conference papers
Structure and dynamics of thin polymer films using synchrotron Xray scattering
^{a}Department of Physics, University of California San Diego, La Jolla, CA 92093, USA, ^{b}Department of Physics and Interdisciplinary Program of Integrated Biotechnology, Sogang University, Seoul 121742, Republic of Korea, ^{c}Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA, ^{d}Department of Materials Science and Engineering, SUNY at Stony Brook, Stony Brook, NY 11794, USA, ^{e}Intense Pulsed Neutron Source, Argonne National Laboratory, Argonne, IL 60439, USA, ^{f}Advanced Photon Source, Argonne, IL 60439, USA, and ^{g}LANSCE, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
^{*}Correspondence email: ssinha@physics.ucsd.edu
Recent measurements of the scattering function and of the dynamics of surface and interfacial fluctuations in thin supported molten films and bilayers using synchrotron Xray diffuse scattering and photon correlation spectroscopy in reflection geometry are reported. The results for monolayer films thicker than four times of the radius of gyration of polystyrene show behavior of normal overdamped capillary waves expected for the surface fluctuations of a viscous liquid. However, thinner films show deviations indicating the need to account for viscoelasticity. The theory has been extended to the surface and interfacial modes in a bilayer film system. The results are discussed in terms of surface tension, viscosity and shear modulus. Also recent experiments to measure the isothermal compressibility of supported polystyrene films by studying `bulk' scattering from the interior of the films is discussed.
1. Introduction
Thin polymer films are not only important in many technological applications but are also of great interest for basic science. The objective of our investigations was to use surface Xray scattering from thin molten polymer films on substrates to examine how their properties might differ from those of bulk polymers. The dynamical properties of entangled chains in polymer melts have been the subject of much intense study over the last several years (de Gennes, 1979; de Gennes & Leger, 1982; Doi & Edwards, 1986; Milner & McLeish, 1998). Attempting to account for the manner in which the viscosity and the elastic moduli depend on quantities such as the molecular weight, entanglement length, frequency, etc. presents a considerable challenge for theory to explain in detail. The corresponding problem for polymer melts adsorbed on substrates in thin film form does not appear to have been addressed, although experimental data have been obtained using atomic force microscopy or the surface force apparatus. In this paper, we show that measurements of the wavevector dependence of the relaxation time for overdamped capillary waves on thin molten polystyrene (PS) films (thickness ≤ 2R_{g}, radius of gyration) provide a noninvasive method for measuring the surface tension, viscosity and shear modulus of these films and discuss the trends in the latter two quantities with molecular weight. In discussing the dynamics, we show that our current model also explains why earlier measurements of the static scattering function S(q) (q being the inplane component of the wavevector transfer k_{f}−k_{i} of the scattering, where k_{i} and k_{f} are the incident and outgoing wavevectors of the Xray photons, respectively) had to be interpreted in terms of brushlike models when the film thicknesses were comparable to R_{g}.
Certain physical properties of polymers such as the molecular mobility (Liu et al., 1997; Kajiyama et al., 1997) appear to differ from those of bulk polymers (de Gennes, 1979), in particular, when the thicknesses of the polymer films are comparable with the polymer radius of gyration (R_{g}). Thus Keddie et al. (1994) report a decrease of T_{g} for polymer films with thicknesses below ~2R_{g}. However, contrary results have been found to indicating no change or even an increase of T_{g} for thin polymer films (Wallace et al., 1995; Zheng et al., 1995; Frank et al., 1996; Zanten et al., 1996; Ge et al., 2000).
2. Surface fluctuations of monolayer films
In an earlier series of grazingincidence diffusescattering Xray measurements of supported polystyrene (PS) films quenched to room temperature from the molten state, Wang et al. (1999) reported that for film thicknesses comparable to R_{g}, the surface height–height correlation function was more characteristic of a film possessing a shear modulus than that of liquid capillary wave fluctuations. Thus these films exhibited a low inplane q cutoff of the surface fluctuations which appeared to scale with the film thickness as h^{−3/4} instead of the h^{−2} behavior characteristic of a cutoff due to van der Waals interactions of a normal liquid film with the substrate. This behavior could be understood in terms of a model developed earlier for a polymer brush (Fredrickson et al., 1992). For much thicker PS films with h >> R_{g}, however, normal capillary wave behavior was observed in S(q).
We have now followed up those earlier measurements by carrying out dynamical measurements using Xray photon correlation spectroscopy (XPCS) (Kim et al., 2003). This emerging technique applies the principles of dynamic light scattering in the Xray regime. The short wavelength and slow time scales characteristic of XPCS provide us with a unique opportunity for measuring dynamics selectively from the surface fluctuations and extending the phase space accessible to scattering studies beyond what is accessible by light and neutron scattering.
