Previously at ]Department of Physics, Stanford University.

# Frank Elastic Constants of Semiflexible Polymer Solutions

Quinn MacPherson  Science Division, College of the Sequoias  [

###### Abstract

We derive the Frank elastic constants for nematic solutions of semiflexible
polymers. We plot these results as a function of the coarse-grained Maier-
Saupe quadrupole aligning strength and polymer stiffness ranging from rigid to
highly flexible. The derivation uses the random phase approximation and
combines the exact results for the statistics of a worm-like-chain with
polymer field theory using a spherical harmonic basis. The results are
evaluated using a numerical inverse Laplace transform. We present the results
in terms of microscopic features such as hairpins and polymer ends so the
trends can be understood independently from the derivation. Key findings are
that for rigid polymers $K_{bend}>K_{splay}>K_{twist}$ while for flexible
polymers $K_{splay}>K_{bend}>K_{twist}$. For rigid polymers, the Frank elastic
constants grow with the polymer length. For flexible polymers the elastic
constants grow with the persistence length, which becomes the characteristic
length scale, with the exception of $K_{splay}$ at high alignment strengths
which grows with polymer lengths due to the elimination of hairpins.

Semiflexible, Frank Elastic Constants, Liquid Crystal

##  I Introduction

When oblong molecules are sufficiently concentrated they tend to align. This
liquid crystalline alignment results from the molecules being much easier to
pack together when the molecules are oriented in roughly the same direction as
their neighbors. That is, the alignment is an entropic effect, though an
enthalpic contribution is also possible. Polymers, which are characterised by
a length that is much larger than their diameter, are oblong in the extreme.
Solutions and melts of semiflexible and rigid polymers are particularly prone
to alignment because their structural rigidity tends to align monomers along
the polymer backbone, compounding the aligning tendency from packing.

Figure 1: Pictorial representation of bend, twist, and splay. The axes of the
cylinders represent the direction of nematic ordering, $\vec{n}$. We cite [1]
for the format of this figure.

Even for highly packed solutions of rigid polymers, imperfections in the
packing will cause the direction of alignment to wander so that distant
positions in the solution will have different local directions of alignment.
The orientation of alignment is critical to the behavior of polymer solutions
ranging from conducting polymers [2] to the packaging of DNA withing a viral
capsid [3]. The deviations of the local direction of liquid crystal alignment
over large distances can be classified into bend, twist, and splay which are
depicted in figure 1.

We formally define bend as $\vec{n}\times\nabla\times\vec{n}$, twist as
$\vec{n}\cdot\nabla\times\vec{n}$, and splay as $\nabla\cdot\vec{n}$ where the
unit vector $\vec{n}$ denotes the direction of alignment (we will formally
define $\vec{n}$ in equation 88). If deviations in the direction of alignment
occur slowly – i.e. over large spatial distances – then bend, twist, and splay
will be small due the presence of the spatial derivative $\nabla$. Because
these deformations are small, we can write the energy of these deformation
modes as a quadratic in bend, twist, and splay

$\displaystyle E_{Frank}$ $\displaystyle=$
$\displaystyle\frac{1}{2}K_{bend}\left\langle\left(\vec{n}\times\nabla\times\vec{n}\right)^{2}\right\rangle+\frac{1}{2}K_{twist}\left\langle\left(\vec{n}\cdot\nabla\times\vec{n}\right)^{2}\right\rangle+\frac{1}{2}K_{splay}\left\langle\left(\nabla\cdot\vec{n}\right)^{2}\right\rangle$
(1)

where $K_{bend}$, $K_{twist}$, and $K_{splay}$ are the Frank elastic constants
for the (achiral) solution. In writing Eq. 1, we have included only long
distance (small k) terms and hence ignore higher order derivative terms such
as the saddle-splay interaction.

The central theme of material science is to predict macroscopic material
properties from microstructure, which in turn is generated by the molecular
properties of the constituents. Here we consider the latter connection between
the molecular properties (in this case, the polymer length, concentration,
Kuhn length and monomer cross sectional area) and the microstructure (bend,
twist, and splay). We are particularly interested in the large effects that
the flexibility or rigidity of the polymer backbone can have on the Frank
elastic constants.

The first step to building a theoretic model of the elastic response of a
solution is to model the thermodynamics of the solution. In other words, when
should we expect the aligned phase to even exist? The phase behavior of
solutions of semiflexible polymers have been investigated by a number of
authors [4, 5, 6, 7, 8, 9, 10, 11]. The interested reader is referred to these
references for the resulting phase diagrams. These dictate the minimum
possible degree of alignment where a nematic phase can be found.

Another important step is deriving the elastic constants for solutions in the
rigid rod limit [12, 13, 14]. This is the limiting case for a polymer of
infinite Kuhn length, to which we will compare our results. There has also
been considerable work on deriving the Frank elastic constants of semiflexible
polymer solutions under various approximations and conditions [15, 16, 9, 17,
18, 19]. Shimada [16] makes the approximation of a weak aligning field.
Petschek and Terentjev [18] estimated Frank elastic constants of a chain of
monomers connected by stiff joints using a ground-state-dominance
approximation111It appears that this approximation roughly corresponds to
including only the leading pole in the complex integral we perform in section
XVI. Santo and Teramoto [19] found the Frank constants for a freely jointed
chain and tabulated results of previous papers.

Exact results for the propagator of the wormlike chain in an aligning field
were found by Spakowitz and Wong [21, 22] and enable a better determination of
polymer statistics. We use this propagator in conjunction with polymer field
theory to more accurately determine the Frank elastic constants. While a
connection between spherical harmonics and the Frank elastic modes has long
been recognized [23], we introduce spherical tensor fields and compact
summation notation which simplify and clarify the field theory derivation (see
equations 7 and 10). The derivation of the Frank elastic constants we present
remains analytical up to an inverse Laplace transform, which we perform
numerically. The results and their interpretation, which are intended to be
intelligible without reference to the derivation, are presented in section
XVII.

##  II Model and Assumptions

###  II.1 Solution of WLC’s

We wish to study deformations in the alignment of liquid crystal polymer
solutions and melts. To this end we propose the following mathematical model
and approximations. The point(s) at which an approximation is used in the
subsequent derivation will be noted for those wishing to relax these
assumptions.

  1. 1.

We will assume all polymers have the same length, $L$, i.e. no polydispersity.

  2. 2.

We describe the $j^{th}$ polymer as a continuous path in space
$\vec{r}_{j}\left(s\right)$, given as a function of path length $s\in[0,L]$.
We define the orientation direction of the polymer at point $s$ as the unit
vector
$\vec{u}_{j}\left(s\right)=\frac{\partial\vec{r}_{j}\left(s\right)}{\partial
s}$.

  3. 3.

Each polymer will be assumed to obey the statistics of an inextensible Worm
Like Chain (WLC). That is, the mechanical energy of $n_{p}$ polymers is given
by

$E_{poly}\equiv\frac{1}{2}\ell_{p}\sum_{j=1}^{n_{p}}\int_{0}^{L}ds\left(\frac{\partial\vec{u}_{j}}{\partial
s}\right)^{2}$ (2)

where $\ell_{p}$ is the persistence length of the polymer, which sets the
polymer stiffness. In other words, the energy is proportional to the square of
the curvature of the polymers. Along with all other energies in this paper,
$E_{poly}$ is assumed to be in units of $k_{b}T$.

  4. 4.

The cross sectional area of the polymer is sufficiently small that its
diameter is smaller than the other length scales of interest in the problem.

  5. 5.

The polymers are achiral. That is, they do not exhibit a natural twist
direction that affects the solution properties. As an example of how this
could be violated, imagine a naturally twisted polymer and further imagine
that the alignment interactions between adjacent chains depend on this twist,
such that a natural twist is conferred to the polymer solution.

  6. 6.

The system is unchanged by reversing any polymer by swapping its $s=0$ and
$s=L$ ends. An example of how this could be violated would be a solution of
polymers with a negative charge on one polymer end and a positive charge on
the other. Such polymers would have dipole-dipole interactions and/or
experience the aligning effect of an external field.

  7. 7.

The system is translationally invariant. With no boundary conditions the
material can be thought of as being infinite in extent.

  8. 8.

We will assume the solvent is good enough that the solution does not segregate
into a polymer rich and polymer lean phase. The range of applicability of this
assumption in the presence of an alignment interaction is studied in [11].

  9. 9.

Polymer interactions are assumed to be two body interactions, that is, the
energy is quadratic in polymer density.

  10. 10.

Polymer interactions are local, meaning that their range is well below the
length scales of relevance to the liquid crystal alignment fluctuation under
study.

  11. 11.

There are only two polymer interactions. The first of these is the Flory-
Huggins interaction which governs the energy of mixing of polymer and solvent.
The second is a Maier-Saupe interaction which captures the tendency of
adjacent polymer segments to align with each other by assigning an energy
proportional to $\left(\vec{u}_{1}\cdot\vec{u}_{2}\right)^{2}$ for two polymer
segments in close spatial proximity.

  12. 12.

The preferential direction of alignment is approximately in the $z$ direction.
The density and the preferred direction of alignment of the polymers will only
make small deviations from their mean field values. Note that the orientation
of particular polymers can differ substantially from the $z$ direction.

  13. 13.

The effective potential experienced by a single polymer, created by the
others, undergoes only small fluctuations from its mean value. This allows the
Random Phase Approximation (RPA).

  14. 14.

The average alignment field will be azimuthally symmetric with respect to the
aligning axis. In other words, there is not a second axis of alignment that
could result from ribbon-like polymers with an asymmetric cross-section.

  15. 15.

We are primarily interested in coordinated deviations of the alignment over
large distances (small $\vec{k}$).

###  II.2 Fuzzball and Rigid Rod

The properties of the WLC are made more evident by comparison to the rigid rod
and an object I will refer to as a fuzzball. The rigid rod is a straight rod
of length $L$ and cross sectional area $A$. The rigid rod is a WLC in the
limit as $\ell_{p}\to\infty$. Because it has no flexibility, the rigid rod has
no relevant internal energy $E_{poly}$. The fuzzball, like the rigid rod, has
a single direction and no internal energy. However, the density of the
fuzzball is spread out in a symmetric Gaussian about a point. The fuzzball can
be thought of as a crude model of a small molecule or an electric quadruple
whose field falls off as a Gaussian.

##  III Summary of Derivation

The results of this paper are intended to be intelligible without consulting
the derivation. However, the derivation contains a number of interesting
tricks and insights that the reader may find useful for similar problems. In
particular, we describe a real spherical harmonic tensor distribution,
extended summation notation for describing orientation over space, mean field
polymer field theory, fluctuations away from this mean field, and WLC
propagators derived via stone fence diagrams.

In this summary we provide a brief description of the approach we use to
calculate the Frank elastic constants for a solution of semiflexible polymers.
A similar procedure is followed for the rigid rod and fuzz-ball examples. We
begin by describing the polymer solution with the partition function

$Z\propto\int\mathscr{D}\left[\vec{\mathrm{r}}_{j}\left(s\right)\right]\exp\left(-E_{poly}-E_{FH}-E_{MS}\right)$
(3)

where the polymer bending energy $E_{poly}$ is given in equation 2, the
$E_{FH}$ is the Flory-Huggins interaction and $E_{MS}$ is the Maier-Saupe
energy representing the preference for nearby polymers to align. The latter
two are given by the sum of pairwise interactions

$E_{FH}=\frac{A^{2}}{2}\sum_{i=1}^{n_{p}}\int_{0}^{L}ds_{1}\sum_{j=1}^{n_{p}}\int_{0}^{L}ds_{2}\chi\delta\left(\mathrm{\vec{r}_{i}}\left(s_{1}\right)-\mathrm{\vec{r}_{j}}\left(s_{2}\right)\right)$
(4)

and

$\displaystyle E_{MS}=$
$\displaystyle\frac{A^{2}}{3}\sum_{i=1}^{n_{p}}\int_{0}^{L}ds_{1}\sum_{j=1}^{n_{p}}\int_{0}^{L}ds_{2}a\left(\vec{u}_{i}(s_{1})\cdot\vec{u}_{j}\left(s_{2}\right)\right)^{2}\delta\left(\mathrm{\vec{r}_{i}}\left(s_{1}\right)-\mathrm{\vec{r}_{j}}\left(s_{2}\right)\right).$
(5)

where $A$ is the cross sectional area of the polymers. A factor of $1/2$ is
included to account for double counting 222without the factor of $1/2$ the
$E_{MS}$ would have a $2/3$ rather than a $1/3$ per the customary definition
of $a$.

Equations 3-5 are then simplified significantly by using the local density and
orientation distribution, $\phi$, which we define in equation 7 and use
throughout the remainder of the derivation. In this notation $E_{FH}$ and
$E_{MS}$ are replaced with $\phi V\phi$ where $V$ is the interaction matrix
defined in equation 11.

To make the system mathematically tractable we first solve the partition
function for the mean field of $\phi$ which we write as
$\left\langle\phi\right\rangle^{MF}$. The self consistent equation 21 requires
that $\left\langle\phi\right\rangle^{MF}$ is the density that would be
expected in a field $\left\langle\phi\right\rangle^{MF}V$ which
$\left\langle\phi\right\rangle^{MF}$ generates and applies to itself. The
Frank elastic constants are defined in terms of fluctuations about this mean
field solution, as will be discussed shortly.

In order to calculate $\phi$ for a WLC in the presence of an aligning mean
field we follow [25, 11, 21] in using stone fence diagrams. These diagrams
allow us to calculate the Laplace transform (from chain length $N$ to Laplace
variable $p$) of $\left\langle\phi\right\rangle$ as well as the Laplace
transforms of products, e.g. $\mathscr{\mathcal{L}}_{N\to
p}\left\langle\phi_{1}\phi_{2}\right\rangle$. We are able to numerically
invert the Laplace transforms via path integration in the complex plane. The
construction of expectation values is described in section VIII and the
description of the diagrams is given in section VII.

In section IX we investigate fluctuations about the mean field solution. The
fluctuations are divided into Fourier modes via a Fourier transform from
position $\vec{r}$ to wave vector $\vec{k}$. To first approximation, the
energy of each mode $\tilde{\phi}\left(k\right)$ is quadratic,

$E\approx\tilde{\phi}\left(\vec{k}\right)\Gamma\left(\vec{k}\right)\tilde{\phi}\left(-\vec{k}\right)$
(6)

where the “spring constants” for each of these modes are combined in the
matrix $\Gamma$.

In section XIII we define the nematic director and the Frank elastic constants
and manipulate them to be written in terms of particular long-wavelength
elements of $\Gamma$. This allows us to write expressions for the Frank
elastic constants in terms of numerical inverse Laplace transforms in section
XV. The numerical evaluation of the inverse transform is described in section
XVI.

##  IV Summation Notation & System Description

The material properties at a point $\vec{r}$ in space are determined by the
position and orientation of polymers in the vicinity of $\vec{r}$. For a
particular configuration $\left\\{\vec{\mathrm{r}}_{j}\left(s\right)\right\\}$
of the system, the local density and orientation distributions can be
mathematically described by

$\hat{\phi}_{l,m}\left(\vec{r}\right)\equiv\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j=1}^{n_{p}}A\int_{0}^{L}dsY_{l,m}\left(\vec{\mathrm{u}}_{j}\left(s\right)\right)\delta\left(\vec{r}-\vec{\mathrm{r}}_{j}\left(s\right)\right)$
(7)

where $Y_{\ell,m}$ is the real spherical harmonic function described in
appendix A. Note that we will use the roman $\vec{\mathrm{r}}$ and
$\vec{\mathrm{u}}$ for position and orientation of points along a polymer and
italicized $\vec{r}$ and $\vec{u}$ for generic position in space. These
continuous polymers (assumption 2) could be replaced with discrete beads by
replacing the integral in equation 7 with a sum. While $\hat{\phi}$ is
formally defined as sum of infinitely thin space curve delta distributions, it
is easier to interpret after integrating over a coarse grained volume $\Delta
V$ in the vicinity of the position $\vec{r}$ in the material. For example, the
scalar component of the density $\hat{\phi}_{0,0}$ is interpreted as the local
volume fraction of polymer; integrating $\hat{\phi}_{0,0}$ over the region of
space $\Delta V$ gives the volume of polymer within that region. The values of
$\hat{\phi}_{l,m}\left(\vec{r}\right)$ for $\ell>0$ describe the degree and
direction of alignment of the polymers. In equation 7 the units of volume from
$Ads$ cancel the volume units from the delta function so that $\hat{\phi}$ is
unitless, as would be expected from a volume fraction.

We write the partition function describing the set of configurations that
$\hat{\phi}$ can take on as

$Z=\frac{1}{n_{p}!}\int\mathscr{D}\left[\vec{\mathrm{r}}_{j}\left(s\right)\right]\exp\left(-E_{poly}-\frac{1}{2}\hat{\phi}_{1}V_{12}\hat{\phi}_{2}\right)$
(8)

The $1/n_{p}!$ prefactor accounts for the indistinguishably of the polymers
but is not consequential the following discussion. The functional integral
$\int\mathscr{D}\left[\vec{\mathrm{r}}_{j}\left(s\right)\right]$ refers to an
integral over all possible configurations of the system, that is, it
integrates the position of each point of each polymer over all space

$\int\mathscr{D}\left[\vec{\mathrm{r}}_{j}\left(s\right)\right]=\lim_{\Delta
s\to
0}\prod_{j=1}^{n_{p}}\int_{-\infty}^{\infty}d\vec{\mathrm{r}}_{j,s=0}\int_{-\infty}^{\infty}d\vec{\mathrm{r}}_{j,s+\Delta
s}....\int_{-\infty}^{\infty}d\vec{\mathrm{r}}_{j,L}$ (9)

The WLC polymer energy $E_{poly}$ defined in equation 2 effectively reduces
the region of integration to continuous smooth polymers. The term
$\frac{1}{2}\hat{\phi}_{1}V_{12}\hat{\phi}_{2}$ in equation 8 describes the
interaction potential between polymers. The interaction is quadratic in
$\hat{\phi}$ in keeping with assumption 9. The subscripts in
$\hat{\phi}_{1}V_{12}\hat{\phi}_{2}$ refer to an extended summation notation
that sums over $\ell$ and $m$ indices and integrates over space such that

$\displaystyle\hat{\phi}_{1}V_{12}\hat{\phi}_{2}\equiv$
$\displaystyle\sum_{\ell_{1}=0}^{\infty}\sum_{m_{1}=-\ell_{1}}^{\ell_{1}}\sum_{\ell_{2}=0}^{\infty}\sum_{m_{2}=-\ell_{2}}^{\ell_{2}}\int
d\vec{r_{1}}d\vec{r_{2}}\hat{\phi}_{\ell_{1},m_{1}}(\vec{r}_{1})V_{\ell_{1},\ell_{2}}^{m_{1},m_{2}}(\vec{r}_{1},\vec{r}_{2})\hat{\phi}_{\ell_{2},m_{2}}(\vec{r}_{2})$
(10)

Combining the concise summation notation with the spherical harmonic indices
for orientation allows us to manipulate the tensor field $\hat{\phi}$ in much
the same way as scalar fields in previous publications [26, 27]. Note that
when $\phi$ has a single subscript this is understood to be the subscript from
the summation notation and when it has two as in $\phi_{\ell,m}$ these refer
to $\ell$ and $m$ as should also be clear from context. The notation
$\hat{\phi}_{1}V_{12}\hat{\phi}_{2}$ is analogous to the vector notation
$\vec{\phi}^{T}V\vec{\phi}$ for matrix $V$ and vector $\vec{\phi}$ which have
an infinite number of indices in order to capture all $\ell,m,$ and $\vec{r}$.
While this notation allows for a wide variety of orientationally dependent and
ranged potentials, we will restrict our discussion to local potentials
(assumption 10) with $V\propto\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)$. We
will also restrict $V$ to be non-zero only for a Flory-Huggins scalar
potential ($\ell=0$) and a Maier-Saupe tensor potential $(\ell=2)$ in keeping
with assumption 11. Interactions of order $\ell=1$ violate the assumed
reversal symmetry (assumption 6). Including interaction terms of order
$\ell>2$ would relax assumption 11 to systematically refine the rotational
detail of inter-chain interactions with the interaction constants depending on
the chemistry and sterics of the polymer in question.

