Dynamics of Surface Fluctuations on Macrocyclic Melts

Jul 20, 2012 - A hydrodynamic continuum theory (HCT) of thermally stimulated capillary waves describing surface fluctuations of linear polystyrene mel...
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Dynamics of Surface Fluctuations on Macrocyclic Melts Shih-Fan Wang,† Zhang Jiang,‡ Suresh Narayanan,‡ and Mark D. Foster*,† †

Department of Polymer Science, The University of Akron, Akron, Ohio 44325, United States X-ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, United States

‡

S Supporting Information *

ABSTRACT: A hydrodynamic continuum theory (HCT) of thermally stimulated capillary waves describing surface fluctuations of linear polystyrene melts is found to describe surface fluctuations of sufficiently thick films of unentangled cyclic polystyrene. However, for cyclic polystyrene (CPS) films thinner than 10Rg, the surface fluctuations are slower than expected from the HCT universal scaling, revealing a confinement effect active over length scales much larger than Rg. Surface fluctuations of CPS films can be slower than those of films of linear polystyrene analogues, due to differences between the glass transition temperatures, Tg, of the linear and cyclic chains. The temperature dependences of the surface fluctuations match those of bulk viscosities, suggesting that whole chain dynamics dictate the surface height fluctuation dynamics at temperatures 25−60 °C above Tg. When normalized surface relaxation rates of thicker films are plotted as a function of T/Tg, a universal temperature behavior is observed for linear and cyclic chains.

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INTRODUCTION Research on cyclic polymers has received considerable attention in the past decade because these intriguing molecules present the opportunity to study behavior in the absence of chain-end effects. However, because of the difficulty of synthesizing cyclics, most studies have focused on the synthesis and molecular characterization of cycles,1−5 and computer simulations of their behavior in the bulk and solution,6−8 with publications on synthesis and simulations outnumbering those investigating the physical properties in the bulk state.9−17 Both diffusion18,19 and melt viscosity10−12,15−17 have been studied some for cyclic chains and we make connection with these studies, but in this contribution we consider primarily the surface fluctuations of thin film melts of cyclic chains. Since applications of polymers as thin films continue to grow in importance, the behaviors of such films as regards chain conformations, chain dynamics, and crystallization behavior are technologically important. All these behaviors in thin films can be significantly different from the behaviors in the bulk state due to confinement by both the polymer film surface and the substrate. However, we are aware of no experimental work on films of cyclic polymers so far. Recently, rapid developments in X-ray photon correlation spectroscopy (XPCS) have made it possible to probe the length scales and time scales characteristic of the cooperative motion of thermally stimulated capillary waves controlled by the surface tension of polymer melts and shear viscosity.20 Kim et al.21 found that the surface relaxation times, τ, of entangled (M = 123K) linear polystyrene melt films varied with film thickness, h, and in-plane wave vector, q∥ , as anticipated by a hydrodynamic continuum theory (HCT).22 The intensity autocorrelation functions, g2, were reasonably well described by single-exponential decays, and the relaxation times varied with q∥ in agreement with the contention that the dynamics © 2012 American Chemical Society

were those of overdamped capillary waves. Viscosities of the linear polystyrene films obtained from the XPCS data assuming the applicability of the HCT agreed well with literature values for bulk viscosities measured by conventional rheometry for films of thickness, h, greater than 4Rg. The effect on polystyrene melt surface fluctuations of van der Waals interactions with the substrate was first studied by investigators using X-ray reflectivity and off-specular (transverse) scattering measurements.23−26 Tidswell et al.27 argued that the scattering from surface roughness of simple liquids due to capillary waves, σ, could be described as arising from two terms: σ 2(h) = σin 2 +

⎤ ⎡ q 1 u,c ⎥ B ln⎢ 2 ⎢⎣ qvdW (h) ⎥⎦

(1)

where σin is the intrinsic roughness on the order of the molecular dimensions, qu,c = 2π/κ ∼ 0.1−1 (1/Å) is the upper wave vector cutoff, and κ is the size of the molecule in the liquid. The parameter B is given by kBT/(πγ), where kB is the Boltzmann constant, T the absolute temperature, and γ the surface tension. qvdW is the van der Waals cutoff wave vector. For simple liquids the van der Waals cutoff wave vector is a measure of the length scale over which van der Waals interactions with the substrate can suppress fluctuations on the surface. For very thin films, surface fluctuations will be suppressed over an experimentally accessible range of q. For much thicker films, the value of qvdW is sufficiently large that one may not observe any perturbation of the fluctuations in the Received: December 30, 2011 Revised: June 26, 2012 Published: July 20, 2012 6210

