Collapse of polyelectrolyte networks induced by their interaction with oppositely charged surfactants

Makromol. Chem., Theory Simul. 1, 105 -118 (1992) 105

Collapse of polyelectrolyte networks induced by their interaction with an oppositely charged surfactant. Theory

AIexei R. Khokhlov: EIena Yu. Kramarenko, Elena E. Makhaeva, Sergei G. Starodubtzev

Physics Department, Moscow State University, Moscow 1 17234, USSR

(Date of receipt: November 25, 1991)

SUMMARY A very simple theory of swelling and collapse of weakly charged polyelectrolyte networks in the

solution of an oppositely charged surfactant has been developed. The following contributions to the free energy were taken into account: free energy of volume interaction and of elastic deformation of the network chains, free energy connected with micelle formation and free energy of translational motion of all mobile ions in the system (translational entropy). Both the cases of a solution of charged surfactant and that of a mixed solution of charged and neutral surfactant components have been taken into account. It has been shown that the behaviour of the network depends on the total surfactant concentration in the system and corresponds to one of the three following regimes: At low concentration, micelles inside the network are not formed and the behaviour of the polymer network is similar to that of a network swelling in the solution of a low- molecular-weight salt (regime 1). In the second regime, surfactant concentration inside the network exceeds the critical micelle concentration and micelles are formed; in this regime the network collapses because surfactant molecules, aggregated in micelles, cease to create “exerting” osmotic pressure in the network sample. In the third regime, at very high surfactant concentration, formation of additional micelles inside the network ceases, and the network dimensions coincide with those of the corresponding neutral network.

Introduction

Polymer networks in a solvent can undergo a sharp decrease in volume under change of external parameters (temperature, quality of solvent etc.). The phenomenon of collapse, as an example of phase transition in polymer systems with extremely high cooperativity, is the subject of intensive studies ‘ -6). The cooperative properties of this phase transition are especially pronounced for polyelectrolyte networks 3,4).

The aim of the present paper is to study theoretically a new situation: collapse of polyelectrolyte networks is realized due to their interaction with an oppositely charged surfactant capable to form micelles. One can expect that in this case as a result of ion exchange the concentration of surfactant inside the network can become very high and can exceed the critical concentration of micelle formation. The surfactant molecules will aggregate in micelles and the osmotic pressure will decrease; as a result, collapse of the network will take place.

The corresponding experimental situation has been studied recently; the results are discussed in ref.’) In particular, it is emphasized that this phenomenon can find interesting practical applications because network-surfactant complexes are very effective sorbents for insoluble organic substances.

Makromol. Chem., Theory Simul. 1, No. 3, May 1992

0 1992, Hiithig & Wepf Verlag, Base1 CCC 101 8-5054/92/$03.00

106 A. R. Khokhlov, E. Y. Kramarenko, E. E. Makhaeva, S. G. Starodubtzev

The present paper is organized as follows: In the next section we consider the case of a polyelectrolyte network swollen in the

solution of oppositely charged surfactants. We begin with the free energy of the system; then the critical micelle concentration (CMC) inside and outside of the network is determined. The last subsection is devoted to the analysis of the results of minimization of the free energy and to subsequent discussion.

In the following section we consider a more general case when the network is swollen in a mixed solution of both neutral and charged surfactants. Such mixtures are often studied in the literatureg>'O), usually micelles which are optimal from the point of view of free energy include both charged and neutral components (due to the presence of neutral surfactants, the energy of Coulomb interactions in the micelle decreases con- siderably).

Free energy of a network in a solution of an oppositely charged surfactant

Let us consider a sample of a charged polymer network swollen in a solvent containing an oppositely charged surfactant that is capable to form micelles.

We will use the following notation: Tis the temperature expressed in energy units (i. e. kT), N i s the total number of monomeric units in the network, rn is the average number of monomeric units in the chain between two nearest-neighbour branch points, a is the average number of monomeric units in the chain between two nearest charged units. We will assume that a % 1, i. e. only a small part of links is charged. This corresponds, in particular, to the experimental situation studied in ref. ') In our notation the total number of charges in all network chains is equal to N/a. The number of counter ions is the same because of the network electroneutrality.

