Linear Viscoelastic and Dielectric Properties of Strongly Hydrogen

Nov 22, 2016 - Key Laboratory of Bio-inspired Smart Interfacial Science and Technology of Ministry of Education, School of Chemistry and Environment, ...
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Linear Viscoelastic and Dielectric Properties of Strongly HydrogenBonded Polymers near the Sol−Gel Transition Zhijie Zhang,† Chang Liu,† Xiao Cao,† Longcheng Gao,*,‡ and Quan Chen*,† †

State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China ‡ Key Laboratory of Bio-inspired Smart Interfacial Science and Technology of Ministry of Education, School of Chemistry and Environment, Beihang University, Beijing 100191, China S Supporting Information *

ABSTRACT: Linear viscoelastic and dielectric measurements were conducted for a model associating polymer system, n-butyl acrylate (PnBA)-based copolymers containing 2-ureido-4[1H]-pyrimidinone (UPy) groups as stickers. The number of stickers per chain was varied from less than one to more than two, which covered a sol-to-gel transition region. Fitting the linear viscoelasticity (LVE) to an analytical model developed in our previous study, we found necessity of distinguishing the intra- and interchain association in the model, with only the latter contributing to the gel formation. For the PnBA-Upy sample slightly above the gel point, the relaxation processes due to the Rouse motion and the sticker dissociation were commonly detected in the same range of T covering from 20 to 60 °C. This feature enabled us to make the time−temperature superposition separately for respective relaxation processes, and two sets of shift factors, reflecting the temperature dependence of the Rouse time τ0 and dissociation time τs, were obtained accordingly. The T dependence of the ratio of these shift factors, being identical to the dependence of τs/τ0, enabled determination of activation energy of the sticker dissociation. This activation energy was found to be consistent with that determined from the model fitting of the LVE data. The dielectric measurements detected both segmental α relaxation and an ionic α2 relaxation processes. The α2 relaxation time was found to be shorter than τs, and this result was discussed in relation to a difference between fluctuation and dissociation of ionic pairs. sulfonate ionomers,15,16 and polysiloxane-based ionomers with a random mixture of phosphonium and poly(ethylene oxide) side chains.17 The sticky Rouse-type linear viscoelastic relaxation was observed for the PEO-, PTMO-, and polysiloxanebased ionomers, all having an average number of ions per chain greater than two. In contrast, for sulfonated styrene oligomers, a sol−gel transition was observed at the average number of ions per chain close to one.12,13 This physical gelation point found for sulfonated styrene oligomers, ∼1 ion per chain, can be rationalized by considering the analogue between physical and chemical cross-linking.13 Analysis for the Bethe lattice, assuming the gel formation through cross-linking of the precursor molecules that proceeds in a hierarchical way (i.e., generation after generation), suggests that each molecule of the functionality F must have one functional group connected to the previous generation and F − 1 groups that could connected to the next generation. Then, the ratio of the number of molecules between the next and current generations would be p(F − 1) with p being the probability of

1. INTRODUCTION Any specific reversible noncovalent interchain attraction can result in associating polymer systems, including the ionic association,1−3 hydrogen bonding,4−7 metal−ligand coordination,8,9 host−guest complexation,10 and so on.11 A random sticky polymer system has stickers randomly attached to the chain backbone. The structure and dynamics of the random sticky polymers have been subjected to extensive researches, and it is now widely accepted that the association energy of the stickers controls the terminal relaxation behavior. The stickers provide an additional source of friction to retard the viscoelastic terminal relaxation and diffusion of the chain. Therefore, the sticky polymers with a sufficient number of stickers per chain behave as reversible gels. Nevertheless, two basic questions have yet to be answered with respect to the sticky polymer system behaving as reversible gel: How is the gel formed at a low density of stickers, and how does the gel behavior change at high density of stickers? Our previous study has suggested some answers to the first question. We conducted linear viscoelastic measurements on model ionomer systems with various ionic contents, which included sulfonated styrene oligomers,12−14 short poly(ethylene oxide) (PEO)- or poly(tetramethylene oxide) (PTMO)-based polyester © XXXX American Chemical Society

Received: September 15, 2016 Revised: November 8, 2016

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the gel point.13 For sulfonated styrene oligomer, the ionic groups aggregate strongly, leading to a certain distribution of the lifetime of ionic association. This distribution is attributable to the difference in the location of the ionic groups within the aggregate: those groups at the center of the aggregate should have a longer lifetime than those near the boundary of the aggregate, thereby giving broad viscoelastic relaxation mode distributions.13,14 In this study, we focus on the dynamics of hydrogen bonding system, i.e., n-butyl acrylate (PnBA)-based copolymers containing quadruple hydrogen bonding side chains based on 2-ureido-4[1H]-pyrimidinone (UPy). Feldman et al. have proven that the Upy system is a good model sticky system in a sense that the sticky points form association with more uniform lifetime.4−7 As a result, the PnBA-Upy chains exhibit the well-defined sticky Rouse region characterized by G′ ∼ G″ ∼ ω1/2 narrower than that observed for the ionomers, given that the chains are not entangled and have more than two stickers per chain on average. This sticky Rouse behavior emerges at frequencies lower than the dissociation frequency of the stickers (and higher than the terminal relaxation frequency).15,17,20 In this study, we synthesized the nonentangled PnBA-Upy chains with numbers of stickers varying from less than one to more than two. Those numbers cover the sol−gel transition region.12−14 We performed viscoelastic measurements on these samples and found that a ratio between intra- and interchain associations plays an important role in the linear viscoelasticity. We also noted that the energy of the association, stemming from multiple hydrogen bonding, can be determined consistently for a sample slightly above the gel point through (1) a delay of the lifetime of ionic association with respect to the Rouse time of Kuhn segment and (2) their temperature dependences. Details of these results are discussed in this article together with the results of dielectric measurements detecting the local fluctuation of stickers.

reaction, and the corresponding chemical gelation point is specified by18 pc = 1/(F − 1)

(1)

