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\begin{document}

\title{Phase equilibria in associating rodlike and flexible chains}

\date{\today}

\author{R. Stepanyan$^{\dagger}$,
        A. Subbotin$^{\dagger ,\sharp}$, 
        O. Ikkala$^{\ddagger}$, 
        G. ten Brinke$^{\dagger}$}

\address{$^{\dagger }$
Department of Polymer Science and Material Science Center,\\
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands;}

\address{$^{\sharp }$
Institute of Petrochemical Synthesis, Russian Academy of\\
Sciences, Moscow 119991, Russia;}

\address{$^{\ddagger }$
Department of Engineering Physics and Mathematics, \\
Helsinki University of Technology, P.O. Box 2200,\\
FIN-02015 HUT, Espoo, Finland}

\date{\today}

\maketitle

\begin{abstract}
Abstract goes here
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\section{Introduction}

Introduction......

\cite{MB1091,BF525,bookdeGennesScalingConcepts,Leibler}.
%-----------------------------------------------------------------

\section{The model and the free energy of the reference system}

Let us consider a melt consisting of rigid rods of length $L$ and diameter 
$d$ and flexible coils consisting of $N$ beads of volume $\nu$ and statistical
segment of length $a$. The coil size is $R_c = a \sqrt{N}$. We will assume that
each rod contains $M$ associating groups (an average distance between two
succesive groups is $b=L/M \ll R_c$) which can form bonds with the
associating end of the coil (FIGURE). It is assumed that each coil has only one
associating end. The energy of association between rod and coil equals
to $-\epsilon $. The concentration of rods in the melt is $c$ and their volume
fraction is $f=(\pi /4)Ld^2c$.

The interactions between rods and coils can be introduced in the
following way. It is well known that rods and polymer coils in the
molten state are practically incompartible and separate on the nematic phase
consisting of rods and isotropic phase consisting of the flexible 
polymers \cite{Flory:MML:11:1138,AbeBallauff}.
Let us consider the interface between the nematic and isotropic phases
(FIGURE fig.1) 
which is assumed to be sharp so that the polymer segments can not
penetrate into the nematic phase, and introduce the interfacial tension 
$\gamma$ corresponding to planar orientation of rods at the interface 
($k_B \equiv 1$)
%
\be{eq0}
\gamma =(w+sT)/d^2 
\ee
%
where $w$ is the energetic part of the surface energy and $s$ is the
entropic part 
(here $T$ is temperature, ??we will also assume that $s \sim 1$??). 
According to the defenition \reff{eq0} if a rod penetrates into the polymer
melt its  energy loss approximatly equals
%
\be{eq01}
\mu _r\simeq 2Ld\gamma =\frac{2L}d\left( w+sT\right) 
\ee
The free energy of the isotropic phase with small amount of rigid rods
therefore is given by

\be{eq02}
{\cal F}_I^{*} = 
T V c 
\ln \left( \frac{f}{e} \right) + 
T V \frac{1-f}{N\nu}
\ln \left( \frac{1-f}{e} \right) +
V c \frac{2L}{d} \left( w+sT \right)  
\ee
%
Here we omitted interaction between the rods. $V$ is the volume of the
system. In \reff{eq02} the first two terms imply the translational
energy of the rods and coils correspondingly and the last term is the energy of
rods.

The coils can also penetrate into the nematic phase where they become
stretched. In order to write the free energy of the nematic phase with small
amount of coils we introduce a chemical potential of the coil in the
nematic phase $\mu _c$ which includes both energetic and entropic 
parts and  limits to infinity,

\be{eq05}
\mu _c/T\rightarrow \infty 
\ee
for arbitrary $T$. As we will see below it means that the coils
practically do not penetrate in the nematic phase.

