Linear Viscoelasticity and Dielectric Spectroscopy ... - ACS Publications

Article pubs.acs.org/Macromolecules

Linear Viscoelasticity and Dielectric Spectroscopy of Ionomer/ Plasticizer Mixtures: A Transition from Ionomer to Polyelectrolyte Quan Chen,†,‡ Nanqi Bao,† Jing-Han Helen Wang,† Tyler Tunic,† Siwei Liang,† and Ralph H. Colby*,† †

Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ‡ State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China S Supporting Information *

ABSTRACT: For ionomers, unfavorable interaction between highly polar ion pairs and the low polarity polymer medium leads to ion aggregation. In contrast, for polyelectrolytes, the counterions prefer solvation in the polar medium to leave the chain charged and accordingly stretched due to the charge repulsion. In this study, linear viscoelastic and dielectric properties of mixtures of two ionomers with high dielectric constant low volatility plasticizers were examined. The ionomer chains having bulky side chains are not entangled. Upon increasing the plasticizer content, the terminal relaxation is significantly accelerated due to two effects: (1) a plasticizing effect lowering the Tg and (2) a higher dielectric constant that softens the ionic interactions, leading to ionic dissociation into isolated pairs that further boosts the static dielectric constant at low frequency/long time. A model incorporating these two mechanisms and utilizing a dielectric constant εC, after the nonionic segmental α relaxation as the relevant dielectric constant for ion dissociation, predicts quantitatively the accelerated dynamics, as ionomers transition to polyelectrolytes on dilution. are defined as “polymers in which bulk properties are governed by ionic interactions in discrete regions (ionic aggregation)”, while polyelectrolytes are “polymers in which solution properties in solvents of high dielectric constants are governed by electrostatic interactions over distances larger than typical molecular dimensions”. To describe the detailed morphology of ionomers, Eisenberg proposed a restricted region model that regards ionomers as heterogeneous materials containing three regions: (1) multiplets or aggregates of ions, (2) restricted regions near the multiplets/aggregates where the segmental motion is restricted, and (3) nonrestricted regions sufficiently far from the multiplets/aggregates. When the ion content is high, the restricted regions overlap and fully occupy the space. Recently, ionomers and polyelectrolytes are widely studied due to various energy related applications like ion batteries, fuel cells, supercapacitors, sensors, and ionic actuators.18 Nevertheless, some studies lack a clear quantification of the ionic status critical in distinguishing ionomer from polyelectrolyte. Therefore, it is informative to test quantitatively an ionomer− polyelectrolyte transition, in particular the polymer and ion dynamics of ion-containing polymer chains during this transition. To this end, we should look into the molecular scale differences between ionomer and polyelectrolyte. In particular, the driving forces for ions to associate into pairs and for ion

1. INTRODUCTION Polymers with one type of ion covalently bonded to them can be classified as two types: ionomer and polyelectrolyte. Historically, polymers with less than 10% ionic monomers were termed ionomer, while those containing much higher (∼100%) content of ionic monomers as polyelectrolyte.1−3 Since polymers usually have dielectric constant lower than small polar molecules, polar ionic groups usually aggregate and the ionomer dynamics are strongly related to dissociation of ion pairs from the aggregates. In contrast, bulk polyelectrolyte usually exhibits extremely high glass transition temperature that limits its application. Therefore, polyelectrolytes are usually dissolved in a high dielectric constant solvent, where the ion pairs can dissociate to some extent. As a result, the polyelectrolyte chain is highly charged and locally extended.4 Without salt present, the extension leads to higher reduced viscosity with dilution because the chain gets larger, a phenomenon referred to as the “polyelectrolyte effect”.5 The above definition based on ion content faces problems in some cases. For example, the same ion-containing polymer may exhibit ionomer-like behavior in lower polarity solvent, like tetrahydrofuran and dioxane, while polyelectrolyte behavior in a more polar solvent, like water, dimethyl sulfoxide, and dimethylformamide.6−16 Even in the same solvent, the ioncontaining polymer may exhibit ionomer-like behavior at high concentration but polyelectrolyte behavior at lower concentration where the dielectric constant is higher. To address this problem, Eisenberg and Rinaudo proposed definitions based on their physical properties rather than ion content:17 Ionomers © XXXX American Chemical Society

Received: September 6, 2015 Revised: October 27, 2015

A

DOI: 10.1021/acs.macromol.5b01958 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules pairs to aggregate, as the polarity of the environment around the ions is changed. For polyelectrolytes in a sufficiently polar medium, there is an important length scale called the Bjerrum length lB that specifies a distance where the electrostatic energy between ions equals the thermal energy kT. If the charge density along the chain is much sparser than one elementary charge per lB, all the ions are able to dissociate. In contrast, if the charge density is larger than one elementary charge per lB, there is an Oosawa/Manning condensation; some of the counterions condense on the chain to form ion pairs, lowering the electrostatic energy of the system but losing their translational entropy. After the condensation, the density of effective charges along the chain (ions that have their counterion dissociated) is of the order of one elementary charge per lB.2,19,20 In media with lower dielectric constant, the condensed ions may exist in different states, for example, ion pair, quadrupoles, or larger ion aggregates. The evolution of ion states with dielectric constant has not been fully understood at the molecular scale but decreasing the dielectric constant clearly drives ions to associate more.1,21,22 For example, it was noted that the dielectric constant of poly(ethylene oxide)-based sulfonate ionomers with sodium counterions obeys the Onsager prediction at low temperatures where most of the ions are paired, but the dielectric constant decays abruptly below the Onsager prediction as temperature is raised, indicating ion aggregation on heating.21,22 The driving force for the pairs to aggregate can be discussed quantitatively with respect to the interaction energy between ion pair permanent dipoles, i.e., the Keesom energy. We assume that the relevant dielectric constant for the Keesom energy between ion pairs is the dielectric constant after the α relaxation of surrounding smaller dipoles but before the considerably slower α2 relaxation of the ion pairs. The Keesom energy gives another important length scale where the interaction energy between isolated ion pairs becomes comparable to kT, which we term the Keesom length. A combination of Bjerrum length and Keesom length allows a molecular interpretation of the linear viscoelasticity (LVE) and dielectric relaxation spectroscopy (DRS) of poly(ethylene oxide) (PEO)-based ionomers along the ionomer-to-polyelectrolyte transition upon adding higher polarity EO-based plasticizer.23 Both the ionomer and the plasticizer are EObased, allowing good miscibility over a wide T range. The dielectric constant of the plasticizer is enhanced significantly by incorporating polar cyclic carbonate end groups. This molecular design allows dielectric constant of the ionomer/plasticizer mixture to vary in a wide range, facilitating investigation of the ionomer−polyelectrolyte transition.

