GRANT D. SMITH, DMITRY BEDROV Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah

Transcription

1 HIGHLIGHT Relationship Between the a- and b-relaxation Processes in Amorphous Polymers: Insight From Atomistic Molecular Dynamics Simulations of 1,4-Polybutadiene Melts and Blends GRANT D. SMITH, DMITRY BEDROV Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah Received 23 March 2006; revised 19 July 2006; accepted 7 November 2006 DOI: /polb Published online in Wiley InterScience ( ABSTRACT: Amorphous polymers exhibit a primary (glass, or a-) relaxation process and a lowtemperature relaxation process associated with polymer backbone motion usually referred to as the b-relaxation process. The latter process can be observed below the glass transition temperature of the polymer and usually merges with the a-relaxation process at temperatures somewhat above the glass transition temperature. While it is widely held that both the a-relaxation and b-relaxation processes are engendered by localized (segmental) motions of the polymer backbone, and that there is a strong mechanistic connection between them, the molecular mechanisms of the a- relaxation and b-relaxation processes in amorphous polymers are not well understood. Recently, atomistic molecular dynamics simulations of melts and blends of 1,4- polybutadiene have provided insight into the relationship between the a- and b-relaxation processes in glassforming polymers and an improved understanding of their molecular origins. VC 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: , 2007 Keywords: amorphous; blends; computer modeling; dielectric properties; glass; melt; molecular modeling; polybutadiene; relaxation Correspondence to: G. D. Smith ( Gsmith2@ gibbon.mse.utah.edu), Vol. 45, (2007) VC 2007 Wiley Periodicals, Inc. 627

2 628 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) GRANT SMITH DMITRY BEDROV Grant Smith is an Associate Professor of Materials Science and Engineering at the University of Utah. His group focuses on multiscale simulations of polymer melts, solutions, nanocomposites, and selfassembling nanomaterials. He joined the faculty at Utah in 1997 following three years as an Assistant Professor of Chemical Engineering at the University of Missouri-Columbia. He also held positions as a staff scientist at the NASA-Ames Research Center in Mountain View California and as a post-doctoral fellow at the IBM Almaden Research Center. He received his Ph.D. under Richard Boyd at the University of Utah in Dmitry Bedrov is a Research Assistant Professor at the Department of Materials Science & Engineering at the University of Utah. In 1995, he received the B.S. degree in Thermophysics from Odessa State Academy of Refrigeration (Ukraine) while in 1999 he obtained his Ph.D. in Chemical Engineering from University of Utah under the supervision of G.D. Smith. After his graduation, he joined the Department of Materials Science & Engineering as research faculty and became involved with the Center for Simulation of Accidental Fires and Explosions (University of Utah). His primary area of expertise is a multiscale modeling of complex polymeric materials including polymer melts, blends and solutions, polymer nanocomposites, micellar solutions, polymer brushes, and polymer networks. INTRODUCTION MD SIMULATIONS OF 1,4-POLYBUTADIENE Experimentalists, theorists, and simulators have long sought to understand the molecular origins of the primary a- (glass) relaxation process and the secondary (subglass) main-chain b-relaxation process in amorphous polymers. Correlations between the a-relaxation and b-relaxation processes deduced from experimental relaxation studies 1,2 reveal that the main-chain b-relaxation process involves local motions of the polymer chain backbone that do not require cooperative motion of surrounding chains and that the main-chain b-relaxation process is a precursor for the primary a-relaxation process. However, the precise nature of these local motions, whether they are spatially and temporally homogeneous or heterogeneous, and how these motions ultimately lead to complete relaxation of the polymer, are not well understood. Molecular dynamics (MD) simulations, which have been demonstrated to accurately reproduce the structural and dynamic properties, including relaxation processes, for a wide variety of amorphous polymers, 3 are in principle well suited to help elucidate the mechanisms of the a- and b-relaxation processes in amorphous polymers. From a simulator s point of view, 1,4-polybutadiene (PBD) is one of the most appealing glass-forming polymer because of its simple chemical structure, lack of bulky side groups, and nonpolar nature. These features allow PBD to be simulated accurately utilizing a united atom model where the hydrogen atoms are subsumed into effective potentials centered on the carbon atoms. Furthermore, the nonpolar nature of PBD eliminates the need to include expensive long-range Coulomb interactions in the simulations. However, even after years of simulation of PBD melts utilizing a quantum chemistrybased united atom potential, our so-called chemically realistic chain model for PBD (CR-PBD model), the simulation trajectories generated to date are not sufficiently long to allow us to fully resolve the a- and b-relaxation processes. As illustrated in Figure 1, the dielectric a- and b-relaxation processes in PBD 4,5 are well resolved only on time scales significantly longer than the multiple microsecond trajectories we have generated. The much longer simulations needed to clearly resolve the a- and b-relaxation processes in PBD for purposes of elucidating their molecular mechanisms remain a daunting com-

