VOLUME 111, NUMBER 21, MAY 31, 2007
© Copyright 2007 by the American Chemical Society
ARTICLES Effect of a Reduced Mobility Layer on the Interplay between Molecular Relaxations and Diffusion-Limited Crystallization Rate in Ultrathin Polymer Films Simone Napolitano* and Michael Wu1 bbenhorst Laboratory of Acoustics and Thermal Physics, Department of Physics and Astronomy, Katholieke UniVersiteit LeuVen, Celestijnenlaan 200D, B-3001 LeuVen, Belgium ReceiVed: December 19, 2006; In Final Form: March 31, 2007
In ultrathin polymer films, the coupling between the segmental mobility, precursor of the molecular diffusion, and the crystallization rate is broken down because of interfacial interactions. In particular, in the presence of a reduced mobility layer at the interface with the substrate, the crystallization kinetics slow down at a length scale bigger than the one connected with the deviation from bulk behavior of the structural relaxation. By modeling the influence of the substrate interactions on the parameters governing the temperature evolution of the main relaxation time, it was possible to reproduce the effect of geometrical confinement on the quantities connected to the diffusion-limited crystallization rate. Upon reduction of the thickness or increasing of the substrate interaction, the films show an apparent higher glass stability in terms of an increase of the cold crystallization temperature and of the crystallization time. The deviations from bulk behavior were found to vanish above a crossover temperature as already observed for the phenomena connected to the glass transition.
Introduction In a recent study on the interplay between molecular relaxations and crystallization rate in ultrathin polymer films, a paradox was identified.1 Upon reduction of the thickness, the crystallization kinetics slowed down, in terms of increase of more than 1 order of magnitude of the crystallization time, as previously reported for other systems,2-4 while the dynamic glass transition temperature, Tg, remained constant within the experimental errors. This implies that the transport of material and thus the molecular diffusion is reduced without any effect on the segmental mobility, violating the Stokes-Einstein5 (SE) and Debye-Stokes-Einstein6 (DSE) relations. The same discrepancies characterize experimental results on the glass transition of ultrathin polymer films assigned by different techniques. In samples of polystyrene, PS, capped between aluminum layers, the dielectric dynamic Tg, defined as τR(Tg) ) 100 s (where τR is structural relaxation time related to the segmental mobility), * Corresponding author. E-mail:
[email protected].
remains constant down to 10 nm; while Tg defined as the kink of the temperature dependence of the capacitance, a probe of the thermal expansivity, shows a reduction by almost 20 °C for films of the same thickness.7 The relaxation scenario is furthermore complicated in systems like ultrathin films of poly(ethylene terephthalate) (PET) on gold, where the crystallization kinetics slows down upon reduction of the glass transition temperature,8 as well as in systems, such as free-standing films of PS, simultaneously showing huge Tg’s reductions9 and no change in the diffusion coefficients of small probes within the polymer matrix.10 These apparent inconsistencies can be solved invoking the presence of different length scales over which the diffusion and its precursor, the segmental mobility, feel the reduction of thickness and the interfacial interactions. Obviously, even the temperature dependence of these quantities should be taken into account. As a general trend, in fact, it was observed that the confinement effect on the phenomena connected to the glass transition extinguish at sufficiently high temperatures.11
10.1021/jp068721t CCC: $37.00 © 2007 American Chemical Society Published on Web 05/05/2007
5776 J. Phys. Chem. B, Vol. 111, No. 21, 2007 In this article, we present a model aiming to solve the apparent discrepancies in the molecular mobility as probed by relaxation (segmental mobility) and by crystallization experiments (diffusion) in the case of an attractive substrate. On the basis of the model calculations, the reduction of crystallization rate in ultrathin films is explained in terms of the effects of a reduced mobility layer, RML, at the interface with the substrate, on the thermal evolution of the main relaxation time. The existence of regions characterized by higher glass transition temperatures,12 lower molecular mobility,13 and almost null expansion coefficients14 at the very interface with an absorbing substrate has been demonstrated by both simulations and experimental work. In particular, it was proposed15 that the interaction with the substrates is temperature dependent and, in the polymer melt, extends over a layer as thick as the gyration radius, Rg, while in deeply supercooled liquids the length scale