Linear Viscoelasticity and Swelling of Polyelectrolyte Complex

6 days ago - Figure 1. Schematic of the reversible polyampholyte gel coacervate formed by mixing ..... at 0.09 M NaCl, meaning that all four data sets...
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Linear Viscoelasticity and Swelling of Polyelectrolyte Complex Coacervates Fawzi G. Hamad, Quan Chen, and Ralph H. Colby* Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States

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S Supporting Information *

ABSTRACT: Mixing oppositely charged hydrophilic polyelectrolytes is the simplest path to constructing a polyampholyte gel that is useful as a soft tissue scaffold for binding enzymes in their native state. The swelling and viscoelastic properties of such a synthetic polyampholyte gel coacervate, constructed from polyions of different charge density, are reported in water with various amounts of NaCl salt. When constructed, this coacervate is roughly 70% water and 15% of each polyion, nearly charge balanced. If salt is removed from the surrounding supernatant, the gel swells owing to the weak charge imbalance because small amounts of salt screen electrostatic repulsions. If instead more salt is added to this coacervate, the gel behaves as any polyampholyte gel, swelling as salt is added because the excess salt screens the electrostatic attractions and eventually this leads to redissolving the coacervate. The amount of salt needed to redissolve the coacervate increases with polyion molecular weight. To our surprise, we discovered that the small charge imbalance within the coacervate grows with the molecular weight of the more strongly charged polyion.

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actually anticipates ∼80% water with ∼10% of each polyion in the coacervate if the two-body nonelectrostatic terms in the free energy can be safely ignored (i.e., the limit where all χij = 0), leaving the Debye−Huckel electrostatic attractions and the Flory−Huggins entropic terms. There is considerable discussion in the recent literature adding other terms for stronger specific interactions and chain connectivity effects, but since eq 1 both is simpler and predicts the coacervate water content that we observe, it suffices for our purposes. This soft coacervate is the simplest way to create a polyampholyte gel that is nearly charge-balanced. As such, these soft materials find many uses, for instance, as scaffolds for enzymes by effectively binding them but also allowing enough water and NaCl salt inside the scaffold to let the enzymes function in their usual ways. Addition of salt lowers the Debye length and favors the mixed state, whereas lower salt concentration stabilizes the coacervate. One cautionary note is in order for this ultrasimple description because it is already known from experiment that with the same two polyions, different choices of counterions matter, which surely means that the entropy of the counterions in the supernatant is not as simple as the mean-field description above, presumably because solvation (hydration by water molecules in the first shell around these ions) matters,13 in the usual way for the

INTRODUCTION When two polyelectrolytes of opposite charge are mixed in nearly stoichiometric proportions, a precipitate is formed as a high-concentration phase,1 driven by electrostatic attractions and the entropy of counterion release,2 since the counterions of both polyelectrolytes primarily exist as salt in the lowconcentration supernatant.3 With hydrophobic polyelectrolytes, the precipitate can be a dense hard solid, but if more hydrophilic polyelectrolytes are used, the high-concentration phase can be a soft coacervate with 5−40 wt % polymer and 60−95 wt % water that find uses as tissue scaffolds for holding enzymes and drugs for later release.4−6 The 1957 Overbeek− Voorn lattice model7,8 provides the simplest thermodynamic description9 where, following Veis, we have added two-body nonelectrostatic Flory−Huggins interaction terms:10−12 F̅ = kT

∑ i

ϕi Ni

ln ϕi +

− ∑ ∑ χij ϕϕ i j i

j>i

v0 12πrD3

(1)

Here F̅ is the free energy per site, ϕi is the volume fraction of component i, Ni is the number of solvent-sized lattice sites occupied by component i, χij are the Flory−Huggins interaction parameters between components, v0 is the site volume (usually taken to be the volume of a water molecule), and rD is the Debye length in the coacervate. The first sum is the usual mixing entropy term, the second (double) sum represents the two-body nonelectrostatic Flory−Huggins interaction terms, while the last term is the Debye−Huckel electrostatic attraction (kT per Debye volume).7,8 Equation 1 © XXXX American Chemical Society

Received: February 22, 2018 Revised: July 1, 2018

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DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematic of the reversible polyampholyte gel coacervate formed by mixing two hydrophilic polyelectrolytes of opposite charge (red is polycation and black is polyanion, with higher charge density along the chain). Each network strand is an almost charge-compensated double strand (polyanion and polycation, one of three possible conformations of the coacervate suggested by Michaels in his 1965 review1), and these strands locally (on the scale of their Kuhn lengths) bind to each other forming a four-Kuhn-cylinder quadrupole. This picture is strongly speculative and surely oversimplified but provides a useful image to keep in mind to understand the model of the next section and our data analysis.

