Polymer Depletion Interaction Between Two Parallel Repulsive Walls

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2001

Polymer depletion interaction between two parallel repulsive walls

F. Schlesener

Andreas Hanke The University of Texas Rio Grande Valley

R. Kimpel

S. Dietrich

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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.

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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 between 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 determined 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-

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. Since in the A. Grand canonical ensemble thermodynamic limit the leading contribution to the first in- In a dilute polymer solution the interaction between N tegral of Eq. ͑2.4͒ is independent of the slit width D we find different chains can be neglected so that the total free energy for Eq. ͑2.1͒ of the system is N times the free energy of a single chain. We consider the polymer solution within the slit to be in equilib- Zˆ ͑z͒ ␦ ϭϪ ͵ d ͩ I Ϫ ͪ rium contact with an equivalent polymer solution in a reser- F n p kbTͭ d r 1 V Zˆ voir outside the slit so that there is exchange of polymer coils I b between the slit and the reservoir. The free energy of inter- Zˆ ͒ I͑z action between the walls in such a grand canonical ensemble Ϫ ͵ ddrͩ Ϫ1 ͪͯ ͮ , ͑2.5͒ is given by V Zˆ I b Dϭϱ

Z͉͉͑D͒ Z͉͉͑Dϭϱ͒ ␦ ϭϪ ͩ ͪ Ϫ ͩ ͪ ͑ ͒ with the number density n ϭN/V of the polymer chains in F kBTNͭ ln Z ln Z ͮ , 2.1 p the bulk solution. The second integral in Eq. ͑2.5͒ reduces to the sum of two half-space ͑HS͒ integrals which yield contri- where Z͉͉(D) is the partition function of one polymer chain butions proportional to the area A of the walls ͓25͔: in a large volume V containing the walls and Z is the corresponding partition function of one polymer chain in the vol- Zˆ ͑z͒ Zˆ ͑z͒ ume V without the walls. In the thermodynamic limit ͵ d ͩ I Ϫ ͪͯ ϭ ͵ d ͩ HS Ϫ ͪ d d r 1 2 d r 1 V→ one has ͓24–26͔ V Zˆ HS Zˆ R I b Dϭϱ b Z→VZˆ ͑2.2͒ ⌬␴ b ϭϪ2A , ͑2.6͒ n pkBT Zˆ ϭ͐ d ЈZ Ј Z Ј with b Rdd r b(r,r ) and where b(r,r ) denotes the partition function of one polymer chain in the unbounded where we have introduced the surface tension ⌬␴ between solution with its ends fixed at r and rЈ. Correspondingly, the polymer solution and the confining wall ͓compare Eqs. Z͉͉(r,rЈ) denotes the partition function of one polymer chain ͑1.7͒ and ͑2.41͒ in Ref. ͓25͔͔. Note that the mean polymer V within the volume containing the parallel walls and with density within the slit is determined by the bulk density n p , VϭV ϩV its ends fixed at r and rЈ. The volume I O can be i.e., in the grand canonical ensemble the chemical potential

