## Wednesday, July 6, 2016

### Force from Pairwise Potentials

Consider a pairwise potential, such as the Lennard Jones potential, $U(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right].$Question: How can we get the force (magnitude and force) from such a potential?

For simplicity let us assume that we have only two particles A and B. Specifically, what is the force on particle A due to particle B?

We know that the force is the negative gradient of the potential; therefore, \begin{align*} \mathbf{f}_{AB} & = -\dfrac{dU(r_{AB})}{d\mathbf{r}_A} \\ & = -\dfrac{dU(r_{AB})}{dr_{AB}} \color{blue} {\dfrac{dr_{AB}}{d\mathbf{r}_A}} \end{align*} How do we evaluate the term in blue?

Let $\mathbf{r}_A = x_A \mathbf{e}_x + y_A \mathbf{e}_y + z_A \mathbf{e}_z$, and $\mathbf{r}_B = x_B \mathbf{e}_x + y_B \mathbf{e}_y + z_B \mathbf{e}_z$ be the positions of the two particles. Let $\mathbf{r}_{AB} = \mathbf{r}_B - \mathbf{r}_A$ be a vector pointing in a direction from A to B. The distance between the two particles is:
$r_{AB} = \sqrt{x_{AB}^2 + y_{AB}^2 + z_{AB}^2},$ where $x_{AB}^2 = (x_B - x_A)^2$, etc.

We can now try to tackle the blue term: \begin{align*} \dfrac{dr_{AB}}{d\mathbf{r}_A} & = \dfrac{dr_{AB}}{dx_A} \mathbf{e}_x + \dfrac{dr_{AB}}{dy_A} \mathbf{e}_y + \dfrac{dr_{AB}}{dz_A} \mathbf{e}_z\\ & = \dfrac{1}{2 r_{AB}} \left(\dfrac{dx_{AB}^2}{dx_A} \mathbf{e}_x + \dfrac{dy_{AB}^2}{dy_A} \mathbf{e}_y + \dfrac{dz_{AB}^2}{dz_A} \mathbf{e}_z \right) \\ & = \dfrac{1}{2 r_{AB}} \left(-2 x_{AB} \dfrac{dx_A}{dx_A} \mathbf{e}_x - 2 y_{AB} \dfrac{dy_{A}}{dy_A} \mathbf{e}_y - 2 z_{AB} \dfrac{dz_{A}}{dz_A} \mathbf{e}_z \right) \\ & = -\dfrac{\mathbf{r}_{AB}}{r_{AB}} \\ & = -\hat{\mathbf{r}}_{AB} \end{align*} Thus, the force is, $\mathbf{f}_{AB}(r) = \dfrac{dU(r)}{dr} \hat{\mathbf{r}}_{AB}.$ If $U^{\prime}(r)$ is negative (repulsive regime of LJ, for instance), the direction of the force is along $-\hat{\mathbf{r}}_{AB} = \hat{\mathbf{r}}_{AB}$; this force points in the direction of A from B, trying to push particle A away from particle B. If $U^{\prime}(r)$ is positive (attractive part of LJ), the force acts along $\hat{\mathbf{r}}_{AB}$.