charged guest polycation (red) adsorbs. With ample water present, the ± adsorption energy of a single contact pair is weak, a few kT, with individual associations having lifetime on the scale of nanoseconds. Consequently, each individual association is only weakly bound, leaving the polycations able to change conformations easily as long as they remain adsorbed to some polyanion. In this view, the underlying host polyanion network bears the stress in linear viscoelastic response, while the adsorbed polycations that hold this network together are free to migrate along the host polyanion scaffold. As prepared, the coacervate is at its equilibrium swelling in its supernatant, for which we expect the osmotic pressure (kT per counterion inside the coacervate) to equal the shear modulus (kT per strand) making roughly one free counterion per strand, assuming interfacial tension is negligible. So the coacervate reversible polyampholyte gel formed by mixing hydrophilic polyelectrolytes of opposite charge is not expected to be charge-balanced, but nearly all counterions are released into the supernatant, making the (first) entropic term in eq 1 play a central role in both swelling and the small charge imbalance of this polyampholyte gel. When two such double strands approach each other, there is a natural tendency to form a four-strand quadrupolar association on the scale of the Kuhn length, which is held together by multiple ionic interactions (see exploded view in Figure 1). Precisely how many ionic interactions in each network junction dictates the viscoelastic character of the polyampholyte gel, and this number of associations n depends on the charge density of each polyion (number of charges per Kuhn length) and exactly how the two strands wrap around each other locally; such details are unknown and best determined by experiment. The primary association of a single specific positive charge on the polycation and the corresponding negative charge on the polyanion have some formation energy E that is set by how close those elementary charges can be and how much hydration persists when they are in contact. With n such primary associations per network junction, there is an association lifetime τs = τ0 exp(nE/kT), of order microseconds, that accounts for the reversible gel viscoelastic character of the coacervate, which on long time scales always flows as a viscoelastic liquid. Consequently, herein we show

Hofmeister series (which is also not understandable using eq 1) where smaller counterions bind water more strongly.14 As either of the polyions become more hydrophobic (some larger positive χij), the coacervate increases in polyion concentration (staying roughly charge-balanced) and eventually makes a hard glassy solid with very little water15 that is not of prime interest herein. With hydrophilic polyelectrolytes, the coacervate has 60−80% water and is a reversible gel that ultimately flows like a viscoelastic liquid at low frequencies (or long time scales).16 The modulus of this gel can be measured at high frequencies, and the number density of strands in this gel can be estimated by dividing this modulus by the thermal energy kT. Each strand of the gel is presumably composed of both polyelectrolyte chains and is expected to be a nearly charge-balanced “double strand”; see the schematic in Figure 1. The details of the double strand conformation are not known, but the two chains strive to have cations on the polycation as close as possible to the anions of the polyanion. With a large mismatch in charge density of the two polyions, this might even lead to helix formation17 of the lower charge density chain wrapping around the polyion of higher charge density. The coacervates studied herein are formed by mixing dialyzed aqueous solutions of poly(diallyldimethylammonium chloride) (PDADMA-Cl) having one ammonium with chloride counterion in every repeat unit that has four backbone carbons with poly(isobutylene-alt-maleate sodium) (IBMANa) having two carboxylates with two sodium counterions in every repeat unit that has four backbone carbons (molecular structures are shown later in Figure 2). Hence, the charge density along the polyanion chain is twice the charge density along the polycation chain. This charge imbalance plays a vital role: The more strongly charged polyanion (black in the schematic of Figure 1) is the host, to which the less strongly

Figure 2. Molecular structure of (a) PDADMA-Cl and (b) IBMA-Na. B

DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

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the sticky Rouse model18 with longest relaxation time τ(N/ Ns)2 known as the sticky Rouse time, where N is the number of Kuhn monomers in the chain and Ns is the average number of Kuhn monomers between stickers. For the picture in Figure 1 this makes N/Ns the number of quadrupolar Kuhn-segment associations per polyanion chain, but more generally N/Ns is the number of dominant associations per chain. On the other hand, if the polyanions are long enough to be entangled in their 15% solution, the sticky reptation time of the chains is a factor of N/Ne longer,25 where Ne is the number of Kuhn monomers in an entanglement strand. Consequently, entanglement can be identified from the point where the longest relaxation time measured in linear viscoelasticity changes from scaling with polyanion chain length as N2 to scaling as N3.

how to make use of the viscoelastic response of the coacervate (measuring the longest relaxation time as a function of added salt and fitting to a simple sticky Rouse model18) to effectively determine the number of primary associations n in each network junction.

