Theory of Microphase Separation in Block Copolymers

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1602 Macr omolecules 1980, 13 , 1602-1617P. Wagli of th e ETH-Zurich an d J. A. Logan of IBM-San Jose for excellent SEM micrographs, and P. G. deGennes for helpful discussions. D.S.C. acknowledges withappreciation the sup por t provided by a G uggenheim fel-lowship and expresses his gratitude for the hospitalityextended to him a t the College de France.Re fe r e nc es and N ote s

(1) de Gennes, P. G. Scaling Concepts in Polymer Physics;Cornell University Press: Ithaca, N.Y., 1979.(2 ) Daoud, M.; de Gennes, P. G. J. Phys. (Paris) 1977, 38, 85 .Brochard, F.; de Gennes, P. G. J. Phys. Lett. 1979, 40, 399.(3 ) Brochard, F.; de Gennes, P. G. J. Chem. Phys. 1977, 67, 52 .Daoudi, S.; Brochard, F. Macromolecules 1978, 11, 751.(4 ) Satterfield, C. N.; Colton, C. K.; Pitcher, W. H., Jr. AIChE J.1973, 18, 628.(5 ) Colton, C. K.; Satterfield, C. N.; Lai, C. J. AIChE J. 1975,21,289.(6 ) Happel , J . ; Brenner , H. Low Reynolds NumberHydrodyna mics; Prentice-Hall: Englewood Cliffs, N.J., 1965;p 285.(7 ) Haller, W. Macromolecules 1977, 10, 83.(8 ) Satterfield, C. N.; Colton, C. K.; de Turckheim, B.; Copeland,T. M. AIChE J. 1978 , 24 , 937 .(9 ) Renkin, E. M. J. Gen. Physiol. 1954, 38, 225.

(10) This means tha t 70 % of th e chains are within f30% of thenumber-average molecular weight. It is thus not a truly mon-odisperse distribution.(11) Munch, J. P.; Candau, S.; Herz, J.; Hild, G. J. Phys. (Paris)1977, 38, 971.(12) Forrester, A. T. ; Parkins, W. E.; Gerjuoy, E. Phys. Rev. 1947,7.7 70 01.4, 1 . 40 .(13) Dubin, S.B.; Lunacek,J. H.; Benedek, G. B. h o c . N atl. Acad.Sci. U.S.A. 1967, 57 , 1164.(14) Cummins. H. Z.: Swinnev. H. L. Prop. O D t . 1970. 8. 133.(15) Chu, B. Laser Light Scattering; Acd em ic Press: New York,

(16) Adam, M.; Delsanti, M. Macrom olecules 1977, 10, 1229.(17) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, N.J., 1962; p 87.(18) Beck, R. E.; Schultz, J. S. Biochim. Biophys. Acta 1972,255,273.(19) Fo r a discussion, see: Villars, F. M. H.; Benedek , G. B.Physics with Illustrative Examples from Medicine andBiology; Addison-Wesley: Reading, Mass., 1974; pp 2-80.(20) Flory, P. J. Principles of Polymer Chemistry; Cornell Uni-versity Press: Ithaca, N.Y., 1971.(21) Unfortunately, scaling laws are not able to predict exact nu-merical coefficients.(22) Daoud, M.; C otton, J. P.;Farnoux, B.; Jannink, G.; Sarma, G.;Benoit, H.; Dup lessix, R.; Picot, C.; de Gennes, P. G. Macro-molecules 1975 ,8, 804.

1974.

Theory of Microphase Separation in Block CopolymersLudwik LeiblertCommissariat 6 %nergie Atom ique, Division de la Physiqu e, Service de Physiqu e du Solideet de Resonance Magnetique, C.E.N. Saclay, 91190 Gif-slYvette, France.Received November 28, 1979

ABSTRA CT: A microscopic statistical theory of phase equilibria in noncrystalline block copolymers of typeA-B is developed. In particular, the onset of an ordered mesophase from a homogeneous melt is studied an da criterion of the microphase separatio n is found. Th e only relevant parame ters of the theory tu rn o ut tobe the product xN (x is th e Flory parame ter characterizing A-B interac tions; N is th e polymerization index)and f , the fraction of monom ers A in a chain. It is shown tha t un der ce rtain critical conditions a specificunstable mode appears in th e homogeneous copolymer melt; this announces the microphase separation transition(MST). After the MST a mesophase with the periodicity equal to the wavelength of the unstable mode an dwith the symmetry of a body-centered-cubic (bcc) lattice should appear. For a large range of compositionsa bcc mesophase is expected to be a metast able phase. Only two other ordered phases, a hexagonal mesophasean d a lamellar mesophase, may be stable near the MST. Th e regions of stability are calculated and the ph asediagram of the system is provided. First-order transitions are predicted between different ordered phases.A new method of measuring of the parameter x, crucia l for verification of th e theo ry, is proposed.

I . Introduction of monomers A and B, forming a copolymer molecule, areTh e particular chemical struc ture of block copolymermaterials is reflected in the most fundamental and in-teresting way by in com patibility effects. These effects giveblock copolym ers a num ber of specific, new morpho logiesand original physical and mechanical properties which haveled to valuable technological applications (for generalreferences see ref 1). The purpose of the present paperis to formulate a microscopic theory w hich, on the basisof statistical phy sics, would explain how and un der w hichconditions th e incomp atibility of molten cop olymer con-stituents leads to the formation of different orderedstructures (mesophases). Thi s general theory is appliedto provide a phase diagram of the simplest copolymersystem-molten diblock copolymer A-B. Both sequen ces

supposed not t o be able to crystallize.Th e most characteristic feature of a block copolymer isth e strong repulsion between unlike s e q u e n c e s even whenthe repulsion between unlike m o n o m e r s is relatively weak.As a result sequences tend to segregate, but as they a rechemically bonded even the com plete segregation cannotlead to a m acroscopic phase sepa ration as in m ixtures oftwo homopo lymers. How ever, in the case of a sufficientlystrong incompatibility, microphase separation occurs:microdomains rich in A (in B) are formed.Such an effect was first observed in co ncentrated solu-tions of block copolymers and afterward in many othercopolymer systems, particularly in molten block co-polym ers (see, e.g., review artic les 3-8 and references giventherein ). Wh en microDhase seDaration occurs, the m i-crodomains are not located a t r&dom but they m ay form

67083 Strasbourg, France (macrolattice). Several techniques (e.g., annealing, ex-Present address: Centre de Recherches sur les MacromolBcules,a regular arra nge me nt giving rise to a periodic structu re

0024-9297/80/2213-1602$01.00/0 0 1980 American Chemical Society


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Val. 13, No . 6, November-December 1980 Theo ry of Microphase Separa tion in Block Copolymers 1603trusion) enable samples to be obta ined which exhibit a veryregular, oriented macrolattice extended over large vol-

Mesophases with three types of lattice have been ob-served: cubic lattice of spheres, two-dimensional hexagonallattice of cylinders, and one-dimensional lamellar struc-tures.Various ex perimental techniques such as small-angleX-ray scattering (SAXS), lectron microscopy, and electrondiffraction have given relatively rich data abo ut moltenblock copolymer mesophases. Sym metries, periodicities,an d microdomain dimensions and their dependencies onmolecular param eters such as molecular mass, fraction ofmonomers A in the chain (composition), and chemicalnatu re of copolymer monom ew have been studied in detail.Th ere is relatively less information ab out t he conditionsof microph ase separation . Neve rtheless th e existence ofth e disordered (homogeneous) phase has been establishedfor copolymers with a low p olymerization index by severalexperimental methods (rni~rocalorimetry, '~- '~AXS,'4J6J9electron microscopyz0) an d the microphase separationtransition (M ST) from the disordered phase to a meso-p ha se h a s be en d ir ec tl y o b ~ e r v e d . ' ~ J ~ J ~

