Thermodynamics of phase equilibrium for systems containing N-isopropyl acrylamide hydrogels

Fluid Phase Equilibria 296 (2010) 140–148

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Thermodynamics of phase equilibrium for systems containing N-isopropyl acrylamide hydrogels Viktor Ermatchkov, Luciana Ninni, Gerd Maurer ∗ Department of Mechanical and Process Engineering, University of Kaiserslautern, P.O. Box 30 49, D-67653 Kaiserslautern, Germany

a r t i c l e

i n f o

Article history: Received 3 December 2009 Received in revised form 15 February 2010 Accepted 8 March 2010 Available online 15 March 2010 Keywords: Synthetic non-ionic and ionic hydrogels Swelling behavior N-isopropyl acrylamide N-IPAAm hydrogels Anionic Cationic and zwitterionic comonomers AMPS DMAPMA MPSA Thermodynamic model

a b s t r a c t Hydrogels are crosslinked polymers of hydrophilic monomers. Hydrogels can swell and shrink in aqueous solutions. The swelling behavior of hydrogels and the encountered phase behavior are of interest in many areas, e.g., in biotechnology, membrane science and controlled drug release. This contribution presents the criteria for such phase equilibria and a previously developed thermodynamic model for correlating/predicting the swelling and shrinking of hydrogels. The application of the method is demonstrated by describing the swelling equilibrium of some synthetic, nonionic N-isopropyl acrylamide (N-IPAAm) hydrogels in aqueous solutions of sodium chloride at 298 K. Furthermore, new experimental results are presented for the degree of swelling of synthetic hydrogels that contain – besides the non-ionic monomer N-IPAAm – either a combination of a cationic comonomer (here, N-[3-(dimethylamino)propyl]methacrylamide (DMAPMA)) and an anionic comonomer (here, 2-acrylamido-2-methyl-1-propanesulfonic acid (AMPS)) or a zwitterionic comonomer (here, [3-(methacryloylamino)propyl]dimethyl(3-sulfopropyl)ammonium hydroxide inner salt (MPSA)). These gels were equilibrated with aqueous solutions of sodium chloride at 298 K. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Hydrogels are elastic networks composed of polymers with hydrophilic segments. There are several methods to classify hydrogels. For example, one can distinguish between natural and synthetic hydrogels or between covalently crosslinked and physically crosslinked (entangled) hydrogels or between homopolymeric, copolymeric or interpenetrating networks [1]. Crosslinking the polymer chains results in some elasticity, i.e., hydrogels behave like liquids as well as like solids. For example, hydrogels can absorb (desorb) solvent and solute components from a coexisting liquid phase. The encountered dilution reduces the Gibbs energy of the hydrogel phase. The simultaneously occurring increase of the stress of the elastic network increases the Helmholtz energy of the network. In thermodynamic equilibrium an infinitesimal increase (decrease) of the Gibbs energy is just compensated by infinitesimal decrease (increase) of the Helmholtz energy of the network. Absorption/desorption of components is typically accompanied by a large volume change (swelling/shrinking). Furthermore, solutes can partition between the liquid phase and the

∗ Corresponding author. Tel.: +49 631 205 2410; fax: +49 631 205 3835. E-mail addresses: [email protected], [email protected] (G. Maurer). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.03.014

coexisting hydrogel phase and external pressure can be used to squeeze solvents and solutes out of the hydrogel phase. Natural hydrogels are widespread (e.g., plants are hydrogels). But the particular properties of hydrogels make them also suitable for many applications. Applications of hydrogels are based on their physico-chemical properties, among which the swelling behavior is one of the most important. For example, hydrogels are used as biocompatible materials, as for example, in soft contact lenses [2], superabsorbers in diapers, in drug delivery systems [3,4], tissue engineering [5] and gel permeation chromatography. Hydrogels respond to externally applied triggers, such as, for example, temperature, composition of the coexisting liquid phase, solvent polarity, electric/magnetic field or light [6–10]. The properties of hydrogels and consequently their responses to such external triggers can be influenced by the hydrogel’s design, i.e., the chemical nature of the components of the polymer chains and the crosslinking density. The development of thermodynamic methods to correlate (and eventually even predict) the responses of hydrogels to external stimulations needs not only a model for the encountered phenomena, but also a large number of experimental data on the swelling behavior of well-characterized hydrogels in various surroundings. Some methods to describe the swelling equilibrium of hydrogels are available in the literature. A short review on models that were available in 1990 was given by Vasheghani-Farahani et al. [11].

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Examples for more recently published models include the models of Marchetti et al. [12], Backer et al. [13], Hino and Prausnitz [14], De et al. [15], C¸aykara et al. [16], Wu et al. [17], Victorov et al. [18], Huang et al. [19] and Wallmersperger et al. [20]. In the present investigation we follow a method described by Maurer and Prausnitz [21]. That method combines an expression for the Gibbs energy of aqueous solutions of (non-crosslinked) polymer chains with an expression for the Helmholtz energy of an elastic network, that accounts for the phenomenon that is caused by crosslinking the polymer chains. The Gibbs energy of an aqueous solution of polymers and salts is described using the VERS-model that has been shown to be able to reliably describe the properties of aqueous solutions of neutral polymers and the liquid–liquid equilibrium of aqueous two-phase systems that might be observed when a hydrophilic polymer and a salt are simultaneously dissolved in water [22–28]. That model was recently extended to describe the Gibbs energy of aqueous polyelectrolyte solutions [29]. In previous works [8,30–35], the phase equilibrium and swelling properties of some synthetic hydrogels based on N-isopropyl acrylamide (NIPAAm) in aqueous solutions of strong electrolytes, polymers and organic solvents were determined and quantitatively described with the semiempirical thermodynamic model that is also used in the present work. Experimental data are required to verify as well as to parameterize a particular model. Publications on swelling properties of hydrogels are mostly restricted to qualitative descriptions for poorly characterized hydrogels, whereas model development requires quantitative experimental results for well-characterized hydrogels. The present investigation starts with new experimental results for the swelling behavior of some ionic hydrogels and continues with a previously developed thermodynamic model to describe the swelling equilibrium. The main constituent of the synthesized and investigated hydrogels is the neutral monomer N-IPAAm. Ionic comonomers are used to produce ionic hydrogels. In previous investigations an anionic comonomer (sodium methacrylate) was used to study the influence of a single ionic comonomer on the degree of swelling of synthetic N-IPAAm hydrogels [8,31]. In the present work, a combination of an anionic comonomer (2acrylamido-2-methyl-1-propanesulfonic acid (AMPS)), and a cationic comonomer (N-[3-(dimethylamino)propyl]methacrylamide (DMAPMA)) as well as a zwitterionic comonomer [3(methacryloylamino)propyl]dimethyl(3-sulfopropyl)ammonium hydroxide inner salt (MPSA)) are introduced into the network.

