Single Chain Dynamic Structure Factor of Poly(ethylene oxide) in

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Single Chain Dynamic Structure Factor of Poly(ethylene oxide) in Dynamically Asymmetric Blends with Poly(methyl methacrylate). Neutron Scattering and Molecular Dynamics Simulations Martin Brodeck,† Fernando Alvarez,*,‡,§ Juan Colmenero,‡,§,∥ and Dieter Richter† †

Jülich Centre for Neutron Science JCNS (JCNS-1) and Institute for Complex Systems (ICS-1), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany ‡ Centro de Física de Materiales (CSIC-UPV/EHU) and Materials Physics Center, Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain § Departamento de Física de Materiales, UPV/EHU, Apartado 1072, 20080 San Sebastián, Spain ∥ Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain ABSTRACT: We have investigated the dynamically asymmetric polymer blend composed of short (Mn ≈ 2 kg/mol) poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA) chains focusing on the collective dynamics of the fast PEO component. Using neutron spin-echo (NSE) spectroscopy, the single chain dynamic structure factor of PEO was investigated and compared to results from molecular dynamics simulations. After a successful validation of the simulations, a thorough analysis of the RPA approximation reveals the composition of the experimentally measured total scattering signal S(Q,t). Using the simulations, we show and calculate two contributions: (1) the relaxation of hydrogenated PEO against deuterated PEO, yielding the single chain dynamic structure factor of PEO, and (2) the relaxation of the PEO component against the PMMA matrix. For the short chains presented here the second contribution shows a significant decay at higher temperatures while it was previously shown that, in the case of long chains, no relaxation is found. This difference is related to a decrease of the glass transition temperature which takes place with decreasing chain length. In a second step we analyze the approximations that are used when calculating the single chain dynamic structure factor using the Rouse model. For a system like pure PEO, where the dynamics follow the predicted Rouse behavior, excellent agreement is achieved. In the case of PEO in PMMA, however, the slow PMMA matrix strongly influences the PEO dynamics. As a result, the distribution functions show a strong non-Gaussianity, and the calculation of S(Q,t) using the Rouse approximation fails even considering nonexponential Rouse mode correlators.

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INTRODUCTION Polymer blends composed of thermodynamically miscible components offer the possibility to engineer new materials with very specific properties. The investigation of dynamic miscibility in polymer blends and in particular in asymmetric blends showing strong dynamic contrast ΔTg (ΔTg = TgA − TgB, A and B being the two blend components) is a very active area of research (see, for example, the reviews1,2). A typical example of an asymmetric blend is that composed of poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA) (ΔTg ≈ 200 K). The dynamics of the PEO/PMMA system, and in particular in the case of the compositions rich in the slow component PMMA, have been extensively investigated by means of different experimental techniques and molecular dynamics (MD) simulations as well (see as representative refs 3−11). In a previous paper,10 we have investigated the chain dynamics of unentangled PEO in a PEO/PMMA of 20/80% © 2011 American Chemical Society

weight composition by means of fully atomistic simulations. The results obtained were compared with those obtained also by MD simulations in a simple bead−spring model of asymmetric blends. It was shown that the chain dynamics of unentangled PEO in the blend strongly deviate from the expected Rouse behavior that was observed for pure PEO. In summary, we found (i) large deviations from the exponential decay of the Rouse correlators, (ii) unusual behavior of the exponent x relating the characteristic times τp of the different Rouse mode correlators (τp ∝ 1/px), and (iii) a gradual increase of x and the dynamic asymmetry with decreasing temperature. At high temperature, x ≈ 2 as it is expected for pure Rouse behavior. Similar trends were also observed for the fast component in bead−spring model simulation results, suggestReceived: July 22, 2011 Revised: November 4, 2011 Published: December 22, 2011 536

