Hard body models for molecules and colloids already produce
most of the bulk properties: isotropic -> nematic -> smectic -> crystalline
phases
No single standard model for elongated particles exists; there are
fused spheres, ellipsoids, spherocylinders, HGO particles, etc., see
my lecture on
Hard Body Modeling
Qualitative difference btw hard ellipsoids (ELL) and spherocylinders (SC):
SC can develop a smectic phase, ELL can not
Desirable: a model that has few parameters, yet can be transformed
continuously btw ellipsoid and spherocylinder
SUPs are not suitable, because no overlap criterion for randomly oriented pairs exists
Let's do SpheroEllipsoids:
Truncated ellipsoids with spherical end caps.
Only two shape parameters
Continuous transformation between a pure ellipsoid and a spherocylinder
Use DFT (later MC) to locate the onset of smecticity
Generic model particles for anisotropic molecules?
Overlap criterion for a pair in a general, non-parallel configuration is
given below
MODEL
The Model
$a,c$ ... short and long semiaxes; $z_{0}$ ... truncation height
Complete the truncated ellipsoid body by spherical segments such that the
tangents change continuously at the junction
this requirement places the cap centers at $\pm z_{c}$, with
$z_{c}=z_{0}(1-a^{2}/c^{2})$; the cap radius is
$r_{c}=a \sqrt{1-(z_{0}^{2}/c^{2})(1-a^{2}/c^{2})}$
Total length of spheroellipsoid: $L=2l=2(z_{c}+r_{c})$
Volume: $v=v_{body}+2v_{cap}$, where
$v_{body}=2 \pi a^{2}z_{0}(1-z_{0}^{2}/3c^{2})$ and
$v_{cap}=(2\pi/3)\,r_{c}^{3}-\pi \Delta z \left(r_{c}^{2}-(\Delta z)^{2}/3\right)$,
with $\Delta z \equiv z_{0}-z_{c} = z_{0}a^{2}/c^{2}$
Note: the only parameters are $c/a$ and $z_{0}/a$; all other quantities follow!
PARAMETER PLANE
Parameter Plane
All possible spheroellipsoids are situated below the diagonal $z_{0}=c$; points
on the diagonal represent ellipsoids
Blue: Shapes with fixed aspect ratio $l/a$. Given $c,$ $a$ and $l$, the required value of $z_{0}$ is
$
z_{0}(c,a,l)=l\left[ 1-\sqrt{(1/l^{2}-1/c^{2})/(1/a^{2}-1/c^{2})}\right]
$
Green: Shapes with fixed particle volume $v_{0}$; $z_{0}$ is determined
numerically
CLOSE PACKING
Close Packing
Simple parallel Ellipsoids, hexagonal configuration:
$\eta_{cp}=v/V_{cp}$, where $V_{cp}$ is the unit cell volume:
The point of contact between a particle and its nearest neighbor in the adjacent upper
layer has a $z$ coordinate of $z^{*}= c \sqrt{2/3}$, yielding a unit cell volume
of $V_{cp}'=8 a^{2} c/\sqrt{2}$.
Spheroellipsoids, hexagonal packing:
If the cutoff height $z_{0}$ is larger than $z^{*}= c \sqrt{2/3}$ the geometry of the
hexagonal unit cell is the same as for ellipsoids, and so $V_{cp}'=8 a^{2} c/\sqrt{2}$
If $z_{0} < z^{*}$ the contact occurs no more between the ellipsoidal
trunks but between the end caps; in this case the unit cell volume changes to
$V_{cp}''=4 a^{2}\sqrt{3}\, \left( z_{c}+\sqrt{r_{c}^{2}-4a^{2}/3} \right)$
DENSITY FUNCTIONAL THEORY
Density Functional Theory
We wish to determine the density at the nematic - smectic transition.
Second virial approximation:
$\phi_{ex}\left\{\rho(\mathbf{r})\right\} \equiv F_{ex}\left\{\rho(\mathbf{r})\right\}/NkT =
-(1/2N) \int d\mathbf{r} \int d\mathbf{r}\,'
\, \rho(\mathbf{r})\, \rho(\mathbf{r}\,') f(\mathbf{r}\, ,\mathbf{r}\,')$
where
$f(\mathbf{r}\, , \mathbf{r}\,') $ is the Mayer overlap function: $f=0$
except for the overlap region of the two objects, where $f=-1$
Try the ansatz $\rho(\mathbf{r}) = \bar{\rho} ( 1 + a\, cos kz )$
to detect a periodic density profile
Variation of the excess free energy density:
$\delta_{2} \phi_{ex} = ( a^{2} \bar{\rho}^{\,2}/4) \, I(k) $
where
$I(k) = \int_{\textstyle v_{exc}} d\mathbf{r} \, cos kz$
Adding the variation of the ideal gas free energy density
$ \delta_{2} \phi_{id} = a^{2}/4 $
we find
$\delta_{2} \phi = ( a^{2}/4 ) \left[ 1+\bar{\rho} I(k) \right]
= (a^{2}/4) \left[ 1+ \eta I(k) / v \right]$
where $\eta=\bar{\rho}v$ is the packing fraction
If at some $\eta = \eta_{s}$ and $k=k_{s}$ the term
$\left[ 1+ \eta I(k) / v \right]$ crosses zero,
the homogeneous (nematic) phase becomes unstable: $\Rightarrow$ smectic!
