modeling the rheology of polymer solutions by dissipative particle dynamics

7/30/2019 Modeling the Rheology of Polymer Solutions by Dissipative Particle Dynamics

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Modeling the rheology of polymer solutions by dissipative particle

dynamics

Y. Kong a, C.W. Manke a, W.G. Madden a and A.G. Schlijper b

a Departmentof Chemical Engineeringand MaterialsScience,WayneStateUniversity, Detroit,MI 48202, USAb ShellResearch andTechnologyCentreThornton, POBox 1,ChesterCH1 3SH,UK

Dissipative particle dynamics (DPD), a new simulation technique that focuses on intermediate time and length scales, is evalu-

ated for systems in which polymers are used as viscosity index improvers. Model studies of simple non-Newtonian fluids show that

DPD reveals the expected shear thinning. DPD solutions confined between solid surfaces, predict anisotropic conformational

relaxation dynamics. In addition, the effects of thermodynamic solvent quality on the configurations and rheological behavior of

dissolved polymers, essential to theperformance of multigrade lubricants,are representedby theDPD model.

Keywords: computersimulation, polymersolutions, polymersolution rheology,dissipative particledynamics (DPD)

1. Introduction

Dilute solutions of polymer additives in oil-based

lubricants are of great utility as viscosity index (VI)

improvers and as anti-wear additives. These applica-

tions depend not only on how the polymer affects the

rheological properties of the bulk solution, but also ^

and perhaps more importantly ^ on how the solution

behaves in very thin layers, when confined between

lubricated surfaces. To design new polymers targeted at

specific applications, it is necessary to understand the

link between microscopic fluid characteristics (i.e. the

detailed chemical structures of both the polymer and the

base lubricant) and the macroscopic rheological and tri-bological behavior. Because the former involves dis-

tances at the atomic level and the latter time scales of

mechanical motion e.g. in automobile engines, no single

simulation methodis adequate to this task.

Traditional rheological simulations are based on

continuum mechanics and focus on the longer time

scales. The lubricant is represented as an undifferen-

tiated substance whose properties enter the calculation

as simple property parameters and as complex, empirical

constitutive equations. The origin of these parameters

and the form of constitutive equations is beyond the

scope of the approach. At the other extreme, the molec-

ular dynamics (MD) methods of chemists and physicistsallow one to explore the motion of individual atoms and

molecules by integrating Newton's equations of motion

subject to realistic intermolecular forces. While some

attempts have been made toapply the MDmethod tothe

dynamics of bulk polymer solutions, these have gener-

ally been limited to isolated studies on highly idealized

molecules and require months of supercomputer CPU

time to execute. Even so, they do not extend to time

scales at which continuum mechanics simulations

become applicable. The difficulty is that the time step in

ordinary MD is constrained to values much smaller thanthe shortest characteristic relaxation time of the smallest

mobile entity in the system. For atoms, molecules or

chemical groups, this is short indeed (measured in femto-

seconds).

If continuum mechanics simulations are thought to

apply at some macroscopic level and MD simulation at a

microscopic level, what is needed is an intermediate

simulation method that operates at some mesoscopic

level with its associated time scales. This is particularly

important for polymer solutions, since a coarse graining

in space and time that eliminates individual solvent

molecules and their irrelevant short time scale motions

does not eliminate the essential chain character of highmolecular weight polymers. Indeed, nearly all approxi-

mate theories for the dynamics of polymers in solution

are based on a mesoscopic viewpoint that dates to the

classic theories of Rouse and Zimm [1,2]. In these the-

ories, the solvent is represented as a continuum fluid but

the polymer molecule is represented as an explicit chain,

not of individual atoms or mers, but of ``blobs'' repre-

senting collections of mers. The problem with these

approximate theories is that they are unable to properly

incorporate the effects of neighboring velocity fields on

one another or to include the presence of solid^fluid

interfaces in a self-consistent fashion.

