Effective force between colloidal particles with end-grafted long polymer chains dissolved in a high-molecular weight solvent

Abstract

We aim at the determination of the effective force between colloidal particles clothed each by f end-grafted long polymer chains of polymerization degree N, which are immersed in a melt of short chains of polymerization degree P< N. We assume that short and probe chains have the same chemical nature. The clothed particles are small enough to be considered as star-polymers. To compute the expected force, we consider two regimes in the ( P, f)-plane: (i) high-grafting density f>f*= P ; and (ii) small-grafting density ( f< f*). We compute, in any case, the repulsive effective force, F( h), as a function of the inter-particle distance h. This force originates from the excluded volume interactions, where long probe chains are swollen by the presence of shorter ones of P units. For regime (i), we find that the force decays as k B TA fh −1· aP f <h<R F , with the universal amplitude A f ∼ f 3/2, as for low-molecular weight solvents case. Here, R F∼ aP −1/5 f 1/5 N 3/5 is the common Flory radius of the two star-polymers. For regime (ii), we show that, for higher distances aP 1/4< h< R F, the force decreases according to the same power law as for regime (i). However, below the cross-over distance aP 1/4·( aP 1/6 f 1/6< h< aP 1/4), the excluded volume interactions are strongly screened by the presence of mobile P-chains, and the force decays rather as k B TB f h −2 (mean-field behavior), with the new universal amplitude B f ∼ f 4/3. For f= f*, the mean-field distance-regime disappears, and then, the two forces relative to regimes (i) and (ii) are the same.