Under the above assumptions we write

$V_{12}=\left(\chi\delta_{\ell_{1},0}-\frac{2}{3}a\delta_{\ell_{1},2}\right)\delta_{m_{1},m_{2}}\delta_{\ell_{1},\ell_{2}}\delta\left(\vec{r}_{1}-\vec{r}_{2}\right)$
(11)

where $\chi$ is the widely used Flory-Huggins mixing parameter. The second
term in the parentheses provides an energetic benefit to polymer segments that
align $\vec{u}_{1}\approx\vec{u}_{2}$ or antialign
$\vec{u}_{1}\approx-\vec{u}_{2}$. The factor of $2/3$ maintains the meaning of
the Maier-Saupe parameter, $a$, from [10] and [11]. The negative sign on the
$\ell=2$ term in equation 11 is chosen so that a positive $a$ corresponds to a
preferential alignment. The equivalence to the Maier-Saupe formulation [10,
11] in terms of $\vec{u}\otimes\vec{u}-\frac{1}{3}I$ rather than
$Y_{2}^{m}\left(\vec{u}\right)$ results from the relation

$\displaystyle\sum_{i,j=1}^{3}\left(\vec{u}_{1}\otimes\vec{u}_{1}-\frac{1}{3}I\right)_{ij}\left(\vec{u}_{2}\otimes\vec{u}_{2}-\frac{1}{3}I\right)_{ij}=\left(\vec{u}_{1}\cdot\vec{u}_{2}\right)^{2}-\frac{1}{3}=\frac{2}{3}P_{2}\left(\vec{u}_{1}\cdot\vec{u}_{2}\right)=\frac{2}{3}\sum_{m=-2}^{2}Y_{2,m}\left(\vec{u}_{1}\right)Y_{2,m}\left(\vec{u}_{2}\right)$
(12)

for all unit vectors $\vec{u}_{1}$ and $\vec{u}_{2}$.

### Fuzzball and Rigid Rod

In contrast to equation 7, the density distribution for solution of fuzzballs
is given as

$\hat{\phi}_{\ell,m}^{FB}=\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j=1}^{n_{p}}\frac{v_{p}}{\left(2\pi\sigma^{2}\right)^{3/2}}\exp\left(-\frac{\left(\vec{r}-\vec{\mathrm{r}}_{j}\right)^{2}}{2\sigma^{2}}\right)Y_{\ell,m}\left(\vec{u}_{j}\right)$
(13)

where $\sigma$ is the standard deviation of the field falloff and defines the
size of the fuzzball. $v_{p}$ is the effective volume (or amplitude if you
prefer) of the fuzzball. For rigid rods the density distribution is

$\hat{\phi}_{\ell,m}^{RR}\left(\vec{r}\right)=\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j=1}^{n_{p}}Y_{\ell,m}\left(\vec{\mathrm{u}}_{j}\right)A\int_{0}^{L}ds\delta\left(\vec{\mathrm{r}}_{j}+s\vec{\mathrm{u}}_{j}-\vec{r}\right)$
(14)

where $\vec{r}_{j}$ is the location of one end of the rod and $\vec{u}_{j}$ is
the orientation of the rod. For the rigid rod, the integration in the above
equation simplifies to

$\hat{\phi}_{\ell,m}^{RR}\left(\vec{r}\right)=\sqrt{\frac{4\pi}{2\ell+1}}\sum_{j=1}^{n_{p}}Y_{\ell,m}\left(\vec{\mathrm{u}}_{j}\right)AL\delta\left(\vec{\mathrm{r}}_{j}+L\vec{\mathrm{u}}_{j}-\vec{r}\right)$

The partition functions for these two solutions are similar

$\displaystyle Z^{RR/FB}=\int d\vec{\mathrm{r}}_{j}\int
d\vec{\mathrm{u}}_{j}\exp\left(\frac{1}{2}\chi\left(\phi_{0,0}^{RR/FB}\left(\vec{r}\right)\right)^{2}\right.\left.+\frac{1}{3}a\int
d\vec{r}\sum_{m=-2}^{2}\left(\phi_{2,m}^{RR/FB}\left(\vec{r}\right)\right)^{2}\right)$
(15)

and, in contrast to the WLC, simply require integrating over $n_{p}$ positions
$\vec{r}_{j}$ and $n_{p}$ orientations $\vec{u}_{j}$.

##  V Field Manipulations

We introduce a delta function at every point in space and orientation
$\delta\left(\phi-\hat{\phi}\right)$. Extending the Fourier representation of
a delta function
$\delta\left(a-b\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dp\left(ip\left(a-b\right)\right)$
to a delta function for a tensor distribution $\phi$ we have

$\delta\left(\phi-\hat{\phi}\right)\propto\int\mathscr{D}W\exp\left(iW_{1}\left(\phi_{1}-\hat{\phi}_{1}\right)\right)$
(16)

where the functional integral $\int\mathscr{D}W$ integrates
$W_{\ell}^{m}\left(\vec{r}\right)$ for each $\vec{r}$, $\ell$, and $m$
combination. Like $\phi$, the Fourier variable $W$ is also a tensor
distribution. Inserting this delta function into the partition function
(equation 8) in the fashion of $f(x)=\int dyf(y)\delta\left(x-y\right)$ we
have

$Z\propto\int\mathscr{D}W\mathscr{D}\phi\mathscr{D}\left[\vec{\mathrm{r}}^{\left(1\right)}\left(s\right)\right]e^{-E_{poly}-\frac{1}{2}\phi_{1}V_{12}\phi_{2}+iW_{1}\left(\phi_{1}-\hat{\phi}_{1}\right)}$
(17)

Note that the $E_{poly}$ and $\hat{\phi}$ only depend on $\vec{r}$ while
$\phi_{1}V_{12}\phi_{2}$ and $W_{1}\phi_{1}$ only depend on $\phi$ so these
terms can be separated

$\displaystyle Z\propto\int\mathscr{D}W\mathscr{D}\phi
e^{-\frac{1}{2}\phi_{1}V_{12}\phi_{2}+iW_{1}\phi_{1}}\left[\int\mathscr{D}\vec{\mathrm{r}}^{\left(1\right)}e^{-E_{poly}^{\left(1\right)}-iW_{1}\hat{\phi}_{1}^{\left(1\right)}}\right]^{n_{p}}$
(18)

This separation is quite useful, and is why we introduced the delta function
16 and the $W$ field.

The second integral in 18 contains no inter-chain interactions other than
through the $W$ field and all chains are identical (assumption 1) so it has
been factored into a product of $n_{p}$ terms. The superscript $\
{}^{\left(1\right)}$ on $\vec{\mathrm{r}},$ $E_{poly}$, and $\hat{\phi}$
indicates a single chain. Defining the contents of the square bracket in
equation 18 as $z_{p}$ we have

$Z\propto\int\mathscr{D}W\mathscr{D}\phi
e^{-\frac{1}{2}\phi_{1}V_{12}\phi_{2}+iW_{1}\phi_{1}+n_{p}\ln\left(z_{p}\right)}$
(19)
$z_{p}=\int\mathscr{D}\left[\vec{\mathrm{r}}^{\left(1\right)}\left(s\right)\right]e^{-E_{poly}^{\left(1\right)}-iW_{1}\hat{\phi}_{1}^{\left(1\right)}}$
(20)

While strictly speaking $W$ is simply an integration variable, we can assign a
physical meaning to it as a chemical potential. The term $iW_{1}$ in equation
20 is the effective potential experienced by a single polymer. Meanwhile, the
term $iW_{1}$ in equation 19 represents the potential that the single polymers
impose to deform the density field $\phi$. In this way single polymers don’t
communicate directly with each other, but communicate though the local
chemical potential $W$. We will later integrate out $W$ between equations 64
and 66.

##  VI Mean Field Solution

###  VI.1 Mean Field Solution for Polymer Solutions

In this section we will solve for the homogeneous mean field solution
$\hat{\phi}$ of partition function 8 so that we can expand about $\hat{\phi}$
in section IX. The essence of the mean field solution is that rather than
calculating all interactions inside and between polymers, we calculate the
statistics of a single polymer with $E_{FH}$ and $E_{MS}$ replaced by the
average field the polymer experiences. The self consistency requirement sets
the mean field strength to that generated by $n_{p}$ such single polymers. The
field experienced by a single polymer can be found by differentiating the
potential $\frac{1}{2}\hat{\phi}_{1}V_{12}\hat{\phi}_{2}$ with respect to the
field generated by a single polymer, $\hat{\phi}^{\left(1\right)}$, to get
$\left<\phi_{1}\right>^{MF}V_{12}$. The self consistent equation sets the
expectation value of
$n_{p}\left\langle\hat{\phi}^{\left(1\right)}\right\rangle$ to be the mean
field strength $\left<\phi\right>^{MF}$.

$\left<\phi\right>^{MF}=\frac{\int\mathscr{D}\left[\vec{\mathrm{r}}^{\left(1\right)}\right]n_{p}\hat{\phi}^{(1)}\exp\left(-E_{poly}-\left<\phi\right>_{1}^{MF}V_{12}\hat{\phi}_{2}^{(1)}\right)}{\int\mathscr{D}\left[\vec{\mathrm{r}}^{\left(1\right)}\right]\exp\left(-E_{poly}-\left<\phi\right>_{1}^{MF}V_{12}\hat{\phi}_{2}^{(1)}\right)}$
(21)

Equation 21 can alternatively be derived by introducing $W$ as was done in
equations 19 and 20 and then minimizing the integrand of equation 19 with
respect to a constant $\phi$ and $W$ which gives $\left<\phi\right>^{MF}$ and
$iW_{1}^{MF}=V_{12}\left<\phi\right>_{2}^{MF}$.

For values of the Maier-Saupe parameter $a$ below its critical value
$a^{\star}$, the homogeneous mean field solution will simply be a constant
density with no net alignment
$\left<\phi\right>_{\ell,m}^{MF}=\left<\phi\right>_{0,0}^{MF}\delta_{\ell,0}$.
Above $a^{\star}$ the tendency for polymers to align will overpower entropy
and break rotational symmetry creating a direction of preferred orientation.
We will align this direction with the $\hat{z}$ axis. Because the mean field
solution is constant throughout all space (assumptions 7 and 8) the $\ell=0$
component of
$\left<\phi\right>_{\ell,m}^{MF}=\left<\phi\right>_{0,0}^{MF}\delta_{\ell,0}+\left<\phi\right>_{2,0}^{MF}\delta_{\ell,0}\delta_{m,0}$.
Because $\left<\phi\right>_{0,0}^{MF}$ is a constant scalar offset it can be
canceled from equation 21 with no consequence. However, there will be a non-
zero $\phi_{2,0}$ which will need to be solved for. Under the potential 11,
the aligning field energy for a single polymer is
$\left<\phi\right>_{1}^{MF}V_{12}\hat{\phi}_{2}^{(1)}=-\gamma\int_{0}^{N}dsY_{2,0}\left(\vec{\mathrm{u}}\left(s\right)\right)$
where $N=L/2\ell_{p}$ is chain length nondimensionalized by the Kuhn
length333For WLCs the Kuhn length is twice the persistence length. and

$\gamma=\frac{2}{3}\sqrt{\frac{4\pi}{5}}2\ell_{p}Aa\left<\phi\right>_{2,0}^{MF}$
(22)

is the strength of the aligning field which follows from equations 11 and 7.
The value of gamma can roughly be thought of as the aligning energy in $kT$’s
per persistence length of polymer.

The procedure for calculating the value of $\left<\phi\right>_{2,0}^{MF}$ as a
function of $\gamma$ will be derived below, resulting in formula 59 with the
inverse Laplace transform being performed numerically as described in section
XVI. In figure 2 we plot the mean field strength $\gamma$ in terms of $a$.
These are plotted by solving 22 for $a$ as a function of $\gamma$.

$a=\frac{\gamma}{\frac{2}{3}\sqrt{\frac{4\pi}{5}}2\ell_{p}A\left<\phi\right>_{2,0}^{MF}\left(\gamma\right)}$
(23)

When working with relatively stiff polymers $L<2\ell_{p}$ it is natural to
nondimensionalize $a$ by the polymer volume $LA$. When working with flexible
polymers it is natural to nondimensionalize $a$ by the volume of a Kuhn length
$2\ell_{p}A$. Because a stronger $a$ is needed in dilute solutions, it is also
convenient to multiply $a$ by the volume fraction of polymer $\phi_{0,0}$.

Figure 2: Mean field strength $\gamma$ as a function of Maier-Saupe $a$. The
lower branch is nonphysical; the nose corresponds to the limit of meta-
stability of the aligned phase. Curves are colored by the length of the
polymer in Kuhn lengths, $N$. The left plot is nondimensionalized by polymer
length, which is convenient for rigid polymers. The right plot is
nondimensionalized by $2\ell_{p}$, which is convenient for flexible polymers
The dashed curve represents the rigid rod.

###  VI.2 Mean Field Solution for Rigid Rod and Fuzzball

As the fuzzball and rigid rod have no Kuhn length, we instead define their
field strengths $\gamma$ in terms of their respective volumes

$\gamma^{FB}=\frac{2}{3}a\left\langle\phi_{2,0}\right\rangle^{MF}\sqrt{\frac{4\pi}{5}}v_{p}$
(24)
$\gamma^{RR}=\frac{2}{3}a\left\langle\phi_{2,0}\right\rangle^{MF}\sqrt{\frac{4\pi}{5}}LA$
(25)

so that the energy associated with each object is $E^{\left(1\right)}=-\gamma
Y_{2,0}\left(\vec{u}_{i}\right)$. The partition function for the orientation
of a single fuzzball or rigid rod is given by

$z^{FB/RR}=\int d\vec{u}_{i}\exp\left(\gamma
Y_{2,0}\left(\vec{u}_{i}\right)\right)$ (26)

To calculate expectation values for this partition function it is useful to
define the matrix

$\mathbf{M}_{\ell_{1},\ell_{2}}^{m}=\int
d\vec{u}_{i}Y_{\ell_{1},m}\left(\vec{u}_{i}\right)\exp\left(\gamma
Y_{2,0}\left(\vec{u}_{i}\right)\right)Y_{\ell_{2},m}\left(\vec{u}_{i}\right)$
(27)

The numerical values of $\mathbf{M}$ can be obtained by matrix exponentiation

$\mathbf{M}^{m}=\exp\left(\gamma\mathbf{J}_{\left(2\right)}^{m,0,m}\right)$
(28)

where

$\left(\mathbf{J}_{\left(2\right)}^{m,0,m}\right)_{\ell_{1},\ell_{2}}=\int
d\vec{u}Y_{\ell_{1},m}\left(\vec{u}\right)Y_{2,0}\left(\vec{u}\right)Y_{\ell_{2},m}\left(\vec{u}\right)$
(29)

which is given in appendix A. As an example of how $\mathbf{M}$ can be used to
calculate an expectation, consider

$\displaystyle\left\langle\phi_{\ell,m}^{RR/FB}\right\rangle^{MF}$
$\displaystyle=n_{p}\frac{\int d\vec{\mathrm{r}}\int
d\vec{\mathrm{u}}\exp\left(\gamma
Y_{2,0}\left(\vec{\mathrm{u}}_{j}\right)\right)\phi_{\ell,m}^{\left(1\right)RR/FB}}{\int
d\vec{\mathrm{r}}\int d\vec{\mathrm{u}}\exp\left(\gamma
Y_{2,0}\left(\vec{\mathrm{u}}_{j}\right)\right)}$ (30)

which, with the help of $Y_{0,0}=\frac{1}{\sqrt{4\pi}}$, simplifies to

$\left\langle\phi_{\ell,m}^{RR}\right\rangle^{MF}=\frac{n_{p}AL}{\mathscr{V}}\sqrt{\frac{1}{2\ell+1}}\frac{\mathbf{M}_{0,\ell_{2}}^{0}}{\mathbf{M}_{0,0}^{0}}\delta_{m,0}$
(31)
$\left\langle\phi_{\ell,m}^{FB}\right\rangle^{MF}=\frac{n_{p}v_{p}}{\mathscr{V}}\frac{1}{\sqrt{2\ell+1}}\frac{\mathbf{M}_{0,\ell}^{0}}{\mathbf{M}_{0,0}^{0}}\delta_{m,0}.$
(32)

From the above we can verify the densities,
$\left\langle\phi_{0,0}^{RR}\right\rangle^{MF}=\frac{n_{p}AL}{\mathscr{V}}$
and
$\left\langle\phi_{0,0}^{FB}\right\rangle^{MF}=\frac{n_{p}v_{p}}{\mathscr{V}}$,
and get the mean field alignment
$\left\langle\phi_{0,2}^{RR/FB}\right\rangle^{MF}$ and then solve for the
Maier-Saupe parameter

$a^{RR}LA\phi_{00}=\frac{15\gamma^{RR}\bm{M}_{0,0}^{0}\left(\gamma^{RR}\right)}{4\sqrt{\pi}\mathbf{M}_{0,2}^{0}\left(\gamma^{RR}\right)}$
(33)
$a^{FB}v_{p}\phi_{00}=\frac{15\gamma^{FB}\mathbf{M}_{0,0}^{0}\left(\gamma^{FB}\right)}{4\sqrt{\pi}\mathbf{M}_{0,2}^{0}\left(\gamma^{FB}\right)}$
(34)

plotted in figure 2.