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BACKGROUND THEORY The electrostatic interaction between a single anion and a single cation on oppositely charged polyelectrolytes is described by the Coulomb energy. E=

l −e 2 = − B kT 4πεε0σ σ

(2)

■

where e is the elementary charge of the ions, σ is the separation between the ions in a contact ion pair, ε is the dielectric constant, ε0 is the vacuum permittivity, and lB is the Bjerrum length (7.1 Å for water). lB =

e2 4πεε0kT

(3)

Salt screening electrostatic interactions relies on the Debye screening length rD =

1 8π ·lB·CS·NA

(4)

Here CS is the molar salt concentration of the solution and NA is Avogadro’s number. At low CS, rD is large meaning that only long-range repulsive interactions are screened. In contrast, small rD at high CS can screen short-range attractions, consequently weakening the ionic cross-links to facilitate polyelectrolyte diffusion. The activation energy for dissociation of one ionic pair is the energy to separate one ionic contact pair from bond length σ to Debye length rD where the electrostatic interaction is screened. ij l l yz i1 Ea = kT jjj B − B zzz = kT ·lBjjj − jσ z r kσ D{ k

y 8π ·lB·CS·NA zzz {

(5)

For densely charged polyelectrolytes with multiple charges per Kuhn length, each binding site has multiple ± associations of n consecutive ionic pairs. The n pairs need to break simultaneously to allow large-scale motion and relaxation of the host polyanions.19,20 Thus, the energy for ionic dissociation can be written in general as nEa (by way of example, Spruijt et al.21 assumed the binding site to be an elementary quadrupole with n = 2), and the association lifetime is i nE y τ = τ0 expjjj a zzz k kT {

where τ0 is the attempt time. i i1 τ = τ0expjjjjn·lBjjj − k kσ

α≡

yy 8π ·lB·CS·NA zzzzzzz {{

[COO−] [COOH[ + [COO−]

(8)

Polyelectrolyte complexes were prepared by dropwise addition of one polyelectrolyte into the oppositely charged polyelectrolyte stirred continuously by a magnetic stir bar at room temperature, resulting in a turbid mixture that when the stirring is stopped allows the coacervate to settle to the bottom. Both viscosity measurements and ζ-potential measurements allow estimation of the stoichiometry of the coacervate; these methods are described below, with some details in the Supporting Information.

(6) 22

EXPERIMENTAL SECTION

Preparation of Polyelectrolyte Complexes. Poly(diallyldimethylammonium chloride) (PDADMA-Cl) with Mw = 300 kg/mol and poly(isobutylene-alt-maleate sodium) (IBMA-Na) with different molecular weights (11, 82, 196, and 410 kg/mol) were chosen for the formation of polyelectrolyte complexes, with structures shown in Figure 2. [The molecular weight of PDADMA-Cl by SigmaAldrich was extrapolated using two different grades of PDADMA-Cl by Lubrizol with known molecular weights (10 and 100 kg/mol). Viscosity measurements at a 4 wt % concentration (chosen to be in the semidilute unentangled regime) inferred the molecular weight of the Sigma-Aldrich grade to be 300 kg/mol.] PDADMA-Cl (Sigma-Aldrich) was dialyzed using an Amicon 8400 dialysis cell for around 1 week, until the dialyzate conductivity reached ∼9 μS/cm, using a 30 000-MW cutoff membrane and many liters of Milli-q deionized water at 40 psi to remove any excess ions and impurities from the solution. PDADMA-Cl was dissolved in deionized water at a concentration of 4 wt % (pH = 7.2) for later preparation of coacervates. Maleic anhydride/isobutylene copolymers were purchased from Kuraray Co., Ltd. (Kuraray America, Inc., NY). IBMA-Na with molecular weights 11, 82, and 196 kg/mol had been previously hydrolyzed, dialyzed, and freeze-dried.26 A maleic anhydride/isobutylene alternating copolymer with 410 kg/mol molecular weight was dissolved in deionized water and hydrolyzed for 24 h at 50 °C with excess NaOH, and the product was eventually dialyzed. All dialyzed polyanions were dissolved in deionized water at a concentration of 2.5 wt % for preparation of coacervate. The fractions of acid groups with sodium counterions (α) shown in Table 1 were calculated from the pH of the polyanion solutions with a concentration of 5.00 mg/mL, using Figure 6b in Sauvage et al.26 as a guide, and the following definition of α:

Combining eqs 5 and 6 gives (7)

This association lifetime τ directly controls all polymer dynamics on longer time scales. As such, eq 7 immediately explains the time−salt superposition introduced by Spruijt;21,23 the sole viscoelastic effect of added salt is to lower the association lifetime which is on the microsecond time scale and consequently oscillatory shear only sees the consequences of that lifetime at much lower frequencies than 1/ τ.21,23,24 If the polyions are short enough, polymer dynamics are described by

X≡

mPDADMA ‐ C1·αPDADMA ‐ C1 M 0 ‐ PDADMA ‐ C1 ·α mPDADMA ‐ C1·αPDADMA ‐ C1 m + 2 MIBMA‐ Na M 0 ‐ PDADMA ‐ C1 0 ‐ IBMA ‐ Na

(9)

