How to Force Polymer Gels to Show Volume Phase Transitions - ACS

Letter Cite This: ACS Macro Lett. 2019, 8, 272−278

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How to Force Polymer Gels to Show Volume Phase Transitions Miroslava Dušková-Smrčková and Karel Dušek* Institute of Macromolecular Chemistry, Czech Academy of Sciences, Heyrovského nám. 2, 162 06 Prague 6, Czech Republic

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S Supporting Information *

ABSTRACT: Relatively few polymer gels are known to show volume phase transition where the gels undergo an abrupt change in the degree of swelling by passing through a three-phase equilibrium. Characteristic for such transition is the existence of van der Waals (vdW) loop on the dependence of solvent chemical potential versus polymer concentration. For the χinduced transition, the existence of vdW loop is determined by the concentration dependence of the interaction function. It is shown that expansive mechanical strains can assist in development of the vdW loop. Systems characterized by continuous change of the degree of swelling transform upon such strain into ones where the degree of swelling changes much and abruptly. Also, expansive modes of strain can make the transition wider and more robust in gels where transition is already observed under free swelling condition. The possibility to induce the volume phase transition by external stresses can be utilized for finding other stimuli sensitive gels, strengthening of gel response, and in modeling of properties of gel constructs by Finite Element Method. shift the loop to the region of Δμ1 = 0, the “hidden” transition becomes observable. Already in the prediction paper,1 it was found that uniaxial extension can facilitate the volume phase transition. Later, the effect of mechanical strain was extensively studied in various laboratories (cf., e.g., refs 4−12). Among these recent studies of the effect of strain on swelling of Gaussian networks (cf., e.g., refs 7−12), methodically the most close to the present work are fundamental studies summarized in refs 9−12. Several authors13−18 analyzed the effect of concentration dependence of interaction functions χ(ϕ2) or g(ϕ2) on phase equilibria and came to the conclusion that in the limit of infinite molecular weight the critical polymer concentration ϕ2,c∞ was not zero, as it usually is (Θ temperature), but quite higher, and the coexistence curves were rather flat if the coefficient χ1 in χ(ϕ2) = χ0 + χ1ϕ2 was higher than 1/3. A more detailed thermodynamic analysis of such systems18 (called “systems with off-zero critical concentration”) revealed that demixing was more intricate if also the quadratic term in the power expansion was considered. The analysis delimited regions of three types of phase coexistence for an un-crosslinked polymer of very high molecular weight:18 Type I demixing is characterized by the conventional zero critical concentration (ϕ2,c∞ = 0), Type II by one off-zero critical concentration ((ϕ2,c∞ > 0), and Type III of two off-zero critical concentrations (more details in Supporting Information on off-zero regions, Section A).

F

ifty years have elapsed from the prediction of existence of volume phase transition in swollen polymer gels1 and 40 years from its experimental discovery.2 The number of published papers dealing with this phenomenon reached the order of 104. In cross-linked gel−solvent systems, the transition is a result of a delicate balance between attractive and repulsive forces contributing positively or negatively to the chemical potential of the solvent Δμ1. Such balance can be tuned by adjusting the coiling state of network chains (by supercoiling,1 prestretching), introduction of charges,3 and specific polymer− solvent interactions3 (hydrogen bonding, hydrophobic association). The specific interactions are effective already in uncross-linked systems and poly(N-isopropylacrylamide) (PNIPAm)−water system is a striking example. This type of volume phase transition is called χ-transition, referring phenomenologically to the concentration-dependent Flory−Huggins parameter. Despite the large number of publications on volume phase transition in gels, there are not very many chemical systems known to show stimuli-induced transition. For most of the systems, the dependence of equilibrium swelling degree on temperature is more or less steep but smooth and corresponds to a two-phase equilibrium between swollen gel and pure solvent. The volume phase transition is characterized by passage through three-phase equilibrium between pure solvent and two polymer phases of different degrees of swelling. For such systems, the chemical potential of solvent in the gel, Δμ1, versus polymer concentration dependence is characterized by the so-called van der Waals (vdW) loop. The aim of this contribution is to show that for some gels the vdW loop is located inconveniently in the positive or negative parts of the Δμ1 dependence on ϕ2. If one is able to © XXXX American Chemical Society

