Langmuir 1993,9, 1160-1161
1160
Equilibrium Distribution of Linear Micellar Aggregates in a Gravitational or Centrifugal Field Alexander Duyndam and The0 Odijk' Department of Polymer Technology, Faculty of Chemical Engineering and Materials Science, Delft University of Technology, P.O.Box 5045, 2600 G A Delft, The Netherlands Received November 2, 1992. In Final Form: January 21, 1993
The behavior of linear micelles subjected to external fields is a topic of current interest (see e.g. refs 1-10]. Aggregates sedimenting under gravity or in a centrifugal field have apparently not been treated theoretically so we focus on the equilibrium case which is relatively straightforward. We first recall the thermodynamics of a dilute incompreasiblesolution (whosesoluteneed not be micellar) in a gravitational field which allows us to formulate the relevant variational principle for a suitably chosen thermodynamic potential. We then calculate the equilibrium size distribution and the concentration profile of a dilute suspension of linear aggregates under gravity or in a centrifuge. We present a modified version of Kirkwood and Oppenheim's analysis11of a dilute suspension sedimenting in an external field (either gravitational or centrifugal). The potential W z ) of the field is defined per unit mass and acta in the z direction of our Cartesian coordinate system whose origin is chosen at the upper surface of the solution. The positive z half-space is situated above the surface. We chose W O ) equal to zero, i.e., \k = gz, for a gravitational field, with g being the gravitational acceleration or \I! = l/w(R2(0)- R 2 ( z ) )in a centrifuge where w is the angular velocity of rotation and R(z)the distance of a point in the solution to the axis of rotation. The solution is in equilibrium with a reservoir which has a constant temperature and exerta a constant pressure P e x on the solution (Figure 1). We divide the solution into a very large number " o f layers aligned perpendicularly to the field. If the thickness of each layer is macroscopically small so that it is much smaller than the scale of variation of the potential Wz), we may regard each layer as homogeneous. This enables us to write the differential of the internal energy uk of one such layer labeled k as where Tk is the absolute temperature, sk the entropy, Pk the pressure, and vk the volume of the layer. The chemical PObntidpk = pk@k,Tk,Zk)= p'k@k,Tk) + m!??(Zk)Of aSOlUte molecule of mass m is the sum of the chemical potential p'in the absenceof the field and the increasein ita potential energy in the gravitational field; likewise we have an analogous relation pw,k = p'w,k@k,Tk) + m,\k(zk) for the (1) Rehage, H.; Wunderlich, 1.; Hoffmann, H. Prog. Colloid Polym. Sci. 1986, 72, 51. (2) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (3) Hoffmann, H.; K r h e r , U.; Thum, H. J. Phys. Chem. 1990, 94,
2027. (4) Ruckenetein, E.; Brunn, P. 0.; Holweg, J. Langmuir 1988,4,350. (5) Cates, M. E.;Candau, S. J. J. Phys. Condens.Matter 1990,2,6869. (6)Wang, S. Q.; Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1990, 94, 2219. (7) Wang, S. Q. Macromolecules 1991,24,3004. (8) Turner, M. S.; Cates, M. E. J . Chem. Soc., Faraday Trans. 1991, 87, 2073. (9) Turner, M. S.;Catee, M. E. J . Phys. Condens. Matter 1992,4,3719. (10) Bruinema,R.; Gelbart, W. M.;Ben-Shaul,A. J. Chem. Phys. 1992, 96,7710. (11) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McCraw-Hill Press: New York, 1961; Chapter 15.
Figure 1. A reservoir which fixes the temperature and the external pressure on a solution. The solution is divided into N homogeneous layers perpendicular to the direction ( 2 ) of the field.
chemical potential of a solvent molecule. The number of solute molecules in the kth layer is Nk and the number of solvent molecules Nwb. For the reservoir of entropy S, and volume Vr, the differential of the internal energy U, reads dU, = T dS, - pexdV, (2) Since the internal energy of a closed system in stable equilibrium is a minimum, we obtain the equilibrium conditions of the solution plus reservoir by equating the first variation of the total internal energy (= Ur c uk)to zero. This variation is subject to the following constraints: the total entropy (=& + csk) and the total number of particles (N = Nw= CNwh)are conserved; the volume of each layer is fixed except for that of the first layer which is allowed to vary so that the system may adjust to the fixed external pressure (dV1= -dVr, dVk = 0 for k # 11). The differential of the total internal energy Ubt is now written as
+
mk,
pw,k ww,k
following conditions for equilibrium p1= pex; Tk = T; pk = p and hw,k = pw are uniform (4)
These conditions should be supplemented with stability criteria to ascertain that the equilibrium is stable, but they will ,not be discussed here. In the theory of micellar growth it is often useful to employ a variational principle, especially when there are complicated interactions pre~ent.'~-'~ For the system at hand we have a minimum principle for a: @
= @(N.NW,~,,,T,W))NCC + N&w
- - I
(6)
If we keep in mind that all the volumes of the layers are constrained except the first, we have on using eqs 1,2, and 4 (12) Gelbart, W. M.; McMullen, W. E.; Ben-Shaul, A. J. Phys. (Paris) 1986,46, 1137. (13) Od!jk, T. J..Phys. (Paria) 1987,48, 125. (14) Odijk, T. Btophys. Chem. 1991,41, 23.
0743-746319312409-1160%04.00/0 .. ~ ~ -----. . Q 1993 American Chemical Society ~
(3)
As dUbt must vanish for arbitrary variations we have the
Notes
Langmuir, Vol. 9, No. 4, 1993 1161
X(s,z) = S" exp(-K
The differentiald@is thereforeequal to zero in equilibrium provided the uppermostlayer of the solution has a constant pressure Pex. We will not pause to formulate the consequences of d2@> 0 which ensures that the extremum is indeed a minimum. At this stage we take the continuum limit (Nextremely large) so that the chemical potential becomes
~ @ ( z ) , T , X ( z ) , * ( z= ) )r('@(z),T,X(z))+ m*(z) = /do@(L),T)+ kBT log X(Z) + m*(Z) p'O@e,,T) + @ ( z ) - Pex)U + kBT log x ( z ) + m*(z) = P ' O @ ~ ~ , T+) kBT log x(Z) (m- p,U)*(Z) (7) with k B being Boltzmann's constant. Here we have used the fact that the solution is dilute (mole fraction X ( z ) N ( z ) / ( N ( z+ ) Nw(z))
