Franz J. Vesely > Square Well Line Mixture

### Square Well Line Mixture: Parallel Particles

Consider two parallel lines of finite lengths $L_{1,2} = 2h_{1,2}$ containing a homogeneous density of square well centers. Let $\vec{r}_{1,2}$ and be the position of the two line centers, and write $\vec{r}_{12}$ for the vector between the centers. By assumption both direction vectors are $\vec{e}_{1,2}= \vec{e} \equiv (0,0,1)$; therefore, taking advantage of symmetry, we will now use cylindrical coordinates, writing the relative vector between particle centers as $\vec{r}_{12} = \left( \rho, z \right)$. For convenience we may always assume that $z \geq 0$; otherwise we let $z \rightarrow -z$, with no change of potential. We also require $h_{2} \geq h_{1}$, with no loss of generality.

Let $\lambda, \, \mu$ be the parameters giving the positions of the interacting points along $1$ and $2$. The squared distance between any two such points is given by $r^{2}(\lambda,\mu)= \rho^{2}+ \left( z + \mu - \lambda \right)^{2}$. The total interaction energy between the two lines is then

$u(1,2) \equiv u(\rho, z) = \frac{\textstyle 1}{\textstyle L_{1} L_{2}} \int \limits_{\textstyle -h_{1}}^{\textstyle h_{1}} d \lambda \int \limits_{\textstyle -h_{2}}^{\textstyle h_{2}} d \mu \, \, u_{SW}\left[ r(\lambda,\mu) \right] \;\;\;\;\;\;(1)$

with $u_{SW}(r)= \infty$ if $r < s_{1}$, $= - \varepsilon$ if $s_{1} < r < s_{2}$, and $=0$ for $r > s_{2}$. Typically, $s_{1}=1.0$ and $s_{2}=1.5 - 2.0$.

In the $(\lambda, \mu )$ plane the integration region is represented by a rectangle $R$ with sides $L_{1,2}$ around $(0,0)$. However, the integrand is non-zero, and constant, only for $r^{2}(\lambda, \mu ) < s_{2}^{2}$. In other words, the integral gives the area shared by the rectangle $R$ and a region $E$ between the two parallels described by $\mu^{\pm}(\lambda) = (\lambda - z) \pm \sqrt{s_{2}^{2}-\rho^{2}}$.

 The potential between two SWL particles is given by the area of the overlap region between the rectangle $R: \{ \lambda \, \epsilon \, [\mp h_{1} ] , \, \mu \, \epsilon \, [\mp h_{2} ] \}$ and the region $E$ between the lines $\mu^{\pm}(\lambda) = (\lambda - z) \pm \sqrt{s_{2}^{2}-\rho^{2}}$.

 Figure 1: Square Well Lines, parallel: The potential is given by the overlap area of the rectangle and the region between the red lines defined by $\mu(\lambda) =$ $(\lambda - z) \pm$ $\sqrt{s_{2}^{2}-\rho^{2}}$. The figure refers to a pair with lengths $L_{1}=2.4$, $L_{2}=2.$, and $s_{2}=1.5$; the $z$ displacement is $z = 0.4$, and the perpendicular distance $\rho=1.3$. Points $1$ and $2$ are defined thus: find the intersection of the horizontal line $\mu = \mu_{l} = -L_{2}/2$ with the upper and lower red line, respectively; if that intersection is outside the lower rectangle side, move to the nearest end of that side. Points $3$ and $4$ are defined similarly but referring to the upper rectangle border, $\mu = \mu_{u} = L_{2}/2$.

The task of computing the shaded area of Fig. 1 may seem trivial, but we have to define a procedure that comprises all possible configurations of the interacting sticks, and thus of $R$ and $E$. The following box presents this procedure in a self-contained formulation.

