Alcohol

Article pubs.acs.org/Macromolecules

Modeling Poly(N‑isopropylacrylamide) Hydrogels in Water/Alcohol Mixtures with PC-SAFT Markus C. Arndt and Gabriele Sadowski* Laboratory of Thermodynamics, Department of Biochemical and Chemical Engineering, TU Dortmund, Emil-Figge-Strasse 70, 44227 Dortmund, Germany

ABSTRACT: The PC-SAFT equation of state is used for thermodynamic modeling of poly(N-isopropylacrylamide) PNIPAAm in water and alcohols (methanol, ethanol, 1-propanol, and 2-propanol). For calculating the swelling behavior of cross-linked PNIPAAm gels, an additional contribution to the Helmholtz energy considering elastic forces is implemented and the resulting pressure difference in the gel is taken into account. The model is used to describe the gel-phase composition and the degree of swelling as a function of both the temperature and the solvent composition in good agreement with experimental data. In particular, the re-entrant phenomenon of the swelling transition in the ternary mixtures is modeled correctly and data from the literature correspond well with the computed results, suggesting a significant predictive capability of the model.

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INTRODUCTION Hydrogels are cross-linked hydrophilic polymers, which have the ability to store large amounts of water or other hydrophilic solvents and have a low interface tension with these solvents. Promising new or already well-established industrial applications vary from superabsorbers and stabilizers over matrices for bioreactions and biological separation processes to chemical sensors/actuators and pharmaceutical drug depots with controlled release.1−3 Figure 1 gives the chemical structure of a poly(Nisopropylacrylamide) (PNIPAAm) chain, one of the most

single phase becomes instable and a two phase liquid−liquid equilibrium (LLE) is established with a broadening miscibility gap.4 One phase of nearly pure water corresponds with a phase of high polymer concentration and only low water content. Reason of this phase separation is that the previously prevailing interactions between water and the acrylamide side chains of PNIPAAm are broken, and polymer−polymer interactions are preferred. This leads to a significant increase of the polymer hydrophobicity and thus the polymer aggregates in one phase with little water only. Experimental values from the literature4 show that the lower critical solution temperature (LCST) in which both phases merge is little above 300 K, as can be seen later in Figure 6. By cross-linking loose polymer chains, an interconnected hydrogel network is obtained. In excess water, the equilibrium behavior of the gel and the water phase reveals a comparable thermodynamic behavior to the LLE (Figure 2). At elevated temperatures, the network chains aggregate to interact predominantly with each other and hence the water storage capability is rather low; the gel is in a shrunken or collapsed state. Lowering the temperature and crossing a critical value corresponding to the LCST, a sharp transition in the swelling

Figure 1. Chemical structure of PNIPAAm.

interesting hydrogel polymers which is often referred to as smart polymer due to its sensitivity toward external stimuli, as will be outlined in the following. Most prominent external stimulus in the literature is the temperature: observing a binary mixture of not cross-linked PNIPAAm polymer chains in water at low temperatures, one homogeneous phase is found. By rising the temperature, the © XXXX American Chemical Society

