Polymers in plane Poiseuille flow: Dynamic Monte Carlo simulationEdgardo Duering and Yitzhak Rabin

Citation: Journal of Rheology (1978-present) 35, 213 (1991); doi: 10.1122/1.550252 View online: http://dx.doi.org/10.1122/1.550252 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/35/2?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Local and chain dynamics in miscible polymer blends: A Monte Carlo simulation study J. Chem. Phys. 124, 174907 (2006); 10.1063/1.2189244 Chain folding in polymer melt crystallization studied by dynamic Monte Carlo simulations J. Chem. Phys. 115, 4395 (2001); 10.1063/1.1389860 Monte Carlo simulation of homopolymer melts in plane Poiseuille flow J. Chem. Phys. 112, 6073 (2000); 10.1063/1.481188 Structure and dynamics of grafted polymer layers: A Monte Carlo simulation J. Chem. Phys. 95, 9288 (1991); 10.1063/1.461158 The flow of dilute polymer solution in a narrow channel. II. Plane Poiseuille flow J. Chem. Phys. 82, 3442 (1985); 10.1063/1.448921

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Polymers in plane Poiseuille flow: DynamicMonte Carlo simulation

Edgardo Duering

Max-Planck-Institut fur Polymerforschung, 6500 Mainz,Federal Republic of Germany

Yitzhak Rabin

Department of Chemical Physics, Weizmann Institute ofScience, Rehovot 76100, Israel

(Received 9 July 1990; accepted 10 October 1990)

Synopsis

We present the results of a dynamic Monte Carlo simulation of macromoleculesin a dilute solution subjected to plane Poiseuille flow in a channel betweenimpenetrable walls, in two dimensions. The polymer dimensions parallel andperpendicular to the flow direction and the polymer distribution in the flowchannel are calculated as a function of the distance from the walls and thedifference with previous work on self-avoiding chains in simple-shear flow isdiscussed. In agreement with our previous analysis of the simple-shear flow casewe find that Poiseuille flows also produce narrowing of wall-depletion layers.

INTRODUCTION

Following recent evanescent-wave-induced-fluorescence studies byRondelez et al. (1987) of polymer distributions near impenetrable, non­attractive walls and, in particular, the work of Ausserre et al. (1990) onthe effect of flow on wall-depletion layers, Duering and Rabin (1990)and Duering (1989)performed dynamic Monte Carlo simulations offlexible, self-avoiding polymers confined to a narrow channel and sub­jected to simple-shear flow. We found that the shear produces stretchingand compression of the polymers in the directions parallel and perpen­dicular to the walls, respectively, and that in the presence of flow poly­mers are more likely to approach the walls than in equilibrium. Thelatter result does not agree with the experiments of Ausserre et al.

© 1991 by The Society of Rheology, Inc.J. Rheol 35(2), February 1991 0148-6055/91/020213-07$04.00 213

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214 DUERING AND RABIN

(1990) in which shear thickening of the wall-depletion layers was re­ported. Possible reasons for the discrepancy were proposed including:the neglect of hydrodynamic interactions in our simulation [see, how­ever, Rabin (1990)], the flexible nature of the polymer in the simulationas opposed to the rodlike polymers (xantan) used in the experiments,and the difference between the flow fields in our simple shear and ex­perimental (plane Poiseuille) work. The latter possibility provided theinitial motivation for the present contribution.

In this work we use the method described in Duering and Rabin(1990) and study polymers in dilute solution, subjected to Poiseuilleflow in a channel, in two dimensions. We calculate the conformationalproperties (i.e., the radius of gyration, parallel and perpendicular to theflow axis) and the monomer and center-of-mass distributions, as a func­tion of the distance from the walls. In order to quantify the wall­depletion effect, we calculate the surface excess in the absence and in thepresence of flow and conclude that the effects of simple-shear and Poi­seuille flow are rather similar in this respect.

SIMULATION AND RESULTS

Since a detailed description of the simulation method is given in aprior work by Duering and Rabin (1990), we only present the detailsparticular to the present application. The polymer is modeled as a chainoften spherical beads of unit diameter each that can be moved freely upto a given maximal distance dmax between successivebeads. In this workwe neglect excluded-volume effects (assume e solvent conditions) thatwere considered in Ref. 3 and thus different beads are allowed to overlapwith each other but are not allowed to penetrate the (nonadsorbing)walls. Bead-bead and bead-wall hydrodynamic interactions are alsoneglected. In order to ensure equilibration of the polymer distribution inthe flow channel and, at the same time, maintain a large enough bulkregion in which most polymer configurations do not experience theeffectof the walls, the separation between the walls is taken to be of theorder of the fully stretched length of the chain. We take dmax = 1.97andthe walls (parallel to the x axis) are located at y = 0 and y = 20, respec­tively.