Let us suppose that the network swells in the solution of a surfactant and that the charge of the surfactant is opposite to that of the network chains (Fig. 1). Let c,j+ be the concentration of surfactant in the solution, then the total number of surfactant molecules in volume V of the system is equal to ci . V . Of course, the same number c; . Vof counter ions corresponding to surfactant molecules are present in the system as well (Fig. 1); the sign of their charge coincides with that of the network chains. When the network swells in the surfactant solution, as a result of an ion exchange, part of the surfactant molecules enters the network while counter ions of the network move to the outer solution (Fig. 1). It is clear that a certain number of counter ions of the surfactant enters the network as well. Let us denote the number of surfactant ions within the network as P + , the corresponding number of counter ions of the surfactant molecules as N-, the number of the network counter ions as N+ (to be definite we will consider the case when the signs of all charges correspond to Fig. 1). In this case the solution outside the network will contain c t . V - P + surfactant molecules, c$ * V - N- counter ions of surfactant and N/a - N+ counter ions of the network. So the condition of total electroneutrality of the network and of the solution outside the network can be written in the following form

N + + P + = N - + N/a (1)

We will denote also the volume of the network as V, so that the volume of the outer solution will be V - V , .

Collapse of polyelectrolyte networks induced by their interaction with . . . 107

In order to understand all main features of the system under consideration it is necessary to take into account that the surfactant molecules in the network either exist in the free state or are aggregated in micelles (Fig. 1). We will consider the case of a sufficiently small concentration of surfactant in the outer solution so that micelles cannot be formed outside the network. The micelle formation leads to a gain in free energy. Let us denote the corresponding value per surfactant molecule as AE Let Pz be the total number of surfactant molecules joined in micelles (P; < P').

Fig. 1. The model of a charged polymer network swelling in a solution that contains an oppositely charged surfactant

L e

The free energy of a polymer network, F; can be written as a sum of four terms

F = Fin, + F,, + Fo + F, (2)

where Finl is the free energy of the volume interaction of monomer links; F,, is the free energy of elastic deformation for the network chains; Fo is the free energy of motion of ions within the network; F, is the energy gain connected with the micelle formation. Electrostatic interactions of charges inside the network are not taken into account in the sum (Z), since, as analysis shows, the corresponding term is always much smaller than the other terms.

Let us suppose that the network is swollen in a good solvent. This means that the free energy Fint can be written as follows4)

Fin, = NTBn (3)

where B is the second virial coefficient of the interaction of monomeric units and n = N/Vn is the average concentration of monomeric units within the network. For the cases considered in this work, accounting for interactions of monomeric units with the surfactant molecules is not necessary (corresponding terms are small).

Let us assume further that network polymer chains are flexible so that I f )

B - a3r (4)

where a is the characteristic size of a monomeric unit; T = (T - 0) /T is the relative temperature deviation from the 0-point (in a good solvent z = 1 and B - a3).

The free energy F,, of elastic deformation of the network can be written in a familiar form 12)


1 N a ' - 1 2

m f F,, = 3- T . [y - - h a

where f is the functionality of branching points; a is the swelling ratio with respect to a reference state in which all the network chains are most close to Gaussian coils j 3 ) . If the average concentration of monomeric units within the network in the reference state is denoted as no then a = (n0/n)"3. For the network prepared in the dry state no * a3 = 1; for the network prepared in the presence of a large amount of diluent no - a3 6 1 13).

There are Pi ions of surfactant, N- counter ions of surfactant and N+ network counter ions inside the network. If micelles are not formed in the network the term of the free energy accounting for the translational motion of all these ions can be written as follows

If P: ions of surfactant have formed micelles then

+ N+ In - + P i In - [TI El1 where the last term is the entropy of translational motion of surfactant ions within the micelle; V, is the volume of micelles inside the network; from the condition of dense packing of surfactant molecules inside the micelles we can expect that V, - P: . a3.

