Comparison of Gel Relaxation Times and End-Block Pullout Times in

Sep 30, 2016 - The gel relaxation times of two different poly[styrene-b-(ethylene-alt-propylene)-b-styrene] (SEPS) ABA triblock copolymers in squalane...
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Comparison of Gel Relaxation Times and End-Block Pullout Times in ABA Triblock Copolymer Networks Andrew J. Peters† and Timothy P. Lodge*,†,‡ †

Department of Chemistry and ‡Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States S Supporting Information *

ABSTRACT: The gel relaxation times of two different poly[styrene-b-(ethylene-altpropylene)-b-styrene] (SEPS) ABA triblock copolymers in squalane at various concentrations has been measured by rheology. These relaxation times were compared with the results of previous time-resolved small-angle neutron scattering (TR-SANS) experiments, which measured chain exchange kinetics in SEP diblock and SEPS triblock micelles in squalane. The gels relaxed up to four orders of magnitude faster than expected based on the chain exchange measurements of equivalent diblock polymers. By accounting for two factorsa bias toward shorter end-block lengths in the gel relaxation, and a reduction in the energy barrier to chain pullout caused by the triblock architecturea model is constructed that reconciles the surprisingly short gel relaxation times with the chain exchange times measured via TR-SANS.

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INTRODUCTION When ABA triblock polymers are dissolved in a midblock selective solvent at sufficiently high concentration, they form a network of bridged micelles.1−3 The solvent is excluded by the A blocks which aggregate to form micelles, while the B blocks can bridge between the micelles and form a percolating network. These systems are desirable because of their ability to form thermoreversible elastic solids,4 while combining the favorable properties of both the liquid and solid components.2 Such gels have been adapted for numerous applications including pressure-sensitive adhesives,3,5 scaffolds in tissue engineering,6 polymer actuators,7 membranes in CO2 filtration,8 and controlled drug release.9,10 They have also been recently featured in ion gels,11,12 where they marry the high ionic conductivity of the ionic liquid component with the tunable mechanical strength of the network. ABA triblock copolymer networks have also been studied extensively theoretically. Transient network theory13−17 has been a popular tool to explain both relaxation kinetics and viscoelastic properties. A series of papers by Rubinstein and Semenov18−21 delved into a range of theoretical aspects of ABA triblock copolymers and their viscoelastic properties. More recent work has studied these systems using molecular dynamics simulations22 or dissipative particle dynamics.23 Some of these models have been compared with experimental results. For example, Annable et al. used transient network theory to explain aspects of their experimental results on aqueous solutions of poly(ethylene oxide) with short hydrophobic end-blocks.14 Vega et al. used a model for entangled star polymers to explain the relaxation of poly[styrene-b-(ethylenealt-propylene)-b-styrene] and poly(styrene-b-isoprene-b-styrene) gels.1 Seitz et al. suggested that for a poly[methyl methacrylate-b-(n-butyl acrylate)-b-methyl methacrylate] sys© XXXX American Chemical Society

tem the activation energy for end-block pullout is determined by the interaction between the solvent and the end-block segments. Clearly, these are interesting systems that have been studied in a variety of contexts. Nevertheless, the relaxation process is not fully understood. Not only is it often difficult to separate the effect of various contributions to relaxation, but a quantitative understanding of the kinetics of the pullout of a single end-block, which is required for relaxation of the percolating network, has been lacking. The question we wish to address is: how does the longest relaxation time of the gel relate to the pullout time of a single end-block? Recent studies focused on the kinetics of micelle equilibration have made it possible to answer this question. A time-resolved small-angle neutron scattering (TR-SANS) technique has significantly improved our understanding of the kinetics of the pullout of a single chain and provide an excellent measure of end-block pullout time.24−30 In particular, such experiments have resulted in direct measurement of chain exchange times for diblock- and triblock-based micellar systems. By making rheological measurements using the same poly[styrene-b-(ethylene-altpropylene)-b-styrene] (SEPS) in squalane (a midblock selective solvent) system used in the aforementioned TR-SANS studies, a direct comparison of chain pullout time to gel relaxation is possible.

