Franz J. Vesely > Lectures > Hard body shapes

 F VESELY EXP PHYS U VIENNA

THE SHAPE OF THINGS
Morphology of hard bodies and excluded volumes

CECAM Workshop "Simulations of hard bodies"
Lyon, April 2007

Sponsored by: Cecam, Simbioma, ESF, COST

Lectio neque in urbe neque Lugduno recitata

 Figure 1: Surface-surface distance $d$ as function of time (From A. Scala/De Michele) We seek an economical (quick) method to determine the first zero-line crossing of $d(t)$. The function $d(t)$ depends on the shapes $r_{s}$ and orientations $\Omega(t)$ of the bodies. Alternative: Let $r_{c}(\Omega_{1},\Omega_{2})$ denote the contact (center-center) distance at the given orientations; then the function $\tilde{d}(t) \equiv r(t)-r_{c}(t)$ has the same zeros as $d(t)$. However, $r_{c}$ may be computed in advance, before the actual simulation. Here is what we need: Ways to describe the shape of a particle, either implicitly via a constraint equation, or explicitly by some expression for the surface points, $\vec{r}_{s}(\vec{\alpha})$ or $r_{s}(\vec{\alpha})$, where $\vec{\alpha}$ denotes some internal coordinates (usually angles.) Ways to describe the "shape" of $r_{c}(\Omega_{1},\Omega_{2})$, the contact (center-center) distance at the given orientations, which is a characteristic of the mutually excluded volume. $(\Omega_{1},\Omega_{2})$ may amount to 2,3, or 5 arguments. Ways to describe the evolution of $\Omega$ in time. There are beautiful papyri on this in the literature.
I
• A PLETHORA OF SHAPES

 Fused Hard Spheres (also in more general arrangements) Hard Spherocylinder (HSC) Hard Ellipsoid Superellipsoid, or Spheroellipsoid Hard Gaussian Overlap (HGO), or Hard Lennard-Jones Lines (HLJL) Weird shapes, 2D Weird shapes, 3D etc. etc.
 Reminder: One particle has no meaning; two make a substance. The only relevant probe to determine the shape of a particle is another particle. The problem $r_{s} \longrightarrow r_{c}$ is always solvable, if not always uniquely. The inverse problem $r_{c} \longrightarrow r_{s}$ is overdetermined. The "ghostlike" HGO and HLJL models are actually defined via their pair contact distance; they have no individual shape. But: We like to visualize individual particles Often a simple geometric shape is desirable as input to theory. Therefore: let us discuss $r_{s}$ for a while, keeping in mind that it is mostly just assumed and may be taken with a grain of salt. And for the treatment of the actual collision event we will need some $r_{s}$ after all: point of contact; transfer of momenta etc.
II
• SHAPE DESCRIPTORS

1. Constraint equations

• Prime example: Ellipse/Ellipsoid

$\sigma(\vec{r},C) \equiv \vec{r}^{T} \cdot C \cdot \vec{r} - 1 = 0$

may be understood as a constraint eqn.; useful e.g. for finding the normal vector at some boundary point $R(\vec{r})$: $\vec{n} = \lambda \nabla_{R} \sigma$.

• Spherocylinder
Again, the requirement of equal (a) point-line distance from axis (for cylinder wall) or (b) point-point distance from end points (for caps) may be seen as 3 constraint eqns.,

$\sigma \equiv \left | \vec{r} - \vec{p} \right| = 0, \;\;\;\;\;\; \sigma \equiv \left| \vec{r}-\vec{r}_{a,b} \right| = 0$

where $\vec{p}$ is a point on the axis in a normal plane through $\vec{r}$, and $\vec{r}_{a,b}$ are the end points of the axis.

• Others:
Superellipsoids / Spheroellipsoids (F. V.) / Fused spheres / etc.
may be described this way.

To find the surface-surface distance, we may start from the constraint eqns. and apply one of the available strategies:
() Perram-Wertheim (for ellipsoids)
() Newton-Raphson à la De Michele
() "Gliders" method (F. V.):
• two virtual particles constrained to remain on their respective ellipsoid surfaces but attracting each other by a harmonic force;
• apply SHAKE to let them find the closest postions (proxy points)
Pretty quick; probably equivalent to Newton-Raphson; likely to be restricted to convex or at least starlike particles.

2. Series expansion of rs or rs2

Usually done in circular or spherical harmonics;
for symmetric tops a Chebysheff polynomial expansion along the $z$ axis is also possible, see Smectic demixing: Koda theory

• Ellipse / Circular Harmonics:
Let the long axis $a$ point up ($y$) and count polar angle $\theta$ from $y$ axis; then

$r_{s}(\theta) = \frac{\textstyle a b}{\textstyle \sqrt{a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta}}$

may be expanded as Fourier series in $\theta$: $r_{s}(\theta)=\sum_{k=0}^{K} a_{k} \cos k\theta$.