In th e last decade a considerable effort has been m adeto develop a microscopic theory of the molten block co-polymer (see, e.g., ref 9 an d ref 21-29). Th e approacheshith erto adopted considered a specified mesophase (i.e.,with a specified symm etry and shap e of the microdomainpatt ern) a nd evaluated a nd minimized the free energy ofthe system in this mesophase in order to obtain the pe-riodicity and m icrodomain dimensions. Th e calculationof the free energy of molten block copolymer encounteredessen tial difficulties-it required very restrictive assum p-tions abo ut the microdomain interfaces and also the in-troduction of phenomenological parameters not directlyavailable from experim ents (compare, e.g., review articles30-32). Recen tly, Helfand ha s proposed a self-consist-ent-field method to o btain th e free energy of th e moltenblock copolymer solely in terms of molecular parame terscharacterizing copolymer chains (molecular mass, com-position, monomer lengths, and monomer interaction en-ergyhn Th e application of the m ethod even to the simplestpossible lamellar phase requires complicated numericalcomputations. Th e assumption tha t the interphase be-tween dom ains is narrow compared w ith domain size sim-plifies th e theory a nd p ermits valuable predictions to bemad e of domain periodicity and size for th e lamellar andsph erical (close-packed cubic) m e s o p h a ~ e . ~ . ~ ~ J ~ ~ ~ ~n theprese nt pap er we shall be concerned with a very differentregime, namely, the onse t of a microphase from a homo-geneous melt. Th e narrow interphase approximationcompletely breaks down in this regime: here even thenotion of an interphase d isappears since the con centrationof monomers A (B) does not change abruptly over a smalldistance but, on the contrary, varies smoothly over thewhole period of the m icrophase st ruc tur e (see section 11;compare ref 29, 30, an d 33). Therefore, the Helfand ap-proximation cannot be used and another approach isneeded.A complete understanding of th e microphase separationtransition (MS T) in m olten block copolymers requires notonly a statistical study of microdomain structures (meso-phases) appearing after the transition b ut also a study ofth e homogeneous (disordered) phase. In particular oneshould find a quantity which may be used to introduce anorder param eter (see section 11). Here we study the mo-nomer A density correlation functions in a disorderedphase, using a generalization of the "random-phase

ume.9,7,1&13approximat ion" (RPA) method of de G e n n e ~ , ~ ~hich isknown to work well in other m olten polymer system^.^^^^We find th at un der certain specific critical conditions anunstable mode with a nonvanishing wave vector appearsin the system; i.e., a certain Fourier component of themonomer den sity fluctuations diverges. Thi s instabilityannounces the microphase separation transition. It isinteresting to note tha t the microphase separation tran-sition is in many respects analogous to th e crystallizationof a liquid. Th e latter has generated a renewal of interestrecently (e.g., ref 37 and 38).Once correlation functions have been found, we makeuse of general statistical physics relations w hich provideth e free energy of th e block copolymer in terms of the o rderparame ter of the microphase separation transition. Th eapplied approach has the advan tage of taking into accountthe connectedness of the blocks in a block copolymermolecule in a very simple and natu ral way. Only twoquantities turn out to be relevant parameters for thecharacterization of phase equilibria in molten diblock co-polymers: the com position f (the fraction of mo nomers Ain a chain) and the product x N (where N is the polym-erization index a nd x is the Flory parameter characterizingth e effective interac tion of monomers A-B; for the defi-nition of x see eq 11-2). Hence, th e determ ination of thephase diagram of the system consists of calculating theregions of stab ility of different p hases (disordered phase,mesophases) in a plane with coordinates x N and f . Ouranalysis of t he free energy leads to predictions concerning(i) both t he sym metry an d the periodicity of phases ap -pearing just after the microphase separation transition, (ii)the criterion for the microphase separation transition ( x Nas a function of f for which microdomain structu res occurin th e system), (iii) the phase diagram in th e vicinity ofthe microphase separation transition line, and (iv) exist-ence of transitions between mesophases with differentsymm etry. Detailed com parisons of the theory with ex-periments will depend on our knowledge of th e interactionparameter x , which is difficult to measure experimentally(compare, e.g., ref 39). We show that the small-angle X-rayscatterin g by block copolymers is strongly depe nden t onmonomer-monomer interactions and may provide a newmethod of determining x .In th e next section we present a qu alitative descriptionof th e m icrophase separation tran sition which follows fromthe present theory and which indicates the guidelines alongwhich this theory is constructed.11. T h e r m o d y n a m i c D e s c r i p ti o n of M i c r o p h a s eS e p a r a t i o nIn this paper, we will consider a molten diblock co-polymer where all chains have the same index of polym-erization N an d the sam e composition f =N A / N NA >>1 and N B >>1denote th e number of monomers of type Aand B, respectively; N =NA +NB); .e., we neglect thepolydispe rsity of th e chains. For th e sake of simplicity wewill assume th at b oth blocks have the same Kuhn statis-tical segment length a. We will limit our analysis to th esituations where thermal equilibrium is achieved.Molten polymers have very low com pressibilities. Thismean s th at an y local compression of th e system will leadto a huge increase of the free energy which can be com-pensated neither by the decrease of monomer interactionsnor by entrop y chan ges (cf. ref 27). Th us it is justified toconsider the limit of zero compressibility an d to assumetha t a t any point in the copolymer l iquid

P A ( 8 +P d 8 =1 (11-1)where p A ( 8 and p B ( 8 denote th e reduced density of mo-


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1604 Leibler Macromoleculesnomers A an d B, respectively. pA(8 s defined as the ratioof the local density of monomers A at the point 7 to theoverall monomer d ensity averaged over the sample.Although th e overall density of monomers is constant(eq 11-l), A and p B themselves may exhibit local fluctua-tions an d it is these fluctuatio ns which will be a t the veryroot of the microphase separation. Th e species A and Btend to realize an equ ilibrium de nsity profile PA(?) an d pB(?)which minimizes the free energy of th e system. Th e energyof interaction of different chemical species (per monomer)may be written as (see, e.g., ref 35 and 40)

W A P B k T X P A P B (11-2)In th e lattice model, th e effective interaction param etera may be related to th e energy of interactions between twomonomers A-A (eAA), B-B (EBB) , and A-B (tAB): a =em- /Z(tuQB). Experiments show tha t x is positive41 ofth e o rde r of 10-3-10-1); mo nom ers A an d B effectively repe leach other. In fact, since x is positive there is a tendencyto decrease the number of contacts between monomers Aand B in order to lower the contribution of the interactionenergy contribution to t he free energy (eq 11-21. However,a decrease in th e numb er of conta cts A-B diminishes th eentrop y of the system and co nsequently increases th e freeenergy. Hence, PA(?) and p B ( 7 ) profiles are the result ofa "competition" between these two opposite trends.For x =0 or x finite but sufficiently small, the e ntropyeffects are dom inant, and they favor mixing: the systemexhibits an isotropic phase with sequen ces A and B of thechains interpenetrating each other so that for all points