The combination of cationic and anionic comonomers was used in order to study the influence of the charge density parameter ϑ of the hydrogel’s polymeric chains on the swelling behavior. The charge density parameter ϑ is here defined as:

2. Experimental

(b) Mole fraction of comonomer i:

2.1. Synthesis and characterization of hydrogels The quantitative swelling behavior of hydrogels containing N-IPAAm and ionic comonomers was experimentally determined in aqueous solutions of sodium chloride at 298 K. A detailed description of the method to produce the hydrogels is available elsewhere [34,35]. That procedure is only briefly described here. All hydrogels were synthesized in oxygen-free, deionized water at 298 K under nitrogen atmosphere by free radical polymerisation of N-IPAAm and the respective ionic comonomers using N,N methylenebisacrylamide (MBA) as crosslinker. The polymerization and crosslinking reactions were initiated by ammonium peroxodisulfate (as starter) and sodium disulfite (as accelerator) both at a concentration of approximately 2 × 10−4 g g−1 in the polymerizing aqueous solution. Two types of ionic hydrogels were synthesized: hydrogels that also contained a combination of the anionic comonomer AMPS and the cationic comonomer DMAPMA, and hydrogels with the zwitterionic comonomer MPSA.

ϑ=

y+ − y− y+ + y−

(1)

where y+ and y− are the mole fractions of the cationic and anionic monomers, respectively, in the mixture of polymerizable monomers (cf. below). In a typical synthesis, a reactor (250 ml volume) was charged step by step with about 120 g of an aqueous solutions of N-IPAAm, with about 50 g of an aqueous solution of MBA, with about 30 g of an aqueous solution of either (AMPS + DMAPMA) or MPSA, with about 15 g of an aqueous solution of the starter ((NH4 )2 S2 O8 ) and finally with about 15 g of an aqueous solution of the accelerator (Na2 S2 O5 ). All single solutions were previously stripped with water-saturated nitrogen to remove all oxygen. Nitrogen stripping was maintained during the mixing period in the reactor. After the last addition, the solution was mixed for about 2 min before the already polymerizing solution was filled into 40 ml Teflon tubes that were closed immediately. The tubes were thermostated in a water bath at 298 K for 24 h to complete the polymerization/crosslinking process. Afterwards, the hydrogel blocks were removed from the tubes. Each block was cut into slices of approximately 5 mm thickness. All upper and lower slices of each block as well as the outer 3 mm of each slice were discarded. The remaining cores were cut into cubes of about 5 mm length. These hydrogel cubes were washed with deionized water for about 2 weeks and afterwards dried (at first at ambient conditions and finally under vacuum at 323 K). The average mass of a dried particle was about 20 mg. We characterize a hydrogel by its production process together with the following properties that are all related to the composition of the polymerization solution: (a) Total mass fraction of polymerizable material: gel =

˜ MBA + ˜ N-IPAAm + m m



˜ coi m

(2)

˜ coi + m ˜ Na2 S2 O5 + m ˜ (NH4 )2 S2 O8 + m ˜w m

(3)

i

˜ feed m

where, ˜ N-IPAAm + m ˜ MBA ˜ feed = m m



+

i

ycoi =

ncoi nN-IPAAm +



(4)

n j coj

(c) Mole fraction of crosslinker: yMBA =

nMBA nN-IPAAm + nMBA +



n i coi

(5)

(d) Mass fraction of the starter (=mass fraction of the accelerator): s =

˜s m ˜ feed m

(6)

Table 1 gives a summary on all synthesized hydrogels that contain AMPS as well as DMAPMA. All such gels are characterized by (y+ + y− ) = 0.05 mol mol−1 . But the mole fractions of the single electrolyte comonomers were varied between 0 and 0.05. Table 2 gives a similar summary on all hydrogels with the zwitterionic comonomer MPSA. The total mass fraction of polymerizable material was about 0.1. The mole fraction of the crosslinker MBA was about 0.02–0.05 and the mass fraction of the starter was about 0.0002.

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Table 1 Characteristic parameters of the investigated N-IPAAm-AMPS-DMAPMA hydrogels.  gel (g g−1 )

yMBA (mol mol−1 )

yAMPS (mol mol−1 )

yDMAPMA (mol mol−1 )

 s 10−4 (g g−1 )

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.02 0.02 0.02 0.02 0.02

0.040 0.016 0.035 0.025

0.010 0.035 0.015 0.025 0.050

1.88 2.02 2.04 2.08 2.02 2.09 2.25 2.01 2.12 2.07 2.10 1.98 2.05

0.050 0.020 0.010

0.032 0.042 0.050 0.040 0.035 0.030 0.020

0.010 0.015 0.021 0.032

2.2. Materials The main constituents of the polymer network are the neutral monomer N-isopropyl acrylamide (N-IPAAm) (97%, CAS 2210-25-5), the ionic comonomers 2-acrylamido-2methyl-1-propanesulfonic acid (AMPS) (99%, CAS 15214-89-8), N-[3-(dimethylamino)propyl]methacrylamide (DMAPMA) (99%, CAS 5205-93-6), [3-(methacryloylamino)propyl]dimethyl(3sulfopropyl)ammonium hydroxide inner salt (MPSA) (96%, CAS 5205-95-8) as well as the crosslinking component (N,N methylenebisacrylamide (MBA) (≥99%, CAS 110-26-9). The chemical structures of these monomers are shown in Fig. 1.