of interaction exceeds by far Rg itself. It was then proven that bounding interfaces may introduce long-range perturbation effects on the diffusion coefficients of the film that decay over length scales bigger than even 10 Rg’s.16 This evidence can explain the slowing down in the crystallization kinetics in the presence of an absorbing substrate: considering the length scale of the substrate interaction, we find that reduction of the mobility of the chains at the interface is sufficient to inhibit the material transport of the whole film, without altering significantly the segmental mobility. In fact, by quantifying the linear response of a generalized modulus in a multilayer system, we recently demonstrated that the presence of a layer with a different mobility acts first on the static properties and then on the dynamical properties of the film itself.13 In the particular case of a dielectric response, upon thickness reduction, an RML first causes the reduction of the dielectric strength, without altering the position of the maxima of the relaxation peaks (first regime), and then (second regime) leads to an increase of the relaxation time and thus a higher dynamic glass transition temperature at thicknesses comparable to the extension of the RML (7 nm for PET on Al,13 20 nm for poly(2-vinylpyridine), PV2P, on SiO212). In the first regime, the mixing of the contributions of the two layers leads to a discrepancy between the dynamic Tg and the effective glass transition of the system taking into account the contributions from both the reduced mobility and the bulk layers, see Figure 1. As a consequence, the apparent paradox on the molecular mobility in thin films can be solved by considering the different ways in which a layer with a reduced or enhanced mobility affects the experimental observables: in the first regime, the crystallization rate cannot be related to the molecular mobility probed on the time and length scale of the dynamic glass transition (segmental mobility).1 The crystallization rate has to be connected to an effective glass transition temperature given by the distribution of relaxation times along the film thickness,17,18 similar to the one estimated via measurements of the diffusion coefficients.16 If this assumption is correct, in the regime of cold crystallization where the diffusion limits the crystallization rate, an increase of the effective Tg of the system should correspond to a reduction of a fictive temperature T*(h) without appreciably altering the molecular mobility. T*(h) is defined here as the temperature at which the crystallization time (or the linear growth factor) of bulk samples equals the one of an ultrathin film of thickness h, in symbols τcry(h,T*) ) τcry(∞,T). This scenario is confirmed by our previous work and the experimental investigation reported here. Similarly, it was already proposed that a reduction of the linear growth factor
Napolitano and Wu¨bbenhorst
Figure 1. Effect of a reduced mobility layer of thickness δ on the reduced dynamic and effective glass transition temperatures of a film of total thickness h. The data relative to the dynamic Tg have been calculated using eq 6 of ref 13. The effective glass transition of the same system has been calculated from eq 5 of this work. The lines are a guide for the eyes.
connected with the slowing down of the crystallization kinetics could be related to an increase of the effective glass transition temperature,3,4 which anyway should not be confused with the dynamic glass transition temperature connected to segmental mobility. In the proposed model, we modify the parameters governing the thermal evolution of the structural relaxation time in order to mimic an ideal increase of the effective Tg. By modeling a bilayer system consisting of a bulk-like layer and an RML, it was possible to reproduce the confinement effects on the diffusion-limited crystallization rate of ultrathin polymer films: an augment of the crystallization time, a reduction of the confinement effects upon increase of the crystallization temperature, and a modest increase in the crystallization temperature probed in non-isothermal conditions. In the following discussion, after introducing the principles of the model, the deviation from bulk behavior on the crystallization rate in the presence of an RML is estimated in section two, and the experimental evidence of the model’s validity is given in sections three and four. Finally, the results of the model calculations are analyzed and framed in a more general discussion on the effect of geometrics in ultrathin polymer films 2. Effects of a Reduced Mobility Layer on the Crystallization Rate In its simplest formulation, the classical crystallization rate, G(T), can be written as the product of two exponential terms f(T) and D(T), respectively related to the nucleation/growth free energy term and the molecular diffusion. The latter is related to the shear viscosity and to the main relaxation time via fractional SE and DSE relations through a temperature-dependent parameter ξ e 1:
