do not indisputably reveal a higher density close to the substrate. Even though the fits to the observed data are fairly good, it is important to point out that decent fits could be also obtained with slightly different EDPs raising the classical question of the uniqueness of the solution. Due to the phase loss of the scattered amplitude and the finite range of angles over which reflectivity is measured, the model is not inherently unique.47−49 For an unambiguous analysis of film structures by X-ray (or neutron) reflection, one should determine the full complex reflection coefficient, that is, not only its modulus square, the reflectivity, but also its phase. Reiss and Lipperheide49 show that it is possible to induce seemingly arbitrary changes in the electron density profile, which all preserve the fit to the measured reflectivity, and some of these reflectivity equivalent profiles may not appear reasonable from a physical point of view. To get from XRR a correct absolute electron density of an ultrathin film that is not homogeneous at the atomic scale, is pretty complex and could explain the various conflicting density profiles shown in the literature. Other possible causes which can influence the adsorption of the polymer to the substrate, thus the density profile, are the influence of the time of annealing50 and the hydrophobicity of the substrate due to the chemical treatment used.51 Regardless of the adsorbed layer, another driving force which could lead to densification is the confinement effects. Krauss et al.52 have studied with diffuse neutron scattering the chain

Figure 5. Effect of film thickness on Tg for PS as measured by ellipsometry.

temperature measured on PS thin films deposited on a HF treated silicon surface decreases as a function of the film thickness. This depression in the Tg is in good agreement with the results from Keddie et al.4 To interpret such a drop, it is common to associate the Tg reduction to either (a) an increase in the mobility of the polymer chains for the thinner films or (b) an increase in the free volume. Concerning the mobility, Reiter and Khanna55 revealed that if polymer chains are grafted by one end at a high areal density onto a solid substrate and covered by a thin film of chemically identical molecules, this would lead to an autophobic behavior;56 that is, the free 11603

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molecules dewet the grafted layer. This is due to a reduction of entropy of the grafted layer with respect to free molecules and interfacial free energies at the interface of brush and melt. Recently Lee et al.57 have shown that the Tg of PS films on grafted PS layers of the same chemical identity decreases with decreasing film thickness. Then inside a PS thin film, the difference of entropy between the bound layer to the substrate and a free layer may play a role of accelerator of the chains mobility. This effect is not incompatible with the effect played by the free surface enhancing the mobility of the chains which lowers Tg.12 Statement (b) associating the increase in the free volume with the Tg reduction seems to be contradictory with thinner films having a higher density. We need now to address how such observations can be conciliated with the drop in the Tg. Free-volume theory was used to explain the anomalous behavior in glass transition temperature for thin confined films.11,13,14 Most of the traditional free-volume concepts of the diffusion of small molecules in liquids start with the theory of Cohen and Turnbull.58 The authors assume that molecular motion in a fluid of hard spheres is conditioned by the presence of located interstitial holes where there are vacancies of atoms. The specific free volume is defined by Vf = V − Vocc, where V is the total specific volume and Vocc is the occupied volume including the interstitial free volume and the van der Waals volume of molecules. The movement of a molecule into a hole within a cage delineated by the immediate neighbors of the moving particle becomes only possible when the hole has a greater size than some critical value v*. The critical void was postulated to arise from the redistribution of free volume that is temperature dependent. This theory is also successfully applied for the motion of polymer chain segments or cooperative motions that become allowed at Tg. On the basis of this general concept, on increasing the temperature, we claim that thinner films reach this critical volume v* at lower temperatures than thicker films. Hence, the thermal expansion of PS films should be much bigger for thin films than for thick films. Ellipsometric measurements shown in Figure 6a clearly evidence this statement. Indeed, one can see in this figure that the 13 nm thick film expands much more than the 83 nm thick film. This expansion is higher both in the glassy and rubbery states. The calculated thermal expansion coefficients in the glassy and rubbery states for all films are presented in Table 2 (second column). These coefficients were determined by fitting the thickness h(T) versus temperature plot with two straight lines for T < Tg and T > Tg. The slope or gradient of these two lines is then normalized by h(T0) where T0 is the first temperature (30 °C) according to the equation: ∝ vg,vr =

1 ⎛ Δh ⎞ ⎜ ⎟ h(T0) ⎝ ΔT ⎠g,r

Figure 6. (a) Percentage of expansion of PS films as a function of temperature. Thicknesses are 13 nm (triangles), 26 nm (circles), and 83 nm (squares). (b) Schematic representation showing that the critical volume is reached at a lower temperature for the thinner film.