In our notation the term F, can be written as follows

F,,, = -AF.Pz (7)

It is necessary to write also the expression for the free energy of the outer solution. Here the translation entropy of all ions has to be taken into account

+ (N/a - N + ) In

+ (cO+ . V - N-) In [ cO+v'-v:-]]

Critical concentration of micelle formation (CMC)

In order to investigate the expression for the free energy of the system it is necessary to know the critical concentrations for micelle formation both inside and outside the network.

Let us consider first the micelle formation in the solution of a surfactant (in the absence of the network). When a micelle is formed in such a solution, immobilization of counter ions of surfactant molecules takes place, because these counter ions tend to


neutralize the charge of the micelles. Thus, in our notation, the free energy of the surfactant solution is

I AF c; . v - cz)ln(c$ - c;) - c; - + 2c; In-

T Vm (9)

where V is the total volume of the solution, c; = P,' / V is the average concentration of surfactant molecules aggregated in micelles (and simultaneously the concentration of immobilized counter ions). Factor 2 before the first and the third terms is connected with the fact that the translational entropy contributions of surfactant molecules and of their counter ions are equal. The equilibrium value of concentration c; can be determined by means of minimization of the free energy: W B C ; = 0. From the condition of dense packing it follows that the ratio c; * V V , is of order and does not depend on c, . Let us denote c; * V/V, = b/u3, where b is a numerical constant. Thus, the equality aF/aci = 0 can be rewritten in the form

At low concentration of surfactant in the solution, Eq. (10) gives a non-physical result, c,' < 0. Hence at these concentrations micelles do not appear. The value of c; becomes positive when c i is greater than

b AF a3

c,,, = - exp (- 2T - 1)

The expression (1 1) defines the critical concentration of micelle formation in the surfactant solution.

Now let us calculate the critical concentration of micelle formation inside a network. It is important that there is no immobilization of counter ions when micelles are formed in the network: the charge of micelles is neutralized by initially immobilized network charges which do not contribute to translational entropy (Fig. 2b). We will not consider, in the main approximation, the conformational changes of the network which

Fig. 2. Micelles of surfactant in the solution (a) and in the network (b)


are connected with the micelle formation. For this case, the free energy of surfactant solution inside the network can be written in the form (cf. Eq. (9))

AF c; * v (c$ - c;) h(c$ - c;) - c$ ln(c$) - c; - + 2c; In-

T vnl where Vis the volume of the network. The second term in Eq. (12) is the translation entropy of counter ions of the surfactant. Minimizing the expression (12) with respect to c; we obtain

Arguments analogous to that used earlier to derive expression (11) lead to the following value of CMC inside the network

b AF ci,, = 7 exp (- T - i)

Comparing expressions (1 1) and (14) one can conclude that c',,, < c,,, . Usually A F $- 7; hence the difference between two concentrations may be as large as several orders of magnitude. This is due to the fact that the micelle formation inside the network does not lead to a significant loss of translation entropy of counter ions.

Collapse of a network swollen in the solution of a surfactant

One can expect from the above consideration that the conformational behavior of a polymer network corresponds to one of the three cases below depending on the value of surfactant concentration c$ .

1 ) At low values of c$ the surfactant concentration P/V, inside the network is less than c',,, (see Eq. (14)). Thus, micelles in the network do not appear and the free energy F,, is defined by Eq. (6), and F,,, = 0. The network dimensions and the concentrations of ions of different kinds inside the network are defined by the usual equilibrium conditions (cf. ref. 5) ) . To write them down one has to use Eq. (1 ) and to express the network free energy F(see Eqs. (2), (3), (5) and (6)) and the free energy of the outer solution (Eq. (8)) in terms of variables P + , N+ and V,,. The equilibrium values of these variables can be found from the condition of the minimum of the full free energy of the system F + Foul. This means that the derivatives of F + Foul with respect to P + , N+ and V,, have to be equal to zero. The corresponding system of three equations defines the values of the variables P + , N+ and V , . The results are shown in Figs. 3 - 6 (range c$ < c*).

