Experimental Test for Viscoelastic Relaxation of Polyisoprene

Sep 2, 2016 - Dynamics of Rouse chains undergoing head-to-head association and dissociation: Difference between dielectric and viscoelastic relaxation...
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Experimental Test for Viscoelastic Relaxation of Polyisoprene Undergoing Monofunctional Head-to-Head Association and Dissociation Yumi Matsumiya* and Hiroshi Watanabe Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan

Osamu Urakawa and Tadashi Inoue Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan S Supporting Information *

ABSTRACT: A viscoelastic test was made for end-carboxylated polyisoprene (PICOOH) of the molecular weight M = 30.5 × 103 that underwent the interchain association and dissociation through hydrogen bonding of the COOH groups at the chain end. As a reference, the test was made also for neat PI unimer (with no COOH group at the chain end) and for PI2 dimer (with M = 61.0 × 103), the latter being synthesized through end-coupling of PI− anions (precursor of the PI-COOH sample). The PI-COOH, neat unimer, and dimer samples were diluted in oligomeric butadiene (oB) to a concentration of 10 wt %. The neat unimer and dimer exhibited nonentangled Rouse behavior at this concentration, as expected from their molecular weights. At low temperatures (T ≤ 0 °C) the PI-COOH sample relaxed slower than the reference unimer but faster than the dimer, whereas the relaxation of PI-COOH approached that of the unimer with increasing T > 0 °C, and this change of the relaxation time of PI-COOH was associated with changes in the angular frequency (ω) dependence of the dynamic modulus. This behavior of PI-COOH was well described by a recently proposed theory considering motional coupling between the end-associating unimer and its dimer at chemical equilibrium. On the basis of this result, an effect of the polymeric character of PI-COOH chain on the viscoelastically detected association/dissociation of the hydrogen bonding of the COOH groups was discussed.

1. INTRODUCTION Associating polymers have been attracting remarkable research interest because of their scientifically new aspect(s) as well as of industrial applications.1−7 Dynamics of those polymer chains is determined by competition between the association/dissociation reaction kinetics and the intrinsic chain motion (the motion in the absence of reaction). When the reaction is much slower than the large-scale intrinsic chain motion, the terminal relaxation of the chain is strongly retarded by the reaction. For this case, the fast relaxation of the chain occurs through its local, intrinsic motion between the associated sites and the slow terminal relaxation occurs through the large-scale motion activated by the dissociation reaction. In contrast, when the reaction is faster than (or at least as fast as) the intrinsic chain motion, the situation becomes more complicated because the reaction strongly couples the motion of associated and dissociated chains for this case. Focusing on this fast reaction case, we have theoretically analyzed the viscoelastic relaxation for the simplest model system free from the entanglement effect, Rouse chains undergoing the monofunctional association and dissociation at the chain ends.8 The key in the analysis was the mapping of © XXXX American Chemical Society

the conformation of associating unimers onto the resulting dimer and the reverse mapping of the dissociating dimer conformation onto the resulting unimers. With an assumption of no change of the chain tension on association of two unimers, the mapping/reverse mapping allowed formulation of the motional coupling of the unimer and dimer in a tractable form, and the normalized relaxation moduli of the unimer and dimer, g1(t) and g2(t), were calculated analytically. The calculation indicated that in the fast reaction limit both g1(t) and g2(t) coincide with the intrinsic Rouse relaxation function of the unimer, g1,R(t).8 From this result, one might consider that the unimer dynamics is not affected by the fast reaction and that all dimers are converted into unimers through the fast dissociation, thereby relaxing simply as the unimers to exhibit g2(t) = g1,R(t). However, this is not the case. The motional coupling due to the fast reaction splits the intrinsic Rouse modes of the unimer (and dimer) and also creates new relaxation modes not deduced from the intrinsic Rouse Received: June 19, 2016 Revised: August 15, 2016

A

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules dynamics, as fully described in the previous paper8 and briefly explained in the following section. Combination of all these effects results in the coincidence of g1(t) and g2(t) with g1,R(t) in the fast reaction limit. This result of the analysis for the Rouse unimer and dimer in turn suggests a possibility of extracting the information for the chemical reaction from the viscoelastic data of the nonentangled unimer/dimer system at equilibrium. It is of interest to experimentally test the theoretical calculation mentioned above and examine the feature of the chemical reaction. Thus, we have synthesized a model polyisoprene (PI) having an associative carboxyl group at one end (PI-COOH) and also two reference samples, neat PI of the same molecular weight but without the carboxyl group and PI2 dimer undergoing no dissociation. These PI-COOH and reference samples were diluted in an oligomeric butadiene so as to eliminate the entanglement effect. Comparison of the viscoelastic behavior of those diluted PI-COOH and reference samples suggested that the dynamics of the associating/ dissociating PI-COOH is in accord with the theoretical calculation and that the viscoelastically detected association/ dissociation is largely affected by the polymeric character of the PI-COOH chain. Details of these results are presented in this article.

