Viscoelastic and Orientational Relaxation of Linear and Ring Rouse

Article pubs.acs.org/Macromolecules

Viscoelastic and Orientational Relaxation of Linear and Ring Rouse Chains Undergoing Reversible End-Association and Dissociation Youngdon Kwon School of Chemical Engineering, Sungkyunkwan University, 300 Cheoncheon-dong, Jangan-gu, Suwon, Gyeonggi-do 440-746, Korea

Yumi Matsumiya and Hiroshi Watanabe* Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan S Supporting Information *

ABSTRACT: For dilute telechelic linear and ring Rouse chains undergoing reversible end-association and dissociation, the time (t) evolution equation was analytically formulated for the bond vector of the subchain (or segment), u[c](n,t) with n being the subchain index and the superscript c specifying the chain (c = L and R for the linear and ring chains). The end-association of the linear chain (i.e., ring formation) occurs only when the ends of the linear chain come into close proximity. Because of this constraint for the ring formation, the time evolution equation for u[L](n,t) of the linear chain was formulated with a conceptually new, two-step expansion method: u[L](n,t) was first expanded with respect to its sinusoidal Rouse eigenfunction, sin(pπn/N) with p = integer and N being the number of subchains per chain, and then the series of odd sine modes is re-expanded with respect to cosine eigenfunctions of the ring chain, cos(2απn/N) with α = integer, so as to account for that constraint. This formulation allowed analytical calculation of the orientational correlation function, [c] [c] S[c](n,m,t) = b−2⟨u[c] x (n,t)uy (m,t)⟩ (c = L, R) with b being the subchain step length, and the viscoelastic relaxation function, g (t) ∝ ∫ N0 S[c](n,n,t) dn. It turned out that the terminal relaxation of g[R](t) and g[L](t) of the ring and linear chains is retarded and accelerated, respectively, due to the motional coupling of those chains occurring through the reaction. This coupling breaks the ring symmetry (equivalence of all subchains of the ring chain in the absence of reaction), thereby leading to oscillation of the orientational anisotropy S[R](n,n,t) of the ring chain at long t with the subchain index n. The coupling also reduces a difference of the anisotropy S[L](n,n,t) of the linear chain at the middle (n ∼ N/2) and end (n ∼ 0).

1. INTRODUCTION

Nevertheless, it is also interesting to examine the relaxation of associating polymers in the opposite case of fast reaction. (This fast-reaction case may be found when low-Tg polymers having associative groups of high activation energies are examined at high temperatures T ≫ Tg. For this case, the acceleration with increasing T would be much more significant for the reaction than for the intrinsic chain motion, so that the reaction at high T could be faster even compared to the chain motion over a large length scale.) With this motivation, we recently focused on the simplest model system, linear Rouse chains (unimers) undergoing monofunctional end-association and dissociation, and attempted to formulate the viscoelastic relaxation functions g(t) for the associated dimer and dissociated unimer.23 This model system is free from the effect(s) of entanglement and cooperativity in the dissociation of successive association sites, thereby allowing us to formulate g(t) analytically. A key in

Polymer systems undergoing reversible association/dissociation reaction, such as solutions of telechelic chains having associative groups at the chain ends1−8 and bulk polymers having more than two associative sites per chain,9−20 have been attracting extensive research interest because of their novel scientific aspect. The relaxation behavior of those associative polymers dramatically changes according to the competition between the polymer chain motion and the reaction. If the dissociation reaction is much slower than the intrinsic chain motion, the local relaxation between two neighboring association sites is completed well before the dissociation occurs. For this case, the relaxation process splits into two branches, the fast relaxation activated by such local motion and the slow relaxation due to the dissociation that activates larger scale motion of the chain. Most of the systems so far examined1−20 belong to this category of slow dissociation, and sticky chain models considering the cooperative dissociation and relaxation21,22 have been proposed for those systems. © XXXX American Chemical Society

Received: February 27, 2016 Revised: April 8, 2016

A

DOI: 10.1021/acs.macromol.6b00424 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules this formulation was the “mapping” (or projection) of the conformation of associating unimers onto the dimer and the reverse mapping of the conformation of the dissociating dimer onto unimers. Assuming no change of the chain tension on the unimer association, we were able to calculate analytically the viscoelastic g(t) of the unimer and dimer. The motional coupling between the unimer and dimer, occurring through the association/dissociation reaction (at equilibrium), yields new series of relaxation modes not existing for the pure Rouse dynamics, and those new modes govern the terminal relaxation of both unimer and dimer.23 The reaction rate in the above linear unimer/dimer system is not explicitly correlated with the conformation of respective chains. For example, the reaction can be (and was) assumed to occur at the same rate irrespective of the end-to-end distance of a focused unimer because the reaction is governed by the distance between the reacting ends of two unimers. This feature enabled us to straightforwardly formulate the time evolution of the orientational correlation functions averaged for all unimers and/or dimers and calculate g(t) of these chains.23 The situation is quite different for a dilute telechelic chain that forms a ring chain through the end-association (cf. Figure 1).

2. MODEL 2.1. System. We consider linear and ring Rouse chains at association/dissociation equilibrium (Figure 1). Each chain is composed of N (≫1) subchains (segments), and the linear chain becomes the ring chain on the end-association at the 0th and Nth subchains (cf. purple circles). The chains are assumed to be dilute so that the linear chain is not associated with the other linear chains and forms neither linear nor ring multimer. An experimental method for detection of the relaxation of such dilute chains is discussed at the end of this paper. For simplicity, the molar concentrations of the linear and ring chains, [L] and [R], are assumed to obey the simplest rate equations for reversible end-association and dissociation d 1 1 [L] = − [L] + [R] dt τas τds

(1)

d 1 1 [R] = − [R] + [L] dt τds τas

(2)

The characteristic times of association and dissociation, τas and τds, being equivalent to the rate constants k and k′ (τas = 1/k′ and τds = 1/k), are defined as the ensemble averages for all linear and/or ring chains. Specifically, the end-association time of the linear chain, τas, includes an averaged probability that the two ends of the linear chain come into close proximity. The equilibrium concentrations of the linear and ring chains are specified by K=

Figure 1. Schematic illustration of linear and ring Rouse chains at association/dissociation equilibrium. A small step shear is applied at time 0, and the following viscoelastic/orientational relaxation is focused.

[R]eq [L]eq

=

τds τas

(3)

where K is the equilibrium constant. At equilibrium, the total concentrations of the linear and ring chains remain constant (independent of time t). However, a given linear chain undergoes the end-association to become a ring chain with a probability pL→R, and a given ring chain becomes a linear chain through the dissociation with a probability pR→L. In a short interval of time from t to t + Δt, the probability pL→R is related to the characteristic time of association, τas, as

The ring is formed only when two ends of the chain come into close proximity, indicating that the reaction rate is strongly correlated with the linear chain conformation. Formulation of the orientational correlation function S and viscoealstic relaxation function g for such conformation-dependent reaction is of our interest, and we have challenged it. Fortunately, we were able to find analytical expressions for those functions. The idea/strategy underlying this formulation and the resulting features of S and g are presented in this paper. This paper is organized in the following way. At first, sections 2.1 and 2.2 explain the system to be examined, the linear and ring Rouse chains at the association/dissociation equilibrium, and define the average quantities of our interest, the orientational correlation function and viscoealstic relaxation function. Then, section 2.3 briefly revisits the stochastic Rouse equation of motion for the subchain bond vector u in the absence of reaction, for convenience of later formulation in the presence of reaction. Readers who are familiar with the Rouse dynamics can skip section 2.3 and directly forward to section 2.4. Section 2.4 explains a conceptually new, two-step expansion of u of the linear chain that enables us to incorporate the conformation-dependent end-association of this chain into the time evolution equation of u. This equation and the accompanying time evolution equation for the ring chain are solved analytically to calculate the orientational correlation function and viscoelastic relaxation function. Finally, section 3 discusses features of the orientational anisotropy and viscoealstic relaxation in relation to the motional coupling between the linear and ring chains, with an emphasis being placed on the “ring symmetry breaking” due to the coupling.

pL → R (Δt ) = 1 − pL → L (Δt ) → Δt /τas

(for Δt /τas → 0) (4a)

pL → L (Δt ) = exp(−Δt /τas) → 1 − Δt /τas

(for Δt /τas → 0) (4b)

Here, pL→L(Δt) is a probability that the linear chain survives (not converted into the ring chain) in the interval Δt. (This pL→L is straightforwardly obtained from eq 1 without the second term representing the linear chain formation. It gives pL→L = [L(t+Δt)]/[L(t)] = exp(−Δt/τas).) Similarly, pR→L and the survival probability of the ring chain pR→R are given by pR → L (Δt ) = 1 − pR → R (Δt ) → Δt /τds

(for Δt /τds → 0) (5a)

pR → R (Δt ) = exp(−Δt /τds) → 1 − Δt /τds

(for Δt /τds → 0) (5b)

As noted from eqs 3, 4a, and 5a, the probabilities pL→R and pR→L satisfy the mass balance relationship that represents the coincidence of the concentrations of the linear and ring chains converted from/to each other: pL → R [L]eq = pR → L [R]eq B

(at equilibrium)

(6)


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vector u[c](n,t) is expressed, in the continuous limit for N ≫ 1, as24,25

2.2. Orientational Relaxation and Viscoelastic Relaxation. We consider a step shear strain γ applied to the linear/ring equilibrium system at time 0. The shear and shear gradient directions are specified as x and y directions, respectively. The strain is assumed to be infinitesimal (γ ≪ 1) so that it gives no effect on the chemical equilibrium between the linear and ring chains. The strain distorts the chain conformation (by an infinitesimal amount) and activates the orientational and viscoelastic relaxation at t > 0. This relaxation is described by the orientational correlation function defined for nth and mth subchains:23 1 S (n , m , t ) ≡ 2 ⟨ux[c](n , t )u[c] y ( m , t )⟩ b [c]

