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Non-exponential Rouse correlators and generalized magnitudes probing chain dynamics
Journal of Non-Crystalline Solids 407 (2015) 302–308
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Non-exponential Rouse correlators and generalized magnitudes probing chain dynamics J. Colmenero 1 Centro de Física de Materiales (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
a r t i c l e
i n f o
Article history: Received 21 May 2014 Received in revised form 8 September 2014 Accepted 10 September 2014 Available online 7 November 2014 Keywords: Polymer dynamics; Molecular dynamics simulations; Neutron scattering; Dielectric spectroscopy; Polymer blends
a b s t r a c t Inspired by the generalized Langevin equation (GLE) formalism with some approximations, we have proposed a method to generalize the Rouse expression of the magnitudes probing chain dynamics, taking into account the non-exponential character of the Rouse mode correlators. In this way, generalized expressions for the incoherent neutron scattering function and for the End-to-End relaxation – which give rise in the frequency domain to the dielectric ‘normal mode’ relaxation – have been obtained. These magnitudes have been checked by means of fully atomistic molecular dynamics (MD) simulations corresponding to pure poly(ethylene oxide) (PEO) and to PEO in an asymmetric blend with poly(methyl methacrylate) (PMMA). Moreover, the results obtained also seem to give support to the unified scenario recently proposed for explaining the non-exponential relaxation of the longwavelength Rouse modes in polymer systems in terms of the effect of density fluctuations (α-process) on chain dynamics. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Chain dynamics dictate the viscoelastic properties of melts of polymer systems. Nowadays it is generally assumed that the Rouse model [1] provides a suitable description of chain dynamics of unentangled polymer melts, i.e., melts of chains of molecular weight lower than a certain limit (‘entanglement mass’). The Rouse model can also be seen as a ‘fixed point’ of all theories of polymer dynamics in which linear connected objects are ideal and subjected to local dissipation. For instance, in the chain dynamics theories of entangled polymers based on the ‘tube’ concept [2,3], the Rouse model is a basic ingredient because chains move ideally (Rouse behavior) until they feel the effects of the confining tube. In the framework of the Rouse model, a tagged chain is represented as a string of N beads of equal mass connected by harmonic springs of constant 3kBT/ℓ2, with ℓ the bond length and kB the Boltzmann constant. The effective interaction experienced by the chain is given by a ! friction coefficient ξ and a set of stochastic forces f j . Excluded volume ! interactions are neglected. The chain normal (Rouse) mode X p ðt Þ of ! index p = 1 … N − 1 (wavelength N/p) is defined as X p ðt Þ ¼ N−1 N ! ! ∑ j¼1 r j ðt Þ cos ½ð j−1=2Þpπ=N, with r j the vector giving the position ! of the j-bead in the chain. X p ðt Þ follows the equation of motion
2Nξ
! d X p ðt Þ ! ! ¼ −K p X p þ g p dt
ð1Þ
E-mail address: [email protected]. Also at Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain. 1
http://dx.doi.org/10.1016/j.jnoncrysol.2014.09.033 0022-3093/© 2014 Elsevier B.V. All rights reserved.
with K p ¼ 24NkB Tℓ−2 sin2 ðpπ=2NÞ . The external force for the p-th N ! ! mode is g p ¼ 2∑ j¼1 f j ðt Þ cos ½ð j−1=2Þpπ=N. The Rouse model fully neglects spatial and time correlations of D E ! ! the stochastic forces, i.e., g p ðt 0 Þ g q t ″ ¼ 12NξkB Tδpq δðt−t 0 Þ. The former two approximations yield, respectively, orthogonality of the Rouse modes and exponential relaxation of the Rouse correlators, leading to the expression for the correlation function of the p-mode, D! E pπ − t ! ℓ2 C p ðt Þ ¼ X p ðt Þ X p ð0Þ , C p ðt Þ ¼ 8N sin−2 2N e τp with the relaxation 2 ℓ ξ pπ time given by τp ¼ 12k sin−2 2N . Due to the effects of local potenBT tials and chain stiffness at short length scales in real systems, Rouse model results are usually discussed in the long wavelength (N/p) pπ pπ 2 ≈ 2N . limit. In this range sin2 2N Orthogonality of the Rouse modes and exponentiality of the Rouse correlators are the two main assumptions of the Rouse model and are the basis of the derivation of the magnitudes probing chain relaxations [3]. These magnitudes (incoherent and coherent scattering functions, dielectric ‘normal mode relaxation’, etc.) are thereby constructed in terms of exponential Rouse mode correlators. For homopolymer melts this seems to be a rather good approximation in the high temperature range (with respect to the glass-transition temperature Tg) were most of the measurements of the chain dynamics are usually carried out and taking into account that the relevant modes are the slowest ones. However, significant deviations from the exponentiality of these mode relaxations have been recently reported in the low temperature range approaching Tg [4]. This range was accessible by using a thermostimulated depolarization current (TSDC) technique, which, on the
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
other hand, allows isolating experimentally the dielectric response of the slowest Rouse mode (p = 1) relaxation. As usually, deviations from exponentiality were accounted for by the well known h using β i stretched exponential function C p ðt Þ≈ exp − t=τ^p . The β values characterizing these deviations resulted to be correlated with the ratio τ^R =τ α being τ^R ¼ τ^p¼1 and τα the characteristic relaxation time of the α-relaxation in the system. This process is a general feature of glassforming systems and it is associated with the evolution of density fluctuations in the system at the inter-macromolecular scale (first maximum of the static structure factor S(Q) in the case of polymers). In the high temperature range, τα uses to be well separated from that of the slowest chain mode relaxation (τR), thereby implying that the statistical independence of both processes could be taken as a good approximation. However, as soon as the temperature decreases approaching the Tg of the polymer melt, both processes approach each other, implying some kind of coupling between both. This would break the Rouse assumption of uncorrelation of the random forces acting on a polymer chain and would explain the non-exponential behavior of the Rouse mode relaxation. Although the possible coupling between chain dynamics and α-relaxation has been widely discussed in the literature concerning the problem of the different temperature dependence of both processes approaching Tg (see, e.g. [5–8]), the effect of local density fluctuations (α-relaxation) on the non-exponentiality of the Rouse mode correlators in homopolymers has started to be recognized only recently [4]. On the other hand, in the case of the chain dynamics of the fast component of asymmetric polymer blends, strong deviations from the exponential relaxation of the slowest Rouse modes are well documented [9–11]. Asymmetric blends are miscible polymer systems A/B where the two components display very different segmental mobility or Tgs. In this case, the deviations from the exponentiality of the Rouse mode relaxation also depend on temperature, being negligible at high temperature where the dynamic contrast between the two components vanishes. As soon as the temperature decreases, the α-relaxation of the slow component of the blend will slow down and its relaxation time will approach the characteristic times of the slowest Rouse mode relaxation of the fast component in the blend. This situation seems to be similar to that described above for homopolymer melts, being now the relevant density fluctuations those associated with the freezing of the slow component in the blend. In fact, based on experimental results of polyisoprene (PI) and PI in different blends, it has been reported [4] that the β values corresponding to the slowest (p = 1) Rouse mode of the fast component in a blend follow the same trend with respect to the ratio τ^R =τ α than those corresponding to the homopolymer case, being τα the characteristic time of the relevant density fluctuations in the system: the α-process for the homopolymer melt and the α-process of the slow component in the blend. We note that in the case of the asymmetric blends we can face situations where τ^R =τ α ≤ 1, something that would not be possible in the case of homopolymers. Thus, in general, we should expect stronger deviations from the exponentiality of Cp = 1(t) [called CR(t)] – i.e., lower values of β – in the case of blends. As it has been mentioned above, in the framework of the Rouse model the non-exponentiality of the Rouse mode correlators will directly affect the magnitudes probing chain dynamics because these magnitudes are written in terms of such correlators. We have also to take into account that in the tube-reptation theories for well entangled polymers, chain dynamics at times t b τe – where τe is the so-called entanglement time – is also described in terms of the Rouse model, thereby being affected as well by the non-exponential behavior of Rouse correlators. Moreover, in the case of the End-to-End correlation function, the mathematical expression is formally the same in the framework of both, Rouse and reptation theories. With these ideas in mind, in a recent paper [12] we have generalized the Rouse expressions for the mean
303
squared displacement of a polymer segment 〈r2s (t)〉 and that of the center of mass of the chain 〈R2CM(t)〉, by considering the non-exponential character of the Rouse mode relaxation. Starting from these expressions and taking into account the Gaussian approximation, we have also proposed equations for the corresponding incoherent scattering functions s CM Fself (Q, t) and Fself (Q, t). The results obtained were checked by using molecular dynamics (MD) simulations data of poly(ethylene oxide) (PEO) in asymmetric blends with poly(methyl methacrylate) (PMMA). In the present work, after briefly summarizing the main approximations used to generalize the Rouse expressions for 〈r2s (t)〉 and 〈R2CM(t)〉, we will extend this approach to the case of the End-to-End correlation function. Finally, we will apply the results obtained to MD-simulation data of both, bulk PEO and PEO in PEO/PMMA blends. 2. A simple approach inspired by the GLE formalism As it has been commented in the Introduction section, the exponential nature of the relaxation of the Rouse modes arises as a consequence of neglecting time correlations of the random forces acting on a polymer segment. In a general formulation (generalized Langevin equation, GLE, formalism) for the time evolution of the Rouse mode correlators, the correlation between the random forces can be introduced within a memory kernel in the Langevin equation. This term effectively produces an extra friction non-local in time and thereby a non-exponential decay of the mode relaxation. The standard Rouse behavior and the corresponding constant friction are recovered when the memory function decays to zero at very short times in comparison to those characteristic of the mode relaxation. In the framework of a simplified GLE formalism [13], we can write an integro differential equation for the time evolution D! E ! of a Rouse correlator, C p ðt Þ ¼ X p ðt Þ X p ð0Þ dC p ðt Þ 1 þ dt ξo
