dissipative particle dynamics simulation of polymer drops in a periodic shear flow

J. Non-Newtonian Fluid Mech. 118 (2004) 65–81

Dissipative particle dynamics simulation of polymerdrops in a periodic shear flow

Shuo Chenb, Nhan Phan-Thiena,∗, Xi-Jun Fanc, Boo Cheong Khoob

a Division of Bioengineering, National University of Singapore, Singapore 117576, Singaporeb Singapore-MIT Alliance, 4 Engineering Drive 3, Singapore 117576, Singapore

c Cooperative Research Centre for Polymers, 32 Business Park Drive, Notting Hill, Vic. 3168, Australia

Abstract

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particledynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE)bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles,respectively, perpendicular toZ-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions areimplemented in theX andY directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour”method is employed to model immiscible fluids according to Rothman–Keller method; theχ-parameters in Flory–Huggins-type models arealso analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shearfield, the relationship between the deformation parameter (Ddef) and the capillary number (Ca) can be delineated into a linear and nonlinearregime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study,Ca< 0.22, in thelinear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescencemay occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-sizedistribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease inthe first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis showsthat the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer dropunder extension due to the memory of the FENE chains.© 2004 Elsevier B.V. All rights reserved.

Keywords:Droplets; Suspension; Dissipative particle dynamics; FENE chain

1. Introduction

Probing the micromechanics of suspensions and emul-sions requires techniques at mesoscopic length and timescales, of which molecular dynamics simulation is unableto deliver to date due to the vast computing resources re-quired. A number of “coarse-grained” approaches have beensuggested and developed to simplify the underlying micro-scopic model while retaining the essential physics. Amongthese various methods, the dissipative particle dynamics(DPD) [1–3] has emerged as a promising new techniquefor modelling rheologically complex liquids[4], includinginterfaces.

∗ Corresponding author.E-mail address:[email protected] (N. Phan-Thien).

As several flowing systems of industrial or biologicalrelevance, such as emulsions[5], polymer blends[6], andbiofluids [7], can be described as suspensions of droplets,the deformation of an isolated drop in an immiscible liquidphase undergoing a flow has been extensively studied, boththeoretically and experimentally[8,9]. The physics of thedrop’s motion and its shape and stability have not been fullyunderstood, partly due to the inadequacy of conventionaltheoretical and computational techniques for treating multi-phase systems, and partly due to the lack of well-controlledexperimental studies[10].

Guido and Villone[11] investigated experimentally thethree-dimensional deformation of an isolated drop in animmiscible liquid phase under a simple shear flow using aparallel-plate apparatus. The steady-state drop shape waswell described within experimental errors by an ellipsoid.The deviation from ellipsoidal shape at higher deformations

0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jnnfm.2004.02.005

66 S. Chen et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 65–81

was also characterised quantitatively. Guido et al.[12] alsostudied the shape evolution of a liquid drop in an immiscibleliquid under a shear-flow reversal. Both the drop and the ex-ternal liquid are Newtonian. The drop axes and orientationwithin the shear plane were independently measured. Theorientation angle has been observed and correctly predicted.

A number of simulations of Lennard–Jones (LJ) fluids inwhich the interface ruptures or coalesces have been reportedrecently. Using molecular dynamics (MD), Koplik and Ba-navar[13] numerically simulated the coalescence of two LJliquid drops in Couette flow, in which a background shearflow drove the coalescence of two liquid drops. The rup-ture of the fluid interface can be simulated in a manner verysimilar to the coalescence[14]. The molecular details of therupture are roughly the time-reversed version of the coales-cence process. The break-up process of a viscous drop wassimulated by Li et al.[15] using a volume-of-fluid method.With a novel computational technique, dissipative particledynamics, Jones et al.[10] studied the dynamics of a dropat a liquid/solid interface in a simple shear field. The mainobjective in the simulation is to establish the mode of de-formation, the motion and possible break-up of the drop atflow rates corresponding to Reynolds number of the orderof 1–10. Clark et al.[16] simulated a pendant drop and adrop in simple shear flow to validate the DPD for the meso-scopic modelling of multiphase fluid–fluid systems in exter-nal fields. In these DPD simulations, only Newtonian dropsin an immiscible Newtonian fluid were considered. Therewas no information about polymer drops in a shear fieldwith the DPD method. The dynamics of polymer drops un-dergoing an external flow field are more complex.

In the present study, the DPD method is used to investigatethe dynamic behaviour of polymer drops in a microchannelunder a shear flow, in both transient and steady states. Pe-riodic boundary conditions are applied, and therefore oneeffectively deals with a periodic suspension of drops. Thepolymer drop is made up of many finitely extensible non-linear elastic (FENE) chains randomly distributed initially.The shear thinning and the first normal stress difference arestudied for this polymer solution. Another important param-eter of the drop, interfacial tension, is also investigated. Theevolution and rupture processes of the drop are simulatedunder differentCa numbers. The transient response of thepolymeric drop has also been analysed.

2. Dissipative particle dynamics

2.1. DPD system

In the DPD system, the basic unit is a set of discretemomentum carriers called particles, moving in continuousspace and in discrete time-steps. The momentum carrier isa coarse grained entity of massm in a three-dimensionalsimulation domain. The particle’s motion represents the col-lective dynamic behaviour of a large number of molecules

(a fluid “particle”). Three interparticle forces act upon theparticles, and these are the dissipative, random and conser-vative forces. Each particle moves along its new velocityfor a time-step after a possible collision of two particles.The computation is carried out by solving Newton’s equa-tions of motion for each particle for a large number of timesteps, sufficient to get convergence for the system proper-ties, such as viscosity, pressure and interfacial tension; theyare obtained by relevant statistical averages of the positions,velocities or forces for each particle at each time step. DPDconserves not only the number of particles but also the totalmomentum of the system, and satisfies Galilean invariance[17], and the detailed balance equations[18].

2.2. DPD model

A set of interaction particles is considered, whose timeevolution is governed by Newton’s equations of motion. Fora simple DPD particlei [19],

dridt

= vi,dvidt

= fi + FS, (1)

whereri andvi are its position and velocity vectors,fi theinterparticle force on particlei by all of other particles (ex-cept itself),FS the spring force used to model the polymerchain. The dynamic interactions between the particles arecomposed of two parts, dissipative and stochastic, comple-menting each other to ensure a constant value for the meankinetic energy of the system. The unit of mass is taken tobe the mass of a particle, so that the force acting on a parti-cle equals its acceleration. The forcefi contains three parts,each of which is pairwise additive:

fi =∑j �=i

(FCij + FD

ij + FRij ), (2)

where the sum runs over all other particles within a certaincut-off radiusrc. The radiusrc can be set at any value andit could be different for different forces.

