Partitioning of polymer chains in solution with a square channel: lattice Monte Carlo simulations Peter Cifra a , Iwao Teraoka b, * a Polymer Institute, Slovak Academy of Sciences, Du Âbravska  cesta 9, 842 36 Bratislava, Slovak Republic b Herman F. Mark Polymer Research Institute, Polytechnic University, Six Metro Tech Center, 333 Jay Street, Brooklyn, NY 11201, USA Abstract Cubic lattice Monte Carlo simulation studies were conducted to examine the effect of con®nement on dilute and non-dilute solutions of polymer chains in a channel with a square cross section. In dilute solutions, the partition coef®cient K c with channels of different widths d followed the scaling-law prediction, and was close to the square of the partition coef®cient K s withaslitofthesame d.Thechainwithitsbulk radius of gyration greater than ,d/2 adopted a conformation extending along the channel and, with a decreasing channel width, the chain ends were forced to face outside. The chain conformation in broader channels was a compressed random coil. The K c increased with an increasing polymer concentration f E in the exterior solution equilibrated with the channel. In a weak con®nement, K c closely followed K s 2 of the same f E and d. The chains contracted at higher concentrations as they did in the bulk solutions. In a strong con®nement, K c was smaller than K s 2 at the same f E in the semidilute regime, and, at higher concentrations, sharply increased to the value close to K s 2 . q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Con®nement; Partitioning; Semidilute solution 1. Introduction Polymer chains, when con®ned to a small space, exhibit properties distinctly different from those in the uncon®ned space [1,2]. Understanding thermodynamics of con®ned polymer solutions is important in separation techniques at high concentrations such as high osmotic pressure chroma- tography [3] and phase ¯uctuation chromatography [4]. Because of limited accessibility to a model porous medium in experiments, thermodynamics of the con®ned polymer solutions have been studied primarily by using computer simulations [5±11]. In Monte Carlo simulations on a cubic lattice, we studied the partitioning of monodisperse polymer chains with a slit between two parallel impene- trable walls and static properties of the chains in the slit over a wide range of concentrations [12±18]. The focus was on the partition coef®cient de®ned as the ratio of the polymer concentration in the slit to the one in the surround- ing solution, the density pro®le in the slit, and the aniso- tropic chain dimension. Employing direct equilibration between the slit space and the free space was effective in evaluating the partition coef®cient [12,15±17]. Below we summarize the thermodynamics of athermal chains in dilute and non-dilute solutions con®ned to a slit: 1) At low concentrations, the partition coef®cient K 0 for the polymerchainsofaradiusofgyration R g0 withtheslitofwidth d isgivenby 2ln K 0 , R g0 =d b [12,13,17]with b closetothe value1=v  1:695predictedinthescalingtheory[19].2)The partitioning of semidilute solution is determined approxi- mately by the ratio of the blob size j correlation length of monomerdensity¯uctuations)totheslitwidthjustaspolymer chains in dilute solutions are partitioned according to R g0 /d [12,15]. As a result, the partitioning exhibits a diffuse transi- tion from a weak penetration to a strong penetration as the decreasing j becomes smaller than d [20]. 3) The chain dimension in the direction parallel to the slit walls decreases with an increasing concentration. The contraction exhibits a cross-over from that of a two-dimensional chain to that of a three-dimensionalchainas j becomes smaller than d [13,14]. 4) The monomer density decays to zero at the sites on the wall, forming a depletion layer. The layer becomes thinner with an increasing concentration [12,15,18]. 5) To bring the monomer density pro®le near the wall in agreement with the scaling theory prediction, a positive constant needs to be added to the distance to the wall [9±11,18]. In effect, the polymer chain sees a theoretical wall behind the physical wall on the lattice points. The constant, called the penetration depth, is 0.13±0.20 of the lattice unit at low concentrations and increases to ,0.36 in the semidilute solutions [18]. Polymer 43 2002) 2409±2415 0032-3861/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0032-386102)00040-X www.elsevier.com/locate/polymer * Corresponding author. Tel.: 11-718-260-3466. E-mail address: teraoka@poly.edu I. Teraoka).