Figure 1. (a) Chemical structure of the random copolymer ionomers and their nonionic homopolymer counterpart ( f = 0). (b) Chemical structure of the plasticizer. (c) Schematic illustration of size and conformation of the ionomer and plasticizer samples. The side chains are not stretched, but the backbones of the ionomer are stretched. reaction was maintained at 80 °C for 9 h for ∼70% conversion under the protection of argon. The ionomer sample was thoroughly dialyzed with deionized water using a Slide-A-Lyzer G2 dialysis cassette with 3500 molecular weight cutoff. The dialyzed solution was concentrated first via rotavap and then vacuum-dried at 80 °C for more than 48 h before preparing solutions. The chemical structure of the ionomer sample is shown in Figure 1a. Plasticizer. The PEO oligomer plasticizer was synthesized by condensation reaction with triethyl amine as acid scavenger. Cyclic [(allyloxy) methyl] ethylene ester carbonic acid (10.86 g, 0.068 mol), anhydrous CH3CN (20 mL), and chlorodimethylsilane (7.67 g, 0.081 mol) were added into a predried flask. The mixture was cooled in an ice bath before 0.3 mL of Pt catalyst (2% solution in xylene) was added. The mixture was allowed to react overnight to complete the reaction. The solvent was evaporated, and the residue was vacuum distilled to obtain 4-((3-(chlorodimethylsilyl)propoxy)methyl)-1,3dioxolan-2-one as a colorless liquid (15.2 g, 89%). 4-((3-(Chlorodimethylsilyl)propoxy)methyl)-1,3-dioxolan-2-one (8.7 g, 0.034 mol) was added dropwise into the mixture of triethylamine (7.5 g), tetraethylene glycol (6.67g, 0.034 mol), and 20 mL of dry THF over 30 min. The reaction mixture was allowed to stir overnight to complete the reaction. The chemical structure of the plasticizer sample is shown in Figure 1b; the plasticizer contains five EO, similar to the ionomer that also contains a large fraction of EO units. Although the ionomer having large enough M contributes negligibly to the mixing entropy, the plasticizers have low M, so the mixing entropy is sufficient to allow the ionomers and the plasticizer to achieve molecular scale miscibility. The polar cyclic carbonate end groups are chosen due to their high polarity. The flexible C−Si−O linkages enable the cyclic carbonate groups to respond easily to an applied electric field. These two features ensure a high dielectric constant of the plasticizer, and accordingly the ionomer to polyelectrolyte transition as plasticizer is added. However, the cyclic carbonate end groups can degrade in the presence of water so care is taken to maintain all samples in a dry state. Solutions. A predetermined amount of fully dried ionomer and plasticizer (or nonionic counterpart and plasticizer) was dissolved in their common solvent methanol and rotovapped until most of the methanol was evaporated. The solution was then kept in a vacuum oven at 80 °C at least for 2 days. Extra plasticizer (more than a 20% excess) was used in preparing the methanol solutions because the plasticizer slowly evaporates at 80 °C in vacuum. The final weight fraction of ionomer is calculated from the masses of the ionomer and the sample as a whole after the vacuum drying. 2.2. Measurements. Solutions of ionomers were optically transparent and studied using differential scanning calorimetry (DSC), linear viscoelasticity (LVE), and dielectric relaxation spectroscopy (DRS) measurements. DSC measurements were conducted on a TA Q100 differential scanning calorimeter. The samples were heated to 140 °C and cooled

2. EXPERIMENTAL SECTION 2.1. Samples. Ionomer. Reversible addition−fragmentation chain transfer (RAFT) polymerization was used to synthesize PEO-based ionomers and a nonionic counterpart. A dry glass reactor with a magnetic stir bar was charged with a mixture of PEO9M or PEO2M (see Figure 1a), the ionic monomer (sodium sulfonated styrene), 2cyano-2-propyldodecyl trithiocarbonate (RAFT agent), 2,2′-azobis(2methylpropionitrile) (AIBN), and 30 mL of dimethylformamide (DMF) (A reference nonionic sample was synthesized by polymerizing only the PEO9M monomer.). RAFT agent and AIBN were at a 5:1 ratio. The degree of polymerization was preset to be ∼400 by ratio of monomers to initiators for both the EO9 and EO2 ionomers. Before reaction, three freeze−pump−thaw cycles were applied to the reactant solution with liquid nitrogen under vacuum to remove oxygen and moisture from the reaction chamber.24 The temperature of the B


Article

Macromolecules Table 1. Characteristics of EO-Based Ionomers sample

Nbb

NEO

f

Ma (kg mol−1)

θb (%)

rpackc (nm)

REOd (nm)

C∞,bbe

Rbbe (nm)

pf (nm)

af (nm)

P0g (nm−3)

EO2 EO9

800 800

9 30

0.12 0.3

76 164

4.5 4.3

0.67 1.1

0.78 1.4

31 57

16 21

0.46 0.66

9 13

0.42 0.49

a

Molecular weight of the sample is calculated as M = (1/2){f Nbbm0,ion + (1− f)Nbbm0,neutral}, where m0,nentral and m0,ion are the molecular weights of neutral and ionic monomers, respectively. The prefactor 1/2 is because each monomer contains two backbone bonds. bThe ratio of ion and ether oxygen is θ = f/[(1 − f)(q + 1)], where q is the number of EO units per neutral monomer (q = 2 or 9). These ion contents were chosen to have very similar θ. cAssuming that a side chain fully occupies a slice surrounding the backbone, the radius of such a fully packed slice is rpack = (V0,neutral/ 2πl)1/2, where V0,neutral = m0,neutral/ρNAV is the volume per neutral monomer, with density ρ = 1.1 g/cm3. dMean end-to-end distance of the side chain assuming a random walk is ⟨REO⟩ = (C∞,EONEOl2)1/2, with C∞,EO = 6.7 being the characteristic ratio of PEO. eThe end-to-end distance of the backbone is estimated to be ⟨Rbb⟩ = (C∞,bbNbbl2)1/2, where the characteristic ratio for the backbone is estimated as C∞,bb = b/l ≈ 4r/l, where b (≈ 4r) is the Kuhn length.25 fThe entanglement length is estimated as 20 times the packing length: a ≈ 20p.26 The packing length is estimated as p = V0,EO/ (2C∞,bbl2). The factor 2 is because each neutral monomer contains two backbone bonds. gThe number density of ions P0 is calculated as fρNAV/ {f m0,ion + (1 − f)m0,neutral}. to −80 °C at 10 °C/min to remove thermal history and heated to 140 °C at 10 °C/min again, during which the DSC traces were recorded. DRS measurements were conducted on a Novocontrol GmbH Concept 40 broadband dielectric spectrometer. Samples were loaded fast on a freshly polished brass electrode and then heated fast up to 80 °C in vacuum before loading the top brass electrode; this step degassed the sample effectively. The increase of sample content due to loss of plasticizer during this step should be less than 0.3 wt % (estimated from the rate of loss of plasticizer with time in vacuum). After charging the vacuum oven with high purity argon, the samples were equilibrated at 80 °C for 3−5 h before being transferred fast to the spectrometer. Before the measurements, samples were annealed in the instrument at 120 °C for 1 h under nitrogen for drying. Isothermal frequency sweeps from 107 to 10−2 Hz were conducted in 10 or 5 K steps from 120 °C downward to below the DSC Tg, with precise temperature control within ±0.05 K. LVE measurements were conducted on an Advanced Rheometric Expansion System (ARES)-LS1 rheometer (Rheometric Scientific). 7.9 and 3.0 mm diameter parallel plates were utilized. Before the measurements, 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. All DSC, LVE, and DRS measurements were performed under nitrogen protection.