041803-2 POLYMER DEPLETION INTERACTION BETWEEN TWO . . . PHYSICAL REVIEW E 63 041803

␮ of the polymer coils is fixed instead of the number NI of C. Free energy of interaction and mean force ͑ ͒ polymer coils in the slit see Sec. II B . We employ the polymer magnet analogy ͓18,21–25͔ in According to Eqs. ͑2.5͒ and ͑2.6͒ the total grand canonical order to calculate the partition functions Zˆ (z), Zˆ (z), and free energy ⍀ of the polymer solution within the slit, HS I Zˆ ͑ ͒ ͑ ͒ ͑ ͒ b as needed in Eqs. 2.5 , 2.6 , and 2.8 for a single chain ⍀ϭϪ ␻ ͑ ͒ n p kBTAD , 2.7 with EV interaction. They are functions of the parameter u0 which characterizes the strength of the EV interaction and L0 with the dimensionless quantity which determines the number of monomers of the chain such R 2ϭR 2 that L0 equals g x / 2 for an ideal chain in the bulk, i.e., D Zˆ ͑ ͒ ϭ 1 I z for u0 0. The well-known arguments of the polymer mag- ␻ϭ ͵ dz , ͑2.8͒ ͓ ͔ D 0 Zˆ net analogy 18,21–23 imply for the present case the corre- b spondence can be decomposed as Z͉͉͑r,rЈ;L ,D,u ͒ϭL → ͗⌽ ͑r͒⌽ ͑rЈ͉͒͘Nϭ 0 0 t0 L0 1 1 0 ⍀ ͑2.14͒ ϭ ϩ ϩ␦ ͑ ͒ Dfb 2 f s f . 2.9a n pkBTA between Z͉͉(r,rЈ) and the two-point correlation function ͗⌽ ⌽ Ј ͘ N On the right-hand side ͑rhs͒ of Eq. ͑2.9a͒ appear the reduced 1(r) 1(r ) in an O( ) symmetric field theory for an N ⌽ϭ ⌽ ⌽ bulk free energy per unit volume -component order parameter field ( 1 ,..., N)in V ͑ ͒ the restricted volume I . In Eq. 2.14 the operator ϭϪ ͑ ͒ f b 1, 2.9b 1 L ϭ ͵ L0t0 ͑ ͒ t →L dt0 e 2.15 the reduced surface free energy per unit area 0 0 2␲i C ⌬␴ f ϭ , ͑2.9c͒ acting on the correlation function is an inverse Laplace trans- s C n pkBT form with a path in the complex t0 plane to the right of all singularities of the integrand. The Laplace conjugate t of L and the reduced free energy of interaction 0 0 and the excluded volume strength u0 appear, respectively, as ␦ the ‘‘temperature’’ parameter and as the prefactor of the F ⌽2 2 ␦ f ϭ . ͑2.9d͒ ( ) term