text:
\section{Introduction}

The quest for soft matter systems which exhibit spontaneous symmetry breaking
in terms of a polar order parameter, analogous to ferromagnetism in solids, has a
long history.
The first class of systems investigated under this perspective are suspensions
of magnetic nanoparticles in simple liquids, which exhibit particularly rich
structural and dynamical properties generated by the intricacies of the dipolar
character of their basic mutual interactions \cite{1995_Jonsson, 1997_Groh, 2000_Camp, 2001_Mueller, 2004_Huke, 2004_Mendelev, 2005_Holm, 2008_Rablau, 2008_Trasca, 2015_Szalai}.
Moreover, they offer a broad range of application prospectives such as in
medicine \cite{2002_Alexiou, 2003_Pankhurst, 2002_Brusentsov} and technology \cite{2005_Scherer, 2015_Yao, 2008_Ravaud, 2010_Kim}.
However, whereas such ferrofluids, i.e., colloidal suspensions of magnetic
particles in isotropic liquids, display fascinating behaviors in the presence of
an external magnetic field, actual systems exhibit only zero net magnetization
once the external field is switched off, i.e., there is no occurrence of spontaneous symmetry
breaking \cite{2007_Odenbach, 2012_LopezLopez, 2008_Dery, 2012_Zakinyan}.

A class of soft matter systems, which are indeed able to exhibit nonzero net
magnetization even in the absence of an external magnetic field, are
ferronematics, i.e., magnetic colloidal particles suspended in anisotropic
liquids, such as a nematic liquid crystal (NLC).
Whereas this type of system has been studied theoretically almost half a
century ago \cite{1970_Brochard}, its experimental realization has been achieved only recently
\cite{2013_Mertelj}.
The remarkable property of ferronematics is caused by the broken rotational
symmetry of the solvent which implies that the colloids prefer certain
orientations with respect to the nematic director, thus restricting their
individual magnetic moments to certain directions.

Alternatively, it may be conceivable to suspend colloidal particles with
an \emph{electric} instead of a \emph{magnetic} dipole moment in an NLC and to
study their properties in external \emph{electric} instead of \emph{magnetic}
fields (see Refs.~\cite{2003_Reznikov,2018_Emdadi} and references therein).
However, in contrast to the case of magnetic colloidal particles and magnetic
fields, strong distortions of the NLC are expected to occur in the electric
analogue, because colloidal particles with electric dipoles strongly polarize their
liquid crystaline environment \cite{2007_Li}, and the molecules of the NLC are highly
susceptible to external electric fields, too \cite{book_deGennes}.
Hence it appears advantageous to focus on ferronematics instead of the more
complicated suspensions of colloidal particles with electric dipole moments in
an NLC.

Exploiting the full range of properties of ferronematics requires a reliable
theoretical description which allows one to infer the mesoscopic structures
formed by these colloidal suspensions from microscopic molecular properties of
the liquid crystalline and colloidal materials.
So far, such a formalism has not been established. Accordingly, the goal of the
present work is to introduce a systematic approach to solve this multi-scale
problem for the case of \textit{dilute} suspensions of magnetic colloids.

In order to describe ferronematic phases an expression for the free
energy of the suspension of magnetic anisotropic colloids in an NLC is required.
The authors of Ref.~\cite{2013_Mertelj} have proposed a phenomenological form
of such a free energy density in terms of the local magnetization field and the
local nematic director field.
Here, a similar form of the free energy density is derived by
starting, however, from a microscopic model.
This enables one to relate the corresponding expansion coefficients of the free energy to material
properties of the colloids and of the liquid crystal.
In order to achieve this goal, a microscopic description of the
interaction between a single colloidal particle and the surrounding liquid is
considered.
As an illustration the focus is on a simplified model of a single circular
disc-shaped colloidal particle suspended in an NLC.
Here, the quantity of interest is the free energy as a function of particle
orientation with respect to the nematic director far away from the colloid.
The theory is formulated in terms of a dimensionless \textit{coupling constant} $c$, which
is proportional to the particle size and which is small ($c<0.1$) for the colloids
used in the experiment reported in Ref.~\cite{2013_Mertelj} (platelet radius
$\approx 35\,\text{nm}$).
Here, analytical expressions of the perturbations of the nematic director
profile up to first order and of the corresponding free energy up to second
order in the coupling parameter $c$ are derived (Sec.~\ref{sec:Theory1}
and Appendix~\ref{sec:A1}).
Numerical calculations are used in order to assess the accuracy of the proposed
perturbation expansion.

This microscopic expression for the free energy of a single colloidal particle in an
NLC can be interpreted from the mesoscopic point of view as an external
one-particle potential the NLC medium exerts onto each colloid.
This one-particle potential can be incorporated into a classical density functional
description of a fluid of magnetic discs suspended in the NLC.
In agreement with the experimental set-up in Ref.~\cite{2013_Mertelj}, the
present work is restricted to the case of dilute colloidal suspensions, which
allows one to neglect the {effective} interactions between two colloidal particles in order
to gain calculational advantages.
The resulting mesoscopic free energy density is a second degree polynomial of
the local magnetization $\mathbf{M}(\mathbf{r})$ and of the local nematic director
$\mathbf{n}(\mathbf{r})$ (Sec.~\ref{sec:Theory2} and Appendix~\ref{sec:A2})
which can be directly compared with the corresponding form proposed in
Ref.~\cite{2013_Mertelj}.

The article is organized as follows.
In Sec.~\ref{sec:Theory} and in the Appendices \ref{sec:A1} and \ref{sec:A2} the mathematical models are
introduced in order to be able to investigate the effective one-particle potential of a single, arbitrarily
thin disc immersed in the NLC and to establish a mesoscopic theory of a dilute
ferronematic.
In Sec.~\ref{sec:Res} the results of a numerical assessment of the proposed
effective one-particle potential are presented and the free-energy
functional of a ferronematic as derived here is compared with the one proposed in
Ref.~\cite{2013_Mertelj}.
Conclusions and final remarks are given in Sec.~\ref{sec:Discus}.

\section{\label{sec:Theory}Theory}

\subsection{\label{sec:Theory0}Noninteracting particles in a nematic liquid crystal}

\begin{figure}[t]
   \includegraphics[scale=1]{colloid.eps}
   \caption{Frame of reference used in the calculation of the free energy of
           a single colloidal disc immersed in an NLC.
           An arbitrarily thin disc of radius $R$ is placed with its center in
           the origin $O$ and is oriented with its normal $\bs{\nu}$ parallel to the $z$-axis, i.e., the
           outer normal to the surface of the disc is denoted as $\bs{\nu}$.
           A point magnetic dipole of magnitude $m$ and direction $\bs{\omega}||\bs{\nu}$ is placed in the center of the disc.
           The angle between the disc normal $\bs{\nu}$ and the far-field director
           $\mathbf{n}_0$ is denoted as $\theta$, whereas the angle between the disc
           normal $\bs{\nu}$ and the nematic director $\mathbf{n}(\mathbf{r})$ at any point
           $\mathbf{r}$ is denoted as $\alpha(\mathbf{r})$.}
   \label{fig:Colloid}
\end{figure}

\begin{figure}[t]
   \includegraphics[scale=0.7]{potential.eps}
   \caption{Illustration of the procedure used in the present study in order to model the one-particle elastic potential $\beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))$ in Eq.~\eqref{eq:MicroF}. (a): Sideview of a disc-like colloidal particle of radius $R$ at position $\mathbf{r}$ (red rod of length $2R$) is immersed in the NLC with the director field varying slowly on microscopic scales (black solid lines). The region of the slow variation of the director is indicated by the dashed square within which $|\nabla\mathbf{n}(\mathbf{r})|R \ll 1$ is satisfied. (b) The approximately constant field $\mathbf{n}(\mathbf{r})$ (i.e., the black solid lines within the dashed square in (a)) is extrapolated to everywhere and now plays the role of the far-field uniform director field $\mathbf{n}_0$ (black, solid, parallel, and straight lines) used in the complementary problem of a single disc immersed in an NLC (see Subsec.~\ref{sec:Theory1}). The microscopic distortion of the nematic director due to its coupling to the surface of the colloid is not shown.}
   \label{fig:Rational}
\end{figure}

As a first step, we consider a collection of magnetic colloids immersed in an NLC.
Each colloid is represented by an arbitrarily thin disc of radius $R$ with the outer normal $\bs{\nu}$ to the surface (see Fig.~\ref{fig:Colloid}). A point magnetic dipole of strength $m$ and direction $\bs{\omega}||\bs{\nu}$ is placed in the center of the disc. (Note that the direction of $\bs{\nu}$ depends on which side of the disc is considered whereas the direction of $\bs{\omega}$ does not.) The position of the colloid is the position of its center and the orientation of the colloid is the direction of its magnetic dipole $\bs{\omega}$. (In a more general model $\bs{\omega}$ and $\bs{\nu}$ form a nonzero angle.)

There are four main types of interaction betwen such colloids: the magnetic
dipole-dipole interaction, the effective interaction induced by the
elasticity of the NLC medium, steric hard-core interactions, and the van der Waals interaction.
Here we consider very dilute suspensions of magnetic particles, the volume
fractions $\phi \approx 3 \times 10^{-4}$ of which are comparable with those in
Ref.~\cite{2013_Mertelj}.
For such small densities, the dipole-dipole interaction between two colloids
with magnetic moments $m \approx 3 \times 10^{-18}\,\mathrm{Am^2}$ (see
Ref.~\cite{2013_Mertelj}) and the van der Waals interaction can be neglected
\cite{footnote1}. Moreover, the steric interaction is disregarded due to its short range and hence the very small impact on the properties of such a dilute solution.
Here, the effective colloid interaction induced by the NLC elasticity can also be
neglected due to the high dilution of the suspension and
the weak coupling of colloids to the NLC matrix (see Sec.~\ref{sec:Theory1}).
Therefore, as direct colloid-colloid interactions are negligible for the
type of systems considered here, on the
mesoscopic level the colloidal fluid can be described as an ideal gas in an external field generated by the NLC. The corresponding grand potential functional in
terms of the number density $\rho(\mathbf{r},\bs{\omega})$ of colloids at
position $\mathbf{r}$ and with orientation $\bs{\omega}$ is given by
\begin{align}
   \beta \Omega[\rho,\mathbf{n}] =
   &\ \!\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^3 r\!\!\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,
   \rho(\mathbf{r},\bs{\omega})
   \Big[\ln\big(\rho(\mathbf{r},\bs{\omega})\Lambda^3\big) - 1 - \beta\mu
   \notag\\
   &\ + \beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))
   - \beta m\bs{\omega} \cdot \mathbf{B}\Big],
   \label{eq:MicroF}
\end{align}
where $\Lambda$ denotes the thermal de Broglie wave length, $\mu$ is the
chemical potential of the colloids, $\mathbf{B}$ describes a uniform external magnetic
field acting on a magnetic dipole of strength $m$ and orientation $\boldsymbol{\omega}$ and $\beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))$ is the one particle external field which describes the coupling of a colloid at position $\mathbf{r}$ and with orientation $\bs{\omega}$ to the NLC with the local director $\mathbf{n}(\mathbf{r})$ at the position of the colloid \cite{footnote2}.

The form of $\beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))$ is not known a priori; therefore we adopt certain assumptions in order to model it:
whereas the nematic director field $\mathbf{n}(\mathbf{r})$ may be nonuniform
on mesoscopic length scales, it is assumed to vary slowly on
the scale of the colloid: $|\nabla\mathbf{n}(\mathbf{r})|R \ll 1$. The colloids are separated far from each other due to their low number density. Thus it is assumed that the interaction of a particular colloid with the NLC is determined by the director field in the close vicinity of the colloid. Accordingly, in order to obtain an expression for the one particle elastic potential $\beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))$ (see, c.f., Eq.~\eqref{eq:VNLC}), the particular case of an isolated, disc-like colloid immersed in a uniform director field $\mathbf{n}(\mathbf{r})=\mathbf{n}_0=\text{const}$ (see Fig.~\ref{fig:Rational}) is considered in Sec.~\ref{sec:Theory1} below.
Hence, a colloid with orientation $\bs{\omega}$ placed at position $\mathbf{r}$
experiences a one-particle potential (see, c.f., Eq.~\eqref{eq:VNLC}) which is obtained from, c.f., Eq.~\eqref{eq:FWC} by replacing the
microscopically homogeneous far-field director $\mathbf{n}_0$ with
the mesoscopic local director $\mathbf{n}(\mathbf{r})$.

