J. Phys. Chem. 1995, 99, 10971-10975
10971
Theoretical Analysis of the Cross-linking Effect on the Polyelectrolyte-Surfactant Interaction J. P. Gong* and Y. Osada Division of Biological Sciences, Graduate School of Science, Hokkaido University, Sapporo 060, Japan Received: February 13, 1995; In Final Form: May 2, 1 9 9 9
:The cooperative binding of a linear as well as a cross-linked polyelectrolyte with an oppositely charged surfactant has been theoretically analyzed. The hydrophobic interaction has been treated using the nearestneighbor interaction model, while the electrostatic interaction has been calculated using the rodlike model. The general formulas derived on the basis of the free energy minimum principle predicted that the crosslinkage enhances the initiation process but strongly suppresses the cooperativity due to the osmotic pressure in the network domain. The theoretical results showed fairly good agreement with the experimental data, confirming the essential features of the theory.
Introduction The stoichiometric binding of surfactant molecules with the charged polymer network is characterized by two processes. One is the electrostatic interaction between the surfactant ion and the oppositely charged network to form salt-like bridging, thus initiating the binding. The other is the hydrophobic interaction between bound surfactants, which stabilizes the aggregation in such a way as to settle the adjacent to the already occupied site along the polymer chain. The former is called “initiation process” and the latter is called “cooperative process”. A number of researches regarding the surfactant-linear polyelectrolyteinteraction have been made.’-6 The cooperative binding process of surfactant with macromolecules has been well described by Zimm-Bragg7 theory, which was proposed for the helix-coil transition of biopolymers.’*8 However, only a few studies have been made on the surfactant-polymer network i n t e r a c t i ~ n . ~Khokhlov .’~ et al.9 proposed a theory of weakly charged polyelectrolyte networks immersed in the solution of an oppositely charged surfactant. In their theory, the swelling and collapse of the network was well described, but the electrostatic interaction between the network and the surfactant was not taken into account. Tanaka et al.’O studied the swelling equilibrium of N-isopropylacrylamide (NIPA) gel as a function of temperature in aqueous solutions of surfactant and developed the volume-phase transition theory to the case of surfactant. We have made a comparative study on the surfactant binding with oppositely charged linear and network polymers”-I4 and found that the initiation of the binding of the network occurs at a surfactant concentration as low as mom. This is 1 order of magnitude lower than that of linear polyelectrolyte, although the propagation process of the network was significantly suppressed (the cooperativity parameter for linear polymer was 630, while that of the polymer network was -1) and the cooperativity was practically prohibited.’ This paper attempts to theoretically ascribe the role and effect of the cross-linkage on the surfactant binding. The contribution of the hydrophobic interaction has been treated using the nearest neighbor interaction model originally employed for polyelectrolyte titration by Marcus.I5 General formulas for the binding of surfactant with a linear polyelectrolyte as well as with a
* To whom correspondence @
should be adressed. Abstract published in Advance ACS Abstracts, June 1, 1995.
0022-3654/95/2099-10971$09.00/0
charged network have been derived. The drived equation showed that the low cooperativity observed for the chxged network is due to the high osmotic pressure created by the mobile counterions, which tend to expand in competition with the conformational shrinkage on binding. The calculated results showed a fair agreement with the experimental data.