dx.doi.org/10.1021/ma2016634 | Macromolecules 2012, 45, 536−542

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PEO/PMMA ratio. Using a total of 10% protonated chains yields sufficient contrast to measure the collective dynamics while keeping the incoherent background low. The measurements have been conducted at the J-NSE (JCNS, FRM-II) at a wavelength of λ = 8 Å measuring Fourier times up to 80 ns. Activated charcoal was used as reference. The protonated PEO was obtained from Merck with Mn = 2 kg/mol and Mn/Mw = 1.03. The deuterated components were synthesized in our laboratories (dPEO: Mn = 1.7 kg/mol, Mn/Mw = 1.02; dPMMA: Mn = 2.0 kg/mol, Mn/Mw < 1.1). For preparation, all polymers have been solved in chloroform to provide a thoroughly mixed blend. After the extraction of the solvent the material was filled into niobium cells and heated several times in a vacuum oven. Sealing of the sample has taken place in an argon environment. Figure 1a shows the results of our measurements (full symbols) at Q = 0.1, 0.2, and 0.3 Å−1 . The hollow symbols in

ing that the observed features are of generic nature for asymmetric blends. In this paper we want to discuss the properties of the single chain dynamic structure factor, Schain(Q,t)which is habitually measured by neutron spin echo (NSE)in asymmetric blends. We will take advantage of the atomistic simulations to gain a deeper understanding about its composition and the different contributions that are seen in the experiment. Larger scale dynamics of PEO in the PEO/PMMA blend have recently been addressed experimentally by measuring the single chain dynamic structure factor of PEO using NSE spectroscopy.7 In such a study, however, the chains were well within the entanglement regime (Mn ≈ 23 kg/mol), making a direct comparison with the simulated system presented here difficult. Therefore, we have conducted new NSE experiments on the ternary polymer blend (hPEO−dPEO−dPMMA, h means protonated and d deuterated) using chains of similar length as in the simulations (Mn ≈ 2 kg/mol) to validate the simulated system. It is well-known that the Tg of PMMA strongly depends on the molecular weight (see, e.g., ref 12). Because of this fact, the Tg of the new PMMA sample here considered was of the order of 50 K lower than that previously used. However, the dynamic contrast with PEO was still rather high (of the order of 150 K). The dynamics of unentangled PEO chains in the blend at 400 K was now faster than in the previously investigated case due to the mobility of PMMA matrix which was found to be stiff in the blend sample with Mn ≈ 23 kg/mol. As a next step, we introduce the details about the simulation and validate the system by a comparison with the experiments. We will show that the correct chemical composition and the ratio of protonated/deuterated polymers is of significant importance to understand the measurements in the case of these short chains. After the validation we will discuss the contributions to the total scattering signal of the investigated system in the context of the random phase approximation (RPA). Here we can use the full spatial information from the simulations to gain a deeper understanding of the meaning of the experimentally measured dynamic structure factor. In a next step we will use the results from the Rouse analysis to describe the single chain dynamic structure factor of PEO Schain(Q,t). We will see that this approach fails due to the non-Gaussianity of the PEO motion which is mainly caused by the strong heterogeneity of the system. Using stretched exponential functions to describe the Rouse correlators, however, yields the correct behavior of the mean square distance between different beads in the chain, but not that of Schain(Q,t). Finally, a summary of the results is given.

Figure 1. (a) Full symbols represent results from J-NSE measurements on samples containing 10% hPEO, 10% dPEO, and 80% dPMMA with low molecular weights (Mn ≈ 2 kg/mol). The hollow symbols show results from a similar system (slightly higher PEO concentration of 25%) with long chains (Mn ≈ 23 kg/mol). Note that the scale of the Yaxis ranges from 0.3 to 1. The dashed area represents the elastic contribution of 0.44 which was calculated using the dynamic random phase approximation for the case of blends with long chains (see section about the RPA). (b) Comparison of the NSE results corresponding to the short chain blend (full) with the simulated experiment (hollow) at T = 400 K.