3 HIGHLIGHT 629 Figure 1. Relaxation times for the dielectric a-relaxation process (small circles) and dielectric b-relaxation process (triangles) obtained from experimental measurements on melts of PBD as a function of inverse temperature. 4,5 Also shown are relaxation times for the (apparent) combined relaxation process obtained from MD simulations (large circles). The tendency of the relaxation time for the combined process as obtained from MD simulations to follow the experimental b-relaxation process at the lowest simulated temperature is due to our inability to adequately sample the slow a-relaxation process. putational challenge. Fortunately, recent dielectric spectroscopy experiments on glass forming liquids under high pressure 6 have shown that increasing pressure can result in a large shift in the relaxation time of the co-operative a-relaxation process relative to that of the local b-relaxation process, which is relatively insensitive to pressure. As a result, the bifurcation of the relaxation processes, i.e., the separation of time scales for the a- and b-relaxation processes, is moved into a shorter time window (at higher temperature). In MD simulations, the effect of increasing pressure (and simultaneously temperature) can be approximated by reducing intramolecular energy barriers for dihedral transitions of the polymer backbone. In such an exercise, the intramolecular energetic barriers to relaxation are reduced while leaving the intermolecular packing essentially unperturbed, similar to the effect of raising both pressure and temperature in such a manner as to leave the density largely while reducing the internal barriers to rotation (relative to kt). Reducing dihedral barriers should therefore move the bifurcation into a shorter time window. We have undertaken this approach, creating PBD melts with faster conformational dynamics by reducing dihedral barriers as described below. Models and Methods A detailed description of the CR-PBD model and the quantum chemistry-based united atom force field used in simulating the CR-PBD melt can be found elsewhere. 7 Extensive comparison of CR-PBD melt simulations with NMR spin-lattice relaxation, 8 dynamic neutron scattering, 9,10 and dielectric relaxation 11 measurements can also be found in the literature. To create melts with faster dynamics, the rotational energy barriers for all backbone dihedrals (except for double bonds) were reduced by a factor of four relative to the CR-PBD model yielding the low barrier, or LB-PBD, model. 12 For example in Figure 2, we compare the dihedral energy profiles for the CR-PBD and LB-PBD models for the backbone C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) dihedrals. The choice of this scaling was motivated by our desire to move the bifurcation of the a-relaxation and b-relaxation processes into an accessible time window, as was found in simulations of a 1,4-polybutadiene (PBD) model with no dihedral barriers, 13 while at the same time maintaining clearly identifiable conformation states corresponding to those found in CR-PBD. All other bonded and nonbonded interactions in the LB- PBD are identical to those in the CR-PBD model. Based upon Vogel-Fulcher fits to the temperature dependence of the a-relaxation time obtained from the torsional autocorrelation function (see below), we obtain glass transition temperatures 14 of 170 and 102 K for the CR-PBD and LB-PBD melts, respectively. MD simulations were carried out on melt ensembles of 40 PBD chains. The chains are random copolymers of 30 repeat units each with a microstructure of 40% 1,4- Figure 2. The conformational energy of the C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) dihedrals as a function of dihedral angle for the CR-PBD and LB-PBD models. Also shown is the division of alkyl dihedral angles into gauche þ, trans, and gauche conformational states as well as schematic illustrations of the trans allyl and alkyl dihedral in PBD.

4 630 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) cis, 50% 1,4-trans, and 10% 1,2-vinyl units. Simulations were conducted using the MD simulation package Lucretius 15 as described in detail elsewhere. 16 Equilibrium volumes were obtained at each temperature from 50 to 100 ns of NPT simulation followed by NVT simulations of up to 1 ls for sampling purposes. When not shown, error bars for all plots are smaller than the size of the symbols except for autocorrelation functions at long time and low temperature and for relaxation times where estimated uncertainties are less than 620% of the values themselves except where otherwise noted. We also investigated blends of LB-PBD with CR-PBD chains. 17 Original configurations were taken from existing well-equilibrated CR-PBD melts trajectories. The desired fraction of chains was converted to LB-PBD and subsequently equilibrated over 40 ns followed by production run over ns using NVT ensemble MD simulations. LB-PBD/CR-PBD blends with 10, 25, 50, and 75% LB-PBD were studied. Probes of Segmental Relaxation Segmental relaxation in the LB-PBD melt and LB-PBD/ CR-PBD blends was probed by monitoring the torsional autocorrelation function and the dipole moment autocorrelation function. The torsional autocorrelation (TACF), defined as TACF i ðtþ ¼ hjhðtþjjhð0þj i h jhð0þj hjhð0þj 2 i hjhð0þji 2 i2 ð1þ was determined for i ¼ C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) and i ¼ C(sp 2 ) cis C(sp2 ) C(sp 3 ) C(sp 3 ) (cis allyl) dihedrals illustrated in Figure 2. Here h(t) is the (absolute) value 18 of the conformational angle for a given dihedral at time t and the ensemble average is taken over all dihedrals of the given type (alkyl or cis allyl). The TACF decays as backbone dihedrals explore conformational space through conformational transitions. We have shown previously that the decay of the TACF in PBD closely reflects segmental relaxation as probed by neutron scattering, NMR T 1 measurements, and dielectric relaxation Linear response theory allows us to obtain the complex dielectric permittivity e(x) ¼ e 0 (x) þ ie@(x) using the relationship 19 e 0 ðxþþie 00 ðxþ De ¼ 1 ix Z 1 0 DACFðtÞ expð ixtþ dt ð2þ where the dipole moment autocorrelation function (DACF) is given as DACFðtÞ ¼ hmð0þmðtþi hmð0þmð0þi ¼ ffi P N i;j¼1 P N i;j¼1 P N i¼1 P N i¼1 M ið0þm j ðtþ M ið0þm j ð0þ M ið0þm j ðtþ M ð3þ ið0þm j ð0þ Here M(t) and M i (t) are the dipole moment of the system (melt or blend) and the dipole moment of chain i at time t, respectively, and N is the number of chains. For purposes of determining M(t), we assume that the dipole moment of each polymer chain is uncorrelated with that of the other polymer chains yielding the right-hand expression in eq 3. Note that all intramolecular correlations between dipoles (primarily cis units) are maintained in this representation. In calculating the DACF from MD trajectories using eq 3, hydrogen atoms were added to the united atom trajectories in the manner described previously, 20 and partial atomic charges were subsequently assigned to all atoms. 