Directed path models of adsorbing and pulled copolymers

Abstract

In this paper we examine the phases of a directed path model of a copolymer attached to a surface under the influence of a pulling force. The simplest model of an adsorbing directed polymer, attached at the one end to a surface and pulled from the surface by the other end, is reviewed-its phase diagram includes free, adsorbed and ballistic phases. In contrast to this model, we consider an adsorbing directed block copolymer attached at both endpoints to the adsorbing surface, and pulled away from the surface at its middle vertex. The phase diagram of this model includes ballistic, adsorbed and free phases, as well as a quasi-ballistic phase which has characteristics of both an adsorbed and a ballistic phase. We also generalize our methods to a model of directed block star copolymers and show that such directed copolymers have numerous adsorbed and quasi-ballistic phases. The phase transitions in this system are continuous in some cases and first order in others.

Force-induced desorption of uniform branched polymers

“…The phase diagram can then be determined from lemma 8 and theorem 3 and it is shown for y > 1 in figure 7. This phase diagram is similar to the phase diagram calculated for a Dyck path model of an adsorbing block copolymer pulled in the middle (see figure 6 in reference [11]). There are 6 distinct phases, namely a ballistic phase, two ballistic-adsorbed mixed phases (A-mixed and B-mixed), and three adsorbed phases (one Aadsorbed, another B-adsorbed, and the third AB-adsorbed).…”

“…For the case of a block copolymer we do know that the blocks behave quasi-independently in that the free energy is the sum of the free energies of the separate blocks [32]. There is however rigorous work on the force-induced desorption for directed walk models of copolymers [11]. The model that was considered is Dyck paths pulled at their mid-point, so these are similar to unfolded loops with both vertices of degree 1 constrained to be in the surface.…”

“…In the directed version of this model [11], Dyck paths (these are loops) are pulled from the middle vertex and both the A-and B-blocks have an endpoint in the surface. This model exhibits a mixed adsorbed-ballistic phase where one block is considered "adsorbed" and the other "ballistic".…”

Exact solutions of directed walk models of polymeric zipping with pulling in two and three dimensions

“…Experimental techniques able to micro-manipulate single polymers [1,19,20] and the connection to modelling DNA denaturation [3,5,6,7,8,9,11] have provided the impetus for studying models of polymer adsorption, pulling and zipping. In the pursuit of exact solutions, idealised two-dimensional directed walk models have been constructed to capture the effects of adsorption, where a polymer grafts itself onto a surface at low temperature [2,4,12,16]; as well as zipping, where two polymers are entwined with one another (again at low temperature) [10,15,18]. Recently extensions of these models to include multiple effects in two-dimensional exactly solved models of directed walks [13,14,21,22,23] have provided rich mathematical results that display key physical characteristics of these polymer systems.…”

Adsorption of symmetric random copolymer onto symmetric random surface: the annealed case

“…In particular, directed walk models of polymers [1][2][3] played an important role in studying homopolymer adsorption [4][5][6][7][8][9][10][11][12] and collapse [13][14][15][16][17] and related problems of force-induced desorption [18][19][20][21][22][23][24][25][26] and unfolding of a collapsed macromolecule [27,28]. Random [29][30][31] and periodic [32][33][34] copolymer adsorption and mechanical desorption [35][36][37] were also studied with the aid of directed models. The advantage of directed models consists in their simplicity; polymer directedness allows one to obtain an exact solution in most cases, typically in the long chain limit.…”