Z
0 C p ðt Þ 0 0 dC p t dt Γ t−t ¼− o : 0 τp dt 0 t
ð2Þ
We note that in order to arrive to Eq. (2), the fast decaying ‘Rouse contribution’ has been extracted from the general memory function of the GLE formalism [13,14]. In this way, the memory function Γ(t) in Eq. (2) only contains the non-Rouse contribution. As consequence of this partition, ξo and τop in Eq. (2) are respectively the ‘bare’ constant friction coefficient connected with the fast local motions and the relaxation time of the Rouse correlator corresponding to pure Rouse behavior. Moreover, as it was described in ref. [12], we considered two additional approximations. First, Γ(t) was assumed to be independent of the length scale (wavelength of the mode, N/p) due to the fact that we were interested in the long wavelength (N/p) limit. Moreover, based on the arguments widely discussed in ref. [12], a second step was to consider a ‘pseudo-Marcovian’ approximation and to replace the convolution integral of Eq. (2) by a time local product. As it was discussed in ref. [12], this strong approximation would be justified under the assumption that in the long wavelength (N/p) limit, Γ(t) decays faster than Cp(t). This is equivalent to say that Γ t ¼ τ^p ≈0, where τ^p is the relaxation time of Cp(t) (see below). In ref. [12], this assumption was numerically checked by means of MD-simulations of PEO/PMMA system, also considered in this work. It was obtained that Γ t ¼ τ^p ≈0 for the range N/p ≥ 10 (see Fig. 7 in ref. [12]), i.e., the relevant long wavelength range considered here (see Section 5). In the framework of the above-described approximations and defining a time dependent effective friction coefficient as ξ(t) = ∫t0dt′Γ(t′) the solution of Eq. (2) can be expressed as φp ðt Þ ¼
" Z C p ðt Þ ξ ¼ exp − oo C p ð0Þ τp
# dt 0 0 : 0 ξo þ ξðt Þ t
ð3Þ
To proceed further with Eq. (3) we have two possibilities. The first, and more rigorous one, would be to construct a microscopic theory for
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J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
Γ(t) which allows to solve the integral equation. As it was commented in ref. [12] this means a formidable theoretical effort, involving many approximations as well (see, e.g. refs. [13,14]). The other possibility is to follow a phenomenological procedure based on: (i) the knowledge that φp(t) has to be non-exponential due to the memory function; (ii) the experimental [4,11] and simulation [9,10] results, which indicate that a stretched exponential function " φp ðt Þ ≈ exp −
t τ^p
!β # ð4Þ
with β = β(T) uses to be a good approximation to describe the nonexponentiality of φp(t). In this context, the procedure followed reduces to the question: what should it be the simplest expression for ξ(t) giving rise by solving Eq. (3) to a stretched exponential φp(t) and assuring that in the Rouse limit (β = 1) φp(t) = exp [−t/τop]. It is straightforward to see that for the time regime so that ξ(t) ≫ ξo the answer to this question is ξðt Þ ¼
ξo t 1−β β tc
ð5Þ
which, taking into account the definition of ξ(t) above-mentioned, is equivalent to a memory function Γ(t) depending on time as an inverse power law. The time tc in Eq. (5) emerges in a natural way keeping the dimensionality of ξ(t) correct and setting the time scale for the non-exponential behavior of φp(t). Now, introducing Eq. (5) into Eq. (3) we obtain for the trivial case β = 1 the Rouse limit and for β b 1 a stretched exponential function (Eq. (4)) with 1=β o ðβ−1Þ=β τ^p ¼ τp tc :
ð6Þ
It is noteworthy that Eq. (6) looks exactly like the basic equation of the ‘coupling model’ (CM) [15–18]. Although we will comment about this question in the Discussion section, we would like to mention here that in the framework of Eq. (3), tc could be, in principle, temperature and N-dependent — in contrast with the constant crossover time of the ‘coupling model’. However, the MD-simulation results discussed in ref. [12] suggest that considering in practice a constant tc can be a good approximation. Readers interested in this question are referred to the recent ‘Comment & Reply’ papers [19,20]. As τop ∝ (N/p)2 (Rouse scaling), the (N/p) dependence of τ^p results to be τ^p ∝ðN=pÞx with x = 2/β. Even without having a microscopic theory for the memory function – i.e., a microscopic theory for β(T) – this expression established a direct an nontrivial correlation between the wavelength (N/p) scaling of τ^p and the nonexponentiality (β) of the corresponding Rouse correlator. For p = 1 – i.e., the large (N/p) limit – this correlation predicts the molecular weight (M) dependence of the generalized Rouse time τ^R ∝M 2=β , as it was shown in [21] where this expression was deduced for the first time in the framework of a formalism similar to that explained in this section. Later, this expression was also deduced by a direct application of the ‘coupling model’ [18]. However, we have to recall that Eq. (6) has been obtained in the framework of a long wavelength (N/p) approximation, where the ‘pseudo-Markovian’ approximation seems to work as it has been mentioned above. Moreover, only in that range the β-value can be considered as independent of (N/p) (see Section 5). Then, the expression x = 2/β should be considered only as an asymptotic law valid for the large (N/p) limit. Deviations from this law are expected at lower values of (N/p) as it was shown in [20] from MD-simulation data of a ‘bead-spring’ generic model for asymmetric blends [9,10]. It is worthy of remark that the correlation found between the wavelength (N/p) dependence of τ^p and the nonexponentiality of φ(t) – quantified in the law x = 2/β – is formally similar to that reported for the first time in ref. [22] for the momentum (Q) transfer dependence
of the relaxation time τ s of the incoherent scattering function Fself(Q, t) corresponding to a sub-diffusive process 〈r2(t)〉 ∝ tβ and the nonexponentiality of Fself (Q, t) [τs ∝ Q−x; x = 2/β]. We note that the dimensionality of Q is just the inverse of that of N/p. In the case of the scattering function, the expression τs ∝ Q−2/β was explained in ref. [23] in terms of the ‘coupling model’. However, later it was demonstrated (see, e.g. [24]) that this law is just a consequence of the Gaussian approximation used to construct Fself (Q, t) from 〈r2(t)〉. This formal analogy suggests that the approximations used in this work to calculate φ(t) – in particular the pseudo-Markovian convolution approximation – play a similar role than the Gaussian approximation does in the case of Fself(Q, t). Moreover, we also note that the above-mentioned deviations from the law τ^p ∝ ðN=pÞ2=β at lower values of (N/p) are just equivalent to the deviations from the law τs ∝ Q−2/β which are well documented at higher values of Q (see, e.g., [24,25]) Finally, though in this approach we are not dealing with a microscopic theory for the memory function Γ(t), taking into account the results discussed in the Introduction section a plausible assumption is that density fluctuations in the system are the relevant mechanism driving the decay of this function. However, it is worthy of remark that the generalized expressions that are deduced in the next section do not depend, in principle, on the origin of non-exponential Rouse mode relaxations, providing Γ(t) decays as an inverse power law. 3. Generalized End-to-End correlation function Taking into account the orthogonality of the chain Rouse modes – i.e., the spatial uncorrelation of the random forces acting on a given polymer segment – together with Eq. (6), generalized Rouse expressions for the mean squared displacement of a polymer segment 〈r2s (t)〉 and the chain center of mass 〈R2CM(t)〉 were deduced in ref. [12]. Starting from these expressions, the corresponding incoherent scattering funcs CM tions Fself (Q, t) and Fself (Q, t) were also calculated in [12] in the framework of the Gaussian approximation. Here Q is the momentum transfer. The total incoherent scattering function resulted to be:
s
CM
F self ðQ ; t Þ ¼ F self ðQ; t Þ F self ðQ; t Þ
ð7Þ
where "
CM F self ðQ ; t Þ
W o ℓ4 ð1−βÞ β ¼ exp −Q tc t 3R2E
#
2
ð8Þ
and " s
F self ðQ ; t Þ ¼ exp −Q
2
W o ℓ4 9π
!1=2
# ð1−β Þ=2 β=2
tc
t
:
ð9Þ
In these expressions Wo is the so-called elementary Rouse rate Wo = 3kBT/(ℓ2ξo). Here we will follow the same scheme to deduce a generalized Rouse expression for the correlation function of the End! to-End vector, R E of a polymer chain. We note that this is a very important magnitude concerning chain dynamics because it is experimentally accessible by dielectric spectroscopy in the case of the so-called A-type polymers in the Stockmayer classification [26]. These polymers contain dipole moments along the chain backbone that do not cancel at the whole chain, giving rise to an ‘End-to-End’ net polarization vector. The fluctuations of this vector produce in the frequency domain the socalled ‘normal mode relaxation’ peak of the dielectric spectra. ! Taking into account that R E can be expressed in terms of the discrete pπ ! ! as well as the Rouse modes as R E ¼ −4∑p:odd X p cos 2N
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
! orthogonality D! E of the Rouse modes, the correlation function of R E , ! R E ðt Þ R E ð0Þ can be expressed as: D! E 2ℓ2 ! 2 pπ ∑ cot R E ðt Þ R E ð0Þ ¼ φ ðt Þ: 2N p N p:odd Taking into account that for a Gaussian chain write for the long (N/p) limit
ð10Þ !2 R E ¼ Nℓ2 , we can
D! E ! R E ðt Þ R E ð0Þ 8 X 1 ¼ 2 φp ðt Þ: ΦEE ðt Þ ¼ D!2 E π p2 RE p:odd
ð11Þ
This equation indicates that the motion of the End-to-End vector is ! mainly governed by the first mode X 1. In the case of the pure Rouse beo havior, φp = exp [−t/τo] with τp = τoRp−2. Then, Eq. (10) reads: Rouse
ΦEE
ðt Þ ¼
8 X 1 2 t exp −p : o τR π 2 p:odd p2
ð12Þ
However, Eq. (10) is a general expression also valid for nonexponential φp(t). Therefore, taking into account a stretched exponential function for φp(t) (Eq. (4)) together with expression (6) for τ^p and the Rouse scaling τop = τoRp−2. We finally obtain: " # ð Þ 8 X 1 p2 t c1−β t β exp − ΦEE ðt Þ ¼ 2 : τoR π p:odd p2
ð13Þ
This expression reduces to Eq. (12) for β = 1 (pure Rouse).