Table 2 Characteristic parameters of the investigated N-IPAAm-MPSA hydrogels.  gel (g g−1 )

yMBA (mol mol−1 )

yMPSA (mol mol−1 )

 s 10−4 (g g−1 )

0.1 0.09 0.1 0.1 0.1 0.1 0.12 0.1 0.1 0.1 0.1

0.02 0.02 0.02 0.02 0.02 0.04 0.02 0.02 0.02 0.02 0.05

0.05 0.05 0.05 0.05 0.1 0.05 0.05 0.05 0.20 0.40 0.40

1.95 1.84 2.22 2.01 1.99 2.07 2.06 3.00 2.01 1.99 2.06

The polymerization was initiated using ammonium peroxodisulfate ((NH4 )2 S2 O8 ) (≥98%, CAS 7727-54-0) and sodium disulfite (Na2 S2 O5 ) (≥98%, CAS 7681-57-4). Sodium chloride (NaCl, ≥99%) was used to prepare the aqueous salt solutions for the swelling experiments. All these chemicals were purchased from Sigma–Aldrich, Taufkirchen, Germany and used as delivered, i.e., without any further purification. Oxygen-free, deionized water was used in all experimental work (for synthesis as well as for swelling experiments). 2.3. Swelling experiments In a typical swelling experiment a dried hydrogel particle was equilibrated with about 100 ml of an aqueous solution of known composition – either pure water or a (10−4 to 5) molal aqueous solution of NaCl – at (298.15 ± 0.1) K for about 2 weeks. Then the gel particle was taken out of the solution and the surface water was removed using fluff-free paper towels and its mass was determined with a microbalance (type MX5, Mettler Toledo GmbH, Giessen, Germany) and the degree of swelling was calculated for each particle. Each experiment was repeated at least 10 times with hydrogel particles from the same production charge. The degree of swelling ˜ gel ) to that q is the ratio of the amount of mass of the swollen gel (m (dry)

˜ gel ): of the dry gel (m q=

˜ gel m (dry)

˜ gel m

(7)

The arithmetic average of the degree of swelling was determined and the standard deviation was calculated. In a typical experiment that standard deviation was about 2%. This number is an indicator for the homogeneity of a hydrogel. However, that number might also depend on some loss of the hydrogel particle during its handling. Therefore, some of the particles that were swollen in pure water were allowed to dry – as described above for the production process – and the mass of the dried particles was determined again. The loss of mass was typically below 1%. As the amount of solution (around 100 g) that coexists with the hydrogels is always much larger than the mass which was absorbed by the hydrogel particle (less than about 0.8 g), it is assumed that the composition of the salt-containing liquid phase remains practically unchanged by the swelling of the hydrogel particles. 3. Experimental results 3.1. N-IPAAm-AMPS-DMAPMA gels in water and in aqueous solutions of NaCl

Fig. 1. Monomers for hydrogel synthesis.

Fig. 2 shows the experimental results for the degree of swelling of N-IPAAm-AMPS-DMAPMA hydrogels in pure water at 298 K as a

V. Ermatchkov et al. / Fluid Phase Equilibria 296 (2010) 140–148

Fig. 2. Influence of the charge density parameter ϑ on the degree of swelling q of some N-IPAAm-AMPS-DMAPMA hydrogels in water at 298 K (characteristic hydrogel parameters:  gel = 0.1 g g−1 ;  s = 2 × 10−4 g g−1 ; yAMPS + yDMAPMA = 0.05 mol mol−1 ; () yMBA = 0.02 mol mol−1 and () yMBA = 0.05 mol mol−1 .

Fig. 3. Influence of the molality of NaCl on the degree of swelling q of some N-IPAAm-AMPS-DMAPMA hydrogels in water at 298 K (characteristic hydrogel parameters:  gel = 0.1 g g−1 ;  s = 2 × 10−4 g g−1 ; yMBA = 0.05 mol mol−1 ; () non-ionic gel; () yAMPS = 0.01 and yDMAPMA = 0.04 mol mol−1 ; () yAMPS = 0.05).

function of the charge density parameter ϑ. The degree of swelling is smallest when the hydrogel carries no net charge (i.e., for ϑ = 0) and increases by a factor of up to three when there is only a single ionic comonomer (either only anionic or only cationic monomers, i.e., for ϑ = ±1). Furthermore, the degree of swelling increases with decreasing concentration of the crosslinker in the polymeric network. Fig. 3 shows the influence of sodium chloride in the aqueous solution on the degree of swelling of two ionic N-IPAAm-AMPSDMAPMA hydrogels and one non-ionic N-IPAAm hydrogel at 298 K. All hydrogels are characterized by the same numbers for the total mass fraction of polymerizable material ( gel = 0.1 g g−1 ), the mass fraction of the starter ( s = 2 × 10−4 g g−1 ) and the mole fraction of crosslinker (yMBA = 0.05 mol mol−1 ). The two ionic hydrogels differ only in the charge density parameter ϑ. One hydrogel contains no cationic comonomers (i.e., ϑ = −1), whereas both types of ionic comonomers are present in the other and there is a surplus of cationic comonomers (ϑ = 3/5). As long as the salt molality is below about 0.003 mol (kg water)−1 all three hydrogels are in a swollen state and the degree of swelling is largest for the gel with the highest absolute number for the charge density parameter and lowest for the non-ionic gel. The non-ionic hydrogel remains in a swollen state up to salt molalities of about 0.3 molal where it reveals a