Received: April 4, 2012 Revised: July 18, 2012

A

dx.doi.org/10.1021/ma300683k | Macromolecules XXXX, XXX, XXX−XXX

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polymer−polymer interactions alternate with polymer−solvent type regions leading to a pearl-necklace-type conformation. Regardless of the assumed molecular mechanism, there is broad consent that at low temperatures analogue to the loose linear chains the interconnected PNIPAAm fibres of the gel are driven to dissolve in the aqueous surrounding medium, a trend which is precluded by the cross-linkers. The tendency of the polymer to dissolve and the water molecules diffusion into the gel network thus lead to a stretching of the polymer chains. Figuratively comparable to an elastic spring, a restoring force counteracts this trend, tending to contract the gel back to its relaxed state, and consequently an elevated pressure in the gel phase is induced in a swollen gel. In comparison with the calculation of the standard phase behavior of LLE mixtures, the modeling of swelling hydrogels has to account for three new constraints: • The polymer cannot distribute to the two phases; the surrounding liquid phase is polymer-free • In the network additional elastic forces exist caused by volume changes • Resulting from the elastic force, the pressure difference in both phases must be taken into account Several modeling approaches for the description of the swelling of hydrogels have been published in the recent years. Breaking on the field of hydrogel modeling with focus on thermodynamic derivation was the work by Maurer and Prausnitz.23 In their model, the elastic forces within the gel are ascribed to a theoretical elastic membrane sheathing the gel phase in a solvent phase. The equilibrium state between both phases is characterized by the minimum of the sum of the elastic Helmholtz energy of the membrane and the Gibbs energy of the phases. Pressure differences between the phases are disregarded due to the liquid character of the system. Inomata et al.24 explicitly considered the elastic pressure in addition with the osmotic pressure in their model to describe the swelling pressure of aqueous poly(ethylene glycol). In a more recent approach by Zhi et al.25 a close-packed lattice model is taken to calculate the Gibbs energy of the mixture adding an affine- network elastic contribution. Tanaka et al.14 propose another theoretical model for modeling the cononsolvency effect with particular focus on the competitive hydrogen bonds in the system. The work of Poschlad and Enders12 is based on two thermodynamic models for the description of the excess Gibbs energy, namely the Koningsveld-Kleintjens approach26 as well as the UNIQUAC model, each of these are combined with different elasticity contributions. Importantly, with UNIQUACas before proposed by Maurera general and established thermodynamic model for various thermodynamic applications such as vapor−liquid equilibria calculations is taken and expanded, whereas most other models are specially designed for the hydrogel phenomena. The literature shows that particularly the range of the theories on the description of the elastic forces is broad. General phenomena of rubber elasticity and theoretical approaches for elastic forces in polymer networks have been widely discussed in the last century.27−33 The most followed theories for the elastic energy are based on the affine network34−38 and phantom network32,39 assumptions. In these, the elastic energy of an idealized network of monodisperse polymer chains results in the summation of the elastic response of the individual chains to an isotropic

Figure 2. Swelling data of PNIPAAm gel in water as gel weight m over dry polymer weight m0 at different temperatures. The experimental data is taken from the literature.5

behavior is found and from the shrunken state the gel undergoes significant swelling. In analogy to the homogeneous phase at lower temperatures, the dense polymer network softens and polymer−water interactions prevail. Thus, a high amount of water is found in the now hydrophilic gel, being stored directly associated with the chains or in the interstitial regions between the meshes of the network. Not only the temperature dependence on aqueous PNIPAAm, but also the influence of third substances, cosolvents or solutes, has been widely examined. An intriguing phenomenon is referred to as re-entrant behavior of the gel: in pure water as well as in the pure cosolvent such as ethanol, PNIPAAm exhibits a strong swelling whereas it undergoes an abrupt shrinkage when exposed to a certain mixture of both.6−12 Solutions of not cross-linked PNIPAAm chains behave alike, labeled cononsolvency: in both pure solvents the polymer is completely soluble but in the ternary mixture water/ PNIPAAm/cosolvent, cloud points and demixing are observable. Most observations of cononsolvency (or re-entrancy) have been made with PNIPAAm in aqueous alcohol solutions,3,10,13−17 but the effect of other substances added to the binary system with water such as acetone and DMSO,18 salts19 or large molecules such as PEG7,8,20 gives a comparable shrinkage of the gel phase. Reason for this phase separation of linear polymers and for the swelling transition of the gel is generally accepted to be found in the preferential intermolecular interactions of the species and in particular the hydrogen bonding. The actual mechanism and thermodynamic causation have not been completely understood, however, and the theoretical explanations follow different approaches. One assumption discussed in the literature18,21 includes complexation of water and cosolvent molecules which is preferred toward hydrogen bonds with the polymer in the miscibility gap; these complexes then act as poor solvent for the polymer. Another explanation for the gel shrinkage following a similar reasoning is dehydration of the polymer chains caused by the hydration shell of the cosolvent.13 Tanaka and co-workers14,15,22 provide a different point of view to the hydrogen bonding: their explanation is based on a competition of possible hydrogen bonds to the polymer chain: polymer/water, polymer/cosolvent, and polymer/polymer. The behavior of swelling transitions and cononsolvency is explained with what is termed “cooperative hydration”, meaning that a successful hydrogen bond of one type leads to preferential bonds of the same type in its vicinity. Coils of the chain with B


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hydrogels: in the case of network calculations, an expression for the elastic energy adds up to the sum of perturbation terms as new contribution for the Helmholtz energy of the system. The residual Helmholtz energy Aresidual, as difference of the real Helmholtz energy to the reference state of the ideal gas, then consists of four additive parts (eq 2). The hard chain contribution accounts for repulsive forces and the dispersion as well as association terms for attractive interactions. The newly introduced elastic-energy contribution refers to the network elasticity.