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experimentally accessible range of q. The value of qvdW is given by qvdW (h) =

about by eliminating ends. We compare viscosities inferred by analyzing the XPCS data using the overdamped capillary wave theory with values obtained from bulk rheometry and consider the effect of confinement on the surface relaxations for cyclic films. Even diffusion in bulk melts of cyclic chains remains the subject of intense interest,7 since the unusual conformational properties of cyclic chains6 lead many to expect their dynamics to have unusual features as well. In the study of diffusion in bulk samples it has already been shown by both experiments19 and simulations6−8 that for chain molecular weight greater than the critical molecular weight for entanglement, Mc, cyclic chains diffuse faster than do their linear analogues due to less interpenetration of the cyclic chains. However, for M < Mc, the detailed diffusional behavior is quite interesting. For alkanes (CnH2n)32,33 with n < 16, cyclic chains diffuse more slowly than do their linear analogues. Viscosities of linear and cyclic analogues, quantities closely related to the diffusion coefficients, have been studied as well. For unentangled chains one expects from Rouse dynamics that viscosity will be inversely proportional to the diffusion coefficient. Orrah and co-workers12 find that for cyclic poly(dimethylsiloxanes) (CPDMSs) the bulk viscosity of CPDMS is lower than the viscosity of the linear analogue for M greater than a crossover molecular weight, Mcross. The bulk viscosity of CPDMS is higher than that of linear PDMS (LPDMS) for M lower than Mcross. A similar crossover in viscosity has also been observed for cyclic poly(ethylene oxides) (CPEOs) and their linear analogues.34 Chien and co-worker10 observed a crossover behavior for cyclic polystyrene at M ∼ 13K. However, after correction for the free volume contribution of chain ends to the viscosity for linear polystyrene, Rouse behavior was recovered for linear polystyrene and the crossover in viscosity disappeared. McKenna and co-workers observed the same phenomena after correction for free volume.15,16 So, if free volume corrected data are considered, the viscosity of linear polystyrene is larger than that of cyclic polystyrene over the whole M range studied to date. Here we also investigate if the variation in viscosity with molecular weight of unentangled cyclic polystyrene inferred from XPCS measurements is consistent with the bulk behavior seen with conventional rheometry.

(Aeff /2πγ )1/2 h2

(2)

where Aeff is the effective Hamaker constant. For low molecular weight liquids, the value calculated from this expression agrees with what is seen experimentally when a conventional value of the Hamaker constant is used (5.9 × 10−20 J).27 This expression has also been found to be correct within a factor of 2 for films of PS of M < 45K. However, for films of higher M Seo et al.26 found that to describe the experimentally observed values of qvdW they had to invoke unphysically large values of the effective Hamaker constant, revealing that some additional confinement mechanism was at play. Later, using XPCS, Jiang et al.28 showed that the surface dynamics of films of chains of these higher molecular weights do not have the form expected for simple viscous films for film thicknesses less than 4Rg for linear PS. For films of thickness h ∼ 2Rg, the surface fluctuations were slower than expected from the HCT, and those authors argued that it was necessary to incorporate viscoelasticity into their model of the film to rationalize both the slowing down of surface fluctuations and a change in the dependence of relaxation time on in-plane scattering vector. They ascribed this behavior to the presence of “pinning” or physisorption to the substrate of the entangled chains lying adjacent to the substrate. Their results have been contrasted with those of Tsui et al.29 which suggest that for extremely thin films (2.3 −11 nm) of PS on silicon a more highly mobile surface layer dominates the behavior of the film, so that the apparent viscosity of the film is lower than that of the bulk. Recent results of Ediger et al.30 seem also to support the contention that there is a more mobile layer at the surface. Further work by Jiang et al.31 revealed that the variation with molecular weight, M, of the surface dynamics probed by current XPCS instrumentation was different in different temperature regimes. At temperatures at least 51 °C above Tg,bulk, the surface relaxation times increased with increasing M in a manner suggesting that motion of the whole chain determines the surface motion. When temperature decreased toward Tg,bulk, both overall chain motion and segmental motion began to manifest themselves in the time window accessible to XPCS, and the autocorrelation function became stretched. It could no longer be represented by a single relaxation time. When the temperature was close to Tg,bulk (11 °C above Tg,bulk), the autocorrelation function became a single-exponential decay function again, suggesting that the segmental motion dictated the surface fluctuations in the time window at those temperatures. Molecular weight, or chain length, is only the simplest element of chain architecture that could conceivably play a role in dictating the surface dynamics of a polymer melt film. Of central interest in this contribution is the additional factor of chain topology or how the parts of the chain are joined together. We consider cyclic polystyrenes (CPS) as a model compound that eliminates the architecture element of chain ends or, alternatively, introduces an element of tethering, in that the chain is tethered to itself, end to end. We systematically investigate the surface relaxation times of cyclic polystyrene films using XPCS, controlling the molecular weight and film thickness, and considering the special case of low molecular weight cyclic polystyrene to emphasize the changes that come