necessarily involves the comparison of the β-values obtained by fitting ΦEE(t) and Fself (Q, t) with Eqs. (7) and (13), with the ‘actual ones’, i.e., those obtained directly from the fitting of φp(t) by a stretched exponential function (Eq. (4)). The actual β-values so obtained in the case of PEO in the blend were already reported and discussed in ref. [12]. Here, we have also obtained those corresponding to pure PEO. On the other hand, the direct fitting of φp(t) (for p = 1) by a stretched exponential function delivers the time τ^R . By means of Eq. (6) and with the actual values of β, we can deduce from τ^R the corresponding values of τoR. These values will be compared with those obtained by fitting ΦEE(t) and Fself(Q, t) with Eqs. (7) and (13) respectively. Fig. 1 shows the fitting results corresponding to the available data of ΦEE(t) for pure PEO (a) and for PEO in PMMA/PEO blend (b). The temperatures are indicated in the figure. As can be seen, the fitting curves perfectly describe ΦEE(t) in all temperatures. The values obtained for the fitting parameters, β and τoR, for the different systems and temperatures are included in Figs. 3 and 4 respectively and they will be discussed in the Discussion section. On the other hand, Fig. 2 displays the fitting results corresponding to the available data of Fself(Q, t) for pure PEO at Q = 0.3 Å−1. Again, the fitting curves perfectly describe these data (see also the inset with a logarithmic time-scale highlighting the short time range). The obtained values of β are also included in Fig. 3. We note that Eq. (7) does not explicitly depend on τoR but on the so-called Rouse variable Woℓ4. However, taking into account that Woℓ4 = R4E/(τoRπ2) [29] and the R2e value above-mentioned, we can easily deduce the corresponding values of τoR. These values are also included in Fig. 4. It is worthy of remark that the values obtained for β and τoR, not only describe Fself (Q, t) for Q = 0.3 Å−1 but for the full low Q-range (0.1 Å−1 to 0.4 Å−1) where the incoherent scattering function characterizes chaindynamics.
4. Application to MD-simulation results of PEO and PEO in PMMA/ PEO blend EE
(t)
1 0.8
350 K
0.6
375 K
0.4
400 K
0.2
(a)
(t)
0
500 K
1
350 K
EE
In the following we will check the validity of Eq. (13) by means of MD-simulation results corresponding to melts of pure PEO and PMMA/PEO blend, which were previously published [10,27]. We note that these simulations were carried out at fully atomistic level and that they were carefully validated by direct comparison with neutron scattering results [27,28]. The blend PMMA/PEO is considered as a canonical example of dynamically asymmetric blends, being PEO the fast component. The simulated blend system had a concentration rich in PMMA (80% in weight). Details about the simulation procedure and the simulation results can be found in refs. [10,27]. Here, first of all, we will fit with Eq. (13) the simulation data of ΦEE(t) at different temperatures and for pure PEO and PEO in the blend with PMMA. Moreover, in order to complete the results previously published [12] and corresponding to the incoherent scattering function Fself (Q, t) of PEO in the blend, we will also fit with Eq. (7) the same type of data but now for pure PEO. In this case, the value of R2e included in Eq. (8) was R2e ≈ 1000 Å2 [27]. As we did in ref. [12], we will fix the value of tc to 20 ps in all fits. We recall that according to expression (5), tc marks the time range (t N tc) of applicability of Eqs. (7) and (13). Thereby, the fitting of the simulation data by these equations was always carried out for t N 20 ps. We note, however, that in the framework of the approximations described in ref. [12] and here, tc should be considered as a crossover time-range more than a well-defined time. As it has been widely discussed in refs. [12,20], a variation of tc within a factor three or four does not affect significantly the results of the parameters β and τoR obtained by fitting, providing that the fitting range is always taken as t N tc. By means of the fitting procedure above-sketched, we will obtain values of the fitting parameters, β and τoR, corresponding to each case. We recall that the β parameter included in Eqs. (7) and (13), actually corresponds to the non-exponential parameter of the correlation function of the Rouse normal modes, φp(t), in the long wavelength (N/p) limit (low-p range). Thus, a proper validation of these equations
305
0.8
400 K
0.6 0.4 0.2 0
500 K
(b) 0
2
4
6
8
t(ns)
10
Fig. 1. ΦEE(t) corresponding to (a) pure PEO and (b) PEO in PMMA/PEO blend at the temperatures indicated in the figure. Continuous lines are fitting curves with Eq. (13) (see the text).
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-1
1
0.8
0.8
0.6
self
F
log[ (ns)]
(Q=0.3Å ,t)
1
0.6
1.5
(a)
350K
0.4
1
400K 0.2
0.4 0 10-2
10-1
100
101 t(ns)
0.2
0.5
350K 0 0
2
4
6
8 t(ns)
Fig. 2. Fself(Q, t) corresponding to pure PEO at Q = 0.3 Å−1 and the temperatures indicated in the figure. Continuous lines are fitting curves with Eq. (7) (see the text). The inset shows the same representation but in a logarithmic time-scale to highlight the short time range.
log[ (ns)]
400K
5 4 3
5. Discussion
2
The ‘actual values’ of β for both PEO and PEO in PMMA/PEO blend are included in Fig. 3 for the different temperatures considered in the MDsimulations. As it was discussed in ref. [12] and mentioned above, these values correspond to the long wavelength (N/p) limit of φp(t) and they were calculated as the average of the β(N/p) values in the long (N/p)-range (typically (N/p) N 10) where the PEO chains can be considered as ideal chains (i.e., no influence from local potentials). The error bars in the figure were estimated as the standard deviation of the values of β at each temperature in that (N/p)-range. As can be seen, the uncertainty is larger in the case of PEO in the blend than in the case of pure PEO. In particular, this uncertainty is rather large at the lowest temperature investigated (300 K), mainly due to the small decay of φp(t) at this low temperature. Fig. 3 also includes the values of β obtained by fitting either ΦEE(t) or Fself(Q, t) data of pure PEO and PEO in the PMMA/PEO blend by means of the generalized Rouse expressions described in this work (Eqs. (13) and (7) respectively). β-Values obtained from the fitting of 〈r2s (t)〉 and 〈R2CM(t)〉 corresponding to PEO in the blend and which were discussed in ref. [12], are also included in
1
1 0.9
PEO
0.8 0.7 0.6
PEO in PMMA/PEO 0.5 0.4 300
350
400
450
500
T(K) Fig. 3. Temperature dependence of the stretching parameter β corresponding to the Rouse mode correlators in the long (N/p)-range for PEO (empty symbols) and PEO in PMMA/PEO blend (filled symbols). (Circles) ‘actual β-values’; (triangles) β-values obtained by fitting Fself(Q, t) to Eq. (7); (inverted triangles) β-values obtained by fitting ΦEE(t) to Eq. (13); (filled squares) β-values obtained from the long-time range of 〈r2s (t)〉 for PEO in PMMA/ PEO blend; (filled diamonds) β-values estimated from the long-time range of 〈R2CM(t)〉 for PEO in PMMA/PEO blend. Lines through the ‘actual β-values’ are only to guide the eye.
(b)
0 1.8
2.2
2.6
3
3.4
1000 / T(K) Fig. 4. Arrhenius plot for τ^R and τoR. Empty symbols in (a) correspond to pure PEO; filled symbols in (b) correspond to PEO in PMMA/PEO blend. (Circles) τ^R from the fitting of φp = 1(t) by a stretched exponential; (squares) τoR obtained from τ^R by Eq. (6); (triangles) τoR obtained by fitting ΦEE(t) to Eq. (13); (diamonds) τoR obtained by fitting Fself(Q, t) to Eq. (7). Dashed line in (a) and continuous line in (b) are Arrhenius fits delivering E a ≈ 4 kcal/mol and E a ≈ 5 kcal/mol respectively. The dashed line in (a) is also plotted in (b) for comparison.
