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Structure and properties of spatial polyelectrolytes on styrene–divinylbenzene matrixes
Reactive & Functional Polymers 54 (2003) 63–84 www.elsevier.com / locate / react
Structure and properties of spatial polyelectrolytes on styrene–divinylbenzene matrixes V.S. Soldatov a,b , *, V.M. Zelenkovskii a , T.V. Bezyazychnaya a a
Institute of Physical Organic Chemistry of the National Academy of Sciences of Belarus, 13 Surganov St., Minsk 220072, Belarus b Institute of Environmental Engineering, Technical University of Lublin, Nadbystrzycka 20 -618, Lublin, Poland
Abstract A mathematical model establishing relations between the selectivity, degree of ion exchange and structure of spatial polyelectrolytes has been developed. The structure is expressed through two parameters: radius of interaction (that is a maximal distance at which the exchange sites influence each other) and the number of exchange sites situated in the sphere of interaction (neighbors). A larger number of neighbors should result in more complicated shape of property–composition dependencies of the ion exchange system. A computer model allowing estimation of the probability of the presence of different numbers of neighbors in different elements of the polymer network as a function of distance from the exchange site was suggested. The polymer network is represented as a body obtained by integration of fragments with eight to ten monomeric units of different shapes: linear chains, H and T shape units and small rings. The theory was applied to interpreting dependencies of selectivity of cation exchange on sulfonic styrene–divinylbenzene cation exchangers and strong base anion exchangers on the same matrix. Dependencies of selectivity on degree of ion exchange and cross-linkage of the resins are in good agreement with the prediction of the theory if the predominant number of neighbors in different particular cases is ranging between two and four. The non-equivalence of exchange sites of spatial polyelectrolytes is their inherent property because different exchange sites have different amounts of the neighbors. The difference in the selectivity dependence on the cross-linkage of cation and anion exchangers on styrene–divinylbenzene matrix finds a simple and natural explanation. 2002 Elsevier Science B.V. All rights reserved. Keywords: Polyelectrolytes; Ion exchangers; Mathematical modeling; Computer modeling; Selectivity
1. Introduction Establishing quantitative relations between properties, ionic composition and structure of spatial polyelectrolytes requires clear definition of concept ‘structure’ in mathematical terms. In *Corresponding author. Tel.: 1375-17-284-2338; fax: 137517-284-2338. E-mail address: [email protected] (V.S. Soldatov).
relation to spatial polyelectrolytes this concept is not as certain as that for simple substances whose molecular structure is strictly determined. Spatial polymers consist of mixtures of molecules of different sizes and shapes forming in the process of statistical co-polymerization of the initial monomers. Therefore, it is impossible to relate their properties to a substance with some definite type of molecules or mixture of the molecules with a certain composition.
1381-5148 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S1381-5148( 02 )00183-9
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The properties of interest in this paper, Y, are those related to equilibrium of binary ion exchange I 1 2I 2 between the spatial polyelectrolyte and the external solution. They are thermodynamic functions, equilibrium and selectivity coefficients and water uptake by the polyelectrolyte. They depend on the ionic composition of the polyelectrolyte, expressed by the ¯ thereequivalent fraction of the entering ion x; fore Y 5 Y(x¯ ). A number of theoretical models were suggested for relating the ion exchange properties and structure of spatial polyelectrolytes. Some of them, such as the Gregor [1] and Glueckauf models [2], indirectly account for the structure of polyelectrolyte by introducing separate pressure-swelling term into the property Y(x¯ ). In the other group of models structure of the polyelectrolyte was expresses in more certain ways. Katchalsky and Lifson [3] represent the spatial polyelectrolyte as a collection of small rigid rods, joined into continuous flexible chains forming a spatial net. Applying the DebyeHuckel distribution equation, some statements of Flory theory and equation of Donnan equilibrium to such a system, some of properties Y(x¯ ) were found for weakly dissociating polyacids of different cross-linkage. Harris and Rice [4] suggested a model of polymeric ion exchanger representing it as a set of electrically charged plates joined by springs into a continuous three-dimensional structure. Hoell [5] developed a model describing the ion exchanger as an electrical condenser in which the counterions form the Nernst-type layers at different distances from the matrix chain depending on their charge and water content in the polyelectrolyte. Structural parameters of Flory theory were used to describe properties of ion exchangers by Trostianskaya [6]. In spite of poor correlation of properties of ion exchangers with the statements of the Flory theory, reasonable agreement between theory and properties of weakly cross-linked ion exchangers was observed in many cases. In the mentioned theoretical models some
features of structure of the spatial polyelectrolyte were explicitly considered, while the others were not accounted for at all. These models do not account for real molecular structure of polyelectrolytes and contain parameters with unclear physical meaning, which can be arbitrarily varied for obtaining agreement between the theory and experiment. Therefore their interpretive ability is extremely limited. In our recent publications we described a model for interpretation of dependence of properties of ion exchange systems Y(x¯ ) on the structure of the polyelectrolyte, expressed as a set of parameters determined by its molecular structure. The present paper is a review of these works [7–16]. Our approach to the problem in question is based on the concept of interaction between the nearest neighbors and includes two steps. In the first one we developed a mathematical model defining dependence of property Y on the ionic composition of the polyelectrolyte and the num¯ i). In the ber of the nearest neighbors i, Y 5 Y(x, second step a method for determination of the structural parameters needed for the Y 5 Y(x¯ i ) function was developed. It is based on a computer modeling of the molecular structure of the spatial polyelectrolyte. It was applied to styrene–divinylbenzene based spatial polyelectrolytes with sulfonic and benzyltrimethyl ammonium groups, being the most spread types of ion exchange resins.
2. Mathematical model
2.1. Definitions Exchange site—a functional group on a polymer matrix whose charge is compensated by a counterion. The location of the exchange site is defined to coincide with the center of the sulfur or nitrogen atoms in the functional groups. Interaction radius, R—the maximal distance between two exchange sites where the interaction between a counterion and the exchange
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site is regarded as being significantly influenced by counterion exchange at the neighboring exchange site. Sphere of interaction—a sphere with radius R drawn around an exchange site. Microenvironment—a part of a sphere of interaction except for the central exchange site. Nearest neighbors (neighbors)—the neighboring exchange sites at a distance less or equal to the radius of interaction from the central site. Property Y—any additive property of the macroscopic system which can be obtained by summation of the same properties of all its micro-states. For example, Y can be specific Gibbs’ energy, enthalpy in a process when infinitely small amounts of ion 1 are replaced by ion 2. It also can be specific volume or water contents in the ion exchange, logarithm of the selectivity or equilibrium coefficient. Property y–property Y related to the central exchange site and its microenvironment. Ionic composition—expressed by equivalent fraction of the entering ion (x¯ 2 ) in ion exchange I 1 –I 2 ; x¯ 1 1 x¯ 2 5 1. In the following we shall use ¯ definition x¯ 2 5x. Local ionic composition—ionic composition in the microenvironment. For simplicity we consider only exchange of monovalent ions. An ion exchange equilibrium between the spatial polyelectrolyte and the solution is formulated as follows: I¯1 1 I2 ↔ I1 1 I¯2
(1)
where the barred symbols are related to the resin phase. Equilibrium coefficient, k is defined by Eq. (2). For homovalent ion exchange it coincides with selectivity coefficient.1 x¯ (1 2 x) k 5 ]]]. x(1 2 x¯ )
1
(2)
These terms were recommended by The International Workshop on Uniform and Reliable Nomenclature, Formulations and Experimentation for Ion Exchange, Helsinki, Finland, May 30–June 1, 1994 [17].
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2.2. Statements Different exchange sites in the spatial polyelectrolyte may have different numbers of neighbors, i. It can vary from zero to n depending on the choice of the interaction radius and the structure of the matrix. Property y is characterized by the number of the neighbors and a local ionic composition. It is independent of the position of the neighbors relative to the central exchange site. A property Y can be represented as a weight sum of properties y.
2.3. Property–ionic composition equation In the present approximation of the model two parameters were chosen to describe the state of the central ion in the microenvironment: a number of neighboring counterions of type 2 denoted as j and that of ions 1 equal to i 2 j. Then each of the possible states of ion I can be expressed as I(i 2 j, j) (Table 1) and property Y related to this state is denoted as y(i 2 j, j). Property Yi can be obtained as a sum of properties y(i 2 j, j) proportional to the probabilities of the existence of exchange sites in the relative states p(i 2 j, j): j51
Yi 5
O y(i 2 j, j) p(i 2 j, j)
(3)
j50
In its turn, a macroscopic property Y of the overall system can be obtained from the Yi and the probability of existence of i neighbors of the exchange site in a real system, Pi : Table 1 States of the central ion in the microenvironment i, number of neighbors
States of ion I
0 1 2 3
I(0,0) I(1,0), I(0,1) I(2,0), I(1,1), I(0.2) I(3,0), I(2,1), I(1,2), I(0,3)
i
I(i 2 j, j), j502i
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O YP.
i 5n
Y5
i
(4)
i
i 50
The values p(i 2 j, j) are proportional to the molar fraction of each of the counterions raised to a power equal to the number of the neighbors of a relative type. It is also proportional to a number of permutations of the counterions 1 and 2 within a given combination of i counterions with j counterions of type 2. If the probabilities are normalized so that their sum is equal to 1 then i! p(i 2 j, j) 5 ]]] (l 2 x¯ )(i 2j ) x¯ j (i 2 j)!j!
(5)
and j51
Yi 5
i! O ]]] y(i 2 j, j) (l 2 x¯ ) (i 2 j)!j!
