University of Central Florida STARS Faculty Bibliography 2000s Faculty Bibliography 1-1-2002 Polymer depletion interaction between a colloid particle and a wall: A Monte Carlo study Andrey Milchev Aniket Bhattacharya University of Central Florida Find similar works at: https://stars.library.ucf.edu/facultybib2000 University of Central Florida Libraries http://library.ucf.edu This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 2000s by an authorized administrator of STARS. For more information, please contact [email protected]. Recommended Citation Milchev, Andrey and Bhattacharya, Aniket, "Polymer depletion interaction between a colloid particle and a wall: A Monte Carlo study" (2002). Faculty Bibliography 2000s. 3365. https://stars.library.ucf.edu/facultybib2000/3365 Polymer depletion interaction between a colloid particle and a wall: A Monte Carlo study Cite as: J. Chem. Phys. 117, 5415 (2002); https://doi.org/10.1063/1.1499717 Submitted: 25 April 2002 . Accepted: 19 June 2002 . Published Online: 27 August 2002 Andrey Milchev, and Aniket Bhattacharya ARTICLES YOU MAY BE INTERESTED IN On Interaction between Two Bodies Immersed in a Solution of Macromolecules The Journal of Chemical Physics 22, 1255 (1954); https://doi.org/10.1063/1.1740347 Colloids dispersed in polymer solutions. A computer simulation study The Journal of Chemical Physics 100, 6873 (1994); https://doi.org/10.1063/1.467003 Theory of the depletion force due to rodlike polymers The Journal of Chemical Physics 106, 3721 (1997); https://doi.org/10.1063/1.473424 J. Chem. Phys. 117, 5415 (2002); https://doi.org/10.1063/1.1499717 117, 5415 © 2002 American Institute of Physics. JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 11 15 SEPTEMBER 2002 Polymer depletion interaction between a colloid particle and a wall: A Monte Carlo study Andrey Milcheva) Institute for Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Aniket Bhattacharyab) Department of Physics, University of Central Florida, Orlando, Florida 32816 ͑Received 25 April 2002; accepted 19 June 2002͒ An off-lattice bead–spring model of a polymer solution in a container with impenetrable walls is used to study the depletion interaction of a colloid particle with the planar wall by means of a Monte Carlo simulation. As expected, this interaction is found to depend essentially on the ratio ␳ ϭ R/Rg of the particle radius R to the mean radius of gyration Rg of the polymer chains in the case of dilute and semidilute solutions. For large particle to polymer size ratio ␳Ͼ1 this effective force is attractive and decreases steadily with growing distance D of the colloid from the wall. It is found to scale linearly with ␳ in agreement with recent theoretical predictions. In the opposite case of ␳Ͻ1 the depletion force is found to change nonmonotonically with D and go through a maximum at a р particular distance Dmax Rg . In both cases, however, local variations of the polymer density profile, which we detect at higher polymer concentrations, are found to influence the depletion force and even to change it locally from attraction to repulsion. The monomer density distribution far away from/or around the colloid in the vicinity of the wall is also investigated and related to the observed behavior of the depletion force. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1499717͔ I. INTRODUCTION In this paper we investigate the depletion interaction be- The polymer-induced depletion interaction between me- tween a spherical particle and a planar wall in dilute and soscopic colloid particles in a solution of nonadsorbing poly- semidilute monodisperse solutions of free flexible polymer mer chains is of fundamental interest in colloid physics.1 For chains using an off-lattice Monte Carlo bead–spring model. Earlier the problem was treated theoretically by Bringer entropic reasons the chains avoid the space between two 8 close particles, or between a particle and a planar wall, and et al., who based their treatment on the well-known diffu- create an effective attraction among the colloid particles, or sion equation satisfied by the partition function of a random push the particles toward the walls of the container. This walk, or, alternatively, on minimization of a Ginzburg– depletion interaction has been used to explain phase dia- Landau functional. In the case of ideal chains they derived a grams of colloid-polymer mixtures2 and is believed to be number of analytic results demonstrating that the depletion important for a variety of interesting colloid systems such as interaction depends crucially on the polymer to particle size casein micelles,3 hemoglubine,4 and globular proteins.5 Not ratio. In contrast, in the present simulational study the ex- surprisingly, the problem has attracted much scientific atten- cluded volume interactions of the