This is a homework of computational nanomechanics. The basic requirement is to use the Lagrangian function to describe the motion of particles. In homework, it requires 5 particles. As an enhancement, I rewrote the code with Qt and use GNU Scientific Library(GSL) to finish the task.
Concept
The Lagrangian equation of motion describe the particle motion with energy method, the equation is:
$$ \displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_k}-\frac{\partial L}{\partial x_k}=0$$
$ L$ is the Lagrangian function of the system, $ k$ denotes the degree of freedom.
$$ \displaystyle L = T – U$$
$ T$ is the kinetic energy and $ U$ is the potential energy.
Since the derivative of $ U$ with respect to $ x$ is the force
$$ \displaystyle f = -\frac{dU(r)}{dr}$$
the Lagrangian equation of motion uses another way to illustrate Newton’s second law.
Single particle example
For a single particle in a chamber, two degree of freedom $ x_1 = x, x_2 = y$, the kinetic energy is:
$$ \displaystyle T =\sum \frac{1}{2} m \left({\dot{x}}^2 + {\dot{y}}^2\right)$$
$ k$ is the degree of freedom. $ m$ denotes the mass of the particle.
The potential energy of the particle in a chamber is expressed as:
$$ \displaystyle U = \alpha e^{- \beta(R-r)}$$
Here the exponent function is used to describe the repulsion from the chamber wall. when the distance $latex R-r$ close to zero, the repulsion potential will be much large and prevent the particle penetrating the chamber wall. The Lagrangian function and equations are:
$$ \displaystyle L = \frac{m}{2} (\dot{x}^2+\dot{y}^2)-\alpha e^{-\beta (R-\sqrt{x^2+y^2})}$$
and for each degree of freedom:
$$
\displaystyle \left\{\begin{matrix}
m \ddot{x}(t) = -\alpha \beta e^{-\beta (R-\sqrt{x^2+y^2})} x(t) / \sqrt{x^2+y^2} \\
m \ddot{y}(t) = -\alpha \beta e^{-\beta (R-\sqrt{x^2+y^2})} y(t) / \sqrt{x^2+y^2}
\end{matrix}\right.
$$
For this example, we take the following parameters:
$$ \displaystyle \alpha = 10^{-27}, \beta = 4nm^-1, R = 10nm, m = 10^{-9}kg$$
The initial conditions are:
$$ \displaystyle x(0) = -3nm, \dot{x}(0) = 10nm/s, y(0) = 5nm, \dot{y}(0) = 20nm/s$$
Using Mathematica (FF01Y5.nb) to plot the trajectory or use this script (YR29Q8.nb) to plot the animation:
Lennard-Jones potential function
Lennard-Jones potential function is almost the world-famous potential function in nano mechanics. It describes the interaction between atoms. The function was proposed through quantum perturbation theory in which the attraction should follow a $latex r^{6}$ relation. It combines the attraction and repulsion together.
$$ \displaystyle U(r_{ij})=4 \varepsilon \left[ \left(\frac{\sigma}{r_{ij}} \right)^{12} – \left( \frac{\sigma}{r_{ij}}\right)^{6} \right]$$
where: $ \varepsilon$ denotes the depth of the energy well, and $ \sigma$ denotes the equilibrium spacing of atoms.
Take $ \sigma = 3, \varepsilon = 0.2$, the result is shown below:
As mentioned before, the derivative of $ U$ with respect to $latex r$ is the force. The force from Lennard-Jones interaction is simply get the derivative of $ U$:
$$ \displaystyle F=-U^{\prime}(r_{ij}) = -4 \varepsilon \left(\frac{6 \sigma ^6}{r^7}-\frac{12 \sigma ^{12}}{r^{13}}\right) $$
The equilibrium distance between two atoms is the position $ F=0$.
$$ \displaystyle \sigma \sqrt[6]{2} = r_0$$
$$ \displaystyle r_0 = 1.225 \sigma$$
Two particles example
If two particles are put into the chamber, Lennard-Jones potential should be considered in the calculation. Here, I try to use GNU Scientific Library(GSL) to solve the problem in three dimensions.
The Explicit 4th order (classical) Runge-Kutta(RK4) method is chosen for this problem. Two particles are denoted as $i$ and $ j$ respectively. The Lagrangian function is ($ k$ denotes the degree of freedom):
$$ \displaystyle L=T-U $$
$$ \displaystyle T = \sum \frac{1}{2}m_i (\dot{x}_{ik}^2)+\sum \frac{1}{2}m_j (\dot{x}_{jk}^2) $$
$$ \displaystyle U = E_{pi} + E_{pj} + W_{LJ}$$
$ E_{pi}$ and $ E_{pj}$ denotes the repulsive potential from the chamber wall, the equations of them are similar to Single particle example. Here, I just analyze the Lanner-Jones potential:
$$ \displaystyle W_{LJ-ijk} = 4 \varepsilon \left[ \frac{\sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^6} – \frac{\sigma ^ 6}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^3} \right]$$
For the derivative $ W_{LJ-ijk}$ with respect to $ x_{ik}$ and $ x_{jk}$:
$$ \displaystyle \frac{\partial W_{LJ-ijk}}{\partial x_{ik}} = 4 \varepsilon \left[ \frac{6 (x_{ik}-x_{jk})\sigma ^ {6}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^4} – \frac{12 (x_{ik}-x_{jk}) \sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^7} \right]$$
$$ \displaystyle \frac{\partial W_{LJ-ijk}}{\partial x_{jk}} = 4 \varepsilon \left[ -\frac{6 (x_{ik}-x_{jk})\sigma ^ {6}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^4} + \frac{12 (x_{ik}-x_{jk}) \sigma ^ {12}}{\left(\sum_{k=1}^3 \left( x_{ik} – x_{jk}\right)^2 \right)^7} \right]$$
With these two equations, we can simply calculate the motion of these two particles:
大神,原来的内容有没有备份?最近发现个人主页好像改版了。。
有备份,最近比较忙,暂时还没有时间整理以前的东西。