For p > pc, each generation would have larger number of molecules than its previous generation, enabling the formation of gel. In analogy with this result, the physical gelation point for ionomer having N monomers would be given by pc = 1/(N − 1), where we have regarded each monomer as a functionalizable site through ionization. For long ionomers (N ≫ 1), this physical gelation point practically agrees with a very low ion concentration, one ion per chain on average. It is worth to mention the assumption in the Bethe lattice: i.e., the hierarchical growth without formation of loops and multiple associations between neighboring chains. Correction of gel point should be made when this assumption is not fully satisfied. In addition to the newly defined physical gel point explained above, our previous study focused on two important transition points emerging as the ion content approaches the physical gelation point. The first transition point is the Ginzburg point specifying a transition from the mean-field percolation to critical percolation that occurs when the large sol clusters newly formed (through association of ionic groups for ionomers) are not overlapping in space.18 The Ginzburg transition should occur near the gelation point pc because the precursor chains are strongly overlapping (there are ∼N1/2 precursor chains overlapping in the pervade volume of a precursor chain) but a gel does not (only one gel molecule in the pervade volume of itself). Both the gel point and the Ginzburg point were defined originally for the chemically cross-linking. In contrary to nonbreakable chemical sol/gel, the physical sol/gel can relax through a competition between “chain relaxation” and “breakup of physical cross-linking”. The precursor chains or small sols should exhibit chain relaxation while the gel can only relax through breakup of cross-linking. Therefore, there is a second transition point, the Rubinstein−Semenov effective breakup point, for the physical gel.19 Let us first define “effective breakup”. In general, there are two types of breakups: one occurring at the side branches would result in a large daughter cluster having the similar size as the mother cluster and a small satellite cluster. This type of breakup hardly contributes to the stress relaxation and thus being defined as the “noneffective” breakup. In contrast, the breakup occurring at the main backbone of the large cluster would result in two daughter clusters of similar sizes, which is defined as the “effective” breakup. The Rubinstein−Semenov effective breakup point specifies a critical sol cluster for which the effective breakup time is the same as the chain relaxation time. Sol clusters smaller than this criterion have relaxation time shorter than the effective breakup time, which would exhibit mainly the chain relaxation. In contrast, sol clusters larger than this criterion have chain relaxation time larger than its effective breakup time; it would exhibit effective breakup continuously until the daughter clusters have the same size as the critical sol cluster, where the daughter clusters relax immediately. Since both transitions should occur as p approaches pc, the relative distance of these transition points to pc would play an important role. We found that for associative polymers with strong stickers, the effective breakup transition is usually closer to the gel point than the Ginzburg transition. This molecular picture has been tested to be valid for sulfonated styrene oligomers neutralized with different alkali ions close to and above

2. EXPERIMENTAL SECTION 2.1. Materials. Chlorobenzene (analytical purity, Beijing Chemical Reagents Co.) was dried with powdered CaH2 and was distilled before use. CuBr was synthesized from CuBr2.21 1,1,4,7,10,10-Hexamethyltriethylenetetramine (HMTETA, 97%, J&K) was used as received. nBA (analytical purity, Beijing Chemical Reagents Co.) was distilled prior to use. The initiator of nBA polymerization, phenyl 2-bromo-2methylpropanoate, was prepared in our lab.22 6-(tert-Butoxycarbonylamino)hexyl acrylate (Boc-acrylate) was synthesized according to the literature.4−6 2-(1-Imidazolylcarbonylamino)-6-methyl-4-[1H]pyrimidinone (IPy) was prepared by following the reported method.4−6,23 Trifluoroacetic acid (TFA, analytical purity, Beijing Chemical Reagents Co.) and DMF (analytical purity, Beijing Chemical Reagents Co.) were used as received. PnBA homopolymers of different M were purchased from Polymer Source, Inc. 2.2. Synthesis and Characterization. Polymerization of PnBANHBoc: 70.5 mg of initiator phenyl 2-bromo-2-methylpropanoate (0.3 mmol), 2.4 mg of CuBr (0.02 mmol), 10.0 g of n-butyl acrylate (nBA, 78.1 mmol), 64.4 mg of Boc-acrylate (0.24 mol), and 60 μL of HMTETA (0.23 mmol) were mixed in a glass tube, degassed by three freeze−pump−thaw cycles, and sealed under vacuum. This reaction tube was placed in an oil bath at 70 °C for 11 h to allow nBA to be polymerized. The polymerization was quenched by dipping the tube into an ice/water bath, and the tube was open to take out the reaction mixture. This mixture was diluted with CH2Cl2, passed through a basic alumina column, and precipitated into methanol. The liquid-like sample, recovered as the precipitant, was redissolved in dichloromethane and precipitated again in methanol for removal of unreacted monomers. B

