The Role of Environment in Structural Relaxation ... - ACS Publications

6598

Macromolecules 2005, 38, 6598-6609

The Role of Environment in Structural Relaxation of Miscible Polymer Blends Arun Neelakantan, Andrew May, and Janna K. Maranas* Department of Chemical Engineering, The Pennsylvania State University, Univeristy Park, Pennsylvania 16802 Received December 22, 2003; Revised Manuscript Received May 27, 2005 ABSTRACT: A system of six miscible blends formed from four saturated hydrocarbon polymers is investigated using molecular dynamics simulation. Using this system, we isolate the role of the environment in the dynamic response of each material. Variations due to features inherent to each material are minimized by maintaining the same effective concentration for all components in all blends. This results from the four components having similar self-concentrations arising from chain connectivity and using 50/50 composition for the blends. Variations due to concentration fluctuations are minimized because the chain lengths accessible in simulations are limited. The dynamic response is measured via the selfintermediate scattering function, and its counterpart measuring collective motion, as would be observed in incoherent and coherent quasielastic neutron scattering experiments. For self-motion, significant variations between materials in the environmental effect on dynamic response are observed. These variations are linked to intermolecular packing over the length scale of a Kuhn segment. When collective motion is considered, variations in dynamic response to environment between materials are greatly reduced.

Introduction Component dynamics in miscible polymer blends have been well explored recently using NMR,1-3 neutron scattering,4-6 oscillatory shear,7-10 and dielectric spectroscopy.11-14 It is widely accepted that the two components in such systems, despite being homogeneous in a thermodynamic sense, often retain some of their individual mobility characteristics. Their relaxations are broadened compared to those of the constituent polymers and shifted toward one another. Theories for describing these observations are generally based on the idea that a distribution of local environments will broaden and shift the dynamic response.3,9,15-20 The prevailing idea behind the molecular origin of distinct component dynamics in a miscible blend is that the local composition around a given polymer segment differs from that of the mixture as a whole. Two controlling factors have been suggested in this regard: selfconcentration arising from intramolecular connectivity,19,20 and concentration fluctuations, as assessed by molecular weight and the Flory χ parameter.15 The response of a given material to blending depends on its inherent mobility and the nature of the environment provided by the blend partner. Although it is possible to investigate these two factors individually, little effort has been made to separate the two effects. Inherent mobility arises from the pure component properties of the component of interest. If we are concerned with the mobility of A in an A/B blend, this would include the self-concentration of A and the dynamic behavior of pure A (e.g., the glass transition temperature and Vogel parameters). Inherent mobility can be assessed by constraining the environment to be primarily B: experiments on blends dilute in the component of interest or simulations with only a single chain of A. The environmental effect depends on the bulk composition, the dynamic behavior of B, and mixture-specific effects including concentration fluctuations and local intermolecular packing. To isolate the environmental effect, one must follow the dynamics of a given material in a range of host environments. This

is difficult in practice because the low miscibility of long chain polymers reduces the number of pairs that may be studied. This work isolates mixture specific environmental effects in blend dynamicssin particular the role of intermolecular packing. To do so requires observation of a single material [so all inherent factors are held fixed] in miscible blends with several second components. Further, the bulk composition and dynamic properties of the second components should vary as little as possible. Finally, concentration fluctuations should be minimized by considering short, rather than long, chain polymers. Such a system is formed from four saturated hydrocarbon polymers: poly(ethylene propylene) [PEP], poly(ethylene butene) [PEB], isotactic polypropyene [PP], and head-to-head polypropylene [hhPP]. Of the six blends possible, only one, PEB/PP, is immiscible.21 We present molecular dynamics simulations of low molecular weight blends [∼850 g/mol], where this blend is also miscible. Each component can thus be observed in three different environments. The self-concentrations as predicted by the Lodge/McLeish model and Kuhn lengths of all components at the temperature used in the simulations are given in Table 1. All have a similar fraction of intramolecular contacts (between 30 and 35%). The intramolecular packing, as assessed by the intramolecular pair distribution function, is similar for all four materials, and perhaps more importantly, no changes occur when mixed with the remaining three.22 The effective concentration of a given component is related to the intramolecular, or self-concentration, φs, by φeff ) φs + (1 - φs)φbulk, where φbulk is the bulk volume fraction.20 As we consider blends with φbulk ∼ 0.50 (we use 25 chains each of A and B, resulting in volume fractions varying from 0.46 to 0.49), and φs does not vary between components, φeff ∼ 0.65 for all components in all blends in this study. The effective concentrations as defined in the Lodge/McLeish model are provided in Table 2. This provides the added advantage that the potential influence of environment (controlled by φeff) does not vary between materials: the reaction of PEP

10.1021/ma035980x CCC: $30.25 © 2005 American Chemical Society Published on Web 07/01/2005

Macromolecules, Vol. 38, No. 15, 2005

Structural Relaxation of Miscible Polymer Blends 6599

Table 1. Properties of Pure Componentsa material

Kuhn lengthb (Å)

self-concentration

Tg (K)

PEP PEB hhPP PP

9.1 9.7 9.5 9.2

0.30 0.30 0.32 0.35

213 223 253 272

a Kuhn lengths and self-concentrations are at 150 °C. The selfconcentration is calculated as in ref 20. b Data taken from: Graessley, W. W. Polymeric Liquids & Networks: Stucture and Properties; Garland Science: New York, 2004.