2) When the concentration of surfactant c$ exceeds some critical concentration c*, the value of P; that corresponds to the minimum of the full free energy F + F,,,, (see Eqs. (2) - (8)) becomes positive. This means that the surfactant concentration within the network P + / V , , reaches the value of d',, and micelle formation begins. The full free energy is written by means of Eqs. (2), (3), ( 5 ) , (6a) and (8), and it should be minimized with respect to four variables: P + , P i , N+ and V,, . The solutions of the obtained system of four equations are represented in Figs. 3 -6 (range c** < c$ < C**).

Collapse of polyelectrolyte networks induced by their interaction with . . .

c

Fig. 3. Dependence of the swelling ratio a on the concentration c: of charged surfactant molecules in a solution with the following parameters: B = 0,4a3; f = 4; m = 100; no = 0,i;

V/Vo = 1 OOO (1) and 40 (2) ( Vo is the volume of the network in the reference state; Vo = N/no)

M / T = 7; CT = 10;

a 3.0

2.5

2.0

1.5

-4

111

Fig. 4. K + = lg(c&$cA:Ft) (~22 and c&:Ft are the surfactant concentrations inside and outside of the network, respectively) as a function of 6 = P + * a/N with the following parameters: B = 0,4a3; f = 4; m = 100; no = 0,l; M / T = 7; CT = 10; V/Vo = lOOO(1)and 40 (2)

0

1.0

Fig. 5 . 0 = P + . d N a s a function of c; with the following parameters:

m = 100, no = 0.1;

V/Vo = lo00 (1) and

0.5 B = 0,4a3; f = 4;

U / T = 7; a = 10;

40 (2) 0

0 0 0.5 1.0

0

C f f

-1.8 -1.1 -4.0 -3.6 -3.2


1.0 -

0.5 -

I I I I

no = 0,l; AF/T = I ; u = lo; V/V, = 1000 (1)

c** C** , \

Fig. 6. a function of c: with the following parameters: B =

Bm = PG . rr/N as

' I 0,4a3;f = 4; m = 100,

3) When the surfactant concentration becomes greater and reaches some other critical value c** > c* the number of surfactant molecules in micelles inside the network becomes equal to the total number of network charges N/a. This means that all ions of the network are already used for neutralization of the charge of micelles. Thus under further increase of c,j+ micelle formation inside the network comes to an end. (In principle, the micelle formation could take place according to the mechanism illustrated in Fig. 2a, but this way can be realized if the concentration of the free surfactant within the network (P+ - P i ) / V , , exceeds the value of c,,, given by Eq. (1 1). The analysis shows that this is not the case in the range of concentration under consideration, c$ < ccmc .) Thus, in the region c: > c** we have to put P + = N/a and to minimize the free energy F + F,,, with respect to P + , N' and V, . The results of minimization are shown in Figs. 3-6 (range c: > c**).

We present in Figs. 3-6 the dependences of the swelling ratio a , K' =

lg(c&$/c&;;t), 8 = P + a/N and 8, = P; . a/N on Ig(c,+ . u3); here c s is the surfactant concentration within the network, and c;;;' is the surfactant concentration in the outer solution. The following values of parameters were chosen in the calculations: B = 0,4u3;f = 4; m = 100; no = 0,l; M / T = 7; a = 10; V/Vo = 40 and 1000 (V, is the volume of the network in the reference state; V, = N/n,).

Analysis of the curves in Figs. 3-6 leads to the following conclusions. When c$ grows in regime 1 (c,' < c*) the network volume slightly decreases and the

concentration of surfactant within the network increases. The behavior of the polymer network in this regime is completely analogous to that of the network swelling in the solution of a low-molecular-weight salt ') (since micelles are not formed). From Figs. 3 and 4 it can be seen that the value of transition concentration c* depends on the ratio V/V,; the greater this ratio, the less is c*.