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NEUTRON SPIN-ECHO (NSE) EXPERIMENTS Before we use the simulations to gain a deeper understanding about the dynamic structure factor of the PEO/PMMA system, we need to validate the simulated system by a comparison with experiments. The NSE method is ideal to measure the dynamics of polymer blends at the appropriate length and time scales to allow the observation of the chain relaxation at distances close to the chain size. In addition, the scattering functions are directly observed in the time domain, allowing an easier comparison with the data obtained from the MD simulations without a Fourier transformation from the energy to the time domain. To investigate the collective dynamics of PEO chains in the PMMA environment, we have measured samples containing 10% hPEO, 10% dPEO, and 80% dPMMA, giving a 20/80

Figure 1a were taken from ref 7 and show results of a very similar system with a 25/75 PEO/PMMA ratio. In that case, however, the molecular weight of both PEO and PMMA was ∼10 times higher (Mn ≈ 23 kg/mol) and, therefore, well within the entanglement regime. It is clearly visible that the dynamic structure factor of the short chains relaxes significantly faster 537


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than that of the long chains even though the content of PEO in the blend is lower. Using differential scanning calorimetry (DSC), we have determined the average glass transition temperature Tg for both systems (each with a weight-ratio of 20/80). It was found that Tg for the short chain blend is significantly lower (311 K) than for the longer chains (347 K). Therefore, the dynamics at equal temperatures are significantly slower in the case of long chains. This effect must be attributed to the PMMA matrix. At the investigated temperature (400 K) the matrix of the long chains is rather stiff (which is also visible using neutron scattering, see section IV), whereas a strong relaxation is found for the short chains. This allows faster PEO dynamics since no segments of the chain are stuck in the PMMA matrix for significant amounts of time.

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1

Scoh(Q , t ) − Sinc(Q , t ) S(Q , t ) 3 = 1 S(Q , 0) Scoh(Q , 0) − Sinc(Q , 0) 3

Using the simulations, we can calculate the exact absolute incoherent scattering contribution and subtract the corresponding intensity (mainly created by the protonated chains) from the coherent contribution. In addition to the incoherent scattering, we have to account for the actual chemical composition of our samples. During the synthesis of the PEO and PMMA chains an initiator is necessary to start the reaction. This initiator is always protonated, even for the synthesis of deuterated chains, and its size roughly corresponds to two monomers. Because of the short chain length of our polymers, a protonated headgroup has a significant effect on the overall scattering behavior of the system (coherent as well as incoherent). For a fully deuterated PEO chain with Mn = 2 kg/mol the headgroup accounts for 6% of the polymer while the ratio for PMMA at the same molecular weight lies at 14%. For long chains with Mn > 20 kg/mol this effect is below 1% and, therefore, negligible. Since half of the PEO has to be protonated and our system contains 5 PEO chains, we have averaged the scattering of 2 and 3 protonated chains. Since there are 10 different combinations for protonating 2 (or 3) out of 5 chains, a total of 20 different scattering functions was averaged. The coherent scattering function now is a complicated mixture of cross-terms between the headgroups, the protonated chains, the deuterated chains, and the deuterated matrix. However, the possibility to include all these contributions into the simulated function allows a direct comparison between experiment and simulation. Figure 1b shows the excellent agreement obtained. The consistency between the two sets of data validates our MD simulations also for the chain dynamics and collective motions and on the other hand highlights the importance of the protonated headgroups for the scattering of systems with such short chains.

MD SIMULATIONS AND VALIDATION

We have used the software package Materials Studio 4.1 and the Discover-3 module (version 2005.1) from Accelrys with the COMPASS force field to carry out the MD simulations. A cubic cell containing 5 PEO molecules (43 monomers each) and 15 PMMA molecules (25 monomers each) was simulated under periodic boundary conditions, giving the same weight ratio of PEO to PMMA as in the experiment (20/80). Temperatures of 300, 350, 400, and 500 K were simulated for up to 200 ns. The simulated cell has a length of a ≈ 42 Å, which in principle limits the lowest accessible Q values to about 0.15 Å−1 . For the comparison with the experiment we have calculated and Fourier-transformed the spatial correlation functions of the different labeled groups. The total coherent scattering intensity is then calculated by weighting the different scattering functions with the product of the corresponding scattering lengths:

S(Q , t ) =

∑ bαcohbβcoh α, β

NαNβ NtV

Sα, β(Q , t ) (1)

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with Nα the number and bα the scattering length of atoms of type α, Nt the total number of atoms, and

Sα, β(Q , t ) =

∫0

∞

dr (gα, β(r , t ) − 1)

sin(Qr ) Qr

(3)

DISCUSSING OUR RESULTS IN CONTEXT OF THE RPA When measuring the sample described above at low Q values, the scattering signal is created by the contrast between the protonated and deuterated components. Since only a part of the PEO chains if protonated, we measure the relaxation of the protonated PEO chains against the deuterated PEO chains and the deuterated PMMA matrix. The contributions to the total scattering can in principle be calculated using the dynamic random phase approximation (RPA) (see refs 13−16). As a consequence of incompressibility in ternary systems, the scattering intensity of the labeled component can be described as a superposition of two decaying modes (hPEO against dPEO and dPMMA). Assuming an exponential decay for these modes, it can be shown that (valid in the regime where S(Q,t) ∝ Q−2)

(2)

the partial dynamic structure factors (radial averaging due to the assumed isotropy of the system). The breakdown of the total scattering function in the corresponding contributions is of particular importance when examining different levels of deuteration (e.g., protonating 50% of the PEO chains) as will be shown in the following section. Additional technical details about our MD simulations can be found in ref 10. Comparison with Experiments. For the comparison of the simulated system with the experiment we have to calculate the signal that would be measured by a NSE instrument. Therefore, we have to include in the calculation the contributions from incoherent scattering. Moreover, we have to take into account the actual chemical composition of our sample. During an incoherent scattering process the spin of 2/3 of all neutrons is flipped while it is unchanged for the rest. For the NSE setup this means that neutrons with an inversed spin orientation have to be subtracted from the final signal and the measured intensity is proportional to

12φPEO (φdPEOe−λ1t S(Q , t ) = 2 2 PEOh PEO l Q (φ h + φd ) PMMA −λ 2t + φPEO e ) h φd

(4)

with the volume fractions ϕ for the three different components. The relative contribution of the two relaxations can be derived 538


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by considering two specific cases. If the PMMA fraction approaches 0 (setting ϕdPMMA = 0), the total scattering S(Q,t) only contains the relaxation of the PEO component (λ1). In this case, the prefactor describes the contrast created between the protonated and deuterated PEO chains. If, on the other hand, the deuterated PEO component is 0 (by setting ϕdPEO = 0), only the relaxation of PEO against the PMMA matrix is seen (λ2). In a mixed scenario both relaxations are added with the corresponding weights ϕdPEO for λ1 and ϕhPEOϕdPMMA for λ2. Therefore, the relative contributions of the two relaxations is given by the amplitudes

aPEO =

φdPEO PMMA φdPEO + φPEO h φd

aPMMA =

(5)

PMMA φPEO h φd

PMMA φdPEO + φPEO (6) h φd For the investigated system (10/10/80) we get aPEO = 0.56 and aPMMA = 0.44. However, rather than assuming an exponential decay of the two modes described above, we can use the simulations to calculate the real decay. The first contribution (aPEO) corresponds to the single chain dynamic structure factor of PEO. To obtain this contribution, we calculate the scattering of the unfolded PEO chains (using absolute coordinates rather than the coordinates after folding the chain back into the periodic cell):

1 Schain(Q , t ) = N

Figure 2. Contributions to the total dynamic scattering function calculated using the RPA. The upper 56% of the plot display the relaxation of the single chain dynamic structure factor of PEO (hollow triangles), the lower part shows the relaxation of PEO against PMMA (hollow circles), and the full symbols show the total scattering function (T = 350 and 400 K).