Films were prepared by dissolving PS of several averaged molecular weights [M_{w} = 65, 123, 400, 650 and 900 kg mol^{−1} (M_{w}/M_{n} < 1.08)] in toluene and then spincasting onto optically flat silicon substrates. The substrates were prepared to be hydrophobic. The thinfilm samples were then annealed in vacuum for 24 h at 443 K to ensure complete solvent removal and afterwards quenched to room temperature. For each molecular weight, the thicknesses of the PS films were R_{g}, 2R_{g} and 4R_{g}. R_{g} of each molecular weight is shown in Table 1.

XPCS experiments were performed at beamline 8ID at the Advanced Photon Source (APS), Argonne National Laboratory, and employed monochromatic radiation with an Xray energy of 7.5 keV. The schematic experimental setup is shown in Fig. 1(a) and the schematic reflectivity geometry in XPCS is shown in Fig. 1(b). The offspecular diffuse scattering of the polymer surface was recorded with a directillumination chargecoupled device (CCD) camera located 3472 mm downstream of the sample. The beam dimensions were 20 × 20 µm^{2} and the speckle size at the detector was comparable to the CCD pixel size of 22.5 × 22.5 µm^{2}. The polymer surface was thus partially coherently illuminated, giving rise to a speckled scattering pattern which varies in time as the surface modes experience random thermal fluctuations. The normalized intensity–intensity time autocorrelation function, g_{2} at a particular inplane wavevector q, characterizing a particular surface fluctuation mode with wavelength 2π/q, is obtained from (Sutton, 2002; Grubel & Zontone, 2004)
For highly viscous liquid films, g_{2} = 1 + βexp[−2t/τ(q)] is used to fit the above autocorrelation function, where β is the speckle contrast and τ(q) is the relaxation time of the overdamped thermally induced capillary wave fluctuations of wavevector q.
Our early set of XPCS studies (Kim et al., 2003) involved supported PS films of M_{w} 123 kg mol^{−1} at various thicknesses and temperatures between 423 K and 463 K, well above the glass transition temperature of 373 K. The behavior of the relaxation time τ as a function of q could be well explained by an expression derived by taking the small frequency limit of an expression for the imaginary part of the susceptibility for height fluctuations derived by Jackle for a liquid film of finite thickness and finite viscosity (Jackle, 1998). This expression is given by (Kim et al., 2003)
where η is the viscosity of the liquid, γ is the surface tension, and the function R is given by
From equation (2), one notices that there is a scaling relation, i.e. the quantity τ/h is only a function of the dimensionless quantity (qh) and the ratio of η/γ for all film thicknesses. This scaling relation was obeyed reasonably well (Kim et al., 2003) for all the thicknesses studied (which were all greater than 4R_{g}). Further, the obtained film viscosity appeared to be consistent with that obtained from bulk measurements. Thus, it appeared that the capillary wave dynamics of molten polymer films exhibited the normal behavior expected of a normal viscous liquid.
However, more recent XPCS measurements have been carried out on the dynamics of a series of thinner polymer films of various molecular weights with thicknesses ranging from 4R_{g} to R_{g} (Jiang et al., 2007). Fig. 2 shows the function g_{2}(q,t) obtained for PS films of M_{w} 123 kg mol^{−1} at 468 K and for different thicknesses of R_{g}, 2R_{g}, and 4R_{g} respectively and a given inplane q. For h = R_{g}, no dynamics is observable, i.e. the chains are confined and `stuck' on the surface.
Fig. 3 shows the derived relaxation times τ as a function of q for capillary waves on 2R_{g} and 4R_{g} films together with fits (dashed lines) using equation (2). While equation (2) fits the data for h = 4R_{g}, the results show deviations from equation (2) when the thickness is equal to 2R_{g}, regardless of molecular weight. This shows that it is not the absolute thickness but rather the ratio of thickness to R_{g} which affects the dynamics.