ζ

(10)

κ=

(0 ≤ n , m ≤ N )

∫0

[c]

d n S (n , n , t )

(c = L, R)

(b2 = ⟨u 2⟩at equilibrium )

b2

(11)

denotes the thermal Brownian force acting on nth subchain. This force is modeled as an isotropic white-noise characterized by its dyadic24,25 [c] ⟨F[c] B (n , t )F B (m , t ′)⟩ = 2IζkBTδ(n − m)δ(t − t ′)

(12)

with I being the unit tensor. 2.3.1. Linear Chain. For the linear chain, the boundary condition representing lack of external force acting on the chain ends is given by u[L](n , t ) = 0

for n = 0, N

(13)

Combining this condition with the Rouse equation of motion, eq 10, we can expand u[L](n,t) as

(8)

(The prefactor of 3/Nγ results from the affine deformation of the subchains at time 0, as explained later for eqs 18 and 24 in more detail.) The relaxation modulus G(t) of the system containing the linear and ring chains is simply given as a weighed sum of g[L](t) and g[R](t) G(t ) = NkBT {νLg [L](t ) + νR g [R](t )}

3kBT

F[c] B (n,t)

Here, u[c] ξ (n,t) is the ξ component (ξ = x, y) of the nth subchain bond vector u[c](n,t) at time t (cf. Figure 1), and b denotes the equilibrium step length of the subchain. The superscript c specifies the chain, c = L (linear) or R (ring), and ⟨...⟩ represents the ensemble average. The viscoelastic relaxation function g[c](t), normalized to unity at time 0, is related to the orientational anisotropy of individual subchains, S[c](n,n,t) (diagonal part of S[c](n,m,t)), as24,25 N

(c = L, R)

Here, ζ is the subchain friction coefficient, and κ is the spring constant for the subchain that is expressed in terms of the step length b, Boltzmann constant kB, and the absolute temperature T as

(7)

3 g [c](t ) = Nγ

∂ [c] ∂2 ∂ [c] u (n , t ) = κ 2 u[c](n , t ) + F B (n , t ) ∂t ∂n ∂n

N

u[L](n , t ) =

⎛ pπ n ⎞ ⎟ N ⎠

∑ Ξp(t ) sin⎜⎝ p=1

(for linear chain) (14)

Ξp(t) is the amplitude of pth eigenmode (p ≥ 1) associated with the Rouse eigenfunction for the linear chain, sin(pπn/N), that satisfies the boundary condition, eq 13. The time evolution of Ξp(t) is specified from eq 10 as

(9)

⎛ p2 t ⎞ 2 ⎟+ Ξp(t ) = Ξp(0) exp⎜ − ⎝ τ1 ⎠ ζN

where kB is the Boltzmann constant, T is the absolute temperature, and νc (c = L, R) represents the equilibrium number densities of the linear and ring chains (νR/νL = [R]eq/[L]eq = K; cf. eq 3). In the absence of the reaction, the orientational correlation function S[c](n,m,t) defined by eq 7 is most easily obtained from calculation of eigenmode expansion coefficients of this function, as explained in Appendix A1 (cf. Supporting Information). However, this method of calculation faces a difficulty when the linear and ring chains undergo the association/dissociation reaction because the end-association of the linear chain (ring formation) occurs only when the two ends of the linear chain come into close proximity (in other words, the association reaction is correlated with the linear chain conformation), as explained in more detail in Appendix A2 (cf. Supporting Information). Thus, in this study, we first expand the subchain bond vector u[c](n,t) of the linear and ring chains into respective Rouse modes, then re-expand a part of those modes of the linear chain with respect to the eigenfunctions of the ring chain, and finally utilize the expanded series to calculate S[c](n,m,t). For convenience of the later formulation of this two-step expansion in the presence of reaction, the first expansion in the absence of reaction (for pure Rouse dynamics) is briefly summarized in section 2.3. Readers familiar with the pure Rouse dynamics can skip that section and directly forward to section 2.4. 2.3. Rouse Relaxation without Association/Dissociation Reaction. For both linear and ring chains in the absence of reaction, the Rouse equation of motion for the subchain bond

⎛ p2 {t − t ′} ⎞ ⎟ × exp⎜ − τ1 ⎠ ⎝

∫0

N

∫0

t

dt ′

⎛ pπn ⎞ ∂ [L] ⎟ dn sin⎜ F (n , t ′) ⎝ N ⎠ ∂n B (15)

where τ1 is the slowest Rouse relaxation time given by ζ ⎛⎜ N ⎞⎟ ζb2N 2 = κ⎝π ⎠ 3π 2kBT 2

τ1 =

(16)

From eqs 14 and 15, the orientational correlation function is straightforwardly calculated as 1 S (n , m , t ) = 2 b [L]

N

∑

⟨Ξp(0)Ξq(0)⟩xy

p,q=1

⎛ {p2 + q2}t ⎞ ⎛ pπn ⎞ ⎛ qπm ⎞ ⎟ sin⎜ ⎟ ⎟ sin⎜ × exp⎜ − τ1 ⎝ ⎠ ⎝ N ⎠ ⎝ N ⎠

(17)

where ⟨Ξp(0)Ξq(0)⟩xy indicates the initial value of the xy component (shear component) of the dyadic ⟨ΞpΞq⟩ at time 0. A term linear to the Brownian force F[L] B (n,t) is included in eq 15, but the corresponding term vanishes in eq 17 because of the isotropic white-noise character of this force (no correlation between x and y components; cf. eq 12). Finally, assuming that the bond vector at equilibrium is affinely C


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the bond vector at t = 0, we find the initial condition for the shear components (cf. Appendix B1 in Supporting Information):

deformed on imposition of the step strain γ at time 0, we find the initial value of the shear component of the dyadic;24−26 see Appendix B1 in the Supporting Information for more details. 2b2γ δpq ⟨Ξp(0)Ξq(0)⟩xy = 3N

(18)

This simple expression of ⟨Ξp(0)Ξq(0)⟩xy corresponds to the Gaussian distribution function of Ξp at equilibrium (cf. Appendix B1). The initial condition, eq 18, is determined by this equilibrium distribution and not affected by the association/dissociation reaction considered later. From eqs 17 and 18, we find the well-known expression for the orientational correlation function of the linear Rouse chain 2γ 3N

S[L](n , m , t ) =

(24a)

⟨A α(0)Aβ(0)⟩xy =

2b2γ δαβ 3N

(24b)

S[R](n , m , t ) =

(19)

The normalized viscoelastic relaxation function (defined by eq 8) is obtained from this S[L] as g [L](t ) =

1 N

N

⎛ 2p2 t ⎞ ⎟ ⎝ τ1 ⎠

g [R](t ) = (20)

∫0

N

(21a)

dn u[R](n , t ) = 0

(21b)

From this condition and the Rouse equation of motion, eq 10, the subchain bond vector of the ring Rouse chain is expanded as27 N /2

u[R](n , t ) =

⎛

⎞

∑ Ψα(t ) sin⎜⎝ 2απn ⎟⎠

α=1 N /2

+

N

⎛

⎞

∑ A α(t ) cos⎜⎝ 2απn ⎟⎠

α=1

N

(22)

Here, Ψα(t) and Aα(t) denote the amplitudes of αth eigenmode (1≤ α ≤ N/2) associated with the Rouse eigenfunctions of the ring chain, sin(2απn/N) and cos(2απn/N). These eigenfunctions, satisfying the boundary condition eq 21, have the same eigenvalue (and are degenerated). For those eigenmode amplitudes, the time evolution is determined by the Rouse equation of motion, eq 10, as

⎛ 4α 2t ⎞ A α(t ) = A α(0) exp⎜ − ⎟ + Lα[A](t ; F[R] B ) ⎝ τ1 ⎠

N /2

⎛ 8α 2t ⎞ ⎛ 2απ {n − m} ⎞ ⎟ ⎟ cos⎜ ⎠ N ⎝ τ1 ⎠ ⎝

∑ exp⎜− α= 1

2 N

N /2

⎛ 8α 2t ⎞ ⎟ ⎝ τ1 ⎠

∑ exp⎜− α= 1

(26)

⎛ p2 Δt ⎞ ⎟ Ξp(t + Δt ) = pL → L (Δt )Ξp(t ) exp⎜ − τ1 ⎠ ⎝

(for ring chain)

⎛ 4α 2t ⎞ Ψα(t ) = Ψα(0) exp⎜ − ⎟ + Lα[Ψ](t ; F[R] B ) τ ⎝ 1 ⎠

2γ 3N

As noted from eq 25 (with m = n), the orientational anisotropy of respective (nth) subchains, specified by S[R](n, n, t), is independent of the subchain index n. This feature is a natural consequence of the equivalence of all subchains of the ring chain in the absence of reaction. 2.4. Rouse Relaxation at Association/Dissociation Equilibrium. 2.4.1. Basic Formulation. We consider the dynamics in a short interval of time, from t to t + Δt. In this interval, each linear chain is converted to a ring chain with the probability pL→R(Δt) (eq 4a) or survives as the linear chain to exhibit its Rouse dynamics according to eq 15 with the probability pL→L(Δt) (eq 4b). Similarly, each ring chain becomes a linear chain with the probability pR→L(Δt) (eq 5a) or survives to exhibit the ring Rouse dynamics (eq 23) with the probability pR→R(Δt) (eq 5b). Thus, in this interval, the time evolution of the eigenmode amplitudes of the linear and ring chains is described by

2.3.2. Ring Chain. For the ring Rouse chain in the absence of reaction, the calculation can be made in the same way except for the boundary condition. For the ring chain, the continuity at n = 0, N is cast in the boundary condition27 u[R](0, t ) = u[R](N , t )

(24c)

(25)

