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Langmuir 1999, 15, 7918-7924
Swelling and Collapse of Physical Gels Formed by Associating Telechelic Polyelectrolytes Valentina V. Vasilevskaya,†,‡ Igor I. Potemkin,‡ and Alexei R. Khokhlov*,†,‡ Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow, 117823, Russia, and Physics Department, Moscow State University, Moscow, 117234, Russia Received August 18, 1998. In Final Form: July 12, 1999 A theory is proposed for the swelling and collapse of physical gels formed by associating telechelic polyelectrolytes. It is shown that increasing the of degree of ionization makes the formation of an associative gel less favorable and increases the density of the supernatant phase. This leads to the contraction of the associative gel, in comparison with an analogous chemically cross-linked gel. The results of calculation show that the destruction of the associative gel happens monotonically in good solvent, while in poor solvent it can proceed in a jump-like fashion accompanied by the first-order phase decollapse transition of the associative gel. The addition of a low molecular weight salt to the solution facilitates the association of telechelic molecules which produces the associative gel and a further increase of salt concentration leads to the contraction of the associative gel.
1. Introduction Most associating polymers used in industry for water control (e.g., viscosity modification of aqueous media, stabilization of aqueous colloidal dispersions, etc.) are polyelectrolytes. However, theoretical works describing the properties of associating polyelectrolytes are practically absent. In the present paper we will develop the simplest theory of phase equilibria in solutions of telechelic polyelectrolytes with associating end-groups (stickers). Due to the attraction, the stickers can aggregate, form multiplets of different morphology, and, generally speaking, can connect all chains or part of them into a gellike cluster, or associative gel.1 The literature on the theory of the formation of associative gels in concentrated nonpolylelectrolyte polymer solution is abundant.2-10 However, in dilute solution the associative gel normally does not spread through the whole volume, but rather occupies part of it, while the remaining part is essentially a very dilute solution of chains (supernatant phase). In other words, there is a two-phase equilibrium (gel phase and supernatant phase). The swelling of the associative gel in the presence of its own supernatant phase has been much less studied.6,8 Generally speaking, it is possible to distinguish two limiting situations. If the attraction between stickers is very strong (i.e. the corresponding characteristic energy, , is much higher than the thermal energy, . kT), the * Corresponding author. † Nesmeyanov Institute. ‡ Moscow State University. (1) Goethals, E. J. Telechelic Polymers: Synthesis and Applications; CRC Press: Boca Raton, FL, 1989. (2) Coniglio, A.; Stanley, H. E.; Klein, W. Phys. Rev. Lett. 1979, 42, 518. (3) Coniglio, A.; Stanley, H. E.; Klein, W. Phys. Rev. B 1982, 25, 6805. (4) Tanaka, F. Macromolecules 1989, 22, 1988. (5) Tanaka, F.; Ishida, M. Macromolecules 1996, 29, 7571. (6) Tanaka, F.; Ishida, M. Macromolecules 1997, 30, 3900. (7) Semenov, A. N.; Rubinshtein, M. Macromolecules 1998, 31, 1373. (8) Semenov, A. N.; Joanny, J. F.; Khokhlov, A. R. Macromolecules 1995, 28, 1066. (9) Ermoshkin, A. V.; Erukhimovich, I. A. J. Chem. Phys. 1999, 110, 781. (10) Erukhimovich, I. A.; Ermoshkin, A. V. JETP 1999, 115, 979.
