THE JOURNAL OF CHEMICAL PHYSICS 143, 014101 (2015)

Transport dissipative particle dynamics model for mesoscopicadvection-diffusion-reaction problems

Zhen Li,1 Alireza Yazdani,1 Alexandre Tartakovsky,2 and George Em Karniadakis1,a)1Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA2Computational Mathematics Group, Pacific Northwest National Laboratory, Richland,Washington 99352, USA

(Received 19 February 2015; accepted 18 June 2015; published online 1 July 2015)

We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopicproblems involving advection-diffusion-reaction (ADR) processes, along with a methodology for im-plementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPDis an extension of the classic dissipative particle dynamics (DPD) framework with extra variablesfor describing the evolution of concentration fields. The transport of concentration is modeled by aFickian flux and a random flux between tDPD particles, and the advection is implicitly considered bythe movements of these Lagrangian particles. An analytical formula is proposed to relate the tDPDparameters to the effective diffusion coefficient. To validate the present tDPD model and the boundaryconditions, we perform three tDPD simulations of one-dimensional diffusion with different boundaryconditions, and the results show excellent agreement with the theoretical solutions. We also performedtwo-dimensional simulations of ADR systems and the tDPD simulations agree well with the resultsobtained by the spectral element method. Finally, we present an application of the tDPD model tothe dynamic process of blood coagulation involving 25 reacting species in order to demonstrate thepotential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solutionof this comprehensive 25-species coagulation model is only twice as computationally expensive asthe conventional DPD simulation of the hydrodynamics only, which is a significant advantage overavailable continuum solvers. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4923254]

I. INTRODUCTION

Many biological processes take place at the cellularand subcellular levels, where the continuum deterministicdescription is no longer valid and hence stochastic effectshave to be considered.1 To this end, mesoscopic methodswith stochastic terms are attracting increasing attention asa promising approach for tackling challenging problems inbioengineering and biotechnology.2 As one of the currentlymost popular mesoscopic methods, dissipative particle dy-namics (DPD) drastically simplifies the atomistic dynamicsby using a single coarse-grained (CG) particle to representan entire cluster of molecules,3 and the effects of unsolveddegrees of freedom are approximated by stochastic dynamics.4

Similarly to the molecular dynamics (MD), a DPD systemconsists of many interacting particles and their dynamicsare computed by time integration of Newton’s equationof motion.5 However, in contrast to MD, DPD has softinteraction potentials allowing for larger integration timesteps. With larger spatial and temporal scales, DPD modelingcan be used to investigate hydrodynamics in larger systems,which are beyond the capability of conventional atomisticsimulations.6

DPD was initially proposed by Hoogerbrugge and Koel-man3 for simulating the mesoscopic hydrodynamic behavior

a)Electronic mail: george_karniadakis@brown.edu

of complex fluids. The interactions between DPD particlesoccur pairwise so that the total momentum of the DPDsystem is strictly conserved. Moreover, since these interactionsdepend only on the relative positions and velocities, the result-ing DPD fluids are Galilean invariant.5 By using the Fokker-Planck equation and applying the Mori projection operator,Español7 and Marsh et al.8 reported that the hydrodynamicequations of a DPD system recover the continuity and Navier-Stokes equations. In other words, DPD can be considered asa particle-based Lagrangian representation of the continuityand Navier-Stokes equations at mesoscopic level. Therefore,DPD can provide the correct hydrodynamic behavior of fluidsat the mesoscale, and it has been successfully applied tostudy DNA suspensions,9,10 blood flows and cell interac-tions,11,12 platelet aggregation,13 and nanoparticles across lipidmembranes.14

However, the classic DPD system has only equationsgoverning the evolution of density and velocity fields but noevolution equations for describing the concentration fields,which precludes DPD from modeling problems involvingdiffusion-reaction processes, i.e., two of the most fundamentalprocesses in biological systems.1 Proteins in an aqueoussolution diffuse in a living cell due to Brownian motion,and some collisions of appropriate proteins may lead tochemical reactions. Moreover, most biological processesdepend on the concentrations of specific proteins, ions, orother biochemical factors.15 For example, blood coagulation,a crucial process involved in thrombus formation and growth,

0021-9606/2015/143(1)/014101/14/$30.00 143, 014101-1 © 2015 AIP Publishing LLC

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014101-2 Li et al. J. Chem. Phys. 143, 014101 (2015)

can be modeled by advection-diffusion-reaction (ADR) pro-cesses of 25 biological reactants.15 As another example,the sickle-cell disease is initiated by the deoxygenation-induced polymerization of sickle hemoglobin in the red bloodcells, and the polymerization dynamics relies heavily on theconcentration of oxygen.16 Therefore, it is important to includethe ADR equation in the DPD model when diffusion andreactions are present.

Since the ADR equation has been extensively inves-tigated by continuum solvers, in which smoothed particlehydrodynamics (SPH) is a Lagrangian discretization of thecontinuum equations,17 it is necessary to clarify the differencesof the mesoscopic stochastic method, DPD, from continuumdeterministic methods, i.e., SPH. Although the hydrodynamicbehavior of a DPD fluid can recover the Navier-Stokesequation, the DPD model does not come from continuumequations. Instead, DPD is usually considered as a CG MDmodel, and its formulations can be obtained from applying theMori-Zwanzig projection to MD systems.4 Table I presentsa qualitative comparison between DPD and SPH methods.It shows that the DPD method has advantages on modelingcomplex fluids and materials, which may even not have aknown constitutive equation.

In the present paper, we develop a transport dissipativeparticle dynamics (tDPD) model for mesoscopic problemsinvolving ADR processes. Similarly to the classic DPDmethod, each tDPD particle is associated with extra variablesfor carrying concentrations in addition to other quantitiessuch as position and momentum. Using the same strategyfor discretization of the heat conduction equation,18,19 thetransport of concentration can be modeled by a Fickianflux and a random flux between particles.20,21 An analyticalformula is proposed to relate the tDPD parameters to theeffective diffusion coefficient. Furthermore, we develop amethodology for implementation of the correct Dirichlet andNeumann boundary conditions in tDPD simulations. It isworth noting that the particle-based tDPD method satisfies theconservation of concentration automatically and provides aneconomical way to solve ADR equations with a large number

of species. For instance, in a problem with 23 ADR equationsinvolving 25 chemical species, the additional computationalcost of tDPD is the same as the flow solver, unlike thecontinuum model requiring 23 Poisson/Helmholtz solversmaking the additional computational cost over ten times higherthan a pure Navier-Stokes solver.22

The remainder of this paper is organized as follows.In Section II, we describe the details of tDPD formulationand its parameters, including an analytical formula torelate the mesoscopic concentration friction to the effectivediffusion coefficient (Section II A) and a methodologyfor imposing Dirichlet and Neumann boundary conditions(Section II B). In Section III, we validate the tDPD modelwith one-dimensional diffusion (Section III A) and two-dimensional advection-diffusion (Section III B) and present anapplication of the tDPD model to ADR processes involving25 species in coagulation and fibrinolysis in flowing blood(Section III C). Finally, we conclude with a brief summary inSection IV.

II. GOVERNING EQUATIONS

tDPD is an extension of the classic DPD framework. Withextra variables for describing concentration fields, the timeevolution of a tDPD particle i with unit mass is governed bythe conservation of momentum and concentration, which isdescribed by the following set of equations:

d2ridt2 =

dvi

dt= Fi =

i, j

(FCij + FD

ij + FRij ) + Fext

i , (1)

dCi

dt= Qi =

i, j

(QDij +QR

ij ) +QSi , (2)

where t, ri, vi, and Fi denote time, and position, velocity,force vectors, respectively. Fext

i is the force on particle ifrom an external force field. The pairwise interaction betweentDPD particles i and j consists of the conservative forceFC

ij , dissipative force FDij , and random force FR

ij , which are

TABLE I. Comparison between DPD and SPH methods. The abbreviation “Diff.” denotes differences, “Adv.” foradvantages, and “Disadv.” for disadvantages.