11 Fitting of the Relaxation Functions We assume that the contribution of a relaxation process in the PBD melt and blends to the (partial) decay of an autocorrelation function (eqs 1 and 3), resulting from segmental motion associated with the relaxation process, can be represented by the Kohlrausch-Williams-Watts (KWW) function 21,22 f ðtþ ¼A exp t b s ð4þ where s is an apparent relaxation time, b is a stretching exponent, and A is an amplitude. We represented the autocorrelation functions (eqs 1 and 3) with the best fit obtainable using a single relaxation process and a sum of two processes, labeled b (short-time) and a (longtime) using ACFðtÞ ¼A b f b ðtþþa a f a ðtþ ð5þ Here f a (t) and f b (t) are KWW functions (eq 4) representing the a- and b-relaxations, respectively, while A a and A b are amplitudes of these processes with the constraint A a þ A b 1.0. When fitting with a single relaxation process we set A b ¼ 0. Relaxation times for the a-relaxation (s a ) and b-relaxation (s b ) processes were determined from the time integral of the corresponding relaxation function, f a (t) and f b (t), obtained from fitting and were described by either a Vogel-Fulcher temperature dependence 23,24

5 HIGHLIGHT 631 ln½sðtþš ¼ C 1 þ C 2 T T 0 ð6þ or an Arrhenius temperature dependence ln½sðtþš ¼ C 1 þ C 2 T ð7þ The choice of KWW functions to represent the relaxation processes, as opposed to other functional forms in the time domain, was a matter of convenience. The a- relaxation in polymers is commonly represented with a stretched exponential function the time domain and an asymmetric (e.g., Havriliak-Negami) function in the frequency domain. 25 The b-relaxation process in polymers, however, is usually considered to be relatively symmetric in the frequency domain and hence is represented with a Cole-Cole function, although asymmetry in the b-relaxation process and the use of a Havriliak- Negami function to represent the process are not unknown. 25 As described below, we find that the b- relaxation process in the LB-PBD melt and LB-PBD component in CR-PBD/LB-PBD blends is typically not very broad (i.e., the stretching exponent b (eq 4) is usually greater than 0.5). A KWW function with b 0.5 can be represented reasonably well in the frequency domain with a (symmetric) Cole-Cole function. MD SIMULATIONS OF LOW BARRIER POLYBUTADIENE Relaxation Functions Figure 3(a) shows the TACF for the alkyl dihedral in LB-PBD melts over a wide range of temperature obtained from MD simulations as well as the best obtainable representations of each TACF assuming a single relaxation process and a sum two relaxation processes. Only data for t > 1.0 ps have been used for in fitting the both the TACF and DACF to exclude decay of the autocorrelation functions due to librational motions. It can be clearly seen that a single relaxation process fails to provide an accurate description of the decay of the TACF. Significant improvement in the description of the TACF can be obtained when a sum of two relaxation processes is utilized. Similar behavior is seen for the TACF for the allyl dihedrals (not shown). The DACF obtained from MD simulations of the LB-PBD melts is shown in Figure 3(b). As with the TACF, it is clear that a much better description of the DACF can be obtained using a sum of two processes. In Figure 4, the dielectric loss obtained from eq 2 using the DACF obtained from fitting MD simulation results with a single process and a sum of two processes is compared for the LB-PBD melt Figure 3. (a) The torsional autocorrelation function (TACF) for alkyl dihedrals and (b) the dipole moment autocorrelation function (DACF) for LB-PBD melts at selected temperatures (symbols). Also shown are fits of the autocorrelation functions utilizing a single (solid lines) and a sum of two (dashed lines) relaxation processes. at 140 K. The presence of two relaxation processes in the LB-PBD melt is even clearer in the frequency domain than in the time domain. Temperature Dependence of the a- and b-relaxation Times The relaxation times for the a- and b-relaxation processes obtained from fitting the decay of the TACF for the alkyl and cis allyl dihedrals as well as from fitting the decay of the DACF in the LB-PBD melts are shown as a function of inverse temperature in Figure 5. As anticipated, lowering the dihedral barriers in the PBD melt has moved the bifurcation of the a- and b-relaxation processes in the time window accessible to MD simulations. At the lower end of the temperature range investigated, the separation of the a- and b-relaxation times is greater than four orders of magnitude in time. We find that we can resolve the a- and b-relaxation processes in LB-PBD melts for the dihedrals as well as the

6 632 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) Figure 4. The DACF and its representation with a single and a sum of two relaxation processes as a function of time for the LB-PBD melt at 140 K. Also shown is the dielectric loss as a function angular frequency obtained from eq 2 using the fits of the DACF. dipole moment at all temperatures investigated, i.e., the processes do not merge completely even at higher temperatures. The a-relaxation times can be described well by a Vogel-Fulcher temperature dependence (eq 6) while the b-relaxation times can be described well by an Arrhenius temperature dependence (eq 7), as shown in Figure 5. Note that at temperatures greater than 222 K, we were unable to completely resolve the b-relaxation process from librations for the cis allyl dihedrals and hence report only properties of the a-relaxation of the cis allyl dihedrals for T > 222 K. Figure 5 reveals that the b-relaxation time for the cis allyl dihedrals is shorter than the b-relaxation time alkyl dihedrals for all temperatures, consistent with the lower conformational energy barriers for rotation about the allyl dihedrals compared with the alkyl dihedrals. 