= (V0,EO/2πl)1/2, where V0,EO is the volume per nonionic monomer. The mean end-to-end distance of the side chain in a random walk conformation is ⟨REO⟩ = (C∞,EONEOl2)1/2, where C∞,EO = 6.7 is the characteristic ratio of PEO. Table 1 lists rpack and ⟨REO⟩ for both ionomers: rpack < ⟨REO⟩ for both the EO2 and EO9 ionomers, suggesting that the side chain is not crowded enough to be stretched. Therefore, it is reasonable to take ⟨REO⟩ as the effective radius r of the linear ionomer chain. It is reported25 that the persistence length for such “thick” linear chains is ∼2r. Then, it is possible to estimate the characteristic ratio as C∞,bb = b/l ≈ 4r/l, with b the Kuhn length, and estimate the packing length as p = VEO/(2C∞,bbl2); C∞,bb = 57 and 31 for EO9 and EO2, respectively, which are much larger than C∞ = 8.2 for PMMA, meaning that the backbone is stretched due to a steric effect of the side chains. On one hand, the entanglement length is ∼20 times the packing length26 for flexible chains, i.e., a ∼ 20p. On the other hand, the end-to-end distance of the backbone is estimated to be ⟨Rbb⟩ = (C∞,bbNbbl2)1/2. ⟨Rbb⟩ < 2a for EO9 and EO2 (see Table 1), suggesting that neither ionomer is long enough to be entangled. To sum up, the backbones are stretched while the side chains are not. The main backbones are not long enough to be entangled, which is in accordance with a Rouse-type dynamics observed for LVE of both ionomers and all solutions, explained later in the discussion of Figure 4. 3.2. DSC Traces. Figures 2a and 2b show DSC traces obtained for EO9 and EO2 solutions having ionomer contents as indicated. The two systems show qualitatively similar behavior: The bulk ionomer samples exhibit broad glass transitions covering wide T ranges. Tg reduces and the glass transition narrows with decreasing ionomer content, and the ionomer to polyelectrolyte transition completes at low weight fraction w: the DSC trace of the 11 wt % samples is almost indistinguishable from that of the plasticizer. The quantitative difference is EO2 ionomer bulk and 88 wt % samples show extremely broad glass transition, covering T ranges of ∼150 and ∼100 °C, respectively. In comparison, the EO9 bulk ionomer shows a somewhat narrower glass transition covering a range of ∼60 °C. These broad glass transitions are attributed to a distribution of ions within Kuhn volumes b3. Considering the local Tg should depend on a localized ion/EO ratio within b3 (higher ratio leading to higher Tg), a broad Tg is naturally expected for the chemical structure of the ionomers. Tg increases and the distribution becomes broader when ions form quadrupoles and ion chains, as expected for the sulfonate−Na pair in the presence of sufficient EO.1,27,28 The EO2 system shows much broader glass transition than the EO9 system at high w, suggesting that the distribution of

3. RESULTS 3.1. Structure. Figure 1 shows the ionomers and the plasticizer. The ionomer samples are named as EO2 and EO9 with respect to the number of EO units in their nonionic side chains (see Figure 1a). EO2 and EO9 have a fraction of ioncontaining monomer f = 0.12 and 0.3, respectively, allowing them to have similar ion/EO ratio, θ = f/[(1 − f)(q + 1)], with q being the number of EO units per nonionic monomer (q = 2 or 9 gives θ = 4.5% and 4.3%, respectively). It is well-known that the dynamics for such a branched chain relies on the conformation of both the side chain and main backbone. Therefore, it is informative to estimate (1) whether the side chain is stretched and (2) whether the main chain is entangled. As a rough estimation, we may neglect any chemical difference between covalent bonds and assume each bond has a length of l = 1 Å. EO2 and EO9 have number of bonds of backbone and side chain as Nbb (≈800 for the EO9 and EO2 ionomers) and NEO (≈30 for the EO9 side chain and ≈9 for the EO2 side chain), respectively. The plasticizer sample has number of bonds Np (≈30) comparable to that of the side chain of EO9 (see Figure 1c). Each side chain corresponds to one monomer that has two backbone bonds. Assuming that a side chain fully occupies a slice surrounding the backbone, the radius of such a slice is rpack C


Article

Macromolecules

Figure 3. Temperature derivative of Cp with respect to T, dCp/dT, of (a) EO9 and (b) EO2 ionomers and their mixtures with plasticizer, with weight percent of ionomer w as indicated. For these curves, the data have been multiplied by a factor K as indicated and shifted vertically for clarity. The filled diamonds indicate the peak of the derivative curve; the unfilled diamond symbols indicate Tg′ estimated from analysis of the viscoelastic shift factors in Figure 5.

Figure 2. DSC traces of (a) EO9 and (b) EO2 ionomers and their mixtures with plasticizer, with weight percent of ionomer w, indicated by the number above each curve.

localized ion concentration is much broader for the ionomer with shorter EO side chain length. First, it is reported that the sulfonate−Na pair can be coordinated by five ethylene oxides.29 Then, this effect should be saturated for sulfonate−Na pairs buried in the EO9 side chains, but not in the short EO2 side chains. Therefore, ions of the latter are more likely to associate. Second, it is obvious that the steric hindrance for the formation of aggregates, in a sense that the side chain should have less favorable conformation, is stronger for the EO9 ionomer than for the EO2 ionomer. In summary, ionic association is both enthalpically and entropically more favorable for the EO2 ionomer with shorter EO side chains. To further show the effect of ion, a nonionic reference EO9 homopolymer (with only EO9 monomer but no ion) was blended with the same plasticizer. Narrow glass transitions (0.3 MPa, still considerably larger than the real plateau (∼0.1 MPa). This difference should be partly caused by a large fraction of intrachain associations that do not contribute to the plateau. Neither the strong narrowing of the glassy mode distribution nor the significant delay of terminal relaxation can be observed for the nonionic EO9 homopolymer blended with the same plasticizer. The glass transition (see Figure S1) and viscoelastic moduli (see Figure S3) are very similar for the nonionic homopolymer and its mixtures of various contents from 11 to 89 wt %. Figure 5 summarizes shift factors aT for the master curves shown in Figure 4. The temperature dependence of these frequency scale shift factors for the ionomers and their mixtures weakens as ionomer content is decreased. However, when Tr + ΔTg′ is chosen as the reference temperature, the reduced shift factor aT/ΔaT of all the other samples exhibit T dependence very similar to that of the 11 wt % sample, which is chosen as the reference (see Figure 6). Here, ΔTg′ is an adjusting parameter that reflects an enhancement of chain friction for higher ionomer content mixtures in comparison with that of the reference sample (w = 11 wt %); ΔTg′ is adjusted to allow the curve aT/ΔaT against T − Tr − ΔTg′ to overlay best with that of the 11 wt % sample. (For each adjusting parameter ΔTg′, ΔaT is chosen to allow the curve aT/ΔaT against T − Tr − ΔTg′ to pass through the point (0, 1), so ΔTg′ is the only adjusting parameter here.)

