Reactive & Functional Polymers 45 (2000) 145–153
www.elsevier.com/ locate / react

N.B. Ferapontov*, L.R. Parbuzina, V.I. Gorshkov, N.L. Strusovskaya, A.N. Gagarin
Department of Chemistry, M.V. Lomonosov Moscow State University, Vorobievy Gory, 119899 Moscow, Russia
Received 27 September 1999; received in revised form 25 February 2000; accepted 7 March 2000
Abstract
We offer a method for calculation of the composition and the amount in solution of cross-linked polyelectrolyte. Methods
to determine amounts and properties of the components in a solution of cross-linked polyelectrolyte are formulated from the
viewpoint of a two-phase model and the law of interphase equilibria. An algorithm to calculate the composition of a solution
of cross-linked polyelectrolyte was developed. We present the results of comparison of theoretical and experimental values
and show that the heterophase model of the swollen gel can serve as a tool for description of such systems. Ó 2000
Elsevier Science B.V. All rights reserved.
Keywords: Two-phase model; Swollen cross-linked polyelectrolyte; Properties of the components; Theory of solutions

1. Introduction A somewhat different situation is observed
when the cross-linked polyelectrolyte comes
Equilibrium between cross-linked polyelec- into contact with the liquid phase. If the gel, in
trolytes and water vapor was studied considera- equilibrium with the water vapor, is placed into
bly in the 1950s to 1970s. The so-called isopies- a solution with the same activity of water, the
tic method of studying the sorption of water by mass of the gel increases. This occurs because
ionites is given along with the experimental the inside volume that cannot be filled by
data, which is traditionally presented as iso- absorbing water vapor becomes filled due to the
therms of sorption of water, in Refs. [1–4]. In swelling of the cross-linked polyelectrolyte in
these studies the influence of different parame- water or solution. This illustrates the difference
ters (such as the nature of polarity of exchange in the condition of water in the gel and,
groups, the counter-ion, and the number of therefore, allows us to consider phase heterocross-
links) on the amount of water sorbed from geneity.
the vapor, was analyzed. Further development of the heterophase
model occurs in Refs. [5–7], where the definition
of a solution of cross-linked polyelec-
*Corresponding author. Tel.: 17-95-939-0283; fax: 17-95- 939-4019. trolyte, as consisting of cross-linked polyelec-
E-mail address: fer@physch.chem.msu.su (N.B. Ferapontov). trolyte, water, and low-molecular-weight elec-
1381-5148/00/$ – see front matter Ó 2000 Elsevier Science B.V. All rights reserved.
PII: S1381-5148(00)00023-7
146 N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153
trolyte, was fully developed. This solution linked polyelectrolyte seems unlikely because of
forms a new phase that is in equilibrium with the high concentration of the latter, but upon a
that of the solution of low-molecular-weight more detailed consideration of the matter, the
electrolyte, which exists on both the inside and reason that this event is possible becomes clear.
the outside of the gel. It is important to note that At equilibrium, there are two solutions in water,
not only ‘isopiestic water’ can get into the and the boundary between the phases does not
solution of cross-linked polyelectrolyte, but the prevent the low-molecular-weight electrolyte
solution of low-molecular-weight electrolyte as from crossing over it. If the low-molecularwell.
In this case, the solution of cross-linked weight electrolyte can interact with the polar
polyelectrolyte must be in contact with the group, it can cross the phase boundary and
solution of low-molecular-weight electrolyte, remain in the solution of cross-linked polyelecrather
than with water vapor. It is evident that trolyte. There are several kinds of interactions
mixtures which, in the general case, consist of that can take place: hydrogen bonds, Van der
water, electrolyte and cross-linked polyelec- Waals interactions, coordination interactions,
trolyte, exhibit phase heterogeneity. At the same and others. Such equilibrium will be described
time, the position of the border between the in detail further on in this work.