\subsection{\label{sec:Theory1}Single disc-like colloid immersed in a nematic liquid crystal}

In this subsection we consider a single disc-like colloidal particle of radius $R$ which is suspended in an NLC described by a mesoscopic director
field $\mathbf{n}(\mathbf{r})$.
According to Fig.~\ref{fig:Colloid}, a frame of reference is attached to the
colloid such that the $z$-axis is parallel to the  normal of the particle.
Here, we study the case of homeotropic boundary conditions on the surface of the colloid, i.e., it is energetically favorable for the director field
$\mathbf{n}(\mathbf{r})$ at the surface to be parallel to the normal of the
colloid (i.e., the $z$-axis).
It is the aim of the present subsection to determine the free energy of the system
as function of the colloid orientation, which is described by
an angle $\theta$ between the $z$-axis and the uniform director $\mathbf{n}_0$
far away from the colloid (see Fig.~\ref{fig:Colloid}).

Due to the local inversion symmetry of the nematic phase (i.e., due to the nematic
directors $\mathbf{n}(\mathbf{r})$ and $-\mathbf{n}(\mathbf{r})$ describing the
same thermodynamic state of the NLC \cite{book_deGennes}), both the angles $\theta=0$ and
$\theta=\pi$ correspond to the free-energetic ``ground'' state, i.e., the director field
$\mathbf{n}(\mathbf{r})$ at any point $\mathbf{r}$ of the colloid surface points
along its surface normal. Moreover, since there are no deformations in the bulk NLC, the
director field $\mathbf{n}(\mathbf{r})$ is uniform \textit{everywhere}, and the elastic
free energy of the NLC attains its minimum as a function of $\theta$.
In order to determine the free energy as function of $\theta\in [0,\pi/2]$ (the
behavior in the range $\theta\in [\pi/2,\pi]$ follows from the symmetry of the
free energy with respect to $\theta=\pi/2$ due to the local
inversion symmetry of the NLC), it is obviously necessary to include the
coupling of the director field $\mathbf{n}(\mathbf{r})$ to the particle surface.

In the limit, which is called ``infinite anchoring'', in the following the
director field $\mathbf{n}(\mathbf{r})$ at the colloid surface is kept fixed to a
certain (here the normal) direction called ``easy axis'' \cite{book_deGennes}.
In the language of boundary value problems this limit corresponds to a
Dirichlet boundary condition at the surface of the particle.
If this constraint is relaxed and the director field $\mathbf{n}(\mathbf{r})$
at the surface can deviate from the easy axis, the free energy acquires an extra
contribution penalizing deviations from the easy direction:
\begin{equation}
   F_{\text{s}} =
   W\int_{\text{disc}}\!\!\!\!\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2 s\,\,
   [\mathbf{n}(\mathbf{s})\times\boldsymbol{\nu}(\mathbf{s})]^2,
   \label{eq:FS}
\end{equation}
where $\mathbf{n}(\mathbf{s})$ is the director field at the point $\mathbf{s}$ of the disc surface, $\boldsymbol{\nu}(\mathbf{s})$ is the easy axis which here is taken to be normal to the surface, and
$W=\text{const}>0$ is the anchoring strength with the dimension energy per surface
area.

In the present context, the infinite anchoring limit has been investigated
before.
Within the infinite anchoring (IA) limit the free energy has the form
\cite{1976_Hayes}
\begin{align}
   \frac{F^\text{IA}}{KR} &= 4\theta^2\text{, }\theta\in[0,\pi/2],
   \label{eq:Hayes}
\end{align}
where $K$ denotes the Frank elastic constant of the NLC with dimension energy
per length (i.e., force) within the one-constant approximation
\cite{book_deGennes}.
The opposite limit,~i.e., the limit of weak anchoring, has not been investigated systematically in the case of discs. This limit, which we shall refer to as ``weak anchoring``, is the relevant one for the colloids used in the experiment reported in Ref.~\cite{2013_Mertelj} (see below).
It turns out (see below) that it is beneficial to formulate the description of the weak anchoring limit as an expansion of the free energy in terms of the dimensionless
coupling constant
\begin{equation}
  \label{eq:C}
  c:=\frac{WR}{K}.
\end{equation}
In the following, the contributions to the
free energy up to and including the order $\propto c^2$ (see, c.f., Eq.~\eqref{eq:FWC}) are determined.

In order to obtain a systematic expansion of the free energy of the NLC with a
colloidal inclusion in terms of powers of the coupling constant $c$\rt, one can start from
the Frank-Oseen functional of the nematic director field $\mathbf{n}$:
\begin{equation}
  \label{eq:FOFunc}
  \frac{F[\mathbf{n}]}{KR}\!=\!
  \frac{1}{2R}\int_{\mathcal{V}}\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,
  \frac{\partial n_i(\mathbf{r})}{\partial x_j}
  \frac{\partial n_i(\mathbf{r})}{\partial x_j} \!+\!
  \frac{c}{R^2}\int_{\partial\mathcal{V}}\!\!\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2\! s\,
  [\mathbf{n}(\mathbf{s})\times\boldsymbol{\nu}(\mathbf{s})]^2,
\end{equation}
where summation over repeated indices is assumed, $\mathcal{V}\subseteq
\mathbb{R}^3$ is the space filled by the NLC, and $\partial\mathcal{V}$ denotes
the boundary of the NLC (colloid + cell walls).
If all lengths are measured in units of $R$ (i.e., $\ensuremath{\mathop{}\!\mathrm{d}}^3 r/R^3=
\ensuremath{\mathop{}\!\mathrm{d}}^3\widetilde{r}$, $R\partial/\partial x_j=\partial/\partial
\widetilde{x}_j$, $\ensuremath{\mathop{}\!\mathrm{d}}^2\!s/R^2=\ensuremath{\mathop{}\!\mathrm{d}}^2\widetilde{s}$, $\mathcal{V}/R^3=\widetilde{\mathcal{V}}$ and $\partial\mathcal{V}/R^2=\partial\widetilde{\mathcal{V}}$) the
dimensionless parameter $c$ is (up to a numerical factor) the ratio of the surface energy (second term on
the right-hand side of Eq.~\eqref{eq:FOFunc}) and the bulk elastic energy
(first term on the right-hand side of Eq.~\eqref{eq:FOFunc}).
Alternatively, $c$ can be viewed as the ratio of the particle radius $R$ and
the extrapolation length $l:=K/W$ \cite{book_deGennes}.
Therefore, the coupling constant $c$ measures the cost of free energy for the
director field $\mathbf{n}$ to deviate at the colloid surface from the easy
axis compared to the cost of free energy for an elastic distortion of the director
field $\mathbf{n}$ in the bulk.
We define the ``weak anchoring'' regime by the condition $c \ll 1$
and note that according to this definition the notion of ``weak'' does not necessarily mean that
the surface anchoring $W$ is small, but rather that the product $WR$ is small
compared to $K$. The latter of which is a material parameter of the particular
NLC, independent of the colloid material or size.
This implies that for large values of $W$ one can still find $c \ll 1$ for
sufficiently small particles.
As a numerical example we consider, in line with Ref.~\cite{2013_Choi}, the realistic range $W\in[0,10^{-4}]\,
\text{N/m}$ of anchoring strengths, the
particle size $R=35\,\text{nm}$, which is roughly the mean of the size
distribution in Ref.~\cite{2013_Mertelj}, and $K=10^{-11}\,\text{N}$ for the liquid crystal 5CB.
For these material parameters the coupling constants are in the range
$c\in[0,0.35]$.

Next, one observes in the case of an arbitrarily thin disc, with homeotropic anchoring of arbitrary strength and with normal $\bs{\nu}=\mathbf{e}_z$,
immersed in an NLC with far-field director $\mathbf{n}_0$, that
the nematic director field $\mathbf{n}(\mathbf{r})$ anywhere inside the
NLC is parallel to the plane spanned by $\mathbf{e}_z$ and $\mathbf{n}_0$ \cite{footnote3},
which, in the following, is, without restriction of generality, taken to be
the $x$-$z$-plane spanned by the unit vectors $\mathbf{e}_x$ and
$\mathbf{e}_z$.
This allows one to express the nematic director field $\mathbf{n}:\mathcal{V}
\to\mathbb{R}^3$ in terms of a scalar field $\alpha:\mathcal{V}\to\mathbb{R}$
according to
\begin{equation}
  \mathbf{n}(\mathbf{r}) =
  \mathbf{e}_x\sin(\alpha(\mathbf{r}))+\mathbf{e}_z\cos(\alpha(\mathbf{r})).
  \label{eq:NAnsatz}
\end{equation}
At large distances from the colloid, $|\mathbf{r}|\gg R$, one has the Dirichlet
boundary condition $\alpha(\mathbf{r})\simeq\theta$.
In terms of the scalar field $\alpha$ the free energy functional in
Eq.~\eqref{eq:FOFunc} reads
\begin{equation}
  \label{eq:AFunc}
  \frac{F[\alpha]}{KR} \!=\!
  \frac{1}{2R}\int_{\mathcal{V}}\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,
  \left[\boldsymbol{\nabla}\alpha(\mathbf{r})\right]^2 \!+\!
  \frac{c}{R^2}\!\int_{\partial\mathcal{V}}\!\!\!\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2\! s\,
  [\sin(\alpha(\mathbf{s}))]^2.
\end{equation}
The equilibrium state minimizes
$F[\alpha]/(KR)$ with respect to variations of $\alpha$ which preserve the
Dirichlet boundary condition at large distances. This corresponds to the
Euler-Lagrange equations
\begin{equation}
  \label{eq:BP}
  \begin{cases}
      \boldsymbol{\nabla}^2\alpha(\mathbf{r})=0
    & \text{, $\mathbf{r}\in\mathcal{V}$}\\
      \displaystyle\boldsymbol{\nabla}\alpha(\mathbf{s})\cdot\boldsymbol{\nu}
      (\mathbf{s})=\frac{c}{R}\sin(2\alpha(\mathbf{s}))
    & \text{, at the disc surfaces}\\
      \alpha(\mathbf{r})\simeq\theta
    & \text{, $|\mathbf{r}|\gg R$}.
  \end{cases}
\end{equation}
The boundary problem posed in Eq.~\eqref{eq:BP} is difficult to solve analytically,
in particular due to the nonlinear expression on the right-hand side of the
second line in Eq.~\eqref{eq:BP}.
However, for small values of the coupling parameter $c$ it is promising to consider an expansion of
the scalar field $\alpha$ in terms of powers of $c$:
\begin{equation}
  \label{eq:Expan}
  \alpha(\mathbf{r})
  = \sum_{n=0}^{\infty} c^n \alpha^{(n)}(\mathbf{r})
  = \alpha^{(0)}(\mathbf{r}) + c\alpha^{(1)}(\mathbf{r}) + \cdots.
\end{equation}
By inserting the above expansion into Eq.~\eqref{eq:BP} and by comparing
corresponding orders of $c$ one infers boundary problems for $\alpha^{(n)}(\mathbf{r}),
n\in\{0,1,2,\dots\}$.
It turns out that the boundary problems for $\alpha^{(0)}(\mathbf{r})$ and $\alpha^{(1)}(\mathbf{r})$
can be solved analytically (see Appendix \ref{sec:A1}). Accordingly, here we restrict the following
discussion to these two terms of the expansion in
Eq.~\eqref{eq:Expan}.
Inserting $\alpha^{(0)}(\mathbf{r})$ (Eq.~\eqref{eq:A0}) and $\alpha^{(1)}(\mathbf{r})$ (Eq.~\eqref{eq:A1}) into Eq.~\eqref{eq:AFunc} leads
to the weak anchoring (WA) limit of the free energy (Eq.~\eqref{eq:OneParticleFromA}):
\begin{equation}
  \label{eq:FWC}
  \frac{F^\text{WA}}{KR} =
  -\left(2\pi c+\frac{32}{3}c^2\right)(\mathbf{n}_0\cdot\bs{\omega})^2
  +\frac{32}{3}c^2(\mathbf{n}_0\cdot\bs{\omega})^4.
\end{equation}
It is worth noting that the term $\propto c^1$ in Eq.~\eqref{eq:FWC} can be written in the form
\begin{equation}
-2\pi c(\mathbf{n}_0\cdot\bs{\omega})^2=-2\pi c[\cos\theta]^2=\text{const}+2\pi\frac{WR}{K}[\sin\theta]^2,
\end{equation}
which is equivalent to the expression obtained in Ref.~\cite{2014_Tasinkevych} for the case of a thin rod with tangential anchoring. This fact is related to the topological similarity between the arbitrarily thin disc with homeotropic anchoring and the arbitrarily thin rod with planar anchoring.