DPD SPH

Major Diff.Particle stochastic approach Continuum deterministic approachCG force field (bottom-up) governing DPD particles Discretization of continuum equations

(i.e., Navier-Stokes)

Inputs Particle interactions Equation of state, viscosity, thermalconductivity, etc.

OutputsEquation of state, diffusivity, viscosity, thermal Close to the inputsconductivity, etc.

Adv.No requirements for constitutive equation, which is Clear physical definition of parameterssuitable for modeling complex systems and materials in continuum equations

Disadv.No clear physical definition of parameters; thus, it Must know the constitutive equation andneeds mapping DPD units to physical units based onoutput properties

macroscopic properties given as inputs

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014101-3 Li et al. J. Chem. Phys. 143, 014101 (2015)

expressed as5

FCij = aijωC(rij)eij, (3)

FDij = −γijωD(rij)(eij · vij)eij, (4)

FRij = σijωR(rij)ξij∆t−1/2eij, (5)

where rij = |rij| with rij = ri − r j, eij = rij/rij is the unit vectorfrom particle j to i, and vij = vi − v j the velocity difference.ωC(r), ωD(r), and ωR(r) are the weight functions of FC,FD, and FR, respectively. Also, aij is the conservative forceparameter, γij the dissipative coefficient, and σij the strengthof random force. Moreover, Ci represents the concentrationof one species defined as the number of a chemical speciescarried by a tDPD particle i and Qi the correspondingconcentration flux. Since tDPD particles have unit mass, thisdefinition of concentration is equivalent to the concentrationin terms of chemical species per unit mass. Then, the volumeconcentration, i.e., chemical species per unit volume, is ρCi

where ρ is the number density of tDPD particles. We notethat Ci can be a vector Ci containing N components, i.e.,{C1,C2, . . . ,CN}i when N chemical species are considered.

Based on Fick’s law,23 the diffusion driving force ofeach species is proportional to the concentration gradient,which corresponds to a concentration difference between twoneighboring tDPD particles. Therefore, in the tDPD model, thetotal concentration flux on particle i accounts for the Fickianflux QD

ij and random flux QRij , which are given by

QDij = −κijωDC(rij) �Ci − Cj

�, (6)

QRij = ϵ ijωRC(rij)∆t−1/2ζij, (7)

where κij and ϵ ij determine the strength of the Fickianand random fluxes. The symbols ξij in Eq. (5) and ζij inEq. (7) represent symmetric Gaussian random variables withzero mean and unit variance.5 A similar DPD transportmodel was proposed in Ref. 20 by drawing an analogy withthe DPD model of heat conduction.18 It is worth notingthat the parameter κ plays the same role of dissipatingthe concentration differences between tDPD particles asthe dissipative coefficient γ does on momentum. To avoidconfusion with γ, we use “Fickian friction coefficient” for κto denote the dissipative effect of concentration. In general,the Fickian friction coefficient κ is an N × N matrix when theinterdiffusivities of N different chemical species are involved.In simple cases, i.e., considering N chemical species indilute solutions and neglecting the interdiffusivities of differentspecies, κ becomes a diagonal matrix.24

The random force FRij and the random flux QR

ij correspondto the irreversible dissipative force FD

ij and dissipative Fickianflux QD

ij , respectively. The fluctuation-dissipation theorem(FDT) is employed to relate the random terms to the dissipativeterms25,26

σ2ij = 2kBTγij, ωD(rij) = ω2

R(rij), (8)

ϵ2ij = m2

sκijρ(Ci + Cj), ωDC(rij) = ω2RC(rij), (9)

where kB is the Boltzmann constant, T the temperature, andms the mass of a single solute molecule. Also, Ci and Cj arethe concentrations on particles i and j, respectively. Simplifiedderivations for obtaining Eq. (9) are presented in Appendix A.

Since ms is small with respect to the mass of a tDPD particlem, which is usually chosen as mass unit, the magnitude of ϵ issmall. Therefore, the contribution of the random flux QR

ij to thetotal diffusion coefficient D is negligible unless ms becomescomparable to m in nanoscale systems. In the present work,we consider ms/m ≪ 1 and neglect the contribution of therandom flux QR

ij to the total diffusion coefficient.The third part in Eq. (2) QS

i represents a source termof concentration due to chemical reactions. Since the sourceterm QS

i relies on specific chemical reaction equations, herewe do not present a general form of QS

i . In the applicationsof the tDPD model in Section III, the reaction equations of 23chemical species are involved and the expressions of QS

i willbe presented. We note that a random rate of chemical reactionscorresponding to the deterministic reaction rate QS

i can be alsoconsidered by applying FDT to particular chemical reactionequations.26 Although we do not consider the random reactionterm in the present work, it can be readily included in a fluidsystem of interest involving specific chemical reactions.

A. Mesoscopic friction of concentration

The macroscopic properties including viscosity anddiffusivity of a tDPD system are output properties rather thaninput parameters. Since the stochastic forces on tDPD particlesyield random movements, the effective diffusion coefficient Dis the result of contributions from the random diffusion Dξ andthe Fickian diffusion DF. In general, the random contributionDξ is a combined result of the random movements of tDPDparticles and random flux QR

ij in Eq. (7). However, the varianceof random flux QR

ij has a small prefactor m2s as given by Eq. (9).

The contribution of the random flux QRij to Dξ is negligible,

which has been confirmed by Kordilla et al.21 Thus, in themathematical derivations in this section, we consider that Dξ

is due to the random movements of tDPD particles.Considering a tDPD system in which the particles are

governed by the interactions given by Eqs. (3)-(5), thediffusion coefficient Dξ induced by the random movementsof tDPD particles can be calculated by5

Dξ =3kBT

4πγρ · rc

0 r2ωD(r)g(r)dr, (10)

where rc is the cutoff radius for forces. Also, the macroscopicdiffusion coefficient DF due to Fickian concentration flux canbe computed by27

DF =2πκρ

3

rcc

0r4ωDC(r)g(r)dr, (11)

where κ is the Fickian friction coefficient and rcc is thecutoff radius for concentration flux. The forces between tDPDparticles include indirect effects of interactions while theexchanges of chemical species need direct contact betweenparticles. Therefore, we suggest to use a cutoff radius rcc lessthan or equal to rc. Specifically, when the radial distributionfunction of ideal gas g(r) = 1.0 is employed, both Dξ and DF

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014101-4 Li et al. J. Chem. Phys. 143, 014101 (2015)

FIG. 1. (a) Concentration profiles obtained using amethod analogous to the reverse Poiseuille flow fordifferent Fickian friction coefficients κ. The lines arequadratic fitting curves for each case. (b) Dependenceof the diffusion coefficient D on κ, in which the linearfitting function is D = 0.0187+0.771κ.

can be evaluated analytically and we have

D = Dξ + DF

=3kBT(s1 + 1)(s1 + 2)(s1 + 3)

8πγρr3c

+16πκρr5

cc

(s2 + 1)(s2 + 2)(s2 + 3)(s2 + 4)(s2 + 5) , (12)

where s1 and s2 are the exponents of the weighting func-tions given as ωD = (1 − r/rc)s1 and ωDC = (1 − r/rcc)s2,respectively. Equation (12) provides a relationship betweenthe macroscopic effective diffusion coefficient D (which canbe experimentally measured) and parameters in the tDPDmodel. Specifically, Eq. (12) defines scaling of Dξ and DF

with temperature and the particle size (i.e., rc and rcc).Equation (12) indicates that the effective diffusion coef-

ficient D is a linear function of the parameter κ, and theminimum value of the effective diffusion coefficient Dmin = Dξ

is obtained at κ = 0. We note that tDPD simulations shouldhave smaller time steps for larger κ to avoid instabilityissues. Considering a tDPD particle i with its concentration Ci

surrounded by other particles with concentration of Ci + ∆C,the concentration flux on particle i can be computed by aintegral of QD

ij in a spherical domain with radius of rcc. Then,the variation of concentration on particle i during one timestep in an explicit time integration must be smaller than ∆C,which results in a necessary condition for limiting the timestep ∆t given by

4πκρ rcc

0r2ωD(r)g(r)dr · ∆t < 1. (13)

By assuming the radial distribution function g(r) = 1corresponding to ideal gases,5,19 Eq. (12) can be used topredict the effective diffusion coefficient of a tDPD systemroughly, but the accurate value of D can only be obtainedby computations in tDPD simulations. For a tDPD fluid withconstant diffusion coefficient, the ADR equation is given by

dCdt= D∇2C +QS, (14)

where D is the diffusion coefficient and QS a source term. Forsteady state problems, Eq. (14) is simplified to D∇2C = −QS,which is the same as the governing equation of Poiseuilleflow driven by a body force υ∇2V = −g. Therefore, we canuse a method analogous to the reverse Poiseuille flow28 forcomputation of the effective diffusion coefficient.