Table 2. Blend Densities and Effective Concentrations of Both Componentsa material

density (g /cm3)

φbulk 1st listed comp

φeff 1st comp

φeff 2nd comp

PEP-HHPP HHPP-PEB HHPP-PP PEP-PEB PEP-PP PP-PEB

0.7702 0.7770 0.7698 0.7696 0.7624 0.7692

0.474 0.489 0.495 0.463 0.469 0.494

0.630 0.652 0.657 0.624 0.628 0.671

0.678 0.658 0.678 0.676 0.695 0.654

a Effective concentrations are calculated following ref 20 using bulk volume fraction, φbulk, and self-concentrations in Table 1.

to environment may be compared directly with that of PP. The dynamic properties do varyswe try to minimize this by choosing a temperature, 423 K, well above the Tgs (given in Table 1) of any of the blend components. Within the mixture specific effects, we isolate packing by the low molecular weight (∼850 g/mol) of the simulated chains, which displaces the system further from a phase boundary than high molecular weight mixtures and greatly reduces concentration fluctuations. Dynamics are observed via structural relaxation as assessed by the self-intermediate and distinct intermediate scattering functions. These are the observables that would be obtained from incoherent and coherent neutron scattering experiments. The conditions of our simulations suggest that shifts in dynamics would be minimized. Although they are small (the largest shifts in component mobility are a factor of 2), we observe changes on blending that mirror other observations: the response is asymmetric, with the slow component experiencing the largest changes, and the size of the response depends on the difference in pure component mobilities. We link the observed dynamic behavior to shifts in local intermolecular packing: the relative packing of self- (A/A) and cross- (A/B) segments parallels shifts in dynamics and shifts from pure component dynamics emerge at the spatial scale of the first intermolecular contacts. Background The dynamics of polyolefin blends have not been extensively investigated. To date, two studies are available, both on the hhPP/PEP blend.23,24 The available evidence suggests that this blend is characterized by distinct component mobilities on both the terminal and segmental levels. As mentioned above, the observables calculated in the present study are those measured by incoherent and coherent neutron scattering experiments, respectively. Neutron scattering has been used to investigate three miscible blend systems: poly(ethylene oxide)/poly(methyl methacrylate) [PEO/ PMMA],25,26 polystyrene/poly(vinyl methyl ether) [PS/PVME],6,27 and polyisoprene/poly(vinylethylene) [PI/PVE].5,28,29 These studies show that over the time

Figure 1. Repeat units for the four polymers in the study. The six blends possible from these pure components are considered.

range observable with chemically realistic simulations (up to tens of nanoseconds) shifts in the dynamic response of blend components are detectable. Spatial resolution is also possible in these studies, where spatial scaling similar in form to the pure components is observed. In pure polymers, a fast, simple exponential process is observed,30,31 followed by a slower stretched process which is normally the target of investigation. Although the fast process in miscible blends could be followed using a time-of-flight spectrometer, to our knowledge, this has not yet been done. Simulation Details A. Model. The force field used for the simulations is the optimized force field for liquid hydrocarbons (OPLS) introduced by Jorgensen et al.32 The united atom method is used, which represents the molecules as a series of carbon containing groups such as C, CH, CH2, and CH3. Constraint forces on each united atom represent covalent bonds, maintaining a fixed carbon-carbon bond length of 1.54 Å. Bonded interactions are taken into account through bending and torsional potentials, while Lennard-Jones (LJ) interaction sites characterize nonbonded interactions located at the center of mass of each united atom. A full discussion of the model can be found in ref 37. The equations of motion were integrated by means of the standard velocity Verlet algorithm33 using a time step of 5 fs, and the bond length was kept constant using the RATTLE algorithm.33 Initial configurations for all the systems considered in this study were generated as described previously.22 The repeat units of the materials are given in Figure 1. The number of backbone carbon atoms is fixed at 42, so that the overall chain dimensions remain roughly the same between chain architectures. As a result, the materials have different total carbon counts and molecular weights. A full discussion can be found in ref 39. The simulations were run in the canonical [N,V,T] ensemble with a system size of N ) 50 chains. A constant temperature of 150 °C was maintained using the velocity rescaling algorithm of Berendsen et al.34 The composition of all blends is 50/50-25 chains of A and 25 chains of B. The blend density used to set the simulation volume is calculated from polymer melt densities at 150 °C and 1 atm, scaled down to oligomeric size.39 Volume changes on mixing are neglected, as experimental data support this approximation for these systems.35 The densities used as simulation input are given in Table 1.

6600

Neelakantan et al.


Figure 2. Statistical error in measured quantities. (a, b) The self-intermediate scattering function for hhPP in PEP calculated from three 3 ns blocks of time coordinates. Four spatial scales as indicated in the figures spanning the range of our calculations are illustrated. (c, d) Fit parameters τ and β resulting from fitting the long time decay in each of the three 3 ns decay curves to a stretched exponential function as a function of spatial scale.