Fig. 3 for comparison. First of all, it is worthy of remark the rather good agreement – within the uncertainties involved in the different data – among the values of β obtained by fitting different dynamic magnitudes by means of the corresponding generalized expressions. Moreover, the agreement among these β-values and the ‘actual-ones’ is remarkable, taking into account the level of uncertainty of the actual values (see the error bars) and the approximations involved in the calculation of the generalized expressions for the different dynamic magnitudes. This agreement gives support to the approach used to deduce the generalized expressions of ΦEE(t) and Fself(Q, t). Now we can discuss the values of τoR included in Fig. 4. First of all, in general, there is a good agreement between the values of τoR obtained by fitting, either ΦEE(t) or Fself(Q, t) data, by means of the corresponding generalized expressions (Eqs. (13) and (7) respectively). Moreover, these values also agree with those directly obtained from τ^R by means of Eq. (6). These results again give support to the generalized expressions for the dynamic magnitudes deduced in this work and in ref. [12]. On the other hand, we note that in this framework, the difference between the values of τ^R and τoR – and its temperature dependence – is mainly due to the values of β and its temperature dependence as well. In the case of pure PEO, as β (≈0.9) is close to one and hardly dependent of temperature in the range here discussed, τoR values result to be rather close to τ^R. However, this is not the case for PEO in the PMMA/PEO blend where the difference between τ^R and τoR dramatically increases as the temperature (and β) decreases. An Arrhenius fit of τoR in the temperature range here considered (see dashed line in Fig. 4(a)) delivers for pure PEO an activation energy Ea ≈ 4 kcal/mol. In the case of PEO in the blend, although the absolute values of τoR are close to those corresponding to pure
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
PEO, the activation energy is slightly higher: Ea ≈ 5 kcal/mol (see continuous line versus dashed line in Fig. 4(b)). We note that in the formalism here presented, τoR means the relaxation time corresponding to pure Rouse behavior. Thereby, we should expect for PEO in the blends values close to those of pure PEO but with a slightly higher activation energy (higher friction coefficient) due to the presence of the less flexible PMMA chains. This is exactly what we found. Conversely, as it has been mentioned in the Introduction section and widely discussed in refs. [10,12], the actual values of β, corresponding to PEO in the blend strongly depend on temperature. In the case of pure PEO the exponentiality of φp(t) is less marked (β ≈ 0.9) and hardly dependent on temperature, at least in the relatively high temperature range here explored. This, in principle, different behavior can be rationalized in the light of the ideas proposed in ref. [4] and that we have commented in the Introduction section. In that framework, the nonexponential character of the long wavelength Rouse mode relaxation – which is reflected in the values of β – is due to the influence of the density fluctuations on the chain dynamics and it will be higher as soon as the ratio τ^R =τ α decreases. We recall that τα is the characteristic time of the relevant density fluctuations, i.e., of the α-process in the pure homopolymer and of the effective α-process of the slow component (PMMA in our case) in the blend. Therefore, within this unified scenario, the relevant parameter controlling the non-exponential character of φp(t) in the long wavelength (N/p) limit results to be the ratio τ^R =τ α instead of being directly the temperature. For a given polymer, this would allow comparing data of β corresponding to the homopolymer case and to the case of this polymer in different blends. In order to test these ideas with the β-values of PEO and PEO in the PMMA/PEO blend, we need to estimate from the simulations the values of τα (density fluctuations) in each case. To do that, we have considered τα as the time scale of the corresponding incoherent scattering function at Q ≈ 1 Å−1. We have followed this criterium because it is well documented that relaxation times from incoherent neutron scattering data at this Q-range use to match the α-relaxation times determined by spectroscopic methods like, for instance, dielectric spectroscopy (see, e.g. [30]). We recall that in the case of PEO in the PMMA/PEO blend, the relevant value of τα is that corresponding to the α-process of PMMA in the blend. Taking into account the values of τα so obtained, together with those reported for τ^R in Fig. 4 for pure PEO and PEO in the blend, we have obtained the ratio τ^R =τ α . Now, the actual values of β shown in Fig. 3 are represented in Fig. 5 as a function of the parameter Δ ¼ log ðτ^R =τα Þ. First, we realize that the values of Δ, corresponding to the pure PEO at different
1 0.9 0.8 0.7 0.6 0.5 -3
-2
-1
0
1
2
3
4
Fig. 5. β as a function of the parameter Δ ¼ log ðτ^R =τ α Þ (see the text). (Filled circles) values correspond to PEO in PMMA/PEO blend; (filled diamonds) values correspond to pure PEO. Empty symbols correspond to data from ‘bead-spring’ model systems: (circles) homopolymer in the blend; (diamonds) pure homopolymer. Error bars in β are those included in Fig. 3. Error bars in Δ mainly reflect the uncertainties in calculating the values of τα, in particular, in the case of the τα of PMMA in the blend. Dashed line through filled symbols is a sigmoidal fit to guide the eye.
307
temperatures, are almost identical and rather high (Δ ≈ 3.5). This would explain why the corresponding β-values are close to one and hardly dependent on temperature. Moreover, the data of β for PEO and PEO in the blend seem to follow a unified trend in terms of the Δparameter. It is worthy of remark that data corresponding to a generic fully flexible ‘bead-spring’ model for a homopolymer and for the same polymer in an asymmetric blend [9] also seems to follow the same qualitative trend taking into account the simplicity of this model. In conclusion, we can say that the data reported here also give support to the scenario that has been recently proposed in ref. [4]. Finally, as it has been mentioned in Section 2, the Eq. (6), deduced in ref. [12] in the framework of the formalism and the approximations widely described in Section 2 of this work, looks exactly like the basic equation of the ‘coupling model’ (CM) developed a long time ago by Ngai for the crossover from non-cooperative to cooperative relaxation processes (CM-terminology) in general [15–17]. In fact, in a recent paper Ngai and Capaccioli have also applied the CM to chain dynamics of PEO in asymmetric blends [18] although using a very large value of tc (tc ≈ 2 ns). This in fact implies that these authors are assuming pure Rouse behavior of PEO in the blend at times shorter than about 2 ns. However, we note that the MD-simulation data of PEO in PMMA/PEO blend [10,12] clearly indicate a non-exponential behavior of φp(t) in the long wavelength (N/p) range (see Fig. 3), thereby invalidating the assumption of ideal Rouse behavior in such a time regime. However, it is worthy of remark that the law τ^p ∝ ðN=pÞx with x = 2/β remains valid independently of the value of the crossover time tc. In any case, readers interested in the debate about the actual value of tc for this problem are referred to the two recent (‘Comment’ and ‘Reply’) papers [31, 32]. The fact that the CM-equation can be deduced from Eqs. (2) and (3), suggests that, at least in this particular case, the CM seems to be a consequence of the GLE formalism and the approximations made here, in particular, the pseudo-Markovian approximation. This point of view has been criticized by Ngai and Capaccioli in a comment [19] to our previous paper [12] (see also the accompanying reply [20] for this question). They claim that, in fact, we are not solving Eqs. (2) and (3) in the usual way. As it is clearly recognized in ref. [12] and in Section 2 of this manuscript, to solve these equations in the standard way would imply to construct a microscopic theory for the memory function Γ(t), something that is beyond of this approach. However, we note that Eq. (6) is nothing more than a formal relationship between τ^p , τop and β, which in the formalism developed here and in refs. [12,21], is independent of the particularities of the memory function, providing the time decay of this function is given by the power law Γ(t) ∝ t−β. Therefore, our procedure is just equivalent to solve Eq. (3) by using this type of memory function. Then we could conclude that Eq. (6) – the basic equation of the ‘coupling model’ – is obtained in our approach as a consequence of the pseudo-Markovian approximation – which allows to obtain Eqs. (3) from (2) – and of a power law decay of the memory function. On the other hand, we note that, as it has been discussed in Section 2, in our framework Eq. (6) is an asymptotic equation only valid for the long wavelength (N/p) limit, where we can consider the pseudo-Markovian as a good approximation, and where β does not depend on (N/p) as well. Certainly this is not the case for lower values of (N/p) [9,10,20]. We wonder whether or not this asymptotic character of Eq. (6) could be incorporated in a direct CM-deduction of this expression, which in principle should work for any value of (N/p). 6. Summary and conclusions Magnitudes probing polymer chain dynamics as, for instance, incoherent and coherent neutron scattering functions or dielectric ‘normal mode relaxation’, are constructed in terms of exponential Rouse mode correlators. The reason for that is that the Rouse model fully neglects the time correlation of the random forces acting on a polymer chain segment. However, experimental and MD-simulations as well clearly show
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that Rouse mode relaxation use to deviate from the exponential behavior even in the long wavelength (N/p) range where polymer chains can be considered as ideals (i.e., no influence from local potentials). These deviations have been observed in homopolymers at low temperature approaching the glass-transition of the system and, in particular, when we follow the chain dynamics of the fast component of the socalled dynamically asymmetric blends. Based on these results it has been proposed that deviations from exponentiality are in fact due to the effect of the density fluctuations in the system at inter-macromolecular level (the so-called α-relaxation) on chain dynamics. In the framework of the generalized Langevin equation (GLE) formalism, the correlation between the random forces acting on a polymer chain can be introduced as a memory kernel in the Langevin equation for the time evolution of the Rouse mode correlators. This term effectively produces an extra friction non-local in time and as consequence, a non-exponential behavior of the mode relaxation. Inspired by this formalism and considering some approximations, we have proposed a procedure for generalizing the Rouse expressions of the typical magnitudes probing chain dynamics, which takes into account a stretched exponential decay for the Rouse mode relaxations. Here we have applied this procedure to obtain an expression for the End-to-End correlation function, which in the frequency domain give rises to the well known ‘normal mode relaxation’. By means of fully atomistic MD-simulations of pure PEO and PEO in a PMMA/PEO dynamically asymmetric blend, we have checked the generalized Rouse expressions obtained here and in a previous work. The simulation data can be nicely described by the proposed expressions for the chain dynamics magnitudes, and give support to the unified scenario recently proposed for explaining the non-exponential relaxation of the Rouse modes in polymer systems in terms of the effect of density fluctuations (α-process) on chain dynamics. Acknowledgments The author would like to thank M. Brodeck, F. Alvarez and A. J. Moreno for providing data of MD-simulations previously published. He also acknowledges the helpful discussions with A. Alegría, A. J. Moreno, A. Arbe and S. Arrese-Igor. Support from the projects IT-654-13 (GV) and
MAT2012-31088 (Ministerio de Economia y Competitividad) is also acknowledged.