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EXPERIMENTAL SECTION

Materials. All chemicals were purchased from Sigma-Aldrich unless otherwise noted. Two SEPS triblock polymers were used in this work. The first, designated SEPS (45−144−45), was synthesized previously, Received: August 31, 2016 Revised: September 20, 2016

A

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Macromolecules where sequential anionic polymerization of styrene, then isoprene, and then styrene was performed, followed by the selective hydrogenation of the poly(isoprene) to produce the PEP blocks.24 The second, SEPS (17−53−17), was obtained from Kraton Performance Polymers, Inc., as unsaturated poly(styrene-b-isoprene-b-styrene). The polyisoprene block was hydrogenated using hydrogen gas and a homogeneous Ni/ Al catalyst in cyclohexane.24,31 The catalyst solution was prepared prior to use from triethylaluminum and nickel 2-ethylhexanoate in cyclohexane. Polymer and catalyst solutions were injected into a stainless steel reactor purged with low-pressure argon to avoid exposure to air and charged with hydrogen to 400 psi at room temperature. The temperature was raised to 77 °C, and the reaction was allowed to progress for 24 h, followed by deactivation of the catalyst with sufficient 8 wt % citric acid in water to render the solution colorless. The saturated polymers were recovered by filtration through activated alumina and precipitation in methanol. 1H NMR spectroscopy indicates essentially complete saturation (>99%). The molecular weight and overall dispersity (Đ) of the SEPS polymers were determined using size exclusion chromatography (SEC) with a light scattering detector (Wyatt DAWN). 1H NMR spectroscopy was used to determine composition, and the molecular weights of the endblocks are assumed to follow the same distribution. Table 1 summarizes the molecular weight, composition, and dispersity of the SEPS polymers.

Figure 1. Time−temperature superposed curves of storage and loss moduli for 6 wt % SEPS (17−53−17) at a reference temperature of 70 °C. azimuthally integrated to give one-dimensional scattering data in the form of intensity (I) versus wave vector (q).

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Table 1. Polymer Characteristics

SEPS (17−53−17) SEPS (45−144−45)a a

MPS (kg/mol)

MPEP (kg/mol)

MPS (kg/mol)

Đ

17 45

53 144

17 45

1.05 1.06

RESULTS AND DISCUSSION Rheology and Structure of SEPS (17−53−17). A representative time−temperature superposed master curve is shown for 6 wt % SEPS (17−53−17) in Figure 1. The master curve shows a plateau at high frequency followed by terminal relaxation indicated by the crossover of G′ and G″ at 4 × 10−2 rad/s and by the low frequency power law exponents of 2 and 1 for G′ and G″, respectively. The structure and rheological behavior of ABA triblock polymers in a B selective solvent have been studied previously.1,2,12,32,33 The high frequency plateau is due to a networked structure, just as would be found in an entangled melt. When the chain ends do not pull out on the experimental time scale, as would be the case at high frequencies and/or low temperatures, the system cannot relax. Terminal relaxation therefore indicates that the chain ends must have pulled out. Consequently, the chain end pullout time τpullout must either correspond to or be smaller than the terminal relaxation time. Figure 1 is representative of samples with 120 °C superpose well with the high frequency plateau and the intermediate relaxation, but not with the low frequency plateau. This suggests the presence of either an order−disorder transition (ODT) or a critical micelle temperature (CMT) between 120 and 130 °C. Thus, the terminal relaxation at 2 × 10−2 rad/s for the 10 wt % sample in