The free energy of the nematic phase contains also a term connected with
orientational ordering of rods. The last one can be estimated as
\cite{KhokhlovTBOA,SemenovKhokhlov} 
$T \ln ( 4 \pi /\Omega )$, 
where $\Omega$ is the characteristic fluctuation angle, 
$\Omega \simeq 2\pi (d/L)^2$. Thus the free energy is given by
%
\be{eq03}
{\cal F}_N^{*}=
T V c \ln \left( \frac{f}{e} \right) +
T V \frac{1-f}{N\nu}
\ln \left(  \frac{1-f}e \right) +
2 T V c \ln \left( \frac{L}{d} \right) +
V \frac{1-f}{N\nu} \mu _c
\ee
The phase equilibrium between the nematic and isotropic phases can be found
in a usuall way by equating the chemical potentials and osmotic pressures in
both phases.
%
\begin{eqnarray}
\mu_I^{*}  & = & \mu _N^{*}; \quad 
\mu _{I,N}^{*} = \frac{1}{V} \frac{\dd {\cal F}_{I,N}^{*}}{\dd c}  
  \nonumber\\
%
P_I &=&P_N; \quad 
P_{I,N}=\frac{1}{V}
\left( 
     c \, \frac{\dd {\cal F}_{I,N}^{*}}{\dd c} - {\cal F}_{I,N}^{*}
\right)  
\lbl{eq04} 
\end{eqnarray}
Considering limit \reff{eq05}, solution of these equations is given by
%
\be{eq06}
f_N \simeq 1,\quad 
f_I \simeq 
    \left( \frac{L}{d} \right) ^2
           \exp \left( -\frac{2L}{d}\left( \frac wT+s \right) 
    \right) \ll 1  
\ee

\section{Nematic-isotropic liquid phase coexistence: effect of association}
%
In this section we study the influence of association between rods and 
coils on the macrophase separation described above.
We start from the free energy of association between
rods and coils, ${\cal F}_{bond}$, assuming that they are
ideal (without excluded volume). Let us introduce the probability of bond 
$p$. The total number of bonds in the system is $VMcp$ and 
equals to the number of associated coils. 
Therefore the number of free coils in the system  is 
$(V/N\nu)(1-f-f\kappa pN)$, where $\kappa \equiv \nu/(\pi b d^2/4)$. The free
energy of bonds can be written through the partition function $Z_{bond}$ as
\cite{SemenovRubinstein1,Erukhimovich:Gel}
%
\be{eq3}
{\cal F}_{bond}=-T\ln Z_{bond} 
\ee
where
%
\be{eq4}
Z_{bond} = 
 P_{comb} 
 \left( \frac{v_b}V \right)^{V M c p}
 \exp \left( \frac{\epsilon \, V M c p}{T} \right)  
\ee
and $P_{comb}$ is the number of different ways to bond  rods and coils
for a fixed probability of bond $p$; $v_b$ is a bond volume. If we denote
the number of rods in the system as ${\cal N}_r=Vc$, and the number of coils
as ${\cal N}_c=V(1-f)/N\nu$ then the number of ways to choose ${\cal N}_rMp$
coils for bonds formation is a binomial coefficient
%
\be{eq5}
C_{{\cal N}_c}^{{\cal N}_rMp}=\frac{{\cal N}_c!}{({\cal N}_rMp)!({\cal N}_c-%
{\cal N}_rMp)!} 
\ee
%
On the other hand there are
%
\be{eq6}
\frac{({\cal N}_rM)!}{({\cal N}_rM(1-p))!}
\ee
different ways to select ${\cal N}_rMp$ bonds from ${\cal N}_rM$
associating groups. Therefore

\be{eq7}
P_{comb} = C_{{\cal N}_c}^{{\cal N}_rMp}
           \frac{ ({\cal N}_rM)! }{ ({\cal N}_rM(1-p))! } 
\ee
and the free energy of bonds is given by
%
\begin{eqnarray}
{\cal F}_{bond} & = & 
VMcp 
\left[ 
  T \ln \left( \frac{N\nu}{v_b} \right) - \epsilon
\right] +
TVcM
\left[ 
  p\ln p + (1-p) \ln (1-p)
\right] \nonumber\\
%
 & & +
TV \frac{\left( 1-f-f\kappa Np\right) }{N\nu}
   \ln \left( \frac{1-f-f\kappa Np}{e} \right) -
TV \frac{(1-f)}{N\nu}
   \ln \left( \frac{1-f}{e} \right)  
\lbl{eq8}
\end{eqnarray}

Thus the free energy of the isotropic phase can be presented as the following
%
\be{eq9}
{\cal F}_I = {\cal F}_I^{*} + {\cal F}_{bond} + {\cal F}_{el}  
\ee
%
where ${\cal F}_{el}$ is the elastic free energy of the side chains 
of the hairy
rod when the density of association is high enough. We approximate it
by \cite{3dFlex,2sorts}

