Comment pubs.acs.org/Macromolecules
Reply to “Comment on ‘A Generalized Rouse Incoherent Scattering Function for Chain Dynamics of Unentangled Polymers in Dynamically Asymmetric Blends’” J. Colmenero* Centro de Física de Materiales (CSIC-UPV/EHU) and Donostia International Physics Center, Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
Macromolecules 2013, 46 (13), 5363−5370. DOI: 10.1021/ma400309c Macromolecules 2013, 46, DOI: 10.1021/ma401732c
T
As it was stated from the Introduction in ref 1, the approach followed for deducing eq 1 (also in the above-mentioned paper5) was based on the combination of a simplified version of the generalized Langevin equation (GLE) formalism and the phenomenological result of the nonexponential decay of the Rouse mode correlators for the fast component (B in the following) in asymmetric A/B polymer blends. The procedure started from a simplified GLE for the Rouse mode variable X⃗ p, which in fact was proposed in another context by Schweizer.6 This equation transforms into an integro differential equation for the corresponding Rouse correlator, Cp(t) = ⟨X⃗ p(t)X⃗ p(0)⟩
he main goal of the work described in the paper published by Colmenero in Macromolecules1 was to find analytical expressions for the mean-squared-displacement for a polymer segment ⟨r2(t)⟩ and for the chain center of mass ⟨RCM2(t)⟩ and the corresponding incoherent neutron scattering functions Fs(Q,t)which generalized those well established in the case of the Rouse model2 and which could be used in the case of unentangled asymmetric polymer blends. In their comment3 on the paper,1 the authors discussed neither the expressions found for ⟨r2(t)⟩, ⟨RCM2(t)⟩, or Fs(Q,t) nor the consistency of the results obtained by fitting with these expressions the molecular dynamics simulations data available4 for poly(ethylene oxide) (PEO) in the asymmetric blend PEO/PMMA (PMMA: poly(methyl methacrylate). Ngai and Capaccioli focus the discussion on the question of the deduction made of the correlation previously found5 between the wavelength (N/p) dependence of the relaxation time τ̂p corresponding to the Rouse correlator Φp(t) and the nonexponentiality of Φp(t), measured by the stretched exponential parameter β: ⎛ N ⎞x τp̂ ∝ ⎜ ⎟ , ⎝ p⎠
with x =
2 β
dCp(t ) dt
1 ξo
∫0
t
dt ′ Γ(t − t ′)
dCp(t ′) dt ′
=−
Cp(t ) τpo
(2)
Γ(t) is a memory function and ξo and τop are respectively the constant friction coefficient and the relaxation time of the Rouse correlation corresponding to the case of Γ(t) decaying to zero at microscopic times, i. e., the case of pure Rouse behavior. As it was clearly stated in ref.,1 a first approximation was to consider Γ(t) independent of the length scale (wavelength (N/ p)) due to the fact that we were only interested in the long wavelength (N/p) limit. Moreover, based on the arguments widely discussed in ref.,1 a second step was to consider a ’pseudo-Markovian’ approximation and to replace the convolution integral of eq 2 by a time local product. Under such approximations we arrived to the equation mentioned by Ngai and Capaccioli in their comment:
(1)
Here N is the number of segments in the polymer chain and p is the Rouse mode number; p = 1, ..., N − 1. Although a detailed deduction of expression 1 was included in ref 1, it is worthy of remark that this expression was first deduced in the same theoretical framework in a previous publication5 for the particular case of p = 1, i.e., for the so-called Rouse time. In this way, the comment of Ngai and Capaccioli appears to be a comment on our previous publication5 as well, and not only on ref 1. The main point raised by Ngai and Capaccioli in their comment is that the procedure used to deduce the expression 1 in ref 1 (and obviously also in ref 5) “introduces no physics” and thereby this translates into some unavoidable ambiguity in the result obtained. These authors claim that, in contrast, deriving expression 1 by a direct application of the “coupling model” (CM) does not involve any ambiguity because the “CM is based on physical principles and constant tc, which corresponds to the onset of chaos in phase space”. Let us sketch here the procedure followed by us in ref 1 to deduce expression 1, and the approximations involved, in order to clarify some of the particular questions raised by Ngai and Capaccioli. © XXXX American Chemical Society
+
Φ Bp (t ) =
⎡ ξ = exp⎢ − oo Cp(0) ⎢⎣ τp Cp(t )
∫0
t
⎤ dt ′ ⎥ ξo + ξ(t ′) ⎥⎦