∑ exp⎜− p=1

for any α , β

Equation 24 is deduced from the Gaussian distribution of Ψα and Aα at equilibrium, as was the case also for Ξp explained for eq 18. Combining eqs 7 and 8 with eqs 22−24, we obtain the wellknown expressions of the orientational correlation function and normalized viscoelastic relaxation function of the ring Rouse chain27

∑ exp⎜− p=1

2b2γ δαβ 3N

⟨Ψα(0)Aβ(0)⟩xy = 0

⎛ 2p2 t ⎞ ⎛ pπn ⎞ ⎛ pπm ⎞ ⎟ sin⎜ ⎟ ⎟ sin⎜ ⎝ τ1 ⎠ ⎝ N ⎠ ⎝ N ⎠

N

⟨Ψα(0)Ψβ(0)⟩xy =

+ pR → L (Δt )Ξ[created] (t + Δt ) p

(27)

⎛ 4α 2Δt ⎞ Ψα(t + Δt ) = pR → R (Δt )Ψα(t ) exp⎜ − ⎟ τ1 ⎠ ⎝ (t + Δt ) + pL → R (Δt )Ψ[created] α

(28a)

⎛ 4α 2Δt ⎞ A α(t + Δt ) = pR → R (Δt )A α(t ) exp⎜ − ⎟ τ1 ⎠ ⎝

(23a)

+ pL → R (Δt )A[created] (t + Δt ) α

(28b)

Here, Ξ[created] (t+Δt) indicates the eigenmode amplitude of the p linear chain (at time t + Δt) created from the ring chain in the interval of time Δt. Similarly, Ψ[created] (t+Δt) and A[created] (t+Δt) α α are the eigenmode amplitudes of the ring chain created from the linear chain.

(23b)

[R] [A] [R] where L[Ψ] α (t;FB ) and Lα (t;FB ) indicate terms (functions of t) linear to the Fourier components of the Brownian force F[R] B similar to that appearing in eq 15. For the affine deformation of

D


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Δt → 0, we obtain the time evolution equations of the mode amplitudes in a differential form:

Strictly speaking, eqs 27 and 28 also include the terms linear to the Brownian force FB, as shown in eqs 15 and 23. However, those terms have no contributions to the shear orientational correlation function S(n,m,t) and to the viscoelastic relaxation function g(t), as explained for eq 17. Thus, for simplicity of the equations, the terms linear to FB have been omitted in eqs 27 and 28. This omission is made also in all equations explained in the followings. (Note, however, that the Brownian terms cannot be omitted if the diagonal components of the orientation tensor such as ux2 and uy2 are to be calculated.) The conformation of the created linear chain at time t + Δt coincides with the conformation of the ring chain at time t (just before the dissociation), except for small conformational changes (of the order of Δt) occurring in the interval Δt through the Rouse dynamics of the created linear chain. Thus, from eqs 14 and 22, Ξp[created](t+Δt) can be related to the mode (t) amplitudes Ψ(t) α and Aα of the ring chain at time t as

N /2

∑ ηpβ Aβ(t ) β=1

(35) p = even integer:

⎛ p2 1⎞ 1 Ξ̇ p(t ) = − ⎜ + ⎟Ξp(t ) + Ψp /2(t ) τas ⎠ τas ⎝ τ1 (36)

⎛ 4α 2 1⎞ 1 + Ξ2α (t ) α = integer (≥1): Ψα̇ (t ) = − ⎜ ⎟Ψα(t ) + τ τ τ ⎝ 1 ds ⎠ ds (37)

α = integer (≥1): ⎛ 4α 2 1⎞ 1 Ȧ α (t ) = −⎜ + ⎟A α (t ) + τds ⎠ τds ⎝ τ1

N /2 ⎛ pπn ⎞ [R]eq ⎧ ⎛ 2απn ⎞ ⎟ = ⎟ ⎨ (t + Δt ) sin⎜ ∑ Ξ[created] ∑ Ψα(t ) sin⎜ p ⎝ N ⎠ [L]eq ⎪ ⎝ N ⎠ ⎩ α=1 p=1 N

⎛ p2 1⎞ 1 Ξ̇ p(t ) = −⎜ + ⎟Ξp(t ) + τas ⎠ τas ⎝ τ1

p = odd integer:

N

∑

ηqα Ξq(t )

q = odd

⎪

N /2

+

⎞⎫

⎛

∑ A α(t ) cos⎜⎝ 2απn ⎟⎠⎬ + O(Δt ) ⎪

N

α=1

(38)

Here and after, the upper dot represents the first-order time derivative; for example, Ȧ α = dA α /dt . Equations 35 and 38 need to be reformulated according to the correlation between the linear chain conformation and ring formation rate, as explained later for eq 46. 2.4.2. Solution of Eqs 36 and 37. As noted from eqs 36 and 37, the amplitude of the even sine mode of the linear chain, Ξp(t) with p = even integer, is coupled only with the amplitude of a particular sine mode of the ring chain, Ψp/2(t), and vice versa. Thus, eqs 36 and 37 can be easily solved irrespective of the amplitudes of the odd sine modes of the linear chain and the cosine modes of the ring chain, Ξp(t) with p = odd integer and Aα(t). The solution is summarized as p = even integer = 2α:

⎪

⎭

(29)

In eq 29, the order of Δt term, O(Δt), indicates the minor effect of Rouse dynamics in the interval Δt explained above, and the front factor [R]eq/[L]eq, expressed in terms of the equilibrium concentrations of the linear and ring chains, normalizes the number of dissociated ring chains per linear chain. (This normalization is necessary because eq 29 represents the conformational change of one linear chain occurring through the ring dissociation.) Making Fourier sine integral of both sides of eq 29, we find p = odd:

Ξ[created] (t + Δt ) = p

[R]eq [L]eq

N /2

Ξ2α (t ) = Ξ2α (0)

∑ ηpα A α(t ) + O(Δt ) α=1

(30)

+ Ψα(0)

with ηpα =

2 N

∫0

N

p ⎛ pπn ⎞ ⎛ 2απn ⎞ 4 ⎟ cos⎜ ⎟ = dn sin⎜ ⎝ N ⎠ ⎝ N ⎠ π p2 − 4α 2

[R]eq [L]eq

λα+ =

Similarly, we find expressions of Ψ[created] (t+Δt) and A[created] α α (t+Δt) of the created ring chain in terms of Ξp(t) of the linear chain + Δt ) =

[L]eq [R]eq

+ Δt ) =

[L]eq [R]eq

Ξ2α (t ) + O(Δt )

∑

{

}

4α 2 , τ1

λα− =

4α 2 + ra + rd 4α 2 + (K + 1)rd = τ1 τ1

and ra =

(33)

τ1 , τas

rd =

τ1 τds

(42)

(Note that the equilibrium constant K defined by eq 3 is identical to the ra/rd ratio.) The initial condition for the shear components of the dyadic of the mode amplitudes is not affected by the association/ dissociation reaction and is given by eqs 18 and 24. In addition, analysis of the equilibrium conformation of the linear and ring chains, similar to that made in Appendix B (cf. Supporting Information), indicates that these reacting chains have no

N p = odd

(39)

(41)

and A[created] (t α

K {exp(− λα+t ) − exp( − λα−t )} 1+K

with

Ψp /2(t ) + O(Δt ) (32)

Ψ[created] (t α

− α

1 K exp( − λα+t ) + exp( − λα−t ) 1+K 1+K 1 {exp(− λα+t ) − exp(− λα−t )} + Ξ2α (0) (40) 1+K

Ψα(t ) = Ψα(0)

and Ξ[created] (t + Δt ) = p

+ α

α = integer: (p = odd)

(31)

p = even:

{ 1 +1 K exp(−λ t) + 1 +K K exp(−λ t)}

ηpα Ξp(t ) + O(Δt ) (34)

Considering eqs 3−5 and eqs 30−34 to retain the terms of the order of Δt in eqs 27 and 28, and further taking the limit of E


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eqs 35 and 38, thereby satisfying eq 46 on the occasion of ring formation. This difficulty can be avoided if a contribution of the odd sine modes to the subchain bond vector u[L](n,t) of the linear chain, already expanded with respect to its Rouse eigenfunctions sin(pπn/N), is re-expanded with respect to cosine functions as

correlation in their initial mode amplitudes when the subchain is affinely deformed by the step strain at t = 0. Namely, we have ⟨Ψα(0)Ξp(0)⟩xy = ⟨A α(0)Ξp(0)⟩xy = 0

for any α , p (43)

Thus, the contributions of the even sine modes of the linear chain and the sine modes of the ring chain to respective orientational correlation functions are calculated from eqs 39 and 40 combined with the initial conditions, eqs 18, 24, and 43, irrespective of the time evolution of the amplitudes of the other modes, the odd sine modes of the linear chain and the cosine modes of the ring chain. The results are summarized as [L] Seven (n , m , t ) =

2γ 3N

N

u[L] odd(n , t ) ≡

p = odd

⎧ 1 + K2 exp( −2λα+t ) 2 + (1 ) K ⎩ α=1 N /2

2K (1 − K ) 2K 2 exp( −2λα−t ) + 2 (1 + K ) (1 + K )2 ⎫ × exp( −{λα+ + λα−}t )⎬ ⎭

2γ m, t) = 3N

N

α = 0:

B0(t ) =

2 ⎧ ∑ ⎨ 1 + K 2 exp(−2λα+t ) α = 1 ⎩ (1 + K ) N /2

p = odd

p

α=0

2 Ξ p (t ) pπ

(48a)