To obtain the accurate value of the effective diffusioncoefficient, we perform tDPD simulations in a computational

domain of 30 × 6 × 30 with periodic boundary conditions.The parameters in the tDPD system are defined as ρ = 4.0,kBT = 1.0, a = 75kBT/ρ, γ = 4.5, σ2 = 2kBTγ, rc = rcc= 1.58, ωC(r) = (1 − r/rc), ωD(r) = ω2

R(r) = (1 − r/rc)0.41,and ωDC(r) = (1 − r/rcc)2 and κ ranges from 0 to 10. Asshown in Fig. 1(a), the fluid system is subdivided into twoequal domains in z-direction, and a concentration source +QS

is applied in the domain of z > 0 while a concentration sinkwith same magnitude −QS is applied in the other domainz < 0. Because of the periodic boundary conditions, theconcentration of the tDPD fluid is constrained to be invariableat the plane z = 0. When the diffusion coefficient D is constant,the steady state solution of the concentration profile is given by

C(z) = QSz2D

(d − |z |) + C0, (15)

where QS = 1.0 × 10−3 is the source term, C0 = 1.0 the initialconcentration of the tDPD system, and d = 15.0 the half lengthof the computational domain in z-direction. The concentrationprofiles shown in Fig. 1(a) were obtained by subdividing thecomputational domain into 51 bins along the z-direction. ThetDPD simulations were run for 500 time units and the symbolsin Fig. 1(a) are the concentration profiles of the last time stepin the simulations. Then, the effective diffusion coefficientcan be determined by fitting the concentration profile withthe analytical solution given by Eq. (15). The solid lines inFig. 1(a) are the best-fitting parabolas. It is obvious that theeffective diffusion coefficient D can be significantly changedby varying the Fickian friction coefficient κ.

Figure 1(b) shows the dependence of the effectivediffusion coefficient D on the Fickian friction coefficient κ. Asindicated by Eq. (12), D should be a linear function of κ, whichis verified by the computed results shown in Fig. 1(b). Insertingthe tDPD parameters in the approximate theoretical prediction,Eq. (12) gives D = 0.0195 + 0.786κ, while the linear fitting ofcomputed data shown in Fig. 1(b) gives D = 0.0187 + 0.771κ.Hence, the analytical formula of Eq. (12) predicts roughly thediffusion coefficients with approximately a 5% deviation fromthe computed results. In practice, the accurate value of Dshould be computed by performing tDPD simulations withthe method analogous to the reverse Poiseuille flow as shownin Fig. 1(a).

B. Boundary conditions

Boundary conditions are crucial for the investigationof diffusion-reaction processes in wall-bounded systems.

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014101-5 Li et al. J. Chem. Phys. 143, 014101 (2015)

FIG. 2. (a) Radial distribution function g(r) of the tDPDparticles and (b) distance dependent functions ϕ(h) inEq. (17) and normalized Φ(h) in Eq. (18). The lines arebased on a parameter set ρ = 4.0, kBT = 1.0, a = 18.75,rc = rcc = 1.58, ωC(r )= (1−r/rc), and ωDC(r )= (1−r/rcc)2.

Usually, defining stationary particles to represent solid objectsis a common treatment in classic DPD simulations.29 However,those solid walls made up by discrete frozen particles induceunwanted temperature and density fluctuations in the vicinityof the walls.30,31 Alternatively, Lei et al.32 reported that theeffects from a solid wall can be evaluated by integratingthe interactions between fluid and solid particles, and hence,the presence of solid wall boundaries was replaced byeffective boundary forces. Subsequently, Li et al.19 extendedthis method to non-isothermal systems for applying thermalboundary conditions and they found negligible temperatureand density fluctuations near the wall boundaries. In thepresent work, we adopt the idea of effective boundary fluxesto impose Dirichlet and Neumann boundary conditions forconcentration in the tDPD systems.

1. Dirichlet boundary condition

Since the fluid particles do not penetrate into wallboundaries, the random movements of fluid particles do nothave any contributions to the boundary concentration flux.Therefore, the effective boundary concentration flux is inducedonly by the Fickian flux. For a fluid particle i in the vicinityof a wall boundary, the effective Fickian concentration flux onparticle i from the wall can be calculated19 by

QWD, i(h) = 2πρ

rcc

z=h

√rcc2−z2

x=0QD(r)g(r)xdxdz

= 2πρ rcc

z=h

√rcc2−z2

x=0−κ · ωDC(r)

· zh(Ci − Cw) · g(r)xdxdz

= 2πρκ (Cw − Ci) · ϕ(h), (16)

where Ci is the concentration of particle i, Cw the expectedconcentration at the wall surface, and h the distance of theparticle i away from the wall surface. Here, ϕ(h) is a functionof h, which is defined as

ϕ(h) = rcc

z=h

√rcc2−z2

x=0ωDC(r)g(r) zx

h· dx · dz. (17)

Equation (16) reveals that the boundary concentration flux isdetermined by the concentration difference between particle iand the wall, and also their distance.

For a given tDPD system with parameters ρ = 4.0, kBT= 1.0, a = 18.75, γ = 4.5, σ = 3.0, rc = 1.58, ωC(r) = (1

− r/rc), and ωD(r) = ω2R(r) = (1 − r/rc)0.41, the radial distri-

bution function g(r) of the tDPD particles is displayed inFig. 2(a). Then, the function ϕ(h) can be evaluated throughEq. (17) with ωDC(r) = (1 − r/rcc)2 and rcc = rc. We notethat though rcc is usually set to be rc for convenience, rcccan be different from rc. The dependence of ϕ(h) on thedistance h is presented in Fig. 2(b), which is approximated byϕ(h) = 4.9562 × 10−2/h · [1+ 4.1517h/rcc + 7.1697(h/rcc)2](1 − h/rcc)4.09 for h < rcc. As the distance h approaches tozero, the magnitude of ϕ(h) goes to infinity. Here, we use atruncation of ϕ(h) at small distances and set ϕ(h < 0.01rcc)= ϕ(0.01rcc).

2. Neumann boundary condition

To consider the effective flux along the normal direction ofwall surface, we integrate the effect of concentration flux fromthe wall boundary and define a distance dependent functiongiven by

Φ(h) = rcc

z=h

√rcc2−z2

x=0ωDC(r)g(r) zx

rdxdz. (18)

The normalizedΦ(h) is defined asΦ(h) = Φ(h)/ rcc0 Φ(x)dx.

Then, the integral ofΦ(h) is equal to one. Using the computedg(r) shown in Fig. 2(a) and ωDC(r) = (1 − r/rcc)2, the func-tion Φ(h) can be obtained through Eq. (18), as shown inFig. 2(b). The computed Φ(h) is approximated by Φ(h)=

6i=0 pi(1 − h/rcc)7−i with the coefficients p = {22.745 45,

−62.6256,55.1365,−19.2584,6.2662,−0.502 42,0.0101} forh ≤ rcc and Φ(h) = 0 for h > rcc.