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EXPERIMENTAL SECTION

Materials. Cyclic polystyrenes with narrow molecular weight distributions were synthesized by anionic polymerization and ringclosing metathesis.35 For a separate batch of chains, the residual double bond in the cyclic chain left by this synthetic route was hydrogenated using Wilkinson’s catalyst/triphenylphosphine/toluene in 800 psi of hydrogen pressure to create hydrogenated cyclic polystyrene (hyCPS) in order to check the sensitivity of the double bonds to the X-ray beam. Analysis with 1H NMR, matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF MS), and tandem MS confirmed the structure of the 2K hydrogenous cyclic polystyrene. A detailed description of the characterization techniques and results are provided in the Supporting Information. There is no evidence of ring catenation for any of the molecular weights, and the formation of catenated rings is highly improbable for the cyclization conditions used here (Supporting Information). McKenna and Plazek14 first showed the effect of contamination by linear chains on the viscosity of cyclic PS materials of molecular weights of 106K−185K using fractions of cyclic chains having polydispersity indices (1.1−1.16) larger than those of the materials in this work. They found a dependence of zero shear viscosity on cyclic chain weight fraction of η0 ∼ φ−5.6. For a weight fraction of linear contaminant of 1 wt % this corresponds to an error in η0 of 6%. Later, 6211

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Kapnistos et al.17 also considered the effect of linear contaminants on the rheological properties of entangled cyclic polystyrenes of molecular weights 161K and 198K. For our much smaller, unentangled cyclic chains, we would anticipate less strong effects on the rheological behavior due to linear contaminants. In any case we take McKenna and Plazek’s result of a 6% error in η0 for 1 wt % linear chains as a bound on the error in our work, as we estimate the linear contamination in all of our samples to be less than 1 wt %. For the central assertions of this report this error is insignificant. Considering the synthetic route used, three sorts of linear impurities could, in principle, be present: dimer from halogen exchange, linear precursor, and monofunctional polystyrene. Fortunately, in the case of the 2K macrocyclic chains, dimers (∼5K) in the precursor are converted to cyclic polymer because of the short chain length. These cyclics of twice the target molecular weight are readily removed by fractionation. The SEC chromatogram after fractionation and the MALDI-TOF mass spectrum of CPS2K were published previously.35 No signals from precursor and monofunctional polystyrene are recognized in the spectrum, and the amount of linear impurity is less than 1%. In the case of hyCPS2K, a monomodal peak with narrow distribution (Mn = 2800 g/mol, PDI = 1.08) is observed in the SEC chromatogram after fractionation (Figure S1), indicating that any dimer was removed. The amounts of precursor and monofunctional polystyrene have been estimated using the MALDI-TOF mass spectrometry results (Figure S3). In the MALDI-TOF mass spectrum, it is difficult to detect signals from any precursor or monofunctional polystyrene. Based on the intensity of the linear impurities in the spectrum, the purity of hyCPS2K is at least 99%. The SEC chromatogram for hyCPS7K after fractionation is shown in Figure S5 of the Supporting Information. A monomodal peak with narrow distribution (Mn = 7700 g/mol, PDI = 1.03) indicates that any dimer has been removed. The absence of any clear signal from other impurities in the reflectron mode MALDI-TOF mass spectrum (Figure S6) indicates the purity of hyCPS7K is at least 99%. Precise determination of the level of any linear impurities is more challenging for the largest cyclic, due to the limitations of MALDITOF MS analysis. Once again the monomodal SEC chromatogram (Mn = 15 100 g/mol, PDI = 1.03) seen for hyCPS14K after fractionation (Figure S7) assures us that any dimer was removed. Because of the higher molecular weight, the MALDI-TOF spectrum can only be run linear mode, while for the lower molecular weights data were collected in both linear and reflectron modes. Because of the resolution of the instrument, it is difficult to resolve the signal from precursor and monofunctional polystyrene in linear mode. However, the shape we see is normal for a well-defined, high molecular polymer using linear mode (Figure S8a,b). For the 7K cyclic we know from reflectron mode data that the fraction of linear impurity is 1% or less. Since the SEC chromatogram for the 14K cyclic looks very good and the shape of the MALDI-TOF spectrum is the same as that for the 7K molecule and the macrocycles of all three molecular weights were made using the same methodology, the amount of linear contaminant in the 14K sample may likewise be estimated at 1% or less. Linear polystyrene analogues, also synthesized by anionic polymerization, each contain a sec-butyl group at one end and a hydrogen at the other end. The molecular characterizations of cyclic polystyrene, hydrogenated cyclic polystyrene and their linear polystyrene analogues are summarized in Table 1 along with values of Tg,bulk measured with differential scanning calorimetry. Each film sample was made by dissolving polymer in toluene (EMD, 99.5%) at a concentration of 1− 3 wt %, filtering the solution with a 0.45 μm filter (Whatman, PTFE) four times and spin-casting onto a silicon wafer at 2000 rpm. Each film was annealed under high vacuum (ca. 1 × 10−7 Pa) at 50 °C above Tg,bulk for 12 h. Actual film thicknesses at room temperature varied slightly about the target nominal values, and then thicknesses varied additionally with temperature. Variation in the thickness of a film over the entire sample (ca. 2 cm2), as determined with ellipsometry before XPCS measurements, was ±5 Å for the films of nominal thickness 210, 400, and 900 Å. For the film of nominal thickness 155 nm the variation was ±10 Å. The uncertainty in determining the thickness from a

