Water molecule bond length: Difference between revisions
No edit summary |
|||
| Line 56: | Line 56: | ||
p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} | p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} | ||
\exp\left[- \frac12 \frac{mv^2}{k_B T} \right] | \exp\left[- \frac12 \frac{mv^2}{k_B T} \right] | ||
</math> | |||
The mean speed is (for 3D?) | |||
<math> | |||
v_\text{mean} = sqrt{ \frac{8k_B T}{\pi m}} | |||
</math> | </math> | ||
Revision as of 22:10, 12 October 2020
Introduction
Classical Mechanics
Newton Equations
where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=-\nabla V} . The system is 1D, thus gradient will be differential, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=-{\frac {dV}{dx}}} .
The mass should be the reduced mass of the oxygen--hydrogen system, but we use mass here.
Potential Function
Lennard--Jones potential with dimensionless parameters for TIPS model:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V(x)=4\left[\left({\frac {1}{x}}\right)^{12}-\left({\frac {1}{x}}\right)^{6}\right]}
where the distance is given in Ångstroms.
Integration
The finite differences (Euler method) are
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}v(t)&={\frac {x(t+\Delta t)-x(t)}{\Delta t}}\\a(t)&={\frac {v(t+\Delta t)-v(t)}{\Delta t}}\end{aligned}}} and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t) = -\frac{\frac{d}{dx}(V(x(t)))}{m} }
Velocity Verlet Algorithm
A very good and easy to implement integration method is velocity Verlet:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x(t + \Delta t) &= x(t) + v(t) \Delta t + \frac12 a \Delta t^2 \\ v(t + \Delta t) &= v(t) + \frac12\left( a(t) + a(t+\Delta t) \right) \Delta t \end{align} }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t) = a(x(t)) = - \frac1m \frac{d}{dx}V(x(t))} is given at Section . . .
Temperature/ Initial distribution
The initial velocity of the hydrogen atom is chosen randomly from the Maxwell-Boltzmann distribution at given temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} \exp\left[- \frac12 \frac{mv^2}{k_B T} \right] }
The mean speed is (for 3D?)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_\text{mean} = sqrt{ \frac{8k_B T}{\pi m}} }
Results
Issues
1D statement
References
D.T.W. Lin and C.-K. Chen: A molecular dynamics simulation of TIP4P and Lennard-Jones water in nanochannel, acta Mechanica 173, 181.194 (2004).