glassy modulus, leading to broadening of the relaxation mode distribution as a whole at high w (from glassy to terminal relaxation as ionomer content increases). Both the mode distribution broadening and the delay of the terminal relaxation are much stronger for the EO2 system; for example, even the 88 wt % EO2 mixture exhibits broader mode distribution and slower terminal relaxation than the EO9 bulk ionomer. The EO2 bulk ionomer shows a plateau ∼0.1 MPa at low ω. This LVE response should be partly related to its unique thermal behavior, i.e., the two Tgs clearly seen in Figure 3, indicating the existence of a high Tg ion aggregate phase that acts as physical cross-links.1,32 If all the strands between ions contribute to LVE, the number density of strands would be ∼P0 and the characteristic modulus would be P0kTr = 1.5 MPa. On the other hand, kT per chain can be estimated as vkTr = ρRTr/M = 0.032 MPa. The plateau modulus of ∼0.1 MPa suggests ∼3 effective network strands per chain and ∼15 ions per effective network strand. To understand why the strand density is much lower than the ion density, it is informative to look into the ionic states. FTIR allows a classification of SO3− groups associated with no cation, one cation, and two cations, corresponding to free, paired, and aggregated states, respectively. Our previous study suggested that EO9 has 23 mol % SO3− groups in aggregation states that serve as physical cross-links.32 This fraction would be larger for EO2 that can aggregate more easily. Therefore, assuming all the cross-linked strands are elastically effective network strands, the plateau E


Article

Macromolecules

The composition dependence of Tg′ obtained from analyzing the LVE shift factors is shown in open symbols in Figure 7, and

Figure 5. Viscoelastic shift factors aT vs T − Tr for (a) EO9 and (b) EO2 ionomers and their mixtures for the master curves in Figure 4, with Tr = −10 °C the reference temperature, with weight percent of ionomer w as indicated. Curves are fits of WLF equations for shift factors of the two 11 wt % samples (eq 1).

Figure 6. Viscoelastic frequency scale shift factors aT′ for (a) EO9 and (b) EO2 ionomers and their mixtures reduced to T − Tr − ΔTg′ with Tr = −10 °C the reference temperature, with weight percent of ionomer w as indicated. ΔTg′ is the Tg difference between a sample and the 11 wt % mixture. Curves are fits of WLF equations for shift factors of the two 11 wt % samples (eq 1). The cross hairs represent a point (0, 1) that all data sets go through.

Here, ΔaT reflects a deceleration of other samples compared to the reference due to the Tg increase, and we can estimate a Tg′ for higher ionomer content mixtures as ΔTg′ + Tg,r, with Tg,r being Tg of the reference sample. Tg′ thus estimated is shown in unfilled diamond symbols in Figure 3, which locates within the broad glass transition region. The solid curves in Figures 5 and 6 are fits to the Williams− Landel−Ferry (WLF) equation log a T = −

7.0(T − Tr) 95 + T − Tr

also in Figure 3. For comparison, Tg obtained from the peak of dCp/dT vs T is shown as filled symbols in Figure 7. Note the difference between these two Tgs: the shift factors at T > Tg

(1)

with Vogel temperature −105 °C. The EO9 and EO2 bulk ionomers and the 88 wt % EO2 mixture show deviations from eq 1 at T close to Tg, where the temperature dependence becomes Arrhenius and weaker than that expected from eq 1. Similar behavior was reported in miscible polymer blends where one component has been locally quenched but the other has not; the latter component exhibits Arrhenius T dependence.33−36 The ionomer might be similar to miscible polymer blends in a sense that the segments are not quenched simultaneously: the segments out of the restricted region are quenched at T lower than the segments within the restricted regions. Then, the segments out of the restricted region exhibit Arrhenius T dependence at T lower than Tg of the segments inside the restricted region.

Figure 7. Composition dependence of the glass transition temperature Tg. Curves are fits of the Fox equations to the mixtures with ionomer content less than 0.6, with plasticizer Tg = −60 °C, EO9 ionomer Tg = −45 °C, and EO2 ionomer Tg = −27 °C, that are significantly lower than the actual ionomer Tg values but in the broad range of their DSC glass transitions (see Figure 3). F


Article

Macromolecules reflect the temperature dependence of friction averaged over a chain. The friction of a chain is influenced by friction of both nonionic and ion-containing monomers, with the latter having higher friction. The peak of dCp/dT vs T, on the other hand, is biased toward the low T end of the glass transition, which is dominated by the nonionic monomers and regions of lower ion content. Tg′ based on the WLF analysis is indistinguishable from that from the peak of dCp/dT at low w, for which the glass transition is narrow, but Tg′ becomes significantly larger for high w, where the glass transition broadens significantly. In Figure 3, it is obvious that when the glass transition is broad, Tg′ obtained from the WLF analysis locates in the middle of the glass transition, significantly higher than the peak of dCp/dT, suggesting that Tg obtained from the WLF analysis reflects an averaged friction of ionomer chains. The solid curves in Figure 7 are calculated by the Fox equation

1 w 1−w = + Tg Tg,A Tg,B

(2)

where Tg,B = −60 °C is glass transition temperature of the plasticizer and Tg,A = −45 °C for EO9 and −27 °C for EO2 are two fitting parameters chosen to fit the data for w < 0.6. Tg from both dCp/dT (filled symbols) and Tg′ from WLF analysis agree reasonably with the Fox equation for low w < 0.6. Nevertheless, for w > 0.6, ion association leads to an enhancement of Tg not anticipated by the Fox equation (fit to the lower ionomer contents), and this enhancement is much stronger for Tg from the WLF analysis. This feature suggests a polyelectrolyte−ionomer transition at intermediate ionomer content. 3.4. DRS Spectra. Our previous study shows that for PEObased ionomers three dielectric processes are detectable above Tg: a high frequency α process associated with segmental motion, an intermediate frequency α2 process associated with ionic dissociation, and the low frequency electrode polarization (EP) process.37,38 The conductivity usually screens the dielectric α2 process in dielectric loss ε″. Then, the derivative formalism, εder(ω) = −π ∂ε′(ω)/2 ∂ ln ω, was used to analyze the low frequency α2 process (corresponding ε′(ω) data are shown in Figure S5).39,40 Figure 8 shows εder(ω) plotted against ω for each ionomer and their mixtures at T = −10 °C. (The dielectric data are not shown for the EO2 bulk ionomer because the sample cannot flow at T ≤ 180 °C (where the sample is safe from degradation), making full contact between sample and electrode difficult, consistent with Figure 4.) For the plasticizer, two processes can be seen: an α process at high ω and EP process at low ω. Once the ionomer is included, an extra α2 process appears in an intermediate frequency range. The α2 processes can be fit with the Havriliak−Negami equation * (ω) = εHN

Δε [1 + (iω/ωHN)β ]γ

Figure 8. Derivative formalism of dielectric spectra, εder(ω) = −π ∂ε′(ω)/2 ∂ ln ω, plotted against angular frequency ω for (a) EO9 and (b) EO2 ionomers and their mixtures at T = −10 °C, with weight percent of ionomer w as indicated.