( PA ( ? ) ) =f ( P B ( 7 ) ) =1 f (11-3)(( ) denotes th e thermal average). Such a phase will becalled the disordered phase. On the other hand , when thesystem is cooled (or comp osed of longer chains) so that xNis larger than a certain value (XN), to be computed insection V), the enthalpic term (eq 11-2) in the free energydom inates and th e system exhibits a microphase separa-tion; i.e., ( P A ) and ( p B ) are not uniform: there are domains(A rich) where (pB) s smaller than 1- and others (B rich)where ( p A ) is smaller than f. Then at all points theproduct p u B s small and th e enthalpic term is low. Aphase of the system in which a microdomain structureoccurs wll be called an ordered p hase (or mesophase). Asit has been already mentioned, depending on t he value ofcomposition f and on th e interaction parameter x (tem-perature T),several different mesophases may be observed.In order to characterize t he differen t phases of a liquidcopolym er as well as th e transitio ns betw een them , we willintroduce an order parameter $(7), defined asIn th e disordered phase $(i)s uniform and vanishes atall points. In different ordered phases (with differentdomain patterns) $ does not vanish. In a single domainsample (macrolattice) $(7) is a periodic function. As in -compressibility of th e system (eq 11-1)has been assumed,$(?) fully describes its thermodynamic state . It should benoted th at (from eq 11-1)

$(?) = ((1- ?PA(?) - P B ( 7 ) ) (11-4)

$(8= ( 6 PA ( F ) ) E (PA(?) - ) (11-5)i.e., J / describes the average deviation from th e uniformdistribution of monomers A and B.Let us consider the system where xN is so small tha tth e disordered phase occurs: th e average value of 6 p A =PA - vanishes at all points. However, when x becomeslarger (as a result of cooling the system) so t h a t xN ap-proaches the value (xn3, (for which the microphase sep-aration transition occurs) th e local values of 6 p ~ave large

fluctuations. These fluctuations may be characterized bya density-density correlatio n functio n(11-6)

Th e Fourier transform $4') of the correlation function (eq11-6) can be stud ied by e lastic radiation scattering exper-iments: light, X-ray, or neutron scattering (4' deontes thescattering vector with length 4 ~ [ s i n0/2)]/X, where X isthe w avslength and 0 th e scattering angle). T he scatteringpower S($) can be calculated with the help of the RPAmethod.% This method assumes tha t th e chains are nearlyTh is is a good approximation for homopolymermelts, where RP A predictions have been compared withneutron experimen ts using partially deuterated ch ains withparticularly small effective interactions between markedand unmarked monomers ( x 5 o-,) an d polymerizationindices not too high ( N 5 o3). When segregation effectsbecome important the RPA must be modified to take intoaccount the monomer interactions (compare studies ofcorrelations in the probJem of homopolymer m i x t u r e ~ ~ ~ p ~ ) .Th e scattering power S(q') is calculated and discussed insection IV.From a thermodynamic sta ndpo int, this calculationleads to the im portant conclusion tha t only certain fluc-tuations become anomalously large when th e interactionearameter x tends toward th e transition po int value xt :S(+) has a very narrow maximum f?r a certain value of 141=q* # 0. (In the isotropic phase S depends only on th elength of th e 4' vector.) A remarkable property of moltencopolymers is tha t q* is independent of the interactionparameter x. As xN becomes greater than ( x N tand isequal to a certain value the fluctuations with wavevector 141=q* diverge: S(q*)- . This is the spinodalpoint. Th e spinodal point determines th e limit of me-tastability of the system. When th e system is quenchedto the metastability region ( ( x N ) , < xN < (xN,) h emicrophase separation can take place by the nu cleationof droplets of mesophases inside the disorder phase. Th einterface energy is positive an d th e droplets a re formedby a slow thermoactivated process. On the contrary, whenN =(xN),he interface energy vanishes: the system be-comes unstable.Th e simplest way to analyze qualitatively th e M ST isto exam ine the expansion of the free-energy density F ofan ordered p hase (occurring below the transition: xN >(xIV)J in powers of th e order param eter $. Th e quadraticterm has th e form (see sections I11 and V)

Fz =Y-'CS1(G)$(Cj)$(-ij) (11-7)awhere $44')denotes th e Fourier transform of $47) and Ythe volume of th e system. As we have pointed ou t, nearthe phase transition S-l(cj)has a deep minimum for thewave vectors i j with 14'1 =q*, but the higher order termsF,, F4 , ...have no singular behavior for q*. As a conse-quence th e only Fourier components of J / which are im-po rtan t (near the transition) are those with 14'1 =q* . Fo rxN >(xMt he free energy is lowered by the presen ce ofa finite $(Q) (14'1 = q*): a periodic ordered mesophaseappears. Th e periodicity of th e ordered phase is equal to2u/q* and is of the order of the radius of gyration ofcopolymer molecules. T he detailed structure and mag-nitude of the coefficients # ( G ) (14'1=q*) for various ori-entations of 4' will depend crucially on the higher o rderterms F,, F4 , .... It is important to note tha t for f # 0.5the third-order term F3 in the expansion of F does notvanish. As pointed ou t long ago by Landau:' th is im pliesth at the transition should be of th e first order: a t (xMt


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Vol. 13, No. 6, November-December 1980$(g*) has a jump . Hence, the microphase separationtransitio n is analogous to th e solidification of the liquid,bu t g* for the M ST is much sm aller; Le., the periodicityof molten block copolymer mesophases is much larger tha nthe periodicity of a crystal lattice.General sym metry considerations show tha t the exist-ence of the third-order term in the expansion of thefree-energy density F allow s for xN =(xNthe onset ofonly two periodic microdomain structures: one with thesymm etry of th e planar triangular lattice and a nother withth at of the body-centered-cubic lattice. In order to decidewhich of these two structur es will actually appear as wellas to calculate (xNt,.e., to predict a criterion of themicrophase sep aration, it is necessary to f ind the coeffi-cients of the th ird an d the fourth order in the expansionof F. The calculation of these coefficients also permitsexamination of the stability of different mesophases forxN close to (xNtnd prediction of phase transitions be-tween them (see sections V and VI).In section I11 we will formulate a theory leading to th efree-energy expansion in powers of $.111. Free Energy of Molten Block Copolymers

T he microphase separation transition in molten blockcopolymer A-B essentially differs from demixing in th emixture of two homopolymers A an d B. Th e lat ter is agas-liquid-type phase transition (with instability for 4 =0), whereas, as we have pointed ou t in section 11,MST isa liquid-solid-type transitio n (w ith instability for a finitewave vector =g*). Moreover, in block copolymers thelow-temperature ordered phase is a mesophase with non-homogeneities occurring a t distances of the ord er of mo-lecular dimensions (radius of gyration). For these reasonsthe simple lattice-type Flory-Huggins theory developedfor homopolymer mixtures is inadaptable to stud y MS Tin copolymers. In fact, such a theory would require thecalculation of th e num ber of configurations available forcopolymer chains (disorientation entropy@). Th e inclusionof constraints which prevent sep aration of copolymer chainsequences gives rise to enormous difficulties.We propose here a quite different approach based onth e RPA. This approach avoids the problem of calculatingseparately th e entropy of disorientation and leads directlyto th e free-energy density F of the molten copolymer asa function of the ord er parameter $(A. Th e essential pointof the me thod is a theorem w hich relates th e coefficientsof the exp ansion of th e free energy of the system in powersof $(?) to th e response functions (section 111.1). To getequations for response functions of the copolymer melt wemake use of the random-phase approximation (section111.2). The n R PA equ ations are solved and the free-energydensity F is obtained in terms of response functions of idea l(Gaussian) chains (calculated in Appendix B).