When c: exceeds c* (regime 2) the network collapses because the surfactant molecules aggregated in micelles cease to impose osmotic pressure which causes additional expansion of the network sample. At relatively small values of the ratio V/Vo the collapse is continuous (see Figs. 3-6) so that the number of surfactant molecules in micelles smoothly increases from zero, starting at the concentration c* (Fig. 6). However, when the ratio V/V, is sufficiently large a discrete first-order phase


transition takes place. It is caused by the fact that in the expanded state cGf < c‘‘,, while in the compact state c z f > c‘,,, .

At c* < c$ < c** further shrinking of the network occurs and both the total concentration of surfactant molecules inside the network and the number of micelles increases (Figs. 3 - 6). Simultaneously the concentration of free surfactant molecules (P - P,) /V, remains unchanged and equal to c,,, (i.e. only those surfactant ions which correspond to the concentration surplus over c‘,,, are organized in the micelles). The value of 0 at c$ = c** exceeds unity because of the presence of free surfactant molecules in the network.

As one can see from Fig. 6 at c$ = c** the value of 6, becomes equal to unity and does not change with further growth of c$ (i.e. the total number of surfactant molecules in micelles remains fixed).

It should be pointed out that at c i = c** the concentration c;:/‘ becomes equal to the concentration of free surfactant molecules in the network (i. e. surfactant molecules which are not organized in micelles). This equality holds also at c$ > c** (in regime 3). Due to this fact the network volume remains unchanged (Fig. 3) in this regime. In fact, in regime 3 concentrations of all kinds of mobile ions inside the network and in the outer solution do not differ, thus internal “exerting” osmotic pressure is absent. Hence the network dimensions in regime 3 coincide with those of the corresponding neutral network.

Free energy of a network swollen in a mixed solution of a charged and a neutral surfactant

Let us consider now a more general case when the network swells in the solvent containing both charged and neutral surfactant molecules. Let co be the concentration of the neutral surfactant in the system. As to other parameters, we will use the same notation as in the previous sections.

A specific feature of the present system is that the gain in free energy due to the incorporation of a given surfactant molecule in the micelle is different for different species (AF for a charged surfactant molecule and AF, for a neutral one). We will assume that AFo > AF because the incorporation of a charged surfactant molecule in the micelle is connected with the additional energy of Coulomb repulsion. This means that in the absence of a network in most cases micelles are composed mainly of neutral surfactant. In the presence of a network the situation becomes much more complex since charged surfactant molecules are preferentially absorbed on the network and included in micelles. Thus micelle composition depends on the structure of the network. A quantitative analysis of this situation is presented below.

We suppose as before that there is no micelle formation in the solvent, i. e. the total surfactant concentration in the system co + c$ is less than the critical concentration of micelle formation; at the same time mixed micelles (consisting of charged and neutral surfactant molecules) can be formed inside the network. Let us denote the total number of neutral surfactant molecules in the network as P and the number of neutral surfactant molecules in micelles as P , .


In the presence of neutral surfactant in the system the electroneutrality condition can still be written in the form (1).

As before, the free energy of the network can be written as a sum of four terms (see Eq. (2)). Moreover, the expressions for F,, and Fin, remain the same (see Eqs. (3)- (5 ) ) . As to the translation entropy terms (Eqs. (6) and (6a)), the presence of neutral surfactant molecules should be taken into account. If micelles inside the network are not formed

If P; charged surfactant molecules and P , neutral ones have aggregated in micelles then

where V, is the total volume of micelles, V, - (P , + PA)a3. The expression for F, for the present case should be written as follows

F, = - LW.P; - AF,.P, (16)

Thus the full free-energy gain of micelle formation depends on the ratio of the

A new term appears in the free energy of the outer solution as well. This term is number of neutral surfactant molecules in micelles to that of charged ones.

connected with the translation entropy of neutral surfactant molecules. Thus,

C,' v - P

v - vn - " - 1 + (c,. V - P) In [ ]] (17) [ c;v- v, + (c; . V - N - ) In

Critical concentration of micelle formation

Let us calculate first of all critical concentrations of micelle formation both inside and outside the network using the same arguments as before.