G(T) ) f(T) ‚η-ξ(T)(T) ≈ f(T) ‚τR-ξ(T)(T)
(1)
In the temperature regime just above Tg, the molecular transport becomes predominant, the condition ∂f(T)/∂T , ∂D(T)/ ∂T is achieved, ξ(T) assumes a constant value (typically ∼0.75), and eq 1 is written into an analog form referring to the crystallization time τcry, an experimentally easily accessible quantity:
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1 1 τcry(T) ≈ G-1(T) ) τξ(T) (T) ≡ u [τR(T)] f R f
(2)
The temperature evolution of the structural relaxation time, τR(T), is well-described by a Vogel-Fulcher-Tamman, VFT, equation
τR(T) ) τ∞ exp
( )
(
BT0 EV ) τ∞ exp T - T0 R(T - T0)
)
(3)
where T0 is the Vogel temperature, B is a positive parameter related to the fragility of the system, R is the universal gas constant and τ∞ and EV are respectively the relaxation time and activation energy associated with the process in the limit of infinite temperatures. For a generic dynamic mode coupled to the main relaxation at a given temperature T, the probability to relax due to the average energy kBT arising from the thermal fluctuations is given by the exponential term in eq 3. By rescaling the temperatures over T0, the parameter B equals the apparent activation energy in units of R (B ) EV/RT0). Thus, for given values of B, T0 is related to the depth of the potential energy minimum connected to the relaxation mode. Various experimental evidence12,13,16 suggests that the substrate interaction alters only the value of T0: at a given temperature, a reduction in molecular mobility in the proximity of an interface can be described as a layer characterized by a higher T0 (or Tg, as the ratio Tg/T0 is constant if B and τ∞ are constant, by imposing τR(Tg) ) 100 s in eq 3 the condition Tg/ T0 ) 1 + B/ ln(100/τ∞) is reached) corresponding to slower dynamics in the same temperature range. The overall dynamics of the sample will then be governed by effective higher glass transitions and Vogel temperatures, respectively Tgeff and T0eff, that in the regime of cold crystallization reflects the bulk behavior observed at a lower temperature T*. In addition to the changes in the crystallization rate imposed by the presence of an RML, in very thin films, the role of nucleation has to be taken into account in terms of a size dependent reduction of the density of active nuclei Λ(h) and thus a further increase of the crystallization time. Indirect evidence of a reduction of the nucleation rate in ultrathin films is given by an analysis of the crystallization kinetics in terms of the Avrami theory. We have already shown19 that upon reduction of the thickness an increase of τcry is accompanied by a reduction of the Avrami exponent, n. These results were related to a reduction of density of active nuclei contributing to the crystallization rate.20 In fact, given a confined space, both the ensemble of crystals nucleated outside of the considered region and the portion of crystals not contained within the space’s borders, even if arising from nuclei within the space itself, cannot contribute to the crystallization rate. Thus, this last quantity will always be lower than that in bulk. It was experimentally shown that the nucleation rate in small droplets of poly(ethylene oxide) with section down to 10-4 µm3, the product of the characteristic time associated to the nucleation in the droplet, and the volume of the droplet itself are constants of the system consistent with the classic nucleation theory.21,22 Consequently, the temperature and thickness dependence of the nucleation rate can be quantified as a simple product of the temperature-dependent term indicated in eq 1 and Λ(h), scales with h.23 A simple modeling of Λ(h) in ultrathin films is not yet achievable: the experimental conditions reproduced in the droplet experiments21,22 permitted the investigation within an impurity-free environment. In spin-coated films, the presence of solvent residuals and substrate roughness could promote
heterogeneous nucleation events. The probability of these events is enhanced during the severe annealing procedures required to remove the solvent and equilibrate the polymer chains. In the following discussion, focusing on the role of diffusion, we will consider the influence of nucleation as a correction factor, and in the calculations, we fix Λ(h) ≡ 1. Taking into account these effects just described, we can write a thickness dependence of the crystallization time and the crystal’s growth rate:
τcry(h,T) ≈
1 1 τξ(h,T) (h,T) ≈ ‚u [τR(h,T)] ≈ f(h,T) R Λ(h)‚f(T) u [τR(T*)] f(T)
G(h,T) ) f(T)‚u
-1
≡
[τ˜ R(T)]‚Λ(h) ) f(T)‚u
u [τ˜ R(T)] f[(T)] -1
(4a)
[τ˜ R(T)] (4b)
