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+\documentstyle[aps,preprint]{revtex}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\newcommand{\lbl}[1]{\label{#1}}
+\newcommand{\reff}[1]{eq~\ref{#1}}
+
+
+\newcommand{\be}[1]{ \begin{equation} \lbl{#1} }
+\newcommand{\ee}{\end{equation}}
+
+\newcommand{\bea}[1]{ \begin{eqnarray} \lbl{#1} }
+\newcommand{\eea}{\end{eqnarray}}
+
+\newcommand{\dd}{\partial}
+
+\newcommand{\vk}{\bf k}
+\newcommand{\vb}{\bf b}
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}