J. Phys. Chem. 1994,98, 6187-6194
6187
Statistical Mechanical Theory of the Coagel-Gel Phase Transition in Ionic Surfactant/Water Systems Masaru Tsuchiya' and Kaoru Tsujii Kao Institute for Fundamental Research, 2606, Akabane, Ichikai-machi, Haga-gun, Tochigi, 321 -34, Japan
Kazuo Maki Kao Institute for Knowledge and Intelligence Science, 2-1 -3, Bunka, Sumida-ku, Tokyo, 131, Japan
Toyoichi Tanaka Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 Received: February 9, 1994"
Some ionic surfactant/water systems undergo the coagel-gel phase transition. Both coagel and gel phases consist of alternating layers of bilayer membranes of surfactants and water. The hydrocarbon chains are in a crystalline state, and thickness of water layer is in the order of 10 A in the coagel phase. Upon the transition from coagel to gel, the hydrocarbon chains become able to rotate around the chain axis, and the thickness of water layer discontinuously increases to the order of 1000 A. We present a statistical mechanical theory to explain this phase transition. The mean field Gibbs free energy of the system is written as a function of three-order parameters which are concerned with the rotational motion of the hydrocarbon chains, the degree of dissociation of counterions of the surfactant molecules, and the distance between neighboring membranes. The area occupied by one hydrophilic head group on the membrane surface in the rotating state of the hydrocarbon chain is assumed to be larger than that in its fixed state. The theory successfully explains the order4isorder phase transition of the hydrocarbon chains and the simultaneous separation of bilayer membranes. The expansion in the area of one molecule leads to a coupling between three-order parameters. The theoretical calculation predicts that the strength of this coupling results in the three types of phase transition: (1) coagel/gel phase transition, (2) pseudogel (the hydrocarbon chains are in crystalline state, and the membranes separate)/gel phase transition, (3) coagel/pseudocoagel (the rotation of hydrocarbon chains around the chain axis is released, and the membranes do not separate) phase transition.
Introduction The gel-liquid crystalline(or micellar) phase transitions (Krafft phenomena) of surfactantlwater systems have been extensively studied and well understood. However, the coagel-gel phase transition, that appears in the temperature range lower than Krafft point, has been rarely investigated.14 During the past ten years, some experimental studies on this transition and the structure in each phase have been made.S-' Both coagel and gel phases consist of alternating layers of bilayer membranes of surfactant and water (so-called lamellar structure). The conformational state of the surfactant molecules in each phase is different, as illustrated schematically in Figure 1. In the coagel phase, the rotation of the hydrocarbon chain of surfactant molecules around the chain axis is restricted, and the chains are ordered in an orthorhombic crystal lattice. In addition,the hydrophilic groups of the molecules are ordered as a crystalline state. The hydrocarbon chains in the gel phase, on the other hand, are packed in a hexagonal lattice and rotateor vibratearound thechain axis. Thedistance between neighboring bilayer membranes is also much different, which is in the order of 10 8, in the coagel phase and of 1000 8, in the gel phase. Similar rotational transitions of hydrocarbon chains to the coagel-gel phase transition have been observed in some kinds of normal paraffins.8-10 They show a phase transition from an orthorhombic or a monoclinic phase to a hexagonal one. The conformationalchange of the hydrocarbonchains at this transition occurs essentially at the same point as the coagel-gel phase transition. Both experimental and theoretical studieson this phase @
Abstract published in Aduance ACS Abstracrs, June 1, 1994.
0022-3654/94/2098-6 187$04.50/0
transition of n-paraffin chains have been extensively Kobayashi calculated the lattice energy in details and succeeded to explain the latent heat at the release of rotational motion of each length of hydrocarbon chains.") These results are helpful for us to make our calculations on the coagel-gel phase transition. One purpose of this paper is to propose a simple statistical mechanical model to explain the mechanism of this phase transition and find essential parameters which determine the stability of the gel phase against the coagel phase. The other purpose is to make clear the reason why membrane separation takes place simultaneously at the rotational phase transition of the chains.
Model and Expression of the Gibbs Free Energy Let us consider a system composed of bilayer membranes and a water layer between them in which a fixed number of surfactant molecules are contained, as indicated by the dotted line in Figure 2. The water layer is assumed to be a continuous media. The total Gibbs free energy per molecule, a, can be written as the sum of three terms
0 = G, + G,+i
+U
(1)
Each term represents the Gibbs free energy per hydrocarbon chain (GE), that per one pair of a head group and a counterion (Gh+i), and the interaction energy of the van der Waals attraction 0 1994 American Chemical Society
Tsuchiya et al.
