Article pubs.acs.org/Macromolecules
Reversible Gelation Model Predictions of the Linear Viscoelasticity of Oligomeric Sulfonated Polystyrene Ionomer Blends Chongwen Huang,† Chao Wang,† Quan Chen,*,‡ Ralph H. Colby,§ and R. A. Weiss*,† †
Department of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022 China § Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ‡
S Supporting Information *
ABSTRACT: The linear viscoelastic (LVE) behavior of oligomeric sulfonated polystyrene ionomers (SPS) and binary blends of two SPS ionomers with different sulfonation levels and cations was compared to the predictions of the reversible gelation model for the rheology of ionomers [Macromolecules 2015, 48, 1221−1230]. Binary blends had the same gel point as the neat ionomer components if a linear mixing rule was used to calculate an average sulfonation level for the blend. The binary blends, however, exhibited a broader relaxation time distribution than the neat ionomers having the same number density of ions. A linear mixing rule for the ionic dissociation frequency of the blend was proposed, and when incorporated into the reversible gelation model, reasonable predictions of the terminal relaxation time of the blends were achieved.
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INTRODUCTION Ionomers are predominantly nonpolar polymers containing a small amount of covalently attached ionic groups. The polar ionic groups tend to form nanometer-sized ionic aggregates in nonpolar media due to strong intermolecular ionic or dipolar associations that behave as reversible cross-links. These intermolecular interactions and the supramolecular microstructure are responsible for the excellent physical and mechanical properties of ionomers, which have produced significant academic and industrial interest for nearly six decades. 1 Ionomers have applications in a variety of technologies, such as membranes,2 thermoplastic elastomers, compatibilizers for polymer blends,3 and more recently as shape memory polymers and self-healing materials.1 Polymer blending is an economical and efficient method for developing new polymeric materials with properties that are not easily achieved with a single polymer.4 For example, blends of two different ionomers, a zinc salt and a sodium salt of poly(ethylene-co-methacrylic acid), Zn-PEMA and Na-PEMA, possess a unique combination of elasticity and toughness5 that makes them suitable for the cover of golf balls. Although there is extensive literature concerning blends of neutral polymers with ionomers, there has been little published research on blends of different ionomers except for a limited literature on blends of two different salts of PEMA (or poly(ethylene-coacrylic acid),5−8 which are used in golf ball applications. The general subject of ionomer dynamics and melt rheology was reviewed by Register and Prud’homme.9 With the exception of a 2005 paper by Nishio et al.,10 there have been no published studies concerned with the dynamics of ionomer blends. Nishio et al.10 reported that time−temperature © XXXX American Chemical Society
superposition (TTS) was obeyed by the linear viscoelastic (LVE) properties of the blends and the zero shear viscosity of PEMA ionomer blends was generally lower than expected due to effects of the residual, non-neutralized, carboxylic acid groups that are present in PEMA ionomers. The results of Nishio et al.10 emphasize a significant problem with generalizing rheology results for PEMA (or PEAA) ionomers is that they are not fully neutralized. They are actually terpolymers of ethylene, methacrylic acid (or acrylic acid), and a metal methacrylate (or acrylate). Although it is a weaker interaction than the ionic and dipolar interactions, hydrogen bonding of the acid groups complicates the answer to the question of how the ionic groups affect the dynamic response. In addition, carboxylic acid component can also plasticize or solvate the ionic or dipole−dipole interactions of the salt groups, which weakens the effect of the interactions of the ionic species.11 With sulfonate ionomers, it is easier to achieve complete neutralization of the acid groups because of the much lower pKa of the sulfonic acid (∼1.0) compared with pKa of carboxylic acids (∼4−5). The viscosities and relaxation times of sulfonate ionomers are also orders of magnitude greater than carboxylate ionomers even when the latter is fully neutralized.12 In addition, for high molecular weight ionomers, the influence of trapped entanglements can be significant and even comparable to the ionic interactions, which obfuscates the specific effects of the ionic species. Received: March 26, 2016 Revised: April 26, 2016
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DOI: 10.1021/acs.macromol.6b00620 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules
MSPS-p, where M denotes the metal cation (Na+, K+, Rb+, Cs+) and p represents the ionic content in mol %. All the ionomer blends used in this study were prepared from the MSPS-2.7 and NSPS-0.20 ionomers by codissolution of the two in a mixed solvent of 90/10 (v/v) toluene and methanol. The blend samples were isolated from solution using the same procedures described above for isolating the neat ionomers. The sample nomenclature for the blend samples is MxNy, where M and N are the metal cations of MSPS-2.7 and NSPS-0.20 ionomers and x and y are the weight percentage of MSPS-2.7 and NSPS-0.20 ionomer. For example, Na60Cs40 denotes a blend of 60 wt % NaSPS-2.7 and 40 wt % CsSPS-0.20. The neat ionomers, MSPS-2.7 and NSPS-0.20, are denoted as M100N0 and M0N100, respectively; e.g., Na100Na0 and Na0Na100 represent the NaSPS-2.7 and NaSPS-0.20 ionomers Materials Characterization. Dynamic oscillatory shear experiments were carried out with a TA Instruments ARES-G2 rheometer. Dynamic isothermal frequency sweeps were performed over a temperature range of 100−250 °C and a frequency range of 0.16− 250 rad/s using either 25 or 8 mm parallel plates. All dynamic measurements were made within the linear viscoelastic (LVE) regime, which was determined from strain sweep experiments. TTS master curves were constructed using the same reference temperature, Tr = 140 °C, for all of the blends and the same shift factors for both G′ and G″ that were calculated by the TA Instruments’ TRIOS software. The glass transition temperatures for the ionomers varied by less than 10 °C, so the error introduced by using the same Tr for comparing the LVE of the ionomers was expected to be small.