References [1] P.E. Rouse, J. Chem. Phys. 21 (1953) 1272. [2] P.G. de Gennes, J. Chem. Phys. 55 (1971) 572. [3] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. [4] S. Arrese-Igor, A. Alegría, J. Colmenero, Phys. Rev. Lett. 113 (2014) 078302. [5] D.J. Plazek, E. Schlosser, A. Schönhals, K.L. Ngai, J. Chem. Phys. 98 (1993) 6488. [6] K.L. Ngai, A. Schönhals, E. Schlosser, Macromolecules 25 (1992) 4915. [7] A. Schönhals, Macromolecules 26 (1993) 1309. [8] D.J. Plazek, C. Bero, S. Neumeister, G. Floudas, G. Fytas, K.L. Ngai, J. Colloid Polym. Sci. 272 (1994) 1430. [9] A.J. Moreno, J. Colmenero, Phys. Rev. Lett. 100 (2008) 126001. [10] M. Brodeck, F. Alvarez, A.J. Moreno, J. Colmenero, D. Richter, Macromolecules 43 (2010) 3036. [11] S. Arrese-Igor, A. Alegría, A.J. Moreno, J. Colmenero, Macromolecules 44 (2011) 3611. [12] J. Colmenero, Macromolecules 46 (2013) 5363. [13] K.S. Schweizer, J. Chem. Phys. 91 (1989) 5802. [14] N. Fatkullin, R. Kimmich, M. Kroutieva, J. Exp. Theor. Phys. 91 (2000) 150. [15] K.L. Ngai, Comment Solid State Phys. 9 (1979) 127. [16] K.Y. Tsang, K.L. Ngai, Phys. Rev. E. 54 (1996) R3067. [17] K.L. Ngai, K.Y. Tsang, Phys. Rev. E. 60 (1999) 4511. [18] K.L. Ngai, S. Capaccioli, J. Chem. Phys. 138 (2013) 054903. [19] K.L. Ngai, S. Capaccioli, Macromolecules 46 (2013) 8054. [20] J. Colmenero, Macromolecules 46 (2013) 8056. [21] S. Arrese-Igor, A. Alegría, A.J. Moreno, J. Colmenero, Soft Matter 8 (2012) 3739. [22] J. Colmenero, A. Alegría, A. Arbe, B. Frick, Phys. Rev. Lett. 69 (1992) 478. [23] K.L. Ngai, J. Colmenero, A. Alegría, A. Arbe, Macromolecules 25 (1992) 6727. [24] J. Colmenero, F. Alvarez, A. Arbe, Phys. Rev. E. 65 (2002) 041804. [25] A. Arbe, J. Colmenero, F. Alvarez, M. Monkenbusch, D. Richter, B. Farago, B. Frick, Phys. Rev. Lett. 89 (2002) 245701. [26] W.H. Stockmayer, Pure Appl. Chem. 15 (1967) 539. [27] M. Brodeck, F. Alvarez, A. Arbe, F. Juranyi, T. Unruh, O. Holderer, J. Colmenero, D. Richter, J. Chem. Phys. 130 (2009) 094908. [28] M. Brodeck, F. Alvarez, J. Colmenero, D. Richter, Macromolecules 45 (2012) 536. [29] D. Richter, M. Monkenbusch, A. Arbe, J. Colmenero, Neutron Spin Echo in Polymer Systems, Advances in Polymer Science, vol. 174, Springer Verlag, Berlin Heidelberg New York, 2005. [30] J. Colmenero, A. Arbe, A. Alegría, J. Non-Cryst. Solids 172–174 (1994) 126. [31] J. Colmenero, J. Chem. Phys. 138 (2013) 197101. [32] K.L. Ngai, S. Capaccioli, J. Chem. Phys. 138 (2013) 197102.
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Non-exponential Rouse correlators and generalized magnitudes probing chain dynamics J. Colmenero 1 Centro de Física de Materiales (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
a r t i c l e
i n f o
Article history: Received 21 May 2014 Received in revised form 8 September 2014 Accepted 10 September 2014 Available online 7 November 2014 Keywords: Polymer dynamics; Molecular dynamics simulations; Neutron scattering; Dielectric spectroscopy; Polymer blends
a b s t r a c t Inspired by the generalized Langevin equation (GLE) formalism with some approximations, we have proposed a method to generalize the Rouse expression of the magnitudes probing chain dynamics, taking into account the non-exponential character of the Rouse mode correlators. In this way, generalized expressions for the incoherent neutron scattering function and for the End-to-End relaxation – which give rise in the frequency domain to the dielectric ‘normal mode’ relaxation – have been obtained. These magnitudes have been checked by means of fully atomistic molecular dynamics (MD) simulations corresponding to pure poly(ethylene oxide) (PEO) and to PEO in an asymmetric blend with poly(methyl methacrylate) (PMMA). Moreover, the results obtained also seem to give support to the unified scenario recently proposed for explaining the non-exponential relaxation of the longwavelength Rouse modes in polymer systems in terms of the effect of density fluctuations (α-process) on chain dynamics. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Chain dynamics dictate the viscoelastic properties of melts of polymer systems. Nowadays it is generally assumed that the Rouse model [1] provides a suitable description of chain dynamics of unentangled polymer melts, i.e., melts of chains of molecular weight lower than a certain limit (‘entanglement mass’). The Rouse model can also be seen as a ‘fixed point’ of all theories of polymer dynamics in which linear connected objects are ideal and subjected to local dissipation. For instance, in the chain dynamics theories of entangled polymers based on the ‘tube’ concept [2,3], the Rouse model is a basic ingredient because chains move ideally (Rouse behavior) until they feel the effects of the confining tube. In the framework of the Rouse model, a tagged chain is represented as a string of N beads of equal mass connected by harmonic springs of constant 3kBT/ℓ2, with ℓ the bond length and kB the Boltzmann constant. The effective interaction experienced by the chain is given by a ! friction coefficient ξ and a set of stochastic forces f j . Excluded volume ! interactions are neglected. The chain normal (Rouse) mode X p ðt Þ of ! index p = 1 … N − 1 (wavelength N/p) is defined as X p ðt Þ ¼ N−1 N ! ! ∑ j¼1 r j ðt Þ cos ½ð j−1=2Þpπ=N, with r j the vector giving the position ! of the j-bead in the chain. X p ðt Þ follows the equation of motion
2Nξ
! d X p ðt Þ ! ! ¼ −K p X p þ g p dt
ð1Þ
E-mail address: [email protected]. Also at Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain. 1
http://dx.doi.org/10.1016/j.jnoncrysol.2014.09.033 0022-3093/© 2014 Elsevier B.V. All rights reserved.
with K p ¼ 24NkB Tℓ−2 sin2 ðpπ=2NÞ . The external force for the p-th N ! ! mode is g p ¼ 2∑ j¼1 f j ðt Þ cos ½ð j−1=2Þpπ=N. The Rouse model fully neglects spatial and time correlations of D E ! ! the stochastic forces, i.e., g p ðt 0 Þ g q t ″ ¼ 12NξkB Tδpq δðt−t 0 Þ. The former two approximations yield, respectively, orthogonality of the Rouse modes and exponential relaxation of the Rouse correlators, leading to the expression for the correlation function of the p-mode, D! E pπ − t ! ℓ2 C p ðt Þ ¼ X p ðt Þ X p ð0Þ , C p ðt Þ ¼ 8N sin−2 2N e τp with the relaxation 2 ℓ ξ pπ time given by τp ¼ 12k sin−2 2N . Due to the effects of local potenBT tials and chain stiffness at short length scales in real systems, Rouse model results are usually discussed in the long wavelength (N/p) pπ pπ 2 ≈ 2N . limit. In this range sin2 2N Orthogonality of the Rouse modes and exponentiality of the Rouse correlators are the two main assumptions of the Rouse model and are the basis of the derivation of the magnitudes probing chain relaxations [3]. These magnitudes (incoherent and coherent scattering functions, dielectric ‘normal mode relaxation’, etc.) are thereby constructed in terms of exponential Rouse mode correlators. For homopolymer melts this seems to be a rather good approximation in the high temperature range (with respect to the glass-transition temperature Tg) were most of the measurements of the chain dynamics are usually carried out and taking into account that the relevant modes are the slowest ones. However, significant deviations from the exponentiality of these mode relaxations have been recently reported in the low temperature range approaching Tg [4]. This range was accessible by using a thermostimulated depolarization current (TSDC) technique, which, on the
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
other hand, allows isolating experimentally the dielectric response of the slowest Rouse mode (p = 1) relaxation. As usually, deviations from exponentiality were accounted for by the well known h using β i stretched exponential function C p ðt Þ≈ exp − t=τ^p . The β values characterizing these deviations resulted to be correlated with the ratio τ^R =τ α being τ^R ¼ τ^p¼1 and τα the characteristic relaxation time of the α-relaxation in the system. This process is a general feature of glassforming systems and it is associated with the evolution of density fluctuations in the system at the inter-macromolecular scale (first maximum of the static structure factor S(Q) in the case of polymers). In the high temperature range, τα uses to be well separated from that of the slowest chain mode relaxation (τR), thereby implying that the statistical independence of both processes could be taken as a good approximation. However, as soon as the temperature decreases approaching the Tg of the polymer melt, both processes approach each other, implying some kind of coupling between both. This would break the Rouse assumption of uncorrelation of the random forces acting on a polymer chain and would explain the non-exponential behavior of the Rouse mode relaxation. Although the possible coupling between chain dynamics and α-relaxation has been widely discussed in the literature concerning the problem of the different temperature dependence of both processes approaching Tg (see, e.g. [5–8]), the effect of local density fluctuations (α-relaxation) on the non-exponentiality of the Rouse mode correlators in homopolymers has started to be recognized only recently [4]. On the other hand, in the case of the chain dynamics of the fast component of asymmetric polymer blends, strong deviations from the exponential relaxation of the slowest Rouse modes are well documented [9–11]. Asymmetric blends are miscible polymer systems A/B where the two components display very different segmental mobility or Tgs. In this case, the deviations from the exponentiality of the Rouse mode relaxation also depend on temperature, being negligible at high temperature where the dynamic contrast between the two components vanishes. As soon as the temperature decreases, the α-relaxation of the slow component of the blend will slow down and its relaxation time will approach the characteristic times of the slowest Rouse mode relaxation of the fast component in the blend. This situation seems to be similar to that described above for homopolymer melts, being now the relevant density fluctuations those associated with the freezing of the slow component in the blend. In fact, based on experimental results of polyisoprene (PI) and PI in different blends, it has been reported [4] that the β values corresponding to the slowest (p = 1) Rouse mode of the fast component in a blend follow the same trend with respect to the ratio τ^R =τ α than those corresponding to the homopolymer case, being τα the characteristic time of the relevant density fluctuations in the system: the α-process for the homopolymer melt and the α-process of the slow component in the blend. We note that in the case of the asymmetric blends we can face situations where τ^R =τ α ≤ 1, something that would not be possible in the case of homopolymers. Thus, in general, we should expect stronger deviations from the exponentiality of Cp = 1(t) [called CR(t)] – i.e., lower values of β – in the case of blends. As it has been mentioned above, in the framework of the Rouse model the non-exponentiality of the Rouse mode correlators will directly affect the magnitudes probing chain dynamics because these magnitudes are written in terms of such correlators. We have also to take into account that in the tube-reptation theories for well entangled polymers, chain dynamics at times t b τe – where τe is the so-called entanglement time – is also described in terms of the Rouse model, thereby being affected as well by the non-exponential behavior of Rouse correlators. Moreover, in the case of the End-to-End correlation function, the mathematical expression is formally the same in the framework of both, Rouse and reptation theories. With these ideas in mind, in a recent paper [12] we have generalized the Rouse expressions for the mean
303