To impose a Neumann boundary condition dC/dn = Λat a wall boundary, it is equivalent to apply a concentrationflux QW = DΛ across the boundary. In practice, the flux QW

is distributed onto the fluid particles in the vicinity of thewall weighted by Φ(h). Let ρ be the number density of thefluid particles, the volume concentration is ρC because theconcentration C in tDPD is defined as the number of a chemicalspecies per particle. Then, any fluid particle i close to the wallobtains a concentration flux from the wall boundary given by

QWi (h) = DΛρ · Φ(h). (19)

In addition to effective boundary concentration flux, theinteractions between wall and fluid particles are also integratedand replaced by effective boundary forces. In particular, byusing the boundary method proposed in Refs. 19 and 32, the

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014101-6 Li et al. J. Chem. Phys. 143, 014101 (2015)

effective conservative and dissipative forces are given by

FC(h) = 2πaρen rc

z=h

√rc2−z2

x=0ωC(r)g(r) zx

rdxdz

= fC(h)en, (20)

where the unit vector en represents the normal direction ofwall surface. With the computed g(r) as shown in Fig. 2(a),we have fC(h) = 6

i=0 pi(1 − h/rc)7−i with the coefficients p= {525.6666,−1457.0412,1485.7229,−927.0881,453.3782,−7.2616,0.177 58},

FD(h) = −πργ rc

z=h

√rc2−z2

x=0ωD(r)g(r)

·�vτeτx3 + 2vnenxz2� z

hr2 dxdz

= −γτ(h) · vτeτ − γn(h) · vnen, (21)

where vnen = (∆v · en)en is the projected component ofvelocity difference ∆v = vi − vwall onto the normal directionof wall surface, and vτeτ = ∆v − vnen is the tangentialcomponent of velocity difference ∆v. Here, the symbolsτ and n represent the tangential and normal directions ofwall surface, respectively. Using the computed g(r) andωD(r) = (1 − r/rc)0.41, both γτ(h) and γn(h) can be computedthrough Eq. (21). In practice, the computed γτ(h) and γn(h)are approximated by γτ(h) = 10.3101/h · [1 + 3.6212h/rc+ 2.2090(h/rc)2](1 − h/rc)3.45 and γn(h) = 20.7789/h ·[1 + 2.1245h/rc + 5.4566(h/rc)2](1 − h/rc)2.30 for h < rc,respectively.

In addition, the effective boundary forces include also thefluctuating part to satisfy the fluctuation-dissipation theorem,which is given by

FR(h) = στ(h)ξτeτ + σn(h)ξnen, (22)

where ξτ and ξn are Gaussian random variables with zero meanand unit variance. The variances of random forces are relatedto the dissipative forces by the fluctuation-dissipation theoremin the form of σ2

τ(h) = 2kBTγτ(h) and σ2n(h) = 2kBTγn(h).

When the distance h approaches to zero, both γτ(h) and γn(h)become infinity. In practice, we set a maximum magnitude forγτ(h) and γn(h) given as γτ(0.05rc) and γn(0.1rc) to avoidinstability of tDPD simulations induced by some extremelylarge random numbers.

The boundary method based on the effective boundaryforces has been well validated by Lei et al.32 and Li et al.19

They have shown that the density and temperature fluctuationsin the vicinity of wall surface are successfully eliminated byusing those effective boundary forces. In the present paper, wewill not recheck the density and temperature fields and willfocus on the concentration fields instead.

III. RESULTS

A. Validation for one-dimensional diffusion

The first test case is solving the diffusion equation withthe initial condition given by the Heaviside step function.Considering the diffusion in a system containing two compo-nents in dilute solution, by neglecting the interdiffusivities

of different species, the system is reduced to two uncoupleddiffusion equations with diffusivities independent betweenspecies. Then, the diffusion equation takes the relatively sim-ple form of the linear second-order partial differential equation

∂

∂t

C1

C2

=

D11 00 D22

∇2

C1

C2

, (23)

where C1 and C2 are the concentration of species 1 and 2,respectively. D11 and D22 are the corresponding diffusioncoefficients. In general, the diffusion coefficient can be afunction of concentration, time, and direction to considerheterogeneous diffusions. However, here we consider onlythe simplest case in which D11 and D22 are constant.

As shown in Fig. 3(a), it is a one-dimensional diffusionalong x-direction. In practice, a tDPD system is definedin a computational domain of 100 × 10 × 10 with periodicboundary conditions in y- and z-directions and solid wallsin x-direction. The concentration of the tDPD system isinitialized as C1 = 1.0 and C2 = 0.0 in the domain of x < 0 andC1 = 0.0 and C2 = 1.0 in the domain of x ≥ 0. The solid wallsprovide no-slip boundary condition for velocity and no-fluxboundary condition for concentrations. At wall boundaries,we apply the effective boundary forces given by Eqs. (20)-(22) and no concentration flux Λ = 0 shown in Eq. (19).The parameters of the tDPD system are given as ρ = 4.0,kBT = 1.0, a = 75kBT/ρ, γ = 4.5, σ2 = 2kBTγ, rc = rcc= 1.58, ωC(r) = (1 − r/rc), ωD(r) = ω2

R(r) = (1 − r/rc)0.41,and ωDC(r) = (1 − r/rcc)2. Unless otherwise specified, allthe simulations in the present paper are based on this setof parameters. Also, κ11 = 0.04 results in D11 = 0.05 andκ22 = 0.62 yields D22 = 0.50. The velocity-Verlet algorithm5

is utilized for the numerical integration of the tDPD equationsof Eqs. (1) and (2). The time step ∆t = 0.01 is used forSection III A and ∆t = 0.001 for Sections III B and III C,which satisfy the necessary condition given by Eq. (13) forlimiting the time step ∆t.

FIG. 3. (a) Initial condition and boundary conditions for the one-dimensionaldiffusion involving two chemical speciesC1 andC2. (b) Time evolution of theconcentration profile and comparison with theoretical solution at t = 10, 50,and 150. The diffusion coefficients are D11= 0.05 and D22= 0.50.

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For one-dimensional diffusion with initial condition ofstep function, the time-evolution of concentration profilesbefore concentrations diffuse to the walls on either side can bedescribed by

Ck(x, t) = 12

er f( (−1)kx√

4Dkkt

)+

12, (24)

where k = 1,2 represent the two different species, ander f (x) = 2/

√π ·

x

0 e−τ2dτ is the error function.

The concentration profiles shown in Fig. 3(b) wereobtained by subdividing the computational domain into 100bins along the x-direction. The local concentration is obtainedby averaging all the particle information in each bin based on asingle time step. The lines in Fig. 3(b) represent the theoreticalsolutions given by Eq. (24). The transient concentrationprofiles in the tDPD simulation are in excellent agreementwith the theoretical solution, which validate the present tDPDmodel for solving the diffusion equation.

The second test case is used to validate the boundarymethod for implementation of the Dirichlet boundary condi-tion. Figure 4(a) illustrates the initial condition C(x,0) = 0and boundary conditions. Similarly to the first test case, thetDPD simulation is performed in a computational domain of100 × 10 × 10 with periodic boundary conditions in y- and z-directions and no-slip solid walls in x-direction. By solving aone-dimensional diffusion equation with boundary conditionsof C(0, t) = 0 and C(100, t) = C0, an analytical solution for thetransient concentration profile can be obtained,

C(x, t) = C0xLx+

∞n=1

2C0

nπ(−1)n sin (βnx) exp

�−D β2

nt�, (25)

where βn = nπ/Lx with Lx being the length of the computa-tional domain in the x-direction, D the diffusion coefficient.The Fickian friction coefficient is κ = 5.0 and the corre-sponding diffusion coefficient is D = 3.87. The computationaldomain is divided into 100 bins along the x-direction forobtaining local concentrations. Figure 4(b) shows a compar-ison between the concentration profiles obtained using tDPDand theoretical solution Eq. (25) at several times includingthe steady state solution. The results are in good agreement,

FIG. 4. (a) Initial condition and boundary conditions for the one-dimensionaldiffusion. (b) Time evolution of the concentration profile and comparison withtheoretical solution at t = 5, 20, 100, and 300 and at steady state.

FIG. 5. (a) Initial condition and boundary conditions for the one-dimensionaldiffusion. (b) Time evolution of the concentration profile and comparison withtheoretical solution at t = 20,150,500, and 1000 and at steady state.

which validate the boundary method for imposing the correctDirichlet boundary condition in the tDPD simulation.