7 At higher temperatures, the a-relaxation time for the cis allyl dihedrals is also shorter than that for the alkyl dihedrals. However, with decreasing temperature, the a-relaxation times for the two types of dihedrals appear to merge. We believe this behavior reflects the increasingly important role of cooperative matrix motion on segmental relaxation in polymer melts with decreasing temperature. We speculate that at higher temperatures, complete relaxation (i.e., the a-relaxation process) for the cis allyl dihedrals occurs more rapidly than that for the alkyl dihedrals because the lower dihedral barrier for the former allows relaxation to occur with less cooperative motion of the matrix. At lower temperatures (and higher density) significant cooperative matrix motion is required for the a- relaxation to occur in either the cis allyl or alkyl dihedrals and little difference is seen in their relaxation times. Interestingly, while the b-relaxation time for the DACF in the LB-PBD melts closely follows that of the alkyl dihedrals for all temperatures, the a-relaxation time for the DACF is systematically shorter than that for the alkyl dihedrals at higher temperatures. At lower temperatures, where matrix effects begin to dominate relaxation behavior, the a-relaxation time for the DACF merges with that for the alkyl dihedrals. Dipole relaxation in PBD is primarily due to angular reorientation of the cis H C ¼C H groups that have a net dipole moment (unlike trans H C ¼C H groups, which have no net dipole moment due to symmetry). Therefore, we can anticipate that dipole relaxation in PBD will be influenced by the relaxation behavior of both the cis allyl and alkyl dihedrals. Figure 5 clearly reveals that the b- relaxation process for the DACF is dominated by relaxation (b-relaxation process) of the relatively slowly relaxing alkyl dihedrals. However, the a-relaxation times for the DACF lies between those for the slowly relaxing alkyl and rapidly relaxing cis allyl dihedrals, indicating that both dihedrals play a role in complete orientational relaxation of the cis H C ¼C H groups, as anticipated. Since the a-relaxation time for the fast cis allyl dihedrals merges with that for the slow alkyl dihedrals at low temperatures, the a-relaxation time for the DACF also merges with that for the alkyl dihedrals. Conformational Motion and the b-relaxation Process The gauche þ, trans, and gauche conformational states of the alkyl dihedrals are depicted in Figure 2. We define P 3-state (t) as the probability that a dihedral has not visited Figure 5. Various relaxation times obtained for the LB-PBD melts as a function of inverse temperature. The lines are Vogel-Fulcher (solid) and Arrhenius (dash) representations of the temperature dependence of the a-relaxation and b-relaxation times, respectively.

7 HIGHLIGHT 633 temperature dependence correspond closely with the b- relaxation time for the alkyl dihedrals. Figure 6. The fraction of C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) dihedrals that have not accessed two or three conformational states as a function of time for the LB-PBD melt at 120 K. Also shown are the relaxation functions obtained from fitting the TACF for the LB-PBD melt at 120 K with a sum of two relaxation processes (eq 5). The vertical solid lines denote important time scales. Width of the b-relaxation and a-relaxation Process The b-relaxation process in glass forming polymers is typically broad in the time/frequency domain and the width of the process, while not universally, is often found to decrease with decreasing temperature. 25 In the time domain, the width of the process is characterized by the stretching exponent for the b-relaxation process, b b,inthe KWW fit. Figure 7(a) shows the stretching exponent for the b-relaxation process from fits of eq 5 to the TACF for the alkyl and cis allyl dihedrals as well as the DACF for the LB-PBD melts as a function of inverse temperature. For the TACFs, b b decreases with decreasing temperature. To better understand the origin of the broadening of the b-relaxation process with decreasing temperature in the LB-PBD melts, we have investigated the distribution of times required for alkyl dihedrals to visit all three conformational states. As illustrated in Figures 5 all three conformational states and P 2-state (t) as the probability that a dihedral has not visited both trans and gauche (gauche þ or gauche ) states after time t, respectively. Hence P 3-state (t) decays completely when all alkyl dihedrals have visited all three conformational states and P 2-state (t) decays completely when all alkyl dihedral have visited both the trans and at least one of the gauche states. P 2-state (t) and P 3-state (t) as well as the a- and b- relaxation components of the TACF for the alkyl dihedrals (f a (t) and f b (t) from eq 5) are shown in Figure 6 for the LB-PBD melt at 120 K. The close correspondence between P 2-state (t) and the b-relaxation process is clear, revealing that large-scale conformational motions (conformational transitions between trans and gauche states) occur on the timescale of the b-relaxation. Furthermore, both P 2-state (t) and P 3-state (t) decay almost completely on time scales greater than the b-relaxation time s b, say 5s b, but before significant a-relaxation occurs. Hence, when the a- and b-relaxation processes are well separated in time, all (or nearly all) dihedrals visit all three conformational states before significant a-relaxation occurs. The relationship between the b-relaxation process and conformational transitions is further illustrated when the average time needed for alkyl dihedrals to visit both trans and gauche (gauche þ or gauche ) states (h2- state access timei) and the average time needed for alkyl dihedrals to visit all three conformational states (h3-state access timei) are compared with the b-relaxation time scale as shown in Figure 5. These times as well as their Figure 7. The width (a) and strength (b) of the b-relaxation process in the LB-PBD melts obtained from fitting eq 5 to the TACF of alkyl and cis allyl dihedrals and DACF as a function of inverse temperature. Also shown are the linear fits of the data. Deviation of the data points from the fits indicate the magnitude of the error bars for these parameters.