Randomly sulfonated polystyrene oligomers (SPS) have recently been used to isolate the effects of the ionic groups on the rheology of ionomers, specifically the effects of the ionic concentration and the choice of cation on the shear and extensional dynamics of ionomer melts.13−17 The use of oligomers precludes any effects of chain entanglements on the rheology. The gel point of neat ionomers occurs at one ionic group per chain on average, and two power law regions are observed in their LVE response, with scaling behaviors typical of mean-field and critical percolation.16 Above the gel point, a supramolecular network (gel) is formed, as evident from a plateau observed in the storage modulus that is independent of cation.13,14,16 The relaxation time of the ionomers increases with decreasing cation size,13 which for alkali metal cations corresponds to increasing Coulomb energy of the ion pair. A reversible gelation model for the ionomer dynamics was developed16 based on the mean-field theory of Rubinstein and Semenov18,19 by incorporating a transition from mean field to critical percolation at the Ginzburg point.20 The LVE behavior of SPS samples with different cations and ionic concentration agreed well with this reversible gelation model, which requires only two inputs: the Rouse time of a Kuhn segment, τ0, and the ionic dissociation time, τs.16 In the present communication, the validity of the reversible gelation model is discussed for higher molecular weight SPS ionomers and their binary blends using different cations and ionic concentrations. The higher molecular weight ionomer, which was still below the entanglement molecular weight for PS, was used to test the reversible gelation model in SPS ionomers with sulfonation levels below the gel point, which was not considered in the previous study16 of the scaling behavior of SPS ionomers. Shortcomings of the reversible gelation model are discussed, and a simple mixing rule is proposed for using the reversible gelation model to estimate the LVE response of ionomer blends.
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RESULTS AND DISCUSSION Reversible Gelation Model. The details of the reversible gelation model are described in ref 16, but the equations for the model are provided in the Appendix. Neat Ionomers. The gel point for an associating polymer, where each chain on average has one sulfonate group,17 occurs at a sulfonate concentration (mol %) defined by eq 1 pc =
1 × 100 mol % N−1
(1) 16
where N is the degree of polymerization. Equation 1 assumes that each sulfonate group interacts with another sulfonate group to provide a physical cross-link. For the SPS13.5 ionomers, pc = 0.78 mol %. Below the gel point (p < pc), the polymer system is a sol without the formation of percolated network. At the gel point (p = pc), a percolated network (gel) exists, but the fraction of gel in the sample is very small. Above the gel point (p > pc), the sol and gel are in equilibrium and the fraction of gel increases with p, until the system is completely gelled at p = 2pc.16 The sol is composed of non-cross-linked, branched chains (clustersnot to be confused with the term used for ionic nanodomains that is prevalent in the ionomer literature) of different sizes formed by the physical bonds between linear precursor chains (Scheme 1). As p approaches pc from the sol state, the size of the clusters increases and two transitions occur in the relaxation behavior. A transition from mean field to critical percolation behavior occurs at the Ginzburg point, p = pg1, where the clusters are at their overlap concentration20
EXPERIMENTAL DETAILS
Materials. Oligomeric polystyrene (PS13.5) with Mw = 13.5 kg/ mol and a polydispersity index (PDI) of 1.06 was obtained from Pressure Chemical Co. (Pittsburgh, PA). The sulfonation of PS was carried out in a 1,2-dichloroethane (DCE) solution with acetyl sulfate following the procedure of Makowski et al.21 Acetyl sulfate was prepared through the reaction of concentrated sulfuric acid and a 60 mol % excess of acetic anhydride in DCE at 0 °C. The freshly prepared acetyl sulfate was added to an ∼10 wt % PS/DCE solution over a period of ∼1 min at ∼50 °C with continuous stirring. After 1 h the reaction was terminated by adding ∼2 mL of 2-propanol. The sulfonation reaction is an electrophilic substitution reaction that occurs primarily at the para-position of the phenyl ring.22 The sulfonation product, the sulfonic acid derivative of sulfonated polystyrene (SPS13.5), was precipitated in boiling deionized water, washed three times with deionized water, dried in air at 70 °C for 1 day, and finally dried at 120 °C in a vacuum oven for 1 week. The sulfonation level was determined by sulfur elemental analysis at Robertson Microlit Analysis (Madison, NJ). This sulfonation procedure for polystyrene produces random substitution of the polymer.23 SPS13.5 samples were prepared with ionic concentrations ranging from 0.066 to 2.7 mol % (i.e., percent of styrene moieties that were sulfonated). Fully neutralized alkali metal salts of SPS13.5 were prepared by adding a 50 mol % excess of an appropriate metal hydroxide or acetate to a 15 wt % solution of SPS in a 90/10 (v/v) mixture of toluene and methanol. After mixing the solution at room temperature for 30 min, the neutralized SPS salt was recovered by precipitating in boiling deionized water, washed, and dried using the same procedures as described above for the SPS. The sample nomenclature used herein is
pg1 = (1 − NX −1/3)pc
(2) 16
where NX is the number of Kuhn segments per chain. For SPS13.5, pg1 = 0.47 mol %. The clusters become very large as p → pc, and a second transition occurs at p = pe1, where a critical cluster size is achieved, i.e., where the relaxation time of the cluster becomes longer than the effective breakup time. Effective breakup refers to fragmentation of the backbone of B
DOI: 10.1021/acs.macromol.6b00620 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules Scheme 1. Dissociation of a Cluster: (a) Effective Breakup Occurring at an Ionic or Dipolar Association (Purple Circle) in the Backbone (Orange) and (b) Noneffective Breakup Occurring at an Association (Green Circle) Involving a Side Branch (Blue)a
a
All connections of the orange and blue lines represent associated ionic groups from different chains (strands).
the clusters,16 which results in two subclusters of comparable size and relaxation times much shorter than their mother cluster (Scheme 1a).18,19 In contrast, noneffective breakup occurs within the side branches of the cluster and result in a large cluster of similar size and relaxation time as the mother cluster and a small satellite cluster (Scheme 1b). Similarly, as p → pc from the gel state, the two transitions in the LVE behavior occur, but at pg2 = (1 + NX−1/3)pc = 1.1 mol % and p = pe2, and the breakup occurs in the strand of the gel network formed by the largest clusters. For pe1 < p < pe2, the effective breakup continues until the clusters reach a critical size where the relaxation time of the subclusters/strands becomes equal to the effective breakup time, and there is no further breakup. The critical cluster controls the terminal relaxation of the ionomer near the gel point. The values of the characteristic sulfonate concentrations for SPS13.5 are listed in Table 1.