structure arising is similar to a chemically cross-linked gel. The swelling behavior of high sticker attraction associative gels obeys the well-known swelling law for chemically cross-linked gels. In the opposite case, when the energy of stickers attraction is rather weak ( < kT), an associative gel can only be formed when the polymer volume fraction is high enough that spreads throughout the whole volume. The most interesting case is the intermediate situation, when the transition of free polymer from solution to the associative gel is accompanied by two-phase separation. In this intermediate case the phase of the associative gel occupies part of the total volume and coexists with the dilute solution of macromolecules (supernatant). The swelling of the associative gel in the intermediate case is influenced by the osmotic pressure due to the free-moving macromolecules of the supernatant. In this paper we present a simple theoretical approach to describe the transition from the polymer solution to the associative gel and to the two-phase separation in the solution of telechelic polyelectrolyte macromolecules. As mentioned above, this is the most practically important situation. Also, we consider the case of polyelectrolyte macromolecules since in the case of chemical gels it is the polyelectrolyte gel that undergoes the most pronounced conformational changes and exhibits an extremely wide set of different possibilities. In particular, such a gel can undergo a collapse transition, which is a sharp jump-wise contraction induced by very small changes in external conditions.11-12 The model used and the free energy of the system are given in next section. Section 3 contains numerical results. The influence of the addition of a low-molecular salt is discussed in Section 4. In Section 5 the conclusions are described. 2. Model and Free Energy In accordance with the Introduction, let us consider a physical gel formed by telechelic polyelectrolytes which coexists with the dilute supernatant solution of these molecules (see Figure 1). Let φ1 be the volume fraction of (11) Shibayama, M.; Tanaka T. Adv. Polym. Sci. 1993, 109, 1. (12) Khokhlov, A. R.; Starodubtzev, S. G.; Vasilevskaya, V. V. Adv. Polym. Sci. 1993, 109, 123.
10.1021/la981057q CCC: $18.00 © 1999 American Chemical Society Published on Web 09/23/1999
Collapse of Telechelic Polyelectrolyte Gels
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The term Fel is the elastic free energy describing the entropy losses due to possible extension of chains with respect to the state, in which the chain conformation is close to ideal. In our simplest theoretical development we will use the following expression proposed by Flory14
(( )
Fel 3 φ1 φ0 ) T 2 P φ1
2/3
-
)
2 φo ln 3f φ1
(2)
where φo is the volume fraction of polymer in the reference state of associative gel (in this state the gel chains most closely resemble ideal chains). For φo the following estimate is valid15 Figure 1. Schematic picture of two-phase system: associative gel of polyelectrolyte molecules and supernatant. Aggregates are shown by large circles. Counterions (shown by smaller circles) move freely within each phase.
polymer in the associative gel phase, and let us denote as φ2 the volume fraction of polymer in the phase of supernatant. For the values φ1 and φ2 the following inequality is valid: φ1 > φ2. Let P be the degree of polymerization of the telechelic molecules. Let us suppose that P/σ monomer links can dissociate into charged monomer units connected with network chains and low-molecular-weight oppositely charged counterions (i.e., σ is the average number of monomer units between two consecutive charged monomer units along the chain and, therefore, the value of 1/σ determines the degree of ionization). Both macroscopic coexisting phases12 (associative gel and phase of supernatant) are electroneutral, thus, the concentration of counterions φicount kept in the each phase i is directly proportional to the volume fraction of polymer in this phase φicount ) φi/σ and the transition of counterions from one phase to another is directly connected with the motion of the macromolecule from phase to phase. Therefore, we have in fact only one independent concentration in each phase. Let be the gain in energy per chain due to the formation of a physical cross-link (aggregate). If the attraction between the chain ends is strong enough, it is natural to assume that the fraction of telechelic molecules forming a gellike structure in the associative gel phase is close to unity; in other words, the presence of free chains, as well as dangling ends within the gel phase, can be neglected. Let f be the average aggregation number of the arising associates (i.e., number of stickers entering into one associate). In other words, f is the average effective functionality of cross-links in the forming associative gel. Generally speaking, the degree of association f is sensitive to the energetic and electrostatic interaction13 in the system. However, it can be shown that the value of f has only a slight influence on the main results of the present consideration; to simplify the calculations we propose that parameter f does not depend on the degree of charging of the macromolecule σ or on the energy . Finally, let us denote as χ the Flory-Huggins parameter of polymer-solvent interaction and let us assume that T is the temperature (expressed in energetic units). The free energy, Fnet, of the associative gel (per unit of volume) can be written as a sum of four contributions
Fnet ) Fel + Fint + Fgain + Ftr-ent
(1)
(13) Shusharina, N. P.; Nyrkova, I. A.; Khokhlov, A. R. Macromolecules 1996, 29, 681.