Simple-shear and Poiseuille flows are introduced such that the veloc­ity fields between the walls are given by Vx = g(y - Yo) and Vx= g'(y - YO)2, respectively. Here Yo = 10 is the position of the mid­point between the walls, g is the shear rate, and the value g' =g/20 ischosen such that the local shear rate in Poiseuille flow varies from zero

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CONCENTRATION CHANGES IN POLYMER FLOWS 215

201612

Yc rn

84o0.0 -+1-----,------,------,------,------,-

3.0

0.6

1.2

<0p..

onP:::

1.8

2.4

FIG. 1. The projection of the polymer radius of gyration on the flow direction, Rg,pa, isplotted as a function of the distance from the wall y for equilibrium (broken line), simpleshear with g = 4 (solid line), and Poiseuille flow with g' = 0.2 (double solid line). Shearrates are measured in units of inverse equilibrium relaxation time of the polymer.

at the center of the pipe to a maximal value g at the walls. Thus, weexpect that when any polymer variable is plotted as a function of dis­tance from the wall, the Poiseuille flow results should approach those ofthe equilibrium in the bulk region and those of the simple-shear case inthe wall region. This is indeed observed in Figs. I and 2 where theparallel Rg,pa and the perpendicular Rg,pe components (to the flowdirection) of the radius of gyration are plotted as a function of thecenter-of-mass distance from the wall.

Unlike the case of a self-avoiding chain analyzed by Duering andRabin (1990), where Rg,pa increases in the wall region, in the absence ofexcluded-volume, Rg,pa is constant across the cell in the equilibriumcase and decreases near the wall in the presence of flow (Fig. 1). Thereason is that in the absence of excluded-volume effectsour model of themacromolecule behaves as a Gaussian random walk in which excursionsin perpendicular directions (x and y) are independent of each other[see, for example, Flory (1953)]. Thus, in equilibrium, the presence ofthe walls affects only the polymer dimensions along the y direction (Fig.

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216

1.3

1.2

1.1

1.0

Q)c;on 0.9 -

P::::0.8

0.7

06

DUERING AND RABIN

~ I

1\1J I f\{

~ ~/ t~1II

05 -+-1-------,----.,...-------,----------,-------,

o 4 8

Yc rn

12 16 20

FIG. 2. Same as in Fig. I, for the projection of the radius of gyration normal to the flowdirection, Rg,pe'

2); the dimensions along the x direction remain unchanged. The de­crease in Rg,pe close to the walls (Fig. 2) follows from the fact that thepolymer dimensions in the direction normal to the impenetrable wallcannot exceed its center-of-mass distance from the wall. In the presenceof flow, the chain undergoes a deformation along the x axis, the mag­nitude of which is proportional to its y dimensions. When one considersconfigurations in which the polymer distance from the wall is smaller orequal to its bulk radius of gyration, the y dimensions of the polymer areobviously smaller than the bulk values and consequently its coupling tothe flow is reduced, resulting in the observed decrease of Rg,pa near thewall. The decrease of Rg,pa in the Poiseuille (relative to simple shear)case in the bulk region is, of course, due to the fact that the effectiveshear rate vanishes in this region.

In Fig. 3 we plot the monomer and center-of-mass distributions as afunction of the distance from the wall. Within our accuracy, both dis­tributions are flat in the bulk region, in agreement with the predictionsby Aubert and Tirrell (1980) and by Brunn (1986)that diffusion acrossstream lines does not take place in plane Poiseuille flow, since suchdiffusion would be associated with inhomogeneous concentration pro-

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CONCENTRATION CHANGES IN POLYMER FLOWS 217

O. II i

16.

12.

~

'"8.

0...-

4.

o 4 8

y12 16 20

FIG. 3. The polymer center-of-mass (broken line) and the monomer (double solid line)distributions as a function of the distance from the wall y for Poiseuille flow with g' = 0.2.

files in the bulk. The above observation should not be interpreted as aproof that such bulk diffusion do not take place in real systems wherehydrodynamic interactions (neglected in our work) play an importantrole.