Figure 1. Master curves of storage modulus (G′, filled symbols) and loss modulus (G″, open symbols) for NaSPS ionomers with different ionic contents (Tr = 140 °C). The solid lines are the model predictions. The colored regions correspond to (I) mean field percolation behavior, (II) critical percolation behavior, and (III) terminal behavior.
expected to be more pronounced for the higher sulfonated ionomer. That explanation is consistent with the greater deviation observed for the experimental data and model predictions for NaSPS-0.52 than for NaSPS-0.20 and NaSPS0.066. For p ∼ pc (NaSPS-0.76), the LVE behavior showed two distinct power law regions before terminal relaxation, meanfield behavior (G′ ∼ G″∼ ω1) at higher frequencies, followed by critical percolation behavior (G′ ∼ G″∼ ω2/3), which is consistent with the model prediction.16 When p > pc (NaSPS1.2 and NaSPS-2.7), a plateau in G′ is clearly observed in Figure 1 and the plateau modulus, GN, increased with increasing ionic content due to the increasing effective cross-link density. When p > 2pc (NaSPS-2.7), where the system is completely gelled, the model overpredicted GN, which is probably a consequence of the model assumption that all strands (i.e., the chain between sulfonate groups) are stress-bearing. That is not the case because defects in the network, e.g., dangling ends, loops, or unassociated (isolated) ionic groups, do not support stress. The experimental GN was only about half of the reversible gelation model prediction, which indicates that nearly half the polymer strands are not active components of the gel network. A similar inefficiency of physical cross-linking in a block copolymer was also reported for polystyrene-block-polyisoprene (PI)-block-polystyrene copolymers with short PS blocks and long PI blocks.24 In that case, the fraction of stress bearing strands was estimated to be 40%. In general, it is difficult to achieve an equilibrium state of the interactions of the ionic groups in ionomers, which may explain, in part, the lower than predicted GN. That is due to the
Table 1. Summary of Characteristic Sulfonation Concentrations for SPS13.5
a
p
pg1
pe1a
pc
pe2a
pg2
mol %
0.47
0.68−0.75
0.76
0.81−0.88
1.1
The actual value depends on the cation type.
Figure 1 shows the LVE master curves and the model predictions for the storage modulus, G′, and loss modulus, G″, at Tr = 140 °C for the different NaSPS-p ionomers. For p < pc, a plateau in G′ was not observed and TTS worked well. For p < pg1, the reversible gelation model predicts mean-field behavior, G′(ω) ∼ G″(ω) ∼ ω1, in the intermediate frequency, which is not observed in both NaSPS-0.066 and NaSPS-0.20 ionomers. The absence of the linear power law behavior in those ionomers may indicate the absence of large clusters at such a low sulfonate concentration.16 The reversible gelation model fit the G″ data well for p < pc but underpredicted G′ in the lower frequency range. The discrepancy in G′ may be due to the failure of the model to consider a molecular weight distribution and the assumption that only the effective breakup of the backbone influenced the relaxation of the cluster. Noneffective breakup of the side-chain branches may also produce higher frequency relaxations that would broaden the overall relaxation of the clusters and is C
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Macromolecules extremely high viscosities that develop as the system gels. The decrease of mobility may frustrate the chains from reaching equilibrium, such that the ideal situation of all functional groups being active components of the network is not attained. For example, Chun and Weiss25 showed that for an elastic ionomer at temperatures far above Tg, an equilibrium microstructure is not easily achieved in the melt and significant physical aging effects of the ionic aggregation occurs. For the present oligomeric SPS system, annealing the ionomer at elevated temperature had little effect on the viscoelastic properties, which were reproducible. Thus, nonequilibrium effects appear to have been negligible for the oligomeric SPS ionomers, which may be a consequence of the lower viscosities of these ionomers near pc, ∼105−106 Pa·s, compared with much higher molecular weight, highly entangled ionomers TTS failed in the terminal region of G′ for NaSPS-0.76 where p ∼ pc and in the valley preceding the terminal region of G″ for the two ionomers with p > pc (NaSPS-1.2 and NaSPS2.7). The failure of TTS was due to two relaxation processes,26 characterized by the Rouse relaxation time τX (relaxation time of precursor chain), and the ionic dissociation time τs, that have different temperature dependences. The terminal relaxation of the ionomer near the gel point (p ∼ pc) is governed by the effective breakup of the clusters into smaller clusters of comparable size, where the effective breakup time16,18,19 is τc ∼ τX1/4τs3/4, where τX is the Rouse relaxation time for the chain. The thermorheological complexity of NaSPS-0.76 can be attributed to the different temperature dependences of Rouse relaxation at high frequency and effective breakup in the terminal region. For p > pc, TTS failed in the frequency region near the minimum in G″, where two relaxation mechanisms, Rouse relaxation and ionic dissociation, overlap. G′ and G″ master curves at Tr = 140 °C for SPS-0.20 (p < pc) and SPS-2.7 (p > 2pc) ionomers with different alkali metal cations are shown in Figure 2. For SPS-0.20 neutralized with Na, K, Rb, and Cs, no plateau was observed in G′ (Figure 2a) and TTS worked well for each ionomer. Increasing the Coulomb energy of the ion pair (Ec ∼ 1/a, where a is the ionic radius of the sulfonate cation) did not significantly change the shape of G′ and G″, but it increased their values slightly, especially at low frequency. The changes of G′ and the melt viscosity (G″/ω) by varying the Coulomb energy of the cation were smaller than those reported in previous studies of SPS ionomers.13−16,27 That is because the sulfonate concentrations used in the previous studies of SPS ionomers were near or above the gel point, where the ionic dissociation time becomes a significant factor in determining the terminal relaxation. For the SPS-0.20 ionomers (p < pc), the sulfonation level was far below the gel point where the terminal relaxation is determined mainly by the Rouse relaxation time of the clusters without dissociation. Although the model does not account for the small effect of Coulomb energy on the terminal region of G′ for p < pc, the model reasonably predicted the LVE behavior of all the SPS-0.20 samples. For p > pc (Figure 2b), TTS failed in G″ for all four alkali metal salts of SPS-2.7 due to the different T dependences of the ionic dissociation and the Rouse relaxation times.13−16 GN was independent of the cation which agrees with the model prediction for p > pc,16 though the model overpredicted the value of GN as discussed above. Increasing Ec shifted the terminal relaxation to lower frequency by more than 2 orders of magnitude, which was not unexpected, since the terminal relaxation is governed mainly by ionic dissociation.
Figure 2. Master curves of G′ (filled symbols) and G″ (open symbols) with Tr = 140 °C for (a) MSPS-0.20 and (b) MSPS-2.7 ionomers. The solid curves are the model predictions.