X is the positive molar charge fraction in the coacervate (X = 0 is pure polyanion, X = 1/2 is charge balanced, and X = 1 is pure polycation), m is the mass of the polyelectrolytes used, and M0 is the molecular weight of a repeat unit. It was assumed that the PDADMACl repeat units were all quaternized amines, making αPDADMA‑Cl = 1, while the α of the polyanions is determined from the titration curve of C

DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

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To determine the swelling of the as-prepared coacervate in its supernatant Qref, the coacervates were freeze-dried for 24 h and further dried in vacuum at 75 °C for approximately 48 h. Coacervate Linear Viscoelasticity. A Rheometric Scientific Dynamic Spectrometer (RDS-II) with parallel plate configuration and small amplitude oscillatory shear was used to measure the linear viscoelastic (LVE) response of the coacervates. Polyelectrolyte complex samples that are fully equilibrated at a given salt concentration were squeezed into a thick film (1−2 mm) using two flat surfaces and then shaped into the desired geometry using a circular shaped cutter. The resulting cylindrical shaped coacervate sample was placed between the circular rheometer plates and was left to relax and equilibrate with an excess of surrounding salt solution before starting a LVE test. Coacervates with lower viscosities (in high salt concentration solution) were just carefully poured onto the lower plate. Depending on the viscosity of the polyelectrolyte complexes, 25 or 50 mm diameter plates were used for LVE measurements. The bottom plate also served as a cup containing around 20−30 mL (submerging the sample) of desired salt concentration aqueous solution to keep the polyelectrolyte complexes from drying out; see schematic in Figure 3. After the sample was sandwiched between the two plates, the cup was covered with parafilm to reduce evaporation during the measurements.

Table 1. pH of the Four Individual IBMA-Na Polyanion Molecular Weights, Their Corresponding Degree of Neutralization Values α, Polydispersity Index, Swelling Extent in the Supernatant (0.09 M NaCl) Qref, and the Positive Charge Fraction of the Coacervate X Obtained from Titrations Measuring the Change in Sign of the ζPotential and the Minimum Viscosity of the Supernatant

pH α Mw/Mn Qref X−ζ-potential titrating polycation X−ζ-potential titrating polyanion X−viscosity titrating polycation X−viscosity titrating polyanion

IBMA-Na 11 kg/mol

IBMA-Na 82 kg/mol

IBMA-Na 196 kg/mol

IBMA-Na 410 kg/mol

8.537 0.75 1.75 2.86 0.45

9.443 0.85 2.60 3.57 0.47

9.105 0.83 3.04 2.94 0.41

8.680 0.77 2.40 2.43 0.39

0.46

0.47

0.41

0.39

0.45

0.46

0.41

0.37

0.44

0.43

0.43

0.37

Sauvage et al.26 assumed to be unaffected by the presence of the polycation. Zeta-Potential. Zeta-potential titration was run at different polycation/polyanion fractions to determine the isoelectric point. Around 1.6 mL of supernatant, composed of nonstoichiometric soluble polyelectrolyte complexes and sodium chloride ions, was placed in a cuvette using an 800 μL pipet. The zeta-potential measurements were made using a Brookhaven ZetaPALS. The cuvette was placed in the analyzer, and all required data were input into the computer software (temperature, solvent type, etc.) to run the experiment. Dielectric Relaxation Spectroscopy. To prepare a DRS sample, a coacervate gel was washed with deionized water to remove any excess salt. After polishing and cleaning both the 15 mm diameter bottom and 10 mm diameter top brass electrodes, about 0.3 g of coacervate was squeezed flat with a spatula onto the bottom electrode. Three triangular 1.02 mm spacers were placed flat on top of the coacervate gel, and then the smaller electrode was placed on top. The sandwiched sample was placed into a liquid cell along with approximately 10 drops of deionized water to keep the system from drying out, making sure the top brass plate is not submerged. The liquid cell was subsequently placed into a Novocontrol Broadband Concept 40 Dielectric Spectrometer, and all the necessary information was inputted into the software (geometry, operating temperature, voltage, frequency range, etc.). Measurements were all run at 25 °C using a 0.2 V amplitude with frequency ranging from 1 × 107 to 1 × 10−2 Hz. Swelling Measurements. After formation, each coacervate was washed with deionized water, divided into several batches, and placed into separate tared vials, each with different concentrations of NaCl solution. All salt solutions were prepared in deionized water. The complexes were left to equilibrate in solution for at least 48 h. Eventually the salt solution was poured out, and the coacervates were weighed to determine their swelling value in terms of mass, using the following equation

Q [CS] /Q Ref =

m[CS] m[CSupernatant ] m[CS] / = mDry mDry m[CSupernatant ]

Figure 3. Polyelectrolyte coacervate sandwiched between the 25 mm diameter top plate and the 65 mm diameter bottom plate, immersed in water (blue) at various salt concentrations: (a) side view and (b) top view. Measurements were taken at room temperature (around 23 °C). Frequency sweep tests on all polyelectrolyte complexes with different salt concentrations covered a range from 0.01 to 100 rad/s.

