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The exact solution of directed self-avoiding walks confined to a slit offinite width and interacting with the walls of the slit via an attractivepotential has been calculated recently. The walks can be considered to modelthe polymer-induced steric stabilisation and sensitised floculation ofcolloidal dispersions. The large width asymptotics led to a phase diagramdifferent to that of a polymer attached to, and attracted to, a single wall.The question that arises is: can one interpolate between the single wall andtwo wall cases? In this paper we calculate the exact scaling functions for thepartition function by considering the two variable asymptotics of the partitionfunction for simultaneous large length and large width. Consequently, we findthe scaling functions for the force induced by the polymer on the walls. Wefind that these scaling functions are given by elliptic theta-functions. Insome parts of the phase diagram there is more a complex crossover between thesingle wall and two wall cases and we elucidate how this happens.

Finite-size scaling functions for directed polymers confined between attracting walls