143

Fig. 4. Influence of the molality of NaCl on the degree of swelling q of some N-IPAAm-MPSA hydrogels in water at 298 K (characteristic hydrogel parameters:  gel = 0.1 g g−1 ;  s = 2 × 10−4 g g−1 ; yMBA = 0.02 mol mol−1 ; () non-ionic gel; () yMPSA = 0.05 mol mol−1 ; (♦) yMPSA = 0.20 mol mol−1 and () yMPSA = 0.40 mol mol−1 ).

phase transition into a shrunken state. That transition is completed when the salt molality reaches about 0.9 molal. One explanation for this transition is the decrease of “free water”, i.e., water that is available for the hydrophilic groups of the network. That decrease might be caused by chemical interactions (hydration of the ions) or by strong physical interactions between the ions and water. The hydrogel with the highest charge density shows two transitions. The first transition starts when the salt molality is increased beyond about 0.003 molal. That transition occurs over a rather broad range of salt molality and reduces the degree of swelling to that of the non-ionic hydrogel. We assume that this first transition is caused by the screening of the ionic groups of the hydrogel by free ions in the aqueous phase. That screening results in a behavior that is similar to that of a non-ionic hydrogel. The second transition is very similar to the transition of the non-ionic hydrogel, but it occurs at somewhat higher salt molalities. The hydrogel that carries both negative and positive charges reveals a somewhat less pronounced first transition and behaves at high salt molalities just like the anionic hydrogel. 3.2. N-IPAAm-MPSA gels in water and in NaCl solutions The experimental results of the degree of swelling of N-IPAAmMPSA hydrogels in water at 298 K are given in Table 3 together with their corresponding standard deviations. All hydrogels have almost the same characteristic gel parameters ( gel = 0.1 g g−1 ;  s = 2 × 10−4 g g−1 ; yMBA = 0.02 mol mol−1 ; yMPSA = 0.05 mol mol−1 ) but are from different production charges. The experimental data for the degree of swelling of hydrogels from different production charges differ by less than 10 percent. Some of that difference is certainly caused by the handling of the gels. Nevertheless the data shown in Table 3 indicate a good reproducibility of the method of hydrogel synthesis. The experimental results for the degree of swelling of three NIPAAm-MPSA hydrogels at 298 K in aqueous solutions of sodium chloride are shown in Fig. 4. For comparison Fig. 4 also shows the degree of swelling of a non-ionic N-IPAAm hydrogel. The hydrogels differ only in the mole fraction of the ionic comonomer (that varies from zero to 40 mole percent), but all other characteristic parameters are the same ( gel = 0.1 g g−1 ;  s = 2 × 10−4 g g−1 and

Table 3 Experimental results for the degree of swelling q of N-IPAAm-MPSA hydrogels (from different production charges) in water at 298 K.  gel (g g−1 )

yMBA (mol mol−1 )

yMPSA (mol mol−1 )

 s 10−4 (g g−1 )

q (g g−1 )

0.1 0.1 0.1 0.1

0.02 0.02 0.02 0.02

0.05 0.05 0.05 0.05

1.95 2.22 2.01 3.00

18.45 16.37 17.94 18.39

± ± ± ±

0.38 0.15 0.12 0.22

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yMBA = 0.02 mol mol−1 ). With increasing mole fraction of MPSA the degree of swelling also increases. At salt molalities up to about 0.2 molal, all gels are in a swollen state and the salt molality has practically no influence on the degree of swelling. However, there is a distinguished difference at higher salt molalities. With increasing comonomer concentration the transition from a swollen to a shrunken state is shifted to higher salt molalities. Furthermore, the transition’s range broadens and at very large comonomer mole fractions the hydrogel even swells more before it starts shrinking. This phenomenon is called “antipolyelectrolyte swelling behavior”. It is probably due to the screening of electrostatic attractions between fixed charges of opposite signs on the zwitterionic comonomers. 4. Theory

When a liquid phase I and a hydrogel phase II coexist in thermodynamic equilibrium at constant temperature (T = TI = TII ) and constant external pressure (pI = p), the sum of the Gibbs energies of the liquid phase (I) and the gel phase (II) have to attain a minimum: GI + GII = min

(8)

The Gibbs energy of the gel phase is assumed to be the sum of two contributions. GII = G(T, p, nIIj ) + A(T, V II )

(9)

The first term on the right hand side of Eq. (9) describes the Gibbs energy of a liquid mixture where the polymer network is cut off at its crosslinking positions, i.e., it is the Gibbs energy of a liquid phase that has the same stoichiometric composition as the hydrogel phase, but the network is replaced by its non-crosslinked polymer chains. The second term describes the contributions from crosslinking these chains. It is approximated by the Helmholtz energy A of an elastic network. In deriving the conditions for phase equilibrium between the hydrogel phase and the coexisting liquid phase one has to distinguish between species that can partition between both phases and species that are only present in one of the coexisting phases. Water is an example for a component that can equilibrate between both phase, the non-crosslinked network chains are an example for a component that is only present in the hydrogel phase. The condition of phase equilibrium for a non-ionic species i that is able to partition between both phases is (cf. Maurer and Prausnitz [21]):



∂A ∂V II



(10)



∂G ∂ni

= ref + RT ln ai (T, p, nj ) i

ln

= ln

aIIi (T, p, nIIj ) +

∂A ∂V II

 + T

(11)

vIIi

RT



∂A ∂V II



(12) T

zi F  RT

(13)



∂A ∂V II

 (14) T

4.2. Gibbs energy of a liquid mixture The Gibbs energy of a liquid phase (the surrounding liquid as well as the “virtual” liquid hydrogel phase) is expressed using the molality scale and the molality based reference chemical potentials for each solute species: G = Gid.mix + GE