Table 1. Molecular Weight Characterization for the Macrocyclic Polystyrenes and Their Linear Analogues sample name

Mna (g/mol)

PDIa

Tg,bulk (°C)b

LPS2K CPS2K hyCPS2K LPS7K hyCPS7K LPS14K hyCPS14K LPS17K

2200 2650 2800 6300 7700 14000 15100 17000

1.05 1.05 1.08 1.05 1.03 1.05 1.03 1.05

61.0 85.4 84.3 86.6 96.2 97.1 101.4 99.6

a

Determined by size exclusion chromatography coupled with light scattering (±5%) in THF at 30 °C. bValue for bulk sample, determined by DSC (±1 °C) at rate of 10 °C/min and collected from the second heating scans. measured XR curve was ±1 Å. This curve was measured on one specific spot, which was typically within 0.16 mm of the spot at which the XPCS data were collected. To limit beam damage, we measured the thickness we used on a fresh spot and then collected the XPCS data from neighboring fresh spots. The variation in thickness over the region of the sample from which we measured XPCS data for a given temperature was generally 4Rg.28

RESULTS AND DISCUSSION Examples of g2 obtained at various values of in-plane wave vector for films of CPS2K and hyCPS2K at 130 °C are shown in Figure 2a,b. A single-exponential decay function g 2 ( t ) = 1 + β e − 2t / τ

γq [cosh(q h) sinh(q h) − q h]

(4)

with β being the coherent contrast, was found to represent well the g2 data for all the films of cyclic PS at all temperatures studied. It is essential to eliminate artifacts due to X-ray beam damage in studying any changes in surface fluctuations caused by 6213