phases, and, therefore, the composition of the
phase of a solution of cross-linked polyelectrolyte
are still a topic of great discussion. 2. Theory
Without describing other approaches, we concentrate
on the heterophase model of a swollen 2.1. Behavior of water in solutions of crosscross-
linked polyelectrolyte [5]. This model linked polyelectrolytes
allows for the presence of a volume, inside the
swollen gel, where there is water or a solution Traditionally, the main characteristic in the
of the same composition as that on the outside description of the properties of a solution is its
of the gel. Davies and Yeoman first suggested concentration. This applies to solutions of crossthis
possibility in 1953 [8]. The next step in that linked polyelectrolytes as well. The easiest way
direction was made by Arhangelskiy [9,10]. In to determine the concentration of exchange
his studies with the example of sulfopolystyrene groups in a solution of cross-linked polyeleccationites
he showed that the specific amounts trolyte is by using isopiests. They give suffiof
water for a given polyelectrolyte (moles of ciently accurate information about the amount
H O/ equivalent of exchange groups) depend on of water per gram equivalent of polar groups for 2
the activity of water in the outside phase and a given activity of water on the outside. As Ref.
not from its physical phase (vapor or liquid). [7] points out, the properties of polar groups in
Arhangelskiy observed that the amount of water a solution of cross-linked polyelectrolytes and
sorbed by one group changes depending on the its monomer are different only because in the
activity of water in the outside solution, in cross-linked polymer there is a certain starting
accordance with the isotherm sorption of water concentration of these polar groups due to the
for this ionite, as obtained by the isopiestic presence of cross-links. Without changing the
method. He explained the presence of elec- properties of the polar groups, cross-links fix the
trolyte in the gel by the existence of pores, distance between groups (i.e. concentration),
which are filled by the solution of low-molecu- and thus change the structure of the system,
lar-weight electrolyte that comes from the out- making a phase of the solution of cross-linked
side. polyelectrolyte.
At a first glance, the entrance of low-molecu- At interphase equilibrium, the chemical polar-
weight electrolyte into the solution of cross- tentials of components that are present at the
N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153 147
same time in both phases are equal, and,
therefore, the influence of the number of crosslinks
on properties of water can be accounted
for by introducing the constant of water disw
tribution, K , which characterizes the inter- distr
dependence of the activities of water in phases
in contact:
w a¯ w K 5] (1) distr aw
where a¯ is the activity of water in solution of w
cross-linked polyelectrolyte and a is the activi- w
ty of water in solution of low-molecular-weight
electrolyte.
w The value of K for a given solution of distr Fig. 1. Dependence of n (mole H O/mole electrolyte) on the w 2 cross-linked polyelectrolyte can be calculated if activity of water for a solution of benzene sulfoacid (monomer).
¯a is known. The difference of this water w
w activity from one determines K and de- distr
scribes the influence of the cross-links on the tions, that correspond to many of the commonly
concentration of the solution of cross-linked used ionites, can be found in the literature.
polyelectrolyte. The value a¯ is obtained from Using these data, the dependencies n 5f(a ) w w w
the isopiestic data and the dependence n 5 can be determined in order to use them in w
f(a ) for the corresponding monomer. For the calculations of the parameters of components in w
desired degree of cross-linkage, one finds the the solutions of cross-linked polyelectrolytes.