in the Ginzburg-Landau Hamiltonian, n pkBTA 1 t0 u0 H͕⌽͖ϭ ͵ ddrͭ ͑ ⌽͒2ϩ ⌽2ϩ ͑⌽2͒2ͮ , B. Canonical ensemble V 2 “ 2 24 ␮ ͑2.16͒ If instead of the chemical potential the number NI of polymer coils in the slit is used as independent variable, the which provides the statistical weight exp(ϪH͕⌽͖) for the total free energy F of the polymer solution within the slit in ͑ ͒ the canonical ensemble follows from ⍀ as the Legendre field theory. The requirement in Eq. 1.1 describing the re- transform pulsive character of the walls imposes the Dirichlet condition ͒ϭ⍀ ␮ ͒ ϩ␮ ͒ ͑ ͒ ⌽͑r͒ϭ0, zϭ0,D, ͑2.17͒ F͑NI ͓ ͑NI ͔ ͑NI NI , 2.10

where ⍀ is given by Eq. ͑2.7͒. The chemical potential ␮ is on both walls. This corresponds to the fixed point boundary ͓ ͔ ͓ ͔ related to n p via 30 condition of the so-called ordinary transition 31,32 for the field theory. For the renormalization group improved pertur- ␮ϭ ⌳d͒ ͑ ͒ kB T ln͑n p , 2.11 bative investigations we use a dimensionally regularized continuum version of the field theory, which we shall renor- where ⌳ is the thermal de Broglie wave length of the par- malize by minimal subtraction of poles in ␧ϭ4Ϫd ͓33͔. The ticles, i.e., polymer coils. Equation ͑2.7͒ implies for dilute basic element of the perturbation expansion is the Gaussian solutions two-point correlation function ͑or propagator͒ ⌽ ⌽ ϭ ͗ i(r) j(rЈ)͘[0] where the subscript ͓0͔ denotes u0 0 ͔ ͒ ͑ ͓ ⍀ Ϫ⍀ץ ⍀͑␮͒ Ϫnץ ϭϪ ϭ p ϭ ͑ ͒ see Eq. A1 in the Appendix . 2.12 . ץ ␮ץ NI kBT n p kBT The loop expansion to first order in u0 and the renormalization of ␻ in Eq. ͑2.8͒ are completely analogous to the Thus F(NI) is given by outline in Sec. II A of Ref. ͓25͔. Some key results of this procedure relevant for the present case are given in the Ap- NI pendix. The final results for f and ␦ f on the rhs of Eq. F͑N ͒ϭϪk TNϩk TN lnͩ ⌳d ͪ ͑2.13͒ s I B I B I AD␻ ͑2.9a͒ are given by Eqs. ͑A5͒ and ͑A6͒ in the Appendix, where Nϭ0 for the present polymer case and ␧ϭ1ind with ␻ from Eq. ͑2.8͒. ϭ3.