##  VII Stone Fence Diagrams

In this section we derive a propagator – equations 51 and 52 – which we will
use to calculate $\left\langle\phi\right\rangle^{MF}$ in the next section.

To obtain a numerical value of $\left<\phi\right>^{MF}$ we follow the approach
of [11, 21, 22] beginning with the propagator for a semi-flexible WLC
(assumption 3) without an aligning field [21] (i.e. as if
$\left<\phi\right>_{2}^{MF}=0$):

$G_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{\ell,m}\left(\vec{\mathrm{u}}\right)Y_{\ell,m}\left(\vec{\mathrm{u}}_{0}\right)e^{-\ell\left(\ell+1\right)\frac{L}{2\ell_{p}}}$
(35)

which gives the probability distribution for the orientation
$\vec{\mathrm{u}}$ at one end of a chain of length $L$, given that the other
end is orientated in the $\vec{u}_{o}$ direction. (Note that in [21] complex
spherical harmonics were used. Because
$\sum_{m=-\ell}^{\ell}Y_{\ell}^{m}\left(\vec{\mathrm{u}}\right)Y_{\ell}^{m*}\left(\vec{\mathrm{u}}_{0}\right)=P_{\ell}\left(\vec{u}\cdot\vec{u}_{0}\right)=\sum_{m=-\ell}^{\ell}Y_{\ell,m}\left(\vec{\mathrm{u}}\right)Y_{\ell,m}\left(\vec{\mathrm{u}}_{0}\right)$
equation 35 also holds for real spherical harmonics.) $G_{o}$ ranges between
the rigid rod $\lim_{L/2\ell_{p}\to
0}G_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)=\delta\left(\vec{\mathrm{u}}-\vec{\mathrm{u}}_{0}\right)$
and flexible chain
$\lim_{L/2\ell_{p}\to\infty}G_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)=\frac{1}{4\pi}$
and has the property
$G_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)=\int
d\vec{\mathrm{u}}_{1}G_{o}\left(\vec{\mathrm{u}}_{1}|\vec{\mathrm{u}}_{0};L_{1}\right)G_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{1};L-L_{1}\right)$
where $\int d\vec{\mathrm{u}}_{1}$ is meant as integration over the unit
sphere. Going forward we will nondimensionalize the polymer length by the Kuhn
length $2\ell_{p}$ to give $N=L/2\ell_{p}$.

The propagator $G_{o}$ allows us to calculate statistics about any point along
a free chain. For example, suppose we have a WLC constrained to start from
orientation $\vec{u}_{0}$ and end at orientation $\vec{u}_{f}$ and there is
some function $f$ that depends on the orientation $\vec{u}_{1}$ at a point
$N_{1}=L_{1}/2\ell_{p}$ along a polymer. We can calculate $f$’s expectation
value via

$\left\langle f\left(\vec{u}_{1}\right)\right\rangle_{o}=\frac{\int
d\vec{u}_{1}G_{o}\left(\vec{u}_{1}|\vec{\mathrm{u}}_{0};N_{1}\right)f\left(\vec{u}_{1}\right)G_{o}\left(\vec{\mathrm{u}}_{f}|\vec{u}_{1};N-N_{1}\right)}{G_{o}\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};N\right)}$
(36)

This expectation value is precisely the type of quantity that we need to
calculate to obtain the propagator $G$ for a WLC which is in an aligning field
$\phi_{1}^{MF}V_{12}$ from equation 21. In particular, if we use the
expectation value in equation 36 to include the aligning field $\gamma$ as a
re-weighting of $G_{o}$ then the partition function for the end orientation is

$G\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0},N\right)=G_{o}\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};N\right)\left\langle\exp\left[\gamma\int_{0}^{N}dsY_{2,0}\left(\vec{\mathrm{u}}\left(s\right)\right)\right]\right\rangle_{o}$
(37)

where the prefactor
$G_{o}\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};N\right)$ cancels the
denominator of 36. Please note that $G$ is an unnormalized distribution that
is proportional to the probability distribution of $\vec{\mathrm{u}}_{f}$ for
a given $\vec{\mathrm{u}}_{0}$,

$P\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};N\right)=\frac{G\left(\vec{\mathrm{u}}_{f},\vec{\mathrm{u}}_{0},N\right)}{\int
d\vec{\mathrm{u}}^{\prime}_{f}G\left(\vec{\mathrm{u}}^{\prime}_{f},\vec{\mathrm{u}}_{0},N\right)}.$
(38)

$P$ and $G$ differ in that $\frac{1}{4\pi}\int
d\vec{\mathrm{u}}_{0}P\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};N\right)$
gives the probability distribution of $\vec{\mathrm{u}}_{f}$ assuming
$\vec{\mathrm{u}}_{0}$ is uniform over the unit sphere, while
$\frac{1}{4\pi}\int
d\vec{\mathrm{u}}_{0}G\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0},N\right)$
is proportional to the probability distribution of $\vec{\mathrm{u}}_{f}$ for
a polymer of length $N$ in an aligning field whose ends are free to rotate.
Like $G_{0}$, the propagator $G$ has the property
$G\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)=\int
d\vec{\mathrm{u}}G\left(\vec{\mathrm{u}}_{1}|\vec{\mathrm{u}}_{0};L_{1}\right)G\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{1};L-L_{1}\right)$.
We can think of $G\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)$ as the
partition function for $\vec{u}$ and $\vec{u}_{0}$. The quantity
$G\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right)$ is useful because it
allows us to calculate chain statistics in the presence of the $\gamma$ field.
For example, the expectation of $f\left(\vec{\mathrm{u}}_{1}\right)$ for a
chain in field of strength $\gamma$ and freely rotating ends is given by

$\left\langle f\left(\vec{\mathrm{u}}_{1}\right)\right\rangle=\frac{\int
d\vec{\mathrm{u}}_{0}\int d\vec{\mathrm{u}}_{1}\int
d\vec{\mathrm{u}}_{f}G\left(\vec{\mathrm{u}}_{1}|\vec{\mathrm{u}}_{0};L_{1}\right)f\left(\vec{\mathrm{u}}_{1}\right)G\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{1};L-L_{1}\right)}{\int
d\vec{\mathrm{u}}_{0}\int d\vec{\mathrm{u}}_{1}\int
d\vec{\mathrm{u}}_{f}G\left(\vec{\mathrm{u}}_{1}|\vec{\mathrm{u}}_{0};L_{1}\right)G\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{1};L-L_{1}\right)}\\\
$ (39)

where the denominator can be simplified to $\int d\vec{\mathrm{u}}_{0}\int
d\vec{\mathrm{u}}_{f}G\left(\vec{\mathrm{u}}_{f}|\vec{\mathrm{u}}_{0};L\right)$.

We now turn to calculating
$G\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};L\right).$ Whereas equation 36
depended only on the orientation at $s_{1}$, the expectation value in equation
37 depends on the orientation at every point along the chain. In order to
expand $G$ in terms of $G_{o}$, we Taylor expand the exponential and reorder
the integrals

$\displaystyle\exp\left[\gamma\int_{0}^{N}dsY_{2,0}\left(\vec{\mathrm{u}}\left(s\right)\right)\right]$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\gamma\int_{0}^{N}dsY_{2,0}\left(\vec{\mathrm{u}}\left(s\right)\right)\right)^{n}$
(40)
$\displaystyle=\sum_{n=0}^{\infty}\left(\gamma^{n}\int_{0}^{N}ds_{n}...\int_{0}^{s_{3}}ds_{2}\int_{0}^{s_{2}}ds_{1}\prod_{i}^{n}Y_{2,0}\left(\vec{\mathrm{u}}_{i}\right)\right)$
(41)

where we have introduced the short hand
$\vec{\mathrm{u}}_{i}\equiv\vec{\mathrm{u}}\left(s_{i}\right)$. The factor of
$\frac{1}{n!}$ canceled the $n!$ from ordering the integrals. To calculate the
the $n=2$ term, for example, we have

$\displaystyle\left\langle\gamma^{2}\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}s_{1}Y_{2,0}\left(\vec{\mathrm{u}}_{2}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{1}\right)\right\rangle_{o}$
$\displaystyle\hskip
28.45274pt=\gamma^{2}\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}ds_{1}\int
d\vec{\mathrm{u}}_{2}\int
d\vec{\mathrm{u}}_{1}G_{o}\left(\vec{\mathrm{u}}_{3}|\vec{\mathrm{u}}_{2};N-s_{2}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{2}\right)G_{o}\left(\vec{\mathrm{u}}_{2}|\vec{\mathrm{u}}_{1};s_{2}-s_{1}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{1}\right)G_{o}\left(\vec{\mathrm{u}}_{1}|\vec{\mathrm{u}}_{0};s_{1}\right)$
(42)

Figure 3:  Schematic illustrating the right hand side of equation 42. The
thick black curve represents a polymer $N$ Kuhn lengths long. Arrows represent
unit vectors $\vec{u_{0}}$, $\vec{u_{1}}$, $\vec{u_{2}}$, $\vec{u_{3}}$ at
path lengths $0$, $s_{1}$, $s_{2}$, and $N$ along the polymer respectively.

To simplify equation 42, it is convenient to perform a Laplace transform from
$N\to p$ . Defining the Laplace transform of equation 35 as
$\breve{G}_{o}\left(p\right)\equiv\mathcal{L}\left[G\right]=\int_{0}^{\infty}dNG_{o}\left(N\right)e^{-pN}$
gives

$\breve{G}_{o}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};p\right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{\ell,m}\left(\vec{\mathrm{u}}_{f}\right)Y_{\ell,m}\left(\vec{\mathrm{u}}_{0}\right)\frac{1}{p+\ell\left(\ell+1\right)}$
(43)

The convolution rule for Laplace transforms says that
$\mathcal{L}\left(f\right)\mathcal{L}\left(g\right)=\mathcal{L}\left[f*g\right]=\mathcal{L}\left[\int_{0}^{t}d\tau
f\left(\tau\right)g\left(t-\tau\right)\right]$. In equation 42 we have
something of the form $\mathcal{L}\left[f*g*h\right]$. By plugging the
convolution rule into itself recursively we arrive at the general relation
$\mathcal{L}\left[f_{1}*f_{2}*...f_{n+1}\right]=\prod_{j=1}^{n+1}\mathcal{L}\left(f_{j}\right)$.
Taking the Laplace transform of the $n=2$ example in equation 42 and
substituting into 43 we arrive at

$\displaystyle\mathcal{L}\left[\left\langle\mathrm{...}\right\rangle_{o}\right]=$
$\displaystyle\sum_{\ell_{i},m_{i}}Y_{l_{1},m_{1}}\left(\vec{\mathrm{u}}_{0}\right)$
$\displaystyle\times\breve{G}_{o}\left(p|\ell_{1}\right)\gamma\int
d\vec{\mathrm{u}}_{1}Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{1}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{1}\right)$
$\displaystyle\times\breve{G}_{o}\left(p|\ell_{2}\right)\gamma\int
d\vec{\mathrm{u}}_{2}Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{2}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{2}\right)Y_{\ell_{3},m_{3}}\left(\vec{\mathrm{u}}_{2}\right)$
$\displaystyle\times\breve{G}_{o}\left(p|\ell_{3}\right)Y_{l_{3},m_{2}}\left(\vec{\mathrm{u}}_{f}\right)$
(44)

where
$\left\langle\mathrm{...}\right\rangle_{o}=\left\langle\gamma^{2}\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}s_{1}Y_{2,0}\left(\vec{\mathrm{u}}_{2}\right)Y_{2,0}\left(\vec{\mathrm{u}}_{1}\right)\right\rangle_{o}$
and
$\breve{G}_{o}\left(p|\ell\right)\equiv\frac{1}{p+\ell\left(\ell+1\right)}$
from equation 43. The integrals over the spherical harmonics in the middle can
be written in terms of $A_{\ell}^{m}$ and $\beta_{\ell}^{m}$ from appendix A.
In particular equation 139 gives us

$\displaystyle\int
d\vec{u}_{1}Y_{\ell_{1},m_{1}}\left(\vec{u}_{1}\right)Y_{2,0}\left(\vec{u}_{1}\right)Y_{\ell_{2},m_{2}}\left(\vec{u}_{1}\right)\hskip
5.69046pt=\left(A_{\ell_{1}+2}^{m_{1}}\delta_{\ell_{1}+2,\ell_{2}}+\beta_{\ell_{1}}^{m_{1}}\delta_{\ell_{1},\ell_{2}}+A_{\ell_{1}}^{m_{1}}\delta_{\ell_{1}-2,\ell_{2}}\right)\delta_{m_{1},m_{2}}$
(45)

so that

$\displaystyle\mathcal{L}\left[\left\langle\mathrm{...}\right\rangle_{o}\right]=$
$\displaystyle\sum_{\ell_{1},m}Y_{l_{1},m}\left(\vec{\mathrm{u}}_{0}\right)\breve{G}_{o}\left(p|\ell_{1}\right)$
$\displaystyle\times\left(A_{\ell_{1}+2}^{m}\delta_{\ell_{1}+2,\ell_{2}}+\beta_{\ell_{1}}^{m}\delta_{\ell_{1},\ell_{2}}+A_{\ell_{1}}^{m}\delta_{\ell_{1}-2,\ell_{2}}\right)$
$\displaystyle\times\breve{G}_{o}\left(p|\ell_{2}\right)$
$\displaystyle\times\left(A_{\ell_{2}+2}^{m}\delta_{\ell_{1}+2,\ell_{2}}+\beta_{\ell_{1}}^{m}\delta_{\ell_{1},\ell_{2}}+A_{\ell_{1}}^{m}\delta_{\ell_{1}-2,\ell_{2}}\right)$
$\displaystyle\times\breve{G}_{o}\left(p|\ell_{3}\right)Y_{l_{3},m}\left(\vec{\mathrm{u}}_{f}\right)$
(46)

which distributes to 9 terms for each $m$ and $\ell_{1}$ (though some are
disallowed by the requirements that $\ell_{1}\geq\left|m\right|$).

We would like to apply the reasoning of equations 44 and 45 to all term in the
sum 40, the Laplace transform of
$G\left(\vec{\mathrm{u}}_{f},\vec{\mathrm{u}}_{0},N\right)$. However,
multiplying 45 by itself for each of the terms in 40 will result in a very
large number of terms. The enumeration of these terms is greatly aided by the
use of stone fence diagrams [25, 21, 22]. Each diagram represents a term, with
the sum of all diagrams equaling
$G\left(\vec{\mathrm{u}}_{f},\vec{\mathrm{u}}_{0},N\right)$. The simplest such
diagram – and only type at the $n=0$ level of 40 – is the orientation
propagator without an aligning field $\left\langle\
\right\rangle_{o}=\breve{G}_{o}\left(p|\ell_{3}\right)$ and represented by a
dot:

There is one of these diagrams for each of the values of $\ell$ \- indicated
by the $\ell$ to the left of the diagram. In this case, the incoming
$\ell_{0}$ and outgoing $\ell_{f}$ are always the same, as indicated by the
same $\ell$ for the spherical harmonics of $\vec{u}_{f}$ and $\vec{u}_{0}$ in
equation 43. However, diagrams for $n>0$, which include factors of $\gamma$,
may connect different $\ell$ values. The form of equation 44 suggests that we
should connect these dots with products of spherical harmonics. There are
three types of connections in equation 45 corresponding to keeping the $\ell$
value the same, increasing $\ell$ by 2, or decreasing $\ell$ by 2. These are
represented by horizontal, upwardly slanted, and downwardly slanted
connections respectively:

and

in this way the vertical axis corresponds to $\ell$ as is indicated by the
expressions to the left of the diagrams. The three possible diagrams starting
at $\ell$ and having a single $\gamma$ are

and 4 of the 9 possible diagrams proportional to $\gamma^{2}$ are

These nine diagrams correspond to the nine terms in equation 46. For each
eigenvalue set $\left\\{m_{0},\ell_{0}\right\\}$ describing the orientation of
$\vec{\mathrm{u}}_{0}$, there are $3^{n}$ such diagrams from the $\gamma^{n}$
term of the expansion 40; though many of these will be zero due to the
requirement that $\ell\geq\left|m\right|$ which is enforced by
$A_{\ell}^{\ell}=A_{\ell}^{\ell+1}=0$ and disallows diagrams that go below
$\left|m\right|$ anywhere along their path. Our task is to sum the allowed
diagrams with any number, $n$, of sections. We define a new symbol to denote
the sum of all such diagrams that don’t contain any factors of $A$ so that it
starts and ends at $\ell$.

which can be evaluated via a geometric series

$\sum_{n=0}^{\infty}\left(\frac{1}{P_{\ell}}\right)^{n+1}\left(\gamma\beta_{\ell}^{m}\right)^{n}=\frac{1}{P_{\ell}-\gamma\beta_{\ell}^{m}}$
(47)

The critical insight made in [21] is to sum all of the diagrams that start and
end at $\ell$ but never stray below $\ell$ in between. We define this new
quantity as

which can be summed with a geometric series

$W_{\ell}^{\left(+\right)m}=\sum_{n=0}^{\infty}\left(\frac{1}{P_{\ell}-\gamma\beta_{\ell}^{m}}\right)^{n+1}\left(\left(\gamma
A_{\ell+2}^{m}\right)^{2}W_{\ell+2}^{\left(+\right)m}\right)^{n}=\frac{1}{P_{\ell}-\gamma\beta_{\ell}^{m}-\left(\gamma
A_{\ell+2}^{m}\right)^{2}W_{\ell+2}^{\left(+\right)m}}$ (48)

which can calculated recursively by truncating at a sufficiently high $\ell$
value.

Likewise

$W_{\ell}^{\left(-\right)m}=\begin{cases}\frac{1}{P_{\ell}-\gamma
B_{\ell}^{m}-\left(\gamma
A_{\ell}^{m}\right)^{2}W_{\ell-2}^{\left(-\right)m}}&\ell\geq\left|m\right|\\\
0&\mathrm{otherwise}\end{cases}$ (49)

where we have explicitly cut off the recursion when $\ell$ dips below
$\left|m\right|$.

Also of interest is the diagram similar to $W^{\left(+\right)}$ but without
the left dot and requiring at least one up-down step

To get all the diagrams that begin and end at $\ell$ we need to include all
possible sets of up steps and down steps. We group these diagrams into four
groups: 1, up first down last; 2, up first up last; 3, down first up last; and
4, down first down last.

where the final $-1/\left(P_{\ell}-\gamma B_{\ell}^{m}\right)$ is to remove
the double counting of .

