Bead–bead interaction parameters in dissipative particle dynamics: Relation to beadsize, solubility parameter, and surface tension Amitesh Maiti and Simon McGrother Citation: The Journal of Chemical Physics 120, 1594 (2004); doi: 10.1063/1.1630294 View online: http://dx.doi.org/10.1063/1.1630294 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/120/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modeling the temperature dependent interfacial tension between organic solvents and water using dissipative particle dynamics J. Chem. Phys. 138, 094703 (2013); 10.1063/1.4793742 Study of interfacial tension between an organic solvent and aqueous electrolyte solutions using electrostatic dissipative particle dynamics simulations J. Chem. Phys. 137, 194701 (2012); 10.1063/1.4766456 Complete mapping of the morphologies of some linear and graft fluorinated co-oligomers in an aprotic solvent by dissipative particle dynamics J. Chem. Phys. 124, 064905 (2006); 10.1063/1.2162885 Mixtures of charged colloid and neutral polymer: Influence of electrostatic interactions on demixing and interfacial tension J. Chem. Phys. 122, 244911 (2005); 10.1063/1.1940055 Dissipative particle dynamics for interacting systems J. Chem. Phys. 115, 5015 (2001); 10.1063/1.1396848
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 3
15 JANUARY 2004
Bead–bead interaction parameters in dissipative particle dynamics: Relation to bead-size, solubility parameter, and surface tension Amitesh Maitia) and Simon McGrother Accelrys Inc., San Diego, California 92121
共Received 28 July 2003; accepted 7 October 2003兲 Dissipative particle dynamics 共DPD兲 is a mesoscale modeling method for simulating equilibrium and dynamical properties of polymers in solution. The basic idea has been around for several decades in the form of bead-spring models. A few years ago, Groot and Warren 关J. Chem. Phys. 107, 4423 共1997兲兴 established an important link between DPD and the Flory–Huggins -parameter theory for polymer solutions. We revisit the Groot–Warren theory and investigate the DPD interaction parameters as a function of bead size. In particular, we show a consistent scheme of computing the interfacial tension in a segregated binary mixture. Results for three systems chosen for illustration are in excellent agreement with experimental results. This opens the door for determining DPD interactions using interfacial tension as a fitting parameter. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1630294兴
and the Flory–Huggins -parameter theory.5 There are established theories in polymer science, which link the -parameter to solubilities and mixing energies of the polymeric components.6,7 The latter can be obtained either from atomistic simulations or from experiments. Thus, the Groot– Warren work provided a sound basis on which the conservative repulsion used in DPD could be derived starting from an atomistic description. In this article, we re-examine the Groot and Warren derivation of the conservative repulsion 共referred to henceforth, simply, as ‘‘DPD interaction’’兲 in terms of Flory–Huggins’ -parameter. The emphasis is on a ‘‘bottom-up’’ derivation of the DPD interaction parameters, and on how they should scale as a function of the bead-size. An internal consistency is established by comparing computed surface tension between segregated components with experimental surface tension, as a function of bead-size. We have considered three systems for illustration: water-benzene, water-octane, and water-tetrachloromethane.
I. INTRODUCTION
Many academic and industrial ‘‘soft’’ materials of interest are designed to have length scales ranging from tens to thousands of nanometers. Such structures consist of millions to billions of atoms, and an atomic-level description becomes prohibitively expensive. One way to circumvent this problem is to coarse-grain a group of atoms into a single ‘‘bead’’ and to replace the atomic-level interactions by simpler bead– bead interactions. In the early 1990s Hoogerbruge and Koelman1 introduced a simulation technique called dissipative particle dynamics 共DPD兲, aimed at simulating soft spherical beads interacting through a simple pair-wise potential, and thermally equilibrated through hydrodynamics. The method was soon extended to polymeric systems by introducing bead-and-spring type models.2 In 1995, Espanˇol and Warren3 expressed the total force on each DPD particle as a sum of three pair-wise additive terms: 共1兲 a conservative force, which is taken to be a soft repulsion; 共2兲 a dissipative force, proportional to the relative velocity of the beads; and 共3兲 a random force, necessary to maintain the system temperature. For polymeric systems, one has to include an additional interaction due to the ‘‘springs.’’ Espanˇol and Warren3 carried out a detailed analysis using the fluctuationdissipation theorem, and showed that the relative amplitudes of the dissipative and random forces have to satisfy a certain relation in order to ensure that the hydrodynamic simulation follows the canonical ensemble. While the dissipative and random forces act in unison as a thermostat for the simulation, it is the conservative soft repulsive force that embodies the essential chemistry of the system. Ideally, one would like to derive the conservative force from detailed atomistic interactions. In 1997, Groot and Warren4 made an important contribution on this front by establishing a relation between a simple functional form of the conservative repulsion in DPD
II. DISSIPATIVE PARTICLE DYNAMICS BASICS: GROOT–WARREN REVISITED
In the DPD formalism, the beads simply follow Newton’s equations of motion: drជ i ⫽ vជ i ; dt
共1兲
where all the masses are normalized to 1 for simplicity, and the force ជf i on bead i contains three parts, each of which is a pairwise additive
ជf ⫽ i
兺j ⬘共 ជf ci j ⫹ ជf Di j ⫹ ជf Ri j 兲 ,
共2兲
where 兺 j ⬘ denotes summation over index j with i⫽ j excluded. In the above equation, f D and f R are, respectively, the dissipative and random forces, which effectively act as a thermostat and result in fast equilibration to the Gibbs–
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2004/120(3)/1594/8/$22.00
d vជ i ជ ⫽fi , dt
1594
© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 120, No. 3, 15 January 2004
Bead-bead interaction parameters in DPD
Boltzmann canonical ensemble. The significance of these terms is investigated in detail elsewhere3 and will not be discussed further in this article. Instead, we focus on ជf ci j , which is modeled as a soft repulsion between beads i and j. The force acts along the line joining the two beads, and is therefore conservative 共i.e., momentum conserving兲. Groot and Warren4 assumed the following functional form for f c : f ci j 共 rជ 兲 ⫽a i j 共 1⫺r/R c 兲 , ⫽0,
g共 r 兲⫽
共3兲
p⫽
ai jRc 共 1⫺r/R c 兲 2 , 2
⫽0,
r⭐R c 共4兲
r⬎R c .