Received: December 26, 2018 Accepted: February 19, 2019

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DOI: 10.1021/acsmacrolett.8b00987 ACS Macro Lett. 2019, 8, 272−278

Letter

ACS Macro Letters In uncharged cross-linked gels, the “off-zero behavior” is a necessary condition for the existence of the vdW loop and volume phase transition, but it is not sufficient. The presence of cross-links and consideration of finite extensibility of network chains are necessary. The beneficial effect of the concentration dependence of χ(ϕ2) on the realization of the phase volume transition was already mentioned in the “prediction” paper, but was not elaborated at that time.19 The analysis of the swelling behavior revealed that water solutions or networks of PNIPAm belonged to Type II,20−22 of poly(vinyl methyl ether) (PVME) to Type III,23,24 and of poly(N-vinylcaprolactam) (PNVC) to Type I (zero critical concentration)25 (cf. also refs 26 and 27). For modeling of the effect of mechanical constraints on phase volume transition through the Gibbs energy change, ΔG, we used the Flory−Huggins type contribution for ΔGmix with concentration dependent interaction function g(ϕ2) and a nonGaussian model for the elastic contribution, ΔGel,n, which respects the finite extensibility of network chains, based on Kovac’s28,29 model. The additivity ΔG = ΔGmix + ΔGel,n

The chemical potential of solvent and conditions for free swelling, uniaxial and biaxial extension/compression as well as for equitriaxial expansion and compression are given in Supporting Information, section B, Gibbs energies, final eqs S-9−S-12. To demonstrate the effect of strain, we employed the Maxwell construction used before by Tanaka and others (cf., e.g., ref 3), which visualizes the equality of chemical potentials of either component in coexisting phases by plotting Δμ1/ϕ22 versus ϕ2, as shown in Figure 1.

(1)

was assumed with

ij yz f −2 ΔGmix = RT jjjjn1 ln ϕ1 − e ne ln ϕ2 + g (ϕ2)n1ϕ2zzzz j z fe k {

Figure 1. Schematic dependence of the reduced chemical potential of solvent on swelling showing van der Waals loop: (A) case of volume phase transition in equilibrium with pure solvent; (B, C) hidden phase transitions at Δμ1 > 0 and Δμ1 < 0.

(2)

where n1 and ne are the number of moles of solvent molecules and elastically active network chains (EANC), respectively, and fe is the elastically effective functionality of cross-links. In this formulation, the entropy change due to coupling chain segments into junctions is included in ΔGmix because it is also operative when the network chains are in their unstretched, reference state, as explained in ref 30. This negative entropy change is also the driving force for liquid−gel phase separation during network formation when the network chains are assumed to be in their reference states.31 The elastic contribution, ΔGel,n, is reserved for the effect of stretching of EANCs relative to the reference state on a lattice occupied by chain segments and solvent molecules, and ΔFel(λ) to the net effect of stretching of ne moles of EANCs ΔGel,n ≈ ΔFel,n = ΔFel(λ) − RTne ln λxλyλz

Ä ne(n − 1) ÅÅÅÅ ij 1 yz ij 1 yz ÅÅlnjj1 − λx2 zz+lnjj1 − λy2 zz ÅÅÅ k n − 1{ k n − 1{ 2 Ç É 1 yz 1 yzÑÑÑÑ i i zzÑÑ zz − 3 lnjjj1 − + lnjjj1 − λz2 n − 1{ n − 1 {ÑÑÑÖ k k

The condition for a three-phase equilibrium with pure solvent is given by equality of the integral I to zero I=

(Δμ1 /ϕ22)dϕ2 = 0

(6)

2 1

If I > 0, the right root (3′′) of Figure 1 is relevant; if I < 0, the equilibrium is determined by the left root (1′). Root 2 is located in the absolutely unstable region. In terms of the Gibbs energy change, the roots correspond to extremes in the ΔG(ϕ2) dependence. The minimum that is deeper is relevant; in the case of the three-phase equilibrium (transition; Figure 1, curve A) both minima are of the same depth. The dependence C can be shifted to the position A by compressive mechanical strains, while the dependence B can be shifted to the position of A by expansive mechanical strains. The concentration dependence of the interaction function is the main factor controlling the shape of the Δμ1(ϕ2) dependence. Most frequently used are power series dependences

(3)

ΔFel(λ) = − RT

(4)

where λx, λy, and λz are deformation ratios with respect to the (isotropic) reference state, and n is the average number of statistical segments in an EANC; n is related to the concentration of EANCs, νe, as n = ρ2 /νeMs

(ϕ2)3

∫(ϕ )

g (ϕ2) = g0 + g1ϕ2 + g2ϕ22 + ... χ (ϕ2) = χ0 + χ1 ϕ2 + χ2 ϕ22 + ...