Bα (t ) =

∑

ηpα Ξp(t )

p = odd

with ηpα =

p 4 (p = odd) 2 π p − 4α 2

(48b)

where ηpα is the coefficient already defined by eq 31. Comparing eqs 46 and 48, we note that the ring can be formed only when B0 = 0 no matter what the values of Bα with α ≥ 1 are, because B0(t) is identical to the normalized end-to-end vector of the linear chain, N−1∫ N0 u[L](n,t) dn. In this sense, the coefficient B0 serves as a simple parameter effectively serving as the Lagrange factor mentioned above. (In the formulation shown below, a function Q(t ) = −2τ1Ḃ 0(t ) serves more directly as the Lagrange factor.) As shown in Appendix B2 (cf. Supporting Information), eq 48 gives ∑∞ α=0Bα(t) = 0 irrespective of the t dependence of Ξp(t). ∞ Thus, the second expansion series ∑α=0 Bα(t)cos(2απn/N) introduced for the linear chain (eq 47) is always equal to 0 for n = 0 and N and satisfies the boundary condition for the linear chain, eq 13, as explained in Appendix B2. Utilizing this result, ∑∞ α=0Bα(t) = 0, in eq 47, we find

2.4.3. Reformulation of Eqs 35 and 38. As noted from eqs 35 and 38, the odd sine modes of the linear chain and the cosine modes of the ring chain exhibit coupled time evolution of their amplitudes, Ξp(t) (with p = odd) and Aα(t), without being influenced by the even sine modes of those chains. Thus, superficially, eqs 35 and 38 may look like a complete set of time evolution equations for Ξp(t) and Aα(t). However, actually, these equations are not the complete set because the ring creation (end-association of linear chain) occurs only when the two ends of the linear chain come into close proximity in space. Namely, the ring formation is strongly correlated with the linear chain conformation. In the strict case, the end association requires the linear chain to satisfy the boundary condition for the ring chain, eq 21b. For this case, the amplitudes of the odd sine modes of the linear chain is subjected to an extra constraint required for the ring formation (cf. eqs 14 and 21b): Ξ p (t )

⎛ 2απn ⎞ ⎟ N ⎠

N

α ≥ 1:

2(K − 1) 2 exp( − 2λα−t ) + (1 + K )2 (1 + K )2 ⎫ × exp( − {λα+ + λα−}t )⎬ ⎭ ⎛ 2απn ⎞ ⎛ 2απm ⎞ ⎟ sin⎜ ⎟ × sin⎜ ⎝ N ⎠ ⎝ N ⎠ (45)

N

∑ p = odd

(44)

+

∑

∞

∑ Bα(t ) cos⎜⎝

(47)

+

[R] Ssine (n ,

⎛ pπ n ⎞ ⎟ = Ξp(t ) sin⎜ ⎝ N ⎠

The cosine function is the Rouse eigenfunction of the ring chain but not of the linear chain. For this reason, the summation in the second expansion needs to be made up to α = ∞ so as to analytically reproduce the first expansion. (In actual numerical calculation shown later, we need to truncate the second summation while keeping good convergence, as explained later for Figures 2−6.) Making the Fourier cosine integral of both sides of eq 47, we can express the coefficients Bα in terms of Ξp as (cf. Appendix B2 in Supporting Information)

∑⎨

⎛ 2απn ⎞ ⎛ 2απm ⎞ ⎟ sin⎜ ⎟ × sin⎜ ⎝ N ⎠ ⎝ N ⎠

∑

∞

u[L] odd(n , t ) =

⎧

⎫ ⎛ 2απn ⎞ ⎟ − 1⎬ ⎭ N ⎠

∑ Bα(t )⎨⎩cos⎜⎝

α=1

(49)

Thus, the linear chain conformation is specified with the expansion coefficients Bα(t) with α ≥ 1. For this reason, we use Bα(t) with α ≥ 1 to formulate the dynamics of the linear chain in the followings. This formulation allows us to describe the time evolution of Bα(t) in the presence of association/ dissociation reaction, but the initial condition needs to be carefully examined because Bα with α ≥ 1 are subjected to the constraint ∑∞ α=1Bα = −B0 (due to eq 48) and no longer have the Gaussian distribution at equilibrium, as explained in Appendix B3 (cf. Supporting Information). Because the cosine functions introduced in eq 47 are not the Rouse eigenfunctions for the linear chain satisfying the boundary condition eq 13, the time evolution of the coefficients Bα for the pure Rouse dynamics (in the absence of reaction) is not straightforwardly

=0 (46)

(No constraint emerges for the amplitudes Ξ2p of the even sine modes of the linear chain because ∫ N0 sin(2pπn/N) dn = 0, which ensures validity of eqs 39 and 40.) It is difficult to incorporate this constraint in the ring formation term in eq 38 (second term on the right-hand-side) as well as in the linear consumption factor in eq 35 (the −Ξp(t)/τas factor). This incorporation should require us to add a complicated Lagrange factor (Lagrange multiplier) to F


Article

Macromolecules

N appearing in eqs 35 and 38, ∑N/2 β=1 ηpβAβ(t) and ∑q=oddηqαΞq(t). Thus, formally, the time evolution is coupled only between the coefficients Bα(t) (for the linear chain) and Aα(t) (for the ring chain) of the same order α ranging from 1 to N/2 (see eqs 53a and 54), and no coupling occurs for Bα(t) with α > N/2 because the ring conformation is described by Aα(t) with α ≤ N/2 (cf. eq 53b). Nevertheless, actually, all coefficients are strongly coupled through the Q(t) factor (eq 51) appearing in eqs 52 and 53. More importantly, eqs 52−54 specify how the conformations of the dissociating ring chain and end-associating linear chain, respectively, are mapped onto the created linear and ring chains. The mapping onto the created linear chain is represented by the Aα(t)/τas term in eq 53a, and the −Aα(t)/τds term in eq 54 indicates the corresponding disappearance of the ring chain. Similarly, the Bα(t)/τds term in eq 54 represents the mapping onto the created ring chain, and the −Bα(t)/τas term in eq 53a represents disappearance of the linear chain. In principle, eq 52 for B0(t) should include similar terms representing the mapping. However, the ring is formed only when B0(t) = 0 no matter what the values of Bα(t) with α ≥ 1 are, as explained for eq 48. Thus, a −B0(t)/τas term representing changes of B0(t) on the ring formation is not explicitly included in eq 52. In addition, the cosine expansion series describing the ring conformation, ∑N/2 α=1 Aα(t)cos(2απn/N), includes no zeroth order A0(t) term, so that the linear chain creation (ring dissociation) does not change B0(t) and eq 52 includes no A0(t)/τas term. Similarly, eq 53b includes no linear creation term (Aα(t)/τas) due to the ring dissociation because the cosine expansion for the ring conformation does not include the higher order coefficients, Aα(t) with α > N/2. The decay of the higher order Bα(t) term (with α > N/2), to be represented as −Bα(t)/τas, should be in balance with this lack of the linear creation term, so that the decay term is not included in eq 53b. Namely, in eq 53b, the part of the conformation of the linear chain described by Bα(t) with α > N/2 is formally unaffected by the ring dynamics, although this dynamics actually affects those Bα(t) through the Q(t) term therein: The lower order Bα(t) (α ≤ N/2), included in Q(t), is affected by the ring dynamics (cf. eq 53a), and this effect is transferred to the higher order Bα(t) (α > N/2) through the Q(t) term in eq 53b. One may argue that the linear chain dynamics should be differently modeled by explicitly adding the decay term, −Bα(t)/ τas, to eq 53b. However, eq 53b with this term leads to an unreasonable situation for a case of very fast reaction (τas, τds ≪ τ1). In this case, the Rouse motion is effectively quenched for both linear and ring chains. However, these chains are still chemically equilibrated through disruption/formation of the local bond between the 0th and Nth subchains. Then, the linear chain conformation just before the bond formation, mapped onto the created ring chain, should be mapped back, as it is, to the linear chain on the bond disruption (ring dissociation). Namely, those chains should have the same, frozen conformation. In fact, for τas, τds ≪ τ1, eqs 53a and 54 correctly give Bα(t) = Bα(0) = Aα(t) = Aα(0) for α = 1 − N/2, and eq 50a, B0(t) = B0(0). However, if the −Bα(t)/τas term is added to eq 53b, Bα(t) with α > N/2 exhibits a nonfrozen decay. Thus, eq 53b with this term is conceptually unreasonable. (Nevertheless, for completeness, the time evolution of Aα(t) and Bα(t) is specified later for both cases of eq 53b with and without the decay term; see eq 60 and explanation for it.) Here, it should be emphasized that the end-association time of linear chain, τas, already includes an averaged probability that the two ends of the linear chain come into close proximity, as explained earlier for eqs 1 and 2. Thus, eqs 52−54 have been formulated on the basis of this preaveraging approximation of the end-encounter

deduced from the Rouse equation of motion, eq 10. This makes a contrast from the situation for the eigenmode amplitude Ξp for which the Rouse time evolution is directly obtained as shown in eq 15, or as an equivalent differential form, Ξ̇ p = −{p2 /τ1}Ξp+ a term linear to FB. However, after some calculation, we can find relatively simple, Rouse time evolution equations also for Bα (with α ≥ 0), as explained in Appendix C (cf. Supporting Information). The results are summarized as follows: For linear chain in the absence of reaction. α = 0:

1 Ḃ 0(t ) = − Q(t ) 2τ1

(50a)

α ≥ 1:

4α 2 1 Ḃα (t ) = − Bα (t ) − Q(t ) τ1 τ1

(50b)

with Q(t ) =

8N π2

∞

∑ Bβ (t ) (51)