Since the time average of the dissipative and fluctuationforces is zero, it does not feature in the equilibrium behaviourof the system, which is governed solely by conservativeforces. The conservative forceFC

ij is a soft repulsion actingalong the line of centres and is given by

FCij =

aij

(1 − rij

rc

)r, rij < rc,

0, rij ≥ rc,

(3)

whereaij is a maximum repulsion between particlesi andj,andrij = ri − rj, rij = |rij |, rij = rij/|rij |. For conservativeforce, we take the cut-off radius to be 1 unit of length.

The dissipative or drag force,FDij , on particlei by particle

j, is given by

FDij = −γwD(rij )(rij · vij )rij , (4)

wherewD is an r-dependent weight function vanishing forr > rc, vij = vi−vj, γ a coefficient which controls the extent

S. Chen et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 65–81 67

of dissipation in a simulation time step. The negative signin front of γ indicates that the dissipative force is oppositeto the relative velocityvij.

The dissipative force, acting against the particle motion,reduces the kinetic energy of the system. This is compen-sated by the random motion produced by the stochastic forceFR

ij , given by

FRij = σwR(rij )ξij rij , (5)

wherewR is also anr-dependent weight function vanishingfor r > rc, andξij a Gaussian variable with zero mean andvariance equals�t−1. Here�t is the time step, andσ is acoefficient characterising the strength of the random forces.These forces also act along the line of centres.

Español and Warren[20] showed that one of the twoweight functions appearing inEqs. (4) and (5)can be chosenarbitrarily; the other weight function is determined by

wD(r) = [wR(r)]2, (6)

σ2 = 2γkBT, (7)

wherekBT is the Boltzmann temperature of the system. Thisis analogous to the fluctuation–dissipation theorem for thesystem[21]. TakingkBT as the unit of energy, we have

σ2 = 2γ, (8)

We use the following weight function to improve on theSchmidt number for the system[22], instead of quadraticfunction,(1 − r/rc)

2 that is usually adopted,

wD(r) = [wR(r)]2 =

√1 − r

rc, r < rc,

0, r ≥ rc.

(9)

This weight function yields a stronger dissipative force be-tween particles than that from the standard quadratic force,for a given configuration of particles and interaction strength.In the present study the cut-off radiusrc forwD(r) andwR(r)

is also set at unity unless otherwise mentioned.

2.3. FENE chain model

By introducing bead-and-spring type particles, polymerscan be modelled by the DPD method[23,24]. Thermody-namic interactions between the polymer and surroundingfluid are represented by simple modifications of the DPD in-terparticle forces[4]. The finitely extensible nonlinear elasticmodel is designed to produce a non-Newtonian fluid that istractable for computer simulation. The FENE rheology andconformational changes under shear are well documented[25–27].

The bead-spring model for polymer chains is constructedby linking a series of DPD particles together with springforces acting between adjacent beads of the polymer struc-ture [4], as shown inFig. 1. In the present study, the FENEspring force is used to impose a finite maximum extension

Fig. 1. FENE bead-spring chain model. The chain segments are fictitious,i.e., phantom chains.

for the chain segment and should be added to the right-handside ofEq. (1).

In the FENE chain[28], the force on beadi due to beadj is

FSij = − Hrij

1 − (rij/rm)2, (10)

whereH is the spring constant, andrm the maximum per-missible length of one chain segment. The spring force in-creases drastically withrij/rm and becomes infinitely largeasrij/rm approaches unity. This model can capture the finiteextensibility of the polymer chains. The mass of the beadsis assumed to be unity, the same as that of other simple par-ticles.

The time constant is important in characterising polymermolecular motion and can be formed from the model pa-rameters. Two constants with dimension of time can be ob-tained for the FENE spring[28]. One (λH) is defined belowfor the Hookean dumbbell,

λH = ζ

4H, (11)

whereζ is the friction coefficient of a bead. The second one(λQ) is defined as follows for the rigid dumbbell,

λQ = ζr2m

12kBT, (12)

The FENE parameter,b, is the ratio of these two constants

b = 3λQ

λH= Hr2

m

kBT, (13)

Chain models usually have a spectrum of relaxation times.There is no closed-form expression for the relaxation timespectrum for FENE chains. However, a modification of theFENE chain, called the FENE-PM chain, has the same spec-trum as the Rouse chain (bead and Hookean spring chain)[28]. A time constant can be defined for the FENE-PM chainas[29]

λFENE = b

b + 3λH

N2b − 1

3, (14)

whereNb is the number of beads in the chain.The equilibrium length is an appropriate parameter to

represent the size of the polymer chain. IfLC denotes theequilibrium length of one segment of a polymer chain, thenthe total equilibrium length of one polymer chain is simply(Nb − 1)LC. For the FENE spring, the equilibrium lengthof a segment may be estimated as follows[28]:

LC =√

3r2m

b + 5. (15)


2.4. Binary immiscible fluids

Immiscible fluid mixtures exist because individualmolecules attract similar and repel dissimilar molecules[30,31]. The miscibility of the two fluids is controlledmainly by the repulsive parametera between the drop andthe surrounding fluids. In order to model immiscible fluids,a new variable is introduced, called the “colour” accordingto Rothman–Keller[30]. Here, for example, red representsthe polymer drop and blue represents the surrounding New-tonian fluid. When two particles of different colours inter-act, we increase the conservative force, thereby increasingthe repulsion, that is,

aij ={a0 if particlesiandj are the same color,

a1 if particlesiandj are different colors.(16)

The two phases would be completely miscible ifa1 ≈a0 and almost entirely immiscible ifa1 exceeds a0significantly.

Groot and Warren[32] have made a link between the re-pulsive parameteraandχ-parameters in Flory–Huggins-typemodels. They pointed out that DPD model cannot produceliquid–vapor coexistence, since the repulsive pressure is sosoftly increasing with density (ρ), leading to an apparentabsence of aρ3 term at high densities. But one can simulateliquid–liquid and liquid–solid interfaces, and in this way themethod is similar to the Flory–Huggins theory of polymers,and can in fact be viewed as a continuous version of thislattice model.