(15)

where Gid.mix = nw [w (T )pure liq. + RT ln aw,id.mix ] +

Eq. (10) can be expressed in terms of the activity ai of component i resulting in aIi (T, p, nIj )

RT



where subscript q denotes the arbitrarily selected ionic species. The calculation of the swelling equilibrium (including the calculation of the degree of swelling) requires expressions for the Gibbs energy of the aqueous solution that coexists with the hydrogel and of the “virtual” hydrogel phase (i.e., a liquid phase that represents the hydrogel but where the network has been replaced by its non-crosslinked polymer chains) as well as an expression for the Helmholtz energy of an elastic network. The following sections give a summary for the simple case where the network consists only of non-ionic monomers and the hydrogel phase is equilibrated with an aqueous solution of NaCl. Then, the “virtual” hydrogel phase consists of the solvent (water) and the solutes s (sodium ions, chloride ions and polymer chains, i.e., noncrosslinked chains of N-IPAAm monomers), while the coexisting liquid is an aqueous solution of sodium chloride.





ns ref s (T ) + RT ln

all solutes s

ms mo

 (16)

and GE is the excess Gibbs energy. The ideal mixture is defined on the molality scale. Therefore, the activity of water in an ideal mixture (aw,id. mix ) is: ln aw,id.mix. = −

Mw 1000

 all solutes s



T,p,nj = / i

vIIi

where zi , F and  are the charge number of species i, Faraday’s constant and the electrical potential difference between the gel and the liquid phase, respectively. When there are q ionic species that partition between both phases, Eq. (13) only holds for (q − 1) components and the chemical potentials of the arbitrarily chosen ionic species q and the Helmholtz energy of the network are used to calculate the electrical potential difference:

T

where Ii and IIi are the chemical potentials of component i in phases I and II, respectively, vIIi is the partial molar volume of species i in the gel phase and VII is the volume of the gel phase. It is particularly worth to mention that in Eq. (10) there is no difference in pressure p between the coexisting phases, i.e., the method used here is not based on the “osmotic pressure concept” that has been used in other methods. If the same reference states are chosen for the chemical potential of component i in both phases, i (T, p, nj ) =

ln aIi (T, p, nIj ) = ln aIIi (T, p, nIIj ) +

vIIq zq F  = ln aIq (T, p, nIj ) − ln aIIq (T, p, nIIj ) − RT RT

4.1. Phase equilibrium conditions

Ii (T, p, nIj ) = IIi (T, p, nIIj ) + vIIi

where R is the universal gas constant. For an ionic species, the condition of phase equilibrium has to be modified as in each phase the condition of electroneutrality has to be fulfilled as a side condition:

ms mo

(17)

Mw is the relative molar mass of water, ms is the molality of solute s, mo = 1 mol (kg water)−1 and superscript ref indicates the reference state of a solute (that is a “virtual” one molal solution of that solute in pure water). The activity of a solute species as is the product of its molality ms and its activity coefficient s (on molality scale): as = ms s

(18)

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The activity coefficient of a solute species s is calculated from the excess Gibbs energy:



RT ln s =

∂GE ∂ns



(19) T,p,nj = / s

The activity of the solvent (water) is calculated from the expression for the excess Gibbs energy and the activities of the solutes: ln aw

Mw = 1000



GE − RT

 all solutes s

ms ln s − mo

 all solutes s

ms mo



(20)

For the liquid phase the excess Gibbs energy is approximated by Pitzer’s equation for aqueous electrolyte solutions [36]. For the “virtual” liquid hydrogel phase the excess Gibbs energy is approximated by contributions from the VERS-model (Virial Equation with Relative Surface fractions) [22,24] and a “freevolume” contribution: GE = GE,VERS + GE,FV

(21)

Details are given in Appendix A. 4.3. Helmholtz energy of the elastic network A modification of the phantom network theory (James [37]; James and Guth [38,39]) is used to express the Helmholtz energy of the elastic network. The deformation from a stress-free state is expressed by the ratio of the actual volume V to the volume in the stress-free state V0 : A =C RT

 2/3 V V0

−1

(22)

where C is an empirical adjustable network parameter that characterizes the elastic properties of the network. The volume V is calculated from the molar volumes of the pure components vi assuming an ideal mixing behavior: V = nMBA vMBA + npoly vpoly + nw vw +



ni vi

(23)

i

where nMBA , npoly , nw and ni are the number of moles of crosslinker, polymer chains, water and further solute species i in the “virtual” hydrogel phase and vi is the molar volume of component i. The number of moles of polymer chains is calculated assuming a perfect tetrafunctional network: npoly =

1 nMBA 2

(24)

and the molar volume of the polymer chain vpoly is calculated from the molar volumes of monomers and the number of chains. For a non-ionic hydrogel that consists only of N-IPAAm monomer that volume is: v n vpoly = N-IPAAm N-IPAAm (25) npoly The volume V0 of the hydrogel in its stress-free state is the volume of the hydrogel after polymerization: V0 = nMBA vMBA + npoly vpoly + n0w vw

(26)

n0w

is the amount of water (i.e., number of moles) in the where polymerizing solution. 4.4. Model parameters The model requires structural parameters (e.g., the number, composition and size of the polymer chains in the “virtual” liquid hydrogel phase, cf. above), pure component data (e.g.,

Fig. 5. Comparison of calculation results (correlation results: continuous line; prediction results: dashed lines) and experimental data for degree of swelling of some non-ionic N-IPAAm hydrogels at 298 K (characteristic hydrogel parameters:  gel = 0.08 g g−1 ,  s = 1.81 × 10−4 g g−1 ; () yMBA = 0.01 mol mol−1 ; () yMBA = 0.015 mol mol−1 ; () yMBA = 0.02 mol mol−1 ) [30].

