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\title{Comb copolymer brush with chemically different side chains}
\date{\today}
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\newcommand{\vA}{v_{AA}}
\newcommand{\vB}{v_{BB}}
\newcommand{\vAB}{v_{AB}}
\newcommand{\vx}{v}
\newcommand{\mnu}{\nu}
\newcommand{\khi}{\chi}
\newcommand{\dd}{\partial}
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\begin{document}

\maketitle
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\begin{abstract}
An investigation of side chain microphase separation
within a single comb copolymer molecule containing chemically different
$A$ and $B$ side chains has been carried out. Expressions for the
transition point $\chi_{AB}^*$ in a 
good     ($\chi_{AB}^* \sim N^{-3/8}$),
marginal ($\chi_{AB}^* \sim N^{-1/2}$),
$\theta$ ($\chi_{AB}^* \sim N^{-2/3}$), 
and poor ($\chi_{AB}^* \sim N^{-1}$)
solvent are derived
both by a mean field calculation and by scaling arguments.
Properties of the system below and above the transition point are
described.
Some unusual ``bow-like'' conformations are predicted for a single
molecule in the microphase separated state in a good solvent.
\end{abstract}
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\section{Introduction}

The conformational characteristics of individual comb copolymers with a 
high grafting density of side chains in solution has been addressed in a
series of theoretical papers 
\cite{Birshtein,WangSafran,Fredrickson,Rouault1,3dFlex,3dRigid,Potemkin,2dComb}
to begin with the original work of 
Birshtein et al \cite{Birshtein}. 
Irrespective of the solvent quality, be it a good 
solvent or a $\theta$-solvent, all theories predict a cylindrical 
brush-like structure for sufficiently long side chains. The pertinent 
parameters are the side chain grafting density, the side chain length, 
the intrinsic stiffness of the backbone and the side chains (the 
respective Kuhn segments) and the solvent quality with respect to the 
side chains and the backbone. The conformation is characterized by a 
number of quantities, the persistence length of the comb copolymer brush 
being most important. For sufficiently long side chains the persistence 
length is predicted to exceed the backbone length, thus resulting in a 
characteristic cylindrical "bottle-brush" structure. Subsequent computer 
simulations using a freely jointed hard sphere model essentially 
confirmed this picture 
\cite{Mika4,BrinkeIkkala,Rouault2,Mika1,Mika3,Mika2}.

The experimental investigation of comb copolymers with a high grafting 
density has assumed large proportions after the successful 
polymerization of macromonomers, yielding degrees of polymerization 
significantly exceeding the length of the macromonomer itself, by 
Tsukahara and coworkers 
\cite{Tsukahara2,Tsukahara3,Schmidt,Tsukahara0}.
Besides polymerization of macromonomers 
alternative routes have been developed recently using grafting from a 
macroinitiator prepared by either atom-transfer radical polymerization
\cite{Beers1} 
or by living cationic polymerization 
\cite{Schappacher}. 
Using atom transfer 
radical polymerization molecular brushes with block copolymer side 
chains have been prepared as well.

The experimental characterization of the comb copolymer brush 
conformation in dilute solution is a highly nontrivial issue. It was 
achieved recently by Schmidt and co-workers 
\cite{Borner1,Wintermantel1,Wintermantel2,Wintermantel3,DziezokSheiko,Sheiko,GerleRoos,FischerGerleSchmidt} 
using a combination 
of light scattering experiments and theoretical modeling. Since the high 
grafting density is supposed to lead to a stiff molecular structure, the 
modeling has been based on the Kratky-Porod worm-like chain model.
For high molar mass polymacromonomers based on methacryloyl 
end-functionalized oligo methacrylates ($M_n = 2410~g/mole$) in the good 
solvent THF, the Kuhn statistical segment length, which is twice the 
persistence length, turned out to be $120~nm$. For polymacromonomers 
($M_n = 3624~g/mole$) consisting of polystyrene main and side chains this 
value was $190~nm$ in the good solvent toluene and $120~nm$ in the 
"$\theta$-solvent" cyclohexane.

At the same time one of the most challenging problems in
the polymer physics is a description of a microphase
formation in copolymer systems.
Theoretically, self-organization in block copolymer systems has 
attracted
a lot of attention during the last decades and a fairly complete picture 
has emerged for the
relatively simple diblock copolymers.
\cite{Helfand1,Semenov,Leibler,FredricksonHelfand}
As a consequence, the interest gradually shifts towards more
complicated architectures such as comb or graft copolymers
\cite{Dobrynin1,Cruz1,Foster1,WernerFredrickson,Nap1}.
The discussion of structure formation in comb copolymers using the weak 
segregation limit has been presented in some detail. Compared to diblock 
copolymers the description is only slightly complicated by the fact that 
the single chain correlation functions are more involved. Phase diagrams 
of various comb copolymer systems have been published. Although 
different in details, the general trends are the same as for diblock 
copolymers. Of course, rather than the overall chain length, it is the 
length of the "repeat unit" that determines the order-disorder 
transition temperature as well as the characteristic length scale of the 
ordered structures.

The application of the weak segregation approach, however, requires a 
relatively low
grafting density such that the distance between two
consecutive grafting points along the backbone is at least of the order 
of the Flory radius of the side chains. If the grafting density of comb 
copolymers is very high, the structure in the melt will usually involve 
segregation between individual molecules. Even if the incompatibility 
between backbone and side chains is high, the high grafting density may 
well prevent segregation of several backbones. Furthermore, a high 
grafting density combined with long side chains implies the volume 
fraction of the backbone to be of the order of 0.1 or lower, not 
necessarily the most interesting part of the melt phase diagram.

Still, microphase separation may occur provided chemically different 
side chains are used. In this case unfavorable interaction between the 
side chains may lead to a micro domain
structure within a single molecule. The present paper is devoted to this 
subject.
The main objective is to identify conditions for "microphase separation" 
of side chains of two
different types within a single comb copolymer molecule under different 
solvent conditions.

The paper is organized as follows. The next section describes the
self-consistent field approach to a molecule with a straight
backbone and chemically different side chains. We show the possibility 
of side chain separation
within the molecule and discuss the limits of the theory's 
applicability.
The subsequent section is devoted to possible unusual behavior
of comb copolymer molecules with a flexible backbone and
microphase separated side chains. Then all results are summarized and
discussed in the last section.

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\section{Useful literature (not from the article)}
Very interesting and useful books on this subject (polymers) are 
\cite{bookdeGennesScalingConcepts,GrosKhokh,bookCloizeaux} or more specific 
books devoted to peculiarities if comb copolymers behavior 
\cite{bookPlate,bookMcArdle}.
Still unpublished works, like \cite{FischerSchmidt}, can be very helpful too.

Thesis \cite{thesisHyvarinen} is a good reading too!
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\begin{references}


\end{references}





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