o activity of water, to which the value of n Knowing the amount of water and its activity in w
(amount of water sorbed by this ion exchange the solution of cross-linked polyelectrolyte, it is
resin at equilibrium with water) in the expres- possible to determine the activities of ion-exsion
for a monomer, n 5f(a ), corresponds change groups. w w
(see Fig. 1). This activity of water is the value
of a¯ . The amount of cross-links in the poly- 2.2. Determination of the activity of an ion w
electrolyte is denoted byC, a parameter that has exchange group in the absence of sorption of
units of number of cross-links / amount of mono- low-molecular-weight electrolytes
mer. In polyelectrolytes that are not cross-linked
C51, while in cross-linked polyelectrolytes The Gibbs–Duhem equation for solutions of
this value is in the range of 1.00,C ,1.25. cross-linked polyelectrolytes is:
The values of a¯ and no for sulfopolystyrene w w n dm¯ 1n dm¯ 50 (2) w w AR AR cationite in hydrogen form with 2, 4, 8, 10, and
13% of divinylbenzene are presented in Fig. 1, where n and n are the number of moles of w AR
and the corresponding values ofC areC51.02, the components of the solution of cross-linked
1.04, 1.08, 1.10, and 1.13, respectively. Know- polyelectrolyte, and m¯ and m¯ are their w AR
w ing the value of K and the dependency chemical potentials. AR is the ionic form A of distr
n 5f(a ), it is possible to determine a¯ and n cross-linked polyelectrolyte R. w w w w
for a given polyelectrolyte and any activity of As a reference system for the solution of
water. cross-linked polyelectrolyte, we choose an infi-
The data concerning the activity of water in nitely dilute solution of the monomer of the
solutions of monomers of different concentra- corresponding type. Where
148 N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153
a¯ 51, a¯ 50, g¯ 51. the solution of the monomer of the corre- w AR 6AR
w sponding kind, K , as well as the activity of distr
Integrating Eq. (2), the following is obtained: water in the outside equilibrium solution.
a¯ w(m) n 2.3. Equilibrium between electrolytes and the w ln a¯ 5 2 ]] ¯ E d ln a (3) solution of cross-linked polyelectrolyte AR n w AR
a¯ w(m®0) 2.3.1. Distribution of electrolytes between
where m is the molality of the solution of phases
polyelectrolyte. As in the case of equilibrium of the solution
At equilibrium with water, the solution of of cross-linked polyelectrolyte with water, we
cross-linked polyelectrolyte has its minimal assume that the distribution of electrolyte beconcentration,
m (c). Thus, it is reasonable to min tween the solution of cross-linked polyelec-
divide the interval of integration (from a¯ w51, trolyte (a¯ ) and the solution of low-molecular- el as m®0 to a¯ at m) in this fashion: w weight electrolyte (a ) can be described by the el
I – from a¯ (m®0) to a¯ (m5m ) constant of electrolyte distribution: w w min
¯aand Kel 5]el . (6) distr ael II – from a¯ (m5m ) to a¯ (m). w min w
el To find K value the activity of the low- distr
This way Eq. (3) becomes molecular-weight electrolyte in the solution of
cross-linked polyelectrolyte a¯ must be deter- el ln a¯ 5 AR mined. For that, the composition of the equilib-
a¯ w(m5mmin) rium solution of the low-molecular-weight elecn
w w trolyte and its properties, as well as K for 2 ]] ¯ distr E d ln an w 1 AR investigating the polyelectrolyte must be known. a¯ w51(m®0) In addition n (the determination of n is a¯ (m) el el w n shown below) is found from experiment. The w] 1 ] ¯ E n d ln aw . (4) value nel is the amount (in moles) of the low- 2 AR
a¯ w(m5mmin) molecular-weight electrolyte in the solution of
cross-linked polyelectrolyte containing 1 mole
As the limits of integration, the literature of the polar groups.
values of the dependency n 5f(a ) for the w w Calculation of the electrolyte activity a¯ in monomer (Fig. 1) are used. el the solution of cross-linked polyelectrolyte is
Since the number of cross-links in a given the same as in a ternary solution of low-molecupolyelectrolyte
is a constant value, c5const, lar-weight electrolytes. For convenience we use
the first member of Eq. (4) for this case is also a the method of calculation for parameters of
constant. This constant is called S (c). Thus, 0 ternary mixtures (Zdanovskiy’s method [11] or
Eq. (4) becomes: Pitzer’s method [12]). To do so we must know
a¯ w(m) concentrations of both the low-molecularn
ln a¯ 5 2S (c)2 ]w] ¯ weight electrolyte and the polyelectrolyte in the E d ln a . (5) AR 0 n w mixture being investigated. In other words we AR
a¯ w(m5mmin) must know the amount of water in the solution
It follows from the above that in order to of cross-linked polyelectrolyte.