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Macromolecules The molecular weights were determined by GPC (see Figure S1), with monodisperse polystyrene used as elution standard. The difference of molecular weight determined by using PS and PnBA as elution standards was reported to be less than 5%,24 and the content of the amino groups was determined by 1H NMR (see Figure S2). Preparation of PnBA-UPy: The procedure for the transformation of PnBA-NHBoc into PnBA-Upy was adapted from the literature.4−6 Typically, PnBA-NHBoc was dissolved in dichloromethane, and excessive TFA was added. After the complete reaction, the PnBA-NH2 was obtained by evaporation. PnBA-NH2 (1.0 g), triethylamine (3 equiv), and IPy (1.5 equiv) were dissolved in DMF (10 mL). The mixture was heated to 100 °C overnight. After the removal of DMF, the polymer was precipitated into methanol to give PnBA-UPy. The chemical structures of the PnBA-UPy were confirmed by 1H NMR (see Figure S3). The PnBA homopolymers, purchased from Polymer Source Inc., were measured with GPC to confirm the molecular weight distribution. The PnBA-150K and -300K samples were nearly monodisperse and used as received. The PnBA-14K and -61K samples showed high M tails, which were removed via fractionation in methonal/water solution. Differential Scanning Calorimetry (DSC). DSC measurements were conducted on a TA Q20 differential scanning calorimeter. The samples were heated to 140 °C and cooled to −80 °C at a speed of 10 °C/min to remove thermal history. Then these samples was heated again to 140 °C at 10 °C/min, and the DSC traces were recorded. Dielectric Response Spectroscopy (DRS). DRS measurements were conducted on a Novocontrol GmbH Concept 40 broadband dielectric spectrometer. The samples were loaded on a freshly polished brass electrode with a diameter of 25 mm and heated quickly up to 80 °C in vacuum for more than 12 h before loading the top brass electrode of diameter 20 mm. With this operation, the sample was degassed effectively. A gap of 1 mm was kept in between the top and bottom electrodes by 1 mm silica spacers. Before the DRS measurements, the samples were annealed in the instrument at 120 °C for 1 h under nitrogen to get rid of the possible humidity uptake during the sample loading. Isothermal frequency sweeps from 107 to 10−2 Hz were conducted in 10 °C steps from 120 to −80 °C, with precise temperature control within ±0.05 °C. Linear Viscoelastic Measurements (LVE). LVE were conducted on MCR302 of Anton Paar, 8 mm plate/plate, and 25 mm cone and plate were utilized. Before the measurements, the samples were annealed in the instrument at 120 °C for 1 h under nitrogen for drying. During the measurements, the strain amplitude was kept in the linear region, verified by strain amplitude sweeps. X-ray Scattering. Experiments were performed on a SAXsess system from Anton-Paar (Austria). A sealed tube X-ray generator (using a copper anode) was operated at 40 kV and 50 mA, generating X-ray wavelength of 0.154 nm. In each measurement, isotropic intensity distribution was recorded on an image plate for the sample sealed in aluminum foil at room T and converted into one-dimensional profile through analysis software of the apparatus.

Figure 1. Model predictions of LVE at different extent of gelation ε. The storage and loss moduli, G′ and G″, are normalized by the characteristic modulus of the precursor chain GX = vkT, with v the chain density and kT the thermal energy, and plotted against the normalized frequency ωτX, with τX being the Rouse relaxation time of the precursor chain.

Here, ε = −1 corresponds to the precursor chain and ε = 0 corresponds to the gel point. For −1 < ε ≤ 0, the precursor chains associate to form sol with size increasing with ε. Two transitions occur with the increase of ε: (1) the mean field to critical percolation transition at the Ginzburg point −εG, characterized by transition from power law behavior G′ ∼ ω to G′ ∼ ω2/3 before the terminal relaxation, and (2) the effective breakup transition at −εc. The effective breakup controls the relaxation behavior in a region of −εc < ε ≤ 0. At 0 < ε ≤ 1, the gel has formed and the relaxation can take place only through breakup of network strands. The size is the same for the strands of the network at 0 < ε ≤ 1 and the clusters at −1 ≤ ε < 0 if the absolute values of ε are the same, reflecting a well-known symmetric nature of gelation. Therefore, there are also two transitions at 0 < ε ≤ 1: For 0 < ε ≤ εc, the effective breakup time of the strands is shorter than its intrinsic Rouse time. Then, the strands exhibit effective breakup continuously until the strands have the same size as the critical cluster, where the strands relax immediately. For εc < ε ≤ 1, the effective breakup time of the strands becomes longer than the intrinsic Rouse time, leading to a plateau between these two time scales. The difference between εc < ε ≤ εG and εG < ε ≤ 1 is the power law behavior of the moduli higher than the plateau, i.e., G′ ∼ ω2/3 for εc < ε ≤ εG and G′ ∼ ω for εG < ε ≤ 1. There is an important transition at ε = 1 where all the precursor chains are incorporated into the network. Namely, the average size of the network strands is larger than that of the precursor chain at ε < 1, while becomes smaller than that of the precursor chain at ε > 1. In the latter case, the majority of chains containing more than two stickers would contain more than one network strands. These chains would relax through continuous breakup of the stickers to exhibit the sticky-Rouse type relaxation,

3. THEORY In our previous study, we considered the case that the effective breakup transition is closer to the gel point than the Ginzburg transition and modified the Rubinstein and Semenov model to predict LVE along with the sol−gel transition.13,19,25 Figure 1 shows an example of LVE predicted by our model with the Rouse relaxation time of the precursor chain of τX = 1 s, and the lifetime of the ionic association of τs = 106 s. The storage and loss moduli, G′ and G″, are normalized by the characteristic modulus of the precursor chain GX = vkT, with v the chain density and kT the thermal energy. It was plotted against the normalized frequency ωτX for samples of different relative extent of gelation that is defined as ε = (p − pc )/pc

(2) C

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Macromolecules as exampled at ε = 3 and 10 in Figure 1 where well-defined sticky-Rouse behavior G′ ∼ G″ ∼ ω1/2 manifests before the terminal relaxation.

4. RESULTS AND DISCUSSION 4.1. Thermal Behavior. Figure 2 shows DSC traces of the PnBA homopolymer samples having Mn = 14K and 150K

Figure 2. Comparison of DSC traces of PnBA bulk samples (0%) having Mn = 14K and 150K, and those of PnBA-pUpy with p = 1.0%, 2.0%, 2.4%, and 3.4%. Inset: plots of Tg against p for the PnBA-pUpy samples in this study (squares) and in ref 4 (circles). Figure 3. Test of time−temperature superposition (tTs) of the storage and loss moduli, G′ and G″, of the PnBA-3.4% sample measured at T as indicated. tTs works for data at T ≤ 10 °C, with the superposed data shown in the black symbols. In panel a, the raw G′ and G″ data corrected by a temperature factor Tr/T are shifted by a factor of aT′ until the high ω G″ data are superposed on those at T ≤ 10 °C. In panel b, the raw G′ and G″ data corrected by a temperature factor Tr/T are shifted by a factor of aT different from aT′ until the low ω G′ data are superposed on those at T ≤ 10 °C.