• Ellipsoid, symmetric or asymmetric / Spherical Harmonics:
$r_{s}$ as above; may be expanded in spherical harmonics. Since for symmetric ellipsoids the azimuthal angle $\phi$ is irrelevant, Legendre polynomials $P_{k}(\theta)$ suffice:
$r_{s}(\theta) = \sum \limits_{k=0}^{K} c_{k} P_{k} (\theta) \;\;\;\;{\rm with} \;\;\;\; c_{k} = \frac{1}{..}\int d cos\theta \; r_{s}(\theta) P_{k} (\theta)$
otherwise, use $Y_{k}^{m}$:
$r_{s}(\theta, \phi) = \sum \limits_{k=0}^{K} \sum \limits_{m=-k}^{k} c_{km} Y_{k}^{m} (\theta, \phi) \;\;\;\;{\rm with} \;\;\;\; c_{km} = \frac{1}{..}\int d cos\theta \int d\phi \; r_{s}(\theta) Y_{k}^{m} (\theta,\phi)$
where $1/..$ denotes the appropriate normalization.

As a specific example, here are an ellipse and a superellipse with $n=6$ together with their Fourier representations.

 Figure 2: Ellipse/Superellipse, original vs. Fourier reproduction (8 terms)

The task of finding the surface-surface distance and the related contact distance $r_{c}(\Omega_{1},\Omega_{,2})$ is painful:
• Choose plausible $\theta$ $\phi$ on both contours;
• compute $r_{s}^{(1,2)}(\theta,\phi)$ from the Fourier series; insert these $r_{s}$ in $x'=r_{s} \sin \theta \cos \phi$ etc. and transform $\vec{r}'=\{x',y',z'\}$ to lab coordinates $\vec{r}=\{x,y,z\}$;
• improve $\vec{r}^{(1,2)}$ by Newton-Raphson or similar such that $\left| \vec{r}^{(2)} -\vec{r}^{(1)} \right|$ is minimized.

3. Elliptical Fourier Analysis (EFA)

Starting from the parametric representation of an ellipse,

$\begin{eqnarray} x(t)&=& x_{0}+ a_{1,1} \cos \frac{2\pi t}{T} + a_{1,2} \sin \frac{2\pi t}{T} \\ y(t)&=& y_{0}+ a_{2,1} \cos \frac{2\pi t}{T} + a_{2,2} \sin \frac{2\pi t}{T} \end{eqnarray}$

where $T$ is the perimeter of the ellipse, we generalize these expressions, writing

$\begin{eqnarray} x(t)&=&x_{0}+\sum \limits_{k=1}^{K} \left[ a_{1,2k-1} \cos \frac{2\pi k t}{T} + a_{1,2k} \sin \frac{2\pi k t}{T} \right] \\ y(t)&=&y_{0}+\sum \limits_{k=1}^{K} \left[ a_{2,2k-1} \cos \frac{2\pi k t}{T} + a_{2,2k} \sin \frac{2\pi k t}{T} \right] \end{eqnarray}$

or

$\vec{r}_{s} = \vec{r}_{0} + A \cdot \vec{w}$

with $A$ the matrix of coefficients, and

$\vec{w} \equiv \left\{ w_{2k-1}, w_{2k} ; \, k=1, ..\right\}^{T} = \left\{ \cos (2 k \pi t/T), \sin(2 k \pi t/T) ; \, k=1, ..\right\}^{T}$

Note that the argument $2 \pi k t / T$, although by definition an angle, is not identical to the polar angle in the ellipse.

Inversion: Given a table of points $\{x_{i},y_{i}, i=1,N \}$ on an arbitrary closed contour the elements of $A$ may be computed according to

$\begin{eqnarray} a_{1,2k-1}&=& \frac{\textstyle T}{\textstyle 2k^{2}\pi^{2} } \sum \limits_{i=1}^{N} \frac{\textstyle \Delta x_{i}}{\textstyle \Delta t_{i} } \left[ \cos \frac{\textstyle 2 \pi k t_{i} }{\textstyle T } -\cos \frac{\textstyle 2 \pi k t_{i-1} }{\textstyle T } \right] \\ a_{1,2k}&=& \frac{\textstyle T}{\textstyle 2k^{2}\pi^{2} } \sum \limits_{i=1}^{N} \frac{\textstyle \Delta x_{i}}{\textstyle \Delta t_{i} } \left[ \sin \frac{\textstyle 2 \pi k t_{i} }{\textstyle T } -\sin \frac{\textstyle 2 \pi k t_{i-1} }{\textstyle T } \right] \end{eqnarray}$
et mut. mut.