B. Calculated Quantities. The self-intermediate and collective intermediate scattering functions are calculated directly in reciprocal space:

S ˜ (q,t) )

S(q,t) S(q,0)

)

∑ij exp[iq‚{ri(t0 + t) - rj(t0)}] ∑ij

(1)

exp[iq‚{ri(t0) - rj(t0)}]

where q is the momentum transfer and ri(t) and rj(t) are the positions of united atoms i and j at time t or a reference time t0. The momentum transfer, q, is often associated with the length scale r ) 2π/q. In an isotropic melt, eq 1 can be reduced to36

S ˜ (q,t) )

∑ij

S(q,t)

) S(q,0) sin{q[ri(t0 + t) - rj(t0)]}/q[ri(t0 + t) - rj(t0)]

∑ij sin{q[ri(t0) - rj(t0)]}/q[ri(t0) - rj(t0)]

(2)

The self-intermediate scattering function, S ˜ self(q,t), is obtained when i ) j, and the collective intermediate scattering function, S ˜ coll(q,t), is obtained when i * j. In mixtures, we determine the scattering functions individually for each component by requiring that the atomic positions used are those of the material in question. C. Equilibration and Model Assessment. As described previously,39 the criteria used to assess equili-

bration in our systems is that the mean distance moved by the united atoms is greater than two Rg and that the end-to-end vector autocorrelation functions have decayed to zero. Based on these criteria, required equilibration times vary from 1.5 to 2 ns. We have equilibrated all systems for 2 ns, the time required for the least mobile component in any of the blends to move a distance of 2Rg. At the temperature of our simulations the largest segmental relaxation times obtained are of order 100 ps, roughly 20 times smaller than our assigned equilibration times. To test whether further equilibration time would affect the results, we have calculated the self-intermediate scattering functions following equilibration times of 2-8 ns. The results (see Figure 2 for an example) reveal no drift and indicate that further equilibration has a negligible effect on Ss(q,t). Following equilibration, data acquisition runs of 3 ns are performed on all systems. The intermediate scattering functions are calculated with time zeros located at 500 ps intervals. This model has been previously tested for a variety of polyolefins against experimental data for thermodynamic properties,37 chain packing,38 and dynamics.39 In all cases the model provides a reasonable correlation of experimental data, in particular the subtle changes that sometimes occur as chain architecture and tacticity is varied. Results In previous contributions, we have investigated both self-motion39 and collective motion40 of pure PEP, PEB, hhPP, and PP at spatial scales ranging from 2 to 14 Å. Before presenting our results on mixing, we briefly review these findings. Both self-motion and collective



motion are characterized by a simple exponential decay at times less than 1 ps, followed by stretched exponential behavior at longer times. The stretching parameters range from 0.4 to 0.6 depending on spatial scale, with the lowest values found near the peak in the intermolecular pair distribution function and the highest at the largest spatial scales investigated. The relative contribution of the two decays is spatially dependentsat spatial scales less than the closest intermolecular contacts [the “intramolecular” region], the amplitude of the fast exponential process decays to 0.2 or less before the slow process intervenes. As the spatial scale increases, the relative contributions of the two processes shifts, with the fast exponential process decaying to at most 0.8 for spatial scales greater than 8 Å. In the intramolecular region, collective motion is faster and is more stretched than self-motion: for PEP at r ) 3 Å, the collective relaxation time (β ) 0.45) is nearly half the self-relaxation time (β ) 0.55). Throughout this region, the dependence of relaxation times on spatial scale goes as τ ∼ q-2. At the spatial scale of closest intermolecular contacts the spatial dependence of relaxation times shifts to τ ∼ q-2/β, and collective relaxation times become larger than self-relaxation times. For PEP at r ) 7 Å, the collective relaxation time is twice the self-relaxation time. Collective relaxation times increase above the observed scaling (τ ∼ q-2) in phase with the static structure factor (de Gennes narrowing) for all four pure materials. We now consider the dynamic response to blending of each material. Changes in both self-motion and collective motion with environment are addressed. As each type of motion is characterized by fast exponential and slow stretched exponential processes, there are four individual relaxations to follow through various environments. The self-motion of one component in a blend may be isolated in neutron scattering experiments designed to measure incoherent scattering because the incoherent scattering cross section for hydrogen is much larger than the other elements. Deuteration can thus be used to “hide” the second component. No such labeling scheme is available for collective motion, which is assessed from coherent scattering. In addition to the labeling difficulty, coherent measurements require larger, perdeuterated samples and thus are more costly and less frequently undertaken, although they yield important information. We calculate the intermediate scattering functions for momentum transfers corresponding to spatial scales between 2 and 14 Å. The upper value was chosen to be well within the limit of half the periodic box size: L/2-25 Å in all cases. Within this region, we consider three spatial ranges. The first, 2.0 Å < r < 3.7 Å, is termed intramolecular because this is the only possibility: it is impossible for two united atoms on different chains to be separated by these distances, as evidenced by the intermolecular pair distribution functions,22 which are zero up to 3.7 Å. The second region, termed intermolecular, 3.7 Å < r < 8 Å, begins at the closest intermolecular contacts and continues to the radius of gyration, Rg. Thus, the intermolecular region represents spatial scales of nearest chain packing, as assessed by ginter(r) or the first (amorphous) peak in the static structure factor, S(q).39 The final region, r > 8 Å, is termed whole chain because it covers distances larger than the average coil size (Rg ∼ 9 Å in all cases). We note that all of these length scales contain intramolecu-