Reproduced from Lu et al.24

Dispersions of SEPS in squalane were prepared using dichloromethane or THF as a cosolvent. Both solvent and cosolvent were used as received. The dichloromethane cosolvent was removed at room temperature and the tetrahydrofuran cosolvent was removed at 40 °C until constant weight was achieved. Rheology. Measurements were conducted on a Rheometrics Scientific ARES rheometer with 25 mm parallel plate geometry. Sufficient material was loaded onto the lower plate to form a 1 mm gap. The sample was then heated to allow for relaxation of the gel, and the upper plate was lowered to the desired gap. The normal force was allowed to completely relax, and excess sample was removed. The samples were then quenched to the lowest desired temperature (usually ∼40 °C) and equilibrated thermally, and a frequency sweep between 10−1 and 102 rad/s was performed at strain values (usually ∼1%) confirmed to lie within the linear viscoelastic regime. The samples were heated to the next desired temperature (typically in increments of 10 °C) and allowed to equilibrate before performing successive frequency sweeps. Master curves were then constructed by time−temperature superposition along the frequency axis; no vertical shift was applied. Longest relaxation times were identified by the point at which the storage modulus (G′) and loss modulus (G″) intersect or by the maximum in tan(δ) if G′ and G″ did not cross. The samples were kept under continuous nitrogen flow except when lowering the plate and trimming excess material. No degradation was observed when samples were tested subsequently using SEC for the times and temperatures used here. Small-Angle X-ray Scattering (SAXS). Experiments were conducted on the equipment maintained by the DuPont−Northwestern−Dow Collaborative Access Team (DND-CAT) at Argonne National Laboratory. The X-ray wavelength was 0.7293 Å, and the sample-to-detector distance was 8.503 m. Samples were sealed in DSC pans and annealed before measurement at 110 °C overnight in the case of SEPS (17−53−17) and at 180 °C for 30 min in the case of SEPS (45−144−45). Samples were equilibrated thermally for at least 5 min for each temperature measured. Two-dimensional scattering patterns were recorded by a Mar-CCD area detector and then B

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Figure 2. Time−temperature superposed curves of storage and loss moduli for 10 wt % SEPS (17−53−17) at a reference temperature of 70 °C. (a) includes temperatures only up to 120 °C (approximately the ODT for this ordered micellar system) while (b) plots temperatures over 120 °C. The solid line in (b) refers to G′ for temperatures below 120 °C, and the dashed line refers to G″ for temperatures below 120 °C. The data in (b) superpose well with the high frequency plateau and relaxation at 2 × 10−2 rad/s but not with the low frequency plateau.

Figure 2 can be attributed to the loss of the networked structure, just as for samples without a low frequency plateau as in Figure 1. SAXS measurements were on SEPS (17−53−17) in order to further understand and verify the interpretation of the rheological measurements, as shown in Figure 3. The clear spherical form factor is indicative of the expected micellar structure. Figure 3a shows data for a 10 wt % sample at various temperatures. A well-defined BCC structure is evident at 120 °C and below, giving way to broader structure factor peaks at higher temperatures. These are due to liquid-like packing of micelles, and the transition from sharp BCC peaks to the broad peaks seen in Figure 3a indicates an ODT, as seen in prior work.36−39 The existence of this ODT as measured by SAXS is consistent with the interpretation of the low frequency plateau in G′ as due to the BCC structure because the low frequency plateau disappears at temperatures above the ODT. In addition, Figure 3b indicates the presence of a BCC structure at 10 wt % and higher, but the absence of such a structure at lower concentrations, again consistent with the interpretation that the low frequency plateau is caused by the BCC structure because no low frequency plateau is found at concentrations below 10 wt %. In addition, the low frequency plateau modulus matches very well with what would be expected based on Kossuth et al.34 and Sebastian et al.,40 where the spacing of a cubic structure in a sphere forming system was related to the resulting plateau modulus. This comparison can be seen in the Supporting Information (Figure S2). The CMT occurs at ∼190 °C as measured by dynamic light scattering (DLS), well above the temperatures used in the rheological measurements. Details of the DLS measurement can also be found in the Supporting Information. Solutions without an ordered structure (e.g. Figure 1) exhibit a terminal relaxation associated with τpullout because terminal relaxation cannot occur before end-block pullout has occurred. For solutions that do form ordered structures (Figure 2) a low frequency plateau occurs because of that ordered structure. This plateau is eventually lost at high temperature. The relaxation between the two plateaus must also be associated with τpullout because when the ordered structure is lost at high temperature and the low frequency plateau disappears, terminal

Figure 3. SAXS for SEPS (17−53−17). (a) shows data for 10 wt % polymer at various temperatures. Samples were heated to desired temperature and held for at least 2 min. Triangles correspond to expected intensity maxima for BCC at q/q* values of 1:√2:√3:√4:√5:√6:√7. (b) SAXS for SEPS (17−53−17) at room temperature for various concentrations A BCC ordered structure exists at 10 wt % and higher concentrations. The data are vertically shifted for clarity.

