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Polymer Depletion Interaction Between Two Parallel Repulsive Walls University of Texas Rio Grande Valley ScholarWorks @ UTRGV Physics and Astronomy Faculty Publications and Presentations College of Sciences 2001 Polymer depletion interaction between two parallel repulsive walls F. Schlesener Andreas Hanke The University of Texas Rio Grande Valley R. Kimpel S. Dietrich Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac Part of the Astrophysics and Astronomy Commons, and the Physics Commons Recommended Citation Schlesener, F., et al. “Polymer Depletion Interaction between Two Parallel Repulsive Walls.” Physical Review E, vol. 63, no. 4, American Physical Society, Mar. 2001, p. 041803, doi:10.1103/ PhysRevE.63.041803. This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact [email protected], [email protected]. PHYSICAL REVIEW E, VOLUME 63, 041803 Polymer depletion interaction between two parallel repulsive walls F. Schlesener,1,2 A. Hanke,3 R. Klimpel,1 and S. Dietrich2,4 1Fachbereich Physik, Bergische Universita¨t Wuppertal, D-42097 Wuppertal, Germany 2Max-Planck-Institut fu¨r Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 4Institut fu¨r Theoretische und Angewandte Physik, Universita¨t Stuttgart, D-70550 Stuttgart, Germany ͑Received 23 October 2000; published 27 March 2001͒ The depletion interaction between two parallel repulsive walls confining a dilute solution of long and flexible polymer chains is studied by field-theoretic methods. Special attention is paid to self-avoidance be- tween chain monomers relevant for polymers in a good solvent. Our direct approach avoids the mapping of the actual polymer chains on effective hard or soft spheres. We compare our results with recent Monte Carlo simulations ͓A. Milchev and K. Binder, Eur. Phys. J. B 3, 477 ͑1998͔͒ and with experimental results for the depletion interaction between a spherical colloidal particle and a planar wall in a dilute solution of nonionic polymers ͓D. Rudhardt, C. Bechinger, and P. Leiderer, Phys. Rev. Lett. 81, 1330 ͑1998͔͒. DOI: 10.1103/PhysRevE.63.041803 PACS number͑s͒: 61.25.Hq, 61.41.ϩe, 68.35.Rh, 82.70.Dd I. INTRODUCTION the EV interaction and is amenable to a field-theoretical treatment via the polymer magnet analogy. The basic ele- Z [0] Ј In polymer solutions the overlap of depletion zones for ments in this expansion are partition functions seg(r,r ) for monomers due to repulsive confining walls or mesoscopic chain segments that have no EV interaction ͑as indicated by particles dissolved in the solution induces an important and the superscript ͓0͔) and with the two ends of the segment tunable effective interaction potential ͓1͔. For example, this fixed at r and rЈ. This perturbative treatment has to be car- depletion interaction successfully explains phase diagrams of ried out in the presence of confining geometries. We con- colloid-polymer mixtures ͓2–4͔. Recent experimental tech- sider two structureless parallel walls in d dimensions and a niques facilitate even the measurement of the depletion force distance D apart so that in coordinates rϭ(rʈ ,z) the surface between a wall and a single colloidal particle ͓5–7͔.Inthe of the bottom wall is located at zϭ0 and rʈ comprises the context of such solutions confined to thin films and porous dϪ1 components of r parallel to the walls. The surface of materials the geometry of two parallel walls has been exten- the upper wall is located at zϭD. The interaction of the sively studied as a paradigmatic case ͓8–14͔. polymer with the nonadsorbing walls is implemented by the For strongly overlapping chains as realized in a semidilute boundary condition that the segment partition function and polymer solution, chain flexibility is taken into account thus the partition function for the whole chain vanishes as within self-consistent field theory or within the framework of any segment approaches the surface of the walls ͓18,23͔, i.e., phenomenological scaling theory ͓15–17͔. On the other Z [0]͑ Ј͒! Ј! ͑ ͒ hand, in a dilute polymer solution different chains do not seg r,r 0, z,z 0,D. 1.1 overlap so that the behavior of the polymer solution is deter- mined by the behavior of a single chain. To a certain extent For the present purpose the only relevant property that char- and under certain circumstances, a single chain can be mod- acterizes one of the polymer chains is its mean square end- eled by a random walk without self-avoidance ͑ideal chain͒. to-end distance in the bulk solution, which we denote by R 2 ͓ ͔ In three dimensions this situation is closely realized in a d x 24–26 ; for convenience