041803-3 F. SCHLESENER, A. HANKE, R. KLIMPEL, AND S. DIETRICH PHYSICAL REVIEW E 63 041803

R ӷ ior for D/ x large, i.e., y 1. As expected the depletion potential and the resulting force are weaker for chains with EV interaction than for ideal chains, because the EV interaction effectively reduces the depletion effect of the walls. The inset of Fig. 1 shows both ⌰ and ⌫ for ideal chains. The narrow slit limits read

2 ⌰͑y→0͒ϭϪ ϩy ͑2.21͒ ͱ␲

and

⌫͑y→0͒ϭϪ1. ͑2.22͒

Deviations from the linear behavior in Eq. ͑2.21͒ become տ 1 →ϱ visible only for y 2 . In the opposite limit y the leading behavior for ideal chains is given by

2 1 2 ⌰͑y→ϱ͒ϭϪ4ͱ eϪy /2 ͑2.23͒ ␲ y 2

and

2 1 2 ⌫͑y→ϱ͒ϭϪ4ͱ eϪy /2. ͑2.24͒ ␲ y

The depletion potential in terms of the scaling function ⌰(y) FIG. 1. Scaling functions ͑a͒⌰(y) for the depletion interaction is attractive. But whether the bulk contribution Dfb has to be potential and ͑b͒⌫(y)ϭϪd⌰/dy for the depletion force between taken into account in addition depends on the ‘‘experimen- two repulsive, parallel plates at a distance D confining a dilute tal’’ setup. In the case that the force is measured between ϭ R ͓ plates immersed in a container filled with the dilute polymer polymer solution in terms of the scaling variable y D/ x see ͑ ͒ ͑ ͔͒ R ϭͱ R R solution such that solvent and polymer coils can freely enter Eqs. 2.18 and 2.19 . x 2 g for ideal chains and x ϭ R ͓ ͑ ͒ ⌰ 1.444 g in a good solvent for a bulk solution see Eqs. 4.2 and the slit from the reservoir, only the scaling functions and ͑ ͒ ͔ R ⌫ 4.3 below , where g is the radius of gyration. For ideal chains are relevant. This is also true for the particle-wall geom- the full expression, which is valid in the whole range of y ͑solid etry discussed in Sec. IV. But if no exchange is allowed, as line; see also the inset͒, and an expansion of this expression, which for the Monte Carlo simulation discussed in Sec. III, the is valid for yտ1 ͑dashed line͒, are shown. The same expansion is force K defined in Eq. ͑3.2͒ is needed. shown for chains in a good solvent ͑dotted line͒. The dotted line stops where the dashed line starts to deviate appreciably from the III. COMPARISON WITH MONTE CARLO SIMULATIONS solid line. Monte Carlo simulations of polymers are well established Figure 1 shows the universal scaling function for the free both for ideal chains and for chains with self-avoidance. In energy of interaction this section we compare our results with the simulation of a polymer chain between two repulsive walls by Milchev and 1 ⌰͑ ͒ϭ ␦ ͑ ͒ Binder ͓14͔ which corresponds to the case studied theoreti- y R f 2.18 x cally here. These authors use a bead spring model for the self-avoiding polymer chain with a short-ranged repulsive ␦ ͑ ͒ with f from Eq. 2.9d and the corresponding scaling func- interaction between the beads. tion for the force In Refs. ͓9͔ and ͓14͔͑see in particular Fig. 4 in Ref. ͓14͔͒ it is stated that the total force K between the two walls is d⌰͑y͒ ⌫͑y͒ϭϪ ͑2.19͒ repulsive and diverges in the narrow slit limit, i.e., dy DK D Ϫ1/␯ ϰ ͑ ͒ in terms of the scaling variable ͩ R ͪ , 3.1 kBT g yϭD/R . ͑2.20͒ x ␯ϭ 1 ␯ϭ where 2 for ideal chains and 0.588 for chains with R Figure 1 shows both the behavior for ideal chains and for EV interaction, and g is the radius of gyration of the poly- ͓ ͔ ͓ R chains with EV interaction. For chains with EV interaction mer chain in unbounded space 14 . For the definition of g the present theoretical approach can only capture the behav- see Eqs. ͑4.2͒ and ͑4.3͒ in Sec. IV.͔ The qualitative differ-