In this work we investigate the applicability of a newmesoscopic simulation method, dissipative particle

dynamics (DPD) [3,4], to polymer solutions of interest

as lubricants. The level of coarse graining is such that

atoms or mers are not represented individually, but are

collected into more massive particles. For the polymer,

these beads are the blobs of the standard classical the-

ories, linked together into chains of appropriate topolo-

gical character. For the solvent, they are collections of

molecules analogous to the polymer blobs. Essentially

these are local ``packets'' of fluid, able to move indepen-

Tribology Letters3 (1997) 133^138 133

J.C. Baltzer AG, Science Publishers


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dently. They influence one another (and are influenced

by the polymer beads) through a set of interaction rules

guaranteed to give rise to the standard fluid mechanical

equations of motion at sufficiently long times. The inter-

actions are sufficiently soft and the density sufficiently

high that there is substantial interpenetration of parti-

cles and near continuity of mass. The particles move by

rules that are analogous to the simplest algorithms for

ordinary molecular dynamics but use a time step that is

two orders of magnitude larger. Complex velocity fields

are represented by the local velocities of individual sol-

vent or polymer particles. Only basic parameters are

required, and no constitutive equations are needed. For

application to polymer solutions, DPD employs a much

more consistent coarse graining than has been used in

the past. Unlike the standard theories, it is designed for

numerical implementation and contains no approxima-

tions motivated by purely mathematical concerns

inserted into the theory merely to produce an ultimately

analytic equation.

DPD is precisely the kind of mesoscopic theoryrequired to bridge the gap between the molecular view-

point and that of continuum mechanics. It fills an inter-

mediate role in a hierarchy that starts with the quantum

mechanics of individual molecules or a few molecules at

a time. This effort predicts the intermolecular forces that

serve as input to ordinary molecular dynamics for short-

time averaging and simple thermodynamics. The MD

results are then mapped onto the parameters of DPD,

which can in turn be used to suggest appropriate consti-

tutive equations and transport parameters for introduc-

tion into macroscopic continuum mechanics

simulations. In such a hierarchical approach, one may

have full detailin thefundamentalmodel without havingto carry that detailed information along when it is no

longer relevant at thetime scales of interest. This enables

one to envision the computer aided design of new lubri-

cants in which molecularstructure of thelubricant's con-

stituents can be directly translated into the ultimate

rheological and tribological consequences. If this sce-

nario seems ambitious, it is also directly analogous to

the computer design of drugs, which seemed equally

ephemeral a mere 15 years ago. Today, computer design

is standard practice in thepharmacological industry, not

so much because these simulations unerringly predict

the winning formulations, but rather because they focus

expensive laboratory investigations on the most promis-inglines of research.

The dissipative particle dynamics (DPD) technique,

developed by Hoogerbruggeand Koelman [3], was moti-

vated by the highly efficient lattice^gas automata meth-

ods for the simulation of complex fluid flow. DPD

extends lattice^gas automata to motion of particles in

continuous space, ensuring Galilean invariance and spa-

tial isotropy. The motion of the DPD particles involves

both stochasticand dissipative termsthat guarantee evo-

lution of the system toward equilibrium and long-term

consistency with macroscopic fluid mechanical equa-

tions of motion (see Bird et al. [5]). Because ineffectual

high-frequency motion is eliminated, the DPD particles

move relatively large distances in a single time step. As a

result, the method probes long-time behavior two orders

of magnitude more efficiently than ordinary MD [3]. It is

easy to introduce bead-and-spring-type polymer chains

into thebasic simulation scheme, which results in a suita-

ble model for a dilute polymer solution. Schlijper et al.

[4] have examined the static and dynamic scaling rela-

tionships for this DPD polymer solution model. For

athermal solutions at rest, they have shown that the

dependence of both radius of gyration and relaxation

times on molecular weight follow the classical Zimm

model [1,2] closely.

DPD has not yet been employed in realistic simula-

tions of lubricating flows of fluids containing polymeric

VI-improvers. However, a number of fundamental stud-

ies relevant to this objective have been performed. In

what follows, we present features of the DPD polymer

model that would be needed for realistic mesoscopicrepresentations of lubrication flows. DPD predictions of

bulk solution rheological properties are presented, and

the ability of the model to represent rheological

responses to solvent^polymer thermodynamic interac-

tions is examined. Finally, the DPD representation of

the static and dynamic behavior of linear polymer chains

in the presence of confining walls is reviewed. Except for

the confined chain study, the DPD simulations pre-

sented here are new calculations that have not been

reported previously in the literature. Our purpose is to

provide an overview of promising new results of DPD

polymer simulations relevant to lubrication. Complete

details of these simulations, which are far too lengthy toinclude in the present letter, will be presented in subse-

quentfull-length publications.

