Viscoelastic and Dielectric Relaxation of Reptating Type-A Chains

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Viscoelastic and Dielectric Relaxation of Reptating Type-A Chains Affected by Reversible Head-to-Head Association and Dissociation Hiroshi Watanabe* and Yumi Matsumiya Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan

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Youngdon Kwon School of Chemical Engineering, Sungkyunkwan University, 300 Cheoncheon-dong, Jangan-gu, Suwon, Gyeonggi-do 440-746, Korea S Supporting Information *

ABSTRACT: For entangled linear polymer having type A dipoles and undergoing head-to-head association and dissociation reaction, viscoelastic and dielectric behavior is theoretically analyzed on the basis of the reptation dynamics combined with the reaction kinetics. Specifically, for the dissociated unimer and associated dimer (indexed with j = 1 and 2, respectively), the normalized complex modulus gj*(ω) and the normalized complex dielectric permittivity ε̃j*(ω) are analytically calculated via eigenfunction expansion of the orientational anisotropy and orientational memory defined in terms of the bond vectors u of entanglement segments. The reaction activates mutual conformational transfer between the unimer and dimer. Multiple coupling occurs for the anisotropy decay modes of the unimer and dimer due to this transfer, and the viscoelastic g1* and g2* of the unimer and dimer, respectively, exhibit considerably retarded and accelerated relaxation compared to the pure reptation case. In contrast, the memory decay modes of the unimer and dimer are only pairwisely coupled, so that the reaction-induced acceleration and retardation for the dielectric ε̃1* and ε̃2* are much weaker than those seen for the viscoelastic g1* and g2*. The orientational anisotropy is the tensorial, second-moment average of u associated with no cancellation in the conformational transfer, whereas the orientational memory is the vectorial, first-moment average accompanied by partial cancellation, which results in the difference between gj* and ε̃j*. This difference between gj* and ε̃j* is noted also for the associating/dissociating Rouse chains. Nevertheless, the reaction-induced retardation of the viscoelastic relaxation is stronger for the reptating unimer than for the Rouse unimer, whereas the reaction-induced acceleration is similar, in magnitude, for the reptating dimer and Rouse dimer. These features of gj* of the unimer and dimer are discussed in relation to the motional coherence along the chain backbone being present and absent in the reptation and Rouse dynamics. their ends.6 In melts, the terminal relaxation of telechelic polymers and/or end-associative polymers is further broadened, partly due to constrained relaxation of entanglement occurring through reorganization of the associative network.8−11 Another example is found for ionomers having strongly associating ionic groups along the chain backbone.12−15 The ionomers also exhibit widely separated time scales for the local relaxation (between ionic groups) and the dissociation of the ionic groups, which results in two-step relaxation activated by fast chain motion and slow dissociation. Sticky chain models16,17 may apply to such ionomers.

1. INTRODUCTION Associative polymers exhibit a wide variety in their dynamics and have been attracting recent research interest.1−20 For example, telechelic polymers in semidilute solutions form transient networks dynamically crosslinked at the polymer chain ends, and their terminal relaxation, being much slower than the local relaxation between the crosslinks, is well described as single-Maxwellian behavior.1−5 This fact has been related to a wide separation of the time scales of the local chain motion and dissociation of dynamic crosslinks. Nevertheless, detailed analysis6 suggests that the single-Maxwellian behavior reflects dissociation of junctions in a superbridge (linear sequence of pairwisely end-associated telechelic chains); in fact, broadened terminal relaxation is noted in relatively concentrated solutions where the superbridge is a minor component and most of chains have multiple association at © XXXX American Chemical Society

Received: April 2, 2018 Revised: July 22, 2018

A

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Considering the above point, we have conducted theoretical analysis for competition of the reptation dynamics and the reaction kinetics with the method developed in the previous study, the eigenfunction expansion of the basic conformational functions underlying the viscoelastic and dielectric relaxation. It turned out that the associated dimer and dissociated unimer respectively exhibit acceleration and retardation in their viscoelastic relaxation compared to the pure reptation case. Qualitatively similar effects of reaction were found also for the dielectric relaxation of the dimer (having symmetrically inverted dipoles) and the unimer, but the effects were much weaker than those for the viscoelastic relaxation because of the difference in the underlying conformational functions (the first- and second-moment averages of conformation for the dielectric and viscoelastic relaxation, respectively). Details of these findings are presented/discussed in this article. We also compared the viscoelastic behavior of the Rouse unimer/dimer undergoing the head-to-head association/ dissociation with the behavior of the reptating unimer/dimer. It turned out that the reaction-induced retardation of the viscoelastic relaxation is stronger for the reptating unimer than for the Rouse unimer whereas the reaction-induced acceleration is similar, in magnitude, for the reptating dimer and Rouse dimer. These features of the viscoelastic relaxation of the unimer and dimer are discussed in relation to the motional coherence along the chain backbone being present and absent in the reptation and Rouse dynamics, respectively.

The understanding of associative polymers explained above has significantly contributed to recent progress in polymer physics. Nevertheless, this understanding was limited for the case of dissociation much slower than the (local) chain motion, and it has been desired to investigate the other case where the dissociation and chain motion are occurring in a comparable time scale. Thus, we recently made theoretical analysis of the dynamics of monofunctionally head-to-head associating Rouse chains by combining the Rouse dynamics and the kinetics of association and dissociation reaction.18 It turned out that the viscoelastic relaxation of those chains is strongly influenced by the reaction-induced motional coupling between the associated dimer and dissociated unimer. In addition, the analysis was extended to the dielectric relaxation of the same Rouse chains but having so-called type A dipoles parallel along the chain backbone.19 The symmetrical inversion of the dipoles of the dimer plays a key role in the analysis. The analysis indicated lack of the reaction effect on the dielectric relaxation of the Rouse unimer/dimer system, which makes a sharp contrast to the significant reaction effect on the viscoelastic relaxation of the same system. Obviously, the dielectric and viscoelastic relaxation processes are governed by the same chain dynamics and the same reaction kinetics. Nevertheless, the difference emerges because of a difference in the averaging order of chain conformation in respective processes, the first-moment vectorial averaging in the dielectric process and the second-moment tensorial averaging in the viscoelastic process.19 Head-carboxylated cis-polyisoprene (PI-COOH), having type A dipoles aligned from the tail to the head and undergoing the head-to-head association through hydrogen bonding of the COOH groups, can be easily synthesized through anionic polymerization followed by termination with carbon dioxide, thereby serving as a good material for testing the above features deduced from the theoretical analysis. In fact, we synthesized a relatively low molecular weight PICOOH sample 20 and examined its viscoelastic 20 and dielectric19 behavior in an unentangled, semidilute solution in oligomeric butadiene (marginal solvent for PI). The theory described the data very well for both viscoelastic20 and dielectric19 relaxation, which in turn demonstrates the importance of the averaging moment of chain conformation in macroscopically measured physical quantities that relax through the chain motion. The above success of the analysis combining the unentangled Rouse dynamics and reaction kinetics encourages us to extend the analysis to entangled linear chains that undergo the head-to-head association/dissociation reaction. This analysis is experimentally testable for high molecular weight PI-COOH. The entanglement relaxation of linear chains is believed to be basically described by the reptation dynamics but is also contributed from additional mechanisms such as the contour length fluctuation (CLF) and constraint release (CR).21−24 However, as the first attempt, it is highly desired to rigorously analyze the competition just between the reptation dynamics and the reaction kinetics: Self-consistent incorporation of the CR mechanism into the reptation framework, often made by replacing the CR concept by a concept of dynamic tube dilation, has not been fully settled and is indeed a subject of current research25,26 (because CR can affect the CLF path). Thus, it is too early to include the CR and CLF mechanisms in the analysis of head-to-head associating chains.

2. MODEL AND ANALYSIS 2.1. System and Chain Dynamics. We focus on entangled linear chains at equilibrium, as illustrated in Figure 1. These chains undergo head-to-head association and

Figure 1. Illustration of associating unimers and dissociating dimer at equilibrium. Purple circle at Nth segment of the unimer indicates its associating head. Unimer has type A dipoles μ uninverted along the chain backbone (μ ∝ u1 for n1 = 0−N), whereas the dimer has onceinverted type A dipoles (μ ∝ u2 for n2 = 0−N and μ ∝ −u2 for n2 = N−2N).

dissociation and, at the same time, obey the reptation dynamics (in the linear viscoelastic regime). The dissociated unimer and associated dimer are composed of N and 2N entanglement segments, respectively, and have type A dipoles parallel along the chain backbone. Following the previous studies,18,19 we start with the simplest rate equations for the association/dissociation reaction. At equilibrium, the equations read d [P]eq = 0 = −k′[Peq ]2 + 2k[P2]eq dt B

(1a)

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules d k′ [P2]eq = 0 = −k[P2]eq + [Peq ]2 dt 2

contributes to the time evolution. This conformational transfer is described by a simple mapping rule:18,19

(1b)

for 0 < n2 < N and 0 < n1 < N :

Here, [P]eq and [P2]eq are the molar concentrations of the unimer and dimer at equilibrium, and k′ and k denote the reaction rate constants for association and dissociation reactions, respectively. The numerical factors of 2 (in eq 1a) and 1/2 (in eq 1b) reflect the stoichiometry, i.e., two unimers being created on dissociation of one dimer, and vice versa. In the rhs of eq 1a, the first term indicates the rate of consumption of unimers due only to their association, and at equilibrium this consumption is exactly compensated by the unimer creation on the dimer dissociation (second term). Similarly, in eq 1b, the first term in the rhs represents the rate of dimer consumption due only to its dissociation (being compensated by the dimer creation due to unimer association). Thus, the characteristic times of the unimer association and dimer dissociation at equilibrium, τas and τds associated with the first terms in eqs 1a and 1b, are specified as τas =

1 , k′[P]eq

τds =

1 k

u 2(n2 , t ) = u1(n1 , t ) with n1 = n2

for N < n2 < 2N and 0 < n1 < N : u 2(n2 , t ) = −u1(n1 , t ) with n1 = 2N − n2

[P2]eq [P]eq

2

=

τds 2[P]eq τas

(4b)

As noted for the thick green arrows shown in Figure 1, the dimer bond vector at N < n2 < 2N and the corresponding unimer bond vector are oriented in the opposite directions (cf. eq 4b) because of the head-to-head fashion of the unimer association. In contrast, the unimer and dimer bond vectors at n1 = n2 < N have the same orientation, as noted for the thick blue arrows in Figure 1. 2.2. Viscoelastic Relaxation. 2.2.1. Orientational Anisotropy and Relaxation Modulus. Linear viscoelastic relaxation of polymers reflects decay of strain-induced orientational anisotropy occurring through thermal motion of the chains. For a small step shear strain γ imposed on the system of unimer and dimer at a time t = 0, this anisotropy is described by the shear orientation function defined in terms of the bond vector u as Sj(nj,t) ≡ a−2⟨uj(nj,t)uj(nj,t)⟩xy (j = 1 and 2 for the unimer and dimer), where x and y denote the shear and sheargradient directions, ⟨uu⟩xy is the ensemble average of the shear component of the dyadic uu, and a2 = ⟨u2⟩eq (mean-square segment size at equilibrium). For the unimer and dimer that obey the reptation dynamics and undergo the association/ dissociation reaction, this function is conveniently expanded with respect to the sinusoidal reptation eigenfunctions:

(2)

Correspondingly, the equilibrium constant K is written as K≡

(4a)

(3)

Here, a comment needs to be made for the association and dissociation times τas and τds (eq 2) that are later utilized for description of the viscoelastic and dielectric relaxation processes of the polymer chains depicted in Figure 1. These processes are unequivocally related to conformational changes of the chains, and the association/dissociation reaction is just a trigger for local motion of the chain nearby the associating/ dissociating segments. This local motion further triggers the global conformational changes reflected in the viscoelastic and dielectric relaxation. Thus, in the analysis of relaxation processes presented in this paper, τas and τds (and the underlying reaction rate constants k′ and k) are not of purely chemical nature but contributed from the chain motion, and τas and τds cannot be shorter than the local relaxation time. In fact, for unentangled chains, this contribution of the chain motion has been confirmed from the data of activation energy of τds.15,20 Although τas and τds might change with the molecular weights of the unimer and dimer, these τ’s for the given unimer and dimer at equilibrium are undoubtedly independent of time and are “constant” in this sense. Thus, τas and τds for the given unimer and dimer should be interpreted as ef fective association and dissociation time “constants” that preaverage the local chain motion.20 In the followings, we keep this point in our mind to analyze the combined effect(s) of the reaction kinetics and chain dynamics on the relaxation of the unimer and dimer having the given entanglement numbers (N and 2N; cf. Figure 1). The chain dynamics is described as the time (t) evolution of the bond vector of the entanglement segments, uj(nj,t) (arrows in Figure 1), with nj being the segment index for the unimer (j = 1) and dimer (j = 2). For both unimer and dimer, this evolution is assumed to occur through the reptation dynamics. In addition, the conformation of associating unimers is transferred to the created dimer and the conformation of the dissociating dimer, to the created unimers, which also

for unimer: S1(n1 , t ) =

∑ Xp(t ) sinijjj

pπn1 yz zz (0 < n1 < N ) k N {

(5a)

i απn2 yz zz (0 < n2 < 2N ) k 2N {

(5b)

p≥1

for dimer: S2(n2 , t ) =

∑ Yα(t ) sinjjj α≥1

This sinusoidal form of the eigenfunction reflects very rapid orientational randomization at free ends of the chain (that is cast as the boundary condition, S = 0 at chain ends).21−23 At t = 0 where the small step strain γ is applied, the unimer and dimer are oriented uniformly along respective backbones, and their orientation functions are specified as Sj(nj,0) = S0 (constant being independent of nj). The corresponding initial condition for the amplitudes of the orientational relaxation modes, Xp and Yp appearing in eq 5, is given by22−24 4

for unimer: X p(0) = S0 pπ (p = odd), X p(0) = 0 (p = even)