1. Free-Energy Expansion an d Re sponse Fun c -tions. In th e theory of m agnetic phase transitions generalrelations between response functions (connected Greenfunctions) an d coefficients in the expans ion of a free-energyfunctional in powers of an order param eter are often used(e.g., ref 43-45). In th is section we estab lish similar rela-tions between monom er density correlation functions an dthe free energy of th e m olten copolymer. Since the m ethodof derivation of such relations is analogous to th at in th emagnetic problem, we only sketch here t he essential pointsand recall some definitions which will be of use later(section 111.2).Let us consider the liquid copolymer under the con-str ain t of the existence of a long-range order. T he averagevalue of 6 p ~ ( ? )=PA(?) - do es not vanish (+(?) = ( 6 p A ( ? ) )# 0). T he tot al free-energy density of th e system F may

Theo ry of M icrophase Separation in Block C opolymers 1605be divided into two parts. Th e first one, Fo, he free energyof th e copolymer liquid in the absence of th e constraintof +(?) # 0, includes the naturally occurring correlationsof the liquid state. T he second one, F , is due to the con-straint $(?) # 0 and i t vanishes when $=0. When thesystem is below th e M ST , i.e., xN >(xNt,he microphaseseparation appears spontaneously, F is negative, an d th esystem occurs in a stable ordered phase. Its $(?)minimizesF. On the contrary, for xN


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1606 Leibler Macromoleculesr3(al, , , a3 )=-OW,, a ,, a3)/h~(2)(ai,ai) (111-15)r 4 ( a 1 , a3,d4) =@(4)($1, a2,a3 ,G ~ )r=l

~ - 1 ~ 8 ( 2 ) - 1 ( ; : -i=j?[@3)(ijl, a2 , j :)G(3)(-tt, i j 3 , a, ) +6(3)(~1,3,G ? @ ~ ) ( - F ,,, a4)+

8(3)(ij1,4, i j9@3)(-a', a, , d 3 ) ] ) / t@2)-1( i j i , ai ) (111-16)It is very imp ortant to n ote tha t, from the definition of$, i t tu rns out tha t$( i j =0) = Jd3 r $(A = Jd3 r ( P A - ) =0 (111-17)This implies tha t 8(2)(0, ) =0 as well as &3)(tj1,j 2 , 0) =0 for ijl+ j2=0. In turn, when Idl +&I=la3+ij41=0,from (111-17) it follows that on the right-hand side of(111-16) th e second te rm does no t ap pear .In th e following y e use the RPA to calculate the cor-relation functions G(")(q',,...,a,,)(n=1,...,4). Th en, therelatio ns (111-14) to (111-16) per m it c alculation of th efree-energy density of the system, F , and a n investigationof the transition region.2. Random -Phase Approximation for Block Co-polymers. Free Energy near the MST. n this sectionwe generalize a standard RPA method34 n order to cal-culate both linear and nonlinear response functions ofblock copolymers and the free-energy expansion in powersof th e order parame ter.We denote by $i th e average value of 6 p i , where i =1,2 correspond to the species A and B, respectively. If wesuppose tha t two different fields Vi act on the monomers,then the response of th e system (Le., the average changeof 6pi) is equal to (see eq 111-12)

4"

t = l

$ l ( G l ) =-@3iJ(a1)uJ(G1)1-P 'Y -~ Qij'i(~1, ~ 2 ,~uj(~,)uk(~S)- P ~ Y - ~ Q ! # I ( G ~ , ~ 2 , 3 ,i 4 )uj(a2) k(~3)l(44)

& d 313! 4 2 d 9 ~ 6 4

E i(l) +$i(2) +$ i ( 3 )(111-18 )

where @:!,,,in(TjJ,..,a, ) denote t he correlation functions ofmonomer densities 6pi,(T1), ..., pi (A (see eq 111-9) and Sij(G)=VIG$)( i j , 3 ) . (l), A2), an d $y3) denote, respectively, th econtribution to $ of th e first, second, and th ird order inexternal p otentials V i . In (111-18) and in t he following thesum ma tion convention over th e repeated indices is used.It should be stressed that from the incompressibilitycondition Ci=12$i0, i t turns o ut tha t there is only oneind_epe_ndentorrelatio_n unction of th e n th order, e.g., Sll=S, G$\ =G(3) nd Gf;Il =&4) (cf. eq 111-12).Th e basic point of th e RPA method is the fact tha t inpolymer m elts chains are nearly ideal on th e scale of onecoil (e.g., ref 35 an d 40). I t may be suppo sed th at whenthe interactions between m onomers are switched off th emonomer den sity correlation functions are equal to thoseof the ideal (Gaussian) independent copolymer chains(which will be den oted by Si j ,G$A, G$J . In the RPA onecalculates th e response of th e system to t he external po-tentials U , as if th e response functions w ere those of th eideal chains, but the potentials acting on monomers Urff(different from t he external potentials Vi)were correctedto take into account monomer interactions. These cor-rections are of two different types (cf. ref 42). Fir st of all,to ensure th e constant overall monomer density (incom-pressibility) a self-consistent potential V acting on all

I t is easy to check (from eq 111-5) th at this definition givesth e w ell-known formulas for correlation functions (cf. ref46)( 6 p A ... &A(?")) =EIIG(l'J'(71,..,7I,I) (111-8)ILl a

where for th e sake of simplicity we have deno ted by I th eset 11,...,n), by (I,] ny partition of I , and by 14the cardinalof the se t I . As an exam ple we use (111-8) to w rite dow nthe lowest order correlation functions which we will needlater@ l ) ( i l)= ( h p A ( 7 1 ) ) =o (111-Sa)

@ 2 ) ( ~ i ,2) = ( ~ P A F J ~ P A V ~ ) ) (111-9b)@3)(71,72, 73 ) = (6PA(71)6PA(72)8pA(73)) (III-gC)

@4)(71,-7Z, 73 , 74 ) = ( ~ P A ( ~ l ) ~ P A ( ~ 2 ) ~ p A ( 4 ) 6 p A ( F h )-G'2'(71, 7&G(')(73, 74) - G'"'(71, ?3)G (2)( F2, 74 ) -@V1,4)G(2)(?2,F3) (111-9d)(In (111-9) th e fact th at t he system is disordered when U=0 (i.e., ( 6 ~ ~ )0 ) has been used.)It should be noted tha t the average of 6pA in the presenceof the field U , which we call + is equal to

+(7) =2-' T r exp[-B(Ho + S U 6 p A d3r)]]=6F/6U(?) (111-10)

Hence, (-@)"-l@)(il , ...,Tn)/(n- )! may be interpreted asth e response function of the nt h order. Actually, from(111-6), th e respon se J . of th e system to th e external po-tential U reads

d3r1...d3r, (111-11)(Yvon's theorem ). I t will be convenient in the followingto use th e Fourier tr ansfo rm of (111-11)

m(&,.., Ij")U(&) ... U(ij,) (111-12)where Oca) =SO (7) eivd 3r enotes the Fourier transformof a function 0 and th e use of th e translational invarianceof th e system has been made.In order to get F(+) we need U as a function of $, soweshould inv ert (111-12). This m ay be done by t he iterationme thod. The n, by integ rating (111-10) we get the expansionof F as a power series of +