(3)
where ξ(t) is not “an unspecified time-dependent effective friction coefficient” as Ngai and Capaccioli mentioned but it has a clear definition in terms of the memory function Γ(t): ξ(t) = ∫ t0 dt′ Γ(t′). It is noteworthy that as in our approximation Γ(t) does not depend on (N/p), ξ(t) results in it also being independent of (N/p). It is also important to emphasize that eq 3 is a consequence of the GLE, eq 2, and the approximations Received: August 28, 2013 Revised: September 6, 2013 Accepted: September 13, 2013
A
dx.doi.org/10.1021/ma4017983 | Macromolecules XXXX, XXX, XXX−XXX
Macromolecules
Comment
indicate that a variation of tc within a factor three does not affect significantly the results obtained by fitting for instance Fs(Q,t) or ⟨r2(t)⟩ to the simulation data of PEO/PMMA. As Ngai and Capaccioli mention, eq 5 deduced in our framework, “looks exactly like the signature equation of the CM”. In fact it can be obtained by a direct application of the CM in the usual way: assuming that ΦBp (t) = exp(−t/τop) at t ≤ tc and ΦBp (t) = exp[−(t/τ̂)β] at t ≥ tc and considering the continuity of ΦBp (t) at t = tc. Ngai and Capaccioli claim that by using directly the CM any ambiguity is removed because “as tc corresponds to the onset of chaos in phase space it is temperature insensitive”. We are not experts in either the CM or in the chaos theory. However, we could not find in the references suggested by Ngai and Capaccioli any clear microscopic proof of the temperature independence of tc beyond saying that “its magnitude is determined by the interaction”. In any case, as it has been discussed above, at least for the problem considered here tc ≈ constant seems to be a good approximation supported by both experiments and simulations. There is another important aspect concerning eq 1 that is not mentioned by Ngai and Capaccioli but which deserves a comment. The simulation results corresponding to PEO/ PMMA blend4 as well as those corresponding to a generic “bead−spring” model of asymmetric blends4,8 show that x = 2/ β in expression 1 should be considered only as an asymptotic law valid for the long wavelength (N/p) limit. For smaller values of (N/p), the β-values strongly depend on (N/p),4,8 thereby implying strong deviations from the law x = 2/β. This is shown as an example in Figure 1 constructed with the data
above-mentioned. In this way, it is formally correct within these approximations in the long (N/p) limit. To proceed further with eq 3, we have two possibilities. The first, and more rigorous one, would be to have a microscopic theory for Γ(t) which allows to formally solve the integral equation. As it was commented in refs 1 and 5, even for simple entangled polymers (not blends) this means a formidable theoretical effort, involving many approximations as well (see, e.g., refs 6 and 7). The method followed in ref 1 was just to take advantage of the knowledge (from both experiments and simulations, see, e.g., refs 4, 5, and 8) that ΦBp (t) in asymmetric blends is nonexponential and can be well-described by a stretched exponential function ΦBp (t) ≈ exp[−(t/τ̂)β] with β decreasing with decreasing temperature. The procedure followed in refs 1 and 5 can be synthesized in the following question: what should be the simplest expression of ξ(t) giving rise by solving eq 3 to a stretched exponential ΦBp (t), and assuring that in the Rouse limit (β = 1) ΦBp (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 ξ(t ) =
1−β ξo ⎛ t ⎞ ⎜ ⎟ β ⎝ tc ⎠
(4)
The time tc in eq 4 emerges in a natural way keeping the dimensionality of ξ(t) correct and setting the time scale for the nonexponential behavior of ΦBp (t). Now introducing eq 4 into eq 3, we obtain for the trivial case β = 1 (Rouse limit) ΦBp (t) = exp(−t/τop) and for β < 1 a stretched exponential function with
τp̂ = (τpo)1/ β tc(β − 1)/ β
(5)