deformation of the network. The main difference of the theories lies in the treatment of network cross-link points: Whereas in the affine model they are fix embedded in the network, a phantom network features a certain fluctuation of cross-links without hindrance of entanglements and crossingover of chains. One example of their many modifications which have been discussed is the decrease of entropy that is associated with stretched chains.40 These models for elastic energy, however, are based on a common assumption of the polymers being capable of infinite stretching, which infers a limitless possible volume. This lacking in physical reason has been shown by Miao et al.41 to be plausibly corrected with a finiteextensibility augmentation, which has particular significance for, e.g., short chain polymers. The statistical association fluid theory (SAFT) developed by Chapman et al.,42 a generally accepted and powerful equation of state based on statistical thermodynamics, provides an excellent framework for a variety of extensions and modifications. Thus, a branched family of further developments has evolved in the recent more than two decades, ranging from changes in the molecular potential (e.g., soft-SAFT43,44) and accounting for variable attractive ranges (SAFT-VR45,46), over numerous methods for including further contributions to the Helmholtz energy such as for polar compounds47,48 or electrolytes,49,50 and to group-contribution methods for the parameter estimation (GC-SAFT51). The perturbed-chain SAFT (PC-SAFT) equation of state is prominent among the many advanced variations of the SAFT family. PC-SAFT, as developed by Groß and Sadowski,52,53 has widely been proven to be a state-of-the-art tool for thermodynamic modeling of complex mixtures. It has been shown to be well capable of precisely describing sophisticated phenomena, with emphasis on polymer systems due to its unique approach of particularly regarding chain-like molecular structures,50,54−58 whereas the focus of most other SAFT derivatives lies on rather small molecules. Thus, in this work, we use the framework of the PC-SAFT EOS in combination with the elasticity approach given by Miao et al. and apply the modifications matching the above-mentioned three constraints.

Equation 3 gives the expression of the association contribution accounting for the effect of hydrogen bonds, which was already part of the SAFT equations as given by Chapman et al.42 Conveniently the expression is set in relation to the total molecule number N, Boltzmann constant kB, and temperature T. xi represents the mole fraction of component i, and in the inner summation over all association site types A of the component i the expression XAi gives the fraction of notbonded association sites of the bonding type A at component i. XAi can be calculated using the implicit relation given in eq 4 with ρ being the number density and ΔAiBj an expression of the bonding strength between the association site of type B of component j and the association site type A of component i. Detailed derivation of the expression for ρ and ΔAiBj are given elsewhere.59 Computing 1 − XAi then allows access to the evaluation of the extent of hydrogen bonding of a substance. Aassoc = NkBT

Ai

j

⎝

X Ai 1⎞ + ⎟ 2 2⎠

(3)

Bj

(4)

On the basis of the physically feasible elasticity term of Miao et al.41 including a finite stretchability the elastic energy of the network, Aelastic is calculated in this work as contribution to the residual Helmholtz energy according to eq 5, conveniently in relation to the total molecule number N, Boltzmann constant kB, and temperature T.

MODELING APPROACH For modeling arbitrary mixtures with PC-SAFT, five universal and physically meaningful pure component parameters have to be fitted for every species. These are the segment number miseg and diameter σi to describe the hard chain model of the molecules, as well as dispersion-energy parameter ui/kB, association-energy parameter εAiBi/kB and association-volume parameter κAiBi for the description of attractive molecular interactions, conveniently dividing the energies by Boltzmann constant kB.52 In case of nonassociating compounds, the total number of pure component parameters even reduces to three. Taking these pure component parameters and applying combining rules, all possible binary interactions of molecules in complex mixtures are calculable. In some cases, however, a binary interaction parameter kij has to be introduced to adjust the binary dispersion energy between component i and j according to eq 1. Notably, i and j are commutative, i.e. kij = kji. uiui (1 − kij)

i

i

X A i = (1 + ρ ∑ xj ∑ X BjΔA iBj )−1

■

uij =

⎛

∑ xi ∑ ⎜ln(X A ) −

⎡ V V0 A Φ − 2 ⎢⎢ 3 = xpolymer × NkBT Φ ⎢2 1− ⎢⎣ elastic

2/3

( ) −1 ( ) V Vmax

2/3

⎤ ⎛ V ⎞⎥ − ln⎜ ⎟⎥ ⎝ V0 ⎠⎥ ⎥⎦ (5)