[P2]eq [P]eq

2

=

τds 2[P]eq τas

g1 − 2o(t ) = +

1 N

4rard

∑ ∑

2

π N (ra + rd) N

4ra2

2N

p = 1 q = odd

π N (ra + rd)

p {exp(−Λ pt ) − exp(−Θqt )} {p2 − q2 /4}2 {q2 + 2rd − 4p2 }

p2 {exp(−Λ− pt ) − exp(−Θqt )}

2N

∑ ∑

2

+

2

{p2 − q2 /4}2 {q2 − 2ra − 4p2 }

p = 1 q = odd

(2d)

(3a)

with ⎧

⎫ ra rd exp(−Λ+ pt ) + exp(−Λ− pt )⎬ r + rd ra + rd ⎭ p=1 ⎩ a N

1 2N

g2e(t ) =

∑⎨

(3b) N

rd N (ra + rd)

g2e − 1(t ) =

∑ {exp(−Λ+ pt ) − exp(−Λ− pt )} p=1

(3c) g2e − 2o(t ) = −

N

4rard π 2N (ra + rd) N

4rard π N (ra + rd)

2N

∑ ∑ p = 1 q = odd

2N

∑ ∑

2

g2o(t )=

p = 1 q = odd

+

2

p {exp(−Λ pt ) − exp(−Θqt )} {p2 − q2 /4}2 {q2 + 2rd − 4p2 }

p2 {exp(−Λ− pt ) − exp(−Θqt )} {p2 − q2 /4}2 {q2 − 2ra − 4p2 }

(3d)

2N

1 2N

∑

exp( −Θpt ) (3e)

p = odd

The functions g1eo, g2e, and g2o (eqs 2b, 3b, and 3e) represent the Rouse modes modified (and split in the first two) by the motional coupling due to the reaction, and the other functions indicate new relaxation modes created by the coupling. The parameters involved in those functions are given by

(1)

Λ+ p = ra = Θp =

2p2 , τ1

τ1 , τas

Λ− p =

2p2 + ra + rd τ1

(4a)

τ1 τds

(4b)

1 2 (p + 2rd) 2τ1

(4c)

rd =

where τ1 (∝ N2) is the longest Rouse relaxation time for the end-to-end fluctuation of the unimer in the absence of reaction. The longest viscoelastic Rouse relaxation time of the unimer is given by τ1/2. As noted from the eqs 2−4, the relaxation behavior of g1(t) and g2(t) is governed by the parameters ra and rd (eq 4b) that represent the characteristic frequencies of the association and dissociation normalized by the Rouse relaxation frequency. g1(t) and g2(t) reduce to the pure Rouse relaxation functions of the unimer and dimer, g1,R(t) and g2,R(t), in the limit of slow reaction, i.e., for ra, rd → 0.

(2a)

⎫ rd ra exp(−Λ+ pt ) + exp(−Λ− pt )⎬ r + rd ra + rd ⎭ p=1 ⎩ a N

N

g2(t ) = g2e(t ) + g2e − 1(t ) + g2e − 2o(t ) + g2o(t )

with g1e o(t ) =

p=1

For dimer:

where [P]eq and [P2]eq represent the equilibrium molar concentrations of the unimer and dimer, respectively. The viscoelastic relaxation is equivalent to the orientational relaxation of subchains (or chain segments) averaged along the chain backbone.8−10 With an assumption that the chain tension does not change on association of two unimers, the unimer conformation is directly mapped onto the resulting dimer.8 For this case, the zero-force boundary condition at the unimer chain ends is transferred to the midpoint of the created dimer backbone. Correspondingly, the conformation of a dissociating dimer is mapped onto the unimers, thereby transferring the orientational anisotropy at the midpoint of the dimer backbone to the ends of the created unimers. These conformational mapping operations are combined with the Rouse relaxation of the orientational anisotropy to give the relaxation moduli g1(t) and g2(t) normalized to unity at time t = 0. For the Rouse unimer and dimer composed of N and 2N subchains, respectively, the results can be summarized as8 For unimer: g1(t ) = g1e o(t ) + g1 − 2e(t ) + g1 − 2o(t )

∑ {exp(−Λ+ pt ) − exp(−Λ− pt )} (2c)

2. THEORY For convenience of later discussion of the viscoelastic relaxation of PI-COOH chains, we here briefly summarize the previously obtained expression of the normalized relaxation moduli of the unimer and dimer Rouse chains, g1(t) and g2(t), at the associating/dissociating equilibrium.8 The characteristic times of the association and dissociation reactions, τas and τds, are included in the equilibrium constant K as8 K≡