relaxation is achieved. By identifying these relaxations with τpullout, a comparison can be made with the TR-SANS experiments. Before such a comparison is made, however, the C

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but the Rouse time is over 4 orders of magnitude shorter than the relaxation time for even the lowest concentration of SEPS (45−144−45). This calculation can be found in the Supporting Information. Second, this relaxation time increases with polymer concentration (see Figure 7, discussed later). Concentration should not have any significant effect on internal core relaxation but would be expected to have an effect on τ pullout . Lu et al. demonstrated that the addition of homopolymer PEP to SEP diblock solutions slowed chains exchange significantly.27 This was attributed primarily to corona screening and a change in the dependence on χ as the matrix transition from a solvent to a solvent polymer blend. The TRSANS experiment by Choi et al.28 demonstrated that chain exchange in ordered systems at high concentration (15 vol %) was more than an order of magnitude slower than the dilute case, similar to the effect seen here. Thus, for SEPS (45−144− 45) the relaxation between the two plateaus is also taken to be a measure of τpullout, which will also be compared with TR-SANS experiments. Comparison of Relaxation Times between TR-SANS and Rheology. The end-block pullout times as measured via TR-SANS and rheology are now compared. TR-SANS measurements for SEPS triblock polymers are only available at dilute concentrations, where the mechanism of chain exchange requires both end-blocks to pull out (the so-called “double activation” process42,43). More concentrated solutions should exhibit “walking diffusion”,42,43 where only one endblock is pulled out at a time, but these experiments are not available. The rheological experiments performed here at higher concentrations require that only one end-block of a triblock be pulled out to allow stress relaxation, and so the comparison between the dilute triblock TR-SANS experiment and the rheological experiment is not straightforward. Chain exchange in the SEP diblock case only requires a single end-block to be pullout out regardless of concentration, so the time scale of stress relaxation for the triblock, and diblock chain exchange as measured by TR-SANS, should be comparable. The mechanisms for the diblock TR-SANS experiment, dilute triblock TRSANS experiment, and the stress relaxation experiment are shown schematically in Figure 6. In the case of SEPS (45−144−45) the time scales found for 1 and 15 wt %26,28 can be compared directly since the diblock equivalent (essentially half of the triblock polymer) of SEPS (45−144−45) was used in TR-SANS. However, in the case of SEPS (17−53−17) no diblock equivalent was used, and so the TR-SANS relaxation time must be calculated. The time scale of relaxation in the diblock experiment was modeled via26

Figure 4. Time−temperature superposed curves of storage and loss moduli for 4 wt % SEPS (45−144−45) at a reference temperature of 70 °C.

rheology of SEPS (45−144−45) and its interpretation is discussed. Rheology and Structure of SEPS (45−144−45). Figure 4 shows master curves for 4 wt % SEPS (45−144−45), which are representative of various concentrations of SEPS (45−144− 45). The larger Mw polymer SEPS (45−144−45) shows similar rheological features to that of SEPS (17−53−17): a high frequency plateau and a low frequency plateau with an intermediate relaxation, which suggests the same interpretation of the rheology. However, the low frequency plateau is present even when no ordered structure is apparent. Figure 5 shows SAXS data for SEPS (45−144−45) at various concentrations. No clear BCC structure is evident, though it is likely that the system simply has not been annealed long enough to form a BCC structure, as it is known that such structures can sometimes take a very long time to form, especially in triblock systems.40 Since the low frequency plateau is not due to a BCC structure as in the SEPS (17−53−17) system, presumably it reflects congestion of a disordered micellar system where the micelles cannot easily flow past one another.41 It is also conceivable that the relaxation in Figure 4 is caused by chain relaxation within the micelle cores.12 However, this is unlikely for two reasons. First, the time scale of relaxation in the core should be approximately the Rouse time for the core blocks,

τpullout = τRouse × e αχNcore τRouse =

(1)