(p = 0%) and the PnBA-pUpy copolymer samples with p = 1.0%−3.4%. The PnBA 14K sample shows glass transition Tg lower than that of the 150K sample due to its lower degree of polymerization and accordingly larger free volume contributed from the chain ends. Incorporation of Upy units would further enhance Tg. This trend can be more clearly seen in the inset, where the glass transition temperature Tgs of the PnBA-pUpy samples are plotted against the fraction of monomers containing the Upy stickers. The square and sphere symbols correspond to samples studied here and earlier by Feldman et al., respectively.4 Our data are in accordance with those by Feldman et al.,4 and these data together reveal an increase of Tg with a rate of ∼3 °C every 1% of Upy, suggesting that the sticky points assigned an extra restriction to segmental motion to efficiently enhance Tg. 4.2. Viscoelastic Behavior. 4.2.1. Time−Temperature Superposition. To test the time−temperature superposition for the PnBA and PnBA-pUpy samples with p = 1.0%−3.4%, the storage and loss moduli, G′ and G″, are multiplied by a temperature factor of bT = Tr/T and shifted by a factor of aT until achieving the best superposition with data at Tr. (In principle, rubbery part of moduli having entropic origin should be multiplied by bT = ρrTr/ρT. However, a change of ρ with T much weaker than a change of T and can be neglected.) We found that tTs works for all the PnBA homopolymer samples (see Appendix) and PnBA-pUpy samples with p = 1.0%, 2.0%, and 2.4%. The master curves of these samples will be shown later in Figure 5 and the Appendix. In Figure 3, tTs is tested for the PnBA-3.4%Upy sample. We found that tTs works for G* of the PnBA-3.4%Upy sample at T ≤ 10 °C, with the master curves shown in black diamond symbols. This may be attributable to a single mechanism of stress relaxation therein. Namely, the stress relaxation is almost completely contributed from the motion of the Rouse segments

over the whole chain. At Tr = 20 °C, G′ transits from powerlaw-like behavior at high ω to a plateau at low ω; the latter stems from physical cross-linking of stickers. Then, we observed a failure of tTs at T ≥ 20 °C because the stress relaxation is contributed from both the Rouse motion and the sticker dissociation that exhibit different temperature dependences. G′ shows a plateau much lower than vkT expected for ε = 1 (dashed line in Figure 3a), indicating εc < ε ≤ 1, where the Rouse time of network strand is shorter than its effective breakup time. As a result, a strand would exhibit orientational relaxation immediately after its effective breakup (where a motional coupling effect is negligible26−28). Panels a and b attempt to shift the data guided by the Rouse relaxation and sticker dissociation, respectively. In Figure 3a, we selectively superpose the high ω G″ data that is more sensitive to the faster Rouse relaxation modes. (In general, the storage and loss modulus can be written as multiple Maxwell modes as G′(ω) = ∑pω2τp2/(1 + ω2τp2) and G″(ω) = ∑pωτp/(1 + ω2τp2), G′(ω) scaling with τp2 is more sensitive to slow modes (with large τp) than G″(ω) scaling with τp. For the same reason, G″(ω) is more sensitive to fast modes (with small τp) than G′(ω).). Namely, we shift bTG* by a factor of aT′ until the high frequency G″ data at T ≥ 20 °C agree with that at T ≤ 10 °C. Upon this shift, the terminal relaxation associated with the ionic dissociation shows strong thermal-rheological complexity: the D

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Macromolecules terminal relaxation at higher T appears at higher ω after this shift. This feature should be attributable to the stronger temperature dependence for ionic dissociation than the Rouse motion. In other words, increasing T accelerates modulus contributed from the ionic dissociation more than that from the Rouse motion. In Figure 3b, we attempt to shift the data on a basis of the ionic dissociation. For this purpose, we selectively superpose the low ω G′ data that are more sensitive to the slow ionic dissociation mode; we shift the data by a factor of aT until the low ω G′ data agree. This shift leads to significant failure of tTS in an intermediate frequency region where stress relaxation is contributed from both Rouse motion and ionic dissociation, i.e., −1 < log ωaT < 1. The two shifts give two different sets of shift factors. Figure 4 compares shift factor aT against T − Tr for the PnBA

are formed for these samples and the stress relaxation is governed by the Rouse motion, resulting in temperature dependence indistinguishable from that of the PnBA homopolymer samples. Nevertheless, for the PnBA-pUpy samples of p ≥ 5.5%, the stress relaxation at high T > Tr is governed by the dissociation of stickers, with temperature dependence higher than that of the first group. For the PnBA-3.4%Upy sample, the filled sphere symbols show aT′ against T − Tr obtained from shifting the high ω G″ data, which agree reasonably with those of the PnBA homopolymer samples, supporting the argument that this shift is on a basis of the Rouse motion. In contrast, the unfilled sphere symbols show aT against T − Tr obtained from shifting the low ω G′ data, whose temperature dependence is stronger than the filled sphere symbols and agrees reasonably with those of PnBA-5.5%Upy and PnBA-13.3%Upy well above the gel point,4−6 meaning the low ω G′ data are governed by the dissociation of stickers. 4.2.2. Linear Viscoelasticity. Figure 5 compares the LVE master curves of the storage and loss moduli, G′ and G″, of the PnBA-pUpy samples. Since our focus is placed on the dissociation of stickers, we choose the master curves in panel b of Figure 3 for the PnBA-3.4%Upy sample. LVE of PnBA-5.5% and PnBA13.3% reported by Feldman et al. are added for comparison.4−6 The reference Tr = Tg + 65 °C was chosen so that the Rouse regions where G′ ∼ G″ ∼ ω1/2 coincide for these samples. Since Mn is slightly different for the PnBA-pUpy samples, the gel points pc = 1/(N − 1) would also be slightly different, which are summarized in Table 1. Comparison of p and pc in Table 1 reveals that PnBA-2.0%Upy is above the gel point. Nevertheless, lack of plateau in LVE suggests that p is close to or below the gel point. This conflict suggests that only a certain fraction of the stickers are “effective” in the network formation. This point is tested through comparison of the experimental results and theoretical prediction in Figure 5. Panel a shows the prediction of LVE through inputting the real p into the analytical model developed in our previous work.13 There are two fitting parameters in the model. One of them is the characteristic time for Kuhn segment τ0 = 1.4 μs is chosen to give proper prediction of the Rouse region G′ ∼ G″ ∼ ω1/2 at ω > 104 rad/s. The other parameter, the lifetime of sticker τs = 1.0s, is chosen according to peak frequency of G″ for PnBA-5.5% and 13.3%Upy that corresponds to onset of the sticky Rouse relaxation. The only modification of the analytical model in current study is to incorporate a distribution of number of stickers per chain into the sticky Rouse modulus at ε ≥ 2: The possibility P(n,p) of n stickers per chain can be calculated from the binomial distribution:29