The last one-dimensional test case is performed to validatethe boundary method for implementation of the Neumannboundary condition. As shown in Fig. 5(a), it has a similarsetup as the second test case but different wall boundaryconditions. Considering a concentration flux at the left wallx = 0, we apply the Neumann boundary condition dC/dn= Λ = 0.01 at x = 0. Also, the right wall at x = 100 has a fixedconcentration C(100, t) = 0. By solving a one-dimensionaldiffusion equation with boundary conditions of dC(0, t)/dx= Λ and C(Lx, t) = 0, we have the theoretical solution for thetransient concentration profile given by

C(x, t) = Λ(x − Lx) + 4ΛLx

∞n=1,odd

β−2n sin2(nπ

4)

× cos (βnx) exp�−D β2

nt�, (26)

where βn = nπ/(2Lx) with Lx being the length of thecomputational domain in the x-direction, D the diffusioncoefficient. tDPD is used to simulate the time evolution of theconcentration profile for D = 3.87. The Neumann boundarycondition is implemented based on Eq. (19). In Fig. 5(b), wecompare the transient concentration profiles obtained usingtDPD with the theoretical solution Eq. (26). An excellentagreement between the tDPD results and the theoreticalsolution confirms the validity of the boundary method forimposing the correct Neumann boundary condition.

B. Validation for two-dimensional advection-diffusion

The one-dimensional diffusions validated the tDPD modeland boundary methods for implementation of the correctDirichlet and Neumann boundary conditions. However, thesecases are only one-dimensional and do not consider the floweffect, i.e., advection. A more challenging test for the presenttDPD model is solving two-dimensional advection-diffusionequations.

Figure 6 illustrates the geometry and boundary conditionsof the problem. A fully developed plane Poiseuille flow isconsidered between two stationary walls. Both walls have

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014101-8 Li et al. J. Chem. Phys. 143, 014101 (2015)

FIG. 6. Computational domain and boundary conditions for the two-dimensional advection-diffusion.

no-slip boundary conditions for velocity. For the concentrationfield, the upper wall has fixed concentration C = 1 while onthe lower wall we prescribe the boundary condition dC/dn = 0for concentration.

Three tDPD simulations with different Peclet numbersPe = 10,100,1000 are performed in a computational domainof 50 × 10 × 10. The tDPD parameters are the same as thoseused in one-dimensional tests. The kinematic viscosity of thetDPD fluid is 6.62. The wall boundary conditions are imposedbased on the boundary method described in Section II B. Togenerate a fully developed Poiseuille flow, we use periodicboundary condition in x-direction and apply a body force ofgx = 3.50 to drive the fluid, which yields a Reynolds number ofRe = UmaxH/ν = 10, where Umax is the centerline velocity andH is the height of the channel. However, for the concentration,we apply non-periodic inflow/outflow boundary conditions inx-direction based on the methodology proposed by Lei et al.32

In practice, to impose a correct inflow boundary condition, weuse a buffer region of rc × 10 × 10 and prescribe C = 0 in thebuffer region. The outflow boundary condition is implementedwith a maximum buffer length of 2rc in which dC/dx = 0 isapplied. We note that outflow buffer regions with length of

2rc and rc yield same results when the Peclet number (Pe= UmaxH/D) is high, e.g., Pe = 100 and 1000, but a shorterbuffer region at Pe = 10 results in distortions of the concentra-tion field in the vicinity of the outlet due to stronger diffusion.

The concentration fields obtained by the tDPD simula-tions at three Peclet numbers are displayed in Figs. 7(a1),7(b1), and 7(c1). The Reynolds number of the channel flowis fixed at Re = 10, and various Peclet numbers are obtainedthrough the relation Pe = Re · Sc by varying Schmidt numbersgiven by Sc = ν/D, in which the diffusion coefficient D isa linear function of Fickian friction coefficient κ shown inEq. (12).

Advection-diffusion equations are also solved in the samesystem using spectral element method (SEM),33 where high-order hierarchical Jacobi polynomials of order 5 have beenused for the solution of velocity, pressure, and concentrationfields. Comparisons between the results of tDPD and SEMare presented in Figs. 7(a2), 7(b2), and 7(c2). Concentrationprofiles shown in Figs. 7(a2), 7(b2), and 7(c2) are extractedfrom the concentration fields along x-direction at threedifferent heights. The results of tDPD simulations indicatevery good agreement with SEM results for various Pecletnumbers, especially for smaller Peclet numbers. However, athigher Pe = 1000, a very thin boundary layer is formed on theupper wall shown by the concentration field of Fig. 7(c1)resulting in small differences between tDPD and SEM asshown in Fig. 7(c2). The small discrepancy is due to a lownumber density of particles ρ = 4 in our tDPD simulationsthat cannot provide enough spatial resolution in the thin layer.

C. Application to biochemical systems

In this section, we present an application of tDPDto a biological system involving many reactant species todemonstrate the capability of the tDPD model. Here, we look

FIG. 7. Contours of concentration field in two-dimensional flows obtained by tDPD simulations with different Peclet numbers of (a1) Pe= 10, (b1) Pe= 100,and (c1) Pe= 1000. The lines in (a2), (b2), and (c2) are the concentration profiles extracted from (a1), (b1), and (c1) along the channel at three heights, whichagree well with the results (symbols) of spectral element method (SEM). h represents the distance away from the upper wall surface.

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014101-9 Li et al. J. Chem. Phys. 143, 014101 (2015)

TABLE II. Constituents involved in the coagulation cascade and fibrinolysisand their initial concentrations C0

i and Schmidt numbers.

Index Species Name C0i Sc

1 IXa Enzym IXa 0.09 15 9492 IX Zymogen IX 90 17 7623 VIIIa Enzym VIIIa 0.0007 25 5104 VIII Zymogen VIII 0.7 32 0515 Va Enzym Va 0.02 26 1786 V Zymogen V 20 17 7627 Xa Enzym Xa 0.17 13 5688 X Zymogen X 170 17 7629 IIa Thrombin 1.4 15 45610 II Prothrombin 1 400 19 19411 Ia Fibrin 7.0 40 48612 I Fibrinogen 7 000 32 25813 XIa Enzym XIa 0.03 20 00014 XI Zymogen XI 30 25 18915 ATIII AntiThrombin-III 2 410 17 95316 TFPI Tissue factor pathway

inhibitor2.5 15 873

17 APC Active protein C 0.06 18 18218 PC Protein C 60 18 38219 α1 AT α1-AntiTrypsin 45 000 17 18220 tPA Tissue plasminogen

activator0.08 18 939

21 PLA Plasmin 2.18 20 28422 PLS Plasminogen 2 180 20 79023 α2 AP α2-AntiPlasmin 105 19 04824 Z Tenase 1.125×10−4 . . .25 W Prothrombinase 0.034 . . .

at the coagulation cascade, which is one of the major processesin thrombosis. The main product of the cascade is thrombin,which is mainly responsible in transforming fibrinogen to highconcentrations of fibrin at the site of injury forming a fibrinnetwork to increase the structural stiffness of thrombus. Itis also known that thrombin enhances platelet activations byapproximately 20%.34 It should be noted that there are severalmathematical models describing the coagulation cascadeleading to a system of partial differential equations (PDEs) andordinary differential equations (ODEs). Among those models,Anand et al.35 proposed a set of 23 coupled ADR equations fordescribing the evolution of 25 biological reactants involvedin a combined model of intrinsic and extrinsic pathways ofblood coagulation process and fibrinolysis, which are listedin Table II. We will be focusing on this model since it takesboth blood flow and spatial heterogeneity into consideration.We would like to emphasize that, for simplicity, we arelooking at blood coagulation cascade in simple Newtonianfluid. Hence, we are not considering the physiological bloodflow in small arteries, where non-Newtonian constitutive lawfor the DPD fluid is needed. Further, a detailed analysis of themodel accuracy and effectiveness in capturing the coagulationdynamics is beyond the scope of this study.

The advection-diffusion-reaction equations are in theform of

dCi

dt= ∇(Di∇Ci) +QS

i i = 1, . . . ,23, (27)

where Ci denotes the concentration of ith reactant and Di thecorresponding diffusion coefficient. QS

i is the nonlinear sourceterms representing the production or depletion of Ci due to theenzymatic cascade of reactions. To simplify the formulas inthis section, we number the 25 species as shown in Table IIand use symbol [i] to denote the concentration of ith reactant.