with τ˜ R(T) ) τ∞ exp(BT0eff/T - T0eff) proposing that the thickness dependence of the interplay of crystallization and relaxation times can be written in terms of the analogy between the dynamics of the thin film and the one of the bulk at lower temperatures in the regime of cold crystallization, or at higher temperatures if the crystallization is achieved from the melt. The effective Vogel temperature of a film of thickness h and an RML δ was calculated by weighting the volume contribution of the two layers:
T0RML‚T0BULK δ ) h δ BULK δ ‚T + 1 - ‚T0RML h 0 h
()
T0eff
(
)
(5)
with T0RML ) RT0BULK. The value of T0RML was perceived as an average value of the Vogel temperatures in the layers with different mobility from the one at the very interface with the substrate to the layer characterized by bulk mobility. The constant R was varied from 1.01 to 1.3 in order to mimic different increasing substrate interactions and reproduce the experimental conditions extrapolated from refs 12, 13, and 16; the form of eq 5 leads to a reasonable thickness dependence with T0eff and consequently with Tgeff. The relative increase of both quantities scales with h-1 as proposed in the case of an increase of the glass transition temperature for grafted layers of low mobility.24 Finally, the validity of the model hypothesis is ensured even in the case of relatively high substrate interactions as in the case of PET on Al; the stable chemical bonds present at the polymer/metal interface correspond to an increase of the glass transition temperature of at least 8 °C and a value of R ) 1.03 in our calculations.13 3. Experimental Section Poly(3-hydroxybutyrate), PHB, purchased from Sigma (Mw ) 170 k, powder), was heated for 5 min at 15 °C above its melting point, estimated by differential scanning calorimetry (TmDSC ) 170 °C), between two brass circular electrodes. Amorphous samples were obtained by quenching (cooling rate qc > 50 °C/s) the molten layer between two cold plates held at a temperature below the glass transition temperature (TgDSC ) 2 °C). The final thickness of the sample was given by the diameter of the glassy fibers (ø ) 50 µm) used to separate the brass electrodes. Dielectric spectra of the quenched samples exhibited a strong peak, labeled R, attributed to the structural relaxation (not present in the semicrystalline samples) and a weaker peak, β, related to local conformational fluctuations in
5778 J. Phys. Chem. B, Vol. 111, No. 21, 2007 the same temperature and frequency ranges as reported for amorphous samples of this polymer. Ultrathin films of PHB were spin coated at room temperature from solutions of the polymer in chloroform on cleaned glass slides. An aluminum strip, used as a lower electrode, was thermally evaporated in an ultrahigh vacuum chamber on the slide itself, before spin coating. Samples as prepared were kept for 2 h at 45 °C in order to remove any solvent residuals. A second strip of Al was finally deposited onto the polymer surface, following the procedure described above. Film thicknesses were evaluated from the electrical capacity of the sample.1 The thin polymer layers were melted and quenched as reported above for bulk samples. Successful amorphizations resulted in samples showing an R relaxation peaked in the same frequency range and with a similar intensity as that observed for bulk samples. Dielectric spectra were recorded (10 mHz-10 MHz) with a high-resolution dielectric analyzer (ALPHA-A from Novocontrol Technologies) immediately after amorphization in order to avoid nucleation during the storage of the sample above its glass transition temperature (below room temperature). Spectra of the complex dielectric permittivity, e*(ω) ) ′(ω) - i′′(ω), were recorded under a continuous nitrogen flow in isothermal conditions with a temperature stability better than 0.1 °C. The thermal evolution of the structural relaxation time was followed via isothermal scans, at temperatures separated by no more than 2 °C during heating and cooling programs. The R process was fitted in the frequency domain in terms of an Havrialiak-Negami function. The structural relaxation time τR was assigned to the maximum of the R peak, being the most probable of the relaxation times associated to the process itself (in terms of a distribution function). Because of the flexibility of the polymer chains, τR is constant within the crystallization time1 both in bulk and ultrathin samples. Isothermal crystallization kinetics were monitored in real time by choosing the appropriate low-frequency limit in order to ensure the validity of the condition tSS , τcry, where tSS is the time necessary to record a single spectrum and τcry is the crystallization time. By analyzing the time evolution of the reduction of the dielectric constant ′, connected via a linear relation to the relative crystalline content, the crystallization time was associated to the minimum of the function ∂′(t)/∂ ln(t), corresponding to the mean characteristic time of the process. Further details on the data analysis are available in Supporting Information.
