Molecular Dynamics of Water, Xenon, and Zinc Membrane
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This simulation models the transport of xenon gas through a zinc membrane, surrounded by water molecules. The process is governed by several key physical chemistry principles:
The O-H bonds in water are modeled using a harmonic potential:
\[ V_{bond}(r) = \frac{1}{2} k_{bond} (r - r_{eq})^2 \]
Where \(k_{bond} = 50000\) kJ/(mol·nm²) and \(r_{eq} = 0.09572\) nm.
The H-O-H angle is also modeled using a harmonic potential:
\[ V_{angle}(\theta) = \frac{1}{2} k_{angle} (\theta - \theta_{eq})^2 \]
Where \(k_{angle} = 500\) kJ/(mol·rad²) and \(\theta_{eq} = 104.52^{\circ}\) (1.8242 radians).
Electrostatic interactions are calculated using Coulomb's law:
\[ F_{coulomb} = k_e \frac{q_1 q_2}{r^2} \]
Where \(k_e = 138.9355\) kJ·nm/mol and charges are \(q_O = -0.834 e\), \(q_H = 0.417 e\), and \(q_{Xe} = -1 e\).
The simulation uses the Velocity Verlet algorithm to integrate the equations of motion:
\[ \mathbf{r}(t + \Delta t) = \mathbf{r}(t) + \mathbf{v}(t) \Delta t + \frac{1}{2} \mathbf{a}(t) \Delta t^2 \]
\[ \mathbf{v}(t + \Delta t) = \mathbf{v}(t) + \frac{1}{2}[\mathbf{a}(t) + \mathbf{a}(t + \Delta t)]\Delta t \]
Temperature control is implemented using a velocity rescaling thermostat, which scales velocities to maintain the target temperature:
\[ \lambda = \sqrt{\frac{T_{target}}{T_{current}}} \]
\[ \mathbf{v}_i \rightarrow \lambda \mathbf{v}_i \]
Xenon transport through the zinc membrane occurs via several mechanisms:
The rate of xenon transport through the membrane can be approximated using a modified Arrhenius equation:
\[ \text{Rate} = A \cdot e^{-\frac{E_a}{RT}} \cdot C_{Xe} \cdot (1 - \theta) \]
Where \(E_a\) is the activation energy, \(C_{Xe}\) is xenon concentration, and \(\theta\) is the fractional occupancy of membrane sites.
The simulation parameters are derived from quantum chemistry calculations that provide:
These quantum-level calculations ensure accurate representation of molecular interactions and dynamics at the atomic scale.
The simulation combines several computational approaches:
Total energy in the NVE ensemble should be conserved:
\[ E_{total} = E_{kinetic} + E_{potential} = \text{constant} \]
Small fluctuations may occur due to numerical integration and thermostat effects.