(6a)

4

for dimer: Yα(0) = S0 απ (α = odd), Yα(0) = 0 (α = even)

(6b)

The initial orientational anisotropy, S0, is proportional to γ and specified as S0 = fγ/3, where f is a numerical factor that changes with the condition at t = 0: For example, f = 1 for the affine deformation and f = 4/5 after the Doi−Edwards local equilibration.21−23 However, this change does not affect the C

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules viscoelastic relaxation function calculated from Sj, as explained below. The shear stress per chain is given as an integral of 3kBTSj(nj,t) with respect to nj along the chain backbone,21−24 where kB and T denote the Boltzmann constant and absolute temperature, respectively. Thus, the normalized relaxation moduli of the unimer and dimer, g1(t) and g2(t) both satisfying gj(0) = 1, are expressed as 1 2 for unimer: g1(t ) = X p(t ) ( = 1 at t = 0) ∑ S0 p = odd ≥ 1 pπ

(cf. eqs 1 and 3), where the factor of 2 accounts for the stoichiometry that two unimers are created from one dimer. Similarly, in the right-hand side of eq 9b, the first and second terms indicate decay of S2 due to the dimer reptation and consumption (dissociation into unimers), and the third term indicates growth of S2 due to the dimer creation (unimer association) normalized to the dimer concentration [P2]eq. It would be informative to further examine a consequence of this normalization of the creation rate constant: If the reptative motion described by the first term in eq 9 is artificially quenched, the unimer and dimer should keep their initial, uniform orientation (i.e., Sj(nj,t) = S0 and Screated (nj,t) = S0 at t j > 0) to exhibit no relaxation (∂Sj/∂t = 0). The normalization gives the same rate constant to the second and third terms in eq 9, thereby guaranteeing this lack of relaxation in the absence of reptative motion. The creation terms in eq 9 can be specified from the conformational mapping rule, eq 4, as

(7a)

for dimer:

1 g 2 (t ) = S0

∑ α = odd ≥ 1

2 Yα(t ) ( = 1 at t = 0) απ (7b)

It should be emphasized that gj(t) is not contributed from the even modes of unimer/dimer because the sinusoidal eigenfunctions (cf. eq 5) of the even modes are antisymmetric with respect to the midpoint of the chain so that the integral of those eigenfunctions along the chain backbone vanishes irrespective of the t dependence of the mode amplitudes, Xp(t) and/or Yα(t). The relaxation modulus G(t) of the system as a whole is given as a weighed average of g1(t) and g2(t): G(t ) = G N{υ1g1(t ) + υ2g2(t )} (t > 0)

for unimer creation: 1 S1created(n1 , t ) = {[S2(n2 , t )]n2 = n1 2 + [S2(n2 , t )]n2 = 2N − n1 } (n1 = 0 − N )

(10)

for dimer creation:

(8)

S2created(n2 , t ) = [S1(n1 , t )]n1= n2 (n2 = 0 − N )

Here, GN is the (experimentally observed) entanglement plateau modulus, and υ1 and υ2 represent the volume fraction of the unimer and dimer, respectively. On first glance, the product GNgj(t) may appear to be proportional to 1/f because S0 = fγ/3. However, this is not the case because Xp(t) and Yα(t) contributing to gj(t) (cf. eq 7) are proportional to S0 through their initial condition, eq 6. Thus, gj(t) is independent of S0 (and of f). 2.2.2. Time Evolution Equation of Sj and Its Solution. The time evolution of the orientation function Sj, determining the t dependence of gj(t), is described as

(11a)

S2created(n2 , t ) = [S1(n1 , t )]n1= 2N − n2 (n2 = N − 2N ) (11b)

It should be noted that Sj is the tensorial, second-moment average of uj, and thus the minus sign in eq 4b is squared to give eqs 10 and 11. (This feature of Sj is in sharp contrast to the feature of the averaged bond vector ⟨uj(nj,t)⟩ underlying the dielectric relaxation, as discussed later in more detail.) Substituting eq 5 into eqs 9−11 and making the Fourier integral, we can recast eq 9 as the time evolution equation for the orientational relaxation mode amplitudes Xp and Yα defined in eq 5: unimer: for all p (= 1, 2, ...)

for unimer: 2 ∂ 1 ji k T zy ∂ S1(n1 , t ) = jjj B2 zzz 2 S1(n1 , t ) − S1(n1 , t ) ∂t τas k ζa N { ∂n1 1 created + S1 (n1 , t ) τas (9a)

ij p2 d 1 yz 1 + zzzzX p(t ) − X p(t ) = −jjjj j τrep‐uni τas z τas dt k {

∑

Yα(t )ξp , α

α = odd ≥ 1

(12)

for dimer: ij k T yz ∂ 2 1 ∂ S2(n2 , t ) = jjj B 2 zzz 2 S2(n2 , t ) − S2(n2 , t ) ∂t τ ds k 2ζa N { ∂n2 1 created + (n 2 , t ) S2 τds (9b)

with τrep‐uni =

ζa 2N3 (pure reptation time of unimer) π 2kBT

ξp , α = −

Here, ζ denotes the friction coefficient of the entanglement segment. In the right-hand side of eq 9a, the first and second terms indicate decay of S1 due to the unimer reptation and unimer consumption (association into dimer), respectively, and the third term indicates growth of S1 due to the unimer creation (dimer dissociation). The creation term (third term) should have the rate constant of 1/τds if this term is normalized to the equilibrium molar concentration of the dimer, [P2]eq. However, eq 9a is defined for the unimer having the equilibrium concentration [P]eq, so that the rate constant for the creation term is given by 1/τds × 2 × {[P2]eq/[P]eq} = 1/τas

=

2 N

∫0

N

i pπn yz ij απn yz zz sinjj zz dn sinjjj k N { k 2N {

i απ y sinjjj zzzcos pπ (α = odd) (4p − α )π k 2 {

(13)

8p

2

2

(14)

dimer: for α = odd

ji α 2 d 1 zyzz 1 + Yα(t ) = −jjjj zzYα(t ) − j τrep‐dim τds z τds dt k {

∑ Xp(t )ξp,α p≥1

(15)

with D

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules τrep‐dim =

8ζa 2N3 (pure reptation time of dimer) π 2kBT

B and C in the Supporting Information. X̃ p(s) and Ỹ α(s) thus calculated can be summarized as

(16)

for all p ≥ 1:

dimer: for α = even

ij α 2 d 1 yzzz + Yα(t ) = −jjjj zYα(t ) j τrep‐dim τds zz dt k {

1

∫0

Yα̃ (s) ≡

∫0

−

(21)

for α = odd: 2 2 4 α + 4Wx + 4(rd,dim/r1) ̃ Yα(s) = S0τrep‐dim π α(α 2 − A+2 )(α 2 − A −2 )

1

4rd,dim tanh(πWx /2) sin(απ /2) (πWx /2) (α 2 − A+2 )(α 2 − A −2 ) r1 32ra,dimrd,dim (πWx) coth(πWx) − π 2r1 −

×

∞

(18b)

For X̃ p(s) and Ỹ α(s), eqs 12 and 15 are rewritten as for all p (= 1, 2, ...): 2 ij 1 yz 1 jjs + p + zzzzX̃ p(s) = X p(0) − jj j τrep‐uni τas z τas k {

∑

l ra,dim [X ] o 1 [X ] 1o H [X ](s) = − o h 2 (s ) + h 3 (s ) m o πo r1 4r1 o n | yz o ra,dim 1 ijj o 1 o [X ] z j z + − 1 h ( s ) } j z 1 2j o j z o z 2r1 Wy cosh(πWy/2) o k { ~ −1 | l o ra,dimrd,dim tanh(πWy /2) [X ] o o o ×m − 1 h s ( ) } 1 o o o o 2r1 (πWy/2) o o n ~

(23)

l o 1 1 o (πP+) cot(πP+) m− 2 2 2o 2 P+ − P − o P+ + Wy 2/4 o n | (πP −) cot(πP −) o o + } 2 2 o P − + Wy /4 o o ~ 2r1 + (πWy/2) coth(πWy/2) ra,dimrd,dim

(24a)

with Yα̃ (s)ξp , α

h1[X ](s) =

α = odd ≥ 1

(19)

for α = odd (≥1):

2 ij 1 yzzz ̃ 1 jjs + α + jj zzYα(s) = Yα(0) − j z τrep‐dim τds τds k {

(22)

[Y]

The factors H and H appearing in eqs 21 and 22 serve as the kernels for solving eqs 19 and 20 (for details, see sections B and C in the Supporting Information) and are functions of the Laplace transformation variable s, as summarized below.

(18a)

Yα(t ) exp( −st ) dt

sin(απ /2) H [Y ] (α 2 − A+2 )(α 2 − A −2 ) [X]

∞

X p(t ) exp( −st ) dt

| l 4p2 + Wy 2 o 1 o o X p(0) o m } o S0 o o 4r1 (p2 − P+2)(p2 − P −2) o n ~

l oij 1 o p cos pπ 2 ra,dim 1 yzzz jj o + m j z j 2 o π r1 (p2 − P+2)(p2 − P −2) o Wy 2 zz ojk 4p { n | o o 1 1 1 1 o − 2 − } 2 o 4p cos(pπ ) Wy cosh(πWy/2) o o ~ πWy /2) [X ] tanh( ra,dimrd,dim p cos pπ +2 H r1 (p2 − P+2)(p2 − P −2) (πWy/2)

S0τrep − dim (17)

Equations 12−17, being combined with eqs 7 and 8, fully specify the viscoelastic relaxation of the system of the associating unimer and dissociating dimer at equilibrium. For the relaxation of the orientational anisotropy underlying the viscoelastic relaxation, we note that the odd modes of dimer and all modes of unimer are mutually coupled through the last terms in the right-hand side of eqs 12 and 15, whereas no coupling occurs for the even modes of the dimer (cf. eq 17). These features reflect the symmetry of Screated and Screated noted 2 1 in eqs 10 and 11: Namely, Screated of the created dimer is 2 symmetric with respect to the dimer midpoint (Screated (N + n,t) 2 = Screated (N − n,t) for any n = 0−N; cf. eq 11) so that each of 2 the dimer odd modes, having this symmetry, is coupled with all unimer modes mapped onto Screated . Correspondingly, each of 2 the unimer modes is coupled with all odd modes of the dimer because Screated of the created unimer, mapped from the dimer 1 modes (cf. eq 10), is neither symmetric nor antisymmetric with respect to the unimer midpoint (n1 = N/2) thereby containing the Fourier amplitudes of all unimer modes. In contrast, the dimer even modes are antisymmetric with respect to the dimer midpoint (n2 = N), thereby being coupled with none of the unimer modes contributing to the symmetric Screated . 2 Equation 17, being combined with eq 6b (initial condition), straightforwardly gives Yα(t) = 0 for α = even. Namely, the even modes of orientational relaxation of the dimer are not activated by the step strain at time 0 and are not coupled with the unimer modes (as noted in eq 17), so that the dimer even modes never grow during the relaxation process. In contrast, the dimer odd modes and all unimer modes are mutually coupled, as noted from eqs 12 and 15. This multiple coupling can be conveniently managed for the Laplace transformation of the mode amplitudes defined by X ̃ p (s ) ≡

X ̃ p (s ) =

∑ X̃p(s)ξp,α p≥1

(20)

where Xp(0) and Yα(0) are the initial values specified by eq 6. Equations 19 and 20 include the coupling terms for X̃ p(s) and Ỹ α(s) but can be solved analytically after a little long but standard calculation; for details of this calculation, see sections

h2[X ](s) =

| l tan(πP /2) o tan(πP −/2) o 1 π2 + o o − m } 2 2o o o 8 P+ − P − o ( P /2) ( P /2) π π + − n ~

(24b) E

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Macromolecules h3[X ](s) =

−

P −2

l o tan(πP+/2) π2 1 1 o m 2 2 2o 2 8 P+ − P − o P + Wy /4 (πP+/2) o n + | tan(πP −/2) o o 1 } o + Wy 2/4 (πP −/2) o o ~ tanh(πWy /4) r1

π2 − 2 ra,dimrd,dim

(πWy/4)

l o 16 jij 2 rd,dim zyz [Y ] o 4 [Y ] H [Y ](s) = m o π h2 (s) + π jjjWx + r zzzh3 (s) o 1 { k n | 4rd,dim tanh(πWx /2) [Y ] o o h1 (s)} − o o r1 (πWx /2) ~ |−1 l 32rd,dimra,dim o o o [Y ] 1 ( W ) coth( W ) h ( s ) ×o + π π m } x x 1 2 o o o o r π 1 n ~

with

P+(s) =

P −(s) =

1 2

2

l o o −(Wx 2(s) + Wy 2(s)/4) m o o n −

1/2 ra,dimrd,dim | o o (Wx (s) − Wy (s)/4) + } o o r1 ~ 2

2

2

(29b) (25)

A+(s) = 2P+(s)

(30a)

A −(s) = 2P −(s)