lar contacts, as gintra(r) does not approach zero until near the end-to-end distance, which ranges from 20 to 25 Å, depending on material.22 Here the term intramolecular is used to indicate those length scales over which intramolecular contacts are the only type possible. Before presenting typical decay spectra and relaxation times, we discuss the statistical errors in our measurements. Shown in Figure 2a,b are the scattering functions S ˜ self(q,t) of hhPP in PEP for four spatial scales encompassing the intramolecular, intermolecular, and whole chain regions. The three curves shown in the figure are obtained from three different blocks of time, 3 ns in duration (the same as data collection runs), and thus indicate the level of statistical variation obtained from repeating our data collection procedure. Very little variation is observed, indicating that the statistical error is of comparable size to the data markers. The fit parameters obtained from fitting the slow portion of the decay (see eq 3 and surrounding discussion below) in each block to a stretched exponential are shown in Figure 2c,d. Variations in relaxation times are negligible, indicating that any differences outside the size of the data markers are statistically relevant. Variations in stretching exponents are larger because the distribution of relaxation times varies more over time than the average derived from this distribution. This is consistent with the idea of dynamic heterogeneitysin this picture, spatially correlated regions of slow or fast mobility change with time, causing variations in the distribution of relaxation times. ˜ coll(q,t) with Representative changes in S ˜ self(q,t) and S environment are illustrated in Figure 3. The scattering functions of pure HHPP and HHPP in PEP, PEB, and PP are shown at four spatial scales: intramolecular (q ) 1.96 Å-1, r ) 3.2 Å), intermolecular (q ) 1.43, r ) 4.4 and q ) 0.9 Å-1, r ) 7 Å), and whole chain (q ) 0.52 Å-1, r ) 12 Å). HHPP is the least mobile of the materials we have considered,39 and thus all environments in which it is placed have greater mobility. At intramolecular spatial scales, both collective and self-motion are insensitive to environment. Differences only emerge past the spatial scale of the closest intermolecular contacts (Figure 3c-h) which suggests that if the length scale of observation does not extend beyond a single chain, a variation with environment is not possible. Other indicators of mobility that are intramolecular in nature, such as the torsional autocorrelation function, are also insensitive to blending and produce curves similar to those in Figure 3a-d. This is not surprising as the length scales associated with torsional angles are comparable to 3.2-4.4 Å. We note that the cage size, as assessed by the crossover of mean-square displacement from ballistic to subdiffusive motion, is ∼1.5 Åsmuch less than the 3.2 Å spatial scale illustrated in Figure 3a,b. Thus, motion remains insensitive to environment even as the united atoms move out of their cages. One could imagine the insensitivity is due to the similarity in the decay curves of three host materials in the pure state. However, even at 2.6 Å, variations in decay curves between the four pure materials are larger than those in Figure 3a,b.39 Differences induced by mixing emerge as the intermolecular region is entered, as seen in Figure 3c-f. Here the decay in all blends is faster than in the melt, indicating that the faster environments of the host materials increase the mobility of HHPP. This is consistent with the usual observation that the slower component becomes more mobile upon

6602

Neelakantan et al.


Figure 3. Intermediate scattering functions for hhPP in the pure state and in all possible blends at intramolecular (r ) 3.2 Å), intermolecular (r ) 4.4 Å, r ) 7.2 Å), and whole chain (r ) 12 Å) spatial scales: (a) self-intermediate scattering function at 3.2 Å; (b) collective intermediate scattering function at 3.2 Å; (c) self-intermediate scattering function at 4.4 Å; (d) collective intermediate scattering function at 4.4 Å; (e) self-intermediate scattering function at 7.2 Å; (f) collective intermediate scattering function at 7.2 Å; (g) self-intermediate scattering function at 12.0 Å; (h) collective intermediate scattering function at 12.0 Å. Symbols are as in figure legends.



Figure 4. Best fit to a single stretched exponential function for hhPP in PEP at 4.4 Å. The inset shows a fit to a simple exponential at short times and a stretched exponential at long times, with both fit lines extended throughout the range of the data.

blending.46-49 The influence of environment on the dynamics of HHPP continues to increase as whole chain spatial scales are entered (Figure 3g,h). Interestingly, here the collective dynamics of HHPP appear to be less sensitive to environment than self-dynamics. As with our investigations of the pure components in this study39,40 and neutron scattering on other polymers,30,31 the spectra are characterized by a two-step decay:

S ˜ (q,t) )

{

Af exp[-t/τf] t < tc As exp[-(t/τs)β] t > tc

(3)

where τf and τs are characteristic times for the fast and slow processes, Af and As are the amplitudes of the fast and slow processes, and β is the extent to which the slow process in nonexponential in character. The failure of a single, stretched exponential to fit the data is illustrated in Figure 4, where the best such fit is shown for HHPP in HHPP-PEP at 4.4 Å. The single fit is not able to capture the variation of the data: instead, the two functions are required, as shown in the inset where the same data is fitted with eq 3, and each fit line is extended over the entire time range. As has been observed previously, the crossover time, τc, is around 1 ps. Further, there does not appear to be an overlap region where the data cannot be described by either function alone. We now consider the effect of blending on the fit parameters, τfself(q), τsself(q), and βself, obtained from fitting S ˜ self(q,t) to eq 3. The same is done for the collective decay curves. Considering Figure 3, the portion of the decay that is insensitive to environment occurs on time scales less than 1 ps. Since the crossover time between the fast and slow processes is also ∼1 ps, it is likely that the fast process is not sensitive to environment. This is indeed the case, as evidenced by Figure 5, in which is plotted the variation of τfself(q) and τfcoll(q) with environment for all materials. Relaxation times obtained from fits to both self-motion and collective motion are shown, with values for self-motion shifted down along the y-axis for clarity. As data markers are overlapping, it is apparent that for both self-motion and collective motion changes in environment have no effect on relaxation times over all spatial scales. This holds for the de Gennes narrowing region of collective motion, i.e., the spatial range over which the peak in S(q) is located. As with the pure components,

