Contents

1. Introduction 2

2. Preceding diagram technique 4

3. Irreducible chemical correlators 13

4. Modified diagrammatic technique 17

5. Higher-order vertex functions 24

6. Binary multiblock copolymers 29

7. Conclusions 33

Appendix A 33

Appendix B 34

Appendix C 35

References 39

1. Introduction

It is hard to overestimate the importance of the Landau theory of phase transitions for the statistical physics of condensed matter. By virtue of its universality, this macroscopic theory finds a wide utility in the description of the phase behavior of magnetics, crystals and low-molecular-weight liquids [1]-[3]. To construct a phase diagram in the framework of the Landau theory, it is sufficient to know the expressions for the first several vertex functions representing the coefficients of the free energy expansion in powers of small-order parameters {фа(г)}. Finding of the vertex functions suggests the recourse to a microscopic approach which, however, in the case of polymer liquids differs drastically from that traditionally employed for the description of ordinary liquids. This is because a specimen of any real synthetic heteropolymer consists of an enormous number of macromolecules varying in number of monomeric units of chemically distinct types and the pattern of their arrangement along a polymer chain. The configuration of every such linear macromolecule is isomorphic to a certain 'word' in an m-letter alphabet, where m is the number of types of units in a heteropolymer. An exhaustive description of a heteropolymer specimen implies specifying the probability measure on the set of macromolecule configurations. Since the chemical structure of macromolecules is known to form at the stage of a polymer synthesis, the problem of determining the configurational probability measure is apparently relevant to statistical chemistry rather than statistical physics of polymers [4, 5]. This problem has been solved for many heteropolymers formed in the course of various processes of their obtaining. In particular, it has been established that macromolecules constituting the products formed for some processes of free-radical copolymerization or copolycondensation can be envisaged as the realization of a certain Markov chain with states S1,..., Sm, each of which, Sa, corresponding to a a-type monomeric unit. The probability measure on

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Landau theory of phase transitions in heteropolymer liquids

the set of such realizations is well known in the theory of Markov chains [6,7]. Many classes of synthetic copolymers are also known where the distribution of monomeric units in macromolecules is described by a Markov stochastic process, provided, however, that these units are supplied by some labels [4,5]. Nowadays substantial results have been achieved in the statistical chemistry of such quasi-Markovian heteropolymers [5].

Any macromolecule in a heteropolymer liquid is also characterized, along with its configuration, by its conformation, which is specified by the positions of all its monomeric units in three-dimensional space. To find the free energy of the melt or solution of a specimen of a synthetic heteropolymer, one should perform a two-step averaging operation. The first step represents the averaging over conformations of a macromolecule with fixed configuration, whereas the second one is the averaging over configurations. In other words, to find any thermodynamic potential of a heteropolymer liquid it is necessary, along with the traditional averaging over the heat disorder characterized by the Gibbs distribution, to additionally perform the averaging of this potential over the 'quenched' structural disorder [8] characterized by the configurational probability measure. Therefore, when dealing with the problem of the thermodynamic description of solutions or melts of heteropolymers, recourse should be made to the methods of the statistical physics of disordered systems. Among these methods the 'replica trick' [9] is likely to be the most popular one.

Relying on the ideas of the replica theory, the authors of [10] proposed an original diagram technique for the solution of the general problem of finding the vertex functions of the Landau free energy expansion for a heteropolymer liquid containing linear macromolecules. Essentially, the employment of this technique imposes no restrictions on the number of types of monomeric units as well as on the pattern of their arrangement along polymer chains.

The key peculiarity of the Landau theory of phase transitions in melts or solutions of polymers is the appearance in the expansion of the free energy of some nonlocal terms [11]—[15], absent in the traditional Landau theory of ordinary liquids [1]-[3]. Allowance for the presence of such terms due to the polydispersity of a polymer specimen enables the explanation of a number of intriguing specific features in the thermodynamic behavior of heteropolymer liquids with macromolecules consisting of long blocks of identical units. Among these features should be mentioned the emergence on the phase diagram of such block copolymers of spatially periodic coexisting mesophases with various morphology whose period falls in the nanoscale range [16]. This unique property of heteropolymers is used in nanotechnology to design advanced materials with extraordinary service properties [17]. Among major challenges for the statistical thermodynamics of polymers is the construction of the phase diagrams of systems of such a kind, permitting theoretical prediction of the morphology and periodicity scale of the mesophases formed. An indispensable prerequisite for the solution of this problem in the framework of the Landau theory evidently consists in preliminarily obtaining the expressions for vertex functions in the expansion of the free energy in power series of the order parameters.