(30b)

2.2.3. Calculation of Complex Modulus. As can be noted from eq 7, the Laplace-transformed g̃1(s) and g̃2(s) of the unimer and dimer are fully specified in terms of X̃ p(s) and Ỹ α(s) shown in the previous section (cf. eqs 21−26). After some calculation explained in the Supporting Information (section D), the expression of g̃1(s) and g̃2(s) can be obtained from eqs 21−26 as for unimer:

∑

g1̃ (s) =

p = odd ≥ 1

+

l o π 1 1 1 1 o h3[Y ](s) = m 2 2o 2 2 2 o 4 A+ − A − n A+ + 4Wx A+ cos(A+π /2) | | o o o r1 π l 1 1 1 o o o − − } m } 2 2 2 o o o ra,dimrd,dim o o 64 A − + 4Wx A − cos(A −π /2) o ~ n ~ 2 | l yz o o r1 1 ijj 1 o Wy zz + π o × m } j1 − z 2j o o j z o 2 2 cosh(πWx) { 16 o Wx k n ra,dimrd,dim ~ A+ A −

(26b)

with

ra,dim 4

τrep‐dim 2 X ̃ p (s ) = pπ S0 r1

f2[X ] (s) −

πra,dimrd,dim 2

{

f1[X ] (s)

f3[X ] (s)

}

l o W 2 yz o tan(πP+/2) ijj 1 o jj1 + y zzz m j o P+2 − P −2 o 4P+2 zz o (πP+/2) jk { n | 2 2 Wy yzzo o 1 ijj Wy yzz tan(πP −/2) ijjj z zzo j − } jj1 + o + 4 jjj P 2P 2 zzz (πP −/2) j 4P −2 zzo o + − { k {~ k

(31)

f1[X ] (s) =

(26c)

l tan(πP /2) 1 o 1 1 o + + 2 m 2 2o ( /2) π P P+ P − P+ − P − o P+2 + n o tan(πP −/2) 1 | 1 π2 o − + } 2o 2 o (πP −/2) P − ~ 4 P+ − P −2 tanh(πWy /2) tanh(πWy /4) l tan(πP+/2) o o × m o o (πWy/2)(πWy/4) n (πP+/2)

In eqs 21−26, τrep‑dim is the pure reptation time of the dimer (cf. eq 16), and the remaining parameters and factors are given by τrep‐dim τrep‐dim τrep‐dim rd,dim = , ra,dim = , r1 = ( = 8) τds τas τrep‐uni

f2[X ] (s) =

(27)

1/2

Wy(s) = {τrep‐dims + rd,dim}

2

(29a)

h2[Y ](s) =

1/2

1/2 ra,dimrd,dim | o o (Wx (s) − Wy (s)/4) + } o o r1 ~ 2

(24c)

l o tan(πA+ /2) π2 1 1 o m 2 2o 2 2 8 A+ − A − o (πA+ /2) + 4 W A + n x o tan(πA −/2) | 1 o − } 2 2 o 4Wx + A − (πA −/2) o ~ tanh(πWx) r1 π2 − 32 ra,dimrd,dim (πWx) (26a)

l ra,dim | o o τrep‐dim o Wx(s) = o s+ m } o o o o r r 1 ~ n 1

l o o −(Wx 2(s) + Wy 2(s)/4) m o o n +

h1[Y ](s) =

l o π 1 1 1 o m 2 2 2o 2 A 4 A+ − A − o cos( + A W 4 +π /2) x n + | o 1 1 o − } 2 2 o A − + 4Wx cos(A −π /2) o ~ r1 π 1 − 16 ra,dimrd,dim cosh(πWx)

1 2

(28a)

− (28b) F

(32a)

2

o tan(πP −/2) | o } o (πP −/2) o ~

(32b) DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules f3[X ] (s)

=

tanh(πWy /2)

for dimer:

∑

g2̃ (s) =

α = odd ≥ 1

−

l o tan(πP+/2) 1 o m 2 2o P+ − P − o n (πP+/2)

o tan(πP −/2) | o − }H [ X ] o (πP −/2) o ~ (πWy/2)

2rd,dim r1

Article

gj″(ω) = ω

16ra,dimrd,dim 2

π r1

(33)

with f1[Y ] (s)

| o o 4Wx 2 + 4(rd,dim/r1) tan(πA+ /2) l 1 o o 1 + m } 2 2 2 o o o o A+ − A − (πA+ /2) n A+ ~ 2 | l o o 4Wx + 4(rd,dim/r1) tan(πA −/2) o 1 o 1 − 2 + m } 2 2 o o o o A− A+ − A − (πA −/2) n ~ 4Wx 2 + 4(rd,dim/r1) + A+2 A −2 (34a) =

f2[Y ] (s) =

(34c)

The normalized relaxation moduli, g1(t) and g2(t), are obtained by inversely Laplace transforming g̃1(s) and g̃2(s) through the standard analysis of poles and residues. However, the resulting expression of g1(t) and g2(t) includes a very large number of viscoelastic relaxation modes (due to the motional coupling between the unimer and dimer) and is much more complicated even compared to eqs 31−34 and thus not shown here. Instead, we focus on the normalized complex moduli of the unimer and dimer being equivalent to g1(t) and g2(t): These complex moduli are defined as gj*(ω) = ω∫ ∞ 0 gj(t){sin ωt + icos ωt}dt with i = √−1 (j = 1, 2), where ω is the angular frequency. Thus, gj*(ω) is simply related to the Laplace transformation, g̃ j(s) = ∫ 0∞gj(t)exp(−st)dt, as gj*(ω) = iω[g̃j(s)]s=iω (j = 1, 2). Consequently, the storage and loss moduli (often measured in experiments), gj′(ω) and gj″(ω), are calculated from g̃j(s) as

∫0

∑p ≥ 1 IG , p

(36)

where τG,p and IG,p are the characteristic time and intensity of the pth viscoelastic relaxation mode. This average relaxation time, introduced by Graessley (and termed the “numberaverage” relaxation time in his review),27 coincides with a ratio of the viscosity to the entanglement plateau modulus; see section E of the Supporting Information for further details. From eq 35b, we note that Re{[g̃j(s)]s=iω→0} equals to {gj″(ω)/ ω}ω→0, and thus Re{[g̃j(s)]s=iω→0} coincides with the viscosity normalized by the plateau modulus, i.e., with ⟨τG⟩n itself. Thus, ⟨τG⟩n of the unimer and/or dimer can be straightforwardly calculated from X̃ p(s) and Ỹ α(s) at s = 0 (that determine [g̃j(s)]s=iω→0; cf. eqs 31−34). Fortunately, this calculation gives a relatively simple expression of ⟨τG⟩n, as explained in section E of the Supporting Information. The results, expressed in terms of parameters v = ra,dim/r1 , rd,dim, r1, and τrep‑dim, are summarized below. for unimer:

f3[Y ] (s) = (πWx) coth(πWx)H [Y ]

gj′(ω) = ω

∑p ≥ 1 IG , pτG , p

⟨τG⟩n =

l ij 1 tanh(πWx /2) o 1 1 o jj m jj 2 2 2 o o (πWx /2) cos(A+π /2) n A+ − A − k A+ | yz 1 1 1 o o zz + − } zz 2 2 2o cos( A /2) π A− A+ A − o − (34b) { ~

l ij 1 o 1 1 o jj ×m j 2 2 2 o j o A+ − A − A+ cos(A+π /2) k n | 1 1 1 o zyz o zz + 2 2 } − 2 o A − cos(A −π /2) z{ A+ A − o ~

gj(t ) cos ωt dt = ω Re{[gj̃ (s)]s = iω }

The factors appearing in eqs 32 and 34, Wx, Wy, P+, P−, A+, and A−, are functions of the Laplace transformation variable s (cf. eqs 28−30), and the kernels H[X] and H[Y] are also functions of s (cf. eqs 23−26). Considering the s dependence of g̃1(s) and g̃2(s) occurring through these factors and kernels, we can straightforwardly convert g̃j(s) into gj′(ω) and gj″(ω) as shown in eq 35. These gj′(ω) and gj″(ω), shown later in Figures 2 and 4, clearly demonstrate the effect(s) of the association/dissociation reaction on the viscoelastic relaxation of the unimer and dimer. 2.2.4. Terminal Viscoelastic Relaxation Time. The unimer and dimer undergoing the reaction exhibit considerably sharp distribution of the terminal relaxation mode distribution, as shown later in Figures 2 and 4. Thus, the terminal relaxation time of the unimer and/or dimer can be evaluated as the firstmoment average relaxation time ⟨τG⟩n defined by

(32c)

| o f3[Y ] (s)o } o o ~

∞

(35b)

l o [Y ] 2 Yα̃ (s) = τrep‐dimo f (s ) m o o1 απ S0 n

f2[Y ] (s) −

∫0

τG

uni n

=

2 4v 2(4 − r1) π 2 (r1 v + rd,dim) + 12 r1(4v 2 + rd,dim) r1(4v 2 + rd,dim)2

| l o tanh(π 4v 2 + rd,dim /4) o o o o o ×m 1− } o o 2 o o o (π 4v + rd,dim /4) o o o n ~

τrep − dim

| l o C [X ] + C2[X ] + C3[X ] o o [X ] o +m 1− 1 L } o o [ ] X o o D n ~

(37)

with π 1 2 2 r1(4v + rd,dim) | l o tanh(π 4v 2 + rd,dim /4) o o o o o ×m − 1 } o o 2 o o o (π 4v + rd,dim /4) o o o n ~

C1[X ] = −

∞

gj(t )sin ωt dt = −ω Im{[gj̃ (s)]s = iω } (35a) G

(38a)

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

l o o π2 4 v2 1 o − m 2 o π rd,dim(4v + rd,dim) o 8 r o d,dim n | ij yzo 1 zzo o × jjjj1 − zz} j o cosh(π rd,dim /2) zo k {o ~

C2[X ] = −

C3[X ] =

C3[Y ] = −

| l o tanh(π 4v 2 + rd,dim /4) o o o o o tanh(π v/2) ×m − } o o 2 o (π v/2) o o (π 4v + rd,dim /4) o o o n ~

(38b)

l o o rd,dim π2 1 o D = m 2 2 o 2 rd,dim(4v + rd,dim) o 4v o o n | tanh(π 4v 2 + rd,dim /2) o o o + } o 2 o (π 4v + rd,dim /2) o o ~

coth(π 4v 2 + rd,dim /2) π 1 4 4v 2 + rd,dim (π 4v 2 + r d,dim /2)

| o o o + } o 2 o cosh(π 4v + rd,dim /2) o o ~ 1

π2 1 1 −2 2 2 4 4v + rd,dim (4v + rd,dim)2 | l o o o o 1 o o ×m − 1 } o o 2 o o o cosh(π 4v + rd,dim /2) o o o n ~

(38c)

| l o coth(π 4v 2 + rd,dim /2) o o o v2 o o 16 ×m + } o π 2 r (4v 2 + r o 2 o o o (π 4v + rd,dim /2) o d,dim d,dim) o o n ~ | l o o o 2v 2 1 o − 1 m } 2 o o o π cosh( r /2) rd,dim(4v + rd,dim) o o o d,dim n ~ | l 2 o o tanh(π 4v + rd,dim /4) o o o o ×m 1− } o o 2 o o o (π 4v + rd,dim /4) o o o n ~ dim n

τrep − dim

=

(39)

(40)

ij y π 2 rd,dim 1 jj1 + r1 v 2zzz jj z 2 rd,dim z{ 3 r1 4v + rd,dim k

ij y j1 − 4 zzz + jj 2 2j r1 zz{ (4v + rd,dim) k | l o tanh(π 4v 2 + rd,dim /2) o o o o o ×m } o1 − o 2 o o o (π 4v + rd,dim /2) o o o n ~

P1(t ) = μ

lr o o d,dim tanh(π v/2) −m + o o r1 (π v/2) n

+

C2[Y ] [Y ] D

l o o o π2 1 1 o = 2 − 2 m o 8 + 4v + rd,dim o 4v rd,dim o o n ij yz| o jj zzo o 1 zzo ×jjj1 − zz} o 2 jj o cosh(π 4v + rd,dim /2) zo k {o ~

∫0

u1(n1 , t ) dn1 N

2N

u 2( n 2 , t ) d n 2 −

∫N

| o u 2( n 2 , t ) d n 2 } o ~

(45a)

+

Here, μ denotes the magnitude of type A dipole per unit length of the segment bond vector. (In eq 45, we have adopted the continuous treatment to represent the summation over entanglement segments as the integral, ∫ N0 ...dn and ∫ 2N N ...dn.) At equilibrium with no external field, the segment bond vectors are isotropically oriented so that ⟨uj(nj)⟩eq = 0 and ⟨Pj⟩eq = 0 (j = 1, 2). In contrast, after application of a weak, constant electric field E over an infinitely long interval of time − ∞ < t < 0, the segments are oriented, on average, in the field direction that is hereafter specified as the y direction. This average orientation of the unimer segment, calculated from its Boltzmann distribution function that involves the elastic and electrostatic energies, is given by19

| C3[Y ] o o [Y ] L } o o ~

C1[Y ]

C2[Y ]

N

(45b)

(41)

with

∫0

l o P2(t ) = μm o n

rd,dim

C1[Y ]

(44)