 Figure 3: Ellipse/Superellipse, original vs. EFA (5 harmonics)

Finding the surface-surface distance and the related contact distance $r_{c}(\Omega_{1},\Omega_{,2})$:
"Gliders" method is simplified, as we need not enforce the constraint (" stay on the contour!"); the matrix $A$ defines the contour. Here is the procedure:
• Choose initial $t$ on both contours; since $\vec{r}=\vec{r}(\vec{w})$ is not invertible for $\vec{w}$ and thus for the parameter $t$, the basic ellipse with $k=1$ may be used in this step;
• assume harmonic force $\vec{f}_{12}$ between $\vec{r}_{1}$ and $\vec{r}_{2}$; project this force onto the local tangent $\vec{v}$:

$f_{v} \equiv \vec{f}_{12} \cdot \vec{v}/|\vec{v}| \;\;\;{\rm where} \;\;\; \vec{v} \equiv \frac{\textstyle d\vec{r}}{\textstyle d t} = A \cdot \frac{\textstyle d\vec{w}}{\textstyle d t}$

• change $t$ by a small $\Delta t$ proportial to $f_{p}$;
• iterate to minimize $\left| \vec{r}_{2} -\vec{r}_{1} \right|$.

 Figure 4: Gliding along a weird contour (red). The blue external point attracts a glider confined to the red outline. The estimated starting point is drawn in light blue.

Extension to 3D? $\Longrightarrow$ Use $Y_{k}^{m}$ in place of cos/sin.
Just playing around with a few low order terms produces this:

 Figure 5: A weird 3D surface

No inversion formulae in the literature, but may be worked out using orthonormality.

"Gliders" method may surely be extended to parametric surfaces.
III
• EXCLUDED VOLUME SURFACE
For given $\Omega_{1,2}$ the contact distance - measured between the particle centers - is called $r_{c}(\Omega_{1},\Omega_{2})$. The number of variables may be 2 (as in 2 dimensions), 3 (cylindrically symmetric particles in 3D), or 5 (asymmetric particles, 3D).

 Figure 6: $r_{c}$, the contact distance of two ellipses, original and reproduced from a double circular harmonics (Fourier) expansion

$r_{c}$ may have been calculated from the individual particles' shapes, or it may be given in the first place, without reference to $r_{s}$. The most prominent example for the latter case is the HGO (Hard Gaussian Overlap) "particle" defined via

$r_{c}(\vec{s},\vec{e}_{1},\vec{e}_{2}) = r_{0} \left[ 1-\chi \frac{\textstyle f_{1}^{2}+f_{2}^{2}-2 \chi f_{1} f_{2} f_{0}}{\textstyle 1-\chi^{2} f_{0}^{2}} \right]^{-1/2}$

where $f_{0}=\vec{e}_{1} \cdot \vec{e}_{2}$, $f_{1}=\vec{s} \cdot \vec{e}_{1}$, $f_{2}=\vec{s} \cdot \vec{e}_{2}$, with the center-center direction vector $\vec{s} \equiv \vec{r}_{12}/|r_{12}|$ and $\chi \equiv (l^{2}-d^{2})/(l^{2}+d^{2}))$, where $l$ and $d$ are the "length" and "width" of the HGO particle.

No earthly shape $r_{s}$ would reproduce this $r_{c}$ for all orientations.

The same holds for the hard body correlate to the Lennard-Jones Lines model potential. However, a glimpse at the "shape" of such HLJL particles is possible by requiring both partners to be mutually parallel.

 Figure 7: The "shape" of an HLJL particle, as scanned by second, parallel HLJL body. An ellipse with the same axes is drawn in green.

The above expression for $r_{c}$ of HGO particles is a special case of the "S function expansion" of pair properties for nonspherical objects (Stone, Zewdie). In the most general case the S-functions are combinations of three Wigner functions, but symmetry often reduces this complexity (as here.)
IV
• OUTLOOK (Pipedreams, rather)
• The time evolution of orientations $\Omega_{1,2}$ is known even for asymmetric tops (van Zon, Schofield)
• The dependence of $r_{c}$ on $\Omega_{1,2}$ is known (see above) so that $r_{c}$ may be reproduced quickly at simulation time
• Thus $\tilde{d}(t)$ is known and may be used to diagnose or predict events
• Maybe one could approximate the non-algebraic function $\tilde{d}(t)$ by a polynomial, Taylor or other? Then the root-finding should be easier. Here is a preliminary attempt at a Taylor expansion:

 Figure 8: Taylor expansion of $\tilde{d}(t)$ for two rotating ellipses

• $r_{c}$ recovered from series expansions is of limited accuracy. If that is not sufficient, the proposed method may at least be used for container dynamics.
• Containers may also be tailored to closely fit complicated particles, sharing the inertial tensor of their content.

... and if all else fails:

vesely apr-2007