The parameterχ is positive when A and B are two im-miscible components; when they are miscible over AA orBB contacts, then it is negative. Ifχ is very small and pos-itive, no segregation will take place, but when it exceeds acritical value, A-rich and B-rich domains will occur. Thecritical point in terms of theχ-parameters can be calculatedby Eq. (19) of[32]. In the present study the parameters willbe chosen where segregation takes place, i.e.,χ > χcrit.The Flory–Huggins parameter for monomers is obtained byGroot and Warren. The calculatedχ-parameter for two den-sities (ρ = 3 and 5) is expressed as a function of the excessrepulsion parameter�a, where�a = a1 − a0. It is shownthat forχ > 3 there is a very good linear relation betweenχ and�a. Explicitly, it is

χ = (0.286± 0.002)�a, ρ = 3,

χ = (0.689± 0.002)�a, ρ = 5. (17)

We fix the density at 4 for our DPD system, and useEq. (17)as an effective mean for extrapolation to estimatethe Flory–Huggins parameterχ.

The Flory–Hugginsχ-parameter for polymers was alsoobtained via the observed segregation by Groot and Warren.The best estimate for 2< N < 10 is

χNkBT

�a= (0.306± 0.003)N. (18)

In Eq. (18), both of the two phases are similar-length poly-mers, i.e.,N = NA = NB; they are different for our case,where the solvent is Newtonian (N = 1) and the polymerphase hasNFENE = 16.

2.5. Computational algorithm for DPD

In a previous study[1], a simple Euler-type algorithm wasused to advance the positions and velocities of the particles.A modified version of the velocity-Verlet algorithm[32] isused here:

ri(t + �t) = ri(t) + �tvi(t) + 12(�t)

2fi(t),

vi(t + �t) = vi(t) + λc�tfi(t),

fi(t + �t) = fi(r(t + �t), v(t + �t)),

vi(t + �t) = vi(t) + 12(�t)(fi(t) + fi(t + �t)), (19)

wherev(t+�t) is the prediction of the velocity of the particleat the instantt + �t. In this algorithm, the force is stillupdated once per iteration (after the second step) thus thereis virtually no increase in computational cost. Hereλc is anempirically introduced parameter, which accounts for someadditional effects of the stochastic interactions. If the totalforce is velocity independent, the standard velocity-Verletalgorithm is recovered forλc = 1/2. Groot and Warren[32]found the optimum value ofλc is 0.65. For this value, thetime step can be increased to�t = 0.06 without a lossof temperature control in simulating an equilibrium systemwith ρ = 3 andσ = 3.

3. Simulation procedure

The computational domain consists of three phases: New-tonian fluid, polymer drop and solid boundary, as shown inFig. 2.

The volume of drop equals to 523.6 units with drop radiusof 5 units located at the centre of the computational domain.This implies a volume concentration of about 1.31% for thedrop if the size of computational domain is 40×20×50. Thepolymer drop can be simulated in DPD by placing FENE

YX

ZX

Y Z

∆∆

∆

Fig. 2. Polymer drop undergoing shear flow between two parallel planes.


-20 -15 -10 -5 0 5 10 15 20

X

-15

-10

-5

0

5

10

15Z

Fig. 3. A polymer drop made up of FENE chains after equilibration.

chains in a spherical region. After equilibration, this willform a rough spherical drop, as shown inFig. 3.

In the present study, the polymer drop consists of 118FENE chains, and each chain is made up of 16 beads. BesideFENE chains, there are 208 particles representing the solventwithin each drop; each comprises of 2096 particles—thusthe polymer drop is a solution of about 90% chains and 10%solvent particles.

The fluid occupies the remaining space in the channel, rep-resented by 157 838 simple DPD particles when the channelsizes are 40× 20× 50 units. All three phases are assumedto be of the same density, equal to 4 units. A shear flow isgenerated by sliding the plates in opposite directions. Peri-odic boundary conditions are applied on the fluid boundaryof the computational domain in theX andY directions.

The solid wall is usually represented by using frozenparticles. Due to the soft repulsion between DPD parti-cles, it is difficult to prevent fluid particles from penetratingthe wall particles. Near-wall particles may not be sloweddown enough and slip may then occur. To prevent this,higher-density wall particles and larger repulsive forces canbe used to strengthen the wall effects. This, however, resultsin large density distortions in the flow field, similar to whathappens in MD. In the present study, frozen particles arestill used to represent the wall and the density of wall parti-cles is increased to 6. Near the wall a thin layer is assumedwhere the no-slip boundary condition holds. We enforce arandom velocity distribution in this layer with zero meancorresponding to a given temperature plus the velocity dis-tribution induced by shear flow. Similar to the reflection lawof Revenga et al.[33,34], we further require that particlesin this layer always leave the wall. The velocity of particlei in the layer is

vi =

vR + n(√

(n · vR)2 − n · vR

)+ D · z, i = x,

vR+n(√

(n · vR)2−n · vR

), i = y or z.

(20)

wherevR is the random vector andn the unit vector nor-mal to the wall and pointing to the fluid,D the shear rate

Table 1Parameters adopted in the present simulation

Density ρ 4.0External fluid-drop repulsion coefficient aed 56.25External fluid-external fluid repulsion coefficient aee 18.75polymer chain-solvent repulsion coefficient aps 11.25polymer chain-polymer chain repulsion coefficient app 18.75Random force coefficient σ 3.0

and z the position in theZ-direction of the particle in thethin layer. The thickness of this layer and the strength ofthe repulsion between wall and fluid particles are chosen tominimise the velocity and density distortion. In this paper,the layer thickness is chosen to be the minimum between0.5% of channel width and 0.5 of the cut-off radius. Thethin layer is necessary to prevent the frozen wall from cool-ing down the fluid and this method is more flexible whendealing with a complex geometry. Using this method, thedensity profiles was shown in[19], for a simple DPD fluidin Poiseuille flow, to be almost uniform across the channelexcept in the region near the wall, where a fluctuation indensity still exists but not as severe as that predicted by MDsimulation. In detail, ifi andj both denote fluid particles orbeads in the polymer chains we chooseaij = afluid = 18.75to satisfy the compressibility of water suggested by Grootand Waren[32]. For wall particles, we assumedawall = 5.0andaij = √

afluidawall = 9.68 when calculating the interac-tion between fluid and wall particles. The density distortionnear the wall can be further reduced by softening the repul-sion between wall and fluid particles.