041803-4 POLYMER DEPLETION INTERACTION BETWEEN TWO . . . PHYSICAL REVIEW E 63 041803

the present field-theoretical approach. The deviation of the R Monte Carlo simulation data at large values of D/ g from R ͑ the power-law behavior at small values of D/ g dashed ͒ K R line occurs because D /kBT tends to 1 for large D/ g , which is not captured by Eq. ͑3.1͒. Figures 5–9 of Ref. ͓14͔ show density profiles for the simulated chains with EV interaction, including a comparison with the analytical result for the profile of ideal chains in the slit ͑Fig. 9 in Ref. ͓14͔͒. We want to mention here that the field-theoretical treatment of the monomer density for chains with EV interaction requires a perturbation expansion involving integrals over GϫGϫGϫG instead of GϫG ϫG ͑see the Appendix͒ and thus in view of the technical challenges is beyond the scope of the present study. Alterna- tive information about the monomer density profiles beyond the ideal behavior can be found in Ref. ͓34͔ in which a self-consistent mean-field approximation is used to obtain the monomer density profiles for a single polymer chain be- FIG. 2. Depletion force K ͓Eq. ͑3.2͔͒ between two parallel walls R tween two repulsive walls. at distance D confining repulsively a dilute polymer solution. g is the radius of gyration of the chains ͓see Eqs. ͑4.2͒ and ͑4.3͔͒. The solid circles correspond to the Monte Carlo simulation data in Ref. IV. COMPARISON WITH EXPERIMENT ͓ ͔ ͑ ͒ 14 . The force obtained from Eq. 3.2 is shown for ideal chains Rudhardt, Bechinger, and Leiderer ͓6͔ have measured the ͑solid line͒ and self-avoiding chains ͑dotted line͒. The latter line depletion interaction between a wall and a colloidal particle stops where the expansion for large D/R turns out to become g immersed in a dilute solution of nonionic polymer chains in unreliable. The dashed line represents the asymptotic behavior at small distances for chains in a good solvent ͓see Eq. ͑3.1͔͒, with a a good solvent by means of total internal reflection micros- R →ϱ copy ͑TIRM͒. They monitored the fluctuations of the relative fit for the amplitude of the power-law behavior. For D/ g the K distance of the colloid particle from the wall induced by reduced force D /(kBT) tends to 1. Brownian motion. From the resulting Boltzmann distribution ence from the scaling function ⌫(y) presented for the force one can infer the corresponding effective depletion potential in Sec. II C is explained by the following arguments. ͑a͒ The between the repulsive wall and the particle. total force between the walls is repulsive due to the contri- In order to compare these experimental data with our re- ͓ ͔ bution from the cost in free energy caused by the loss of the sults we apply the Derjaguin approximation 35 . In the limit total available space for the polymer chains within the slit, that the radius R of the spherical particle is much larger than Ϫ ͑ ͒ both R and the distance a of closest approach surface to i.e., the bulk pressure f b contributes to the total force. b x The total force K diverges due to the change from the grand surface between the particle and the wall, the particle can be ͑ ͒ regarded as composed of a pile of fringes. Each fringe builds canonical to the canonical ensemble see Sec. II B . In the 2 ͓ ͔ a fringelike slit with distance Dϭaϩr͉͉/2R, where r͉͉ is the simulation in Ref. 14 the number NI of polymers in the slit is given by one polymer in the slit volume. Therefore we radius of the fringe. Thus the interaction between the particle obtain the total force K by differentiating Eq. ͑2.13͒ with and the wall is given by respect to the slit width D and by setting N ϭ1, i.e., I ⌽ ͒ ϱ ␷2 depl͑a a ϭ2␲R R 2 ͵ d␷␷⌰ͩ ϩ ͪ , ͑4.1͒ DK 1 d n k T x R 2 ϭ ͑ ␻͒ ͑ ͒ p B 0 x ␻ D , 3.2 kBT dD where ␷ is a dimensionless variable and ⌰(y) is the scaling ␻ϭϪ⍀ where D /(n p kBTA) is given by the rhs of Eq. function for the free energy of interaction for the slit geom- ͑2.9a͒ in conjunction with Eqs. ͑A5͒ and ͑A6͒ in Appendix etry ͓see Eq. ͑2.18͔͒. ͓ ͔ ⌽ A. The lhs of Eq. ͑3.2͒ corresponds to the quantity Df in In Ref. 6 the interaction potential depl(a) is measured Ref. ͓14͔. for nonionic polymer chains in a good solvent for polymer ϭ ␮ Ϫ3 Figure 2 shows comparison of the simulation data of Ref. number densities n p 0, 7.6, 10.2, 12.7, and 25.5 m . All ͓14͔ and the corresponding theoretical result from Eq. ͑3.2͒, these polymer number densities represent a dilute polymer ⌽ ͓ the latter for both ideal chains and chains with EV interac- solution so that depl(a) is a linear function of n p see Eq. tion. Note that the theoretical curve for chains with EV in- ͑2.1͔͒. Therefore our results are applicable and Fig. 3 shows R ⌽ teraction is valid only for large D/ g . The curve for chains depl /n p as a function of a. The crosses in Fig. 3 correspond with EV interaction is closer to the simulation data than the to those values of a for which the above mentioned linear ⌽ ϰ ͑ ͒ curve for ideal chains, in agreement with the fact that the behavior depl n p not shown is in good agreement with polymer chain in the simulation is a self-avoiding one. One the experimental data. Deviations from the linear behavior ⌽ ϰ ͑ possible reason for the remaining deviation might be that the depl n p for small and large particle-wall distances a not chain in the simulation is too short to be fully described by shown͒ can be explained by the experimental method TIRM