Ncore 2b2ζ 6π 2kT

(2)

where αχNcore is the energy barrier to chain pullout, b is the statistical segment length, ζ is the monomeric friction factor, k is the Boltzmann constant, and T is the temperature. The energy barrier was taken to be equal to αχNcore because the extraction of a core block carries a thermodynamic penalty proportional to the product χNcore due to the creation of enthalpically unfavorable segment−solvent contacts. The parameter α is an unknown prefactor.44−46 The parameter αχ was found to be independent of Ncore for SEP in squalane26 and also in a poly(methyl methacrylate)-b-poly(n-butyl methacry-

Figure 5. SAXS for SEPS (45−144−45) at various concentrations. D

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Figure 6. Schematic illustration showing the relaxation process in each type of experiment. (a) In a diblock chain exchange experiment using TRSANS, a single end-block must pull out, followed by chain transportation and insertion, which are assumed to be rapid.26 (b) In a dilute A−B−A triblock chain exchange experiment both ends must pull out essentially simultaneously followed by transportation and insertion.24 A concentrated solution (not shown) would exhibit primarily “walking diffusion” where the pullout of one end-block controls the chain exchange. (c) In an A−B−A triblock rheology experiment, a single end-block must pull out followed by micellar relaxation.

late)/ionic liquid system.47 Three values are needed to calculate τpullout: b, αχ, and ζ. The statistical segment length b = 0.67 nm was taken from the literature;48 the value for αχ at 1 and 15 wt % was taken from the previous TR-SANS work,26,28 and the value for ζ was estimated based on published rheological data for polystyrene.49 The diblock τpullout values from TR-SANS experiments are compared with the ABA triblock τpullout values from rheology in Figure 7. Remarkably, the time scale of relaxation is much shorter in rheology than in diblock TR-SANS experiments, by as much as 4 orders of magnitude in the case of SEPS (45− 144−45). This result is initially very surprising, and the remainder of this work will be devoted to explaining this important result. Similarly, SEPS (17−53−17) relaxes approximately 3 orders of magnitude faster than the TR-SANS experiments suggest. The fact that the two measures of relaxation are closer for SEPS (17−53−17) can be explained by the result that increasing the relative corona block length reduces the chain exchange time.50,51 Thus, the shorter midblock in SEPS (17−53−17) would be expected to retard chain exchange for this system. The rapid rheological relaxation in the triblock gels may be understood better in the light of the triblock TR-SANS

experiments. On the basis of two reasonable assumptions, the triblock TR-SANS experiments could be anticipated from the diblock results. These assumptions are that the energetics of a single end-block pullout are identical in the diblock and triblock case and that the two required pullouts for a triblock are independent and must occur nearly simultaneously because a single “dangling” end-block will quickly reinsert. This leads to a “double activation” mechanism that has been effectively used to model diffusion of triblocks in bulk systems.42,43 These assumptions yield a new equation for the pullout time (τABA) of an ABA triblock polymer24 τABA = τRouse × e αχNcore × e αχNcore

(3)

A relaxation function R(t) that describes the extent of chain exchange (R(t) = 1 corresponds to no mixing, while R(t) = 0 corresponds to complete exchange) was modeled using R (t ) =

∫0

∞

P(Ncore)K (t , Ncore) dNcore

(4)

where P(Ncore) is the Zimm−Schultz distribution function for the core block length, and K(t,Ncore) characterizes the kinetics of core block extraction. K(t,Ncore) is expressed as E

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slower than the experiment. Just as in the rheological experiment, the triblock system relaxes many orders of magnitude faster than expected based on diblock TR-SANS experiments. It is evident that at least one of the assumptions used to produce eq 3 is invalid. However, the only way to explain both the ABA triblock TR-SANS pullout time and the ABA rheological pullout time being orders of magnitude faster than expected based on diblock TR-SANS results is to remove the first assumption that the energetics of a single end-block pullout are identical in the diblock and triblock case. Failure of the second assumption, that the two pullouts are independent and simultaneous, although it could explain the triblock TRSANS result, could not explain the rheological result because only one end-block needs to pull out for the network to relax. This strongly suggests that the energy barrier of a single endblock pullout is not equivalent in the diblock and triblock case. This conclusion is consistent with the previously mentioned result that the corona block length plays an important role in the energy barrier to chain pullout in both AB diblocks50 and BAB triblocks.24 These accelerating effects of a longer corona block can be explained by an increase in the energetic benefit of fully solvating the corona block(s), as suggested in earlier theoretical studies.52,53 An ABA triblock would similarly have an increased benefit of solvating the corona block as compared to its diblock counterpart because the total corona block is twice as long. Additionally, the lack of any free chain ends in the inserted state may result in a decreased entropy in the inserted state. Reconciling Diblock, Triblock, TR-SANS, and Rheology Experiments. The effect of the triblock architecture on the energy penalty for end-block pullout can be included by using the rheological experiment to recalculate the energy penalty of pullout. By rearranging eq 1, we obtain an equation for αχ