The parameters a i j , henceforth referred to as bead-bead repulsion parameters, or simply as DPD interaction parameters depend on the underlying atomistic interactions. The interaction range R c in the above equation sets the basic length-scale of the system, and is defined as the side of a cube containing an average number of ¯ beads. Therefore R c ⫽(¯ v b ) 1/3, where v b is the volume of a bead. One can think of ¯ as a dimensionless bead-density, related to the average bead-density ⫽1/v b by the equation ¯ ⫽ R 3c . Even in a heterogeneous system consisting of several different species, the basic assumption is that all bead-types (each representing a single species) are of the same volume v b . This assumption is necessary in order to conform to the Flory– Huggins -parameter theory.8 The pressure of the bead system can be obtained from the virial theorem,9 as follows 共具...典 below denotes ensemble average兲: p⫽ kT⫹ ⫽
¯ R 3c
1 具 rជ • ជf c 共 rជ 兲 典 6V i⫽ j i j i j i j
kT⫹
兺
1 6V i⫽ j
兺
冕
drជ 具 ␦ 共 rជ ⫺rជ i j 兲 典 rជ • ជf ci j 共 rជ 兲 ,
共5兲
where V is the total volume of the system, T is the equilibrium temperature, and k the Boltzmann constant. The first term is due to the kinetic energy of the beads, while the second term arises out of bead-bead interaction and is called the virial term. For a typical liquid system, the virial term is the dominant contribution to pressure. The summation representing the virial term is over both i and j indices, with i ⫽ j terms excluded 共i.e., over all bead-pairs兲. A. Case 1. Single-component homogeneous solution
In this case, there is only one DPD interaction parameter, i.e., a i j ⫽a, for all i and j. Substituting Eq. 共3兲 simplifies Eq. 共5兲 to p⫽
¯
¯ 2
Rc
6R 6c
kT⫹ 3
a
冕
Rc
r⫽0
4 r 2 drg 共 r 兲 r 共 1⫺r/R c 兲 ,
where g(r) is the pair-correlation function defined by
共6兲
共7兲
¯
¯ 2
Rc
R 2c
kT⫹ ␣ a 3
,
共8兲
¯r 3 共 1⫺r ¯ 兲¯g 共¯r 兲 dr ¯,
共9兲
with
The corresponding potential energy function 共within an additive constant兲 is given by V pot i j 共 r 兲⫽
兺 具 ␦ 共 rជ ⫺rជ i j 典 .
N 2b i⫽ j
In Eqs. 共6兲 and 共7兲, r⫽ 兩 rជ 兩 , and N b ⫽¯ V/R 3c is the total number of beads in volume V. Equation 共6兲 can be simplified to
兩 rជ 兩 ⫽r⭐R c
r⬎R c .
V
1595
␣⫽
2 3
冕
1
0
where ¯r ⫽r/R c is the reduced coordinate, and ¯g is the pair correlation function expressed as a function of ¯r . In the actual DPD code, the interaction parameters a are expressed as reduced 共i.e., dimensionless兲 quantities ¯a ⫽
aR c . kT
共10兲
In terms of the reduced interaction parameter, expression 共8兲 for p becomes p⫽
¯
¯ 2
Rc
R 3c
kT⫹ ␣¯a kT 3
共11兲
,
which can be rewritten in terms of a dimensionless pressure ¯p as follows: p⫽
kT R 3c
where ¯p ⫽¯ ⫹ ␣¯a¯ 2 .
¯p ,
共12兲
¯ ), and therefore ␣, is in general a function of Note that ¯g (r the density ¯ , as well as of the interaction parameter a. However, as the Appendix indicates, ␣ varies very slowly as a function of both ¯ and a within the ranges of our interest 共3⭐⭐5, a⭓15). Therefore, for simplicity in the following discussion, ␣ has been considered to be a constant ⬃0.1, independent of ¯ and a.10 The Appendix estimates errors in interaction parameters due to the use of a constant ␣, and determines them to be only a few percent. 1. Free energy density for a pure component
Let F be the Helmholtz free energy 共NVT ensemble兲, and f the free energy density. Then F⫽V f , and p⫽(⫺ F/ V) T,N b ⫽⫺V f / V⫺ f ¯⫺f ⫽¯ f /
冋
⬗¯ ⫽N b
⬖¯
R 3c V
, and N b and R c are fixed for derivatives
¯ ¯ 2 f ⫺ f ⫽ 3 kT⫹ ␣¯a kT 3 ¯ R R c
⇒
册
关 using Eq. 共 11兲兴
c
冉冊
kT 1 ␣¯a kT f ⫽ 3 ⫹ ¯ ¯ R c ¯ R 3c 共 upon dividing both sides by 1/ 2 兲 .