The initial configurations of fluid and wall particles aregenerated separately by a pre-processing program and readin as input data[19]. All fluid particles are initially locatedat the sites of a face-centred cubic (fcc) lattice. The initialvelocities of fluid particles are set randomly according to thegiven temperature but the wall particles are frozen. At the be-ginning of the simulation the particles are allowed to move,with shear rate set to zero, until a thermodynamic equilib-rium state is reached. Then the upper and lower plates startto slide in opposite directions and the non-equilibrium sim-ulation starts. The parameters adopted in the present studyare shown inTable 1.

FromTable 1, we can see the excess repulsive parameter�a = aed − aee = 37.5 between the Newtonian drop andthe external Newtonian matrix. Hence we can estimate theχ-parameter is within the range of 10.65–25.91 according toEq. (17). For the polymer drop, as the two phases are quitedifferent in chain lengths,Eq. (18)may not be suitable tocompute theχ-parameter.

4. Stress tensor

Stress tensor components are calculated by the Irving–Kirkwood method [35,36]. Here, the contribution ofeach particle to the stress tensor consists of two parts, a


configuration part and a kinetic part:

Sαβ = − 1

V

⟨ Np∑i

miuiαuiβ +Np∑i

Np∑j>i

rijαFijβ

⟩, (21)

wheremi is the particle mass (mi = 1 unit), Np the numberof particle,uiα anduiβ the peculiar velocity components ofparticle i, for example,uiα = viα − vα(x), with v(x) beingthe stream velocity at positionx, and〈· · · 〉 denotes the en-semble average.Fijβ is theβ-component of the force exertedon particlei by particle j. The first sum in the right-handside ofEq. (21)denotes the contribution to the stress fromthe momentum transfer of DPD particles. The second sumrepresents the contribution from the interparticle forces. Fora bead on a polymer chain the force termFijβ will consistof both the interparticle’s interactions and also the springforce on the bead which are due to the spring stretching[37]. The constitutive pressure can be determined from thetrace of the stress tensor:

p = −1

3tr S. (22)

5. Viscosity and normal stress differences

We use the constitutive equation to find the viscosity,

η = Sxz

D, (23)

the first and second normal stress differences,

N1 = Sxx − Szz, N2 = Szz− Syy. (24)

Normal stress differences for Newtonian fluids are zero butviscoelastic fluids may exhibit non-zero normal stress dif-ferences.

6. Computation of the interfacial tension, ΓΓΓAB

The interfacial tensionΓ AB between fluid A and fluid Bcan be computed using the Irving–Kirkwood equation[36],

ΓAB =∫ [

pzz− 1

2(pxx + pyy)

]dz, (25)

wherepxx, pyy, pzz are the three diagonal components ofthe pressure tensor (−S). The interface is parallel to thex–yplane. The interfacial tension can be further expressed as

ΓAB = L[〈pzz〉 − 12(〈pxx〉 + 〈pyy〉)], (26)

whereL is the height of the simulation channel.Eq. (26)will be applied to the flat interface geometry shown inFig. 9. The angular brackets denote the average overthe simulation run. Groot and Warren[32] gave the bestfit of the surface tension as a function ofχ-parameterandN:

ΓAB = (0.75± 0.02)ρkBTrcχ0.26±0.01

×[1 − 2.36± 0.02

χ

]3/2

forN = 1, (27)

ΓAB = (0.583± 0.004)ρkBTrcχ0.4

×[1 − 2

χN

]3/2

forN > 1, (28)

where N is the number of segments per molecule. Formonomers,N = 1.

7. Results and discussion

In DPD simulations, all of the parameters are scaled byDPD units. The DPD units are not defined explicitly as inMD simulation. In the present study, the scaling is muchlarger than that in MD. For detailed information, one canrefer to[19].

7.1. Linear viscoelastic behaviour of FENE chain

In the present study, we choose the spring constant of theFENE chain asH = 6.0 and the maximum length of onechain segment asrm = 3.0. The value ofb in Eq. (13)is 54.The linear relaxation modulusG(t) for a solution of polymerchains can be evaluated directly from the equilibrium shearstress autocorrelation function through the relationship[38]

G(t) = 1

nkBT〈Sp

xz(t)Spxz(0)〉, (29)

whereSpxz(t) is the instantaneous contribution of the poly-

mer molecule to the shear stress,n is the polymer numberconcentration, and the symbol〈〉 indicates a long-time av-erage. Here we compute the linear relaxation modulusG(t)using equilibrium (no flow) simulations according to Panand Manke[38], in which a single polymer chain is placedin the simulation box.

Fig. 4 shows the linear relaxation modulusG(t) forthe 16-bead FENE chain varying with the timet. Differ-ent values ofape/aee represent different solvent qualities.Whenape/aee< 1.0, we have the good solvent case, whileape/aee > 1.0 represents poor solvent, whereape is the re-pulsive coefficient between the external fluid and polymerparticles,aee is the repulsive coefficient between the exter-nal fluid particles. The ratioape/aee = 1.0 is the near-thetasolvent case. TheG(t) curve for the good solvent is higherthan the curves for the near-theta solvent and the poor sol-vent, reflecting a greater extent of chain expansion in thegood solvent according to Pan and Manke[38]. It is alsoclear from Fig. 4 that a spectrum of relaxation times isrequired to fitG(t) (i.e. the FENE chain solution is not asingle relaxation time fluid).


0 2 4 6 8 10 12 14 16 18 20

t

0.001

0.01

0.1

1

G(t

)

a(pe)/a(ee)0.613

Fig. 4. Relaxation modulusG(t) vs. time.

7.2. Viscosity and normal stress differences of polymersolution

Fig. 5shows the polymer solution viscosity as a functionof the shear rateD (left). The size of the computationaldomain is 40× 10× 30, corresponding to a total number of52 800 particles in the computational domain including 4800wall particles, 43 200 polymer beads (2700 FENE chain,and each chain is made up of 16 beads) and 4800 solventparticles. This gives rise to the number concentration of 90%for the polymers. It can be clearly seen from this figurethat the fluid is slightly shear-thinning for this shear raterange. The shear-thinning behaviour can be described by apower-law relationship,

η = C1Dq−1, (30)

with the power-law indexq = 0.861 andC1 = 6.96.For a comparison,Fig. 5 (right) shows the relationship

between shear stress and shear rate for the solvent withoutpolymer chains—the fluid is Newtonian, the slope is equalto 2.53 units, the solvent viscosity.

