Statistical thermodynamics of block copolymer adsorption

The aim of this study was to develop a statistical thermodynamic theory for the adsorption of linear flexible block copolymers from a multicomponent solution. This has been accomplished by a more general derivation of the self-consistent field theory of Scheutjens and Fleer for adsorption of homopolymer from a binary mixture, introducing local segment potentials for any type of segment.In chapter 1 the statistical thermodynamic analysis for a multicomponent mixture (including block copolymers) near a surface is given in detail. Near the surface, a density gradient for every type of molecule is found due to spatial restrictions and mutual interactions between segments and between segments and the surface. Every individual segment is subjected to a local (segment) potential, which depends on the distance from the surface and on its chemical nature. We use a lattice model to evaluate the contact energies and the conformation count. The segment potential is derived from the maximum term in the canonical partition function. Like in the original derivation of Scheutjens and Fleer we maximize the canonical partition function with respect to the number of molecules in each particular conformation. However, to perform the necessary partial differentiations under the appropriate boundary conditions we apply the method of Lagrange multipliers. From the segment potentials we can calculate for every particular conformation its statistical weight as a multiple product of Boltzmann factors (one for each segment) and its contribution to the overall segment density profile. In Appendix III of chapter 1 a set of equations is formulated from which the segment potentials can be found in a self-consistent manner by standard numerical techniques.A number of results on the segment distribution of di- and triblock copolymers is given. Diblock coplymers are found to adsorb with the adsorbing block rather flat on the surface and the less or non-adsorbing block in one dangling tail protruding far into the solution. A comparison with terminally anchored chains shows overall agreement but also typical differences.In chapter 2 the physical background of the theory is briefly reviewed. Results on the adsorbed amount and the hydrodynamic layer thickness of adsorbed di- and triblock copolymers are given. We find a strong dependence of these parameters on the chain composition. When the total length and bulk solution volume fraction of a diblock copolymer are kept constant, a maximum is found in the adsorbed amount as a function of the fraction of adsorbing segments. The fraction of adsorbing segments corresponding to this maximum could be named "the optimal fraction"; it is found to decrease with increasing chain length, increasing bulk solution volume fraction, increasing surface affinity of the more strongly adsorbing block, and decreasing surface affinity of the weakly adsorbing block. From these results we have been able to relate in a simple way the adsorbed amount of an AB-diblock copolymer (where A adsorbs more strongly than B) to the adsorbed amount of an A-homopolymer of equal length. A linear relation is obtained between the adsorbed amount of AB-diblock copolymer (as compared with an A-homopolymer) and the block length ratio r B /r A . where r A and r B are the lengths of the A-block and the B-block, respectively. Usually, diblock copolymers form thick adsorbed layers, with a hydrodynamic layer thickness that depends strongly on the adsorbed amount. This thickness is of the order of 10 to 30 % of the length of the B-block. For BAB-triblock copolymers with adsorbing A-segments and non- adsorbing B-segments we find lower adsorbed amounts as compared to an AB-block copolymer with the same total number of A- and B-segmentsThe interaction between adsorbed layers of block copolymers is examined in chapter 3. The calculation of the free energy of interaction is straightforward. We elaborate the concept of full equilibrium and that of restricted equilibrium for a multicomponent mixture. Full equilibrium refers to the case that all molecules in the mixture are free to diffuse out or into the gap between the surfaces. Hence, in full equilibrium all molecules have a constant chemical potential when the surfaces are brought closer. If one or more of the components are unable to leave the gap when the surfaces come closer we have a restricted equilibrium and the chemical potentials of those molecules will not be constant. Usually, the interaction between adsorbed layer of adsorbed diblock copolymers at full equilibrium is found to be repulsive. in contrast to the case of homopolymers where the interaction is always attractive. At full equilibrium, when the surfaces are brought closer, homopolymers desorb and form bridges resulting in attraction between the surfaces. Since diblock copolymers hardly form any bridges when the surface affinities of both blocks differ enough, no attraction is found at full equilibrium. For the same reason we find always repulsion in a good solvent when the amount of diblock copolymer is kept constant (restricted equilibrium). The onset of the repulsion increases with increasing adsorbed amount and with increasing length of the non-adsorbing block. The interaction curves at various lengths of the adsorbing- and non-adsorbing block could be scaled onto approximately one master curve. When the solvent quality for the non-adsorbing block becomes poor (χ>0.5). there is an attraction at large separation as a result of osmotic forces (phase separation), even at restricted equilibrium. In fact, adsorbed diblock copolymers behave like infinitely long homopolymer chains in solution, which phase separate when χis above 0.5. For ABA-triblock copolymers with adsorbing A-segments and non-adsorbing B-segments, we find attraction at not too small separations in a good solvent for the B-blocks, because now bridging is again possible: adsorbing segments are found at both extremities of the chains.This model has provided a detailed insight in the properties of adsorbed block copolymer layers and should be a useful tool for the development and optimization of experiments and products in which copolymer adsorption plays a role.