In Sec.~\ref{sec:Res} an interval $c\in [0,c_\text{weak}]$ with $c_\text{weak}
\approx 0.1$ is determined numerically such that within this interval Eq.~\eqref{eq:FWC} is a
quantitatively reliable approximation of the exact free energy $F/(KR)$.

\subsection{\label{sec:Theory2}Mesoscopic functional}

\begin{figure}[t!]
   \includegraphics[scale=0.29]{model.eps}
   \caption{Illustration of the coarse-graining used to merge the microscopic
           and the mesoscopic theory.
           The mesoscopic theory is based on the magnetization field $\mathbf{M}(\mathbf{r})$
           (black arrows in the lower left part) and on the director field $\mathbf{n}(\mathbf{r})$ (pale
           red rods in the lower left part).
           The mesoscopic magnetization at the position of a volume element
           (red square in the lower left part) is obtained from the magnetic
           moments of individual colloids (blue arrows normal to the black projected discs in the
           upper right part) averaged over the microscopic distribution inside
           the volume element (red square in the upper right part).
           Within the mesoscopic picture the local director field corresponds to the far-field director $\mathbf{n}_0$ within the microscopic
           one-particle picture of individual colloids (pale red hatching, see
           Sec.~\ref{sec:Theory1}). The microscopic distortion of the nematic director due to its coupling to the surface of the colloid is not shown.}
   \label{fig:Model}
\end{figure}

In order to enable a comparison with Ref.~\cite{2013_Mertelj} we aim for replacing the functional $\Omega[\rho,\mathbf{n}]$ of the number density profile
$\rho$ and the nematic director profile $\mathbf{n}$ in Eq.~\eqref{eq:MicroF}
by a functional $\bar{\Omega}[\mathbf{M},\mathbf{n}]:=\Omega[\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}]),\mathbf{n}]$ of the magnetization
field $\mathbf{M}$ and the nematic director profile $\mathbf{n}$
where $\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])$ is the
number density profile of colloids at position $\mathbf{r}$ with orientation
$\bs{\omega}$ for a prescribed magnetization field $\mathbf{M}(\mathbf{r})$ and
a nematic director field $\mathbf{n}(\mathbf{r})$; $\bar{\rho}$ minimizes the functional in Eq.~\eqref{eq:MicroF},
i.e., it is a solution of the Euler-Lagrange equation
\begin{align}
   m(\bs{\omega}\cdot\bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}]))
   &=
   \frac{\delta\ \beta\Omega}{\delta \rho(\mathbf{r},\bs{\omega})}
   [\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}]),\mathbf{n}]
   \label{eq:ELEq}\\
   &= \ln(\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])\Lambda^3)
      - \beta\mu
   \notag\\
   &\phantom{=}\ + \beta V(\bs{\omega},\mathbf{n}(\mathbf{r}),\mathbf{B}),
   \notag
\end{align}
where $\beta V(\bs{\omega},\mathbf{n}(\mathbf{r}),\mathbf{B}):=
\beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r})) - \beta m
\bs{\omega}\cdot \mathbf{B}$ and where $\bs{\lambda}(\mathbf{r},[\mathbf{M},
\mathbf{n}])$ are the Lagrange multipliers which implement the constraint
\begin{equation}
  \label{eq:MDef}
  \mathbf{M}(\mathbf{r}) =
  \int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,
  m\bs{\omega} \bar\rho(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}]).
\end{equation}
According to Subsecs.~\ref{sec:Theory0} and \ref{sec:Theory1}, the external field which the NLC exerts on the fluid of colloidal discs, is described by
\begin{align}
  &\ \beta V_\text{NLC}(\bs{\omega},\mathbf{n}(\mathbf{r}))
  = -(a+b)(\bs{\omega} \cdot \mathbf{n}(\mathbf{r}))^2 +
     b(\bs{\omega} \cdot \mathbf{n}(\mathbf{r}))^4,
  \notag\\
  &\ a:=2\pi\beta KRc, \quad
     b:=\frac{32}{3}\beta KRc^2.
  \label{eq:VNLC}
\end{align}
The solution of Eq.~\eqref{eq:ELEq} is given by
\begin{align}
  &\phantom{=}\ \ \bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])
  \label{eq:rhobar}\\
  &= \zeta\exp\Big(-\beta V(\bs{\omega},\mathbf{n}(\mathbf{r}),\mathbf{B})
  +m\bs{\omega}\cdot\bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}])\Big)
  \notag
\end{align}
with the fugacity $\zeta:=\exp(\beta\mu)/\Lambda^3$.
Upon inserting Eq.~\eqref{eq:rhobar} into Eq.~\eqref{eq:MicroF} one obtains
\begin{align}
  \beta \Omega[\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}]),\mathbf{n}] =
  &\ \!\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^3 r\!\!\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,
  \bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])
  \Big[\beta\mu-\beta V(\bs{\omega},\mathbf{n}(\mathbf{r}),\mathbf{B})
  \notag\\
  &\ +m\bs{\omega}\cdot\bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}]) - 1 - \beta\mu + \beta V(\bs{\omega},\mathbf{n}(\mathbf{r}),\mathbf{B})\Big]
  \notag\\
  &\ =\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\,\Big[\bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}])
  \cdot\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,m\bs{\omega}\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])
  \notag\\
  &\ -\int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,\bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])\Big],
\end{align}
from which it follows that (see Eq.~\eqref{eq:MDef})
\begin{equation}
  \label{eq:NewMicroF}
  \beta \bar{\Omega}[\mathbf{M},\mathbf{n}]=
  \int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\,\Big[\mathbf{M}(\mathbf{r})\cdot
  \bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}])
  -\rho_0(\mathbf{r},[\mathbf{M},\mathbf{n}])\Big],
\end{equation}
where $\rho_0[\mathbf{M},\mathbf{n}]$ is the orientation-independent number
density profile of the discs:
\begin{equation}
  \label{eq:Rho0}
  \rho_0(\mathbf{r},[\mathbf{M},\mathbf{n}]):=
  \int\!\! \ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,
  \bar{\rho}(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}]).
\end{equation}
Equation \eqref{eq:MDef} provides the important link which allows one the formulation of the functional of the density profile $\bar\rho(\mathbf{r},\bs{\omega},[\mathbf{M},\mathbf{n}])$ in terms of the mesoscopic magnetization field $\mathbf{M}(\mathbf{r})$. The illustration of this idea is shown in Fig.~\ref{fig:Model}.

In order to derive an explicit expression for $\beta\bar{\Omega}[\mathbf{M},
\mathbf{n}]$ it is convenient to introduce an effective magnetic field
\begin{equation}
  \mathbf{h}(\mathbf{r},[\mathbf{M},\mathbf{n}]):=
  \bs{\lambda}(\mathbf{r},[\mathbf{M},\mathbf{n}])+\beta\mathbf{B}
  \label{eq:EffMF}
\end{equation}
and the generating function
\begin{align}
  &Z(\mathbf{h}):=\int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,
  \exp\bigg(A_1(\bs{\omega}\cdot \mathbf{n}(\mathbf{r}))^2 +
            A_2(\bs{\omega}\cdot \mathbf{n}(\mathbf{r}))^4
  \notag\\
  &\phantom{Z(\mathbf{h}):=\int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^2\omega\,\,\exp\bigg(}
  + m\mathbf{h} \cdot \bs{\omega}\bigg)
  \label{eq:ExactZ}
\end{align}
with $A_1:=a+b$ and $A_2:=-b$ denoting the coefficients of two powers of $(\bs{\omega}\cdot \mathbf{n}(\mathbf{r}))$.
This leads to (see Eqs.~\eqref{eq:rhobar} and \eqref{eq:Rho0})
\begin{equation}
   \rho_0(\mathbf{r},[\mathbf{M},\mathbf{n}])
   = \zeta Z(\mathbf{h}(\mathbf{r},[\mathbf{M},\mathbf{n}]))
   \label{eq:rho0Z}
\end{equation}
and
\begin{equation}
   \mathbf{M}(\mathbf{r})
   = \zeta \frac{\partial Z}{\partial\mathbf{h}}
   (\mathbf{h}(\mathbf{r},[\mathbf{M},\mathbf{n}])).
   \label{eq:MZ}
\end{equation}
With this notation Eq.~\eqref{eq:NewMicroF} reads
\begin{align}
  \beta \bar{\Omega}[\mathbf{M},\mathbf{n}]
  \!=\!\!\int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\bigg[
  &\zeta
  \left(\mathbf{h}(\mathbf{r})\cdot
  \frac{\partial Z}{\partial\mathbf{h}}(\mathbf{h}(\mathbf{r})) -
  Z(\mathbf{h}(\mathbf{r}))\right)
  \notag\\
  & - \left.\beta\mathbf{M}(\mathbf{r})\cdot\mathbf{B}\bigg]
  \right|_{\mathbf{h}=\mathbf{h}(\mathbf{r}, [\mathbf{M},\mathbf{n}])}.
  \label{eq:NewNewMicroF}
\end{align}

In the experiments described in Ref.~\cite{2013_Mertelj} the sample is prepared
by dispersing a number density $\rho_\text{iso}$ of colloids in the isotropic
high-temperture phase of the solvent, followed by a quench of the solvent into
the low-temperature nematic phase.
In the absence of an external magnetic field ($\mathbf{B}=0$), the
magnetization vanishes ($\mathbf{M}=0$) before and after the quench, which
corresponds to the effective magnetic field $\mathbf{h}=0$ \cite{footnote4}.
Noting that the number density of colloids does not change during the quench,
one obtains from Eq.~\eqref{eq:rho0Z}
\begin{equation}
   \zeta = \frac{\rho_0}{Z(0)} = \frac{\rho_\text{iso}}{Y_{00}},
   \label{eq:Fugacity}
\end{equation}
where $Y_{00}$ is defined in Eq.~\eqref{eq:Y00}.