β=0

2 Note that ∑∞ β=0Bβ(t) = 0, but the front factor in eq 51, 8N/π

with N ≫ 1 (or N → ∞) in our formulation, allows Q(t) to remain nonzero. In this sense, Q(t) serves as the Lagrange factor. (The normalized end-to-end vector of the linear chain, B0(t) = N−1∫ N0 u[L](n,t) dn, does not exhibit the Rouse fluctuation if we erroneously set Q(t) = 0, as noted from eq 52.) In fact, solving eqs 52−54 without presetting the value of Q(t), we obtain nonzero Q̃ (s) together with B̃α (s) and Ã α (s) (Laplace transformations of Q(t), Bα(t), and Aα(t)), as explained in detail in Appendix D (cf. eq D16 in Supporting Information). This type of result is characteristic to a problem of calculating the Lagrange factor and main variables consistently. (In our case, Q̃ (s) was consistently calculated together with the main variables, B̃α (s) and Ã α (s).) Considering the Rouse evolution of the cosine mode amplitudes specified by eqs 50 and 51 and adding the contribution from the association/dissociation reaction (via mapping of the chain conformation explained below), we can rewrite eq 35, in the presence of the association/dissociation reaction, as For linear chain: 1 Ḃ 0(t ) = − Q(t ) 2τ1

(52)

⎛ 4α 2 1⎞ 1 1 Ḃα (t ) = −⎜ + ⎟Bα (t ) − Q(t ) + A α(t ) τas ⎠ τ1 τas ⎝ τ1 (1 ≤ α ≤ N /2)

4α 2 1 Ḃα (t ) = − Bα (t ) − Q(t ) τ1 τ1

(53a)

(α > N /2)

(53b)

Similarly, eq 38 is rewritten as For ring chain: ⎛ 4α 2 1⎞ 1 + Ȧ α (t ) = −⎜ Bα (t ) ⎟A α (t ) + τ τ τ ⎝ 1 ds ⎠ ds (1 ≤ α ≤ N /2)

(54)

Here, several comments need to be made for eqs 52−54. First of all, the expansion for the linear and ring chains has been made with respect to the same base function, cos(2απn/N). For this reason, eqs 52−54 do not include the summation similar to those G


Article

Macromolecules probability. Within this approximation, eq 52 properly accounts for the constraint B0 = 0 required for the ring formation and the other constraint, the lack of change of B0 on the ring dissociation. A more rigorous treatment without the preaveraging should require us to introduce a fully conformation-dependent, stochastic endassociation time, τas({Bα(t)}). With such a treatment, eqs 52−54 should be rewritten in an intractably nonlinear form. Those nonlinear time evolution equations can be managed with a numerical Brownian simulation, but such numerical calculation is beyond the scope of this study aiming at analytical examination of the chain dynamics in the presence of the reaction. 2.4.4. Solution of Eqs 53 and 54. The Q(t) term appearing in eqs 52 and 53 is linear to all amplitudes Bβ(t) (and expressed as a simple sum; cf. eq 51). Thus, we can analytically solve eqs 53 and 54 with the Laplace transformation method to obtain the expression of Bα(t) and Aα(t) with α ≥ 1, as explained in Appendix D of the Supporting Information. (B0(t) is given by −∑∞ α=1Bα(t).) The results can be summarized as the expansion series shown below. Readers who like to avoid bulky expansion series can skip the following equations and directly forward to section 3. For linear chain: N /2

+

k=1

Kλβ−Ω(t ;

λα*, Λk , λβ−)}

(α > N /2)

(57a)

∞

k=1

(α > N /2) 2K K+1

[B] aαβ (t ) =

−

(57b)

∞

∑ R k{λβ+Ω(t ; λα*, Λk , λβ+) k=1

λβ−Ω(t ;

λα*, Λk , λβ−)}

(a > N /2)

(57c)

For ring chain: N /2

A α (t ) =

∞

[A] ∑ Bβ (0)bαβ (t ) + ∑

*[A](t ) Bβ (0)bαβ

β > N /2

β=1 N /2

*[B](t ) Bβ (0)bαβ

[A] (t ) ∑ Aβ(0)aαβ

+

β > N /2

β=1

∞

∑ R k{λβ+Ω(t ; λα*, Λk , λβ+)

*[B](t ) = δαβ exp( −λα*t ) + 2λβ* ∑ R k Ω(t ; λα*, Λk , λβ*) bαβ

∞

[B] ∑ Bβ (0)bαβ (t ) + ∑

Bα (t ) =

2 K+1

[B] bαβ (t ) =

(1 ≤ α ≤ N /2) (58)

β=1

N /2 [B] (t ) ∑ Aβ(0)aαβ

+

with

(α ≥ 1) (55)

β=1

[A] bαβ (t ) =

with [B] bαβ (t ) =

1 δαβ {exp(−λα+t ) + K exp(−λα−t )} K+1 ∞ 2λβ+ + ∑ R k{Ω(t ; λα+ , Λk , λβ+) 2 (K + 1) k = 1

− Ω(t ; λα− , Λk , λβ+)} +

+ K Ω(t ;

Λk , λβ−)}

×

2λβ* K+1

∑ R k{Ω(t ;

(1 ≤ α ≤ N /2)

Λk , λβ*) (1 ≤ α ≤ N /2)

[A] (t ) = aαβ

K δαβ {exp(−λα+t ) − exp(−λα−t )} K+1 ∞ 2K + R k{λβ+Ω(t ; λα+ , Λk , λβ+) ∑ (K + 1)2 k = 1 − +

K+1

λβ−Ω(t ;

Λk , λβ−)} ∞ 2K 2 R k{λβ+Ω(t ; 2 (K + 1) k = 1

− λβ−Ω(t ; λα− , Λk , λβ−)}

∞

∑ R k{Ω(t ; λα+ , Λk , λβ*) k=1

(1 ≤ α ≤ N /2)

(59b)

1 δαβ {K exp(−λα+t ) + exp(−λα−t )} K+1 2Kλβ+ ∞ ∑ R k{Ω(t ; λα+ , Λk , λβ+) + 2 (K + 1) k = 1 − Ω(t ;

λα+ ,

∑

(59a)

− Ω(t ; λα− , Λk , λβ*)}

(56b) [B] aαβ (t ) =

2λβ*

*[A](t ) = bαβ

k=1

+ K Ω(t ; λα− , Λk , λβ*)}

∑ R k{Ω(t ; λα+ , Λk , λβ−) − Ω(t ; λα− , Λk , λβ−)} (1 ≤ α ≤ N /2)

∞

λα+ ,

(K + 1)2

k=1

(56a)

*[B](t ) = bαβ

2Kλβ−

∞

+ K Ω(t ; λα− , Λk , λβ+)} 2Kλβ− ∞ + ∑ R k{Ω(t ; λα+ , Λk , λβ−) (K + 1)2 k = 1 λα− ,

1 {exp(−λα+t ) − exp(−λα−t )}δαβ K+1 ∞ 2λβ+ ∑ + R k{Ω(t ; λα+ , Λk , λβ+) (K + 1)2 k = 1

λα− ,

Λk , λβ+)}

−

2Kλβ− (K + 1)2

∞

×

λα− , Λk , λβ+)

∑ R k{Ω(t ; λα+ , Λk , λβ−) − Ω(t ; λα− , Λk , λβ−)} k=1

(1 ≤ α ≤ N /2)

(1 ≤ α ≤ N /2)

(59c)

In eqs 56, 57, and 59, rd (= τ1/τds) is the normalized dissociation rate specified by eq 42, and K (= ra/rd = [R]eq/[L]eq) is the equilibrium constant defined by eq 3. λ+α and λ−α are the relaxation

(56c)

and H


Article

Macromolecules rates specified by eq 41. In the analysis using eq 53b in the current form, λ*α is a relaxation rate defined by λα* =

4α 2 (=λα+) τ1

with R k1 =

(60)

+

Rk2

(62a)

Ω(t ; λ* , λ , λ*) = [Ω(t ; λ* , λ , λ′)]λ ′→ λ* t exp( − λ*t ) exp( − λt ) − exp( − λ*t ) + λ − λ* (λ* − λ)2 (62b)

1 2 t exp( −λt ) 2 (62c)

As explained in Appendix D (cf. Supporting Information), the above results were obtained with the Laplace transformation method, and Λk and Rk appearing in eqs 56, 57, and 59 are the relaxation rate and residue of a kernel function involved in this method. Specifically, a normalized eigenvalue μk of the kernel function is determined by

−

μk

+

{

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

N

8 K+1 Rk = 2 τ1π R k1 + R k 2

(66b)

⎛ pπ n ⎞ ⎟ Ξ*p (t ) sin⎜ ⎝ N ⎠

(67)

with N

}

Ξ*p (t ) =

N /2

∑

Ξq(0)ξpq(t ) +

q = odd

ξpq(t ) = (63)

∞

4q pπ 2

∑ α=1

∑

+

β > N /2

(64)

χpβ (t ) =

1 pπ

∑ Aβ(0)χpβ (t ) (68a)