Theory
Binding with a Linear Polyelectrolyte. We consider the electrostatic interaction between a solubilized linear polyelectrolyte with negative charges and positively charged surfactants capable of forming micellar-like c~mplexes.’~ The free energy of the system, F, can be written as a sum of three terms:
where Fint is the free energy of the volume interaction of monomer links, Fmobilethe free energy of motion of micro ions in the solution, and Fcomp the free energy of surfactants bound with polyions. According to Flory-Huggins theoryI6
where Vis the volume of solution, I$ the volume fraction of the polymer, x the Flory -Huggins parameter, m the polymer chain length, and vc the molar volume of solvent or monomer, assuming that they have a same specific volume. R and T are the gas constant and absolute temperature, respectively. We denote Nand M as the total molar number of monomeric units of the polymers and surfactants, respectively, /3 the degree of binding that is the ratio of the molar number of bound surfactants to N , = (M - N/3)/V the equilibrated molar concentration of surfactant in the polymer solution. In the following text, superscripts p and g denote the binding with a linear polymer and a cross-linked gel, respectively. Since the free energy of the microions largely attributes to the translational entropy of mobile ions in the solution, the molar numbers of counterions of surfactant and polymer are M and N, respectively, we have 0 1995 American Chemical Society
10972 J. Phys. Chem., Vol. 99, No. 27, 1995 Fmobile
Gong and Osada
=
As described in a previous paper,I4 the binding of surfactant to a polyelectrolyte should be attributed to the electrostatic interaction between oppositely charged macroions and the surfactant ions. The cooperative binding, which indicates the energetic advantage of binding adjecently to an already bound microion, should be associated with the hydrophobic interactions. We denote A F e as the free energy change due to the electrostatic binding, which equals the electrostatic potential energy on the surface of the macroion, and m h as that through the hydrophobic interaction. For simplification, we supposed that AFe does not change with respect to the progress of binding. AFh depends on the chemical structure of the surfactant, particularly on the size of alkyl chains. If we take account of the nearest-neighbor interaction between surfactant molecules, the binding of surfactant should occur in a continuous sequence but not in a random distribution along the polymer chain. Theoretical considerations of the nearest neighbor interaction for the titration curve of a linear polymer chain was given by MarcusI5 and Lifson” using the Ising model.I8 When the polymer chains have N/3 already bound groups and N(l - /3) unbound groups at a surfactant concentration the total free energy of such an assembly is associated with Naa, the number of pairs of nearest-neighbor surfactant-surfactant binding. Using Marcus’s theory,I5 we have
binding is determined not only by the electrostatic interaction but also by the hydrophobic interaction, while the cooperative process of the binding is determined only by the hydrophobic interaction, which is obtained from the slope of the binding curve at /3 = 0.5. It is known that if the oppositely charged polyelectrolyte is presented in the surfactant solution, the critical transition concentration (cmc) of the surfactant is 2 or 3 orders of magnitude smaller than that without polyelectrolyte. Equation 6 indicates that two factors contribute to the lowering of the transition concentration. One is the enhanced electrostatic interaction between surfactant ions and the macroions. Since the electrostatic potential valley on the polymer chains is much deeper than that of corresponding monomer ion, the polyelectrolyte is able to effectively bind the surfactant molecules and favors the aggregation through hydrophobic interaction. The other is a decreased entropy loss of counterions. When the surfactant molecules form the aggregates, it does not lead to a loss of translational entropy of the polymer, although the surfactant aggregation would result in a significant loss in the translational entropy of counter ions without the presence of polyelectrolytes. From eq 6 we have
e,
Thus, the slope of the isotherm curve increases exponentially with an increase in AFh. If we denote the cooperativity parameter u and the initiation constant KO as
-= NAN(AFe+ AFh) + aFCOlllp
as
we have
where NA is the Avogadro number, Nab = 2(Np - N,) is the number of nearest-neighbor pairs of bound and unbound groups, g[Nj3,N(1 - P),Nab] is the number of ways of binding for a given j3 and Nab. The equilibrium value of can be determined by minimization of the total free energy of the system aF/a,6 = 0. Thus, we have
44p(11/48(1 - /3)[ exp( -
gj-+ + 11
s) + - 11
1 1
2p - 1
+ 1 - 28
(6)