Table 2 summarizes the values of the Rouse relaxation time of an unsulfonated Kuhn segment (τ0) and the ionic Table 2. Rouse Relaxation Time (τ0) and Ionic Dissociation Time (τs) Used for NaSPS-p, SPS-0.20, and SPS-2.7 Ionomers at 140 °C NaSPS-p τ0 (μs) τs (s)
PS13.5
0.066
0.20
0.52
0.76
20
29
34
27
35 3300
SPS-0.20 τ0 (μs) τs (s)
1.2 96 4340 SPS-2.7
2.7 463 4650
Na
K
Rb
Cs
Na
K
Rb
Cs
34
30
26
27
463 4650
489 430
437 93
415 21
dissociation time (τs) used in the model predictions in Figures 1 and 2 (solid curves). Note that both τs and τ0 were determined from the nonlinear least-squares fit of dynamic moduli. For NaSPS-p ionomers, the τ0, which was higher than that of the PS bulk, as obtained from fitting the PS data to the Rouse model, increased with the sulfonation degree p, presumably due to the increased fraction of segments whose mobility is restricted by the ionic interactions. This increasing dependence of τ0 is consistent with the increase of dynamic modulus in the Rouse region (G′ ∼ G″ ∼ ω1/2) at frequency higher than the mean-field region for ionomers with higher p D
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Macromolecules (see Figure 1). τs for the NaSPS-p ionomers, also increased with p, which may be a consequence of better packing of the ionic associations at the higher sulfonation levels. Decreasing Ec (i.e., increasing the cation radius) had only small effect on τ0 for the SPS-0.20 and SPS-2.7 ionomers, but it reduces τs by 2 orders of magnitude as the cation was changed from Na to Cs. The segment relaxation time τ0 depends mainly on the sulfonation degree p, while the ionic dissociation time τs is more sensitive to the Coulomb energy of the ion pair. τ0 exhibited an exponential dependence on p, eq 3 (see Figure S1) log τ0 (μs) = 1.3 + 0.49p
p ̅ = pM x /100 + pN (1 − x /100)
(4)
where x is the concentration of SPS-2.7 (wt %) and pM and pN are the ionic contents (mol %) of the SPS-2.7 and SPS-0.20, respectively. The gel point for the neat ionomers, pc = 0.78 mol %, was also used for the blends, since the molecular weights of the two components of the mixture were the same. It is important to note that for the SPS ionomer blends it does not appear that cation exchange between the two components of the blend occurs in the melt, at least during the time of the experiments described herein. The evidence for that conclusion will be discussed later in this paper, but that conclusion marks an important distinction between PEMA ionomer blends, where cation exchange in the melt has been reported, facilitated by the unneturalized carboxylic acid groups.30 Blends of Ionomers with the Same Counterion. Master curves of ionomer blends NaxNay are shown in Figure 3. The p
(3)
which was used to estimate τ0 for the ionomer blends using p = p,̅ where p̅ is the average sulfonation degree of the ionomer blends. The values τs obtained for the SPS13.5 ionomers with the same cation, but different p, were averaged, and the averaged value (Table 3) was used for estimating τs of the ionomer blends. Table 3. Average Ionic Dissociation Time τs for the Different Cations at 140 °C τs (s)
Na
K
Rb
Cs
4050
380
93
21
The temperature dependence of the shift factors, aT(T), used to construct the master curves in Figure 2 was fit to the Williams−Landel−Ferry (WLF) equation (Table S1 and Figure S2). For each sulfonation level a T (T) was relatively independent of the cation, but the shift factors for SPS-0.20 had a weaker temperature dependence than those for SPS-2.7. For the SPS-0.20 ionomers, the WLF constants were C1 = 7.3 ± 0.2 and C2 = 102 ± 3.0 K, and for SPS-2.7, C1 = 11.4 ± 0.2 and C2 = 129 ± 2.2 K. The shift factors for the ionomer with the lower sulfonate concentration are consistent with the fit of the parent PS, which is consistent with the fact that the Rouse motions of the polystyrene backbone dominate the LVE behavior when p < pc. However, for p > 2pc the ionic dissociation time becomes important28 and influences the WLF fit. The free volume theory upon which the WLF equation is derived predicts that C1 ∝ 1/f r and C2 ∝ f r/αf, where f r is the fractional free volume at Tr and αf is the free volume expansion coefficient.26 Thus, the changes in the WLF constants for the two ionomers indicate that the fractional free volume and the volume expansion coefficient decreased by ∼35% and ∼50%, respectively, as the sulfonation level was increased from 0.20 to 2.7 mol %. That result is in qualitative agreement with the densification of SPS and PSMA ionomers due to the strong associative interactions that arise from the strong ionic or dipole−dipole interactions.27,29 Binary Ionomer Blends. The LVE behavior of ionomer blends was studied using binary mixtures of SPS-0.20 and SPS2.7 ionomers, where the former component had a sulfonation level far below the gel point, p ≪ pc, and the latter ionomer had a sulfonation level far above the gel point, p > 2pc. The properties of blends where the two ionomers had the same counterion (Na) were first evaluated to compare their LVE behavior to a neat ionomer with a comparable total sulfonate concentration, and then mixtures of ionomers with different counterions were studied to determine how the properties were influenced by the two separate cations. The average ionic content (p)̅ of the mixture MxNy was calculated from a mass average
Figure 3. Master curves of G′ (filled symbols) and G″ (open symbols) for NaSPS-0.20 (Na0Na100), NaSPS-2.7 (Na100Na0), and blends of NaSPS-0.20 and NaSPS-2.7 ionomers (NaxNay). Tr = 140 °C. The numbers in the parentheses of the legend are p̅, and the solid curves are the model predictions. The colored regions correspond to (I) mean field percolation behavior, (II) critical percolation behavior, and (III) terminal behavior.