The terminal viscoelastic relaxation time ⟨τG⟩n given above is later shown in Figure 8 and discussed in relation to the reaction-induced coupling of the relaxation modes of the unimer and dimer. 2.3. Dielectric Relaxation. 2.3.1. Orientational Memory and Dielectric Relaxation Function. As explained for Figure 1, the unimer is composed of N entanglement segments, and each segment has the type A dipoles parallel to its bond vector u1(n1,t). Thus, dielectrically, the head (n1 = N) and tail (n1 = 0) of the unimer are clearly distinguished. Correspondingly, the head-to-head associated dimer has the dipoles once inverted at the middle segment at n2 = N. The microscopic polarization P1(t) and P2(t) of such unimer and dimer at time t can be expressed in terms of uj(nj,t) as19,28

L[X ] =

τG

(43)

L[Y ] =

π tanh(π rd,dim /2) tanh(π rd,dim /4) 4rd,dim

for dimer:

(42c)

[Y ]

l o o 4v 2 4v 2 + rd,dim 1 ×o − m o o π cosh( r r rd,dim /2) o d,dim d,dim o n

D[X ] =

tanh(π 4v 2 + rd,dim /2) π2 1 2 r1(4v 2 + rd,dim) (π 4v 2 + r d,dim /2)

l rd,dim tanh(π v/2) | o o π2 = 1− m } 2 2 o o o ( v/2) π 8 r1 v (4v + rd,dim) o n ~

(42a)

for unimer: (42b)

ij 0 yz jj zz jj z 2 ⟨u1(n1 , 0)⟩ = jj Eμa /3kBT zzz with a 2 = u 2 jj zz j z k0 {

eq

(46) H

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

and P2(t) of all dimers in the system. Because of the association/dissociation reaction, the cross-correlation between the unimer and dimer, ⟨P1(t)·P2(0)⟩eq and ⟨P2(t)· P1(0)⟩eq, does not vanish, so that the GK expression of ε(t) requires us to calculate this cross-correlation as well as the autocorrelation of the unimer and dimer, ⟨P1(t)·P1(0)⟩eq and ⟨P2(t)·P2(0)⟩eq.19 This GK calculation gives the same ε(t) as obtained from eqs 50 and 51, but includes complicated analysis of the cross-correlation. For this reason, we start our calculation with eqs 49−51. 2.3.2. Time Evolution of Averaged Bond Vector. No electrostatic force is exerted on the unimer and dimer at t ≥ 0 (after removal of the electric field), and the random Brownian force vanishes on averaging. Thus, the time evolution equations of the averaged bond vectors of the unimer and dimer at t > 0 can be cast in a simple form shown below. for unimer:

For the dipole-inverted dimer, the corresponding average is specified as19 for dimer: ⟨u 2(n2 , 0)⟩0 < n2 < N = −⟨u 2(n2 , 0)⟩N < n2 < 2N = ⟨u1(n1 , 0)⟩

(47)

From eqs 45−47, the average polarization of the unimer and dimer in the y direction (electric field direction at t < 0) is given by ⟨P1, y(0)⟩ =

Na 2μ2 E (unimer) 3kBT

(48a)

⟨P2, y(0)⟩ =

2Na 2μ2 E (dimer) 3kBT

(48b)

∂⟨u1(n1 , t )⟩ ijj kBT yzz ∂ 2⟨u1(n1 , t )⟩ 1 = jj 2 zz − ⟨u1(n1 , t )⟩ 2 ∂t τ ζ ∂ a N n as 1 k { 1 created + ⟨u1(n1 , t )⟩ (0 ≤ n1 ≤ N ) τas (52a)

After removal of the electric field at t = 0, ⟨P1,y(t)⟩ and ⟨P2,y(t)⟩ decay with time through the reptative motion of the unimer and dimer affected by their association/dissociation reaction, thereby activating the dielectric relaxation. This decay is characterized by the normalized dielectric relaxation function Φj(t) (= 1 at t = 0) that describes the memory of the initial polarization (memory of the initial orientation of u): Φj(t ) =

⟨Pj , y(t )⟩ ⟨Pj , y(0)⟩

∂⟨u 2(n2 , t )⟩ ijj kBT yzz ∂ 2⟨u 2(n2 , t )⟩ 1 − ⟨u 2(n2 , t )⟩ = jj z 2 z 2 ∂t τ ζ ∂ 2 a N n ds 2 k { 1 + ⟨u 2(n2 , t )⟩created (0 ≤ n2 ≤ 2N ) τds (52b)

for dimer:

(j = 1 and 2 for unimer and dimer) (49)

The un-normalized dielectric relaxation function ε(t) of the system as a whole, which corresponds to the experimental dielectric data of a mixture of the unimer and dimer at equilibrium, is simply given as a sum weighed with their equilibrium number densities, ν1 and ν2:19 ε (t ) =

The parameters appearing in these equations are the same as those in eq 9. Equations 52a and 52b are formally identical to eqs 9a and 9b for the orientation function underlying the viscoelastic relaxation, and the meaning of the terms in the right-hand side (and the parameters therein) is the same as in eqs 9a and 9b: The first and second terms represent decay of ⟨uj(nj,t)⟩ due to reptation and consumption (unimer association in eq 52a and dimer dissociation in eq 52b), respectively. The third term indicates growth of ⟨uj(nj,t)⟩ due to creation (dimer dissociation in eq 52a and unimer association in eq 52b), with the rate constant being normalized to the equilibrium concentrations of the unimer and dimer as explained for eqs 9a and 9b. Despite this similarity, there is an essential difference between the creation terms in eqs 52 and 9. From the conformational mapping rule, eq 4, the creation term in eq 52 is specified as unimer creation (in eq 52a):

4π {ν1⟨P1, y(t )⟩ + ν2⟨P2, y(t )⟩} E

= Δε{w1Φ1(t ) + w2 Φ2(t )}

(50)

with Δε =

4πμ2 ν⟨R12⟩eq 3kBT

(51)

The factors w1 and w2 appearing in eq 50 are the equilibrium weight fractions of the unimer and dimer: w1 = [P1]eq/{[P1]eq + 2[P2]eq} and w2 = 2[P2]eq/{[P1]eq + 2[P2]eq} where [P1]eq and [P2]eq indicate the molar concentrations of the unimer and dimer at equilibrium (cf. eq 1). The dielectric relaxation intensity Δε (eq 51) is expressed in terms of the total number density of the unimer unit (that includes the two half fragments of the dimer), ν = ν1 + 2ν2, and the mean-square end-to-end distance of the unimer at equilibrium, ⟨R12⟩eq = Na2. (Equation 51 assumes a negligibly weak dipole−dipole interaction and includes no correction for the internal electric field strength.19 The numerical prefactor of 4π appearing in the first line of eq 50, converting the polarization into the MKS unit, is absorbed in Δε as shown in eq 51.) A comment needs to be made for the dielectric relaxation function ε(t) (eq 50). This function is frequently expressed in the Green−Kubo (GK) form,29−31 ε(t) = Δε⟨Ptotal(t)· Ptotal(0)⟩eq/⟨{Ptotal(0)}2⟩eq with Ptotal(t) being the microscopic polarization of the system at equilibrium (in the absence of the electric field): Ptotal(t) is given as a sum of P1(t) of all unimers

⟨u1(n1 , t )⟩created =

1 {⟨u 2(n2 , t )⟩n2 = n1 2

− ⟨u 2(n2 , t )⟩n2 = 2N − n1 } (0 ≤ n1 ≤ N )

(53)

dimer creation (in eq 52b): ⟨u 2(n2 , t )⟩created = ⟨u1(n1 , t )⟩n1= n2 for 0 ≤ n2 ≤ N

(54a)

⟨u 2(n2 , t )⟩created = −⟨u1(n1 , t )⟩n1= 2N − n2 for N ≤ n2 ≤ 2N (54b)

The minus sign in the mapping rule for N < n2 < 2N (eq 4b) is preserved in eqs 53 and 54b because ⟨uj(nj,t)⟩ is the vectorial, I

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules first-moment average of uj(nj,t). In contrast, the orientation function Sj(nj,t) is the tensorial, second-moment average of uj(nj,t), and the minus sign in eq 4b is squared to give a plus sign in eqs 10 and 11 for Screated (nj,t). This simple difference in j the creation terms for ⟨uj(nj,t)⟩ and Sj(nj,t) results in huge differences in the effects of the association/dissociation reaction on the dielectric and viscoelastic behavior, as discussed later in detail. 2.3.3. Calculation of Dielectric Relaxation Function and Dielectric Permittivity. At t ≥ 0, the electric field is removed and no external electrostatic force acts on the unimer and dimer, so that the y component of the averaged bond vector of the unimer and dimer, ⟨uj,y(nj,t)⟩, can be conveniently expanded with respect to the standard, sinusoidal reptation eigenfunctions: for unimer:

⟨u1, y(n1, t )⟩ =

for dimer:

⟨u 2, y(n2 , t )⟩ =

for unimer, p ≥ 1:

| l 2 o d 1o 1 o o p θp(t ) = −m + }θp(t ) + ψ2p(t ) o o τrep‐uni o dt τas o τas n ~ | l o d 1o 1 o o α2 ψα(t ) = −m + θα /2(t ) }ψα(t ) + o o τrep‐dim o dt τds o τds n ~

απn2 yz zz k 2N {

α≥1

(0 ≤ n2 ≤ 2N )

| l o d 1o o o α2 ψα(t ) = −m + }ψα(t ) o o τrep‐dim o dt τds o n ~

ψα(t ) = 0 for α = odd

(In the continuous treatment adopted in this study, the orientation at the chain ends is considered to be randomized very rapidly in the absence of the external force acting on the ends.22,23 Thus, the standard, sinusoidal reptation eigenfunctions are utilized in eq 55.) From eqs 46 and 47 specifying ⟨uj,y(nj,0)⟩ at time t = 0, the initial values of the expansion coefficients appearing in eq 55, θp and ψα, are obtained as19 for unimer:

θp̃ (s) = 0 for p = even

(60c)

and

2 2 4 p + (Wy + ra,dim)/4 ijj μa 2 yzz ̃ θp(s) = j zEτrep − dim πr1 p(p2 − P+2)(p2 − P −2) jjk 3kBT zz{

for p = odd (56b)

From eq 49 combined with eqs 45, 55, and 56, the normalized dielectric relaxation function Φj(t) is expressed as

for dimer:

Φ2(t ) =

1 ijj 3kBT yzz j z E jjk a 2μ zz{

∑

2 θp(t ) pπ

1 ijj 3kBT yzz 4 ψ (t ) j z ∑ E jjk 2a 2μ zz{ α = odd ≥ 1 απ 2α p = odd ≥ 1

(60b)

ψα̃ (s) = 0 for α = double‐even (α = 2q with q = even)

(56a)

li 2 o o j Eμa yzz 4 o o for α = 2q with q = odd zz o jjjj z ψα(0) = m o k 3kBT { qπ o o o o0 for other α n

Φ1(t ) =

(60a)

In contrast, the dimer even modes and the corresponding unimer modes are pairwisely coupled, as noted in eqs 58 and 59a. This pairwise feature of coupling, combined with the initial condition (eq 56), allows us to easily solve eqs 58 and 59a. The results, expressed for the Laplace transformation ∞ θ̃p(s) = ∫ ∞ 0 θp(t) exp(−st) dt and ψ̃ α(s) = ∫ 0 ψα(t) exp(−st) dt, are summarized as

for dimer:

for unimer:

(59b)

The dimer odd modes are coupled with none of the unimer modes (eq 59b) because the odd eigenfunctions of the dimer, sin(απn2/2N) with α = odd (cf. eq 55b), is symmetric with respect to the dimer midpoint n2 = N whereas the dimer creation term, ⟨u2(n2,t)⟩created (eq 54), is antisymmetric. Because of the initial condition eq 56b, eq 59b gives

(55a)

(55b)

l o ij Eμa 2 yz 4 o o jj z o jj 3k T zzz pπ for p = odd θp(0) = o m B { k o o o o o for p = even o n0

(59a)

for dimer, α = odd:

pπn1 yz zz k N {

∑ ψα(t ) sinijjj

(58)

for dimer, α = even:

∑ θp(t ) sinijjj p≥1

(0 ≤ n1 ≤ N )

The time evolution equations of the expansion coefficients θp and ψα are straightforwardly deduced from eqs 52 and 55 (after the Fourier integral of both sides of eq 52) as

ψα̃ (s) =

2 2 8 α + 4(Wx + rd,dim/r1) ijj μa 2 yzz j zEτrep‐dim π α(α 2 − 4P+2)(α 2 − 4P −2) jjk 3kBT zz{

for α = double‐odd (α = 2q with q = odd)

(61a)

(61b)

Here, E is the magnitude of the electric field applied at t < 0, and parameters r1, ra,dim, rd,dim, and τrep‑dim are the same as those appearing in the calculation of orientation function (cf. eqs 16 and 27). The factors Wx, Wy, P+, and P− are functions of the transformation variable s as indicated in eqs 28 and 29, although this s dependence is not explicitly shown in eq 61 for simplicity of equations. The Laplace transformation of the normalized dielectric relaxation function of the unimer, Φ̃1(s) = ∫ ∞ 0 Φ1(t) exp(−st) dt, is calculated by summing the 2θ̃p/pπ terms for odd p (cf. eq 57a). With the aid of a mathematical identity,32 ∑p=odd≥1(p2 − x2)−1 = (π/4x) tan(πx/2), we can make this summation analytically to find

(57a)

(57b)