Prediction of transition density, 2nd Virial Approximation:
$\eta_{s} = - v / I(k_{s})$, where $k_{s}$ minimizes
$I(k)= \int_{\textstyle v_{exc}} d\mathbf{r} \, cos kz $
$\lambda_{s} \equiv 2 \pi / k_{s}$ is the layer distance
Parsons-Lee correction:
In the expression $\left[ 1+ \eta I(k) / v \, \right]$
replace $\eta$ with the Carnahan-Starling term
$\chi(\eta) \equiv (\eta-3\eta^{2}/4)/(1-\eta)^{2}$
and require that $\chi(\eta_{s}) I(k)/v = -1$
Prediction of transition density, Parsons-Lee:
$\eta_{s}=\left[(2w-1)-\sqrt{1-w} \right]/(2w-3/2)$,
with $w \equiv v/I(k)$
$\Rightarrow$ Central task: Calculation of $I(k)$
CALCULATE I(k)
Calculation of $I(k)$
For parallel particles the excluded volume is again a spheroellipsoid,
with axes $A=2 a$, $C=2 c$ and also $Z_{0}=2 z_{0}$,
$Z_{c}=2 z_{c}$ etc.
$
I_{cap}(k) \equiv \pi \int_{Z_{c}+\Delta}^{Z_{c}+R_{c}}
dz \, [ R_{c}^{2}-\left(z-Z_{c} \right)^{2} ] \, cos \, kz
=\pi \, K_{1}\,cos \, kZ_{c} -\pi k K_{2}\,( sin \, kZ_{c}/k) ,
$
where $\Delta \equiv Z_{0}-Z_{c}$ and
$
K_{1} \equiv \int_{\Delta}^{R_{c}}dz \, \left( R_{c}^{2}-z^{2} \right) \, cos \,kz
$
$\qquad = 2\, ( sin \, kR_{c}-kR_{c} \, cos \, kR_{c})/k^{3}
- 2\, ( sin \, k\Delta-k\Delta \, cos\, k\Delta)/k^{3}
- \left( R_{c}^{2} - \Delta^{2} \right) \,sin \, k \Delta /k
$
$ k \, K_{2} \equiv \, k \, \int_{\Delta}^{R_{c}}dz \,
( R_{c}^{2}-z^{2} ) \, sin \, kz
$
$= - 2\, ( R_{c} \, sin \, kR_{c}-\Delta \, sin \, k \Delta )/k
+ ( R_{c}^{2} \, cos \, k R_{c} - \Delta^{2} \, cos \, k\Delta )-
$ $\qquad \qquad \qquad\qquad
- 2\, ( cos \, kR_{c}- cos \, k\Delta )/k^{2}
- R_{c}^{2} \, ( cos \, kR_{c} - cos \,k \Delta )
$
All terms in these expressions are well-behaved when $k \rightarrow 0$
$I(k)$ for two spheroellipsoids with aspect ratio $l/a=4$
red: $(c/a, z_{0}/a)=(4.320, \, 3.612)$
green: $(c/a, z_{0}/a)=(10.000, \,3.079)$
DFT RESULTS
Results of DFT
Single Aspect Ratio:
For fixed $l/a=4$ we vary $c/a$ (with the respective $z_{0}/a$),
and determine the packing density at the N-S transition.
Aspect ratio $l/a=4$:
$\lambda/2l$ ... layer distance;
$\eta_{cp}$ ... close packing density
$\eta_{s}$ ... transition density
Dotted: true spherocylinders
How can DFT stability analysis predict smecticity?