the de Gennes narrowing region is evident, but it is no more sensitive to environment than other spatial scales. The fast process has not previously been observed in blends, and its invariance with environment reinforces the idea that it is intramolecular in origin. We have observed in simulations of PMMA that when the nonbonded potential is turned off, the resulting decay resembles the fast process alone, further emphasizing this idea. The variation of mobility with spatial scale is similar to observations on pure polymers. In this case, neutron scattering studies indicate a power law dependence of relaxation times on momentum transfer: τ ∼ q-x.27,41-44 The value of x (usually 2 for the fast process and 2 or 2/β for the slow decay) depends on the material and the observable q window. As with the pure component relaxation times,39 fast relaxation times in mixtures also show -2 scaling (τ ∼ q-2). We now turn our attention to fit parameters for the slow portion of the decay. Figure 6 shows the stretching parameters, βself and βcoll, for the pure components and in different environments as a function of spatial scale. This parameter is thought to represent the extent to which different atoms within the system have different mobilities as well as the extent to which a single atom experiences a change in mobility over time, although it has been interpreted as the inherent nonexponential character of the relaxations of all atoms in the system.45 Although one might expect that β would be decreased for both components on blending due to the wider range of local environments, β values tend to shift toward one another as with relaxation times. Thus, a material, when mixed with a second component with a narrower distribution of relaxation times, will narrow its own distribution in response, despite the introduction of a new local environment provided by the blend partner. The spatial dependence of the stretching parameters in mixtures is similar to the pure components, and values of βself vary outside of error only in some cases. For collective motion, statistical variations in the stretching parameter are larger, and it is difficult to draw conclusions about its response to mixing. The variation of τs(q) with environment is illustrated in Figure 7. The sensitivity of both self-motion and collective motion to environment in the slow portion of the decay curves is larger than that of the fast portion (Figure 3). Considering self-motion, on first glance it appears the influence of environment on dynamics is correlated with pure component mobility. The ordering of the pure component relaxation times (self-motion) is τ(HHPP) > τ(PEB) > τ(PP) > τ(PEP). Self-relaxation times for PEP show minimal changes (Figure 7a), whereas self-relaxation times for hhPP vary the most with environment (Figure 7g). The implication is that in a given blend the slow component would experience a larger shift in dynamics than the fast component. This asymmetric response has been observed using dielectric spectroscopy, neutron scattering, rheology, and 2D proton NMR on PI/PVE3,19,27,46-50 and PEO/PMMA25,26 blends. The reasons for the different responses to environment between the four materials are now considered. In our model system, concentration fluctuations are minimal due to the short chains. We now consider the ability of the chain connectivity model to explain our observations. The model predicts that the ratio of mixture to pure relaxation times of component A is proportional to the Vogel parameter BA and the differ-

6604

Neelakantan et al.


Figure 5. Variation of fast decay times with environment for both self-motion and collective motion. Each figure shows one material in the pure state and in blends with the three possible blend partners, as a function of spatial scale. Times for selfmotion (lower curve) have been shifted downward by 0.5 for clarity. (a) PEP; (b) PEB; (c) PP; (d) hhPP. Symbols are as in figure legends.

ence between the effective and pure Tgs, defined as δA(φeff) ) TgA(φeff) - TgA

ln(τAmix/τApure) ) BA

[

δA(φeff)

(∆T0 - δA(φeff))(∆T0)

]

(4)

where ∆T0 ) T - T0A is the difference between the measurement temperature and the Vogel temperature of pure A. Variations in shifts from pure component relaxation times thus depend on BA, ∆T0, and δA(φeff). The first two depend on properties of pure A, whereas the Tg of pure B enters in the third through the Fox expression used to compute TgA(φeff). In our system, the effective concentrations of the blend components are approximately equal, but δA(φeff) can differ from δB(φeff) because shifts in effective Tgs away from their pure component values are not always symmetric. The range of δA(φeff) computed using the Fox equation with φeff ) 0.65 are comparable for each component in the other three environments: PEP (16, 12, and 3 K), PEB (15, 10, and 3 K), aPP (21, 19, and 7 K), and hhPP (16, 11, and 6 K). Since PEP does not have significantly smaller values of δA(φeff) than hhPP, this does not provide an explanation for our observations. Vogel parameters (BA and T0) are available for PEP and aPP,51 so we consider the predicted response for this blend. The prediction of eq 4 for the PEP/aPP blend is 0.46 for A ) PEP (B ) 2274, ∆T0 ) 287) and -0.51 for A ) aPP (B ) 1394, ∆T0 ) 224). Clearly, our data do not suggest such a symmetric response in relaxation times at any spatial scale. We will return to this blend below. We now investigate the possibility that variations in response to environment occur due to variations in

interchain packing with mixing. Changes such as this could alter the local (within ∼10 Å) effective concentration of each species independent of connectivity effects and concentration fluctuations. Variations in self-packing (A with A) with mixing may also be different than those in cross-packing (A with B), which could further skew local compositions. We have recently assessed the changes in intra- and intermolecular packing for this same set of mixtures.22 As mentioned in the Introduction, changes in intramolecular packing as environment is altered by blending are virtually nonexistent. Intermolecular packing can be characterized by determining the pair distribution function, g(r), including only pairs on different chains. In a mixture of components A and B, the A/A, B/B, or A/B distributions may be considered. The A/A and B/B distributions vary from the pure state considerably with mixing, most normally such that the packing characteristics of the two blend components move further apart. The A/B distribution normally falls between the A/A and B/B distributions. Since the more efficient packer (component A) is typically the more mobile, the implication is that more mobile component would see a self-concentration (A/A) that is larger than that with the slow component (A/B), leading to a larger than expected effective concentration and a smaller dynamic response. For the slow component (B), the opposite is true. In our system, the materials where the varying environments have the smallest effect, such as PEP, have enhanced self-packing.22 The packing of PEP with itself in the mixtures is greater than self-packing in pure PEP and is always equal to or more efficient than the packing with its blend partner (cross-packing). Conversely,