8 634 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) Figure 8. The distribution of times required for C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) dihedrals to access all three conformational states for LB-PBD melts at various temperatures. The time scale has been normalized by the most probable access time for each temperature. and 6, there is a close correlation between the b-relaxation process and the time scale on which dihedrals visit each conformational state. Figure 8 reveals that the distribution of times on which individual alkyl dihedrals visit each conformational state is broad and broadens substantially with decreasing temperature, consistent with the temperature dependence of the stretching exponents for the b-relaxation process shown in Figure 7(a). The full-width at half maximum of the conformational access time distribution is less than an order of magnitude at 293 K and broadens to almost two orders of magnitude at 120 K. Figure 8 also indicates that the b- relaxation process is quite heterogeneous in the sense that individual dihedrals accomplish the exploration of conformational states corresponding to the b-relaxation process at very different rates. In contrast to the b-relaxation, the width of the a- relaxation in the LB-PBD melts is observed to be independent of temperature within estimated uncertainties for all probes of segmental relaxation investigated. Specifically, values of b a ¼ 0.36, 0.33, and 0.29 were obtained for the alkyl TACF, DACF, and allyl TACF, respectively. Temperature-independent widths for the a- relaxation process are required for time temperature superposition of segmental relaxation to be valid. Such behavior has been observed in dielectric and coherent dynamic neutron scattering on PBD melts 4 as well as in molecular dynamics (MD) simulations of PBD melts. 16 Strength of the b-relaxation Process The strength of the b-relaxation process in LB-PBD melts is characterized by A b for the TACF and DACF from eq 5. The b-relaxation in LB-PBD increases significantly in strength with increasing temperature, as shown in Figure 7(b), becoming the dominant relaxation process at temperatures well above the glass-transition temperature. These results are consistent with dielectric measurements on PBD melts 4 that also reveal a dominant b-relaxation process at high temperatures whose strength decreases with decreasing temperature. To better understand the mechanism of the (main chain) b-relaxation process and the temperature dependence of the strength of the b-relaxation process in LB- PBD melts, it is constructive to consider what must happen in order for segmental relaxation, as monitored by the TACF, to occur. The TACF decays completely only on the time scale over which all dihedrals populate each conformational state with near-equilibrium probability. In other words, when the conformation of individual dihedrals is monitored over the time scale of the complete decay of the TACF, each dihedral will occupy each conformational state for a fraction of this time equal (or nearly equal) to the equilibrium populations (e.g., 52% gauche þ and gauche, 48% trans for the alkyl dihedrals of LB-PBD at 140 K) obtained by averaging over all dihedrals. This complete relaxation occurs on time scales longer than the a-relaxation time. On times scales longer than the b-relaxation time, say five b-relaxation times (5s b ), but short compared to the a-relaxation time, partial relaxation corresponding to the b-relaxation process is observed. On this time scale, individual dihedrals do not occupy each conformational state for a fraction of this time equal to the equilibrium populations, and the TACF decays only partially. The contribution of individual alkyl dihedrals to the decay of the TACF in LB-PBD melts after 5s b, corresponding to a time that is sufficiently long for the b- relaxation process to be largely complete and short enough such that significant a-relaxation has not occurred at the lower temperatures, has been investigated. This was done by monitoring the conformational state of the individual alkyl dihedrals over 5s b and sorting the dihedrals based upon the fraction of this time spent in the trans state (P trans ). Figure 9 shows the fraction of alkyl dihedrals with P trans smaller than a given value for LB-PBD melts at several temperatures. For example, over a time 5s b at 140 K, only a small fraction, specifically 0.14 ¼ ( ), of the alkyl dihedrals have P trans near the equilibrium value, specifically P trans ¼ , where 0.48 is the equilibrium probability. While most alkyl dihedrals have visited both trans and gauche states during the 5s b time window, i.e., 0 < P trans < 1 for the majority of dihedrals, the distribution of P trans over 5s b becomes increasingly broad (further from equilibrium) with decreasing temperature. Those dihedrals with P trans & P eq over 5s b will contribute sig-

9 HIGHLIGHT 635 Figure 9. The fraction of C(sp 2 ) C(sp 3 ) C(sp 3 ) C(sp 2 ) (alkyl) dihedrals that spend a smaller fraction of time in the trans conformational state over a period of 5s b (P trans ) than a given value for the LB-PBD melts at various temperatures. Also shown is the distribution obtained over s a for the 140 K melt as well as the equilibrium trans probability at 140 K (solid vertical line). The dashed and dotted lines illustrate the fraction of dihedrals with near equilibrium populations measured over 5s b and s a, respectively, for the 140 K melt. contains about 4% of the dihedrals. hp trans i and hp gauche i from the low P trans and high P trans ends of the dihedral distribution, respectively, contributing to each subpopulation for the LB-PBD melt at 140 K are shown in Figure 10. Examination of the decay of the TACF (eq 1) after 5s b, or TACF (t ¼ 5s b ), for each subpopulation shown in Figure 10 reveals that the dihedrals on this time scale can be divided into three classes as indicated by the solid lines in the figure. The first class consists of those dihedrals that do not contribute significantly to the decay of TACF (t ¼ 5s b ), consisting of 30 35% of the dihedrals at 140 K. The majority of these dihedrals are not quiescent since they have visited both trans and gauche states, be it very asymmetrically, i.e., they spend a large fraction of the 5s b time window in one state or the other, but not all of it. Only around 4% of the alkyl dihedrals have not visited both trans and gauche states at 140 K after 5s b. The second class consists of dihedrals that have partially relaxed after 5s b comprising 50 55% of the dihedrals at 140 K. Finally, about 15% of the dihedrals have completely relaxed after 5s b at 140 K. Figure 10 (dotted lines) reveals that dihedrals must spend more than about 1% of their time in their minority state to contribute to the relaxation of the TACF, and that dihedrals that spend nificantly to the decay of the TACF on this time scale (i.e., to the b-relaxation process) and those that spend most of their time in either the trans (P trans & 1) or gauche states (P trans & 0) will contribute little to the b- relaxation process. For the purpose of investigating the contribution of individual dihedrals to the decay of the TACF, we have grouped the alkyl dihedrals into subpopulations whose average P trans (over 5s b ) is equal to P eq, i.e., hp trans i subpopulation ¼ P eq, where the average is taken over P trans of all member dihedrals of the subpopulation. The subpopulations were constructed by taking dihedrals corresponding to the first 2% of the P trans distribution (those dihedrals with the smallest P trans ) and adding to this subpopulation, the number dihedrals from the other end of the distribution (largest P trans, or equivalently the smallest P gauche ) required to achieve hp trans i subpopulation ¼ P eq. These are the dihedrals that are furthest from equilibrium in their occupation of conformational states over 5s b, i.e., the dihedrals that spend the largest fraction of their time in either the trans or the gauche states. This process was repeated, taking the next 2% of the P trans distribution and adding dihedrals from the other end of the distribution until all dihedrals were divided into subpopulations. When so constructed, each subpopulation Figure 10. Characteristics of the alkyl dihedral subpopulations for the LB-PBD melt at 140 K. The squares show the average trans (hp trans i) and gauche (hp gauche i) probabilities for the two sets of dihedrals contributing to each subpopulations. The circles show the value of the TACF for each subpopulation after 5s b. The solid lines delineate three classes of dihedrals (nonrelaxing, partially relaxed and fully relaxed), while the dotted lines associate conformational characteristics of the dihedrals with their relaxation behavior (e.g., transition from nonrelaxing dihedrals to partially relaxing dihedrals occurs at hp trans i & hp gauche i & 0.01).