25% hPEO and 75% dPMMA (Figure 3). In this case, no decay of the dynamic structure factor is observed, confirming the

N

∑ ⟨exp(iQ⃗ ( rn⃗ (t ) − rm⃗ (0))⟩ n,m

(7)

Here, the center of mass of each monomer was defined as a scattering particle. Using every atom as scattering particle adds no additional information since the observed Q values are below the atomistic resolution. The difference obtained by means of these two procedures is within the size of the symbols of the figure for the Q values shown. Using the absolute coordinates allows the calculation of Schain(Q,t) for Q values even below the limit given by the size of the cell (Q ≈ 0.15 Å−1). The second decay in eq 4 describes the collective relaxation of PEO against PMMA. For a fully rigid PMMA matrix this term will not decay (λ2 = 0), and an elastic contribution of relative intensity aPMMA is expected in addition to the single chain dynamic structure factor of PEO. Using the simulations, this contribution is calculated by creating full contrast between the PEO and PMMA chains. This is achieved by assigning a scattering length of 1 to all PEO monomers of all chains in the system while the PMMA monomers are not regarded. For this calculation the coordinates of each bead have to be folded back into the unit cell since correlations between different chains have to be included (therefore, the limit for the Q values has to be taken into account). Figure 2 shows the different contributions at two different temperatures calculated from the simulations. The upper part contains the PEO relaxation (hollow triangles). At the bottom, the relaxation of PEO against PMMA is shown (hollow circles). At T = 400 K a significant relaxation of the PMMA matrix is observed. Experimentally, the corresponding signal can be accessed using a sample of fully protonated PEO and fully deuterated PMMA. For chains with Mn ≈ 23 kg/mol this was investigated by Niedźwiedź et al. at T = 375 K for a ratio of

Figure 3. NSE data for a system containing 25% hPEO and 75% dPMMA at T = 375 K. In this experiment the molecular weight of PEO and PMMA was in the range of Mn ≈ 23 kg/mol.

assumption of a fully frozen/immobile PMMA matrix (λ2 = 0) at the investigated time and length scale. Our simulations show that this assumption is no longer valid once the chain length decreases due to the lower glass transition temperature. As a result, the signal measured by NSE no longer contains the PEO dynamics on top of a constant elastic contribution, but rather a combination of both decays (full symbols in Figure 2). As a next step we can compare the predictions of the RPA (combined relaxation of two correlations) with the simulated scattering of our sample. Rather than comparing the RPA prediction with the real experimental results or the simulated experiment, which contains the protonated headgroups and the incoherent contribution to the NSE signal, we use coherent scattering of an ideally labeled sample containing 10% hPEO, 10% dPEO, and 80% dPMMA. 539


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Figure 4a shows this comparison for T = 400 K at two Q values. The two curves for the RPA are calculated as shown in

simulation (protonated headgroups), this experimental curve could also be described. However, the measurement was not suited for the correction of the fast decay. Figure 4b shows the comparison of the RPA with the coherent scattering signal where the fast decay has been removed. In this case excellent agreement is found for times up to about 20 ns. Small deviations between the two functions are seen for t > 20 ns. A similar behavior was found for these times studying the self-motion of PEO in the PMMA environment and was connected to the motion of PEO monomers exploring their local environment.10 For T = 350 K similarly good agreement as shown in Figure 4 has been observed. The comparison using exponential functions as seen in eq 4 fails because the two decaying channels do not follow an exponential decay. This comparison has allowed us to study the origin of the coherent scattering signal of a ternary polymer blend in the context of the RPA. Two relaxation channels of PEO are observed, and the RPA allows the calculation of the two corresponding amplitudes. In the case of a rigid PMMA matrix the single chain dynamic structure factor of PEO is measured by NSE on top of the elastic contribution caused by the stiff PMMA matrix. However, our calculations show that for shorter chains the final signal is a mixture of both relaxations. In this case the knowledge about the relaxation of the PMMA matrix is necessary for a full interpretation of the experimental NSE data.