It turns out (see Appendix~\ref{sec:A2}) that this part of the
integrand in Eq.~\eqref{eq:NewNewMicroF}, which depends on $Z$, is an even function of
both $H:=m|\mathbf{h}|$ and $u:=\mathbf{n}\cdot\mathbf{h}/|\mathbf{h}|$ (Eq.~\eqref{eq:EffMF}), i.e., a function of $m^2|\mathbf{h}|^2$ and $(\mathbf{n}\cdot\mathbf{h}/|\mathbf{h}|)^2$.
Moreover, it can be shown (see Appendix~\ref{sec:A2}) that the quantities
$T:=|\mathbf{M}|/(m\zeta)$ and $t:=\mathbf{n}\cdot\mathbf{M}/(m\zeta)$ are both
functions of $H$ and $u$, too.
If one can invert the map $(H,u)\mapsto(T,t)$, the integrand in
Eq.~\eqref{eq:NewNewMicroF}, which equals the grand potential density, can be
expressed as function of $|\mathbf{M}|^{2}$ and $(\mathbf{M}\cdot\mathbf{n})^{2}$.
However, in general inverting this map is very challenging.
Therefore, the following considerations are restricted to a quadratic
approximation which includes only terms up to $\sim H^2$ in $Z(\mathbf{h})=: \bar{Z}(H,u)$ (see Eq.~\eqref{eq:quadraticZ}):
\begin{equation}
  \label{eq:ApproxZ}
  \bar{Z}(H,u) \simeq Y_{00} + Y_{10}H^2 + Y_{12}H^2u^2.
\end{equation}
(Note the absence of a term $\propto u^2$.) From Eq.~\eqref{eq:Tandt} in Appendix \ref{sec:A2} one obtains the following system
of equations (see Eq.~\eqref{eq:tTHu}):
\begin{equation}
  \begin{cases}
    \big(t(H,u)\big)^2&=4(Y_{10}+Y_{12})^2 H^2 u^2\\
    \big(T(H,u)\big)^2&=4(2Y_{10}Y_{12}+Y_{12}^2)H^2 u^2+4Y_{10}^2 H^2,
  \end{cases}
  \label{eq:tT}
\end{equation}
which readily can be inverted.
This renders the grand potential functional within the quadratic approximation:
\begin{align}
  \beta \bar{\Omega}[\mathbf{M},\mathbf{n}]
  &=\int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\,\Big[
  \zeta(C_{00} + C_{20}T(\mathbf{r})^2 + C_{02}t(\mathbf{r})^2)
  \notag\\
  &\phantom{= \int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\,\Big[}
  -\beta\mathbf{M}(\mathbf{r})\cdot\mathbf{B}\Big]
  \label{eq:OmegaTt}
\end{align}
with (see Eqs.~\eqref{eq:C00}-\eqref{eq:C02})
\begin{align}
  C_{00}=-Y_{00},\
  &C_{20}=\frac{1}{4Y_{10}},\
  C_{02}=-\frac{Y_{12}/Y_{10}}{4(Y_{10}+Y_{12})},
\end{align}
or, if written explicitly in terms of $|\mathbf{M}|$ and
$\mathbf{M}\cdot\mathbf{n}$:
\begin{widetext}
  \begin{equation}
    \label{eq:FinalF}
    \beta\bar{\Omega}[\mathbf{M},\mathbf{n}]
    =\int\!\!\ensuremath{\mathop{}\!\mathrm{d}}^3 r\,\,\Big[\rho_\text{iso}\left(
    \frac{C_{00}}{Y_{00}}+
    Y_{00}C_{20}\left|\frac{\mathbf{M}(\mathbf{r})}{m\rho_\text{iso}}\right|^2+
    Y_{00}C_{02}\left(\frac{\mathbf{M}(\mathbf{r})}{m\rho_\text{iso}}\cdot
    \mathbf{n}(\mathbf{r})\right)^2\right)
    -\beta\mathbf{M}(\mathbf{r})\cdot\mathbf{B}\Big].
  \end{equation}
\end{widetext}

\section{\label{sec:Res}Results}

\subsection{Limits of reliability for using the one-particle potential}

In the present study we use the expression in Eq.~\eqref{eq:VNLC} for the
one-particle potential $V_\text{NLC}$ corresponding to the weak anchoring regime described by the energy $F^\text{WA}/(KR)$ in Eq.~\eqref{eq:FWC} of a single colloidal particle
with orientation $\bs{\omega}$ at position $\mathbf{r}$, which is immersed in the NLC with
the nematic director $\mathbf{n}(\mathbf{r})$.
In order to assess the accuracy of the expression in Eqs.~\eqref{eq:FWC} or \eqref{eq:VNLC} as
function of the coupling constant $c$, for comparison the full expression for the free energy
in Eq.~\eqref{eq:AFunc} is minimized numerically by using a Galerkin finite element
method \cite{1985_Wait}, because analytical solutions of the boundary value
problems for $\alpha^{(j)}, j\geq 2$, are not available.
The specific set-up, which is considered, consists of a cubic box of dimension $30R \times
30R \times 30R$ which contains a single arbitrarily thin disc in its center.
The interior of the box is decomposed into tetrahedra and the boundary due to the disc
is decomposed into triangles.
Within each finite element the unknown function $\alpha(\mathbf{r})$ is
approximated by linear functions which interpolate between its values at the
corners, i.e., the vertices of the triangulation.
Within the finite-dimensional subspace of functions, which are piecewise linear with
respect to the given triangulation, both the volume and the surface integral in
Eq.~\eqref{eq:AFunc} can be calculated explicitly for each finite element. This
allows for a numerical minimization within this finite-dimensional
subspace.
The described Galerkin method can be performed for arbitrary values of the
coupling constant $c$, i.e., one is not restricted to the weak anchoring
regime.

\begin{figure}[t!]
   \includegraphics[scale=1.0]{weak.eps}
   \caption{Comparison of the free energy $F/(KR)$ obtained by numerical minimization of
           Eq.~\eqref{eq:AFunc} (symbols) and the corresponding analytical result $F^\text{WA}/(KR)$ (Eq.~\eqref{eq:FWC}, solid lines) for three values of the
           coupling constant $c$.
           Already for $c=0.22$ (blue, squares) the numerical data noticeably
           deviate from the analytic result.}
   \label{fig:Weak}
\end{figure}

Figure \ref{fig:Weak} compares the numerically obtained free energy (symbols)
with the one obtained within the weak anchoring limit (Eq.~\eqref{eq:FWC},
solid lines) as function of $\theta$ for three values $c\in\{0.02, 0.12,
0.22\}$.
For very small coupling constants (see $c=0.02$) the weak anchoring limit
Eq.~\eqref{eq:FWC} agrees very well with the numerical results, whereas there are small but
visible deviations for larger values of $c$ (see Fig.~\ref{fig:Weak},
$c=0.22$).

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{criterion1.eps}
   \caption{Quadratic norm $\delta_1(c):=\sqrt{\int_0^{\pi/2}d\theta|
           \frac{F^\text{WA}}{KR}-\frac{F}{KR}|^2}$ as function of the coupling
           strength $c$, where $F$ represents the numerically exact free energy obtained from
           Eq.~\eqref{eq:AFunc} and $F^\text{WA}$ its weak anchoring limit
           (Eq.~\eqref{eq:FWC}).
           The black solid line is a fit of the power law $\alpha c^{\zeta}$ to the data; $\alpha=6.00\pm 0.01$ and $\zeta=2.73\pm 0.01$.
           The inset is a close-up of the range $c\in [0,0.1]$ illustrating
           the criterion in Eq.~\eqref{eq:Criterion1} for $\varepsilon=10^{-2}$.}
   \label{fig:Criterion1}
\end{figure}

In order to quantify the deviation of the weak anchoring approximation
$F^\text{WA}/(KR)$ in Eq.~\eqref{eq:FWC} from the exact expression $F/(KR)$ in
Eq.~\eqref{eq:AFunc}, the following criterion is introduced
\cite{footnote5}:
For a given $\varepsilon>0$ the weak anchoring approximation $F \approx F^\text{WA}$
is considered to be sufficient to describe the free energy $F$ for a fixed value
of $c$, if $\varepsilon$ is an upper bound of the quadratic norm
\begin{equation}
  \label{eq:Criterion1}
  \delta_1(c) :=
  \sqrt{\int_0^{\pi/2}\ensuremath{\mathop{}\!\mathrm{d}}\theta\left|\frac{F^\text{WA}}{KR}-
                                   \frac{F}{KR}\right|^2}<\varepsilon.
\end{equation}
The particular choice of $\varepsilon$ is somewhat arbitrary.
Figure \ref{fig:Criterion1} shows $\delta_1$ as a function of the coupling
strength $c$.
The resulting curve can be fitted by a power law $\alpha c^{\zeta}$ with the amplitude
$\alpha=6.00 \pm 0.01$ and the exponent $\zeta=2.73 \pm 0.01$.
This fit function allows one to determine a value $c_{\text{weak}}$ such
that the criterion in Eq.~\eqref{eq:Criterion1} with a given tolerance $\varepsilon$
is fulfilled for $c<c_{\text{weak}}$:
\begin{equation}
  c_{\text{weak}} =
  \left(\frac{\varepsilon}{\alpha}\right)^{1/\zeta} \approx
  \left(\frac{\varepsilon}{6}\right)^{0.366}.
\end{equation}
A value of, e.g., $\varepsilon=10^{-2}$ implies $c_{\text{weak}} \approx 0.1$
(see the inset in Fig.~\ref{fig:Criterion1}).

Considering Eq.~\eqref{eq:OneParticleFromA}, the contributions to which, up to quadratic order in
$c$, are given by Eq.~\eqref{eq:FWC}, it is tempting to speculate
that the term of cubic order in $c$ is of the form $\sim c^3[\sin(3\theta)]^2$.
In order to assess this presumption one can use the fact that the free energy
$F(\theta)/(KR)$ in Eq.~\eqref{eq:AFunc} is an even function of $\theta$ with
period $\pi$, which allows for an expansion into a Fourier series
\begin{equation}
  \label{eq:FSeries}
  \frac{F(\theta)}{KR} = \frac{a_0}{2}+\sum_{n=1}^\infty a_n \cos(2n\theta),
\end{equation}
with the Fourier coefficients
\begin{equation}
  \label{eq:FCoeff}
  a_n =
  \frac{4}{\pi} \int_0^{\pi/2}\!\!\ensuremath{\mathop{}\!\mathrm{d}} \theta\,\,
  \frac{F(\theta)}{KR}\cos(2n\theta), \quad n=0,1,2,\dots\ .
\end{equation}
In the context of actual numerical schemes only a finite number $N$ of free energy values
$F(\theta_i)/(KR)$ for the angles $\theta_i$, $i\in\{0,1,\dots,N-1\}$, are
available.
Hence, instead of using Eq.~\eqref{eq:FCoeff} by applying a suitable quadrature,
one can --- as an alternative approximation scheme --- restrict the sum in Eq.~\eqref{eq:FSeries} to $n \leq
N-1$ and determine the coefficients $a_n$, $n\in\{0,1,\dots,N-1\}$, via fitting the
numerical data $F(\theta_i)/(KR)$, $i\in\{0,1,\dots,N-1\}$, by trigonometric
polynomials, i.e., superpositions of terms $\cos(2n\theta)$, $n\in\{0,1,\dots,
N-1\}$.

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{f_coeffs_multi.eps}
   \caption{Absolute values $|a_n|$ of the first seven Fourier coefficients of
           the expansion of the exact free energy
           (Eq.~\eqref{eq:FSeries}, full dots) along with the three corresponding coefficients obtained within the
           weak anchoring limit (Eq.~\eqref{eq:FWC}, open circles) for the coupling
           constant $c=0.02$ (upper panel) and $c=0.22$ (lower panel).}
   \label{fig:FCoeffs1}
\end{figure}

Figure~\ref{fig:FCoeffs1} shows the absolute value of the coefficients $a_n$ as functions of $n$
(full dots) for $c=0.02$ (upper panel) and $c=0.22$ (lower panel).
In addition, Fig.~\ref{fig:FCoeffs1} also displays the corresponding
coefficients obtained within the weak anchoring limit (see Eq.~\eqref{eq:FWC} and the open circles):
$a_0/2 = \pi c - 4c^2/3 + \mathcal{O}(c^3)$, $a_1 = -\pi c + \mathcal{O}(c^3)$,
and $a_2 = 4c^2/3 + \mathcal{O}(c^3)$.
For small values of the coupling constant $c$ (see the case $c=0.02$ in the upper panel)
the agreement between the weak anchoring coefficients and the exact ones is
excellent.
On the other hand, for large coupling constants (see $c=0.22$ in the lower
panel) one finds (i) that modes appear with comparatively large amplitudes $|a_n|$, $n\geq 3$, which signals that the exact data cannot be strictly described within the
weak anchoring limit given by Eq.~\eqref{eq:FWC}, and (ii) that the exact coefficients $|a_n|$, $n\in\{0,1,2\}$, are not perfectly reproduced by those calculated
within the weak anchoring limit according to Eq.~\eqref{eq:FWC}. On a logarithmic scale these features are not conspicuous. However, they are much more apparent on a linear scale (not
shown here).
Both of these observations suggest that upon increasing the coupling constant
$c$, Eq.~\eqref{eq:FWC} has to be modified in a way that (i) higher-order
terms proportional to $\cos(2n\theta),n\geq3$, occur and (ii) terms of order
$c^3$ or higher modify the coefficients $a_n$, $n=0,1,2$.