Fig. 6shows the first and second normal stress differencesagainst the shear rate. The first normal stress difference(N1) is positive, whereas the second normal stress differ-

0.01 0.1 1 10

Shear rate

1

10

100

Vis

co

sity

0 0.5 1 1.5 2 2.5 3 3.5

Shear rate

0

1

2

3

4

5

6

7

8

9

Shear

stre

ss

Fig. 5. Polymer solution viscosity (left) and solvent shear stress (right) as functions of shear rateD.

0.01 0.1 1 10

Shear rate

-20

0

20

40

60

80

100

120

No

rmal

stre

ssdif

fere

nce

N1

an

dN

2

N1

N2

Fig. 6. First and second normal stress differences.

ence (N2) is almost zero and slightly negative, in agreementwith [29]. N1 can be fitted to a power-law in the formN1 = C2D

m, with m ∼ 1.56 andC2 ∼ 55 in the presentstudy. The ratio of first normal stress difference to the squareof the shear stress is weakly dependent on the shear ratehere.

In order to validate the present DPD model of FENEchains, inFigs. 7 and 8the dimensionless viscosity anddimensionless first normal stress coefficient for steady-stateshear flow predicted by DPD are compared with theFENE-PM model and the BDJ model for dilute FENE chainsolutions; here in the DPD computation the bead numberN = 10 andb = 100 according to[29]. The viscosityand the first normal stress coefficient are normalised withthe zero-shear-rate properties. It shows the dimensionlessviscosity is in good agreement with the FENE-PM andBDJ models, while the differences in the dimensionlessfirst normal stress coefficient between the DPD, BDJ andFENE-PM model may be due to the various approximationsin the models. All three models predict the same power-lawbehaviour at high shear rates, so that on a log–log plot ofη versusD, the viscosity approaches a straight line of slope−2/3 and on a log–log plot ofΨ1 versusD, the first nor-mal stress coefficient versus shear rate approaches a slopeof −4/3.


0.01 0.1 1 10 100

Dimensionless shear rate

0.001

0.01

0.1

1

10

Dim

ensi

on

ale

ss v

iscosi

ty

FENE-PM

BDJ

present

Fig. 7. The dimensionless viscosity(η− ηs)/(η0 − ηs) vs. dimensionlessshear rateλFENED for the DPD, FENE-PM and BDJ models, whereb = 100 andN = 10.

7.3. Interfacial tension

In order to compute the interfacial tension, the channelof 40 × 10 × 30 units is divided into two layers of thick-ness of 15 units each, as shown inFig. 9. The upper layer isfilled with simple DPD solvent particles and the lower layeris filled with our polymer solution, made up of 90% poly-mer chains and 10% solvent particles. Each layer consistsof 24 000 DPD particles, totalling 48 000 particles in thesystem.

From Groot and Warren’s formula,Eq. (27), we obtainthe following interfacial tension for monomers with�a =37.5: (1) ΓAB = 3.81 for ρ = 3 and (2)ΓAB = 6.06 forρ = 5. The extrapolation value for our case (ρ = 4) is 4.94,which is close to our simulation result,Fig. 9, of 5.25 for amonomer with�a = 37.5, a/a0 = 3.0 andρ = 4.

The interfacial tension increases with increasinga/a0(wherea0 = 18.75) between the two fluids, shown as inFig. 9 (right). For the same value ofa, the interfacial ten-sion of the fluid with polymer chains is always larger thanthat of the solvent.

1 1.5 2 2.5 3 3.5 4 4.5 5

a / a0

0

1

2

3

4

5

6

7

8

Inte

rfacia

lte

nsi

on

90% solution

Newtonian

Fluid A

Fluid B

Fig. 9. Two immiscible fluids and their interfacial tensions.

0.1 1 10 100

Dimensionless shear rate

0.001

0.01

0.1

1

10

Dim

ensi

onle

ss 1

st n

orm

al

stre

ss c

oeff

icie

nt

BDJ

FENE-PM

present

Fig. 8. The dimensionless first normal stress coefficientΨ1/Ψ1,0 vs. di-mensionless shear rateλFENED for the DPD, FENE-PM and BDJ models,whereb = 100 andN = 10.

7.4. Drop in a steady-state shear field

The dynamics of drop deformation and break-up isgoverned by the competition between hydrodynamicstresses, which act to deform the drop, and surface ten-sion, which opposes any surface area increase, and thustends to restore a weakly deformed object to a sphericalshape.

The first theoretical analysis of drop shape in shear flowwas proposed by Taylor[39,40], who presented a small de-formation analysis restricted to Newtonian fluids. The shapeof the sheared drop would be governed by the capillary num-ber, Ca, which represents the ratio between the hydrody-namics force to interfacial tension,

Ca = ηeDR0

ΓAB. (31)

Hereηe is the external fluid viscosity,D is the shear rate,R0is the radius of the undeformed drop, andΓ AB is the inter-facial tension. Another dimensionless parameter governingthe shape is the viscosity ratio,λ = ηd/ηe, whereηd is thedrop viscosity.


Within the limit of small deformations according toTaylor’s theory, the drop is an ellipsoid with a major radiusRa at an angleθ of 45◦ with respect to the velocity gradi-ent direction. The degree of deformation is represented bythe parameterDdef = (Ra − Rb)/(Ra + Rb), whereRb isthe smallest radius of the ellipsoid. Taylor’s theory yieldedthe following well-known relation between the deformationparameterDdef as a function ofλ andCa,

Ddef = 19λ + 16

16λ + 16Ca. (32)

The theoretical prediction is valid only in the limit ofCa �1 [11].

7.4.1. Effect of the size of computational domainIt is necessary to explore the effects of computational

domain sizes, i.e.,�X, �Y and�Z as shown inFig. 2. Thedrop with original diameter ofd = 10 DPD units is placedat the centre of the computational domain.

As the channel’s upper and lower walls are in close prox-imity, hydrodynamic wall effects should be accounted for[41]. The effect of the walls is investigated by changing thechannel height�Z with �X and�Y held fixed.Fig. 10ashowsDdef as a function of�Z/d, whered is the originaldiameter of the drop, and�X and�Y are fixed at 40 and20, respectively.Ddef decreases monotonically as the chan-nel height�Z increases. If the drop is too close to the wall,the wall exacerbates the deformation of the drop[42–44].

