Scaling theory for radial distributions of star polymers in dilute solution in the bulk and at a surface, and scaling of polymer networks near the adsorption transition

摘要

Monomer density profiles rgr;(r) and centerndash;end distribution functionsg(rCE) of star polymers are analyzed by using a scaling theory in arbitrary dimensionsd, considering dilute solutions and the good solvent limit. Both the case of a free star in the bulk and of a centerhyphen;adsorbed star at a free surface are considered. In the latter case of a semihyphen;infinite problem, a distinction is made between repulsive walls, attractive wallsmdash;where for large arm lengthlthe configuration of the star is quasihyphen;(dminus;1) dimensionalmdash;, and lsquo;lsquo;marginal wallsrsquo;rsquo; where forlrarr;infin; the transition fromdhyphen;dimensional structure occurs. For free stars, rgr;(r) behaves asrminus;d+1/ngr;for smallr, where ngr; is the exponent describing the linear dimensions of the star, e.g., the gyration radiusRgyrsim;lngr;. For centerhyphen;adsorbed stars at repulsive or marginal walls, rgr;(rpar;,z) behaves as rgr;(rpar;,0) sim;rminus;d+lgr;(thinsp;fthinsp;)par;and rgr;(0,z)sim;zminus;d+1/ngr;, whererpar;andzdenote the distances parallel and perpendicular to the surface, respectively; the new exponent lgr;(thinsp;fthinsp;) depends explicitly on the number of arms thinsp;fthinsp; in general. For centerhyphen;adsorbed stars at attractive walls, rgr;(rpar;,z) behaves as rgr;(rpar;,0)sim;rminus;(dminus;1)+1/ngr;(dminus;1)par;, ngr;(dminus;1)being the exponent describing (dminus;1)hyphen;dimensional stars, while rgr;(0,z) decays exponentially.On the other hand, the centerndash;end distribution function at short distances is described by nontrivial exponents. For free stars with thinsp;fthinsp; arms,g(rCE)sim;(rCE)thgr;(thinsp;fthinsp;)for smallrCE, where thgr;(thinsp;fthinsp;) is expressed in terms of the configurationhyphen;number exponent ggr;(thinsp;fthinsp;) and the exponent ggr; of linear polymers as thgr;(thinsp;fthinsp;) =lsqb;ggr;minus;ggr;(thinsp;f+1) +ggr;(thinsp;fthinsp;)minus;1rsqb;/ngr;. For centerhyphen;adsorbed stars, at repulsive or marginal wallsgs(rCEpar;,ze) behaves asgs(rCEpar;,0) sim;(rCEpar;)thgr;par;(thinsp;fthinsp;),gs(0,zE) sim;(zE)thgr;perp;(thinsp;fthinsp;)with thgr;par;(thinsp;fthinsp;) =lsqb;ggr;1minus;ggr;s(thinsp;f+1) +ggr;s(thinsp;fthinsp;)minus;1rsqb;/ngr; and thgr;perp;(thinsp;fthinsp;) =lsqb;ggr;minus;ggr;s(thinsp;f+1) +ggr;s(thinsp;fthinsp;)minus;1rsqb;/ngr;, ggr;1being the exponent of a linear polymer with one end at the surface. The scaling theory of general polymer networks at the adsorption transition is also presented.The configurationhyphen;number exponent ggr;Gfor a polymer networkGwithnhhfunctional units in the bulk,nrsquo;hhhyphen;functional units at the surface and totally composed of thinsp;fthinsp; linear polymers with the same length is given by ggr;SBG=agr;minus;1minus;f+ngr; +sum;infin;h=1lsqb;nhDgr;h+nhDgr;SBhrsqb;. Dgr;hand Dgr;SBhare related, respectively, to the exponents of star polymers as ggr;(thinsp;fthinsp;)=agr;minus;1+(ggr;minus;agr;)f/2+Dgr;fand ggr;SBs(thinsp;fthinsp;) =agr;minus;1+ngr;+(ggr;minus;agr;)f/2 +Dgr;SBf, with agr; given by agr;=2minus;ngr;d. The exponent ggr;SBs(thinsp;fthinsp;) is evaluated by means of the renormalizationhyphen;group egr;=4minus;dexpansion to the first order.