Figure 4. LVE shift factor aT against T − Tr for PnBA bulk samples (0%) having Mn = 14K and 150K, and those of PnBA-pUpy with p = 1.0%, 2.0%, 2.4%, and 3.4% with Tr = Tg + 65 °C. For PnBA-3.4%Upy, the shift factor aT′ obtained from shifting high ω G″ data (panel a of Figure 3) is added as filled sphere symbols for comparison. The curve is WLF eq 3.

homopolymer samples of 14K and 150K, and PnBA-pUpy samples with p = 1.0%, 2.0%, 2.4%, 3.4%, 5.5%, and 13.3%, with the data of samples having p = 5.5% and 13.3% taken from ref 4. Since Tg increases with M for the PnBA hompolymer samples (p = 0%) and with p for the PnBA-pUpy samples, we choose T = Tg + 65 °C to ensure the consistent comparison for all the samples. The plots of aT against T − Tr are clearly divided into two groups. The first group is the bulk PnBA samples and the PnBA-pUpy samples with p ≤ 2.4%, for which the temperature dependence can be well described by the WLF equation shown in solid curve: log a T = −

6.0(T − Tr) 125 + T − Tr

P(n , p) =

(3)

N! pn (1 − p)N − n (N − n) ! n!

(4)

with p being an average fraction of sticker-containing monomers. In Figure 5a, we see that the prediction based on real p consistently overestimated the relative extent of gelation. To solve this problem, we chose to use a content of effective stickers peff = fp, where the fitting parameter f represents the fraction of effective stickers forming interchain cross-linking. We choose f in a way so that the predicted zero-shear viscosity and accordingly G″ ∼ ω terminal tails agree with those in the experiments, with f values summarized in Table 1. Remarkably, incorporating an effective f in panel b greatly improves the prediction.

The second group is the PnBA-pUpy samples with p ≥ 5.5% reported in the literature by Feldman et al.,4 for which the temperature dependence becomes stronger than that of the first group. The 3.4% sample shows an interesting transition behavior between groups 1 and 2. The shifts based on high ω G″ data and low ω G′ data yield shift factors aT′ and aT belonging to groups 1 and 2, respectively. The PnBA-pUpy samples of p ≤ 2.4% belongs to the first group because they are below the gel point, as suggested from their linear viscoelasticity explained later in Figure 6. Therefore, sols E

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Figure 5. Comparison of LVE master curves of storage and loss moduli, G′ and G″, reduced at Tr = Tg + 65 °C, and predictions given by (a) real p and (b) effective f p with f being a fitting parameter to characterize the fraction of stickers forming interchain association.

Table 1. Key Properties of the PnBA Homopolymers and PnBA-pUpy Samples

a

samples

Mn (g/mol)

Mw (g/mol)

Mw/Mn

Upya (mol %)

PnBA-0%(14K) PnBA-0%(150K) PnBA-1.0%Upy PnBA-2.0%Upy PnBA-2.4%Upy PnBA-3.4%Upy PnBA-5.5%Upya PnBA-13.3%Upya

14 000 150 000 7 000 8 600 9 400 9 800 21 000 21 100

15 500 170 000 9 400 12 000 13 000 13 000

1.11 1.13 1.34 1.38 1.34 1.33

0 0 1.0 2.0 2.4 3.4 5.5 13.3

pc (%)

1.9 1.5 1.4 1.3 0.61 0.61

f

Tg (°C)

1 0.50 0.64 0.69 0.50 0.65

−52.0 −47.5 −46 −45.5 −43 −41 −35 −10

Data reported in ref 4.

The value of f changes from f = 1 for PnBA-1.0%Upy to f = 0.6 ± 0.1 for the samples having higher p. For PnBA-1.0%Upy, p < pc and the average number of stickers per chain is smaller than one. Therefore, the intrachain associations are unlikely to form, thereby giving f = 1. The intrachain association should start to play a role when the average number of stickers per chain becomes two or higher. Here, we do not try to solve the ratio between inter- and intrachain hydrogen bonding theoretically. Instead, we can make a very rough mean-field estimation. In a monodisperse sample, the number of chains share a pervade volume which is P, and P is proportional to N1/2. Obviously, P increases with the chain size and approaches a value of ∼20 when the chains become entangled.30 P = 20 corresponds to a probability of interchain bonding of f = 95% (= 19/20). For PnBA in this study, the entanglement molecular weight is estimated to be Me = 18 000, as explained in the Appendix. Therefore, the PnBA-pUpy samples having M ≤ 21 100 in Figure 5 are not entangled to have f ≤ 0.95 estimated under the mean-field assumption. The connectivity of stickers with the same chain may further facilitate the intrachain association to reduce f to 0.6 ± 0.1 in this study. Another possibly contribution to lower f is the multiple associations between neighboring chains. The necessity of incorporating f in the model

should be partly related to the binary nature for the interaction of the Upy groups.31,32 Because of the binary interaction, the intrachain-paired stickers do not contribute to either sol or gel formation. It was reported that the Upy groups could aggregate or even exhibit microphase separation under certain conditions.33−36 To test this point, the scattering profiles are compared for the PnBA bulk and PnBA-pUpy samples with p = 1.0%, 2.0%, 2.4%, and 3.4% at room T in Figure S4. The scattering profiles show commonly a high q amorphous halo and a correlation peak at q ≈ 5 nm−1 (of a spacing ≈1.3 nm) that corresponds to a backbone-to-backbone spacing of PnBA. No clear sign of aggregation or microphase separation is seen at q < 5 nm−1. 4.2.3. Activation Energy. It is well-known that the hydrogen bonding assigned the sticky chains an additional frictional source. In particular, when physical gel forms, its relaxation strongly depends on sticker dissociation with characteristic time: τs = τ0 exp(Ea /kT )