Napolitano and Wu¨bbenhorst
Figure 2. Crystallization time as a function of the structural relaxation time for bulk and ultrathin films of poly(3-hydroxybutyrate). The dashed line is the best linear fit (R ) 0.999) for the data reported. The arrows indicate the relaxation times of bulk samples that would correspond to the crystallization times of the confined systems, according to the best linear fit. The fictive temperatures T*(h), defined in the text, were assigned to the temperatures corresponding to those relaxation time, via a VFT fit, see Supporting Information.
identical, ξ ) 1, for T > 1.1-1.3Tg. At lower temperatures, approaching Tg, the temperature dependence of D becomes smaller than that of η; as a consequence, ξ decreases and assumes the typical values of ∼0.7-0.8, even if data in the range 0.5 e ξ e 0.95 have been reported. The nature of the fractional exponent is not yet clear (special thermal fluctuations, different averaging over that observably investigated,25,29 strong temperature dependence of the relaxation distributions in polymers30), but it was already pointed out that its value depends on the experimental method and path used to calculate it.26 Anyway, as the results of the model are independent from the numerical values of the parameters used, the same qualitative trend can be obtained from all of the series of data satisfying the requirements ensuring the physical meaning of the variables associated. From the comparison between the values of τcry in ultrathin films and those in bulk, it was possible to estimate the values of ∆T* ≡ T - T* for the samples considered, ∆T*(50 nm) ≈ 7 °C and ∆T*(26 nm) ≈ 8 °C, corresponding to an increase of the effective Tg of the system of 4 °C and 6 °C, respectively. 5. Discussion
4. Interplay between Crystallization and Structural Relaxation Times To prove the validity of eq 2 and to estimate a value for the reduction of the effective diffusive temperature T* in eq 4, we carefully measured the crystallization and structural relaxation times of bulk and ultrathin amorphous films of PHB following the experimental procedure just reported. The linear relationship between log τcry and log τR expected from eq 2 is validated; see Figure 2. The relation connecting the diffusion and shear viscosity, D ‚ηξ ) constant, implies, through eqs 1 and 2, that in the regime of temperatures investigated τcry ‚τR-ξ ) constant′. The linear fit of the bulk data of PHB reported in Figure 2, in the temperature range below 1.1 Tg, led to ξ ) 0.748 ( 0.044, a value similar to those reported for low molecular weight and polymer glass formers, via analog relationships in the same range of reduced temperatures.25-30 Usually, the dependence of the diffusion coefficients and the inverse of the average rotational relaxation time (or similarly the shear viscosity) are
The variations of the cold crystallization temperature TCC(1 s) as a function of reduction of the bulk component, given by the ratio δ/h and the increase of T0eff, is shown in Figure 3. TCC was calculated as the temperature at which the crystallization time equals 1 s, corresponding to the crystallization temperature measured during a scan with a heating rate of ∼1 K/s. A positive variation of TCC is achieved both upon reducing the thickness and upon increasing the substrate interaction, as experimentally certified.8 The ratio TCC(h)/TCCBULK scales with (Tgeff(h)/Tg)ξ and thus indicates the deviation from bulk behavior; in the case of the absence of free surfaces, TCC(h)/TCCBULK is a direct measure of dependence of the substrate interactions. Moreover, as ξ < 1, the relative variation of the crystallization temperature is expected to be smaller than the variation of the effective Tg. The calculations proposed here can explain the results recently reported for an ellipsometric investigation of the variation of thermal transitions in ultrathin films of conjugated polymers:31 the non-monotone trend of TCC reflects the same trend of the
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Figure 3. Normalized cold crystallization temperature, as a function of the ratio of the thickness of the RML over the total thickness, at different ratios of the Vogel temperature of the RML and the Vogel temperature of the bulk. The lines are a guide for the eyes.