Next to the mole fraction of polymer chains xpolymer, the network functionality Φ as well as the current gel volume V, its reference state V0, and the volume of the maximally swollen gel Vmax influence the elastic energy. The extensive volume V is obtained directly from the molar volume of the mixture, which is calculated with PC-SAFT as described elsewhere,52 and the number of polymer chains nchain, see eq 8. Aiming at experimental accessibility, the reference volume V0 is the well-defined state of dry and pure polymer; with the assumption that the density of pure and not crosslinked chains is in par with the density of the pure polymer network. The ideal maximum volume Vmax at the stretching limit is calculated based on the volume of nchain polymer chains

(1)

Advantageously, the additive principle of perturbation contributions for the residual Helmholtz energy within PCSAFT conveniently can be used for an extension for modeling C


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Table 1. Physical Data and Parameters Determined in This Work Physical Data M Mcross‑linker

g mol−1 g mol−1

molecular weight molecular weight

113.15 154.16

General Data PNIPAAm

water

Mmonomer miseg σi

molecular weight segment number segment diameter

g mol−1 − Å

Mchain (see eq 9) Mchain/Mmonomer 5.38

ui/kB Niassoc εAiBi/kB κAiBi Φ ref

dispersion-energy parameter number of association sites association-energy parameter association-volume parameter network functionality

K − K − −

297.343 2/monomer 175 0.045 2.24 this work

18.015 1.204 66 2.7927 +10.11 × e(−0.01775T/K) − 1.417 × e(−0.01146T/K) 353.9449 2 2425.67 0.045 09 − 61

Mchain

MeOH 32.042 1.525 52 3.23

208.42 2 2253.9 0.024 68 − 55

233.397 2 2276.78 0.015 27 − 55

198.237 2 2653.39 0.032 38 − 55

188.905 2 2899.49 0.035 18 − 55

(10)

As stated before, the elastic forces in the gel phase counteracting the swelling cause a pressure difference to the ambient pressure pI. pII = pI − pelastic

(11)

The respective pressures can be derived from the expression of extensive Helmholtz energy A, taking one of the characteristic functions of thermodynamics into use. ⎛ ∂A ⎞ −⎜ ⎟ = p ⎝ ∂V ⎠T , n i

(12)

Since this coherence is valid for every single contribution of the residual Helmholtz energy in eq 2, pelastic is explicitly calculable. The overall pressure in a phasein analogy to the Helmholtz energyconsists of the ideal gas pressure pideal and the additive residual pressure presidual.

Now the number of chains in the network nchain as well as their molecular weight Mchain can be calculated depending on the network functionality Φ.

+ nlinker Mlinker m = monomer nchain

EtOH 46.069 2.382 67 3.177 06

xiIφi IpI = xiIIφi IIpII

Assuming all calculations and extensive quantities related to a mass of monomer mmonomer = 1 g and knowing its molecular weight M, the number of cross-linker molecules becomes m y nlinker = monomer M monomer 1 − y (7)

Φ nlinker 2

1-PrOH 60.096 2.999 71 3.252 21

loops or dangling chain ends obviously depend on the experimental conditions and procedure. To account for such insufficiencies as well as deviations from the ideal monodispersity, Φ was considered an adjustable parameter in this work. A similar approach is followed by Maurer and colleagues,8,60 in whose work the whole coefficient (Φ − 2)/ Φ merges in an adjustable network parameter. For thermodynamic equilibrium calculations of a gel phase (II) in a solvent phase (I), the isofugacity criterion f iI = f iII has to be fulfilled for every substance i distributed in both phases I and II. Here, the fugacity f can be written as the product of mole fraction x, fugacity coefficient φ and pressure p in the respective phase.

connected with cross-linking agents to form a tetrahedral network, which in ideal monodisperse phantom networks is a feasible assumption. To obtain the maximal length of these chains flanked by cross-linkers, the theoretical number of spherical segments in each chain is multiplied with their temperature-dependent segment diameter, which is obtained as described by Groß and Sadowski.52 The number of monomer units in between the cross-linker molecules is used to obtain the molecular weight of the polymer chains. The factual molecular weight of the whole network reaches arbitrary quantities, but we found the model appropriate considering the chains as free entities and account for their interconnections with the elastic terms only. Their chain lengths thus depend directly on the amount of crosslinker in relation to the amount of monomers, which is why within this work an experiment-compatible nomenclature has been taken to identify the molecular weight of the ideally monodisperse chains. As proposed elsewhere,5,16 a feasible quantity to consider is y, the molar amount of the cross-linker with respect to all feed molecules of the network: nlinker y= nlinker + nmonomer (6)

nchain =

2-PrOH 60.096 3.0929 3.2085

p = pideal + p residual (8)