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films. The film of thickness equivalent to approximately 10Rg (RghyCPS14K = 22 Å) behaved as though it were more viscous than were the thicker films. While the data for the 209 Å film lie above those for the other thicknesses, the shape of the curve for 209 Å is essentially the same as the shapes of the curves for the other thicknesses. Thus, to explain the data for 209 Å thickness, it is necessary only to postulate a larger viscosity than seen at larger thicknesses. It is not necessary to invoke the presence of a modulus in the film as found by Jiang et al.28 for films of entangled linear chains of thickness less than 4Rg. We find the length over which this deviation from the expected universality of the HCT behavior is seen to be surprisingly large (i.e., many times Rg) and conjecture that its size is due to differences in the near surface dynamic behavior arising from the nonlinear architecture of the chains. We should, however, consider whether this may alternatively be attributed simply to van der Waals interactions with the substrate. Seo et al.26 found behavior that was liquid-like in some respects for melt films of linear PS of molecular weights 20.8 and 45. For M = 20.8K the ql,c for a film of thickness 4Rg was 10−4 Å−1. The value of ql,c was somewhat larger for the 20.8K chains than for 45.4K chains, but for both the value of ql,c decreased with h/Rg as ql,c∼ (h/Rg)−2. Therefore, for a 209 Å, or ca. 6Rg, film of 14K linear chains, the cutoff could be approximately 0.44 × 10−4 Å−1, neglecting any influence of M, but could be somewhat higher if the M dependence suggested by the data of Seo et al.26 persists to lower M. In the region where the deviation from universal behavior is seen for the cyclic chains, q∥h = 0.08−0.12, which corresponds to q∥ ≈ (3.6−5.5) × 10−4 Å−1. Exactly how the van der Waals cutoff might be different for cyclic chains has not yet been demonstrated experimentally. However, since the dynamics of the cyclic chains are otherwise similar to those of linear chains (once the difference in viscosity is accounted for), we estimate that the ql,c cutoff for our 14K cyclic chains can be estimated as well from the behavior observed for linear chains. For the cyclic 14K chains the 209 Å film thickness corresponds to roughly 10Rg thickness. The cutoff for a 10Rg film of linear 20K would be about 10−4 Å−1 × (10/4)−2 = 0.16 × 10−4 Å−1, which is more than an order of magnitude below the range of q∥ probed in these experiments. Therefore, we suggest that the confinement effect seen for the 209 Å film is not simply that due to the van der Waals interactions with the substrate. This is an intriguing point of the present contribution and will be further investigated in future work. The fact that the surface fluctuations of the thinnest film behave as though it were more viscous than the bulk must also be discussed in comparison to the assertion of Tsui et al.29 that a layer more mobile than the bulk is to be found at the surface of such thin films. This mobile layer becomes evident in their measurements of the evolution of the surface of an unstable film only when the film has nominal thickness less than 11 nm. One difference is that the XPCS measurement is done on a thermally equilibrated surface. In addition, as has been discussed by Kim et al.21 and Jiang et al.,37 XPCS as currently available is not able to resolve, on films of ca. 100 nm thickness, the influence of a mobile surface layer with viscosity less than one-tenth of the bulk value and thickness less than 10 nm, even if this layer exists. Any “mobile layer” would be detected by XPCS only when the total film thickness is comparable to the thickness of the mobile layer so that the surface really dominates the dynamics. For measurements of τ as a function

The variation in surface relaxation time with thickness for the hyCPS14K film is shown in Figure 4a, and the model is found

Figure 4. τ vs q∥ (a) for hyCPS14K films having thicknesses 209, 431, 899, and 1195 Å as marked, all at 140 °C and (b) for a 899 Å thick hyCPS14K film at T = 120, 130, 140, and 150, and 160 °C, as marked. (c) τ/h vs q∥h for hyCPS14K films at 120, 130, 140, 150, and 160 °C. The solid curves represent least-squares fits to the HCT using a nonslip boundary condition. For distinguishing the curves, nominal thicknesses of h = 209, 431, 899, and 1195 Å have been used. Measured film thicknesses vary with temperature within ranges of ±3, ±2, ±10, and ±10 Å about these nominal values as tabulated in the Supporting Information.

to capture very well the shape of the data when η/γ is allowed to vary as a free parameter. Likewise, when temperature is varied, all the data can be described well with the HCT, as shown in Figures 3 and 4b for the specific examples of ca. 431 Å thick films of CPS2K and hyCPS2K and a 899 Å thick film of hyCPS14K. However, when the data for films of various thicknesses are plotted together in Figure 4c in the form of τ/h vs q∥h, it becomes clear that the data for the 209 Å film do not collapse onto the universal behavior for any of the four temperatures. That is because in order to fit the data for that film at a given temperature a value of η/γ had to be used that was larger than the value used to fit all the data from thicker 6214

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of q∥, one would only be able to resolve the presence of a ca. 2 nm surface layer by going to sufficiently large values of q∥ (>0.01 Å−1), which is not currently possible due to the limited coherent X-ray flux density provided by synchrotron technologies. Molecular Weight Dependence of Surface Relaxation and Viscosity. The relaxation of surface fluctuations is substantially slower on melts of cyclic chains than on melts of their linear analogues for the lowest molecular weight chains studied. Figure 5 compares τ/h as a function of q∥h at 110 and

Figure 5. Comparison of τ/h vs q∥h for LPS2K and hyCPS2K films at 110 and 120 °C. The solid curves represent least-squares fits to the HCT using a nonslip boundary condition. The measured thicknesses of the LPS2K film at 110 and 120 °C are 409 ± 3 and 412 ± 3 Å, respectively. Those of the hyCPS2K film at 110 and 120 °C are 411 ± 3 and 412 ± 3 Å.