When the α process is visible in the frequency window of the measurements, the α process was fit in the same way as the α2 process via utilizing eqs 3 and 4, and accordingly ε′(ω) = ε∞ + εα′(ω) + εα2′(ω). Nevertheless, at high T, the α peak is out of the frequency window of the measurements. Then, ε′(ω) = εC + εα2′(ω) is used to fit the data in the frequency window, with εC = ε∞ + Δεα. Here, the subscript “C” is used because εC is the relevant dielectric constant of the “Coulomb energy” for the dissociation of ion pairs from aggregates detected as the α2 process. In other words, the electrostatic interaction of ion pairs is mediated by a mixture of polymer with large density of ethylene oxides and plasticizer with five EO, two cyclic carbonates, and two very flexible C−Si−O linkages.23 The method explained above is only applicable when the α2 and α processes can be resolved. For high w (= 100 wt % for EO9 and 88 wt % for EO2), the two processes broaden and merge into an extremely broad process. For this case, it is impossible to determine εC (= ε∞ + Δεα) but still possible to determine εs (= εC + Δεα2) by fitting the data at ω higher than EP with one broad HN process (β = 0.45, γ = 1). Therefore, for these two samples, εC is not shown (in Figure 9) while εs is shown (in Figure 11). For samples exhibiting separated α2 and α processes that can be resolved with the HN analysis, the Coulomb dielectric constant εC determined as εC = ε∞ + Δεα is plotted against 1000/T in Figure 9. An increase of εC with the content of plasticizer reflects an increasingly polar surrounding for the ionic dissociation (the α2 process). This increase can be more clearly seen in Figure 10, where εC at the same T = −10 °C (=

(3)

Here, β and γ are shape parameters and ωHN is the characteristic frequency. The EP process of εder can be fitted by a simple power law function. From this analysis, the peak frequency of the α2 process is obtained easily.39 ωmax

−1/ β 1/ β ⎛ βπ ⎞ ⎛ βγπ ⎞ = ωHN⎜sin ⎟ ⎜sin ⎟ ⎝ 2 + 2γ ⎠ ⎝ 2 + 2γ ⎠

(4) G


Article

Macromolecules

Figure 11. Temperature dependence of the static dielectric constant εs for (a) EO9 and (b) EO2 systems, with weight percent of ionomer w as indicated. Solid lines show Onsager fit for εs at low T. T dependence of εs stronger than the Onsager lines means ions aggregate on heating.21,22,32,43

Figure 9. Temperature dependence of the Coulomb dielectric constant before the α2 relaxation, εC, for (a) EO9 and (b) EO2 ionomers and their mixtures with the plasticizer, with weight percent of ionomer w as indicated.

(5)

Figure 11.39,42 For the mixture data, εs ∼ 1/T holds at low T, but not at high T where εs becomes more reduced compared to the solid line Onsager predictions. This effect has been noted in previous studies, which appears to be related to ion aggregation induced by reduction of dielectric constant with increasing T,21,22,32,43 lowering the number density of highly polarizable ion pairs. For the 90 and 100 wt % EO9 samples, εs increases with decreasing T but never achieves εs ∼ 1/T even at the lowest T, suggesting that the number density of dipoles has not been saturated even at the lowest T of the measurements. The EO2 system exhibits concentration dependence similar to that of EO9, with however, lower εs than EO9. This result is in accordance with an expectation that ions can aggregate more strongly for EO2, leading to a smaller number density of isolated ion pairs that are free to respond to the AC electric field. Moreover, by regarding εs below the Onsager prediction as a sign of ion aggregation as temperature is raised, it seems that the EO2 system starts to aggregate at lower T, in accordance with the easier aggregation expected for the ionomer with shorter EO side chains.21,22,32,43

with εC,plasticizer = 49 and εC,ionomer = 7 at −10 °C; the latter is εC of the EO9 nonionic homopolymer. This result shows a big role of introducing two carbonate groups at the two ends of the plasticizer: increasing εC and softening ionic interactions.31 Figure 11 plots static (low frequency) dielectric constant εs = εC + Δεα2 against 1000/T (εs = εC for the bulk plasticizer exhibiting no α2 process). εs exhibits w and T dependences quite different from that of εC. For the EO9 system at 11 wt %, εs is much larger than that of the plasticizer, reflecting a solvation effect, i.e., breakup of the ion aggregates into more polarizable ion pairs. When the dielectric constant is governed by the orientation of permanent dipoles, i.e., εs ≫ ε∞, we should have εs ≈ Δεα + Δεα2 (ε∞ ≈ 3 is small). Then, εs ∼ 1/T is predicted by the Onsager equation if the number density of dipoles does not change with T, as indicated by solid lines in

4. DISCUSSION 4.1. Dielectric Processes and Conductivity. Ionic conductivity σDC obtained from the ionic conduction region, where ε″ ∼ ω−1 and σDC = ωε0ε″, is plotted against Tg/T in Figure 12, with Tg from the peak of dCp/dT (σDC vs. 1000/T and Tg′/T are shown in Figures S7 and S8, respectively). After the Tg normalization, σDC for low w samples are reduced to common curves. Nevertheless, high w ionomer samples, i.e., EO2 with w = 72 and 88 wt % and EO9 with w = 90 and 100 wt %, show lower conductivity, indicating enhanced ionomer behavior: ions are increasingly condensed and aggregated as ionomer content increases. Our previous studies show that the ionic conductivity σDC is controlled by the α2 process,37,44 as expected by the Barton,

Figure 10. Composition dependence of the Coulomb dielectric constant εC. The curve is the Landau and Lifshitz mixing rule,41 eq 5.

Tr for LVE master curves) is plotted against w. The curve is a fit to Landau and Lifshitz’s mixing rule.23,41 1/3 1/3 εC1/3 = wεC,ionomer + (1 − w)εC,plasticizer

H


Article

Macromolecules

and G″ of the EO9 system are divided by volume fraction of the ionomer, ϕ, and plotted against a reduced frequency ωΔ in Figure 14a. The reduction factor Δ is chosen to best

Figure 12. Ionic conductivity σDC plotted against Tg/T for (a) EO9 and (b) EO2 ionomers and their mixtures with the plasticizer, with weight percent of ionomer w as indicated.

Nakajima, and Namikawa (BNN) equation44−49 σDC = ωmaxεsε0, tested in Figure 13 by plotting σDC/ε0 against

Figure 14. (a) G′ and G″ in Figure 4 are normalized by volume fraction of ionomer ϕ and plotted against a reduced shift factor ωΔ, with weight percent of ionomer w as indicated. Here, Δ is chosen so that the G′ of the mixtures at low ω can be superimposed best on that of the reference (w = 11 wt %). Curves are predictions of the Rouse model by assuming monodisperse ionomer. (b) Glassy modulus Gg′ and Gg″ has been fit with a KWW equation and subtracted from the overall modulus. G′ − Gg′ and G″ − Gg″ are normalized by φ plotted against the same reduced shift factor ωΔ. Curves are Rouse model predictions by considering an M distribution.

superimpose the G′ of mixtures to that of the reference sample (w = 11 wt %). After the reduction, the terminal tails for G″ cannot be well superimposed for low w samples, attributable to a contribution of glassy modulus to the terminal relaxation. For these samples, the complex glassy modulus Gg* can be fit with a KWW equation and subtracted from G* (details are in Figures S9 and S10);38,50 G* − Gg* of the EO9 system is plotted against the same reduced frequency ωΔ in Figure 14b. After this subtraction, a power law region G′ ∼ G″ ∼ ω1/2 appears before the terminal relaxation for high w samples, suggesting Rouse dynamics, in accordance with nonentangled status of the ionomers as suggested by the backbone length smaller than twice the tube diameter. Similar superposition is noted for G* and G* − Gg* of the EO2 system, as shown in Figure S11. Considering that ionic groups impart extra friction for the chain motion, it is possible to regard each ionic group as a sticky point, and thus the local chain hopping is activated by the dissociation of the sticky points.51,52 When chains are not entangled, relaxation is Rouse-like, with local hopping time the lifetime of sticky points, defined as the sticky-Rouse relaxation. To check the sticky-Rouse relaxation mechanism, it is

Figure 13. Plots of DC conductivity rate σDC/ε0 against ωmaxεs, with weight percent of ionomer w as indicated. Solid line corresponds to the BNN equation σDC = ωmaxεsε0.