041803-5 F. SCHLESENER, A. HANKE, R. KLIMPEL, AND S. DIETRICH PHYSICAL REVIEW E 63 041803

R ϭ R ϭ ͑ ͒ g 0.6927 x , d 3. 4.3

Figure 3 shows that the experimental data of Ref. ͓6͔ deviate from the theoretical result derived here. Note that this deviation is not visible if the polymer size is used as a freely adjustable parameter in order to gain agreement between the experimental and theoretical data. The theoretical curve for chains with EV interaction, as realized in the experiment, is even further away from the experimental data than the theoretical curve for ideal chains. The latter obser- vation confirms the necessity of reinterpreting the experimental data, e.g., by adjusting the radius of gyration. This can be done such that agreement with the theoretical data for chains with EV interaction is gained for the intermediate ⌽ values of the distance where the linear relationship depl ϰ ͓ n p allows for a direct comparison with the theory. The ⌽ remaining differences outside this intermediate regime (᭺) FIG. 3. Depletion interaction potential depl(a) between a spherical colloidal particle immersed in a dilute polymer solution pose a separate issue as discussed in the paragraph following ͑ ͒ ͔ and the container wall as a function of the distance a of closest Eq. 4.1 . However, the difference between the two theoret- approach surface to surface between the sphere and the wall ͓see ical curves for ideal chains and chains with EV interaction is ͑ ͔͒ small compared to the deviation from the experimental data. Eq. 4.1 . The interaction potential is given in units of kBT and of the polymer number density n p . The circles and crosses indicate Any attempt to use the Monte Carlo data obtained in Ref. the experimental data from Ref. ͓6͔. Crosses show the range where ͓14͔ for predicting the depletion interaction between a col- ⌽ ϰ K the linear relationship depl n p allows for a direct comparison with loidal particle and a wall would require two integrations of the theoretical results. The theoretically calculated depletion inter- and a change to the grand canonical ensemble, which poses action is shown for ideal chains ͑solid line͒ and chains in a good prohibitive accuracy problems. Nonetheless, a qualitative solvent ͑dotted line͒ as realized in the experiment. These curves statement can be given easily. Figure 2 shows that the force R ϭ ␮ correspond to the value g 0.101 m, which has been determined obtained in the Monte Carlo simulation is weaker than the by independent experiments ͓see Eqs. ͑4.2͒ and ͑4.3͔͒. Using the force calculated for chains with EV interaction. This is also radius of gyration as a fit parameter yields the dashed line corre- expected to hold for the depletion interaction between a par- R ϭ ␮ sponding to g 0.13 m and ideal chains. ticle and the wall. used in Ref. ͓6͔: a higher interaction potential implies a V. CONCLUDING REMARKS AND SUMMARY lower probability of finding the particle at the corresponding distance, causing lower accuracy. It turns out that the total Based on field-theoretical techniques we have determined interaction potential as the sum of depletion potential, elec- the effective depletion interaction between two nonadsorbing trostatic repulsion, and gravity is highest for short and large walls confining a dilute solution of long flexible polymer distances, which are those where the linear relationship chains. Our main results are the following. ⌽ ϰ depl n p is not confirmed experimentally. Figure 3 also ͑1͒ The field-theoretical calculation yields the universal shows the corresponding theoretical predictions for the ex- scaling functions of the depletion interaction potential and perimental data, for both ideal polymer chains and chains the corresponding force for ideal chains and for chains with ϭ R ӷ with EV interaction, i.e., chains in a good solvent as realized excluded volume interaction in the limit y D/ x 1, where R in the experiment. Note that all parameters entering these D is the separation between the walls and x is the projected theoretical predictions are fixed by available experimental end-to-end distance of the chains ͓see Eqs. ͑2.18͒, ͑2.19͒, R data for n p , a, and g , i.e., there are no freely adjustable and ͑4.3͒ and Fig. 1͔. The depletion potential is weaker for R parameters. In particular, the radius of gyration g of the chains in a good solvent than for ideal chains. polymer chains used in Ref. ͓6͔ has been measured fairly ͑2͒ For yտ1 we find fair agreement with corresponding accurately by means of small angle scattering of x rays ͓36͔, Monte Carlo simulation data ͓14͔ if the excluded volume R ϭ ␮ resulting in g 0.101 m. According to the definition of interaction is taken into account ͑see Fig. 2͒. We surmise that R ͓ ͔ g as measured in small angle scattering experiments 37 , remaining discrepancies are due to higher order terms in the i.e., field-theoretical calculation which are not yet taken into account, and due to the possibility that the length of the simulated polymer chain has not yet reached the scaling limit for ͵ d3r ͵ d3rЈ␳͑r͒␳͑rЈ͉͒rϪrЈ͉2 which the field theory is valid. R 2ϭ ͑ ͒ ͑ ͒ g , 4.2 3 Using the Derjaguin approximation we have compared 2 ͵ d3r ͵ d3rЈ␳͑r͒␳͑rЈ͒ our theoretical results with the experimental data ͓6͔ for the depletion potential between a spherical colloidal particle and a wall ͑see Fig. 3͒. We obtain a fair agreement only if the ␳ ϭ ͓ ͔ R where (r) is the monomer density, in d 3 one has 38 radius of gyration g of the polymers is used as a fit param-