N

ra 2N (ra + rd)

g1 − 2e(t ) =

⎧

∑⎨

g1,R (t ) =

(2b) B

1 N

N

⎛ 2t ⎞ ⎟⎟ ⎝ τp ⎠

∑ exp⎜⎜− p=1

(5a) DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules g2,R (t ) =

1 2N

2N

⎛

t ⎞⎟ ⎟ ⎝ 2τp ⎠

∑ exp⎜⎜− p=1

inert unimer and dimer were utilized as the reference materials for the associative PI-COOH synthesized in the following way. The PI-DPE− anion in the third (main) flask was allowed to react with CO2 to convert the chain end into COOLi group. No detectable coupling occurred (as confirmed from GPC) because the DPE− anions, having fully dissociated countercation (Li+) in a polar solvent (THF), were allowed to react rapidly with much excess (∼20 mol equiv) of CO2 that was dissolved in THF in advance. The PI-DPECOOLi sample thus obtained were dissolved in THF and allowed to react with ∼5 mol equiv of HCl to fully convert the COOLi group at the chain end into the COOH group. Details of these reaction procedures, the GPC data confirming no coupling, and 7Li NMR data confirming the full conversion into the −COOH group are shown in the Supporting Information. The PI sample having the acid (COOH) group at the chain end as well as the neat PI unimer and PI2 dimer samples were characterized with GPC (CO-8020 and DP-8020; Tosoh) equipped with a refractive index (RI) monitor (RI-8020, Tosoh). The elution solvent was THF, and the previously obtained monodisperse linear PI samples11,12 were utilized as the elution standards. The molecular weight and polydispersity index were Mw = 30.5 × 103 and Mw/Mn = 1.02 for the acid-end and neat PI samples and Mw = 61.0 × 103 and Mw/Mn = 1.02 for the dimer sample. These samples are hereafter designated as PI30-COOH, PI30, and (PI30)2, respectively, with the sample code number representing the molecular weight in unit of 1000. (For simplicity, the DPE unit included in the chain backbone is not shown in the sample code.) Because this study attempts to examine the effect of association/ dissociation reaction on the relaxation of nonentangled (Rouse) chains, viscoelastic measurements were conducted for the PI30COOH, PI30, and (PI30)2 samples diluted with a 1,2-rich oligomeric butadiene (oB), a good solvent for PI. An oB sample purchased from Polymer Source Co. (1,2 content = 85% and Mn = 3.0 × 103; manufacture’s designation) was utilized after purification through dissolution and precipitation in benzene and methanol, respectively. This purified oB sample was characterized with GPC to determine its Mw (3.5 × 103) and Mw/Mn (= 1.05) and is hereafter designated as oB3. For viscoelastic measurements, 10 wt % PI/oB3 solutions were prepared by first dissolving prescribed masses of oB3 and one of the PI samples in benzene. Then, these solutions were cast in thin films, and the films were fully dried under vacuum at 40 °C for 1 week to thoroughly remove benzene and a trace amount of water. The resulting PI/oB3 solutions (or blends) were kept in a desiccator filled with pure argon at room temperature and were dried again under vacuum at 40 °C for a day just prior to use. The PI molar concentration in the 10 wt % solution was 3.0 mmol/dm3 for PI30 and PI30-COOH and 1.5 mmol/dm3 for (PI30)2. 3.2. Measurement. The PI/oB3 solutions prepared as above were subjected to linear viscoelastic (dynamic) measurements in a range of angular frequency ω covering from 102.5 to 10−2.5 s−1 at several temperatures T between −20 and 25 °C. The measurements were conducted with a strain-controlled rheometer, ARES-G2 (TA Instruments), utilizing a parallel plate fixture of the diameter of 7.9 mm. The oscillatory strain amplitude γ0 was kept small (γ0 ≤ 0.1) to ensure the linearity of the storage and loss moduli, G′ and G″, obtained from the measurements. For evaluation of the association/dissociation equilibrium constant K in the PI30-COOH/oB3 solution, the infrared (IR) absorption spectrum AIR was measured for this solution as well as for a reference PI30/oB3 solution, with a spectrometer (Bio-Rad laboratories Excalibur FTS 3000) equipped with a thermal controller utilizing ethanol as a coolant. A homemade cell having a 1 mm spacer sandwiched between CaF2 windows of diameter of 25 mm was utilized. The IR spectrum AIR was measured at the temperatures where the viscoelastic measurements were conducted. Difference of the spectra of the PI30-COOH/oB3 and PI30/oB3 solutions, ΔAIR = AIR(PI30-COOH) − AIR(PI30), was utilized to evaluate the equilibrium constant K.