We may write the sum

$\displaystyle W_{\ell}^{\left(\pm\right)m}$
$\displaystyle=\left(\frac{1}{P_{\ell}-\gamma\beta_{\ell}^{m}}\right)\sum_{n=0}^{\infty}\left[\left(\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(+\right)m}-1\right)\left(\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(-\right)m}-1\right)\right]^{n}$
$\displaystyle\ \ \ \
\times\left(2+\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(+\right)m}-1+\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(-\right)m}-1\right)-\left(\frac{1}{P_{\ell}-\gamma\beta_{\ell}^{m}}\right)$
$\displaystyle=\frac{W_{\ell}^{\left(+\right)m}W_{\ell}^{\left(-\right)m}}{W_{\ell}^{\left(-\right)m}-\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(+\right)m}W_{\ell}^{\left(-\right)m}+W_{\ell}^{\left(+\right)m}}$
(50)

The term $W_{\ell}^{\left(\pm\right)m}$ gives all the diagrams starting and
ending at $m$ and $\ell$. The sum of diagrams that connect different $\ell$
values

can be calculated recursively

$G_{\ell_{0}\ell_{f}}^{m}=\begin{cases}0&\mathrm{if\ }\ell_{f}-\ell_{0}\
\mathrm{is\ odd}\\\ W_{\ell_{0}}^{\left(\pm\right)m}&\mathrm{if\
}\ell_{f}=\ell_{0}\\\
G_{\ell_{0},\ell_{f}-2}^{m}W_{\ell_{f}}^{\left(+\right)m}A_{\ell_{f}}^{m}\gamma&\mathrm{if\
}\ell_{f}>\ell_{0}\\\
G_{\ell_{0},\ell_{f}+2}^{m}W_{\ell_{f}}^{\left(-\right)m}A_{\ell_{f+2}}^{m}\gamma&\mathrm{if\
}\ell_{f}<\ell_{0}\end{cases}$ (51)

with the application of the flanking harmonics we have our desired propagator

$\breve{G}\left(\vec{\mathrm{u}}|\vec{\mathrm{u}}_{0};p\right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{\ell,m}\left(\vec{\mathrm{u}}_{0}\right)G_{\ell_{0}\ell_{f}}^{m}Y_{\ell,m}\left(\vec{\mathrm{u}}\right)$
(52)

##  VIII Evaluating Expectations

In this section we will show how the propagator $\breve{G}$ can be used to
calculate expectation values of various quantities (e.x.
$\left\langle\phi_{2,0}\right\rangle^{MF}$ or
$\left\langle\phi_{2,1}\phi_{2,1}\right\rangle^{MF})$ for WLC’s in an aligning
mean field.

In the mean field solution, the $n_{p}$ polymers are distributed over a volume
$\mathscr{V}$. Taking the expectation value of equation 7 we have

$\left\langle\phi_{2,0}\right\rangle^{MF}=\left\langle
n_{p}\frac{1}{\mathscr{V}}A\sqrt{\frac{4\pi}{5}}2\ell_{p}\int_{0}^{N}dsY_{2,0}\left(\vec{u}\left(s\right)\right)\right\rangle^{MF}$
(53)

or

$\left\langle\phi_{2,0}\right\rangle^{MF}=\sqrt{\frac{4\pi}{5}}\frac{\phi_{0,0}}{N}\int_{0}^{N}ds\left\langle
Y_{2,0}\left(\vec{u}\left(s\right)\right)\right\rangle^{MF}$ (54)

The probability the chain will point in direction $\vec{u}$ at path length $s$
along the chain is weighted by the propagator from the beginning of the chain
to $s$, i.e. $G\left(\vec{u}|\vec{u}_{0};s\right)$, and the propagator from
$s$ to the other end of the chain $G\left(\vec{u}_{f}|\vec{u};L-s\right)$. The
Laplace transform from $N$ to $p$ of
$\left\langle\phi_{2,0}\right\rangle^{MF}$ can be preformed using the
convolution rule (see discussion of equation 44)

$\mathcal{L}\left[\int_{0}^{N}dsG\left(\vec{u}_{f}|\vec{u};p\right)G\left(\vec{u}|\vec{u}_{0};s\right)\right]=\breve{G}\left(\vec{u}_{1}|\vec{u}_{0};p\right)\breve{G}\left(\vec{u}_{f}|\vec{u}_{1};p\right)$
(55)

To calculate $\left\langle\phi_{2,0}\right\rangle^{MF}$ we want to integrate
over beginning, middle, and end orientations as in

$\left\langle\phi_{2,0}\right\rangle^{MF}=\frac{\phi_{0,0}}{N}\sqrt{\frac{4\pi}{5}}\frac{\mathcal{L}^{-1}\left[\int
du_{0}\int du_{1}\int
du_{f}\breve{G}\left(\vec{u}_{1}|\vec{u}_{0};p\right)Y_{2,0}\left(\vec{u}_{1}\right)\breve{G}\left(\vec{u}_{f}|\vec{u}_{1};p\right)\right]\left(N\right)}{\mathcal{L}^{-1}\left[\int
du_{0}\int
du_{f}\breve{G}\left(\vec{u}_{f}|\vec{u}_{0};p\right)\right]\left(N\right)}$
(56)

where the denominator provides the appropriate normalization. We can then
substitute equation 52 into 56 to change basis from $\vec{u}$ to $Y_{\ell,m}$.
The sums over $\ell$ can be made implicit by writing the result in matrix form
where the rows and columns run over $\ell$ values so that

$\left(\mathbf{G}^{m}\right)_{\ell_{1},\ell_{2}}=G_{\ell_{0}\ell_{f}}^{m}$
(57)

and the product of three real spherical harmonics is defined as

$\left(\mathbf{J}_{\left(\ell\right)}^{m,m^{\prime},m^{\prime\prime}}\right)_{\ell_{1},\ell_{2}}\equiv\int
d\vec{u}Y_{\ell_{1},m}\left(\vec{u}\right)Y_{\ell,m^{\prime}}\left(\vec{u}\right)Y_{\ell_{2}m^{\prime\prime}}\left(\vec{u}\right)$
(58)

The values of $\mathbf{J}_{\left(\ell\right)}^{m,m^{\prime},m^{\prime\prime}}$
are derived in appendix A. After the change of basis, equation 56 becomes

$\phi_{2,0}^{MF}=\frac{\phi_{0,0}}{N}\sqrt{\frac{4\pi}{5}}\frac{\mathcal{L}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(2\right)}^{0,0,0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]\left(N\right)}{\mathcal{L}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]\left(N\right)}$
(59)

were $\bm{e}_{0}=\left[1,0,0,0...\right]$ is the unit vector so that
${e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}=\mathbf{G}^{0}_{1,1}$. The pinning
to $\ell=0$ at the chain ends results from inserting
$Y_{0,0}=\frac{1}{\sqrt{4\pi}}$ and applying $\int
d\vec{u}Y_{0,0}Y_{\ell,m}=\delta_{0,\ell}\delta_{0,m}$. Furthermore, the
requirement that $\left|m\right|\leq\ell$ required $m_{1}=m_{2}=0$ in equation
59. The inverse Laplace transforms can be performed numerically as described
in section XVI.

The process of connecting sequential propagators together can be extended to
the expectation value of several spherical harmonics

$\displaystyle\int_{0}^{N}ds_{n}...\int_{0}^{s_{2}}ds_{1}\left\langle\prod_{j=1}^{n}Y_{\ell_{j},m_{j}}\left(s_{j}\right)\right\rangle^{MF}=...$
$\displaystyle\frac{\mathcal{L}^{-1}\left[\sum_{\mathscr{M},...\mathscr{M}^{\prime\prime\prime}}\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(\ell_{n}\right)}^{0,m_{n},\mathscr{M}}\cdot\mathbf{G}^{\mathscr{M}}\cdot\mathbf{J}_{\left(\ell_{n-1}\right)}^{\mathscr{M},m_{n-1},\mathscr{M}^{\prime}}\cdot...\cdot\mathbf{G}^{\mathscr{M}^{\prime\prime\prime}}\cdot\mathbf{J}_{\left(\ell_{1}\right)}^{\mathscr{M}^{\prime\prime\prime},m_{1},0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]}{\mathcal{L}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]}$
(60)

##  IX Fluctuations

Having solved for the average field values $\left<\phi\right>^{MF}$ (and by
extension $W_{1}^{MF}=-iV_{12}\left<\phi\right>_{2}^{MF}$) we now investigate
local fluctuations about these values by substituting
$\phi=\left<\phi\right>^{MF}+\delta\phi$ and $W=W^{MF}+\delta W$ into 19.
Resulting constant terms such as $W^{MF}\left<\phi\right>^{MF}$ can be
absorbed into the proportionality constant which only offsets the reference
energy. Furthermore, the mean field solution $W^{MF}$ found above was uniform
in space so the term $W_{1}^{MF}\delta\phi_{1}$ must be zero. Equation 19 can
further be simplified by using $iW_{1}^{MF}=V_{12}\left<\phi\right>_{2}^{MF}$
yielding:

$Z\propto\int\mathscr{D}\delta
W\delta\phi\exp\left[-\frac{1}{2}\delta\phi_{1}V_{12}\delta\phi_{2}+i\delta
W_{1}\delta\phi_{1}+n_{p}\log\left(z_{p}\right)\right]$ (61)
$z_{p}=\int\mathscr{D}r_{i}^{(1)}\exp\left[-\beta
E_{poly}-\hat{\phi}_{1}^{(1)}V_{12}\left<\phi\right>_{2}^{MF}-i\delta
W_{1}\hat{\phi}_{1}^{(1)}\right]$ (62)

Per assumption 12, the field $\phi$ only makes small deviations from its mean
field value $\left<\phi\right>^{MF}$ so $\delta\phi$ is small. Also, under RPA
(assumption 13) $W$ only makes small deviations from its mean value $W^{MF}$
constraining $\delta W$ to be small. Taylor expanding $z_{p}$ about $\delta
W=0$ gives

$z_{p}\approx 1+\left\langle-i\delta
W_{1}\hat{\phi}_{1}^{(1)}\right\rangle^{MF}+\frac{1}{2}\left(\left\langle-i\delta
W_{1}\hat{\phi}_{1}^{(1)}\right\rangle^{MF}\right)^{2}$ (63)

(up to an additive constant of no consequence). We expand the logarithm
$\log\left(1+x\right)=x-\frac{1}{2}x^{2}$ in equation 61 and substituting in
$z_{p}$ we have

$\displaystyle Z\propto\int\mathscr{D}W\delta\phi$ $\displaystyle
e^{-\frac{1}{2}\delta\phi_{1}V_{12}\delta\phi_{2}+i\delta
W_{1}\delta\phi_{1}-n_{p}\frac{1}{2}\delta W_{1}S_{12}\delta W_{2}}$ (64)

where

$S_{12}\equiv\left\langle\hat{\phi}_{1}^{(1)}\hat{\phi}_{2}^{(1)}\right\rangle^{MF}-\left\langle\hat{\phi}_{1}^{(1)}\right\rangle^{MF}\left\langle\hat{\phi}_{2}^{(1)}\right\rangle^{MF}.$
(65)

We then perform a Gaussian integral over $\delta W$ to arrive at

$Z\propto\int\mathscr{D}\delta\phi\exp\left[-\frac{1}{2}\delta\phi_{1}\left(V_{12}+\frac{1}{n_{p}}S_{12}^{-1}\right)\delta\phi_{2}\right]$
(66)

where $S_{12}^{-1}$ is defined via
$S_{12}^{-1}S_{23}=\delta_{\ell_{1},\ell_{3}}\delta_{m_{1},m_{3}}\delta\left(\vec{r}_{1}-\vec{r}_{3}\right)$.
Appendix C discusses how to perform this inverse.

The energy expression in 66, which is often written with $\Gamma_{12}\equiv
V_{12}+\frac{1}{n_{p}}S_{12}^{-1}$, describes the quadratic fluctuations of
$\phi$ around the mean field solution. The bigger an element of $\Gamma$ is,
the smaller the corresponding fluctuation will be. Given the critical
importance of 66 we will devote considerable effort to calculating $S^{-1}$.

Because of translational invariance (assumption 7), the Fourier transform
$\vec{r}\to\vec{k}$ of $S_{12}$ is much easier to invert than the real space
$S_{12}$. Following the conventions in appendix B,

$Z\propto\int\mathscr{D}\delta\tilde{\phi}\exp\left[-\frac{1}{2}\delta\tilde{\phi}_{1}\left(\tilde{V}_{12}+\frac{1}{n_{p}}\tilde{S}_{12}^{-1}\right)\delta\tilde{\phi}_{2}\right]$
(67)

where
$\tilde{V}_{12}=\left(4\pi\chi\delta_{\ell_{1},0}-\frac{8\pi}{15}a\delta_{\ell_{1},2}\right)\delta_{m_{1},m_{2}}\delta_{\ell_{1},\ell_{2}}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)$.
In other words, the Fourier modes of fluctuations in $\phi$ are decoupled from
each other (at least to quadratic order in $\phi$). Another advantage of the
Fourier transform is that by assumptions 7, 8, and 12 and as discussed in
section VI the mean field solution is uniform in space so the latter term in
equation 65,
$\left\langle\hat{\phi}_{1}^{(1)}\right\rangle^{MF}\left\langle\hat{\phi}_{2}^{(1)}\right\rangle^{MF}$,
contributes only at $\vec{k}=0$. While we will be interested in the limit as
$\vec{k}\to 0$, the value at precisely $\vec{k}=0$ will not be used so we can
drop this constant term. We will refer to the first term on the right of
equation 65 as $\delta
S_{12}\equiv\left\langle\hat{\phi}_{1}^{(1)}\hat{\phi}_{2}^{(1)}\right\rangle^{MF}$.

##  X Calculating S for Fuzzball and Rigid Rod

Calculating $\delta S$ for a fuzzball or rigid rod can be accomplished using
$\mathbf{M}$ from equation 27 which amounts to the expectation of a product of
two spherical harmonics in an applied field. Inserting equation 13 into
$\delta
S_{12}\equiv\left\langle\hat{\phi}_{1}^{(1)}\hat{\phi}_{2}^{(1)}\right\rangle^{MF}$
for the fuzzball the spatial and rotational terms, $\rho$ and $v$
respectively, factor because the position and orientation of a fuzzball are
unrelated.

$\delta\tilde{S}_{12}^{FB}=\rho\left(\vec{r}_{1},\vec{r}_{2}\right)v\left(\ell_{1},m_{1},\ell_{2},m_{2}\right)$
(68) $\rho\left(\vec{r}_{1},\vec{r}_{2}\right)=\frac{1}{\mathscr{V}}\int
d\vec{r}_{i}\frac{v_{p}^{2}}{\left(2\pi\sigma^{2}\right)^{3}}\exp\left(-\frac{\left(\vec{r}_{1}-\vec{\mathrm{r}}_{i}\right)^{2}}{2\sigma^{2}}-\frac{\left(\vec{r}_{2}-\vec{\mathrm{r}}_{i}\right)^{2}}{2\sigma^{2}}\right)$
(69)
$v\left(\ell_{1},m_{1},\ell_{2},m_{2}\right)=\frac{4\pi}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}\frac{\mathbf{M}_{\ell_{1},\ell_{2}}^{m_{1}}\delta_{m_{1},m_{2}}}{4\pi\mathbf{M}_{0,0}^{0}}$
(70)

where the denominators come from the single fuzzball partition function
$z^{FB}=\mathscr{V}\int d\vec{u}\exp\left(\gamma
Y_{2,0}\left(\vec{u}_{i}\right)\right)=\mathscr{V}4\pi M_{0,0}^{0}$. Taking
the Fourier transform of a Gaussian yields a Gaussian, hence

$\delta\tilde{S}_{12}^{FB}=\frac{v_{p}^{2}}{\mathscr{V}}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)e^{-\sigma^{2}\vec{k}^{2}}\frac{\mathbf{M}_{\ell_{1},\ell_{2}}^{m_{1}}\delta_{m_{1},m_{2}}}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}\mathbf{M}_{0,0}^{0}}$
(71)

where the $\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)$ arises from
translational invariance.

The position along a rigid rod is inherently coupled to the orientation of the
rod so the factorization into $\rho$ and $v$ is not possible. In particular,
writing out
$\delta\tilde{S}=\left\langle\hat{\phi}_{1}^{(1)}\hat{\phi}_{2}^{(1)}\right\rangle^{MF}$
explicitly from equation IV we have

$\displaystyle\delta S_{12}^{RR}=$
$\displaystyle\frac{1}{z^{RR}}\frac{A^{2}}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}$
$\displaystyle\times\int d\vec{\mathrm{r}}_{j}\int
d\vec{\mathrm{u}}_{j}e^{\gamma
Y_{2,0}\left(\vec{u}_{j}\right)}\int_{0}^{L}ds_{1}\delta\left(\vec{\mathrm{r}}_{j}+s_{1}\vec{\mathrm{u}}_{j}-\vec{r}_{1}\right)\int_{0}^{L}ds_{2}\delta\left(\vec{\mathrm{r}}_{j}+s_{2}\vec{\mathrm{u}}_{j}-\vec{r}_{2}\right)Y_{\ell_{1,}m_{1}}\left(\vec{u}_{j}\right)Y_{\ell_{2},m_{2}}\left(\vec{u}_{j}\right)$
(72)

where the single rod partition function is $z^{RR}=\mathscr{V}\int
d\vec{u}\exp\left(\gamma
Y_{2,0}\left(\vec{u}_{i}\right)\right)=\mathscr{V}4\pi\mathbf{M}_{0,0}^{0}$.
Applying the Fourier transform from $\vec{r}\to\vec{k}$ takes

$\delta\left(\vec{\mathrm{r}}_{j}+s_{1}\vec{\mathrm{u}}_{j}-\vec{r}_{1}\right)\delta\left(\vec{\mathrm{r}}_{j}+s_{2}\vec{\mathrm{u}}_{j}-\vec{r}_{2}\right)\to\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)\exp\left(i\vec{k}\cdot\vec{\mathrm{u}}_{j}\left(s_{1}-s_{2}\right)\right).$
(73)

The exponential can be expanded to quadratic order and the $s$ integrals
evaluated directly

$\int_{0}^{L}ds_{1}\int_{0}^{L}ds_{2}\exp\left(i\vec{k}\cdot\vec{\mathrm{u}}_{j}\left(s_{1}-s_{2}\right)\right)\approx
L^{2}-\frac{1}{2}\left(\vec{k}\cdot\vec{\mathrm{u}}_{j}\right)^{2}\frac{L^{4}}{6}.$
(74)

Introducing $\kappa$ defined in equation 79 and doing some simplification we
arrive at

$\displaystyle\delta\tilde{S}_{12}^{RR}=\frac{A^{2}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}$
$\displaystyle\left[\frac{L^{2}}{\mathscr{V}}\frac{\mathbf{M}_{\ell_{1},\ell_{2}}^{m_{1}}}{\mathbf{M}_{0,0}^{0}}\delta_{m_{1},m_{2}}-\frac{4\pi}{z^{RR}}\frac{L^{4}}{12}\sum_{m=-1}^{1}\sum_{m^{\prime}=-1}^{1}\kappa_{1,m}\kappa_{1,m^{\prime}}\sum_{\mathscr{M}}\left(\mathbf{J}_{\left(1\right)}^{m_{1},m,\mathscr{M}}\cdot\mathbf{M}^{\mathscr{M}}\cdot\mathbf{J}_{\left(1\right)}^{\mathscr{M},m^{\prime},m_{2}}\right)_{\ell_{1},\ell_{2}}\right]$
(75)

where the integer $\mathscr{M}$ runs over all values allowed by the selection
rules of $\mathbf{J}$. We now have an explicit formula for the low $k$
expansion of $\delta\tilde{S}^{RR}$.