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SINGLE CHAIN DYNAMIC STRUCTURE FACTOR WITHIN THE ROUSE MODEL In a previous work,10 we have presented a detailed analysis of the simulated PEO/PMMA system in terms of the Rouse modes and Rouse mode correlators. It was shown that the Rouse mode correlators corresponding to PEO in the blend were significantly stretched in particular at low temperatures. The averaged Rouse times ⟨τp⟩ were obtained from KWW fits to the Rouse mode correlators. With decreasing temperature, a stronger dynamic asymmetry between the PEO and PMMA components (separation between the segmental relaxation time of PEO and PMMA) was created, resulting in strong deviations from the usual p−2 dependence of the Rouse times with the mode number p. For T = 400 K an exponent of 2.45 was found; at 350 K it increased to 2.7. In contrast, ideal Rouse behavior was found for pure PEO once the length scale exceeded the atomic detail involved in the MD simulations.17 Now we address the question of how the strong deviations from the Rouse behavior found for the chain dynamics of PEO in the blend affect the Rouse expression for the single chain dynamics structure factor, Schain(Q,t), which was first deduced by de Gennes18,19 and which is habitually used to analyze neutron scattering results. Such an expression is based, first of all, on the so-called Gaussian approximation; i.e., it assumes that the atomic displacements distribution function is Gaussian. Under this assumption, Schain(Q,t) (eq 7) can be written as

Figure 4. (a) Comparison of the dynamic scattering functions calculated using the RPA (full symbols) and directly calculated from the h/d scattering contrast for Q = 0.1 and 0.2 Å−1. In (b) the fast dynamics of the coherent scattering signal have been removed.

Figure 2 (a superposition of the two relaxations with the corresponding amplitudes). The hollow symbols are calculated from the scattering of hPEO dPEO and dPMMA (HDD, 10/ 10/80). A fast decay at very low times (t < 0.5 ps) is observed for the coherent scattering of the HDD sample. At t = 0.5 ps the dynamic scattering function has already decayed to 0.96 at Q = 0.3 Å−1 (for higher Q values this effect increases). Because of this fast decay at low times, the HDD function lies below the values of the RPA for times up to about 100 ps. The origin of this fast decay is the collective fast relaxation of the interchain correlations. Experimentally, it was investigated in pure dPEO in ref 17 where a relaxation of the coherent scattering function within the first picoseconds was observed. The peak of these collective correlations lies at Q ≈ 1.5 Å−1. A small contribution, however, can also be seen at much lower Q values (with a significantly reduced intensity). This fast decay, which is caused by the atomic structure of the sample, is obviously not seen in the RPA calculations. There, only the scattering of the monomer blobs is considered, and the fast dynamics are invisible. Experimentally, this fast contribution is removed by subtracting the scattering from samples with fully deuterated polymers, where the scattering function decays to zero within the first few picoseconds. This sample has been measured during the experiments described above. However, due to the high content of protonated material (the initiators of the chain), additional contributions arising from the h/d contrast have been observed. Using the correct chemical labeling in the

S( Q , t ) =

1 N

N

∑ m,n

⎛ 1 ⎞ exp⎜ − Q 2⟨Δrnm2(t )⟩⎟ ⎝ 6 ⎠

(8)

with the mean-square segment correlation function ⟨Δrnm2(t)⟩ = ⟨(rn(0) − rm(t))2⟩. Taking into account the definition of Rouse modes (Rouse coordinates) and the orthogonality of these modes, Δrnm2(t) can be expressed as 540


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2

2

⟨Δrnm (t )⟩ = 6DR t + |n − m|l +

4Nl 2 π

2

N

∑ p=1

1 p2

⎛ pπn ⎞ ⎛ pπm ⎞ ⎟ cos⎜ ⎟ cos⎜ ⎝ N ⎠ ⎝ N ⎠ (1 − ⟨X⃗p(t )X⃗p(0)⟩)

(9)

where DR is the center-of-mass diffusion and ⟨X⃗ p(t)X⃗ p(0)⟩ are the Rouse mode correlators. If we now consider simple exponential decay of these correlators, eqs 8 and 9 give N