Another point to consider is the physical aging which can also influence the thermal behavior of the PS films. Despite the fact that the set of studies cited in refs 4, 59, and 60, evidence an increase of the CTE in the glassy state when the film thickness is reduced, there is some disparity in the reported values. Glassy polymers are inherently in nonequilibrium and can evolve over time toward thermodynamic equilibrium, a process termed physical aging. Aging effects appear as change in temperature dependence of material properties such as volume, CTE, heat capacity, and so on. Temperature dependence of these properties changes with the aging condition such as temperature and time. Hence, reliable comparison between CTE is often difficult because the material properties are sensitive to the thermal history. In addition, the larger density of thinner films might also be due, at least in part, to aging affects. Some reports62,63 have shown that physical aging can be very sensitive to sample sizes and it has been observed via gas permeability measurements that thin glassy polymer films physically age more rapidly than their bulk counterpart. Kawana and Jones64 used ellipsometry to measure the overshoot of the CTE of ultrathin PS films that had been aged for 7 days at either 70 or 80 °C. Studies of 10−200 nm thick films supported on silicon reveal that the thinner film which has a higher CTE than that of thick films no longer shows signs of physical aging after 1 week of aging below bulk Tg. This implies that the thinner film has been aged faster than the thick films, leading to a smaller free volume. Finally, it should be noted that CTE and physical aging rate both increase for the thinner films. An in depth discussion about the influence of the physical aging on CTE in thin films requires further study with cooling rate dependent experiments which is not the purpose of this paper.

(6)

These expansion coefficients are consistent with those obtained by ellipsometric studies for polystyrene thin films by others authors.4,59,60 However, unlike the work of Kawana and Jones,59 the expansion in the rubbery state depends on thickness. A possible cause which can explain the origin of such a discrepancy is the nature of the interactions between the substrate and the PS thin film. Pochan et al.61 have shown by using neutron reflectivity a strong dependence of the coefficients of thermal expansion (CTE) above Tg on the thickness due to the coupling between the PS film and the solid interface. 11604

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Table 2. Coefficients of Thermal Expansion for Free Volume and Specific Volumea volume V thickness [nm] 83 26 13 bulk 1200 22

αgv [K‑1] −4

2.3 × 10 4.9 × 10−4 6.9 × 10−4 2.25 × 10−4 1.75 × 10−4 3.9 × 10−4

free volume Vf αrv [K‑1]

αgf [K‑1]

−4

αrf [K‑1] −3

7 × 10 10 × 10−4 18.6 × 10−4 6.36 × 10−4 6.21 × 10−4 13 × 10−4

1.8 × 10 3.9 × 10−3 5 × 10−3 1.8 × 10−3 1.4 × 10−3 3.2 10−3

measurement technique −3

9.2 × 10 1.32 × 10−2 2.37 × 10−2 8.4 × 10−3 8.2 × 10−3 1.72 × 10−2

ellipsometry “our work”

PVT and PALS66,67 EVPALS65

a

The coefficients marked in bold type are extracted directly from the experiments. The others are estimated from the link between volume V and free volume Vf in refs 66 and 67.

particular, one can see that the coefficients obtained by ellipsometry for the 83 nm thick film are in perfect agreement with the ones measured for the bulk (fourth line in the table). Similarly the results obtained for 26 nm thick film are remarkably closed to that obtained for 22 nm thick film. This implies that either the expansion of the free volume is mainly directed normal to the substrate or alternatively the ellipsometric CTEs are a little bit overestimated. Another issue in this comparison is the coefficient fα which is calculated for the bulk and used to deduce the CTE in thin films. This assumes that the ratio fα is the same in thin and thick films which implicitly considers that Vocc has the same temperature dependence in thin and thick films. Whatever the absolute values of these coefficients, it can be seen that their values increase for the thinner films and do not change the behavior of the free volume with temperature. This means that the free volume is temperature dependent and as it increases with temperature one can draw the following conclusions: (1) The free volume of a thin film is smaller than the one of a thick film below the glass transition temperature; it reaches a characteristic value v* at a lower temperature which is near the glass transition temperature as shown schematically in Figure 6b. (2) As the density of the films scales as the free volume, thinner films are denser than thicker ones. (3) Since the free volume increases much faster for thin films than for thick films, we should expect that the density of a thin film should decay faster than the one of a thick film when temperature is raised. For each thickness, the film density should be the same at Tg since the density is dictated by the value of the free volume. These two last points are clearly consistent with the results shown in Figure 7 in which a characteristic critical density close to 1.05 g/cm3 is found whatever the film thickness at Tg. This means that the onset of the glass to rubbery transition occurs whenever a critical free volume v* is reached. It should be noted that if there is different packing as a function of the thickness, a constant critical density (or volume) is not necessarily expected. Then the determined critical density is more certainly an average density than an absolute value. In any case, the important point is that thinner films reach this critical volume at a lower temperature. As a result, this point is fully supporting the fact that Tg is lower for thinner films