(5)

and the closed form dependences

where ρ2 is the specific gravity of the polymer and Ms is the molecular weight of the statistical segment. eq 4 was slightly modified compared to that in ref.29 in that the factor 1/n was changed to 1/(n − 1) to comply with the correct limit of full extensibility when n = 1 (one statistical segment) is reached instead of the original n = 0.

g (ϕ2) = a +

b 1 − cϕ2

(7)

32

χ (ϕ2) = a +

b(1 − c) (1 − cϕ2)2

(8)

33−37

where a, b, and c are constants. Wolf et al. analyzed the concentration dependences in detail and proposed several variants; the favored one reads 273


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Figure 2. Effect of expansive strain and negative hydrostatic pressure on the dependence of reduced chemical potential of solvent: (a) closed interaction function (eq 8), a = 0.127, b = 0.69, c = 0.47, νe = 1 × 10−4 mol/cm,3 ϕ02 = 0.6, λ = 3; phase transition attained for equibiaxial strain; (b) negative hydrostatic pressure (equitriaxial extension), closed interaction function (eq 8), a = 0.20, b = 0.69, c = 0.47, νe = 1 × 10−3 mol/cm3, pressure in kPa indicated.

Figure 3. Effect of compressive strain and positive hydrostatic pressure on the dependence of reduced chemical potential of solvent: (a) free swelling (black), uniaxial compression (red), biaxial compression (blue); power series interaction function (eq 3), g0 = 0.782, g1 = 0.560, g2 = 0, λ = 0.7, νe = 1 × 10−4 mol/cm;3 (b) positive hydrostatic pressure (equitriaxial compression), power series interaction function (eq 7), g0 = 0.782, g1 = 0.560, g2 = 0, νe = 1 × 10−4 mol/cm;3 pressure in kPa indicated. χ (ϕ2) =

χ0 + ζλ (1 − νϕ2)2

− ζλ(1 + 2ϕ2) =

α0 (1 − νϕ2)2

While positive pressure can be generated easily, for instance, by letting the gel swell under triaxial confinement, generation of negative pressure is experimentally more complicated and, according to our knowledge, has not been reported as yet. One can try to prepare a gel in a (meso)porous mold enabling transport of solvent through the pores while anchoring the gel in the wall pores. Under compressive modes of strain and positive hydrostatic pressure, the vdW loops become smaller and eventually transform into inflection. However, these compressive modes can be applied for shifting the reduced chemical potential dependence to a position where it meets the Maxwell condition at solvent activity a1 = 1. Both cases are visualized in Figure 3. Gel−solvent systems potentially able to generate van der Waals loop and phase volume transition: In the literature, some data on concentration dependence of the interaction functions are available which can be analyzed as to whether the

− ζλ(1 + 2ϕ2)

(9)

The values of the constants are tabulated in a review chapter.37 The concentration dependences of the interaction function g(ϕ2) in ΔGmix can be transformed into χ(ϕ2).16 Figure 2 shows that expansive strain can transform a dependence with only an inflection and without the vdW loop, into one showing a distinct vdW loop and shift it toward the region of three-phase equilibrium when one of the phases is pure solvent. As can be expected, biaxial strain is more effective than the uniaxial strain. Even more effective for vdW loop development can be a triaxial expansion. An equitriaxial expansion is equivalent to negative hydrostatic pressure (Treloar38). Hydrostatic pressure in this context is understood a pressure acting on the swollen network while the solvent in equilibrium is under hydrostatic pressure zero (atmospheric pressure). 274


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Letter

CROSS-LINKED POLY(N-ISOPROPYLACRYLAMIDE (PNIPAM)−WATER Strains have also an effect on systems known to show phase transition. PNIPAm gel−water: The concentration and temperature dependences of the interaction function by Afroze et al.22 were used. The strain effect is shown in Figure 5. The

polymer−solvent systems are potentially able to show volume phase transition. The data were obtained either for solutions of linear or branched polymers, or from swelling measurement. Branched (cross-linked) polymer−solvent systems show somewhat higher χ(ϕ2) or g(ϕ2).34,39