First we will consider the solution of neutral and charged surfactants outside the network. The following expression for the free energy can be written for the case under consideration

2(c; - c;) h(c; - c;) + (c, - c,) In(co - c,)


This is just the generalization of expression (9). Here, V, - (c, + c;)Va3. As before we denote (c, + c i )V/V, = b/a3.

The equilibrium value of concentrations c; and c, can be found by minimization of F with respect to c; and c, . Writing down the equations aF/ac, = 0 and aF/ac; = 0 we obtain the following system

(c, - c,)(c, + c;)/c, = @/a3) exp[-AFo/T - 1 - c,/(c, + c;)]

(c; - c;) (c, + c;)/c: = (b/a3) e ~ p [ - AF/T - 1 - c,/(c, + ~;)/2]

(19)

(20)

This system of equations was solved numerically. The results of the calculation are presented in Fig. 7 for b/a3 = 0,4, AWT = 7 , M 0 / T = 9. Curve A in Fig. 7 divides the plane (c; , c,) into two regions. The values of c, and c; are positive in the region above this curve and negative in the region below it. Thus, curve A corresponds to the line of points of critical micelle concentration.

Concentrations c; and c, become positive simultaneously. Thus, in the region below curve A neither charged nor neutral surfactant molecules are aggregated in mixed micelles.

0 5

uo In' L 0

0

3 ,

2 .

1.

0

Fig. 7. The curve of CMC in the solution (A) and in the network (B); b/a3 = 0,4; AF/T = 7 ; AFo/T = 9

c;. a3

At low concentration of neutral surfactant molecules CMC tends to the value given by Eq. (1 1); at low concentration of charged surfactant molecules CMC is close to that expressed by Eq. (14) with AF = AFo. This is quite understandable because the formation of neutral micelles in the solution is somewhat similar to the formation of charged micelles in the network (no counter ion immobilization in both cases).

For the free energy of the network swelling in the solution of two kinds of surfactant molecules we obtain the following expression (cf. Eq. (12))

(c; - c;) In (c$ - c;) + (c, - cm) In(co - c,) + c$ In@$)


From the conditions aF/ac, = 0 and aF/ac; = 0 we obtain a system of two equations; its solution defines the equilibrium values of c, and c: . Curve B in Fig. (7) is the curve of CMC in the network (parameters are the same as for curve A). One can see that the CMC in the network is much smaller than that in the solution for all values of concentrations co and c$ .

Collapse of the network

Minimization of the free energy for the case under consideration leads to the following conclusions.

As before the conformational behaviour of the network corresponds to one of three regimes which are analogous to those described above; but now the realization of each of these regimes depends on two variables: co and c$ .

1) At low values of co and c$ , micelles inside the network are not formed (regime 1). In this regime the presence of neutral surfactant molecules has no influence on the network dimensions. The concentration of neutral surfactant molecules inside the network P , / V, in our approximation is equal to co . The conditions of equilibrium and equilibrium values of variables coincide with those in the absence of neutral surfactants (regime 1; co + c$ < cT); see Figs. 8 and 9 (range c: < c*) and Figs. 10 and 11 (range co < c*); c* (and c**) denote the critical value of concentration of charged surfactants in Figs. 8 and 9, and the critical value of concentration of neutral surfactants in Figs. 10 and 1 1 .

2) The values of P , and P i become positive when the total concentration co + c$ exceeds some critical value cf* (regime 2). In this regime mixed micelles appear. Minimizing the full free energy (see Eqs. (3), (5) , (15a), (16) and (17)) with respect to P, , P i , I: P + , N - and V,, we obtain a system of six equations. Results of the analysis of this system of equations are represented in Figs. 8 and 9 (range c* < c$ < c**) and in Figs. 10 and 11 (range c* < c,,).