(5b)

In the other limit of fast reaction, ra, rd → ∞, both of g1(t) and g2(t) coincide with pure Rouse function of unimer, g1,R(t), as shown analytically in ref 8. For comparison of eqs 2−4 with the viscoelastic data of the PI-COOH system, we need to consider a fact that the unimer and dimer coexist in the system. The relaxation modulus of the system, G(t), is expressed as a weighed sum of g1(t) and g2(t)8 G(t ) = RT {N[P]eq g1(t ) + 2N[P2]eq g2(t )}

(6)

where [P]eq and [P2]eq denote the molar concentration of the unimer and dimer at equilibrium, R is the gas constant, and T is the absolute temperature. (The factors N and 2N in the parentheses indicate that the unimer and dimer are composed of N and 2N subchains.) Thus, eq 6 combined with eqs 2−4 is to be directly compared with the data. As can be noted from eq 6 combined with eqs 2−4, the slow relaxation behavior of the theoretically obtained G(t) is determined only by a few parameters: τ1, ra, and rd. The number of Rouse subchains per unimer N (= M/MR with MR being the molecular weight of Rouse subchain) is known experimentally. However, this N factor included in eq 6 is compensated by the front factor 1/N appearing in eqs 2b−2d and 3b−3e, so that G(t) at long t (where the higher order modes have relaxed) is not dependent on N, except for the N2 factor included in the end-to-end fluctuation time of the unimer, τ1 (∝ N2). τ1 can be experimentally determined from the viscoelastic data of the neat PI unimer (without the COOH group), as shown in the Supporting Information. In addition, the ra/rd ratio is identical to 2[P2]eq/[P]eq (cf. eqs 1 and 4b) and can be evaluated from spectroscopic data, e.g., infrared absorption data explained later. Furthermore, the values of [P]eq and [P 2 ]eq are separately determined from the spectroscopic data and the known value of [P]eq + 2[P2]eq (= total molar concentration of the unimer in either dissociated or associated form). Thus, the comparison of theory and experiment requires us to utilize just one fitting parameter, say rd. We make this fitting to discuss the feature of the association/dissociation reaction in relation to the polymeric character of PI-COOH, as explained in detail later in section 4.3.

3. EXPERIMENTAL SECTION 3.1. Material. High-cis linear polyisoprene (PI) having the associative carboxylic group at the chain end was synthesized via living anionic polymerization and successive termination with CO2. sec-Butyllithium and benzene (Bz) were utilized as the initiator and solvent for the polymerization. Details of the synthesis are described in the Supporting Information, and a brief summary is given below. At first, the polymerized PI− anions were allowed to react with diphenylethylene (DPE) in tetrahydrofuran (THF)/benzene mixture to convert their isoprenyl anion ends into less reactive DPE− anions. Then, the PI-DPE− anion solution was split into three flasks. The PIDPE− anions in the first flask were terminated with methanol to recover the neat PI (unimer) sample. The anions in the second flask was allowed to react with α,α′-dichloro-p-xylene (bifunctional coupler) to obtain the head-to-head coupled PI2 dimer. The amount of the coupler was set to be ∼90% equimolar to the anion to ensure full reaction of the coupler, and the unreacted unimer (∼10%) was removed through repeated fractionation from benzene/methanol mixed solvents to recover the pure PI2 dimer. These chemically C

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

4. RESULTS AND DISCUSSION 4.1. Overview of Viscoelastic Data. The top panel of Figure 1 shows storage and loss moduli, G′ and G″, measured

Figure 2. Temperature dependence of terminal relaxation time, ⟨τ⟩w, determined for PI30-COOH, (PI30)2, and PI30 chains in the oB3 solution. −d in the 10 wt % solutions is estimated to be Msoln = Mbulk = e e υ 3 bulk 3 13 100 × 10 , where υ (= 0.1) and Me (≅ 5.0 × 10 ) are the PI volume fraction in the solution and the entanglement molecular weight in bulk PI, respectively, and d (≅ 1.3)14 is a dilution exponent determined for similar PI/oB solutions. This Msoln is e considerably larger even compared to the molecular weight of the (PI30)2 dimer (61.0 × 103), indicating that the PI30 unimer and (PI30)2 dimer are in the nonentangled state. Furthermore, the ΔG′ and ΔG″ data of the unimer and dimer are well described by the simple Rouse model, as expected from the data in the bottom panel of Figure 1 and quantitatively verified in the Supporting Information (cf. Figure S4). All these results in turn suggest that the ΔG* and ⟨τ⟩w data of PI30-COOH, located between the unimer and dimer data (cf. Figures 1 and 2), reflect the effect(s) of the association/dissociation of the hydrogen bonding between the PI30-COOH chains on the viscoelastic relaxation of these chains. 4.2. Infrared Spectra. Figures 3a−c show the difference of the IR absorption spectra of the PI30-COOH/oB3 and PI30/ oB3 solutions having the same PI concentration (10 wt %), ΔAIR = AIR(PI30-COOH) − AIR(PI30), at representative temperatures, −20, 0, and 25 °C, respectively (see black curves). The stretching vibration of carbonyl (CO) group is detected in the range of wavenumber shown in Figures 3a−c, ν/cm−1 = 1685−1780. The spectra were fitted with a sum of three Lorentzian terms