(13)

Also, the fugacity coefficients φ for the respective components and phases can be derived from knowledge of the residual Helmholtz energy Aresidual given in eq 2 as described elsewhere.52 With the input variables of the solvent phase composition xiI, the ambient pressure pI as well as the temperature, the composition of the gel phase xiII with Z components can be obtained by iteratively solving the mechanical equilibrium of eq 11 and Z − 1 equations derived from eq 10. Trivially, for the

(9)

Comparison of experiments suggests5 that the swelling behavior of hydrogels is not independent of the production procedure, a conclusion which is strongly supported by considerations about inhomogeneities of the polymerized network and network defects. Occurrence of these such as D


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water with y = 0.01 underlie the parameter fitting, the other gel systems are predictions. The graph shows clearly that the experimental swelling transition temperature is well met for all three values of y. The model predictions confirm that the chain length has only very little impact on the transition temperature ( 0.5 a reswelling of the gel occurs until, in pure EtOH, a very high degree of swelling is reached again. It can be seen that the modeled curve follows this behavior

Figure 7. Modeled prediction of binary mass swelling of PNIPAAm hydrogels in alcohols depending on the temperature. Black, full line: methanol. Black, dotted line: ethanol. Gray, full line: 1-propanol. Gray, dotted line: 2-propanol. F


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qualitatively well, overestimates the swelling in pure ethanol however. The latter is not surprising, since as stated before, the binary mixtures PNIPAAm/alcohol are predicted and no parameter has been adjusted to these. The introduction of such a binary kij obviously could correct the overestimation and still allow for good modeling of the ternary mixture, but must be handled with care, as a comparison of references of experimental work suggests for the example of PNIPAAm/ ethanol at 25 °C: A cross-linker content y = 0.005 results in a swelling of m/m0 ≈ 20 (Zhi et al.25), shortening the chains with y = 0.01 results in m/m0 ≈ 30 (Althans et al.63), whereas other researchers measure values around 15 (Mukae et al.13), and for further shortening the chains with y = 0.015 degrees of swelling between 20 and 25 are reported (Hüther16). This comparison clearly uncovers the difficulty in accessing consistent quantitative results for the swelling in alcohols, when comparing different researcher groups and their synthesis procedures, which should, however, lead to comparable gel properties and similar measuring results. The PC-SAFT model allows not only the calculation of the degrees of swelling but, as eq 10 indicates, assesses the gel phase composition. Figure 9 shows such equilibrium results,

Figure 10. Modeled mass-swelling of PNIPAAm in aqueous alcohol solutions at 25 °C depending on the molar alcohol fraction in the solution. Black, full line: methanol. Black, dotted line: ethanol. Gray, full line: 1-propanol. Gray, dotted line: 2-propanol.

pure water, twice as much methanol is needed for reaching the completely shrunken state than in case of larger alcohols. In the reswelling section, the trend says that the higher the carbon number in the alcohol is, the earlier the re-entrancy starts. This modeling supports the observations of Mukae et al.13 who experimentally found that higher amounts of methanol than of ethanol, propanol, or tert-butanol are required for initiation of both the collapse of the gel and its reswelling. Only 1-propanol fails to follow the theoretical trend of our predictions, requiring more alcohol in phase I for reswelling. Having reliable experimental data, this could easily be readjusted by introducing a nonzero kij between polymer and alcohol. At 20 °C the similar observations can be made. Figure 11 as an example gives phenomena for swelling in aqueous methanol

Figure 9. Composition of the gel phase (II) in weight fraction depending on the ethanol weight fraction in the corresponding solvent phase (I) at 25 °C with y = 0.015. Circles (water), squares (PNIPAAm), and triangles (EtOH) represent experimental data from the literature;16 the lines give the respective modeled results of this work.

