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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)}