ωmaxεs (ωmax of the α2 process are plotted against 1000/T in Figure S6). The strong correlation between σDC/ε0 and ωmaxεs indicates that the ion dissociation controls the ion diffusion over wide ranges of temperature, ionomer content, and time scales. 4.2. Viscoelastic Relaxation. To quantify the delay of the terminal relaxation with increasing ionomer content in the master curves of storage and loss moduli at Tr = −10 °C, G′ I


Article

Macromolecules informative to compare the low ω relaxation behavior of complex modulus G* with that predicted directly from the sticky-Rouse model:37,38 G*(ω) =

∑ i

Ns

+

∑ p=1

ionic dissociation. In general, the Coulomb energy of ionic interaction is

Ea =

N ⎧ ⎪ iωτ0Ni 2/p2 ρwRT i ⎨ ∑ 2 2 Mi ⎪ ⎩ p = Ns + 1 1 + iωτ0Ni /p

⎫ ⎪ ⎬ 2 2⎪ 1 + iωτsNs, i /p ⎭

(9)

indicating that the activation energy Ea of ionic dissociation is inversely proportional to the Coulomb dielectric constant εc of the surrounding medium governing the ionic associations. Since the delay of relaxation due to the ionic dissociation is ∼exp(Ea/ kT), a change of this delay due to a change of dielectric constant after introducing plasticizer is

iωτsNs, i 2/p2

(6)

Here, ρ is the density, R is the gas constant, wi and Mi are the weight fraction and molecular weight of the ith component, τ0 and τs are the characteristic times of the Rouse segment and the lifetime of the ionic association, respectively, and N and Ns are the number of segments and sticky Rouse segments per chain, respectively. τ0N2 is the Rouse time, and τsNs2 is the sticky Rouse time. For PEO-based ionomers, the strong interaction between ions and ethylene oxide should lead to motional coupling between ions and polymer segments, delaying the Rouse relaxations.37,38 In addition, the glassy modulus broadens significantly with increasing ionomer content. The broad glassy modulus further complicates the analysis of the Rouse relaxations. Therefore, the focus is placed on the low ω modulus where sticky-Rouse dynamics is expected, corresponding to the second term in the large brackets of eq 6. In Figure 14a, the curves are calculated on assuming the monodispersity with M listed in Table 1, with τs = 0.76 ms and Ns = 120 for EO9 of 11 wt % (for EO2 of 11 wt %, τs = 2.2 ms and Ns = 48). Obviously, the relaxation of the samples is broader than that expected for a monodisperse sample, indicating an importance of incorporating the M distribution. Unfortunately, wi as a function of Mi is not known for these ionomer samples but is known for the neutral EO9 homopolymer. Here, we use an approximation that the fraction of wi for ionomer of Mi equals to wi,neutral of the neutral sample of MiMn,neutral/Mn, wi(Mi) = wi,neutral(MiMn,neutral/Mn), where wi,neutral as a function of Mi is determined from gel permeation chromatography (GPC), and Mn of the ionomers are listed in Table 1. The association lifetime τs can be written as ⎛E ⎞ τs = τ0 exp⎜ a ⎟ ⎝ kT ⎠

e2 4πεCε0r

⎧ E (w , T ) E (w , T ) ⎫ ⎬ Δ Ea = exp⎨ a − a r ⎩ kT ⎭ kT ⎧ E (w , T ) ⎛ ε ( w , T ) ⎞⎫ = exp⎨ a r − 1⎟⎬ ⎜ C r ⎝ εC(w , T ) ⎠⎭ ⎩ kT ⎪

⎪

⎪

⎪

(10)

Here, εC is the relevant dielectric constant for the Coulomb energy. From eqs 7 to 10, the terminal relaxation time τ ∼ τsNs2 due to a change of τ0 and Ea is τ = τr Δτ0Δ Ea =

⎧ E (w , T ) ⎛ ε ( w , T ) ⎞⎫ τr exp⎨ a r − 1⎟⎬ ⎜ C r Δa T ⎝ εC(wr , T ) ⎠⎭ ⎩ kT ⎪

⎪

⎪

⎪

(11)

where τr is the terminal relaxation time of the reference solution having w = 11 wt %. Figures 15a and 15b show respectively terminal relaxation time τ = [G′/ωG″]ω→0 and the steady state recoverable compliance J = [G′/(G″)2]ω→0 obtained at Tr =

(7)

where Ea is an activation energy for ionic dissociation and k is the Boltzmann constant. Obviously, τs decreases with decreasing w, which reflects at least two effects: plasticizing and ion-softening effect of the polar plasticizer. The focus is first placed on a plasticizing effect. In general, the plasticizing effect at T > Tg can be absorbed into a change of the characteristic time of one Rouse segment as a function of ϕ and T. Namely, the plasticizing effect can be quantified by the ratio of segmental time τ0 for the sample of interest and the reference sample Δτ0 =

τ0(w , T ) 1 = Δa T τ0(wr , T )

(8) Figure 15. (a) Terminal relaxation time τ vs weight fraction of ionomer w for EO9 and EO2 systems at −10 °C. τ predicted from Tg change and a combination of Tg and εs changes are shown in dashed and solid curves, respectively. (b) Recoverable compliance J vs w for EO9 and EO2 systems at −10 °C. The monodisperse Rouse predictions, J = 2M/[5wρRT], are shown as solid lines.

with wr = 11 wt %. Here, ΔaT is determined in Figure 6, which reflects a shift due to an enhancement of Tg of a mixture relative to that of the reference w = 11 wt % mixture. The softening of ionic interaction with increasing dielectric constant can be incorporated into the activation energy for J