Figure 7. Comparison of measured relaxation times (in black) and calculated expectation based on diblock TR-SANS experiments (in red) at a reference temperature of 70 °C. (a) shows results for SEPS (17−53−17) and (b) shows results for SEPS (45−144−45). Expected relaxation time is calculated from the data in refs 26 and 28 as described in the text.

K (t , Ncore) = exp( −t /τ )

(5)

The relaxation function R(t) for the triblock and diblock equivalents of SEPS (45−144−45) at dilute concentrations are shown in Figure 8.24,26 The diblock model using eq 1 for τ and the triblock model using eq 3 for τ are also shown. It is clear that the triblock model using the above assumptions does not accurately describe the data and in fact is 5 orders of magnitude

αχ =

τpullout

( )

ln

τRouse

Ncore

(6)

This could then be substituted into eq 3, and a new prediction for the triblock TR-SANS data would be produced. However, another important factor must first be accounted for. Because only one end-block needs to pull out in order for stress relaxation to occur, the rheological experiment will be biased toward pulling out shorter end-blocks. This effect is magnified by the double exponential dependence on Ncore through eqs 3 and 5. To include this bias, we create a new “effective” distribution of end-block lengths Peff(N). In the simplest approximation, we will assume that the shorter chain will always dominate the pullout time of a single triblock chain. We will also assume that the dispersity and average degree of polymerization are the same for each end-block. Finally, we will assume that there is no correlation between the length of each end-block on a given polymer chain. These are all reasonable assumptions given the synthetic method. In the work by Choi et al., the dispersity was modeled as a Schultz−Zimm distribution.26 To create Peff(N), two end-block lengths are randomly selected from a Schultz−Zimm distribution, and the shorter of the two, which dominates relaxation, is binned into the new distribution Peff(N). This process is repeated (∼105 times) to produce Peff(N). Peff(N) was calculated for SEPS (45−144−45), for which the number-

Figure 8. Relaxation function R(t) reproduced from ref 3 for SEP-2 (diblock) and SEPS-2 (triblock, equivalent to SEPS (45−144−45)) at 1 and 0.25 vol %, respectively. The diblock model is fit to the diblock data while the triblock model is a prediction based on assumptions given in the text. The model overestimates the relaxation time by ∼5 orders of magnitude at R(t) = 0.5. F

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Figure 10. Relaxation function for SEPS-2 (equivalent to SEPS (45− 144−45)) at 0.25 wt % taken from ref 24 along with models predicting the relaxation. The improved model (dashed line) incorporates a change in the energy penalty for end-block pullout and accounts for the short end-block biasing in bulk rheological relaxation fits with experiment very well while the model which neglects these effects (solid line) significantly over predicts the relaxation time.

Figure 9. Probability distribution for the original (solid line) and the effective (dashed line) distributions for a triblock polymer with end blocks with Nn = 433 with Đ = 1.06, equivalent to SEPS (45−144− 45). Both Nn and Đ decrease when converting to the effective distribution. The inset shows the effect of Đ on Nn and τ.