calculate the activity of an ion-exchange group, It is helpful to look at how the solution of
the dependency n 5f(a ) must be known for cross-linked polyelectrolyte absorbs the elec- w w
N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153 149
trolyte, according to our model. The composi- solution of cross-linked polyelectrolyte. The
tion of a swollen grain of cross-linked polyelec- amount of water necessary for this can be easily
trolyte is shown schematically in Fig. 2. Here determined, since it is known [11] that the
the volume of the solution of cross-linked mixture of two binary solutions with equivalent
polyelectrolyte ;V is divided conditionally activities of water results in a ternary solution 123
w el into three volumes, V , V , and V . with the same activity of water as the original 12 3 3
As mentioned earlier, in a solution of cross- two solutions. The proportion in which the
linked polyelectrolyte at equilibrium with a original solutions are taken does not matter,
solution of low-molecular-weight electrolyte, therefore, the low-molecular-weight electrolyte
the activity of water differs from the activity of enters the solution of cross-linked polyelecwater
in the surrounding solution. The ‘isopies- trolyte as a solution, in which the activity of
tic’ amount of water ensures this equilibrium. water is the same as that of the solution of
The low-molecular-weight electrolyte that pene- cross-linked polyelectrolyte.
trates the solution of cross-linked polyelec- Since the activity of water is known, the
trolyte should not change the value of a¯ . It is concentration (in this case, molality) m¯ in V el w 3i 3 obvious that this is only possible when the can be determined. The amount of water in this
low-molecular-weight electrolyte enters the so- volume is determined according to the formula:
lution of cross-linked polyelectrolyte together 55.55 el with a certain amount of water that compensates nw 5]m¯]nel. (7) 3 its contribution to the activity of water in the
Then we determine the total amount of water
in the solution of cross-linked polyelectrolyte:
el nO 5nw 1nw
and calculate molal concentration of exchanging
groups and low-molecular-weight electrolyte:
55.55
m¯ 5]] AR nO 55.55 (8) 5 m 5]]] n 3 el el n 1n w w
Taking into account that the average molal
coefficients of activity is the function of activity
of water, we write:
Fig. 2. Structure of a swollen grain of cross-linked polyelectrolyte
(scheme drawing). V is the volume of polymer matrix with ¯ 2 ¯ 2 12 a mg w el el 3 6 counter-ions; V is the volume of water associated with the polar K 5]5]]. (9) 3 distr 2 2 group of polyelectrolyte (isopiestic water); V el is the volume of ael m g 3 6
solution of electrolyte inside the solution of cross-linked polyelec- el trolyte; and V4 is the solution of low-molecular-weight electrolyte Here all values besides Kdistr are known.
inside the grain; a¯ is the activity of water in solution of el w Thus, if K is known we can calculate the distr
cross-linked polyelectrolyte and a is the activity of water in w amount and the composition of the solution of solution of low-molecular-weight electrolyte; n and n are the w el number of moles of the components of the solution of cross-linked cross-linked polyelectrolyte when it contains the
el el polyelectrolyte; n is the number of moles of water that is in V ; low-molecular-weight electrolyte. w 3
¯a is the activity of electrolyte in solution of cross-linked el The amount of electrolyte in solution of the
polyelectrolyte and a is the activity of electrolyte in the solution el of low-molecular-weight electrolyte; a¯ is the activity of the cross-linked polyelectrolyte, Q, is determined AR
polar groups of the polyelectrolyte. by using the results of analyses, in the column
150 N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153
at equilibrium with the solution of sorbed chemical potential of components of the soluelectrolyte
BX and label AX. The last is an tion of cross-linked polyelectrolyte, respectiveelectrolyte
that has no ability to penetrate into ly.