3) When the sum co + c$ exceeds some higher critical value cf** the number of charged surfactant molecules in the micelles remains constant and equal to N/a (regime 3). As before we let P = N/a and minimize the free energy with respect to I: P , , N - and V, . The results are shown in Figs. 8 and 9 (range c$ > c**).

From Fig. 8 one can see that, as in the absence of neutral surfactants, collapse of the network can be induced by increasing the concentration of ionic species. The critical concentration c* for this case is smaller because of partial inclusion of neutral surfactants in the micelles. Calculations show that at the values of the parameters corresponding to Fig. 8 the fraction of neutral molecules in the micelles is of order

Fig. 9 illustrates the fact that at c$ > c* the concentration of surfactants of both kinds inside the network is essentially higher than in the outer solution. At the values of the parameters adopted in Fig. 9 the surplus of neutral surfactant inside the network is even higher than that of the charged species (although at c; < c* there is no preferential adsorption of neutral surfactant on the network).

Collapse of a network with micelle formation can be induced also by increasing the concentration of neutral surfactants, as it can be seen from Fig. 10. This fact is due to

0,20-0,25.


a 3,5 -

3.0 - 2.5 -

2.0 - 1.5 -

1.0 -

2 -

0

11 12 L !

I

:L I I . I

C **

L Y

0.5 I -5.5 -5.0 -L,5 -L,O -3.5 -3.0

lg cc;. 031

Fig. 8.

3.0

1.5

0

C **

0.8 1.2 0

Fig. 9.

Fig. 8. Dependence of the swelling ratio a on the concentration c$ of charged surfactant molecules in the solution, with the following parameters: B = 0,4u3; f = 4; rn = 100; no = 0,l; AF/T = 7; AF,/T = 9; a = 10; V/Vo = 1o00, c0-u3 = lo-’ (1) and co * u3 = 0 (2)

Fig. 9. K+ = l g ( c $ ~ / c ~ ~ ~ t ) (1) and K = lg(c&f/c::if) (2) as functions of 0, with the following parameters: B = 0,4u3; f = 4; rn = 100, no = 0.1; M/T = 7; AFo/T = 9; a = 10; V/Vo = 1000; c 0 . u 3 = lo-’

~g (c,. a31 Fig. 10.

I I 1

-5.0 -1.5 Ig(c,.u3)

Fig. 11.

Fig. 10. Dependence of the swelling ratio a on the concentration co of neutral surfactant molecules in the solution, with the following parameters: B = 0,4u3; f = 4; rn = 100, no = 0,l; LLF/T=7;AFo/T=9;a= 10;V/Vo= 1 0 0 0 ; ~ ~ - 1 ~ ~ = 2 . 1 0 - ~

Fig. 11. Dependence of the fraction of neutral molecules in micelles, z, on the concentration c of neutral surfactant molecules in the solution, with the following parameters: B = 0,4u{ f = 4 ; r n = 1 0 0 ; n , = 0 , 1 ; ~ / ~ = 7 ; ~ ~ ~ / ~ = 9 ; a = IO;V/V,= i ~ , c o + . u 3 = 2 . 1 0 - ’


the dependence of the critical micelle concentration on the value of co: the greater co the lower is the CMC because neutral surfactants are more and more included in the micelles. This fact can be seen also from Fig. 11: the fraction of neutral surfactants in the micelles increases with increasing co.

Conclusion

We see that the interaction of polyelectrolyte networks with oppositely charged surfactants gives rise to a number of interesting phenomena including intensive adsorption of surfactants on the network and the following collapse of the network accompanied with micelle formation. It is thus interesting to study this system experimentally. This was the subject of recent experimental investigations7, @.

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Collapse of polyelectrolyte networks induced by their interaction with oppositely charged surfactants

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