In any case, the CTE in the glassy state is found to increase sharply as film thickness is decreased below approximately 10 times the polymer radius of gyration (Rg). It is then clear that if thinner films thermally expand more rapidly than thicker ones, they reach the critical volume v* at a lower temperature as schematically depicted in Figure 6b. It is interesting to compare our results to those obtained by other techniques used to probe the free volume at the nanometer scale in polymer thin films like variable energy position annihilation spectroscopy (EVPALS)65 and pressure volume temperature experiment (PVT).65−67 To do this comparison, it is necessary to link the coefficients of thermal expansion of the thickness h(T) to the one of the specific volume V(T) and of the free volume Vf(T). For thin films for which the surface of the films is constrained by the substrate, films preferentially expand in the direction perpendicular to the substrate.68 Thus, the volume expansion is equal or very close to the thickness expansion according to the relationship: αv =

1 ∂h 1 ∂V ≃ h ∂T V ∂T

(7)

Under this hypothesis for a constant mass of PS, it is expected that the specific volume V(T) should follow the same trend as h(T). At present, we want to know how an expansion of the specific volume impacts the free volume. At a fixed temperature, the free volume Vf is proportional to the volume V according to the equation.66,67 Vf = fV = V − Vocc

(8)

where f is the hole fraction or the fractional (excess) free volume69 and Vocc is the occupied volume. In order to correlate the coefficients of thermal expansion of free volume and that of the specific volume, one can define the ratio: fα =

αv dV dV = f (T0) / f αf dT dT

(9)

where f(T0) is the fractional free volume at T0 = 30 °C equal to 3.6% from the work of Dubleck et al.67 The derivatives dV/dT and dVf/dT correspond to the slope of the specific volume and free volume, respectively. As the CTE for specific volume and free volume are given in the literature58,59 for bulk PS, fα can be estimated to be 0.125 for T < Tg and 0.0758 for T > Tg. By using the relationship αv = fααf, it is therefore possible to estimate the CTE for free volume from our CTE and conversely deduce the CTE of specific volume from the CTE of free volume as presented in Table 2. As shown in this table, one can appreciate the very good agreement between the values of CTE measured from ellipsometry and the ones obtained by other techniques which are measuring the volume expansion. In

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CONCLUSION In conclusion, we have shown in this paper how the change in density and the Tg depression of polystyrene thin films evolve as a function of their thickness. First, using ellipsometry and XRR, we have shown that thin PS films on HF treated surface 11605

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become increasingly denser when the thickness is reduced below 50 nm reaching a density 30% higher than the bulk one. The increase in the density is attributed to the influence of a compacted adsorbed layer on the substrate whose proportion in the average density of the whole film increases by decreasing the thickness. Second, we confirm that the Tg of PS films on the same substrates were depressed relative to their bulk counterpart when their thickness was reduced as reported by most of the research groups. These two seemingly incompatible results were analyzed in the framework of the free volume theory. The thermal evolution of the free volume is much more important for thin films than for thick ones. This effect explains why thinner films reach a critical free volume more rapidly than thicker films and therefore a lower Tg. Another sticking feature that we evidence is that all films exhibit the same characteristic mass density around their respective Tg(h). As the density of films scales as the free volume, this characteristic critical mass density can be directly related to the critical free volume. Finally, the dependence of the thermal expansion according to the initial film thickness is the key parameter to be considered for progress in the understanding of the Tg in polymer thin films.

ASSOCIATED CONTENT

S Supporting Information *

Figures showing surface density as a function of temperature, rocking scans, and X-ray reflectivity of specular scan. This material is available free of charge via the Internet at http:// pubs.acs.org/.

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REFERENCES

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Figure 7. Average film density versus temperature for PS films with thicknesses of 13 nm (triangles), 26 nm (circles), and 83 nm (squares). The dotted line represents the critical density at which all the films undergo a change in their thermal expansion.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] 33-2 97 87 45 55. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the regions of Bretagne and Pays de la Loire, the European Community and the CEFIPRA program for their financial support. J. K. Bal thankfully acknowledged to Department of Science and Technology (DST), Government of India, for providing research grant through INSPIRE Faculty Award (IFA13-PH-79). 11606

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