The second term of the above equation is a transition function that becomes steeper with an increase in the value of A&. Equation 6 indicates that the binding isotherm of the surfactant onto the linear polyelectrolyte consists of two terms: the f i s t term characterizes the transition concentration (initiation process), and the second term characterizes the steepness of the transition (cooperative process). The initiation process of the
(1 1)
KO= ev, exp(-AFelkT)
(12)
Since the overall equilibrium constant K of the surfactantpolyelectrolyte interaction can be written as
from eq 6 , we have
K = K,,u
1 +In
u = exp(-AFhlkT)
- P)(U - 1) -I- 1 + 1 - 2p] (1 - p)[44p(1 - B)(u - 1) + 1 + 2/3 - 11 p[J4/?(1
(14)
It is clear that KO characterizes the electrostatic interaction of the surfactant with the polyelectrolyte that corresponds to the binding constant of the surfactant molecule to sit to an isolated binding site on a polymer. u characterizes the aggregation process of surfactant molecules already bound to the polyelectrolyte and is equivalent to the cooperative parameter defined in the theory of Satake and Yang.‘ Since AFh < 0 for the hydrophobic interaction, u > 1 when the hydrophobic interaction contributes to the binding. For the case of binding through a simple salt formation, AFh = 0 and u = 1. Equation 14 shows that the equilibrium constant K is no longer a constant but a function of j3 if the hydrophobic
Polyelectrolyte-Surfactant Interaction
J. Phys. Chem., Vol. 99, No. 27, 1995 10973
interaction contributes to the binding. At /3 = 0.5, K = KOU. Eq 14 also shows that the binding isotherm neither depends on the polymer concentration nor on the molar ratio of polymer to surfactant. Binding with a Cross-Linked Polyelectrolyte. In the case of a cross-linked polyelectrolyte immersed in an oppositely charged surfactant solution, the free energy of the system, F, is the sum of the free energy of the outer solution, F,, and the free energy of the polymer network, F,:
N(l
+ y - a - /?)InN ( l + y - a - P)vc vg
The elastic free energy for the deformation of the network, Fel, is expressed asI6 F,,=-RTv, 3
2
F, can be expressed as follows: Fg
+ Fmobile + Fcomp + Fel
= 'int
(16)
where Fin, and Fcomp are the same as those used before, Fmobile is the free energy of motion of microions in the network, and Fel the elastic free energy of the network. If we denote ST, Sr, and P+ as the molar numbers of surfactant ions, their counterions, and network counterions in the outer solution (i = s), and in the network (i = g), we have
[(:y -
- 1 - ~n($)''~]
(25)
where 40is the volume fraction of the network at the reference state. The free energy of the surfactant binding with the polymer network Fcomp should be the same as eq 4. By minimizing the total free energy of the system aFlap = 0, we get the equilibrium value of p:
In e v C=
AFe
+ AF,, - 1 +
kT
4 4 / 3 ( 1 - /3)[exp(-
In 4 4 / 3 ( 1 - /3)[exp( -
z)
- 11 + 1
$)-
11
+ 2/3 - 1
+
+ 1 + 1 - 2/3
from the neutrality condition. From the ion number conservation condition,
+
where = [M - N(a p)]l[V - V,] is the molar concentration of the surfactant in the outer solution. a and y can be determined from the condition of aFlaa = 0 and aF/@ = 0, which give rise to
N = Pi +Pi
(21)
+
If we denote a = $IN, y = S i / N , then S t = M - N ( a p), Si = M - N y , and Pg' =N[1 y - (a + p ) ] , a n d P t =N[(a
+
+ P) - rl.
Since the free energy of the outer solution is equal to that of the translational motion of micro ions in outer solution, we have
+
[M- N ( a ,@]In
[M- N ( a + /3>1vc
v - vg
+
The equilibrium value of the network volume fraction 4 can be determined by aF/a# = 0:
where V, is the equilibrated volume of network in the surfactant solution. We consider only the situation that no micelle formation occurs in the outer solution. The free energy of the volume interaction of monomer links, Fint,is obtained from eq 2 by putting m and substituting the volume of network V, for the volume of solution V:
-
where the volume fraction of polymer 4 = VdVgand VO= Nv, is the volume of the polymer network in the dry state. The free energy due to the mobile ions in the network is
Combining eqs 26-29, the binding isotherms of the surfactant with the polymer network can be obtained. The differentiation of the isotherm curve at ,8 = 0.5 now becomes
10974 J. Phys. Chem., Vol. 99, No. 27, 1995
Gong and Osada
where V,, a,and y are the values at @ = 0.5. In a manner similar to the linear polymer, we define (d In q/@)B=o,5 = 4/&
Q
(32)
0
For the surfactant with strong hydrophobic interaction exp(AFh/2kT) and a - y
= 0.
0
(34)
We found that (35)
lnC,v, Figure 1. Binding isotherms of dodecylpyridiniumchloride (C12PyCl) with linear and cross-linked poly(2-acrylamido-2-methylpropanesulfonic acid) (PAMPS). Solid lines are calculated curves using the following M, VO= 6 x parameters: N = 3 x L, V = 0.01 L, vc = 0.018LA4,AFdkT= -6.2, q = 50. Open circles (cross-linkedpolymer) and triangulars (linear polymer) are experimental data quoted from ref 11.
