Previously, we described how to interface two SDPD fluids featuring a different resolution using a transition zone that is divided into three parts: 1) splitting, 2) overlap, and 3) combining subregions [37] (see Chapter 4). In this approach, the coarse region is discretized into large fluid volumes, whereas the finely-resolved one is composed of a collection of less massive particles at a higher number density. If a large particle is transported to the refining region, it splits into fine SDPD particles. Similarly, a fine particle that crosses into the combining region is merged with another nearby fine particle into a single coarse one (see Fig. 5.1). The fine and coarse particles coexist in the overlap region, whose presence ensures a smooth transition of fluid properties across the interface between the two different scales [52]. In this section, we focus on the refining region. Note that in the prior work described in Chapter 4, we used a uniform continuum description for the entire problem domain, and combining/splitting rules were straightforwardly determined from conservation laws, i.e. such that the total mass, momentum, and solute remain constant. The rules in Ref. [37] are still used for the particles’ masses and momenta upon splitting and combining, and in this chapter we focus on the concentration variable alone, and how concentration values are assigned upon splitting and combining of particles. Unlike Chapter 4, we now consider the case where fine particles assume discrete identities (solute/solvent), whereas the coarse particles adopt a continuum of concentration values for the solute. Hence, when a continuum particle splits into two or more discrete particles (or vice versa), some of the dissolved species may be lost or gained, and will not be precisely conserved globally.
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Therefore, in the following, we do not preserve the total solute number exactly, and instead enforce a constant chemical potential.
For simplicity, we first consider the case where coarse particles are twice as massive as the fine ones (Fig. 5.1), i.e. we have fine particles with mass and smoothing length m1 and
h1, and large particles with m2 = 2m1 and h2 > h1. Thus, when a coarse particle enters the
refining region, it splits into two particles, each having half the mass of the parent. Later, we generalize this approach for situations involving large fluid volumes that are n times more massive that the fine ones, where n is an integer greater than one. For the present case, when a large particle i enters the refining region, it divides into two discrete particles j and k. When assigning concentrations to the daughter particles j and k, we have the following possibilities
Fig. 5.1. Illustration of discrete-continuous coupling. The left-hand-side of the simulation box features a “discrete” SDPD fluid with mass and smoothing length m1 and h1, respectively, where concentrations can
only assume values of 1 or 0, and hence each particle is either solute (orange) or solvent (green). On the right-hand-side, we have a normal SDPD fluid where particles are twice as massive (m2 = 2m1, h2 > h1)
and can have any concentration Φ. The large, continuous SDPD particles are able to split into fine, discrete ones when crossing into the refining region. Similarly, fine, discrete particles can combine into large particles upon entering the coarsening region.
h1 m1
interface
refining overlap coarsening
k j h2 m2 Φi j k Φ i “discrete” SDPD “continuous” SDPD
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(Fig. 5.2): 1) The large particle i splits into two solute particles ( j k 1), 2) the large particle splits into one solute and one solvent particle (e.g. j 1 and ), and 3) the k 0 large particle splits into two solvent ( j k 0). Note that for case 2), there are actually two possibilities: either j 1 and , or k 0 j 0 and . Therefore, this splitting k 1 move has a degeneracy of two ( 2 2). For the other two splitting options, there is no degeneracy ( ). 1 3 1
Next, we construct rules for selecting from these three possibilities such that a constant chemical potential is enforced. Consider a mixture of two types of particles (i.e. one where the jth particle has concentration of either j 1 or j 0) with average bulk solute
concentration . From simple combinatorics, we know that the probability of drawing a solute particle ( j 1 ) from the box is proportional to the average concentration,
j 1
P , i.e. a lower concentration of solute particles corresponds to a lower
probability of drawing a solute particle at random from the box. Therefore, the probability for drawing two solute particles (i.e. particles with concentration j k 1) is given by
Fig. 5.2. Three different possibilities when splitting a continuous particle into two discrete ones. For case 1, the large particle divides into two solute particles. In case 2, the particle splits into one solute and one solvent, and finally for case 3, the large particle breaks up into two solvent particles.
Φ
iΦ
iΦ
iΦ
icase 1
case 2
case 3
1 j 1 k 0 j 1 k 1 j 0 k 0 j 0 k
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j 1
k 1
2P P . Note that this is only exactly true if the first particle is replaced before drawing the second. However, for large systems comprised of many particles, this is a good approximation without replacement as well. Moreover, when the chemical potential is fixed this is exact, since drawing a particle would not affect the average concentration, which is maintained constant by contact with a bath. Hence, the probability for case 1) is:
2
1 1 .
P (5.8)
Similarly, the probability for drawing a solvent molecule is P
j 0
1 .Therefore, the probability for drawing one solvent and one solute is P
j 1
P
k 0
, and thus for case (2) we have
2 2 1 .
P (5.9)
Finally, the probability for a large particle splitting into two solvent particles [case (3)] is:
23 3 1 .
P (5.10)
Therefore, Eqs.(5.8), (5.9), and (5.10) with degeneracies and 1 3 1 give 2 2 the probabilities for each of the three cases in Fig. 5.2. When a particle crosses into the refining region and divides into two daughter particles, we have to choose from one of these three possibilities, and therefore the total probability that we pick one of these three outcomes must be unity. Due to the definition of the concentrations as mass fractions, the probabilities are normalized, i.e. l l 1
l P
. Summarizing, P1 gives the probability to splitinto two solute particles relative to the probabilities for the other two options, P2 is the
probability to split into one solute and one solvent relative to the other possibilities, and P3 is
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from one of these three outcomes is done using a Monte Carlo algorithm, where a uniformly distributed random number X is generated [X ~U
0,1
]. Then, if X P1, the particle splits into two solutes, P1 X P1P2 results in a solute and solvent, and X P1P2 gives two solvent particles.It is straightforward to extend these rules for large fluid volumes splitting into more than two small particles. For the general case of a coarse particle splitting into n fine particles, the probabilities become
0 1 1 1 1 2 2 2 2 3 3 0 1 1 , 1 , 1 , ... 1 . n n n n n n P P P P (5.11)Here, the lth degeneracy factor is the 1 n l
binomial coefficient. By coupling to a fluid with
coarse particles significantly more massive than the fine ones, there is greater reduction in the total number of particles used, giving further computational savings. However, note that if the discrepancy in the particle masses is too large, this leads to artifacts in properties such as the density distribution, as well instability due to heating upon particle insertion [52,37]. We have performed tests with n = 2, 4, and 6, and in this work present results from the n = 2 and 6 cases.
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