Only odd modes contribute to Φ1(t) of the unimer (eq 57a) because of the cancellation in the integral of the sinusoidal eigenfunctions for the even modes, ∫ N0 sin(pπn1/N) dn1 = 0 for p = even (cf. eqs 45a and 55a). For the dimer (eq 57b), only a half of the even modes (double-odd modes) having the mode indices α′ = 2α with α = odd contribute to Φ2(t) because of the cancellation ∫ N0 sin(α′πn2/2N)dn2 − ∫ 2N N sin(α′πn2/2N) dn2 = 0 for other α′ (cf. eqs 45b and 55b). J

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules l tan(πP+/2) o o tan(πP −/2) | 1 o o − m } 2 2o o o r1 P+ − P − n (πP+/2) (πP −/2) o ~ l 1 i tan(πP /2) y τrep‐dim Wy 2 + ra,dim o j z o + + − 1zzz m 2 jjj z o P+ jk (πP+/2) 4r1 P+2 − P −2 o { n | yo 1 ijj tan(πP −/2) o − − 1zzzz} j 2j o P − k (πP −/2) {o (62) ~ Similarly, the Laplace transformation for the dimer, Φ̃2(s) = ∫∞ 0 Φ2(t) exp(−st) dt, is obtained from eqs 57b and 61b as for dimer:

reaction, the unimer and dimer relax through pure reptation dynamics to exhibit quite simple relationships:25,28

for unimer: Φ̃1(s) =

τrep‐dim

for unimer (pure reptation case): Δε1̃ ′(ω) = g1′(ω), Δε2̃ ′(ω) = g2′(ω/4),

2

⟨τε⟩n =

Δεj

=ω

∫0

εj̃ ″(ω) ≡

εj″(ω) Δεj

=ω

∫0

= ω Re{[Φ̃j(s)]s = iω }

∑p ≥ 1 Iε , pτε , p ∑p ≥ 1 Iε , p

(67)

for unimer: 2 ⟨τε⟩nuni 4v 2(r1 − 4) π 2 v + (rd,dim/r1) = − τrep‐dim 12 4v 2 + rd,dim r1(4v 2 + rd,dim)2

| l o tanh(π 4v 2 + rd,dim /4) o o o o o ×m 1− } o o 2 o o o (π 4v + rd,dim /4) o o o n ~

(68)

for dimer:

∞

2 (4 − r1)rd, ‐dim ⟨τε⟩dim π 2 v + (rd,dim/r1) n = − 2 τrep‐dim 12 4v + rd,dim r1(4v 2 + rd,dim)2

Φj(t ) sin ωt dt

= −ω Im{[Φ̃j(s)]s = iω }

(66)

In eq 67, τε,p and Iε,p denote the characteristic time and intensity of pth dielectric relaxation mode. This ⟨τε⟩n, being analogous to the viscoelastic ⟨τG⟩n (cf. eq 36), coincides with {ε̃j″(ω)/ω}ω→0 = Re{[Φ̃j(s)]s=iω→0} (cf. eq 64b) and can be straightforwardly calculated from θ̃p(s) and ψ̃ α(s) at s = 0 (that determine [Φ̃j(s)]s=iω→0). The results of this calculation, expressed in terms of parameters v = ra,dim/r1 , rd,dim, r1, and τrep‑dim, are summarized below.

The dielectric relaxation function Φj(t) can be obtained by inversely transforming Φ̃j(s) through the standard analysis of poles and residues. Φ̃j(s) (eqs 62 and 63) is much more simply expressed than the viscoelastic g̃j(s) (eqs 31−34), and the inversion can be easily made for Φ̃j(s). However, for direct comparison with the viscoelastic moduli gj*(ω) (cf. eq 35), we here focus on the complex dielectric permittivity of the unimer and dimer (j = 1 and 2), εj*(ω) ≡ Δεj{1 − iω∫ ∞ 0 Φj(t) exp(−iωt) dt} where i = −1 , ω is the angular frequency, and Δεj denotes the dielectric intensity of the unimer and/or dimer. This εj*(ω) is related to the Laplace transformation, ̃ Φ̃j(s) = ∫ ∞ 0 Φj(t) exp(−st) dt, as εj*(ω)/Δεj = 1 − [sΦj(s)]s=iω. Thus, the normalized decrease of dynamic dielectric constant from its static value, Δε̃j′(ω), and the normalized dielectric loss, ε̃j″(ω), are directly obtained from Φ̃j(s) as Δεj̃ ′(ω) ≡

ε2̃ ″(ω) = g2″(ω/4)

We can straightforwardly conclude that the association/ dissociation reaction affects the relaxation if these purely reptative relationships are violated. 2.3.4. Terminal Dielectric Relaxation Time. The unimer and dimer exhibit considerably sharp distribution of the terminal dielectric relaxation modes, as shown later in Figures 3 and 5. Thus, the first-moment average dielectric relaxation time ⟨τε⟩n defined below can be utilized as the terminal dielectric relaxation time of the unimer and/or dimer.

τrep‐dim

εj′(0) − εj′(ω)

(65)

for dipole-inverted dimer (pure reptation case):

l o o tan(πP+/2) tan(πP −/2) | 1 o o − m } 2o o o π π 4 P+ − P − o ( /2) ( /2) P P + − n ~ 2 l yz τrep − dim Wx + rd,dim/r1 o o 1 ijj tan(πP+/2) + − 1zzz m 2 jj 2 2 o j z 4 P+ − P − o { n P+ k (πP+/2) | o yo 1 jij tan(πP −/2) − − 1zzzz} j 2j o P − k (πP −/2) {o (63) ~

Φ̃2(s) =

ε1̃ ″(ω) = g1″(ω)

| l o tanh(π 4v 2 + rd,dim /4) o o o o o ×m − 1 } o o 2 o o o (π 4v + rd,dim /4) o o o n ~

(64a) ∞

Φj(t ) cos ωt dt

(69)

The terminal dielectric relaxation time ⟨τε⟩n thus obtained is later shown in Figure 8 and discussed in relation to the reaction-induced coupling of the relaxation modes of the unimer and dimer.

(64b)

The factors Wx, Wy, P+, and P− determining Φ̃j(s) (cf. eqs 62 and 63) and thus governing Δε̃j′(ω) and ε̃j″(ω) are dependent on the Laplace transformation variable s, as shown in eqs 28 and 29. Considering this s dependence, we can straightforwardly evaluate Δε̃j′(ω) and ε̃j″(ω) through eq 64, as shown later in Figures 3 and 5. Here, it should be noted that eqs 64a and 64b are formally identical to eqs 35a and 35b for the viscoelastic moduli, gj′(ω) and gj″(ω), so that direct comparison of Δε̃j′(ω) and ε̃j″(ω) with those moduli allows us to examine a difference(s) in the effects of association/dissociation reaction on the viscoelastic and dielectric relaxation. Specifically, in the absence of the

3. RESULTS AND DISCUSSION 3.1. Overview of Reaction Effect on Viscoelastic and Dielectric Relaxation. The effect of association/dissociation reaction on the viscoelastic and dielectric relaxation can be examined through comparison of the normalized storage and loss moduli, gj′(ω) and gj″(ω), and the normalized dielectric permittivity decrease and dielectric loss, Δε̃j′(ω) and ε̃j″(ω), calculated for several values of the normalized association and dissociation rates, ra,dim and rd,dim (defined by eq 27). For semidilute, unentangled (Rouse) cis-polyisoprene having K

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules associative carboxylic group at the chain head (PI-COOH), experiments20 indicated that the ra,dim/rd,dim ratio evaluated from the unimer and dimer concentrations at equilibrium (ra,dim/rd,dim = τds/τas = 2[P2]eq/[P]eq; cf. eq 3) is not significantly different from unity at temperatures T ranging from −20 to 25 °C and also suggested that the rates ra,dim and rd,dim at those T have values in a range between 1 and 5. Thus, for the reptating unimer and dimer, we assumed similar reaction rates to have chosen ra,dim/rd,dim = 1 and examined their gj*(ω) and Δε̃j*(ω) for several rd,dim values between 0.1 and 30 (in a range wider than experimentally observed for unentangled PI-COOH.) For ra,dim/rd,dim = 1, the volume fractions of the unimer and dimer are υ1 = υ2 = 0.5 ([P]eq = 2[P2]eq). In Figures 2−5, the viscoelastic gj*(ω) (eqs 31−35) and dielectric Δε̃j*(ω) (eqs 62−64) calculated for those rd,dim values are double-logarithmically plotted against a reduced angular frequency ωτG‑uni, with τG‑uni being the longest viscoelastic relaxation time of the unimer in the absence of the reaction. (This τG‑uni is identical to τrep‑uni given by eq 13.) In Figures 2−5, black curves indicate the viscoelastic and dielectric behavior of the unimer and dimer in the absence of the association/dissociation reaction. The reptation relationships, eqs 65 and 66, obviously hold for these black curves.

Figure 3. Effect of association/dissociation reaction on dielectric behavior of reptating unimer (colored plots). Black curves indicate the pure reptation behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other.

Figure 2. Effect of association/dissociation reaction on viscoelastic behavior of reptating unimer (colored plots). Black curves indicate the pure reptation behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other. Figure 4. Effect of association/dissociation reaction on viscoelastic behavior of reptating dimer (colored plots). Black curves indicate the pure reptation behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other.

For the unimer, the reaction progressively retards the relaxation on an increase of rd,dim up to 1, and this retardation is stronger for the viscoelastic relaxation (Figure 2) than for the dielectric relaxation (Figure 3). This difference between the viscoelastic and dielectric behavior indicates failure of eq 65, which allows us to conclude changes of the unimer dynamics due to the reaction even without any detailed analysis. For larger rd,dim > 1, the unimer relaxation remains slower than the purely reptative relaxation. Nevertheless, delicate

features are noted in the insets of Figures 2 and 3 that magnify the g1′(ω) and Δε̃1′(ω) curves in the terminal relaxation regime: The terminal viscoelastic relaxation of the unimer is moderately accelerated on an increase of rd,dim from 1 to 30, L

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

In a future experimental test of this reaction effect (achievable for high molecular weight PI-COOH), the viscoelastic and dielectric data can be measured for the unimer/dimer mixture as a whole but not for the isolated unimer and/or dimer. Thus, it is informative to theoretically examine how the reaction affects the viscoelastic moduli and dielectric permittivity of the mixture. As noted from eqs 8 and 35, the storage and loss moduli of the mixture normalized by the entanglement plateau modulus GN are straightforwardly obtained from gj′ and gj″ shown in Figures 2 and 4 as G′(ω)/ GN = υ1g1′(ω) + υ2g2′(ω) and G″(ω)/GN = υ1g1″(ω) + υ2g2″(ω), with υ1 and υ2 being the unimer and dimer volume fractions. Similarly, the dielectric permittivity decrease and dielectric loss normalized by the dielectric relaxation intensity Δε are obtained from Δε̃j′(ω) and ε̃j″(ω) (Figures 3 and 5) as Δε′(ω)/Δε = υ1Δε̃1′(ω) + υ2Δε̃2′(ω) and ε″(ω)/Δε = υ1ε̃1″(ω) + υ2ε̃2″(ω) (cf. eqs 50 and 64). Figures 6 and 7 show G*(ω)/GN and Δε*(ω)/Δε thus calculated for the parameters, ra,dim/rd,dim = 1, rd,dim = 0.1−30, and υ1 = υ2 = 0.5 (identical to those utilized in Figures 2−5).

Figure 5. Effect of association/dissociation reaction on dielectric behavior of reptating dimer (colored plots). Black curves indicate the pure reptation behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other.

whereas no such acceleration is noted for the terminal dielectric relaxation. This behavior is further discussed later for Figure 8 in relation to the coupling between the unimer and dimer. For the dimer, the reaction progressively accelerates the terminal relaxation with increasing rd,dim, and the acceleration is much stronger for the viscoelastic relaxation (Figure 4) than for the dielectric relaxation (Figure 5). Thus, the viscoelastic and dielectric behavior violates eq 66, confirming a significant deviation of the dimer dynamics from the pure reptation dynamics. This reaction effect on the dimer dynamics is further discussed later for Figure 8 in relation to the coupling of the unimer and dimer relaxation modes. For both unimer and dimer in the range of rd,dim examined (rd,dim ≤ 30), the effect of the association/dissociation reaction is more significant for the viscoelastic relaxation than for the dielectric relaxation, as explained above. This result reflects the difference in the basic structural functions underlying the viscoelastic and dielectric relaxation, the orientation function Sj(nj,t) (eq 5) and the average bond vector ⟨uj,y(nj,t)⟩ (eq 55). The mutual conformational transfer between the unimer and dimer due to the reaction (eq 4) leads to multiple coupling of the relaxation mode amplitudes of their orientation functions (as noted for the ξ terms in eqs 19 and 20) because these functions are the tensorial second-moment average of the bond vector and the conformational transfer includes no cancellation, as explained for eqs 10 and 11. In contrast, for the vectorial first-moment average ⟨uj,y(nj,t)⟩, the transfer includes partial cancellation, as explained in detail for eqs 53 and 54 (see the minus sign therein that results in the partial cancelation). Thus, the behavior shown in Figures 2−5 confirms that a simple difference in the averaging moment results in a significant difference in the reaction effect on the viscoelastic and dielectric relaxation.

Figure 6. Changes of normalized viscoelastic moduli, G′/GN and G″/ GN, of the associating/dissociating unimer/dimer mixture as a whole with the normalized dissociation rate rd,dim (chosen to be identical to ra,dim). Black curves indicate the behavior of a simple mixture in the absence of reaction.

