Ring polymers in confined geometries

Abstract

The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution of colloidal particles of big size was performed. Taking into account the correspondence between the field theoretical φ 4 O(n)-vector model in the limit n → 0 and the behaviour of long-flexible polymers in a good solvent, the correspondent depletion forces and the forces which exert phantom ideal ring polymers and ring polymers with EVI on the walls were obtained in the framework of the massive field theory approach at fixed space dimensions d=3 up to one-loop order. Besides, taking into account the Derjaguin approximation, the depletion forces between big colloidal particle and a wall and in the case of two big colloidal particles were calculated. The obtained results indicate that phantom ideal ring polymers and ring polymers with EVI due to the complexity of chain topology and for entropical reasons demonstrate a completely different behaviour in confined geometries compared with linear polymers.Key words: colloidal systems,critical phenomena, polymers, phase transitions PACS: 64.70.pv, 61.25.he, 67.30.ej, 68.35.Rh, 64.70.qd As it was shown in a series of the atomic force spectroscopy (AFM) experiments [1, 2], biopolymers such as DNA very often present a ring topology. Such a situation takes place, for example, in the case of Escherichia coli (E.coli) bacteria with a chromosome which is not a linear polymer, but has a ring topology [3]. The biopolymers of DNA of some viruses such as bacteriophages λ that infect bacteria oscillate between linear and ring topology [4,5]. The physical effects arising from confinement and chain topology play a significant role in the shaping of individual chromosomes and in the process of their segregation, especially in the case of elongated bacterial cells [6]. The behaviour of linear ideal and real polymers with excluded volume interaction (EVI) in a good solvent confined in a slit of two parallel repulsive [7,8], inert or mixed walls is well understood [8]. Unfortunately, the physics of confined ideal ring polymers and ring polymers with EVI effects is still unclear. Ring polymers with specified knot type were chemically synthesized a long time ago [9]. Ring topology of polymers influences the statistical mechanical properties of these polymers, for example the scaling properties [10,11] and shape [2,12,13] because it restrains the accessible phase space. An interesting point which was confirmed by numerical studies in [14] is that longer ring polymers are usually knotted with higher frequency and complexity. In [15], it was established that ring polymers with more complex knots are more compact and have a smaller radius of gyration and this decreases their ability to spread out under confinement. The results of Monte Carlo simulations performed in [16] suggest that the knotted ring polymers will exert higher entropic forces on the walls of the confini...

Linear and ring polymers in confined geometries

“…The question of influence of the polymer chain topology on the depletion interaction potentials and the depletion forces was investigated in a series of our papers [56][57][58], where we performed the investigation of a dilute solution of ring polymers immersed in a confined geometry like slit of two parallel walls with different boundary conditions and in a solution of mesoscopic colloidal particles of one sort or two sorts which can be both attractive, both repulsive for polymers or discuss the mixed case when one sort of mesoscopic colloidal particles is attractive for ring polymers and the other one is repulsive. It should be mentioned, that our investigation and calculation of the respective depletion interaction potentials and the depletion forces for linear and ring polymers with the EVI in a good solvent immersed in confined geometries is based on one of the powerful analytical method: the massive field theory approach in fixed space dimensions d = 3 (see [25,29,30,59]) up to one-loop order, which demonstrates quite good agreement with the results of numerical calculations and experimental data.…”

“…In a series of our papers (see [25,29,30,[56][57][58]) the investigations of a dilute solution of linear or ring polymers confined in a slit geometry of two parallel walls and in a solution of mesoscopic spherical colloidal particles of one sort or two different sorts were performed. We assumed that the solution of polymers is sufficiently dilute so that the interchain interactions and the overlapping between different polymers can be neglected and it is sufficient to consider the behaviour of a single polymer chain.…”

“…As it is known [15,16,61,62], the behaviour of long flexible polymers is determinated by entropy and characterized by detail independence, universality and scaling. The properties of universality and scaling also took place for critical systems with surfaces [64,65] and can be applied for the case of polymer solution in confined geometries [24,25,56,57,62].…”

“…It should be mentioned that in a series of our papers [25,[56][57][58] we followed the thermodynamic description of the problem as it was proposed in [24] and performed at the beginning the calculations in the framework of the grand canonical ensemble where the chemical potential μ is fixed and later obtained the results for the canonical ensemble via the Legendre transform. We performed the investigation of a dilute solution of linear or ring polymers immersed in the slit geometry of two parallel walls and allowed for the exchange of polymer coils between the slit and the reservoir.…”

“…Following the thermodynamic description proposed in [24,25,56,57] the corresponding reduced free energy of interaction δf per unit area A = 1 for the case of linear or ring polymers confined in the slit geometry of two parallel walls after performing Fourier transform in the direction parallel to the surfaces and integration over d d−1 r can be obtained:…”