(i2j )
x¯ j
(6)
j50
The general equation ‘property–ionic composition’ for a real system is j51
i! O P O ]]] y(i 2 j, j) (l 2 x¯ ) (i 2 j)!j!
i5n
Yi 5
i
i50
(i2j )
equations are polynomial presented in a special form (Bernstein polynomial). They have features making them especially convenient if x¯ varies in interval [0,1], i.e. for consideration of ‘property–composition’ dependencies. Coefficients of the first and the last terms of Eqs. (8) are equal to values of Yi at x¯ 5 0 and x¯ 5 1. The intermeddle coefficients control the curvature of the line Yi 5 f(x¯ ). More detailed analysis of properties of the functions expressed by Eq. (8) is given in Ref. [9]. Eq. (7) shows that ‘property–ionic composition’ dependence should have a power equal to maximal number of neighbors and can be used for interpretation of experimental data by comparing them with relative theoretical dependencies. Quantities i and y(i 2 j, j) should be determined independently or found as fitting parameters. At the present state of development of described model we can evaluate value i by computer modeling of the polyelectrolyte structure. A method for their estimation is described in the following section.
x¯ j
j50
(7) The number of neighbors in real systems is rarely more than four. Thus, Eq. (6) represents the following set of equations: Y0 5 y(0,0)
(8a)
Y1 5 y(1,0)(1 2 x¯ ) 1 y(0,1)x¯
(8b)
2
Y2 5 y(2,0)(1 2 x¯ ) 1 2y(1,1)(1 2 x¯ )x¯ 1 y(0,2)(1 2 x¯ )x¯
2
(8c)
3 2 Y3 5 y(3,0)(1 2 x¯ ) 1 3y(2,1)(1 2 x¯ ) x¯
1 3y(1,2)(1 2 x¯ )x¯ 2 1 y(0,3)x¯ 3
(8d)
4 3 Y4 5 y(4,0)(1 2 x¯ ) 1 4y(3,1)(1 2 x¯ ) x¯ 2 2 3 1 6y(2,2)(1 2 x¯ ) x¯ 1 4y(1,3)(1 2 x¯ )x¯
1 y(0,4)x¯
4
(8e)
Since all y(i 2 j, j) values are constant, these
3. Computer model
3.1. Elementary fragments We assume that a statistical spatial polymer matrix can be represented by a set of elementary fragments with known structure. They are randomly integrated into a matrix and a property of the integrated system can be found as a weight average of the properties of the fragments. The functional group is assumed to be situated in para-position related to the chain. The following types of fragments were chosen to represent the polymer structure: linear chain (L), H- and Tshaped fragments, and small loops (rings, R). Any other structures can be obtained as combinations of these fragments. The choice of size of the elementary fragments is important. We accepted it as a junction of eight to ten monomeric units, since with the large fragments it would be impossible to represent the structure of highly cross-linked resins. The smaller fragments do not have statistical meaning.
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84 Table 2 The letter codes for different fragments of the spatial polymer net Type of the
Letter code
fragment
No funct. group in DVB
Linear chain
L(ht) L(hh) L(tt)
T-fragment
T(ht) T(hh) T(tt)
T(ht)* T(hh)* T(tt)*
H-fragment
H(ht) H(hh) H(tt)
H(ht)* H(hh)* H(tt)*
Rings a
R(ht)N R(hh)N R(tt)N
R(ht)N* R(hh)N* R(tt)N*
Funct. group in DVB
a In place of N in each specific case should stay the number of the monomer units in the ring. * Presence of the functional group in the DVB residue.
Structure of the elementary fragments can be different. ‘Head to tail’ (ht) junctions are predominant, but there are no convincing arguments to discriminate the hh and tt junctions. We name hh and tt junction irregularities. Several varieties of all types of fragments, differing in the type of junctions, the cross-bridge type (para- or metadivinylbenzene (DVB)), presence or absence of the functional group in the DVB fragment, were considered. We suggest assigning letter codes to different types of fragments for easier referring. Their list is given in Table 2. When referring to a particular structure, it should be additionally specified which crossagent (in our case meta or para DVB) and which initiator of polymerization were used. The letter is important for specifying the Tfragment structure. In our case it was benzoil peroxide.
3.2. Method for the probabilities evaluation Knowledge of the probabilities of different numbers of neighbors, pi , in each type of
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representative fragments k is required for identification of structural elements of the polyelectrolyte matrix responsible for degree of complexity of the Y 5 f(x¯ ) dependence. We describe our method for their evaluation with the example of one of the structures, namely H(ht)*-pDVB and sulfonic group as an exchange site. The structural formula and the optimized structure of this fragment, obtained by computer modeling, are presented in Fig. 1. The structure in Fig. 1 was calculated by the molecular mechanics method with the force field MM2 [18]. The procedure of these calculations was as follows. Since the fragment does not represent a separate molecule and is a part of the larger structure, as a first step it is necessary to optimize geometric size and the structure of the fragment by variation of the distance of the each functional group from the central sulfonic group or the center of the cross agent residual. The angle between the sub-fragments in the H- and T-fragments was also varied. In each case more than 200 conformations were examined. From all considered conformations only those with low energies were taken for further consideration, whose introduction in the total polymer structure could be meaningful accounting for their Boltzman’s population. The number of meaningful conformations was usually |20–30. The first step of these calculations was computing the total energy of the fragment at small variations of the initially chosen distance to search for the energy minimum. The energy appeared to be highly sensitive to the distance value. The distance corresponding to the energy minimum (the equilibrium value) could be different from the initially chosen one by several tenths of an angstrom. This distance was taken for the energy calculations. The structure and energy characteristics of different conformers have been computed in molecular mechanics approximation using force field MM2. If the elementary fragments are different only in conformation, the probability of existence of i neighbors in a specified type of the fragments with 1, 2 . . . 3 . . . q . . . s conformers depends
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Finally, the probabilities Pi of the presence of i neighbors in a network can be found by summing of the pi proportionally to the probabilities of existence fragments themselves. Pi 5
Op
i
(10)
4. Results of the computer simulation of the polyelectrolyte structure
4.1. Evaluation of the radius of interaction
Fig. 1. Schematic drawing (a) and computer image (b) of an optimized H-shaped fragment of sulfonated styrene–divinylbenzene copolymer. The distances between the central S atom and its three nearest neighbors are shown.
only on the energy of the conformers according to Boltzman’s distribution: exp(2Eq /RT ) pi 5 ]]]]]] . s exp(2Es /RT )
O
q 51
(9)
Choosing the value for the interaction radius is the most uncertain operation in the whole model. It can be expected that this value may depend on the structure of the ion exchanger and the nature of the exchangeable ions. Therefore such an estimate has to be done individually on the basis of knowledge of properties of any specific type of the system. The experimental data on the heat effects of ion exchange of selectivity of ion exchange of simple inorganic ions on resins with low crosslinkage (Fig. 9) are rather simple and require the model Eq. (8c) of the second power at most. Hence, in these cases two neighbors have to be accounted for. As seen from the probability histograms (Figs. 3 and 4) this is possible only if in linear fragments of the polymer networks ˚ the radius of interaction is not greater that 10 A. At its higher values the probability of more than two neighbors becomes high and the properties should obey Eq. (8d)–(8e) of the power higher than two. This does not agree with the experimental data. On the other hand, the probability of the presence in the sphere of interaction of more than two neighbors in the nonlinear fragments becomes substantial at the ˚ Redistances already slightly less than 10 A. spectively, properties of the sulfonic ion exchangers with a high cross-linkage as a function of their ionic composition are described by Eq. (8)d of the third power. Based on these considerations the radius of interaction for sul-
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fostyrene ion exchangers in forms of simple monovalent inorganic ions was taken equal to ˚ 10 A.
Fig. 2. Computer images of three stable conformations of Na salt of a dimer of sulfostyrene hydrated with 20 water molecules, corresponding to minima of the energy. The distances between the centers of the S atoms and hydrogen bonds forming bridges between the sulfonic groups are shown as lines.
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Additional arguments in favor of this value were obtained with the help of quantum chemical calculations of the most probable configurations of a dimer of p-sulfostyrene (ht junction) in the presence of 20 water molecules. The charges of the sulfonic groups were compensated by Na 1 ions. The calculations of the structure of this fragment were done with the help of non-empirical method SCF MO LCAO and program GAMESS [19,20] using basis set MINI [21]. Initial distances between the sulfur atoms were chosen by variation of torsion angles formed by four carbon atoms of the hydrocarbon chain in the fragment. Optimizing of the coordinates of all atoms in the system was done to find the total energy minimum. The calculations have shown that only three stable states of the fragment exist. They are presented in Fig. 2. The first configuration corresponds to the ˚ distance between the sulfur atoms (6.28 A) interacting with each other via a water molecule and having common hydrate structure with water molecules bound by hydrogen bonds. The second local minimum of energy was found at a ˚ between the S atoms. distance of 7.98–8.20 A In this case the groups interact via a chain of two water molecules and also form one common hydrate structure. Increasing the distance ˚ leads between the sulfonic groups above 10 A to spontaneous separation of the sulfonic groups ˚ Their hydrate to a distance of 11.5–11.6 A. structure splits into two independent structures. These facts indicate absence of significant interaction between sulfonic groups situated at a ˚ At such distances distance greater than 10 A. electrostatic interaction between sulfonic groups in the presence of water molecules is weaker than the conformation strength in the hydrocarbon chain. In this case the latter determines geometric shape of the structure. At distances ˚ a decisive role belongs to shorter than 10 A, electrostatic interaction in the common hydration structure, the sulfonic groups spontaneously approach each other until they form stable structures 2 and 1 with a lower energy for the
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latter structures. The least energetically profitable is structure 3.