In Figures 6 and 7, the black curves indicate the behavior expected for a simple mixture of the reptating unimer and dimer in the absence of reaction, Grep*(ω)/GN = υ1g1,rep*(ω) + υ2g2,rep*(ω) and Δεrep*(ω)/Δε = υ1Δε̃1,rep*(ω) + υ2Δε̃2,rep*(ω). Δε̃j′(ω) and ε̃j″(ω) of the reacting unimer/dimer mixture are close to those of this simple mixture (cf. Figure 7) because the reaction affects the dielectric behavior just moderately as M

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 8. Changes of terminal relaxation time of the unimer and dimer with the normalized dissociation rate rd,dim (chosen to be identical to ra,dim). The terminal relaxation time is evaluated as the first-moment average relaxation time.

Figure 7. Changes of normalized decrease of dielectric permittivity and normalized dielectric loss, Δε′/Δε and ε″/Δε, of the associating/ dissociating unimer/dimer mixture as a whole with the normalized dissociation rate rd,dim (chosen to be identical to ra,dim). Black curves indicate the behavior of a simple mixture in the absence of reaction.

theless, for completeness of discussion, Figure 8 shows plots of normalized ⟨τG⟩n and ⟨τε⟩n up to unrealistically large rd,dim (= 104). As noted in Figure 8, the reaction hardly affects the terminal relaxation times of the unimer and dimer for small rd,dim < 0.1. Thus, the pure reptation behavior, ⟨τG⟩rep‑uni = ⟨τε⟩rep‑uni = (π2/ n n 2 rep‑dim rep‑dim 12)τrep‑uni and ⟨τG⟩n = 4⟨τε⟩n = (π /12)τrep−dim = 8 × (π2/12)τrep−uni (with the factor of 4 attached to ⟨τε⟩rep‑dim n reflecting the symmetrical dipole inversion of the dimer), is recovered for sufficiently small rd,dim. On an increase of rd,dim > 0.1, ⟨τG⟩n and ⟨τε⟩n of the dimer monotonically decrease to approach their high-rd,dim (fast reaction) asymptotes; see blue circles in the top and bottom panels. For the unimer, ⟨τε⟩n (red triangles in the bottom panel) monotonically increases to approach its asymptote on an increase of rd,dim > 0.1, whereas ⟨τG⟩n exhibits this approach after an overshoot, as noted in the top panel. These changes of ⟨τG⟩n and ⟨τε⟩n well correspond to the changes of gj*(ω) and Δε̃j*(ω) with rd,dim seen in Figures 2−5 (for rd,dim ≤ 30) that include the retardation of the viscoelastic relaxation of the unimer followed by moderate acceleration (cf. inset of Figure 2), and are related to the reaction-induced conformational transfer between the unimer and dimer that results in coupling of their relaxation modes. For further discussion of the retardation and acceleration of the unimer and dimer relaxation, it is informative to quantify the asymptotic values of ⟨τG⟩n and ⟨τε⟩n in the fast reaction limit. The compact expressions of ⟨τG⟩n (eqs 37 and 41) and ⟨τε⟩n (eqs 68 and 69) allow us to analytically evaluate the asymptotic value as

explained earlier. In contrast, we note a significant reaction effect for the viscoelastic moduli (Figure 6). The reaction progressively accelerates and narrows the terminal viscoelastic relaxation of the reacting unimer/dimer mixture on an increase of rd,dim from 0.1 to 30. These results encourage us to experimentally examine the reaction effect, the coupling of the relaxation modes of the unimer and dimer, in a future study through viscoelastic and dielectric tests for high molecular weight PI-COOH. 3.2. Effect of Association/Dissociation on Terminal Relaxation. For the unimer and dimer undergoing the association/dissociation reaction, the viscoelastic and dielectric terminal relaxation times have been analytically obtained as the first-moment average relaxation times, ⟨τG⟩n (eqs 37 and 41) and ⟨τε⟩n (eqs 68 and 69). In Figure 8, these ⟨τG⟩n and ⟨τε⟩n are normalized by the terminal relaxation time of the reptating unimer in the absence of reaction, ⟨τG⟩rep‑uni = ⟨τε⟩rep‑uni = (π2/ n n 12)τrep‑uni with τrep‑uni being the longest reptation time given by eq 13, and plotted double-logarithmically against the normalized dissociation rate, rd,dim (= τrep‑dim/τds; eq 27). The normalized association rate, ra,dim (= τrep‑dim/τas), has been chosen to be identical to rd,dim, as done in Figures 2−7. The association and dissociation times τas and τds determining these rates are not of purely chemical nature but contributed from the local chain motion, as explained earlier in section 2.1. For this reason, rd,dim cannot be very large in an actual association/ dissociation system such as the PI-COOH system. NeverN

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules for both unimer and dimer: ⟨τG⟩n , ⟨τε⟩n →

4 τG 3

= ra,dim → ∞

i

rep‐uni j jj j n

k

=

yz π2 τrep‐uni zzz 9 {

Obviously, the association/dissociation reaction is just a trigger of the extra motion of the unimer and dimer not emerging in the absence of reaction, and this motion determines the viscoelastic (and dielectric) relaxation. Thus, even in the fast reaction limit, the relaxation occurs with a finite characteristic time τ* not falling below the unimer relaxation time in the absence of reaction (= shortest possible relaxation time): For the case of rd,dim = ra,dim, this τ* is longer than the reptation time of the unimer by the factor of λ = 4/3. Now, we turn our attention to the overshoot of ⟨τG⟩n of the unimer noted at rd,dim (= ra,dim) ≅ 10 in Figure 8. For this rd,dim value, the dimer is in an intermediate state before entering the asymptotic (fast reaction) regime, and its viscoelastic relaxation is still considerably “slow” (by a factor of ≅ 1.5; cf. top panel of Figure 8) compared to the relaxation in that regime. Then, the viscoelastic relaxation of the unimer appears to be retarded beyond its asymptote through the strong coupling with this “slow” dimer, which should naturally result in the overshoot of ⟨τG⟩n of the unimer. In contrast, the dielectric relaxation modes of the unimer and dimer are rather weakly coupled, as discussed for Figures 3 and 5. In addition, the dielectric ⟨τε⟩n of the dimer is longer than ⟨τε⟩n of the unimer only by a factor of 2 even in the absence of reaction (because of the dipole inversion of the dimer). For these reasons, no overshoot is visible for the dielectric ⟨τε⟩n of the unimer (cf. bottom panel of Figure 8). The retardation of the viscoelastic relaxation of the unimer followed by the acceleration (observed also as the overshoot of its ⟨τG⟩n) is not unique to the reptating unimer undergoing the head-to-head association. In fact, the associating Rouse unimer also exhibits a qualitatively similar retardation and acceleration on an increase of rd,dim (= ra,dim)18 for the same reason as explained above. However, the terminal relaxation of the Rouse unimer in the fast reaction limit exactly coincides with its purely Rouse-type relaxation (as noted from analytical calculation),18 whereas that of the reptating unimer is slower than its purely reptative relaxation (by the factor of 4/3 for the case of rd,dim = ra,dim; cf. eq 70). Thus, the reaction effect on the viscoelastic (and dielectric) relaxation changes its magnitude according to the type of the chain dynamics, e.g., reptation or Rouse dynamics. This point is further discussed below. 3.3. Comparison with Rouse Unimer/Dimer Undergoing Head-to-Head Association/Dissociation. The characteristic features seen in Figures 2−5, the reactioninduced retardation and acceleration of the unimer and dimer relaxation and this reaction effect being much less significant for the dielectric relaxation, have been noted also for the Rouse chains undergoing head-to-head association and dissociation.18,19 In fact, we can analytically find that the reaction does not affect the dielectric relaxation of the Rouse unimer and dimer, the latter having symmetrically inverted dipoles.19 Namely, these unimer and dimer obey eqs 58 and 59 with τrep‑uni and τrep‑dim therein being replaced by respective Rouse relaxation times for the end-to-end fluctuation, τRouse‑uni and τRouse‑dim = 4τRouse‑uni, so that their memory decay mode amplitudes exactly coincide with each other (i.e., θp(t) = ψ2p(t) in eq 58 and ψα(t) = θα/2(t) in eq 59a)19 to raise no reaction effect on the dielectric relaxation. Despite this qualitative similarity of the reaction effect, we also note a quantitative difference that reflects a difference in the basic features of the reptation and Rouse dynamics. This difference can be most clearly tested for the viscoelastic moduli

for rd,dim (70)

Namely, in the fast reaction limit, the terminal relaxation time becomes identical for the unimer and dimer and for the viscoelastic and dielectric relaxation, as visually noted in Figure 8 where the horizontal green lines indicate the asymptotic value of ⟨τG⟩n and ⟨τε⟩n (= (4/3)⟨τε⟩rep‑uni for the case of rd,dim n = ra,dim). In this limit, the unimer and dimer should mutually transfer their conformation very rapidly (even before the terminal relaxation occurs significantly) to effectively behave as the same species thereby sharing the same ⟨τG⟩n and the same ⟨τε⟩n. This limiting behavior can be confirmed in Figure 9 where the viscoelastic gj*(ω) and dielectric Δε̃j*(ω) of the unimer

Figure 9. Normalized storage and loss moduli, gj′ and gj″, and the normalized decrease of dielectric permittivity and dielectric loss, Δε̃j′ and ε̃j″, of the unimer and dimer calculated for rd,dim = ra,dim = 300 (almost in the fact reaction limit). Solid green curves indicate the reptation modulus of the unimer with a delay of relaxation by a factor of 4/3.

and dimer, calculated for rd,dim = ra,dim = 300 (almost in the fast reaction limit; cf. Figure 8), are plotted against ωτG‑uni. Black dashed curves indicate the modulus g1,rep*(ω) of the reptating unimer in the absence of reaction. (The dielectric Δε̃1,rep*(ω) of this unimer coincides with g1,rep*(ω); cf. eq 65.) For rd,dim = ra,dim = 300, gj*(ω) and Δε̃j*(ω) of the unimer and dimer relax more slowly compared to this g1,rep*(ω), as already noted for ⟨τG⟩n (cf. eq 70). Nevertheless, all of g1*(ω), Δε̃1*(ω) (of the unimer) and g2*(ω), Δε̃2*(ω) (of the dimer) coincide with each other to satisfy the reptation relationship for the unimer, eq 65 (the remaining small differences noted in Figure 9 disappear in the limit of rd,dim, ra,dim → ∞) and agree with the reptation modulus g1,rep*(λω) of the unimer (solid green curve) with a delay of the relaxation by a factor of λ = ⟨τG⟩n/ ⟨τG⟩rep‑uni = 4/3 (cf. eq 70). Thus, the unimer and dimer in the n fast reaction limit commonly behave like the unimer relaxing through pure reptation but with the delay by this factor of λ. O

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules being sensitive to the reaction. The results of the test are summarized below. For the Rouse chains undergoing the association and dissociation, our previous paper analyzed the time evolution of the orientational anisotropy to calculate the normalized viscoelastic relaxation function, gj(t) with j = 1 and 2 for the unimer and dimer. The corresponding Laplace transformation, g̃j (s) = ∫ ∞ 0 gj(t)exp(−st)dt, is summarized in the Appendix, and the storage and loss moduli, gj′(ω) and g j″(ω), are straightforwardly obtained from this g̃j(s) (cf. eq 35), as shown in Figures 10 and 11.

Figure 11. Effect of association/dissociation reaction on viscoelastic behavior of Rouse dimer (colored plots). Black solid curves indicate the pure Rouse behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other.

by the end-to-end fluctuation time of the dimer, ra,dim and rd,dim, were chosen to be equal to each other, and their values were set identical to those utilized for the reptation case (Figures 2−5), ra,dim = rd,dim = 0.1, 0.3, 1, 3, and 30. Thus, for each value of rd,dim, the reaction is equally fast (in comparison with the chain motion) for the Rouse dimer and reptating dimer examined in Figures 11 and 4. Consequently, for those values of ra,dim, the reaction is effectively faster for the Rouse unimer compared to the reptating unimer, as can be noted from the association rate normalized by τuni of the unimer (not of the dimer), ra,uni = ra,dim/4 = 0.025−7.5 for the Rouse unimer and ra,uni = ra,dim/8 = 0.0125−3.75 for the reptating unimer. In Figures 10 and 11, black curves indicate the moduli of the Rouse unimer and dimer in the absence of the reaction. As noted in Figure 10 and its magnified inset, the viscoelastic relaxation of the Rouse unimer is a little retarded on an increase of rd,dim (= ra,dim) up to 1 and then accelerated on a further increase of rd,dim. Comparing Figures 10 and 2, we note that this behavior of the Rouse unimer is qualitatively similar to the behavior of the reptating unimer, but the retardation is much weaker for the Rouse unimer despite the fact that the reaction for a given rd,dim value is effectively faster for the Rouse unimer (as explained above in relation to the ra,uni value). In addition, the modulus of the Rouse unimer for rd,dim = 30 (bright green circle in Figure 10) is indistinguishable from its pure Rouse modulus (black curve), indicating that the fast reaction limit is almost attained at rd,dim = 30 and the modulus in this limit exactly coincides with the pure Rouse modulus (as noted from analytic calculation18). Neither such rapid approach to the fast reaction limit nor disappearance of the reaction effect on the unimer modulus in this limit (i.e.,

Figure 10. Effect of association/dissociation reaction on viscoelastic behavior of Rouse unimer (colored plots). Black curves indicate the pure Rouse behavior of the unimer and dimer. The normalized association and dissociation rates, ra,dim and rd,dim, are set identical to each other.