dependence of the LVE behavior is similar to that observed for the neat ionomer melts that was discussed earlier in this paper. When the blend contained 10 wt % of NaSPS-2.7 ionomer (Na10Na90), both G′ and G″ exhibited power law behavior, G′(ω) ∼ G″(ω) ∼ ω1, in the intermediate frequency range, which is consistent with the model prediction for p ̅ < pg1.16 When the concentration of NaSPS-2.7 increased to 20 wt % (Na20Na80), p ̅ ∼ pc, similar to the LVE behavior of the neat ionomers, the blend exhibited a clear transition from mean field behavior, G′(ω) ∼ G″(ω) ∼ ω1, to critical percolation behavior, G′(ω) ∼ G″(ω) ∼ ω2/3. The appearance of the critical percolation region indicates that the composition of the ionomer blend was close to the gel point, which validates the assumption that the same gel point is applicable for the neat ionomer and the binary blends corresponding always to an average of one sulfonate group per chain. When p ̅ > pc (Na30Na70), a weak plateau in G′ was observed (Figure 3a), E
DOI: 10.1021/acs.macromol.6b00620 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules which became more distinct as p̅ increased, i.e., increasing MSPS-2.7 concentration. GN for the blend also increased with increasing p,̅ which is consistent with increasing the effective cross-link density. Figure 4 compares the LVE behavior of two blends with that of two neat ionomers with similar sulfonation level: (1)
Figure 5. Time−temperature shift factor aT for NaSPS-0.20 (Na0Na100), NaSPS-2.7 (Na100Na0), and blends of NaSPS-0.20 and NaSPS-2.7 ionomers (NaxNay) with Tr = 140 °C. The solid lines show the WLF fits.
increased with increasing p. The changes in the constants corresponded to decreases of the fractional free volume by ∼30% and the volume expansion coefficient by ∼50% when the blend composition changed from p ̅ < pc to p ̅ ≫ 2pc. Those changes are similar to the densification and change in the volume coefficient of expansion of the neat ionomers discussed earlier in this paper, and they are consistent with the explanation that for p > pc the relaxation associated with the ionic dissociation time of the ionic or dipolar interactions dominate the viscoelastic behavior of the melt. Figure 4 indicates that when p > pc, a binary blend of two ionomers has a broader distribution of relaxation times than a neat ionomer with the same value of p. This difference may be related to the distribution of ionic groups per chain. Because the sulfonation reaction is random,23 each polymer chain may contain different numbers of sulfonate groups. Some chains have more than, and some chains less than, the average sulfonation level. The distribution of ionic groups per chain can be represented by a binomial distribution23
Figure 4. Comparison of LVE behavior for binary blends and neat ionomers with similar ionic contents at 140 °C. The colored regions correspond to (I) mean field percolation behavior, (II) critical percolation behavior, and (III) terminal behavior.
Na20Na80 (p̅ = 0.70 mol %) and NaSPS-0.76 (p = 0.76 mol %) and (2) Na40Na60 ( p ̅ = 1.2 mol %) and NaSPS-1.2 (p = 1.2 mol %). The compositions of the first pair are close to the gel point, and for the second pair pc < p < 2pc. For the p ∼ pc, the LVE behavior of the neat ionomer and the ionomer blend were similar, though there was a small difference in the values of G′ and G″ at intermediate frequencies. For p > pc the qualitative trends of the LVE behavior were similar, but GN for the blend was distinctly lower than for the neat ionomer and the data in the peak region of G″ differed considerably. These difference are attributed to differences in the sulfonate distribution in the blends and the neat ionomers, as is discussed below. In general, for the neat ionomers and blends TTS worked well when p < pc, but for p ∼ pc superposition of G′ failed in the terminal region and for p > pc G″ failed in the intermediate frequencies prior to the terminal region. When p > pc, the failure of TTS for G″ was more apparent at the frequencies above the peak in G″, but the G′ data appeared to obey TTS. The success of TTS for G′, but not G″, is comparable to the results of Shahamy and Eisenberg31 and Earnest and MacKnight32 for carboxylate ionomer melts, where they also found that TTS worked for their G′ data, but not for the G″ data. TTS failed only when the viscous dissipation associated with the ionic aggregates became important. Earnest and MacKnight32 attributed this behavior to changes in the shift factors for short and long relaxation times, which were associated with two relaxation process with different temperature dependences.26 Figure 5 shows the temperature dependence of the shift factors for the master curves in Figure 3. For x ≤ 30 wt % ( p ̅ ≤ 0.95), the shift factors for the PS13.5 and the NaxNay blends were fitted with the WLF equation with C1 = 7.3 ± 0.2 and C2 = 99.7 ± 3.7 K (Table 4), which were similar to the values used for the neat ionomers for p < pc as was discussed earlier. For x ≥ 30 wt %, the constants needed to fit the WLF equation began to deviate from those values (Table 4). In general, C1 and C2
P(x) =
N! px (1 − p)N − x x ! (N − x )!
(5)
where P(x) is the probability that for an ionomer with ionic concentration p a chain with a degree of polymerization N will have x ionic groups. Figure 6 compares the distribution of ionic groups per chain, P(x), for two binary blends of NaSPS-0.20 and NaSPS-2.7 and the neat ionomer with a similar sulfonation level. The calculations assumed a constant value of N = 130 for each chain (note that the polydispersity index was 1.06). The distribution of ionic groups per chain is different for the binary blend and neat ionomer with similar p, but both materials had similar concentrations of chains with x = 0 and x = 1, which do not contribute to the gel network. That may be why the dynamic modulus for the blends and neat ionomer overlapped at the lower melt temperatures where short time relaxation processes should dominate. It also explains the lower GN predicted by the reversible gelation model, since the chains with x ≤ 1 cannot form stress-bearing strands of the network. However, the sulfonate distributions of the blends were skewed to higher values of P(x) for x > 2. For example, although Na20Na80 blend had fewer functional chains (i.e., P(x ≥ 2)) F
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Macromolecules Table 4. WLF Constants for NaxNay Ionomers NaxNay C1 C2 (K)
PS13.5
x=0
x = 10
x = 20
x = 30
x = 40
x = 60
x = 80
x = 100
7.5 106
7.3 99.9
7.1 98.4
7.1 95.4
7.2 98.2
8.4 107
10.5 128
10.7 127
11.3 130
between the model predictions and the experimental data is clearly seen in the terminal region of G′. With the exception of the ionomer with the lowest sulfonation level, NaSPS-0.066, a similar disagreement between the theory and experiment was observed for the neat ionomers with p < pc (see Figure 1). As was discussed earlier for the neat ionomer data, the lack of fit of the model may be due to the failure to account for the noneffective breakup of the clusters. The reversible gelation model and the experimental data for the blends agreed when p̅ ∼ pc (Na20Na80), but for p̅ > pc the model overpredicted G″ in the terminal region. Although the model predicted a plateau region in G′, it overpredicted GN and predicted a sharper onset of terminal response than observed. The higher GN prediction is due to the assumption that all the strands between ionic groups are stress bearing. However, strands with two ionic groups that join the same clusters form loops and chains with only one or no ion pairs form dangling chains that do not contribute to GN. The onset of terminal behavior for the blends was more gradual than for the neat ionomers (cf. Figures 1 and 4). That difference and the failure of the model to predict the slower relaxation response are also likely due to the differences in the sulfonate distribution shown in Figure 6 and the failure of the model to account for difference in the sulfonate distribution. Blends of Ionomers with Different Counterions. The effect of using different cations for the two ionomer components on the LVE behavior of a blend was studied using three blend compositions: M10N90 ( p ̅ < pc), M20N80 (p̅ ∼ pc), and M60N40 ( p ̅ > 2pc). For each blend composition, the cation of one ionomer component was varied (Na, K, Rb, and Cs), while the cation of the other ionomer component was fixed. It did not appear that any significant cation exchange occurred in these ionomers. That possibility was assessed by mixing ionomers with two different sulfonation levels and cations. As will be shown and discussed below, the viscosity of a
Figure 6. Distribution of sulfonate groups per chain calculated from eq 5 for the binary blends and neat ionomers used for Figure 4.