 Computing $u(1,2)$ for two unequal, parallel SW Lines: Let $\vec{r}_{1,2}$ be given, as well as the lengths $L_{1,2} \equiv 2h_{1,2}$ and the square well limits $s_{1}$ and $s_{2}$; the particles are assumed to point in the $z$ direction. The particle index $2$ is reserved for the longer stick, if any. To calculate $u(1,2)$ proceed as follows: Make sure that $z \; (= z_{12}) \geq 0$; otherwise let $z \rightarrow -z$. Make sure that the SW particles have no overlap, i.e. contain no points with a mutual distance below $s_{1}$: if $\rho^{2} > s_{1}^{2}$, there can be no overlap; else we have to discern the cases (a) $z \leq h_{1}+h_{2}$ (overlap) and (b) $z > h_{1}+h_{2}$ which leads to overlap if $\rho^{2}+(z-(h1+h2))^{2} \leq s_{1}^{2}$. Using $s_{2}$ in place of $s_{1}$ we can ascertain if there is any non-zero interaction at all. From now on we assume that there is an interaction but no hard overlap. Determine points $1$ and $2$ by computing the intersection points of the lower rectangle side $\mu = \mu_{l} \equiv -h_{2}$ with the upper/lower red lines (i. e. interaction range limits) given by $\mu^{\pm}(\lambda) = (\lambda - z) \pm$ $\sqrt{s_{2}^{2}-\rho^{2}}$; the desired points are either these intersections or the nearest endpoints of the lower rectangle side. The same relations are used to determine points $3$ and $4$ which refer to the intersection between the upper rectangle border $\mu = \mu_{u} \equiv h_{2}$ and the red lines: $\lambda_{1,2} = min ( h_{1}, \, max (-h_{1}, z-h_{2} \mp \sqrt{s_{2}^{2}-\rho^{2}}\, ) \, ) \;\;\;\;\;\;\;\;\;\;(2.1)$ $\lambda_{3,4} = min ( h_{1}, \, max (-h_{1}, z+h_{2} \mp \sqrt{s_{2}^{2}-\rho^{2}}\, ) \, ) \;\;\;\;\;\;\;\;\;\;(2.2)$ Compute the overlap area $A$ (grey in Fig. 1) according to $A = \int \limits_{\textstyle \lambda_{1}}^{\textstyle \lambda_{4}} \left(\mu^{+}(\lambda)-\mu^{-}(\lambda) \right) + \int \limits_{\textstyle \lambda_{1}}^{\textstyle \lambda_{2}} (\mu^{-}(\lambda)-\mu_{l} ) - \int \limits_{\textstyle \lambda_{3}}^{\textstyle \lambda_{4}} (\mu^{+}(\lambda)-\mu_{u} ) \equiv I_{0} + I_{12}^{-} - I_{34}^{+} \;\;\;\;\;\;\;(3)$ with $\begin{eqnarray} I_{0} & = & 2 \, \sqrt{s_{2}^{2}-\rho^{2}} \, \left(\lambda_{4}-\lambda_{1}\right) &(4.1) \\ I_{12}^{-} &=& \frac{\textstyle 1}{\textstyle 2 } \left( \lambda_{2}^{2}-\lambda_{1}^{2} \right) - \left( z - h_{2} + \sqrt{s_{2}^{2}-\rho^{2}} \right) \left(\lambda_{2}-\lambda_{1} \right) &(4.2) \\ I_{34}^{+} &=& \frac{\textstyle 1}{\textstyle 2 } \left( \lambda_{4}^{2}-\lambda_{3}^{2} \right) - \left( z + h_{2} - \sqrt{s_{2}^{2}-\rho^{2}} \right) \left(\lambda_{4}-\lambda_{3} \right) \;\;\;\;\;&(4.3) \end{eqnarray}$ The potential is then given by $u(1,2)=u(\rho,z)=- A \varepsilon/L_{1}L_{2}\, \;\;\;\;\;\;\;\;(5)$ A Java Applet is here      and a Fortran77 subroutine is here

### Area Integrals of $u(1,2)$

To apply density functional theory to parallel SWL mixtures we need the quantity

$J(z) \equiv 2 \pi \int d\rho \, \rho \, u(\rho, z) \;\;\;\;\;\;\;\;(6)$

for given values of the $z$ displacement. Equs. 2-5, while useful for the calculation of specific pair energies, do not contain the explicit forms of $\lambda_{i}(\rho)$ and are therefore not suited to a formal integration over $\rho$. Here we derive an alternative expression for $u(\rho,z)$, making the $\rho$-dependence explicit.

The terms $I_{0}$, $I_{12}^{-}$ and $I_{34}^{+}$ depend on pairwise combinations of $\lambda_{1-4}$, namely (1,4), (1,2), and (3,4). The functional forms of $\lambda_{i}(\rho)$ change between different regions of $\rho$. For example, $\lambda_{1}(\rho) = -h_{1}$ (constant) for all configurations in which $z-h_{2}-\sqrt{s_{2}^{2}-\rho^{2}} \leq -h_{1}$, or $\rho^{2} \leq s_{2}^{2}-(z-h_{2}+h_{1})^{2}$. In Figure 1 this corresponds to those situations in which the upper red line crosses the base of the rectangle left of its left boundary - as in the case sketched there. On the other hand, the form $\lambda_{1}(\rho) =$ $z-h_{2}-\sqrt{s_{2}^{2}-\rho^{2}}\,$ will hold when the intersection occurs within the base line boundaries, $\pm h_{1}$. Quantifying these considerations for all $\lambda_{i}(\rho)$ we may easily identify the various $\rho$ intervals. First we introduce the following parameters:

$\begin{eqnarray} e &\equiv& z-h_{2}-h_{1} & \;\;\;& e_{0} &\equiv& min(\, 0,\, e) & \;\;\;& e_{1} &\equiv& max(\, 0,\, e) \;\;\;\;\;\;\;\;(7.1) \\ f &\equiv& z-h_{2}+h_{1} & \;\;\;& f_{0} &\equiv& min(\, 0,\, f) & \;\;\;& f_{1} &\equiv& max(\, 0,\, f) \;\;\;\;\;\;\;\;(7.2) \\ u &\equiv& z+h_{2}-h_{1} & & \geq 0 & & & & & \;\;\;\;\;\;\;\;(7.3) \\ v &\equiv& z+h_{2}+h_{1} & & > 0 & & & & & \;\;\;\;\;\;\;\;(7.4) \\ \end{eqnarray}$

For given values of $z \geq 0$, $h_{1}$ and $h_{2} \geq h_{1}$ we have $f_{1} \geq e_{1}$ and $f_{0} \leq e_{0}$. The limits of hard overlap and of outer interaction range are $\rho_{min}^{2} \equiv max \left(\,0, \, s_{1}^{2}-e_{1}^{2} \, \right)$ and $\rho_{max}^{2} \equiv max \left(\,0, \, s_{2}^{2}-e_{1}^{2} \, \right)$. Within that relevant interaction region we define the following interval boundaries:

$\begin{eqnarray} \rho_{1}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - f_{1}^{2}) \,) \\ \rho_{2}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - e_{1}^{2}) \,) \;\;=\rho_{max}^{2} \\ \rho_{3}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - f_{0}^{2}) \,) \\ \rho_{4}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - e_{0}^{2}) \,) \\ \rho_{5}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - v^{2}) \,) \\ \rho_{6}^{2}&=& min(\rho_{max}^{2}, \, max(\rho_{min}^{2}, \, s_{2}^{2} - u^{2}) \,) \;\;\;\;\;\;\;\;(8.1-8.6) \end{eqnarray}$

Obviously, $\rho_{1} \leq \rho_{2}$, $\rho_{4} \leq \rho_{3}$, and $\rho_{5} \leq \rho_{6}$. Using these definitions we find the following scheme:

 Case $\lambda_{1}(\rho)$ $\rho$ interval 1A $-h_{1}$ $\left[\,\rho_{min}, \, \rho_{1} \, \right]$ 1B $z-h_{2}-\sqrt{s_{2}^{2}-\rho^{2}}$ $\left[\, \rho_{1}, \, \rho_{max} \,\right]$ 1C $+h_{1}$ $-$

 Case $\lambda_{2}(\rho)$ $\rho$ interval 2A $-h_{1}$ $\left[\, \rho_{3}, \,\rho_{max}\,\right]$ 2B $z-h_{2}+\sqrt{s_{2}^{2}-\rho^{2}}$ $\left[\, \rho_{4},\, \rho_{3}\,\right]$ 2C $+h_{1}$ $\left[\,\rho_{min}, \, \rho_{4}\,\right]$

 Case $\lambda_{3}(\rho)$ $\rho$ interval 3A $-h_{1}$ $\left[\,\rho_{min}, \,\rho_{5} \,\right]$ 3B $z+h_{2}-\sqrt{s_{2}^{2}-\rho^{2}}$ $\left[\, \rho_{5},\, \rho_{6}\,\right]$ 3C $+h_{1}$ $\left[\, \rho_{6}, \,\rho_{max}\,\right]$

 Case $\lambda_{4}(\rho)$ $\rho$ interval 4A $-h_{1}$ $-$ 4B $z+h_{2}+\sqrt{s_{2}^{2}-\rho^{2}}$ $-$ 4C $+h_{1}$ $\left[\,\rho_{min}, \,\rho_{max}\,\right]$

Table 1: For given $z$ the functions $\lambda_{i}(\rho)$ retain their form over the given intervals. The notation is obvious: 1A means that $\lambda_{1}'$ is left of the base line of the rectangle, 1B...on the base line, 1C...right of the base line.