assigned Tg, the latter being related to Tgeff, because connected to the changes in the thermal expansivity and as in calculations of our model, the relative variations of TCC are smaller than those affecting Tg. By fixing the substrate interactions (R ) 1.01, 1.05, 1.10 in Figure 4), the crystallization rate slows down upon reduction of the thickness, as indicated by an increase of the crystallization time, for different crystallization temperatures, in Figure 4. Obviously, the crystallization time increases much faster for a higher value of T0RML, in analogy with the experimental observations: the crystallization time was found to increase by increasing the substrate interaction.8 Considering the interplay between the structural relaxation and the crystallization time, we find the result resemble the very similar glass transition temperature elevations upon increasing interfacial interactions, as reported in ultrathin films of PS and poly(methyl methacrylate), PMMA.32 The trends obtained for TCC and τcry indicate an apparent enhancement of the glass stability on the nanoscale. The deviations from bulk behavior are temperature dependent, as shown by the changes in the relative variations of the crystallization time at different crystallization temperatures; see Figure 4. The result implies that the reduction in thickness leads to an increase of the temporal stability against devitrification, permitting longer storage periods without a reduction of the material performances. However, even if relatively high values of τcry are indicative for amorphous samples33 (or at least samples that will not crystallize within a given experimental time2,34,35), thinner films are not as stable toward heating. In fact, an increase of the effective Vogel temperature by 5% is sufficient to delay the crystallization times, depending on the crystallization temperature, up to 10 orders of magnitude over the bulk values. The resulting rise in the cold crystallization temperature is instead limited to the 5% itself. Thus, the modest increases of TCC’s and the vanishing of confinement effects on τcry at higher temperatures (see Figure 4) suggest that the thermal stability is not enhanced as much as that of the bulk, as observed experimentally.1,31 Moreover, even in cases of very absorbing substrates, crystallization is always possible if the crystallization temperature is sufficiently high.36 The reduction of the interfacial effects at higher temperatures is connected to a more general discussion on the role of
Figure 4. Deviation from the bulk behavior of the crystallization time as a function of the ratio of the thickness of the RML over the total thickness, at different crystallization temperatures, normalized to the dynamical glass transition temperature and increasing values of the substrate interaction parameter R. The lines are a guide for the eyes.
geometric confinement: the result itself is related to the peculiar form of eq 3: the ratio τ˜ R(T)/τR(T) tends to unity in the limit of infinite temperatures. This implies that, depending on the extent of the confinement effects and the sensitivity of the technique used, the dynamics of a layer with a different mobility will be indistinguishable from the bulk dynamics above a certain crossover temperature.37 This trend predicted by our calculations is thus consistent with the complex scenario characterizing the dynamics of thin polymer films. The temperature at which the experiments are performed is a key parameter for the entity of deviations from bulk behavior.11 Regardless of the substrate interaction and the sign of directions in the mobility, the main cause of the discrepancies between measurements in confined systems is probably the different temperature dependence of the length scales related to the variables investigated. Conclusions The more efficient packing of polymer chains at the very interface with an absorbing substrate, assisted by specific polymer wall interactions, leads to a reduction of the molecular mobility on the scale of several Rg. The role of these reduced mobility layers on the crystallization rate of ultrathin polymer films has been discussed in terms of a bilayer model. In order to account for the substrate interactions, the dynamics of an RML were modeled as a bulk layer characterized by a higher
5780 J. Phys. Chem. B, Vol. 111, No. 21, 2007 Vogel temperature, so that the segmental mobility and the diffusion coefficient governing the crystallization kinetics fitted fractional SE and DSE relations. Through simple calculations, it was possible to reproduce the confinement effects characterizing the tremendous slowing down of the crystallization kinetics on the nanoscale. In particular, the effect of the substrate interaction on the increase of the crystallization time and on the cold crystallization temperature (parameters related to the glass stability of the system) was highlighted. It was finally noted that, in analogy to the glass transition, the deviation from bulk behavior is temperature dependent, vanishing above a crossover temperature limiting the experimental investigation. Acknowledgment. S.N. acknowledges financial support from the European Community’s “Marie-Curie Actions” under Contract No. MRTN-CT-2004-504052 [POLYFILM]. Note Added after ASAP Publication. This article was published ASAP on May 5, 2007. Reference 19 was modified. The corrected version was reposted on May 15, 2007. Supporting Information Available: Assignment of the crystallization time via the drop in the dielectric constant and the estimation of the fictive temperatures via crystallization experiments. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Napolitano, S.; Wu¨bbenhorst, M. Macromolecules 2006, 39, 59675970. (2) Despotopoulou, M. M.; Frank, C. W.; Miller, R. D.; Rabolt, J. F. Macromolecules 1996, 29, 5797-5804. (3) Schonherr, H.; Frank, C. W. Macromolecules 2003, 36, 11881198. (4) Schonherr, H.; Frank, C. W. Macromolecules 2003, 36, 11991208. (5) Einstein A. InVestigation of the Theory of Browinan Motion; Dover: New York, 1956. (6) Debye P. Polar Molecules; Dover: London, 1929. (7) Fukao, K.; Miyamoto, Y. Physical ReView E 2000, 61, 1743-1754. (8) Zhang, Y.; Lu, Y. L.; Duan, Y. X.; Zhang, J. M.; Yan, S. K.; Shen, D. Y. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 4440-4447. (9) Forrest, J. A.; Dalnoki Veress, K.; Stevens, J. R.; Dutcher, J. R. Phys. ReV. Lett. 1996, 77, 2002-2005.