\be{eq10}
{ \cal F}_{el}=
\left[ 
 \begin{array}{cl}
    TVc\frac{3\kappa d^2}{32a^2}Mp^2\ln \left( \kappa Np\right) ,\quad &
        p>\frac{1}{\kappa N} \\ 
    0, \quad &
        \textrm{otherwise}
 \end{array}
\right.  
\ee
Hence the final expression for the free energy of the isotropic phase is
given by (per volume of one rod $(\pi /4)Ld^2)$
%
\begin{eqnarray}
\frac{F_I(f,p)}T &=&
  f\frac{2L}{d} \left( \frac wT+s \right) 
  +Mfp\left[ \ln \left( \frac{N\nu}{v_b}\right) -\frac \epsilon T \right] 
  +fM\left[ p\ln p+(1-p)\ln (1-p)\right]    \nonumber\\
&&
  +f\ln \left( \frac fe \right) 
  +M \frac{\left( 1-f-f \kappa Np \right) }{N\kappa }
     \ln \left( \frac{1-f-f\kappa Np}{e} \right)  \nonumber\\
&&
  +f\frac{3\kappa d^2}{32a^2} Mp^2 
     \ln \left( \kappa Np \right) H\left( p-\frac 1{\kappa N}\right)  
\lbl{eq11}
\end{eqnarray}
%
where
%
$$
H(x)=
\left[ 
 \begin{array}{cl}
   1,\quad & x \geq 0 \\ 
   0,\quad & x <    0
 \end{array}
\right. 
$$
is the Heavyside's function. 
Similarly, the free energy of the nematic phase is
%
\begin{eqnarray}
\frac{F_N(f,p)}T &=&
  2 f \ln \left( \frac Ld \right) 
  +M\frac{1-f}{N\kappa }\frac{\mu _c}T
  +Mfp\left[ \ln \left( \frac{N\nu}{v_b}\right) 
  -\frac \epsilon T\right]
  +fM\left[ p\ln p+(1-p)\ln (1-p)\right]    \nonumber\\
&&
  +f\ln \left( \frac fe\right) 
  +M\frac{\left( 1-f-f\kappa Np\right) }{ N\kappa }
    \ln \left( \frac{1-f-f\kappa Np}e\right)  
\lbl{eq12}
\end{eqnarray}
%
%
The probability of bonding in both phases can be found from the minimization
of the corresponding free energies
%
\be{eq13}
\frac{\dd F_I}{\dd p}=0;
\quad 
\frac{\dd F_N}{\dd p}=0
\ee
%
and is given by ($N^* \equiv N \nu / v_b$)
%
\be{eq14}
p= \frac{1}{2\kappa Nf}
\left[ 
  1-f+\kappa Nf-\epsilon /(TN^{*})-
  \sqrt{
    \left(1-f+\kappa Nf-\epsilon /(TN^{*}) \right) ^2
    -4\kappa Nf(1-f)
  }
\right]
\ee
for the nematic phase and for the isotropic phase when $p<\frac 1{\kappa N}$. 
%Here $N^{*}\equiv N\nu/v_b.$ 
For $p>\frac 1{\kappa N}$ the probability of
bonding in the isotropic phase obeys
%
\be{eq15}
 \ln 
  \left[ 
     \frac{ pN^{*}e^{-\epsilon /T} }
          { \left( 1-p\right) \left(1-f_I-f_I\kappa Np\right) }
  \right] 
 +\frac{3\kappa d^2p}{16a^2}\ln \left( \kappa Npe\right) 
 = 0  
\ee
%
and for a small volume fraction of rods, $f_I \ll 1$, is approximately given by
%
\be{eq24}
p \simeq \frac 1{ 1 + N^{*} e^{-\epsilon^{*}/T}},
\quad 
\epsilon ^{*} = \epsilon -
                \frac{3\kappa d^2T}{32a^2} \,
                 \frac{1}{1+N^{*}e^{-\epsilon /T}}
                 \ln \left( \frac{\kappa N}{1+N^{*} e^{-\epsilon /T}} \right)
\ee
%
Phase equilibrium between the isotropic and nematic phases can be found in a
standard way from the equilibrium equations
%
\begin{eqnarray}
\frac{\dd F_I}{\dd f_I}        &=& \frac{\dd F_N}{\dd f_N}
\nonumber \\
f_I\frac{\dd F_I}{\dd f_I}-F_I &=& f_N\frac{\dd F_N}{\dd f_N}-F_N 
\lbl{eq16} 
\end{eqnarray}
using eqs.~\ref{eq11},\ref{eq12} together with \reff{eq14} and \reff{eq24}. 
When the probability of bonding in the
isotropic phase  $p_I<\frac 1{\kappa N}$ 
(or equivalently $\frac{\epsilon}{T} < \ln \frac{\nu}{\kappa v_b}$),
expression \reff{eq14} can be used giving the volume fraction of rods
%
\begin{eqnarray}
f_N  & \simeq & 1,  
   \nonumber\\
f_I  & \simeq & 
 \left( \frac Ld \right) ^2
 \exp 
  \left( 
     -\frac{2L}{d}
     \left( \frac wT+s\right) 
     +\frac M{1+N^{*}e^{-\epsilon /T}}
      \left( \frac \epsilon T-\ln N^{*}\right) 
  \right) \ll 1  
\lbl{eq17}
\end{eqnarray}
%
However, if
$p_I>\frac 1{\kappa N}$ 
(or  $\frac \epsilon T>\ln  \frac \nu{\kappa v_b}$), 
the volume fraction of rods in the nematic phase 
is still close to the unity whereas $f_I$ obeys the equation
%
\be{eq18}
\ln f_I 
- Mp_I \ln \left( 1-f_I-f_I\kappa Np_I \right) 
\simeq 
2 \ln \left( \frac L d\right) 
+ \frac M{N\kappa} - \frac{2Ls}{d}
-Mp_I\ln N^{*}
+\frac{1}{T} \left( Mp_I\epsilon -\frac{2Lw}d \right)  
\ee
where $p_I$ has to be determined from \reff{eq15}. 
Obviously, for $T \to 0$ $p_I \to 1$ and therefore the last term 
in eq.\ref{eq17} becomes dominant. Depending on its sign two 
characteristical assymptotics can be distinguished
%
\begin{eqnarray}
f_I  \to  0    \qquad\qquad\textrm{if }\quad  M\epsilon <\frac{2Lw}d  
\nonumber\\
f_I  \to  \frac 1{1+N\kappa } \quad\textrm{if }\quad M\epsilon >\frac{2Lw}d  
\label{eq19}
\end{eqnarray}
Thus for $\epsilon /w>2b/d$ rods and coils become partially compartible.
This fact has a clear physical meaning. Negative sign of 
$-\epsilon + \frac{2Lw}{Md}$ corresponds to the negative ``total'' energy
($\epsilon$-part plus $\gamma$-part) 
due to attaching of a coil to a rod, i.e. making it favorable to keep 
\emph{all} coils bonded (for $T\to 0$, of course).
Further on we consider only the case $\epsilon /w>2b/d$, 
where a region of compatibility of rods and coils exists.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Phase equilibria between nematic, isotropic liquid and microphases}