which are the weight fractions of water, PNIPAAm and ethanol in the gel corresponding to the input solution phase concentration. The PNIPAAm concentration runs reciprocally to the degree of swelling: the swollen state gel has very low polymer content, whereas in the collapsed gel a rather high PNIPAAm concentration is found. The course of the water concentration highlights that the loss of water is mainly responsible for the shrinking of the gel. As elucidated before, water is driven out as the gel increases its hydrophobicity. The reswelling at higher ethanol concentrations obviously is attributed to the significant increase of ethanol content in the gel, while the water concentration has only a minor influence and is evenly displaced by ethanol at wEtOHI > 0.7. The calculated shrinkage seems a little bit overestimated, but apart from that, these predictive equilibria with y = 0.015 compare very well with the experimental results from Hüther.16 A comparison of the ternary swelling of PNIPAAm in alcohol solutions shows that all four considered alcohols are comparable in their re-entrancy effect. Figure 10 gives an overview of the modeled results at 25 °C. A comparison of the shrunken regions shows tendencies of the swelling transitions depending on the carbon number of the alcohols. Starting with

Figure 11. Swelling of PNIPAAm in aqueous solutions of methanol and ethanol at 20 °C depending on the alcohol mole fraction in the solution. Spheres (methanol) and squares (ethanol) represent experimental literature data64 of the degree of swelling (DoS) of capillary microgels. The lines show the respective modeling results of PNIPAAm weight fraction in the gel phase (Solid: ethanol. Dotted: methanol).

and ethanol; for the sake of clarity, results for 1- and 2-propanol are not shown in the graph. Miki et al.64 have published experimental swelling results of microgels synthesized in capillaries, using an experimental setup and giving quantitative results which are hardly expressible with our model, but the basic gel response at different alcohol concentrations is highly comparable. As shown in the graph, both alcohols induce a dramatic collapse of the gel at rather low cosolvent concentrations, represented by the decrease of the experimental swelling data and reciprocally by the modeled increase of PNIPAAm concentration in the gel phase. The experimental data confirm our results of the polymer response distinctly shifted to lower alcohol concentrations in the case of ethanol G


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(wEtOHI = 1). This indicates that the interaction of PNIPAAm with water is preferred toward the interaction of PNIPAAm with the alcohol. (iii) The degree of association is considerably lower for PNIPAAm than for the solvent molecules, which have more than 90% of bonded association sites. As in the binary system water/PNIPAAm, sterical hindrances for the polymer chain can be supposed to be causing this behavior. The given example of the ethanol system is representative for all considered alcohols, for which similar graphs can be computed. For modeling the re-entrant behavior of the ternary systems water(1)/PNIPAAm(2)/alcohol(3) at 25 °C, in each case of the above-described systems an additional binary parameter k13 between water and the cosolvent appeared to be required, readjusting the geometric mean interaction obtained from the pure component parameters. Following eq 1, with the binary kij the interaction between two components i and j is corrected. In the present case, negative values for the k13 were required, leading to strengthened attractive forces between the respective alcohol and water molecules. In the case of all four alcohols a comparison of the binary dispersion forces between the two solvents (u13) and the competing forces between PNIPAAm species and the solvents (u12 and u23) showed that k13 had to be chosen to allow for at least (u13)2 ≈ (u12 × u23). If the water/ alcohol interaction is considerably greater than the polymer interactions with the solvents, a fully developed re-entrant phenomenon is guaranteed. Figure 13 demonstrates a parameter study for the ternary ethanol system. For the widest range of k13 no re-entrant behavior of the gel phase is calculated and the energy ratio (u13)2/(u12 × u23) is distinctly less than unity. With k13 = −0.16

compared to methanol. Analogous trends are observable both in the experiment64 and the model (omitted here) for 1- and 2propanol. The described observations support the image of the hydration of cosolvents being responsible for the re-entrant behavior: small amounts of cosolvent added to the binary mixture of PNIPAAm swollen in pure water allow the complete hydration of the cosolvent molecules in phase I. If more cosolvent is added, at some point more water molecules for the hydration shells are required than available in phase I and water molecules leave the gel phase II. Consequently, and facilitated by the PNIPAAm/water system being near its thermal instability point, water diffuses out of the gel which remains collapsed. Since larger alcohol molecules have a higher molecular water/cosolvent ratio for a complete hydration sheath, the collapsed gel state is reached earlier. Consequently, the reswelling can be explained in a similar way. Starting is phase I being literally saturated with cosolvent molecules, so there are approximately two or less shielding water molecules per cosolvent molecule (xalcoholI = 0.3−0.4). Additional cosolvent then begins diffusing into the polymer network, in which it is likely to dissolve completely. It is obvious, that smaller and more hydrophilic cosolvent molecules require less water for this saturated state and tolerate higher alcohol concentrations, before starting to dissolve into the polymer. The analysis of the extent of hydrogen bonds of the three components in the gel phase provides another view on this reentrancy behavior and discloses principles of the phenomenon. Figure 12 gives the fraction of occupied hydrogen bond sites of