Figure 1. Linear viscoelastic behavior of PI/oB3 solutions at −20, 0, and 25 °C (top panel). Storage and loss moduli of PI in the solutions, ΔG′ and ΔG″ shown in the bottom panel, were obtained by subtracting the solvent moduli from the solution moduli.

for the three PI/oB3 solutions at representative temperatures, T = −20, 0, and 25 °C. At a given T, the moduli of the three solutions coincide with each other at high angular frequencies (ω). In particular, at low T (−20 °C), the relaxation of the solvent oB3 dominated the high-ω data in our experimental window, and the data of all solutions approach the solvent data with increasing ω. (The solvent data are not shown here so as to avoid too heavy overlapping of the plots.) In contrast, the G′ data of the solutions deviate from each other on a decrease of ω down to the terminal relaxation regime where the power-law behavior, G′ ∝ ω2 and G″ ∝ ω, is observed. This deviation, noted more clearly at low T, is indicative of a difference in the relaxation behavior of the PI30-COOH, PI30, and (PI30)2 chains in the solutions. For quantitative analysis of this difference, we subtracted the solvent moduli data from the solution data to evaluate the PI moduli in the solutions, ΔG* = ΔG′ + iΔG″. The results are shown in the bottom panel of Figure 1. Clearly, the PI chains in the solutions exhibit the terminal relaxation characterized by the power law, ΔG′ ∝ ω2 and ΔG″ ∝ ω. The terminal relaxation time of the PI chains in the solutions, ⟨τ⟩w = [ΔG′/ωΔG″]ω→0, was obtained from those ΔG′ and ΔG″ data. In Figure 2, ⟨τ⟩w thus evaluated at several different T (including those examined in Figure 1) is plotted against T−1. The T dependence of ⟨τ⟩w excellently agrees for the neat PI30 unimer and (PI30)2 dimer, and the magnitude of ⟨τ⟩w is larger for the latter by a factor of 4; namely, ⟨τ⟩w,dimer = 4⟨τ⟩w,unimer irrespective of T. This feature of ⟨τ⟩w coincides with the feature of Rouse unimer and dimer, both being free from the entanglement effect. In fact, the entanglement molecular weight

3

L (ν ) =

∑ ajLj(ν) (7a)

j=1

with Lj(ν) =

Bj 1 π (ν − νj)2 + Bj 2

(ν1 > ν2 > ν3) (7b)

Here, νj is the wavenumber for the peak of jth normalized Lorentzian function Lj(ν) having the full width at a halfmaximum 2Bj, and aj is the weighing factor for Lj(ν). Satisfactory fitting was achieved (see blue curves), with the factor a1 (for ν1 ≅ 1745 cm−1; cf. green curve) increasing whereas the factors a2 and a3 (for ν2 ≅ 1730 cm−1 and ν3 ≅ 1700 cm−1; cf. dotted and solid red curves) decreasing with increasing T. These absorption bands at ν1, ν2, and ν3 can be assigned in the following way. D

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules 2[P2]eq [P]eq

=

2(a 2 /εs) + (a3/εd) a1

( = ra /rd)

(8)

where εd (= 1.8) and εs (= 2.4) are relative absorption coefficients of double-bonded dimer and single-bonded dimer to that of unimer.15 This mass ratio, being identical to the ratio of the characteristic frequencies of association and dissociation, ra/rd (cf. eqs 1 and 4b), is shown in Figure 4a.

Figure 3. Difference of the IR absorption spectra of the PI30-COOH/ oB3 and PI30/oB3 solutions having the same PI concentration (10 wt %), ΔAIR = AIR(PI30-COOH) − AIR(PI30), obtained at (a) −20, (b) 0, and (c) 25 °C (see black curves). Results of fitting with a sum of three Lorentzian terms (eq 7) are also shown: The green, dotted red, and solid red curves show the terms with j = 1, 2, and 3 in (eq 7), and the blue curve indicates the sum. The absorption bands fitted with these terms with j = 1, 2, and 3 are assigned to the unimer, singlebonded dimer, and double-bonded dimer of carboxyl group, as depicted in (d).

Figure 4. Temperature dependence of (a) mass ratio of unimer and dimer, 2[P2]eq/[P]eq (red circle), and equilibrium constant K (blue square) and (b) equilibrium molar concentrations of unimer and dimer, [P]eq and [P2]eq.