2. Bulkrheological properties of polymer solutions

Steadyshear flow of solutions of rigid dumbbellsmod-

els represents an important category of rheological

behavior where both non-Newtonian viscosity and

shear-thinning first normal stress behavior is known to

occur, andwhere thestresstensor components have been

predicted by standard kinetic theory [6]. Moreover, the

rheological behavior of rigid dumbbell solutions isknown to be sensitive to the presence of inter-bead

hydrodynamic interaction. Thus the known kinetic the-

ory results for this flow can serve as an important bench-

mark for comparison with DPD rheological

predictions.

Rigid dumbbells can be modeled by connecting pairs

of DPD particles with a Fraenkel spring connector hav-

ing a very large spring constant [4], which effectively

constrains the DPD dumbbells to a fixed length.

Viscosity predictions are shown in fig. 1 for DPD strong

Y.Konget al./ DPDsimulationofpolymersolutions134


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Fraenkel spring dumbbells with radius of gyration

rg 1X0rc, where rc is the interaction radius of the DPDparticles forming the two beads of the dumbbell. Thus

the calculations are performed for dumbbells with beads

that are just touching each other. This corresponds to

the case of complete hydrodynamic interaction in con-

ventional polymer kinetic theory treatments, and

Stewart and Sorensen [6] have made correspondingkinetic theory predictions for solutions of rigid dumb-

bells employing a hydrodynamic interaction parameter

h 0X38. The DPD predictions for rg 1X0rc agree verywell with the rigid dumbbell predictions for h 3a8.Comparison of the rigid dumbbell curves for h 0 (no

hydrodynamic interaction) and h 3a8 shows thatshear-thinning becomes less pronounced as hydrody-

namic interaction increases. Hydrodynamic interaction

is an intrinsic feature of the DPD polymer model,

because the flow of the solvent is modeled explicitly.

Variations of the first normal stress coefficient with

shear rateare shown in fig.2 for the same cases displayed

in fig. 1. Like the rigid dumbbell kinetic theory, the DPDstrong Fraenkel spring model predicts shear-thinning of

21. The DPD results for rg 1X0rc are again very close tothe rigid-dumbbell predictions for h 3a8.

The excellent comparison of the DPD strong

Fraenkel spring dumbbell with rg 1X0rc to the compar-able Stewart and Sorensen rigid dumbbell with h 3a8demonstrates that the rheological behavior predicted by

DPD is directly comparable to results given by standard

kinetic theory for a case where the model features are

similar. Moreover, the comparison suggests that the

hydrodynamic interaction effects that emerge naturally

from the explicit DPD representation of the solvent flow

field are comparable to hydrodynamic interaction

effects predicted by conventional bead^connector mod-

els, such as the rigid dumbbell kinetic theory, that model

interbead hydrodynamic interaction through an Oseen

tensor. These comparisons generally strengthen confi-

dence in the rheological predictions of the DPD bead-

and-spring polymer model and provide a rational basis

for extending DPD predictions to multi-bead polymer

chains and other more realistic polymer architectureswhere direct comparisons with kinetic theory are not

possible.

3. Effects of thermodynamicsolvent quality

In multigrad lubricants, the improvement with

increasing temperature of the thermodynamic solvent

quality of a base oilproduces increasedthickening power

of dissolved polymeric VI improvers that partially coun-

teracts the decrease in base oil viscosity with tempera-

ture. This increase in the thickening power of polymeric

components arises from expansions of macromolecularconfigurations with increasing solvent power. These

effects canbe modeled in DPDsimulations by modifying

the repulsive interactions between solvent particles and

polymer beads to make the repulsive forces either stron-

ger (poor solvent case) or weaker (good solvent case)

than repulsions between like particles. We have

employed this DPD solvent^polymer interaction model

to calculate the radius of gyration of chains of different

length at different solvent qualities. The predicted scal-

ingexponents forthe variationof radius of gyration with

Fig. 1. Viscosity of DPD strong Fraenkel spring dumbbells is shown

as a function of reduced shear rate. The case where the beads are sepa-

rated by1X0rc (rg 1)compares very well with the rigidrod kineticthe-ory predictions of Stewart and Sorensen [6] for the comparable case

with interbead hydrodynamic interaction h 3a8. For reference, thekinetic theory case for h 0 (no hydrodynamic interaction) is also dis-

played. In the axes labels, denotes viscosity at shear rate , 0 is theviscosity at zeroshear rate, ands is the viscosity of the solvent.! is the

principal relaxation time for the polymer in solution, measured at zero

shear rate.