which indicates that when the polyelectrolyte is cross-linked to form a network, it behaves like a rigid polymer with respect to the binding, therefore ug=
1
(36)
This shows that the surfactant-polymer network binding is not cooperative regardless of the strong hydrophobic interaction. Comparing eq 26 with eq 6 used for the linear polymer, we can easily see that the cross-linking effect is indicated in the last term in eq 26, which can be rewritten as the ratio of polymer counterion concentration distribution in the solution, [Pz], and within the network, [Pi]:
and by combining eqs 26 and 27, we have
where = NdV, is the equilibrium free surfactant concentration in the network. Thus, it is seen that the presence of the cross-linkage introduces an extra ionic osmotic pressure difference inside and outside the network. When @ is small, [PTI *: [Pi], and the network always tends to swell which is balanced by the elastic force. This swelling ionic osmotic pressure enhances the initial binding process but suppresses the surfactant aggregation compared with those of the linear polyelectrolyte. As pointed out by Kwak et a1.,* the cooperative binding is sensitively influenced by the chemical structure and flexibility of a polymer chain, and the cooperativity parameter becomes very low when the polymer chain lacks flexibility. If we regard the polymer network as a cross-linked three-dimensional polymer chain in water, the presence of the locally concentrated counterions, which originate the swelling osmotic pressure, makes the network expand in competition with the conformational shrinkage on binding and thus strongly reduces the cooperativity. We have previously reported that the cooperativity parameter of the binding for the polymer network can be increased by adding the neutral salt, in sacrifice of the initiation p r o c e ~ s . ' ~
This can be well understood from the present theory. The presence of salt strongly shields the electrostatic repulsion between macroions, thus lowering the electrostatic potential energy AF,. This leads to a shift of the isotherms curve toward a higher equilibrium concentration. On the other hand, a high ionic strength suppresses the network expansion since the addition of salt makes [PT]/[P:] insensitive to @ and enhances the cooperative binding, as can easily be predicted from eq 26. Comparison with the Experimental Results The described theoretical approach was compared with the experimental data for the binding of dodecylpyridinium chloride (CI2PyC1) with a linear and a cross-linked poly(2-acrylamido2-methylpropanesulfonic acid) (PAMPS).I' PAMPS is a strong polyacid with fully ionized sulfonic groups as macroions and H+ as counterions. The electrostatic free energy along the chain of linear PAMPS AF,can been analytically calculated using the rodlike mode1.19-20 From eq 18 in ref 20, the relation between AF,, which equals to the chain surface potential energy eW(ri) in ref 20, and the volume fraction of polymer is AFJkT= a
+ b In 9
(39)
where a and b are constants characterized by the chemical structure and the property of solvent. We have a = 0.1713, b = 0.9627, whereupon the following constants were used: radius of the polymer chain ri = 6.08 x m, monomer segment length b = 2.55 x m, dielectric constant cr = 78. Figure 1 shows the theoretical binding isotherms (solid line) calculated from eq 6 and the observed data (solid triangulars) obtained for C~zPyclbound with a linear PAMPS. The theoretical calculation was carried out using the following values: N = 3 x M, VO= 6 x L, V = 0.01 L, v, = O.O18uM, AFdkT = -6.2, and AFJkT = -7.0 calculated from eq 39. We see that the theoretical curve fits quite well with the observed data when ,f?is less than 0.7. However, a deviation was observed when j3 exceeds 0.7. This could be associated with the drastic conformational change of polymer chain from the extended random coil to globules, which should occur at a certain j3 value. This conformational change would largely
J. Phys. Chem., Vol. 99,No. 27, 1995 10975
Polyelectrolyte- Surfactant Interaction
TABLE 1: Calculated and Observed Data of the Initiation Constant KO and Cooperativity Parameter u for ClZPyCl Bound with PAMPS Gel of Various Degrees of Swelling 4“ 4
50
230
350
490
790
17W
KO [Umol] (calcd)
5.0 x 104 1.3 4.4 x 104 1.3
4.9 x 104 1.6 2.4 x 104 1.9
4.9 x 104 1.7 2.1 x 104 1.7
5.0 x 104
5.4 x 104 1.8 0.8 x 104 1.8
52 490 44 630
u (calcd) KO [Wmol] (obsd) u (obsd)
1.8 1.4 x 104 2.3
The observed data were quoted from ref 11. AFhIkT = -6.2 was used for the calculation. Linear polyelectrolyte solution.
Q
ds’l
0.5t
t
Table 1 shows the calculated and observed data of the initiation constant KO and the cooperativity parameter u for the network with varying degree of swelling. Data for the linear polymer are also shown for a comparison. From Table 1, we find that the calculated values of the cooperativity parameter u fairly agree with those of the observed data, while the initiation constant KO show a discrepancy between the calculated and the observed data when q becomes higher. In the above numerical calculation, we simply assumed that a % 0, and y % 0 which means that all of the surfactant ions in the polymer network have bounded with the polyions, and we did not consider the equilibrium between the free surfactant ions and the bound surfactant in the network domain. This is true only when the surfactant has a very strong hydrophobic interaction and the polymer network has a high cross-linking density. A numerical calculation of the general situation in which the equilibrium of surfactants inside the network is considered should be performed for the polymer network having a high degree of swelling. Nevertheless, the calculated isotherms showed fairly good agreement with the experimental data, which confirms the essential feature of the theory.