φo ≈ p-1/2
(3)
The contribution Fint describes the excluded volume interaction of monomer units with molecules of solvent. This contribution can be written in the framework of the Flory-Huggins approximation14 as
Fint ) χφ1(1 - φ1) + (1 - φ1)ln(1 - φ1) T
(4)
The term Fgain is the energy gain due to the association of the stickers.
Fgain ) -2 φ1 T P
(5)
Finally, the term Ftr represents the contribution to the free energy from the entropy of translational motion of the freely moving counterions. As was shown, in the case of the polyelectrolyte system this contribution is most important and it significantly exceeds the direct electrostatic interaction (cf. ref 12)
Ftr-en φ1 φ1 ) ln T σ σ
(6)
The free energy, Fsup, of the supernatant phase with volume fraction φ2 of the polymer can be written as a sum of the free energy of interaction and the contribution due to the translational motion of the counterions.
φ2 φ2 Fsup ) χφ2(1 - φ2) + (1 - φ2)ln(1 - φ2) + ln + T P P φ2 φ2 ln (7) σ σ In writing eq 7 we neglected the possible associates in the supernatant phase. This is justified when the supernatant phase is very dilute, which is the case of interest in the present paper. Also, we have taken into account that due to the low concentration of polymer in the supernatant phase and the short length of telechelic molecules, practically all counterions move freely and independently from telechelic molecules and occupy the whole volume of supernatant phase.16 The additional analysis of such a system developed by us in ref 17 shows (14) Flory P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (15) Grosberg, A. Yu., Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (16) Kramarenko, E. Yu.; Khokhlov, A. R.; Yoshikawa, K. Macromolecules 1997, 30, 3383. (17) Potemkin, I. I.; Vasilevskaya, V. V.; Khokhlov, A. R. J. Chem. Phys. to be published.
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that this approximation is valid for a wide range of system parameter values. The equilibrium values φ1 and φ2 of polymer volume fractions in the supernatant and associative gel are determined by the usual equilibrium conditions: equality of osmotic pressures
-Fnet +
∂Fnet ∂Fsup φ1 ) -Fsup + φ ∂φ1 ∂φ2 2
(8)
and equality of chemical potentials
∂Fnet ∂Fsup ) ∂φ1 ∂φ2
(9)
of the two coexisting phases. The obtained system of two equations (8)-(9) with free energy (1) and (7) was solved numerically for f ) 4, and different values of P, , σ, and χ. Results are given and discussed in the next section. 3. Numerical Results and Discussion The numerical results of the calculations are presented as phase diagrams which show three different regions: the region of stability supernatant phase, the associative gel phase, and the region of phase separation. Within the phase separation region the homogeneous state is unstable and the solution separates into two phases with two different volume fractions of polymer, the supernatant phase and the associative gel phase. The composition of the coexisting phases φ1 and φ2 for fixed parameter values (energy, , degree of ionization, 1/σ, parameter χ, and degree of polymerization, P) does not depend on the total volume fraction φtot of polymer in solution. However, as the volume fraction of polymer φtot increases, the relative volume of the system occupied by the associative gel increases. The ratio between the volumes of the coexisting phases V1 and V2 is determined according to Maxwell’s rule so that if the volume fraction φtot of polymer is equal to the volume fraction of polymer in the supernatant (φtot ) φ2) then the whole volume V of the system is occupied by the free-moving macromolecules of the supernatant phase (V2 ) V); and in the opposite case (φtot ) φ1) the supernatant phase disappears and the associative gel occupies the whole system volume (V1 ) V)
V2 φ2 - φtot V1 )1) V V φ2 - φ1 Figure 2 shows the phase diagram in the variable vs the volume fraction φ of polymer in solution for the case of good solvent (χ ) 0). In this case the driving force for phase separation is the attractive forces between the stickers. Here and below, curve φ2 is the boundary of the supernatant phase region, while line φ1 shows the boundary of the associative gel phase region. One can see that the associative gel can be formed only at the values of which are higher than some critical value cr: > cr. With an increase of the phase separation proceeds at lower values of φtot and the volume fraction φ2 of polymer in the supernatant decreases. This result corresponds with the experimental data of ref 18 where a strong depression of the lower critical solvent temperature and a shift of the critical concentration toward lower values for hydropho(18) Abrahmsen-Alami, S.; Alami, E.; Francois, J. J. Coll. Interface Science. 1996, 179, 1, 20.