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RESULTS AND DISCUSSION Coacervate Stoichiometry. Measurements of both ζpotential and specific viscosity of the supernatant were made to determine the stoichiometry of the coacervate as 4 wt % polycation solution is titrated into 2.5 wt % polyanion solution and also as 2.5 wt % polyanion solution is titrated into 4 wt % polycation solution. These data are presented in the Supporting Information (Figures S1 and S2). With excess polyanion (X ≪ 1/2) the ζ-potential is negative, and with excess polycation (X ≫ 1/2) the ζ-potential is positive. At some point in the titration the ζ-potential passes through zero, indicating an average charge balance in the soluble complexes present in the supernatant (Figure S2). The viscosity of the supernatant decreases as more coacervate forms near the charge balance point,3 with a minimum viscosity indicating the lowest amount of polymer remaining in the supernatant and the stoichiometry of the coacervate (Figure S1). The results for the stoichiometry of the coacervate from these two methods are summarized in Table 1 and plotted in Figure 4. Due to the varying α values for the four IBMA-Na polyanions, different amounts of polycation are expected to be required to neutralize; IBMA-Na with a higher α value requires more PDADMA-Cl to neutralize. Theoretically, X =

(10)

where Q[CS] and Qref are the swelling ratios of the coacervate (gel mass/dry mass) at a certain salt concentration and supernatant, respectively. The result gives a relationship between the masses of the coacervate at a specific salt concentration m[CS] to its initial mass in its supernatant at CS = 0.09 M. D

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complexes that are further away from zero net charge show stronger polyelectrolyte characteristic compared to net-neutral systems that show predominately charge-balanced polyampholyte behavior. If a coacervate has net charge at the low salt concentration limit (pure deionized water with ∼0.000 01 M salt, obtained through conductivity measurements) osmotic pressure of the counterions trapped inside the coacervate will lead to swelling. Figure 5 shows that coacervates made from high molecular

Figure 4. Comparing positive charge fraction X of the coacervate from ζ-potential and specific viscosity. PDADMA-Cl titrant is shown using black symbols and IBMA-Na titrant using red symbols. Zetapotential technique indicated with X, and specific viscosity with □.

0.5 would indicate the coacervate has a zero net-charge and there are no uncomplexed sites. Obtaining 0.37 < X < 0.47, consistently below 0.5, implies that our coacervates have an overall negative charge compensated by some Na+ counterions that remain inside the coacervate. In addition to the strong charge density mismatch discussed earlier, this net anionic charge might originate from mismatches in the flexibility of the chains or the effective size of the monomer group along the polymer backbone.27 The data in Figure 4 suggest an uncertainty in X of ±0.01, consistent also with the raw specific viscosity and ζ-potential data in Figures S1 and S2, respectively. Given the two assumptions of fully quaternized polycation (αPDADMA‑Cl = 1) and that the α of the polyanions can be determined from the titration curve of Sauvage et al.,26 unaffected by the presence of the polycation, it is easy to rationalize a systematic error in our estimate of X. For instance, one could argue that 0.05 should be added to all X values, making the two coacervates made from low molecular weight polyanions charge balanced. However, even in this scenario, the coacervates prepared from the two higher molecular weight polyanions still must have a net negative charge which increases in magnitude with increasing polyanion molecular weight, an unexpected finding that deserves discussion. Figure S1 shows that the specific viscosity of the 4 wt % polycation solution is 30, indicating that the starting polycation solution is semidilute (the overlap concentration should have a specific viscosity of order unity). The 2.5 wt % polyanion solutions have specific viscosity of 80, 9, 4, and 0.3 for polyanion molecular weights of 410, 196, 82, and 11 kg/mol, respectively, indicating that the three highest molecular weight polyanion starting solutions are semidilute, while the 11 kg/mol polyanion starts in dilute solution. Mixing to near the stoichiometric balance decreases the concentration of each polyion by a factor of order 2. Perhaps the distinction between the high and low molecular weight polyanions is related to their chain overlap in some way; this idea deserves further study. Coacervate Swelling. Swelling measurements allowed us to determine the range of salt concentrations at which these coacervates behave as either polyelectrolytes or polyampholytes. Coacervate swelling observed in deionized water, the low salt limit, is a test that allows us to identify qualitatively how far the system is from net neutrality. In general, polyelectrolyte

