Copolymers consisting of both adsorbing and nonadsorbing segments can show an adsorption behaviour which is very different from that of homopolymers. We have mainly investigated the adsorption of AB diblock copolymers, which have one adsorbing block (anchor) and one nonadsorbing block (buoy). The anchors adsorb from solution onto a surface and the buoys protrude into the solution, Thus, a polymer brush is formed. This name is derived from the resemblance between the protruding chains of B segments and the bristles of a brush. The presence of the adsorbing segments can be neglected when studying the characteristics of such a polymer brush, which is then modelled as (B-) homopolymer molecules which are terminally attached to the surface of a solid interface.In chapter 1 two self-consistent field (SCF) theories are introduced which give a description of such a polymer brush. The first of these theories is a lattice model. It takes into account all possible conformations that can be generated on a lattice; the molecules are treated as freely jointed chains. The overall volume fraction profile (that is, the polymer volume fraction φas a function of the distance z to the surface) is then found by weighting each conformation with an appropriate Boltzmann factor. This theory can both be applied for systems with end-attached polymer molecules and for systems with freely adsorbing chains. The volume fraction profiles for any given system must be found using a complicated numerical procedure.The second theory explicitly assumes that the polymer molecules are strongly stretched. Under this assumption only a fraction of all possible molecular conformations need be taken into account to find the volume fraction profile. Although this approach is less exact than the lattice model, it has as a major advantage that an analytical expression can be derived for the shape of the volume fraction profile. A simple algebraic expressions is also available for the brush height, if only the second and third order terms of a virial expansion of the free energy of mixing polymer and solvent are taken into account. If this free energy is accounted for in a more exact manner, one must (numerically) calculate the brush height from a (simple) integral equation.In the first chapter we make a detailed comparison of the predictions of both theories for a polymer brush at a flat surface in a low molecular weight solvent. In general an excellent agreement is found between the results of both theories. Significant deviations only occur very close to the surface and at the periphery of the grafted layer. In the lattice model there is a small depletion zone near the grafting surface, which is caused by the entropical restrictions imposed upon many polymer conformations by this impenetrable surface. The lattice calculations further show a "foot" of the volume fraction profile, which extends further away than the brush height as calculated from the strong-stretching approximation. The relative importance of these deviations increases with decreasing chain length, decreasing grafting density, and decreasing solvent quality. In order to find good quantitative agreement between the lattice calculations and the strong-stretching theory, one must incorporate the full Flory- Huggins expression for the mixing free energy of polymer and solvent into the latter theory. The derivation of elegant, analytical expressions for the layer structure by expanding this free energy in a virial series is only valid for low grafting densities.In all chapters except the second, the polymer chains are treated as freely jointed chains in a potential gradient. In chapter 2 more elaborate models are introduced for the polymer chains. Chain stiffness is incorporated by reducing the flexibility of the segment bonds. Stiffer chains give larger brush heights. Over a large range of chain stiffnesses the volume fraction profiles agree well with analytical expressions based on the incorporation of chain stiffness into the Gaussian approximation for the local stretching of a polymer chain. A further modification is a first order correction to the excluded volume interactions in the generation of the chain conformations. This correction slightly reduces the brush height. The opposing effects of this correction on the one hand, and chain stiffness on the other, suggest that the freely jointed chain is a good model for "real" polymers.Chapter 3 considers polymer brushes on cylindrical and spherical surfaces with a radius of curvature R. On such surfaces the dependence of the brush height H on the chain length N differs from that of a flat brush. SCF lattice calculations are presented to investigate this dependency as a function of R. For large values of R the scaling law H - N is recovered for both spherical and cylindrical surfaces. For R = 1 good agreement is found with the scaling laws H - N 0.6(spherical surface) and H - N 0.75(cylindrical surface). Polymer brushes on spherical surfaces can be seen as a model for AB diblock copolymers adsorbed