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J. Chem. Phys., Vol. 120, No. 3, 15 January 2004
A. Maiti and S. McGrother
Integration yields f⫽
kT
␣¯a kT
Rc
R 3c
¯ ln ¯ ⫹ 3
¯ 2 ⫹c 1¯ ,
共13兲
where c 1 is an integration constant. Only the ¯ 2 term of f in Eq. 共13兲 is dependent on the DPD interaction parameters, and its generalization for a two-component system establishes a correspondence with the Flory–Huggins theory, as shown below. Let us note that the ¯ 2 term in Eq. 共13兲 arises from integrating the virial term of pressure p virial⫽
where ␣ is defined by Eq. 共9兲. It is important to note that the same interaction cutoff R c has been assumed for ¯a AA , ¯a AB , and ¯a BB , i.e., the same interaction-range for all DPD interactions irrespective of the bead types. The corresponding contribution to free energy density is obtained by integrating p virial in the same way as for the one-component system f virial⫽
␣ kT¯ 2 R 3c
¯ AB x 共 1⫺x 兲 ⫹a ¯ BB 共 1⫺x 兲 2 兴 . 关 ¯a AA x 2 ⫹2a 共15兲
The virial contribution of total free energy (F⫽V f ) is thus given by
1 具 rជ • ជf c 共 rជ 兲 典 . 6V i⫽ j i j i j i j
兺
¯ AB x 共 1⫺x 兲 F virial共 N b ,x 兲 ⫽N b ␣¯ kT 关 ¯a AA x 2 ⫹2a B. Case 2. Mixture of two components A and B
¯ BB 共 1⫺x 兲 2 兴 , ⫹a
N Ab
Let the total number of A-beads and B-beads be and N Bb , respectively, in a total volume V. The total number of beads in volume V is therefore N b ⫽N Ab ⫹N Bb . By construction, the Groot–Warren formalism assumes the same interaction cutoff R c independent of A and B. Therefore the average number of A and B beads within a cube of side R c are given by ¯ A ⫽
N Ab V
¯ B ⫽
R 3c ;
N Bb V
兺
冋
⫹
兺
兺
i苸A, j苸A i⫽ j
i苸B, j苸B i⫽ j
具 rជ i j • ជf i j 典 ⫹2
具 rជ i j • ជf i j 典
¯ 2 kT R 3c
兺
i苸A, j苸B
具 rជ i j • ជf i j 典
册
关 ␣ AA¯a AA x 2 ⫹2 ␣ AB¯a AB x 共 1⫺x 兲
¯ )g ¯ XY (r ¯ )dr ¯, where ␣ XY ⫽2 /3兰 10¯r 3 (1⫺r and g XY (r) X Y ជ ជ ⫽V/N b N b 兺i苸X, j苸Y 具 ␦ (r ⫺r i j ) 典 , with X, Y ⫽A or B. i⫽ j Assuming an ␣ independent of ¯ and a, the above equation can be simplified to
␣ kT¯ 2 R 3c
冋
F m,virial共 N b ,x 兲 ⫽2N b ␣¯ kT ¯a AB ⫺
共17兲
册
¯a AA ⫹a ¯ BB x 共 1⫺x 兲 . 2 共18兲
Recalling the Flory–Huggins -formula8 F m,virial(N b ,x) ⫽N b kT AB x(1⫺x), one can readily make the identification
冋
册
¯a AA ⫹a ¯ BB . 2
¯a AA ⫽a ¯ BB ,
⫹ ␣ BB¯a BB 共 1⫺x 兲 2 兴 ,
p virial⫽
⫺F virial共 N Bb ,0兲 .
共19兲
By construction, the Flory–Huggins theory assumes that all beads are of the same equilibrium volume. This requirement, in conjunction with the fact that the pairwise DPD interactions between all bead-species have the same cutoff R c , implies that ¯ A ⫽¯ B for all pure components A and B under segregation. Now, let us consider two species A and B with sufficiently large Flory–Huggins -parameter such that they segregate in equilibrium. In order to maintain mechanical equilibrium at the interface, the pressure in each species must be equal: p A ⫽ p B . 11 Using Eq. 共11兲 for p, and the fact that R c and ¯ are the same on both sides, we arrive at the important result
as follows: p virial⫽
F m,virial共 N b ,x 兲 ⫽F virial共 N b ,x 兲 ⫺F virial共 N Ab ,1兲
AB ⫽2 ␣¯ ¯a AB ⫺
1 p virial⫽ 具 rជ • ជf c 共 rជ 兲 典 6V i⫽ j i j i j i j 1 6V
where the relation has been used. In order to map onto the Flory–Huggin theory, one needs to determine the virial contribution to the free energy of mixing. This is given by8
Substituting Eq. 共15兲 into Eq. 共16兲, and using N Ab ⫽xN b and N Bb ⫽(1⫺x)N b one obtains
R 3c .