##  XI Calculating S for WLC

We now turn to finding $\delta S_{12}$ for the WLC. Writing $\delta S_{12}$
explicitly

$\displaystyle\left(\delta
S_{12}\right)_{\ell_{1}\ell_{2}}^{m_{1}m_{2}}\left(\vec{r}_{1},\vec{r}_{2}\right)=$
$\displaystyle\frac{4\pi}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}A^{2}\int_{0}^{L}ds_{2}\int_{0}^{L}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}\left(s_{1}\right)\right)\delta\left(\vec{r}_{1}-\mathrm{\vec{r}}\left(s_{1}\right)\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}\left(s_{2}\right)\right)\delta\left(\vec{r}_{2}-\vec{\mathrm{r}}\left(s_{2}\right)\right)\right\rangle^{MF}$
(76)

Note the distinction between the argument $\vec{r}_{1}$ and the unitalicized
$\vec{\mathrm{r}}(s_{1})$ which stands for position of a point $s_{1}$ along
the polymer. We take the Fourier transform $\vec{r}_{1}\to\vec{k}_{1}$ and
$\vec{r}_{2}\to\vec{k}_{2}$ via $\frac{1}{\left(2\pi\right)^{3}}\int
d\vec{r}_{1}d\vec{r}_{2}\exp\left(i\vec{k}_{1}\vec{r}_{1}+i\vec{k}_{2}\vec{r}_{2}\right)$.
The integrals over $\vec{r}$ are eliminated by delta functions to give
$\exp\left(i\vec{k}_{1}\vec{\mathrm{r}}\left(s_{1}\right)+i\vec{k}_{2}\vec{\mathrm{r}}\left(s_{2}\right)\right)$
which can be factored into
$\exp\left[i\vec{k}_{1}\cdot\left(\vec{\mathrm{r}}_{1}-\vec{\mathrm{r}}_{2}\right)\right]\exp\left[i\left(\vec{k}_{1}+\vec{k}_{2}\right)\vec{\mathrm{r}}_{2}\right]$
where we have introduced the notation
$\vec{\mathrm{r}}\left(s_{2}\right)=\vec{\mathrm{r}}_{2}$. By translational
invariance (assumption 7) we can average out $\vec{\mathrm{r}}_{2}$ over the
system volume $\mathscr{V}$ by applying $\mathscr{V}^{-1}\int
d\vec{\mathrm{r}}_{2}$ and recognizing the delta function
$\left(2\pi\right)^{3}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)=\int
d\vec{r}e^{i\vec{r}\left(\vec{k}_{1}+\vec{k}_{2}\right)}$, giving

$\displaystyle\delta\tilde{S}_{12}=$
$\displaystyle\frac{4\pi}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}\frac{A^{2}\left(2\ell_{p}\right)^{2}}{\mathscr{V}}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)\int_{0}^{N}ds_{2}\int_{0}^{N}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)\exp\left[i\vec{k}_{1}\cdot\left(\vec{\mathrm{r}}_{1}-\vec{\mathrm{r}}_{2}\right)\right]Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{2}\right)\right\rangle^{MF}$
(77)

where we nondimensionalized $L=\frac{N}{2\ell_{p}}$ so that
$\int_{0}^{L}ds_{2}$ became $2\ell_{p}\int_{0}^{N}ds_{2}$. It will be
convenient to write $\vec{k}$ in spherical coordinates. Because $\vec{k}$ is a
vector, it is an $\ell=1$ object. Specifically we will define
$\kappa_{\ell,m}$ as

$\kappa_{1,m}\equiv\sqrt{\frac{4\pi}{3}}\begin{cases}k_{y}&m=-1\\\
k_{z}&m=0\\\ k_{x}&m=1\end{cases}$ (78)

which has the property

$\vec{k}\cdot\vec{u}=\sum_{m=-1}^{1}\kappa_{1,m}Y_{1,m}\left(\vec{u}\right)$
(79)

which is useful because
$\vec{k}_{1}\cdot\left(\vec{\mathrm{r}}_{1}-\vec{\mathrm{r}}_{2}\right)$ in
equation 77 can be written as
$\vec{k}_{1}\cdot\left(2\ell_{p}\right)\int_{s_{1}}^{s_{2}}ds\vec{\mathrm{u}}\left(s\right)=\int_{s_{1}}^{s_{2}}ds\sum_{m=-1}^{1}\kappa^{\prime}_{1,m}Y_{1,m}\left(\vec{u}\right)$
where $\kappa^{\prime}=2\ell_{p}\kappa$. It can be shown – or verified
numerically – that $\int_{0}^{N}ds_{2}\int_{0}^{N}ds_{1}$ may be replaced with
$2\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}ds_{1}$ in this expression. To further
simplify the expression we define the nondimensionalized

$\tilde{S}^{\prime}_{12}\left(\vec{k}\right)\equiv\delta\tilde{S}_{12}\left(\vec{k}_{1},-\vec{k}_{1}\right)\frac{\mathscr{V}}{A^{2}\left(2\ell_{p}\right)^{2}N^{2}}$
(80)

By assumption 15 we are interested in the large distance (small $\vec{k}$)
behavior, so we Taylor expand the exponential in 77 which, along with the
above modifications, becomes

$\displaystyle\tilde{S}^{\prime}_{12}$
$\displaystyle\approx\frac{4\pi}{\sqrt{\left(2\ell_{1}+1\right)\left(2\ell_{2}+1\right)}}\frac{2}{N^{2}}$
$\displaystyle\times\left[\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{2}\right)\right\rangle^{MF}\right.$
$\displaystyle+\sum_{m=-1}^{1}i\kappa^{\prime}_{1,m}\int_{0}^{N}ds_{3}\int_{0}^{s_{3}}ds_{2}\int_{0}^{s_{2}}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{1,m}\left(\vec{\mathrm{u}}_{2}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{3}\right)\right\rangle^{MF}$
$\displaystyle\left.-\sum_{m,m^{\prime}=-1}^{1}\kappa^{\prime}_{1,m}\kappa^{\prime}_{1,m^{\prime}}\int_{0}^{N}ds_{4}\int_{0}^{s_{4}}ds_{3}\int_{0}^{s_{3}}ds_{2}\int_{0}^{s_{2}}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{1,m}\left(\vec{\mathrm{u}}_{2}\right)Y_{1,m^{\prime}}\left(\vec{\mathrm{u}}_{3}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{4}\right)\right\rangle^{MF}\right]$
(81)

The expectation values in 81 can be evaluated as described in equation 60. The
first term in the series becomes

$\int_{0}^{N}ds_{2}\int_{0}^{s_{2}}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{2}\right)\right\rangle^{MF}=\frac{\mathscr{\mathcal{L}}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(\ell_{1}\right)}^{0,m_{1},m_{1}}\cdot\mathbf{G}^{m_{1}}\cdot\mathbf{J}_{\left(\ell_{2}\right)}^{m_{1},m_{1},0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]\delta_{m_{1},m_{2}}}{\mathscr{\mathcal{L}}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]}$
(82)

the linear term is killed by selection rules for $\mathbf{J}$ and the
quadratic term is given by

$\displaystyle\int_{0}^{N}ds_{4}\int_{0}^{s_{4}}ds_{3}\int_{0}^{s_{3}}ds_{2}\int_{0}^{s_{2}}ds_{1}\left\langle
Y_{\ell_{1},m_{1}}\left(\vec{\mathrm{u}}_{1}\right)Y_{1,m}\left(\vec{\mathrm{u}}_{2}\right)Y_{1,m^{\prime}}\left(\vec{\mathrm{u}}_{3}\right)Y_{\ell_{2},m_{2}}\left(\vec{\mathrm{u}}_{4}\right)\right\rangle^{MF}$
$\displaystyle=\frac{\mathscr{\mathcal{L}}^{-1}\left[\sum_{\mathscr{M}}\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(\ell_{1}\right)}^{0,m_{1},m_{1}}\cdot\mathbf{G}^{m_{1}}\cdot\mathbf{J}_{\left(1\right)}^{m_{1},m,\mathscr{M}}\cdot\mathbf{G}^{\mathscr{M}}\cdot\mathbf{J}_{\left(1\right)}^{\mathscr{M},m^{\prime},m_{2}}\cdot\mathbf{G}^{m_{2}}\cdot\mathbf{J}_{\left(\ell_{2}\right)}^{m_{2},m_{2},0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]}{\mathscr{\mathcal{L}}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]}.$
(83)

##  XII Cancellation

Having gone though the effort to calculate the low $k$ behavior of
$\delta\tilde{S}$ it is worth taking some time to consider the form of the
result. In this paper we are primarily interested in the density $\ell=0$ and
quadrupole $\ell=2$ interactions. It is convenient two write $\delta\tilde{S}$
in the basis

$\left(\ell_{i},m_{i}\right)=\left[\left(0,0\right),\left(2,0\right),\left(2,-1\right),\left(2,1\right),\left(2,-2\right),\left(2,2\right)\right].$
(84)

In the long wavelength limit there are only correlations between fluctuations
of the same $m$ values, i.e. $\lim_{k\to
0}\delta\tilde{S}\propto\delta_{m_{1},m_{2}}$. As it turns out, in this basis
we have

$\lim_{\vec{\kappa}\to
0}\delta\tilde{S}_{\ell_{1}\ell_{2}}^{m_{1},m_{2}}=\begin{bmatrix}t&h&0&0&0&0\\\
h&x&0&0&0&0\\\ 0&0&y&0&0&0\\\ 0&0&0&y&0&0\\\ 0&0&0&0&z&0\\\
0&0&0&0&0&z\end{bmatrix}$ (85)

where the expressions for $t,\ h,\ x,\ y,\ \mathrm{and\ z}$ depend on the
whether we are talking about a WLC, rigid rod, or fuzzball. In section XIII we
will discuss the meaning of the various fluctuations (see equation 101 and
figure 6). For now, we will point out that $t$ governs density-density
interactions, $x$ governs variations in alignment strength, $y$ governs
variations in alignment direction, and $z$ governs fluctuations in a secondary
direction of alignment. As the Frank elastic constants are of most interest to
us we will focus on $y$. Here we provide plots of $\lim_{\vec{\kappa}\to
0}\delta\tilde{S}_{\ell_{1}\ell_{2}}^{m_{1},m_{2}}$ for a nearly rigid WLC.

Figure 4:  Plot of selected terms of $\lim_{k\to 0}\delta\tilde{S}$ as a
function of field strength. Because $\lim_{k\to
0}\delta\tilde{S}^{1,1}_{2,2}=\lim_{k\to 0}\delta\tilde{S}^{-1,-1}_{2,2}$ and
$\lim_{k\to 0}\delta\tilde{S}^{2,2}_{2,2}=\lim_{k\to
0}\delta\tilde{S}^{-2,-2}_{2,2}$ some curves are the same.

In the isotropic limit, $\gamma\to 0$, all directions $Y_{2,m}$ are identical.
As alignment in the $\hat{z}$ direction (i.e. $Y_{2,0}$) grows, it does so at
the expense of the alignment in the other directions.

When $\delta\tilde{S}$ is inverted and substituted into equation 66 we get the
energy coefficients

$\tilde{\Gamma}_{12}=\left(\chi\delta_{\ell_{1},0}-\frac{2}{3}a\delta_{\ell_{1},2}\right)\delta_{m_{1},m_{2}}\delta_{\ell_{1},\ell_{2}}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)+\frac{1}{n_{p}}\delta\tilde{S}_{12}^{-1}$
(86)

for the various modes where the inverse is discussed in C. Critically, we have
verified numerically, that in the limit that $\vec{\kappa}\to 0$ that $y$
precisely cancels with $\frac{2}{3}a$ term from the original interaction
potential. This cancellation occurs at all field strengths $\gamma$. This
cancellation comes about because of the invariance of the energy to a rotation
of the entire system. Hence, the system will exhibit long distance
deformations in the direction of alignment. It is only the rate of change of
these directions that contribute to the energy, hence the derivatives,
$\nabla$, in the Frank elastic expressions.

##  XIII Frank Elastic Energies

In this section we will define the nematic director and Frank elastic
constants and their relation to the orientation field $\hat{\phi}$. The Frank
elastic energy density depends on the orientations $\vec{u}$ in the vicinity
$\Delta V$. The orientation distribution can be described by the 3x3 matrix

$M=\frac{\int^{\Delta
V}d\vec{r}\sum_{j=1}^{n_{p}}A\int_{0}^{L}ds\delta\left(\vec{r}-\vec{\mathrm{r}}_{j}\left(s\right)\right)\vec{\mathrm{u}}_{j}\left(s\right)\otimes\vec{\mathrm{u}}_{j}\left(s\right)}{\int^{\Delta
V}d\vec{r}\sum_{j=1}^{n_{p}}A\int_{0}^{L}ds\delta\left(\vec{r}-\vec{\mathrm{r}}_{j}\left(s\right)\right)}$
(87)

the denominator of which normalizes out the local polymer density,
$\phi_{0,0}$, leaving only the directional information. The $\Delta V$ is the
coarse-graining volume such that details of the structure smaller than this
volume will not be treated. Going forward we will drop the $\int^{\Delta V}$ .
The eigen-decomposition of the $M$ matrix is [29]

$M=\lambda_{1}\vec{n}\otimes\vec{n}+\lambda_{2}\vec{e}_{2}\otimes\vec{e}_{2}+\lambda_{3}\vec{e}_{3}\otimes\vec{e}_{3}$
(88)

where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ This defines the unit length
nematic director $\vec{n}$ which indicates the general direction of alignment
within $\Delta V$. As $\vec{n}$ is an eigenvector, its sign is arbitrarily
chosen. The eigenvalues of $M$, which obey
$\lambda_{1}+\lambda_{2}+\lambda_{3}=1$ and $\lambda_{i}\geq 0$, designate the
degree of alignment [29]. The interpretation of $\lambda_{i}$ can be
understood in terms of the traceless alignment matrix

$Q\equiv
M-\frac{1}{3}Ic_{1}=c_{1}\left(\vec{n}\otimes\vec{n}-\frac{1}{3}I\right)+c_{2}\left(\vec{e}_{2}\otimes\vec{e}_{2}-\vec{e}_{3}\otimes\vec{e}_{3}\right)$
(89)

where

$c_{1}=\frac{3\lambda_{1}-1}{2},\ \ \ \ c_{2}=\lambda_{2}-\lambda_{1}$ (90)

with $c_{1}\in\left[0,1\right]$ describing the degree of alignment in the
primary direction and $c_{2}$ describing the degree to which a secondary
direction of alignment exists.

The Frank elastic energy density is the sum of bend, twist, and splay energies
and is defined in terms of the nematic director

$\displaystyle E_{Frank}=$
$\displaystyle\frac{1}{2}K_{bend}\left\langle\left(\vec{n}\times\nabla\times\vec{n}\right)^{2}\right\rangle+\frac{1}{2}K_{twist}\left\langle\left(\vec{n}\cdot\nabla\times\vec{n}\right)^{2}\right\rangle+\frac{1}{2}K_{splay}\left\langle\left(\nabla\cdot\vec{n}\right)^{2}\right\rangle$

To see why there are exactly three deformation modes, consider figure 5. Each
mode prescribes a change in orientation associated with a displacement. For
each mode there is a direction $\vec{n}$ of alignment (represented by the
cylinder), a displacement (green arrow), and a change in the direction of
alignment (red arrow). The displacement must be perpendicular to $\vec{n}$ as
it is a differential change. The displacement, therefore, can be in one of
three direction: 1) parallel to $\vec{n}$, in which case the deformation is
bend; 2) parallel to the change in direction, in which case the deformation is
splay; and 3) perpendicular to both, in which case the deformation is twist.

Figure 5: Pictorial representation clarifying the chief difference between
bend, twist, and splay. The axes of the cylinders represent the direction of
nematic ordering, $\vec{n}$. Each green arrow represents a displacement to a
different position in a liquid crystal. Red arrows represent the change in
orientation associated with the change in position. A semitransparent cylinder
represents the new position but with the old orientation as a guide to the
eye.

We will devote the reminder of this derivation to finding the Frank elastic
constants $K_{bend}$, $K_{twist}$, and $K_{splay}$ in terms of polymer
properties. Our first goal will be to solve for the associated expectation
values in terms of $\hat{\phi}_{\ell,m}$. We begin by exploring the relation
between $\hat{\phi}$ and
$\vec{\mathrm{u}}_{j}\left(s\right)\otimes\vec{\mathrm{u}}_{j}\left(s\right)$.
Using the real spherical harmonic for $\ell=2$

$Y_{2,m}\left(\vec{u}\right)=\sqrt{\frac{5}{4\pi}}\begin{cases}\sqrt{3}u_{x}u_{y}&m=-2\\\
\sqrt{3}u_{y}u_{z}&m=-1\\\ \frac{1}{2}\left(3u_{z}^{2}-1\right)&m=0\\\
\sqrt{3}u_{z}u_{x}&m=1\\\
\frac{\sqrt{3}}{2}\left(u_{x}^{2}-u_{y}^{2}\right)&m=2\end{cases}$ (91)

and definition 7 we can write

$\displaystyle Q$
$\displaystyle=\frac{1}{\phi_{0,0}}\sum_{j=1}^{n_{p}}A\int_{0}^{L}ds\delta\left(\vec{r}-\vec{\mathrm{r}}_{j}\left(s\right)\right)\left(\vec{\mathrm{u}}_{j}\left(s\right)\otimes\vec{\mathrm{u}}_{j}\left(s\right)-\frac{1}{3}\mathbb{I}\right)$
(92)
$\displaystyle=\frac{1}{\phi_{0,0}\sqrt{3}}\begin{bmatrix}\hat{\phi}_{2,2}-\frac{1}{\sqrt{3}}\hat{\phi}_{2,0}&\hat{\phi}_{2,-2}&\hat{\phi}_{2,1}\\\
\hat{\phi}_{2,-2}&-\hat{\phi}_{2,2}-\frac{1}{\sqrt{3}}\hat{\phi}_{2,0}&\hat{\phi}_{2,-1}\\\
\hat{\phi}_{2,1}&\hat{\phi}_{2,-1}&\frac{2}{\sqrt{3}}\hat{\phi}_{2,0}\end{bmatrix}$
(93)

The magnitude of $\hat{\phi}_{2,m}$ integrated over a coarse-graining volume
$\Delta V$ depends on the direction and degree of local alignment of $\vec{u}$
vectors and grows with amount of polymer within $\Delta V$.