than NaSPS-0.76, the functional chains with x ≥ 2 in the blend had higher average functionality (x ̅ ∼ 3.7 ) than NaSPS-0.76 (x ̅ ∼ 2.4 ). A comparison of Na40Na60 and NaSPS-1.2 produced similar results. The effect of the sulfonation distribution being skewed toward higher sulfonation levels should be similar to that of a higher molecular weight tail of the molecular weight distribution for an entangled polymer, but the effect of increasing ionic groups per chain on the viscosity and elasticity of an ionomer melt is much stronger than the effect of increasing chain entanglements.13 As a result, the higher “x” end of the sulfonate distribution shown in Figure 6 has a disproportionate influence on the relaxation time distribution and produces longer relaxation times for the blends than for the neat ionomers with similar sulfonation levels. The model predictions of the experimental data for the NaxNay blends shown in Figure 3 capture the qualitative trend for the changes in G′ and G″. Although the model fit well for the G″ data for p< ̅ pc (Na10Na90), an obvious discrepancy
Figure 7. Master curves of G′ (filled symbols) and G″ (open symbols) with Tr = 140 °C for (a) M10Na90 and (b) Na10N90 blends where p ̅ = 0.45 mol % < pc. The solid curves are the model predictions. The colored regions correspond to (I) mean field percolation behavior and (II) terminal behavior. G
DOI: 10.1021/acs.macromol.6b00620 Macromolecules XXXX, XXX, XXX−XXX
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Figure 8. Master curves of G′ (filled symbols) and G″ (open symbols) with Tr = 140 °C for (a) M20Na80 and (b) Na20N80 with p̅ = 0.7 mol % ∼ pc. The solid curves are the model predictions. The colored regions correspond to (I) mean field percolation behavior, (II) critical percolation behavior, and (III) terminal behavior.
Figure 9. Master curves of G′ (filled symbols) and G″ (open symbols) with Tr = 140 °C for (a) M60Na40, (b) Na60N40, (c) K60N40, and (d) Cs60N40 samples with p̅ = 1.7 mol % > 2pc. The solid curves are the model predictions.
blend of a high and low sulfonation level with different cations is dominated by the cation on the ionomer with the higher sulfonation level. Thus, if cation exchange occurred, one would expect that the elasticity and viscosity of the melt would change appreciably with time. To assess that possibility, the storage modulus (G′) and the dynamic viscosity (G″/ω) at 180 °C of a
20/80 (w/w) blend of CsSPS-2.7 and NaSPS-0.20 were monitored during cyclic frequency sweeps for 7 h, and no significant changes in either occurred. One focus in this study was the effect of the mixing of ions. In order to quantify its effect, a hypothesis was proposed that the dissociation rate, i.e., the frequency of dissociation, of a H
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of magnitude by increasing Ec of cation M by 56%, i.e., changing it from Cs to Na. However, for Na60N40 blends, where the cation M of SPS-2.7 was fixed as Na, increasing Ec of cation N by 56% (changing N = Cs to N = Na) only increased the terminal relaxation time by 1 order of magnitude (Figure 9b). When M was fixed as Cs, i.e., Cs60N40 blends (Figure 9d), the same increase of Ec had little effect to the terminal relaxation time. These results indicate that for p ̅ > pc the terminal behavior is dominated by the choice of the cation (M) on the SPS-2.7 ionomer, since the SPS-2.7 contributes to 95% of the total cation. Thus, the LVE behavior of an ionomer blend can be tuned considerably by varying the cation on the component with higher sulfonation (p). With the use of eq 7 to calculate τsb̅ the reversible gelation model predicted the LVE behavior of the M60N40 blends reasonably well, except for an overestimation of GN (Figure 9), which as discussed earlier is believed to be due to the assumption of an ideal supramolecular network. Even with the discrepancies between the reversible gelation model prediction and the experimental data shown in Figures 7 and 8, the model provided a remarkably good prediction of the changes of the terminal relaxation times as Ec was varied for all the M60N40 blends ( p ̅ > pc), where the network was well-developed and the terminal relaxation was governed by the ionic dissociation time. The validity of the mixing rule proposed by eq 7 was further confirmed in Figure 10, which plots τsb̅ against the ionic
given type of ions in a blend was independent of the presence of the other type of ion. Thus, the average frequency of ionic dissociation was assumed to be a weighted average of the frequencies for the different cations ωs̅ = ϕ1ω s1 + ϕ2ωs2
(6)
where ϕi is the mole fraction of each cation in the blend (i.e., the mole fraction of cation i of the total number of sulfonate groups in the blend) and ωsi is its frequency of dissociation for cation i. The average ionic dissociation time for the blend is then ϕ ϕ 1 = 1 + 2 τsb̅ τs1 τs2
(7)