A comment needs to be added for the calculation of g̃j(s) in the Appendix. This calculation includes a compact, analytical summation over the relaxation mode index p ranging from p = 1 up to p = ∞, not up to a cutoff index pc (= Rouse segment number per chain), because the summation up to pc cannot be conducted analytically. Thus, the moduli gj′(ω) and gj″(ω) obtained from this analytical summation have the values >1 at high ω. This problem is well-known but does not affect the low ω behavior of the moduli shown in Figures 10 and 11: At such low ω, the artificial higher order modes with p > pc have fully relaxed, and the moduli shown therein were confirmed to numerically agree with the moduli calculated from the summation up to pc. Figures 10 and 11 respectively show double-logarithmic plots of the moduli of the Rouse unimer and dimer against the reduced angular frequency ωτG‑uni, with τG‑uni being the longest viscoelastic relaxation time of the unimer in the absence of the reaction (τG‑uni = τRouse‑uni/2 for the Rouse dynamics). In the calculation, the number of Rouse segments per unimer was set at N = 50. The association and dissociation rates normalized P

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules recovery of purely reptative behavior) is observed for the reptating unimer, as clearly noted in Figure 8. Thus, we may conclude that the reaction effect on the relaxation is much weaker for the Rouse unimer than for the reptating unimer. For the Rouse dimer, the reaction significantly accelerates the viscoelastic relaxation, and the fast reaction limit (where the dimer modulus exactly coincides with the Rouse modulus of the unimer18) is almost attained at rd,dim = 30, as noted in Figure 11. Namely, for rd,dim = 30, the reaction accelerates the Rouse dimer relaxation by a factor of 4 (= dimer to unimer Rouse relaxation time ratio in the absence of reaction). For the reptating dimer, a similar magnitude of acceleration is noted for rd,dim = 30 (see Figure 8a), although the acceleration in the fast reaction limit is a little stronger (acceleration by a factor of ⟨τG⟩dim‑rep /⟨τG⟩dim n n = 6 for the case of rd,dim = ra,dim → ∞; cf. eq 70). Thus, roughly speaking, the reaction effect on the dimer relaxation is similar (though not identical), in magnitude, for the Rouse and reptation dynamics. The above difference of the reaction effect for the reptating unimer and Rouse unimer should reflect the characteristic features the reptation and Rouse dynamics, presence and absence of motional coherence along the chain backbone. The similarity noted for the reptating dimer and Rouse dimer should also reflect these features. Thus, it is informative to summarize how those features are reflected in the orientational anisotropy distribution along the chain backbone, S(n,t) (being identical to the orientation function). In the pure reptation and pure Rouse dynamics, the orientational anisotropy under step shear strain γ(imposed at time t = 0) decays with t as22,23 pure reptation: S(n , t ) = S0

pure Rouse:

∑ α= odd ≥ 1

| l o 4 i απn zy o o α 2t o zz expm − sinjjj } o o τrep o o απ k N { n ~

S(n , t ) 2 = S0 N

Figure 12. Time evolution of orientational anisotropy distribution along chain backbone for the pure reptation mechanism. This evolution behavior is independent of the number of entanglement segments per chain.

Figure 13. Time evolution of orientational anisotropy distribution along chain backbone for the pure Rouse mechanism. The number of Rouse segments per chain is chosen to be N = 50.

(71)

l o o 2p2 t | i pπn yz o zz expo − m } o o o o τ k N { Rouse n ~

∑ sin 2jjj N

p=1

In contrast, in the Rouse dynamics, the relaxation proceeds through diffusive transport of the anisotropy along the chain backbone and also in the direction perpendicular to it, i.e., through two-dimensional diffusion.33 At short t < τRouse, the segments move incoherently to exhibit essentially uniform decay of the anisotropy throughout the chain backbone thanks to this two-dimensional feature of the relaxation; see the flat S(n,t)/S0 profile at t < τRouse in Figure 13. In other words, the effective source of relaxation at short t is uniformly distributed along the backbone of the Rouse chain.33 Nevertheless, in the time scale of terminal relaxation, t ≥ τRouse, this uniformity vanishes and the anisotropy is the largest at around the chain midpoint. At such long t, eq 72 becomes S(n,t)/S0 ≅ (2/N) sin2(πn/N)exp{−2t/τRouse}, which indicates that the anisotropy is transported (through the two-dimensional diffusion) to the chain ends to vanish.33 Namely, at long t ≥ τRouse, the chain ends become the dominant source of relaxation even in the Rouse dynamics. These features of the anisotropy relaxation in the reptation and Rouse dynamics allow us to qualitatively correlate the difference of the reaction effect seen for the Rouse and reptating unimers (Figures 2 and 10) as well as the similarity noted for the Rouse and reptating dimers (Figures 4, 8, and 11) with the motional coherence, as explained below. At the chemical equilibrium considered in this study, a given mass of unimer is converted into the dimer at every moment

(72)

where τrep and τRouse are the end-to-end fluctuation time of the chain (consisting of N segments) considered in respective dynamics and S0 (∝γ) is the initial anisotropy. (τrep is identical to the longest viscoelastic relaxation time τG of the reptating chain, whereas τRouse = 2τG for the Rouse chain.) For the reptation and Rouse dynamics, respectively, Figures 12 and 13 show semi-logarithmic plots of the normalized anisotropy S(n,t)/S0 calculated from eqs 71 and 72 against the normalized segment index n/N. The horizontal dotted line indicates the initial, uniform anisotropy distribution along the chain backbone. The decay of the anisotropy from this initial state exhibits different features in the reptation and Rouse dynamics, as explained below. In the reptation dynamics, all entanglement segments move coherently along the chain backbone so that the chain ends serve as the only source of orientational relaxation. Thus, the decay of the anisotropy propagates from the chain ends (n/N = 0, 1) to the midpoint (n/N = 1/2), and the segments at around the midpoint essentially preserve their initial orientation at short t < τrep where the terminal relaxation has not started significantly (cf. Figure 12). Namely, the reptative relaxation proceeds through one-dimensional diffusion of the anisotropy along the chain backbone toward the chain ends. Q

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Macromolecules

(without the delay; cf. Figures 10 and 11). This difference can be also understood qualitatively in relation to the difference in the dimensionality of the anisotropy diffusion explained earlier. In the reptation case, the diffusion occurs only one-dimensionally (along the chain backbone) so that the blocking of the relaxation source (unimer heads) due to the reaction unavoidably delays the relaxation even in the fast reaction limit. (The unimer heads are blocked for a finite fraction of time even in the fast reaction limit, thereby allowing the reaction effect to survive in that limit.) In contrast, in the Rouse case, no such delay emerges in that limit because of the two-dimensional nature of the anisotropy diffusion. For this case, the anisotropy is transported toward the chain ends through a route perpendicular to the chain backbone even when the unimer heads are blocked, thereby allowing the relaxation in the fast reaction limit to occur without the delay. It should be emphasized that the above argument provides us with just a qualitative explanation of the reaction effects being different for the reptating unimer and Rouse unimer and being similar for the reptating dimer and Rouse dimer. All quantitative details are to be found in the analytical expressions of the viscoelastic moduli, eqs 31−35 for the reptation case and eqs A7−A13 in the Appendix for the Rouse case. Nevertheless, the qualitative explanation focusing on the creation and removal (blocking) of the relaxation source is still useful because it could serve as a clue for further discussing, in a future study, how the reaction effect on the relaxation changes with the type of chain dynamics.

and the same mass of dimer is converted back to the unimer, as /τas in eq 9a for the described by the terms −S1/τas and Screated 1 reptating unimer. This balance of conversion also occurs for the reptating dimer (cf. the terms −S2/τds and Screated /τds in eq 2 9b) as well as for the Rouse unimer and dimer undergoing the association/dissociation reaction at equilibrium.18 In general, the dimer relaxation is slower than the unimer relaxation, and thus the total anisotropy of the dimer transferred to the unimer at a time t, ∫ N0 Screated dn1, is larger than the total anisotropy of 1 the unimer at that t, ∫ N0 S1 dn1, so that the reaction retards the unimer relaxation (namely, τas−1∫ N0 {Screated − S1} dn1 > 0; cf. eq 1 9a). Correspondingly, the total anisotropy of the unimer created dn2, is smaller than the transferred to the dimer, ∫ 2N 0 S2 S total anisotropy of the dimer, ∫ 2N 0 2 dn2, and the reaction created − S2}dn2 < 0; accelerates the dimer relaxation (τds−1∫ 2N 0 {S2 cf. eq 9b). This argument suggests that the dimer anisotropy transferred to the unimer, Screated , is the key quantity that governs 1 the retardation of the unimer relaxation. Qualitatively speaking, Screated is larger and the unimer relaxation is more strongly 1 retarded when the unimer anisotropy transferred to the dimer, , relaxes more slowly which largely contributes to this Screated 1 through the dimer motion. This is the case for the reptating unimer because the chain end (or head) serves as the only source of relaxation in the coherent reptation dynamics (as explained for Figure 12), and the source at the unimer head is blocked in the created dimer. In contrast, in the incoherent Rouse dynamics, the created dimer (associated unimers) can relax at any portion along the chain backbone before the terminal relaxation of this dimer becomes significant (as explained for Figure 13), namely, even in the time scale of the terminal relaxation of the unimer. Thus, the reaction-induced retardation of the terminal relaxation is stronger for the reptating unimer than for the Rouse unimer, as noted from comparison of Figures 2 and 10. Similarly, we may consider that the unimer anisotropy transferred to the dimer, Screated , is smaller and the dimer 2 relaxation is more strongly accelerated when the dimer anisotropy transferred to the unimer, which contributes to this Screated , relaxes faster through the unimer motion. In the 2 reptation dynamics, the dimer anisotropy remains large at around the dimer midpoint, as explained for Figure 12. However, on the dimer dissociation, the midpoint becomes the unimer head that serves as the relaxation source. Thus, the anisotropy of the reptating dimer transferred to the unimer relaxes rapidly through the unimer reptation with the aid of this source created on the dimer dissociation. This rapid relaxation occurs also for the Rouse dimer on its dissociation because the created unimer head serves as the dominant source of relaxation in the time scale of the terminal relaxation of the Rouse unimer (as explained for Figure 13) and thus in a longer time scale of the terminal relaxation of the Rouse dimer. (In the fast reaction limit, the time scale of terminal relaxation is the same for the Rouse unimer and dimer, but this qualitative argument still holds even for that case.) For this reason, the reaction-induced acceleration of the terminal relaxation is similar (though not identical), in magnitude, for the reptating dimer and Rouse dimer, as noted in Figures 4, 8, and 11. Here, we remember that the relaxation of the reptating unimer and dimer in the fast reaction limit coincides with the delayed reptation of the unimer (cf. Figures 8 and 9) whereas the relaxation of the Rouse unimer and dimer in that limit exactly coincides with the pure Rouse relaxation of the unimer

4. CONCLUDING REMARKS For entangled linear polymer having type A dipoles and undergoing head-to-head association and dissociation reaction, we have combined the reptation dynamics and the reaction kinetics to analyze the viscoelastic and dielectric relaxation behavior. The analysis is made on the basis of the eigenfunction expansion of basic functions representing the averaged chain conformation, the isochronal orientational anisotropy and the orientational memory underlying the viscoelastic and dielectric relaxation. The key in the analysis is the mutual conformational transfer between the unimer and dimer occurring through the reaction. This transfer differently affects the viscoelastic and dielectric relaxation, as explained below. The orientational anisotropy is the tensorial second-moment average of the chain conformation, so that the anisotropy decay modes of the unimer and dimer exhibit multiple coupling due to the reaction-induced conformational transfer. This multiple coupling significantly retards and accelerates the viscoelastic relaxation of the unimer and dimer, respectively. In contrast, the orientational memory is the vectorial firstmoment average of the chain conformation, and thus the conformational transfer just results in a pairwise coupling of the memory decay modes of the unimer and dimer. Because of this pairwise feature of the coupling, the retardation and acceleration of the dielectric relaxation of the unimer and dimer are much less significant compared to those noted for the viscoelastic relaxation. This difference between the viscoelastic and dielectric relaxation is found also for the associating/dissociating Rouse chains. Nevertheless, the reaction-induced retardation of the viscoelastic relaxation is stronger for the reptating unimer than for the Rouse unimer, whereas the reaction-induced acceleration is similar, in magnitude, for the reptating dimer and R

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Macromolecules Rouse dimer. The unimer association blocks the relaxation source at the unimer head, whereas the dimer dissociation creates a relaxation source at the dimer midpoint. These changes of the relaxation source, being combined with the reptation and Rouse dynamics characterized by the presence and absence of the motional coherence along the chain backbone, result in the above reaction effects on the unimer and dimer relaxation. The above results in turn suggest that the relaxation of associative polymers strongly depends on the type of chain dynamics that competes with the reaction. Thus, it is highly desired to analyze the competition between the reaction and the realistic entanglement dynamics, reptation combined with contour length fluctuation (CLF) and constraint release (CR). The CLF mechanism enhances the relaxation near the free ends of the chain.22−24 Thus, the association of the unimer chains should suppress the CLF relaxation near the unimer heads to retard (and narrow) the unimer relaxation. The CR relaxation has the Rouse-like character to activate the anisotropy decay uniformly along the chain backbone in the time scale of unimer reptation (that is a unit process for local CR relaxation).22−24 Considering the reaction effect for the Rouse unimer discussed for Figures 10 and 13, we expect that the unimer relaxation is affected by the reaction less significantly when the CR mechanism contributes more to the relaxation. It is interesting to test these expectations on the basis of the analysis of the competition of the chain dynamics and reaction that first requires self-consistent incorporation of CLF and CR in the reptation framework (i.e., self-consistent specification of the CLF path in the presence of CR). This analysis is considered as a very basic and important subject of future work.