Napolitano and Wu¨bbenhorst (10) Pu, Y.; White, H.; Rafailovich, M. H.; Sokolov, J.; Patel, A.; White, C.; Wu, W. L.; Zaitsev, V.; Schwarz, S. A. Macromolecules 2001, 34, 8518-8522. (11) Fakhraai, Z.; Forrest, J. A. Phys. ReV. Lett. 2005, 95. (12) vanZanten, J. H.; Wallace, W. E.; Wu, W. L. Physical ReView E 1996, 53, R2053-R2056. (13) Napolitano, S.; Prevosto, D.; Lucchesi, M.; Pingue, P.; D’Acunto, M.; Rolla, P. Langmuir 2007, 23, 2103-2109. (14) DeMaggio, G. B.; Frieze, W. E.; Gidley, D. W.; Zhu, M.; Hristov, H. A.; Yee, A. F. Phys. ReV. Lett. 1997, 78, 1524-1527. (15) Baschnagel, J.; Binder, K. Macromolecules 1995, 28, 6808-6818. (16) Zheng, X.; Rafailovich, M. H.; Sokolov, J.; Strzhemechny, Y.; Schwarz, S. A.; Sauer, B. B.; Rubinstein, M. Phys. ReV. Lett. 1997, 79, 241-244. (17) Ellison, C. J.; Torkelson, J. M. Nat. Mater. 2003, 2, 695. (18) Kim, J. H.; Jang, J.; Zin, W. C. Langmuir 2001, 17, 2703-2710. (19) Napolitano, S.; Wu¨bbenhorst, M. J. Phys.: Condens. Matter, 2007, 19, 205121. (20) Schultz, J. M. Macromolecules 1996, 29, 3022-3024. (21) Massa, M. V.; Dalnoki-Veress, K. Phys. ReV. Lett. 2004, 92. (22) Massa, M. V.; Carvalho, J. L.; Dalnoki-Veress, K. Phys. ReV. Lett. 2006, 97, 247802. (23) The result is obtained assuming that Λ(h) has a form like V(h)/ VBULK, where VBULK is the bulk volume and V(h) is the volume of the sample h and considering samples with a similar base surface. (24) Tate, R. S.; Fryer, D. S.; Pasqualini, S.; Montague, M. F.; de Pablo, J. J.; Nealey, P. F. J. Chem. Phys. 2001, 115, 9982-9990. (25) Ngai, K. L. J. Phys. Chem. B 1999, 103, 10684-10694. (26) Ngai, K. L.; Magill, J. H.; Plazek, D. J. J. Chem. Phys. 2000, 112, 1887-1892. (27) Mapes, M. K.; Swallen, S. F.; Ediger, M. D. J. Phys. Chem. B 2006, 110, 507-511. (28) Wu, T.; Yu, L. J. Phys. Chem. B 2006, 110, 15694-15699. (29) Ngai, K. L. J. Phys. Chem. B, 2006, 110, 26211-26214. (30) Hall, D. B.; Dhinojwala, A.; Torkelson, J. M. Phys. ReV. Lett. 1997, 79, 103-106. (31) Campoy-Quiles, M.; Sims, M.; Etchegoin, P. G.; Bradley, D. D. C. Macromolecules 2006, 39, 7673-7680. (32) Fryer, D. S.; Peters, R. D.; Kim, E. J.; Tomaszewski, J. E.; de Pablo, J. J.; Nealey, P. F.; White, C. C.; Wu, W. L. Macromolecules 2001, 34, 5627-5634. (33) Frank, C. W.; Rao, V.; Despotopoulou, M. M.; Pease, R. F. W.; Hinsberg, W. D.; Miller, R. D.; Rabolt, J. F. Science 1996, 273, 912. (34) Capitan, M. J.; Rueda, D. R.; Ezquerra, T. A. Macromolecules 2004, 37, 5653-5659. (35) Ma, Y.; Hu, W.; Reiter, G. Macromolecules 2006, 39, 5159-5164. (36) Reiter, G.; Sommer, J. U. Phys. ReV. Lett. 1998, 80, 3771-3774. (37) Considering the sensitivity of the experimental technique used, the extent of the confinement effects will vanish at the temperature Tlim, that verifies the condition limTfTlim |log[τ˜ R(T)/τR(T)]| ) ∆, where ∆ is smallest variation of τR appreciable by the technique itself.