squared displacement of a polymer segment 〈r2s (t)〉 and that of the center of mass of the chain 〈R2CM(t)〉, by considering the non-exponential character of the Rouse mode relaxation. Starting from these expressions and taking into account the Gaussian approximation, we have also proposed equations for the corresponding incoherent scattering functions s CM Fself (Q, t) and Fself (Q, t). The results obtained were checked by using molecular dynamics (MD) simulations data of poly(ethylene oxide) (PEO) in asymmetric blends with poly(methyl methacrylate) (PMMA). In the present work, after briefly summarizing the main approximations used to generalize the Rouse expressions for 〈r2s (t)〉 and 〈R2CM(t)〉, we will extend this approach to the case of the End-to-End correlation function. Finally, we will apply the results obtained to MD-simulation data of both, bulk PEO and PEO in PEO/PMMA blends. 2. A simple approach inspired by the GLE formalism As it has been commented in the Introduction section, the exponential nature of the relaxation of the Rouse modes arises as a consequence of neglecting time correlations of the random forces acting on a polymer segment. In a general formulation (generalized Langevin equation, GLE, formalism) for the time evolution of the Rouse mode correlators, the correlation between the random forces can be introduced within a memory kernel in the Langevin equation. This term effectively produces an extra friction non-local in time and thereby a non-exponential decay of the mode relaxation. The standard Rouse behavior and the corresponding constant friction are recovered when the memory function decays to zero at very short times in comparison to those characteristic of the mode relaxation. In the framework of a simplified GLE formalism [13], we can write an integro differential equation for the time evolution D! E ! of a Rouse correlator, C p ðt Þ ¼ X p ðt Þ X p ð0Þ dC p ðt Þ 1 þ dt ξo
Z
0 C p ðt Þ 0 0 dC p t dt Γ t−t ¼− o : 0 τp dt 0 t
ð2Þ
We note that in order to arrive to Eq. (2), the fast decaying ‘Rouse contribution’ has been extracted from the general memory function of the GLE formalism [13,14]. In this way, the memory function Γ(t) in Eq. (2) only contains the non-Rouse contribution. As consequence of this partition, ξo and τop in Eq. (2) are respectively the ‘bare’ constant friction coefficient connected with the fast local motions and the relaxation time of the Rouse correlator corresponding to pure Rouse behavior. Moreover, as it was described in ref. [12], we considered two additional approximations. First, Γ(t) was assumed to be independent of the length scale (wavelength of the mode, N/p) due to the fact that we were interested in the long wavelength (N/p) limit. Moreover, based on the arguments widely discussed in ref. [12], a second step was to consider a ‘pseudo-Marcovian’ approximation and to replace the convolution integral of Eq. (2) by a time local product. As it was discussed in ref. [12], this strong approximation would be justified under the assumption that in the long wavelength (N/p) limit, Γ(t) decays faster than Cp(t). This is equivalent to say that Γ t ¼ τ^p ≈0, where τ^p is the relaxation time of Cp(t) (see below). In ref. [12], this assumption was numerically checked by means of MD-simulations of PEO/PMMA system, also considered in this work. It was obtained that Γ t ¼ τ^p ≈0 for the range N/p ≥ 10 (see Fig. 7 in ref. [12]), i.e., the relevant long wavelength range considered here (see Section 5). In the framework of the above-described approximations and defining a time dependent effective friction coefficient as ξ(t) = ∫t0dt′Γ(t′) the solution of Eq. (2) can be expressed as φp ðt Þ ¼
" Z C p ðt Þ ξ ¼ exp − oo C p ð0Þ τp
# dt 0 0 : 0 ξo þ ξðt Þ t
ð3Þ
To proceed further with Eq. (3) we have two possibilities. The first, and more rigorous one, would be to construct a microscopic theory for
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Γ(t) which allows to solve the integral equation. As it was commented in ref. [12] this means a formidable theoretical effort, involving many approximations as well (see, e.g. refs. [13,14]). The other possibility is to follow a phenomenological procedure based on: (i) the knowledge that φp(t) has to be non-exponential due to the memory function; (ii) the experimental [4,11] and simulation [9,10] results, which indicate that a stretched exponential function " φp ðt Þ ≈ exp −
t τ^p
!β # ð4Þ
with β = β(T) uses to be a good approximation to describe the nonexponentiality of φp(t). In this context, the procedure followed reduces to the question: what should it be the simplest expression for ξ(t) giving rise by solving Eq. (3) to a stretched exponential φp(t) and assuring that in the Rouse limit (β = 1) φp(t) = exp [−t/τop]. It is straightforward to see that for the time regime so that ξ(t) ≫ ξo the answer to this question is ξðt Þ ¼
ξo t 1−β β tc
ð5Þ
which, taking into account the definition of ξ(t) above-mentioned, is equivalent to a memory function Γ(t) depending on time as an inverse power law. The time tc in Eq. (5) emerges in a natural way keeping the dimensionality of ξ(t) correct and setting the time scale for the non-exponential behavior of φp(t). Now, introducing Eq. (5) into Eq. (3) we obtain for the trivial case β = 1 the Rouse limit and for β b 1 a stretched exponential function (Eq. (4)) with 1=β o ðβ−1Þ=β τ^p ¼ τp tc :
ð6Þ
It is noteworthy that Eq. (6) looks exactly like the basic equation of the ‘coupling model’ (CM) [15–18]. Although we will comment about this question in the Discussion section, we would like to mention here that in the framework of Eq. (3), tc could be, in principle, temperature and N-dependent — in contrast with the constant crossover time of the ‘coupling model’. However, the MD-simulation results discussed in ref. [12] suggest that considering in practice a constant tc can be a good approximation. Readers interested in this question are referred to the recent ‘Comment & Reply’ papers [19,20]. As τop ∝ (N/p)2 (Rouse scaling), the (N/p) dependence of τ^p results to be τ^p ∝ðN=pÞx with x = 2/β. Even without having a microscopic theory for the memory function – i.e., a microscopic theory for β(T) – this expression established a direct an nontrivial correlation between the wavelength (N/p) scaling of τ^p and the nonexponentiality (β) of the corresponding Rouse correlator. For p = 1 – i.e., the large (N/p) limit – this correlation predicts the molecular weight (M) dependence of the generalized Rouse time τ^R ∝M 2=β , as it was shown in [21] where this expression was deduced for the first time in the framework of a formalism similar to that explained in this section. Later, this expression was also deduced by a direct application of the ‘coupling model’ [18]. However, we have to recall that Eq. (6) has been obtained in the framework of a long wavelength (N/p) approximation, where the ‘pseudo-Markovian’ approximation seems to work as it has been mentioned above. Moreover, only in that range the β-value can be considered as independent of (N/p) (see Section 5). Then, the expression x = 2/β should be considered only as an asymptotic law valid for the large (N/p) limit. Deviations from this law are expected at lower values of (N/p) as it was shown in [20] from MD-simulation data of a ‘bead-spring’ generic model for asymmetric blends [9,10]. It is worthy of remark that the correlation found between the wavelength (N/p) dependence of τ^p and the nonexponentiality of φ(t) – quantified in the law x = 2/β – is formally similar to that reported for the first time in ref. [22] for the momentum (Q) transfer dependence
of the relaxation time τ s of the incoherent scattering function Fself(Q, t) corresponding to a sub-diffusive process 〈r2(t)〉 ∝ tβ and the nonexponentiality of Fself (Q, t) [τs ∝ Q−x; x = 2/β]. We note that the dimensionality of Q is just the inverse of that of N/p. In the case of the scattering function, the expression τs ∝ Q−2/β was explained in ref. [23] in terms of the ‘coupling model’. However, later it was demonstrated (see, e.g. [24]) that this law is just a consequence of the Gaussian approximation used to construct Fself (Q, t) from 〈r2(t)〉. This formal analogy suggests that the approximations used in this work to calculate φ(t) – in particular the pseudo-Markovian convolution approximation – play a similar role than the Gaussian approximation does in the case of Fself(Q, t). Moreover, we also note that the above-mentioned deviations from the law τ^p ∝ ðN=pÞ2=β at lower values of (N/p) are just equivalent to the deviations from the law τs ∝ Q−2/β which are well documented at higher values of Q (see, e.g., [24,25]) Finally, though in this approach we are not dealing with a microscopic theory for the memory function Γ(t), taking into account the results discussed in the Introduction section a plausible assumption is that density fluctuations in the system are the relevant mechanism driving the decay of this function. However, it is worthy of remark that the generalized expressions that are deduced in the next section do not depend, in principle, on the origin of non-exponential Rouse mode relaxations, providing Γ(t) decays as an inverse power law. 3. Generalized End-to-End correlation function Taking into account the orthogonality of the chain Rouse modes – i.e., the spatial uncorrelation of the random forces acting on a given polymer segment – together with Eq. (6), generalized Rouse expressions for the mean squared displacement of a polymer segment 〈r2s (t)〉 and the chain center of mass 〈R2CM(t)〉 were deduced in ref. [12]. Starting from these expressions, the corresponding incoherent scattering funcs CM tions Fself (Q, t) and Fself (Q, t) were also calculated in [12] in the framework of the Gaussian approximation. Here Q is the momentum transfer. The total incoherent scattering function resulted to be:
s
CM
F self ðQ ; t Þ ¼ F self ðQ; t Þ F self ðQ; t Þ
ð7Þ
where "
CM F self ðQ ; t Þ
W o ℓ4 ð1−βÞ β ¼ exp −Q tc t 3R2E
#
2
ð8Þ
and " s
F self ðQ ; t Þ ¼ exp −Q
2
W o ℓ4 9π
!1=2
# ð1−β Þ=2 β=2
tc
t
:
ð9Þ
In these expressions Wo is the so-called elementary Rouse rate Wo = 3kBT/(ℓ2ξo). Here we will follow the same scheme to deduce a generalized Rouse expression for the correlation function of the End! to-End vector, R E of a polymer chain. We note that this is a very important magnitude concerning chain dynamics because it is experimentally accessible by dielectric spectroscopy in the case of the so-called A-type polymers in the Stockmayer classification [26]. These polymers contain dipole moments along the chain backbone that do not cancel at the whole chain, giving rise to an ‘End-to-End’ net polarization vector. The fluctuations of this vector produce in the frequency domain the socalled ‘normal mode relaxation’ peak of the dielectric spectra. ! Taking into account that R E can be expressed in terms of the discrete pπ ! ! as well as the Rouse modes as R E ¼ −4∑p:odd X p cos 2N
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
! orthogonality D! E of the Rouse modes, the correlation function of R E , ! R E ðt Þ R E ð0Þ can be expressed as: D! E 2ℓ2 ! 2 pπ ∑ cot R E ðt Þ R E ð0Þ ¼ φ ðt Þ: 2N p N p:odd Taking into account that for a Gaussian chain write for the long (N/p) limit
ð10Þ !2 R E ¼ Nℓ2 , we can
D! E ! R E ðt Þ R E ð0Þ 8 X 1 ¼ 2 φp ðt Þ: ΦEE ðt Þ ¼ D!2 E π p2 RE p:odd
ð11Þ
This equation indicates that the motion of the End-to-End vector is ! mainly governed by the first mode X 1. In the case of the pure Rouse beo havior, φp = exp [−t/τo] with τp = τoRp−2. Then, Eq. (10) reads: Rouse
ΦEE
ðt Þ ¼
8 X 1 2 t exp −p : o τR π 2 p:odd p2
ð12Þ
However, Eq. (10) is a general expression also valid for nonexponential φp(t). Therefore, taking into account a stretched exponential function for φp(t) (Eq. (4)) together with expression (6) for τ^p and the Rouse scaling τop = τoRp−2. We finally obtain: " # ð Þ 8 X 1 p2 t c1−β t β exp − ΦEE ðt Þ ¼ 2 : τoR π p:odd p2
ð13Þ
This expression reduces to Eq. (12) for β = 1 (pure Rouse).