6188 The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 (side view)
0
0
0
0
(cross sectional view)
0
hydrocarbon chain
(-)
-1 000 A
(T) Figure 1. Schematic illustrations of the rotational state of surfactant molecules in both the coagel and the gel phase. Open circles and filled circles denote hydrogen atoms and oxygen atoms, respectively, in the cross sectional view.
introduced as follows:
-{I, rotating state
+
- - -1, fixed state
They are “spin variables” of a chain on each lattice. We further assume that the area occupied by one surfactant molecule on the membrane surface is larger in the rotating state than the fixed one. We denote the surface area occupied by a chain by
(3-1)
and the difference by Figure 2. The system to be considered which is composed of the bilayer membranes and water layer.
and hydration repulsion per molecule between neighboring membranes (U). Let us consider Gcfirst. A lattice model is employed in order to describe the behavior of the chains. Each chain occupies ”+ lattice point” or “- lattice point” as shown in Figure 3. It is simply assumed that each chain can take only two states, a fixed state or a rotating state. Variables denoting these states are
Af
< A,
(3-2)
Note that the transition from the coagel to the gel phase enlarges the area of the membranes. Thevan der Waals interaction energy between the first nearest chains can be classified into u,, uyt, u&, and utras shown in Figure 4, and the internal entropy of a chain is assumed to take different magnitude corresponding to its state. We define the difference as
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6189
Coagel-Gel Phase Transition in Surfactant/HzO Systems lattice model of surfactants membrane
The critical temperature (Tc)is determined by the first and the second term of eq 6
2 states of a chain
I I
T, = _-2 5
(9)
k
Below this temperature, the first-order phase transition of the chains takes place at TIwhere H(TJ is equal to zero. But these temperatures are shifted by the other terms, Gh+i and U. The interaction energy, U, between membranes may be expressed empirically by I
+ -
U(d)= --dZ+ H e x p ( - f - )
( a cross section 01 a membrane )
Figure 3. Surfactant molecules are on the + and - lattice points. They can take only two states: "rotating state" and "fixed state".
Uff
where
Ud
Ufr
where d is the distance between the membranes. The first term represents thevan der Waals attractive interaction, and the second one is the repulsive interaction energy which comes from "hydration f0rce~.~*J3 The hydration force acts to prevent a direct contact of the membranes. Finally, let us move to consider Gh+i which is the Gibbs free energy per one pair of a head group and a counterion. Neglecting the expanding workagainst pressure, Gh+i is equal to the Helmholtz free energy. Then, Gh+i is written as follows:
Figure4. The van der Waals interaction energy between the first nearest hydrocarbon chains is classified into four categories; fixed chain-fixed
), chain (u~),rotatingchain-fixed chain chain ( u ~ ) rotatingchain-rotating (urf), and fixed chain-rotating chain (ufr).
where S, and Sf denote the internal entropy of a chain in the rotating and in the fixed state, respectively. In thermodynamic equilibrium state, we assume that (a-) = ( u + ) , where ( ) denotes the ensemble average. The order parameter of the chains, u, is defined as u = (a_)= (a+)
=( >0, 0), Fi(u < 0), and U. gel
13 shows the competition between four kinds of minimum in the free energy around the transition temperature from the coagel to the gel phase.
Dependence of the Phase Transition BehaviOr on A&, and AA It is clear from the previous discussions that AA (= A, - Af) plays an essential role in the coagel-gel phase transition. We
- A&
plane. The region in which well-knowncoagel-gel phase transition occurs becomes larger with increasing of A,/Af.
investigate the behavior of the phase transition while changing the value of Ahis, and A,/Af, where Af is fixed at 18.40 A2.The results are summarized in Figure 14. The region of Afdb in which the phase transition between gel and coagel takes place expands as A,/Af increases. This tendency is attributed to the stronger coupling between order parameters. Consequently, we say that L 4 or A,/Afis an essential factor in the separation of membranes which accompanies the rotational transition of the hydrocarbon chain. It is interesting to note that the pseudogel and the pseudocoagel is not necessarily unstable as the result of our calculations, although they have not been experimentally found yet.
Conclusions We have presented a statistical mechanical theory of the coagelgel phase transition and concluded as follows. (1) Analysis of the freeenergy of ionicsurfactant/water systems allows prediction of the four phases: coagel, gel, pseudocoagel, and pseudogel phases. They are characterized by the rotational motion of the hydrocarbon chains around the chain axis and the distance between the bilayer membranes. In the coagel phase, the chains are in the crystalline state, and the membranes do not separate. The gel is the opposite phase in which the rotation of the chains is released, and the membranes separate. On the other hand, the chains are in rotational state, but the membranes do not separate in the pseudocoagel phase, and the chains are in crystalline state but the membranes separate in the pseudogel phase. The pseudocoagel phase and the pseudogel phase have not been experimentally found yet. (2) The simultaneous membrane separation at the transition from the coagel to the gel phase results from the coupling between
~
.
~
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The Journal of Physical Chemistry, Vol. 98, No. 24, 1994
the threeorder parameterswhich areconcemed with the rotational motion of the hydrocarbon chains, the dissociation of the counterions of surfactant molecules, and the distance between the membranes. The strength of this coupling is determined by the parameters A&, the free energy change of a counterionwhen it dissociates from a bound state to an isolated state into water medium, and AA,the change of the area on the membrane surface per surfactant molecule when its rotational motion is released. (3) The phase transition can be classified into the three categories; transition between (i) the coagel phase and the pseudocoagel phase, (ii) the coagel phase and the gel phase, and (iii) the pseudogel phase and the gel phase. Acknowledgment. The authors express their sincere thanks to Drs. J. Maskawa and J. Mino of Kao Knowledge and Intelligence Science Laboratory and also to Mrs. N. Satoh and S.Ueda of Kao Institute for Fundamental Research for their helpful discussions.
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