If x⫽N Ab /N b is the volume fraction of component A, and 1 ⫺x⫽N Bb /N b the volume fraction of component B, then ¯ A ⫽x¯ and ¯ B ⫽(1⫺x)¯ , where ¯ is the average total number of beads in volume R 3c . Following the same algebra as for the one-component system, it is tempting to simplify the virial term of beadpressure for the two-component system
⫽
共16兲
N b ⫽¯ V/R 3c
¯ AB x 共 1⫺x 兲 ⫹a ¯ BB 共 1⫺x 兲 2 兴 , 关 ¯a AA x 2 ⫹2a 共14兲
共20兲
i.e., intraspecies interactions are the same for all interacting species.12 With the above result, Eq. 共19兲 simplifies to ¯ AA 兴 . AB ⫽2 ␣¯ 关 ¯a AB ⫺a
共21兲
At this point, it is important to point out that the derivation of Eq. 共14兲 and subsequent derivations of Eqs. 共18兲, 共19兲, and 共21兲 assume that the spatial variation of the volume fraction x is small, i.e., x/ ln RcⰆ1. In such a case, the mixture is quasihomogeneous, the pair correlation function
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J. Chem. Phys., Vol. 120, No. 3, 15 January 2004
Bead-bead interaction parameters in DPD
g AB (r) is isotropic, and ␣ AB ⬃0.1 is a good approximation 共see the Appendix兲. This scenario holds only for small values of the -parameter, more specifically for ⬍ critical, 13 where Eq. 共21兲 should be applicable. However, in the case of a segregated mixture with a sharp interface such an approximation breaks down. In particular, the pair correlation function g AB (rជ ) 共for r⬍R c ) is nonzero only at the interface and is uniaxially anisotropic with the anisotropy axis perpendicular to the interfacial plane. Even the definition of ␣ AB in terms of a spherically symmetric integral breaks down. This leads to a significant departure from Eq. 共21兲, as pointed out in the original paper by Groot and Warren,4 and probed in more detail by Wijmans et al.14 through Gibbs ensemble Monte Carlo simulations. In both Refs. 4 and 14 the computed volume fraction of the minority component in the homogeneous part of the mixture 共i.e., x at the binodal point兲 were used to fit the ¯ ⫽(a ¯ AB ⫺a ¯ AA ). For a -parameter as a function of ⌬a monomer–monomer mixture, Groot–Warren 共GW兲 determined a linear fit ¯ 共 for ¯ ⫽3 兲 ⬇0.286⌬a ¯ 共 for ¯ ⫽5 兲 , ⬇0.689⌬a while Wijmans–Smit–Groot 共WSG兲 determined a nonlinear fit 共see Fig. 2 of Ref. 14兲. It should be remarked here that the binodal x, which decreases exponentially with increasing , becomes a very small number for moderately large values of , say l⬃10 or so. For ⬎ l, the minority component x at the binodal point cannot be determined accurately from simulations. Thus, both the GW and WSG fits are appropriate only in the range ⬍ l, which roughly corresponds to ¯ ⬍30 or so 共for ¯ ⫽3). ⌬a In this paper, we compute from solubility parameters using the formula:6,7 共22兲
where ␦ A and ␦ B are the solubility parameters of systems A and B, respectively, and V bead is the volume of a bead 共all beads having the same average volume R 3c /¯ ). All DPD interaction parameters can then be determined once one calibrates the value of ¯a AA for a specific system. Following Groot and Warren we choose water as the specific system, and equate the density-derivative of pressure at a constant T and V to the inverse of 共dimensionless兲 isothermal compressibility
冉 冊 ¯p ¯
⫽ ⫺1 ⇒2 ␣¯a AA¯ ⫽ ⫺1 ⫺1.
III. SCALING OF DPD INTERACTION PARAMETERS a ij WITH BEAD SIZE
The solubility parameters are normalized per unit volume, and are therefore independent of the bead size. Thus, Eq. 共22兲 implies that AB is proportional to the size of the bead. Therefore, when m elementary beads are coarsegrained into a single bead, scales as
AB,m ⫽m AB,1 .
共24兲
For a monomer–monomer mixture, and assuming critical⬍ AB,m ⬍ l⬃10, a GW or a WSG fit could be used ¯ . With GW scaling to determine ⌬a ¯ AA,m 兴 ⫽ AB,m /0.286 关 ¯a AB,m ⫺a
for ¯ ⫽3,
共25a兲
¯ AA,m 兴 ⫽ AB,m /0.689 关 ¯a AB,m ⫺a
for ¯ ⫽5.
共25b兲
Now, Eq. 共23兲 used to determine the intraspecies interaction ¯a AA is independent of R c , and therefore, independent of bead-size.15,16 Thus, ¯a AA,m ⫽a ¯ AA,1 .
共26兲
Substituting Eq. 共26兲 into Eq. 共25兲 yields
and
V bead AB ⫽ 共 ␦ A⫺ ␦ B 兲2, kT
1597
共23兲
¯ AA,1兴 ⫽ AB,m /0.286 关 ¯a AB,m ⫺a
for ¯ ⫽3,
共27a兲
¯ AA,1兴 ⫽ AB,m /0.689 关 ¯a AB,m ⫺a
for ¯ ⫽5.
共27b兲
Equations 共25兲 and 共27兲 are to be suitably modified when WSG fitting is employed, or in case of a polymer–monomer or a polymer–polymer mixture, where suitable fitting formulas need to be developed. However, as shown below for the water-octane system, a monomer–monomer treatment could provide suitable DPD interaction parameters for a monomerpolymer system as well. IV. SURFACE TENSION AS A FUNCTION OF BEAD-SIZE
Surface tension is usually computed by integrating the difference between normal and tangential stress across the interface separating the two segregated components. Thus, if the normal to the interface lies along the x-direction, surface tension is given by the formula
calc⫽
冎
1 p xx 共 x 兲 ⫺ 关 p y y 共 x 兲 ⫹ p zz 共 x 兲兴 dx. 2
共28兲
The DPD code actually computes a dimensionless version of the above quantity
DPD⫽
冕冋
册
1 ¯p xx 共¯x 兲 ⫺ 关 ¯p y y 共¯x 兲 ⫹ ¯p zz 共¯x 兲兴 dx ¯. 2
共29兲
From Eq. 共12兲, and the relation ¯x ⫽x/R c it follows that:
T
For water at room temperature, ⫺1 ⬇16. With ␣⬇0.1 共see discussions in the Appendix兲, and typically used values of ¯ ⫽3 and ¯ ⫽5, we have ¯a AA ⬇25 and ¯a AA ⬇15 respectively. Once ¯a AA and are determined, the cross-species interactions ¯a AB can be determined using Eq. 共21兲 for ⬍ critical, 13 a GW or WSG-like fit for critical⬍ ⬍ l, and from interfacial tension for ⬎ l 共see Sec. IV below兲.