As τop corresponds to the pure Rouse case, τop ∝ (N/p)2 and thereby τ̂p ∝ (N/p)2/β; i.e., x = 2/β as in expression 1. Ngai and Capaccioli claim in their comment that the procedure followed to deduce eq 5 −which has been synthesized heredoes not ensure that tc is independent of (N/p). As consequence, the (N/p) dependence of τ̂p could be different from (N/p)2/β. This is not the case. As it is stated in ref 1 and mentioned above, our approach is based on a Γ(t) independent of (N/p). Therefore, ξ(t), i.e., its integral, is also independent of (N/p). Equation 4 shows that tc enters via ξ(t), thereby implying that in our approximation tc has to be independent of (N/p). We note that in ref 1 eq 4 was not included. Instead of we included the equivalent equation for Γ(t) (eq 6 in ref 1). Obviously, eq 4 can easily be obtained by integration of Γ(t). The other question posed by Ngai and Capaccioli is the possible temperature dependence of tc. As was already commented upon in ref 1, the procedure followed by us does not imply anything about a possible temperature dependence of tc. Equation 5 is a general equation encompassing also this possibility. However, as it was also discussed in ref.1 the available experimental and simulations results as well suggest that considering in practice a constant tc can be a good approximation. In their Comment, Ngai and Capaccioli suggest different hypothetical temperature dependences of tc and their consequences on the expected temperature dependence of τ̂p. However, there does not seem to be any evidence that any of these forms are supported by experimental facts or simulations. On the other hand, it is worthy of remark that, at least in the framework of the approximations described in ref 1 and sketched here, tc should be considered as a crossover time range more than a well-defined time. The results shown in ref 1
Figure 1. Wavelength (N/p) dependence of the ratio 2/β where β corresponds to the Rouse mode correlators of the fast component (B) in a generic “bead−spring” model of asymmetric polymer blends. The data (reported in ref 4) were obtained by MD-simulations at three different temperatures in units of ε/KB, being ε the strength of the inter “bead−bead” interaction and KB the Boltzmann constant: T = 1 (filled diamonds); T = 0.6 (empty squares); T = 0.5 (filled circles). The dashed lines mark the x-values corresponding to the different temperatures, giving the (N/p)-dependence of the relaxation times τ̂p of the Rouse mode correlators: τ̂p ∝ (N/p)x. The (N/p) range covered in the figure corresponds to the regime used in ref 4 to determine the x-values from τ̂p(N/p).
reported in ref 4 for the above-mentioned generic “bead− spring” model. Only in the asymptotic long (N/p) limit do the values of 2/β approach the x-values independently determined at the different temperatures (dashed lines in Figure 1). Although the deviations are stronger at lower temperatures, it is noteworthy that they also take place at high temperature (T = B
dx.doi.org/10.1021/ma4017983 | Macromolecules XXXX, XXX, XXX−XXX
Macromolecules
Comment
1) where the system almost follow the Rouse behavior (x = 2.2). This asymptotic character of x = 2/β in eq 1 is well captured in the procedure followed in ref 1 to deduce this equation. This procedure was based on approximations valid for the long (N/p) limit as it was clearly stated in ref 1 and here repeated. We wonder whether or not this asymptotic character of eq 1 could be incorporated in a direct CM deduction of this expression, which in principle should work for any (N/p) value. Ngai and Capaccioli conclude that the procedure followed in ref 1 to deduce eq 1 “introduces no old or new physics...”, in contrast of the CM, “which is based on physical principle and constant tc”. In fact, we believe the opposite to be true. The fact that the “CM signature equation” (eq 5) 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, in particular, the pseudo-Markovian approximation and the independence of the memory term on the wavelength (N/p). To check whether or not this idea can be generalized to other situations will be the subject of future work.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: (J.C.)
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS The author expresses thanks for support from Projects IT-65413 (GV) and MAT2012-31088. REFERENCES
(1) Colmenero, J. Macromolecules 2013, 46, 5363. (2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press:Oxford, 1986. (3) Ngai, K. L.; Capaccioli, S. Macromolecules 2013, 46, DOI: 10.1021/ma401732c . (4) Brodeck, M.; Alvarez, F.; Moreno, A. J.; Colmenero, J.; Richter, D. Macromolecules 2010, 43, 3036. (5) Arrese-Igor, S.; Alegría, A.; Moreno, A. J.; Colmenero, J. Soft Matter 2012, 8, 3739. (6) Schweizer, K. S. J. Chem. Phys. 1989, 91, 5802. (7) Fatkullin, N. F.; Kimmich, R.; Kroutieva, M. J. Exp. Theor. Phys. 2000, 91, 150. (8) Moreno, A. J.; Colmenero, J. Phys. Rev. Lett. 2008, 100, 126001.
C
dx.doi.org/10.1021/ma4017983 | Macromolecules XXXX, XXX, XXX−XXX