" 1F =Fo +kTC V"- - n ( h -, d n ) $ ( d J $ ( a n )n=2 n!dl ..., n (111-13)Th e coefficients I?,, in-expansion III-13 are related to th ecorrelation functions G'"), G("-l),...,G(,). As we will confineou r interest to the tem perature region near th e transition,it is justified to dr op ou t in expansion 111-13 all term scontainin g higher powers of +than the fourth power. Th ecoefficients of interest are given by

r2(a1,2)=W 2 ) ( a l , 2 ) (111-14)


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Vol. 13, No . 6 , Nouember-December 1980monomers (both A and B) is present. Secondly,to expressthe fact tha t external potentials change the average mo-nomer densities and, consequently, change also the ef-fective monom er interaction by the qu antity IZTX$,$~seeeq II-2), the internal potentials kTx$, acting on monomersA and kTxJ/l acting on B are introduced. Hence$ i ( G ) = - P S i j ( G ) u j e f f ( G ) +

(III-19)Ui"ff(ij)=Ui(S')+ +V ( $ ) (111-20)where

with Vll =V , =0 an d V12=V2,=kTx. Equ ations 111-19together with the condition(111-21)

form the RPA equation set. A simple procedure to solvethis se t is presented in Appendix A. Th e proc_edurepro-$des th e correlation functions of the system s, ( 3 ) , an dG ( 4 ) in terms of the correlation functions of the nonin-teracting ideal copolymer chains Sij,Gfi, and G$$Th e coefficients of the free-energy density expan sion inpowers of $ (eq 111-13) may be obtained by putting thesolutions of th e RPA equation s (cf. App endix A) (eq 111-19to 111-21) in to eq 111-14 to 111-16. T h e result rea ds

i= l

F2(ijl, G2)v-lwl+a,, =WG 1 ) =s(al)/w(Gl)- 2x(111-22)[sjl-'(G2)S j2 - ' ( G& ] [SkC1(GJ - k2-'(;3)] (111-23)

r3(G1,G2, G3) =-G!fA(Gl, G 2 , G S ) [ S i l - l ( G l ) - S i2 - ' ( G l ) I X

r 4 ( Q 1 , d2, d 3 , d4) =Yi jk l ( G1 , G2 , G3, G 4) [sil-'(Gl) SiZ-l(G1)l [Sjl-'(G2) -Sj2- ' (+2)1 [Skl- ' (G3) - k 2 - ' ( q 3 ) 1 [Sll-'(G4) - 12- YG4) I(111-24)

withY i j d G 1 , Gz , G3 , G 4) =Y- lC S m , - l ( G? [ G! iA( q ' l , G2 , 4'1X9''GL%(-G', G3, G4) +Glljn(G1, G3, G?G$(-G', G2 , G4 ) +

G$%(Gl, G?G!z$j(-G', a 3 1 d 2 ) l - G!$&(Gl, G z , G3, G4)(111-25)For the case of Gl +4t2 = 0 (or any other Gi +G j =0)formula 111-25 is to be modified (compare the commentbelow eq 111-17). Th e first term on th e right-hand side of(111-25) takes the form

Cs-l(0)G!fA(Gi, 2, O)GL$i(O, G3 , G4) 011-26)mnIn e q III-22to 111-26 th e following notations h as been us edS i . - l ( G ) denotes the (i , j ) th element of the matr ixlld-l(G)ll-the inverse of the m atrix IIS(G))I composed ofthe correlation functions of the ideal independent co-polymer chain Si.(G); (G) denotes the s um of all elementsof the m atrix 11d11 (i.e., S =&Si,); and W( J )s the de-terminant of llSll (i.e., W =SllS22 SI22).A very remarka ble prope rty of th e coefficients r3 ndr4 n the expansion of the free energy in powers of theorder parameter is their independence of the interactionconstant x. Actu ally eq 111-23 to 111-26 reveal th at the se

Theory of Microphase Separa tion in Block Copolymers 1607coefficients depen d solely on correlation functions of idealcopolymer chains, i.e., only on th e comp osition f and thepolymerization index N . This means that th e terms ofhigher order than the second in the expansion of F inpowers of $ are of entropic origin an d have no singularbehavior a t the transition point. It should be also notedthat for f =0.5, Sll=SZ2for ideal chains) an d r3=0 asmight be expected from th e symmetry considerations.Contrary to the coefficients r3 nd r4 he coefficient ofthe second-order r2 eq 111-22) depends on th e interactionconstant x. Th e dependence on x is particularly simple.Th is will facilitate the calcu lation of the spinoda l line (seesection V).Equations III-22 to III-26wll be used to study the phasediagram of th e block copolymer melt (sections V and VI).These equations are very general, they apply for anypolymeric melt comp osed of two m onomers (in particularthey are valid also for a mixture of two homopolymers).Their application to a block copolymer melt requirespreliminary knowledge of correlation functions of idealcopolymer chains. In Appendix B a general method ofcalculating these functions is described.IV . Radiation Scatter in g by Liquid Copolymers

Radiation scattering experiments provide a very at-tractive method for studying the various aspects of themicrophase separation. Th e scattering techniques ofparticular interest are those which give monomer corre-lation functions, i.e., which involve scattering vectorssmaller than u- l and even N-lI2u-l (e.g., light, X-ray(SAXS), and n eutron (SANS) small-angle scattering).Below the MS T, the scattering by th e periodic micro-domain structure gives the characteristic diffractionspectrum. Th e detailed struc ture of the Bragg peaks re-flects the symmetry an d th e periodicity of the system (e.g.,ref 5 an d 8). Above the transition, in th e disordered phase,the scattering spectrum is completely different from th atobserved below th e M ST as well as from th at of a liquidcomposed from small molecules. Th e discussion of thescattering by the disordered phase is the subject of th eprese nt section. Partic ular em phasis will be laid on th erole of monomer interactions. In this respect th e choiceof an appropriate experimental technique should becarefully considered. SAX S is very suitable when thechemical species of t he copolymer chain have significantlydifferent electron densities (e.g., copoly(styrene-isoprene)),which usually im plies a high degree of incom patibility, i.e.,a rather large x. Consequently, th e studied struc ture willexhibit a microdomain architecture (unless chains aresufficiently short) and a diffraction pattern will be ob-served. On the other hand , neutron scattering experimentsmay pe rm it stud y of copolymers formed from two verysimilar species (e.g., copoly(styrene-a-dimethylstyrene) fonly one of them is deuterated. It is even possible to dealwith partially deuterated chains composed of the samemonomers (e.g., polystyrene), in which a case the inter-action co nstan t is very small (x - 0-4-10-3). Therefore,the SANS technique offers a possibility for studying thecopolymers in both cases when a microphase separationoccurs or not, even for long chains.T he scattering intensity for th e disordered phase of theblock copolymer is proportional to the correlation functionS(q'). Within th e RPA this function is given by (see Ap-pendix A, eq A-7)

(IV-1)where S s the s u m of all elements and W the determinantof the matrix composed of th e correlation functions of th eideal independ ent copolymer chains IISijll. The calculation

R G ) =w(G,/[s(a)2xW(G)l


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1608 Leibler Macromoleculesof S,(G) functions with the help of the method exposedin Appendix B is straightforward. For f N >>1 and N(l- ) >> 1 one gets (see B-18)