Figure 5. Swelling ratios of polyelectrolyte coarcervates as a function of salt concentration. The swellings shown here are relative to that in their preparation state Qref (see eq 10) surrounded by supernatant at 0.09 M NaCl, meaning that all four data sets share the point 0.09 M, 1.

weight polyanions are more likely to be nonstoichiometric; swelling getting as large as 1.35 ± 0.1 times greater than the gel surrounded by its supernatant. Swelling in deionized water with all salt removed shows that IBMA-Na 410 and 196 kg/ mol, with the largest net charge (consistent with the coacervate stoichiometry in Table 1 and Figure 4), exhibit the largest swelling in deionized water, while IBMA-Na 82 and 11 kg/mol have significantly lower net charge and do not swell further in deionized water. This suggests that X = 0.45 ± 0.01 for IBMANa 82 and 11 kg/mol are really much closer to chargebalanced (X = 1/2) and there is a systematic error in our estimation of X, which is hardly surprising given the assumptions involved. In that instance, IBMA-Na 410 and 196 kg/mol still each have a net negative charge with Na counterions, consistent with their swelling in deionized water. Such a conclusion is also consistent with data on these four coacervates from dielectric spectroscopy28 shown in Figure S3 in the Supporting Information, which shows that the coacervates formed from the high molecular weight polyanions (IBMA-Na 410 and 196 kg/mol) are orders of magnitude more polarizable than the coacervates made from low molecular weight polyanions (IBMA-Na 82 and 11 kg/mol). The dielectric data in Figure S3 also reveal a MHz relaxation that is consistent with a sticker lifetime of order microseconds. As salt is added to the surrounding water the Debye length decreases, eventually becoming smaller than the spacing between net charges where even with a net charge, screening changes character from screening long-range repulsions to screening the short-range attractions that hold the coacervate E

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Macromolecules together (see the preceding Background Theory section). The swelling goes through a minimum at the point of this shift from polyelectrolyte to polyampholyte behavior.29,30 The minimum swelling found for our coacervates occurs somewhere between 0.15 and 0.2 M NaCl solutions, depending on the polyanion’s molecular weight. These salt concentrations correspond to Debye lengths of 7.9−6.8 Å, as estimated from eq 4, consistent with both the Bjerrum length and the Kuhn length of these flexible polyions. For higher salt concentrations, ranging from 0.4 to 1.3 M NaCl, salt ions partly screen ionic attraction between the oppositely charged polyelectrolytes without dissolving the gel but cause the polyampholyte gel to swell. This polyampholyte feature is similar for the four IBMA-Na molecular weights in the coacervates. However, IBMA-Na 82 kg/mol demonstrates the highest swelling, suggesting that it has the largest amount of ionic interactions, perhaps related to its highest α value. The roughly linear increase in swelling with salt can be understood very simply. The pioneering work of Pfeuty31 showed that the magnitude of the effective excluded volume v of a polyelectrolyte is inversely proportional to salt concentration CS, consistent with modern scaling models of polyelectrolytes.32,33 The coacervate has negative excluded volume (attraction), and the Flory−Huggins style thermodynamics of eq 1 expects the volume fraction of sediment34 ϕ = −v/b3 (b is the Kuhn length) so Q ∼ 1/ϕ ∼ 1/|v| ∼ CS, as observed for CS > 0.4 M for each of the four coacervates in Figure 5, before they redissolve at higher CS. Going to extremely high salt concentrations, larger than 0.7 M NaCl for coacervates with IBMA-Na 11 kg/mol and higher salt concentration for coacervates with larger molecular weight IMBA-Na, leads to sufficient screening of electrostatic attractions that the coacervates resolubilize, as expected by eq 1. This redissolution precludes further swelling of the coacervates, especially for polyelectrolyte complexes with lower polyanion molecular weights. We note that the redissolution salt concentration of each coacervate roughly corresponds to the association energy of eq 5 reaching the thermal energy kT. Coacervate Linear Viscoelasticity. Viscoelasticity of coacervates exhibits reversible behavior compared to regular cross-linked polymers that have permanent cross-links due to the reversible nature of the ionic bonds.35,36 The ionic linkages between polycations and polyanions in polyelectrolyte complexes can break and rearrange. This rearrangement allows the system to flow as a viscoelastic liquid on long time scales. In contrast, at higher frequencies, the system is viscoelastic. The addition of salt screens ionic interactions, lowering the Debye length rD, leading to lower Ea ∼ lB/σ − lB/rD (eq 5) and thus facilitates chain motion. Figure 6 shows the viscoelastic response of one coacervate obtained after equilibrating with different salt concentrations outside the coacervate. At low salt concentrations, polyelectrolyte complexes behave like soft elastomers, whereas at higher salt concentrations the complexes become more liquidlike, as the association lifetime decreases as salt is added (eq 7). Time−salt superposition21,23 allows estimation of LVE covering a wide frequency range, using solutions with different ionic contents. LVE at different salt concentrations shown in Figure 6 can be constructed into master curves after (1) multiplying the modulus scale by Q/ Qref, reflecting the number density of network strands, and (2) shifting along the frequency scale by a factor β. The master curves obtained for the four coacervates with PDADMA-Cl