where τsi is the ionic dissociation time for each cation obtained from the neat ionomers (Table 3). Figure 7 shows the master curves for M10N90 blends. In Figure 7a the cation for the NSPS-0.20 component was fixed as Na (M10Na90 blends), and in Figure 7b the cation for the MSPS-2.7 component was fixed as Na (Na10N90 blends). For high and intermediate frequencies the G′ and G″ data for the four M10Na90 blends superposed and exhibited mean-field behavior, G′(ω) ∼ G″(ω) ∼ ω1, which qualitatively agrees with the model prediction for p ̅ < pc. However, the model overpredicted the two moduli for the intermediate frequencies and underpredicted G′ for the lowest frequencies. In the terminal region, the G″ data for the four M10Na90 blends superposed, but G′ for the four blends were different, decreasing with the decrease of Coulomb energy, Ec. Similar with the predictions for the SPS-0.20 ionomers (cf. Figure 2), the reversible gelation model failed to account for the effect of Ec because it assumes that τs has a negligible effect on the terminal relaxation when p ̅ < pc, at which the relaxation time of the largest cluster is still shorter than τsb̅ . As with the neat ionomer with p ∼ pc, the LVE behavior of the M20Na80 (Figure 8a) and Na20N80 (Figure 8b) blends exhibited three distinct power law relaxation regions: (1) meanfield behavior, G′(ω) ∼ G″(ω) ∼ ω1; (2) critical percolation behavior, G′(ω) ∼ G″(ω) ∼ ω2/3; and (3) terminal behavior, G′(ω) ∼ ω2, G″(ω) ∼ ω1. TTS failed for G′ in the terminal region for both blend compositions and each set of cations used, due to the overlap of the of the ionic dissociation and Rouse relaxation processes.16 Decreasing Ec increased the frequency of the onset of terminal behavior as a consequence of decreasing τs. Whereas, the divergence of only the G′ data for the different cations was observed, for p ̅ < pc in Figure 7 and for p ̅ ∼ pc in Figure 8, both G′ and G″ decreased with decreasing Ec in the terminal region, with larger changes occurring in G′. When p ̅ > pc, see the LVE behavior of the M60N40 blends in Figure 9, G′ exhibited a distinct plateau and TTS failed in the valley of G″ before the terminal relaxation because of the overlap of two relaxation processes with different temperature dependences, the terminal relaxation that is governed by the ionic dissociation time (τs) and the Rouse relaxation time (τX) at higher frequency before the valley. The choice of the cation did not affect GN, which agrees with the results for the neat SPS-2.7 ionomers (Figure 2), and other LVE data for similar SPS ionomers with p > 2pc.13,14 The reversible gelation model correctly predicts that the GN is independent of cation when the gel was formed at p > pc.16 For the M60Na40 blends (Figure 9a), the terminal relaxation time increased by 2 orders
Figure 10. Test of the validity of the linear additivity of breakup frequency for the ionomer blends. τsb was calculated from the mixing rule (eq 7); τs′ was calculated from the measured terminal relaxation time τ using eqs 8−10. The black solid line is the linear fit, and the dashed lines are the 95% confidence limit of the linear fit. The legend denotes the cation pairs as M−N for MxNy blends.
dissociation time of the blend, τs′, calculated from terminal relaxation time, τ, which is determined directly from the experimental data in the terminal region τs′ = (τ /τX1/4)4/3 τs′ = τ /ε
τs′ = τ /ε 2
for pe1 < p ̅ < pe2
for pe2 ≤ p ̅ < 2pc
for p ̅ ≥ 2pc
(8) (9) (10)
where τX is the Rouse relaxation time of the precursor (PS) chain and ε is the relative extent of gelation, which is a measure of how far the system is from the gel point.16 ε = (p − pc )/pc I
(11) DOI: 10.1021/acs.macromol.6b00620 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules τ was calculated from eq 12.26 ⎡ G′(ω) ⎤ τ = lim ⎢ ⎥ ω→ 0⎣ ωG″(ω) ⎦
chains that may create more chains with one or no sulfonate groups that relax more rapidly and/or differences in the nonequilibrium distribution of ionic associations that arise from decreases in molecular mobility as the material gels. A linear mixing rule for the dissociation frequency of the ionic or dipolar interactions in was proposed for binary blends, and it worked well for blends of two SPS ionomers with different sulfonation levels and different cations. The mixing rule can be easily extended for mixtures of greater than two components
(12)
The linear relationship between τsb and τs′, τsb ∝ (τs′)1, confirms the mixing rule proposed in eq 7, which now provides a convenient method for predicting the terminal properties of at least these SPS ionomer blends.
1 = τsb̅
∑
ϕi τsi
(13)
When the mixing rule is incorporated, the reversible gelation model qualitatively predicts the LVE behavior of ionomer blends with various compositions and cations. Although the model fails to quantitatively predict the terminal response of blends with p ̅ < pc, for the identical reasons for why it does not work for neat ionomers, the model provides a good prediction for the terminal relaxation time of ionomer blends with p ̅ > 2pc, where the ionic dissociation frequency controls LVE behavior. Thus, the incorporation of the mixing rule proposed herein for the ionic dissociation time, τsb, for ionomer blends provides a convenient method for predicting a priori the terminal relaxation time of a blend and, at least, qualitatively predicting its LVE behavior. The incorporation of the effects of the distribution of ionized sites into the percolation model remains a problem that needs to be addressed if the model is to provide more quantitative predictions of the LVE behavior of neat ionomers and blends.
Figure 11. Schematic illustration of storage modulus (solid lines) and loss modulus (dashed lines) for p < pc (blue), p ∼ pc (orange), and p > pc (purple) with the scaling rules indicated in the figure. The colored regions correspond to (I) mean field percolation behavior, (II) critical percolation behavior, and (III) terminal behavior.