Article

Macromolecules −10 °C plotted against w (symbols). In panel a, τ r Δ τ 0 incorporating only the plasticizing effect, quantified by ΔaT is shown as the dashed curves, which underestimated the deceleration with increasing w. τ predicted by incorporating both ΔaT (in Figure 6) and εC (in Figure 10) is shown by the solid curves in Figure 15a. It is surprising that the prediction can fully account for the observed delay of viscoelastic terminal relaxation, with the only fitting parameter the activation energy Ea(wr,T) for ion dissociation, which is 10 and 6 kJ/mol for EO2 and EO9 of 11 wt %, respectively. Activation energy is larger for the EO2 system than the EO9 system, which is in accordance with an expectation that ion dissociation is entropically and enthalpically less favorable for the EO2 system. For monodisperse nonentangled chains in bulk or in semidilute solution, the recoverable compliance is expected to be J = 2M/[5wρRT], as shown in the solid lines in Figure 15b.53,54 Nevertheless, J in experiments is higher than this prediction and seems to be insensitive to w. This result suggests a broadening of the relaxation mode distribution with increasing w, increasing J. This broadening is presumably due to a distribution of ion dissociation lifetimes τs that broadens as w is increased, due to a stronger ionic aggregation with a wider variety of ion states, and accordingly ion pairs find themselves in a wider distribution of localized environments when they attempt to escape from ion aggregates. This result does not contradict the rough agreement of the main part of G′/ϕ and G″/ϕ in Figure 14 because J is very sensitive to the terminal tail of G′, which does not agree completely after the normalization in Figure 14 (see the disagreement at G′/ϕ < 102 Pa in Figure 14). 4.3. Molecular Details. To understand the transition from ionomer to polyelectrolyte, it is informative to quantify the interaction energy between ions and between ion pairs. From the Coulomb energy (eq 9), it is possible to derive a Bjerrum length based on the Coulomb dielectric constant, defining a length where the interaction energy between charges becomes comparable to the thermal energy kT. lB ≡ e 2 /(4πεCε0kT )

Figure 16. Comparison of Bjerrum length, lB, Keesom length, rK, and distance between ions, rion, as functions of weight fraction of ionomer w.

condensation and stronger ion aggregation as rion approaches rK, typical ionomer behavior. In fact, rK3 is proportional to the polarizability volume Vp discussed in our previous study by taking εC the relevant dielectric constant (these two parameters differ by a numerical factor √6).37,44,56 rion = rK corresponds to ϕP0Vp = 1/√6.

5. CONCLUDING REMARKS This study tested the ionomer to polyelectrolyte transition of a PEO-based sulfonate ionomer with sodium counterions, plasticized by polar oligomeric plasticizer. The transition shows itself as (1) narrowing and lowering the glass transition, (2) softening of ionic interaction that depends on polarity of the medium, i.e., the Coulomb dielectric constant εC before the α2 relaxation, and (3) dissolving of ion aggregates. The dissolving of associations into isolated ion pairs promotes a significant increase of εs compared to εC. We find the transition from ionomer to polyelectrolyte quite gradual as more polar plasticizer is added. The dielectric constants of the mixtures approach or surpass that of the plasticizer at low ionomer content where the Bjerrum length becomes smaller than the distance between ions, allowing significant ion dissociation.

(12)

On the other hand, if the ion is paired, the interaction energy between the dipoles of these pairs with dipole moment μ can be written as the Keesom energy (Derivation of Keesom energy is briefly explained in the Appendix).55,56 E K = μ4 /(24π 2εC 2ε0 2kTr 6)

■

APPENDIX. DERIVATION OF KEESOM ENERGY The average interaction energy between two identical permanent dipoles rotating at a fixed distance r is55,56

(13)

From this Keesom energy of weakly interacting dipoles, it is possible to define another length scale rK where EK = kT. rK = [μ2 /(2 6 πεCε0kT )]1/3

⟨E K ⟩ = μ2 ⟨f ⟩/4πε0r 3

(A1)

Here, ⟨f⟩ represents the ensemble average of f, a function of particular orientations of two dipoles. The probability of two dipoles to take a particular orientation is given by the Boltzmann distribution p ∝ exp(−EK/kT), with EK being the interaction energy corresponding to a particular relative orientation of the two dipoles. If this EK < kT, one can expand the expression of p and keep the first two terms as p ∝ 1 − EK/ kT + .... The spherical average of f becomes55,56

(14)

Finally, the mean distance between ions is rion = (ϕP0)−1/3, with P0 the number density of ions in the bulk ionomer (see Table 1). In Figure 16, rion is compared with lB and rK; the latter two are calculated with eqs 12 and 14, respectively, from εC shown earlier as the curve in Figure 10. As w increases, rion decreases, but lB and rK both increase because lB ∼ εC−1 and rK ∼ εC−1/3 and εC decreases with increasing w. Then, there is a cross point between rion and lB at w ≈ 30 wt %. Since rion > lB is needed for the ion to be free from Manning condensation, the region w ≤ 30% can be regarded as the polyelectrolyte region. This result is in harmony with a polyelectrolyte behavior for samples of w ≤ 30 wt % estimated earlier based on DSC and LVE. Increasing w above 30 wt % leads to enhanced ion

⟨f ⟩ = ⟨f ⟩0 −

μ2 ⟨f 2 ⟩0 4πε0kTr 3

(A2)

where ⟨f⟩0 represents the unweighted spherical average and equals to zero. (⟨f⟩ is not equal to zero because certain mutual orientations having lower energy are more favored. In other words, the dipoles do not rotate completely freely.) In K


Article

Macromolecules comparison, f 2 is positive at all orientations so ⟨f 2⟩0 > 0. A combination of eqs A1 and A2 gives55,56 ⟨E K ⟩ = −