(5)

Here, τ0 is a local attempt time. For polymeric system, it is reasonable to assign the characteristic time of the smallest Rouse segment, i.e., the Kuhn segment, to τ0. Ea is the activation energy for the sticker dissociation. τs is the time required F

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Macromolecules

corresponding to Ea ≪ 22 kJ/mol at room temperature. For this reason, one frequently adopted method is to construct a master curve that covers a wider frequency range. Nevertheless, to determine τs and τ0 from the master curve strongly relies on how the LVE data are shifted because τs and τ0 usually exhibit different temperature dependences, leading to thermorheological complexity of LVE. The suitable shift should be guided by the dissociation part of moduli, i.e., by τs. The second approach through temperature dependence of relaxation time includes an implicit assumption; i.e., the activation energy Ea does not change with T. We should note that a linear relationship between log aT and 1/T (usually used as an evidence of the validity of the second approach) does not necessarily mean that Ea is insensitive to T (an apparent counterexample is that Ea also changes linearly with 1/T). In addition, an important effect that is frequently overlooked in the second approach is the temperature dependence of τ0. In other words, it is τs/τ0 instead of τs that is directly correlated to Ea. Unfortunately, most experimental approaches that fit viscosity, terminal relaxation, or log aT directly against 1/T have neglected the temperature dependence of τ0, yielding Ea considerably larger than the actual dissociation energy, as evidenced by the SPS-Na and PnBA-pUpy samples examples earlier. Since the moduli of PnBA-3.4Upy is slightly above the gel point, they are contributed from both the Rouse motion and sticker dissociation over a wide T range of ∼50 °C, yielding two sets of shift factors aT′ and aT corresponding to temperature dependences of τ0 and τs, respectively. We may use aT/aT′ to represent the temperature dependence of τs/τ0 to evaluate Ea. Figure 6 plots log(aT/aT′) against 1/T; the linear fit

for the sticker to overcome Ea to dissociate. Equation 5 is quite a general expression applicable for all the associative polymers.19,25 In the literature, there are usually two types of approaches to determine Ea: The first approach bases on a delay of the terminal relaxation of the gel with respect to the Rouse time, i.e., a τs/τ0 ratio.13,15,37 The second approach bases on the temperature dependence of relaxation time (or LVE shift factor).4−6,38−40 The inconsistency is frequently seen for Ea determined from these two approaches. For example, the first approach gives Ea = 70 kJ/mol at T = 140 °C for sulfonated polystyrene with sodium as counterion (with τs and τ0 reported in the literature13,14), while the second approach by Weiss and co-workers gave Ea = 177 kJ/mol. Similarly, we use τs = 1 s and τ0 = 1.4 μs obtained in Figure 5 to determine Ea = 33 kJ/mol by the first approach, which is in sharp contrast to Ea = 90− 120 kJ/mol determined by Feldman et al.4−6 and Ea = 75− 105 kJ/mol by Lewis et al.;7 both were based on the second approach. A natural question is: why cannot Ea be determined consistently when two approaches are essentially based on the same physics incorporated in eq 5? From an experimental perspective, the difficulty of the first approach lies in precise determination of τs and τ0. On one hand, τ0 specifying characteristic time of a Kuhn segment can be evaluated from the Rouse region exhibiting power law, G′(ω) = G″(ω) = G0(ωτ0)1/2, where G0 = ρRT/MK is the characteristic modulus of the Kuhn segment having molecular weight MK. On the other hand, τs specifies the lifetime of stickers, which controls the terminal relaxation τ when the sample is above the gel point. In general, τ can be determined directly from LVE while τs cannot be. Therefore, we should understand a relationship between τs and τ to determine τs. The relationship depends on the extent of gelation ε at 0 < ε: (1) For the samples very close to the gel point, 0 < ε < εC, the network strand is so large that its Rouse time becomes longer than the time for the strand to breakup effectively into two substrands of similar size. Then, the effective breakup would occur continuously until the Rouse time of the substrands becomes the same as their effective breakup time, where the substrands can relax immediately through the Rouse motion. The critical substrands (having the same Rouse time and effective breakup time) has a fixed size, leading to τ being insensitive to τs.13,41 (2) For εC < ε < 1, the Rouse time of network strand becomes shorter than the effective breakup time. Then, any effective breakup occurring at the main backbone would lead to the terminal relaxation of the strand immediately. If each strand has neff effective breakable sites, the frequency for breakup of one of these sites would be neff/τs, leading to a strand relaxation time of τs/neff. Since each strand contains more than one precursor chains on average at εC < ε < 1, neff is larger than one on average to leads to τ ∼ τs/neff < τs. (3) For 1 < ε, each chain would be involved in several network strands. The terminal relaxation time would be controlled by the sticky-Rouse type relaxation of the chain to exhibit a terminal relaxation of τ ∼ τsε2 > τs. Therefore, the difficulty to determine τs lies in a complex relationship between τs and τ. To use τ to approximate τs is valid when ε ≈ 1, corresponding to one network strand per chain averagely. To determine reliable τs and τ0 at the same T is even more challenging, because a usual linear viscoelastic measurement covering a frequency range of about 4 decades is insufficient to detect stress relaxation contributed from both the Rouse motion and sticker dissociation unless exp(Ea/kT) ≪ 104,

Figure 6. Natural logarithm of a ratio of LVE shift factors in panels b and a of Figure 3, aT/aT′, for the PnBA-3.4%Upy sample against inverse of absolute temperature 1/T. The solid line is linear fit whose gradient equals to Ea/k, with Ea the activation energy for dissociation of stickers.