average degree of polymerization of the end-blocks Nn is =433 and Đ = 1.06. The original and effective distributions are compared in Figure 9. The effective distribution shows a substantial drop in both Nn (373 vs 433) and Đ (1.04 vs 1.06). This new Nn,eff results in a shorter relaxation time in the rheological experiment relative to the diblock experiment with the same average end block length. The effect of dispersity can be seen in the inset to Figure 9. As dispersity increases Nn,eff decreases, indicating that polymers with the same Nn but larger dispersity will relax more rapidly than those with smaller dispersity. As the inset shows, for this polymer system a drop in relaxation time of approximately an order of magnitude is expected by increasing the dispersity from 1 to 1.1 and approximately another order of magnitude drop by further increasing dispersity to 1.4. This is a significant effect arising from this bias toward sampling shorter end-blocks. This can be accounted for by substituting Nn,eff (number-average degree of polymerization of the effective distribution) into eq 6 to yield

MxGN (8) cRT where Mx is the molecular weight between entanglements in the midblock, GN is the networked plateau modulus, and c is the concentration of the block polymer in w/v. The fraction of elastically active chains thus obtained for both SEPS (17−53− 17) and SEPS (45−44−45) is plotted versus concentration in Figure 11. There is a significant drop in f for SEPS (17−53− 17) at 4 wt %, but there is no corresponding drop in relaxation time in Figure 7, indicating that connectivity does not affect the f=

τpullout

( )

ln αχ =

ordering. This might contribute to the small difference between the model prediction calculated using the viscoelastic relaxation time and the actual diblock chain exchange time in Figure 10, as it would result in slightly faster gel relaxation but would not contribute in the chain exchange experiments. Effect of Connectivity and Identifying the Longest Relaxation Time as τ pullout . Identifying the longest rheological relaxation time as τpullout neglects the effect of connectivity, but it appears that this is not a significant factor, as we will now discuss. Only elastically active chains contribute to the elasticity and so only elastically active chains will be sampled in the rheological experiments. The fraction of elastically active chains f can be estimated from54,55

τRouse

Nn,eff

(7)

By using Nn,eff instead of Ncore, the αχ is modified to better reflect the bias in the rheological experiment toward short chain ends. Substituting the result from eq 7 into eq 3 results in αχ = 0.026 at a reference temperature of 125 °C, and by calculating R(t) from eq 4, a new model prediction for the triblock TRSANS result is produced. This resulting R(t) is shown in Figure 10. Good agreement is found between this improved model and the triblock TR-SANS data. Incorporating the effect of the triblock architecture on the energy penalty and the bias toward short end-block lengths found in the bulk rheology experiments has produced an effective model for the triblock chain exchange experiment. One factor that has not been considered is the effect of disorder in the micelle packing. Disorder could introduce regions where chains bridge at distances larger than the equilibrium distance, resulting in faster end block pullout for these chains and a more rapid viscoelastic relaxation. Such effects would likely be present even in systems with many sharp SAXS peaks and certainly in systems with only liquid-like

Figure 11. Fraction elastic active chains f as estimated via eq 8. G

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added homopolymer, which have a reduced d* compared to the samples with added homopolymer at the same PEP wt %, would have pullout out significantly slower since they are less stretched. Instead, τpullout is affected the same regardless of d*. This indicates that the bridging chains are not thermodynamically destabilized by stretching in this case. Vega et al.1 concluded that for their SEPS/squalane system the rheological relaxation time was controlled by entanglements in the midblock. This requires the end-blocks to pull out rapidly compared to the midblock entanglement relaxation. The SEPS polymers they used had 8.6 kg/mol styrene endblocks and 119 kg/mol midblocks, with an overall dispersity of 1.03. This small end-block length results in a significantly depressed Tg, which they suggest to be ∼30−50 °C. Indeed it must be significantly depressed or their samples could not have relaxed in the 30−60 °C experimental temperature window. Using the results from this work, τpullout can be calculated, but this drop is Tg must be accounted for in the calculation of the Rouse time. Using Tg ≈ 40 °C, the newly calculated αχ value from eq 7 for SEPS (45−144−45), and Neff = 75 as calculated using Peff described above, τpullout ≈ 4.6 × 104 s at the reference temperature of 30 °C used by Vega et al. A value of τ ≈ 3.1 × 105 s can be extracted for a 6 wt % solution from Figure 11 of that work. The fact that the calculated τpullout is almost an order of magnitude faster than the measured relaxation is therefore consistent with their argument that the relaxation time is governed by midblock entanglements.