Slim=& ? , u t x )s z z m =Ng,(l - , x ) (IV-2)where gl(f, x ) , the Debye function, is defined as

gl(f, x ) =2[fx +exp(-fx) - ] / x 2 (IV-3)and

x =q2Na2 /6=qzR2 (IV-4)with R denoting the radius of gyration of a_n ideal chainwith N monomers. Th e final formula for S readssea, =N / [ F ( x )- 2XN (IV-5)whereF ( x ) =gi(1, x ) / k ! i ( f , X)gi(l - , x ) - 1/4[gi(l, ) -gi(f, x ) - gi(1 - , x)I21 (IV-6)

Equation IV-5 for the scattering function s( i j )showsth at for large G vectors ( q R >> l ) , S(G) tend s to zero likeVI2

S(G) ==2 N f ( l - ) / q 2 R 2 (IV-7)Th is means th at on the scale of a coil the monom er densityfluctuations are like those of a Gaussian chain and areindependent of t he effective repulsion between monom ersA and B. Also for very small 4' vectors ( q R


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Vol. 13, N o . 6 , November-December 19805% )

20

Figure 3. Plot of sl(Q)s a function of x =q2R2or th e samplewith f =0.4 and for three values of xN : (-) xN =0.0; (---)x N=2.0; (-) xN =5.0. The curves for the different xN are parallel.Figure 1 shows that the position of the maximum q*does not depend on the monomer interactions. In fact,expression IV-5 has a maximum for x * =q*2R2 or whichthe function F(x*) (independent of x ) has a m inimum.Th is is due to th e fact tha t the monomer interactions arelocal an d would not be t he case if they had a finite range.Th e shape of the peak and th e position of its maximumdepend on the chains' composition f. Figure 2 shows a plotof x * vs. f . Th e dependence is weak for 0.2


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1610 Leibler Macromoleculesand bcc mesophases. In order to decide which mesophasewill actually appear, it is necessary to analyze the freeenergy of these phases in more detail.B. Free Energy of Hexagonal and bc c Mesophases.Let us c op ide r a mesophase characterized by the set ofvectors (Qi) ( i =1, 2, ..., n). Its free-energy density maybe written in the form

F N / k T =2N(x.9- X ) J , ~an$ +Pn$ ; W-8)The determination of the coefficients a , and p, requiresrather tedious calculations of the correlation functions ofGaussian chains by t he m ethod explained in Appendix B.For the sake of simplicity it is important to use somesymmetry properties of coefficients r3 nd r4 and tocontract the notation.First of all we note th at I'3(G1, G2, G3) vanishes unless G1+ j z +G3 =0 and all nonvanishing I'3(G1, &,&) are equal(since l&l=q * ) an d will be denoted by r3.Th e coefficientI'4(tji7 d2,&,d4) # 0 only if xi=:&=0, so tha t r4 ependsonly on th e two indep endent angles, e.g., between vectorsGl an d a2and between vectors ijl and a4(since lQ i l =q* ) .Whe n these angles are equ al for two-diffecent _sets(&) hecoe[ficie@s q e alto equ al, e.g., r4(Qtl,Qtl, Qt2, -Qtz) =r4(Qbl,Qbl, Qb3, Qb3). Theref ore, it is convenient to adoptthe following notation: I'4({&))=I'4(hl, h 2 ) ,where +q2l2=hlq*2an d IGl +?j4l2=h2q*,. It should be noted tha tr4(hl , h,) =r4(hl, 4 - hl - h2)=r4(h2, 4 - h l - h2) ince1411 +22 = l q 3 +G4l2, Is'i +4'41' = Id 2 +&I2, "d Id1+G3l2= +G4l2=(4 -hl -hz)q*2.All vectors Q1 lie in the sam eplane if hl or h2 s equal to 0 or hl +h2=4.Making use of this notation an d of eq V-4, we get thefollowing formulas for th e coefficients a3 and p3 in ex-pression V-8 for the free-energy density of th e triangularmesophase:

a3 =- ( 2 / 3 f i ) ~ r ~ OS (ptl +pt2+ 'pt3) (v-9)p3 =N [ ~ ~ ( o ,) +4r4( o, 1)1/12 (v-io)

In order to determine the relative phase ptl, (ot2, and cp wof Fourier components $(ati) ( i = 1, 2, 3), one shouldminimize F with respect to pti (i =1,2 ,3) . This gives thecondition cos (ptl +pt2+pt3)=1 as -r3 urns out to bepositive) a nd, the na3=- ( 2 / 3 f i ) ~ r , ( V - 1 1 )

Within the approximation adopted here (dropping outterms F5,F6,) it is not possible to determine the relativephases of $(Qti) with more precision. However, this am -biguity in the detailed form of $(?) is not im port ant forcalculation of the microphase separation criterion.The coefficients (Ye an d 0 6 for the bcc mesophase areequal to(Y g =- ( 1 / 3 ~ % ) N I ' ~ ( c o s +cos p +cos y +cos (a - p +7)) 7-12)&=N(~ , ( o ,) +8r4(0,1)+2r4 (o ,2) +2 r 4 ( i ,2) x[COS a+7) +CO S ( p - y)])/24 (V-13)where a = - p4+'p6. Th e free-energy density takes th e m inimal value fora =p =y =0. (Also for the bcc mesophase the mini-mization procedure does not perm it a determination of allof the relative phases pi ( i =1, 2, ..., 6)). Thus

- p3 - p6, P = cp1- p4- p5,and y =

(Yg =- ( 4 / 3 & i ) ~ r ~ ( V - 1 4 )06 =N ( r 4 ( 0 ,0) +8r4(0,1)+2r4(0, 2) +4r4(1, 2))/24(V-15)

relative phase of different components $ ( a )may be im-portant . Thus, the order parameter $(i)akes the form= Iexp[i(Qk?+ +CC ) (v-4)

T he equilibrium value of th e amplitude_$,, $ for an or -dered phase characterized by the se t (Qi) ( i =1, ..., n ) isto be determined by minimizing the free-energy densityF with respect to $ and (p,). For a given xN the stablephase is that corresponding to the smallest free energyF($,). As far as the M ST is concerned, one can calculatexN a t which the transition to different possible orderedphases may occur. In principle, th e mesophase actuallyappearing is that corresponding to th e smallest xN (thehighest temperature r). Th e calculus shows tha t this value(XN), is close to the spinodal ( x N ) , for a large range ofcompositions f . This justifies the adopted approach (seesection V I).It should be pointed out that the third- and thefqurth-order term s F3 and F4 depend crucially on the set(Qi). For this reason, a prediction of the conditions of theM S T requires a detailed analysis of F3and F4.As we havealready mentioned, for f #0.5 the third-order term F3doesnot vanish identically and the MST is, in general, afirst-order phase transition occurring for (xNt


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Vol . 13, N o . 6 , November-December 1980 Theory of Microphase Separa tion in Block Copolymers 1611107

106

105

104

103

102,

-NT,

I I 1 )0.05 01 0.15 02

0.00.2 025 0 3 0.35 0.4 0 6 0.5fFigure 5. ( - N r 3 )as a. function of composition f .