■

for Rouse dimer:

o rd,uni | o + −} o s + Λp o 2N (ra,uni + rd,uni) p = 1 o s + Λ p o n ~ N l | o o rd,uni ∑ omoo 1 + − 1 − o}oo + N (ra,uni + rd,uni) p = 1 o s + Λ p s + Λp o n ~ N i yz ra,unird,uni j ∑ jjjjj 1 + − 1 − zzzzz + N (ra,uni + rd,uni) p = 1 s + Λ p s + Λp k { 2 | | l l o o o o 8 p tanh(πz /2) o 1 o ×o m }o m1 + 2 } 2 o o 4p2 + z 2 o oo o o (πz /2) o 4 p z + n ~n ~ +

o ra,uni | o + −} o o s s + Λ + Λ o p po p=1 n ~ N l | o o ra,uni ∑ omoo 1 + − 1 − o}oo + s + Λp o 2N (ra,uni + rd,uni) p = 1 o s + Λ p n ~ N l | o o ra,uni r r ∑ omo d,uni + + a,uni − oo}o + N (ra,uni + rd,uni) p = 1 o s + Λp s + Λp o o n ~ 2 | | l l o o o o 8p tanh(πz /2) o 1 o ×o m }o m1 + 2 } 2 o o 4p2 + z 2 o oo o o (πz /2) o 4 p z + n ~n ~ N

l o rd,uni

p=1

z(s ) =

+

1 s + Θ2p − 1

(A2)

rd,uni =

τRouse‐uni , τds

2(sτRouse‐uni + rd,uni)

(A3)

and 2p2

Λ+p =

τRouse‐uni

Θ2p − 1 =

,

Λ−p =

2p2 + ra,uni + rd,uni τRouse‐uni

,

(2p − 1)2 + 2rd,uni 2τRouse‐uni

(A4)

Here, τRouse‑uni is the longest end-to-end fluctuation time of the Rouse unimer. (The longest viscoelastic relaxation time of the unimer is given by τRouse‑uni/2.) For N ≫ 1, the summations appearing in eqs A1 and A2 can be calculated analytically and compactly with the aid of the following mathematical identities.32

For the Rouse unimer and dimer composed of N and 2N segments, respectively, our previous study18 has analyzed time evolution of the mode amplitudes of orientational correlation function (being analogous to Sj(nj,t) utilized in this study but defined for two different segments along the chain backbone). For N ≫ 1, the analysis gave the Laplace-transformed relaxation moduli g̃j (s) (cf. eqs 40, 41, 43, and 45-47 in Ref18.). The results can be summarized as for Rouse unimer:

∑ omoo

∑

τRouse‐uni , τas

ra,uni =

Laplace-Transformed Relaxation Modulus of Rouse Unimer and Dimer

1 N (ra,uni + rd,uni)

N

with

APPENDIX

g1̃ (s) =

N

1 2N

l o ra,uni

∑ omoo

1

g2̃ (s) =

∞

∑ p≥1 ∞

∑ p≥1

π 1 1 =− 2 + coth(πx) 2 2x p +x 2x 2

(A5)

π2 π 1 1 1 = + 3 coth(πx) − 4 2 2 2 2 2 2x (p + x ) 4x sinh (πx) 4x (A6)

After this analytical summation, eqs A1 and A2 become

+

for Rouse unimer: g1̃ (s) =

| ra,uni o o τRouse‐uni l o o uni uni Q ( s ) Q ( s ) + m } 1 2 o o o 2N o r r + a,uni d,uni n ~

(A7)

with Q 1uni(s) = − (A1)

2 π 2 coth(π z − 2rd,uni /2) 2 + 2 (π z 2 − 2r z 2 − 2rd,uni d,uni /2)

(A8) S

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2 ij 1 1 yzz 1 (π z − 2rd,uni /2) zz − + Q 2uni(s) = jjjj rd,uni z{ rd,uni (πz /2) k ra,uni

g2,Rouse (s ) = ̃

2 1 (π z + 2ra, uni /2) × − ra,uni (πz /2) tanh(π z 2 − 2rd,uni /2)

tanh(πz /2)

×

+ 2π

(A9)

for Rouse dimer: g2̃ (s) = +

o τRouse‐uni l o dim Q 1 (s) + Q 2dim(s) m o o 4N n

o π 2 ra,unird,uni tanh(πz /2) dim | o Q 3 (s ) } o o πz /2 2 ra,uni + rd,uni ~

(A10)

with Q 1dim(s) = − +

2 2 π 2 coth(π z − 2rd,uni /2) + 2 (π z 2 − 2r z 2 − 2rd,uni d,uni /2)

π tanh(πz /2) z

■

(A11)

ji 2 zyz 2 zz Q 2dim(s) = jjjj 2 − 2 jz z − 2rd,uni zz{ k | l o o coth(π z 2 − 2r d,uni /2) coth(πz /2) o π2 o o o + − m } o 2 o o (πz /2) o 2o o (π z − 2rd,uni /2) o o n ~

(A15)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00691. (A) Parameters/factors and mathematical identities used in calculation, (B) calculation of X̃ p(s) of unimer, (C) calculation of Ỹ α(s) (α = odd) of dimer, (D) calculation of g̃1(s) and g̃2(s) of unimer and dimer, (E) terminal relaxation time of unimer and dimer (PDF)

(A12) 2 ij 2 z 2 yzzz coth(π z − 2rd,uni /2) − Q 3dim(s) = jjjj z j rd,uni rd,uni 2 zz{ (π z 2 − 2rd,uni /2) k

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AUTHOR INFORMATION

Corresponding Author

2 ij 2 z 2 yzzz coth(π z + 2ra,uni /2) j j +jj + z j ra,uni ra,uni 2 zz{ (π z 2 + 2ra,uni /2) k | l ij 1 ij 1 o coth(πz /2) 1 yzzo 1 yzz 2o o jj zzm z − jjjj + + − 1 z } j z o z j z o o (πz /2) rd,uni {o rd,uni { o o k ra,uni k ra,uni n ~ ji 1 1 zyz 1 zz + jjjj + 2 z ra,uni { sinh (πz /2) (A13) k rd,uni

*E-mail: [email protected] (H.W.). ORCID

Hiroshi Watanabe: 0000-0003-0826-2454 Yumi Matsumiya: 0000-0002-4690-3251 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partly supported by the Grant-in-Aid for Scientific Research (B) from MEXT, Japan (Grant No. 15H03865), Grant-in-Aid for Scientific Research (C) from JSPS, Japan (Grant No. 15K05519), Collaborative Research Program of ICR, Kyoto University (Grant No. 2018-77), Collaborative Research Program of Kyoto University Research Coordination Alliance, and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4006278).

For ra,uni → 0 and rd,uni → 0 (no association/dissociation dim reaction), eqs A7 and A10 retain only the Quni 1 and/or Q1 terms and reduce to the expressions for the pure Rouse relaxation of the unimer and dimer: o τRouse − uni l 1 o − m o o sτRouse − uni 2N o n | o π coth(π sτRouse − uni /2 ) o + } o o 2 sτRouse − uni /2 o ~

| coth(π 2sτRouse − uni ) o o } o o 2sτRouse − uni ~

From g̃1(s) and g̃2(s) given by eqs A7 and A10, the storage and loss moduli of the Rouse unimer and dimer undergoing the head-to-head association/dissociation are straightforwardly evaluated; cf. eq 35. These moduli are shown in Figures 10 and 11. A comment needs to be added for those moduli. In the above calculation, the relaxation modes were analytically summed over the mode index p ranging from p = 1 up to p = ∞, not up to a cut-off index pc = Rouse segment number per chain (as appearing in eqs A1 and A2), because the summation up to pc cannot be conducted analytically. Thus, the moduli gj′(ω) and gj″(ω) obtained from this analytical summation have the values >1 at high ω. This problem is well known but does not affect the low ω behavior of the moduli shown in Figures 10 and 11: In the range of ω examined in Figures 10 and 11, the artificial higher order modes with p > pc have fully relaxed and the moduli shown therein were confirmed to numerically agree with the moduli calculated from the summation up to pc.

tanh(πz /2) tanh(π z 2 + 2ra,uni /2)

o τRouse − uni l 1 o − m o o 4N n sτRouse − uni

Article

(s ) = g1,Rouse ̃

■

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(A14) T

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(22) Watanabe, H. Viscoelasticity and dynamics of entangle polymers. Prog. Polym. Sci. 1999, 24, 1253−1403. (23) McLeish, T. C. B. Tube Theory of Entangled Polymer Dynamics. Adv. Phys. 2002, 51, 1379−1527. (24) Likhtman, A. E.; McLeish, T. C. B. Quantitative Theory for Linear Dynamics of Linear Entangled Polymers. Macromolecules 2002, 35, 6332−6343. (25) Matsumiya, Y.; Kumazawa, K.; Nagao, M.; Urakawa, O.; Watanabe, H. Dielectric Relaxation of Monodisperse Linear Polyisoprene: Contribution of Constraint Release. Macromolecules 2013, 46, 6067−6080. (26) Shivokhin, M. E.; Read, D.; Kouloumasis, D.; Kocen, R.; Zhuge, F.; Bailly, C.; Hadjichristidis, N.; Likhtman, A. E. Understanding Effect of Constraint Release Environment on End-to-End Vector Relaxation of Linear Polymer Chains. Macromolecules 2017, 50, 4501−4523. (27) Graessley, W. W. The Entanglement Concept in Polymer Rheology. Adv. Polym. Sci. 1974, 16, 1−179. The definition of the first-moment average relaxation time (or “number average” relaxation time) is given in p 25 of this review. (28) Watanabe, H. Dielectric Relaxation of Type-A Polymers in Melts and Solutions. Macromol. Rapid Commun. 2001, 22, 127−175. (29) Cole, R. H. Correlation Function Theory of Dielectric Relaxation. J. Chem. Phys. 1965, 42, 637−643. (30) Uneyama, T.; Masubuchi, Y.; Horio, K.; Matsumiya, Y.; Watanabe, H.; Pathak, J. A.; Roland, C. M. A Theoretical Analysis of Rheodielectric Response of Type-A Polymer Chains. J. Polym. Sci., Part B: Polym. Phys. 2009, 47, 1039−1057. (31) Kremer, F.; Schönhals, A. Theory of Dielectric Relaxation. In Broadband Dielectric Spectroscopy; Kremer, F., Schönhals, A., Eds.; Springer: Berlin, 2003; Chapter 1. (32) Moriguchi, S.; Udagawa, K.; Hitotsumatsu, S. Mathematical Formula; Iwanami: Tokyo, 1986. (33) For the orientational correlation function of the Rouse chain, Soc(n,n′,t) ≡ b−2⟨ux(n,t)uy(n′,t)⟩ with uξ(m,t) being ξ component of the mth bond vector having the average size b, the time evolution in the absence of reaction is described by21−23 ∂Soc/∂t = (κ/ζ){∂2Soc/∂n2 + ∂2Soc/∂n′2}, where κ and ζ denote the spring constant and friction of the segment. The orientational anisotropy of respective segments is given by S(n,t) = Soc(n,n,t). As noted from the above time evolution equation, Soc and S relax through two-dimensional diffusion occurring along the chain backbone and in the direction perpendicular to the chain backbone, which indicates the incoherent feature of the Rouse dynamics. (In the coherent reptation dynamics, the diffusion is onedimensional and occurs only along the chain backbone.) The ends of the Rouse chain always have randomized orientation and serve as the relaxation source, as shown in the boundary condition, Soc(n,n′,t) = 0 at n = 0, N and n′ = 0, N. In addition, no orientational correlation emerges for different segments on application of step strain at t = 0, which is cast in the initial condition,21−23 Soc(n,n′,0) = S0 for |n − n′| < 1 and Soc(n,n′,0) = 0 for |n − n′| > 1 (or, Soc(n,n′,0) = {2S0/N}∑p≥1 sin{pπn/N}sin{pπn′/N} in the continuous treatment). Namely, in the initial state at t = 0, the spatial gradient of Soc(n,n′,0) along the chain backbone emerges only at n = n′ = 0, N, but the gradient in the direction perpendicular to the chain backbone emerges uniformly at all n = n′ between 0 and N. Because of this initial condition, the orientational anisotropy of the Rouse chain at short t relaxes dominantly through the diffusion in the latter direction that occurs uniformly at all segments, which corresponds to the effective source of relaxation discussed for Figure 13. In contrast, in the long time scale of terminal Rouse relaxation (t ≥ τRouse), the gradient of Soc(n,n′,t) becomes essentially uniform in all directions, as noted from the asymptotic form at those t, Soc(n,n′,t) ≅ {2S0/N}sin{πn/N}sin{πn′/ N}exp(−2t/τRouse). In this range of t, the chain ends serve as the dominant source of relaxation (and the relaxation proceeds through two-dimensional diffusion of the anisotropy toward this source).

U

DOI: 10.1021/acs.macromol.8b00691 Macromolecules XXXX, XXX, XXX−XXX