041803-6 POLYMER DEPLETION INTERACTION BETWEEN TWO . . . PHYSICAL REVIEW E 63 041803

eter. This value, however, differs from independently deter- with the Gaussian propagator G˜ (p,z,zЈ;t ,D)inp-z repre- R 0 mined values for g . The reasons for these differences are sentation given by ͓32,39͔ not yet understood. The excluded volume interaction between the monomers of the polymer chain plays only a mi- nor role for reaching agreement between theory and experi- ˜ ͒ G͑p,z,zЈ;t0 ,D ment. 1 eϪb(zϪzЈ)ϩeϪb(zЈϪz) ϭ ͫ Ϫb͉zϪzЈ͉Ϫ Ϫb(zϩzЈ)ϩ ACKNOWLEDGMENTS e e 2b e2bDϪ1 We thank A. Milchev and K. Binder for a helpful discus- Ϫb(zϩzЈ)ϩ b(zϩzЈ) sion of their simulation data and C. Bechinger for providing e e Ϫ ͬ, ͑A2͒ the experimental data. This work has been supported by the e2bDϪ1 German Science Foundation through both Sonderforschungs- bereich 237 ‘‘Unordnung und große Fluktuationen’’ and the graduate college ‘‘Feldtheoretische und numerische Meth- ϭͱ 2ϩ where b p t0. The loop expansion of the total suscep- oden der Elementarteilchen- und Statistischen Physik.’’ tibility reads ͓40͔

APPENDIX D u Nϩ2 D The Gaussian two-point correlation function in the slit of ␹͑ ͒ϭ ͵ Ј˜ ͑ Ј͒Ϫ 0 ͵ Ј Љ t0 ,D,u0 dzdz G 0,z,z dzdz dz width D reads 0 2 3 0

͗⌽ ͑r͒ ⌽ ͑rЈ͒͘ ddϪ1p i j [0] ϫ ͵ ˜ ͑ Љ͒˜ ͑ Љ Љ͒ Ϫ G 0,z,z G p,z ,z ϭ␦ ͒ ͑2␲͒d 1 ij G͑r,rЈ;t0 ,D ϭ␦ ˆ ͉ Ϫ Ј͉ ͒ ϫG˜ ͑0,zЉ,zЈ͒ϩO͑u2͒. ͑A3͒ ij G͑ rʈ rʈ ,z,z Ј ;t0 ,D 0

ddϪ1p ϭ␦ ͵ ͓ ͑ Ϫ Ј͔͒ ˜ ͑ Ј ͒ ij exp ip rʈ rʈ G p,z,z ;t0 ,D , The procedure outlined in Sec. II A of Ref. ͓25͔ yields the ͑2␲͒dϪ1 • renormalized total susceptibility in one-loop order as ͑see ͑A1͒ also Appendix B of Ref. ͓40͔͒:

ͱ ͱ ͱ 1 2 1ϪeϪ ␶ Nϩ2 2 eϪ ␶ 1Ϫe ␶ ␹ ͓␶ϭ͑ ␮͒2 ͔ 3ϭ Ϫ ϩ ͫͱ␶ϩ ͱ␶ Ϫ ͬ ren D t,D,u / D ͱ u 2 ͱ 3 ͱ ␶ ␶3/2 1ϩeϪ ␶ 3 ␶3/2 ͭ ͑1ϩeϪ ␶͒2 1ϩe ␶

