One of the principal input parameters to the model is the inter-particle interaction potential. In the current study, the inter-particle potential is modeled by DLVO theory [22,23]:

VTOT = VVDW + VEDL (8)

where *V*_{VDW }is the van der
Waals’ potential and *V*_{EDL }the electrostatic double layer
potential. The attractive van der Waals’ potential between equal-sized
particles can be modeled by many available expressions. In this study, it is
calculated by [24]:

(9)

where *A*_{H
}is the Hamaker constant and *s*=*r*/*a *the dimensionless
center-to-center separation between two particles scaled

by the particle
radius *a*.

The electrostatic interaction potential between the particlesiscalculatedfromthelinearsuperpositionapproximation of Bell et al. [25] and Chew and Sen [26]:

V_{EDL }= (10)

(11)

κ−^{1
}=(12)

AI

where κ−^{1
}is the Debye screening length, *N*_{A }the Avogadro’s
number, *I *the ionic strength of the solution, ε_{r }the dimensionless dielectric
permittivity of water, ε_{0
}the permittivity of free space, *n*_{0 }the ion number
concentration, *k*_{B }the Boltzmann’s constant, *T *the
absolute temperature, *z*_{s }the valence of ions in the bulk
solution, *e *the charge of an electron, and ψ_{s }the surface potential of the
particles which can be substituted by the zeta potential [27].

The second primary physical parameter inputted into the current model is the hydrodynamic drag. The expression for the hydrodynamic drag force derives directly from Stoke’s law for a single particle in an infinitely unbounded medium:

F_{Stokes
}= 6πµav (13)

where µ is the fluid viscosity and v the permeate velocity. For fluid flow through a porous medium composed of equal-sized spherical particles, the true hydrodynamic drag force exerted on each particle within the medium is adjusted by Happel’s correction factor Ω, defined earlier in Eq. (7), [19]:

Fhydro = F_{Stokes}Ω (14)

Fig. 2. Range of particle displacements for Monte Carlo simulation.

Monte Carlo simulation (MC) is a stochastic modeling method that is especially suitable to problems involving the dynamics of particle motion because of its capability to evaluate each discrete particle displacement. A classical method for performing MC begins by casting the process as being Markovian, meaning that the outcome for a particle displacement depends solely on the outcome that immediately precedes it. The system progresses from one state to the next by aid of its transition probability matrix. Metropolis et al. [28] present the now standard Metropolis method for conducting MC simulations using the energy of the system as the criterion to evaluate the acceptance or rejection of a MC step.

The Metropolis method begins by considering a system of particles
as shown in Fig. 2. The particle of interest
*i *is initially located at r_{i}^{m}.
Particle *i *is then proposed to be displaced randomly to a new location r_{i}^{n }using
the algorithm:

r_{i}^{n }= r_{i}^{m
}+ (2a_{0 }− 1)δr_{max }(15)

where *a*_{0 }is a uniform
random number between 0 and 1 and δ*r*_{max
}the maximum allowable displacement. The proposed displacement is next
evaluated for acceptance or rejection. The criterion for acceptance is the
change in potential energy *V _{n}*

) (16)

If *V _{n}*

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