relaxation time in this case. Additionally, the connectivity of SEPS (45−144−45) is low at 15−20%. The aggregation number for this polymer is ∼90,56 so if a micelle has on average ∼18 connections to neighboring micelles and ∼8 nearest neighbors (as would be the case for BCC packing), then there are only 2−3 connections with those neighbors, and it is reasonable to assume that the time scale of stress relaxation is comparable to the time scale of chain pullout. The prior discussion neglects the possibility that the elastically active chains (primarily those chains that bridge between micelles) are thermodynamically destabilized compared to the elastically inactive chains (primarily looped chains). Bridged chains might be stretched between micelles and thus pull out more readily. This phenomenon would affect gel relaxation much more significantly than chain exchange because only the elastically active chains are sampled in gel relaxation, but all chains are sampled in chain exchange. If this were the case, then reducing the average distance between micelles d* by increasing the micelle concentration would reduce the average stretching of the bridges and thus increase τpullout. Increasing the concentration does increase τpullout in both SEPS (17−53−17) and SEPS (45−144−45), but this could also be attributed to the increased PEP concentration in the matrix between the micelle cores. Increasing the concentration of the ABA polymer decreases the distance between micelles as seen in Figure 3b, where d* for the 15 wt % sample (as calculated by d* = 2π/q*) is 37.0 nm as opposed to the 39.5 nm for the 10 wt % sample. However, increasing the PEP concentration in the matrix would actually increase d* by increasing the aggregation number,27 thereby reducing the number of micelles. If the bridges are substantially stretched and destabilized, this would result in a decrease in τpullout. Experimentally, an increased PEP concentration has been shown to decrease chain exchange time via corona screening.27 To explore this in rheology, the effect of added PEP was tested by adding homopolymer (PEP 86 kg/mol) to a SEPS (17−53− 17) 10% sample. Rheology was then used to measure τpullout for these samples just as described for the samples with no added homopolymer, and the results are plotted in Figure 12. The addition of homopolymer increases τpullout at the same rate as the addition of PEP via added triblock. If bridge stretching played an important role in τpullout, then the samples without

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CONCLUSIONS The rheological relaxation times of ABA triblock polymer gels of SEPS in squalane were compared with previously measured chain exchange times of the same system using TR-SANS. The primary result is that the end-block pullout time of ABA triblock polymers is significantly faster (by 3−4 orders of magnitude) than in AB diblock polymers. This is initially surprising, but after analysis of these results and the triblock TR-SANS, the apparent discrepancy can be resolved. This effect is in part due to the effect of dispersity in the end-blocks of the triblocks, as stress relaxation will be biased toward the pullout of the shorter end-block of the two in an ABA triblock, resulting in an overall shorter relaxation time. This effect has been modeled using an effective distribution of end-block lengths. There also appears to be a decrease in the energy barrier to end-block pullout as compared to the diblock case. This is consistent with triblock TR-SANS results and attributable to entropic considerations that have been shown to be important in the effect of corona block length on diblock chain exchange, where the extra corona block length significantly speeds up chain exchange. By extracting a modified energy penalty from rheological experiments, and accounting for the effect of dispersity in triblock gels, the ABA triblock TRSANS results can be consistently modeled.

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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.6b01921.

Figure 12. Comparison of τpullout for SEPS (17−53−17) with (blue points) and without (black points) added PEP homopolymer plotted versus wt % PEP. The samples with added homopolymer were 10 wt % SEPS (17−53−17). Dashed line is a linear fit to the samples without added homopolymer.

Time−temperature superposition shift factors, modulus of cubic structures, and dynamic light scattering experiments (PDF) H

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work was supported in part by Infineum, LLP, and in part by the National Science Foundation Polymers Program (DMR01206459). Portions of this work were performed at the DuPont−Northwestern−Dow Collaborative Access Team (DND-CAT) located at Sector 5 of the Advanced Photon Source (APS). DND-CAT is supported by Northwestern University, E.I. DuPont de Nemours & Co., and The Dow Chemical Company. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Data were collected using an instrument funded by the National Science Foundation under Award No. 0960140. We thank Dr. Jie Lu for the SEPS (45−144−45) polymer previously produced.

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