Figure 6. Variation of slow decay stretching parameters with environment. Each figure shows one material in the pure state and in blends with the three possible blend partners, as a function of spatial scale. Self-motion curves (lower curves) have been shifted downward by 0.2 for clarity. Symbols are as in figure legends.

materials where the environment has a larger effect, such as hhPP, show a decrease in self-packing, which is less efficient than cross-packing. An example is provided in Figure 8, which illustrates the variations of PEP/PEP packing when placed in the other three materials. Figure 9b shows an example of a specific blend, PEP/PP discussed below, where the cross-packing is also provided. Comparing the PEP/PEP distribution with the PEP/PP (cross) distribution gives an idea of the relative number of self- and cross-contacts: in this case, a PEP united atom is surrounded by more PEP atoms on different chains than PP atoms on different chains. The opposite is true for PP united atoms, as the PP/PP distribution falls below that of PEP/PP. Note that there is no difference between the PEP/PP and PP/PEP distributions. We suggest that the relative extent of selfpacking vs cross-intermolecular packing leads to varying dynamic response. The more mobile component, also normally being the more flexible, is the more efficient packer of the two pure components. This packing is enhanced further when mixed and coupled with crosspacking, which is less efficient, creates a local effective concentration, on the order of a Kuhn length, which is enhanced in itself beyond the prediction based on chain connectivity. In support of this suggestion, we observe that the length scale where intermolecular contacts begin to show distinct behavior (i.e., A/A separates from A/B) coincides with that where shifts in dynamics away from pure component values are first observed. Changes in the intermolecular pair distribution functions with mixing are first observed around 5 Å. In Figure 7 we mark this spatial scale with vertical lines. It is apparent

that the onset of changes in the slow decay times with mixing begin at or near these lines. As a specific example of possible intermolecular packing effects, we return to the PP/PEP blend. The selfrelaxation times for each component, in the blend and in the pure state, are shown in Figure 9a. Motion of PP in the PP/PEP blend is accelerated relative to pure PP, whereas motion of PEP shows little change (perhaps a slight slowing can be detected). Shifts from pure component Tgs predicted from the Fox equation are similar: 21 K for PP and 16 K for PEP. The Vogel parameter, B, is 586 K for PP and 987 K for PEP.51 Considering the intermolecular packing in this blend, shown in Figure 9b, the PEP/PEP distribution becomes more peaked in the blend than in the pure state, indicating that locally the effective concentration of PEP segments increases. For PP, the opposite is observed. The PP/PP distribution is suppressed by mixing, which acts to decrease the local effective concentration of PP segments. The PEP/PP distribution lies between the two. A PEP united atom thus has more intermolecular contacts with itself than with PP. As a result, the dynamic behavior changes little from the pure state. In contast, a PP united atom has more intermolecular contacts with PEP than with itself, leading to a decreased effective concentration and a larger response in dynamic behavior. Similar correlations are observed for other blends. The implication is that the selfconcentration, in addition to including chain connectivity effects, must also include effects in local intermolecular packing. To examine this suggestion, we consider those cases where the Lodge-McLeish model can fit the

6606

Neelakantan et al.


Figure 7. Variation of self-motion and collective motion slow decay times with environment. Each figure shows one material in the pure state and in blends with the three possible blend partners, as a function of spatial scale. (a) PEP self-motion; (b) PEP collective motion; (c) PEB self-motion; (d) PEB collective motion; (e) PP self-motion; (f) PP collective motion; (g) hhPP self-motion; (h) hhPP collective motion. Symbols are as in figure legends. Lines indicate the spatial scale of first intermolecular contacts, as determined from intermolecular pair distribution functions.


Figure 8. Intermolecular packing of PEP as a function of environment. Intermolecular pair distribution functions for PEP and for PEP in PEB, PP, and hhPP are shown. Symbols are as in figure legend.

Figure 9. Correlation of shifts in slow self-decay times with intermolecular packing for the PP/PEP blend: (a) slow selfdecay times for pure components and each component in the blend; (b) intermolecular pair distribution functions of pure components, each component in the blend (A/A and B/B), and the cross-correlation (A/B). Symbols are as in figure legends.

data, but the fitted self-concentration differs from the predicted value. If intermolecular packing (such as that observed here for PEP and PP) plays a role, one would expect that the low Tg component, having better intermolecular packing, would require a larger self-concen-


Figure 10. Comparison of the mean-squared displacement, MSD, of pure PEP, and the mean-squared pair separation, MSPS, for pure PEP and PEP in each blend. See the text for explanation of the mean-squared pair separation.