we include the spatial dimen- so-called ␪-solvent ͓18͔. If the solvent temperature is below sion d as a prefactor. The results presented in the following the ␪-point ͑poor solvent͒ the polymer coils tend to collapse are obtained for dϭ3 both for ideal chains and for chains ͓19,20͔. However, in the common case that the solvent tem- with EV interaction. In Sec. II we present our results for the perature is above the ␪-point ͑good solvent͒ the excluded interaction potential and the force between two parallel volume ͑EV͒ interaction between chain monomers becomes walls. We note that for chains with EV interaction these ӷR relevant so that the polymer coils are less compact than the results are valid only in the limit D x because our theo- corresponding ideal chains. The emphasis in this work is on retical approach is not capable of describing the dimensional the latter situation and we investigate the effect of the EV crossover to the behavior of a quasi-(dϪ1)-dimensional sys- ӶR interaction on the depletion interaction between two parallel tem which arises for D x . In Sec. III we compare these walls as compared to the case of confined ideal chains. results with the simulation data of Milchev and Binder ͓14͔. By focusing on long flexible chains in a system of meso- In Sec. IV we apply the Derjaguin approximation in order to scopic size we obtain mostly universal results that are inde- obtain from the results in Sec. II the depletion interaction pendent of microscopic details ͓18,21–26͔. Due to the uni- between a spherical particle and a wall in a dilute polymer versality of the corresponding properties it is sufficient to solution and compare it with the corresponding experimental choose a simple model for calculating these results. For our results of Rudhardt, Bechinger, and Leiderer ͓6͔. investigations we use an Edwards-type model ͓18,21,22͔ for In view of the complexity of the actual experimental sys- the polymer chain, which allows for an expansion in terms of tems involving spatially confined colloidal suspensions dis- 1063-651X/2001/63͑4͒/041803͑9͒/$20.0063 041803-1 ©2001 The American Physical Society F. SCHLESENER, A. HANKE, R. KLIMPEL, AND S. DIETRICH PHYSICAL REVIEW E 63 041803 V V solved in a solution containing polymers, in the past the cor- divided into the volume I within the slit and the volume O responding theoretical descriptions relied on suitable coarse- outside the slit. Since the polymer chain cannot penetrate the grained, effective models and on integrating out less relevant walls, whose lateral extensions are large, Z͉͉(r,rЈ) is nonzero V V degrees of freedom. The gross features of these systems can only if both r and rЈ are in I or in O so that be obtained by mapping the polymers onto effective hard spheres as pioneered by Asakura and Oosawa ͓27͔. In a more d d refined description Louis et al. ͓28,29͔ derived effective in- Z͉͉͑D͒ϭ ͵ d r ͵ d rЈZ͉͉͑r,rЈ͒ V V teraction potentials between polymer coils such that they be- have like soft spheres. This approach allows one to capture ϭ ͵ d Zˆ ͑ ͒ϩ ͵ d Zˆ ͑ ͒ ͑ ͒ the crossover in structural properties to semidilute and dense d r O z d r I z , 2.3 V V polymer solutions. Based on such a model Louis et al. ͓29͔ O I have calculated, among other things, the corresponding d with Zˆ (z)ϭ͐V d rЈZ͉͉(r,rЈ). In the thermodynamic depletion energy between two parallel repulsive plates. Be- O,I O,I d sides presenting the depletion energy for ideal chains in limit V!R the logarithm ln(Z͉͉ /Z) in Eq. ͑2.1͒ can be ex- terms of an expansion introduced by Asakura and Oosawa, panded using Eqs. ͑2.2͒ and ͑2.3͒, i.e., they find the occurrence of repulsive depletion forces in an Z͉͉͑D͒ Z͉͉͑D͒ϪZ intermediate regime of D upon significantly increasing the ͩ ͪ ϭ ͩ ϩ ͪ polymer density. In our present completely analytic study we ln Z ln 1 Z focus on dilute polymer solutions, for which the depletion forces turn out to be always attractive, by fully taking into 1 Z͉͉͑D͒ϪVZˆ ! b account the flexibility of the polymer chains and the self- V Zˆ avoidance of the polymer segments with a particular empha- b sis on long chains. This complementary point of view allows Zˆ ͒ 1 O͑z us to make contact with the Monte Carlo simulation data in ϭ ͫ ͵ ddrͩ Ϫ1 ͪ V V Zˆ Ref. ͓14͔ and with the experimental data in Ref. ͓6͔; these O b comparisons have not been carried out before to our knowl- Zˆ ͒ I͑z edge. ϩ ͵ ddrͩ Ϫ1 ͪͬ, ͑2.4͒ V Zˆ I b II. EFFECTIVE INTERACTION BETWEEN PARALLEL Z ϪZ Z V V WALLS where the ratio ( ͉͉(D) )/ is of the order I / , which tends to zero because the slit width D is fixed.
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