There are two mechanisms of attraction between hairy rods, namely due to
incompartibility of the rods and coils and due to nonhomogeneous
distribution of the free polymer coils which is created by the hairy rods.
These mechanisms ultimately result in formation hexagonal and lamellar
structures in the blend. Moreover we can separate two different hexagonal
phases. In one of the phases (we call it H1) the mechanism connected with
nonhomogeneous distribution of the free polymers is dominant and the
''cylinders'' contain only one rod per unit cell $(Q=1)$. In the second
phase (H2) the surface term becomes important so that rods attract each
other and the cylinders contain $Q>1$ rods per unit cell (fig.2FIGURE). With
decreasing temperature the cylinders first adopts elipsoidal form and
finally transform to the lamellar phase.

\subsection{Separation of the hexagonal phase H1}

Let us start with calculation of the interaction energy between the
cylinders in the hexoganal phases (H1, H2). It is connected with
nonhomogeneous distribution of the free polymer coils and is given by (per
cylinder of unit length)

\be{eq20}
U_H(Q)=
\frac{N\nu(Qp)^2}{2b}
\left[ 
  \frac 2{\sqrt{3} \, \ell ^2}
  \sum_{\{ \vb \}}
    \frac{h^2(\frac{a^2N{\vk}^2}6)}{g(\frac{a^2N{\vk}^2}6)}
  -\frac 1{4\pi ^2}
   \int d{\vk}\frac{h^2(\frac{a^2N{\vk}^2}6)}{g(\frac{a^2N{\vk}^2}6)}
\right] 
\ee
where $\ell $ is the period of the structure, $\{ \vb \}$ are the vectors
of the reciprocal lattice,
$$h(u) =\frac 1u\left( 1-e^{-u}\right) $$
$$g(u) =\frac 2{u^2}\left( u-1+e^{-u}\right)$$

After calculation of the sum and integral in eq.\ref{eq20} we find the
interaction energy per volume $(\pi /4)Ld^2$