Figure 2. Phase diagram in variables: gain in energy vs volume fraction of polymer φ for following values of other parameters P ) 20, σ ) 5, χ ) 0.0. Curve 1 show the phase of the associative gel, curve 2 is the boundary of the supernatant phase.
bically modified poly(ethylene oxide) in comparison with that for homopoly(ethylene oxide) were found. The density of the associative gel coexisting with the supernatant phase decreases with an increase of . At high values of the associative gel swells as the usual polyelectrolyte gel12
φ1 ≈
φo P 3/2 σ
()
because the supernatant phase is extremely dilute and its osmotic pressure is very low. With the decrease of the concentration of the supernatant phase increases and, therefore, the corresponding osmotic pressure of the supernatant created mainly by the freely moving counterions increases. As a result the associative gel in equilibrium with the supernatant phase shrinks. Note that the associative gel volume decreases as the volume fraction φtot of polymer decreases up to the total dissolution of the associative gel. Note also that at
φtot ) φcr ≈
φo P 3/2 σ
()
(see Figure 2) the smallest possible polymer volume fraction in the associative gel phase is reached. Figure 3 shows phase diagrams in the variables φ vs σ for two different values of energy . Here also three different regions are shown (supernatant (I), associative gel (II), and phase separation (III)). One can see that with an increase in the degree of ionization (decrease of σ) the phase separation region becomes more narrow. In particular, the onset of the region of phase separation shifts to higher volume fractions φtot of telechelic molecules in solution, indicating that the formation of the associative gel coexisting with the supernatant becomes less favorable as the number of charges on the polymer chain increases. The concentration of the associative gel coexisting with supernatant phase decreases with an increase of degree of ionization due to the osmotic pressure of counterions of gel chains trapped within gel phase exerted on the associative gel. Note that the stronger the attraction is, the lower the concentration of polymer in the supernatant phase φ2 is, and thus the influence of the supernatant phase on the swelling behavior of the associative gel is reduced. It is interesting to note that at
Collapse of Telechelic Polyelectrolyte Gels
Figure 3. Phase diagrams in variables: σ vs volume fraction of polymer φ for the following values of other parameters: P ) 20, χ ) 0.0, and ) 16(a), 20(b). Shown are the supernatant phase (I), associative gel phase (II), and region of phase separation (III). Curves 1 are the boundary of phase of associative gel, curves 2 show the boundary of the supernatant phase.
Figure 4. Phase diagrams of telechelic polyelectrolyte in poor solvent with χ ) 1.0 in variables σ vs volume fraction of polymer φ for the following values of other parameters: ) 16, P ) 20 (A), ) 20, P ) 25 (B). Curves φ1 show the boundary of the associative gel, curves φ2 are the boundary of the supernatant phase.
high values of σ, the volume fractions of polymer in associative gels formed by telechelic polyelectrolytes with different values of are almost the same, whereas at low values of σ (high degree of ionization) the associative gels formed in a solution of telechelic polyelectrolytes with low values of the volume fractions are more compressed. This fact is connected with the increase in osmotic pressure of the supernatant phase at lower values of . This phase becomes more concentrated with a decrease in . In Figure 4 phase diagrams of telechelic polyelectrolyte immersed in poor solvent (χ ) 1) for two different degrees of polymerization P are presented. One can see that in this case the associative gel undergoes sharp contraction with a decrease in the degree of ionization in a rather
Langmuir, Vol. 15, No. 23, 1999 7921
Figure 5. Swelling curves φ1(σ) of associative gels in poor solvent with χ ) 1.0 for P ) 25 and ) 25.0 (a), 20 (b), 10 (c), 5 (d). The swelling curve of the corresponding chemical crosslinked gel is given by a dashed line. The dotted line shows the point of coil-globule transition for cases (a)-(b).