β=1

∞

μk 2 τ1

∑ p = odd

Λk and Rk are expressed in terms of this μk as Λk =

⎫ ⎞⎪ μk 2 4K ⎛ ⎜ ⎟ 2 − ⎜ ⎟⎬ (K + 1)rd − μk 2 ⎠⎪ μk 2 π 2 ⎝ ⎭

u[L] odd(n , t ) =

(K + 1)rd − μk 2

⎧ 1 1 ⎫ ⎬=0 − 2K ⎨ + 2 μk 2 ⎭ ⎩ (K + 1)rd − μk

}

(For further details, see Appendix D in the Supporting Information.) Equations 55−60 specify the dynamics of the linear and ring chains in the presence of the association/dissociation reaction, except for the initial mode amplitudes, Bα(0) and Aα(0). The cosine mode amplitudes Aα of the ring chain (cf. eq 22) has the Gaussian distribution at equilibrium, so that the dyadic average on the affine deformation at t = 0, ⟨Aα(0)Aβ(0)⟩xy, is expressed simply as in eq 24 (see also Appendix B1 in the Supporting Information). In contrast, the cosine mode amplitudes Bα of the linear chain do not have the Gaussian distribution at equilibrium because Bα are subjected to the constraint ∑∞ α=1Bα = −B0, as explained in Appendix B3. The dyadic average on the affine deformation, ⟨Bα(0)Bβ(0)⟩xy, could be obtained from this distribution function but only after tedious calculation. To avoid this difficulty, we can focus on the amplitudes of the odd sine modes of the linear chain Ξp that obeys the Gaussian distribution at equilibrium to have the simple dyadic average on the affine deformation (eq 18; see also Appendix B1 in Supporting Information). Specifically, with the aid of eq 48b, we can express the initial amplitudes Bα(0) appearing in eqs 55 and 58 in terms of Ξp(0). Then, from eqs 49 and 55, the contribution of the odd sine modes to the bond vector of the linear chain is expressed as

t exp( −λt ) exp(−λt ) − exp(−λ*t ) − λ* − λ (λ* − λ)2

Kπ coth (π /2) (K + 1)rd − μk 2

(K + 1)rd − μk 2

⎫⎧ ⎧ μk 2 ⎪ cot(μ π /2) k ⎬⎨ ⎨ = 1+ 2 ⎪ ( /2) μ π (K + 1)rd − μk ⎭⎩ ⎩ k +

Ω(t ; λ*, λ , λ) = [Ω(t ; λ*, λ , λ′)]λ ′→ λ

π cot(μk π /2)

2

k

(66a)

For some special cases of λ*, λ, and λ′, Ω(t;λ*,λ,λ′) is simplified as

Ω(t ; λ , λ , λ) = [Ω(t ; λ*, λ , λ′)]λ ′ , λ*→ λ =

{(K + 1)r − μ } sinh {(π /2) d

1 ⎧ exp( −λ′t ) − exp( −λ*t ) ⎨ Ω(t ; λ*, λ , λ′) = λ − λ′ ⎩ λ* − λ′ exp( −λt ) − exp( −λ*t ) ⎫ ⎬ − ⎭ λ* − λ (61)

=

Kμk2 2

(If the decay term −Bα(t)/τas is added to eq 53b, λα* is modified as (4α2 + Krd)/τ1. This addition is conceptually problematic, as discussed in the previous section. However, the results shown later in Figures 2−6 were negligibly affected, in a numerical sense, by this modification of λ*α .) The function Ω(t;λ*,λ,λ′) appearing in eqs 56, 57, and 59 is given by

=

1 sin (μk π /2) 2

∞

∑ α=1

⎧ N /2 16α 2 ⎪ 1 [B] ⎨∑ 2 bαβ (t ) 2 2⎪ p − 4α ⎩ β = 1 q − 4β 2 ⎫ ⎪ 1 [B] * ⎬ b ( t ) 2 2 αβ ⎪ q − 4β ⎭

(68b)

16α 2 [B] aαβ (t ) p2 − 4α 2

(68c)

For calculation of eq 68, we have utilized Fourier-sine expansion of the {cos(2απn/N) − 1} term appearing in eq 49:

(65) I


Article

Macromolecules ⎛ 2απn ⎞ ⎟ − 1 = cos⎜ ⎝ N ⎠

⎛ pπ n ⎞ 16α 2 ⎟ sin⎜ pπ (p2 − 4α 2) ⎝ N ⎠

∑ p = odd

Substituting eqs 72, 73 and eqs 44 and 45 in eq 8, we immediately obtain the expressions of the normalized viscoelastic relaxation function: For linear chain:

(69)

Similarly, from eqs 48b and 58, the contribution of the cosine modes to the bond vector of the ring chain (defined in eq 22) is obtained as N /2

u[R] cos(n , t ) =

⎛ 2απn ⎞ ⎟ N ⎠

∑ A*α(t ) cos⎜⎝

α=1

g [L](t ) =

N

2K (1 − K ) 2K 2 exp(− 2λα−t ) + (1 + K )2 (1 + K )2 N N ⎫ 1 × exp(− {λα+ + λα−}t )⎬ + ∑ ∑ {ξpk(t )}2 ⎭ N p = odd k = odd

(70)

N /2

∑

Ξq(0)ραq (t ) +

q = odd

N /2

4 ραq (t ) = π

q

∑

q2 − 4β 2

β=1

4 + π

[A] (t ) ∑ Aβ(0)aαβ

∞

[A] bαβ (t )

q

∑

q2 − 4β 2

β > N /2

+

(71a)

β=1

g [R](t ) =

*[A](t ) bαβ (71b)

The functions b*αβ[c](t), and a[c] αβ (t) (c = A, B) [R] the above expressions of u[L] odd(n,t) and ucos (n,t)

involved in have been specified by eqs 56, 57, and 59, and the initial amplitudes Ξp(0) and Aβ(0) are characterized by eqs 18 and 24b. Thus, eqs 67−71 fully specify the dynamics of the odd sine modes of the linear chain and the cosine modes of the ring chain. Correspondingly, the orientational correlation functions of these chains defined by eq 7, the quantity of our interest, are obtained as For linear chain: [L] [L] S[L](n , m , t ) = Seven (n , m , t ) + Sodd (n , m , t )

N p = odd

⎪

⎪

⎪

⎪

⎛ pπn ⎞ ⎛ qπm ⎞ ⎟ sin⎜ ⎟ × sin⎜ ⎝ N ⎠ ⎝ N ⎠ +

2γ 3N

N

N

N /2 ⎧ ⎪

⎫ ⎪

⎪

⎭

∑ ∑ ⎨ ∑ χpβ (t )χqβ (t )⎬

⎪

⎩ β=1 ⎛ pπn ⎞ ⎛ qπm ⎞ ⎟ sin⎜ ⎟ × sin⎜ ⎝ N ⎠ ⎝ N ⎠ p = odd q = odd

N /2

⎧ 1 + K2 exp(− 2λα+t ) 2 (1 + ) K ⎩ α=1 2 + exp(− 2λα−t ) (1 + K )2 N /2

1 N

∑⎨

+

⎫ 2(K − 1) exp(− {λα+ + λα−}t )⎬ 2 (1 + K ) ⎭

+

1 N

N /2

N

∑ ∑

{ραk (t )}2 +

α = 1 k = odd

(72b)

[R] [R] S[R](n , m , t ) = Ssine (n , m , t ) + Scosine (n , m , t )

(73a)

[R] where S[R] sine(n,m,t) has been specified by eq 45 and Scosine(n,m,t) is given by [R] Scosine (n , m , t ) =

2γ 3N

∑

⎧ N ⎫ ∑ ⎨ ∑ ραk (t )ρβk (t )⎬ ⎭ β = 1 ⎩ k = odd

α=1

⎪

⎪

⎪

⎪

⎛ 2απn ⎞ ⎛ 2βπm ⎞ ⎟ cos⎜ ⎟ × cos⎜ ⎝ N ⎠ ⎝ N ⎠ +

2γ 3N

N /2 N /2

⎧ N /2

⎫

⎪

⎭

⎪

α=1

⎛ 2απn ⎞ ⎛ 2βπm ⎞ ⎟ cos⎜ ⎟ × cos⎜ ⎝ N ⎠ ⎝ N ⎠

N /2 N /2 [A] (t )}2 ∑ ∑ {aαβ α=1 β=1

[B] bαβ (0) = 0 for α > N /2 (76a)

*[B](0) = 0 for 1 ≤ α ≤ N /2, bαβ

*[B](0) = δαβ for α > N /2 bαβ (76b)

[B] aαβ (0) = 0 for 1 ≤ α ≤ N /2

(76c)

[A] *[A](0) = 0 for 1 ≤ α ≤ N /2 bαβ (0) = bαβ

(77a)

[A] aαβ (0) = δαβ for 1 ≤ α ≤ N /2

(77b)

Substituting eqs 76 and 77 respectively in eqs 68 and 71, we recover, from eqs 67 and 70, the correct expressions of contributions of the odd sine modes of the linear chain and the cosine modes of the ring chain to respective bond vectors at t = 0:

⎪

∑ ∑ ⎨ ∑ aα[A]k (t )a[A] βk (t )⎬ β=1 ⎩ k=1

1 N

[B] bαβ (0) = δαβ for 1 ≤ α ≤ N /2,

For ring chain:

N /2 N /2

(74)

p = odd β = 1

3. DISCUSSION 3.1. Analytically Noted Features. It is informative to analytically examine characteristic features of the subchain bond vector u[c](n,t) and the orientational correlation function S[c](n,m,t) of the linear and ring chains (c = L, R) obtained from the formulation explained in previous sections. At first, we note from eqs 56, 57, and 59 that bαβ and aαβ have the following initial values irrespective of the values of K and rd (= τ1/τds). (The function Ω(t;λ*,λ,λ′) appearing in eqs 56, 57, and 59 is equal to zero at t = 0 for any value of λ*, λ, and λ′, as noted from eqs 61 and 62.)