冕再
calc⫽
kT R 2c
DPD .
共30兲
The quantity calc can be directly compared with experimentally measured surface tension. Below we compute calc for three systems: 共1兲 water-benzene; 共2兲 water-CCl4 ; and 共3兲 water-octane. Table I lists some important parameters necessary to compute the DPD interaction parameters for these three systems.
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A. Maiti and S. McGrother
TABLE I. Molecular volume (V molecule), computed solubility parameters 共␦兲, and experimental surface tension ( expt) at water-liquid interface for different liquid components relevant to surface tension calculations in Sec. IV. In column 3, experimental ␦-values20,21 are indicated in parentheses.
Component Water (H2 O) Benzene (C6 H6 ) Tetrachloromethane (CCl4 ) Octane (C8 H18) Ethane (C2 H6 )
V molecule 共Å3兲
共J/cm3兲1/2
␦
expt 共dyn/cm兲
30 148 161
47.9 共47.9兲 18.8 共18.6兲 18.0 共17.8兲
¯ 35.0 45.0
270 87
15.6 共15.6兲 14.8a
TABLE II. Water-benzene system with GW fit: DPD interaction parameters and computed surface tension ( calc) as a function of bead-size 共m兲, defined in terms of water monomers 关Eq. 共33兲兴. is obtained using Eq. 共22兲. Equation 共27兲 is then used to determine DPD interaction parameters ¯a AB,m . Finally, Eq. 共30兲 is used to obtain calc . Values of DPD are in excellent agreement with Eq. 共36兲 of Ref. 4. Experimental surface tension is 35.0 dyn/cm 共see Table I兲.
51.7 ¯
Ethane 共boiling point 185 K兲 is a gas at room temperature, and we compute its solubility parameter at its boiling point density of 0.572 g/cc.
a
The solubility parameters for these systems are obtained from atomistic simulation of water, octane, tetrachloromethane (CCl4 ), and benzene. The method comprises the construction of amorphous fluid structures of each system at experimental densities 共in the absence of such values, NpT dynamics can be performed to equilibrate the density兲. Periodic cells are generated with ⬃25 Å sides using the amorphous cell program.17 All-atom models of the molecules are used. The interatomic interactions are described by the COMPASS force-field,18 known to accurately reproduce experimental structures, densities, and solubility parameters. The discover molecular dynamics engine19 is used with group-based interaction cut-offs to evolve these systems and generate statistically independent structures. For each saved structure 共⬃200 structures for each molecule兲 the cohesive energy is calculated as the difference between the system energy and the sum of the energies of the constituent molecules with the same internal coordinates in isolation. That is E coh⫽E system⫺
兺 E gas .
共31兲
Physically the cohesive energy can be seen as the forces holding the molecules together in the liquid phase. The solubility parameter ␦ is the square root of the cohesive energy density, i.e.,
␦ ⫽ 冑E coh /V,
共32兲
where V is the volume of the cell at equilibrium. The calculated solubility parameters are reported in Table I. They are in excellent agreement with experimental values.20,21 Equation 共22兲 is then used to calculate Flory–Huggins interaction parameters AB for each pair of species. Lastly, Eq. 共21兲 or a GW or a WSG-like fit can be used to convert the -parameter into a DPD interaction ¯a AB . It should be noted that there are alternative routes to the derivation of ¯a AB , including experimental solubility parameters, pair-interaction calculations,22 experimental vapor-pressures,23 or solubilities. We chose atomistic dynamics, since the method is quite general, does not rely on the existence of experimental data and has previously been shown to provide quantitative agreement with experimental data24,25—depending on the quality of the force-field employed. Table II displays the computed interaction parameters and the surface tension calc for the water-benzene system as
m 共bead-size兲
¯a AA,m
¯a AB,m
DPD
calc 共dyn/cm)
1 2 3 4 5
25 25 25 25 25
¯ ⫽3 46.7 68.3 90.0 111.7 133.3
1.76 3.20 4.00 4.51 4.91
35.9 41.1 39.2 36.5 34.3
1 2 3 4 5
15 15 15 15 15
¯ ⫽5 24.0 33.0 42.0 51.0 59.9
2.93 5.25 6.60 7.51 8.23
42.5 48.0 46.1 43.2 40.8
a function of bead-size m, where the GW fit 关Eq. 共27兲兴 has been employed along with Eq. 共22兲. Both for ¯ ⫽3 and ¯ ⫽5, we find that calc increases slightly from m⫽1 to m ⫽2, and then decreases monotonically for larger m. The decrease for large m is a result of using the GW fit for values of ¯ ⬎30 where it is not accurate. Table III attempts to correct ⌬a this by using the nonlinear WSG fit (¯ ⫽3). 