To explain the results for h = 2R_{g}, we calculated the surface dynamic susceptibility by generalizing the theory in Jackle (1998) for viscoelastic liquids. This can be done by replacing the viscosity by a complex frequencydependent quantity η(ω) ≃ η + iμ/ω. Here μ is a shear modulus and η is the frequency. This yields a modified expression for τ(q),
where τ_{0}(q) is the expression for τ(q) given by equation (2), and R is the function given by equation (3). This expression fits the results for thicknesses of 2R_{g} much better as shown in Fig. 3. In equation (4), when τ_{0}(q) << η/μ, τ(q) becomes equal to τ_{0}(q). This result is usually valid for large γ, thick films and small μ. On the other hand, when τ_{0}(q) >> η/μ, τ(q) ~ η/μ. In general, the damping of capillary waves on the surfaces of thin polymer films exhibits intermediate properties between Newtonian liquids and Hookean solids. We define τ_{m} ≡ η/μ, the relaxation time that characterizes the time scale over which the viscoelastic liquid relaxes the stress undertaken. When polymer films are thin enough, the surface wave relaxation becomes wavevector independent and long wavelength capillary waves are completely suppressed. A corresponding calculation of the static susceptibility obtained from the dynamical theory above yields an expression for S(q) which is identical to that given by Fredrickson et al. (1992), although the expression they give for μ is that for a polymer brush which is not necessarily valid for an adsorbed polymer chain. Thus we can understand why for films with h of the order of 2R_{g} or less, the earlier measurements of S(q) for adsorbed polymer films (Wang et al., 1999; Seo et al., 2005) could be better explained by the model in Fredrickson et al. (1992) than by standard capillary wave theory.
The viscosities obtained for the films by fitting to the XPCS data show scaling with molecular weight with an exponent somewhat less than the 3.4 predicted by the reptation theory for entangled polymer melts, probably indicating a lower degree of entanglement as the film becomes thinner. The shear moduli obtained by fitting to the measured τ(q) for the above films were roughly consistent with bulk shear moduli for the same molecular weights. Thus it appeared that the normal models for viscoelastic films could account for the observed dynamics of thin PS films. However, there are indications that these models are not obeyed as the temperature begins to approach the glass transition temperature. Here measurements of the dynamics for PS films with large M_{w} values (up to ∼900 kg mol^{−1}) and also very recent measurements for lower M_{w} values but at larger q values indicate that the effective film viscosity may be considerably lower than the corresponding bulk values. This may be consistent with observed reductions in T_{g} for thin supported polymer films. Further exploration of this regime is in process. These results, incidentally, point to the need for a more intense source of coherent Xrays in order to study dynamics at larger values of q. Such measurements are currently severely intensity limited. However, radiationinduced damage is a major concern and care has to be taken to avoid radiating the sample in one spot for a longer time than it takes to produce damage. This can often be managed by a fast shutter which can be periodically opened and closed if the time scale being measured is longer than the damage time.
3. Surface and interface fluctuations of bilayer films
The dynamics of the interface between two polymer films in a bilayer is also of interest for applications involving polymer blends, adhesives, etc. One can study this by using the grazing angle of incidence α_{i} of the photon beam to probe both the free surface and the interface selectively. Thus for the free surface, if α_{i} is less than the critical angle for total reflection, only the top surface is probed (provided the depth of the buried interface is much larger than the extinction depth), while if α_{i} is chosen so that there is a node of the standing wave made by the incident wave and the reflected wave, while the interface is at, or close to, an antinode then the buried interface is selectively probed. This can be shown using the distortedwave Born approximation (DWBA) (Sinha et al., 1988) for the scattering.
We have carried out experiments (Hu et al., 2006) on a film of PS of M_{w} 200 kg mol^{−1} and thickness 100 nm on a film of poly(4bromostyrene) (PBrS) of M_{w} 350 kg mol^{−1} and thickness 200 nm deposited on a silicon substrate. The bromination of the PBrS was 89%. The results (Fig. 4) show that there are two characteristic relaxation times associated with the free surface: a fast relaxation which is actually faster than the relaxation time for a single PS film of comparable thickness on Si and a slower mode with a qindependent relaxation time. The buried interface between the two films had fluctuations which essentially coincided in their τ vs q relation with the slow mode for the free PS surface.