(72a)

⎧ N ⎫ ⎨ ∑ ξpk(t )ξqk(t )⎬ ⎭ q = odd ⎩ k = odd N

∑ ∑

N

∑ ∑ {χpβ (t )}2

(75)

[L] where S[L] even(n,m,t) has been specified by eq 44 and Sodd(n,m,t) is given by

2γ 3N

1 N

For ring chain:

b[c] αβ (t),

[L] Sodd (n , m , t ) =

⎧ 1 + K2 exp(− 2λα+t ) 2 (1 + ) K ⎩ α=1 N /2

∑⎨

+

with A*α(t ) =

1 N

⎪

N

u[L] odd(n , 0) = (73b)

∑ p = odd

J

⎛ pπ n ⎞ ⎟ Ξp(0) sin⎜ ⎝ N ⎠

(78a)


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u[R] cos(n , 0) =

⎛ 2απn ⎞ ⎟ N ⎠

∑ A α(0) cos⎜⎝

α=1

shown below were obtained from eqs 72−75 based on eq 53b in the current form without the decay term, −Bα(t)/τas. Addition of this term modified the relaxation rate λ*α as explained for eq 60, but this modification negligibly affected those results, in a numerical sense, for the rd and K values examined.) [c] g[c](t) and S ̃ (n , n , t ) (eqs 72−75) include functions ξpq(t), χpβ(t), and ραq(t) that are expressed as single or double summation over indices running from 1 to ∞ (cf. eqs 68b, 68c, and 71b). The numerical calculation utilized the eigenvalues μk (cf. eqs 63−66) up to k = 200 (= 4N), and the summation was truncated at the maximum index of 250 (= 5N), which gave good convergence. The results of the calculation are summarized in Figures 2−6. In Figure 2, green and red circles show g[L](t) and g[R](t) of the linear and ring chains calculated for the rd and K values as

(78b)

In calculation of eq 78a, we have utilized eq 68b combined with the mathematical formulas,28 ∞

∑ α= 1 ∞

∑ α= 1

1 1 π = 2 − cot(πx) 2 2x α −x 2x 2

(79)

1 π2 π 1 = 2 cosec 2(πx) + 3 cot(πx) − 4 2 2 2x (α − x ) 4x 4x 2

(80)

Corresponding to the above results, eqs 72 and 73 correctly reproduce the initial condition of the orientational correlation function: for linear chain: S[L](n , m , 0) =

N

⎛ pπ n ⎞ ⎛ pπ m ⎞ ⎟ sin⎜ ⎟ N ⎠ ⎝ N ⎠

2γ 3N

∑ sin⎜⎝

2γ 3N

∑ cos⎜ 2απ(n − m) ⎟

p=1

(81a)

for ring chain: S[R](n , m , 0) =

N /2

α= 1

⎛ ⎝

⎞

N

⎠

(81b)

In addition to the above general feature related to the initial condition, it is also informative to examine the results of our analysis in the limit of slow dissociation reaction specified by rd = τ1/τds → 0, or, equivalently, by λ−α , λ*α → λ+α = 4α2/τ1. In this limit, the functions given by eq 59 for the ring chain are simplified 2 as b[A] *[A](t) = 0, and a[A] αβ (t) = 0, bαβ αβ (t) = δαβ exp(−4α t/τ1), which correctly reproduces the pure Rouse behavior of the subchain bond vector u[R](n,t) (eqs 22 and 23) and the orientational correlation function S[R](n,m,t) (eq 25) of the ring chain, as explained in Appendix E (cf. Supporting Information). Our formulation also reproduces the pure Rouse behavior of the linear chain (eqs 15 and 19) in the slow reaction limit, as noted after some calculation shown in Appendix E. It should be emphasized that the pure Rouse behavior is recovered when rd → 0 irrespective of the value of the equilibrium constant, K = [R]eq/[L]eq, because in this case the Rouse relaxation completes much faster compared to the chemical equilibration reflected in K. 3.2. Numerically Noted Features. In experimentally achievable conditions, the dissociation reaction time of the ring chain would be comparable to/shorter than the Rouse relaxation time, and the chemical equilibrium would be largely shifted toward the linear chain (namely, K = [R]eq/[L]eq ≪ 1). The features of the linear and ring chains for such cases are numerically examined below. For seemingly plausible values of the parameters, rd = τ1/τds = 0.1−10 and K = [R]eq/[L]eq = 0.1, eq 63 was numerically solved to evaluate the normalized eigenvalues μk. The subchain number per chain was chosen to be N = 50. For comparison, μk was evaluated also for rd = 10 and K = 1. The largest rd value, rd = 10, was chosen to be still moderate because the actual dissociation should involve the motion of the ring chain and cannot be extremely faster than the slowest Rouse relaxation of this chain having the characteristic time, τ1/4. For both linear and ring chains, we utilized those μk values to calculate the normalized viscoelastic relaxation function g[c](t) and the normalized orientational anisotropy of the subchain [c] S ̃ (n , n , t ) ≡ {γ /3}−1S[c](n , n , t ) (c = L and R). (The results

Figure 2. Normalized viscoelastic relaxation function of linear and ring chains with N = 50 calculated for the rd and K values as indicated.

indicated. These g[c](t) are double-logarithmically plotted against the normalized time t/τoG,L, with τoG,L (= τ1/2) being the longest viscoelastic Rouse relaxation time of the linear chain in the absence of the reaction. The black curves show the pure Rouse relaxation functions in the absence of reaction, g[L] Rouse(t) and g[R] (t) (eqs 20 and 26). Rouse [c] In Figures 3−6, S ̃ (n , n , t ) at reduced times t/τ1 = 0, 0.01, 0.1, 0.5, and 1 (t/τoG,L = 0, 0.02, 0.2, 1, and 2) is shown as a K


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Figure 3. Normalized orientation function of linear and ring chains with N = 50 calculated for rd = 0.1 and K = 0.1 at reduced times t/τ1 as indicated.

Figure 4. Normalized orientation function of linear and ring chains with N = 50 calculated for rd = 1 and K = 0.1 at reduced times t/τ1 as indicated.

function of the normalized subchain index, n/N; see red circles. [c] The green curves indicate S ̃ (n , n , t ) for the pure Rouse dynamics (cf. eqs 19 and 25). Starting from the initial value, [c] [c] S ̃ (n , n , 0) = 1, S ̃ (n , n , t ) decays with t significantly (by a factor of >100 at t = τ1), so that its features at long t cannot [c] be clearly resolved if we make linear plots of S ̃ (n , n , t ) against n/N. For this reason, Figures 3−6 show plots of the logarithm of [c] S ̃ (n , n , t ) against n/N. For rd = 0.1 and K = 0.1, g(t) of the linear and ring chains are indistinguishable from the pure Rouse relaxation function g(t) Rouse (see Figure 2a). Correspondingly, S(̃ n , n , t ) of those chains is very close to that for the pure Rouse dynamics (cf. Figure 3). Thus, for rd = τ1/τds = 0.1 (and ra = τ1/τas = Krd = 0.01), the reaction is too slow to significantly affect the viscoelastic and orientational relaxation behavior of the chains. Namely, the results obtained for rd = 0.1 are close to those in the analytical limit of rd → 0 explained earlier. With increasing rd from 0.1 to 1 and further to 10 while keeping K = [R]eq/[L]eq = 0.1, the viscoelastic and orientational relaxation behavior of the linear chain is hardly affected by the reaction because the linear chain is much more populated compared to the reaction counterpart (ring chain) and its motional coupling with the ring chain is not significant. Thus, [L] g[L] (t) and S ̃ (n , n , t ) of the linear chain are almost indistinguishable from the pure Rouse functions, as noted in Figures 2b, 2c, 4, and 5. In contrast, for the ring chain, the increase of rd (up to 10) retards the terminal relaxation of g[R](t) [R] and activates oscillation of S ̃ (n , n , t ) with the subchain index n/N at long t, which reflects strong motional coupling of the ring chain with the much more populated linear chain. The fast relaxation modes of g[R](t) are hardly affected by the coupling whereas the slow (terminal) modes are retarded by the coupling,

Figure 5. Normalized orientation function of linear and ring chains with N = 50 calculated for rd = 10 and K = 0.1 at reduced times t/τ1 as indicated.

so that the g[R](t) plot crosses the pure Rouse curve as shown in [R] Figures 2b and 2c. The oscillation of S ̃ (n , n , t ) reflects split of the relaxation time scale of the sine and cosine branches of L


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Macromolecules [R]

cos2(2πn/N) type oscillation of S ̃ (n , n , t ); see the bottom panel of Figure 6. In relation to this point, we also note in the top panel that the reaction moderately reduces the difference of the orientational anisotropy at the middle and ends of the linear [L] chain, S ̃ (n , n , t ) at n = N/2 and n ∼ 0, compared to the pure Rouse behavior (green curves) at long t. This reduction can be qualitatively understood as a result of the motional coupling between the linear and ring chains. The coupling breaks the ring symmetry, but the anisotropies of the subchains of the ring at n = 0 and N/2 still remain almost equivalent even at long t, as [R] noted from the cos2(2πn/N) type oscillation of S ̃ (n , n , t ) (note that cos2(2πn/N) = 1 at n = 0 and N/2). This equivalence appears to be partly transferred to the linear chain (through the [L] coupling), thereby reducing the difference of S ̃ (n , n , t ) of the linear chain at n ∼ 0 and N/2.