Fig. 2. First normal stress predictions for the cases shown in fig. 1.

The DPD predictions for rg 1 are in very good agreement with the

comparable rigid rod kinetic theory results for h 3a8. In the axeslabels 21 is the first normal stress coefficient; other symbols are as

definedin fig. 1.

Y.Konget al./ DPDsimulationofpolymersolutions 135


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Fig. 3. Simulation snapshot of instantaneous configuration of a ten-bead linear polymer in a sea of DPD solvent particles. Open circles denote

the centers of solvent particles and filled circles denote beads of the polymer structure. The solid lines connecting polymer beads represent strong

Fraenkel spring connectors. DPD interparticle repulsions between solvent particles and polymer beads have been set stronger than like-particles

repulsions to represent thecase of a poorsolvent, which producesthe tightlycollapsed polymerconfiguration shown. Therectangulargrid denotes

theperiodic boxof theDPD simulation andits surroundingimage boxes.

Fig. 4. Simulation snapshot of instantaneous configuration on the same ten-bead linear polymer solution depicted in fig. 3, but with DPD param-

eters set to model the good solvent case. Here the DPD interparticle repulsions between solvent particles and polymer beads are weaker than like-

particles repulsions, resulting in a favorable solvent^polymer interaction. Note that the polymer structure expands into the solvent under these

conditions.(See fig.3 forcomplete explanation of symbolsand lines.)



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molecular weight vary from 0.32 for DPD poor solvents

to 0.60 for DPD good solvents, values which are in

nearly perfect agreement with known experimental and

theoretical scaling behavior [7] for polymer solutions.

Thus the DPD polymer solution simulations model

polymer^solvent interactions with remarkable accu-

racy, even though DPD is not intrinsically a thermody-

namic modeling technique.

Figs. 3 and 4 illustrate the predicted effects of solventquality on theconfiguration of a ten-bead linear polymer

chain. In the case of the poor solvent, fig. 3, the polymer

chain is collapsed into a very tight configuration, in

which thermodynamically unfavorable contacts

between polymer chain beads and solvent particles are

minimized. The good solvent case, fig. 4, produces a

highly expanded chain configuration, in which contacts

between the polymer chain beads and solvent particles,

now thermodynamically favorable, are promoted. The

effects of solvent-induced configurational changes on

rheological properties are examined in figs. 5 and 6. The

polymer in good solvent shows higher viscosities and

higher first normal stress coefficients than the polymer

in a theta solvent, where polymer^solvent repulsions are

identical to solvent^solvent and polymer^polymer

repulsions. (The poor solvent case leads to precipitation

of high molecular weight polymers from solution, and is

not shown here.) The expected shear-thinning behavior

for both viscosity and first normal stress coefficient is

observed in both solvents.

4.Dynamicsof confinedpolymer chains

The effect of wall confinement on the rheological

properties of lubricant films with dissolved polymers is

another key problem that can be approached by meso-

scopic modeling. Here DPD is employed to examine the

effect of wall confinement on the relaxation times of dis-

solved polymer solutes. In this study, mobile solvent and

polymer particles are confined between two impene-

trable walls made up of ``frozen'' (immobile) DPDparti-

cles. The roughness of these glassy walls is less than a

single particle diameter. The effect of the confining walls

on the conformation and dynamics of the chain becomes

significant as the gap approaches the radius of gyrationof thepolymer chainsin bulk solution.

We have shown previously [8] that the behavior of

Fig. 5. Intrinsic velocity () of ten-bead linear polymer is shown as a

function of shear rate for the theta solvent and good solvent cases.The expanded configuration of the polymer in the good solvent pro-

duces higher intrinsic viscosity values. Intrinsic viscosity is reported inunits of r3c /particle, and shear rate is in inverse time step units. (The

points at the lowest shear rates, near 0.0001, suffer high statistical

uncertainty.)