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POLY(2-HYDROXYETHYL METHACRYLATE) (PHEMA)−WATER GELS The interaction parameter χ(ϕ 2 ) was determined by equilibrium swelling of gels of low cross-link densities νe = (0.6−2.0) × 10−4 mol/cm3 and can be described by a linear dependence on ϕ240 (cf., also refs 41−43). The linear dependence χ = χ0 + χ1ϕ2, with χ0 = 0.32, χ1 = 0.90 was modified to χ0 = 0.327, χ1 = 0.886 when water was considered as trimer.44,45 However, the concentration dependence of χ(ϕ2) can also be expressed as a quadratic dependence χ = χ0 + χ1ϕ2 + χ2ϕ22, with χ0 = 0.417, χ1 = 0.549, χ2 = 0.315. Figure 4

Figure 5. Plot of reduced chemical potential of the solvent vs volume fraction of polymer for cross-linked PNIPAm gel−water system for free swelling (black) and samples strained uniaxially (red) and biaxially (blue); each state obeys the Maxwell condition at different transition temperatures indicated; g(ϕ2) = g0 + g0TT + (g1 + g1TT)ϕ2 + (g2 + g2TT)ϕ22; ϕ02 = 0.1, νe = 1 × 10−4 mol/cm3; the values of the coefficients gi and giT from ref 22.

transition temperature does not change much with strain but the width of the transition and especially its magnitude do change. The temperature T = 305 K is in agreement with experiment.22 For higher cross-link density (νe = 1 × 10−3 mol/cm3) the PNIPAm gel does not seem to be able to undergo the transition and only an inflection point develops, but upon application of tensile strain the vdW loop appears again. (Supporting Information, PNIPAm of higher cross-link density, section D).

Figure 4. Plot of reduced chemical potential of the solvent vs volume fraction of polymer for PHEMA gel−water system at 25 °C for free swelling (black) uniaxial extension (red) and biaxial extension (blue), λ = 3; νe = 2 × 10−4 mol/cm3, ϕ02 = 0.6; full lines linear dependence χ = χ0 + χ1ϕ2, with χ0 = 0.327, χ1 = 0.886; dashed lines quadratic dependence χ = χ0 + χ1ϕ2 + χ2ϕ22, with χ0 = 0.417, χ1 = 0.549, χ2 = 0.315.

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shows that in the case of linear dependence of χ(ϕ2), the system would be on the verge of phase transition and a small extensive strain would be sufficient for a phase transition to a highly swollen state. Nothing like that has been observed so far. The PHEMA gel−water system described by the quadratic dependence would require more expansive strain for such a jump to occur. More studies of strained water swollen PHEMA gels are desirable.

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POLY(VINYL METHYL ETHER) (PVME) GEL−WATER The system linear PVME−water belongs to the category III of the off-zero critical concentration group which should show two transitions: one main LCST transition and the other one UCST transition of small size at high degree of swelling.22,23,46,47 Figure 6 was generated using g-values of refs 22 and 23. For this combination of g-coefficients, small negative g1 and large positive g2, a robust vdW loop situated in the region of high polymer concentrations develops. In that region, the effect of strain is almost nonexistent. In Figure 6, all three curves (nonstrained and strained gels) merge in a single dependence. A second, smaller transition develops in the high swelling region when the cross-link density is increased. It may be related to that observed experimentally. In conclusion, the analysis has shown that expansive strains (uniaxial, biaxial, triaxial extensions) can affect swelling of some gels. Gels showing only gradual changes of swelling degree upon action of stimulus (e. g., temperature change), show abrupt

CROSS-LINKED POLYSTYRENE (PS)−CYCLOHEXANE

PS gel−cyclohexane is a popular UCST system with a concentration dependent parmeter χ. We have used Wolff’s model (eq 9) and values tabulated in ref 37. The result is similar to that of PHEMA gel. Graphical result is available in Supporting Information, other systems, section C. There are several other systems potentially being able to develop a vdW loop upon strain; for instance, cross-linked PS−acetone (χ1 = 0.53) or PS−1,2-dichloroethane (χ1 = 0.60). 275