The dynamics of the bilayer was modelled as follows. The viscosity of the PS layer is assumed to be uniform throughout the film, and the PBrS layer is assumed immobile at the time scales of the experiment. A thin mixed layer is placed between the PS and PBrS films to account for the interface. Its viscosity is approximated as uniform. A value of 2.6 × 10^{−3} J m^{−2} was used for the interface tension between the mixed layer and the PS film. This was obtained from the magnitude of the surface roughness due to thermally excited, long wavelength capillary modes as measured by static diffuse Xray scattering (Hu et al., 2005). Static scattering also provides the value of 6.6 nm for the thickness of the mixed layer (assuming the width is 2.35σ, with σ being the Gaussian roughness measured by neutron scattering). To discuss the dynamics of the coupled layers, we generalized the theory for the dynamics of the surface of a single viscoelastic film to the case of a bilayer film (Jiang et al., 2006). The linearized Navier–Stokes equation was used to solve for the fluid flow throughout two coupled layers. A nonslip boundary condition was assumed at the boundary between the mixed layer and the PBrS. At the PS/vacuum interface the surface tension balances the viscous stress. At the interface between the PS and the mixed layer the velocity is continuous and the difference in viscous stress between the two fluids is balanced by the interface tension. The viscoelastic properties of the polymer were taken into account by introducing a frequency dependent viscosity whose imaginary part involves the shear modulus. For the thick polystyrene film, the effect of the shear modulus is negligible and can be ignored. For the thin mixed layer, the addition of a shear modulus significantly changes the dynamics, as discussed above. This yields coupled modes for each surface, i.e., the free surface and the buried interface. The magnitude of the viscosity and shear modulus of the mixed layer in the resulting equation was then varied using nonlinear leastsquares regression to obtain a best fit to the measured experimental relaxation times. This model provides a good fit, shown as the dashed line in Fig. 4, yielding μ ≃ 18 N m^{−2} and η ≃ 327 Ns m^{−2} for the mixed layer. The bestfit viscosity is only around 2% that of the PS layer. Thus, if one only considers the dynamics of the PS fast mode, the mixed layer can be viewed as introducing a finite slip length to the PS/PBrS interface. The dynamics of the slow mode of the top surface, which is nearly identical to the dynamics of the mixed layer, shows almost no dependence on wavevector. This wavevector dependence cannot be explained by a thin viscous layer, since by equation (2), a q^{−4} dependence for τ is predicted for a thin film. A simple scaling argument also predicts the same dependence for a thin film (de Gennes et al., 2002). However, the inclusion of the shear modulus term produces a flattening of the spectrum. The value of μ obtained from the fits was of the same magnitude of the shear modulus for bulk PS at comparable time scales (Graessley, 2004) and also of the same magnitude as the values needed to fit the relaxation times of the thin singlelayer polymer films at comparable temperatures discussed in the previous section.
While the model described above does give a good fit to the data, it is clear that it can only be an approximation to the actual interfacial region, since the mixed layer is unlikely to have a uniform viscosity. Furthermore, the approximation of a surface tension is not precisely applicable over length scales comparable with the thickness of the interface. Nevertheless, the present results should provide a significant improvement over previous rheological results. Here we not only can imply the existence of slip between PS and PBrS, but we have estimates for the thickness, viscosity, surface tension and shear modulus of the mixed region. It seems reasonable to speculate that the origin of the low viscosity in the mixed region is similar to the causes of the reduced glass transition at the free surface of PS first observed by Keddie et al. (1994), since one might expect that the viscosity above T_{g} should decrease if T_{g} decreases. It would be interesting to investigate if thin PS films on top of PBrS show a reduced glass transition temperature closer to a freely suspended film, than a supported one. Finally, we note that experiments on the mobility of gold nanoparticles at the polymer–polymer interface in a bilayer system of PtBA/PtBA (Narayanan et al., 2005) have also shown anomalously high mobility. This may be similarly related to a region of reduced viscosity as we have found here.
4. Interior fluctuations of polymer films
At grazing angles of incidence, the scattering can be treated as coming from the surface. However, if the angle the scattered beam makes with the surface is made large, `bulk' fluctuations from the interior of the polymer film are also significant and can be studied. In fact, in the DWBA (Sinha et al., 1988), the scattering may be written as the sum of two parts: a `surface' part given by
where E(z, in) and E(z, out) are the incident electric field of the incident and outgoing radiation, respectively, at height z in the film, and a `bulk' part given by
where S_{b}(q) is the bulk structure factor and F(K) is the Fourier transform of [E(z, in)E*(z, out)]. S_{b}(q) can be approximated at small values of q by
The sum of these two terms in equations (5) and (6) was fitted to the scattering from a molten PS film (M_{w} 129 kg mol^{−1}) as a function of grazing angle of incidence and of temperatures for a large scattering angle at an inplane q vector of 1.34 nm^{−1}. An example is shown in Fig. 5. It may be seen that while the surface term predominates at small grazing angles of incidence, the `bulk' scattering from the film becomes important at large grazing angles of incidence.
The corresponding obtained values of S_{b}(0) are shown in Fig. 6. It can be seen that a break in the temperature dependence corresponding to a glass transition temperature can be observed in the thickest film but washes out as the film becomes thinner. It may also be seen that the isothermal compressibility κ_{T} (related to S_{b}(0) by equation (7)) increases as the films become thinner. This is presumably due to lower entanglement densities in the thinner films.
Acknowledgements
This work was supported by the US Department of Energy, BES Programs, NSF Grant No. DMR0209542, the Ministry of Science and Technology of Korea (the International Cooperation Research Program) and 2003 Special Research Fund from Sogang University. The Advanced Photon Source is supported by the US Department of Energy, Office of Basic Science, under Contract No. DEAC0206CH11357.
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