14 Since the ¯ ⬍30, we extended the fit original WSG fit was meant for ⌬a ¯ ⭐115 by assuming a linearly varying to the range 15⭐⌬a ¯ as a function of ⌬a ¯ , with a value of 0.3 at ⌬a ¯ ⫽15 and /⌬a ¯ ⫽115(¯ ⫽3). This results in essentially a value of 0.2 at ⌬a the same values of calc at m⫽2 and m⫽3. We did not ¯ , which attempt to extend the WSG fit to larger values of ⌬a would have been necessary to compute for m⬎3. The last two rows of Table III also displays results with the WSG fit26 when the intraspecies interaction ¯a AA scales linearly with bead size.15,16 Table IV summarizes the bead-size dependence for the computed surface tension for all three systems: water-benzene, water-tetrachloromethane, and water-octane using the extended WSG fit as described above. The wateroctane calculation is performed in two different ways, first in which an octane molecule is treated as a monomer 共i.e., as a single bead兲, and second in which it is treated as a polymer of four ethane beads. All calculations reported in Tables TABLE III. Water-benzene system with WSG fit (¯ ⫽3): DPD interaction parameters and computed surface tension ( calc) as a function of bead-size 共m兲, when the extended WSG scaling 共see text兲 is employed. The last two rows correspond to linear scaling of ¯a AA with m.15 Experimental surface tension is 35.0 dyn/cm 共see Table I兲. m 共bead-size兲
¯a AA,m
¯a AB,m
DPD
calc 共dyn/cm)
1 2 3
25 25 25
46.1 71.1 103.6
1.70 3.31 4.33
34.7 42.5 42.5
2a 3a
50 75
96.1 153.6
4.04 6.48
52.0 63.6
Following linear scaling of ¯a AA with bead-size 共Ref. 16兲. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.115.190.38 On: Thu, 11 Jun 2015 17:54:36 a
J. Chem. Phys., Vol. 120, No. 3, 15 January 2004
Bead-bead interaction parameters in DPD
TABLE IV. Computed surface tension ( calc) for three systems: Waterbenzene, water-CCl4 , and water-octane as a function of bead-size 共m兲, employing the extended WSG fit (¯ ⫽3). Column 4 corresponds to simulations for the water-octane system in which the whole octane molecule is defined as a single bead, while column 5 lists results in which an octane molecule is treated as a 4-mer of ethane, but using the same DPD interactions as derived for column 4 共see text兲. Experimental surface-tension values ( expt) are also indicated.
calc 共dyn/cm)
m
Water-benzene ( expt⫽35.0)
Water-CCl4 ( expt⫽45.0)
Water-octane 共octane monomer兲 ( expt⫽51.7)
Water-octane 共ethane 4-mer兲 ( expt⫽51.7)
1 2 3
34.7 42.5 42.5
37.4 44.2 44.2
44.3 49.1 49.2
58.1 54.2 52.1
II–IV were performed with Materials Studio 2.2 version of the DPD code from Accelrys,27 in a 20⫻10⫻10 box containing a total number of 6000 beads for ¯ ⫽3, and 10 000 beads for ¯ ⫽5. For moderate interaction parameters 共⬍100兲 a time-step of 0.05 共in DPD units兲 was used, while larger interactions necessitated smaller time-steps in order to ensure numerical stability. For each -calculation an ‘‘equilibration’’ period of 500 DPD units 共corresponding to 10 000 steps with a step-size of 0.05兲 was followed by a production stage of 2500 DPD units. The bead-size was defined in terms of water monomers, i.e., V bead,m ⫽mV molecule共water)⫽30 m 共 Å3 兲 .
共33兲
The important results are summarized below: 1. For monomer—monomer systems like water-benzene and water-CCl4 , Tables II, III, and IV suggest that computed from pure-component solubilities using Eq. 共22兲, along with a WSG-like fit yields a nearly bead-size-independent calc for m⭓2. The fall off of calc for m⬎2 in Table II is clearly due to the GW-fit, which is not applicable for large ¯ . Tables III and IV attempt to correct this by values of ⌬a using the extended WSG-fit 共see discussion in above paragraph兲, which is valid only up to m⬃3 for our systems. Ideally, both for the water-benzene and water-CCl4 systems, one would like to consider a value m⬃5, which is closer to the size of the larger molecule 共see Table I兲. Assuming that calc at this m is essentially the same as for m⫽2 or 3, one estimates a surface tension that is 20% higher than experiment for the water-benzene system, and to within 2% for the water-CCl4 system. 2. More importantly, a bead-size-independent calc can be used to parametrize the DPD interactions ¯a AB,m from experimental surface tension without requiring the computation of in the first place. This is particularly important for m⫽3 or larger 共a typical case兲 where neither GW nor WSG fits are applicable. As an illustration, for the water-benzene system if we consider m⫽5, ¯ ⫽3, and ¯a AA,m ⫽25, the experimental interfacial tension of 35.0 dyn/cm can be used to fit a cross-interaction parameter of ¯a AB,m ⬃138. 3. Column 4 of Table IV lists the computed values of calc for the water-octane system, where the octane molecule has been treated as a single bead. The values at m⫽2 or 3 are in excellent agreement with experimental interfacial ten-
1599