narrow field of σ. Simultaneous with that contraction, the concentration of the supernatant phase coexisting with the associative gel decreases. The contraction of the associative gel can proceed by a first-order phase transition with a significant jump in concentration φ1 (collapse) similar to the collapse transition of a chemically crosslinked gel.11,12 At high values of σ, the density of the associative gel is determined by nonelectrostatic interactions; therefore, the chains forming the gel stay in a collapsed globular state, due to the strong attractive interaction between monomer units at χ ) 1. With an increase in the degree of ionization (1/σ) the globular state for molecules becomes thermodynamically unfavorable due to the osmotic pressure of counterions. At this point, chains undergo the transition to the coil conformation. Since the physical gel contains charged monomer units, the transition occurs as a first-order phase transition with a significant jump in concentration.15 Therefore, the density of the associative gel drops in a jump-wise manner. Simultaneously, a great part of the telechelic molecules from the associative gel are released to the supernatant. This means that in this case the dissolution of the associative gel can happen in explosive fashion when the associative gel disappears by a jump as the system parameters are changed by an infinitely small value. Such explosion-like destruction of associative gel can lead to the total disappearance of the associative gel if the volume fraction of polymer φtot is rather low (in other words, the case if we cross the jump region on the left boundary of the phase separation region; this is not actually distinguishable in Figure 4b). If the volume fraction of polymer φ is high enough, the associative gel can explode to occupy the whole volume of the system with the decrease of σ (increase of the degree of ionization). The sharpness of the collapse transition of the associative gel increases with an increase of (Figure 5). In the case of high values of energy the swelling curves are sigmoidal, which indicates that the transition of associative gel from swollen coil conformation to the compacted globule state proceeds as a first-order phase transition. The point of this transition is where the free energies corresponding to the two different states become equal to each other. It can be seen (Figure 5) that at rather high values of the swelling curve of the associative gel coincides with the corresponding curve of a chemically cross-linked gel (dashed line in Figure 5) practically at all values of parameter σ. As the attraction becomes weaker, the swelling curve of the associative gel deviates significantly from that of the chemical gel at low values of σ or higher degrees of gel ionization (curves a-b, Figure 5). With
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further lowering of , the amplitude of the collapse transition decreases and this transition shifts to the field of somewhat higher degrees of ionization σ. Indeed, with the decrease of some macromolecules remain in the supernatant rather than forming the associative gel; therefore the concentration of polymer in the supernatant phase increases. This creates strong osmotic pressure which compresses the associative gel. As a result, the associative gel becomes less swollen and the sharpness of the globule-coil transition of this associative gel becomes less pronounced (curve d, Figure 5). 4. Influence of Low Molecular Weight Salt
(
)(
)
(10)
Thus, the total free energy Fnet of the associative gel can be written as
(( )
)
(
) (
Fnet 3 φ1 φo ) T 2 P φ1
2/3
2 φo - ln + χφ1(1 - φ1) + 3f φ1 (1 - φ1) ln (1 - φ1) - 2 φ1 + P φ1 φ1 + n1s ln + n1s + n1s ln n1s (11) σ σ
)
Correspondingly, the translational entropy of the low molecular weight species in the supernatant phase (which provides the last contribution to the free energy Fsup of the supernatant phase (5)) can be rewritten as
(
) (
)
φ2 φ2 F2tr-en ) + n2s ln + n2s + n2s ln n2s T σ σ
(12)
and the total free energy Fsup of the supernatant phase containing the low molecular weight salt takes the form
Fsup ) χφ2(1 - φ2) + (1 - φ2)ln(1 - φ2) + T φ 2 φ2 φ2 φ2 ln + + n2s ln + n2s + n2s ln n2s (13) P P σ σ