necessarily involves the comparison of the β-values obtained by fitting ΦEE(t) and Fself (Q, t) with Eqs. (7) and (13), with the ‘actual ones’, i.e., those obtained directly from the fitting of φp(t) by a stretched exponential function (Eq. (4)). The actual β-values so obtained in the case of PEO in the blend were already reported and discussed in ref. [12]. Here, we have also obtained those corresponding to pure PEO. On the other hand, the direct fitting of φp(t) (for p = 1) by a stretched exponential function delivers the time τ^R . By means of Eq. (6) and with the actual values of β, we can deduce from τ^R the corresponding values of τoR. These values will be compared with those obtained by fitting ΦEE(t) and Fself(Q, t) with Eqs. (7) and (13) respectively. Fig. 1 shows the fitting results corresponding to the available data of ΦEE(t) for pure PEO (a) and for PEO in PMMA/PEO blend (b). The temperatures are indicated in the figure. As can be seen, the fitting curves perfectly describe ΦEE(t) in all temperatures. The values obtained for the fitting parameters, β and τoR, for the different systems and temperatures are included in Figs. 3 and 4 respectively and they will be discussed in the Discussion section. On the other hand, Fig. 2 displays the fitting results corresponding to the available data of Fself(Q, t) for pure PEO at Q = 0.3 Å−1. Again, the fitting curves perfectly describe these data (see also the inset with a logarithmic time-scale highlighting the short time range). The obtained values of β are also included in Fig. 3. We note that Eq. (7) does not explicitly depend on τoR but on the so-called Rouse variable Woℓ4. However, taking into account that Woℓ4 = R4E/(τoRπ2) [29] and the R2e value above-mentioned, we can easily deduce the corresponding values of τoR. These values are also included in Fig. 4. It is worthy of remark that the values obtained for β and τoR, not only describe Fself (Q, t) for Q = 0.3 Å−1 but for the full low Q-range (0.1 Å−1 to 0.4 Å−1) where the incoherent scattering function characterizes chaindynamics.
4. Application to MD-simulation results of PEO and PEO in PMMA/ PEO blend EE
(t)
1 0.8
350 K
0.6
375 K
0.4
400 K
0.2
(a)
(t)
0
500 K
1
350 K
EE
In the following we will check the validity of Eq. (13) by means of MD-simulation results corresponding to melts of pure PEO and PMMA/PEO blend, which were previously published [10,27]. We note that these simulations were carried out at fully atomistic level and that they were carefully validated by direct comparison with neutron scattering results [27,28]. The blend PMMA/PEO is considered as a canonical example of dynamically asymmetric blends, being PEO the fast component. The simulated blend system had a concentration rich in PMMA (80% in weight). Details about the simulation procedure and the simulation results can be found in refs. [10,27]. Here, first of all, we will fit with Eq. (13) the simulation data of ΦEE(t) at different temperatures and for pure PEO and PEO in the blend with PMMA. Moreover, in order to complete the results previously published [12] and corresponding to the incoherent scattering function Fself (Q, t) of PEO in the blend, we will also fit with Eq. (7) the same type of data but now for pure PEO. In this case, the value of R2e included in Eq. (8) was R2e ≈ 1000 Å2 [27]. As we did in ref. [12], we will fix the value of tc to 20 ps in all fits. We recall that according to expression (5), tc marks the time range (t N tc) of applicability of Eqs. (7) and (13). Thereby, the fitting of the simulation data by these equations was always carried out for t N 20 ps. We note, however, that in the framework of the approximations described in ref. [12] and here, tc should be considered as a crossover time-range more than a well-defined time. As it has been widely discussed in refs. [12,20], a variation of tc within a factor three or four does not affect significantly the results of the parameters β and τoR obtained by fitting, providing that the fitting range is always taken as t N tc. By means of the fitting procedure above-sketched, we will obtain values of the fitting parameters, β and τoR, corresponding to each case. We recall that the β parameter included in Eqs. (7) and (13), actually corresponds to the non-exponential parameter of the correlation function of the Rouse normal modes, φp(t), in the long wavelength (N/p) limit (low-p range). Thus, a proper validation of these equations
305
0.8
400 K
0.6 0.4 0.2 0
500 K
(b) 0
2
4
6
8
t(ns)
10
Fig. 1. ΦEE(t) corresponding to (a) pure PEO and (b) PEO in PMMA/PEO blend at the temperatures indicated in the figure. Continuous lines are fitting curves with Eq. (13) (see the text).
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-1
1
0.8
0.8
0.6
self
F
log[ (ns)]
(Q=0.3Å ,t)
1
0.6
1.5
(a)
350K
0.4
1
400K 0.2
0.4 0 10-2
10-1
100
101 t(ns)
0.2
0.5
350K 0 0
2
4
6
8 t(ns)
Fig. 2. Fself(Q, t) corresponding to pure PEO at Q = 0.3 Å−1 and the temperatures indicated in the figure. Continuous lines are fitting curves with Eq. (7) (see the text). The inset shows the same representation but in a logarithmic time-scale to highlight the short time range.
log[ (ns)]
400K
5 4 3
5. Discussion
2
The ‘actual values’ of β for both PEO and PEO in PMMA/PEO blend are included in Fig. 3 for the different temperatures considered in the MDsimulations. As it was discussed in ref. [12] and mentioned above, these values correspond to the long wavelength (N/p) limit of φp(t) and they were calculated as the average of the β(N/p) values in the long (N/p)-range (typically (N/p) N 10) where the PEO chains can be considered as ideal chains (i.e., no influence from local potentials). The error bars in the figure were estimated as the standard deviation of the values of β at each temperature in that (N/p)-range. As can be seen, the uncertainty is larger in the case of PEO in the blend than in the case of pure PEO. In particular, this uncertainty is rather large at the lowest temperature investigated (300 K), mainly due to the small decay of φp(t) at this low temperature. Fig. 3 also includes the values of β obtained by fitting either ΦEE(t) or Fself(Q, t) data of pure PEO and PEO in the PMMA/PEO blend by means of the generalized Rouse expressions described in this work (Eqs. (13) and (7) respectively). β-Values obtained from the fitting of 〈r2s (t)〉 and 〈R2CM(t)〉 corresponding to PEO in the blend and which were discussed in ref. [12], are also included in
1
1 0.9
PEO
0.8 0.7 0.6
PEO in PMMA/PEO 0.5 0.4 300
350
400
450
500
T(K) Fig. 3. Temperature dependence of the stretching parameter β corresponding to the Rouse mode correlators in the long (N/p)-range for PEO (empty symbols) and PEO in PMMA/PEO blend (filled symbols). (Circles) ‘actual β-values’; (triangles) β-values obtained by fitting Fself(Q, t) to Eq. (7); (inverted triangles) β-values obtained by fitting ΦEE(t) to Eq. (13); (filled squares) β-values obtained from the long-time range of 〈r2s (t)〉 for PEO in PMMA/ PEO blend; (filled diamonds) β-values estimated from the long-time range of 〈R2CM(t)〉 for PEO in PMMA/PEO blend. Lines through the ‘actual β-values’ are only to guide the eye.
(b)
0 1.8
2.2
2.6
3
3.4
1000 / T(K) Fig. 4. Arrhenius plot for τ^R and τoR. Empty symbols in (a) correspond to pure PEO; filled symbols in (b) correspond to PEO in PMMA/PEO blend. (Circles) τ^R from the fitting of φp = 1(t) by a stretched exponential; (squares) τoR obtained from τ^R by Eq. (6); (triangles) τoR obtained by fitting ΦEE(t) to Eq. (13); (diamonds) τoR obtained by fitting Fself(Q, t) to Eq. (7). Dashed line in (a) and continuous line in (b) are Arrhenius fits delivering E a ≈ 4 kcal/mol and E a ≈ 5 kcal/mol respectively. The dashed line in (a) is also plotted in (b) for comparison.