■
CONCLUSIONS The reversible gelation model for the LVE properties of ionomers proposed in ref 16 was further tested for a higher molecular weight SPS ionomer and including materials with p < pc and blends of two ionomers with different sulfonation levels and cations. Although the reversible gelation model with only two parameters, the Rouse relaxation time and an ionic dissociation time, predicted the LVE properties of ionomer melts reasonably above the gel point (p > pc), it failed to accurately predict G′ in the terminal region for ionomers with p < pc, though it predicted G″. The failure to predict G′ is believed to be a consequence of the model neglecting to account for noneffective breakup of the side branches of a cluster. The model also does not consider the effects of ionic dissociation on the terminal behavior of ionomers with p < pc. Binary blends of SPS ionomers exhibit a sol−gel transition at the identical gel point, pc, as a neat ionomer if one uses a simple mass averaged mixing rule to calculate an average sulfonation level for the blend. However, even for the same average ionic group concentration, binary SPS ionomer blends show different LVE behavior than a neat ionomer. That is because the random nature of the sulfonation reaction produces different ion-pair distributions for ionomer samples with different values of p, and mixing two ionomers with different values of p produces a broader distribution of relaxation times than either of the neat ionomer components. The broader relaxation distribution of relaxation times produces a longer terminal relaxation time. It also produced a lower value of the plateau modulus, GN, as a consequence of broadening the ion-pair distribution on the
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APPENDIX. REVERSIBLE GELATION MODEL
For different regions of sulfonation level, the complex modulus function G*(ω) is p < pg1: ⎡ NX τX /n2 G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = 2 1 + iωτX /n2 +
∫τ
∞
X
(τX /τ )1 dτ ⎤⎥ iω + 1/τchar + 1/τ τ ⎥⎦
(A1)
pg1 ≤ p < pe1: ⎡ NX τX /n2 G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = 2 1 + iωτX /n2
J
+
NX − 1 NX
+
0.67 NX
∫τ
(τX /τ )1 dτ iω + 1/τG + 1/τ τ X (τG/τ )0.67 dτ ⎤⎥ iω + 1/τchar + 1/τ τ ⎥⎦
∫τ ∞
G
∞
(A2)
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Ginzburg point, εG = NX−1/3, and λ is the fitting parameter of order unity. The schematic of dynamic modulus for ionomers with p < pc, p ∼ pc, and p > pc are shown in Figure 11. The mean-field percolation region is denoted by the blue shaded region, the critical percolation region is denoted by yellow shading, and the terminal region is shaded pink.
pe1 ≤ p < pe2: ⎡ NX τX /n2 G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = 2 1 + iωτX /n2 +
NX − 1 NX
+
0.67 NX
∫τ
(τX /τ )1 dτ iω + 1/τG + 1/τ τ X (τG/τ )0.67 dτ ⎤⎥ iω + 1/(λτc) + 1/τ τ ⎥⎦
∫τ ∞
G
∞
■
S Supporting Information *
(A3)
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00620. Figures S1 and S2; Table S1 (PDF)
pe2 ≤ p < pg2: ⎡ NX τX /n2 N −1 + X × G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = 2 1 + iωτX /n2 NX
∫τ
X
∫τ
■
(τX /τ )1 dτ 0.67 + × iω + 1/τG + 1/τ τ NX
∞
G
*(Q.C.) E-mail
[email protected]. *(R.A.W.) E-mail
[email protected]. Notes
The authors declare no competing financial interest.
■
(A4)
pg2 ≤ p < 2pc:
ACKNOWLEDGMENTS C.H., C.W., and R.A.W. thank the Polymers Program of the Division of Materials Research at the National Science Foundation (Grant DMR-1309853) for support of this research. Q.C. thanks Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, for financial support. R.H.C. thanks the financial support of Polymers Program of the Division of Materials Research at the National Science Foundation (Grant DMR-1404586).
⎡ NX τX /n2 G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = 2 1 + iωτX /n2 +
∫τ
∞
X
⎤ τlife (τX /τ )1 dτ ⎥ + ε3 iω + 1/τstrand + 1/τ τ 1 + iωτlife ⎥⎦ (A5)
■
GLOSSARY OF SYMBOLS p, sulfonation degree in mol %; p̅, sulfonation degree in mol % calculated from eq 4; pc, gel point determined from eq 1; pg1, Ginzburg point in the sol state calculated from pg1 = (1 − NX−1/3)pc (eq 2) where the characteristic clusters no longer overlap; pg2, Ginzburg point in the gel state calculated from pg2 = (1 + NX−1/3)pc where the characteristic network strands no longer overlap; pe1, critical sulfonation degree in the sol state at which the Rouse relaxation time of the characteristic clusters equals to the effective breakup time τc; pe2, critical sulfonation degree in the gel state at which the relaxation time of the characteristic gel network strands equals to the effective breakup time τc; ε, relative extent of gelation obtained from eq 11; NX, the number of Kuhn segments per chain; τ, terminal relaxation time obtained from eq 12; τ0, Rouse relaxation time of a Kuhn segment; τs, ionic dissociation time; τsb̅ , ionic dissociation time of the binary blend calculated from eq 7; τX, Rouse relaxation time of the precursor chain with τX = τ0NX2; τc, effective breakup time; ωs, frequency of ionic dissociation; P(x), the probability of an ionomer chain with x ionic groups calculated from eq 5; ϕi, mole fraction of cation i in the blend.
p ≥ 2pc: ⎡ NX τX /n2 G*(ω) = iωvkT ⎢ ∑ ⎢⎣ n = ε + 1 1 + iωτX /n2 ε
+
∑ n=1
⎤ ⎥ 1 + iωτsε 2 /n2 ⎥⎦
AUTHOR INFORMATION
Corresponding Authors
⎤ τlife (τG/τ )0.67 (ε /εG)2.7 dτ ⎥ + iω + 1/τstrand + 1/τ τ NX 1 + iωτlife ⎥⎦
∞
ASSOCIATED CONTENT
τsε 2 /n2
(A6)
where p is sulfonation degree in mol %, pc is the gel point determined from eq 1, pg1 is the Ginzburg point in the sol state calculated from pg1 = (1 − NX−1/3)pc (eq 2) where the characteristic clusters no longer overlap, pg2 is the Ginzburg point in the gel state calculated from pg2 = (1 + NX−1/3)pc where the characteristic network strands no longer overlap, pe1 is the critical sulfonation degree in the sol state at which the Rouse relaxation time of the characteristic clusters equals to the effective breakup time τc, pe2 is the critical sulfonation degree in the gel state at which the relaxation time of the characteristic gel network strands equals to the effective breakup time τc, NX is the number of Kuhn segments per chain, τX is the Rouse relaxation time of the precursor chain, τc is the effective breakup time, τs is the ionic dissociation time, τchar is the Rouse relaxation time of the characteristic cluster in the sol, τG is the Rouse relaxation time of the characteristic cluster at the Ginzburg point (pg1) with τG = τXNX, τstrand is the characteristic Rouse time of the network strand, τlife is the effective breakup time of a gel network strand, ε is the relative extent of gelation obtained from eq 11, εG is the relative extent of gelation at
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