1 2 3 4 5 6 7 8 9

∆Z/d

0.1

0.12

0.14

0.16

0.18

0.2

Dd

ef

0 2 4 6 8 10 12 14 16 18 20

∆X/d

0.1

0.12

0.14

0.16

0.18

0.2

Ddef

(a) (b)

0 1 2 3 4 5 6 7 8 9

∆Y/d

0

0.05

0.1

0.15

0.2

Ddef

(c)

Fig. 10. Effects of the sizes of computational domain on deformation parameterDdef.

Fig. 10ashows that this effect is small when�Z/d is greaterthan 5. This agrees with the results of Kennedy et al.[43]and Uijittewaal and Nijhof[44].

To study the effects of periodicity,�X and�Y are variedrespectively at fixed channel height�Z. Fig. 10bshows theeffect of�X on Ddef, while �Y and�Z are fixed at 20. ItshowsDdef increases with increasing�X, and approaches asteady value when�X/d is greater than 4. When(�X/d) ≥4, the incident flow has time to recover from disturbancescaused by neighbouring periodic images and the drop iseffectively decoupled hydrodynamically from its periodicneighbours.Fig. 10cshows the effect of�Y on Ddef, whileholding�X and�Z at a fixed separation of 20. Unlike theeffects of�X and�Z, the size of the computational domainin Y direction has little influence onDdef; this is so for�Y/d ≥ 1.2.

In order to ensure that the drop deformation are not in-fluenced by the size of the computational domain, we takethe size of the computation domain as�X = 40,�Y = 20and�Z = 50.

7.4.2. Deformation of the dropIn Fig. 11a and b, Ddef is plotted as a function ofCa.

In Fig. 11athe drop is composed of the Newtonian fluid,with no FENE chains. The capillary number is calculatedwith ΓAB = 5.252 whenλ = 1.0, andΓAB = 5.380 whenλ = 1.4. Fig. 11ashows a linear regime,Ca < 0.14, anda nonlinear regime,Ca > 0.14. A comparison with Guido


0 0.1 0.2 0.3 0.4 0.5

Ca

0

0.2

0.4

0.6

0.8

1

Ddef

Guido, viscosity ratio=1.4

Taylor,viscosity ratio=1.0

present, viscosity ratio=1.4

present, viscotisy ratio=1.0

Li, viscosity ratio=1.0

Rallison, viscosity ratio=1.0

0 0.1 0.2 0.3 0.4 0.5

Ca

0

0.2

0.4

0.6

0.8

1

Dd

ef

(a) (b)

Fig. 11. Variation ofDdef with the capillary number for (a) Newtonian (b) polymeric drop.

et al.’s [11] experimental results (taken at a viscosity ratioof λ = 1.4), some previous numerical investigations[15,45]and Taylor’s theoretical prediction (at viscosity ratio 1.0) arealso shown. In Guido et al.’s experiment, both the drop andthe external fluids are Newtonian; the linear regime persistsup to Ca = 0.35, exceeding beyond Taylor’s assumption(Ca � 1), in qualitative agreement with the present results.

According to Taylor’s theory, the slope inEq. (32) is1.09 whenλ = 1.0 while in the present study it is 1.099for the sameλ value. The computational value of the slopefor viscosity ratio of 1.4 is about 1.10, while the theoreticalvalue is 1.109.

Spenley[46] reported that DPD polymers are “phantomchains” which pass freely through each other, and there isno evidence for entangled behaviour in any of the simula-tions described in his work. The reason is that there is nohard-core repulsion in DPD; it is possible for the chains topass through each other, despite the fact that MD studies ofLennard–Jones and hard sphere polymers did find entangle-ment. In the present study, the polymer drop data are shownin Fig. 11b, whereΓAB = 5.612 andλ = 1.0. The polymerdrop is not easy to deform, thus the linear regime extendsto a large value ofCa ≈ 0.22, larger than that for the New-tonian drop, while the slope inEq. (32)decreases to 0.65for the polymer drop. Similar results have been observedby Hooper et al.[47]; they modelled the deformation of aviscoelastic drop suspended in another Newtonian fluid sub-jected to uniaxial extensional flow using the DEVSSG fi-nite element formulation. Their computational results showa viscoelastic drop in a Newtonian solvent elongates less atsteady-state extension than a Newtonian drop, because ofthe accommodation of stress by elasticity.

7.4.3. Drop break-upAs the drop ellipsoidal shape increasingly distorts with

shear rate, necking may then develop. When this happens,the drop may break up into several droplets. In the presentstudy, the break-up of the polymer drop occurs whenCa>0.38. Fig. 12 shows the break-up process of the drop. Thedrop is first stretched. Att = 300 the drop has not broken upyet; part of the drop appears on the left side of the periodic

box, due to the periodic boundary condition. The drop breaksup into new droplets att = 360; here 2 and 4 are two partsof one drop, and part 1 is the new droplet. Droplet 1 is notstable and it coalesces with part 2 soon after to form part 3.In the mean time, a new droplet 4 is developed from the tipregion of the stretched drop. The two droplets att = 500 arenot yet stable; drop 3 continues to elongate and break up. Thedynamical equilibrium between break-up and coalescenceresults in a steady-state average droplet-size distribution, asshown att = 780, two small drops with approximately samesize are relatively stable. FromFig. 12we also noted somethreads between the droplets. It is the single FENE chainwhich forms the thread when stretched out of the dropletsby the shear flow. Very possibly, these single FENE chainsforming the threads could form some daughter droplets.

For a drop undergoing shear flow there is a critical cap-illary number,Cac, which describes the minimum value ofCa required to initiate the break-up process. In our simula-tion the critical capillary number is 0.38 for the polymericdrop and 0.25 for the Newtonian drop, while the critical cap-illary number is found to be about 0.5 in Guido et al.’s[11]experiment for the Newtonian drop. Li et al.’s[15] simula-tion results show that the critical capillary number for vis-cosity ratioλ = 0.5 lies between 0.38 and 0.40, which isslightly less than that atλ = 1.0. The difference betweenthe DPD method and previous experimental and simulationdata needs to be further investigated.

7.5. Transient deformation of polymer drop undershear-flow reversal

In the present study, the shape evolution and orientationof polymer drops under transient reversal shear flow are alsoconsidered. The shear rate is reversed after a steady-state de-formed drop has been reached. A sketch for the time historyof the reversal shear flow is shown inFig. 13. The initialzero shear rate (region 1) serves to generate an equilibriumstate of the drop.

Fig. 14 shows the drop deformation as viewed from thevorticity direction (Y-direction). At the inversion time,t =300, the drop is at a steady deformed shape. During the shear


Fig. 12. Evolution of a polymer drop during break-up under a steady-state shear field.


