Mapping of dissipative particle dynamics in fluctuating hydrodynamics simulations

0.11 MB·English

by  R. Qiao

Mapping of Dissipative Particle Dynamics in Fluctuating Hydrodynamics
Simulations
R. Qiao∗ and P. He
College of Engineering & Sciences, Clemson University, Clemson, SC, 29634
(Dated: February 2, 2008)
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0 Dissipativeparticledynamics(DPD)isanovelparticlemethodformesoscalemodelingofcomplex
0 fluids. DPD particles are often thought to represent packets of real atoms, and the physical scale
2 probed in DPD models are determined by the mapping of DPD variables to the corresponding
physical quantities. However, the non-uniqueness of such mapping has led to difficulties in setting
n
a upsimulationstomimicrealsystemsandininterpretingresults. Formodelingtransportphenomena
J where thermal fluctuations are important (e.g., fluctuating hydrodynamics), an area particularly
suitedforDPDmethod,weproposethatDPDfluidparticlesshouldbeviewedasonly1)toprovide
5
amedium inwhich themomentumandenergy aretransferred according tothehydrodynamiclaws
1
and 2) to provide objects immersed in the DPD fluids the proper random ”kicks” such that these
] objects exhibit correct fluctuation behaviors at the macroscopic scale. We show that, in such a
h case, the choice of system temperature and mapping of DPD scales to physical scales are uniquely
p determinedbythelevelofcoarse-graining andpropertiesofDPDfluids. WealsoverifiedthatDPD
- simulation can reproduce the macroscopic effects of thermal fluctuation in particulate suspension
m
byshowingthattheBrowniandiffusionofsolidparticlescanbecomputedinDPDsimulationswith
e good accuracy.
h
c
s. Dissipative particle dynamics (DPD) is a method ily by bonding DPD beads together2,3. However,
c developed primarily for the simulation of complex there are two unresolved issues in DPD simulation
i
s fluids at mesoscopic scales1,2,3. While it is often offluctuatinghydrodynamics,namely,howtosetup
y
thought that DPD beads represent packets of real a model for a given physical system and how to in-
h
atoms moving collectively, the statistical mechani- terpretthesimulationresults. Forexample,tostudy
p
cal foundation of such a view for DPD model re- thediffusionofa30nmdiameterparticleinwaterat
[
mains obscure4. Associated with the ambiguity of 300K, what should be the temperature of the DPD