molar volumes) for all components, binary and ternary interaction parameters (for interactions between solute species) and for each hydrogel a parameter that characterizes the elastic properties of the network. As an example, we discuss only the case when a nonionic N-IPAAm hydrogel coexists with an aqueous solution of NaCl. Then there are four species: water, polymer chains, and sodium as well as chloride ions. All pure component parameters were taken from a previous publication (Hüther et al. [30]). The parameters for interactions between sodium and chloride ions were taken from Pitzer [36]. All ternary parameters for interactions where one of the interacting species is the N-IPAAm-group are neglected and the remaining binary interaction parameters were taken from Hüther et al. [30]. The only remaining parameter is the parameter C (cf. Eq. (22)) that characterizes the elastic properties of the hydrogel. That parameter is estimated from the experimental results for the degree of swelling of a hydrogel in pure water (cf. Hüther et al. [30]). Fig. 5 shows a comparison between calculation results and experimental data for the degree of swelling of three non-ionic NIPAAm hydrogels. The comparison reveals that a good agreement is achieved. Successful extensions of the thermodynamic model to describe the swelling equilibrium of non-ionic as well as of ionic hydrogels in aqueous electrolyte solutions as well as in aqueous solutions of an organic solvent (including also organic solvents that are only partially miscible with water) have been described recently [8,31–35]. The extension of the model to describe the new experimental results (cf. Figs. 2–4) for the swelling equilibrium of N-IPAAm hydrogels that also contain a cationic as well an anionic comonomer or a zwitterionic comonomer is currently being developed. 5. Conclusions The swelling equilibrium of synthetic N-IPAAm hydrogels containing cationic (DMAPMA) and/or anionic (AMPS) and zwitterionic (MPSA) comomers was investigated in aqueous solutions of sodium chloride at 298 K. As expected, the degree of swelling of the ionic hydrogels containing AMPS and DMAPMA is higher than that of the non-ionic gels. The gels containing both ionic comonomers reveal two phase transitions in aqueous solutions of sodium chloride. At low salt molalities the gels are swollen. The first phase transition occurs when the salt molality reaches about 0.003 molal. It extends over a rather large salt concentration and reduces the degree of swelling of the ionic hydrogel to that of the otherwise similar, but non-ionic gel. The second transition resembles the phase transition of non-ionic N-IPAAm gels but it occurs at higher salt molalities. The hydrogels containing a zwitterionic comonomer (MPSA), reveal an “antipolyelectrolyte swelling behavior” when the comonomer con-

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tent of the hydrogel becomes large (mole fraction of MPSA in the polymerizing material above 0.2). A thermodynamic framework for the correlation/prediction of the swelling phenomena of non-ionic as well as of ionic hydrogels is presented. In the scope of that framework the hydrogel is considered to behave like an “elastic” liquid. The “liquid-like” behavior is treated by virtually cutting the gel at its crosslinking positions. The “virtual” liquid consists of those polymer chains, water and the species provided from the aqueous solution that coexists with the hydrogel. The thermodynamic properties of that liquid are considered via an expression for its Gibbs energy. The “elasticity” of the hydrogel is taken into account via an expression for the Helmholtz energy of an elastic network. The sum of the Gibbs energy of the “virtual” liquid and the Helmholtz energy of the network is assumed to be a good approximation for the Gibbs energy of the hydrogel phase. The framework is applied here to the swelling of non-ionic N-IPAAm hydrogels in aqueous solutions of sodium chloride. In that particular case the Gibbs energy of the “virtual” liquid phase is described by the VERS-model and the “elasticity” is approximated by the phantom network theory. The model is suited to correlate the swelling equilibrium and to predict the influence of some hydrogel parameters (e.g., the comoner content and the content of crosslinking monomers) on the swelling equilibrium. Currently the model is extended to describe the newly presented experimental data. The extension requires additional information such as, for example, dissociation/protonation equilibrium constants of the comonomers MPSA and DMAPMA to account for the partial dissociation/protonation of these groups in aqueous solutions, and parameters of the VERS-model for further interactions between the new comoners and the other solute species that are present in the “virtual” liquid. List of symbols A Helmholtz energy ai activity of component i AMPS 2-acrylamido-2-methyl-1-propanesulfonic acid Aϕ Debye–Hückel parameter for water (0)

(1)

ad,l , ad,l binary interaction parameter between groups d and l (0) Ai,j ,

(1)

Ai,j

binary interaction parameter between components i and j b Pitzer parameter (b = 1.2) bd,l,m ternary interaction parameter between groups d, l and m size parameter of a segment i bi Bi,j,k ternary interaction parameter between components i, j and k c external degree of freedom of the mixture C network parameter ci external degree of freedom of component i DMAPMA N-[3-(dimethylamino)propyl]methacrylamide F Faraday’s constant f2 (Im ), f3 (Im ) function of Im G Gibbs energy excess Gibbs energy GE Im ionic strength on molality scale mo =1 mol (kg water)−1 MBA N,N -methylenebisacrylamide ˜ feed mass of feed m ˜ gel m mass of swollen gel ˜i m mass of component i mi molality of component i MPSA [3-(methacryloylamino)propyl]dimethyl(3sulfopropyl)ammonium hydroxide inner salt Mw relative molar mass of water (dry) ˜ gel m mass of the dried gel NaCl sodium chloride

Na2 S2 O5 sodium disulfite (NH4 )2 S2 O8 ammonium peroxodisulfate ni number of moles of component i N-IPAAm N-isopropyl acrylamide number of moles of water in the polymerizing solution n0w p pressure q degree of swelling qd surface parameter of group d Qi surface parameter of component i R universal gas constant volume parameter of a segment (monomer) i ri T temperature v molar or partial molar volume V volume v˜ reduced volume of the mixture V0 volume of the stress-free gel VERS Virial Equation with Relative Surface fractions v˜ i reduced volume of component i v∗i hard sphere volume of one segment of component i x mole fraction y− mole fraction of the anionic monomer y+ mole fraction of the cationic monomer yi mole fraction of component i z coordination number (z = 10) zi charge number of species i Greek symbols ˛ Pitzer parameter (˛ = 2) i activity coefficient of component i iFV free-volume contribution to the activity coefficient of component i  electrical potential difference i surface fraction of component i ϑ charge density parameter (i) ϑd relative contribution of group d to the surface parameter of component i i chemical potential of component i number of groups in the mixture i number of groups or segments in component i (i) number of groups of type d in component i d  gel mass fraction of polymerizable material mass fraction of the starter s

geometric factor for the mixture

i geometric factor for segment i Superscripts FV free volume ref reference state I liquid phase II gel phase Subscripts co comonomer d, l, m groups i, j, k, q components id. mix ideal mixture poly polymer chain pure liq. pure liquid s solute or starter w water Acknowledgments The authors appreciate financial support by Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, Ger-