\be{eq21}
U_H(Q)=-\frac 3{32}\frac{\kappa MQp^2fd^2}{a^2N}
       \left[ 
          3.457
          +\ln \left( \frac{a^2Nf}{Qd^2}\right) 
       \right] 
\ee
Thus the free energy of H1 phase is given by
%
\begin{eqnarray}
\frac{F_{H1}}T &=&
  f\frac{2L}{d} \left( \frac wT+s\right) 
- Mfp\left[ \frac \epsilon T - \ln N^{*} \right] 
+ fM\left[ p\ln p+(1-p)\ln (1-p) \right] 
+ 2f\ln \left( \frac Ld \right)  
\nonumber \\
&&
+ M\frac{\left( 1-f-f\kappa Np\right) }{N\kappa }
  \ln \left( \frac{1-f-f\kappa Np}e\right) 
+ f\frac{3\kappa d^2}{32a^2}Mp^2\ln \left( \kappa Np\right)   
\nonumber \\
&&
- \frac{3}{32} \frac{\kappa Mp^2fd^2}{a^2N}
   \left[ 3.457+\ln \left( \frac{a^2Nf}{d^2}\right) \right]  
\lbl{eq22}
\end{eqnarray}
Here we approximated the loss of the orientational energy of rod by the term 
$2Tf\ln \left( \frac Ld\right) $, and omitted the loss of it translational
entropy because it is relatively small. Phase equilibrium between isotropic
phase and H1 phase can be found from the equilibrium equations
%
\begin{eqnarray}
 \frac{\dd F_I}{\dd f_I} =\frac{\dd F_{H1}}{\dd f_{H1}},
 &\quad& 
 \frac{\dd F_I}{\dd p_I}=\frac{\dd F_{H1}}{\dd p_{H1}}=0  
\nonumber\\
 f_I\frac{\dd F_I}{\dd f_I}-F_I 
 &=&
 f_{H1}\frac{\dd F_{H1}}{\dd f_{H1}}-F_{H1}  
\lbl{eq23}
\end{eqnarray}
and the probability of bonding and the binodal lines are
%
\begin{eqnarray}
 & p_1          \simeq & p_{H1}\simeq 1, 
\nonumber \\
 & f_{H1}^{(1)} \simeq & \frac 3{16}\frac{d^2}{a^2 N},  
\nonumber \\
 & f_I          \simeq & 
    \left( \frac Ld\right) ^2
    \exp \left( -\frac 3{16}\frac{d^2p^2\kappa M}{a^2}\right) \simeq 0  
\lbl{eq26}
\end{eqnarray}

Similarly the phase equilibrium between the nematic and H1 phases follow
from equations
%
\begin{eqnarray}
 \frac{\dd F_N}{\dd f_N} =\frac{\dd F_{H1}}{\dd f_{H1}},
 &\quad& 
 \frac{\dd F_N}{\dd p_N}=\frac{\dd F_{H1}}{\dd p_{H1}}=0  
\nonumber\\
 f_N\frac{\dd F_N}{\dd f_N}-F_N 
 &=&
 f_{H1}\frac{\dd F_{H1}}{\dd f_{H1}}-F_{H1}  
\lbl{eq25}
\end{eqnarray}
%
and solution is given by
%
\begin{eqnarray}
& p_N \simeq          &0,  \quad p_{H1}\simeq 1,   \nonumber \\
& f_N \simeq          &1,                          \nonumber\\
& f_{H1}^{(2)} \simeq &\frac 1{1+\kappa N}
  \left[ 
    1-\exp 
      \left( 
        -\frac{\epsilon}{T}
        +\frac{2bw}{Td}
        +\frac{2bs}d+\ln N^{*}
        +\frac{3\kappa d^2}{32a^2}\ln \left( \kappa N\right) 
      \right) 
  \right]  
\lbl{eq251}
\end{eqnarray}
The critical temperature $(\epsilon /T)_c$ can be obtained from the
intersection of the curves $f_{H1}^{(1)}$ and $f_{H1}^{(2)}$, and obeys the
following equation
%
\be{eq27}
(\epsilon /T)_c =
  \frac{1}{1-\frac{2bw}{\epsilon d}}
  \left( \frac{2bs}d+\ln
    N^{*}+\frac{3\kappa d^2}{32a^2} \ln \left( \kappa N \right) 
  \right)
\ee
where the probability of bonding $p_c\simeq 1.$ Thus the hexagonal H1 phase
is stable for $f_{H1}^{(1)}<f<f_{H1}^{(2)}$; for $f_I<f<f_{H1}^{(1)}$ the
system separates on the isotropic and H1 phase and for $f_{H1}^{(1)}<f<f_N$
it separates on the H1 and nematic phase.