μ4 ⟨f 2 ⟩0 2 6 (4πε0) kTr

(12) Hara, M.; Wu, J. L.; Lee, A. H. Macromolecules 1988, 21 (7), 2214−2218. (13) Hara, M. Polyelectrolytes in Nonaqueous Solutions. In Physical Chemistry of Polyelectrolytes; Radeva, T., Ed.; Marcel Dekker: New York, 2001; pp 245−279. (14) Wang, J.; Wang, Z. L.; Peiffer, D. G.; Shuely, W. J.; Chu, B. Macromolecules 1991, 24 (3), 790−798. (15) Jousset, S.; Bellissent, H.; Galin, J. C. Macromolecules 1998, 31 (14), 4520−4530. (16) Sehgal, A.; Seery, T. A. P. Macromolecules 2003, 36 (26), 10056−10062. (17) Eisenberg, A.; Rinaudo, M. Polym. Bull. 1990, 24 (6), 671−671. (18) Page, K. A.; Soles, C. L.; Runt, J. P. Polymers for Energy Storage and Delivery Polyelectrolytes for Batteries and Fuel Cells; American Chemical Society: Washington, DC, 2012. (19) Manning, G. S. J. Chem. Phys. 1969, 51 (3), 924−933. (20) Liao, Q.; Dobrynin, A. V.; Rubinstein, M. Macromolecules 2003, 36 (9), 3399−3410. (21) Wang, W. Q.; Tudryn, G. J.; Colby, R. H.; Winey, K. I. J. Am. Chem. Soc. 2011, 133 (28), 10826−10831. (22) Tudryn, G. J.; O’Reilly, M. V.; Dou, S. C.; King, D. R.; Winey, K. I.; Runt, J.; Colby, R. H. Macromolecules 2012, 45 (9), 3962−3973. (23) Liang, S.; Chen, Q.; Choi, U. H.; Bartels, J.; Bao, N.; Runt, J.; Colby, R. H. J. Mater. Chem. A 2015, 3, 21269. (24) Shaplov, A. S.; Vlasov, P. S.; Armand, M.; Lozinskaya, E. I.; Ponkratov, D. O.; Malyshkina, I. A.; Vidal, F.; Okatova, O. V.; Pavlov, G. M.; Wandrey, C.; Godovikova, I. A.; Vygodskiia, Y. S. Polym. Chem. 2011, 2, 2609−2618. (25) Rathgeber, S.; Pakula, T.; Wilk, A.; Matyjaszewski, K.; Beers, K. L. J. Chem. Phys. 2005, 122 (12), 124904. (26) Fetters, L. J.; Lohse, D. J.; Colby, R. H. Chain Dimensions and Entanglement Spacings. In Physical Properties of Polymers Handbook, 2nd ed.; Mark, J. E., Ed.; Springer: New York, 2007; pp 445−452. (27) Eisenberg, A.; Hird, B.; Moore, R. B. Macromolecules 1990, 23 (18), 4098−4107. (28) Shiau, H.-S.; Liu, W.; Colby, R. H.; Janik, M. J. J. Chem. Phys. 2013, 139 (20), 204905. (29) Borodin, O.; Smith, G. D. Macromolecules 2007, 40 (4), 1252− 1258. (30) Liang, S.; Choi, U. H.; Liu, W.; Runt, J.; Colby, R. H. Chem. Mater. 2012, 24 (12), 2316−2323. (31) Choi, U. H.; Liang, S.; O’Reilly, M. V.; Winey, K. I.; Runt, J.; Colby, R. H. Macromolecules 2014, 47 (9), 3145−3153. (32) Wang, J.-H. H.; Yang, C. H.; Masser, H.; Shiau, H.-S.; O’Reilly, M. V.; Winey, K. I.; Runt, J.; Painter, P. C.; Colby, R. H. Macromolecules 2015, 48 (19), 7273−7285. (33) Lorthioir, C.; Alegria, A.; Colmenero, J. Phys. Rev. E 2003, 68 (3), 031805. (34) Zhao, J. S.; Ediger, M. D.; Sun, Y.; Yu, L. Macromolecules 2009, 42 (17), 6777−6783. (35) Colmenero, J.; Arbe, A. Soft Matter 2007, 3 (12), 1474−1485. (36) Shi, P. L.; Schach, R.; Munch, E.; Montes, H.; Lequeux, F. Macromolecules 2013, 46 (9), 3611−3620. (37) Chen, Q.; Liang, S.; Shiau, H.-S.; Colby, R. H. ACS Macro Lett. 2013, 2 (11), 970−974. (38) Chen, Q.; Tudryn, G. J.; Colby, R. H. J. Rheol. 2013, 57 (5), 1441−1462. (39) Kremer, F.; Schönhals, A. Broadband Dielectric Spectroscopy; Springer: Berlin, 2003. (40) Wubbenhorst, M.; van Turnhout, J. J. Non-Cryst. Solids 2002, 305 (1−3), 40−49. (41) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon Press: Oxford, 1963. (42) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486−1493. (43) Chen, Q.; Masser, H.; Shiau, H.-S.; Liang, S.; Runt, J.; Painter, P. C.; Colby, R. H. Macromolecules 2014, 47 (11), 3635−3644. (44) Choi, U. H.; Mittal, A.; Price, T. L.; Gibson, H. W.; Runt, J.; Colby, R. H. Macromolecules 2013, 46 (3), 1175−1186. (45) Barton, J. L. Verres Refract 1966, 20 (5), 328−335.

(A3)

where ⟨f ⟩0 = 2/3 when integration is carried at all possible orientations, leading eventually to eq 13 by utilizing εC as the relevant dielectric constant. Since the key assumption in derivation of the Keesom energy is E < kT, rK defined at EK = kT is not strict. However, rK thus defined should reflect a qualitative trend in a region rK < rion (i.e., EK < kT): lB ∼ εC−1 and rK ∼ εC−1/3, reflecting the fact that the interaction strength between dissociated ions changes more strongly with polarity of environment than the interaction between isolated ion pair dipoles. 2

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01958. Figures S1−S11 and Table S1 (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (R.H.C.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the Division of Materials Research Polymers Program at the National Science Foundation (Grant DMR1404586) for financial support. Q.C. is grateful for start-up fund of Changchun Institute of Applied Chemistry, Chinese Academy of Sciences. James Runt and Micheal Rubinstein are thanked for helpful discussions.

■

REFERENCES

(1) Eisenberg, A.; Kim, J.-S. Introduction to Ionomers; Wiley: New York, 1998. (2) Dobrynin, A. V.; Rubinstein, M. Prog. Polym. Sci. 2005, 30 (11), 1049−1118. (3) Colby, R. H. Rheol. Acta 2010, 49 (5), 425−442. (4) de Gennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. 1976, 37 (12), 1461−1473. (5) Dubin, P. Macromolecular Complexes in Chemistry and Biology; Springer-Verlag: Berlin, 1994. (6) Lundberg, R. D.; Phillips, R. R. J. Polym. Sci., Polym. Phys. Ed. 1982, 20 (7), 1143−1154. (7) Lantman, C. W.; Macknight, W. J.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Macromolecules 1987, 20 (5), 1096−1101. (8) Lantman, C. W.; Macknight, W. J.; Higgins, J. S.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Macromolecules 1988, 21 (5), 1339− 1343. (9) Lantman, C. W.; Macknight, W. J.; Sinha, S. K.; Peiffer, D. G.; Lundberg, R. D.; Wignall, G. D. Macromolecules 1988, 21 (5), 1344− 1349. (10) Chakrabarty, K.; Shao, P.; Weiss, A. R. Solution Properties of Ionomers. In Ionomers: Synthesis, Structure, Properties and Applications, 1st ed.; Tant, M. R., Mauritz, K. A., Wilkes, G. L., Eds.; Blackie Academic & Professional: London, 1997; pp 158−207. (11) Hara, M.; Wu, J. L. Macromolecules 1986, 19 (11), 2887−2888. L


Article

Macromolecules (46) Nakajima, T. 1971 Annual Report. In Conference on Electric Insulation and Dielectric Phenomena, National Academy of SciencesNational Research Council, Washington, DC, 1972; pp 168−176. (47) Namikawa, H. J. Non-Cryst. Solids 1974, 14 (1), 88−100. (48) Namikawa, H. J. Non-Cryst. Solids 1975, 18 (2), 173−195. (49) Choi, U. H.; Ye, Y. S.; de la Cruz, D. S.; Liu, W. J.; Winey, K. I.; Elabd, Y. A.; Runt, J.; Colby, R. H. Macromolecules 2014, 47 (2), 777− 790. (50) Bohmer, R.; Ngai, K. L.; Angell, C. A.; Plazek, D. J. J. Chem. Phys. 1993, 99 (5), 4201−4209. (51) Baxandall, L. G. Macromolecules 1989, 22 (4), 1982−1988. (52) Leibler, L.; Rubinstein, M.; Colby, R. H. Macromolecules 1991, 24 (16), 4701−4707. (53) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (54) Graessley, W. W. Polymeric Liquids and Networks: Dynamics and Rheology; Garland Science: London, 2008. (55) Keesom, W. H. Proc. Acad. Sci. Amsterdam 1915, 18, 636−646. (56) Atkins, P. W.; DePaula, J. Physical Chemistry, 7th ed.; W.H. Freeman: New York, 2002.

M


Linear Viscoelasticity and Dielectric Spectroscopy ... - ACS Publications

Recommend Documents