There are n c ABC triblock copolymers with polymerization

degree N and n g polymer with selleck chemicals llc polymerization degree P (here, we take P = N) grafting on the two parallel surfaces. buy Omipalisib Each copolymer chain consists of N segments with compositions (average volume fractions) f A and f B (f C = 1 – f A – f B), respectively. The ABC triblock copolymer and the grafted polymers (brush) are assumed to be flexible, and the mixture is incompressible with each polymer segment having a statistical length a and occupying a fixed volume . The two parallel surfaces coated by the polymer brush are horizontally placed on the xy-plane at z = 0 and L z + a, respectively. The volume of the system is V = L x L y L z, where L x and L y are the lateral lengths of the surfaces along the xy-plane and L z is the film thickness. The grafting density is defined as σ = n g a 2/(2L x L y ). The average volume fractions of the grafted chains and copolymers are φ g = n g N/ρ 0 V and φ c = n c N/ρ 0 V, respectively. In the SCFT, one considers the statistics of a single copolymer chain in a set of effective external fields w i , where i represents block species A, B, and C or grafted polymers. These external fields, which represent the actual interactions between different components, are conjugated to the segment density fields, ϕ i , of different species i. Hence, the free energy (in unit of k B T) of the system is given by (1) where χ ij is the Flory-Huggins

interaction parameter between species i and j, ξ selleck chemical is the Lagrange multiplier (as a pressure), η iS is the interaction parameter between the species i and the hard surface S. rs is the position of the hard surfaces. Q c = ∫drq c(r, 1) is the partition function of a single copolymer chain in the effective DOK2 external fields w A, w B, and w C, and Q g = ∫drq g(r, 1)

is the partition function of a grafted polymer chain in the external field w g. The fundamental quantity to be calculated in mean-field studies is the polymer segment probability distribution function, q(r, s), representing the probability of finding segment s at position r. It satisfies a modified diffusion equation using a flexible Gaussian chain model (2) where w(r) is w A(r) when 0 < s < f A, w B(r) when f A < s < f A + f B, w C(r) when f A + f B < s < 1 for ABC triblock copolymer, and w g(r) for the grafted polymer. The initial condition of Equation (2) satisfies q c(r, 0) = 1 for ABC triblock copolymer. Because the two ends of the block copolymer are different, a second distribution function is needed which satisfies Equation (2) but with the right-hand side multiplied by -1 and the initial condition The initial condition of q g(r, s) for grafted polymer is q g(r, 0) = δ(r - rS), where rS represents the position of the substrates, and that of is The periodic boundary condition is used for and along x- and y-directions when z∈ [0,L z ]. and are equal to zero when z ≤ 0 or z ≥ L z.