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{crossover.eps}
   \caption{Coefficient $a_0$ of the leading term in the expansion in Eq.~\eqref{eq:FSeries} of
           the free energy as function of the
           coupling constant $c$.
           The limit $a_0(c)/2 \to \pi^2/3$ for $c\to\infty$ is indicated by
           the red horizontal dashed line.
           The inset displays a semi-logarithmic plot of the deviation of $a_0/2$ from its asymptotic value, which is very well fitted by an exponential
           function $\lambda\exp(-\chi c)$ (solid line), where $\lambda=1.11 \pm 0.02$ and
           $\chi=0.247\pm 0.003$.}
   \label{fig:Cross}
\end{figure}

As a byproduct of the Fourier analysis presented above, the very strong
anchoring limit can be reconsidered.
Figure \ref{fig:Cross} shows the coefficient $a_0/2$ appearing in Eq.~\eqref{eq:FSeries} as
function of $c$.
As expected, for very strong couplings this curve approaches the value
$\pi^2/3$, which is the coefficient $a_0/2$ appearing in the Fourier expansion of
Eq.~\eqref{eq:Hayes}.
This limiting value is attained exponentially (see the inset of
Fig.~\ref{fig:Cross}).

\begin{figure}[t]
   \centering
   \includegraphics[scale=1.0]{criterion2.eps}
   \caption{Quadratic norm $\delta_2$ (see Eq.~\eqref{eq:Criterion2}) as
           function of the coupling constant $c$ (full dots).
           The solid line is a fit of the data in the interval $c\in[4,10]$ by
           an exponential function $\kappa\exp(-\nu c)$; $\kappa=10.84\pm 0.16$ and $\nu=0.246\pm 0.003$.
           The inset provides the same information but in terms of a semi-logarithmic plot.}
   \label{fig:Criterion2}
\end{figure}

In order to obtain an estimate of $c_{\text{strong}}$ such that the infinite
anchoring limit in Eq.~\eqref{eq:Hayes} is reliable for $c>c_\text{strong}$, one
can use a criterion similar to the one in Eq.~\eqref{eq:Criterion1}, based on the
quadratic norm
\begin{equation}
  \label{eq:Criterion2}
  \delta_2(c):=
  \sqrt{\int_0^{\pi/2}\ensuremath{\mathop{}\!\mathrm{d}}\theta\left|\frac{F^\text{IA}}{KR}-
                                     \frac{F}{KR}\right|^2},
\end{equation}
where $F^\text{IA}$ (Eq.~\eqref{eq:Hayes}) is the free energy within the infinite anchoring limit.
Figure \ref{fig:Criterion2} shows $\delta_2$ as a function of $c$.
The tail of the data in the interval $c\in[4,10]$ can be fitted by an
exponential function $\kappa\exp(-\nu c)$ with $\kappa=10.84 \pm 0.16$, and $\nu=0.246 \pm
0.003$ (see the inset of Fig.~\ref{fig:Criterion2}).
Thus, $\delta_2<\varepsilon=10^{-2}$ leads to
\begin{equation}
  c_{\text{strong}}=
  -\frac{1}{\nu}\ln\frac{\varepsilon}{\kappa} \approx
  -\frac{1}{0.246}\ln\frac{\varepsilon}{10.84} \approx
  28.4 \approx
  28.
\end{equation}

Accordingly, the interval $c\in[0,\infty)$ of coupling constants provides three regimes: (i) the weak coupling regime $c\in[0,c_{\text{weak}}]$ in which
the free energy of a disc-like colloid in an NLC is very well described by
Eq.~\eqref{eq:FWC}, (ii) the strong coupling regime $c\in[c_{\text{strong}},
\infty)$ in which the free energy is independent of $c$ and has the form of
Eq.~\eqref{eq:Hayes}, and (iii) the intermediate coupling regime $c\in
[c_{\text{weak}},c_{\text{strong}}]$ in which the crossover between both previous
limits takes place.

\subsection{Free energy functional in quadratic approximation}

\begin{figure}[b]
   \centering
   \includegraphics[scale=1.0]{coeffs.eps}
   \caption{Coefficients $Y_{00}C_{20}$ (Eqs.~\eqref{eq:Y00} and \eqref{eq:C20}) and $Y_{00}C_{02}$ (Eqs.~\eqref{eq:Y00} and \eqref{eq:C02}) in Eq.~\eqref{eq:FinalF2} as
           functions of the coupling constant $c$.}
   \label{fig:Coeffs}
\end{figure}

Disregarding the constant, $\mathbf{M}$- and $\mathbf{n}$-independent term
$\propto C_{00}$ in the integrand on the right-hand side of Eq.~\eqref{eq:FinalF},
one infers the following expression for the free energy density of a fluid of
magnetic discs suspended in an NLC:
\begin{align}
   \label{eq:FinalF2}
   f(\mathbf{M},\mathbf{n})
   &= k_\text{B}T\rho_\text{iso}\bigg(
   Y_{00}C_{20}\left|\frac{\mathbf{M}}{m\rho_\text{iso}}\right|^2 + \\
   &\ \phantom{= k_\text{B}T\rho_\text{iso}\Big(}
   Y_{00}C_{02}\left(\frac{\mathbf{M}}{m\rho_\text{iso}}\cdot\mathbf{n}
   \right)^2\bigg)
   - \mathbf{M}\cdot\mathbf{B}.
   \notag
\end{align}
The dependence of the coefficients $Y_{00}C_{20}$ and $Y_{00}C_{02}$ (see Eqs.~\eqref{eq:Y00}, \eqref{eq:C20}, and \eqref{eq:C02}) on the coupling
strength $c$ is shown in Fig.~\ref{fig:Coeffs}.
In contrast, in Ref.~\cite{2013_Mertelj} the following
\textit{ph}enomenological form of the free energy density has been proposed:
\begin{equation}
  \label{eq:Mertelj}
  f_\text{ph}(\mathbf{M},\mathbf{n}) =
  \frac{a}{2}|\mathbf{M}|^2 + \frac{b}{4}|\mathbf{M}|^4
  - \frac{1}{2}\gamma\mu_0(\mathbf{M}\cdot\mathbf{n})^2
  - \mathbf{M}\cdot\mathbf{B},
\end{equation}
where the first two terms on the right-hand side are part of the Landau
expansion describing the interaction between magnetic dipoles, the third term
represents the coupling between the nematic order and the magnetization, and
the last term is the interaction of magnetic dipoles with an external magnetic
field.

The comparison between Eqs.~\eqref{eq:FinalF2} and \eqref{eq:Mertelj} leads to the
following conclusions:
(i) By identifying the terms proportional to $|\mathbf{M}|^2$ in both
expressions one infers the positive coefficient $a=2k_\text{B}TY_{00}C_{20}/
(m^2\rho_\text{iso})>0$.
(ii) Since $a>0$, the term proportional to $|\mathbf{M}|^4$ is unnecessary in the
phenomenological expression in Eq.~\eqref{eq:Mertelj} and its absence in
Eq.~\eqref{eq:FinalF2} is without consequences.
(iii) In agreement with physical intuition the coefficient $\gamma$ introduced
in Ref.~\cite{2013_Mertelj} is positive and its dependence on the coupling
strength $c$ is given by
\begin{align}
  \gamma
  &=
  -2k_\text{B}T\frac{1}{\rho_\text{iso}}\frac{1}{\mu_0m^2}Y_{00}C_{02}
  \notag\\
  &=\frac{k_\text{B}T}{2}\frac{1}{\rho_\text{iso}}\frac{1}{\mu_0m^2}Y_{00}
  \frac{Y_{12}/Y_{10}}{Y_{10}+Y_{12}}.
  \label{eq:gamma}
\end{align}

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{gamma.eps}
   \caption{Dependence of the phenomenological coupling coefficient $\gamma$
           introduced in Ref.~\cite{2013_Mertelj} (see Eq.~\eqref{eq:Mertelj})
           on the microscopic coupling constant $c$ as it follows from
           Eq.~\eqref{eq:gamma}.
           The red circle denotes the value of $c\simeq 0.019$, which corresponds to the
           estimate of $\gamma\simeq 110$ given in Ref.~\cite{2014_Mertelj}.
           The blue square denotes the value of $\gamma\simeq 203$, which corresponds to the
           estimate of $c\simeq 0.03$ based on the parameters used in Ref.~\cite{2014_Mertelj}.
           Under ideal circumstances, the blue and the red dashed lines coincide.}
   \label{fig:Gamma}
\end{figure}

Figure \ref{fig:Gamma} shows $\gamma$ as function of the coupling constant $c$
(see Eq.~\eqref{eq:gamma}) for the experimentally relevant parameters
$\rho_\text{iso}=1.5 \times 10^{19}\,\mathrm{m^{-3}}$ and $m \approx 3 \times
10^{-18}\,\mathrm{Am^2}$ taken from Ref.~\cite{2014_Mertelj}.
The order of magnitude of the theoretical result (black line) is in agreement
with the values estimated from the experiment ($\gamma_\text{exp}\approx 110$,
see Ref.~\cite{2014_Mertelj}).
Figure~\ref{fig:Gamma} shows that (at least in the regime of weak anchoring)
$\gamma$ increases monotonically with $c$.
Based on the values given in Ref.~\cite{2014_Mertelj} one can, on the one hand,
estimate $c$ from $\gamma$ (red dot and dashed lines) and, on the other hand,
one can estimate $\gamma$ from $c$ (blue square and dashed lines).
Since values of $c$ and $\gamma$ given in Ref.~\cite{2014_Mertelj} belong to
one and the same system the red and blue dashed lines in Fig.~\ref{fig:Gamma}
should coincide.
However, this is not quite the case.
The discrepancy may arise due to the fact that in all calculations the mean value of the particle
size has been used assuming that the discs are monodisperse in size, whereas in the experiment the size distribution of the colloids
has a finite width.
Moreover, elastic interactions between the discs (generated by the nematic director field $\mathbf{n}$), which have been entirely
neglected in the present study, might play a role for the properties of the actual system.

Knowing the explicit dependence $\gamma(c)$ offers the possibility to estimate
the anchoring energy $W$ by performing an experiment similar to the one
described in Ref.~\cite{2014_Mertelj}: Using Fig.~\ref{fig:Gamma},
from an estimate of $\gamma$ one obtains the corresponding value of $c$, which, knowing the
mean size $R$ of the platelets and the elastic constant $K$ of the NLC, renders the value of $W$.