(solid line) enables us to determine Ea = 37 kJ/mol. This value is much closer to Ea = 33 kJ/mol evaluated from the first approach than Ea = 90−120 kJ/mol determined by the second approach of Feldman et al. (from plots of log aT against 1/T).4−6 This result suggests that an ideal model system to determine the activation energy of sticker dissociation could be samples slightly above the gel point, to meet the requirements that the stress relaxations contributed from both the Rouse motion and the sticker dissociation are not too separated so that they can be measured at the same T over certain T range. In fact, we have also observed an agreement on Ea evaluated from two approaches for sulfonated polystyrene with sodium as counterion slightly above the gel point, which will be reported in a following paper. G

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Macromolecules 4.3. Dielectric Behavior. Broadband DRS detects the atomic and electronic polarizations, fluctuation of permanent dipoles, and migration of ions and charged groups. The sticky system usually contains at least two types of dipoles: the dipoles associated with the polymer medium and those associated with the stickers that are usually highly polar. Therefore, the dielectric response of these dipoles provides supplementary dynamic information to those obtained from the rheological measurements. 4.3.1. Overview. The dielectric processes of the PnBA-pUpy system were reported earlier by Shabbir et al.,42 where the dielectric response for samples having p ≥ 3 was characterized by a high-ω α-relaxation associated with the segmental motion of the chain backbone, an α2-relaxation at intermediate ω associated with the fluctuation of sticker dipoles, and ion conduction and electrode polarization at low ω. The dielectric conduction partly masked the low-ω α2-process in the dielectric loss ε″, so Shabbir et al. used the derivative formalism to analyze the dielectric processes.43−45 εder(ω) = −

π ∂ε′(ω) 2 ∂ ln ω

broadens in mode distribution with an increase of p. The increase of Tg and broadening of relaxation mode distribution is consistent with the DSC traces, which should stem from restriction of segmental motion due to the association of stickers. Eisenberg and co-workers have incorporated this effect in the well-known restricted region model.2,46 In this model, the segments in the vicinity of the sticker association are restricted in motion, leading to broader mode distribution of glassy dynamics. When T is raised to 20 °C (panel b), the α process shifts to the higher frequency and lower frequency processes enter the frequency window. The lower frequency processes fully enter the frequency window when T is raised to 80 °C (panel c), where there are clearly two types of behavior: The PnBA-2.4%Upy and 3.4Upy samples seem to exhibit only the α2 process with amplitude increasing with the content of Upy. This is consistent with the results reported for samples having p ≥ 3 by Shabbir et al.42 The PnBA-1.0%Upy and 2.0%Upy samples clearly exhibit one additional dielectric process that is sharper than the α2 process of the PnBA-2.4%Upy and 3.4%Upy samples (as indicated in solid arrows). The molecular origin of this process is not clear so far, but it may be an intrinsic feature for the PnBA hompolymer. (One possible origin is the interfacial polarization for nanophase separated main and side chains, as noted previously for several poly(n-alkyl acrylates) and poly (n-alkyl methacrylates47,48), considering (1) the PnBA homopolymer (p = 0) also show this process and (2) the amplitude of this process is PnBA-0% > PnBA-1.0%Upy > PnBA-2.0%Upy. To explain this trend, we show in Figure 8 a distribution of the

(6)

In this study, we use the same method to analyze the dielectric responses. The derivative formalism εder obtained for PnBA0%-3.4% samples at (a) −20, (b) 20, (c) and 80 °C is plotted against ω in Figure 7. At −20 °C, we can see only the dielectric α process, which shifts to lower frequency and

Figure 8. P(n) of PnBA-pUpy chains having n ions per chain against n, with p value as indicated. n = 0 corresponds to chains containing no stickers.

possibility P(n,p) of n stickers per chain as calculated by eq 4 for the PnBA-1.0%, -2.0%, -2.4%, and -3.4%Upy samples. We found that all the PnBA-pUpy samples in this study contain homopolymer chains with no sticker (n = 0). The fraction of n = 0 decreases from 1 for the PnBA homopolymer to ∼0.6 for the PnBA-1.0%Upy sample and further to ∼0.35 for the PnBA2.0%Upy sample. This may explain a weakening of the low ω process with increasing p if we regard this process as belonging to the homopolymer. 4.3.2. Characteristic Times of Dielectric and Viscoelastic Processes. There are four dielectric processes for these PnBApUpy samples, α, α2, unknown low ω process, and electrode polarization. We can fit the former three processes with the Havriliak−Negami equation

Figure 7. Derivative formalism of dielectric spectra, εder(ω) defined in eq 6, plotted against ω for PnBA bulk (0%) and PnBA-pUpy samples with p = 0%, 1.0%, 2.0%, 2.4%, and 3.4% at (a) −20, (b) 20, and (c) 80 °C. The three dielectric processes, α and α2 for representative samples are indicated in thick unfilled arrows. An unknown low ω process is indicated in filled thin arrow.

* (ω) = εHN

Δε [1 + (iω/ωHN)β ]γ

(7)

where β and γ are shape parameters and ωHN is the characteristic frequency, and the EP process with a power law, εder H

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Macromolecules ∼ Kω−θ, with θ being close to 2. The peak frequency can be obtained accordingly as −1/ β 1/ β ⎛ βπ ⎞ ⎛ βγπ ⎞ ωmax = ωHN⎜sin ⎟ ⎜sin ⎟ ⎝ 2 + 2γ ⎠ ⎝ 2 + 2γ ⎠

Second, τs−1 is lower than ωα2 of α2 relaxation, which was reported also by Shabbir et al.42 This feature differs from our previous observation on the PEO-based ionomers, where τs−1 defined in LVE agrees with ωmax of the α2 process in DRS.15,17 The delay of τs−1 with respect to ωα2 seems to suggest that the fluctuation of Upy groups (detected in DRS) can occur at frequency higher than the effective dissociation of the stickers in LVE. One possible mechanism is that a dissociated sticker should wait for another dissociated sticker to pair.26−28,54 When the density of dissociated pair is low, it is quite possible that the dissociated sticker would reassociate with its old partner several times.26−28,54 During this process, it is possible for a dipole to fluctuate but this would not contribute to the stress relaxation: the sticky Rouse segment would relax only if the dissociated sticker associates with another dissociated sticker different from its original partner. This mechanism should play a role for the difference between τs−1 and ωα2.