the solution of cross-linked polyelectrolyte. As a reference system, as in Section 2.2, we
Only the solution of low-molecular-weight elec- choose an infinitely dilute solution of monomer
trolyte contains this label AX. Then the contents of the corresponding kind. Integrating Eq. (14),
of the column are washed out with water, we get:
collecting all the filtrate into the flask with ln a¯ 5 AR volume V and the amounts of electrolyte in the f a¯ w O filtrate are determined (with normal concen- nw] 2 ] ¯ trations CA9 X and C9BX, respectively): E n d ln aw 1 AR
a¯ w51 Q 5V C9 AX f AX (10) a¯ el QBX 5VfC9BX nel 1 ]] ¯ E d ln a . (15) n el2 We can calculate the volume of the solution AR a¯ el50 of low-molecular-weight electrolyte in the col-
umn (V ) because the value of Q is known: Changing the first integral as in Section 2.2 s AX and taking into account that for interval I, no 5 w
Q o el AX n and for interval II, n 5n 1n , we get V 5]] (11) w w w w s CAX ln a¯ 5 2S (c) AR 0
where CAX is the concentration of electrolyte a¯ w(m) O AX in the initial solution. Then, the amount of nw] 2 ] ¯ E d ln asorbed electrolyte BX in the solution of cross- n w 1 AR
linked polyelectrolyte is calculated: a¯ w(m5mmin)
¯ael
qBX 5QBX 2VsCBX (12) nel 1 ]] ¯ E d ln a . (16) n el2 where C is the concentration of electrolyte AR BX a¯ el50 BX in the initial solution. Knowing the ion
exchange capacity of the ion exchange resin we Determination of the variables included in
Eq. (16) is described in Sections 2.2 (a¯ ) and can calculate the n (the amount of the low- w el 2.3 (no, n , a¯ ). The desired activities of ion- molecular-weight electrolyte in the solution of w el el
cross-linked polyelectrolyte containing 1 mole exchange groups in solution of cross-linked
of the polar groups): polyelectrolyte are obtained from Eq. (16) using
these variables.
qBX n 5]]. (13) el E
2.3.2. Activities of ion-exchange groups in the 3. Results and discussion
solution of cross-linked polyelectrolyte 3.1. The effect of number of cross-links on
containing low-molecular-weight electrolyte w KThe Gibbs–Duhem equation for a solution of distr
cross-linked polyelectrolyte containing low-mo- The results of this work allowed us to comlecular-
weight electrolyte is as follows: pose an algorithm to calculate composition and
nO dm¯ 1n dm¯ 1n dm¯ 50. (14) properties of solution of cross-linked polyelec- w w AR AR el el trolyte, mainly applying the literature values
Here n and m¯ are the number of moles and the with minimal use of experiment. i i
N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153 151
w Since the value of K for this polyelec- Use of these dependencies in order to create distr
trolyte is determined exclusively by the valueC isotherms of sorption of water by ionites gave
w [K 5f(C )], this dependency was studied in data that correlates well with the experimental distr
detail. The analysis of the data obtained for the isotherms, which were determined by the isosolution
of cross-linked polyelectrolyte with piestic method (Fig. 4).
w different C showed that the values of Kdistr
depend practically linearly on the number of
3.2. Calculation of the parameters of a
cross-links forC $1. The results that show this
solution of cross-linked polyelectrolyte
dependency are given in Fig. 3. Here, it is clear
that straight lines with different slopes char- If the properties of an equilibrium solution of
acterize different polyelectrolytes. If the low-molecular-weight electrolyte are known, in w tangents are known, then K can be deter- distr order to obtain full information about the solumined
with a good degree of accuracy for a tion of cross-linked polyelectrolyte the followdesired
cross-linked polyelectrolyte even if ing need to be known:
there are no isopiestic data about its crosslinkage:
· dependency n 5f(a ) for the monomer w w
Kw 512(C 21)tan a (17) from which the cross-linked polyelectrolyte distr is synthesized;
where tan a is the tangent of the angle. There- · the amount of cross-linking agent or C;
fore, in order to determine the amount of water · one isopiestic curve for the given crossin
a solution of cross-linked polyelectrolyte, it is linked polyelectrolyte with any degree of
sufficient to know the dependency nw 5f(aw) cross-linkage;
for the corresponding monomer and the depen- · the constant of distribution of electrolyte.