S(Q , t ) =

⎛ 1 ⎞ 1 exp(− Q 2DR t ) ∑ exp⎜ − |n − m|Q 2l 2⎟ ⎝ 6 ⎠ N m,n

⎧ ⎪ 2 R ee 2Q 2 exp⎨− ⎪ 3 π2 ⎩

∑ p

⎧ 1 ⎪ ⎜⎛ pπm ⎟⎞ ⎜⎛ pπn ⎟⎞ ⎨cos cos ⎝ N ⎠ ⎝ N ⎠ p2 ⎪ ⎩

⎛ ⎞⎞⎫⎫ ⎛ ⎪ ⎜1 − exp⎜− t ⎟⎟⎪ ⎬ ⎜ τ 2 ⎟⎟⎬ ⎜ ⎝ p ⎠⎠⎪ ⎝ ⎭⎪ ⎭

(10)

Using these equations, we now have several possibilities to calculate the single chain dynamic structure factor of the PEO chains from the simulation data: A. With the knowledge about the coordinates of each monomer, we can use eq 7 to directly calculate Schain(Q,t). Technically, we are using radially averaged distribution histograms to include the isotropic nature of the system. This function then corresponds to the actual single chain dynamic structure factor. B. The correlation functions ⟨Δrnm2(t)⟩ can directly be calculated using the coordinates of the monomers. Together with eq 8, this also yields Schain(Q,t) if the segmental correlation functions are Gaussian. C. ⟨Δrnm2(t)⟩ is also accessible using the Rouse correlators together with eq 9. Here we can directly insert the correlators ⟨X⃗ p(t)X⃗ p(0)⟩ from the Rouse analysis of the system or use the fitted stretched exponential functions

X⃗p(t )X⃗p(0) = exp( − (t /τ)pβ )

Figure 5. (a) Single chain dynamic structure factor for pure PEO chains at T = 350 K calculated with the four methods described in the text at Q = 0.2 and 0.3 Å−1. The center-of-mass diffusion has been removed in all cases. This is the reason for the plateau at long times. (b) Single chain dynamic structure factor for PEO chains in the blend at T = 350 K and Q = 0.3 Å−1 calculated with the four methods described in the text. The center-of-mass diffusion has been removed.

distribution of ⟨Δrnm2(t)⟩. Therefore, the Gaussian assumption for pure PEO is valid and allows us to calculate the coherent scattering using eq 8. 2. The calculation of the single chain dynamic structure factor using the Rouse description of the chains also yields excellent agreement. Even using exponential decays with τR as the only parameter allows the correct calculation of Schain(Q,t), confirming the findings that the pure PEO system is fully described by the Rouse model.17 Now we analyze the blend system (Figure 5b). We will only present results for Q = 0.3 Å−1 since other Q values show a similar behavior. This time big differences are found for the different methods. Comparing the direct calculation of Schain(Q,t) (A) with the method using the mean-square segmental correlation functions (B), we see a significantly slower decay for the direct method. Obviously, the assumption of Gaussian distribution functions is false and eq 8 fails. In fact, during the analysis of the distribution function of the atomic displacements in PEO/ PMMA carried out in ref 10 it was shown that the nonGaussianity of PEO in the blend increases drastically as the temperature of the system decreases. Now we compare Schain(Q,t) with the curves that were calculated using the Rouse correlators. For times up to about 20 ns, relatively good agreement is found between methods B and C. Both functions obviously fail to describe the correct single chain dynamic structure factor since they rely on Gaussian distribution functions. Their agreement up to 20 ns, however, confirms the fact that the system is correctly described using the exponentially stretched Rouse correlators. This means that

(11)

Both approaches yield the same result since the decay of the correlators is well described by the KWW fits. D. Finally, for a system with Rouse behavior (as is the case for pure PEO), we can directly use eq 10. We will now compare these four different approaches for the pure PEO system and for PEO in PMMA. The results will be presented for a temperature of 350 K in order to highlight the dynamic asymmetry in the blend. At T = 400 K, the significant relaxation of the PMMA matrix softens the dynamic asymmetry of the system (see Figure 2). For pure PEO at T = 350 K Figure 5a shows the single chain dynamic structure factor at Q = 0.2 and 0.3 Å−1 calculated by the different methods. The center-of-mass diffusion of the chain has been removed for all calculations either by removing the center of mass diffusion in the absolute coordinates (in methods A and B) or by setting the diffusion coefficient DR = 0 (for methods C and D). All four approaches show excellent agreement allowing the following conclusions: 1. The agreement between the direct calculation of Schain(Q,t) (A) and the method using the mean-square segmental correlation functions with eq 8 (B) confirms the Gaussian 541