Equation (27) includes 23 ADR equations involving 25chemical species. Different species affect each other throughchemical reactions. The expressions of the 23 source terms areshown in Table III. The involved reaction kinetics parametersare given in the caption of Table III. The initial conditions ofthe fibrinogen(I), zymogens (II, V, VIII, IX, X, XI), inhibitors(ATIII, TFPI), and other factors (PC, α1AT, tPA, PLS,α2AP) are determined based on normal human physiologicalvalues.15,35,36 We also consider 0.1% initial concentration forthe enzymes (IIa, Va, VIIIa, IXa, Xa, XIa), active protein C(APC), and Plasmin (PLA) as shown in Table II in order toinitiate the coagulation process. The concentrations of twoother chemical species tenase (Z) and prothrombinase (W)are computed through the relations [24] = [3][1]/0.56 and[25] = [5][7]/0.1, respectively.35

TABLE III. Reaction equations for source terms. The parameters aregiven as k9= 2.54×10−2, K9M = 160, h9= 3.74×10−5, k8= 0.449, K8M= 1.12×105, h8= 5.13×10−4, hC8= 2.36×10−2, HC8M = 14.6, k5= 6.24×10−2, K5M = 140.5, h5= 3.93×10−4, hC5= 2.36×10−2, HC5M = 14.6,k10= 5.523, K10M = 160, h10= 8.01×10−4, hTFPI = 1.11×10−3, k2= 3.105,K2M = 1060, h2= 1.65×10−3, k1= 8.177, K1M = 3160, h1= 3.456, H1M= 2.50×105, k11= 1.80×10−5, K11M = 50, hA3

11 = 3.70×10−6, hL111 = 3.00

×10−8, kPC= 9.01×10−2, KPCM = 3190, hPC= 1.52×10−9, kPLA= 2.77×10−2, KPLAM = 18, and hPLA= 2.22×10−4. Mapping of physical units totDPD units is described in Appendix B.

QS1 = (k9[13][2])/(K9M + [2])−h9[1][15]

QS2 =−(k9[13][2])/(K9M + [2])

QS3 = (k8[9][4])/(K8M + [4])−h8[3]− (hC8[17][3])/(HC8M + [3])

QS4 = (k8[9][4])/(K8M + [4])

QS5 = (k5[9][6])/(K5M + [6])−h5[5]− (hC5[17][5])/(HC5M + [5])

QS6 =−(k5[9][6])/(K5M + [6])

QS7 = (k10[24][8])/(K10M + [8])−h10[7][15]−hTFPI[16][7]

QS8 =−(k10[24][8])/(K10M + [8])

QS9 = (k2[25][10])/(K2M + [10])−h2[9][15]

QS10=−(k2[25][10])/(K2M + [10])

QS11= (k1[9][12])/(K1M + [12])− (h1[21][11])/(H1M + [11])

QS12=−(k1[9][12])/(K1M + [12])

QS13= (k11[9][14])/(K11M + [14])−hA3

11 [13][15]−hL111 [13][19]

QS14=−(k11[9][14])/(K11M + [14])

QS15=−(h9[1]+h10[7]+h2[9]+hA3

11 [13])[15]QS

16=−hTFPI[16][7]QS

17= (kPC[9][18])/(KPCM+ [18])−hPC[17][19]QS

18=−(kPC[9][18])/(KPCM+ [18])QS

19=−hPC[17][19]−hL111 [13][19]

QS20= 0

QS21= (kPLA[20][22])/(KPLAM+ [22])−hPLA[21][23]

QS22=−(kPLA[20][22])/(KPLAM+ [22])

QS23=−hPLA[21][23]

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014101-10 Li et al. J. Chem. Phys. 143, 014101 (2015)

We consider a flow with maximum velocity of u = 1 mm/sthrough a micro-channel with width of 20 µm in roomtemperature. The viscosity of the fluid is 1.0 × 10−6 m2/s.Taking the diffusivity of the thrombin as 6.47 × 10−11 m2/s,the corresponding Peclet and Schmidt numbers are Pe = 309and Sc = 1.546 × 104. The corresponding Peclet number ofvarious biological species can be computed in a similarway. Further, noting that Pe = Re · Sc, the value of Reynoldsnumber Re ≈ 0.02 so the flow is approximately in the Stokesregime. The above values are based on the physiologicallyrelevant values of blood velocity and viscosity in arteriolesand venules.

As mentioned in Section II A, the effective diffusioncoefficient D of a tDPD system is a combined result ofthe random diffusion Dξ and the Fickian diffusion DF.Although the magnitude of D can be linearly adjusted byvarying the Fickian friction coefficient κ, D has a minimumof Dmin = Dξ at κ = 0. Therefore, the maximum Schmidtnumber that a tDPD system can achieve is determined bySc(κ = 0) = ν/Dmin. Table II lists the Schmidt numbers fordifferent chemical species, which are in the order of 104. Suchhigh values of Schmidt numbers in tDPD are achieved throughincreasing the momentum cutoff radius rc. We construct atDPD system in a computational domain of 100 × 10 × 20 withthe Schmidt numbers listed in Table II. The parameters of thetDPD interactions are given as ρ = 4.0, kBT = 1.0, a = 18.75,γ = 4.5, σ = 3.0, rc = 2.9, rcc = 1.0, ωC(r) = (1 − r/rc),ωD(r) = ω2

R(r) = (1 − r/rc)0.4, and ωDC(r) = (1 − r/rcc)2.Note that here we set rcc < rc to reduce the computational costof solving advection-diffusion for 25 species. Computationsfor kinematic viscosity and coefficient of self diffusion basedon this system will result in ν = 138.6 and D = 0.002 61.The maximum Schmidt number is Sc = ν/D = 5.31 × 104,which ensures that this tDPD system can cover all theSchmidt numbers in Table II. The diffusion coefficients forall species can be evaluated by using the method analogousto the reverse Poiseuille flow as described in Section II A,which gives Di = 0.002 61 + 0.0645κi. Since the Schmidtnumbers of different species Sci are listed in Table II, thediffusion coefficient Di can be obtained by Di = ν/Sci withthe viscosity ν = 138.6. The length and time scales are chosenas L∗ = 1 µm and t∗ = 1.386 × 10−4 s. The typical time scale

TABLE IV. Flux boundary conditions at the injured vessel wall re-gion. The parameters are given as k7,9= 7.48×10−2, K7,9M = 24,k7,10= 0.238, K7,10M = 240, φ11= 7.85×10−5, Φ11M = 2000, kCtPA= 1.51×10−3, k IIa

tPA= 2.14×10−2e−972.58(t−t0), k IatPA= 1.17×10−8, [TF-VIIa]W

= 0.25, [ENDO]W = 2.0×10−3, [XIIa]W = 375, and L = 10. Mapping ofphysical units to tDPD units is described in Appendix B.

Index j Boundary flux terms Q j

1 (k7,9[2]W [TF-VIIa]W )/(K7,9M + [2]W )L2 −(k7,9[2]W [TF-VIIa]W )/(K7,9M + [2]W )L7 (k7,10[8]W [TF-VIIa]W )/(K7,10M + [8]W )L8 −(k7,10[8]W [TF-VIIa]W )/(K7,10M + [8]W )L13 (φ11[14]W [XIIa]W )/(Φ11M + [14]W )L14 −(φ11[14]W [XIIa]W )/(Φ11M + [14]W )L20 (kCtPA+k

IIatPA[9]W +k Ia

tPA[11]W )[ENDO]WL

FIG. 8. Computational domain and initial and boundary conditions for thetDPD simulation of the coagulation and fibrinolysis dynamics in flowingblood. Flux boundary conditions are imposed at the injured wall region for7 chemical species given in Table IV and non-flux boundary conditions forelse species.

of blood coagulation dynamics is in the order of minutes,hence making the long-time tDPD computation very expensiveand almost impractical. Therefore, in the tDPD modeling,we keep the same Peclet number to capture the correct flowphysics, while we accelerate the reaction processes by 1000times, which compresses the dynamic process of 1 min into60 ms. This adjustment, however, may reduce the size ofthe zone in which the reactions occur, and hence the resultsare qualitative. Although the Peclet number is the relevantnondimensional parameter, we use the physiologic Schmidtand lower Reynolds numbers in the simulations so that thereactions occur at the site of injury before chemical species getadvected downstream. Also, the concentrations are scaled by1.0 nM · (L∗)3/ρ with ρ the number density of tDPD particles,and hence a concentration of C∗ = 1.0 nM is scaled into theconcentration unit in our tDPD system. We note that all theparameters without physical units in Tables II–IV are definedin DPD units. The details of mapping physical quantities totDPD parameters are described in Appendix B.