There are a number of publications (see, for example, [12,13,18,19]), reporting such expressions for the vertex functions of the second, third and fourth orders in the particular case of an incompressible melt of a Markovian copolymer whose macromolecules comprise units of two types. Expressions for the above-mentioned vertex functions have been found for binary multiblock copolymers obtained by

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Landau theory of phase transitions in heteropolymer liquids

condensation of monodisperse oligomers [14]. Later on [20], these results were extended to semi-Markovian multiblock copolymers with any number of types of monomeric units. In macromolecules of such copolymers the distribution in length of blocks of each type of unit is taken to be arbitrary, whereas the statistics of the succession of these blocks along macromolecules is described by some Markov chain.

However, as has lately become clear, the results reported in the above publications are of limited value. The fact is that when constructing the phase diagram, one should consider some fifth- and sixth-order terms in the expansion of the Landau free energy. The necessity of such consideration has recently been revealed [21] for the melt of binary Markovian multiblock copolymer. The prime objective of the present paper is to extend such a consideration to other classes of linear heteropolymers, proceeding from the diagram technique introduced earlier [10]. Below we will analyze the diagrams describing the fifth- and sixth-order vertices to reveal those of them that make the largest contributions to the free energy. To do this, an original approach based on switching from the reducible chemical correlators to irreducible ones has been advanced in this paper. The recourse to this approach enables a maximum simplification of the appearance of the analytic expressions for the vertex functions. This is highly important when calculating the phase diagrams of particular heteropolymer liquids.

The paper is organized as follows. Subsequent to the introduction, the essentials of the preceding diagram technique are formulated. Then expressions are presented that relate the reducible and irreducible chemical correlators. In the next two sections we introduce a modified diagram technique and discuss how this can be employed to calculate the higher-order vertex functions. The main body of the paper is concluded by the summary of the expressions for the contributions to the free energy of the melt of a binary block copolymer whose blocks have arbitrary distributions in length. Most of the intermediate formulas used for the derivation of the final results are summarized in three appendices.

2. Preceding diagram technique

The Landau free energy of a polymer liquid (divided by overall number M of monomeric units and temperature T, expressed in energetic units) in the theory of the self-consistent field can be presented as a sum of two functional [22, 23]. The first of them, JF*, is equal to the difference of the free energy of the subsystem of separate units, FSU, and their ideal gas, FIG, whereas the second one, FCB, is equal to the free energy of the subsystem of chemical bonds representing the ideal gas of macromolecules. It is a common practice to write the functionals T* and FCB as a sum of contributions corresponding to the terms of their expansion in powers of the order parameters (OPs)

DO _, ОС

n=2 n=2

Contributions to the first of these functionals can b

References

[1] Toledano J C and Toledano P, 1987 The Landau Theory of Phase Transitions (Singapore: World Scientific) [2] Toledano P and Neto A F, 1998 Phase Transitions in Complex Fluids (Singapore: World Scientific) [3] Izumov Yu A and Syromyatnikov V N, 1990 Phase Transitions and Crystal Symmetry (Dordrecht:

К luwer- Ac ademic)

[4] Kuchanov S I, 1978 Methods of Kinetic Calculations in Polymer Chemistry (Moscow: Khimia Publishers) [5] Kuchanov S I, Principles of the quantitative description of the chemical structure of synthetic polymers,

2000 Adv. Polym. Sci. 152 157 [6] Lowry G G (ed), 1970 Markov Chains and Monte Carlo Calculations in Polymer Science (New York:

Dekker)

[7] Koenig J L, 1980 Chemical Microstructure of Polymer Chains (New York: Wiley) [8] Ziman J M, 1979 Models of Disorder (Cambridge: Cambridge University Press) [9] Dotsenko V, 2001 Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge:

