Particle-Particle Interactions
Coulomb Force
The Coulomb force on particle i in a system of N particles is defined as:
where
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e = 1.602176634 × 10−19 C is the elementary charge,
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ε0 = 8.854187817 × 10−12 F/m is the permittivity of vacuum,
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ri is the position vector of the ith particle (SI unit: m),
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rj is the position vector of the jth particle (SI unit: m), and
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Z (dimensionless) is the particle charge number.
The Coulomb force is repulsive for particles of the same charge and attractive for particles with opposite charge.
Lennard-Jones Force
The Lennard-Jones interaction is often used as the intermolecular potential function to estimate the transport properties of a gas. The Lennard-Jones potential is given by:
where r is the distance between the particles (SI unit: m), ε is the interaction strength (SI unit: Nm), and σ is the collision diameter (SI unit: m).
Figure 3-7: Plot of the Lennard-Jones potential normalized to the interaction strength versus the nondimensional radial distance.
Figure 3-8: Plot of the Lennard-Jones force versus nondimensionalized radial distance between particles.
Using the fact that the force is computed as:
the expression for the force on the ith particle becomes:
A typical interaction strength is around 1.6·10−21 J and a typical collision diameter is around 3.3·10−10 m (3.3 Å). The first term in the square brackets is a repulsive force due to internuclear repulsion, and the second term is an attractive force known as the London dispersion force (LDF). The variation of the force with distance is plotted in Figure 3-8.
Linear Elastic Force
The linear elastic force Fi on particle i in a system of N particles is defined as
where ks is the spring constant (SI unit: N/m) and r0 is the equilibrium distance between particles (SI unit: m).