10 636 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) strength, is therefore incorrect, at least for LB-PBD melts. The decreasing strength of the b-relaxation process with decreasing temperature in fact indicates that the complete exploration of conformational space (i.e., every dihedral visiting each conformational state) associated with the b-relaxation process becomes less effectual in fomenting relaxation with decreasing temperature because the manner in which individual dihedrals visit trans and gauche states on the time scale of the b-relaxation becomes increasingly heterogeneous, with an increasingly large fraction of dihedrals spending an decreasing fraction of their time (over 5s b ) in their minority state. Figure 11. The value of the TACF for each alkyl dihedral subpopulation after 5s b for LB-PBD melts at various temperatures. Also shown is TACF for each subpopulation after s a for the LB-PBD at 140 K. The solid lines delineate three classes of dihedrals (nonrelaxing, partially relaxed, and fully relaxed). more than 30% of their time in their minority state (i.e. 0.3 < P trans < 0.7) are essentially completely relaxed. The TACF after 5s b, or TACF (t ¼ 5s b ), for each subpopulation of the alkyl dihedrals in the LB-PBD melt is shown in Figure 11 for a range of temperatures. The fraction of dihedrals that are partially relaxed on time scales longer than the b-relaxation time but before onset of significant a-relaxation, indicated by the sloped lines, is approximately independent of temperature. The decrease in the strength of the b-relaxation process, given as the combined contribution of all subpopulations to the decay of the TACF, with decreasing temperature can be attributed to an increasing fraction of dihedrals that do not contribute to the decay of the TACF and a decreasing fraction of dihedrals that are completely relaxed. Note, however, that nearly all dihedrals, even those that do not contribute significantly to the decay of the TACF, visit both trans and gauche states (and hence undergo conformational transitions) on the time scale of the b-relaxation process at all temperatures. The b-relaxation process is hence homogeneous in the sense that it corresponds to large-scale conformational motions of all (or nearly all) dihedrals. A picture of the b-process resulting from motion of a subset of active dihedrals that becomes increasingly small with decreasing temperature, leading to a corresponding reduction in relaxation The Relationship between the a- and b-relaxation Processes The time scale for all dihedrals to simply visit (occupy for some period of time) each conformational state, associated with the b-relaxation process in LB-PBD melts, is much shorter than the time required for every dihedral to achieve equilibrium occupancy of conformational states. As shown in Figure 9, the latter occurs on time scales longer than but comparable to the a-relaxation time, s a. Specifically, Figure 9 shows that when the state of the individual dihedrals is averaged over time s a, a substantial fraction ( ¼ 0.50) of the alkyl dihedrals at 140 K exhibit P trans near the equilibrium value ( ), compared with only 14% of the dihedrals after complete (or nearly complete) b-relaxation (5s b ). Figure 11 reveals that after s a all dihedrals contribute to the relaxation of the alkyl TACF for the LB-PBD melt at 140 K, and that 60% of the dihedrals are completely relaxed, when compared with only 15% after 5s b at the same temperature. A likely explanation for the partial relaxation associated with the b-relaxation process is conformational memory due to restricted chain motion imposed by the surrounding matrix. This conformational memory results in biased (nonequilibrium) population of conformational states for individual dihedrals on time scales smaller than the a-relaxation time. Hence, while all (or nearly all) dihedrals are able to visit each conformational state during the b-relaxation process, an increasing majority of them (with decreasing temperature) have a propensity to return to a preferred conformational state. This biasing of conformational populations on time scales shorter than the a-relaxation time becomes more severe with decreasing temperature because of more restrictive packing (higher density) and hence more heterogeneous environments, resulting in the observed reduction of the strength of the b-relaxation process with decreasing temperature. This mechanistic interpretation is supported by Figure 12, where the mean-square displacement (MSD) of monomers and the

11 HIGHLIGHT 637 Figure 12. The TACF and monomer mean-square displacement (MSD) as a function of time for the lowest temperature LB-PBD melts investigated. The horizontal dashed line denotes the length scale at which deviation from the expected scaling behavior for monomer displacements is observed, indicating the caging length scale. The vertical dotted lines indicate the b-relaxation time for each temperature. TACF are shown for the three lowest temperatures of the LB-PBD melt investigated as a function of time. Also shown are the b-relaxation times for the alkyl dihedrals and the expected scaling behavior for the monomer MSD in the diffusive regime. The onset of the diffusive regime is often considered to coincide with cooperative segmental relaxation (a-relaxation process). Deviation from diffusive motion indicates monomer caging by the matrix, which for the LB-PBD melts occurs on a length scale between 0.5 Å 2 and 5 Å 2 in the temperature range of K, as indicated in Figure 12. On the time scale of the b-relaxation, the monomer MSD is much less than 5 Å 2, indicating that the cage imposed by the matrix remains largely intact during the b-relaxation process. This cage, while not restricting access of individual dihedrals to each conformational state, biases the populations of individual dihedrals (but not the overall population when averaged over all dihedrals!). The final decay of the TACF (equilibration of individual dihedral populations) occurs after significant matrix motion results in disappearance of the original caging-induced biasing of dihedral populations. blends consisting of component polymers with very different glass-transition temperatures in the neat melt. Dielectric measurements on such blends reveal significant dielectric loss at high frequency that exhibits weak concentration dependence and is located in a frequency range similar to the response of the neat melt of the lower T g component as illustrated in Figure 13 for a polyisoprene(pi)/poly(vinylethylene) (PVE) blend. 