Figure 6. (a) Comparison of τ/h vs q∥h for nominally 400 Å films of linear and hydrogenous macrocyclic analogues of molecular weights Mn = 7K and (b) comparison of τ/h vs q∥h for nominally 880 Å films of linear and hydrogenous macrocyclic analogues of molecular weights Mn = 14K at 120, 130, and 140 °C. The solid curves represent leastsquares fits to the HCT using a nonslip boundary condition. Thicknesses of the LCPS7K films at 120, 130, and 140 °C are 412 ± 5, 413 ± 5, and 416 ± 5 Å, respectively. The film thicknesses of the hyCPS7K film at 120, 130, and 140 °C are 374 ± 5, 375 ± 5, and 376 ± 5 Å, respectively. Thicknesses of the LPS14K films at 120, 130, and 140 °C are 862 ± 10, 863 ± 10, and 864 ± 10 Å, respectively. Thicknesses of the hyCPS14K films at 120, 130, and 140 °C are 880 ± 10, 898 ± 10, and 904 ± 10 Å, respectively.

120 °C for the hyCPS2K films of thickness about 400 Å with that of a linear analogue. The normalized relaxation times of the cyclic and linear films differ by 2 orders of magnitude in the available range of q∥h (0.065−0.35). As shown above, this is not an artifact of X-ray beam damage, but rather results from the difference in molecular architecture between the cyclic and linear polystyrenes. It appears that the difference in molecular architecture becomes less important as temperature increases from 110 to 120 °C, suggesting that the molecular architecture may play a more important role closer to Tg. When the molecular weight of the chains is increased to 7K and then 14K, the difference in surface relaxation times due to architecture, at a given temperature, decreases, as shown in Figure 6a,b. For the 7K case the difference is somewhat more than 1 order of magnitude at 120 °C and for 14K a factor of 2. The reduction in the difference with temperature is seen also with the two higher molecular weight cases. As will be shown below, to explain the difference in surface dynamics with chain architecture for films of thickness about 400 Å and higher in this group of molecules, it proves sufficient to consider the effect of chain architecture on the glass transition temperature. The Tg’s of the 2K cyclic and linear analogues differ by 24 °C, while those of the 7K and 14K analogues differ by only 5.5 and 4 °C, respectively. We anticipate that at sufficiently large molecular weight (but still less than the entanglement molecular weight) the difference would disappear completely. Since the surface fluctuations of the thicker macrocyclic PS films are described well by the HCT of overdamped thermal capillary waves, the viscosity of the macrocyclic melt can be calculated if the surface tension is known. For the linear chains, literature values of surface tension38 can be found, but for greater consistency in our comparison we used surface tensions

calculated from a group contribution method39,40 (Table 2) to derive viscosities from the XPCS data, ηXCPS, for both the linear and cyclic chains. These values of ηXCPS for the linear chains Table 2. Calculateda Surface Tensions of Different Cyclic Polystyrenes and Their Analogues Mn

T (°C)

γL linear (mN/m)b

γC cyclic (mN/m)b

2K

110 120 130 140 120 130 140 150 120 130 140 150 160

32.3 31.5 30.8 30.0 32.9 32.3 31.7 31.0 33.6 32.8 32.1 31.4 30.7

34.4 33.6 32.8 31.9 33.7 33.0 32.4 31.7 33.9 33.2 32.5 31.8 31.0

7K

14K

a

References 39 and 40. bThe uncertainty of the surface tension is about ±0.5%.

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special feature of the chain dynamics near the interfaces that impacts the surface fluctuations, it cannot be detected by XPCS in the films of thickness ∼20Rg and higher. There has been a long-standing interest in distinctions between the molecular weight dependences of viscosity for linear and cyclic chains measured with conventional rheometry,15,41 and thus we consider briefly what light may be shed on this by our XPCS measurements. The M dependences for ηXPCS of the unentangled linear and cyclic PSs are compared in Figure 8a−c for three temperatures. Because the surface

agree well with values of viscosity measured in the bulk using conventional rheometry, as shown in Figure 7a, confirming the

Figure 7. Comparison of viscosities obtained from XPCS data and conventional bulk rheometry at various temperatures for (a) LPS14K and (b) hyCPS14K. Bulk rheometry values for LPS14K41 and hyCPS14K15 are from the literature.