onto small colloidal particles. For R = 1 a star-branched polymer molecule in solution is modelled.The volume fraction profile of the brush is also studied as a function of R. For this purpose we focus our attention on spherical brushes immersed in athermal solvents. For large radii of curvature we make the assumption that the potential energy profile of the segments can be approximated by a parabolic function, as for flat surfaces. Applying this approximation, we derived an analytical expression for the volume fraction profile which agrees reasonably well with the lattice calculations. For very small radii of curvature the lattice calculations predict volume fraction profiles which follow the scaling prediction (φ- z -4/3for spherical brushes in athermal solvents). For intermediate curvatures we propose an analytical expression for the volume fraction profile which is a combination of the parabolic potential near the surface, and the scaling form farther away from the surface. Thus, over the whole range of radii of curvature, analytical expressions for the volume fraction profiles are available which give reasonably good correspondence with the lattice calculations.We also studied the "dead zone" from which the free ends are excluded near the grafting surface. The lattice calculations show such a dead zone under all solvency conditions, both for spherical and cylindrical surfaces. The extension of this zone is a non-monotonic function of the surface curvature. The relative size of this zone (with respect to the brush height) is a decreasing function of R. No easy analytical expression is available for the size of the dead zone.In chapter 4 the adsorption equilibrium of AB diblock copolymers is considered for adsorption from solution onto small spherical particles. For adsorption onto flat surfaces it is known that the adsorbed amount shows a maximum as a function of the size of the adsorbing block, if the total chain length is kept constant. The thickness of the adsorbed layer shows a similar behaviour. Assuming that the adsorption energy is independent of surface curvature, we showed that the maximum in the adsorbed amount increases when the surface curvature increases. The hydrodynamic layer thickness of the adsorbed layer decreases strongly with increasing surface curvature. This increase occurs for all ratios of anchor to buoy sizes. On the other hand, the root- mean-square layer thickness changes much less as a function of the surface curvature. Depending on the anchor to buoy size ratio, it may either increase or decrease when the surface becomes more strongly curved.Chapter 5 treats the interaction between two polymer brushes, both in the presence and absence of free polymer in the solution. In this chapter we first study the effect of free polymer chains in solution on the height and volume fraction profile of an isolated polymer brush. Using self-consistent field and scaling arguments, diagrams of state are constructed, which indicate different regimes with different scaling laws for the brush height and for the interpenetration of free and grafted polymer chains, as a function of grafting density, free and grafted chain length, and bulk volume fraction of the free polymer. These scaling laws are again corroborated by SCF lattice calculations. Predictions are also given for the volume fraction profiles of free and grafted chains based on the strong-stretching approximation. In the derivation of these expressions it is explicitly assumed that the free chain length is far smaller than the brush height. When this condition is satisfied, the volume fraction profiles from the lattice calculations agree excellently with those predicted by the strong-stretching theory. When this condition is not satisfied, both approaches still predict the same height, but the strong-stretching theory gives a far too sharp interface between the grafted layer and the free polymer.The repulsive interaction between two compressed brushes starts at slightly larger separations according to the lattice calculations than one would expect from the strong- stretching approximation. This is caused by the "foot" of the volume fraction profile. This phenomenon occurs both in the absence and in the presence of free polymer in the solution. When free polymer is present the free energy of interaction can have an attractive part, caused by the depletion of the free chains.Chapter 6 deals with the interaction between two surfaces bearing adsorbed multiblock copolymer layers. We first study ABA triblock copolymers. Grafted layers of B chains with an end A block ("brushes with stickers") are used to model an adsorbed layer of such polymers. When the A adsorption energy of such a grafted layer is small, the free energy of interaction between two surfaces is purely repulsive. When this adsorption energy increases, a minimum appears, which reaches a limiting value at a certain adsorption energy. The minimum adsorption energy needed to find an attraction increases with increasing grafting density σ, and chain