sion. Although, static properties like surface tension may be well estimated by treating the octane molecule as a single bead, there are many other applications where one would ideally like to treat the octane as a polymer of smaller beads.28 For polymer chains, a recipe has been developed relating a statistical unit 共e.g., a Kuhn segment29,30兲 of the real polymer by a single bead in a DPD chain. The amount of material that each bead comprises is related to the characteristic ratio or persistence length, which can be calculated from RIS theory31 or by Monte Carlo simulations. In this work we have simply assumed the smallest bead to be an ethane molecule, i.e., the octane molecule is considered to be a 4-mer of ethane beads. 4. Column 5 of Table IV lists calc for the water-共ethane) 4 system where the same DPD interaction parameters as for the water-octane 共monomer兲 system have been used. Ideally, one would like to use experimental interfacial tension to fit DPD Interaction parameters for water-共ethane) 4 system. However, as we see for m⫽3 共the ideal value for an ethane bead, see Table I兲, the agreement with experimental interfacial tension is excellent even with ¯a AB,m determined from the extended WSG-fit on the monomer system. This suggests that in general, a monomer— monomer simulation outlined in this article could be used to obtain an initial guess of the interaction parameter, which could be further refined to fit experimental surface tension. 5. The last two rows in Table III display results when ¯a AA,m scales linearly with m.15 In this case, calc clearly increases as a function of m, and is inconsistent with experimental values of interfacial tension for larger values of m. This provides further support for bead-size-independent ¯a AA,m 关Eq. 共26兲兴 as employed in this work. V. SUMMARY AND DISCUSSION
We have returned to the issue of relating a coarsegrained model of a complex fluid to the underlying chemistry. The important work of Groot and Warren created a link between the calculable Flory–Huggins interaction parameter , and the DPD interaction values. In this article, we have tried to address some of the open questions regarding selfinteractions, scaling with bead volume, and relationship with experimental interfacial tension. We conclude the following: 1. For a monomer—monomer system, the use of solubility parameters to obtain 关Eq. 共22兲兴, and then using the ¯ ) to determine DPD interaction WSG fit 共for not too large ⌬a parameters yield interfacial tension that is independent of the bead-size m, and also in great agreement with experiment. Thus, for large bead volumes in strongly segregated systems it would be consistent to determine DPD interactions ¯a AB by fitting computed interfacial tension to experimental values. 2. One can easily extend the above fitting procedure to monomer—polymer systems, with a good initial guess provided from monomer-monomer simulations. 3. In a segregated binary mixture a bead-size¯ AA ) appears to independent self-interaction parameter (a yield the correct interfacial tension. Scaling of ¯a AA with bead-size16 leads to strongly bead-size-dependent interfacial tension that is much higher than experimental values. Of course, even in this case one can fit the DPD interaction
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1600
J. Chem. Phys., Vol. 120, No. 3, 15 January 2004
A. Maiti and S. McGrother
TABLE V. Variation of ␣ with ¯ 共constant ¯a ). ¯a 25.0
300.0
¯
␣
2.0 3.0 4.0 5.0
0.083 0.092 0.096 0.098
2.0 3.0 4.0 5.0
0.086 0.099 0.099 0.100
parameters 共for a given m兲 such that the computed interfacial tension agrees with experimental value. However, such a procedure would not necessarily be consistent with GW or WSG like relations for smaller values of m (m⫽2 or 3兲. For a mixture of more than two species, it is not obvious as to what the best choice for an optimum bead-size should be. One essentially has to look for the largest ‘‘common’’ size that can be representative of the building blocks of various molecular/polymeric components of the mixture, as was the approach adopted by Groot and Rabone.16 In complex cases, it might become necessary to define beads of different sizes to represent different molecular/polymeric components. Such an extension might necessitate different values of ¯ , R c and/or ¯a AA for different bead-types, which would warrant a generalization of the theoretical framework as outlined in the present work.
¯ ⭐300.32 Also the variation of ␣ as a function range 15⭐a tion of ¯ is very small at ¯ ⫽5, especially at large interactions. For ¯ ⫽3 the variation in ␣ is somewhat larger, especially as a function of ¯ at smaller interactions. One of the ¯ is quantities that can be potentially affected by a large ␣ / the calibration of intra-species interaction ¯a AA by Eq. 共23兲, which should get modified to
冉
2 ␣⫹
冊
¯ ␣ ¯a ¯ ⫽ ⫺1 ⫺1. ¯ AA 2
¯ ) has a However, at ¯a AA ⫽25, the quantity ( ␣ ⫹¯ /2 ␣ / ¯ much smaller variation as a function than ␣ itself, being 0.095, 0.100, 0.101, and 0.101 for ¯ ⫽2, 3, 4, and 5, respectively.33 Thus, using an effective value ␣⬇0.1 in Eq. 共23兲 is an excellent approximation for obtaining ¯a AA , especially for ¯ ⫽3 or larger. 1