w dency K 5f(C ), which can be calculated if distr
w at least one value of K is known. The calculations of parameters of a solution distr
w Fig. 3. Dependence of K on the nature of polyelectrolyte and distr
the number of cross-links: (1) strongly acidic polyelectrolytes (of
the type DW-50), and (2) strongly basic polyelectrolytes (of the Fig. 4. Comparison of calculated isotherms (lines with solid dots)
type DW-1). and isopiestic isotherms (open circle dots) of water sorption.
152 N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153
of cross-linked polyelectrolyte are carried out in determined the tangent of the angle with the
the following order. help of Eq. (17) for the desired cross-linked
w polyelectrolyte with a given C, K can be o distr 1. Using the value of nw when aw51.00 and calculated for any degree of cross-linking.
n 5f(a ), the corresponding values of a¯ 3. With a known Kw the activity of water, a¯ , w w w distr w w are found, and then Kdistr is determined for and its amount n in the solution of cross- w
the given degree of cross-linking (Fig. 1). linked polyelectrolyte at a given activity of
2. As shown in Section 3.1 and in Fig. 3, the water in the outside solution, a , can be w w function Kdistr 5f(C ) is linear; thus, having determined. If low-molecular-weight electrolyte
is absent from the solution of crosslinked
polyelectrolyte, then the activity of
the polar group can be calculated immediately
from Eq. (5).
4. The activity of low-molecular-weight electrolyte
when it enters the solution of crosslinked
polyelectrolyte can be calculated with
Eq. (6).
5. The molal concentration of exchanging
groups and low-molecular-weight electrolyte
can be determined by using the system of
Eqs. (8).
6. The final stage is determination of activity of
the polar group in a ternary mixture, which
can be done with the help of Eq. (16).
This algorithm was tested for systems with
strongly basic anionites: ARA-4 – HCl, LiCl;
AV-17×8 – HCl, LiCl, NH Cl; ARA-12 – 4
NH Cl and for systems with sulfopolystyrene 4
cationites: KU-2×4 – HNO , KNO ; KU-2×8 – Fig. 5. Comparison of experimental and calculated isotherms of 3 3 HNO , KNO . Some of the results are shown in electrolyte distribution in a solution of cross-linked polyelec- 3 3
trolyte. Fig. 5 and in Table 1.
Table 1
Parameters of the study systems in accordance with the model
w el No. Ion exchange resin % DVB Ionic form Equilibria electrolyte K K distr distr
2 1 ARA 4 Cl LiCl 0.8861 0.668
2 2 AV-17 8 Cl LiCl 0.7886 0.3368
2 3 ARA 4 Cl HCl 0.8861 1.177
2 4 AV-17 8 Cl HCl 0.7886 2.60
2 5 AV-17 8 Cl NH Cl 0.7886 0.0047 4
2 6 ARA 12 Cl NH Cl 0.6886 0.0541 4
1 7 KU-2 8 K KNO 0.92 0.058 3
1 8 KU-2 4 K KNO 0.96 0.332 3
1 9 KU-2 8 H HNO 0.912 0.648 3
1 10 KU-2 4 H HNO 0.956 0.758 3
N.B. Ferapontov et al. / Reactive & Functional Polymers 45 (2000) 145 –153 153
4. Conclusion Acknowledgements
The effect of the number of cross-links on the This work has been supported by the Russian
properties of the system electrolyte–water was Fund of Fundamental Researches (project 98-
studied in this work. We showed that the 03-32072a) and the Fund of Leading Scientific
peculiarity of systems that include cross-linked School of Russia.
polyelectrolyte is that the polar groups are fixed
in the polymer carcass and cannot fill the entire
volume of the swollen gel. That is, in the References
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