Explicitly writing the relation between $\hat{n}$ and $\phi_{2,m}$ is
complicated as it involves the cubic equation for the eigenvalues of $M$. This
relation can be much simplified by first making some assumptions. By
assumption 19, there is a primary direction of alignment requiring
$\lambda_{1}>\lambda_{2}$ implying $c_{1}>0$. Furthermore, assumption 12
requires the direction of alignment to be approximately in the $\hat{z}$
direction. Excluding terms that are quadratic in the deviation from $\hat{z}$
(i.e. of order $n_{x/y}^{2}$ or higher) we write

$\vec{n}\approx\hat{z}+n_{x}\hat{x}+n_{y}\hat{y}$ (94)
$\vec{e}_{2}\approx\left(\hat{x}-n_{x}\hat{z}\right)\cos\left(\theta\right)+\left(\hat{y}-n_{y}\hat{z}\right)\sin\left(\theta\right)$
(95)
$\vec{e}_{3}\approx\left(\hat{y}-n_{y}\hat{z}\right)\cos\left(\theta\right)-\left(\hat{x}-n_{x}\hat{z}\right)\sin\left(\theta\right)$
(96)

where $\theta$ specifies the orientation of any azimuthal asymmetry.
Substituting into 89

$\vec{e}_{2}\approx\cos\left(\theta\right)\hat{x}+\sin\left(\theta\right)\hat{y}-\left(n_{y}\sin\left(\theta\right)+n_{x}\cos\left(\theta\right)\right)\hat{z}$
(97)
$\vec{e}_{3}\approx-\sin\left(\theta\right)\hat{x}+\cos\left(\theta\right)\hat{y}+\left(-n_{y}\cos\left(\theta\right)+n_{x}\sin\left(\theta\right)\right)\hat{z}$
(98)

and

$Q=\begin{bmatrix}-\frac{1}{3}c_{1}+c_{2}\cos 2\theta&c_{2}\sin
2\theta&c_{1}n_{x}-c_{2}\left(\sin 2\theta n_{y}+\cos 2\theta n_{x}\right)\\\
-&-\frac{1}{3}c_{1}-c_{2}\cos 2\theta&c_{1}n_{y}+c_{2}\left(\cos 2\theta
n_{y}-\sin 2\theta n_{x}\right)\\\ -&-&\frac{2}{3}c_{1}\end{bmatrix}$ (99)

where the dashes indicate a symmetric matrix.

By assumption 14 we will work with systems where $c_{2}\approx 0$ and
therefore $\lambda_{2}\approx\lambda_{3}\approx\left(1-\lambda_{1}\right)/2$.
We write $c_{1}$ and $c_{2}$ in terms of their deviations so that $c_{1}\to
c_{1}+\delta c_{1}$ and $c_{2}\to 0+\delta c_{2}$. Keeping only first order
terms we have

$Q=\begin{bmatrix}-\frac{1}{3}\left(c_{1}+\delta c_{1}\right)+\delta
c_{2}\cos\left(2\theta\right)&\delta
c_{2}\sin\left(2\theta\right)&c_{1}n_{x}\\\ -&-\frac{1}{3}\left(c_{1}+\delta
c_{1}\right)-\delta c_{2}\cos\left(2\theta\right)&c_{1}n_{y}\\\
-&-&\frac{2}{3}\left(c_{1}+\delta c_{1}\right)\end{bmatrix}$ (100)

Comparing to equation 93 we have

$\hat{\phi}_{2,m}=\phi_{0,0}\begin{cases}\sqrt{3}\delta
c_{2}\sin\left(2\theta\right)&m=-2\\\ \sqrt{3}c_{1}n_{y}&m=-1\\\ c_{1}+\delta
c_{1}&m=0\\\ \sqrt{3}c_{1}n_{x}&m=1\\\ \sqrt{3}\delta
c_{2}\cos\left(2\theta\right)&m=2\end{cases}$ (101)

This gives us an interpretation of $\phi_{2,m}$ under the assumption that
alignment only slightly differs from the $\hat{z}$ direction as shown in
figure 6. The value of $\phi_{2,0}$ dictates the magnitude of alignment,
$\phi_{2,1}$ and $\phi_{2,-1}$ denote fluctuations that rotate the direction
of the alignment into the $\hat{x}$ and $\hat{y}$ directions respectively.
Meanwhile $\phi_{2,-2}$ and $\phi_{2,2}$ denote fluctuations that create a
secondary direction of alignment in the $\hat{x}$ and $\hat{y}$ directions
respectively.

Figure 6:  Distortions of the local orientation probability due to changes in
the field strength at various $m$ values. If a small spinner (e.g small
molecule) is placed inside the field $\phi$ then the probability distribution
of the orientation of the spinner will go as
$\propto\exp\left(a^{\prime}\sum_{m}\phi_{2,m}Y_{2,m}(\vec{u})\right)$ where
$a^{\prime}$ depends on the magnitude of the spinner. In the above plots the
radius in each direction is proportional to the probability that a molecule
will point in that direction when it is subject to a field of the form
$\phi_{0,0}$ and for various deviations away from $\phi_{0,0}$.

Solving for $\hat{n}$ we have

$\vec{n}=\frac{1}{\phi_{0,0}\sqrt{3}c_{1}}\phi_{2,1}\hat{x}+\frac{1}{\phi_{0,0}\sqrt{3}c_{1}}\phi_{2,-1}\hat{y}+\hat{z}$
(102)

from which we can write the bend, twist and splay

$\left(\hat{n}\times\nabla\times\hat{n}\right)^{2}=\left(\frac{1}{\phi_{0,0}\sqrt{3}c_{1}}\right)^{2}\left(\left(\frac{\partial}{\partial
z}\phi_{2,1}\right)^{2}+\left(\frac{\partial}{\partial
z}\phi_{2,-1}\right)^{2}\right)$ (103)
$\left(\hat{n}\cdot\nabla\times\hat{n}\right)^{2}=\left(\frac{1}{\phi_{0,0}\sqrt{3}c_{1}}\right)^{2}\left(\frac{\partial}{\partial
x}\phi_{2,-1}-\frac{\partial}{\partial y}\phi_{2,1}\right)^{2}$ (104)
$\left(\nabla\cdot\hat{n}\right)^{2}=\left(\frac{1}{\phi_{0,0}\sqrt{3}c_{1}}\right)^{2}\left(\frac{\partial}{\partial
x}\phi_{2,1}+\frac{\partial}{\partial y}\phi_{2,-1}\right)^{2}$ (105)

In appendix B we show that

$\int dx\left(\frac{\partial}{\partial x}\phi\right)^{2}=\int
k^{2}\tilde{f}\left(k\right)\tilde{f}\left(-k\right)dk$ (106)

allowing us to write the Fourier representation of bend, twist, and splay as

$\left(\hat{n}\cdot\nabla\times\hat{n}\right)^{2}=\left(\frac{k_{x}\tilde{\phi}_{2,-1}-k_{y}\tilde{\phi}_{2,1}}{\phi_{0,0}\sqrt{3}c_{1}}\right)^{2}$
(107)

$\left(\hat{n}\times\nabla\times\hat{n}\right)^{2}=\frac{\left(k_{z}\tilde{\phi}_{2,1}\right)^{2}+\left(k_{z}\tilde{\phi}_{2,-1}\right)^{2}}{\left(\phi_{0,0}\sqrt{3}c_{1}\right)^{2}}$
(108)

$\left(\nabla\cdot\hat{n}\right)^{2}=\left(\frac{k_{x}\tilde{\phi}_{2,1}+k_{y}\tilde{\phi}_{2,-1}}{\phi_{0,0}\sqrt{3}c_{1}}\right)^{2}$
(109)

Using

$\kappa_{1,m}=\sqrt{\frac{4\pi}{3}}\begin{cases}k_{y}&m=-1\\\ k_{z}&m=0\\\
k_{x}&m=1\end{cases}$ (110)

and $\phi_{0,0}c_{1}=\left\langle\phi_{2,0}\right\rangle^{MF}$ we get

$\left(\hat{n}\cdot\nabla\times\hat{n}\right)^{2}=\frac{\left(\kappa_{1,1}\tilde{\phi}_{2,-1}-\kappa_{1,-1}\tilde{\phi}_{2,1}\right)^{2}}{4\pi\left(\left\langle\phi_{2,0}\right\rangle^{MF}\right)^{2}}$
(111)

$\left(\hat{n}\times\nabla\times\hat{n}\right)^{2}=\frac{\left(\kappa_{1,0}\tilde{\phi}_{2,1}\right)^{2}+\left(\kappa_{1,0}\tilde{\phi}_{2,-1}\right)^{2}}{4\pi\left(\left\langle\phi_{2,0}\right\rangle^{MF}\right)^{2}}$
(112)

$\left(\nabla\cdot\hat{n}\right)^{2}=\frac{\left(\kappa_{1,1}\tilde{\phi}_{2,1}+\kappa_{1,-1}\tilde{\phi}_{2,-1}\right)^{2}}{4\pi\left(\left\langle\phi_{2,0}\right\rangle^{MF}\right)^{2}}$
(113)

Equation 1 can be written in matrix form as

$E_{Frank}=\frac{1}{2}\left(\left\langle\phi_{2,0}\right\rangle^{MF}\right)^{-2}\frac{1}{4\pi}\vec{\left(\kappa\phi\right)}\cdot\begin{bmatrix}K_{splay}&0&0&0&0&B\\\
0&K_{twist}&0&0&A&0\\\ 0&0&K_{bend}&0&0&0\\\ 0&0&0&K_{bend}&0&0\\\
0&A&0&0&K_{twist}&0\\\
B&0&0&0&0&K_{splay}\end{bmatrix}\cdot\vec{\left(\kappa\phi\right)}$ (114)

where $A+B=K_{splay}-K_{twist}$ and

$\vec{\left(\kappa\phi\right)}=\begin{bmatrix}\kappa_{1,-1}\tilde{\phi}_{2,-1}&\kappa_{1,-1}\tilde{\phi}_{2,1}&\kappa_{1,0}\tilde{\phi}_{2,-1}&\kappa_{1,0}\tilde{\phi}_{2,1}&\kappa_{1,1}\tilde{\phi}_{2,-1}&\kappa_{1,1}\tilde{\phi}_{2,1}\end{bmatrix}$
(115)

The Frank elastic constants are related to the deformation energy of the
system given in equation 86, in that they comprise the portion of the energy
that comes from $\tilde{\phi}_{2,\pm 1}$ deformations that are quadratic in
$\kappa$. The comparison between equations 111-112 and $\delta\tilde{S}^{-1}$
in equation 86 will allow us to write the Frank elastic constants in terms of
the microscopic details of the system (e.g. $L$, $A$, $\phi_{0,0}$, $a$,
$v_{p}$, etc.).

##  XIV Frank Elastic Constants for Fuzzball and Rigid Rod

Comparing equations 111-112 and 114 to equation 71 we find that the Frank
elastic constants for a fuzzball are identical and equal

$K_{bend}^{FB}=K_{twist}^{FB}=K_{splay}^{FB}=3\phi_{0,0}\frac{\sigma^{2}}{v_{p}}\frac{\left(M_{0,2}^{0}\right)^{2}}{M_{2,2}^{1}M_{0,0}^{0}}$
(116)

Comparing equations 111-112 and 114 to equation 75 we get the Frank elastic
constants for a rigid rod

$K_{bend}^{RR}=\phi_{0,0}\frac{L}{A}\frac{2\pi}{3}\frac{\left(M_{0,2}^{0}\right)^{2}}{M_{2,2}^{1}M_{2,2}^{1}M_{0,0}^{0}}\left(J_{\left(1\right)}^{1,0,1}\cdot
M^{1}\cdot J_{\left(1\right)}^{1,0,1}\right)_{2,2}$ (117) $\displaystyle
K_{twist}^{RR}=$
$\displaystyle\phi_{0,0}\frac{L}{A}\frac{2\pi}{3}\frac{\left(M_{0,2}^{0}\right)^{2}}{M_{2,2}^{-1}M_{2,2}^{-1}M_{0,0}^{0}}$
(118)
$\displaystyle\times\sum_{\mathscr{M}\in\left\\{-2,0\right\\}}\left(J_{\left(1\right)}^{-1,1,\mathscr{M}}\cdot
M^{\mathscr{M}}\cdot J_{\left(1\right)}^{\mathscr{M},1,-1}\right)_{2,2}$
$\displaystyle K_{splay}^{RR}=$
$\displaystyle\phi_{0,0}\frac{L}{A}\frac{2\pi}{3}\frac{\left(M_{0,2}^{0}\right)^{2}}{M_{2,2}^{1}M_{2,2}^{1}M_{0,0}^{0}}$
(119)
$\displaystyle\times\sum_{\mathscr{M}\in\left\\{0,2\right\\}}\left(J_{\left(1\right)}^{1,1,\mathscr{M}}\cdot
M^{\mathscr{M}}\cdot J_{\left(1\right)}^{\mathscr{M},1,1}\right)_{2,2}$

##  XV Frank Elastic Constants for WLC

Comparing equations 111-112 and 114 to equation 81 we find the dimensionless
elastic constants

$K^{\prime}_{splay}=\frac{4\pi\left(B_{1}\right)^{2}B_{4}\left(1,1,1,1\right)}{N^{2}\left(B_{2}\left(1\right)\right)^{2}B_{0}}$
(120)
$K^{\prime}_{twist}=\frac{4\pi\left(B_{1}\right)^{2}B_{4}\left(-1,-1,1,1\right)}{N^{2}\left(B_{2}\left(-1\right)\right)^{2}B_{0}}$
(121)
$K^{\prime}_{bend}=\frac{4\pi\left(B_{1}\right)^{2}B_{4}\left(1,1,0,0\right)}{N^{2}\left(B_{2}\left(1\right)\right)^{2}B_{0}}$
(122)

where

$K^{\prime}=K\frac{A}{\phi_{0,0}\left(2\ell_{p}\right)N}$ (123)
$B_{0}=\mathscr{\mathcal{L}}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]$
(124)
$B_{1}=\mathcal{L}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(2\right)}^{0,0,0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]$
(125)
$B_{2}\left(m_{1}\right)=\mathscr{\mathcal{L}}^{-1}\left[\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(2\right)}^{0,m_{1},m_{1}}\cdot\mathbf{G}^{m_{1}}\cdot\mathbf{J}_{\left(2\right)}^{m_{1},m_{1},0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]$
(126) $\displaystyle
B_{4}\left(m_{1},m_{2},m,m^{\prime}\right)=\mathscr{\mathcal{L}}^{-1}\left[\sum_{\mathscr{M}}\bm{e}_{0}\cdot\mathbf{G}^{0}\cdot\mathbf{J}_{\left(2\right)}^{0,m_{1},m_{1}}\cdot\mathbf{G}^{m_{1}}\cdot\mathbf{J}_{\left(1\right)}^{m_{1},m,\mathscr{M}}\cdot\mathbf{G}^{\mathscr{M}}\cdot\mathbf{J}_{\left(1\right)}^{\mathscr{M},m^{\prime},m_{2}}\cdot\mathbf{G}^{m_{2}}\cdot\mathbf{J}_{\left(2\right)}^{m_{2},m_{2},0}\cdot\mathbf{G}^{0}\cdot\bm{e}_{0}\right]$
(127)

##  XVI Numerical Laplace Inversion

Laplace inversion is accomplished by the numerical integration

$f\left(N\right)=\frac{1}{2\pi
i}\int_{\lambda-i\infty}^{\lambda+i\infty}e^{pN}F\left(p\right)dp.$ (128)

Figure 7:  Diagram of typical path of integration on the complex plane of
Laplace variable $p$. The path goes from bottom to top along the blue lines.
Black dots represent typical poll locations. Not drawn to scale, in practice
$p_{max}\gg w$.

The complex integration is done along the path in figure 7 where the dots are
poles. The position of the first pole is found with binary search using the
facts that $\bm{G}_{0,0}^{0}$ changes sign as you go horizontally through the
poles and that the first pole is always on the real axis. As the pole moves to
the right the $\exp\left(pN\right)$ in the integral becomes extremely large.
To avoid overflow we can multiply by $\exp\left(-p_{max}N\right)$ which will
cancel when we take the ratio of two inverse Laplace transforms, as always
occurs in this derivation.

Python code to do the inverse Laplace transform and generate the plots in this
paper can be found at
https://github.com/SpakowitzLab/wlcsim/tree/master/wlcsim/FrankElastic. The
documentation can be found at
https://wlcsim.readthedocs.io/en/latest/FrankExample.html.

##  XVII Results

The units of the Frank elastic constants are energy times length because they
denote energy per unit volume per rate of change of direction squared. To
compare the magnitudes of the Frank elastic constants we first
nondimensionalize them. Throughout this paper we have dropped the implied
energy units of $k_{b}T$’s, so energy is already dimensionless. For relatively
stiff polymers, it makes sense to nondimensionalize $K$ by multiplying it by
$A/L$ because $L$ and $A$ are the only relevant length scales for the rigid
rod. When nondimensionalizing we also divide the Frank elastic constants by
the volume fraction of polymer $\phi_{0,0}$ because denser solutions
experience proportionally more aligning field. Figure 8 presents the Frank
elastic constants nondemensinalized via $K\left(\frac{A}{L\phi_{0,0}}\right)$.
The Maier-Saupe parameter $a$, which captures the energetic coupling strength
of aligning molecules, has units of energy (also in $k_{b}T$’s) per volume,
therefore we nondimensionalize it via $a\left(LA\phi_{00}\right)$. This
nondimensionalization is convenient because a rigid rod that is twice as long
will experience twice the aligning potential.

Figure 8: Frank elastic constants for stiff polymer solutions for stiff chains
that are $N$ Kuhn lengths long. Dashed curves are for the rigid rod. For the
rigid rod, $K_{splay}=3K_{twist}$.