26 For polymer blends such as polystyrene/poly(methyl phenyl siloxane) 27 or polystyrene/poly(vinyl methyl ether), 28,29 where the T g difference between the component polymers is greater than 80 8C, dielectric spectroscopy has clearly showed two distinct relaxations of the fast component where one relaxation was found to have a relaxation time similar to that of the neat melt of the fast component and another closer to that expected based upon blend-average dynamics. Several theoretical models attempt to explain dynamical behavior of blends and their components based on thermally driven concentration fluctuations, 30 chain connectivity, 31 or a combination of both. 27,32 34 In these models, the local (segmental) dynamics of a polymer chain are thought to be determined by the local environment, which deviates from the average blend composition. The Lodge-McLeish model 31 suggests that segmental dynamics are defined by the environment on the scale of the Kuhn length. On this length, scale chain connectivity results in a local environment for a polymer segment that is richer in segments of its own type than the average blend composition. Another class of models assumes that segmental dynamics are determined by a a- AND b-relaxation PROCESSES IN MISCIBLE POLYMER BLENDS Despite extensive study, the segmental dynamics in miscible polymer blends remain poorly understood. A prime example is the segmental relaxation of the lower glasstransition temperature (dynamically fast) component of Figure 13. Dielectric loss in a 50/50 poly(isoprene)/poly (vinyl ethylene) blend as well as the pure component polymer response at 270 K. The symbols are experimental data are from Arbe et al. 4 Solid lines are HN fits showing the contribution of the a- and b-relaxation processes to the dielectric response of the melts.

12 638 J. POLYM. SCI. PART B: POLYM. PHYS.: VOL. 45 (2007) local region that can be spontaneously rich in segments of either blend component because of concentration fluctuations. The size of this local region is related to cooperatively rearranging volumes associated with the glass transition 35 and (depending on the model) has certain temperature and composition dependences. According to this class of models, a segment of each component can have a broad distribution of local environments ranging from pure melt-like to dilute limits on length scales of several nanometers that consequently leads to a broad dynamical response of the components and the blend. These models attempt to explain the dynamical response of miscible polymer blends and the perplexing behavior of the lower T g component by considering heterogeneity in the composition of the local environment of polymer segments and local structure (e.g., density fluctuations, chain connectivity, etc.). While it has been shown that parameters of these models can be fitted to quantitatively reproduce composition and temperature dependence of segmental relaxation times for variety of miscible polymer blends, 31,32,36,37 there are number of blends in which these models are able to predict only qualitative trends observed in experiments 32,34 or fail completely, 36,38 indicating that some physics of segmental relaxation in polymers blends might be missing in those models. Another model that has been applied to segmental relaxation in polymer blends (as well as melts) is based on a concept of intermolecular coupling of a relaxing segment with its environment. This coupling model 39 (CM) correlates the relaxation time for the a-relaxation process with a primitive relaxation time (s 0 ) and a coupling parameter (n), which can be related to the b parameter (n ¼ 1 b) of the KWW function that describes the a-relaxation process in the time domain. It was suggested that the primitive relaxation time s 0 in this model can be associated with the Johari-Goldstein (JG) relaxation process (b-relaxation process), 40 therefore defining the separation between the primary (a-relaxation) and secondary (b-relaxation) relaxation processes. While the CM is the only model for polymer blends (or melts) that we are aware of that attempts to correlate a- and b-relaxation processes, the mechanisms of correlation (or coupling) of segmental dynamics with surrounding matrix are not rigorously defined, and therefore, the composition dependence of the CM parameters and the influence of blending on relaxation times are not predicted within the model. Nevertheless, the CM has been applied to polymer blends 39 and other glass-forming mixtures, 41 including systems where the LM and CF models have struggled. 42 We believe the segmental relaxation behavior of the lower T g component in miscible polymer blends can be explained, at least for some blends, by considering the possibility that a- and b-relaxation processes of a polymer are likely to be influenced differently upon blending. In miscible polymer blends, the local environment for any polymer segment differs from that in the pure melt reflecting the presence of the second blend component. The models described above attempt to account for the influence of the local environment on the segmental relaxation of the component polymers of a blend by assuming that segmental relaxation of components in the blend is qualitatively (mechanistically) the same as in the pure melts of these components and that only relaxation times (or glass-transition temperatures) are shifted upon blending. Since, as discussed above, the a-relaxation process in amorphous polymers depends upon cooperative motion of the local environment (matrix), while the b-relaxation process is a local process that depends much less on matrix mobility, we anticipate that blending will influence the a-relaxation process of the component polymers much more than their b-relaxation processes. The insensitivity of local relaxation processes to blending has been observed for glass forming molecular mixtures 41,43 and in some miscible polymer blends 29 and has been anticipated by the CM model. 42(b) To investigate this supposition, we have investigated the dynamics of CR-PBD/LB-PBD polymer blends. Taking into account that the CR-PBD and LB-PBD melts have quite different glass transition temperatures (170 and 102 K, respectively) while their structural and conformational properties are very similar, the CR-PBD/LB-PBD blend allows relaxation processes in a miscible blend to be investigated without the complications of mismatch in chemical structure or strength of interactions between blend components. The Influence of Blend Composition on Relaxation Processes in LB-PBD/CR-PBD Blends In Figure 14, we show the DACF of the fast (LB-PBD) and slow (CR-PBD) blend components for several blend compositions at 198 K. Also shown is the DACF for the pure component melts. Signatures of the a- and b-relaxation processes [see also Fig. 3(b)] can be seen for the pure LB-PBD melt and LB-PBD blend component. While separate a- and b-relaxation processes are resolved experimentally for the pure PBD melt at 198 K as shown in Figure 1, our MD simulations are too short to allow definitive resolution of a-relaxation and b- relaxation processes for the DACF in the pure CR-PBD melt and CR-PBD blend component. Hence, while the DACF for the LB-PBD component is represented as a sum of two KWW processes (eq 5), the DACF for the CR-PBD component has been represented as a single relaxation process as shown in Figure 14.