T h e coefficients r3 nd r4(hl, h2)occurring in th e for-mulas or a3, 3, 06, an d &may be calculated with the helpof eq 111-22 to 111-26. One getsr3=-ci jk(i)[si1-l - si2-l][sj1-lSj2-1~[Skl-1 Sk2-l1(V-16)r4(hl,h,) =-fijkl(hl, hJ[Si,.-' - s.24 ] [ S '1 l - S '2-'][skl-l - k 2 - 1 1

(V-17)withYijkl(h1,h2) =Gijrn(hl)S,,-'(hlx*)Gklrn(hl) +

Gi1rn(h,)Srn,-'(h,x*)Gjkn(h2)+Gikrn(4 - h l - h2) xSrnn-'((4 - h l - hz)x*)Gjln(4 - hl - h2) - Gijkl(h1,h2)(V-18)

S..-'1 =Sirl(x*) and Sjy1(O) =1/N. In writing th e corre-lation functions of the ideal copolymer chain, we haveintroduced a notation similar to th at used for coefficientsr3 nd r4 .Therefore %$l(&, G2, G3, G4) is denoted by Gijkl(hi,h2) and G(3d(s'l,&.,43) by Gijk(h),where 1Q1+& I2 =14312=hq*2 (ijl + +a:, =0). The formulas for G+(h) andGi,kl(hl, h 2)are given in Appendix C. Putt ing eq C-1 toC-14 into eq V-16 to V-18 and setting x =x*, one cancalculate r3, 4(0, ) , r4(0,11, r4(0, ) , and r4 ( l , 2) . InFigure 5 we have plotted -Nr3 as a function of the com-position f . Figure 6 shows plots of N r4 (0 , 0), Nr 4( 0, 2),and m 4 ( l , ) vs. the composit ion f. In th e scale of Figure6 it is not possible to distinguish between r4(0, )an d r4(0,2). I t is important to stress that for all f , r4(0, ) >r4(0,1).C. Conditions for the MST. Periodicity of theOrdered Phase . Th e conditions for the transition fromth e disordered phase to the m esophase (characterized byn =3 or n =6) may be written as

Nrlo'[

1

106

10

10

lo5

10

LcI 03 0.2 0.3

n a n LE .Figure 6. Nr4(0,0) (-), Nr4(0,1) (.-), and Nr4(l, ) (---)vs.f . The difference between Nr4(0,1) and Nr4(0, ) cannot beillustrated in the scale of the picture. Fo r f C 0.3 in the scale ofthe figure there is no difference between Nr4(0, ), Nr4(0, ),Nr4(0, ), and Nr4(l, ).Th e first two conditions express the fact t ha t a t equilib-rium the order parameters Ipn take th e value which min-imizes th e free energy F. The last condition says tha t atthe transition point (x =xn),F vanishes (as it vanishesin the disordered phase ). From (V-9) and (V-19)(V-20)pn =3an(1 +T n ) / W n

Y n = [1 - 64Pn(xs- ~ ) N / g a n ~ I " ~

xnN =x&' - a,2/86n

with(V-21)

07-22)Equ ations V-10, V-11, and V-16 to V-18 and eq C-1 to C-3an d C-7 to C-11 show both coe fficients anand Pn do notdepend directly on N (only throug h x * ) , which means tha tthe relevant parameter is the product xN and not x itself.In order to predict the phase which appears after the MSTa32 /p3nd a 6 2 / p 6 should be compared. From (V-12 toV-15) an d (V-22) it follows th at a bcc mesophase appearsfirst if

k =

F($,) =0 requires Gn =1/3 and consequently

> 1(V-23)

4[r4(O9o)+4r4 (o, 1)1r4(0, ) +8r4(o , 1) +2r4 (0 ,2) +4 r 4 ( i ,2)

Otherwise the triangular s tructu re will appear. It is im-portant to note tha t a?


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1612 Leibler Macromolecules

550 t 109 ft

150 1 I I 1 L t0.3 0 35 0 4 0 .45 0 .5 fFigure 7. Plot of Nr4(0, /3 ) (-) an d NI'4(4/3,/3 ) (-) vs. f .the bcc mesophase is expected to appear just after theMS T. Actually, it turn s out (cf. Figures 5 and 6) tha t foran y f r4(0, )


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Vol. 13, No. 6 , November-December 1980 Theory of Microphase Separa tion in Block Copolymers 1613Rh om boh edr ic Mesophase. This mesophase would becharacterized by $(?) tha t has the symm etry of a rho%-bohedric lattice sp anne d by th e following Q vectors: Q1=q*[l,O , 01, Q2 =q*[cos a, in a, 1, and Q 3 =q*[cos a,cos a an (a /2 ) , 1 - cos2 a (1+t an2 (a /2)}] (n= 3) . T hecoefficient 8, equals

p, =N [ r 4 ( 0 ,0) +4r4(o, 4 t an2 (a /2 ) ) ] /12 (V-29)F a c e - C e n t e r e d - C u b i c ( f c c ) M e s o p h a s e ( C l o s e -Pa ck ed H exag onal) . This mesophase is characterizedby I)(?) which has the symm etry of a fcc lattice (or _ahexagonal-close-packed lattice). T he corresponding ,8vectors are Q1=q * _ l ,1 , 1 v 4 3 , Q 2 =q*[-l, 1 ,11 /4 /3 , Q 3=q*[- l , -1 , 11/43 , and Q4 =q*[ l , 1 1 / 4 3 (n=4). T hecoefficient P4 equals

P 4 =N[ r , ( O ,0)+6r4(0, Y3) +2r4(Y3, 73 ) XCOS (PI ~ 2 fi - ~ 4 ) 1 / 1 6

and F takes a minimal value for cos (a-PL +fi - p4) =-1 so t h a tp4=N [ ~ ~ ( o ,) +6r4(o, ) - 2r4(Y3,Y3)1/16 (v-30)Th e coefficients NI',(O, 4/3) and N r 4 ( 4 / 3 ,/3) are plottedvs. f i n Figure 7.It should be noted that the coefficient r4(0, ) has aminimum for h =01 increases with a n increase in h, andtakes a m aximum value for h =2. The n for h- , r4(0,h) ends to r4(0, ) (cf. Figure 6) . It should also be stressedt h a t 1'4(4/3, /3 ) >P4(O, 0) . Hence p2 >3Nr4(0, ) /8 , P,and p, >5Nr4(O, ) /12, and p4>N[7r4(0, ) - 21'4(4/3,4/3)]/16;we see th at th e most stable phase is the lamellarphase for which p1=NT4(O,0)/4 (cf. eq V-25) and the freeenergy F has th e smallest value.T o establish the phase diagram of the liquid block co-polymer ne ar th e spinodal point, we should consider th ephas e equilibria of thre e mesophases: bcc, triangu lar, an dlamellar. We note that p1


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1614 Leibler29). We have shown in section V tha t the fcc mesophasecannot be an equilibrium phase near the MST. Actually,the result of the present theory is not surprising. We havealready stressed that the MST is analogous to the soli-dification of a liquid (cf. sections I1 and 111). For the lattertransition a bcc structure is also favored.% The periodicityof the m esophase occurring just below th e M ST is expectedto be equal to D =2a /q * (Figure 2), i.e., of th e order ofth e radius of gyration of the copolymer chains. Th er e isno shar p interface between th e A- and B-monomer-richdomains bu t the m onomer density varies smoothly overth e whole period (see eq V-4).In section V we have shown th at near the spinodal ofth e copolymer melt, apar t from the bcc mesophase, twoother phases may be stable, namely, the triangular a ndlamellar microdomain structu res. We predict the condi-tions of the first-order transitions betw een these differentphases. Th e transition from the bcc to the triangular phaseis expected to occur for x N = ( x N 1 see eq V-35 andFigure 8). It is of crucial importance to note that (xN) 'is greater than (xN) , ; however, for a large compositionrange (0.2 <


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$1(3) +$2(3) =0 (-4-5)where

uieff(3)Vim$,(3) +V3) (A-6)It is straightforward to find the solution of the_ inear

(A-7)set (A-1) and to obtain t he correlation function S(G)