Fig. 6. First normal stress coefficient 21 divided by polymer concentration & for a ten-bead linear polymer solution is shown as a function of shear

rate for the theta solvent and good solvent cases. The first normal stress coefficient is higher for the good solvent case due to the expansion of the

polymerchain in thesolvent. (The points at thelowestshearrates,below 0.001, suffer highstatistical uncertainty.)

Y.Konget al./ DPDsimulationofpolymersolutions 137


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the components of the radius of gyration of the polymer

chain behave as expected as the gap decreases. DPD

simulations of five- and ten-bead polymer chains show

that the component of the radius of gyration in the direc-tion perpendicular to the confining walls begins to col-

lapse as the gap is reduced to about five times the

unperturbed radius of gyration. The component of the

radius of gyration parallel to the walls remains essen-

tially unaffected by confinement, however, even when

the gap approaches the unperturbed radius of gyration.

This result is consistent with other statistical mechanical

results from theliterature [9].

What is new from the DPDsimulations is the capabil-

ity to analyze the dynamical response of the polymer

chains to wall confinement, as well as the configura-

tional response detailed above. This is accomplished by

calculating the decay curve for a configurational auto-correlation function [3,7], which is initially equal to the

square of the radius of gyration, and decomposing this

decay curve into components representing the decay of

configurational correlations in the directions parallel

and perpendicular to the confining walls (see Kong et al.

[8]). These decay curves are well represented as sums of

exponentials, with each exponential decay contributing

a characteristic relaxation time. Fig. 7 represents the

behavior of the principal relaxation time for the decay of

the perpendicular component of the configurational

autocorrelation function for five- and ten-bead polymer

chains. As the gap is decreased, a strong enhancement of

this relaxation time is observed. This DPD-predictedeffect has important consequences for the rheological

behavior of thin lubricating layers that have been incor-

porated in hierarchical models for lubrication flow

described by Coy[10].

5. Conclusions

We have investigated the applicability of dissipative

particle dynamics (DPD) as a mesoscopic simulation

tool for polymer solutions. From studies of model solu-

tions, we find that it predicts shear thinning and changes

in first normal stress consistent with known theory and

available experimental data. We have demonstrated that

the method can be used to study the changing effects of

solvent quality on polymer dynamics in solutions.

Finally, the dynamics of the chain in the vicinity of con-

fining walls have been analyzed. In each case, DPD

either predicts correctly the accepted results or provides

the first calculation of any sort of the microscopic

relaxation investigated. We conclude that DPD is a very

promising simulation tool for polymer solutions that

focuses directly on the length scales and time scaleswhich are difficult to probe experimentally and which

cannot be easily addressed by more conventional simula-

tion tools.

Acknowledgement

The authors are grateful to Shell Research Ltd. for

financialsupport of this work.

References

[1] P.E. RouseJr.,J. Chem.Phys. 21 (1953)1272.

[2] B.H. Zimm,J. Chem.Phys. 24 (1956)269.

[3] P.J. Hoogerbrugge and J.M.V.A. Koelman, Europhys. Lett. 19

(1992) 155.

[4] A.G. Schlijper, P.J. Hoogerbrugge and C.W. Manke, J. Rheol.

39 (1995) 567.

[5] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport

Phenomena (Wiley,New York, 1960).

[6] W.E.Stewart and J.P.Sorensen,Trans. Soc. Rheol. 16 (1972) 1.

[7] P.-G. de Gennes, Scaling Concepts in Polymer Science (Cornell

University Press, Ithaca,1 979).

[8] Y. Kong, C.W. Manke, W.G. Madden and A.G. Schlijper, Int.

J. Thermophys.15 (1994) 1093.

[9] C.M. Lastoskie and W.G. Madden, in: Computer Simulation of

Polymers, ed.R.J. Roe(PrenticeHall, Englewood Cliffs,1991).[10] R.C. Coy, Modeling of boundary lubrication and its application

to industrially important systems, Limits of Lubrication

Conference,Williamsburg,14^18 April 1996.

Fig. 7. Variation of principle relaxation time for parallel component

of configurational autocorrelation function with gap dimension in 3D

simulations of ten-bead chains. Principal relaxation time in parallel

direction isscaledby itsvalue inbulksolution,seeKonget al.[8].


modeling the rheology of polymer solutions by dissipative particle dynamics

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