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{orders.eps}
   \caption{Comparison of the values of $\gamma$ calculated by using in
           Eq.~\eqref{eq:VNLC} only the first-order term in $c$ (black dashed line, $\mathcal{O}(c^1)$)
           or both the first- and the second-order term in $c$ (red solid line, $\mathcal{O}(c^2)$).
           The larger $c$, the more does the first-order theory overestimate the
           value of $\gamma$.}
   \label{fig:Orders}
\end{figure}

In Fig.~\ref{fig:Orders} two expressions of $\gamma$ as function of $c$ are
compared: one, for which in Eq.~\eqref{eq:VNLC} only the term of first order in $c$ (black dashed
line, $\mathcal{O}(c^1)$) and one, for which terms up to second order in $c$ (red solid line,
$\mathcal{O}(c^2)$) are retained. (Note that the black dashed line is a \textit{nonlinear} function of $c$, even if one has kept only the linear contribution $\mathcal{O}(c^1)$ in $V_\text{NLC}$, because $\gamma$ depends nonlinearly on $V_\text{NLC}$.)
The difference between the two expressions suggests that if one cuts off the expansion in $c$ at too low order, the
resulting phenomenological coupling constant $\gamma$ is overestimated if $c\gtrsim 0.04$. The value of $c$ inferred from the experiment (see Ref.~\cite{2014_Mertelj}) lies below the value $c=0.04$, i.e., in a range within which the two curves in Fig.~\ref{fig:Orders} de facto coincide. Therefore, considering only the term $\propto c^1$ in Eqs.~\eqref{eq:FWC} or \eqref{eq:VNLC} does not change the outcome of the present effective theory in the context of the experimental parameters used in Ref.~\cite{2014_Mertelj}. However, this assessment requires an analysis up to higher orders in $c$, as carried out in the present study.

\section{\label{sec:Discus}Discussion}

Inspired by an expression introduced in Ref.~\cite{2013_Mertelj}, the present
study derives a Landau-like free energy density of ferronematics in terms of the
magnetization $\mathbf{M}$ and the nematic director $\mathbf{n}$.
The derivation starts from a density functional theory (DFT) which
describes colloids suspended in a nematic liquid crystal (NLC).
The coupling between the colloids and the NLC is modeled in terms of a
one-particle potential (see Eq.~\eqref{eq:VNLC}), which in the DFT framework
plays the role of an external field. It depends on the orientation of the
colloid and on the local nematic director field $\mathbf{n}(\mathbf{r})$.
Motivated by the high dilution of the colloidal suspensions under consideration,
a direct colloid-colloid interaction is neglected.
Accordingly, the theory can be formulated in terms of a relatively simple local
density functional.
The one-particle potential in Eq.~\eqref{eq:VNLC} is derived from the
perturbation expansion (Eq.~\eqref{eq:FWC}) of the free energy in terms of the small
parameter $c$, which represents the strength of the coupling of the NLC to the
surface of a single colloidal particle (see Eq.~\eqref{eq:C}).
In the present study the expansion of the free energy in terms of powers of $c$, together with the corresponding
analytical expression for the nematic director profile around a colloidal particle, is determined. The term $\propto c^1$ in the expansion derived here is equivalent to the expression derived elsewhere (see Ref.~\cite{2014_Tasinkevych}) for the different case of arbitrarily thin rods with tangential coupling.
Using numerical methods, the range of values of the coupling constant $c$ is
estimated, within which the weak coupling limit (Eq.~\eqref{eq:FWC}) is accurate.
It is shown that the next-order term is proportional to
$\big(\sin (3\theta)\big)^2$ with $\theta$ introduced in Fig.~\ref{fig:Colloid}.

In the next step, the expression in Eq.~\eqref{eq:VNLC} for the one-particle potential
is used to establish the density functional in Eq.~\eqref{eq:MicroF}
of noninteracting discs subjected to an external field.
Four possible kinds of pair interactions are neglected:
(i) the direct dipole-dipole interaction due to the presence of magnetic moments; (ii) the steric repulsive interaction; (iii) the van der Waals interaction; and
(iv) the effective elastic interaction induced by the NLC.
On one hand, the dipole-dipole and the van der Waals interactions are negligibly small compared to the thermal energy
$k_\text{B}T$ for the mean distances between the disc centers as given in
Ref.~\cite{2014_Mertelj}.
Moreover, the steric interaction is disregarded due to its short-ranged character and therefore due to the low impact onto mesoscopic properties of a very dilute colloidal solution.
On the other hand, the effective elastic interaction might be important even for dilute
solutions because the effective elastic interaction for two discs, which are both
inclined with respect to the far-field director, is described by a long-ranged
Coulomb-like pair potential \cite{2007_Pergamenshchik}.
Here, we have neglected it nevertheless in order to keep the theory analytically tractable.

\begin{figure}[t!]
   \centering
   \includegraphics[scale=1.0]{zcalc.eps}
   \caption{Comparison of the exact generating function $\bar{Z}(H,u)$ given by
           Eq.~\eqref{eq:ExactZ} (black dashed line) and its quadratic
           approximation provided in Eq.~\eqref{eq:ApproxZ} (red solid line) for
           $c=0.05$ and $u=1$.}
   \label{fig:HMax}
\end{figure}

Deriving the free energy density in terms of powers of $|\mathbf{M}|$ and of the
scalar product $\mathbf{M}\cdot\mathbf{n}$ appears to be out of analytic reach because the integral in Eq.~\eqref{eq:ExactZ} and thus $Z(\mathbf{h})$ cannot be calculated analytically.
Hence, in Eq.~\eqref{eq:ApproxZ} the approximation of $Z(\mathbf{h}) =
\bar{Z}(H,u)$ by a second-degree polynomial in terms of the variables
$H:=m|\mathbf{h}|$ and $u:=\mathbf{n}\cdot\mathbf{h}/|\mathbf{h}|$ is
introduced.
The quality of this quadratic approximation is high for small values  of $H$ and
it becomes poorer upon increasing $H$.
Note, however, that in the present study large values of $H$ are not needed.
Indeed, whereas one can show that within the present model one has $|\mathbf{M}|\to\infty$ for $H\to\infty$,
this limit is not realized in the context of the experimental situation under
consideration, because the colloids carry a finite magnetic dipole moment
$m$ and their suspension in the NLC is given by a finite number density
$\rho_\text{iso}$, so that $|\mathbf{M}(\mathbf{r})|\leq m\rho_\text{iso}$ at any
point $\mathbf{r}$ (for reason of simplicity assuming only very small segregation effects).
Therefore one should not consider the whole range of $H$ but only up to a
certain value $H_\text{max}$, defined such that $|\mathbf{M}|/(m\rho_\text{iso})\leq
T(H_\text{max})/Y_{00}:=1$ holds (see the definition of $T$ in the paragraph below Eq.~\eqref{eq:Fugacity}, and see Eq.~\eqref{eq:Fugacity}).
According to the second line of Eq.~\eqref{eq:tT}, $H_\text{max}$ is a function
of $u\in[-1,1]$, i.e., of the cosine of the angle between the effective
magnetic field $\mathbf{h}$ and the nematic director $\mathbf{n}$ of the NLC (see below Eq.~\eqref{eq:Fugacity}).
Figure \ref{fig:HMax} shows a comparison of the exact generating function
$\bar{Z}(H,u)$ (black dashed line) with the one calculated within the quadratic
approximation (red solid line) in the interval $H\in[0,H_\text{max}(u)]$ for
$u=1$, i.e., for the case that the magnetization is parallel to the nematic
director.
It turns out that the difference between the exact generating function and its
quadratic approximation increases with $H$ but that it remains below 10\% for the
whole range of physically reasonable values of $H$.
Thus the quadratic approximation in Eq.~\eqref{eq:ApproxZ} provides a reasonable,
at least qualitatively correct description of the free energy density of ferronematics.

The free energy-density obtained in the present study (Eq.~\eqref{eq:FinalF2}) and
the one in Eq.~\eqref{eq:Mertelj} proposed in Ref.~\cite{2013_Mertelj} share
two main features:
(i) In the absence of the coupling between the magnetization $\mathbf{M}$ and the nematic
director $\mathbf{n}$, the suspension is in the paramagnetic phase, and
(ii) the ferromagnetic properties are generated by a coupling of the magnetization
and the nematic director via a term proportional to
$(\mathbf{M}\cdot\mathbf{n})^2$.
The comparison of the coefficients multiplying $(\mathbf{M}\cdot\mathbf{n})^2$
allows one to obtain an expression for the phenomenological coupling parameter
$\gamma$ (Eq.~\eqref{eq:Mertelj}) in terms of the microscopic coupling constant $c$ (Eq.~\eqref{eq:gamma}).
A slight inconsistency between the estimates for $\gamma$ and for the anchoring
energy $W$ from Ref.~\cite{2014_Mertelj} is found (Fig.~\ref{fig:Gamma}). The reasons for this
discrepancy encompass both the simplifications used in the present study
(such as arbitrarily thin discs, monodisperse disc size, neglect of elastic pair
interactions) as well as those applied in the theoretical model used in
Ref.~\cite{2014_Mertelj}, e.g., the simplified form of the coupling of the director field to
the colloid surface.

redpajama_set_name: RedPajamaArXiv


text:
Církevní oblast Piemonte - Val d'Aosta () je jedna z 16 církevních oblastí, do nichž je rozdělena římskokatolická církev v Itálii. Její hranice se z valné části kryjí s hranicemi italského regionu Piemont a italského autonomního regionu Údolí Aosty.

Rozdělení

Církevní oblast Piemonte – Val d'Aosta tvoří dvě metropole a jejich patnáct sufragánních diecézí:

 Arcidiecéze turínská
 Diecéze Acqui
 Diecéze Alba
 Diecéze Aosta
 Diecéze Asti
 Diecéze Cuneo
 Diecéze Fossano
 Diecéze Ivrea
 Diecéze Mondovì
 Diecéze Pinerolo
 Diecéze Saluzzo
 Diecéze Susa
 Arcidiecéze Vercelli
 Diecéze Alessandria
 Diecéze Biella
 Diecéze Casale Monferrato
 Diecéze Novara

Statistiky
 plocha: 29 544 km²
 počet obyvatel: 4 592 376
 počet farností: 2 249

Biskupská konference oblasti Piemonte – Val d'Aosta
 Předseda: Cesare Nosiglia, arcibiskup turinský
 Místopředseda: Marco Arnolfo, arcibiskup vercellský
 Sekretář: Franco Lovignana, biskup v Aostě

Externí odkazy
  Církevní oblast Piemonte – Val d'Aosta a její mapa na oficiálních stránkách katolické církve v Itálii
  Statistická a historická data na catholic-hierarchy.org