4.2. Linear chains ( L-fragments) We calculated the probabilities of the presence of different numbers of neighbors in linear fragments containing eight repeated units of the polyelectrolyte in which (a) all of them are
joined in the ht structures, (b) all junctions are ht except the one between the fourth and the fifth units where the junctions are hh or (c) tt types. The histograms of the probabilities of presence of zero to five neighbors as a function of distance from the exchange site are presented in Figs. 3 and 4. It appeared that all three sets of histograms are substantially different. In the regular chain
Fig. 3. Linear chain fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere ˚ of interaction with radius 10 A.
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Fig. 4. Calculated probabilities of occurrence of zero to five neighbors in different linear fragments of the sulfonic acid polyelectrolyte as a function of distance from the exchange site.
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the situation when no neighbors are present at a ˚ is impossible. At the distance greater than 5 A same time, the probability of that is rather high in the linear fragments with both types of irregularities. One and two neighbors are most probable in the sphere of interaction while their radial distribution is quite different. The presence of more than two neighbors in the distance ˚ is not very probable in the less than 10 A regular chain and is highly probable for the hh and (somewhat lower) for the tt structure. Presence of more than three neighbors in the sphere of interaction is not very probable in all cases. Four and five neighbors are highly prob˚ which can able at distances greater than 10 A, be important for large organic counterions.
4.3. H-fragments Ion exchangers may contain many different types of H-shaped fragments. The difference between them is caused by the following reasons: • Different cross-linking agent (meta- or paradivinylbenzene); • Sulfonated or non-sulfonated benzene ring of the cross-agent; • Different sequence of junction of the monomeric units in the sulfostyrene chains. The probabilities pi for different cases in ˚ are illustrated by dependence on i and R510 A Figs. 5 and 6.
4.4. T-fragments T-fragments form in the polymerization process when the second double bond of divinylbenzene residue in the polymeric chain is activated by a reaction with an initiator of polymerization. Therefore the structure of this junction depends on the type of the initiator. In our case it was benzoil peroxide. Several possible structures of the T-fragment are possible. The difference between them is illustrated by Fig. 7. Only ht sequence was considered.
4.5. Rings ( R-fragments) We present estimates for structural parameters i and Pi for small rings having a general structure of the type presented in Fig. 8 and Table 3. The calculations were performed for the Rstructures containing two to six molecular fragments in the ring. The monomers were assumed to contact by the ht type only.
4.6. Summary of the probability calculations The radial distribution of the probability of existence of different numbers of neighbors is highly sensitive to the structure of the polymer network. In the sphere of interaction of all considered polymeric fragments of regular structure with non-sulfonated cross agent almost only exchange sites with one or two neighbors present. Any irregularity, hh or tt junctions, leads to increasing the probabilities of appearance of exchange sites with greater number of neighbors. Presence of a sulfonic group in the structure of the cross-bridge increases the probability of existence of higher number of neighbors, while absence may lead to the Pi decrease. Irregular H- and R-fragments with non-sulfonated crossbridges may have a large probability of i .2. We can identify the fragments of the spatial polyelectrolyte net, which are responsible for a complicated shape of dependence Y 5 Y(x¯ ) because of substantial concentration of exchange sites with large number of neighbors in the sphere of interaction. They are: irregular linear chains (L-hh and L-tt); H-fragments with sulfonated cross-bridge (Figs. 3–5); and some rings with three to six sulfonated phenyl radicals (Table 3). 5. Application of the model for interpretation of ion exchange equilibria
5.1. Effect of cross-linkage It is known from the first systematic studies
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Fig. 5. H-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ ht junctions. interaction with radius 10 A;
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Fig. 6. H-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ hh and tt junctions. interaction with radius 10 A;
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Fig. 7. T-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ ht junctions. interaction with radius 10 A;
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Fig. 8. R-fragment with eight styrene units in the ring and calculated probabilities of the occurrence of zero to five neighbors in the sphere ˚ ht junctions. of interaction with radius 10 A;
Table 3 ˚ Probabilities of the presence of n nearest neighbors in various R-structures for the interaction radius R s 510 A N
p-R3
p-R3*
m-R2
m-R2*
p-R5
p-R3*
m-R5
p-R6
p-R6*
p-R6*
m-R6
m-R6*
0 1 2 .2
0.289 0.261 0.347 0.103
0.000 0.086 0.487 0.427
0.023 0.962 0.015 0.000
0.203 0.491 0.268 0.039
0.000 0.848 0.152 0.000
0.000 0.668 0.316 0.017
0.179 0.439 0.382 0.000
0.000 0.443 0.423 0.134
0.000 0.187 0.638 0.176
0.000 0.144 0.528 0.329
0.145 0.468 0.341 0.047
0.143 0.286 0.199 0.373
of selectivity of ion exchange of alkali metal ions with H 1 [22] that increasing cross-linkage of sulfostyrene ion exchangers causes complicated changes in the dependence of selectivity coefficient on the degree of exchange. Increasing percent of DVB leads to increase in the selectivity coefficient at low loadings of the resins with the metal ions and may cause the opposite effect on the selectivity at high loadings; the shape of curves k 5 k(x¯ ) becomes more complicated. In order to explain these phenomena several varieties of hypotheses of non-equivalence of the exchange sites in ion exchangers were suggested. But the reasons for such irregularity were not convincingly explained. Now we can assert that the nonequivalence of exchange sites is a direct consequence of structural non-regularity of statistical polymer networks manifested in different numbers of the neighbors of different exchange sites. Therefore non-equivalence of the ex-
change sites can be considered an immanent property of spatial polyelectrolytes. In the simplest case, corresponding to exchange of small ions and ion exchangers of low cross-linkage dependencies of equilibrium coefficient and the other properties on the degree of ion exchange should be expressed by Eq. (8)c with the second power with a weak curvature because of presence of large amount of groups with one neighbor. Increasing degree of cross-linkage in terms of the model described means increase in concentration of H and R fragments, as well as (possibly) the irregularities. Figs. 9 and 10 illustrate these deductions. Log k 5 f(x¯ ) and DH 5 f(x¯ ) dependencies for the resins with DVB percent below 12 are described with Eq. (8)c of the second power. The same dependencies for the resins with a higher percent of DVB have specific features inherent to polynomials of higher power, for example inflections are clearly seen
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Table 4 Application of the model equation to description of dependencies for the KRSm resins % mDVB
Y
y(3,0)
y(2,1)
y(1,2)
y(0,3)
1 5 5 8 8 12 12 16 25 25
log k log k DH log k DH log k DH log k log k DH
0.154 0.359 21.90 0.500 22.60 0.874 23.33 1.361 1.448 25.59
0.101 0.333 21.18 0.475 22.14 0.390 22.05 0.160 0.518 21.12
0.099 0.165 22.04 0.282 21.60 0.629 22.43 0.904 0.759 23.44
0.104 0.201 20.48 0.350 20.54 0.119 0.010 20.13 20.38 0.28
i53, exchange K 1 –H 1 , t525.0 8C.
for the properties of highly cross-linked resins. Dependence log k 5 f(x¯ ) for the ion exchanger with very high cross-linkage (25% DVB) for all alkali ions requires Eq. (8)d of the third power (Fig. 11). Two additional notes are relevant. (a) In order to determine correctly the power of the polynomial Y 5 Y(x¯ ) property Y should be determined ¯ The regions in a sufficiently wide interval of x. ¯ ¯ near x50 and x 5 1 are very important [9]. (b) Dependence Y 5 Y(x¯ ) is sensitive to the structure, therefore we can expect that ion exchangers of the same chemical composition but obtained under different conditions of formation of the polymer network may have different properties.
Fig. 9. Dependencies of enthalpy of the degree of ion exchange H 1 –K 1 on styrene m-DVB ion exchanger KRSm with different percent of m-DVB: 1, 1%; 2, 5%; 3, 8%; 4, 12%; 5, 16%; 6, 25%; t 5 25 8C. The data were obtained by measuring heat effect of ion exchange. The beginning and the end of the arrows show the direction, starting and end points of the process in the calorimetric measurements [23]. The points are experimental data, the curves are computed from Eq. (6) with parameters given in Table 4.
5.2. Large organic ions Regularities of selectivity of ion exchange with participation of large organic ions are very complicated [6] and are usually qualitatively explained by interplay of several strong factors, such as change in water structure of the external and internal solution in the ion exchange system; interaction of hydrophobic parts of the counterions with the polymeric matrix; hydrogen bonding and other specific interactions of the counterions with the functional groups and between themselves. It is likely that these factors mainly affect the value of the selectivity
measures rather than their dependence on the degree of loading of the ion exchanger. Such dependence should be affected by the structure of the resins, expressed by the above described parameters. The peculiarity of the organic ions in most cases is their large size, exceeding the radius of the sphere of interaction determined for small counterions. This means that in the addition to the neighbors present in the sphere of interaction (defined for simple inorganic ions) additional exchange sites may exist, situated outside this sphere and interacting with the large counterion. So the total number of neighbors
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Fig. 10. Log k 5 f(x¯ ) dependencies. See caption to Fig. 9.
affecting the ion exchange may be larger than that for the inorganic ions in the same ion exchanger. For example, the length of cations of different derivatives of tetracycline significantly ˚ and it is not surprising that exceeds 10 A description of log k 5 f(x¯ ) dependencies re-
Fig. 11. Log k 5 f(x¯ ) dependencies for exchange of alkali metals with hydrogen ions, ion exchanger KU-2X25% DVB, t525 8C. The points are experimental data, the curves are computed from Eq. (6) with parameters in Table 5 [24].
quires Eq. (8)e of the fourth power indicating importance of four neighbors (Fig. 12).