0 100 200 300 400 500 600 700 800

t

-0.15

-0.1

-0.05

0

0.05

0.1

Shear

rate

Region 1

Region 2

Region 3

Region 4

Fig. 13. Shear history regimes imposed.

rate reversal the elongated oriented drop retracts towards anearly spherical shape (t ≈ 360), and then is stretched againby the flow, changing to a new orientation. At timet = 600the flow is turned off, then the drop returns to its sphericalconfiguration. The ability of shape relaxation will impart acharacteristic time scale to a suspension of droplets. Thecapillary number for the data inFig. 14is Ca = 0.225. Thesame behaviour has been observed by Guido et al.[12] underflow conditions where there is a shear rate reversal. In theircase,Ca ≈ 0.20, and the drop is Newtonian. It should benoted that the drop centre of mass moves toward a negativeZvalue with time, due to the random force in DPD simulationexerting a very small non zero total force on the drop.

In Guido et al.’s work[12], another different behaviourwas reported. The drop remains elongated throughout its

-20 -15 -10 -5 0 5 10 15 20

X (t = 300)

-15

-10

-5

0

5

10

15

Z

-20 -15 -10 -5 0 5 10 15 20

X (t = 360)

-15

-10

-5

0

5

10

15

Z

-20 -15 -10 -5 0 5 10 15 20

X (t = 600)

-15

-10

-5

0

5

10

15

Z

-20 -15 -10 -5 0 5 10 15 20

X (t = 800)

-15

-10

-5

0

5

10

15

Z

Fig. 14. Evolution of a polymer drop under flow reversal,Ca = 0.225.

-20 -15 -10 -5 0 5 10 15 20

X (t=350)

-15

-10

-5

0

5

10

15

Z

Fig. 15. Flow reversal atCa = 0.3.

evolution atCa ≈ 0.4; however, the drop’smajor axisro-tates in the opposite sense to that of a rigid elongated bodysubjecting to the same flow field. This is referred to as thetumbling behaviour.Fig. 15shows the minimal deformationof the polymer drop whenCa ≈ 0.3 using the present DPDsimulation. However, we do not observe any tumbling be-haviour here.

Narumi et al.[48] experimentally examined the transientstress response under shear flow of concentrated suspen-sions of non-Brownian spheres. They showed that the nor-mal stress difference (N1) behaves in a similar manner to theshear stress (Sxz) in a shear reversal experiment. If the sub-sequent shearing after a rest period is in the same directionas the initial one,N1 andSxz will jump instantaneously tothe appropriate steady-state values, whereas if the shearing


0 200 400 600 800 1000 1200

t

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Shear

rate

Region 1

Region 2

Region 3

Fig. 16. Stepwise increase in shear rate, followed by a flow reversal.

is reversed there is a transient period inN1 and Sxz. Thisconclusion is also in agreement with the theoretical predic-tion of some constitutive models[49–51]. Minale et al.[52]investigated immiscible polymer blends, which are soft ma-terials in contrast to suspension of solid spheres in[48], intransient experiments consisting of a stepwise increase inshear rate, applied for a time period, followed by a flow re-versal. Two major types of behaviour have been identified,depending on whether the flow has been reversed before orafter the maximum in first normal stress difference (N1) hasbeen reached.

In order to investigate the stress response in the presentDPD system of polymer drops, we also impose a shear his-tory on the computational domain, as shown inFig. 16. Thedrop is subjected to a stepwise increase in shear rate (D =0.2 in region 1 toD = 0.4 in region 2), followed by a flowreversal (D = −0.4 in region 3).

Fig. 17 shows the shear stress response imposed on thefluid. With a stepwise increase of shear rate att = 300, theshear stress gradually rises to a new value, in contrast tothe stress jump observed in hard-sphere suspensions, then itgradually decreases to a negative value after the flow is re-versed (t = 700). The “relaxation” time scales of the shearstress in these two periods are similar. The shear stress re-sponse in the present DPD system is relatively slower than

0 200 400 600 800 1000 1200

t

-1.5

-1

-0.5

0

0.5

1

1.5

Sh

ear

stre

ss

Region 1

Region 2 Region 3

Fig. 17. Shear stress response.

0 200 400 600 800 1000 1200

t

-0.1

0

0.1

0.2

0.3

0.4

0.5

N1

Region 1Region 2 Region 3

Fig. 18. First normal stress difference (N1) response in the polymeric drop.

the more abrupt results of Minale et al.[52]. It also showsthe shear stress response of soft matter is different from thesuspension of solid spheres[48], where the shear stress jumpto the new steady-state values during stepwise increase anddecreases gradually to a stable negative value if the shear-ing is reversed as observed by Narumi et al.[48]. Fig. 18showsN1 versus time in the polymeric drop. Stress jump inN1 is not observed in the stepwise increase of the shear rate.This is due to the shape relaxation ability of the drop. Mi-nale et al.[52] did not observe any jump ofN1 for thepolymer blends when a stepwise increase of shear rate oc-curred. InFig. 18, after a step-up in shear rate,N1 initiallyincreases with time, passes through a maximum and sub-sequently decreases towards the steady-state value. Due tothe possible coalescence of the droplets,N1 will increaseagain, as shown at the end of region 2. During flow reversal,N1 first decreases, and then increases again to a steady-statevalue—this is due to the shape relaxation in the shear rever-sal. There is no apparent tumbling for the polymeric dropduring the flow reversal, and this is in agreement with thebehaviour of single polymeric drop observed inFig. 15.

Fig. 19 shows the transient behaviour of shear stressagainst the shear strain after the start-up of a shear flow.

0 50 100 150 200 250 300

Strain

-0.2

0

0.2

0.4

0.6

0.8

1

( Sxz)

/(S

xz)

max

shear rate=0.133

shear rate=0.266

shear rate=0.4

shear rate=0.533

shear rate=0.667

shear rate=1

Fig. 19. Transient behaviour of normalised shear stress against strain ina start-up of a shear flow.


Here, the shear stress is normalised with the steady value,at different shear rates, from 0.133, 0.266, 0.4, 0.533, 0.667to 1. Unlike sphere suspensions[48], where the normalisedshear stress is observed to be a function of the shear strain,here the shear stress is not just a function of the imposedstrain, but is also dependent on the shear rate imposed.