V. Ermatchkov et al. / Fluid Phase Equilibria 296 (2010) 140–148

many, with the priority program “Intelligente Hydrogele” (SPP 1259). Appendix A. Model for the excess Gibbs energy of aqueous polymer solutions

i =



mi Qi

(A1)

mQ all comp. j j j

where mi and Qi are molality and surface parameter of component i, respectively. (i) When d is the number of groups of type d in component i and qd is the surface parameter of group d, the surface parameter of component i, Qi , becomes:



Qi =

(i)

d qd

(A2)

all groups d

⎡ Mw =− 1000

−



ln sVERS = −Aϕ zs2 +2

√

 Im 2 ln(1 + b Im ) √ + b 1 + b Im

 1000 Q    j s Mw

Qw

j= / w

w

(0)

Qs  Qw

Mw

j= / w k= / w

 j  k



w w

 1000 2

(1)

Aj,k + 3

Bs,j,k



all groups d

all groups l





all groups d

all groups l

(1)

Ai,j =

j= / w

k= / w

(1)



1−



Bi,j,k

(A7)

(i)

(j) (0)

(A8)

(i)

(j) (1)

(A9)

ϑd ϑl ad,l





all groups d

all groups l

all groups m

(i)

(j)

(k)

ϑd ϑl ϑm bd,l,m

(A10)

(i)

(i)

ϑd =

d qd

(A11)

Qi (i)

where ϑd is the relative contribution of group d to the sur(1)

face parameter of species i, while ad,l , ad,l and bd,l,m are binary and ternary interaction parameters between groups, respectively. These parameters are symmetric, for example, bd,l,m = bl,m,d = bm,d,l . Here, it is assumed that all ions (sodium and chloride ions), water as well as N-IPAAm consist of a single group. The surface parameter qd of ion d is replaced by the surface parameter of water. Therefore, the VERS-model for an aqueous solution of sodium chloride simplifies to Pitzer’s model for such an aqueous solution. The free-volume contribution to the excess Gibbs energy is taken from Flory et al. [41]. The free-volume contribution to the activity coefficient of a component i is:



iFV

= i ci ln

 v˜ i

v˜ i





+ 3 ln

1−

v˜ v˜ i



v˜ 1/3 −1 i



v˜ 1/3 − 1 1

v˜ 1/3 − 1

− i c



i



−1

(A12)

(A5)

where i , ci , i and v˜ i are the number of segments, the external degree of freedom, a geometric factor and the reduced volume of component i. xj is the mole fraction of component j (here water, polymer chains, sodium as well as chloride ions) of the “virtual” hydrogel phase. The reduced volume of component v˜ i is calculated from the molar volume of the pure component i, vi , the hard sphere volume of one segment of component i, v∗i , and the number of segments, i :

v˜ i =



w w w

ϑd ϑl ad,l



Bi,j,k =

− c i xj vj (A4)



2 ˛2 Im

w w



Im }

j

  2 1 − (1 + ˛ Im ) exp{−˛ Im } 2 ˛ Im

1

i= / w



(0)

Ai,j =

v

(0)

f3 (Im ) =

j= / w



(1)

The binary and ternary interaction parameters between solute species are expressed using a group-contribution approach:

Mw

Aϕ is the Debye–Hückel constant and As,j , As,j and Bs,j,k are parameters for interactions between two solutes (s and j) and three solutes (s, j and k), respectively. The functions f2 and f3 are

and

i= / w

Mw

(A3)

ms 2 z mo s

all solutes s

f2 (Im ) =

Mw

(0)

Ai,j + Ai,j exp{−˛

 1000 2       j i k

Im is the ionic strength (on molality scale) of the solution: Im

j= / w

1.5 Im − 2Aϕ √ ⎦ mo 1 + b Im

 1000      j i

− 2

ln

w w

j= / w k= / w



(1)

As,j + As,j f2 (Im )

 1000 2     j k

− zs2 f3 (Im )

1 = 2

⎣

⎤

 mj

(0)

The activity coefficient of a solute species s, s , from the VERSmodel is:

×

The activity of the solvent (i.e., water), aw , from the VERS-model is: ln aVERS w

For the “virtual” liquid hydrogel phase the excess Gibbs energy is approximated by contributions from the VERS-model (Virial Equation with Relative Surface fractions) [22,24] and a “freevolume” contribution. The VERS-model is a modification of Pitzer’s equation for the excess Gibbs energy of aqueous electrolyte solutions. It replaces the molality scale by a surface-fraction scale and applies a groupcontribution approach to deal with polymers. The amount of a solute is expressed by its “normalized surface fraction”, that is the ratio of its surface fraction s divided by the surface fraction of the solvent w . The surface fraction of a component i is given by

147



1+˛

Im +

˛2 Im 2





exp{−˛

Im }

(A6)

The parameters b and ˛ were adopted from Pitzer [36,40]: b = 1.2 and ˛ = 2.0.

vi

(A13)

i v∗i

The hard sphere volume of one segment of component i is estimated from the volume parameter ri of a segment i:

v∗i 3 cm /mol

= 15.17bi ri

(A14)