Table 5 Application of the model equation to the description of dependencies for the KU-2X25 Exchange 1
1
Li –H Na 1 –H 1 K 1 –H 1 Rb 1 –H 1 Cs 1 –H 1
y(3,1)
y(2,1)
y(1,2)
y(0,3)
0.159 0.653 1.122 1.327 1.453
20.383 20.120 0.363 0.312 0.446
0.144 0.707 0.660 0.751 0.572
20.561 20.600 20.286 20.464 20.704
Exchange of alkali metal–hydrogen ions, t525.0 8C
Orthotetracycline
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
79
Chlorotetracycline
5.3. Difference in regularities of cation and anion exchange on strongly dissociating resins Log k 5 f(x¯ ) dependencies for styrene–DVB based anion exchangers with quaternary ammonium groups are much simpler than those for the sulfonic acid cation exchangers on the same matrix. They are usually rather close to straight lines. In spite of a strong influence of crosslinkage on the value of the selectivity coefficients, log k 5 f(x¯ ) dependencies remain almost the same with increasing cross-linkage (Figs. 13
Fig. 12. Log k 5 f(x¯ ) dependencies for exchange of orthotetracyclinium with sodium ion on Dowex 50 with different percent of DVB: 1, 0.5%; 2, 2%; 3, 6%; 4, 8%. The points are experimental data, the curves are computed from Eq. (6) with parameters given in Table 6. The data were obtained from Ref. [25].
Table 6 Application of the model equation to the description of log k 5 f(x¯ ) dependencies. Exchange of ortho- and chloro-tetracycline with Na 1 on the sulphostyrene resins (i54) Resin Orthotetracycline Dowex 5030.5 Dowex 5032 Dowex 5036 Dowex 5038 Chlorotetracycline SK-6 3 mm SK-6 22 mm Dowex 5035
y(4,0)
y(3,1)
y(2,2)
y(0,3)
0.83
2.22
20.57
3.38
0.91
1.39
1.71
0.68
3.91
20.61
1.58
1.97
0.63
17.53
1.51
3.69
221.43
2.27
2.04
2.47
1.31
3.22
1.91
1.84
3.79
1.16
3.35
1.89
2.54
1.73
4.40
0.26
180
y(0,4)
2105.1 21228
80
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
Fig. 13. H-fragments of polymer network of strong base resin with quaternary ammonium group with m- and p-DVB and calculated ˚ ht junctions. probabilities of the occurrence of zero to five neighbors in the sphere of interaction with radius 10 A;
and 14). The question arises why the effect of cross-linkage is so different for cation- and anion exchangers on the same matrix. We performed the same probability calculations of the anion exchangers as those described for the cation exchangers. It appeared that the structures in which the tetraalkylammonium group is situated in the cross-bridge are impossible. Incorporation of the quaternary ammonium group into the polymer structure is usually done in two stages. The first one is chloromethylation. As our calculations have shown no obstacles appeared at this stage for introducing of the chloromethyl group in the cross-bridge. This is similar to introduction of sulfonic
groups. At the second stage the chloromethyl group reacts with trimethylamine and converts into the quaternary amine group. The calculations done by MM2 method have shown that formation of such groups in the cross agent is practically impossible because of its large size. It would lead to formation of strongly strained structures having too high energy. Therefore we could only account for structures having no anion exchange groups in the cross-link. Figs. 13 and 14 show histograms of the probabilities in H-fragments of the anion exchangers. The probability of more than two neighbors in the sphere of interaction is zero and log k 5 f(x¯ ) according to the model may not have a power
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
81
Fig. 14. Calculated probabilities of occurrence of zero to five neighbors in H-fragments of polymer network of the strong base resin with quaternary ammonium group with m- and p-DVB as a function of distance from the exchange site.
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
82
Fig. 15. Log k 5 f(x¯ ) dependencies. For anion exchange on Dowex 1 with different cross-linkage. The points are experimental data [26], the curves are computed from Eq. (7) with parameters in Table 7.
greater than 2. Moreover, most of the exchange sites have either no neighbors or one neighbor and only a small fraction of them have two neighbors. A large difference between the cation and the anion exchangers is obvious. This should correspond to a rather weak and close to linear dependence log k 5 f(x¯ ) for this type of ion exchangers. Such a result of the calculations
2 2 Fig. 16. Log k 5 f(x¯ ) dependencies for ion exchange F –Br on Dowex 2 of different cross-linkage. Points are experimental data [27,28], the curves are computed from Eq. (7) with parameters in Table 8.
Table 8 Coefficients of polynomial 8c for curves (1–4) in Fig. 16 Curve no.
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4
0.5 2 4 10
5.58 5.12 7.08 9.97
6.11 6.64 8.47 12.18
7.40 8.19 9.90 14.21
The data are taken from Refs. [27,28]. Table 7 Coefficients of polynomial 8c for curves (1–8) in Fig. 15 Curve no.
Exchange
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4 5 6 7 8
2 NO 2 3 –Cl 2 NO 2 –Cl 3 2 2 Cl –OH 2 NO 2 3 –OH 2 NO 3 –OH 2 Br 2 –OH 2 2 NO 2 3 –OH Br 2 –OH 2
4 10 4 10 4 4 10 10
0.34 0.49 0.70 1.00 1.18 1.18 1.53 1.50
0.52 0.48 0.96 1.20 1.26 1.28 1.65 1.69
0.50 0.6 1.00 1.35 1.66 1.68 2.08 2.10
The data are taken from Ref. [26].
is in good agreement with experimental data, presented in Figs. 15–17.
6. Conclusion The mathematical model described in this paper allows interpretation of selectivity of ion exchange on spatial polyelectrolytes. It is based on the concept of effect of neighbors of the
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
2 2 Fig. 17. Log k 5 f(x¯ ) dependencies for ion exchange Cl –Br on Dowex 2 of different cross-linkage. Points are experimental data [27,28], the curves are computed from Eq. (7) with parameters in Table 9.
Table 9 Coefficients of polynomial 8c for curves (1–9) in Fig. 17 Curve no.
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4 5 6 7 8 9
0.5 1 2 4 6 10 8 24 16
5.53 6.98 9.40 8.38 10.39 11.25 15.74 17.23 16.29
5.41 7.98 9.67 10.13 11.8 13.24 14.55 16.02 15.84
6.29 7.99 9.37 10.63 12.13 13.35 13.98 15.61 16.19
The data are taken from Refs. [27,28].
exchange sites on the energy of ion exchange. Due to statistical nature of the spatial polyelectrolyte network different exchange sites have different numbers of neighbors, which predetermines dependence of selectivity on the ionic composition of the ion exchanger. The general equation ‘property–ionic composition’ is ex-
83
pressed by a special type of polynomial whose power is equal to maximal number of the neighbors. This means that a larger number of neighbors may cause more complicated property–composition dependencies. Computer modeling of representative fragments of the polymer network with eight to ten monomer units (linear chains, H- and T-shaped units and small rings) allows evaluation of the probability of different numbers of the neighbors at different distances from the exchange site. This allowed identification of fragments responsible for complexity of dependence of the selectivity coefficient on the ionic composition of the spatial polyelectrolytes. These fragments are H and ring structures with the functional group in the cross-bridge and different kinds of irregular structures with hh and tt monomer junctions. The models allow easy and natural interpretation of difference in dependencies of selectivity on the degree of ion exchange on spatial polyelectrolytes with different cross-linkage as well as different behavior of cation and anion exchangers on the basis of styrene–divinylbenzene matrixes. Peculiarities of ion exchange behavior of large organic ions are caused by a large number of neighboring exchange sites interacting with the counterion.
References [1] H.P. Gregor, J. Am. Chem. Soc. 70 (1948) 1293; H.P. Gregor, J. Am. Chem. Soc. 73 (1951) 642. [2] E. Glueckauf, Proc. R. Soc. (Lond.) A214 (1952) 207. [3] A. Katchalsky, S. Lifson, J. Polym. Sci. 11 (1953) 409. [4] F.E. Harris, S.A. Rice, J. Chem. Phys. 24 (1956) 1258. [5] J. Horst, W.H. Hoell, S. Eberle, React. Polym. 13 (1990) 209. [6] G.V. Samsonov, E.B. Trostianskaya, G.E. El’kin, in: Ion Exchange. Sorbtion of Organic Substances, Nauka, Leningrad, 1969, pp. 7–218, in Russian. [7] V.S. Soldatov, Dokl. Akad. Nauk. SSSR 34 (1990) 664. [8] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Dokl. AN Belarusi 37 (1993) 143. [9] V.S. Soldatov, React. Polym. 19 (1993) 105. [10] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Dokl. Russ. Akad. Nauk. 337 (1994) 63. [11] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, in: J.A. Greig (Ed.), Ion Exchange. Development and Applications, Royal Society of Chemistry, Cambridge, UK, 1996, p. 240.
