6876 Macromolecules 1992,25, zyxwvu 6876-6884 Self-consistent Field Theory of Micelle Formation by Block Copolymers X.-F. Yuan, A. J. Masters, and C. Price' Chemistry Department, University zyxwv of Manchester, Manchester, M13 9PL zyx U.K. Received September 23, 1991; Reuised Manuscript Received August 25, 1992 ABSTRACT A self-consistent field theory was developed to account for the micellization of block copolymers in selective solvents. The mean-fieldexpression used successfully by Edwards in treating the polymer excluded- volume problem was modified to enable it to be applied to chains in bad solvents without the generation of unrealistically high segment densities. Full consideration was given to the issue of standard states so that a meaningful comparisoncould be made between theoretical and experimentalresults. The theory was able to account in a consistentand satisfactorymanner for the thermodynamics of micellizationof two polystyrene- block-polyisoprenecopolymers in n-hexadecane which is a Selectively bad solvent for polystyrene. 1. Introduction When a block copolymer is dissolvedin an organicliquid that is a bad solvent for one of the polymer components but a relatively good solvent for the other, the copolymer chains associate reversibly to form micelles.'+ For such systems it has been shown experimentally that it is a large negative enthalpy change on association which gives micelle ~tability.~ This thermodynamic behavior is in strong contrast to the association of amphiphilic block copolymers and other synthetic surfactants in aqueous media for which a favorable entropy change arising from the hydrophobic effect is normally responsible for micelle formation occurring.8 From the studies carried out to-date it appears that block copolymers tend to form spherical micelles in dilute solution, althoughmetastable wormlike micelleshave been observed under certain condition^.^ By ultracentrifuga- tion,1°gel permeation chromatography," and transmission electron microscopy,6J1 it has been shown that block copolymers which are fairly uniform in mass and com- position form micelles with very narrow molar mass distributions; the molar mass distribution of the micelles is generally significantly narrower than that of the unassociated block copolymer. The presence of solvent in the core regions of block copolymer micelles in organic solvents has been studied by small-angle scatterings and 'H and 13C NMR methods.12 A number of theoretical treatments of micelleformation by block copolymers have led to predictions of scaling laws for micellar properties. To obtain mathematically tractable expressions, rather severe approximations had to be made about the segment density distributions and the nature of the interfa~e.l~-~' Whilst the scaling predictions might be expected to be valid under special circumstances,in order to obtain a reasonably quantitative account of micellar behavior under more general conditions one must relax the highly simplified assumptions and develop a more comprehensive theory. As the theory must deal with the behavior of a polymer chain both in good and bad solvents, a self-consistent Weld approach would seem to be the theory of choice. Although such theories do not for example predict the exact exponents charac- terizing the behavior of an isolated homopolymer chain in a good solvent, the values obtained are good approxima- tions in three dimension^,^^-^^ and recent studies have shown that with an appropriate choice for the mean-field potential, quite good agreement an be achieved between theory and simulation over a r zyxwvuts ,n" ge of solvent condition^.^^ A theory of this type has been used to study end-adsorbed polymer brushes in polymeric matrices,26 and a lattice 0024-9297/92/2225-6876$03.00/0 version of a mean-field theory has been applied to copolymeric micelles.27 In this paper we present a continuous space self- consistent theory for micelles. Unlike much past work, we treat the micelle and the isolated chain in a unified manner. As micellar properties depend critically upon the difference between the standard Gibbs energies of a micelle and that of the isolated chains, we believe it is importantto treat the isolated chain at the same theoretical level as the micelle.18 In common with all self-consistent theories, however, no assumptions need to be made about the micellar structure-density profiles since these can be predicted. In this paper, we compare our theoretical predictions with experimental results18728 for solutions of two polystyrene-block-polyisoprene copolymers in zy n- hexadecane (which is a selectively bad solvent for poly- styrene), full attention being given to the issueof standard states. In later publications, we hope to present further comparisons with experimental results and also to inves- tigate numerically whether the theory predicts any scaling laws. 2. Chemical Potential of Unassociated Chains The chemical potential of an unassociated block co- polymer chain in solution at the infinitely dilute solution limit can be written as p1 = -kT In zp - kT In zlc - kT In [h1-3p,-'l (1) where zp is the internal partition function of the chain made up of rotational, vibrational, and electronic contri- butions and short-range intramolecular interactions. zy zlc is the configurational partition function of a chain with fixed center of mass; it is termed for reference the coupling partition function since it accounts for long-range in- tramolecular interactions of the unassociated copolymer chain and for the intermolecular interactions of the copolymer chain with the solvent. The last term on the right-hand side of eq 1 accounts for the translational motion of the chain within the volume, V, of the system. The de Broglie wavelength XI for classical fluids depends only on the mass of the chains and the temperature; AI = h/(2~ML-lkT)'/~, where M is the molar mass of the chains, h is the Planck constant, and L the Avogadro number. p1 (=Nf/v) is the number density of the unassociated chains. Expre%sed per mole of chains and in terms of a molar density, p1 (=Nf/Lv), eq (1) becomes i1 = -RT in zp - RT In zlc - RT In [X~3~1-'L-11 (2) If we choose an ideally dilute standard state of unit molar 0 1992 American Chemical Society