P. J. Hoogerbruge and J. M. V. A. Koelman, Europhys. Lett. 19, 155 共1992兲; J. M. V. A. Koelman and P. J. Hoogerbruge, ibid. 21, 363 共1993兲. 2 Y. Kong, C. W. Manke, W. G. Madden, and A. G. Schlijper, Int. J. Thermophys. 15, 1093 共1994兲; A. G. Schlijper, P. J. Hoogerbruge, and C. W. Manke, J. Rheol. 39, 567 共1995兲. 3 P. Espanˇol and P. Warren, Europhys. Lett. 30, 191 共1995兲. 4 R. D. Groot and P. B. Warren, J. Chem. Phys. 107, 4423 共1997兲. 5 P. J. Flory, Principles of Polymer Chemistry 共Cornell University Press, Ithaca, New York, 1953兲. 6 J. H. Hildebrand and R. L. Scott, The Solubility of Non-Electrolytes 共Reinhold, New York, 1949兲. 7 F. Case and J. D. Honeycutt, Trends Polym. Sci. 2, 259 共1994兲. 8 M. Doi, Introduction to Polymer Physics 共Clarendon, Oxford, 1996兲, Chap. 2. 9 J. M. Haile, Molecular Dynamics Simulation: Elementary Methods ACKNOWLEDGMENTS 共Wiley, New York, 1992兲, Appendix B. 10 It should be noted that for small ¯ (Ⰶ1) the value of ␣ is not a constant 共as The authors would like to thank Accelrys, Inc. for supcan be guessed from Table V itself兲, and in fact is strongly ¯ -dependent. port of this work. We would also like to acknowledge useful Therefore, the present theory does not apply to gases. 11 If this is violated, e.g., if p A ⬎p B , then A will compress B until the presdiscussions with Dr. David Rigby. sures become equal. But that would imply that at equilibrium ¯ A ⬍¯ B , which is a contradiction. APPENDIX: DEPENDENCE OF ␣ ON INTERACTION 12 This result can be generalized to mixing species A and B by hypothesizing PARAMETER AND BEAD-DENSITY a third species C, which does not mix with either A or B. By previous ¯ CC and ¯a BB ⫽a ¯ CC , which implies arguments, that would imply ¯a AA ⫽a In this Appendix we estimate the errors involved in us¯a AA ⫽a ¯ BB . 13 critical ing a value of ␣ independent of bead-density ¯ and DPD is the value of at the critical point separating homogeneous and interaction parameter ¯a . Tables V and VI list the computed segregated mixtures, and is given by4 critical⫽1/2(1/冑N A ⫹1/冑N B ) 2 for a binary mixture of polymers with chain lengths N A and N B , respectively. values of ␣ as a function of ¯ 共constant ¯a ) and as a function In particular, for a monomer-monomer mixture, critical⫽2. of ¯a 共constant ¯ ), respectively. The ranges investigated are 14 C. M. Wijmans, B. Smit, and R. D. Groot, J. Chem. Phys. 114, 7644 ¯ ⭐300, which typically covers almost all 2⭐¯ ⭐6 and 15⭐a 共2001兲. simulations of interest. Upon examining Tables V and VI, 15 Groot and Rabone16 considered the possibility of linear scaling of ¯a AA as one immediately discovers that at ¯ ⫽5, the variation of ␣ is a function of bead-size. However, that leads to an increasing surface tension as a function of bead-size, as shown in Sec. IV of the text. very small 共⬍2%兲 as a function of ¯a over the whole interac16 R. D. Groot and K. L. Rabone, Biophys. J. 81, 725 共2001兲. 17 Amorphous cell is a module in the Materials Studio Modeling software package from Accelrys Inc. See, http://www.accelrys.com/mstudio/ TABLE VI. Variation of ␣ with ¯a 共constant ¯ ). msគmodeling/amorphous.html 18 H. Sun, J. Phys. Chem. B 102, 7338 共1998兲. ¯ ¯a ␣ 19 Discover is a module in the Materials Studio Modeling software package 25.0 0.092 from Accelrys Inc. See, http://www.accelrys.com/mstudio/msគmodeling/ 3.0 100.0 0.095 discover.html 20 J. E. Mark, editor, Physical Properties of Polymer Handbook 共AIP, New 200.0 0.098 York, 1996兲. 300.0 0.099 21 A. F. M. Barton, editor, Handbook of Solubility Parameters 共CRC, Boca 15.0 0.098 Raton, FL, 1983兲. 22 5.0 100.0 0.099 C. F. Fan, B. D. Olafson, M. Blanco, and S. L. Hsu, Macromolecules 25, 200.0 0.100 3667 共1992兲. 23 S. J. Mumby and P. Sher, Macromolecules 27, 689 共1994兲. 300.0 0.100 24 D. Rigby and R. J. Roe, J. Chem. Phys. 89, 5280 共1988兲. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.115.190.38 On: Thu, 11 Jun 2015 17:54:36
J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 B. E. Eichinger, D. Rigby, and J. Stein, Polymer 43, 599 共2002兲. We assume the same WSG fit for ¯a AA ⫽50 and ¯a AA ⫽75 as for ¯a AA ⫽25. See Fig. 3 of Ref. 14. 27 See Accelrys page: http://www.accelrys.com/mstudio/msគmodeling/ dpd.html 28 For DPD simulations with polymers, an extra ‘‘spring-interaction’’ term is added between connected beads. A form proportional to the bead-bead separation was used in our simulations of Table V,4 with a spring-constant of 4.0.
Bead-bead interaction parameters in DPD
1601
W. Kuhn, Kolloid-Z. 76, 258 共1936兲. G. Strobl, The Physics of Polymers, 2nd ed. 共Springer, Berlin, 1997兲. 31 P. J. Flory, Macromolecules 7, 381 共1974兲. 32 For larger values of ¯a , one needs to use smaller time-steps of integration in order to avoid numerical instability. Thus for ¯a ⫽25, one can safely use a time-step of 0.05, while for ¯a ⫽300 one needs to reduce the time-step to ⬃0.02 共in DPD units兲. 33 ¯ ) were estimated by fitting the following The values of ( ␣ ⫹¯ /2 ␣ / functional form: ␣ (¯ )⫽ ␣ 1 ⫹ ␣ 2 e ␣ 3¯ at ¯a ⫽25.
25
29
26
30
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