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[12] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Dokl. AN Belarusi 40 (1996) 65. [13] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Polym. Sci. Ser. B 38 (1996) 380. [14] V.S. Soldatov, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Polym. Sci. Ser. B 40 (1998) 268. [15] V.S. Soldatov, T.V. Bezyazychnaya, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Polym. Sci. Ser. A 41 (1999) 380. [16] V.S. Soldatov, T.V. Bezyazychnaya, V.I. Gogolinskii, V.M. Zelenkovskii, A.L. Pushkarchuk, Zhurn. Fiz. Khim. 73 (1999) 1080. [17] R. Harjula, J. Lehto, React. Funct. Polym. 27 (1995) 121. [18] N.L. Allinger, J. Am. Chem. Soc. 99 (1977) 8127. [19] A.A. Granovsky, http: / / classic.chem.msu.su / gran / gamess / index.html. [20] M.W. Schmidt, K.K. Baldridge, J.A. Boatz et al., J. Comp. Chem. 14 (1993) 1347.
[21] S. Huszinaga, J. Andzelm, M. Klobukowski et al., Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam, 1984. [22] D. Reichenberg, D.J. McCauley, J. Chem. Soc. (1955) 2741. [23] V.S. Soldatov, R.V. Martzinkevich, A.I. Pokrovskaya et al., Zhurn. Fiz. Khim. 50 (1976) 480. [24] V.S. Soldatov, R.V. Martzynkevich, A.I. Pokrovskaya et al., Zhurn. Fiz. Khim. 43 (1969) 2889. [25] G.V. Samsonov, A.T. Melenevskij, in: Sorption and Chromatographic Methods of Physical Chemical Biotechnology, Nauka, Leningrad, 1986, pp. 121–127, (in Russian). [26] V.S. Soldatov, V.I. Sokolova, Izv. Akad. Nauk BSSR, Ser. Khim. Nauk. 6 (1969) 31. [27] G.E. Boyd, S. Lindenbaum, G.E. Myers, J. Phys. Chem. 64 (1961) 577. [28] G.E. Boyd, G.E. Myers, J. Phys. Chem. 60 (1961) 521.
Structure and properties of spatial polyelectrolytes on styrene–divinylbenzene matrixes V.S. Soldatov a,b , *, V.M. Zelenkovskii a , T.V. Bezyazychnaya a a
Institute of Physical Organic Chemistry of the National Academy of Sciences of Belarus, 13 Surganov St., Minsk 220072, Belarus b Institute of Environmental Engineering, Technical University of Lublin, Nadbystrzycka 20 -618, Lublin, Poland
Abstract A mathematical model establishing relations between the selectivity, degree of ion exchange and structure of spatial polyelectrolytes has been developed. The structure is expressed through two parameters: radius of interaction (that is a maximal distance at which the exchange sites influence each other) and the number of exchange sites situated in the sphere of interaction (neighbors). A larger number of neighbors should result in more complicated shape of property–composition dependencies of the ion exchange system. A computer model allowing estimation of the probability of the presence of different numbers of neighbors in different elements of the polymer network as a function of distance from the exchange site was suggested. The polymer network is represented as a body obtained by integration of fragments with eight to ten monomeric units of different shapes: linear chains, H and T shape units and small rings. The theory was applied to interpreting dependencies of selectivity of cation exchange on sulfonic styrene–divinylbenzene cation exchangers and strong base anion exchangers on the same matrix. Dependencies of selectivity on degree of ion exchange and cross-linkage of the resins are in good agreement with the prediction of the theory if the predominant number of neighbors in different particular cases is ranging between two and four. The non-equivalence of exchange sites of spatial polyelectrolytes is their inherent property because different exchange sites have different amounts of the neighbors. The difference in the selectivity dependence on the cross-linkage of cation and anion exchangers on styrene–divinylbenzene matrix finds a simple and natural explanation. 2002 Elsevier Science B.V. All rights reserved. Keywords: Polyelectrolytes; Ion exchangers; Mathematical modeling; Computer modeling; Selectivity
1. Introduction Establishing quantitative relations between properties, ionic composition and structure of spatial polyelectrolytes requires clear definition of concept ‘structure’ in mathematical terms. In *Corresponding author. Tel.: 1375-17-284-2338; fax: 137517-284-2338. E-mail address: [email protected] (V.S. Soldatov).
relation to spatial polyelectrolytes this concept is not as certain as that for simple substances whose molecular structure is strictly determined. Spatial polymers consist of mixtures of molecules of different sizes and shapes forming in the process of statistical co-polymerization of the initial monomers. Therefore, it is impossible to relate their properties to a substance with some definite type of molecules or mixture of the molecules with a certain composition.
1381-5148 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S1381-5148( 02 )00183-9
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The properties of interest in this paper, Y, are those related to equilibrium of binary ion exchange I 1 2I 2 between the spatial polyelectrolyte and the external solution. They are thermodynamic functions, equilibrium and selectivity coefficients and water uptake by the polyelectrolyte. They depend on the ionic composition of the polyelectrolyte, expressed by the ¯ thereequivalent fraction of the entering ion x; fore Y 5 Y(x¯ ). A number of theoretical models were suggested for relating the ion exchange properties and structure of spatial polyelectrolytes. Some of them, such as the Gregor [1] and Glueckauf models [2], indirectly account for the structure of polyelectrolyte by introducing separate pressure-swelling term into the property Y(x¯ ). In the other group of models structure of the polyelectrolyte was expresses in more certain ways. Katchalsky and Lifson [3] represent the spatial polyelectrolyte as a collection of small rigid rods, joined into continuous flexible chains forming a spatial net. Applying the DebyeHuckel distribution equation, some statements of Flory theory and equation of Donnan equilibrium to such a system, some of properties Y(x¯ ) were found for weakly dissociating polyacids of different cross-linkage. Harris and Rice [4] suggested a model of polymeric ion exchanger representing it as a set of electrically charged plates joined by springs into a continuous three-dimensional structure. Hoell [5] developed a model describing the ion exchanger as an electrical condenser in which the counterions form the Nernst-type layers at different distances from the matrix chain depending on their charge and water content in the polyelectrolyte. Structural parameters of Flory theory were used to describe properties of ion exchangers by Trostianskaya [6]. In spite of poor correlation of properties of ion exchangers with the statements of the Flory theory, reasonable agreement between theory and properties of weakly cross-linked ion exchangers was observed in many cases. In the mentioned theoretical models some
features of structure of the spatial polyelectrolyte were explicitly considered, while the others were not accounted for at all. These models do not account for real molecular structure of polyelectrolytes and contain parameters with unclear physical meaning, which can be arbitrarily varied for obtaining agreement between the theory and experiment. Therefore their interpretive ability is extremely limited. In our recent publications we described a model for interpretation of dependence of properties of ion exchange systems Y(x¯ ) on the structure of the polyelectrolyte, expressed as a set of parameters determined by its molecular structure. The present paper is a review of these works [7–16]. Our approach to the problem in question is based on the concept of interaction between the nearest neighbors and includes two steps. In the first one we developed a mathematical model defining dependence of property Y on the ionic composition of the polyelectrolyte and the num¯ i). In the ber of the nearest neighbors i, Y 5 Y(x, second step a method for determination of the structural parameters needed for the Y 5 Y(x¯ i ) function was developed. It is based on a computer modeling of the molecular structure of the spatial polyelectrolyte. It was applied to styrene–divinylbenzene based spatial polyelectrolytes with sulfonic and benzyltrimethyl ammonium groups, being the most spread types of ion exchange resins.
2. Mathematical model
2.1. Definitions Exchange site—a functional group on a polymer matrix whose charge is compensated by a counterion. The location of the exchange site is defined to coincide with the center of the sulfur or nitrogen atoms in the functional groups. Interaction radius, R—the maximal distance between two exchange sites where the interaction between a counterion and the exchange
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
site is regarded as being significantly influenced by counterion exchange at the neighboring exchange site. Sphere of interaction—a sphere with radius R drawn around an exchange site. Microenvironment—a part of a sphere of interaction except for the central exchange site. Nearest neighbors (neighbors)—the neighboring exchange sites at a distance less or equal to the radius of interaction from the central site. Property Y—any additive property of the macroscopic system which can be obtained by summation of the same properties of all its micro-states. For example, Y can be specific Gibbs’ energy, enthalpy in a process when infinitely small amounts of ion 1 are replaced by ion 2. It also can be specific volume or water contents in the ion exchange, logarithm of the selectivity or equilibrium coefficient. Property y–property Y related to the central exchange site and its microenvironment. Ionic composition—expressed by equivalent fraction of the entering ion (x¯ 2 ) in ion exchange I 1 –I 2 ; x¯ 1 1 x¯ 2 5 1. In the following we shall use ¯ definition x¯ 2 5x. Local ionic composition—ionic composition in the microenvironment. For simplicity we consider only exchange of monovalent ions. An ion exchange equilibrium between the spatial polyelectrolyte and the solution is formulated as follows: I¯1 1 I2 ↔ I1 1 I¯2
(1)
where the barred symbols are related to the resin phase. Equilibrium coefficient, k is defined by Eq. (2). For homovalent ion exchange it coincides with selectivity coefficient.1 x¯ (1 2 x) k 5 ]]]. x(1 2 x¯ )
1
(2)
These terms were recommended by The International Workshop on Uniform and Reliable Nomenclature, Formulations and Experimentation for Ion Exchange, Helsinki, Finland, May 30–June 1, 1994 [17].
65
2.2. Statements Different exchange sites in the spatial polyelectrolyte may have different numbers of neighbors, i. It can vary from zero to n depending on the choice of the interaction radius and the structure of the matrix. Property y is characterized by the number of the neighbors and a local ionic composition. It is independent of the position of the neighbors relative to the central exchange site. A property Y can be represented as a weight sum of properties y.