4. CONCLUDING REMARKS For dilute telechelic linear and ring Rouse chains undergoing the end-association and dissociation reaction, we have analytically formulated the time evolution equation for the subchain bond vector, u[c](n,t) with n being the subchain index and c = L and R for the linear and ring chains. Considering a fact that the endassociation of the linear chain (i.e., ring formation) occurs only when the linear chain ends come into close proximity, we formulate the equation for u[L](n,t) of the linear chain with the two-step expansion method, within a preaveraging approximation for the end-encounter probability. In this method, u[L](n,t) is first expanded with respect to the sinusoidal Rouse eigenfunctions, sin(pπn/N) with p = integer, and then the series of odd sine modes is re-expanded with respect to cosine functions, cos(2απn/N) with α = integer (= 0−∞). For the ring chain, u[R](n,t) was expanded with respect to its Rouse eigenfunctions, sin(2απn/N) and cos(2απn/N). The time evolution equation of u was analytically solved on the basis of this two-step expansion to calculate the orientational correlation functions S[c](n,m,t) = [c] b−2⟨u[c] x (n,t)uy (m,t)⟩ and the viscoelastic relaxation functions [c] g (t) of the linear and ring chains. It turned out that the terminal viscoelastic relaxation processes of the ring and linear chains are slower and faster than respective pure Rouse processes when the motional coupling due to the reaction is significant for both chains. We also noted that the ring symmetry (equivalence of all subchains of the ring chain in the absence of reaction) is broken by this coupling, thereby leading to oscillation of the orientational anisotropy S[R](n,n,t) of this chain with n at long t. This oscillation was (partly) transferred to the linear chain again through the coupling, thereby reducing, at long t, a difference of the anisotropy of the linear chain at the middle and end, S[L](n,n,t) at n ∼ 0 and N/2. All above results were obtained for linear and ring chains at association/dissociation equilibrium. In other words, the linear chains were assumed to be dilute and form neither linear nor ring multimer. Experimentally, it would not be easy to detect the viscoelastic relaxation of such dilute linear and ring chains, in particular for the ring chain that should be much less populated than the linear chain in usual conditions of the equilibrium constant, K ≪ 1. For K ≪ 1, the measured modulus is dominated by the linear chains (cf. eq 9), and their g[L](t) hardly deviates from the Rouse function (cf. Figures 2a−c). Thus, the measured modulus does not seem to sensitively reflect the effect of the motional coupling (due to the reaction) on the ring chain dynamics. However, with a rheo-optical (dichroism) method, we have a chance to selectively detect the orientation anisotropy of 0th

Figure 6. Normalized orientation function of linear and ring chains with N = 50 calculated for rd = 10 and K = 1 at reduced times t/τ1 as indicated. [R] the orientation function, S[R] sine(n,m,t) (eq 45) and Scos (n,m,t) (eq 73b), due to the coupling with the linear chain dynamics. The slowest mode of the sine branch has the relaxation time τ1/8 (= 1/2λ+α; cf. eq 45) whereas the slowest mode of cosine branch relaxes with a longer time ∼τ1/2 (= 1/2Λ1 = τ1/2μ12 with μ1 ∼ 1 for the parameter values examined, rd ≤ 10 and K = 0.1; cf. [R] eqs 59−62, 71b, and 73b). Thus, the behavior of S ̃ (n , n , t ) at long t is dominated by the slowest mode of the cosine branch that oscillates as cos2(2πn/N); see eq 73b with α = β = 1. This cos2(2πn/N) type oscillation is clearly noted in Figure 5 (in the [R] logarithmic scale) at long t = τ1. This oscillation of S ̃ is smeared [R] in g because of the integral with respect to the subchain index n (cf. eq 8). Nevertheless, the strong coupling that activates [R] significant oscillation of S ̃ can be noted also for g[R] through the retardation of slow modes of g[R] explained above. The above behavior of the ring chain demonstrates an important effect of the reaction that couples the ring chain dynamics with the linear chain dynamics. In the absence of the reaction, all subchains of the ring chain are equivalent so that the orientational anisotropy S[R] Rouse(n,n,t) is independent of the subchain index n; see eq 25 and also green lines in the bottom panels of Figures 3−5. This ring symmetry is broken by the reaction occurring at 0th and Nth subchains, and the subchains behave differently according to their contour distance (n or N − n) from the reacting site thereby exhibiting the oscillation of S[R](n,n,t) explained above. This “ring-symmetry breaking” is the most prominent effect of the reaction on the ring chain dynamics. Finally, for rd = 10 and K = 1, we note that the reaction detectably affects the behavior of both ring and linear chains because they are equally populated. Specifically, the terminal relaxation of g[L](t) of the linear chain is accelerated in a way synchronized with the retardation of the terminal relaxation of g[R](t) of the ring chain (see Figure 2d). Furthermore, the ring symmetry breaking is confirmed also for K = 1 as the

M


Article

Macromolecules Ψα(t)

(or Nth) subchain of the ring chain where the association/ dissociation reaction occurs. This anisotropy, expressed as [A] [A] (2γ/3N)∑α,β≥1∑k{ραk(t)ρβk(t) + aαk (t)aβk (t)} (cf. eqs 45 and 73b with n = m = 0), does not vanish until the viscoelastic relaxation completes. In contrast, for the linear chain, the anisotropy at the chain ends remains zero throughout the relaxation process, as can be noted from eqs 44 and 72b with n = m = 0. Namely, the linear chains are the major component (for K ≪ 1) but do not disturb selective detection of the ring anisotropy. Thus, the orientational anisotropy of 0th subchain of the ring could be well resolved as the dichroism of the specific, associating functional groups existing only at n = 0 and N (such as carboxyl groups forming hydrogen bonds at the linear chain ends), and analysis of those dichroism data could provide us with quantitative information for K and the reduced dissociation frequency rd (= τ1/τds) included in the above expression of the anisotropy of 0th subchain. A rheo-optical test of detecting this anisotropy is considered as an interesting subject of future work.

■

Aα(t) g[c](t)

Parameters Appearing in Calculation of Coupled Relaxation of Linear and Ring Chains

λ+α, λ−α

relaxation rates of even sine modes of linear chain and sine modes of ring chain (cf. eq 41); these rates split because of motional coupling of those chains due to the reaction u[L] contribution of odd sine modes to nth odd(n,t) subchain bond vector u[L](n,t) of linear chain at time t (cf. eq 47) Bα(t) expansion coefficients of u[L] odd(n,t) with respect to cosine functions (cf. eqs 47 and 49) Q(t) parameter serving as Lagrange factor in calculation (cf. eq 51) J(̃ s) kernel function appearing in calculation of Q(t) with Laplace transformation method (cf. eq D17 in Appendix D of the Supporting Information) μk , R k normalized eigenvalue and residue associated with first-order poles of kernel function J(̃ s) (cf. eqs 63, 65, and 66; see also eqs D18−D21 in Appendix D of the Supporting Information) Λk relaxation rate of kernel function J(t) in the time domain; Λk is expressed in terms of μk (cf. eq 64; see also eq D22 in Appendix D of the Supporting Information) [B] [B] b[B] (t), b * (t), a (t) normalized decay factors for αth αβ αβ αβ cosine-expansion coefficient Bα(t) of linear chain affected by motional coupling of linear and ring chains (cf. eq 55); these factors are expressed in terms of λ+α, λ−α , Λk, and an additional relaxation rate λα* (cf. eqs 56, 57, and 60−62) [A] [A] b[A] αβ (t), b* αβ (t), aαβ (t) normalized decay factors for amplitude Aα(t) of αth cosine mode of ring chain affected by motional coupling of linear and ring chains (cf. eq 58); these factors are expressed in terms of λ+α, λ−α , Λk, and an additional relaxation rate λα* (cf. eqs 59 and 60−62) ξpk(t), χpβ(t) normalized decay factors appearing in final expression of contribution [L] Sodd (n,m,t) of odd sine modes to [L] S (n,m,t) of linear chain (cf. eq 72); these factors are expressed in terms of [B] [B] b[B] αβ (t), b* αβ (t), and aαβ (t) (cf. eq 68) ραβ(t) normalized decay factors appearing in final expression of contribution [R] S cos (n,m,t) of cosine modes to [R] S (n,m,t) of ring chain (cf. eq 73); these factors are expressed in terms of b[A] *[A](t) (cf. eq 71) αβ (t) and bαβ

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00424. Appendix A: direct calculation of orientational correlation function; Appendix B: mode expansion of subchain bond vector; Appendix C: Rouse evolution of coefficients Bα appearing in eq 47; Appendix D: solution of eqs 52−54; Appendix E: behavior of linear and ring chains in slow reaction limit (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (H.W.). 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), Grantin-Aid for Scientific Research (C) from JSPS, Japan (Grant No. 15K05519), Collaborative Research Program of ICR, Kyoto University (Grant No. 2016-85), and Collaborative Research Program of Kyoto University Research Coordination Alliance.

■

LIST OF SYMBOLS K equilibrium constant (cf. eq 3) Basic TimeConstants

τas τds τ1 ra rd

amplitude of αth sine mode of ring chain at time t (cf. eq 22) amplitude of αth cosine mode of ring chain at time t (cf. eq 22) normalized viscoelastic relaxation function of linear and/or ring chain (c = L, R) at time t (cf. eq 8)

end-association time of linear chain dissociation time of ring chain slowest Rouse relaxation time of linear chain (cf. eq 16) reduced end-association frequency of linear chain (cf. eq 42) reduced dissociation frequency of ring chain (cf. eq 42)

Basic Quantities Specif ying Chain Conformation/Relaxation

u[c](n,t)

bond vector of nth subchain of linear and/or ring chain (c = L, R) at time t S[c](n,m,t) orientational correlation function defined for nth and mth subchains of linear and/or ring chain (c = L, R) at time t (cf. eq 7) S[c](n,n,t) orientational anisotropy of nth subchain of linear and/or ring chain (c = L, R) at time t Ξp(t) amplitude of pth sine mode of linear chain at time t (cf. eq 14) N


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(22) Rubinstein, M.; Semenov, A. N. Dynamics of Entangled Solutions of Associating Polymers. Macromolecules 2001, 34, 1058−1068. (23) Watanabe, H.; Matsumiya, Y.; Masubuchi, Y.; Urakawa, O.; Inoue, Y. Viscoelastic Relaxation of Rouse Chains undergoing Head-to-Head Association and Dissociation: Motional Coupling through Chemical Equilibrium. Macromolecules 2015, 48, 3014−3030. (24) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (25) Watanabe, H. Viscoelasticity and dynamics of entangle polymers. Prog. Polym. Sci. 1999, 24, 1253−1403. (26) Watanabe, H.; Inoue, T. Conformational dynamics of Rouse chains during creep/recovery processes: a review. J. Phys.: Condens. Matter 2005, 17, R607−R636. (27) Watanabe, H.; Inoue, T.; Matsumiya, Y. Transient Conformational Change of Bead-Spring Ring Chain During Creep Process. Macromolecules 2006, 39, 5419−5426. (28) Moriguchi, S.; Udagawa, K.; Hitotsumatsu, S. Mathematical Formula; Iwanami: Tokyo, 1986.

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Viscoelastic and Orientational Relaxation of Linear and Ring Rouse

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