Figure 6. Linear viscoelasticity of PDADMA-Cl + IBMA-Na 410 kg/ mol coacervates at different salt concentrations. Each salt concentration data set was shifted on the modulus scale by a factor b for visual purposes. Shift constants b: DIW = 10 000, supernatant has b = 1800, 0.2 M NaCl has b = 300, 0.3 M NaCl has b = 100, 0.5 M NaCl has b = 15, 0.7 M NaCl has b = 7.5, 0.9 M NaCl has b = 3, 1.1 M NaCl has b = 1.5, and 1.3 M NaCl has b = 1.

and IBMA-Na of M = 410, 196, 82, and 11 kg/mol are shown in Figure 7. Master curves for coacervate dynamics cover a range of eight decades in frequency for the high molecular weight coacervate (Figure 7). In contrast, at a fixed salt concentration only four

Figure 7. Master curves constructed using time−salt superposition of coacervate samples with different polyanion molecular weight, using the coacervate immersed in deionized water as the reference state. Storage modulus G′ are filled symbols and loss modulus G′’ are open symbols, at 23 °C. IBMA-Na 410 kg/mol (■), 196 kg/mol (blue ◆), 82 kg/mol (red ▲), and 11 kg/mol (green ●). F

DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

explicitly. In comparison, σ and τ0 are two interdependent variables: increasing σ leads to lower Ea and accordingly increases τ0. Nevertheless, we know that in water at room temperature the attempt time22 is the ratio of the viscosity (mPa s) and glassy modulus (GPa) making τ0 ∼ 10−12 s, so we can choose this value for τ0 leading to n = 5 and σ = 1.7 Å, which allow a consistent fitting of the salt concentration dependence of relaxation time for all four coacervates in Figure 8: n = 5 signifies that approximately five primary ionic bonds need to be broken simultaneously to allow chain motion. In general, five repeat units of flexible polymers usually correspond to the length scale of one Kuhn segment,34 consistent with the quadrupolar associations of Kuhn segments shown schematically in Figure 1. The value σ = 1.7 Å is a reasonable length for anion−cation associations in contact, although this also will effectively reflect any changes in hydration of ions when they separate. In our greatly oversimplified model, σ = 1.7 Å really just means that the effective primary association free energy of + and − charges in contact in the coacervate is E = lBkT/σ ≈ 4kT. The crossover relaxation time τc = 1/ωX is the typical (average) terminal time in these polydisperse coacervates and is always simply proportional to the association lifetime τ of eq 6. It is convenient to define τ* such that τc = τ* exp(nEa/kT) and in this way τ* contains all molecular weight dependence. In Figure 8b we demonstrate that τc /τ* vs salt concentration CS reduces all data for the four coacervates, as this is effectively plotting exp(nEa/kT) vs CS, which are related by eq 5 (solid curve in Figure 8b). The deviation of τc /τ* from the prediction at low CS seems to be stronger for the coacervates made with higher molecular weight polyanions; this observation is in accordance with the argument that screening length is controlled by the counterions trapped inside the coacervate since those made from higher Mw polyanions are less chargebalanced. Finally, to explore the molecular weight dependence of terminal relaxation times, τ* was plotted against polyanion molecular weight M in Figure 9. A scaling of τ* ∼ M2 is

decades of frequency were actually measured (Figure 6). The upper limiting factor is from the rheometer reaching its maximum frequency accessible (100 rad/s) for the coacervates immersed in deionized water. The mechanical properties of these polyelectrolyte coacervates are independent of their formation pathway because neither the order of addition nor the polycation/polyanion fraction used for formation had any effect on the coacervate LVE. This feature is only expected to apply to soft coacervates that are viscoelastic liquids and hence can attain equilibrium at any particular salt concentration. The viscoelastic relaxation time can be estimated at τc = 1/ ωX, where ωX is the crossover frequency from liquidlike behavior (G″ > G′) at lower frequencies to solidlike behavior (G″ < G’) at higher frequency. Figure 8a plots the relaxation

Figure 8. (a) Sticky Rouse model comparison to experimental data of salt concentration dependence of relaxation time τc defined as the reciprocal of the frequency at which G′ = G″, for coacervates formed using four polyanions of different molecular weights. (b) Salt concentration dependence of relaxation time τc reduced to a common curve by dividing τc by τ*; the dependence of τ* on polyanion molecular weight is shown in Figure 9. Solid curves are eq 7 with n = 5, τ0 = 10−12 s, and σ = 1.7 Å.