\subsection{Separation of the hexagonal phase H2}

Let us follow along the binodal line $f_{H1}^{(1)}(T)$ decreasing the
temperature. At some temperature H1 phase becomes unstable with respect to
separation of the isotropic phase and the hexagonal H2 phase. The
corresponding triple point can be obtained from the system of equations

\begin{eqnarray}
\frac{\dd F_I}{\dd f_I} 
=\frac{\dd F_{H1}}{\dd f_{H1}}
=\frac{\dd F_{H2}}{\dd f_{H2}} \, ,
\qquad 
\frac{\dd F_I}{\dd p_I}
=\frac{\dd F_{H1}}{\dd p_{H1}}
=\frac{\dd F_{H2}}{\dd p_{H2}}
=0  
\nonumber \\
f_I\frac{\dd F_I}{\dd f_I}-F_I 
=f_{H1}\frac{\dd F_{H1}}{\dd f_{H1}}-F_{H1}
=f_{H2}\frac{\dd F_{H2}}{\dd f_{H2}}-F_{H2}
\lbl{eq23a} 
\end{eqnarray}
%
where the free energy of the H2 phase for $Q<\sqrt{N}$ is given by
%
\begin{eqnarray}
\frac{F_{H2}}T 
&=&
 f\frac Ld\left( \frac wT+s\right) \left( 1+\frac 2Q\right)
 +Mfp\left[ \ln N^{*}-\frac \epsilon T\right] 
 +fM\left[ p\ln p+(1-p)\ln (1-p)\right]   
\nonumber \\
&&
 +2f\ln \left( \frac Ld\right) 
 +M\frac{\left( 1-f-f\kappa Np\right) }{\kappa N}
   \ln \left( \frac{1-f-f\kappa Np}e\right) 
 +f\frac{3d^2\kappa Q}{32a^2}Mp^2\ln \left( \kappa Np\right)   
\nonumber \\
&&
 -\frac 3{32}\frac{\kappa MQp^2fd^2}{a^2N}
   \left[ 3.457+\ln \left( \frac{a^2Nf}{Qd^2}\right) \right]   
\lbl{eq28}
\end{eqnarray}
and the number of rods $Q$ in the cross-section of the cylinder can be
calculated from the minimum condition $\dd F_{H2}/\dd Q=0$,
%
\be{eq28a}
Q \simeq 
\sqrt{
 \frac{64ba^2}{3\kappa p^2d^3 \ln \left( \kappa N \right) }
 \left( \frac wT+s \right) 
} 
\ee
Solution of the eqs.~\ref{eq23a} is given by
%
\begin{eqnarray}
&& p_I \simeq p_{H1} \simeq p_{H2}\simeq 1  \nonumber \\
&& Q_1 \simeq 2+\sqrt{2},
\quad 
f_I\simeq 0,
\quad 
f_{H1}^{(1)} \simeq \frac 3{16} \frac{d^2}{a^2N},
\quad 
f_{H2}^{(1)} \simeq \frac 3{16}\frac{Q_1d^2}{a^2N} 
\lbl{eq28b}
\end{eqnarray}
and the critical temperature is

\be{eq23e}
\frac w{T_{c1}} \simeq -s+\frac{3\kappa d^3 Q_1^2}{64 b a^2}
                       \ln \left( \kappa N \right)  
\ee

Similarly we the binodal line $f_{H1}^{(1)}(T)$ finishes at the triple point
which can be found from the system of equations