Fig. 3 for comparison. First of all, it is worthy of remark the rather good agreement – within the uncertainties involved in the different data – among the values of β obtained by fitting different dynamic magnitudes by means of the corresponding generalized expressions. Moreover, the agreement among these β-values and the ‘actual-ones’ is remarkable, taking into account the level of uncertainty of the actual values (see the error bars) and the approximations involved in the calculation of the generalized expressions for the different dynamic magnitudes. This agreement gives support to the approach used to deduce the generalized expressions of ΦEE(t) and Fself(Q, t). Now we can discuss the values of τoR included in Fig. 4. First of all, in general, there is a good agreement between the values of τoR obtained by fitting, either ΦEE(t) or Fself(Q, t) data, by means of the corresponding generalized expressions (Eqs. (13) and (7) respectively). Moreover, these values also agree with those directly obtained from τ^R by means of Eq. (6). These results again give support to the generalized expressions for the dynamic magnitudes deduced in this work and in ref. [12]. On the other hand, we note that in this framework, the difference between the values of τ^R and τoR – and its temperature dependence – is mainly due to the values of β and its temperature dependence as well. In the case of pure PEO, as β (≈0.9) is close to one and hardly dependent of temperature in the range here discussed, τoR values result to be rather close to τ^R. However, this is not the case for PEO in the PMMA/PEO blend where the difference between τ^R and τoR dramatically increases as the temperature (and β) decreases. An Arrhenius fit of τoR in the temperature range here considered (see dashed line in Fig. 4(a)) delivers for pure PEO an activation energy Ea ≈ 4 kcal/mol. In the case of PEO in the blend, although the absolute values of τoR are close to those corresponding to pure
J. Colmenero / Journal of Non-Crystalline Solids 407 (2015) 302–308
PEO, the activation energy is slightly higher: Ea ≈ 5 kcal/mol (see continuous line versus dashed line in Fig. 4(b)). We note that in the formalism here presented, τoR means the relaxation time corresponding to pure Rouse behavior. Thereby, we should expect for PEO in the blends values close to those of pure PEO but with a slightly higher activation energy (higher friction coefficient) due to the presence of the less flexible PMMA chains. This is exactly what we found. Conversely, as it has been mentioned in the Introduction section and widely discussed in refs. [10,12], the actual values of β, corresponding to PEO in the blend strongly depend on temperature. In the case of pure PEO the exponentiality of φp(t) is less marked (β ≈ 0.9) and hardly dependent on temperature, at least in the relatively high temperature range here explored. This, in principle, different behavior can be rationalized in the light of the ideas proposed in ref. [4] and that we have commented in the Introduction section. In that framework, the nonexponential character of the long wavelength Rouse mode relaxation – which is reflected in the values of β – is due to the influence of the density fluctuations on the chain dynamics and it will be higher as soon as the ratio τ^R =τ α decreases. We recall that τα is the characteristic time of the relevant density fluctuations, i.e., of the α-process in the pure homopolymer and of the effective α-process of the slow component (PMMA in our case) in the blend. Therefore, within this unified scenario, the relevant parameter controlling the non-exponential character of φp(t) in the long wavelength (N/p) limit results to be the ratio τ^R =τ α instead of being directly the temperature. For a given polymer, this would allow comparing data of β corresponding to the homopolymer case and to the case of this polymer in different blends. In order to test these ideas with the β-values of PEO and PEO in the PMMA/PEO blend, we need to estimate from the simulations the values of τα (density fluctuations) in each case. To do that, we have considered τα as the time scale of the corresponding incoherent scattering function at Q ≈ 1 Å−1. We have followed this criterium because it is well documented that relaxation times from incoherent neutron scattering data at this Q-range use to match the α-relaxation times determined by spectroscopic methods like, for instance, dielectric spectroscopy (see, e.g. [30]). We recall that in the case of PEO in the PMMA/PEO blend, the relevant value of τα is that corresponding to the α-process of PMMA in the blend. Taking into account the values of τα so obtained, together with those reported for τ^R in Fig. 4 for pure PEO and PEO in the blend, we have obtained the ratio τ^R =τ α . Now, the actual values of β shown in Fig. 3 are represented in Fig. 5 as a function of the parameter Δ ¼ log ðτ^R =τα Þ. First, we realize that the values of Δ, corresponding to the pure PEO at different
1 0.9 0.8 0.7 0.6 0.5 -3
-2
-1
0
1
2
3
4
Fig. 5. β as a function of the parameter Δ ¼ log ðτ^R =τ α Þ (see the text). (Filled circles) values correspond to PEO in PMMA/PEO blend; (filled diamonds) values correspond to pure PEO. Empty symbols correspond to data from ‘bead-spring’ model systems: (circles) homopolymer in the blend; (diamonds) pure homopolymer. Error bars in β are those included in Fig. 3. Error bars in Δ mainly reflect the uncertainties in calculating the values of τα, in particular, in the case of the τα of PMMA in the blend. Dashed line through filled symbols is a sigmoidal fit to guide the eye.
307
temperatures, are almost identical and rather high (Δ ≈ 3.5). This would explain why the corresponding β-values are close to one and hardly dependent on temperature. Moreover, the data of β for PEO and PEO in the blend seem to follow a unified trend in terms of the Δparameter. It is worthy of remark that data corresponding to a generic fully flexible ‘bead-spring’ model for a homopolymer and for the same polymer in an asymmetric blend [9] also seems to follow the same qualitative trend taking into account the simplicity of this model. In conclusion, we can say that the data reported here also give support to the scenario that has been recently proposed in ref. [4]. Finally, as it has been mentioned in Section 2, the Eq. (6), deduced in ref. [12] in the framework of the formalism and the approximations widely described in Section 2 of this work, looks exactly like the basic equation of the ‘coupling model’ (CM) developed a long time ago by Ngai for the crossover from non-cooperative to cooperative relaxation processes (CM-terminology) in general [15–17]. In fact, in a recent paper Ngai and Capaccioli have also applied the CM to chain dynamics of PEO in asymmetric blends [18] although using a very large value of tc (tc ≈ 2 ns). This in fact implies that these authors are assuming pure Rouse behavior of PEO in the blend at times shorter than about 2 ns. However, we note that the MD-simulation data of PEO in PMMA/PEO blend [10,12] clearly indicate a non-exponential behavior of φp(t) in the long wavelength (N/p) range (see Fig. 3), thereby invalidating the assumption of ideal Rouse behavior in such a time regime. However, it is worthy of remark that the law τ^p ∝ ðN=pÞx with x = 2/β remains valid independently of the value of the crossover time tc. In any case, readers interested in the debate about the actual value of tc for this problem are referred to the two recent (‘Comment’ and ‘Reply’) papers [31, 32]. The fact that the CM-equation can be deduced from Eqs. (2) and (3), suggests that, at least in this particular case, the CM seems to be a consequence of the GLE formalism and the approximations made here, in particular, the pseudo-Markovian approximation. This point of view has been criticized by Ngai and Capaccioli in a comment [19] to our previous paper [12] (see also the accompanying reply [20] for this question). They claim that, in fact, we are not solving Eqs. (2) and (3) in the usual way. As it is clearly recognized in ref. [12] and in Section 2 of this manuscript, to solve these equations in the standard way would imply to construct a microscopic theory for the memory function Γ(t), something that is beyond of this approach. However, we note that Eq. (6) is nothing more than a formal relationship between τ^p , τop and β, which in the formalism developed here and in refs. [12,21], is independent of the particularities of the memory function, providing the time decay of this function is given by the power law Γ(t) ∝ t−β. Therefore, our procedure is just equivalent to solve Eq. (3) by using this type of memory function. Then we could conclude that Eq. (6) – the basic equation of the ‘coupling model’ – is obtained in our approach as a consequence of the pseudo-Markovian approximation – which allows to obtain Eqs. (3) from (2) – and of a power law decay of the memory function. On the other hand, we note that, as it has been discussed in Section 2, in our framework Eq. (6) is an asymptotic equation only valid for the long wavelength (N/p) limit, where we can consider the pseudo-Markovian as a good approximation, and where β does not depend on (N/p) as well. Certainly this is not the case for lower values of (N/p) [9,10,20]. We wonder whether or not this asymptotic character of Eq. (6) could be incorporated in a direct CM-deduction of this expression, which in principle should work for any value of (N/p). 6. Summary and conclusions Magnitudes probing polymer chain dynamics as, for instance, incoherent and coherent neutron scattering functions or dielectric ‘normal mode relaxation’, are constructed in terms of exponential Rouse mode correlators. The reason for that is that the Rouse model fully neglects the time correlation of the random forces acting on a polymer chain segment. However, experimental and MD-simulations as well clearly show
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that Rouse mode relaxation use to deviate from the exponential behavior even in the long wavelength (N/p) range where polymer chains can be considered as ideals (i.e., no influence from local potentials). These deviations have been observed in homopolymers at low temperature approaching the glass-transition of the system and, in particular, when we follow the chain dynamics of the fast component of the socalled dynamically asymmetric blends. Based on these results it has been proposed that deviations from exponentiality are in fact due to the effect of the density fluctuations in the system at inter-macromolecular level (the so-called α-relaxation) on chain dynamics. In the framework of the generalized Langevin equation (GLE) formalism, the correlation between the random forces acting on a polymer chain can be introduced as a memory kernel in the Langevin equation for the time evolution of the Rouse mode correlators. This term effectively produces an extra friction non-local in time and as consequence, a non-exponential behavior of the mode relaxation. Inspired by this formalism and considering some approximations, we have proposed a procedure for generalizing the Rouse expressions of the typical magnitudes probing chain dynamics, which takes into account a stretched exponential decay for the Rouse mode relaxations. Here we have applied this procedure to obtain an expression for the End-to-End correlation function, which in the frequency domain give rises to the well known ‘normal mode relaxation’. By means of fully atomistic MD-simulations of pure PEO and PEO in a PMMA/PEO dynamically asymmetric blend, we have checked the generalized Rouse expressions obtained here and in a previous work. The simulation data can be nicely described by the proposed expressions for the chain dynamics magnitudes, and give support to the unified scenario recently proposed for explaining the non-exponential relaxation of the Rouse modes in polymer systems in terms of the effect of density fluctuations (α-process) on chain dynamics. Acknowledgments The author would like to thank M. Brodeck, F. Alvarez and A. J. Moreno for providing data of MD-simulations previously published. He also acknowledges the helpful discussions with A. Alegría, A. J. Moreno, A. Arbe and S. Arrese-Igor. Support from the projects IT-654-13 (GV) and
MAT2012-31088 (Ministerio de Economia y Competitividad) is also acknowledged.
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