observation of Kim et al.21 with linear chains of other molecular weights. This agreement for these unentangled linear chains is consistent with the contention that the surface fluctuation dynamics probed in the time and q∥ window available here are dictated by the chain dynamics that also manifest themselves in the phenomenon of bulk viscosity. We may expect the surface tensions of the cyclic chains to be similar to those of the linear analogues, with the difference growing smaller with increasing molecular weight. In fact, for comparison of viscosities derived from XPCS and conventional rheology on a log scale the differences in the values of ηXCPS derived using surface tensions from the linear analogues or values estimated for the cyclic chains are hardly noticeable, but for completeness we note briefly how our estimated surface tensions for the cyclic chains were obtained using the group contribution method.39 The only difference between linear and cyclic polystyrene analogues is the presence of the two ends on the linear chain. We therefore estimate the surface tensions of the cyclic melts from those of linear analogue melts using the method of Koberstein et al.40 and assuming that any enrichment in chain ends at the surface by linear chains can be neglected at this level of approximation. The estimated surface tensions for the cyclic polystyrenes are listed in the Table 2. The value of surface tension of the cyclic PS is closer to that of its linear analogue when the molecular weight is larger. Since we have only small amounts of the cyclic chains, comparison of viscosities derived from XPCS with those from bulk rheometry is made in Figure 7b using values of bulk viscosity from the literature.15 We find good agreement for the films of thickness 400 Å and more, suggesting that for the cyclic chains as well the surface fluctuation dynamics manifest the chain dynamics that dictate bulk viscosity. If there is some

Figure 8. Molecular weight dependences of ηXPCS for unentangled linear PS and cyclic PS at (a) 120, (b) 130, and (c) 140 °C.

fluctuations for 2K linear films are too fast to be measured at °C and η140 °C for the 120, 130, and 140 °C, the values of η130 XPCS XPCS 2K LPS are extrapolated from data at temperatures 90−120 °C and below by fitting the data to an WLF expression:41,42 B log η = log A + T − T0 (6) While it is difficult to discuss exponents for power law relationships for these small data sets involving only a single decade of M values, it appears that the variations in ηXPCS with 6216

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M for the linear chain at 120, 130, and 140 °C are both much more rapid (exponents of approximately 3.4 for 120 °C, 3.1 for 130 °C, and 3.0 for 140 °C) than the first-order dependence predicted by the Rouse dynamics.43 This is due to the rapid change in the free volume contribution of ends in this range of molecular weight.41 Flory44 proposed that the zero shear viscosity of flexible chains could be expressed as a product of two functions:

η = F(Z)f (ρ)

(7)

where F(Z) is sensitive to the number of segments Z of a polymer chain and f(ρ) is sensitive to mass density (ρ) of the polymer and represents the free volume contribution. Allen and Fox41 found that for linear polystyrene chains with M < 30 000 the effect of ρ on viscosity dominates over that of Z. The polymer density increases markedly with increasing molecular weight for low molecular weights and then approaches an asymptotic value at high M. The density approaches this asymptotic value closely already for M below Mc. In the case of the linear PS chains, all the values of M considered here are substantially below Mc, and thus the M dependence of η deviates appreciably from the prediction of the Rouse model which only considers the effect of the friction coefficient of the segments. With the same caveats as for the data from the linear chains we note that the apparent dependences of ηXPCS for the cyclic films are approximately M2.2, M1.9, and M1.8 at 120, 130, and 140 °C, respectively. The three exponent values are comparable with the exponent values derived from measurements by McKenna et al.15 of bulk zero shear viscosity of cyclic chains in the molecular weight range 10 000 < M < 40 000. We conjecture that the density of cyclic chains does not change so rapidly with molecular weight as does the density for linear chains, and thus the viscosity is less strongly molecular weight dependent. Finally, we consider how much of the variation in surface fluctuation dynamics with chain architecture may be attributable to the chain architecture effect on Tg. In Figure 9a are plotted all the data for films of thickness 375 Å or higher in the form of τ/h vs T at a particular value of q∥h = 0.25. (The thickness of 375 Å corresponds to ca. 50Rg for 2K chains and 33Rg for the 7K chains.) Two ways of accounting for the distance of the measurement T from Tg were considered. The first was to consider the dimensionless temperature T/ Tg,bulk.45−48 In Figure 9b, the data for both linear and cyclic architectures are seen to nearly collapse onto a single curve when plotted vs T/Tg,bulk, suggesting that Tg is the key parameter for understanding the surface dynamics of both cyclic and linear PS films in the molecular weight (below Mc) and temperature range (20−60 °C above Tg,bulk) studied here. The capillary wave dynamics for the polystyrene polymers with linear and cyclic architectures have very similar temperature dependences.15,41 Guo and Simon49 suggest that plotting relaxation times as a function of the difference between the actual temperature and Tg (i.e., T − Tg) may be helpful. This approach is based on the idea that at Tg the dynamics of the various systems are similar and that they change as the temperature moves away from Tg. Such a scaling with T − Tg can be rationalized on the basis of either the Williams−Landel− Ferry50 function or the Vogel51 − Fulcher52− Tammann− Hesse53 function. When the data are plotted in this way (Supporting Information), they are grouped closely together, but they do not nearly overlap as they do when plotted vs T/ Tg,bulk. Figure 10 shows, using values available in the literature,

Figure 9. Temperature dependence of XPCS relaxation times for linear and macrocyclic PS films at q∥h = 0.25 plotted as a function of (a) T and (b) T/Tg,bulk with values of Tg,bulk noted in the legend.