\begin{eqnarray}
&&\frac{\dd F_N}{\dd f_N} =
\frac{\dd F_{H1}}{\dd f_{H1}} =
\frac{\dd F_{H2}}{\dd f_{H2}},
\quad 
\frac{\dd F_N}{\dd p_N} =
\frac{\dd F_{H1}}{\dd p_{H1}} =
\frac{\dd F_{H2}}{\dd p_{H2}} =0  
\nonumber \\
&&f_N\frac{\dd F_N}{\dd f_N}-F_N = 
f_{H1}\frac{\dd F_{H1}}{\dd f_{H1}}-F_{H1}=
f_{H2}\frac{\dd F_{H2}}{\dd f_{H2}}-F_{H2}
\lbl{eq23c} 
\end{eqnarray}
%
and is characterized by
%
\begin{eqnarray}
&&
p_N \simeq  0, 
\quad 
p_{H1} \simeq p_{H2} \simeq 1  
\nonumber\\
&&
Q_1^{^{\prime }} \simeq  Q_1\simeq 2+\sqrt{2}, \quad f_N \simeq 1,
\nonumber \\
&&
f_{H1}^{(2)}\simeq \frac 1{1+\kappa N}\left[ 1-\exp \left( -\frac \epsilon {%
T_{c1}}+\frac{2bw}{T_{c1}d}+\frac{2bs}d+\ln N^{*}+\frac{3d^2\kappa }{32a^2}%
\ln \left( \kappa N\right) \right) \right] 
\nonumber\\
&&
f_{H2}^{(2)}\simeq \frac 1{1+\kappa N}\left[ 1-\exp \left( -\frac \epsilon {%
T_{c1}}+\left( \frac{2bw}{T_{c1}d}+\frac{2bs}d\right) \left( 1+\frac 2{Q_1}%
\right) +\ln N^{*}+\frac{3d^2\kappa Q_1}{32a^2}\ln \left( \kappa N\right)
\right) \right] 
%
\lbl{eq23d} 
\end{eqnarray}

In the first approximation the corresponding critical temperature coinside
with the critical temperature \reff{eq23e}. Note, the small difference
between these critical temperatures, which we do not consider here, result
in a small area of phase separation between H1 and H2 phases.

The phase equilibrium between the isotropic and the hexagonal H2 phase can
be found based on the equations
%
\begin{eqnarray}
&&
 \frac{\dd F_I}{\dd f_I} =\frac{\dd F_{H2}}{\dd f_{H2}},
 \quad 
 \frac{\dd F_I}{\dd p_I} = \frac{\dd F_{H2}}{\dd p_{H2}} =0  
\nonumber\\
&&
 f_I\frac{\dd F_I}{\dd f_I}-F_I = f_{H2}\frac{\dd F_{H2}}{\dd f_{H2}}-F_{H2}  
\lbl{eq231}
\end{eqnarray}
and for $1\ll Q<\sqrt{N\text{ }}$ the probability of bonding and the binodal
lines are given by

\begin{eqnarray}
  && p_I \simeq p_{H2}\simeq 1,    
\nonumber\\
  && f_I \simeq 0,\quad  
\nonumber \\
  && f_{H2}^{(1)} \simeq 
     \frac 1{1+\kappa N}
      \left[ 
        1-\exp \left( -\frac 3{16}\frac{Qd^2}{a^2N}\right) 
      \right]  
\lbl{eq28b1}
\end{eqnarray}
%
where $Q$ defined by \reff{eq28a}. Similarly the equilibrium between the
nematic and the hexagonal H2 phase obeys equations
%
\begin{eqnarray}
&&
  \frac{\dd F_N}{\dd f_N} = \frac{\dd F_{H2}}{\dd f_{H2}},
  \quad 
  \frac{\dd F_N}{\dd p_N} = \frac{\dd F_{H2}}{\dd p_{H2}}=0
\nonumber \\
&&
  f_N\frac{\dd F_N}{\dd f_N}-F_N = f_{H2}\frac{\dd F_{H2}}{\dd f_{H2}}-F_{H2}  
\lbl{eq232}
\end{eqnarray}
and the corresponding probabilities and binodals are
%
\begin{eqnarray}
&&
 p_N \simeq 0,\quad p_{H2}\simeq 1,  
\nonumber \\
&&
 f_N \simeq 1, 
\nonumber \\
&&
 f_{H2}^{(2)} \simeq \frac 1{1+\kappa N}
 \left[ 
    1-\exp \left( -\frac{\epsilon}{T}
    +\left( \frac{2bw}{Td}+\frac{2bs}d \right) \left( 1+\frac{2}{Q} \right)
    +\ln N^{*}+\frac{3d^2\kappa Q}{32a^2} \ln \left( \kappa N\right) \right) 
 \right]  
\lbl{eq28b21} 
\end{eqnarray}

With further decreasing temperature the number of rods in the cross-section $%
Q$ becomes larger than $\sqrt{N}$ and the cylinders become elongated in one
direction.