ϱ ͱs2Ϫ1 ϫͫ ϩ ͑ ␮͒Ϫ ␶ϩ ͑ ␲͒ϩ Ϫ Ϫ ͵ ͬ 2 f 1 2ln D ln ln 4 1 CE 8 ds ͱ␶ 1 e2 sϪ1 ͱ ͱ 1/2Ϫ1/ͱ3ϩeϪ ␶͑2Ϫͱ3͒ϩeϪ2 ␶͑1/2Ϫ1/ͱ3͒ Ϫ ␲ 4 ͱ ͑1ϩeϪ ␶͒2 ͱ 1ϪeϪ ␶ ϱ ͱs2Ϫ1 2 2 1 1 Ϫ4 ͵ ds ϩ ϩ Ϫ ϩO͑u2͒. ͑A4͒ Ϫͱ␶ 2ͱ␶s Ϫ ϩ 1ϩe 0 e Ϫ1 ͩ 1 1 s 1 s 1 ͪͮ sϩ sϪ 2 2

␮ ␹ Here is the inverse length scale which determines the malized total susceptibility for the unbounded space ren,b . renormalization group flow; t and u are the renormalized and The decomposition into bulk, surface, and finite-size contri- dimensionless counterparts of t0 and u0, respectively; CE is butions is carried out by analysis of the scaling behavior of Euler’s constant; and for the definition of the constant f 1 we these parts of the free energy. The surface and finite-size refer to Ref. ͓25͔. parts of the free energy ͓see Eq. ͑2.9͔͒ at the fixed point of The free energy is obtained via the inverse Laplace trans- the renormalization group and for Nϭ0 with the scaling ␹ ϭ R form of ren(D) and normalization by the transformed renor- variable y D/ x are given by

041803-7 F. SCHLESENER, A. HANKE, R. KLIMPEL, AND S. DIETRICH PHYSICAL REVIEW E 63 041803

2 2 ␧ 3ln2 ␲ ␲ ϩO͑␧ ͒, ͑A6͒ f ϭR ͱ ͭ 1Ϫ ͫ 1Ϫ Ϫ ϩ ͬͮ ͑A5͒ s x ␲ 4 2 2 ͱ 3 ␹ ͑ ͒ where ren in Eq. A4 is expanded for large plate separa- ͱ 2 and tions D up to order O(eϪ D t), because for small distances the correct behavior including the dimensional crossover ␦ f y 2 1 2 cannot be obtained even for the full expression. But using the ϭ4 erfcͩ ͪ Ϫ4 ͱ eϪy /2 D ͱ2 ␲ y expansion has the additional beneﬁt of yielding partly analytical results for the separations D of interest here. For d Ϫy2/2 ϭ R ͑ ͒ ␧ e 4␲ 3, Rx is related to the radius of gyration g by Eq. 4.3 . Ϫ ͫ ͩ ϩ Ϫ ␲ϩ ͑ 2͒Ϫ ͪ ␦ ͑ ͒ 4 4 6ln 2y 6CE The full result for ideal chains for f is given by Eq. A4 4 yͱ2␲ ͱ3 for uϭ0, i.e., y 2␲ ϩ ͩ ͪͩ ␲Ϫ Ϫ ͑ 2͒ϩ ͪ erfc ͱ 2 ͱ 2ln 2y 2CE 4 1 2 3 ␦ f ϭϪDL␶→ 2 ͭ ͮ , ͑A7͒ 1/2y ␶3/2 ϩ ͱ␶ ͱ ͱ 1 e eϪ ␶ ln ␶ eϪ ␶ ln ␶ ϪL 1 ͭ ͮ Ϫ L 2 ␶→ 3 ␶→ ͭ ͮ ͬ 1/2y 2y2 ␶ ␶3/2 which is valid for all y.

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