In the present study, the stepwise increase and reversal ofshear rate are implemented by changing the direction andmagnitude of the velocity on the moving solid boundary. TheSchmidt numberSc= η/ρDdiff characterising the dynamicbehaviour of fluids is an important parameter to consider inthis respect, whereη is the viscosity andDdiff is the diffusionconstant. The Schmidt number is about 0.6805 according toEq. (37) of[32] andη ≈ 1.077 (in a typical fluid, water forinstance,Sc ∼ O(103), reflecting the fact that momentumis transported much more efficiently). In the present DPDsystem, when the modified weight function inEq. (9) isemployed, viscosity andSccan be increased to 2.4023 and6.3987, respectively[22]. This improves DPD simulationswithout increasing computational time, but it may not besufficient. So when DPD particles undergo the flow reversal,it would be still difficult for such soft particles to responseto the rapid momentum change, thus the drop would retractto a spherical configuration first (Fig. 15).

Another effective way to raiseSc is to increaserc [22].When rc was increased from 1.0 to 1.5, together with themodified weight function,Scandη attain values of 164.5 and16.972, respectively, with an improved dynamic response,but at a computational cost, increasing approximately as thecube of the cut-off radius. With the modified weight functionand increasing cut-off radius torc = 1.5, we observed animproved stress response in the fluid for flow reversal, asshown inFig. 20. In the mean time, we indeed observe thetumbling of the drop.Fig. 21shows the comparison of themodified DPD results with Guido et al.’s[12] experiments.

Fig. 21. Tumbling of the drop: (a) Guido et al’s experiment and (b) computational results withrc = 1.5.

150 170 190 210 230 250

t

-25

-20

-15

-10

-5

0

5

10

15

20

25

S xz

(a)

(b)

Fig. 20. Comparison of shear stress response (a)rc = 1.0 and (b)rc = 1.5.

When t = 314 the drop reaches its minimal deformation.It is very clear that the drop remains elongated throughoutthe evolution and its rotation direction is opposite to that ofa rigid sphere, which is in good qualitative agreement withGuido et al.’s[12] experimental results.

In order to investigate both the transient extension of aninitially spherical polymer drop and the transient retractionof a deformed drop, the shear flow withD = 0.2 is appliedfrom t = 40, and aftert = 600 the shear flow is removedby stopping both the upper and lower plates. The value ofthe cut-off radius is set as 1.5 and the capillary number issmaller than the critical value, to ensure a steady drop shape.

Fig. 22shows the extension (strain),Ra/R0, as a functionof time for the polymer drop during start-up of the shearflow, whereRa is the major radius andR0 the radius of un-deformed drop withR0 = 5.0. Although there are differentappropriate time scales for describing drop extension anddrop retraction, according to Hooper et al.[47] we specifythe characteristic time scale as that of the shear flow for both


0 50 100 150 200

t

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Ra

/R0

Newtonian

Polymer

Fig. 22. Drop strain vs. time for Newtonian drop and polymer drop understart-up of shear flow condition.

extension and retraction of the drop and set the polymerdrop Deborah number asDe = λFENED. HenceDe = 2.11for D = 0.2 andλFENE = 10.57. The Newtonian externalfluid is represented by simple DPD particles. For both poly-mer and Newtonian drops, the viscosity ratioλ = 1.0. Forthe Newtonian drop, the extension increases monotonicallywith time until a steady state is reached. However, for thenon-Newtonian drop, an overshot occurs. Such a responsehas also been observed by Hooper et al.[47,53]. In theirstudy, the contours of Tr(τ) show that the overshoot is due tothe memory of the polymeric fluid in the drop and both thepresence of overshoot and the decrease in final deformationsuggest the storage of elastic energy within the drop.

Fig. 23 shows the retraction of the polymer drop afterthe removal of the shear flow compared with the Newtoniandrop, with the polymer drop Deborah numberDe = 2.11.The retraction speeds are the same for both the Newtonianand polymer drop in this DPD study. However, Hooper et al.[47] reported that viscoelastic drop retraction also exhibitsa departure from Newtonian behaviour. They also observedthat the Newtonian drop retracts monotonically. In theirstudy of the polymer drop, it shows a markedly faster rate ofinitial retraction, in spite of possessing the highest viscosity.

580 600 620 640 660 680 700

t

0

0.5

1

1.5

2

2.5

Ra

/R0

Polymer

Newtonian

Fig. 23. Drop strain vs. time for Newtonian drop and polymer dropretraction once the shear flow is removed.

The enhanced retraction rate of the viscoelastic drop resultsfrom the release of elastic energy stored in the drop near thetips. This is supported by the evolution of contours of Tr(τ)in their study. In the present study, the external flow couldnot be instantaneously stopped using the DPD method, so itmaybe the reason for the different behaviour observed.

8. Conclusions

Using the dissipative particle dynamics method, we haveinvestigated the evolution of polymer drops in a shearfield under periodic boundary conditions. The appropriatenon-Newtonian behaviour, shear thinning and normal stressdifferences, are all observed with the FENE chain model inthe DPD system. In the present study, both the polymer vis-cosity and its first normal stress difference obey a power-lawrelationship with the shear rate, with power-law indices of0.861 and 1.56, respectively. The Rothman–Keller “colour”scheme is used to model the immiscibility of two fluids byan appropriate choice of a repulsive parametera, and theresultingχ-parameters are also computed. The interfacialtension of the polymer drop is always greater than that ofNewtonian drop keeping all other conditions the same.

Because of the elasticity of the polymer chains, the linearregime for polymeric drops under a steady-state shear flowis considerably larger than that of Newtonian drop. The dy-namical equilibrium between break-up and coalescence re-sults in a steady-state average droplet-size distribution. Thethreads occurring in the break-up process are observed inthe DPD system.

For a polymeric drop under shear flow reversal, the de-formed polymer drop retracts toward a roughly sphericalshape first, and then it deforms and orients in a new direc-tion. This shape relaxation will impart time scale into thesuspension. The tumbling phenomenon of the polymer dropcould be observed whenSc is high enough, by increasingthe cut-off radius. A stress analysis shows that the stress re-sponse of soft matter is different from that of a suspensionof solid spheres. The overshoot of the extension is observedin the extension process under shear flow due to the elastic-ity of the polymer drop.

Acknowledgements

The authors would like to thank Dr. H.S. Dou (NationalUniversity of Singapore) and Dr. G. Pan (Wayne State Uni-versity) for many fruitful discussions which helped to im-prove this work.

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dissipative particle dynamics simulation of polymer drops in a periodic shear flow

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