2.3. Property–ionic composition equation In the present approximation of the model two parameters were chosen to describe the state of the central ion in the microenvironment: a number of neighboring counterions of type 2 denoted as j and that of ions 1 equal to i 2 j. Then each of the possible states of ion I can be expressed as I(i 2 j, j) (Table 1) and property Y related to this state is denoted as y(i 2 j, j). Property Yi can be obtained as a sum of properties y(i 2 j, j) proportional to the probabilities of the existence of exchange sites in the relative states p(i 2 j, j): j51
Yi 5
O y(i 2 j, j) p(i 2 j, j)
(3)
j50
In its turn, a macroscopic property Y of the overall system can be obtained from the Yi and the probability of existence of i neighbors of the exchange site in a real system, Pi : Table 1 States of the central ion in the microenvironment i, number of neighbors
States of ion I
0 1 2 3
I(0,0) I(1,0), I(0,1) I(2,0), I(1,1), I(0.2) I(3,0), I(2,1), I(1,2), I(0,3)
i
I(i 2 j, j), j502i
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
66
O YP.
i 5n
Y5
i
(4)
i
i 50
The values p(i 2 j, j) are proportional to the molar fraction of each of the counterions raised to a power equal to the number of the neighbors of a relative type. It is also proportional to a number of permutations of the counterions 1 and 2 within a given combination of i counterions with j counterions of type 2. If the probabilities are normalized so that their sum is equal to 1 then i! p(i 2 j, j) 5 ]]] (l 2 x¯ )(i 2j ) x¯ j (i 2 j)!j!
(5)
and j51
Yi 5
i! O ]]] y(i 2 j, j) (l 2 x¯ ) (i 2 j)!j!
(i2j )
x¯ j
(6)
j50
The general equation ‘property–ionic composition’ for a real system is j51
i! O P O ]]] y(i 2 j, j) (l 2 x¯ ) (i 2 j)!j!
i5n
Yi 5
i
i50
(i2j )
equations are polynomial presented in a special form (Bernstein polynomial). They have features making them especially convenient if x¯ varies in interval [0,1], i.e. for consideration of ‘property–composition’ dependencies. Coefficients of the first and the last terms of Eqs. (8) are equal to values of Yi at x¯ 5 0 and x¯ 5 1. The intermeddle coefficients control the curvature of the line Yi 5 f(x¯ ). More detailed analysis of properties of the functions expressed by Eq. (8) is given in Ref. [9]. Eq. (7) shows that ‘property–ionic composition’ dependence should have a power equal to maximal number of neighbors and can be used for interpretation of experimental data by comparing them with relative theoretical dependencies. Quantities i and y(i 2 j, j) should be determined independently or found as fitting parameters. At the present state of development of described model we can evaluate value i by computer modeling of the polyelectrolyte structure. A method for their estimation is described in the following section.
x¯ j
j50
(7) The number of neighbors in real systems is rarely more than four. Thus, Eq. (6) represents the following set of equations: Y0 5 y(0,0)
(8a)
Y1 5 y(1,0)(1 2 x¯ ) 1 y(0,1)x¯
(8b)
2
Y2 5 y(2,0)(1 2 x¯ ) 1 2y(1,1)(1 2 x¯ )x¯ 1 y(0,2)(1 2 x¯ )x¯
2
(8c)
3 2 Y3 5 y(3,0)(1 2 x¯ ) 1 3y(2,1)(1 2 x¯ ) x¯
1 3y(1,2)(1 2 x¯ )x¯ 2 1 y(0,3)x¯ 3
(8d)
4 3 Y4 5 y(4,0)(1 2 x¯ ) 1 4y(3,1)(1 2 x¯ ) x¯ 2 2 3 1 6y(2,2)(1 2 x¯ ) x¯ 1 4y(1,3)(1 2 x¯ )x¯
1 y(0,4)x¯
4
(8e)
Since all y(i 2 j, j) values are constant, these
3. Computer model
3.1. Elementary fragments We assume that a statistical spatial polymer matrix can be represented by a set of elementary fragments with known structure. They are randomly integrated into a matrix and a property of the integrated system can be found as a weight average of the properties of the fragments. The functional group is assumed to be situated in para-position related to the chain. The following types of fragments were chosen to represent the polymer structure: linear chain (L), H- and Tshaped fragments, and small loops (rings, R). Any other structures can be obtained as combinations of these fragments. The choice of size of the elementary fragments is important. We accepted it as a junction of eight to ten monomeric units, since with the large fragments it would be impossible to represent the structure of highly cross-linked resins. The smaller fragments do not have statistical meaning.
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84 Table 2 The letter codes for different fragments of the spatial polymer net Type of the
Letter code
fragment
No funct. group in DVB
Linear chain
L(ht) L(hh) L(tt)
T-fragment
T(ht) T(hh) T(tt)
T(ht)* T(hh)* T(tt)*
H-fragment
H(ht) H(hh) H(tt)
H(ht)* H(hh)* H(tt)*
Rings a
R(ht)N R(hh)N R(tt)N
R(ht)N* R(hh)N* R(tt)N*
Funct. group in DVB
a In place of N in each specific case should stay the number of the monomer units in the ring. * Presence of the functional group in the DVB residue.
Structure of the elementary fragments can be different. ‘Head to tail’ (ht) junctions are predominant, but there are no convincing arguments to discriminate the hh and tt junctions. We name hh and tt junction irregularities. Several varieties of all types of fragments, differing in the type of junctions, the cross-bridge type (para- or metadivinylbenzene (DVB)), presence or absence of the functional group in the DVB fragment, were considered. We suggest assigning letter codes to different types of fragments for easier referring. Their list is given in Table 2. When referring to a particular structure, it should be additionally specified which crossagent (in our case meta or para DVB) and which initiator of polymerization were used. The letter is important for specifying the Tfragment structure. In our case it was benzoil peroxide.
3.2. Method for the probabilities evaluation Knowledge of the probabilities of different numbers of neighbors, pi , in each type of
67
representative fragments k is required for identification of structural elements of the polyelectrolyte matrix responsible for degree of complexity of the Y 5 f(x¯ ) dependence. We describe our method for their evaluation with the example of one of the structures, namely H(ht)*-pDVB and sulfonic group as an exchange site. The structural formula and the optimized structure of this fragment, obtained by computer modeling, are presented in Fig. 1. The structure in Fig. 1 was calculated by the molecular mechanics method with the force field MM2 [18]. The procedure of these calculations was as follows. Since the fragment does not represent a separate molecule and is a part of the larger structure, as a first step it is necessary to optimize geometric size and the structure of the fragment by variation of the distance of the each functional group from the central sulfonic group or the center of the cross agent residual. The angle between the sub-fragments in the H- and T-fragments was also varied. In each case more than 200 conformations were examined. From all considered conformations only those with low energies were taken for further consideration, whose introduction in the total polymer structure could be meaningful accounting for their Boltzman’s population. The number of meaningful conformations was usually |20–30. The first step of these calculations was computing the total energy of the fragment at small variations of the initially chosen distance to search for the energy minimum. The energy appeared to be highly sensitive to the distance value. The distance corresponding to the energy minimum (the equilibrium value) could be different from the initially chosen one by several tenths of an angstrom. This distance was taken for the energy calculations. The structure and energy characteristics of different conformers have been computed in molecular mechanics approximation using force field MM2. If the elementary fragments are different only in conformation, the probability of existence of i neighbors in a specified type of the fragments with 1, 2 . . . 3 . . . q . . . s conformers depends
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
68
Finally, the probabilities Pi of the presence of i neighbors in a network can be found by summing of the pi proportionally to the probabilities of existence fragments themselves. Pi 5
Op
i
(10)
4. Results of the computer simulation of the polyelectrolyte structure
4.1. Evaluation of the radius of interaction
Fig. 1. Schematic drawing (a) and computer image (b) of an optimized H-shaped fragment of sulfonated styrene–divinylbenzene copolymer. The distances between the central S atom and its three nearest neighbors are shown.
only on the energy of the conformers according to Boltzman’s distribution: exp(2Eq /RT ) pi 5 ]]]]]] . s exp(2Es /RT )
O
q 51
(9)
Choosing the value for the interaction radius is the most uncertain operation in the whole model. It can be expected that this value may depend on the structure of the ion exchanger and the nature of the exchangeable ions. Therefore such an estimate has to be done individually on the basis of knowledge of properties of any specific type of the system. The experimental data on the heat effects of ion exchange of selectivity of ion exchange of simple inorganic ions on resins with low crosslinkage (Fig. 9) are rather simple and require the model Eq. (8c) of the second power at most. Hence, in these cases two neighbors have to be accounted for. As seen from the probability histograms (Figs. 3 and 4) this is possible only if in linear fragments of the polymer networks ˚ the radius of interaction is not greater that 10 A. At its higher values the probability of more than two neighbors becomes high and the properties should obey Eq. (8d)–(8e) of the power higher than two. This does not agree with the experimental data. On the other hand, the probability of the presence in the sphere of interaction of more than two neighbors in the nonlinear fragments becomes substantial at the ˚ Redistances already slightly less than 10 A. spectively, properties of the sulfonic ion exchangers with a high cross-linkage as a function of their ionic composition are described by Eq. (8)d of the third power. Based on these considerations the radius of interaction for sul-
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
fostyrene ion exchangers in forms of simple monovalent inorganic ions was taken equal to ˚ 10 A.
Fig. 2. Computer images of three stable conformations of Na salt of a dimer of sulfostyrene hydrated with 20 water molecules, corresponding to minima of the energy. The distances between the centers of the S atoms and hydrogen bonds forming bridges between the sulfonic groups are shown as lines.