tration than the predicted value. The opposite would be expected for the high Tg component. This model has recently been used to correlate data from a large number of miscible blend systems.52 There are five cases where the fitted φs differs significantly from the predicted φs. In four of them, the low Tg component requires a larger φs than predicted (PVME in PS and P2CS, PEO in PMMA and TMPC in PS); in the remaining case, the high Tg component requires a lower φs than predicted (PMMA in PEO/PMMA). All of these exceptions could potentially be explained by a difference in intermolecular packing. If intermolecular packing plays a role, considering collective rather than self-motion should have an effect on the variations in the dynamic response to environment between materials. As can be seen in Figure 7, the response of collective motion to changes in environment is different than that of self-motion. Responses to mixing are more uniformsPEP responds just as much to changes in environment as hhPP. Although the reasons behind this result are not yet clear, it may be that changes in packing are already “built in” to this function because collective motion by definition contains information about packing. In this case, the responses to collective motion are symmetricsPEP in PP is shifted as much as PP in PEP. The same is true for the hhPP/ PEP blend, whereas the response of self-motion in this blend is quite asymmetric. It is reasonable to expect that pairs of atoms at optimal intermolecular contact will hold that contact as they move about the simulation box. To investigate whether this occurs in our system, we have calculated the average pair distance: 〈ri(t) - rj(t)〉2, which we refer to as mean-squared pair separation, MSPS. To enforce optimal contact, the indices i and j define pairs of atoms that are separated by a distance between 5 and 7 Å at t ) 0. This implies that all pairs contributing to the average in MSPS were initially separated by the range of distances defining the first peak in g(r). If the change in MSPS with time is slower than the average motion of the atoms alone (i.e., mean-squared displacement ) MSD), this indicates that pairs at optimal contact tend to move together. Figure 10 shows such a plot for PEP. The MSPS is shown for pure PEP and PEP mixed with PP, hhPP, and hhPP. No significant changes in the holding of optimal contact are observed with blending. This is the case for the other three materials also (not shown). For comparison, Figure 10 also presents the MSD for pure PEP. Clearly, the MSD increases more rapidly than the MSPS. This means that the pair

6608

Neelakantan et al.


Table 3. Diffusion Coefficients for the PEP/PEB Blenda component

Dpure [cm2/s]

Dmix [cm2/s]

PEP PEB

2.2 × 10-6 1.4 × 10-6

1.9 × 10-6 1.7 × 10-6

a Coefficients for each pure component and each component in the mixture are given. Diffusion coefficients are obtained from the long time limit of the center of mass mean-squared displacement: D ) limtf∞MSDcm(t)/6t.

separation between two atoms initially in optimal contact changes more slowly than the distances each atom has traveled from their initial positions. As a result, we conclude that atoms in optimal contact tend to travel together. Although our primary objective is to address the role of intermolecular packing on blend dynamics, for one system (PEP/PEB) we have made longer production runs and obtained diffusion coefficients for the pure components and for each component in the blend. The results are presented in Table 3. As expected, the diffusion coefficients shift closer to one another with blending, and PEP shifts by less (12%) than PEB (26%). Concluding Remarks A model system of the six blends formed by four saturated hydrocarbon polymers has been studied using molecular simulation. The system has been selected to fix factors affecting inherent mobility: effective concentrations arising from chain connectivity and shifts in effective Tg from pure component values are comparable between materials. There are minimal concentration fluctuations due to the short chain length of the simulated polymers. The dynamic responses of the materials to environment as each material is mixed with three blend partners vary considerably. It does not appear that these variations can be explained by chain connectivity effects. We link the response to environment to changes in intermolecular packing on local length scales that occur on mixing. These local packing changes are not accounted for by chain connectivity or concentration fluctuations. Interestingly, when collective motion, rather than self-motion, is considered, similarity of dynamic response is recovered. This may occur because the reference point in the collective correlation function already accounts for mixing induced changes in packing. The implication of this result is that different types of experiments, for example coherent and incoherent quasi-elastic neutron scattering, could lead to different qualitative observations on the dynamic response to mixing. Acknowledgment. The financial support of the National Science Foundation, Polymers Program, through a CAREER grant, DMR0134910, is gratefully acknowledged. References and Notes (1) Min, B.; Qiu, X.; Ediger, M. D.; Pitsikalis, M.; Hadjichristidis, N. Macromolecules 2001, 34, 4466. (2) Ngai, K. L.; Roland, C. M. Macromolecules 1995, 28, 4033. (3) Chung, G. C.; Kornfield, J. A.; Smith, S. D. Macromolecules 1994, 27, 5729. (4) Cendoya, I.; Alegrıá, A.; Alberdi, J.; Colmenero, J.; Grimm, H.; Richter, D.; Frick, B. Macromolecules 1999, 32, 4065. (5) Doxastakis, M.; Kitsiou, M.; Fytas, G.; Theodorou, D. N.; Hadjichristidis, N.; Meier, G.; Frick, B. J. Chem. Phys. 2000, 112, 8687.