redpajama_set_name: RedPajamaWikipedia


text:
Part of HuffPost Business. ©2021 Verizon Media. All rights reserved.
Walmart Is One Of The Biggest Beneficiaries Of Food Stamps
By Jillian Berman
A customer carries a shopping basket during the grand opening of a Wal-Mart Stores Inc. location in the Chinatown neighborhood of Los Angeles, California, U.S., on Thursday, Sept. 19, 2013. Wal-Mart Stores Inc. will phase out 10 chemicals it sells in favor of safer alternatives and disclose the chemicals contained in four product categories, the company announced Sept. 12. Photographer: Patrick T. Fallon/Bloomberg via Getty Images
One of the major beneficiaries of the nation's food-stamp program is actually a hugely profitable company: Walmart.
Americans spend about 18 percent of all food stamp dollars at Walmart, according to company estimates told to the Wall Street Journal and confirmed by The Huffington Post. That's about $14 billion of the $80 billion Congress set aside for food stamps last year. The company's total profits for 2013 were $17 billion.
The news comes just as food-stamp benefits are about to be cut for 47 million Americans. On Friday, a key provision boosting the program is set to expire.
After that, 47 million Americans will struggle even more than usual to afford the basics. Walmart isn't too concerned: When shoppers become more concerned about price, they're more likely to turn to Walmart, Bill Simon, the retailer's U.S. CEO, said at an analyst meeting earlier this month.
"I would say we're cautious but modestly optimistic," Simon said. "When the [food-stamp] benefits expanded, our market share actually went down."
Still, as the nation's largest low-cost retailer, the company has close ties to the food-stamp economy. Walmart has historically lobbied around food stamps, according to a report from Eat Drink Politics, an advocacy group. In addition, the company also worked with First Lady Michelle Obama in 2011 on the Great American Family Dinner Challenge, a push to get families to eat healthy on a food-stamp budget.
"A significant percentage of all SNAP dollars are spent in our stores, and they are used to buy items like bananas, whole milk, Ramen noodles, and hot dogs," then-Walmart executive Leslie Dach said at a conference in 2011, where he discussed the retailer's plans to help shoppers buy healthier meals on food stamps.
That so many people associate Walmart with food stamps was clear earlier this month, when Louisiana food-stamp users realized a computer glitch caused them to temporarily have unlimited food-stamp money to spend. What did they do? They raced to local Walmart stores and stripped the shelves bare.
In the past, companies like Pepsi, Coke and the midwestern grocery chain Kroger, have lobbied around food stamps, indicating how much they rely on the money. Last year, Tony Vernon, then the incoming CEO of Kraft, admitted the mac n' cheese maker opposed food stamp cuts because food stamp users were "a big part of our audience."
762,000 Pounds Of Hot Pockets Recalled For Potential Glass, Plastic Contamination Pivotal Union Election Moves Ahead At Amazon Warehouse In Alabama Coal Company Officials Plead Guilty After Hiding Black Lung Dangers Sheldon Adelson, GOP Megadonor Who Founded Las Vegas Sands, Dies
9 Companies That Really Love Food Stamps
Jillian Berman
Associate Business Editor, The Huffington Post
Business Walmart Food Stamps
PepsiCo lobbied the House of Representatives in 2011 on restrictions on SNAP, according to a June 2012 report, called "Food Stamps: Follow the Money." The company spent $750,000 lobbying Congress overall during the third quarter of 2011. At the same time, Pepsi donated money to anti-hunger groups working to fight health restrictions on SNAP.

redpajama_set_name: RedPajamaCommonCrawl


Thanks for this quick share! I know enough to glean a slight understanding, unfortunately not enough for anything actionable! I was searching for a solution to get my drupal install updated though synology and your blog showed up in the results but with "drush". I've never heard of that so I got curious!
This is really fabulous. Thanks so much. I almost have it working. But I get the "needs a higher bootstrap level" error no matter what directory I run "drush ups" from.
Following is the output of "drush status". Does the drush version have to match the drupal version? I'm also suspicious of the URI value.


text:
Misadventures in kiddie lit (or, why I've been inspired to write a children's book parody called, "Curious George Gets What's Coming to Him")
I love reading to Lily. I really do. It's one of my absolute favorite activities to do with her. But frankly, there are some books I'm less thrilled about reading to her than others.
We have a very truncated board book version, for instance, of "Jack and the Beanstalk," and Lily seems to like its colorful illustrations, but the story drives me batty. Why? Because it might as well be called "The Shameless Goniff and the Beanstalk."
Things start off fine – Jack sells a cow for magic beans, his mother throws them out the window, and the giant beanstalk grows and reaches up through the clouds overnight. All well and good. But then Jack steals a bunch of treasure from the giant's castle and scoots down the beanstalk to safety. And then he decides to go back up the beanstalk for more, because apparently, he's developed a taste for bling. But while trying to make off with an enchanted harp, the giant wakes up, and Jack barely makes it down the beanstalk in time to chop it down and send the giant plummeting to his death.
What exactly is the moral here? When you steal, make sure you get out of there quickly? And are we really expected to cheer for the boy who stole a bunch of stuff from a giant that was minding his own business, living atop the clouds, before having his home invaded? Who exactly is the victim in this story? (Because of my misgivings, I think this book will mysteriously be "lost" in the very near future.)
Similarly, I have issues with some Curious George books. Not so much the early ones – though there are minor problems there, too. (George is kidnapped from his seemingly happy life in Africa and bagged up by the world's most inattentive pet owner, the Man in the Yellow Hat; George smokes a pipe before bed; George is tossed into a prison cell for mistakenly calling the fire department; etc.). Generally, I'm fine with the original George: a well-intentioned monkey whose curiosity causes him to inadvertantly stumble into trouble.
What I'm not fine with are some of the later Curious George books that are apparently based on a film series. In "Curious George and the Dinosaur," for instance, George goes to a museum with a class of kids; he wanders off from the group; he spots a giant dinosaur skeleton and climbs all over it; and although the museum director gets upset with George, the teacher defends George by saying that the monkey's "ride" on the dinosaur got the students interested in talking about prehistoric animals.
George climbs down the skeleton to cheers and the director's forgiveness, so his bad bahavior is ultimately rewarded, and he's treated like a hero.
Oy. Where to begin.
Lily loves the original Curious George story – she asks for a banana each time we read it, since in the first illustration, George is swinging from the trees while eating one – so I had pulled this stupid "Dinosaur" book off the local library's shelf, thinking she'd like a change-up. But as I was reading it to her, I felt like I wanted to shove the words right back into my mouth. Wandering off alone in a strange place? Climbing on a fragile museum exhibit, and being applauded for it? Yikes. (I was so annoyed by this that shortly thereafter, when prompted by my writing group to write a children's story, I thought of composing a parody titled, "Curious George Gets What's Coming to Him.")
Of course, there have always been children's books with twisted messages that you're not exactly sure you want to hand down. Shel Silverstein's "The Giving Tree" depicts selflessness and sacrifice in the name of love, yes, but perhaps to an unhealthy degree, since the tree is reduced to a stump by book's end. (Perhaps "The Codependent Tree" would be a more apt title?)
Check out this Sassy Gay Friend video that makes a similar point, in a much funnier way.
Robert Munsch's "Love You Forever" gets really creepy and weird when the mother's son grows up and moves across town, only to have his mother drive in the dark of night to his house, sneak through his window, crawl across the floor, and rock him, sleeping, in her arms. Boundaries, people, boundaries.
Even Sam McBratney's "Guess How Much I Love You," which I'd previously thought a sweet book that I'd bought for countless couples who were having a baby, now seems somewhat problematic. Yes, it's sweet that a parent and child are enthusiastically declaring their love for each other; but because the father's size and physical dexterity necessarily exceeds his small son's, the father's trumping of each statement of affection, and his insistence upon having the last word on the matter – even though the son is asleep at the time – makes the tone of one-upsmanship far less appealing.
What strikes me about these later examples is that, in my mind, they seem to be written for, and have a far greater impact on, the parents more than the children. I can't tell you how many people have told me they start crying when reading books like "The Giving Tree," "Love You Forever," and "Guess How Much I Love You?" to their child, and I don't think this is a coincidence. The books, at their best, touch on the overwhelming sense of love that we mothers and fathers feel for our children, and this washes over us as we read a story to that child about our willingness to sacrifice much, as well as our desire to help and protect them throughout their lifetime. But from a colder, more objective distance from the material, I think we have to take a hard look at what we're laying out for our children. Yes, they'll likely feel the same way when they become parents years from now, but that's light years away, both temporally and intellectually. And until then, I can't help wondering what they're making of these messages.
Am I overthinking all of this? Perhaps. It would hardly be the first time THAT's happened. And in our living room, there's a giant basket of books that I adore reading to Lily, without such niggling reservations (including "Corduroy," "The Mitten," and "The Foot Book"). But I must nonetheless confess that I've been surprised, as a new parent, by how I read and process children's books very differently now.
And sometimes, I reverse my initial judgments. When I first read "Go, Dog, Go" to Lily, when she was a baby, I thought, "What a pointless book." Yet a few months before she turned two, she came to love this book, and I began to understand why. She was just getting a grasp on colors, objects, spatial relationships, and, well, she liked dogs. So she got noticeably excited when reading it because it was one of the first books where she felt like she was connecting with it and could understand most of the words in it. No small thing. Obviously, I needed to revise my previous assessment in that case.
For after years and years of careful writing and reading, I know only too well the power of stories. They are our earliest lessons, and they are a means of finding and making a place for ourselves in the larger world.
So although Lily is young, and thus guileless in regard to the sometimes-unsettling messages found in children's lit, I nonetheless want to do my best, while her world is largely filtered through me (and her father), to share with her stories that not only don't make me squirm, but make me feel like a have a fighting chance at raising a responsible, productive, thoughtful person.
Just to be safe, I won't be taking Lily to any dinosaur exhibits anytime soon.
She's a good little monkey, but always very curious.
This entry was tagged children's lit, curious george, go dog go, guess how much i love you, jack and the beanstalk, kiddie lit, love you forever, motherhood, parenting, sassy gay friend, the giving tree, toddler, two year olds.
12 thoughts on "Misadventures in kiddie lit (or, why I've been inspired to write a children's book parody called, "Curious George Gets What's Coming to Him")"
I personally like "The Enabling Tree" better.
I can really identify with the giant in the Beanstalk story, btw. 🙂
Jenn McKee says:
I can see that. And since Lily might meet you one day, I don't want her emptying your pockets and running off somewhere…
kerianne says:
You could bring Lily to "Into the Woods" – Jack gets his come-upance there. Mind you, his mother is killed, so that may be just as horrific. 🙂
The nice part is that at this age, Lily can't read and we're able to read ahead fairly well. When I find a new book where I'm not liking some of the story, I'll just modify or skip ahead. I think I've only actually stopped reading one book entirely which was about a circus elephant that runs away to have an adventure and ends up being abused by a farmer. I couldn't believe it.
Totally with Sophie on enabling.
I love "The Mitten"! Also, "Caps for Sale" and "The Fool and the Flying Ship", although the latter definitely has some questionable overtones and undertones. This is a great post! I totally want to read "Curious George Gets What's Coming to Him" – it's about time 🙂
Erik Kuszynski says:
From reading your blogs, I have determined that you DEFINITELY need to have another child. The reasons behind this conclusion are as follows:
1) It will extend the life of the blog.
2) It will be great to see how you view child raising the second time around. (I'm not doing this myself as my wife has a tendency to kick out two kids at a time.)
3) Lilly need a sibling. I don't know why, I she just does.
4) Later in life, 4 is the ideal number for a trombone quartet.
So go ahead and have another child. Remember: Your life you won't be wreckin' if have another Grekin!
Michael Canfield says:
The is excellent, and I'd love to see more. You'd think that someone would have read these books before they were published. A monkey, smoking a pipe???
I can't tell you how shocked I was while reading that first book the first time. ("After a good meal and a good pipe … " – wait a minute. What the hell?)
Courtney Denney says:
Great entry! I do the same thing reading books to Theo. Try reading Beatrix Potter stories for the other extreme. Their morals are, essentially: If you misbehave, such as fail to wear your raincoat in bad weather, YOU WILL DIE.
Love your idea of writing a book in which Curious George gets what's coming to him, and I would love to read it! A friend of mine just wrote a book called _The Taking Tree,_ of course spoofing _The Giving Tree_. It's a bit dark in the end, but quite funny. My favorite lines: "The boy loved the tree. And the tree knew it."
Love your writing.
Trudi Feenstra says:
Just wait 'til you get to Junie B. Jones (1st and 2nd grade reading)… these books should be outlawed. They give kids *terrible ideas*… my first encounter was Junie hiding under a desk in kindergarten and they had to call the police to find her. Not exactly the images I want to give my young children, they come up with enough bad ideas on their own without emulating a character in a book.
Ooh – thanks for warning me! I've had no experience with the Junie B. Jones books, so I'll definitely be on the lookout.
Thanks for reading and for the tip!
Thanks for the feedback, Courtney!
We have a set of Beatrix Potter stories, but I hadn't ventured into them. Maybe I'll just leave those on the shelf, or perhaps we can just look at the pretty illustrations… 🙂