polymers are inherent in tion and has been addressed both theoretically6–10 as well as the model which also takes into account all the fluctuations by means of computer simulations.11–14 In most of the theo- which are neglected as a rule in the analytic mean-field like retical work on the depletion interaction one employs the treatments. simple representation of the polymer coils as penetrable hard The paper is organized as follows: In Sec. II we give a ͑ ͒ 6 brief summary of theoretic results pertaining to limiting spheres PHSs by Asakura and Oosawa. Dilute or semidi- 8 lute solutions of nonintersecting polymer chains are mapped cases where a number of analytic expressions for RW are onto a fluid of ‘‘soft’’ spheres interacting via a concentration- available. The model used in the simulations is introduced in dependent effective potential13,14 although one should be Sec. III. In Sec. IV we discuss the polymer density profiles at aware that the PHS approach fails if the polymers are much the container walls and around the colloid particle, and in larger than the colloids. Such cases in which a long flexible Sec. V present our findings about the polymer-induced deple- chain cannot be reduced to a single degree of freedom have tion force on a sphere for the cases of large and small sphere been considered using scaling and field theories,15 mean to polymer size ratio. We end this report with a brief sum- field,7 and integral equation techniques.16 Recently experi- mary of our conclusions in Sec. VI. mental measurements of the effective interaction between two individual particles or for a single particle near a wall II. SUMMARY OF SCALING PREDICTIONS have been reported17,18 too. For a spherical colloid particle of radius R which is im- mersed in a dilute solution of polymer chains of mean gyra- ͒ a Electronic mail: [email protected] tion radius Rg and kept at a distance D apart from a planar b͒Electronic mail: [email protected] wall the depletion interaction attains a universal form when 0021-9606/2002/117(11)/5415/6/$19.005415 © 2002 American Institute of Physics 5416 J. Chem. Phys., Vol. 117, No. 11, 15 September 2002 A. Milchev and A. Bhattacharya the three lengths, R,D,Rg are much larger than microscopic lengths and persistent length characterizing the degree of stiffness of the chain.8 The free energy of polymer-induced interaction is proportional to the number density nb of chains in the bulk and is independent of the microscopic lengths ͑ ␨ϭ ␳ϭ scaling in terms of the ratios D/Rg , R/RG)as Gϭ d ͒ ͑ ͒ pR Y͑D/Rg ,R/Rg 1 ϭ with p nbkBT the bulk osmotic pressure of the chains, d the dimensionality, and Y(␨,␳) a universal scaling function. Generally for chains with excluded volume interactions one has for the case of large particles:10 G! ͑dϪ1͒/2 ͑dϩ1͒/2Y͑␨͒ ␨Ӷ␳ ͑ ͒ pR Rg , 2 with a scaling function Y yielding a polymer-induced attrac- ␨ which decreases monotonically with theץ/Gץ tive force distance ␨ from the wall. This behavior is in qualitative FIG. 1. Plot of the bonded ͑FENE͒,Eq.͑7͒—dotted line, and nonbonded ͑ ͒ ͑ ͒ agreement with the PHS-approximation of Asakura and Morse ,Eq. 8 —full line, potentials used for the monomer–monomer in- teractions in the simulation. Vertical arrows mark the minimal, l and Oosawa.6 In contrast, for small colloid particles one finds9 min maximal lmax extension of a polymer bond. The colloid particle acts on the polymers by means of truncated and shifted Lennard-Jones ͑LJ͒ potential, G! dϪ͑1/␯͒ 1/␯Y˜ ͑␨͒ ␳Ӷ␨ ͑ ͒ pR Rg , 3 Eq. ͑9͒—dashed line, of which only the repulsive branch is retained. Y˜ ϭ ⌽ ␨ ␯Ϸ with a scaling function A͓ h( )͔ where 0.589 is the 10 ⌽ Flory exponent, A is a known universal number, and h , ⌽ ϱ ϭ Here l is the length of an effective bond, which can vary normalized to h( ) 1, is the monomer density at distance Ͻ Ͻ ϭ ϭ ␨ ⌽ in between lmin l lmax , with lmin 0.4, lmax 1 being the from the wall if the particle is absent. Since h , which ϭ unit of length, and has the equilibrium value l0 0.7, while increases steadily with ␨, has a point of inflection, the force ϭ Ϫ ϭ Ϫ ϭ ␨ ␨ Ϸ R lmax l0 l0 lmin 0.3, and the spring constant K is takenץ Gץ / should have a maximum at certain max 1 and in- ϭ Ͻ␨Ͻ as K/kBT 40. The nonbonded interactions between the ef- crease for 0 1. fective monomers are described by the Morse potential—cf. Gϭ 2Y ␨ ␳ For Gaussian chains pRRg ( , ) and simple ex- Fig. 1, plicit expressions can be obtained analytically in several lim- 8,19 ϭ⑀ Ϫ ␣ Ϫ ͒ Ϫ Ϫ␣ Ϫ ͒ iting cases: For small distances from the wall the forces UM M͕exp͓ 2 ͑r rmin ͔ 2 exp͓ ͑r rmin ͔͖ , on large and small colloid particles albeit attractive, behave ͑8͒ differently, that is, initially they decrease or increase with where r is the distance between the beads, and the param- ␨ ϭ ⑀ ϭ ␣ϭ distance : eters are chosen as rmin 0.8, M 1, and 24.