(
)(
∂Fnet ∂Fnet 1 φ1 + n s) ∂φ1 ∂n1 s
-Fsup +
∂Fsup ∂Fsup 2 φ2 + n s (14) ∂φ2 ∂n2 s
and equality of chemical potential of the polymer
∂Fnet ∂Fsup ) ∂φ1 ∂φ2
(15)
and of the low molecular weight salt
Let us propose now that the solution of telechelic polylelectrolytes contains a low molecular weight salt which is a 1:1, strong, totally dissociated polyelectrolyte. The main question which we would like to answer here is how the position and width of the phase separation region depends on the salt concentration. Let ns be the total salt concentration. In the course of phase separation the salt molecules will be redistributed between the coexisting phases. Let n1s and n2s be the salt concentration within associative gel and in the supernatant phase, respectively. The free energies Fnet and Fsup of the coexisting phases should contain the contribution due to the presence of salt molecule. In ref 12 it was shown that the main contribution to the free energy in the case under consideration is that due to the translational entropy of low molecular weight ions, which is larger than the contribution of Coulombic interactions. The contribution Ftr-en from the translational entropy of the free-moving species (counterions and ions of salt) (eq 7) now takes the form12
Ftr-en φ1 φ1 ) + n1s ln + n1s + n1s lnn1s T σ σ
-Fnet +
)
The equilibrium state of the system is determined now by three equilibrium conditionssequality of osmotic pressure
∂Fnet ∂Fsup ) ∂n1s ∂n2s
(16)
in coexisting phases. After simple calculations combining free energies (11) and (13), eq 16 takes the form
n1s )
x( )
1 2
φ1 σ
2
(
+ 4n2s n2s +
)
φ2 φ1 σ 2σ
(17)
giving the value of salt concentration n1s in the phase of the associative gel as a function of salt concentration n2s in the supernatant phase. In addition to eq 16, the values n1s and n2s are connected with each other and total concentration ns of salt in solution by the condition of conservation of total number of salt molecules in the whole volume V of solution
nsV ) n1sV1 + n2sV2
(18)
Here V1 and V2 are the volumes of the associative gel and supernatant phase, respectively. Using Maxwell’s rule, we could rewrite eq 18 in the variables of volume fraction of the polymer φ1 and φ2 as
φtot - φ2 φtot - φ1 ns ) n1s + n2s φ1 - φ2 φ2 - φ1
(19)
here φtot is the total volume fraction of polymer in solution. Equations (17) and (19) allow us to find the concentrations n1s and n2s of salt in the coexisting phases at given values of salt concentration ns and volume fraction φtot of polymer in whole system as a function of the volume fraction of polymer φ1 and φ2 in the coexisting phases. Since both the concentration of salt n1s and n2s in coexisting phases and, therefore, the corresponding contribution into the osmotic pressure, are determined by the total concentration φtot of polymer in solution, not only relative volumes but also the composition of the coexisting phases (i.e., φ1 and φ2) depend on the total volume fraction of the polymer, φtot. The system of equations (14)-(16), (17), and (19) with free energies (11) and (13) was solved numerically for χ ) 0.0, P ) 20, φtot ) 0.02 and different values , σ, and salt concentration ns. The results of the calculations are presented in Figures 6-9. In Figure 6, the swelling curves of the associative gel for the case of strong attraction between the associative groups ( ) 20) are presented for a few salt concentrations ns. It should be noted that in a salt-free solution the extremely diluted supernatant phase does not influence the associative gel, and the swelling of the associative gel is similar to the swelling of common chemical gel (see Figure 5). As salt is added to the solution, the macro-
Collapse of Telechelic Polyelectrolyte Gels
Figure 6. The volume fraction φ1 of polymer in the associative gel as a function of parameter σ at χ ) 0.0, P ) 20, ) 16, φtot ) 0.02 and different salt concentrations ns ) 0-10-5(a), 10-4(b), 10-3 (c).
Figure 7. The concentrations of salt in the associative gel and the supernatant as function of parameter σ at χ ) 0.0, P ) 20, ) 16, φtot ) 0.02, and ns ) 10-4. The dashed horizontal curve shows the total salt concentration in solution, ns.