In Figure 8 the value of the elastic constants $K$ generally grow with
increased Maier-Saupe nematic coupling strength $a$. The coupling strength
drives the formation of a liquid crystal state, so it is natural that higher
$a$ could induce higher $K$ and thereby reduce deviations from the perfectly
aligned state. Figure 8 shows that liquid crystal solutions of rigid rods and
other stiff polymers ($N\ll 1$) have a relatively high bend modulus
$K_{bend}$. Figure 9A shows rigid rods fit poorly into a bending field with
their elongated nature inevitably spanning regions of different orientation.
In contrast, rods neatly connect together regions of similar orientation
making it easy for a solution of rods to twist and splay (see figure 9B and
C).

The fundamental difference that leads to $K_{bend}$ being greater than the
other $K$’s is that bend is a change in orientation associated with a
displacement in the direction of alignment (see figure 5). This is precisely
the direction in which the rod is elongated. To emphasise this point, compare
the rod to the fuzzball which is equally elongated in all directions and has
$K^{FB}_{twist}=K^{FB}_{bend}=K^{FB}_{Splay}$.

In figure 8 we see that making a rod-like molecule more flexible (i.e.
increasing $N$) reduces all three elastic constants. The dramatic reduction in
$K_{bend}$ reflects the ability of the polymers to bend with a bending field.
Increased flexibility also reduces splay and twist constants, but to a lesser
extent.

Figure 9:  Schematics providing qualitative arguments for the observed trends
in the Frank elastic constants. A-C are for rigid rods or very stiff ($N\ll
1$) polymers. D-H are for long semiflexible polymers ($N\gg 1$).

Note that at the lower left corner of figure 8 the elastic constants turn
downward and end. They do this because the lower left end of the curve
represents the limit of metastability corresponding to the minimum $a$ value
in figure 2. At values of $a$ below this cutoff the solution will revert to an
isotropic state.

For relatively flexible polymers it makes sense to nondimensionalize by the
Kuhn length, ${2\ell_{p}=L/N}$, rather than the length of the entire polymer.
In figure 8 the Frank elastic constants decrease with increasing $N$, in
contrast to figure 10 where they increase. The difference between the two is
their different nondimensionalization of $K$. In 8 the polymer length
${L=2\ell_{p}N}$ is held constant so that increasing $N$ makes the chain more
flexible and decreases $K$. In figure 10 the Kuhn length $2\ell_{p}$ is held
constant so that increasing $N$ makes the chain longer and increases $K$.

Nondimensionalizing by the Kuhn length in figure 10 accentuates the universal
behavior of polymers at large $N$. When comparing a solution of polymers of
length $N=100$ and $N=200$ but equivalent volume fractions of polymer
$\phi_{0,0}$ the solutions look quite similar other than the $N=200$ solution
will have half as many polymer ends. Figure 10 shows that above about $N=10$
the bend and twist no longer depend on $N$. Indeed, for relatively flexible
($N\gg 1$) polymers, physical properties (such as the bend and twist moduli)
depend much more on the Kuhn length $2\ell_{p}$ than the total length $L$.
This is consistent with the intuition from figure 9D.

Figure 10: Frank elastic constants for polymers of different lengths, $N$, in
Kuhn lengths.

The splay modulus $K_{splay}$ for long polymers is larger and more complex. At
low coupling strength splay can be accommodated by hairpins as shown in figure
9E. However, increasing $a$ makes hairpins increasingly energetically
unfavorable leading to a exponential growth in the spacing between hairpins
and the resulting splay modulus [18]. At sufficiently high $a$, all hairpins
are removed from the chain and the splay is accommodated by chain ends (figure
9F). As the number of chain ends depends on the chain length (at a fixed
concentration) rather then $a$, $K_{splay}$ will plateau at a value that
increases with $N$.

Surprisingly, after initially increasing, the twist modulus $K_{twist}$
moderately decreases with increasing $a$ as shown in figure 10. We justify
this by arguing that at low $a$ polymers will mix between twist planes as
shown in figure 9G. Each time a polymer mixes between twist planes it
introduces stress pulling the orientations of the planes into alignment. As
$a$ increases the amount of stress introduced when a polymer mixes between
plains increases but the polymers are much less likely to stray from their
plane as depicted in 9H. The latter effect apparently overwhelms the former at
high $a$.

Figure 11:  Ratio of Frank elastic constants for stiff polymers.

The primary trend presented above is that for rigid polymers $N\ll 1$ the
$K_{bend}>K_{splay}$ and for long flexible polymers $K_{splay}>K_{bend}$.
Naturally, we’d like to know when the crossover between bend and splay occurs.
In figure 11 we plot the ratios of the Frank elastic constants. The crossover
where $K_{bend}=K_{splay}$ depends modestly on $a$ and occurs a lengths a
little above one Kuhn length. The curves in this figure end when the polymers
become flexible to the point where the aligned phase is no longer metastable
and spontaneously transforms to the isotropic state. Note that we have
nondimensionalized $a$ in this plot by the total polymer length. If we instead
had nondimensionalized $a$ by the Kuhn length, the curves would still cross at
$N$ values just above one, but the shape of the curves would be different.

##  XVIII Summary

We have presented a method for evaluating the size of orientation deformations
for nematic solutions of semiflexible polymers. These deformations, described
by the Frank elastic constants, are accurately found for a wide range of chain
flexibilities and alignment interaction strengths (i.e. persistence length,
$\ell_{p}$ and Maier-Saupe parameter, $a$). Our method shows the convenience
of combining a previously-derived, stone-fence-diagram-based, exact statistics
for the worm-like chain in a quadrupole field [21, 22] with a real-spherical-
harmonic orientational density distribution. This method allows us provide
both quantitative plots and qualitative explanations for the Frank elastic
constants over this range. Here we summarize the features of the Frank elastic
constants, some of which have been previously reported.

Below a minimum alignment interaction strength the nematic state will
spontaneously degenerate into the isotropic one. That is, the aligned state is
not even metastable. This minimum interaction is associated with finite - if
small - Frank elastic constants and a large but finite amount of deformation.
The following features are for solutions in a stable (or at least metastable)
nematic state.

We will divide our results into those for relatively rigid polymers
($L\lessapprox\ell_{p}$) and those for relatively flexible polymers. First, we
will make three observations that hold over both ranges. 1) The twist modulus
is always smaller than bend and splay moduli. For most polymer solutions
$K_{twist}$ is order(s) of magnitude smaller than $K_{bend}$ and $K_{splay}$.
This means that the primary deformation mode of polymer solutions will be
twist. 2) At the level of the present theory (no coupling between deformation
modes) the Flory Huggins $\chi$ parameter does not affect the Frank elastic
constants. This means that using a better solvent will not directly444A better
solvent could effect behavior on shorter length scales which could renormalize
into a different Maier-Saupe parameter on longer length scales and thereby
indirectly effect the Frank elastic constants. We have also assumed that all
solvents are good enough to dissolve the polymers. affect the Frank elastic
constants. 3) Frank elastic constants are inversely proportional to polymer
concentration. Higher concentrations lead to more alignment.

For nematic solutions of rigid polymers the Frank elastic constants increase
as $~{}L\phi_{0,0}$ where $L$ is the polymer length. That is, increasing
polymer length and concentration decreases the expected amount of bend, twist,
and splay. Over the range of Maier-Saupe coupling strengths $a$ where liquid
crystal alignment can be found, we find $K_{bend}>K_{splay}>K_{twist}$. When
coupling strength $a$ is just above the limit of metastability of the nematic
phase, then $K_{bend}\approx K_{splay}$ as predicted by [16]. However, with
increasing $a$ away from the metastability limit, we find that $K_{bend}$
increases rapidly so that $K_{bend}\gg K_{splay}$. In other words, solutions
of long rigid molecules exhibit little bend deformation. Interestingly, for
perfectly rigid rods ($N=0$) we have $K_{splay}=3K_{twist}$ independent of
coupling strength. Furthermore, away from the metastability limit, $K_{splay}$
and $3K_{twist}$ are nearly independent of $a$, leaving the amount of splay
and twist primarily determined by polymer length and concentration.

For nematic solutions of semiflexible and flexible polymers the qualitative
elastic behavior differs significantly from that of rigid polymers. For
flexible polymers $K_{splay}>K_{bend}$. The crossover where
$K_{bend}=K_{splay}$ occurs at $N\approx 1.25$ with the crossover value
increasing modestly with aligning strength $a$. Unlike solutions of rigid
polymers where the elastic constants depended primarily on the polymer length,
the elastic constants of semiflexible polymers depend more heavily on the
persistence length. In particular, the twist and bend constants, $K_{twist}$
and $K_{bend}$, grow as $~{}\ell_{p}\phi_{0,0}$ where $\ell_{p}$ is the
polymer persistence length. The splay constant $K_{splay}$ grows as
$~{}\ell_{p}\phi_{0,0}$ at low coupling strengths but as $L\phi_{0,0}$ at high
coupling strengths.

The dependence of the elastic deformations of flexible polymer solutions have
an interesting dependence on coupling strength $a$. The value of $K_{splay}$
increases quickly with $a$ as hairpins are removed from chains as shown by
[18]. The increase flattens off when all the hairpins are removed and splay is
accommodated by chain ends. In contrast, after initially increasing, the twist
modulus actually _decreases_ with increasing coupling strength. Over this
range, an increase in the level of alignment of the polymers is associated
with an increase in the amount of twist deformation.

Taken together, the elastic deformations depend on the the polymer length,
rigidity, and local alignment strength in a varied but predicable fashion.
Armed with the knowledge of these dependencies, the reader can both predict
and rationally design the liquid crystalline behavior of polymer solutions.

###### Acknowledgements.

I thank Andrew Spakowitz for is advising during the research and writing of
this paper. I thank Ashesh Ghosh for his comments during the editing or this
document.

##  Appendix A Real Spherical Harmonics

The real spherical harmonics, $Y_{\ell,m}$, are defined in terms the more
widely know complex spherical harmonics, $Y_{l}^{m}$, by

$Y_{\ell,m}=\begin{cases}\frac{i}{\sqrt{2}}\left(Y_{\ell}^{m}-Y_{\ell}^{m*}\right)&m<0\\\
Y_{\ell}^{0}&m=0\\\
\frac{1}{\sqrt{2}}\left(Y_{\ell}^{-m}+Y_{\ell}^{-m*}\right)&m>0\end{cases}$
(129)

and conversely

$Y_{\ell}^{m}=\begin{cases}\frac{1}{\sqrt{2}}\left(Y_{\ell,-m}-iY_{\ell,m}\right)&m<0\\\
Y_{l,m}&m=0\\\
\frac{\left(-1\right)^{m}}{\sqrt{2}}\left(Y_{\ell,m}+iY_{\ell,-m}\right)&m>0\end{cases}$
(130)

where we note that $Y_{\ell}^{m*}=\left(-1\right)^{m}Y_{\ell}^{-m}$. In
particular

$Y_{0,0}=\frac{1}{\sqrt{4\pi}}$ (131)

$Y_{1,m}=\sqrt{\frac{3}{4\pi}}\begin{cases}u_{y}&m=-1\\\ u_{z}&m=0\\\
u_{x}&m=1\end{cases}$ (132)

$Y_{2,m}\left(\vec{u}\right)=\sqrt{\frac{5}{4\pi}}\begin{cases}\sqrt{3}u_{x}u_{y}&m=-2\\\
\sqrt{3}u_{y}u_{z}&m=-1\\\ \frac{1}{2}\left(3u_{z}^{2}-1\right)&m=0\\\
\sqrt{3}u_{z}u_{x}&m=1\\\
\frac{\sqrt{3}}{2}\left(u_{x}^{2}-u_{y}^{2}\right)&m=2\end{cases}$ (133)

The orthogonality of real spherical harmonics is

$\int
d\vec{u}_{i}Y_{\ell_{1},m_{1}}\left(\vec{u}_{i}\right)Y_{\ell_{2},m_{2}}\left(\vec{u}_{i}\right)=\delta_{\ell_{1},\ell_{2}}\delta_{m_{1},m_{2}}$
(134)

A useful property of products of real spherical harmonics is their relation
the the Legendre polynomial

$P_{\ell}\left(\vec{u}_{1}\cdot\vec{u}_{2}\right)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y_{\ell,m}\left(\vec{u}_{1}\right)Y_{\ell,m}\left(\vec{u}_{2}\right)$
(135)

Furthermore, products of harmonics can be factored into sums of single
harmonics

$Y_{1,0}Y_{\ell,m}=\alpha_{\ell+1}^{m}Y_{\ell+1,m}+\alpha_{\ell}^{m}Y_{\ell-1,m}$
(136) $\displaystyle Y_{2,0}Y_{\ell,m}=$
$\displaystyle\frac{\alpha_{\ell+1}^{m}\alpha_{\ell+2}^{m}}{\alpha_{2}^{0}}Y_{\ell+2,m}+\frac{\alpha_{\ell+1}^{m}a_{\ell+1}^{m}-\alpha_{1}^{0}Y_{0}^{0}+\alpha_{\ell}^{m}\alpha_{\ell}^{m}}{\alpha_{2}^{0}}Y_{\ell,m}+\frac{\alpha_{\ell}^{m}\alpha_{\ell-1}^{m}}{\alpha_{2}^{0}}Y_{\ell-2,m}$
(137)

where

$\alpha_{\ell}^{m}\equiv\sqrt{\frac{3\left(\ell-m\right)\left(\ell+m\right)}{4\pi\left(2\ell-1\right)\left(2\ell+1\right)}}$
(138)

Because it will be used frequently, we will rewrite

$Y_{2,0}Y_{\ell,m}=A_{\ell+2,}^{m}Y_{\ell+2,m}+\beta_{\ell}^{m}Y_{\ell,m}+A_{\ell}^{m}Y_{\ell-2,m}$
(139)

where

$A_{\ell}^{m}\equiv\frac{\alpha_{\ell}^{m}\alpha_{\ell-1}^{m}}{\alpha_{2}^{0}},\
\beta_{\ell}^{m}\equiv\frac{\alpha_{\ell+1}^{m}a_{\ell+1}^{m}-\alpha_{1}^{0}Y_{0,0}+\alpha_{\ell}^{m}\alpha_{\ell}^{m}}{\alpha_{2}^{0}}$
(140)

Furthermore,

$\displaystyle Y_{1,1}Y_{\ell,m}=$
$\displaystyle\alpha_{\ell+1}^{\left(+\right)m}Y_{\ell+1,m+1}f\left(m+1\right)-\alpha_{\ell+1}^{\left(+\right)-m}Y_{\ell+1,m-1}f\left(m\right)$
$\displaystyle+\alpha_{\ell}^{\left(+\right)m-1}Y_{\ell-1,m-1}f\left(m\right)-\alpha_{\ell}^{\left(+\right)-m-1}Y_{\ell-1,m+1}f\left(m+1\right)$
(141) $\displaystyle Y_{1,-1}Y_{\ell,m}=$
$\displaystyle\alpha_{\ell+1}^{\left(+\right)-m}\frac{1}{\sqrt{2}}Y_{\ell+1,-m+1}g\left(m\right)-\alpha_{\ell}^{\left(+\right)m-1}\frac{1}{\sqrt{2}}Y_{\ell-1,-m+1}g\left(m\right)$
$\displaystyle+\alpha_{\ell+1}^{\left(+\right)m}\frac{1}{\sqrt{2}}Y_{\ell+1,-m-1}h\left(m\right)-\alpha_{\ell}^{\left(+\right)-m-1}\frac{1}{\sqrt{2}}Y_{\ell-1,-m-1}h\left(m\right)$
(142)

where

$\alpha_{\ell}^{\left(\pm\right)m}\equiv\sqrt{\frac{3\left(\ell+m\right)\left(\ell+m\pm
1\right)}{8\pi\left(2\ell-1\right)\left(2\ell+1\right)}}$ (143)
$f\left(m\right)=\frac{1}{\sqrt{2}}\cdot\begin{cases}-1&m<-1\\\ -1&m=-1\\\
0&m=0\\\ \sqrt{2}&m=1\\\ 1&m>1\end{cases}\ \ \ \ \mathrm{and}\ \ \
g\left(m\right)=\begin{cases}-1&m<-1\\\ -1&m=-1\\\ 0&m=0\\\ 0&m=1\\\
1&m>1\end{cases}\ \ \ \ \mathrm{and}\ \ \
h\left(m\right)=\begin{cases}-1&m<-1\\\ -\sqrt{2}&m=-1\\\ \sqrt{2}&m=0\\\
1&m=1\\\ 1&m>1\end{cases}$ (144)

Furthermore

$Y_{2,\pm
1}Y_{\ell,0}=\frac{\alpha_{\ell+1}^{\left(+\right)0}\alpha_{\ell+2}^{1}}{\alpha_{2}^{1}}Y_{\ell+2,\pm
1}+\frac{\alpha_{\ell+1}^{\left(+\right)0}\alpha_{\ell+1}^{1}-\alpha_{\ell}^{(-)0}\alpha_{\ell}^{1}}{\alpha_{2}^{1}}Y_{\ell,\pm
1}-\frac{\alpha_{\ell}^{(-)0}\alpha_{\ell-1}^{1}}{\alpha_{2}^{1}}Y_{\ell-2,\pm
1}$ (145) $Y_{2,\pm
2}Y_{\ell,0}=\frac{\alpha_{\ell+1}^{\left(+\right)0}\alpha_{\ell+2}^{\left(+\right)1}}{\alpha_{2}^{\left(+\right)1}}Y_{\ell+2,\pm
2}-\frac{\alpha_{\ell+1}^{\left(+\right)0}\alpha_{\ell+1}^{\left(-\right)-1}+\alpha_{\ell}^{\left(-\right)0}\alpha_{\ell}^{\left(+\right)1}}{\alpha_{2}^{\left(+\right)1}}Y_{\ell,\pm
2}+\frac{\alpha_{\ell}^{\left(-\right)0}\alpha_{\ell-1}^{\left(-\right)-1}}{\alpha_{2}^{\left(+\right)1}}Y_{\ell-2,\pm
2}$ (146)

At several points throughout this derivation we use the product of three real
spherical harmonics combined in matrix form

$\left(\mathbf{J}_{\left(\ell\right)}^{m,m^{\prime},m^{\prime\prime}}\right)_{\ell_{1},\ell_{2}}\equiv\int
d\vec{u}Y_{\ell_{1},m}\left(\vec{u}\right)Y_{\ell,m^{\prime}}\left(\vec{u}\right)Y_{\ell_{2}m^{\prime\prime}}\left(\vec{u}\right)$
(147)

The elements of the matrix are mostly zero except a few diagonals which follow
from the products above and the orthogonality property. The selection rules
for real spherical harmonics differ from that of the complex version. The
selection rules dictate that
$\mathbf{J}_{\left(\ell\right)}^{m,m^{\prime},m^{\prime\prime}}=0$ unless
$m+m^{\prime}-m^{\prime\prime}=0$ or $m-m^{\prime}+m^{\prime\prime}=0$ or
$-m+m^{\prime}+m^{\prime\prime}=0$.

