%% This is an example of a TeX file with formats %% included into the header of the document. %% To run it you have to do something like: %% biblio.pl -o 2sorts_formats.1.tex -d refers.db 2sorts_formats.tex %% or %% biblio.pl -o 2sorts_formats.1.tex -d refers.db --nobibitem --uselist 2sorts_formats.tex %% if you want a really well formatted citation list %% and then %% latex 2sorts_formats.1.tex %% to get 2sorts_formats.1.dvi %%------------------------------------------------------ %% this is copyed from the template for Macromolecules %% no ordering of the references is needed. Otherwise add line %% '%ordering= AUTHORL' %begin{biblio} %cite_format={(%s)} %ref_format=${^{%s}}$ %% normal article: % ARTICLE [$AUTHOR0P~L,~F~M; $ {\it $JOURNAL$} {\bf $YEAR$}, {\it $VOL$}, $PAGE$.] %% article which is submitted or accepted % ARTICLE0[$AUTHOR0P~L,~F~M; $ {\it $JOURNAL$} $STATE$.] %% normal book % BOOK [$AUTHOR0P~L,~F~M; $ {\it $TITLE$}; $PUBL$: $PLACE$, $YEAR$.] %% article in a book AUTHORE, AUTHORG and AUTHORH - last, first %% and middle names of the editor % EDBOOK [$AUTHOR0P~L,~F~M; $ in {\it $BOOKTITLE$}, edited by $AUTHOR0I~E,~G~H;$; $PUBL$: $PLACE$, $YEAR$.] %% article in a book without editor % INBOOK [$AUTHOR0P~L,~F~M; $ in {\it $BOOKTITLE$}, p.$PAGE$ $PUBL$: $PLACE$, $YEAR$.] % EPRINT [$AUTHOR0P~L,~F~M; $ in {\it $BOOKTITLE$} $INFO$, $YEAR$.] %% Thesis % THESIS [$AUTHOR0P~L,~F~M; $ {\it $SORT$}; $UNIV$: $PLACE$, $YEAR$.] %end{biblio} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %REAL DOCUMENT STARTS HERE!!!! \documentstyle[aps,manuscript]{revtex} \title{Comb copolymer brush with chemically different side chains} \date{\today} %----------------------------------------------- \newcommand{\vA}{v_{AA}} \newcommand{\vB}{v_{BB}} \newcommand{\vAB}{v_{AB}} \newcommand{\vx}{v} \newcommand{\mnu}{\nu} \newcommand{\khi}{\chi} \newcommand{\dd}{\partial} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle %-------------------------------------------------- \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} %-------------------------------------------------- \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. %------------------------------------------- \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! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{references} \end{references} \end{document}