From the ra/rd ratio, the number fractions of the unimer and dimer, n1 and n2, were evaluated as

Acetic acid (and similar low-M carboxylic acids) dimerizes in nonpolar organic solvents through hydrogen bonding, as established from extensive studies (see ref 15 and Table 3 therein). Fujii et al.15 conducted spectral investigation for acetic acid in several nonpolar solvents at various concentrations and concluded that the dimers exist as an equilibrium mixture of the single- and double-bonded forms as shown in Figure 3d. They assigned the absorption bands at ν ≅ 1764, 1725, and 1715 cm−1 to the stretching vibration of carbonyl group of the unimer (unassociated COOH), single-bonded dimer, and double-bonded dimer, respectively. The wavenumbers of the three bands of our PI-COOH dissolved in oB3 are close to those observed by Fujii et al.,15 although we also note small differences attributable to a chemical difference between their solvents (mostly alkane solvents) and our oB3. Thus, we assign the three absorption bands of PI-COOH with j = 1−3 as unimer, single-bonded dimer, and double-bonded dimer of the COOH group, respectively, as shown in Figure 3d. With this assignment, we can utilize the weighing factors aj (eq 7) to estimate an equilibrium mass ratio of dimer to unimer at respective T as

n1 =

[P]eq [P]eq + [P2]eq

=

2 , 2 + ra /rd

n2 = 1 − n1

(9)

The molar concentrations of the unimer and dimer at equilibrium, [P]eq and [P2]eq, were obtained from these fractions and the total concentration of PI30-COOH existing in either unimer or dimer form, [P]total = [P]eq + 2[P2]eq = 3.0 mmol/dm3, as n1 n2 [P]eq = [P]total , [P2]eq = [P]total 2 − n1 1 + n2 (10) Finally, the equilibrium constant K was evaluated from [P2]eq and the ra/rd ratio as (cf. eqs 1 and 4b) K=

ra /rd 2[P]eq

(11) −1

The [P]eq and [P2]eq thus obtained are plotted against T in Figure 4b, and K, in Figure 4a. Clearly, K and ra/rd ratio E

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules decrease with increasing T, indicating that the dissociation is enhanced at higher T (as naturally expected). The data of [P]eq and [P2]eq (Figure 4b), directly indicating this temperature effect, are utilized later in the comparison of the theoretical calculation (eqs 2−4 and 6) with the viscoelastic data of the PICOOH chains. Here, a comment needs to be added for the equilibrium constant K. K is related to a change of the standard Gibbs free energy ΔGodim on dimerization of our PI-COOH chains as ln K = −ΔGodim/RT. ΔGodim is evaluated from the Arrhenius plots (Figure 4a) as o ΔGdim = −30 kJ/mol

(12)

ΔGodim

This value is comparable to for dimerization of low-M carboxylic acid in nonpolar organic solvent,15,16 which lends support to our assignment of the absorption bands explained above and the resulting [P]eq and [P2]eq data (Figure 4b). 4.3. Comparison of Theoretical Modulus with Data of PI30-COOH Chain. Theoretically calculated relaxation modulus G(t), given by eq 6 combined with eqs 2−4, is determined only by a few parameters: τ1, [P]eq, [P2]eq, ra, and rd. The endto-end fluctuation time of the neat unimer, τ1, is evaluated from the viscoelastic data of the unimer (and dimer), as shown in the Supporting Information (cf. Figure S4). The equilibrium molar concentrations of the unimer and dimer of PI30-COOH chains, [P]eq and [P2]eq, as well as the ratio of characteristic frequencies of association and dissociation, ra/rd, have been evaluated from IR absorption data (see Figure 4). Thus, the comparison of the theoretical G(t) with the data of PI30-COOH chains requires us to utilize just one fitting parameter, say rd, in the comparison. Here, a comment needs to be made for the number of Rouse subchains per unimer, N, included in eq 6. N can be experimentally evaluated as M/MR, where MR = 190 is the molecular weight of Rouse subchain of PI.17 However, this N factor is compensated by the front factor 1/N appearing in eqs 2b−2d and 3b−3e, as explained earlier. Thus, in a long time scale of our interest, the theoretical G(t) is quite insensitive to N, except for the N2 factor implicitly included in τ1 (∝ N2). (Note that the higher order modes appearing in eqs 2b−2d and 3b−3e have relaxed at such long times so that the upper bound of the mode summation in those equations, specified by the mode index N, does not affect the calculated G(t).) Thus, we utilized rd as a single fitting parameter to compare the theoretical storage and loss moduli, ΔG′ and ΔG″, with the data of PI30-COOH in the oB3 solution: These theoretical moduli are given by the Fourier transformation of G(t) specified by eq 6 combined with eqs 2−4. The results of this comparison are summarized in Figure 5a, and the parameter rd (reduced dissociation frequency; cf. eq 4b) utilized in this comparison is shown in Figure 6a. The accompanying parameter ra (reduced association frequency), calculated from this rd and the data of ra/rd ratio (Figure 4a), is also shown in Figure 6a. Figure 5a demonstrates that the ΔG′ and ΔG″ data of PI30COOH in the entire range of T are excellently described by the theory with the adequately chosen rd (Figure 6a). This result lends support to the basic idea in the theory, motional coupling of the unimer and dimer occurring through the association/ dissociation reaction, and to the formulation of this coupling through conformational mapping8 between the unimer and dimer briefly explained in section 2. Here, it is informative to compare the data of PI30-COOH also with the modulus for the case of simple mixing of the

Figure 5. Comparison of viscoelastic moduli (ΔG*) data of PI-COOH chains in oB3 solution (plots) with (a) theoretical calculation considering motional coupling of unimer and dimer (black curves) and with (b) modulus calculated for the case of simple mixing of unimer and dimer (blue curves). In (a), the normalized dissociation frequency rd in the calculation was utilized as a fitting parameter. (All other parameters were determined from independent experiments.) For details, see the text.