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correlators the dynamic correlation functions ⟨Δrnm2(t)⟩ can be recovered. The calculation of Schain(Q,t) failed due to the strong heterogeneity of the system which results in a strong non-Gaussianity of the distribution functions.

⟨Δrnm2(t)⟩ can be correctly calculated only with the information about the Rouse correlators (time scale τp and stretching parameter βp). This agreement was also observed when directly comparing ⟨Δrnm2(t)⟩ calculated from the simulations with the results using eq 9 and the data from the Rouse analysis. The difference in the decays in Figure 5b for t > 20 ns can be connected to similar deviations we found in the dynamic incoherent scattering functions of PEO in PMMA.10 Up to this point, good fits with KWW functions were possible; at t > 20 ns the actual decay of the incoherent function was slower than the fitted KWW functions (similar to the effect found here). Finally, looking at Figure 5b, we see that the calculation obviously fails if we simply use the pure Rouse expression (D). In this case Schain(Q,t) relaxes much slower than the real system which is caused by the p−2.7 dependence of the relaxation times τp. Only using τR as input parameter results in significantly slower local relaxation times than actually found for the system. In summary, we can say that the Rouse expression for the single chain dynamic structure factor (eq 10) cannot be used in the case of asymmetric blends, even if it is generalized by including stretched exponential decay of the Rouse correlators. The reason is that this expression relies on the Gaussian approximation for the atomic displacements distribution function, an assumption that clearly breaks in the case of asymmetric blends, in particular at low temperatures. It is noteworthy that this is not the case for the correlation function of the end-to-end vector Ree, ⟨Ree(t)Ree(0)⟩, which can be measured by dielectric spectroscopy in the so-called A-type polymers (normal mode relaxation). The Rouse expression for this correlator is only based on the orthogonality of the Rouse modes (as ⟨rnm2(t)⟩) and thereby can be generalized for the case of asymmetric blends20 where the Rouse correlators results to be nonexponential, but the Rouse modes are still orthogonal in a good approximation.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS This research project has been supported by the European Commission NoE SoftComp, Contract NMP3-CT-2004502235, and the “Donostia International Physics Center”. J.C. and F.A. acknowledge support from the projects MAT200763681 and IT-436-07 (GV).

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REFERENCES

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SUMMARY AND CONCLUSIONS 1. Experiments have shown that a decrease of the chain length significantly influences the collective dynamics of PEO in PMMA. This effect is caused by higher mobility of the PMMA matrix (leaving the regime of a completely frozen matrix) and faster motion of the PEO chains. The interpretation of the scattering curves is difficult due to the significant influence of the protonated head groups for the short chains. 2. Using the NSE experiments, we were able to fully validate the simulated PEO/PMMA system for chain dynamics. For a successful comparison it was necessary to include the incoherent contribution to the scattering signal and, even more important, the effects of the protonated initiators of the investigated chains. 3. Using the RPA, we have seen that the measured signal is a combination of the PEO chain relaxation and, in addition, a relaxation of PEO against the matrix. For a fully frozen matrix, as was the case for the long chains due to the higher glass transition temperature, this contribution builds a constant, elastic contribution. For the short chains both effects have to be regarded. In addition, the fast dynamics (at times in the picosecond regime) have to be considered in the experiment (fully deuterated background measurements). 4. Calculating the single chain dynamic structure factor for pure PEO and PEO in PMMA, using the results from the Rouse mode analysis, good agreement was achieved only for pure PEO. For PEO in PMMA significant differences were observed. Using KWW functions to describe the Rouse 542


Single Chain Dynamic Structure Factor of Poly(ethylene oxide) in

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