Before we perform the simulation of blood coagulation,we estimate the extra computational cost for solving 23

FIG. 9. Time evolution of the fibrin (Ia) concentration field during the dy-namic process of blood coagulation in flowing blood. The channel lengthLx = 100 µm and height Lz = 20 µm. The accelerated time of each snapshotis (a) 27.72 s, (b) 55.44 s, (c) 83.16 s, (d) 110.88 s, (e) 138.60 s, (f) 166.32 s,(g) 221.76 s, and (h) 332.64 s.

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014101-11 Li et al. J. Chem. Phys. 143, 014101 (2015)

FIG. 10. Time evolution of the concentrations of se-lected chemical species at the center of the injured wallregion. (a) Thrombin, (b) fibrin, (c) anti-thrombin III, and(d) active protein C.

ADR equations by performing a tDPD simulation and aconventional DPD simulation. A small fluid system with sizeof 30.0 × 33.0 × 10.0 including 39 600 particles is employedto test the computational cost on a single core of Intel XeonE5-2637 CPU running at 3.50 GHz. We find that the tDPDsimulation involving 23 ADR equations (25 chemical species)needs 0.398 s per time step, while the conventional DPDwithout any chemical species needs 0.192 s per time step. Weconclude that the total computational cost of tDPD for solving23 ADR equations is only doubled compared to the flow solverusing the conventional DPD. However, in continuum models,a Poisson solver occupies approximately 60% computationtime,22 which makes the additional cost of 23 Poisson solversover ten times higher than the pure Navier-Stokes solver.

Figure 8 illustrates the initial and boundary conditionsfor the tDPD simulation of the coagulation and fibrinolysisdynamics in blood flows. The velocity field is periodic in xand z directions with a fully developed Poiseuille flow insidethe channel. Concentration of every species at the inlet is setto its initial concentration C0

i and is imposed in a buffer regionof length rc. Further, zero Neumann boundary condition isimposed in a buffer region in the similar way described inSec. II B. We assume that coagulation occurs at the site ofinjury (0.1 < x/Lx < 0.2), and the process is initiated andsustained by imposing a boundary flux for 7 species listedin Table IV. Note that the tissue factor (TF-VIIa complex) isone of the important factors triggering the extrinsic pathway ofcoagulation, whereas other species are involved in the intrinsic(contact) pathway, except for tPA which starts the fibrinolysis(degradation of fibrin) process.

Representative results are shown in Figs. 9 and 10 notingthat concentrations are evolving in a 1000 times acceleratedtime sequence. Figure 9 shows the time evolution contoursof fibrin concentration, which can represent the growth of

thrombus at the site of injury and its diffusion downstream.Here, we assume a threshold value of 3 µM for fibrin torepresent the actual geometry of the clot. More detailedresults can be found in Fig. 10 where we plot time-variationof concentrations for thrombin, fibrin, anti-thrombin, andAPC at the center of the vessel wall injury. These resultsare in qualitative agreement with the previous numericalstudies.35,36 The most prominent observation in coagulationdynamics of blood is the significant increase of thrombinconcentration in the vicinity of the wall during the initiationphase also known as thrombin burst followed by a drop.36 Thisburst is also accompanied by a significant increase in fibrinand APC concentrations and a major drop in anti-thrombinconcentration. Fibrin concentration shows a slight decreasefollowing the initial peak until it levels off to a steady value,which in turn explains the unchanged concentration contoursafter Fig. 9(f). All the observations in tDPD simulation withtime acceleration are consistent with the results of Bodnár andSequeira.36

It is worth noting that the treatment of “time acceleration”will result in quantitative error. However, the main purpose ofthis case is to demonstrate how to apply the tDPD model tocomplex diffusion-reaction systems. With “time acceleration”,the designed objective of Section III C has been achieved. Theerror induced by the “time acceleration” can be eliminated byrealtime simulation using large-scale parallel computations,which are beyond the scope of the present work.

IV. SUMMARY AND DISCUSSION

A tDPD model for simulating mesoscopic fluid sys-tems involving ADR processes has been developed. EachtDPD particle is associated with extra variables for carrying

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014101-12 Li et al. J. Chem. Phys. 143, 014101 (2015)

concentrations to describe the evolution of concentrationfields. The exchange of concentration between tDPD particlesis modeled by a Fickian flux and a random flux, andan analytical formula is provided to relate the mesoscopicparameters to the effective diffusion coefficient D, which istunable by linearly varying the Fickian friction coefficient κ.

A methodology for imposing the correct Dirichlet andNeumann boundary conditions has been developed. In partic-ular, the effects of a real solid wall are replaced by effectiveboundary forces and concentration fluxes, which are extractedfrom the fluid-solid particle interactions using the continuumapproximation. Simulations of one-dimensional diffusion withdifferent boundary conditions were carried out to validate thepresent tDPD model and the boundary methods for imposingthe Dirichlet and Neumann boundary conditions. Furthermore,the tDPD model was used to solve two-dimensional advection-diffusion equations with various Peclet numbers, where tDPDresults showed very good agreement with the results obtainedby the spectral element method.

Since the particle-based tDPD method satisfies theconservation of concentration automatically, it provides aneconomical way to solve ADR equations with a large numberof species. An application of the tDPD model to the dynamicprocess of blood coagulation was performed to demonstratethe promising biological applications of tDPD. Results re-vealed that the tDPD simulation provided qualitatively correctevolution of fibrin concentration initialized by a injuredvessel wall in flowing blood. Moreover, the tDPD simulationconfirmed that the total computational cost for solving 23ADR equations is only doubled compared to the flow solver,unlike the continuum model requiring 23 Poisson/Helmholtzsolvers making the computational cost over ten times higherthan the Navier-Stokes solver.

Although we demonstrated the implementation of tDPDsimulation with uncoupled linear diffusion equations in whichthe diffusion coefficients D are constant and the Fickianfriction coefficient κ is a diagonal matrix, tDPD can beextended to modeling nonlinear systems, where diffusivityis a function of concentration, position, and time, i.e., D= D(C,x, t), because of the linear dependence of D on thetDPD parameter κ. Furthermore, as future works, it is alsointeresting to consider a dense mixture involving multiplecross terms and chemical interactions between differentspecies, where the Fickian friction coefficient κ becomesan N × N matrix to represent interdiffusivities, and there iscoupling between various chemical species.

ACKNOWLEDGMENTS

This work was primarily supported by NIH (Grant No.1U01HL116323-01) and the DOE Center on Mathematicsfor Mesoscopic Modeling of Materials (CM4). Computa-tional resources were provided by the Innovative and NovelComputational Impact on Theory and Experiment (INCITE)program and TACC/STAMPEDE through the XSEDE Grant(Grant No. TG-DMS140007). Z. Li would like to acknowledgehelpful discussions with Dr. Wenxiao Pan and Professor BruceCaswell. A. Yazdani would like to thank Dr. Hessam Babaeefor the support he gave for the spectral element solver.