Figure 6. Temperature dependence of (a) ra and rd, and (b) τas and τds. rd was obtained from the fitting of the modulus data of PI30-COOH with the theory. For comparison, the ⟨τ⟩w data of PI30 and (PI30)2 are also shown in (b).

F

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX

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Ea,as and Ea,ds values in eq 13) reflects the polymeric character of the PI30-COOH chain, as discussed below. First of all, the end segment of the PI30-COOH chain carrying the associative COOH group is connected to the chain backbone, and its motion is synchronized with the motion of the neighboring segments in the backbone and occurs in a highfriction medium, oB3. This polymeric feature of the motion of the end segment would naturally increase the motional barrier for association as compared to that of low-M carboxylic acids in low-M solvents, thereby enlarging Ea,as of the PI30-COOH chain. Similarly, the motional barrier for the middle segment of the dimerized PI30-COOH chain should be larger than that for dissociation of the low-M carboxylic acid dimer, thereby enlarging Ea,ds of the dimerized chain. In addition to this mechanism of enlarging Ea,as and Ea,ds for the PI30-COOH chain, we also note another possible mechanism related to low diffusivity of polymeric segments (as compared to low-M carboxylic acids in low-M solvents). The local dissociation of the COOH groups of the PI30COOH chains, achieved through rearrangements of just a few monomeric segments, may have occurred with just a moderately large Ea,ds value reflecting the polymeric character of the segmental motion discussed above. However, this local dissociation would be rapidly compensated by the local association, and vice versa, thereby resulting in no significant orientational relaxation of the Rouse subchains of the PI30COOH chain. In other words, the viscoelastic relaxation reflecting this orientational relaxation does not necessarily reflect such local association/dissociation in the segmental length scale. Instead, the viscoelastic relaxation may occur after many local association/dissociation events (as similar, in a sense, to the relaxation of sticky Rouse chain7) and thus involve a larger scale motion of the PI30-COOH chain backbone. For this reason, the activation of Rouse subchain motion (reflected in the ⟨τ⟩w data of the neat unimer and dimer shown in Figure 6b) could have considerably contributed to the Ea,as and Ea,ds values, in addition to the contribution from the polymeric character of the local segmental motion discussed above, thereby giving the Ea,as and Ea,ds values (eq 13) significantly larger than those for the low-M carboxylic acids. These kinetic factors, constraint for the segmental motion due to the backbone connectivity and the repeated local association/ dissociation required for the orientational relaxation of the Rouse subchains, do not affect the thermodynamic quantity, ΔGodim (eq 12). This difference between the kinetics and thermodynamics seems to be essential for understanding the dynamics of end-associating chains.

unimer and dimer without the motional coupling due to the reaction, ΔG*(ω) = RT{N[P]eqg1,R*(ω) + 2N[P2]eqg2,R*(ω)} (cf. eq 6) with g1,R*(ω) and g2,R*(ω) being the normalized Rouse modulus of the unimer and dimer (Fourier transformation of g1,R(t) and g2,R(t) given by eq 5). This ΔG*(ω) is straightforwardly calculated from known parameters, N (= M/ MR), [P]eq and [P2]eq (Figure 4b), and τ1 (Figure S4). As shown in Figure 5b, ΔG*(ω) for this simple mixing (blue curves) significantly deviates from the data, confirming that the motional coupling of the unimer and dimer considered in the theory is essential for the relaxation of PI30-COOH. On the basis of the success of the theory seen above, it is informative to examine the characteristic times of association and dissociation, τas and τds, evaluated from ra accompanying rd (Figure 6a) and giving the excellent fit. (Note that τas = τ1/ra (eq 4b) and τds = 2K[P]eqτas (eq 1) are fully specified by this ra value and the K and [P]eq data shown in Figure 4.) The temperature dependence of τas and τds thus obtained is shown in Figure 6b. In Figure 6b, we first note that τas and τds, respectively, are not too much different, in magnitude, from the data of the terminal relaxation times ⟨τ⟩w of the neat unimer and dimer shown with the dotted curves. The end-to-end fluctuation time of the neat unimer, τ1, is related to the ⟨τ⟩w data of the unimer through the Rouse relationship, τ1 = (30/π2)⟨τ⟩w ≅ 3⟨τ⟩w (see section S3 in the Supporting Information for further details of this relationship). Consequently, τas and τds are rather close to τ1, as already noted for the ra and rd values (Figure 6a) being a little larger than unity. Thus, the end-association and dissociation reactions of the PI30-COOH chains are moderately faster than the large-scale motion of the chain, thereby fulfilling the condition considered in the theory. More importantly, we note from Figure 6b that the temperature dependence of the association and dissociation times τas and τds is as strong as that of the ⟨τ⟩w data. Although τas and τds do not exhibit the Arrhenius-type T dependence, apparent activation energies of association and dissociation of our PI30-COOH can be estimated from slopes of tangential lines for the plots of τas and τds as Ea,as ≅ 100 kJ/mol