Cambridge University Press) [10] Aliev M A and Kuchanov S I, Diagram technique for finding of vertex functions in the Landau theory of

heteropolymer liquids, 2005 Eur. Phys. J. B 43 251 [11] Shakhnovich E I and Gutin A M, Formation of microdomains in quenched disordered heteropolymers, 1989

J. Physique 50 1843 [12] Panyukov S V and Kuchanov S I, Non-local nature of the Landau free energy in the theory of polymer

systems, 1991 Sov. Phys. JETP Lett. 54 501 [13] Panyukov S V and Kuchanov S I, New statistical approach for the description of spatial inhomogeneous

states in heteropolymer solutions, 1992 J. Physique II 2 1973 [14] Fredrickson G H, Milner S T and Leibler L, Multicritical phenomena and microphase ordering in random

block copolymers melts, 1992 Macromolecules 25 6341 [15] Kuchanov S I and Panyukov S V, 1996 Comprehensive Polymer Science—Second Supplement ed

G Allen (Oxford: Pergamon) chapter 13 (Statistical Thermodynamics of Copolymers and their Blends) [16] Hamley J W, 1998 The Physics of Block Copolymers (Oxford: Oxford University Press) [17] Balsara N P and Hahn H, Block copolymers in nanotechnology, 2003 The Chemistry of Nanostructured

Materials ed P Yang (Singapore: World Scientific) [18] Sfatos C D, Gutin A M and Shakhnovich E I, Critical composition in the microphase separation transition

of random copolymers, 1995 Phys. Rev. E 51 4727 [19] Angerman H, ten Brinke G and Erukhimovich I, Microphase separation in correlated random copolymers,

1996 Macromolecules 29 3255

[20] Slot J J M, Angerman H J and ten Brinke G, Theory of microphase separation in multiple segment-type statistical multiblock copolymers with arbitrary block molecular weight distribution, 1998 J. Chem. Phys. 109 8677

doi:10.1088/1742-5468/2008/03/P03017 39

Landau theory of phase transitions in heteropolymer liquids

[21] Kuchanov S I and Panyukov S V, A correct account of the nonlocal terms in the Landau theory of phase

transitions in polydisperse heteropolymers, 2006 J. Phys.: Condens. Matter 18 L43 [22] Lifshitz I M, Grossberg A Yu and Khokhlov A R, Some problems of the statistical physics of polymer chains

with volume interactions, 1978 Rev. Mod. Phys. 50 683 [23] Grosberg A Yu and Khokhlov A R, 1994 Statistical Physics of Macromolecules (New York: American

Institute of Physics) [24] Balsara N P, 1996 Physical Properties of Polymers ed J E Mark (New York: AIP Press) chapter 19

(Thermodynamics of Polymer Blends) [25] Aliev M A and Kuchanov S I, Vertex functions in the Landau theory of phase transitions in melts of

Markovian multiblock copolymers, 2008 Physica A 387 363 [26] Harary F, 1969 Graph Theory (London: Addison-Wesley) [27] Ursell H D, 1927 Proc. Camb. Phil. Soc. 23 685

[28] Domb C, On the theory of cooperative phenomena in crystals, 1960 Adv. Phys. 9 149 [29] Hill T L, 1956 Statistical Mechanics (New York: McGraw-Hill) chapter 5 (Theory of Nonideal Gas and

Condensation) [30] Ulenbeck G E and Ford G W, 1963 Lectures in Statistical Mechanics (Providence, RI: American

Mathematical Society) chapter 2 (Theory of Nonideal Gas)

[31] Isihara A, 1971 Statistical Physics (New York: Academic) chapter 5 (Connected Cluster Expansion) [32] Croxton C A, 1974 Liquid State Physics—A statistical Mechanical Introduction (Cambridge: Cambridge

University Press) chapter 1 (Theory of Nonideal Gases) [33] Mayer J E and Goeppert Mayer M, 1977 Statistical Mechanics 2nd edn (New York: Wiley) chapter 8

(Dense Gases) [34] Whittaker E T and Watson G N, 1927 A Course of Modern Analysis 4th edn (Cambridge: Cambridge

University Press) [35] Kuchanov S I and Panyukov S V, Molecular theory of solutions and blends of heteropolymers. I.

Thermodynamics of amorphous multicomponent polymer systems, 1998 J. Polym. Sci. B 36 937

Примечаний нет.