The expressions 4.1-4.3 for $I_{0}$ etc. contain pairs of $\lambda_{i}$. Obviously, the $\rho$ intervals in which these combinations have a certain functional form are the intersections of the respective pair of intervals in the foregoing list. Here follows a complete listing of all non-trivial combinations of $\lambda$ and regions of $\rho$, together with the functional form of $I_{0}$, $I_{12}^{-}$ and $I_{34}$, as well as their respective integral functions; for convenience we use the name $t \equiv \sqrt{s_{2}^{2}-\rho^{2}}$ for that frequently occuring term:

 Case $\rho$ interval *) $I_{0}(\rho)$ $J_{0}(\rho)/2 \pi \equiv \int d\rho' \, \rho' \, I_{0}(\rho')$ 1A-4C $\left[\,\rho_{min}, \, \rho_{1} \right]$ $4 \, h_{1} \, t$ $- \frac{4}{3} h_{1} t^{3}$ 1B-4C $\left[\,\rho_{1}, \, \rho_{max} \right]$ $2\,s_{2}^{2}-2\,e\,t\,-2\,\rho^{2}$ $s_{2}^{2}\rho^{2} + \frac{2}{3} e \, t^{3} - \frac{1}{2} \rho^{4}$ $I_{12}^{-}(\rho)$ $J_{12}^{-}(\rho)/2 \pi \equiv \int d\rho' \, \rho' \, I_{12}^{-}(\rho')$ 1A-2B $\left[\,\rho_{4}, \, min(\rho_{1}, \rho_{3} ) \right]$ $-\frac{1}{2}(s_{2}^{2}+f^{2}) -f \, t + \frac{1}{2} \rho^{2}$ $-\frac{1}{4}(s_{2}^{2}+f^{2})\rho^{2} +\frac{1}{3}f \, t^{3} + \frac{1}{8} \rho^{4}$ 1A-2C $\left[\,\rho_{min}, \, min(\rho_{1}, \, \rho_{4}) \right]$ $2\,h_{1}(z-h_{2})-2\,h_{1} t$ $-h_{1}(z-h_{2})\rho^{2}+\frac{2}{3}\,h_{1} t^{3}$ 1B-2B $\left[\,max(\,\rho_{1},\,\rho_{4}), \, \rho_{3} \right]$ $-2\,s_{2}^{2}+2\,\rho^{2}$ $-s_{2}^{2}\rho^{2}+\frac{1}{2}\,\rho^{4}$ 1B-2C $\left[\,\rho_{1}, \, \rho_{4} \right]$ $- \frac{1}{2} \,(3 s_{2}^{2}- e^{2}) + e \, t +\frac{3}{2} \,\rho^{2}$ $- \frac{1}{4} \,(3 s_{2}^{2}- e^{2})\rho^{2} - \frac{1}{3} e \, t^{3} + \frac{3}{8} \,\rho^{4}$ $I_{34}^{+}(\rho)$ $J_{34}^{+}(\rho)/2 \pi \equiv \int d\rho' \, \rho' \, I_{34}^{+}(\rho')$ 3A-4C $\left[\,\rho_{min}, \, \rho_{5} \right]$ $-2 \, h_{1} \, (z+h_{2})+2 \, h_{1} \,t$ $- h_{1} (z+h_{2})\rho^{2}-\frac{2}{3} h_{1} t^{3}$ 3B-4C $\left[\,\rho_{5}, \, \rho_{6} \right]$ $\frac{1}{2}\,(u^{2}+s_{2}^{2}) - u \, t\, -\frac{1}{2}\,\rho^{2}$ $\frac{1}{4} (u^{2}+s_{2}^{2})\rho^{2} + \frac{1}{3} u t^{3} - \frac{1}{8} \rho^{4}$ 3C-4C $\left[\, \rho_{6}, \, \rho_{max} \right]$ $0$ $0$

Table 2: For given $z$ the relevant $\rho$ intervals are listed (second column). Within these intervals, some of which may be of zero width, the functions $I_{0}(\rho)$ etc. and their integral functions are listed (third and fourth rows.)

* Note: the expressions in column 3 may in principle be used as alternative formulae to equs. 2-5 to compute individual pair energies at specific ($z, \rho$); in that case only non-zero intervals must be considered.

 Here is a short resume: Area integral of the potential for parallel SWL sticks We want to calculate the integral $J(z) \equiv 2 \pi \int d\rho' \, \rho \, u(\rho'; z)$ where $u(\rho; z) = (- \varepsilon/L_{1}L_{2}) ( I_{0} + I_{12}^{-} - I_{34}^{+} )$ as in eq. 3. Since the functional form of $u(\rho; z)$ varies in different $\rho$ intervals we have to identify these intervals before we can formally integrate: () Determine the parameters $\rho_{min,max}$ and $\rho_{1-6}$ (eqs. 8.1-8.6); () insert these as limits in the expressions for $J_{0} \equiv 2 \pi \int d\rho' \, \rho' I_{0}(\rho')$ etc. as in Table 2; () these terms are summed over all intervals (rows) and combined to give $J(z)=J_{0}+J_{12}^{-}-J_{34}^{+}$. A Java Applet is here A Fortran code is deposited here

Franz Vesely, 2010