at 0 °C

(13a)

Ea,ds ≅ 130 kJ/mol

at 0 °C

(13b)

These Ea,as and Ea,ds values and the value of Gibbs free energy difference between the dissociated unimers and associated dimer, ΔGodim (eq 12), can be cast in an energy diagram shown in Figure 7. The ΔGodim value is comparable to that for low-M

5. CONCLUDING REMARKS We have synthesized an end-associative PI30-COOH sample and reference materials, neat PI30 unimer and (PI30)2 dimer, and examined their viscoelastic behavior in an oligomeric oB3 solvent. The PI concentration was 10 wt %, ensuring lack of entanglement effect on the dynamics of those PI chains. Indeed, the Rouse relaxation was confirmed for the neat unimer and dimer at this concentration. The PI-COOH sample underwent end-association (dimerization) and dissociation through the formation and breakage of interchain hydrogen bonding of the COOH groups at the chain ends. Reflecting this situation, PI30-COOH relaxed slower compared to the reference unimer but faster than the dimer at low T ≤ 0 °C, whereas the relaxation of PI-COOH approached that of the unimer with increasing T > 0 °C. With the aid of the data of equilibrium

Figure 7. Energy diagram for viscoelastically resolved association/ dissociation of PI30-COOH chains in oB3.

carboxylic acids in nonpolar organic solvent.15,16 In contrast, the activation energy of dissociation Ea,ds of our PI30-COOH chain is significantly larger than Ea,ds (∼50 kJ/mol)18 of low-M carboxylic acids. This difference suggests that the viscoelastically detected association/dissociation process (that gave the G

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constant K obtained from IR absorption measurement, a recently formulated theory8 well described this behavior of PICOOH, which lent support to the molecular idea in the theory, the motional coupling between the unimer and dimer due to the end-association and dissociation reaction. The decrease of the Gibbs energy on dimerization ΔGodim, evaluated from the T dependence of K, was close to that for low molecular weight (low-M) carboxylic acids. Nevertheless, the (apparent) activation energy for the dissociation of PI-COOH chains was found to be significantly larger than that of low-M carboxylic acids. This difference was related to a polymeric character in the kinetics of PI-COOH chain: Namely, the local segmental motion required for the association and dissociation of the COOH groups is constrained by the neighboring segments in the chain backbone, and this polymeric character of segmental motion (in particular, in the high-friction medium, oB3) possibly enlarges the association/dissociation activation o energies without affecting the thermodynamic ΔGdim . In addition, the dissociation of the COOH groups occurring in the monomeric length scale is just a trigger for the viscoelastic (and orientational) relaxation of the Rouse subchains of the PI30-COOH chain, and the viscoelastically detected dissociation may be associated with the chain backbone motion over a considerably larger length scale. Changes of the characteristic time of this large-scale motion with T would have increased the activation energy, in addition to the increase due to the polymeric character of the segmental motion explained above. These results, suggesting an interesting interplay between the chemical reaction and polymeric kinetics, encourage further studies of the relaxation of associative chains in the nonentangled state with the viscoelastic, spectroscopic, and other available methods. A very recent theory19 formulated the viscoelastic relaxation of dilute, associative telechelic chains that form ring chains on the end-association. An experimental test for such dilute telechelic chains is one of the interesting subjects of future work.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01313. S1: synthesis of PI samples; S2: characterization of PI samples; S3: comparison of viscoelastic data of PI30 unimer and (PI30)2 dimer with Rouse model; and S4: some detail of fitting of viscoelastic data of PI30-COOH with theory (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (Y.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was partly supported by Grant-in-Aid for Scientific Research (B) (Grant No. 15H03865) and Grant-in-Aid for Scientific Research (C) (Grant No. 15K05519), both from JSPS, Japan, and Collaborative Research Program of ICR, Kyoto University (Grant No. 2015-89). We are grateful for Ms. Kyoko Ohmine at ICR, Kyoto University, for her advice and help for NMR measurements. H

DOI: 10.1021/acs.macromol.6b01313 Macromolecules XXXX, XXX, XXX−XXX