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Additional arguments in favor of this value were obtained with the help of quantum chemical calculations of the most probable configurations of a dimer of p-sulfostyrene (ht junction) in the presence of 20 water molecules. The charges of the sulfonic groups were compensated by Na 1 ions. The calculations of the structure of this fragment were done with the help of non-empirical method SCF MO LCAO and program GAMESS [19,20] using basis set MINI [21]. Initial distances between the sulfur atoms were chosen by variation of torsion angles formed by four carbon atoms of the hydrocarbon chain in the fragment. Optimizing of the coordinates of all atoms in the system was done to find the total energy minimum. The calculations have shown that only three stable states of the fragment exist. They are presented in Fig. 2. The first configuration corresponds to the ˚ distance between the sulfur atoms (6.28 A) interacting with each other via a water molecule and having common hydrate structure with water molecules bound by hydrogen bonds. The second local minimum of energy was found at a ˚ between the S atoms. distance of 7.98–8.20 A In this case the groups interact via a chain of two water molecules and also form one common hydrate structure. Increasing the distance ˚ leads between the sulfonic groups above 10 A to spontaneous separation of the sulfonic groups ˚ Their hydrate to a distance of 11.5–11.6 A. structure splits into two independent structures. These facts indicate absence of significant interaction between sulfonic groups situated at a ˚ At such distances distance greater than 10 A. electrostatic interaction between sulfonic groups in the presence of water molecules is weaker than the conformation strength in the hydrocarbon chain. In this case the latter determines geometric shape of the structure. At distances ˚ a decisive role belongs to shorter than 10 A, electrostatic interaction in the common hydration structure, the sulfonic groups spontaneously approach each other until they form stable structures 2 and 1 with a lower energy for the
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latter structures. The least energetically profitable is structure 3.
4.2. Linear chains ( L-fragments) We calculated the probabilities of the presence of different numbers of neighbors in linear fragments containing eight repeated units of the polyelectrolyte in which (a) all of them are
joined in the ht structures, (b) all junctions are ht except the one between the fourth and the fifth units where the junctions are hh or (c) tt types. The histograms of the probabilities of presence of zero to five neighbors as a function of distance from the exchange site are presented in Figs. 3 and 4. It appeared that all three sets of histograms are substantially different. In the regular chain
Fig. 3. Linear chain fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere ˚ of interaction with radius 10 A.
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
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Fig. 4. Calculated probabilities of occurrence of zero to five neighbors in different linear fragments of the sulfonic acid polyelectrolyte as a function of distance from the exchange site.
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the situation when no neighbors are present at a ˚ is impossible. At the distance greater than 5 A same time, the probability of that is rather high in the linear fragments with both types of irregularities. One and two neighbors are most probable in the sphere of interaction while their radial distribution is quite different. The presence of more than two neighbors in the distance ˚ is not very probable in the less than 10 A regular chain and is highly probable for the hh and (somewhat lower) for the tt structure. Presence of more than three neighbors in the sphere of interaction is not very probable in all cases. Four and five neighbors are highly prob˚ which can able at distances greater than 10 A, be important for large organic counterions.
4.3. H-fragments Ion exchangers may contain many different types of H-shaped fragments. The difference between them is caused by the following reasons: • Different cross-linking agent (meta- or paradivinylbenzene); • Sulfonated or non-sulfonated benzene ring of the cross-agent; • Different sequence of junction of the monomeric units in the sulfostyrene chains. The probabilities pi for different cases in ˚ are illustrated by dependence on i and R510 A Figs. 5 and 6.
4.4. T-fragments T-fragments form in the polymerization process when the second double bond of divinylbenzene residue in the polymeric chain is activated by a reaction with an initiator of polymerization. Therefore the structure of this junction depends on the type of the initiator. In our case it was benzoil peroxide. Several possible structures of the T-fragment are possible. The difference between them is illustrated by Fig. 7. Only ht sequence was considered.
4.5. Rings ( R-fragments) We present estimates for structural parameters i and Pi for small rings having a general structure of the type presented in Fig. 8 and Table 3. The calculations were performed for the Rstructures containing two to six molecular fragments in the ring. The monomers were assumed to contact by the ht type only.
4.6. Summary of the probability calculations The radial distribution of the probability of existence of different numbers of neighbors is highly sensitive to the structure of the polymer network. In the sphere of interaction of all considered polymeric fragments of regular structure with non-sulfonated cross agent almost only exchange sites with one or two neighbors present. Any irregularity, hh or tt junctions, leads to increasing the probabilities of appearance of exchange sites with greater number of neighbors. Presence of a sulfonic group in the structure of the cross-bridge increases the probability of existence of higher number of neighbors, while absence may lead to the Pi decrease. Irregular H- and R-fragments with non-sulfonated crossbridges may have a large probability of i .2. We can identify the fragments of the spatial polyelectrolyte net, which are responsible for a complicated shape of dependence Y 5 Y(x¯ ) because of substantial concentration of exchange sites with large number of neighbors in the sphere of interaction. They are: irregular linear chains (L-hh and L-tt); H-fragments with sulfonated cross-bridge (Figs. 3–5); and some rings with three to six sulfonated phenyl radicals (Table 3). 5. Application of the model for interpretation of ion exchange equilibria
5.1. Effect of cross-linkage It is known from the first systematic studies
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Fig. 5. H-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ ht junctions. interaction with radius 10 A;
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Fig. 6. H-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ hh and tt junctions. interaction with radius 10 A;
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
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Fig. 7. T-fragments with different structures and calculated probabilities of the occurrence of zero to five neighbors in the sphere of ˚ ht junctions. interaction with radius 10 A;
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Fig. 8. R-fragment with eight styrene units in the ring and calculated probabilities of the occurrence of zero to five neighbors in the sphere ˚ ht junctions. of interaction with radius 10 A;
Table 3 ˚ Probabilities of the presence of n nearest neighbors in various R-structures for the interaction radius R s 510 A N
p-R3
p-R3*
m-R2
m-R2*
p-R5
p-R3*
m-R5
p-R6
p-R6*
p-R6*
m-R6
m-R6*
0 1 2 .2
0.289 0.261 0.347 0.103
0.000 0.086 0.487 0.427
0.023 0.962 0.015 0.000
0.203 0.491 0.268 0.039
0.000 0.848 0.152 0.000
0.000 0.668 0.316 0.017
0.179 0.439 0.382 0.000
0.000 0.443 0.423 0.134
0.000 0.187 0.638 0.176
0.000 0.144 0.528 0.329
0.145 0.468 0.341 0.047
0.143 0.286 0.199 0.373
of selectivity of ion exchange of alkali metal ions with H 1 [22] that increasing cross-linkage of sulfostyrene ion exchangers causes complicated changes in the dependence of selectivity coefficient on the degree of exchange. Increasing percent of DVB leads to increase in the selectivity coefficient at low loadings of the resins with the metal ions and may cause the opposite effect on the selectivity at high loadings; the shape of curves k 5 k(x¯ ) becomes more complicated. In order to explain these phenomena several varieties of hypotheses of non-equivalence of the exchange sites in ion exchangers were suggested. But the reasons for such irregularity were not convincingly explained. Now we can assert that the nonequivalence of exchange sites is a direct consequence of structural non-regularity of statistical polymer networks manifested in different numbers of the neighbors of different exchange sites. Therefore non-equivalence of the ex-
change sites can be considered an immanent property of spatial polyelectrolytes. In the simplest case, corresponding to exchange of small ions and ion exchangers of low cross-linkage dependencies of equilibrium coefficient and the other properties on the degree of ion exchange should be expressed by Eq. (8)c with the second power with a weak curvature because of presence of large amount of groups with one neighbor. Increasing degree of cross-linkage in terms of the model described means increase in concentration of H and R fragments, as well as (possibly) the irregularities. Figs. 9 and 10 illustrate these deductions. Log k 5 f(x¯ ) and DH 5 f(x¯ ) dependencies for the resins with DVB percent below 12 are described with Eq. (8)c of the second power. The same dependencies for the resins with a higher percent of DVB have specific features inherent to polynomials of higher power, for example inflections are clearly seen
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Table 4 Application of the model equation to description of dependencies for the KRSm resins % mDVB
Y
y(3,0)
y(2,1)
y(1,2)
y(0,3)
1 5 5 8 8 12 12 16 25 25
log k log k DH log k DH log k DH log k log k DH
0.154 0.359 21.90 0.500 22.60 0.874 23.33 1.361 1.448 25.59
0.101 0.333 21.18 0.475 22.14 0.390 22.05 0.160 0.518 21.12
0.099 0.165 22.04 0.282 21.60 0.629 22.43 0.904 0.759 23.44
0.104 0.201 20.48 0.350 20.54 0.119 0.010 20.13 20.38 0.28
i53, exchange K 1 –H 1 , t525.0 8C.
for the properties of highly cross-linked resins. Dependence log k 5 f(x¯ ) for the ion exchanger with very high cross-linkage (25% DVB) for all alkali ions requires Eq. (8)d of the third power (Fig. 11). Two additional notes are relevant. (a) In order to determine correctly the power of the polynomial Y 5 Y(x¯ ) property Y should be determined ¯ The regions in a sufficiently wide interval of x. ¯ ¯ near x50 and x 5 1 are very important [9]. (b) Dependence Y 5 Y(x¯ ) is sensitive to the structure, therefore we can expect that ion exchangers of the same chemical composition but obtained under different conditions of formation of the polymer network may have different properties.