times τc thus obtained as functions of salt concentration CS. This relaxation time decreases as salt is added due to screening attractions between the opposite charges on the two polyions and is fit to eq 7, shown as the four solid curves in Figure 8a. The fitting of the four molecular weights shows a good fit in the polyampholyte regime, where τc increases with polyanion molecular weight and decreases as salt is added. The relaxation time with no added salt is consistently smaller than expected by eq 7 because the screening length would then be controlled by the counterions of the gel, whereas eq 7 only considers the screening length from added salt. During the fitting, parameters n and σ are obtained through fitting τc with eq 7. First and foremost, n determines the gradient of the τc vs CS curve and thus can be determined

Figure 9. τ* = τc exp(−nEa/kT) dependence on IBMA-Na molecular weight. The solid line is τ* = τ0N2 with attempt time τ0 = 10−12 s while the dashed line is τ* = τ0N2M/Me with Me = 105 g/mol.

expected for unentangled semidilute solutions since they should be Rouse like. The solid line shows τ* = τ0N2 with attempt time τ0 = 10−12 s and number of segments N = M/M0, which fits well the lowest three molecular weights of polyanion. Figure 9 shows that the τ* (= τ0N2) dependence has a stronger slope as the molecular weight of the polyanion is increased. The power-law slope of 2 for the lowest three polyanion molecular weights suggests that these coacervates relax by Rouse motion, in agreement with our sticky Rouse model hypothesis.18,19 On the other hand, the two highest G

DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

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molecular weight polyanions roughly exhibit a slope of 3, expected for sticky-reptation;24 the crossover between these two molecular weight dependences allows estimation of the entanglement molecular weight of the reversible gel’s double strand of Me ≈ 105 g/mol. The molecular weight of strands between network junctions can be estimated using the highfrequency value (100 rad/s) of the storage modulus G′ of the three highest molecular weight coacervates with no salt (Figure 7 suggests that the coacervate with the lowest molecular weight polyanion did not get to high enough frequency for a meaningful estimation of this modulus). From rubber elasticity32 G′ =

wf ρRT Mstrand

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00401.

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Specific viscosity of the supernatant; ζ-potential of the supernatant; dielectric permittivity of the coacervates; loss tangent for the time−salt superposition master curves (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: (814)863-3457.

(11)

ORCID

where wf represents the weight fraction of polyelectrolyte in the coacervate, ρ is the density of the coacervate (taken to be that of water, 1 g/cm3), R is the gas constant, T is absolute temperature, and Mstrand is the molecular weight of a coacervate network strand. Table 2 summarizes the estimation of Mstrand using eq 11.

Quan Chen: 0000-0002-7771-5050 Ralph H. Colby: 0000-0002-5492-6189 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the National Science Foundation, Polymers Division, for funding this work via DMR-1404586. We also thank Michael Rubinstein for very helpful discussions.

Table 2. Estimation of Coacervate Network Strand Molecular Weights Using Eq 11 Mpolyanion

G’ at 100 rad/s for the coacervate with no salt (kPa)

wf

Mstrand

82000 196000 410000

21 35 60

0.28 0.34 0.41

33000 24000 17000

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REFERENCES

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Probably the Mstrand estimation for the coacervate with the highest molecular weight polyanion includes some entanglement effects (see Figure 9), and our “best” estimation comes from the second highest molecular weight polyanion, with Mstrand = 24 000. The number density of these strands ν = G′/ kT ≈ 0.01 nm−3, giving a mesh size of ν−1/3 ≈ 4 or 5 nm, consistent with SANS from coacervates by Spruijt37 and also with the correlation length of polyelectrolyte solutions at similar concentration.38

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CONCLUSION We report the swelling and linear viscoelastic response of polyelectrolyte complex coacervates constructed from one polycation PDADMA-Cl (Mw = 300 kg/mol) and various polyanions IBMA-Na (Mw = 11, 82, 196, and 410 kg/mol). Swelling and LVE of coacervates and viscosity/zeta-potential of soluble complexes suggest these materials are reversible equilibrium structures, with properties independent of construction pathway. The coacervates are polyampholyte reversible gels that are viscoelastic liquids with relaxation times set by the association energy. The molecular weight and salt concentration dependences of relaxation time are well described by a sticky Rouse model. Our sticky Rouse model fit, using eq 7, reveals that the free energy of a primary ± contact pair is 4kT and that n = 5 of those primary associations need to simultaneously open to allow Rouse motion, roughly consistent with the number of primary associations per Kuhn length of the higher charge density polycation in these flexible polyelectrolytes. We hope that such simple analyses can help with data interpretation in this rapidly advancing field. H

DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00401 Macromolecules XXXX, XXX, XXX−XXX