1 the exact nature of DPD beads is the ambiguity in system and how to map the results to dimensional
v
mappingofDPDscalestothephysicalscales. While values are not clear.
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the mapping of length is straightforward, the map- To address the above issues, we propose to aban-
5
3 ping of time is more complicated. Depending on don the idea that DPD fluid beads are “clumps”
2 the problems being studied, mapping of DPD time of real fluid atoms, but view them together as a
. to physical time has been based on bead diffusion “media” that provides an arena for the transport
1
rate, bead thermal velocity, or externally imposed ofmomentum,energyandparticulatesthatsatisfies
0
8 time scales5,6,7, to name just a few. In all but a few the fluctuating hydrodynamics laws. This idea is
0 cases7, temperature of the DPD system was cho- inspiredby a recentpaper oncoarse-grainingin col-
: sen arbitrarily. Established procedures for mapping loidalsuspensions4. TobridgetheDPDandphysical
v
i between DPD and physical scales and for choosing systems, we require that
X system temperature are not yet available.
r Our interest in DPD originates from the need to 1. The time scale of diffusional transport of mo-
a
study transport phenomena in particulate suspen- mentum (and energyif in non-isothermalsim-
sion where thermal fluctuations may play a crit- ulations) inside the DPD fluids should match
ical role8,9. Such transport can be described by that of the real fluids;
the fluctuating hydrodynamics theories10. However,
solving the hydrodynamics equations in particulate 2. DPD fluids should provide objects immersed
suspension is computationally demanding and in- in them the proper random ”kicks” such that
troducing thermal fluctuations that rigorously sat- these objects exhibit correct fluctuation be-
isfy the Fluctuation-Dissipation Theorem (FDT) is haviors at the macroscopic scale.
challenging11. DPD is well-suited for studying such
phenomena as momentum/energy conservation and We now consider the application of the above re-
FDTareguaranteedbythewayDPDmodelsarede- quirements in the simulation of colloidal particles
signed, and colloidal particles can be modeled eas- (diameter: d) dispersed in fluids (density: ρ, kine-
matic viscosity: ν, temperature: T). Properties of
corresponding DPD fluids are denoted by the same
symbol as in real fluids but without the bar, e.g.,
∗Correspondingauthor. Email: [email protected] density of DPD fluids is denoted as ρ. We will limit
2
ourdiscussiontoisothermalsimulations,andtheex- enforcing them simultaneously in fluctuating hydro-
tension to non-isothermalsimulations is straightfor- dynamics simulations and thus leading to a unique
ward. The DPD model reads3 choiceofsystemtemperaturehasnotbeenproposed.
To demonstrate the above mapping scheme and
dri =vidt; mdvi =FCi dt+FDi dt+FRi √dt (1) to investigate the ability of DPD in modeling fluc-
tuating hydrodynamics, we study the diffusion of a
where m, ri, and vi are the mass, position, and ve- single nanoparticle immersed in fluids with ρ¯= 103
locity of bead i, respectively. T is the system tem- kg/m3 and ν¯ = 0.89 10−6 m2/s at 300 K. We set
perature. FCi , FDi andFRi arethe conservative,dis- [L]=10 nm and ρ =×6.0. The nanoparticle is built
sipativeandrandomforcesactingonbeadi,respec- bybonding117DPDbeadstogetherandismodeled
tively. These forces are given by3,12 as a rigid body. In Equ. (2), w(r /r )=1 r /r
ij c ij c
−
withr = 1.0anda issetto a =10.0anda =
c ij ff fp
FCi = Xaijw(rij/rc)eij (2) 17.0, where ff and fp denote fluid-fluid and fluid-
j6=i nanoparticle interactions, respectively. In Equ. (3),
FDi = X−γijwd2(rij/rcd)(eij ·vij)eij (3) uwsde(drijw/irtcdh)r=d =p411.0−. rσij/rincd Esuqgug.es(t4e)disintaRkeefn. a1s25.i0s
j6=i c ij
for all bead pairs. Using these parameters, we first
FR = σ w (r /rd)θ e (4)
i X ij d ij c ij ij measured the viscosity of DPD fluids as a function
j6=i oftemperature,andthe temperatureinDPDmodel
ofthe nanoparticle-fluidssystemis then determined
where a is the conservative force coefficient and
ij via Equ. (8) to be 0.3875. To investigate the abil-
r = r = r r . w and w are weighting func-
tiiojnsw| iitjh| cu|toiff−disjt|ancesofrdandrd,respectively. ity of DPD model in reproducing the macroscopic
c c effectsofthermalfluctuations,wecomputedthedif-
e =r /r , and v =v v . θ is a symmetric
ij ij ij ij i− j ij fusion coefficient (Dp) of the nanoparticle, which
random variable with zero mean and unit variance.
is a macroscopic manifest of the thermal fluctua-
γ and σ are lated by the Fluctuation-Dissipation
ij ij tions. Fig. 1(a) shows the mean square displace-
Theorem as γij =σi2j/2kBT, where kB is the Boltz- ment ofthe particle,and a D of (1.59 0.12) 10−4
p
mannconstant. Mass,length,andtimeintheabove ± ×
was obtained. We also computed D independently
p
modelaremeasuredbym,r ,and k T/m,respec-
c p B by using the Einstein-Stokes law Dp = kBT/6πRµ,
tively. We assume that the unit length and time in
where µ is the fluid viscosity. From the particle-
DPD map to physical length [L] and time [t], re-
fluids pair correlation function shown in Fig. 1(b),
spectively. Tosatisfytheproposedrequirements,we
thenanoparticleradiusRwasdeterminedtobe1.65,
enforce
with which a D of 1.55 10−4 was then computed.
p
×
d2/ν[t] = d2/ν (5) Given the ambiguity in the definition of particle di-
ameter and the statistical uncertainty of DPD re-
3 sults, the agreement between DPD simulation and
pkBT/ρd3[L]/[t] = qkBT/ρd (6)
Combining Equs. (5-6)
x 10−3
2
[t] = [L]2ν/ν (7) (a)
1.8
ρν2k T
B
k T = (8) 1.6
B ρν2 [L] ent
m1.4
e
c
a 2
Equs. (7-8) can be used to setup and analyze DPD displ1.2 R=1.65 (b)
spEirmqouvu.ildaet(is8o)nthsienodfmicflaauptcpetsiunatghtioanftgtthihmyedertsoecdmaylpeneairnmatiDucsrP.eDEinqmuDo.dP(e7Dl). an square 0.18 −water RDF1.15
smiminueldatbioynthofe flleuvcetluaotfincgoarhsyed-groradiynninagmi(crsepirsesdeentteerd- Me00..46 Particle0.5
by [L] and ρ) and transport properties of DPD flu- 0.2 0
ids (represented by ν). If ν and ρ are known, then 0 r2adial distanc4e 6
0
0 0.5 1 1.5 2
the system temperature is uniquely determined. In Time
practice, since ν of DPD fluids is a function of their
temperature, the temperature of DPD system can FIG.1: (a). Meansquaredisplacementofthenanoparti-
only be determined after the dependence of ν on cleimmersed in fluids,(b)Particle-fluid paircorrelation
temperatureisknown. WhileEqus. (5)and(6)have obtained from DPD simulation.
been used to map DPD models to physical scales,
3
Einstein-Stokes prediction is very good. This veri- uniquely by the level of coarse-grainingand proper-
fiesthe abilityofDPDincapturingthemacroscopic ties of DPD fluids. Following the proposed method
effectsofthermalfluctuations inparticulatesuspen- of choosing system temperature, we studied the
sions, which has been assumed but not explicitly Brownian diffusion of a nanoparticle and showed
confirmed in the literature. thatDPDsimulationcanreproducethemacroscopic
In summary, we proposed a method of mapping fluctuation of nanoparticle immersed in fluids with
DPD simulation of fluctuating hydrodynamics, and good accuracy.
the temperature in such simulations is determined
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