length N. The absolute value of this minimum also depends on N and σ. It scales as or σ 1/3N -1. The minimum always occurs at a separation d that is larger than the separation 2h at which the brushes are just in contact if the "feet" in the profiles are neglected. The difference d-2h scales as Nσ 1/3. The attraction has an entropic origin. When the surfaces are far apart, the grafted chains form loops, with the A blocks adsorbed to the grafting surface. When the surfaces are brought together, the A block of a grafted chain can either adsorb onto the surface to which this chain is grafted, or it can adsorb onto the other surface. This freedom to choose between two surfaces leads to an entropically driven attraction.The interaction between adsorbed layers of ABA triblock copolymers (where the adsorbed amount is determined by the equilibrium between free and adsorbed chains) has an attractive part if the copolymer chains are symmetric. The interaction curve is the same as that of a grafted layer ("brush with stickers") with a grafting density corresponding to the adsorbed amount of the triblock copolymers. If one of the adsorbing blocks is larger than the other block, the attraction decreases. For a relatively low asymmetry (one block roughly 20% larger than the other) the attraction disappears completely.Multiblock copolymers consisting of more than three blocks can form bridges between two surfaces comprising several blocks. We studied the interaction between two surfaces bearing adsorbed multiblock copolymer layers. The overall composition of the polymer chains was kept constant, but the chains were divided into different numbers of A and B blocks (so that the blocks become shorter when there are more blocks per chain). Chains with smaller blocks give smaller adsorbed layer thicknesses, so that the interaction starts at smaller separations. In all cases an attractive part is found in the interaction curve. Copolymer chains consisting of alternating small blocks of A and B segments very much resemble homopolymers (with properties that are some average of the A and B segments). These copolymers show a strong attraction at small separations (<10 layers), and repulsion at very small surface separations (around 2 layers).So far, we have only considered situations were the solvent is a good solvent for both blocks. The A blocks adsorb preferentially with respect to the B blocks, because the former have a stronger intrinsic affinity for the surface. We also consider the adsorption of an ABA triblock copolymer were both blocks have the same intrinsic affinity for the surface, but where the solvent is poorer for the A block. Now the A blocks adsorb preferentially, because of the selectivity of the solvent. We also pay attention to the interaction between two surfaces bearing adsorbed layers of such copolymers. When the interactions between the A and B segments and the solvent differ only slightly, the interaction curve resembles that of an adsorbing homopolymer, with an attraction at small separations. When these interactions differ a great deal, the interaction resembles that of a "conventional" triblock copolymer, with an attractive part at a large separation and repulsion at smaller surface separations. In the intermediate situation a more complicated interaction curve is found.The subject of chapter 7 is the interaction between two small particles bearing adsorbed polymer layers. An extended version of the lattice SCF theory was introduced, which takes account of gradients in two directions. In this version a cylindrical coordinate system is used, so that the volume fractions can vary both parallel to the axis connecting the centres of both particles, and in planes perpendicular to this axis. Results are presented for terminally attached polymer layers. It is first shown that this cylindrical model gives an isotropic profile around one isolated particle. This profile agrees well with the profile calculated from the "conventional" SCF lattice model, where a concentration gradient can exist in one direction only. Various free energy of interaction curves are presented for two spherical particles with terminally attached chains.If two spherically curved surfaces bearing adsorbed polymer layers interact, then the Derjaguin approximation relates this interaction to that between two similar flat surfaces, as long as the radius of curvature is far larger than the adsorbed layer. In chapter 7 we deal with systems where this condition does not hold. That is why we find interactions that are far less repulsive than the interaction according to Derjaguin's approximation. For increasing radii of curvature R, the interaction does move in the direction of the interaction predicted for very large R by the Derjaguin approximation. On a molecular level the decreased repulsion can be explained by the freedom of the grafted chains to mover laterally out of the gap between the two particles. Whether or not the grafting segments themselves can also move over the surface plays only a minor role.