Title: CASIMIR FORCE IN CRITICAL TERNARY POLYMER SOLUTIONS
Authors: H.Ridouane, E.-K.Hachem, M.Benhamou [m.benhamou@univh2m.ac.ma] (Laboratoire de Physique des Polym\`{e}res et Ph\'{e}nom\`{e}nes Critiques, Facult\'{e} des Sciences Ben M'sik, B.P. 7955, Casablanca, Morocco)

Consider a mixture of two incompatible polymers $A$ and $B$ in a common good solvent, confined between two parallel plates separated by a finite distance $L$. We assume that these plates strongly attract one of the two polymers close to the consolute point (critical adsorption). The plates then experience an effective force resulting from strong fluctuations of the composition. To simplify, we suppose that either plates have the same preference to attract one component (\textit{symmetric} plates) or they have an opposed preference (\textit{asymmetric} plates). The force is \textit{attractive} for symmetric plates and \textit{repulsive} for asymmetric ones. We first exactly compute the force using the blob model, and find that the attractive and repulsive forces decay similarly to $L^{-4}$. To go beyond the blob model that is a mean-field theory, and in order to get a correct induced force, we apply the Renormalization-Group to a $\varphi ^4$-field theory ($% \varphi $ is the composition fluctuation), with two suitable boundary conditions at the surfaces. The main result is that the expected force is the sum of two contributions. The first one is the mean-field contribution decaying as $L^{-4}$, and the second one is the force deviation originating from strong fluctuations of the composition that decreases rather as $L^{-3}$. This implies the existence of some cross-over distance $L^*\sim aN\phi ^{1/2}$ ($a$ is the monomer size, $N$ is the polymerization degree of chains and $\phi $ is the monomer volumic fraction), which separates two distance-regimes. For small distances $\left( LL^{*}\right) $ the fluctuation force is more important.

Key words: ternary polymer solutions, confinement, Casimir force
PACS: 64.75.+g, 68.45.-v, 61.41.+e