Figure 10. Temperature dependence of bulk viscosity according to data from the literature15,41 for a molecular weight range of cyclic and linear PS and temperature range similar to those probed in the XPCS measurements.

the dependence of bulk viscosity on normalized temperature for a similar temperature range and similar range of molecular weight range of cyclic and linear PS. When the temperature is normalized by Tg,bulk, the viscosities of the polymers nearly collapse into one curve no matter what the molecular weight or architecture is. When plotted vs T − Tg,bulk (Supporting Information), the bulk viscosities of the cyclic chains collapse well onto a single curve, but those of the linear chains overlap less well than when plotted vs T/Tg,bulk. The limited temperature range examined readily with XPCS places limitations on the distinctions we can draw between the merits of these two alternatives for understanding the temperature dependence. However, within those constraints we remark that the variation of the bulk viscosity with T/Tg,bulk is consistent 6217

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with that of the relaxation time of the surface fluctuations, suggesting that the capillary wave dynamics are controlled by polymer chain dynamics and the molecular weight dependence of chain dynamics for polystyrene is not so strong. Others have argued that on the length scales of chain motion dynamic heterogeneities are averaged out54 and therefore also for the lateral length scales characteristic of the surface height relaxations measured here. However, in closing, we return to the observation that for a film of hyCPS14K of thickness ∼10Rg a behavior is seen that does not match with the universal scaling expected from the HCT. Apparently there is an effect of confinement. It might be that considering the apparent Tg of the film, as distinguished from that of the bulk, could lead to further understanding. For supported films of low molecular weight linear PS various measurements of Tg have suggested that the Tg,film may be as much as 10 °C lower than Tg,bulk for a film thickness of 400 Å.55 No measurements of Tg depression for thin films of cyclic PS have yet been published but are underway in our laboratories.

AUTHOR INFORMATION

Corresponding Author

*Fax +1-330-972-5290; Tel +1-330-972-5323; e-mail mdf1@ uakron.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Ernst D. von Meerwall, Alexander Agapov, David T. Wu, and Renfeng Hu for helpful discussions. We thank Mr. Aleer M. Yol, and Dr. Chrys Wesdemiotis for MALDI-TOF mass spectroscopy measurements and discussions. This material is based upon work supported by the National Science Foundation under Grant CBET-0730692. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the DOE’s Office of Science under Contract DE-AC02-06CH11357.

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CONCLUSION The surface fluctuations of unentangled cyclic polystyrenes of Mn = 2K, 7K, and 14K are slower than those of linear analogues, with the disparity decreasing with increasing molecular weight. This difference caused by chain architecture can be rationalized on the basis of the behavior of melt bulk viscosity for sufficiently thick films. The variations in viscosities of the linear and cyclic polystyrenes derived from XPCS (and not corrected for free volume effects) over the small range of M probed are consistent with the existence of a crossover from a region at low M in which ηcyclic > ηlinear, to a region in which ηcyclic < ηlinear. The data available here suggest a crossover molecular weight of about 22K at 120 °C, 24K at 130 °C, and 31K at 140 °C. For M ≪ Mc, the local density of polymer segments controls both bulk and surface fluctuation dynamics, resulting in a stronger molecular weight dependence of surface fluctuations on M for films of linear polystyrene than for CPS and slower relaxation of surface fluctuations on the cyclic melt film than on the analogue linear melt film. Universal scaling of the surface dynamics with film thickness is seen for all but the thinnest film studied for hyCPS14K, and viscosities inferred from the XPCS data using the HCT for all the other films agree well with bulk viscosity measurements. Coincidence between ηXPCS and ηbulk for both cyclic chains and the linear analogues, and the universal behavior of the relaxation time with T/Tg,bulk, implies that the surface relaxations are controlled by chain dynamics in the regime studied. For a film of 14K CPS chains, deviation from the universal thickness scaling expected from the HCT is seen at a thickness of 10Rg, a thickness substantially larger (with respect to chain size) than the thickness at which confinement effects on surface fluctuations have been reported28 for entangled linear chains.

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ASSOCIATED CONTENT

S Supporting Information *

Information on synthesis molecular characterization including SEC, NMR, and MALDI-TOF MS characterization data; thicknesses used for XPCS analysis; discussion of catenation; and comparisons using T − Tg,bulk. This material is available free of charge via the Internet at http://pubs.acs.org. 6218

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