Experience shows that a nematic-smectic transition occurs around
$\eta^{*}\equiv\eta_{s}/\eta_{cp} \approx 0.35 - 0.46$
If the instability of the nematic (z-homogeneous) phase occurs at
$\eta^{*} > 0.5$, this indicates a crystalline (not smectic) phase
Hence we choose the limit for acceptable transition densities
$\eta^{*}=0.5$
In the Figure above a sequence of shapes with $\lambda/a=4$ is scanned;
$c/a=4$: ellipsoid; $c/a \geq 10$: spherocylinder (almost)
N-Sm Transition: $(c/a)_{crit}=6.457$, with $z_{0}/a=3.205$
All Aspect Ratios:
Magenta: dividing line between smectogenic and
non-smectogenic spheroellipsoids
Blue lines connect shapes with equal volumes
Green lines connect shapes with equal aspect ratios
The magenta line separates particles that can develop a smectic phase and those that can not
Left: "ellipsoid-like" particles with no smectic phase
Right: "spherocylinder-like" shapes that may undergo a smectic transition
Example: $(c/a, z_{0}/a)= (12.0, 3.054)$...almost spherocylindrical, with $l/a=4$;
$\eta^{*}=0.422$ (near to the SC value of $0.386$). Keeping the aspect ratio
constant but decreasing $c/a$ we find that the $\eta^{*}=0.5$ limit is
reached around $(c/a, z_{0}/a)= (6.46, 3.21)$. For smaller $c/a$, the particles
are too "ellipsoidal" to display a smectic phase
$\Rightarrow$ Simulations!
OVERLAP CRITERION
Overlap Criterion for Two Spheroellipsoids
Two discoellipses in arbitrary configuration.
Red: "Bridge curve" according to Perram and Wertheim.
The "plummet point" (blue) provides an overlap
criterion.
A. Shape Potential
Ellipsoid surface:
$\sigma_{A}^{ell} (\mathbf{r})\equiv (\mathbf{r}-\mathbf{r}_{A})'
\cdot \mathbf{A} \cdot (\mathbf{r}-\mathbf{r}_{A})-1 = 0$
with
$\mathbf{A} = \mathbf{R}^{-1} \cdot \mathbf{A}_{0} \cdot \mathbf{R}$
where $\mathbf{R}$ is a rotation matrix, and $\mathbf{A}_{0}$ the
unrotated ellipsoid matrix
Spherical surface around
$\mathbf{r}_{c}^{\pm}=\mathbf{r}_A \pm z_c \mathbf{c}_{A}\,$:
$\sigma_{A}^{cap}(\mathbf{r}) \equiv
(\mathbf{r}-\mathbf{r}_{c}^{\pm})' \cdot \mathbf{C} \cdot (\mathbf{r}-\mathbf{r}_{c}^{\pm})-1=0$
with $\mathbf{C} = (1/r_{c}^{2}) \, \mathbf{I}$
( $\mathbf{c}_{A}$ is a unit vector along the positive long semiaxis of $A$ )
Shape Potential:
The potential function $\sigma_{A}(\mathbf{r})$ is well-defined everywhere.
The equipotential surface $\sigma_{A}(\mathbf{r})=0$ is continuous, others
are not: $\Rightarrow$ Multiply $\sigma_{A}^{cap}$ by the factor
$\gamma \equiv
\left[\, |\nabla \sigma_{A}^{ell} |/|\nabla \sigma_{A}^{cap} | \,\right]_{z=z_{0}}
= r_{c}^{2}/a^{2}$
if
$ \; \left| s_{A}(\mathbf{r}_{b}) \right| < z_{0} $ and
$ \;\;\; \left| s_{B}(\mathbf{r}_{b}) \right| \geq z_{0} $
or vice versa (A $\leftrightarrow$ B)
C. Overlap criterion:
The bridge points $\mathbf{r}_{b}(\lambda)$ trace a curved path between the
centers of $A$ and $B$; see Figure (above) for an example in two dimensions.
Intersect of the bridge curve with the surface of $A$ to find the
"plummet point" $\mathbf{r}_{p}$ (drawn in blue): use Newton-Raphson
to find $\tilde{\lambda} : \sigma_{A}(\mathbf{r}(\tilde{\lambda}))=0$.
The required Newton-Raphson derivative is given by
$ D \equiv $ $ d\sigma_{A}(\mathbf{r}_{b}(\lambda))/d\lambda
$ $= \mathbf{g}_{A}(\mathbf{r}_{b}) \cdot d\mathbf{r}_{b}/d\lambda\, $
A and B overlap if and only if $\sigma_{B}(\mathbf{r}_{p}) < 0 \,$$ \; \Leftarrow \;$ CRITERION
CONCLUSIONS
Conclusions
Versatile model for a large group of elongated convex particles
Various shapes achieved by changing only two parameters, $c/a$ and $z_{0}/a$
Explore the large variation of mesogenic properties over this class of particles,
using only two parameters
For parallel spheroellipsoids, we have derived a tentative dividing line between
smectogenic and non-smectogenic particle shapes
For MC simulations and DFT of freely reorienting spheroellipsoids we have derived
a fast overlap detection procedure