(6) Hoffman, S.; Willner, L.; Richter, D.; Arbe, A.; Colmenero, J.; Farago, B. Phys. Rev. Lett. 2000, 85, 772. (7) Colby, R. H. Polymer 1989, 30, 1275. (8) Yang, X.; Halasa, A.; Hsu, W.; Wang, S. Macromolecules 2001, 34, 8532. (9) Roland, C. M.; Ngai, K. L. Macromolecules 1991, 24, 2261. (10) Roovers, J.; Toporowski, P. M. Macromolecules 1992, 25, 1096. (11) Alvarez, F.; Alegrıá, A.; Colmenero, J. Macromolecules 1997, 30, 597. (12) Dionisio, M.; Fernandes, A. C.; Mano, J. F.; Correia, N. T.; Sousa, R. C. Macromolecules 2000, 30, 1002. (13) Pathak, J. A.; Colby, R. H.; Floudas, G.; Je´roˆme, R. Macromolecules 1999, 32, 2553. (14) Alegrıá, A.; Go´mez, D.; Colmenero, J. Macromolecules 2002, 35, 2030. (15) Kumar, S. K.; Colby, R. H.; Anastasiadis, S. H.; Fytas, G. J. Chem. Phys. 1996, 105, 3777. (16) Zawada, J.; A. Fuller, G. G.; Colby, R. H.; Fetters, L. J.; Roovers, J. Macromolecules 1994, 27, 6861. (17) Zetsche, A.; Fischer, E. W. Acta Polym. 1994, 45, 168-175. (18) Katana, G.; Fischer, E. W.; Hack, T.; Abetz, W.; Kremer, F. Macromolecules 1995, 28, 2714. (19) Chung, G. C.; Kornfield, J. A.; Smith, S. D. Macromolecules 1994, 27, 964. (20) Lodge, T. P.; McLeish, T. C. B. Macromolecules 2000, 33, 5278. (21) Krishnamoorti, R.; Graessley, W. W.; Balsara, N.; Lohse, D. J. Macromolecules 1994, 27, 3073. (22) Neelakantan, A.; Maranas, J. K. Macromolecules 2004, 37, 8473. (23) Gell, C. B.; Krishnamoorti, R.; Kim, E.; Graessley, W. W.; Fetters, L. J. Rheol. Acta 1997, 36, 217. (24) Pathak, J. A.; Colby, R. H.; Kumar, S. K.; Krishnamoorti, R.; Fetters, L. J.; Floudas, G.; Gell, C. B.; Graessley, W. W., preprint. (25) Chen, C.; Maranas, J. K.; Kamath, S.; Colby, R. H.; Farago, B.; Pasquale, A. J.; Long, T. E., in review. (26) Garcıá Sakai, V.; Chen, C.; Maranas, J. K.; Chowdhuri, Z. Macromolecules 2004, 37, 9975. (27) Arbe, A.; Alegria, A.; Hoffmann, S.; Willner, L.; Richter, D. Macromolecules 1999, 32, 7572. (28) Mukhopadhyay, R.; Alegria, A.; Colmenero, J.; Frick, B. J. Non-Cryst. Solids 1998, 233, 235. (29) Alegrıá, A.; Cendoya, I.; Colmenero, J.; Alberdi, J. M.; Frick, B. Physica B 1997, 234-236, 442. (30) Zorn, R.; Arbe, A.; Colmenero, J.; Frick, B.; Richter, D.; Buchenau, U. Phys. Rev. E 1995, 52, 781. (31) Colmenero, J.; Arbe, A.; Coddens, G.; Frick, B.; Mijangos, C. Reinecke, H. Phys. Rev. Lett. 1997, 78, 1928. (32) Jorgenson, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. Soc. 1984, 106, 6638. (33) Allen, M.and Tildesley, D. Computer Simulation of Liquids; Clarendon Press: Oxford, 1976. (34) Berendsen, H. J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Di Nola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (35) Krishnamoorti, R. PhD. Thesis, Princeton University, 1994. (36) Smith, G. D.; Paul, W.; Monkenbusch, M.; Willner, L.; Richter, D.; Qiu, X. H.; Ediger, M. D. Macromolecules 1999, 32, 8857. (37) Maranas, J. K.; Kumar, S. K.; Debenedetti, P. G.; Graessley, W. W.; Mondello, M.; Grest, G. S. Macromolecules 1998, 31, 6991. (38) Maranas, J. K.; Kumar, S. K.; Debenedetti, P. G.; Graessley, W. W.; Mondello, M.; Grest, G. S. Macromolecules 1998, 31, 6998. (39) Neelakantan, A.; Maranas, J. K. J. Chem. Phys. 2004, 120, 465. (40) Neelakantan, A.; Maranas, J. K. J. Chem. Phys. 2004, 120, 1617. (41) Richter, D.; Arbe, A.; Colmenero, J.; Monkenbusch, M.; Farago, B.; Faust, R. Macromolecules 1998, 31, 1133. (42) Farago, B.; Arbe, A.; Colmenero, J.; Faust, R.; Buchenau, U.; Richter, D. Phys. Rev. E 2002, 65, 051803-1. (43) Arbe, A.; Richter, D.; Colmenero, J.; Farago, B. Phys. Rev. E 1996, 54, 3853. (44) Arbe, A.; Richter, D.; Colmenero, J.; Farago, B. Physica B 1997, 234-236, 437. (45) Colmenero, J.; Arbe, A.; Alegria, A.; Monkenbusch, M.; Richter, D. J. Phys.: Condens. Matter 1999, 11, A297.



(46) Wetton, R. E.; MacKnight, W. J.; Fried, J. R.; Karasz, F. E. Macromolecules 1978, 11, 158. (47) Zetsche, A.;. Kremer, F.; Jung, W.; Schulze, H. Polymer 1990, 31, 1883. (48) Katana, G.; Fischer, E. W.; Hack, T.; Abetz, V.; Kremer, F. Macromolecules 1995, 28, 2714. (49) Shinchi, K.; Yoshiaki, T.; Ichiro, N. Macromolecules 1994, 27, 7397.

(50) Haley, J. C.; Lodge, T. P.; He, Y.; Ediger, M. D.; von Meerwal, E. D.; Mijovic, J. Macromolecules 2003, 36, 6142. (51) Luettmer-Strathmann, J. J. Chem. Phys. 2000, 112, 5473. (52) He, Y.; Lutz, T. R.; Ediger, M. D. J. Chem. Phys. 2003, 119, 9956.

MA035980X

The Role of Environment in Structural Relaxation ... - ACS Publications

Recommend Documents