7 -18 -10
$2
-20
-16
-14
-12
-8
lnCsv, Figure 2. Binding isotherms of surfactant with different alkyl chain length to a network of q = 50. Solid lines are calculated curves using AFJkT = -8.0, -6.2, -4.0 from left to right. Other parameters are the same as in Figure 1. Experimental data (0,ClsPyCl; 0, C12pYCI; 0, CloPyC1) were quoted from refs 11 and 14.
decrease the surface potential energy AF,and alter the binding equilibrium to give decreased binding of the surfactant. Unlike the linear polymer solution, the electrostatic potential on the polymer network cannot be analytically described by the rodlike model. The charged network introduces not only potential valleys but deep potential wells at every cross-linking point.2’ Therefore the binding of the surfactant to the network presumably starts at the cross-linkingpoints which later develops to the chain. However, consideration of the electrostatic potential distribution along the polymer chain would be very complicating and is beyond the scope of this paper, we suppose that AFe of the network is approximately equal to that of the linear polymer at a corresponding polymer concentration and use eq 39 to evaluate AF, for polymer network. To simplify the numerical calculation, we assumed that a % y % 0, which means that all of the surfactant molecules penetrated into the network have bound with the macroions. The theoretical calculation was carried out with v$N = 0.08 (equivalent to a degree of swelling q = 50), & = 1, = -1.5. hF,/kT = -3.6 was calculated from eq 39 using the relation 4 = llq. The results of the theoretical (solid line) and the observed data (open circles) for a polymer network are also shown in Figure 1. Again, the theoretical curve fits well with the observed data in the range of p less than 0.7. Figure 2 shows the theoretical and experimental binding isotherms of the surfactants with different alkyl chain length bound to a network of q = 50. Increasing the alkyl size of the surfactant shifts the binding curves toward lower surfactant concentrations, while it does not enhance the cooperativity. Deviation from the experimental data is also observed at ,8 = 0.7 or higher.
x
Acknowledgment. This research was supported in part by Terumo Life Science Foundation and Grant-in-Aid for the Experimental Research Project “Electrically Driven Chemomechanical Polymer Gels as Artificial Muscle” from the Ministry of Education, Science and Culture (03555 188), Japan. The authors also acknowledge to the Agency of Science and Technology, Minister of International Trade and Industry (MITI) for the financial support. References and Notes (1) Satake, I.; Yang, J. T. Biopolymers 1976, 15, 2263. (2) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Macromolecules 1983, 16, 1642. (3) Shirahama, K.; Tashiro, M. Bull. Chem. SOC.Jpn. 1984, 57, 377. (4) Chu, D. Y.; Thomas, J. K. J. Am. Chem. SOC.1986, 108, 6270. (5) Thalberg, K.; Lindman, B. J. Phys. Chem.. 1989, 93, 1478. (6) Canane, B. J. Phys. Chem. 1977, 81, 1639. (7) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 526. (8) Schwarz, G. Eur. J. Biochem. 1 9 0 , 12, 442. (9) Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodubtzev, s. G. Mukromol. Chem., Theory Simul. 1992, 1, 105. (10) Kokufuta, E; Zhang, Y. Q.;Tanaka, T.; Mamada, A. Macromolecules 1993, 26, 1053. (1 1) Okuzaki, H.; Osada, Y. Macromolecules 1995, 28, 4554. (12) Osada, Y.; Okuzaki, H.; Hori, H. Nature 1992, 355, 242. (13) Osada, Y.; Ross-Murphy, S. B. Sci. Am. 1993, 268 (3,82. (14) Okuzaki, H.; Osada, Y. Macromolecules 1994, 27, 502. (15) Marcus, R. A. J. Phys. Chem. 1954, 58, 621. (16) Flory, P. J. Principles of Polymer Chemistry;Come11 University Press: Ithaca, NY, 1953. (17) Lifson, S. J. Chem. Phys. 1957, 26, 721. (18) Ising E. Physik 1925, 31, 253. (19) Lifson, S.; Katchalsky, A. J. Polym. Sci. 1954, 13, 43. (20) Gong, J. P.; Nitta, T.; Osada, Y . J. Phys. Chem. 1994, 98, 9583. (21) Gong, J. P.; Osada, Y . Chem. Left. Jpn. 1995, 6 , 449. JP950401E
