2.2 Model
2.2.5 Evolution by the Monte Carlo method
Initially, randomly distribute surfactant particles (SP) and
colloidal particles (CP)
Evaporate the droplet by reducing the contact angle and height
Try an attempt to randomly move in a lattice for each SP
Try an attempt to randomly move in a lattice for each CP
Accept each attempt with Pacc=exp(-H
/kT)
Update the time step t = t+ 1
Finally, obtain the final deposition patterns for the
SP and CP At time t
If t< tf
If t = tf Aceept each attempt with
Pacc=exp(-H/kT)
Calculate the velocity with the given SP distribution at surface
Figure 2.2: The algorithm of the Monte Carlo simulation.
and CP movements using a MC procedure described below. For random walks in a drop, the drop height is prescribed at this time step and the velocity field is obtained from the previous time step. Upon the completion of each MC step, we “evaporate” the drop by decreasing the contact angle θ. According to the decrease of θ, the drop height, h(r, t), is computed. Although θ can be computed directly from the mass loss of the drop by evaporation [20], we assume that θ decreases linearly with each MC step, which is the first-order solution forθ. The linear decrease of θ in time is consistent with the full drop dynamics when the initial contact angle is small enough [69]. At each MC step, therefore, we update the drop height, the corresponding cell numbers in the z-direction, and the solvent velocity. In addition, particles outside of the drop at a new MC step due to the decrease of drop height, are forced to move only downward to simulate the downward movement of particles pushed by the drop surface.
In principle, the evaporation rate can be controlled by varying either Nmove ormc, or by changing capillary flow velocity through the velocity scaleR0/tf in eqn 2.23. Increasing Nmove increases the number of attempts of particle motions in a given MC cycle; hence more particles move toward the contact line during the MCS, leading to an effective increase of the evaporation rate. On the other hand, increasing mc increases the MC time to fully dry the drop, which effectively decreases the evaporation rate. In practice, we takeNmove,CP=NCP andNmove,SP =NSP so that all particles have at least one chance to move during ∆. The MC dynamics at each MCS are as follows: since the distribution of SP affects the dynamics of CP, a single move of one SP is supposed to change the velocity field ~u and subsequently the moves of all CP. However, all individual SP move at the same time per MCS rather than sequentially. Therefore, we do not update the flow ~u until all SP have a chance to move based on Hν; then, the dynamics of all CP is computed by using the updated velocity ~u at the MC step. Since the velocity ~u is not affected by the concentration of CP, moves of CP mimic the simple tracers of the velocity flow and the dynamics evolve only via convection. In the absence of SP, the process of CP self-assembly is related to classical coffee ring effects due to the decoupling between the convection of CP and the flow field. Our model should reflect this basic phenomenon.
The dynamics of our model is stochastic both for convection of CP and SP and
for sorption of SP. Initially, the lattice cells except on the surface are randomly filled with CP and SP (Fig. 2.1). The evolution of the configurations computed by Monte Carlo dynamics: the random walk of each particle is conducted in either the horizontal or vertical direction (see Fig. 2.2). During each cycle, the total number of particles is conserved. We attempt to move a randomly-chosen particle at a cell (i, j) to the nearest neighbor (i0, j0), resulting inAij →Aij−1 and Ai0j0 →Ai0j0+ 1, where Aij is eithernij or mij. In particular, SP random walks on the surface may be attempted into the nearest neighbors on the surface: a randomly-chosen particle at a surface cell (i, j =si) attempts a move to a nearest neighbor (i0, j0 = si0) on the surface, resulting in Mi → Mi−1 and Mi0 →Mi0 + 1. These attempts are accepted with Metropolis criteria with a Boltzmann probability, pacc= exp(−∆Hν/kBT), where ∆Hν is the resulting change in Hamiltonian in eqn (2.10).
Finally, the deposition of CP or SP is modeled by introducing a maximum number of particles that can be filled in a cell (i, j):
nmax= the integer rounded down from
φbmaxVij VP
, (2.27)
where 0 ≤φbmax ≤1 is the given maximum volume fraction of CP or SP, the volume of a particle is VP = 4πR3/P/3, and RP is the radius of the particle, CP or SP. The volume of a cell at (i, j) is Vij = dr2dz and dz are the unit lengths of a lattice cell. The deposition phase of a cell (i, j) is defined when nij > nmax. Hence, no MC move may put more CP or SP into the cells above the deposition phases. As discussed above, we also limit the number of SP on the surface by the maximum value to simulate different adsorption characteristics of the surfactant:
Mmax= the integer rounded down from
φsmaxAsj ASP
, (2.28)
where 0 ≤ φsmax ≤ 1 is the maximum surface area fraction of SP, Asi, is the surface area at a cell (i, si), ASP = πR2SP, and RSP is the radius of SP. We assume that in the model the surface area Asi depends only on the contact angle, making it constant with
radius but decreasing with time. Then the maximum number concentration of SP on the surface, Γ∞, used in eqn (2.3), becomes
Γ∞= Mmax
Asi = φsmax
ASP (2.29)