\subsection{Separation of the lamellar phase}

The free energy of the lamellar phase is
%
\begin{eqnarray}
\frac{F_L}T &=&
  f\frac Ld\left( \frac wT+s\right) 
 +Mfp\left[ \ln N^{*}-\frac \epsilon T\right]
 +fM\left[ p\ln p+(1-p)\ln (1-p)\right]   
\nonumber \\
&&
 +2f\ln \left( \frac Ld\right)
 +M\frac{\left( 1-f-f\kappa Np\right) }{\kappa N}
   \ln \left( \frac{2h^{*}}\xi \frac{1-f-f\kappa Np}e\right)
 +f\frac{ 3\pi ^2d^2\kappa ^2}{32a^2}NMp^3  
\nonumber \\
&&
 -0.227f^{*}M \left( \frac{p^2a^2}{\kappa ^2d^2N}\right) ^{1/3}
 -1.312M \frac{(f-f^{*})}{f^{*}}
         \left( \frac{p^2d^2}{\kappa a^2N^2}\right) ^{1/3} 
%
\lbl{eq29}
\end{eqnarray}
where
%
$$
h^{*}=\frac{\pi d}2\left( 1+\kappa Np \right) ;
\quad 
\xi =\frac a{6\pi } \left( \frac{aN}{\kappa pd}\right) ^{1/3} 
$$
%
The phase equilibrium between the isotropic and the lamellar phase can be
found from the equations
%
\begin{eqnarray}
&&
 \frac{\dd F_I}{\dd f_I} =\frac{\dd F_L}{\dd f_L},
 \quad 
 \frac{\dd F_I}{\dd p_I}=\frac{\dd F_L}{\dd p_L}
\nonumber \\
&&
 f_I\frac{\dd F_I}{\dd f_I}-F_I = f_L\frac{\dd F_L}{\dd f_L}-F_L  
\lbl{eq2311}
\end{eqnarray}
and the probability of bonding and the binodals are given by
%
\begin{eqnarray}
&&
  p_I \simeq p_L\simeq 1,   
\nonumber\\
&&
  f_I \simeq 0,  
\nonumber \\
&&
  f_L^{(1)} \simeq \frac{1}{1+\kappa N}
  \left[ 
     1-\frac \xi {2h^{*}}
     \exp \left( -1.312\left( \frac{\kappa ^2d^2N}{a^2}\right) ^{1/3}\right) 
  \right] 
\label{eq28b2}
\end{eqnarray}
Similarly the equilibrium between the nematic and the lamellar phase obeys
equations
%
\begin{eqnarray}
&&
  \frac{\dd F_N}{\dd f_N} = \frac{\dd F_L}{\dd f_L},
  \quad 
  \frac{\dd F_N}{\dd p_N} = \frac{\dd F_L}{\dd p_L}
\nonumber \\
&&
  f_N \frac{\dd F_N}{\dd f_N}-F_N  = f_L\frac{\dd F_L}{\dd f_L}-F_L 
\lbl{eq2321}
\end{eqnarray}
and the corresponding probabilities and binodals are
%
\begin{eqnarray}
&&
  p_N \simeq 0,
  \quad 
  p_L\simeq 1,    
\nonumber\\
&&
  f_N \simeq 1,
  \quad  
\nonumber \\
&&
  f_L^{(2)} \simeq \frac{1}{1+\kappa N}
  \left[ 
      1-\exp \left( -\frac \epsilon T
     +\frac{2bw}{Td}+\frac{2bs}d+\ln N^{*}
     +\frac{3\pi ^2d^2\kappa ^2N}{32a^2}\right) 
  \right] 
\lbl{eq28b3}
\end{eqnarray}

\section{Discussion}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf Literature}


Relevant literature

\cite{SemenovKhokhlov,SemenovRubinstein1,Erukhimovich:Gel}

\cite{AndrikopoulosVlassopoulosVoyiatzis,Benmouna}

\cite{KhalaturKhokhlov1,KhalaturKhokhlov2}

\cite{SemenovNyrkovaKhokhlov,3dFlex}

\cite{Angerman:PhaseAssocDiblock,Dormidontova:PhaseHbondBrush}

Theory:

\cite{Ballauff:CompatHairyRodsCoils,Ballauff:PhaseHairyRodsCoils}

Lattice:

\cite{SemenovBlockHomo,Leibler,bookChaikinLubensky,bookKorn}


Hairy rods: 

Experiment:

\cite{SteuerRehahnBallauff,AdamSpiess,SteuerHorthBallauff}

\cite{GaldaKistnerMartinBallauff,PetekidisVlassopoulosFytas2}

\cite{PetekidisVlassopoulosFytas1}


\begin{references}

\end{references}

\end{document}