We now turn to the question of flow effects on wall-depletion layers.Although the shapes of the distributions in Fig. 3 are very similar tothose of their equilibrium counterparts, there is a systematic decrease inthe bulk and increase near the wall region, compared to the latter cases.In order to obtain a more quantitative estimate of the effectof Poiseuilleflow on wall-depletion layers we consider the surface excess

(20r= Jo [lfr(y) -lfrbJdy,

where lfr(y) is the distribution at position y and lfrb is the correspondingbulk value (since the surface excesses corresponding to center-of-massand monomer distributions are nearly equal, we do not distinguish be­tween them). Similarly to the simple-shear flow case discussed by Duer­ing and Rabin (1990), the surface excess decreases in absolute magni-

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218 DUERING AND RABIN

tude as g' is increased. While in the absence of flow we calculatedr o = - 22± 1, in Poiseuille flow with g' = 0.2 (in units of the inverselargest equilibrium relaxation time of the polymer which is calculatedseparately) the surface excess becomes I' = - 18± 1.

The existence of wall-depletion layers has interesting consequencesfor mixtures of high- and low-molecular weight polymers of radiiRlarge and Rsma11> respectively. Since the equilibrium width of the de­pletion layer is of the order of polymer dimensions, at distancesR small < Y < Rlarge from the wall, one expects to see a reduction in theratio of the high to the low-molecular weight polymer concentrations,compared to the bulk value, as discussed by Hertanto and Dickman(1988). Shear affects the depletion layers when S> 1, where g increaseslinearly with the polymer relaxation time, i.e., with a power of molec­ular weight [Duering and Rabin (1990)]. In the regime where g exceedsunity for the high molecular weight polymers but is smaller than unityfor the low-molecular weight ones, our prediction of flow-induced thin­ning of the depletion layers indicates that the concentration ratio shouldincrease and eventually approach the bulk value in the limit of highshear rates.

DISCUSSION

Due to the nonuniform character of Poiseuille flow, its effect on thepolymer conformation and distribution in the flow channel depends onthe distance from the wall. While in the central bulk region the shearrate is always low and all polymer properties are only weakly affected bythe flow, large effective shear rates can be achieved near the wall. Weexpect that as long as we only focus on a narrow (of the order of theequilibrium dimensions of the polymer) region near the wall, the Poi­seuille flow results should be similar to those of simple-shear flow. Thisexpectation is indeed borne out in our simulations where in both caseswe obtained shear thinning of wall-depletion layers, produced by theshear-induced reduction of polymer dimensions normal to the wall.Therefore, the discrepancy between the simple-shear simulation resultsof Duering and Rabin (1990) and the experimental observation of shearthickening of depletion layers in plane Poiseuille flow by Ausserre et al.( 1990), cannot be attributed to the differencebetween the two flows andis probably due to the rodlike nature of the polymer (xantan) used inthe experiments.

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CONCENTRATION CHANGES IN POLYMER FLOWS 219

ACKNOWLEDGMENTS

This work was supported by a U.S.-Israel Binational Science Foun­dation Grant No. 87-00134. One of us (E.D.) would like to acknowl­edge helpful discussions with S. Redner.

References

Aubert, J. H. and M. Tirrell, "Macromolecules in Nonhomogeneous Velocity GradientFields," J. Chern. Phys, 72, 2694-2701 (1980).

Ausserre, D., J. Edwards, J. Lecourtier, H. Hervet, and F. Rondelez, "HydrodynamicThickening of Depletion Layers in Polymer Solutions," Phys. Rev. Lett. (submitted).

Brunn, P.O., "Diffusion of Polymer Molecules: The Need for a Diffusive Time Scale,"Physica D 20, 403-412 (1986).

Duering, E., "Wall Effects on Polymer Distributions in Shear Flow," in Non-EquilibriumStatistical Mechanics, edited by S. Hernandez (World Scientific, Singapore, 1989).

Duering, E. and Y. Rabin, "Polymers in Shear Flow Near Repulsive Boundaries," Mac­romolecules 23, 2232-2237 (1990).

Flory, P. J., Principles of Polymer Chemistry (Cornell University, Ithaca, NY, 1953).Hertanto, A. and R. Dickman, "Equation of State of Athermal Lattice Chains in Three

Dimensions," J. Chern. Phys. 89, 7577-7584 (1988).Rabin, Y., "Slowing Down of Polymer Diffusion Near a Wall," Macromolecules (Com­

mun.) 23,3194-3196 (1990).Rondelez, F., D. Ausserre, and H. Hervet, "Experimental Studies of Polymer Concen­

tration Profiles at Solid-Liquid and Liquid-Gas Interfaces by Optical and X-RayEvanescent Wave Techniques," Annu. Rev. Phys. Chern. 38, 317-341 (1987).

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