Fig. 9. Dependencies of enthalpy of the degree of ion exchange H 1 –K 1 on styrene m-DVB ion exchanger KRSm with different percent of m-DVB: 1, 1%; 2, 5%; 3, 8%; 4, 12%; 5, 16%; 6, 25%; t 5 25 8C. The data were obtained by measuring heat effect of ion exchange. The beginning and the end of the arrows show the direction, starting and end points of the process in the calorimetric measurements [23]. The points are experimental data, the curves are computed from Eq. (6) with parameters given in Table 4.
5.2. Large organic ions Regularities of selectivity of ion exchange with participation of large organic ions are very complicated [6] and are usually qualitatively explained by interplay of several strong factors, such as change in water structure of the external and internal solution in the ion exchange system; interaction of hydrophobic parts of the counterions with the polymeric matrix; hydrogen bonding and other specific interactions of the counterions with the functional groups and between themselves. It is likely that these factors mainly affect the value of the selectivity
measures rather than their dependence on the degree of loading of the ion exchanger. Such dependence should be affected by the structure of the resins, expressed by the above described parameters. The peculiarity of the organic ions in most cases is their large size, exceeding the radius of the sphere of interaction determined for small counterions. This means that in the addition to the neighbors present in the sphere of interaction (defined for simple inorganic ions) additional exchange sites may exist, situated outside this sphere and interacting with the large counterion. So the total number of neighbors
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78
Fig. 10. Log k 5 f(x¯ ) dependencies. See caption to Fig. 9.
affecting the ion exchange may be larger than that for the inorganic ions in the same ion exchanger. For example, the length of cations of different derivatives of tetracycline significantly ˚ and it is not surprising that exceeds 10 A description of log k 5 f(x¯ ) dependencies re-
Fig. 11. Log k 5 f(x¯ ) dependencies for exchange of alkali metals with hydrogen ions, ion exchanger KU-2X25% DVB, t525 8C. The points are experimental data, the curves are computed from Eq. (6) with parameters in Table 5 [24].
quires Eq. (8)e of the fourth power indicating importance of four neighbors (Fig. 12).
Table 5 Application of the model equation to the description of dependencies for the KU-2X25 Exchange 1
1
Li –H Na 1 –H 1 K 1 –H 1 Rb 1 –H 1 Cs 1 –H 1
y(3,1)
y(2,1)
y(1,2)
y(0,3)
0.159 0.653 1.122 1.327 1.453
20.383 20.120 0.363 0.312 0.446
0.144 0.707 0.660 0.751 0.572
20.561 20.600 20.286 20.464 20.704
Exchange of alkali metal–hydrogen ions, t525.0 8C
Orthotetracycline
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Chlorotetracycline
5.3. Difference in regularities of cation and anion exchange on strongly dissociating resins Log k 5 f(x¯ ) dependencies for styrene–DVB based anion exchangers with quaternary ammonium groups are much simpler than those for the sulfonic acid cation exchangers on the same matrix. They are usually rather close to straight lines. In spite of a strong influence of crosslinkage on the value of the selectivity coefficients, log k 5 f(x¯ ) dependencies remain almost the same with increasing cross-linkage (Figs. 13
Fig. 12. Log k 5 f(x¯ ) dependencies for exchange of orthotetracyclinium with sodium ion on Dowex 50 with different percent of DVB: 1, 0.5%; 2, 2%; 3, 6%; 4, 8%. The points are experimental data, the curves are computed from Eq. (6) with parameters given in Table 6. The data were obtained from Ref. [25].
Table 6 Application of the model equation to the description of log k 5 f(x¯ ) dependencies. Exchange of ortho- and chloro-tetracycline with Na 1 on the sulphostyrene resins (i54) Resin Orthotetracycline Dowex 5030.5 Dowex 5032 Dowex 5036 Dowex 5038 Chlorotetracycline SK-6 3 mm SK-6 22 mm Dowex 5035
y(4,0)
y(3,1)
y(2,2)
y(0,3)
0.83
2.22
20.57
3.38
0.91
1.39
1.71
0.68
3.91
20.61
1.58
1.97
0.63
17.53
1.51
3.69
221.43
2.27
2.04
2.47
1.31
3.22
1.91
1.84
3.79
1.16
3.35
1.89
2.54
1.73
4.40
0.26
180
y(0,4)
2105.1 21228
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Fig. 13. H-fragments of polymer network of strong base resin with quaternary ammonium group with m- and p-DVB and calculated ˚ ht junctions. probabilities of the occurrence of zero to five neighbors in the sphere of interaction with radius 10 A;
and 14). The question arises why the effect of cross-linkage is so different for cation- and anion exchangers on the same matrix. We performed the same probability calculations of the anion exchangers as those described for the cation exchangers. It appeared that the structures in which the tetraalkylammonium group is situated in the cross-bridge are impossible. Incorporation of the quaternary ammonium group into the polymer structure is usually done in two stages. The first one is chloromethylation. As our calculations have shown no obstacles appeared at this stage for introducing of the chloromethyl group in the cross-bridge. This is similar to introduction of sulfonic
groups. At the second stage the chloromethyl group reacts with trimethylamine and converts into the quaternary amine group. The calculations done by MM2 method have shown that formation of such groups in the cross agent is practically impossible because of its large size. It would lead to formation of strongly strained structures having too high energy. Therefore we could only account for structures having no anion exchange groups in the cross-link. Figs. 13 and 14 show histograms of the probabilities in H-fragments of the anion exchangers. The probability of more than two neighbors in the sphere of interaction is zero and log k 5 f(x¯ ) according to the model may not have a power
V.S. Soldatov et al. / Reactive & Functional Polymers 54 (2003) 63–84
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Fig. 14. Calculated probabilities of occurrence of zero to five neighbors in H-fragments of polymer network of the strong base resin with quaternary ammonium group with m- and p-DVB as a function of distance from the exchange site.
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Fig. 15. Log k 5 f(x¯ ) dependencies. For anion exchange on Dowex 1 with different cross-linkage. The points are experimental data [26], the curves are computed from Eq. (7) with parameters in Table 7.
greater than 2. Moreover, most of the exchange sites have either no neighbors or one neighbor and only a small fraction of them have two neighbors. A large difference between the cation and the anion exchangers is obvious. This should correspond to a rather weak and close to linear dependence log k 5 f(x¯ ) for this type of ion exchangers. Such a result of the calculations
2 2 Fig. 16. Log k 5 f(x¯ ) dependencies for ion exchange F –Br on Dowex 2 of different cross-linkage. Points are experimental data [27,28], the curves are computed from Eq. (7) with parameters in Table 8.
Table 8 Coefficients of polynomial 8c for curves (1–4) in Fig. 16 Curve no.
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4
0.5 2 4 10
5.58 5.12 7.08 9.97
6.11 6.64 8.47 12.18
7.40 8.19 9.90 14.21
The data are taken from Refs. [27,28]. Table 7 Coefficients of polynomial 8c for curves (1–8) in Fig. 15 Curve no.
Exchange
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4 5 6 7 8
2 NO 2 3 –Cl 2 NO 2 –Cl 3 2 2 Cl –OH 2 NO 2 3 –OH 2 NO 3 –OH 2 Br 2 –OH 2 2 NO 2 3 –OH Br 2 –OH 2
4 10 4 10 4 4 10 10
0.34 0.49 0.70 1.00 1.18 1.18 1.53 1.50
0.52 0.48 0.96 1.20 1.26 1.28 1.65 1.69
0.50 0.6 1.00 1.35 1.66 1.68 2.08 2.10
The data are taken from Ref. [26].
is in good agreement with experimental data, presented in Figs. 15–17.
6. Conclusion The mathematical model described in this paper allows interpretation of selectivity of ion exchange on spatial polyelectrolytes. It is based on the concept of effect of neighbors of the
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2 2 Fig. 17. Log k 5 f(x¯ ) dependencies for ion exchange Cl –Br on Dowex 2 of different cross-linkage. Points are experimental data [27,28], the curves are computed from Eq. (7) with parameters in Table 9.
Table 9 Coefficients of polynomial 8c for curves (1–9) in Fig. 17 Curve no.
DVB %
y(2,0)
y(1,1)
y(0,2)
1 2 3 4 5 6 7 8 9
0.5 1 2 4 6 10 8 24 16
5.53 6.98 9.40 8.38 10.39 11.25 15.74 17.23 16.29
5.41 7.98 9.67 10.13 11.8 13.24 14.55 16.02 15.84
6.29 7.99 9.37 10.63 12.13 13.35 13.98 15.61 16.19
The data are taken from Refs. [27,28].
exchange sites on the energy of ion exchange. Due to statistical nature of the spatial polyelectrolyte network different exchange sites have different numbers of neighbors, which predetermines dependence of selectivity on the ionic composition of the ion exchanger. The general equation ‘property–ionic composition’ is ex-
83
pressed by a special type of polynomial whose power is equal to maximal number of the neighbors. This means that a larger number of neighbors may cause more complicated property–composition dependencies. Computer modeling of representative fragments of the polymer network with eight to ten monomer units (linear chains, H- and T-shaped units and small rings) allows evaluation of the probability of different numbers of the neighbors at different distances from the exchange site. This allowed identification of fragments responsible for complexity of dependence of the selectivity coefficient on the ionic composition of the spatial polyelectrolytes. These fragments are H and ring structures with the functional group in the cross-bridge and different kinds of irregular structures with hh and tt monomer junctions. The models allow easy and natural interpretation of difference in dependencies of selectivity on the degree of ion exchange on spatial polyelectrolytes with different cross-linkage as well as different behavior of cation and anion exchangers on the basis of styrene–divinylbenzene matrixes. Peculiarities of ion exchange behavior of large organic ions are caused by a large number of neighboring exchange sites interacting with the counterion.
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