Figure 8. State diagram in variables σ vs volume fraction of polymer φ at χ ) 0.0, P ) 20, ) 12, φtot ) 0.02, and different salt concentrations ns )10-6(a), 10-5(b), and 10-4(c). The dashed vertical curve shows the total volume fraction of polymer in solution. Curves 1 show the boundary of the associative gel, curves 2 show the boundary of the supernatant phase.
molecules of the supernatant phase start to precipitate into the associative gel phase more intensively, the concentration φ2 decreases, and finally the supernatant phase disappears totally. With a further increase of salt concentration ns, the associative gel contracts. The physical reason for this contraction is the osmotic pressure created
Langmuir, Vol. 15, No. 23, 1999 7923
Figure 9. Dependencies of the volume fraction of polymer in the associative gel φ1 and in the supernatant φ2 on salt concentration ns at χ ) 0.0, P ) 20, ) 12, φtot ) 0.02, and σ ) 10.
by low molecular weight ions of salt. This pressure is proportional to the difference between the salt concentration in the coexisting phases ≈ n1s - n2s. From eq 17 and the condition (φ1/σ) > (φ2/σ) one can obtain the following limiting estimations: at n2s , (φ1/σ), n1s ≈ 0, at n2s . (φ1/σ), n1s ∼ n2s - (φ1/2σ). Thus, the concentration of salt within the supernatant phase n2s is always higher than that in the associative gel n1s (see also Figure 7) and the addition of salt leads to the gel contraction at all values of σ. The phase diagrams in σ vs φ for the case of weaker attraction ) 12.0, and different salt concentrations are shown in Figure 8. Here the vertical curve shows the total volume fraction of polymer φtot in the system which undergoes phase separation, curves with subscript 1 denote the volume fraction of polymer φ1 of the associative gel phase, while curves with subscript 2 show the volume fraction of polymer in the supernatant phase. It was found that in this case at relatively low salt concentration ns, the phase diagrams practically coincide with the phase diagram of the salt-free solution (see Figure 9). With an increase of salt concentration ns, the field of phase separation becomes somewhat wider. The larger portion of the macromolecules from the supernatant phase precipitates into the associative gel phase. Simultaneously the associative gel contracts due to the compressive osmotic pressure of salt. On the other hand, as the degree of macromolecule charging increases (parameter σ decreases), the fraction of the system volume occupied by the associative gel grows because of macromolecule stretching. At the point where σ ) σ* when the vertical curve φ ) φtot intersects with the curve φ ) φ1 the telechelic molecules only form the associative gel phase since the concentration of polymer exceeds the overlap concentration of solution of such macromolecules. As the overlap concentration decreases with an increase of salt concentration (at a given value of σ), the phase separation at a given total volume fraction φtot of polymer stops at a higher degree of ionization (1/σ) for a more concentrated salt solution. At high values of σ, the supernatant phase occupies a larger part of system volume than the compacted associative gel; the system volume is divided into the equal parts at σ ≈ 6. With a further increase in the degree of ionization (1/σ) the larger part of the system volume belongs to the associative gel. 5. Conclusions A simple theoretical approach was proposed to describe the swelling and collapse of an associative gel of telechelic polyelectrolytes coexisting with a supernatant phase. It
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was shown that this approach allows the description of the dependence of associative gel swelling on the following three parameters of the system: degree of polymerization and degree of ionization of telechelic polyelectrolytes, and the strength of attraction between stickers. It was found that introduction of charges into the chain of telechelic molecules makes the formation of associative gel thermodynamically less favorable. As a rule, the dissolution of the associative gel with an increase in its degree of ionization occurs monotonically. However, in some cases it can proceed in jump-wise manner because of the first order decollapse transition of the associative gel. The swelling curve of the associative gel coincides with the corresponding curve of a chemically cross-linked gel if the
Vasilevskaya et al.
attraction between stickers is very strong. On the other hand, the associative gel is compressed in comparison with chemical gel in the case of relatively weak attraction between the stickers. The addition of a low molecular weight salt to the solution facilitates the association of telechelic molecules to form associative gels and to the contraction of the associative gel with further increase of salt. Acknowledgment. This work is supported by INTAS under grant 96-1193 and by the Program University of RussiasFundamental Research under grant 5261. LA981057Q