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V. Ermatchkov et al. / Fluid Phase Equilibria 296 (2010) 140–148

where bi is another size parameter of segment i. , c, and v˜ are the same properties but for the “virtual” liquid phase that represents the hydrogel phase. =



xi i

(A15)

x c i i i i

(A16)

i

 c=





x i i i i

=

 v˜ = 



xv i i i

x v∗ j j j j

=

mIIs sII,VERS sII,FV

(A17) (A18)

(A19)

aIIw =

(A20)

For the liquid phase I that coexists with the hydrogel phase II the activities of all components are calculated without any “freevolume” contribution (i.e., iI,FV = 1). References [1] [2] [3] [4]

[17] [18] [19] [20] [21] [22] [23] [24] [25]

and the activity of the solvent is: II,FV aII,VERS w w

[12] [13] [14] [15] [16]

Thus the activity of any solute species that exists in the “virtual” liquid hydrogel phase II becomes: aIIs

[8] [9] [10] [11]

J. Kopeˇcek, J. Yang, Polym. Int. 56 (2007) 1078–1098. P.C. Nicolson, J. Vogt, Biomaterials 22 (2001) 3273–3283. Y. Qiu, K. Park, Adv. Drug Deliv. Rev. 53 (2001) 321–339. R.V. Ulijn, N. Bibi, V. Jayawarna, P.D. Thornton, S.J. Todd, R.J. Mart, A.M. Smith, J.E. Gough, Mater. Today 10 (4) (2007) 40–48. [5] G.P. Raeber, M.P. Lutolf, J.A. Hubbell, Biophys. J. 89 (2005) 1374–1388. [6] E.S. Gil, S.M. Hudson, Prog. Polym. Sci. 29 (2004) 1173–1222. [7] T. C¸aykara, S. Kiper, G. Demirel, J. Appl. Polym. Sci. 101 (2006) 1756–1762.

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

Y. Orlov, X. Xu, G. Maurer, Fluid Phase Equilib. 249 (2006) 6–16. M. Zrínyi, Colloid Polym. Sci. 278 (2000) 98–103. A. Suzuki, Adv. Polym. Sci. 110 (1993) 199–240. E. Vasheghani-Farahani, J.H. Vera, D.G. Cooper, M.E. Weber, Ind. Eng. Chem. Res. 29 (1990) 554–560. M. Marchetti, S. Prager, E.L. Cussler, Macromolecules 23 (1990) 1760–1765. J.P. Baker, H.W. Blanch, J.M. Prausnitz, Polymer 36 (1995) 1061–1069. T. Hino, J.M. Prausnitz, Polymer 39 (1998) 3279–3283. S.K. De, N.R. Aluru, B. Johnson, W.C. Crone, D.J. Beebe, J. Moore, J. Microelectromech. Syst. 11 (2002) 544–555. T. C¸aykara, U. Bozkaya, O. Kantoglu, J. Polym. Sci. Part B: Polym. Phys. 41 (2003) 1656–1664. Sh. Wu, H. Li, J.P. Chen, K.Y. Lam, Macromol. Theor. Simul. 13 (2004) 13–29. A. Victorov, C. Radke, J. Prausnitz, Phys. Chem. Chem. Phys. 8 (2006) 264–278. Y. Huang, X. Jin, H. Liu, Y. Hu, Fluid Phase Equilib. 263 (2008) 96–101. T. Wallmersperger, D. Ballhause, B. Kröplin, M. Günther, G. Gerlach, J. Intel. Mater. Syst. Struct. 20 (2009) 1483–1492. G. Maurer, J.M. Prausnitz, Fluid Phase Equilib. 115 (1996) 113–133. C. Großmann, R. Tintinger, J. Zhu, G. Maurer, Fluid Phase Equilib. 106 (1995) 111–138. C. Großmann, R. Tintinger, J. Zhu, G. Maurer, Ber. Bunsen Phys. Chem. 99 (1995) 700–712. C. Großmann, R. Tintinger, J. Zhu, G. Maurer, Fluid Phase Equilib. 137 (1997) 209–228. C. Großmann, R. Tintinger, J. Zhu, G. Maurer, Biotechnol. Bioeng. 60 (1998) 669–711. N. Fedicheva, L. Ninni, G. Maurer, J. Chem. Eng. Data 52 (2007) 1858–1865. N. Fedicheva, L. Ninni, G. Maurer, J. Chem. Eng. Data 52 (2007) 2228–2232. L. Ninni, A.J.A. Meirelles, G. Maurer, Carbohyd. Polym. 59 (2005) 289–303. S. Lammertz, T. Grünfelder, L. Ninni, G. Maurer, Fluid Phase Equilib. 280 (2009) 132–143. A. Hüther, X. Xu, G. Maurer, Fluid Phase Equilib. 240 (2006) 186–196. Y. Orlov, X. Xu, G. Maurer, Fluid Phase Equilib. 254 (2007) 1–10. Y. Orlov, X. Xu, G. Maurer, Fluid Phase Equilib. 235 (2005) 18–25. Y. Orlov, X. Xu, G. Maurer, Fluid Phase Equilib. 238 (2005) 87–94. A. Hüther, G. Maurer, Fluid Phase Equilib. 226 (2004) 321–332. A. Hüther, X. Xu, G. Maurer, Fluid Phase Equilib. 219 (2004) 231–244. K.S. Pitzer, Activity Coefficients in Electrolyte Solutions, second ed., CRC Press, Boca Raton, 1991. H.M. James, J. Chem. Phys. 15 (1947) 651–668. H.M. James, E. Guth, J. Polym. Sci. 4 (1949) 153–182. H.M. James, E. Guth, J. Chem. Phys. 21 (1953) 1039–1049. K.S. Pitzer, J. Phys. Chem. 77 (4) (1973) 268–277. P.J. Flory, R.A. Orwoll, A. Vrij, J. Am. Chem. Soc. 86 (1964) 3507–3514.