##  Appendix B Fourier Transform Conventions

We use the Fourier transform convention

$\tilde{f}\left(\vec{k}\right)=\frac{1}{\left(2\pi\right)^{3/2}}\int
d\vec{r}\exp\left(i\vec{k}\cdot\vec{r}\right)f(\vec{r})$ (148)

and

$f\left(\vec{r}\right)=\frac{1}{\left(2\pi\right)^{3/2}}\int
d\vec{k}\exp\left(-i\vec{k}\cdot\vec{r}\right)\tilde{f}(\vec{k})$ (149)

When we extend the summation notation to integrate over $\vec{k}$ rather than
$\vec{r}$ it has the effect of reversing the latter $\vec{k}$. For real
functions $A$ and $B$

$\displaystyle\tilde{A}_{1}\tilde{B}_{1}$
$\displaystyle\equiv\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int
d\vec{k}A_{l}^{m}\left(\vec{k}\right)B_{l}^{m*}\left(\vec{k}\right)$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int
d\vec{k}A_{l}^{m}\left(\vec{k}\right)B_{\ell}^{m}\left(-\vec{k}\right)$ (150)

which, along with $\tilde{S}_{12}\tilde{\mathbb{I}}_{23}=\tilde{S}_{13}$,
implies that the identity is
$\tilde{\mathbb{I}}_{12}=\delta_{\ell_{1},\ell_{2}}\delta_{m_{1,}m_{2}}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)$.
By plugging 149 into $A_{1}B_{1}$ we find that

$\displaystyle A_{1}B_{1}$ $\displaystyle=\frac{1}{\left(2\pi\right)^{3}}\int
d\vec{k}\int
d\vec{k}^{\prime}\tilde{A}(\vec{k})\tilde{B}^{*}(\vec{k}^{\prime})\int
d\vec{r}e^{-i\left(\vec{k}-\vec{k}^{\prime}\right)\cdot\vec{r}}$
$\displaystyle=\int d\vec{k}\int
d\vec{k}^{\prime}\tilde{A}(\vec{k})\tilde{B}^{*}(\vec{k}^{\prime})\delta\left(\vec{k}-\vec{k}^{\prime}\right)$
$\displaystyle=\tilde{A}_{1}\tilde{B}_{1}$ (151)

Of particular interest to this paper is the forier transform of a derivative
squared. To this end we prove equation 106

$\displaystyle\int dx\left(\frac{\partial}{\partial x}\phi\right)^{2}$
$\displaystyle=\int dx\left(\frac{1}{\sqrt{2\pi}}\frac{\partial}{\partial
x}\int e^{-ikx}\tilde{f}\left(k\right)dk\right)^{2}$ (152) $\displaystyle=\int
dx\left(\frac{1}{\sqrt{2\pi}}\int\left(-ik\right)e^{-ikx}\tilde{f}\left(k\right)dk\right)^{2}$
(153) $\displaystyle=\int dk_{1}\int
dk_{2}\left(-ik_{1}\right)\left(-ik_{2}\right)\tilde{f}\left(k_{1}\right)\tilde{f}\left(k_{2}\right)\times\frac{1}{2\pi}\int
dxe^{-i\left(k_{1}+k_{2}\right)x}$ (154) $\displaystyle=-\int dk_{1}\int
dk_{2}k_{1}k_{2}\tilde{f}\left(k_{1}\right)\tilde{f}\left(k_{2}\right)\delta\left(k_{1}+k_{2}\right)$
(155) $\displaystyle=\int
k^{2}\tilde{f}\left(k\right)\tilde{f}\left(-k\right)dk$ (156)

All interactions discussed in this paper are assumed to be translationally
invariant (assumption 7) meaning that for some interaction
$S\left(\vec{x}_{1},\vec{x}_{2}\right)=S\left(\vec{x}_{1}+\vec{x}_{0},\vec{x}_{2}+\vec{x}_{0}\right)$
for all $\vec{x}_{0}$. Taking the Fourier transform

$\tilde{S}\left(\vec{k}_{1},\vec{k}_{2}\right)=\frac{1}{\left(2\pi\right)^{3}}\int_{-\infty}^{\infty}d\vec{x}_{2}d\vec{x}_{1}e^{i\vec{k}_{1}\cdot\vec{x}_{1}+i\vec{k}_{2}\cdot\vec{x}_{2}}S\left(\vec{x}_{1},\vec{x}_{2}\right)$
(157)

because of the infinite extent of the system we can do a change of integration
variables $\vec{x}_{3}=\vec{x}_{1}+\vec{x}_{0}$

$\displaystyle\tilde{S}\left(\vec{k}_{1},\vec{k}_{2}\right)=$
$\displaystyle\frac{1}{\left(2\pi\right)^{3}}\int_{-\infty}^{\infty}d\vec{x}_{2}d\vec{x}_{3}e^{i\vec{k}_{1}\cdot\left(\vec{x}_{3}-\vec{x}_{0}\right)+i\vec{k}_{2}\cdot\vec{x}_{2}}$
$\displaystyle\times S\left(\vec{x}_{3}-\vec{x}_{0},\vec{x}_{2}\right)$ (158)

using the definition of translation invariance
$S\left(\vec{x}_{3}-\vec{x}_{0},\vec{x}_{2}\right)=S\left(\vec{x}_{3},\vec{x}_{2}+\vec{x}_{0}\right)$
and setting $\vec{x}_{0}=-\vec{x}_{2}$ yields

$\displaystyle\tilde{S}\left(\vec{k}_{1},\vec{k}_{2}\right)=$
$\displaystyle\frac{1}{\left(2\pi\right)^{3}}\int_{-\infty}^{\infty}d\vec{x}_{2}d\vec{x}_{3}e^{i\vec{k}_{1}\cdot\left(\vec{x}_{3}\right)+i\left(\vec{k}_{1}+\vec{k}_{2}\right)\cdot\vec{x}_{2}}$
$\displaystyle\times S\left(\vec{x}_{3},0\right)$ (159)

Recognizing the delta function
$\left(2\pi\right)^{3}\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)=\int_{-\infty}^{\infty}d\vec{x}_{2}\exp\left(i\left(\vec{k}_{1}+\vec{k}_{2}\right)\cdot\vec{x}_{2}\right)$
and defining
$\tilde{S}\left(\vec{k}\right)=\int_{-\infty}^{\infty}d\vec{x}\exp\left(i\vec{k}\cdot\vec{x}\right)S\left(\vec{x},0\right)$
we have

$\tilde{S}\left(\vec{k}_{1},\vec{k}_{2}\right)=\tilde{S}\left(\vec{k}_{1}\right)\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)$
(160)

Being diagonal with respect to $\vec{k}$ makes $\tilde{S}$ easy to invert. We
define the inverse by $\tilde{S}\cdot\tilde{S}^{-1}\equiv\mathbb{I}$ which
written out implies

$\tilde{S}^{-1}(k_{1},k_{3})=\frac{\delta\left(\vec{k}_{1}+\vec{k}_{2}\right)}{\tilde{S}\left(\vec{k}_{1}\right)}$
(161)

Note that if $\tilde{S}$ contains $\ell$ and $m$ indices, it still needs to be
inverted with respect them in the normal way.

##  Appendix C Inverting S

As the deformation energy 86 requires the inverse of $S$, we will take a
moment to discuss how to take the inverse of a power series of matrices. The
derivative of an inverse can be calculated by differentiating the identity
$\partial
I=0=\partial\left(SS^{-1}\right)=S\partial\left(S^{-1}\right)+\left(\partial
S\right)S^{-1}$ which gives

$\partial\left(S^{-1}\right)=-S^{-1}\left(\partial S\right)S^{-1}$ (162)

$\displaystyle\frac{\partial}{\partial\kappa_{1}}\frac{\partial}{\partial\kappa_{2}}\left(S^{-1}\right)=$
$\displaystyle
S^{-1}\left(\frac{\partial}{\partial\kappa_{1}}S\right)S^{-1}\left(\frac{\partial}{\partial\kappa_{2}}S\right)S^{-1}$
$\displaystyle+S^{-1}\left(\frac{\partial}{\partial\kappa_{2}}S\right)S^{-1}\left(\frac{\partial}{\partial\kappa_{1}}S\right)S^{-1}$
$\displaystyle-S^{-1}\left(\frac{\partial}{\partial\kappa_{2}}\frac{\partial}{\partial\kappa_{1}}S\right)S^{-1}$
(163)
$S^{-1}\left(k\right)=\left.S^{-1}\right|_{\kappa=0}+\left.\frac{\partial}{\partial\kappa}S^{-1}\left(\kappa\right)\right|_{\kappa=0}\kappa+\frac{1}{2}\left.\frac{\partial}{\partial\kappa}S^{-2}\left(\kappa\right)\right|_{\kappa=0}\kappa^{2}$
(164)

In other words, if we have a Taylor expansion

$S\left(\kappa\right)=S+S_{\kappa\kappa}\left(\kappa^{2}\right)$ (165)

then

$S^{-1}\left(\kappa\right)=S^{-1}-S^{-1}S_{\kappa\kappa}\left(\kappa^{2}\right)S^{-1}.$
(166)

##  Appendix D List of Variables and Notation

Variables

$A$ – polymer cross sectional area (also matrix element in equation 114)

$A_{\ell}^{m}$ – defined in equations 140

$a$ – Maier-Saupe parameter

$B$ – a particular matrix element in equation 114

$B_{\left(n\right)}$ – products of propagators defined and used in section XV

$c_{i}$ – degree of alignment, expressed as $s_{i}$ in [29], see equation 89

$\bm{e}_{0}$ – unit vector for $\ell=0$, i.e. $[1,0,0,....0]$

$\vec{e}_{i}$ – eigenvector of alignment matrix, see equation 88

$E_{FH}$– Flory-Huggins Separation energy

$E_{Frank}$– Frank elastic energy

$E_{MS}$– Maier-Saupe alignment energy

$E_{poly}$– polymer bending energy

$f$ – used to denote arbitrary function

$\ {}^{FB}$ – super script indicating fuzzball system

$G$ – propagator for wormlike chain in aligning field (not normalized!), see
equation 37

$G_{o}$ – propagator for wormlike chain given in equation 35

$G_{\ell_{0}\ell_{f}}^{m}$ – $\breve{G}$ coefficient, see equation 51

$\text{$\mathbf{G}$}^{m}$ – $\breve{G}$ coefficient matrix representation
$\left(\text{$\mathbf{G}$}^{m}\right)_{\ell_{1},\ell_{2}}\equiv
G_{\ell_{1},\ell_{2}}^{m}$

$h$ – short for $\lim_{k\to 0}S^{\prime}_{02}{00}$, see equation 85

$I$ – 3x3 identity matrix

$\mathbb{I}_{12}$ – the identity in the space defined by our summation
notation

$i$ – $\sqrt{-1}$ or index depending on context

$j$ – polymer or monomer index

$\mathbf{J}$ – product of three spherical harmonics defined in 147

$\vec{k}$ – Fourier conjugate of position

$K$ – Frank elastic content.

$K^{\prime}$ – Nondimensionalized Frank elastic content.

$L$ – path length of polymer

$\mathcal{L}\left[...\right]$ – Laplace transform $N\to p$

$\mathcal{L}^{-1}\left[...\right]\left(N\right)$ – Inverse Laplace transform
$N\leftarrow p$

$\ell$ – angular momentum eigenvalue

$\ell_{p}$ – persistence length of the polymer when placed in isotropic
solution

$\mathbf{M}_{\ell_{1},\ell_{2}}^{m}$ – rigid rod/fuzzball matrix, see equation
27

$M$ – 3x3 alignment matrix defined by equation 87

$m$ – z-component of angular momentum eigenvalue

$\mathscr{M}$ – m value in particular summations.

$N$ – polymer length in Kuhn lengths, $N\equiv L/\left(2\ell_{p}\right)$

$n$ – order in $\gamma$ expansion, see equation 40

$\vec{n}$ – nematic director defined in equation 88

$n_{p}$ – number of polymers

$p$ – Laplace transform variable of $N$

$P$ – Legendre polynomial

$P$ – probability propagator, see equation 38

$P_{\ell}$ – shorthand for $p+\ell\left(\ell+1\right)$

$Q$ – traceless alignment matrix $M-\frac{1}{3}I$

$\ {}^{RR}$ – super script indicating rigid rod

$\vec{\mathrm{r}}_{j}\left(s\right)$ – position of point $s$ along polymer $j$

$\vec{\mathrm{r}}_{1}$ – shorthand for $\vec{\mathrm{r}}\left(s_{1}\right)$

$\vec{r}$ – arbitrary position is space

$s$ – path length along polymer

$S_{12}$ – two point correlation (single polymer structure factor), see
equation 65

$t$ – short for $\lim_{k\to 0}S^{\prime}_{00}{00}$, see equation 85

$\vec{\mathrm{u}}_{j}\left(s\right)$ orientation unit vector
$\frac{\partial\vec{\mathrm{r}}_{j}\left(s\right)}{\partial s}$ for polymer
$j$ at path length $s$

$\vec{\mathrm{u}}_{1}$ – shorthand for $\vec{\mathrm{u}}\left(s_{1}\right)$

$\vec{u}_{1}$ – arbitrary unit vector

$\mathscr{V}$ – volume of space (assumed to be large)

$V_{12}$ – interaction potential defined in equation 11

$v$ – rotational term in equation 68

$v_{p}$ – volume of one polymer, rod, or fuzzball

$W$ – field conjugate to $\phi$, similar to chemical potential, see equation
16

$W_{\ell}^{\left(\pm\right)m}$ – see equation 50

$W_{\ell}^{\left(+\right)m}$ – see equation 48

$W_{\ell}^{\left(-\right)m}$ – see equation 49

$w$ – distance to avoid pole by inverse Laplace transform, see section XVI

$x$ – short for $\lim_{k\to 0}S^{\prime}_{22}{00}$, see equation 85

$\hat{x}$ – Cartesian unit vector

$y$ – short for $\lim_{k\to 0}S^{\prime}_{22}{\pm 1\pm 1}$, see equation 85

$\hat{y}$ – Cartesian unit vector

$Y_{l}^{m}$ – real spherical harmonic. We use the normalization $\int
d\vec{u}Y_{l}^{m}\left(\vec{u}\right)Y_{l^{\prime}}^{*m^{\prime}}\left(\vec{u}\right)=\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}$

$\hat{z}$ – Cartesian unit vector

$Z$ – multi-polymer partition function

$z$ – short for $\lim_{k\to 0}S^{\prime}_{22}{\pm 2\pm 2}$, see equation 85

$z_{p}$– single polymer partition function, see equation 19

$\alpha_{\ell}^{m}$ – raising symbol, defined in equation 138

$\alpha_{\ell}^{\left(\pm\right)m}$– raising symbol with change in $m$,
defined in equation 143

$\beta_{\ell}^{m}$ – defined in equations 140

$\Gamma_{12}$ – the quadratic order fluctuation coefficient, see 66

$\gamma$ – aligning field strength

$\nabla$ – derivative vector
$\hat{x}\partial_{x}+\hat{y}\partial_{y}+\hat{z}\partial_{x}$

$\Delta V$ – coarse-graining volume element

$\delta\left(...\right)$ – delta function or product of delta function at
every point is space as in equation 16

$\delta\phi$ – fluctuations in $\phi$ from it’s mean field value

$\delta W$ – fluctuations in $W$ from it’s mean field value

$\delta S_{12}$ – first term in 65

$\kappa_{1}^{m}$ – spherical representation of $\vec{k}$ as defined in 78

$\lambda$ – real offset in numerical inverse Laplace transform see section XVI

$\rho$ – spatial term in equation 68

$\sigma$ – standard deviation of a fuzzball

$\phi$ – smooth field approximately equal to $\hat{\phi}$

$\hat{\phi}_{l}^{m}\left(\vec{r}\right)$ – polymer position and orientation
field defined in equation 7

$\chi$– Flory-Huggins interaction parameter

Notation

$\int\mathscr{D}\left[\vec{r}_{j}\left(s\right)\right]$ – functional integral
of positions of polymers defined in equation 9

$\int\mathscr{D}W=\prod_{\vec{r}}\left(\int dW\left(\vec{r}\right)\right)$ –
functional integral of $W$ over all space

$\int d\vec{u}$ – Integral over surface of ball
$\int\sin\left(\theta\right)d\theta d\phi$

$\propto$ – proportional to

$\equiv$ – definition

$\vec{u}\otimes\vec{v}$ – outer product matrix
$\left(\vec{u}\otimes\vec{v}\right)_{i,j}=u_{i}v_{j}$

$A_{1}B_{1}\equiv\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int
d\vec{r}A_{l}^{m}\left(\vec{r}\right)B_{l}^{m*}\left(\vec{r}\right)$ –
extended summation notation

$f*g$ – convolution $f*g=\int_{0}^{t}d\tau
f\left(\tau\right)g\left(t-\tau\right)$

$G\left(\vec{u}|\vec{u}_{0};L\right)$ – distribution $G\left(\vec{u}\right)$
given $\vec{u}_{0}$ for polymer of length $L$

$...^{\left(1\right)}$ – superscript denoting single polymer

$...^{MF}$ – superscript denoting mean field

$\left\langle...\right\rangle^{MF}$ – mean field expectation value as in
equation 21

$\left\langle...\right\rangle_{o}$ – expectation value for free chain (no
applied field) with fixed end orientations, see example 36

$\breve{...}$ – indicates Laplace transform $N\to p$

$\tilde{...}$ – indicates Fourier transform $\vec{r}\to\vec{k}$

$\vec{...}$ – Euclidean vector in 3 dimensions

Stone Fence symbols

All symbols are assumed to start from $\ell$ on the left.

$=1/P_{\ell}$

$=\gamma\beta_{\ell}^{m}$

$=\gamma A_{\ell}^{m}$

$=\gamma A_{\ell+2}^{m}$

$=1/\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)$

$=W_{\ell}^{\left(+\right)m}$

$=W_{\ell}^{\left(-\right)m}$

$=\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(+\right)m}-1$

$=\left(P_{\ell}-\gamma\beta_{\ell}^{m}\right)W_{\ell}^{\left(-\right)m}-1$

$=W_{\ell}^{\left(\pm\right)m}$

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