APPENDIX A: CORRELATION OF CONCENTRATIONFLUCTUATIONS

In this appendix, we present simplified derivations for ob-taining the concentration fluctuations given by Eqs. (7) and (9).According to fluctuating hydrodynamics, the fluctuations canbe described by random noise terms whose correlation func-tions are determined by a fluctuation-dissipation theorem.26

By applying the local-equilibrium assumption to theclassical thermodynamics Gibbs-Duhem equation for thespecific entropy s, we have the balance of entropy given by26

Tds = du + pdv −k

µkdωk, (A1)

where T is the temperature, u the energy, p the pressure,and v = ρ−1

0 the specific volume. Also, µk are the chemicalpotentials defined per unit mass, and ωk the correspondingconcentrations in terms of mass fraction. Considering the bal-ance laws in fluids together with phenomenological relations,the correlation of random concentration fluxes QR is given bythe fluctuation-dissipation theorem in the form of

⟨QR(t) · QR(t ′)⟩ = 2kBT · ρ0D(∂µ

∂ω

)−1

· δ(t − t ′), (A2)

where D is the diffusion coefficient. For more details onthe derivations leading to Eq. (A2) from Eq. (A1), we referinterested readers to a book by Ortiz de Zárate and Sengers.26

Considering dilute solutions, the chemical potential of aspecies can be represented by

µ = µ0(P,T) + m−1s kBT ln x, (A3)

where x is the molecular fraction. We note that m−1s appears in

Eq. (A3) because the chemical potential has been defined perunit mass, and ms is the mass of a solute molecule. Let c bethe molar concentration in terms of molecules (or moles) perunit volume, then c is related to the mass fraction ω by

c =ρ0

msω, (A4)

where ρ0 is the density of the solution. By assuming thatthe mole fraction of the solvent in dilute solutions can beapproximated by unity, then we have x ≈ c. SubstitutingEqs. (A3) and (A4) into Eq. (A2) results in

⟨QR(t) · QR(t ′)⟩ = 2m2sDc · δ(t − t ′). (A5)

We note that the correlation of concentration fluctuationgiven by Eq. (A5) includes the equilibrium concentration c.Therefore, it is only valid for fluid in thermodynamic equilib-rium or with the approximation of locally thermodynamic equi-librium. Moreover, the fluctuation-dissipation theorem is validonly for large cell size limit in which fluctuations are small.To this end, Eq. (A5) cannot be used for giant stochastic fluxes.

In the tDPD model, the diffusion coefficient D in Eq. (A5)is replaced by the strength of Fickian friction coefficient andthe volume concentration becomes c = ρC with ρ the numberdensity of tDPD particles. Therefore, we have the fluctuation-dissipation relationship for concentration fluxes in tDPD givenby

ϵ2ij = m2

sκijρ(Ci + Cj), ωDC(rij) = ω2RC(rij). (A6)

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014101-13 Li et al. J. Chem. Phys. 143, 014101 (2015)

TABLE V. Mapping of physical units to tDPD units. The mapping is basedon the length unit L∗= 1.0×10−6 m, time unit t∗= 1.386×10−4 s, andvolume concentration unit C∗= 1.0 nM. The magnitudes of parameters inparentheses are temporally accelerated by 1000 times. The physical unit of tand t0 for k IIa

tPA is second.15

Physical units tDPD units (accelerated)

k9 11 min−1 2.54×10−5 (2.54×10−2)K9M 160 nM 160h9 0.0162 nM−1 min−1 3.74×10−8 (3.74×10−5)k8 194.4 min−1 4.49×10−4 (0.449)K8M 1.12×105 nM 1.12×105

h8 0.222 min−1 5.13×10−7 (5.13×10−4)hC8 10.2 min−1 2.36×10−5 (2.36×10−2)HC8M 14.6 nM 14.6k5 27.0 min−1 6.24×10−5 (6.24×10−2)K5M 140.5 nM 140.5h5 0.17 min−1 3.93×10−7 (3.93×10−4)hC5 10.2 min−1 2.36×10−5 (2.36×10−2)HC5M 14.6 nM 14.6k10 2391 min−1 5.523×10−3 (5.523)K10M 160 nM 160h10 0.347 nM−1 min−1 8.01×10−7 (8.01×10−4)hTFPI 0.48 nM−1 min−1 1.11×10−6 (1.11×10−3)k2 1344 min−1 3.105×10−3 (3.105)K2M 1060 nM 1060h2 0.714 nM−1 min−1 1.65×10−6 (1.65×10−3)k1 3540 min−1 8.177×10−3 (8.177)K1M 3160 nM 3160h1 1500 min−1 3.456×10−3 (3.456)K1M 2.50×105 nM 2.50×105

k11 0.0078 min−1 1.80×10−8 (1.80×10−5)K11M 50 nM 50hA3

11 1.6×10−3 nM−1 min−1 3.70×10−9 (3.70×10−6)hL1

11 1.3×10−5 nM−1 min−1 3.00×10−11 (3.00×10−8)kPC 39.0 min−1 9.01×10−5 (9.01×10−2)KPCM 3190 nM 3190hPC 6.6×10−7 nM−1 min−1 1.52×10−12 (1.52×10−9)kPLA 12.0 min−1 2.77×10−5 (2.77×10−2)KPLAM 18.0 nM 18.0hPLA 0.096 nM−1 min−1 2.22×10−7 (2.22×10−4)k7,9 32.4 min−1 7.48×10−5 (7.48×10−2)K7,9M 24.0 nM 24.0k7,10 103 min−1 2.38×10−4 (0.238)K7,10M 240.0 nM 240.0φ11 0.034 min−1 7.85×10−8 (7.85×10−5)Φ11M 2000 nM 2000kCtPA 6.52

×10−13 nM m2 min−11.51×10−6 (1.51×10−3)

k IIatPA 9.27×10−12e−134.8(t−t0) 2.14×10−5e−972 580(t−t0)

m2 min−1 (2.14×10−2e−972.58(t−t0))k Ia

tPA 5.059×10−18 m2 min−1 1.17×10−11 (1.17×10−8)[TF-VIIa] 0.25 nM 0.25[ENDO] 2.0×109 cells/m2 2.0×10−3

[XIIa] 375 nM 375

APPENDIX B: MAPPING OF UNITS

Appendix B presents the details of mapping physicalquantities to tDPD parameters that are used in Section III C.Table V lists both the physical units and the tDPD units of theparameters, which appear in Tables III and IV.

The length scale is determined by the size of physicalsystem. In Section III C, a channel with length Lx = 100 µmand height Lz = 20 µm is considered and the length unit ischosen as L∗ = 1 µm. Therefore, a tDPD system is constructedin a computational domain of 100 × 10 × 20 to represent thephysical system.

The time unit of tDPD system is determined by

νDPD ·(L∗)2

t∗= ν0, (B1)

where ν0 = 1.0 × 10−6 m2/s is the kinematic viscosity of fluidin physical units. Since the length unit L∗ = 1 µm and thekinematic viscosity of the tDPD system is νDPD = 138.6, wehave the time unit t∗ = 1.386 × 10−4 s.

Then, all the physical units of length can be scaled by L∗

and the physical units of time can be scaled by t∗ to obtaincorresponding tDPD units. For example, 1 m/L∗ = 1 m/1.0× 10−6 m = 1.0 × 106, 1 m3/(L∗)3 = 1.0 × 1018, 1 min/t∗

= 60 s/1.386 × 10−4 s = 4.33 × 105, and 1 min−1 · t∗ = 1.386× 10−4 s/60 s = 2.31 × 10−6.

Moreover, the volume concentration unit is chosen asC∗ = 1 nM, which indicates that the number of moleculesper tDPD particle should be scaled by n∗ = C∗V ∗/ρ whereV ∗ = (L∗)3 is the volume unit and ρ the number density oftDPD particle. Here, the divisor ρ means that the unit volumeis divided equally into ρ particles. Then, a concentration of1.0 nM is scaled into the concentration unit of the tDPDsystem.

For example, in Section III C, species I (fibrinogen)has initial concentration of CI,0 = 7.0 × 103 nM = 7.0× 10−3 mol/m3, the volume unit is V ∗ = (L∗)3=1.0 × 10−18 m3,and the number density of tDPD particle is ρ = 4. Thus,the moles of species I carried by each tDPD particle isCI,0 · V ∗/ρ = 1.75 × 10−21 mol, which should be scaled by n∗

= C∗ · V ∗/ρ = 1.0 nM · V ∗/ρ. Therefore, CI,0 = 7.0 × 103 nMis mapped into tDPD unit as CI = 7.0 × 103.

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