Figure 12. Extent of hydrogen bonding in the gel phase of the ternary system water/PNIPAAm/ethanol at 25 °C. The solid black line gives the fraction of bonded association sites of water, the dotted black line that of ethanol, and the solid gray line that of PNIPAAm.

the components in the gel phase on the example of the ethanol system. Three key aspects are to be focused on: (i) During shrinking, a reduction of associative bonds of all components in the gel phase is calculated. Particularly significant is the reduction of PNIPAAm association, since most of the solvent molecules leave the gel phase. Evidently, the water molecules in the gel have a stronger affinity to the alcohol in the liquid phase than to the polymer chains and thus diffuse out of the gel. In the same way, ethanol prefers the interaction with water rather than with PNIPAAm and starts diffusing into the gel phase in a significant amount only when the solvent phase ethanol molecules are in a saturated water environment. (ii) The amount of “active” PNIPAAm association sites is higher in pure water (wEtOHI = 0) than in pure ethanol

Figure 13. Parameter study of the swelling system water/PNIPAAm/ ethanol at 25 °C. The polymer weight fraction in the gel phase is calculated depending both on the ethanol weight fraction in the solution and on the binary interaction parameter between water and ethanol. The bold line gives the model results for k13 = −0.185 (see also Figure 9). H


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the ratio crosses unity and the first slight gel shrinkage can be calculated. For k13 = −0.185, which is the finally adjusted value as given in Table 2, the gel collapse phenomenon is fully developed (w2,maxII > 0.8) and the energy ratio becomes 1.044, so the water/ethanol interactions clearly dominate over the averaged polymer/water and polymer/ethanol interactions. This supports the suggested theory that during shrinking the water molecules in the gel are facilitated to leave the gel phase for the sake of interacting with the ethanol molecules in the solvent phase. The attempt of introducing binary association interaction parameters (instead of the dispersion energy correcting kij) for adjusting the attractive forces between the solvents failed. Keeping in mind that the degree of active association sites of water as well as of the alcohols are already near 100%, eq 3 makes evident that the association contribution is not the adjusting knob of choice to strengthen the binary attractive interaction. Analogous parameter studies were performed for the other alcohol systems at 25 °C, showing highly similar results: for strong gel shrinkage as presented above, the water/alcohol interaction has to be strengthened with aid of the k13 parameter and exceed the interaction between PNIPAAm and the solvents. Lacking adequate experimental data, the k13 were fitted to fulfill w2,maxII = 0.8 at 25 °C, guaranteeing a significant and fully developed shrinkage. Table 2 comprises the respective values, giving an overview about the constant binary parameters between water and the alcohols as well as the previously determined interaction adjustment between water and PNIPAAm. The magnitude of the kij values indicates that more than only the correction of the dispersive interaction is taken into account, and also the correction of the associative interaction merges into this binary parameter. Notably, all other binary dispersive interactions within PC-SAFT are calculated merely using the respective pure component parameters and thus have predictive character. Figure 14 gives proof of the reasonability of the determined parameter set with a predicted ternary LLE with water, not

not exhibit a phase split, but the direct proximity of the closed miscibility gap to this binary axis gives proof of the cononsolvency effect as soon as PNIPAAm enters the system. At 35 °C, the binary water/PNIPAAm LCST is passed and a binary miscibility gap in the respective axis is formed, matching the results in Figure 6. As expected, the water/methanol axis shows no phase separation. With the model and the PNIPAAm parameters trained only to the polymer density in water and the swelling in pure water as a function of the temperature, and with the k13 interactions fitted to the swelling at constant temperature of 25 °C but depending on a cosolvent concentration, a three-dimensional swelling surface can be predicted. Figure 15 shows this modeled

Figure 15. Modeling results of PNIPAAm gel mass swelling showing the influence of both a wide temperature range and the ethanol weight fraction in the corresponding solvent phase. The bold lines outline the swelling in pure water (see also Figure 4) and pure ethanol (see also Figure 7) depending on the temperature as well as the swelling at 25 °C depending on the solution phase composition (see also Figure 8).