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documented in the monograph.16 At present, no bridging theory is available for gels. An insight into swelling of networks in the semidilute region based on the scaling approach is offered in Rubistein’s monograph.60 Time effects can be another reason of continuity: the phases develop in time and on microscale which affects the transition steepness. The long time needed to reach equilibrium favors continuity of the experimentally observed swelling transition. During gradual deswelling or collapse transition effectuated at finite speed, the system may pass through a metastable state characterized by microphase separation (microsyneresis, opaque gel) to thermodynamically stable macrosyneresis (liquid, clear gel). Such experiments were reported a long time ago when upon surpassing the miscibillity limit the excess solvent separated in the form of submicron droplets.61 The time needed for the microseparated opaque gel to form macroseparated transparent bulk phases reached the order of 104 h, depending on sample size. Recently, such a situation was nicely described by Chang et al.62 in the case of collapse transition: when the transition temperature was reached, the sample was shrinking, while the gel remained transparent. After some time, the gel got opaque and the volume decrease almost stopped. Since expansive strains increase the volume, we do not expect that the situation will be worse; on the contrary, the strains may be beneficial. We hope that this contribution will stimulate work in this interesting area of volume phase transition and enrich the manifold properties and applications of polymer gels.63 Also, volume phase transition is behind the self-assembly of marine biopolymers and gel formation that yields ∼10 gigatons of bacterial substrate for Carbon cycling in the ocean;64−66 the application of this theory opens the way to understand the mechanisms of material storage in secretory cells and the explosive volume expansion by osmotic forces upon release of their cargo.66−69

Figure 6. Plot of reduced chemical potential of the solvent vs. volume fraction of polymer for cross-linked PVME−water system; λ = 3; ϕ02 = 0.1; in thick black curve, dependences for freely swollen and swollen strained gels merge for νe = 1 × 10−4 mol/cm;3 thin curves are calculated for νe = 1 × 10−3 mol/cm3; free swelling (thin black), uniaxial extension (thin red), biaxial extension (thin blue); g0 = 1.38; g1 = −0.0125; g2 = 1.25.

change in swelling when strained. Expansive strains can either develop the vdW loop from a dependence with only an inflection point or widen and strenghten the transition in systems where the vdW loop already exists. The generation of vdW loop and volume phase transition by strain has never been reported before. In addition to the usual modes of strain, very promising seems to employ core−shell structures, namely the expansive strains exerted by the shell on the core. There exists a gradient of the degree of swelling in the shell, whereas swelling of the core is homogeneous. When a core−shell sphere is in equilibrium with solvent good for the shell but intermediate or poor for the core, the polymer network in the core is subjected to expansive stress forcing it to swell more than in free state.9,48 The theory of swelling of core−shell particles is well elaborated.9 If FEM is used to model core−shell and other more complicated constructs from gels of vdW loop type, one has to find the correct minimum of the Gibbs energy change out of the two existing minima. The deeper minimum corresponds to correct solution. For the three-phase equilibrium, both minima are of the same depth. It would be interesting to investigate the structure based models of polar polymer−solvent systems such as the Double Lattice Theory, Perturbed Chain Statistical Associative Fluid Theory (PC-SAFT) and other, see, for example, refs 49−53. In the course of experimental investigation, various problems may be encountered: The discontinuity of the transition predicted by the mean-field theories (used also here) was not observed, which was explained by distribution of length of network chains.54 Indeed, the mean field theory of Flory and Huggins is valid for the concentrated regime where it cannot be distinguished whether the neighboring segments belong to the same or different chains. The collapse of single chains in dilute solution is controlled by its lengths and in the semidilute region, to where to dilute conjugated phase of many systems (like PNIPAm but not PVME) belong, the chain length matters somewhat. The problem of phase equilibria between concentrated and dilute phases is general, including solutions of un-cross-linked chains and several forms of the bridging function were put forward,55−59 but in practical applications, overwhelmingly, the classical F−H approach is used as

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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/acsmacrolett.8b00987.

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Classification of systems with off-zero critical concentration, Gibbs energies and chemical potentials for various deformation modes of swollen gels, and dependences of reduced chemical potential on volume fraction of polymer for gels of various chemical compositions (PDF).

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Miroslava Dušková-Smrčková: 0000-0003-2684-9814 Karel Dušek: 0000-0002-4597-5501 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support by Czech Science Foundation, Project No. 17-08531S. 276


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