13 HIGHLIGHT 639 Figure 14. The DACF for the LB-PBD and CR-PBD components of LB-PBD/CR-PBD blends as a function of the concentration of the LB-PBD component at 198 K. Also shown are the fits using two processes eq 5 (solid lines) for the LB- PBD component and single process eq 4 (dash lines) for the CR-PBD component. In Figure 15(a), the dielectric loss of a 50/50 blend of CR-PBD and LB-PBD at 198 K is shown while in Figure 15(b) the dielectric loss of the corresponding pure melts are shown CR-PBD and LB-PBD melts at 198 K are shown as obtained by Fourier transform of the DACF. The dielectric response of the 50/50 CR-PBD/ LB-PBD blend and the 50/50 PI/PVE blend (Fig. 13) are remarkably similar. Also shown in Figure 15(a) are the contributions of the fast (LB-PBD) and slow (CR-PBD) blend components to the dielectric loss. Figure 15(a) reveals that the low-frequency peak in the dielectric response of the 50/50 CR-PBD/LB-PBD blend is due to relaxation of the CR-PBD component (combined a- and b-relaxation processes) shifted to higher frequency compared with the CR-PBD pure melt [Fig. 15(b)] combined with the low frequency wing of the a-relaxation process of the LB-PBD component, which is shifted to lower frequency and broadened compared with the LB-PBD pure melt [Fig. 15(b)]. The high-frequency peak in the blend response is due largely to the b-relaxation process of the LB-PBD component largely uninfluenced by blending combined with the high frequency wing of the broad a- relaxation process of the LB-PBD component. Figure 15(c) shows the (integrated) relaxation times for the combined dielectric a- and b-relaxation processes in the CR-PBD component and the individual a- and b- relaxation processes in the LB-PBD component of a 50/ 50 CR-PBD/LB-PBD blend at 198 K as a function of blend composition. The insensitivity of the b-relaxation time for the LB-PBD component, the strong dependence of the a-relaxation time for the LB-PBD component, and the strong dependence of the relaxation time for the combined a- and b-relaxation processes for the CR-PBD component on blend composition can be clearly seen. Figure 15(d) shows the amplitude (strength) of the dielectric b-relaxation process of the fast LB-PBD blend component as well as the KWW stretching exponents for all resolvable processes in the blend as a function of blend composition. At 198 K (well above T g of the pure LB-PBD melt), the strength of the b-relaxation process in the LB-PBD melt is comparable to the strength of the a-relaxation process. The strength of the b-relaxation process in the LB-PBD blend component changes very little with blending, indicating that the b-relaxation process of the fast component, while becoming well separated from the a-relaxation process in the blend [Fig. 15(c)], remains strong for all blend compositions and is responsible for a significant fraction of the relaxation of the fast LB-PBD component in the blend. It can also been seen that both the a- and b-relaxation process in LB-PBD broaden (i.e., the b stretching exponent becomes smaller) upon blending with the slow CR-PBD component. The broadening of the a-relaxation process in LB-PBD with blending is much more pronounced than for the b-relaxation process, with b for the a-process decreasing almost 45% in the 10 wt % LB-PBD blend relative to the pure melt, compared with a decrease of only 20% for b for the b-process. For the CR-PBD component, Figure 15(d) reveals significant narrowing of the combined a- and b-relaxation processes upon blending with the fast LB-PBD component, particularly for low LB-PBD content blends. This narrowing is likely due to a combination of reduced separation between the a- and b-relaxation processes with blending and intrinsic narrowing of the individual processes. In summary, while blending with slow CR-PBD increases the separation between the a- and b-relaxation times for the fast LB-PBD component and results in broadening of the relaxation processes (particularly the a-relaxation process), blending of the slow CR-PBD component blending with the fast LB-PBD narrows the combined a- and b-relaxation, consistent with a decrease in the separation between the a- and b-relaxation times and an intrinsic narrowing of the relaxation processes. Both of these results indicate that blending has a stronger influence on the cooperative a-relaxation processes of the component polymers than on the local (mainchain) b-relaxation processes. Relaxation Mechanisms in Polymer Blends We have recently applied the concepts enumerated above regarding the influence of blending on the response of the component polymers in CR-PBD/LB- PBD blends to experimentally measure dielectric loss in PI/PVE blends. 44 The dielectric response of the 50/50