hypersurface as an example for the water/PNIPAAm/ethanol system, giving the degree of swelling in equilibrium to an arbitrary solvent phase composition and over a wide temperature range. The coordinates for the gel being in the collapsed state are precisely distinguishable from the swollen gel states. Notably, the re-entrant behavior of the gel being swollen in both pure solvents and having a strong collapse effect in the mixture can be observed only in the temperature range between 285 and 308 K. At lower temperatures, the gel does react to changes in the solvent phase composition, but keeps certain degrees of swelling and does not collapse completely. At higher temperatures, the swollen state is only found at increasingly high ethanol concentrations in phase I, whereas at high water concentrations the gel is shrunken. By knowledge of this hypersurface, one can see instantly which changes in terms of solvent concentration and temperature lead to transitions in the swelling state.

Figure 14. Ternary LLE of the water/PNIPAAm/methanol system in weight fraction. Circles give experimental data65 at 25 °C. Lines represent the predicted model results: the outer dotted black line at 35 °C, the gray line at 25 °C, and the inner black line at 15 °C including some tie lines.

cross-linked PNIPAAm and methanol. Three temperatures are calculated, viz. 15, 25, and 35 °C. At 15 °C, clearly below the LCST of the binary water/PNIPAAm system, tie lines are given showing a reasonable phase split in a polymer-rich and a polymer-depleted phase. At 25 °C our prediction compares qualitatively well with experimental data from Tao and Young,65 who measured only at very low polymer concentrations and reported major experimental restrictions due to large viscosity. Notably, the binary system water/methanol does

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CONCLUSION The PC-SAFT equation of state was applied to modeling PNIPAAm as both non cross-linked polymer and cross-linked hydrogel. With a new pure component parameter set for the I


Macromolecules

Article

wi(I) XAi xi(I) y

polymer thermodynamic properties such as densities and the LLE miscibility gap could be described in good agreement with experimental data. For considering the elastic forces within the gel polymer, a contribution to the Helmholtz energy explicitly considering the finite stretchability was implemented and the swelling behavior as well as the gel phase composition could be modeled successfully. It was shown that in a wide temperature range experimental data of PNIPAAm in pure water can be met and swelling in alcohols (methanol, ethanol, 1-propanol, and 2propanol) can reasonably be predicted. The re-entrancy effect of PNIPAAm in aqueous alcohol systems with respect to the gel swelling in a mixed solvent was described satisfactorily and the causation of this phenomenon was identified as relative attractive interactions between the participating species. Considering the swelling results in aqueous alcohol solutions, clear indication was found for the hydration sheath being responsible for the re-entrant behavior and explaining the differences in the gel response to the various alcohols. Additionally, the theoretical analysis of the extent of association gave evidence of the principles of the phenomenon and proved that hydrogen bonds have a strong influence on the swelling behavior. Parameter studies showed that particularly interactions between the two solvents (water and alcohol) have crucial influence on the sudden collapse of the gel near its thermodynamic instability conditions. The model clearly emphasizes that the binary attraction between the solvents has to outbalance the combined attractions between polymer and the solvents. On this basis, a predicted hypersurface of the degree of swelling as a function of temperature and corresponding solvent phase composition could be modeled. To our knowledge, for the first time a general equation of state such as the PC-SAFT was used successfully for this kind of hydrogel modeling.

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Greek Symbols

εAiBj κAiBj ρ σi φi(I) Φ

0 Ai (Bj) AiBj I II i j max

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reference association site type A (B) of component i (j) binary association between Bj and Ai phase I (solvent) phase II (gel) component i component j maximal

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AUTHOR INFORMATION

*Telephone: +49 231 7552635. Fax: +49 231 7552572. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the priority programme SPP 1259 “Intelligent Hydrogels” by the Deutsche Forschungsgemeinschaft (DFG). LIST OF SYMBOLS

Latin Symbols

A f i(I) kij kB M m miseg Niassoc N p T ui V

association energy association volume density segment diameter fugacity coefficient of species i (in phase I) network functionality

Indices

Corresponding Author

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weight fraction of component i (in phase I) fraction of not-bonded association sites Ai mole fraction of component i (in phase I) molar content of cross-linker

Helmholtz energy fugacity of component i (in phase I) binary interaction parameter Boltzmann constant molar mass mass segment diameter number of association sites molar amount pressure temperature dispersion energy volume J


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K


Alcohol

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