Dynamics of Bilayer Membrane

The theory combines the Rouse model for a single chain with the liquid crystal theory for bilayer membrane developed by Marcelja. Thus the forces acting on a bead within a chain are spring forces, Brownian force and dissipative or viscous force which appear in Rousetheory plus the liquid crystal and lateral pressure forces. The dynamic viscosity of the system is calculated. 1 . Introduction. Extensive experimental study of both natural and synthetic bilayer membranes has been made in recent years [l, 2, 31. Whereas interpretation of measured data in naturally occurring membranes is complicated, the properties of synthetic bilayers are somewhat simpler to interpret. Due to the fact that the structure of these bilayers is well established, theoretical models can be worked out to further elucidate the nature of physical phenomena. The model of bilayer membrane presented by Marcelja [4, 51 treats the problem of thermodynamics and phase transitions in these systems by introducing the concept of liquid crystal theory. Marcelja was able to calculate the transition temperature from isotropic to nematic phase. The transition temperatures predicted by Marcelja model are in good agreement with experiments. Other experiments indicate that in bilayer systems the liquid chain is in constant motion parallel to the plane of the bilayer. This motion has been studied by spin labelled experiments by McConnel and coworkers [6] and Sackman and Trauble [7,8]. The information obtained from these measurements gives the lateral diffusion coefficient of the chain in the isotropic phase for which both experimental group report a value of the order of 2 X 10-* cm2/s. More recent experiments by Webb and coworkers [g] extended these extended studies using florescence spectroscopy and photobleaching techniques. They have measured the lateral diffusion coefficient in both isotropic and nematic state for bilayers and have found that the diffusion (*) This research has been supported by a grant from the ministry of Science and higher education of Iran. coefficient has a larger value by a factor of three in the nematic phase. These experiments and similar ones require a dynamical theory of membranes for interpretation. Although the increase in the lateral diffusion coefficient in the nematic phase would involue a more complete theory in the phase transition region, a preliminary theory away from the critical region is here developed assuming that the chain behaves as a Rouse chain [10]. 2. The model. -We consider a single polymer chain of N links, each link with effective length b embedded among similar chains in the membrane. One end of the chain is constrained to move only in the X-y plane, the other end being free. Actually a bilayer consist of two such systems where chains are constrained between parallel planes. However for this purpose it is sufficient to consider one of the chains in the bilayer. Consider now the link i in this single chain. This link consist of the carbon (bead) number i 1 at one end and carbon number i at the other. One is interested in writing the equation of motion for the system of N beads. Now one must characterize the forces acting on each bead, this has been done by Rouse and Zimm [10, 111 for a system of beads connected together by harmonic springs. In our problem there is an additional force arising from the surrounding polymers in the membrane, the latter forces also has been studied in the equilibrium case by Marcelja [5]. Our object is to combine these two types of forces. The Langevin equation Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979390 C3-452 A. R. MASSIH AND J. NAGHIZADEH for the jth bead for a free polymer in solution is given by d 2 ~ , m [Rj KB T Vj log + VjU (1) dt2 d . where m is the inertial force, [Rj is the fricdt2 tional force, 5 being the friction coefficient, KB T Vj log $ is the random force and VjU is the spring forces given in more detail with Z, = 0 and K being the spring constant. In order to incorporate the interpolymer interactions we apply the method employed in theory of liquid crystals. One begins by writing the Ising like Hamiltonian [12], the contribution of repulsive forces. Marcelja [5] has discussed these forces in detail and has called them lateral pressure forces. The dependence of the lateral pressure on order parameter V is given by an equation of the form PA, L,/L with L being the total projection of the chain in the z direction and A, and L, are the area and the length respectively, of the chain in the frozen nematic phase. Using this concept and relating it to the order parameter the final equation becomes nonlinear in V. In order to avoid this difficulty one notes that, as far as small oscillations are concerned, the forces exerted by neighbouring polymers are of elastic type and the lateral pressure force should appear as proportional to the order parameter or the square of the projection along the z axis. ~Crther scrutiny reveals that this result can also be obtained by the expansion of non-linear Marcelja form of the lateral pressure. It has also been used by Marcelja in his earlier paper [4] as a logical analogy of the Ising type Hamiltonian eq. (4). We can write the interpolymer forces on the ith X = CAi jq iq j H C q i (3) bead (along z axis) as ij i axi ax, aei where qi is the order parameter associated with a F.= = -p-= czi 82, aei azi (8) bead i on one polymer and qj is that associated with where bead j on a neighbouring polymer, Aij is the coupling constant. The second term in eq. (3) represents the 3 c = ( v o ~ < ~ j ) b2 j + H ) . (9) contribution of external forces which in this case are the elastic (repulsive) interpolymer forces. We now apply the mean field approximation to eq. (3) we Combining eq. (8) with eq. (1) we obtain the ~ a n ~ e v i n obtain the form equation of motion of the ith bead for the system under consideration xi = v0 C < V j ) Vi HVi j (4) m d2Ri --= dt2 mi K, T Vi log + ViU + czi (10) where v, is given by j 3. Reduction of the equation of motion. FollowHere the mean order Parameter ( V j ) is obtained ing the standard procedure given in Yamakawa [13], as a solution of the self-consistent equation we neglect the inertial term in eq. (10) and utilize the continuity equation for the distribution function + [dQVjexP[~vo ( V j ) rti] given by