(8)

Figure 9 summarizes the peak frequency ωmax for α and α2, ωα, and ωα2 of the PnBA homopolymer and PnBA-pUpy samples,

5. CONCLUSIONS In this study, we examined the linear viscoelasticity of the PnBA-pUpy samples, a model sticky system with well-defined pairing strength for stickers. We found that to incorporate a ratio between inter- and intrachain association enables us to predict quantitatively the linear viscoelasticity on a basis of the analytical model developed in our previous study. A fixed τs (and accordingly Ea) enables reasonable prediction of LVE, suggesting that the dependence of τs on density of stickers, if exists, is not strong in the low p range in this study. For the PnBA-3.4%Upy slightly above the gel point, superposing LVE guided by the stress relaxation from the Rouse motion and sticker dissociation yield two sets of shift factors corresponding to temperature dependence of τ0 and τs, respectively. A ratio of them enables determination of temperature dependence of τs/τ0 and accordingly an activation energy of ionic dissociation Ea. DRS shows three dielectric processes, i.e., the α, α2, and unknown low ω process (might be an intrinsic feature of PnBA with no sticker) before the EP process. The relationship between the LVE-detected lifetime of stickers and the DRSdetected α2 relaxation time becomes more complex than that observed in our previous study on the PEO-based ionomers: the α2 relaxation is consistently faster the sticker lifetime determined in LVE. The difference is not well understood so far but may be related to the reassociation of a sticker with its original partner before associating to a new partner. The detailed mechanism is considered as an interesting subject for future study.

Figure 9. Comparison of the peak frequency ωmax obtained for α (black symbols) and α2 (blue symbols) obtained from analyzing DRS data and τ0−1 (green symbols) and τs−1 (red symbols) from analyzing LVE data. ωmax, τ0−1, and τs−1 are plotted against (a) 1000/T and (b) Tg/T.

where panels a and b plot ωmax against 1000/T and Tg/T, respectively. The PnBA homopolymer exhibits only the α process. In contrast, the PnBA-pUpy samples exhibit the α and α2 processes. For comparison, τs−1 and τ0−1 from LVE are added, which are obtained from τs = 1 s and τ0 = 1.4 μs in the model fitting at Tr and extended to other T via the viscoelastic shift factors. For the PnBA-3.4%Upy samples, shift factors aT and aT′ are utilized to extend τs and τ0, respectively. The comparison of panels a and b reveals that to plot against Tg/T reasonably normalizes the ωmax of α process obtained in DRS and τs−1 and τ0−1 obtained in LVE. Comparison of characteristic times in LVE and DRS provides some molecular details of the relaxation behavior of these samples. First, τ0−1 is lower than ωα of the α relaxation, which is not surprising because τ0 is the characteristic time of the Kuhn segment and accordingly reflects an onset of the rubbery modulus, while the α relaxation reflects the fluctuation of the type-B dipoles perpendicular to the chain backbone, which is more associated with the glassy dynamics.49 The molecular expression and length scale are not the same for τ0 and τα, which is also partly responsible to their different values.49−53

■

APPENDIX Figure 10 shows the G′ and G″ as functions of ω for a series of PnBA samples reduced at Tr, with Tr = 15 °C for the PnBA-14K sample and Tr = 20 °C for the higher M samples. Key properties of all the PnBA homopolymers are summarized in Table 2. The 5 °C difference in Tr reflects a difference of their Tg. This choice allows a good superposition of the Rouse region where G′ ∼ G′ ∼ ω1/2. The PnBA-14K sample shows a typical behavior for nonentangled chains, i.e., terminal relaxation following the Rouse region. In comparison, the PnBA-61K, -150K, and -300K samples show entangled chain behavior: the rubbery plateau can be clearly seen before the terminal relaxation. The plateau modulus GN = 105 Pa, determined as the value of G′ at the valley of G″, enables us to calculate the entanglement I

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Donghua Xu at Changchun Institute of Applied Chemistry for helpful discussion. Q.C. acknowledges Natural Science Foundation of China (21674117) and “Thousand Youth Talents Plan” program.

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Figure 10. LVE master curves of storage and loss moduli, G′ and G″, obtained as a functions of angular frequency ω for PnBA bulk samples of Mn as indicated at Tr = 20°C. Data of the 14K sample having Tg 5 °C lower than the other three samples are reduced to 15 °C to ensure an isofrictional status for these samples, as indicated by the agreement in a Rouse region where G′ ∼ G″ ∼ ω1/2. Inset shows viscosity η plotted against weight-average molecular weight Mw, with the lines being η ∼ M and η ∼ M3.4, the cross point of this two lines locates at Mc = 30 000.

Table 2. Key Properties of the PnBA Homopolymers samples

Mn (g/mol)

Mw (g/mol)

Mw/Mn

Tg (°C)

PnBA-14K PnBA-61K PnBA-150K PnBA-300K

14 000 61 000 150 000 300 000

15 500 79 000 170 000 366 000

1.11 1.29 1.13 1.23

−52 −48 −47.5 −48

molecular weight to be Me = ρRT/GN = 20K. To further check this point, we plot the zero-shear viscosity η0 = [G″/ω]ω→0 against M in the inset. The lines associated with the viscosity data are η0 ∼ M and η0 ∼ M3.4 best fitting the η0 vs M for the low and high M samples. The cross point of these two lines locate at Mc = 30K, which usually corresponds to ∼2Me.55 A combination of GN and Mc suggests Me = 18K ± 2K. The value of Me is in accordance with Me = 20K reported by Zosel et al.56 and Me = 17.5K reported by Hawke et al.57 The PnBA-pUpy samples examined in this study having Mn ≤ 21 100 can hardly be entangled.

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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.6b02017. Figures S1−S4 (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*(Q.C.) E-mail [email protected]. *(L.G.) E-mail [email protected]. ORCID

Quan Chen: 0000-0002-7771-5050 J

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DOI: 10.1021/acs.macromol.6b02017 Macromolecules XXXX, XXX, XXX−XXX