Magazine 65 features three design stories using our popular spring summer yarns. Modern Heritage, 12 designs taking inspiration from heritage knitting techniques of the British Isles and Ireland.
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During a speech in 1942, Adolf Hitler said, "The most foolish mistake we could possibly make would be to allow the subject races to posses arms." In other words, Jews and others whom Hitler wanted to wipe from the earth first had to be separated from their guns.
This was not new position for Hitler. Rather, it was something he had put into practice from the moment he came into office.
When Adolf Hitler came to power in 1933, he seized gun registration information that had been compiled by Germany's previous government under the premise that the information would not fall into the wrong hands or be used for an all-out confiscation scheme. But as gun scholar Stephen P. Halbrook makes clear in his new book, Gun Control in the Third Reich: Disarming the Jews and "Enemies of the State," Hitler's hands were the wrong hands. And Hitler and the Nazi movement used all the information they could find–as well as the threat of brute force and death–to force Jews and others to hand over their firearms.
Gun Control in the Third Reich covers gun control under Hitler but, perhaps more importantly, it covers the methodology of that gun control in great detail. In so doing, Halbrook helps the reader sees how registration led to confiscation, and how confiscation turned citizens into subjects and eventually into victims–victims of their own government and/or of an invading force.
Halbrook also shows how propaganda was used to further gun control and how terms like "assault rifle" did not exist until Hitler created them.
1968 was only 23 years after World War II ended and there was some back and forth in Congress on what the Nazis had done. The Library of Congress did a study to see if it could turn up information on any Nazi registration scheme but it could not–the dirty little secret is that it was not intended to. Gun control was going to happen.
Halbrook talked about the implementation of the instant background check under Bill Clinton and how Janet Reno wanted to keep background check records for six months. It was out of fear that those records could be used for devious ends–perhaps a registration and confiscation scheme–that Halbrook litigated a case against Reno's plan and then Congress intervened with a law that required the destruction of said records after 24 hours.
According to Halbrook, this is the fight that has to be fought–the fight to keep such information from being gathered in the first place.
Follow AWR Hawkins on Twitter @AWRHawkin.


The seaside township of Bicheno is conveniently nestled on the east coast of Tasmania and has picturesque white sandy beaches for you to enjoy.
Popular things to do in Bicheno include fishing, scuba diving, golf, sea kayaking, visiting local wineries and glass bottom boat tours. There are many walking tours in the Diamond Island Nature reserve where you can see local wildlife and the famous blowhole.
When you choose to holiday in Bicheno you can choose from a range of hotels, motels and apartment accommodation that is perfect for everyone needs.
Best Western Beachfront at Bicheno Bicheno 3.5 star budget hotel motel accommodation located opposite Bicheno Beach with breathtaking views of Waubs Bay!!


The GARDENA Master Unit 1000 is the starter component of the GARDENA Micro-Drip-System. The Master Unit reduces initial pressure to an operating pressure of approx. 1.5 bar for optimal operation of the connected Drip Heads and Nozzles. The water flow is approx. 1,000 litres per hour. The Master Unit also filters the water. The Master Unit is equipped ready-to-connect. Thanks to the patented "Quick & Easy" connection technology, installation of the Master Unit is especially easy.


John R. Kopp is a 2008 graduate of the John Marshall Law School in Chicago, Illinois. He attended Marquette University in Milwaukee, Wisconsin, where he earned a degree in communications. He grew up in Aurora, Illinois, and attended Marmion Academy.
Mr. Kopp evaluated the strengths and weaknesses of each case he prosecuted for the Cook County State's Attorney Office. When he encountered a weakness, he placed himself in the position of the defense lawyer so that he could best prepare himself for arguing the prosecution's case.
Many of his cases involved preparing police officers for their testimony at trial. Through that process in hundreds of trials, Mr. Kopp gained a thorough grasp of police investigations, warrant executions and arrest procedures. With that knowledge, he can attack the shortfalls and errors the police made in their case against you.
Mr. Kopp is an experienced litigator who is ready to defend your rights.


text:
Q: Why stack grows by 16 bytes in this disassembly, when I only have one 4 byte local variable? I'm having trouble understanding why the compiler chose to offset the stack space in the way it did with the code I wrote.
I was toying with Godbolt's Compiler Explorer in order to study the C calling convention, when I came up with a simple code that puzzled me by its choices.
The code is found in this link. I selected GCC 8.2 x86-64, but am targetting x86 processors and this is important. Bellow is the transcription of the C code and the generated assembly reported by the Compiler Explorer.
// C code
int testing(char a, int b, char c) {
    return 42;
}

int main() {
    int x = testing('0', 0, '7');

    return 0;
}

; Generated assembly
testing(char, int, char):
        push    ebp
        mov     ebp, esp
        sub     esp, 8
        mov     edx, DWORD PTR [ebp+8]
        mov     eax, DWORD PTR [ebp+16]
        mov     BYTE PTR [ebp-4], dl
        mov     BYTE PTR [ebp-8], al
        mov     eax, 42
        leave
        ret
main:
        push    ebp
        mov     ebp, esp
        sub     esp, 16
        push    55
        push    0
        push    48
        call    testing(char, int, char)
        add     esp, 12
        mov     DWORD PTR [ebp-4], eax
        mov     eax, 0
        leave
        ret

Looking at the assembly column from now on, as I understood, line 15 is responsible for reserving space in the stack for the local variables. The problem is that I have only one local int and the offset was by 16 bytes instead of 4. This feels like wasted space.
Is this somewhat related to word alignment? But even if it is, if the sizes of the general purpose registers are 4 bytes, shouldn't this alignment be with regards to 4 bytes?
One other strange thing I see is with respect to the placement of the local chars of the testing function. They seem to be taking 4 bytes each in the stack, as seen in lines 7-8, but only the lower bytes are manipulated. Why not use only 1 byte each?
These choices are probably well intended, and I would really like to understand their purpose (or whether there is no purpose). Or maybe I'm just confused and didn't quite get it.

A: So, by the comments, I could figure out that the stack growth issue is due to the i386 SystemV ABI requirements, as stated by @PeterCordes.
The reason why the chars are word aligned may be due to GCC's default behavior to improve speed, as maybe inferenced from @Ped7g's comment. Although not definite, this is a good enough answer for me.

A: It's common today to acquire stack space in multiples of this size, for several reasons:

*

*cache lines favor this behaviour by maintaining the whole data in the cache.

*space for temporaries is preallocated, avoiding push and pop instructions to be used in case some temporary storage is needed out of the cpu.

*individual push and pop instructions degrade pipeline execution, by requiring data to be updated before next instruction is executed.  This decouples the data dependencies between consecutive instructions and allow them to run faster.

For this reasons, actual compilers specify ABIs to be designed in this way.

redpajama_set_name: RedPajamaStackExchange


Celebrate one of chemistry-teacher-turned-kingpin Walter White's very first cooks with this collectible minifigure. Featuring his tighty whities, classic green shirt, and revolver. Material: Plastic Certification: CE Brand: Kopf Warning: Small parts, choking hazard for young children.
Celebrate one of chemistry-teacher-turned-kingpin Walter White's very first cooks with this collectible minifigure. Featuring his tighty whities, classic green shirt, and revolver.


One of Beada Beada's more socially gregarious resident denizens has compiled a pile o' rhyme, addressing several woefully nefarious world conditions and finding good minds in new friends online.
All original publishing royalties forwarded to various charities in attendance at Tsu, the social media platform that would value what we'd do.
One swoop-sweep and you'll reap, from this tale of unique.
These infused, worldly views, ranging lengthy to terse.
Budding truths stemmed in hues of a palette diverse..
Cutting loose all fish juice, splashed in rhymes unrehearsed.
^^^^^ Five Fins Up For Finesse in Fishosophy. Encore, por favor, as in, "Let there be more!"
English is my third language- to tackle it- no pun intended fishman- is a commitment. So I sit in my dry place- lowest precipitation in the states outside Death Valley- and study water, bubbles and fish antics.
Reward is always a laugh.
... Followed by a space, a time of no- thought.
Then... Much later a head shake! How does he think of that??
Great as a gift too. I bought one for me and one for my cowboy friend to take him from the dusty planes into the cool refreshing pond.
Beada Treasure Hunt 3x4inch Ziplock 6 Oz!
Beada Treasure Hunt Jr. 2x3inch Ziplock 60+ Grams!


I will keep this blog lively for the Archive of all the posts going back to 2008. Whether or not she's dashing to drop her daughter, Vivienne, off in school nonetheless in her pajamas, or hustling to get dinner on the desk, Kristen is as approachable because it will get, and we were thrilled to search out just how receptive she was to divulging her secrets and techniques on parenting, constructing a creative profession and designing a house to be pleased with.
Director Cody Estle uses the smallness of The Artistic Home's area to nice advantage throughout Christopher Hampton's translation of The Seagull." There will be no litter on the stage, so Mary 'Dowd's props are saved to what's fairly necessary: Treplev's manuscripts, Arkadina's umbrella, alcohol for the numerous who imbibe.
Being an actual estate dealer in addition to vendor of manufactured houses and modular houses allows Claude Rancourt to help his consumer by means of every step of buying process, from qualifying for the loan, to specifying the proper home, to finding and securing real estate, to transferring and arrange.
Ya'll understand how Fb is; any time somebody has puppies or kittens, they submit it looking for them new properties. Stop by often to maintain up with the entire occasions, exhibits and classes at the gallery. The pal who held me and hugged me when a stranger handed me $200 and walked away earlier than I may even say thank you, who shares my bizarre inventive bent, who was there the evening of my dad's visitation, and who loves clocks.
Inventive House Enchancment has over 20 years expertise ( 14 years building homes in New York, USA) offering complete constructing companies to clients through Hampshire and London space. I go decide up Vivienne from college at 1 p.m. We've got lunch and playtime together, after which she naps and I get back to work.


text:
/ TV
Star Wars Episode IX
Game of Thrones Star Sophie Turners Take Down of Social Media Influencers Goes Viral
By Patrick Cavanaugh - October 10, 2019 12:19 pm EDT
And that's the tea! ? ? #SophieTurner shades influencer culture in this hilarious satire, and we love her for it! ?? #influencerlife
(?: @SophieT ) pic.twitter.com/fpPH571iro
— Entertainment City (@EntCity) October 8, 2019
In a recent video shared to Instagram, Game of Thrones star Sophie Turner went viral as she imitated social media influencers by calling them out for being irresponsible with the brands they're pushing on their followers. Turner wasn't calling out any influencer in particular, merely mocking the entire concept of sharing messages that are paid sponsorships, especially when those impressionable followers could be harmed by the product that is being promoted. In the months since the final episode of Game of Thrones aired, Turner has been quick to take to social media to call out those engaging in irresponsible or disrespectful discourse on a variety of topics.
"Hey, you guys, just kind of going for my influencer look today," Turner shared in the video while using an American accent. "Today I just wanted to promote this new powdered stuff that you put in your tea. And basically it makes you shit your brains out and is totally really, really bad for me to be promoting to young women and young people everywhere, but I don't really give a fuck because I'm getting paid money for it."
She concluded, "Influencer life."
This is only the latest example of Turner hilariously making fun of viral social media trends, as she contributed to the "Bottle Cap Challenge," in which people attempted to remove the cap of a bottle using a martial arts maneuver, only for her actual submission to be a video of her using her hands to open a bottle and stare into the camera to say, "Stop this. Now."
Following the amount of negativity the actress was subjected to earlier this year when fans voiced their criticisms of the final season of Game of Thrones, Turner clearly had to grow some thick skin to suffer through the harassment. Some fans went so far as to start a petition demanding the final season get a remake, which Turner called out as being disrespectful.
"All of these petitions and things like that — I think it's disrespectful to the crew, and the writers, and the filmmakers who have worked tirelessly over 10 years, and for 11 months shooting the last season," Turner shared with The New York Times earlier this year. "Like 50-something night shoots. So many people worked so, so hard on it, and for people to just rubbish it because it's not what they want to see is just disrespectful."
There are currently multiple Game of Thrones spinoff series being developed, though Turner isn't attached to any of them.
What do you think of the actress' video? Let us know in the comments below!
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