S(G) =Sii(G)=W ( G ) / [ S ( G ) 2xW(G)lwhere

W(G)=Sl,(G)S22(G) - s122(G) (A-8)S(G) =S,l(G)+2S12(G) +s,,ca, (A-9)W denotes the determinant of the matrix IlS1Iwhose ele-ments S i jare th e correlation functions of th e ideal, njn -interac ting copolymer chains. T he expression for S isdiscussed in deta il in section IV.Making use of th e above result an d expressing Urff(l)nthe form (cf. (A-1) and (A-2))Uieff(l)(G)=Sim- l ( f j )& , ,n(~)Un(G) (A-10)

one can solve the second-order set (A-3) an d get the @3 )correlation function@ 3 ) ( ~ 1 ,?, ii3)?-'6(jl+G~ +G ~ )s(al)S(G2)s(G3)Gl(Gl, G2 , G3)[sil-1(G1) -

si2-'(aI)l[sj,--'(a2) - j2-'(G2)] [Sk l - l ( ; 3 ) - Sk2-l(d3)1(A-11)T he same approach as for the second-order set may beused to solve the third-order set (A-5). One expressesUieffc2)n term s of external potential Ui (cf. (A-3) and (A-4),puts th e result into (A-5),and then finds $i(3) as a functionof Vi. The resulting expression for G"(q',, G2 , a3 , G4) israther complicated_. However, relative to the simplestresponse function S directly m easurable in sc$tering _ex-periments the nonlinear response functions G(3)and G(4)ar e of less inte rss t and so we do not give here a n explicitexpression for G(4). More im portan t is expression for thecoefficient r4 n th e expansion of F in pow ers of fi (cf. eq111-16). It tu rn s ou t tha t the second term in (111-16)

Theory of Microphase Separ ation in Block C opolymers 1615(containing 6c3))ancels exactly with a term in @4) andas a result one gets a com pact expression for Fa, eq 111-24.Appendix B. Correlation Function s of Gaussian,Non interact ing Copolymer Chains

In th is appendix we show how to calculate th e correla-tion functions of Gaussian, noninteracting copolymerchains. Th e method presented here is applicable not onlyto block copolymers but also to systems composed ofchains with an arbitr ary distrib ution of monom ers A andB along the chain.Le t us denote by Z ( I =1 , 2 , ...,N) the Ith m onomer ineach chain and by p r ( i ) the density of m onomers Z at pointE i.e., p I ( i ) =1 f there is such a chain that its Zth mono merlies a t point 7 nd p I ( f l =0 otherwise. Th e qua ntity 6 p I=PI(?') - 1/N characterizes the deviation from the average,uniform distribution in the disordered phase. Making useof the fact tha t ( 6 ~ ~ )0, we may write the expressionsfor the monomer density correlation functions as follows(cf. eq 111-9)

GfY(F1, 72) = (6PI(71)6pJ(72)) (B-1)(B-2)~I)K(F;,2, 73) = ( S P ~ ( T ~ ) S P J ( ~ ~ ) S P K ( ~ )

G&CL(?I,29 73, 74 ) = ( 6 p I ( 7 1 ) 6 p J ( 7 2 ) ~ P K ( 7 3 ) 6 p ~ ( 7 4 )-GfYF1, 72)G&(73, 74) - Gf%&, ?3)G$%72, F4)-G jW l , 74)G$%(72,F3) 03-3)In order to calculate GfY we observe tha t

GIYFi, 72) = (~1(71)~~(7z:2))1/W (B-4)Th ere a re two different ways to have PI(?I)pJ(72) =1.First, there is th e Zth monom er of a chain a t point T1an dthe J t h monomer of th e same chain at point 7> Secondly,there is the Ith monomer a t F1a n d th e J t h a t F2but theybelong to different chains. Th us

1(pI(71)PJ(72))= 72) +l /W (B-5)where PIJ(F;, T2) denotes the probability that the chain withthe I th monomer at point T1has he J t h monomer at pointy2 . For G aussian chains PIJ s a function of - F21withthe Fourier transform equal to

PIJ($)= 1 d 3 r iQiP,J(i9=exp(-ylZ - JI) (B-6)where y =q2a2/6. F rom e q B-4 and B-5 we get

GfYCil, 72) =pIJ(F1, 72)/N (B-7)and its Fourier transform equal to

GfY(Gi, G 2 ) =P I J ( G ~ ) Y - ~ ~ ( G ~G2)/N (B-8)For GfyKwe get the result

GfI)K(Fl, 7'2,73) = (PI(71)PJ(72)PK(73)) +2 / p -((P1(71)pJ(72))+ (pJ(72)pK(73)) +(P1(71)pK(73)))/N =PIJK(~I ,2, 4)/N(B-9)where PIJK(i1, F2,4) s the probability tha t the same chainhas the monomer Z, J, and K at points 71, 2, and y3, re -spectively. In writing (B-91,eq B-5 has been used. Th efunction P ~ J Kay be expressed in terms of pair corr elationfunctions PIJ

'IJK(71, 727 73 ) =PIJ(T1, 72)pJK(72, 73 ) (B-10)for Z


15/16

1616 Leibler=P J I ( i 2 , l )P IK( i l ,3). Fouri er tra nsf orm of GfYK rea ds

Gf%dGl, QZ, G3) =V-6(ql +G2 +G3)PZJ(Gl)PJK(G3)/N(B-11)fo r I C J C K

Very similar considerations can be made for thefourth -orde r response functions. One finds easily th atGfQ)KL(?l,2, 73, 74 ) =p I J K L ( F 1 , 72, 73 , ?4)/N (B-12)where PIJKL(?l, 2,i3 ,F4) denotes the probability tha t thechain has th e It h monomer a t point F;, J t h a t F2,K t h a t?3, and L th at 74. In space we getGfQ)KL(&, 2, G 3 , 4 4 ) =

V-g(Gl +G2 +G3 +Q4)PIJ(Gl)PJK(al +Q 2 ) P K L ( G 4 ) / N(B-13)fo r I


16/16

Vol. 13,No. ,November-December 1980g4(f, 0) =(1 - exp[-(l - ) x ] ) ( f x - 1 +exp(-fx))/x3(C-6a)

g4(f, 1.)= 3) =g3(f, 1) (C-6 b)For third-order response functions Gijkl(h1, h2)we get

G1221(hl, h2) =G2112(hl, h2) =4Wf3(f, 4 - hl - h2)f3(1- , 4 - hl - h2) (C-10)(C-11)G1212(hl, h2) =G2121(hl, h2) =4Mf3(f, h2)f3(1 - , h2)

whereflu,h)=V x / h + x exp(-fx)/(h - 1) + (2 h -3) exp(-fx)/(h - +exp(-fhx)/h2(h - -for h # 0, 1, 3, 4

flu,0) =flu, ) =(fLx2/2 - x exp(-fx)- 3 exp(-fx) - 2fx +3) /x4 (C-12a)f l u , ) =f lu , ) =Ifx2exp(-fx)/:! + x +2fx exp(-fx) +3 exp(-fx) - 3}/x4 (C-12b)fd , )=I1 - exp[-(l - )xIl(l /h - (h- 2) exp(-fx)/(h- - exp(-fhx)/h(h - - x exp(-fx) / (h -1)) lx4

(2h +1)/h21/24 (c-12)

(C-13)for h # 0, 1, 3, 4

f&, 0) =11 - exp[-(l - )xl l xf~cf ,) =(1 - e xp[ - ( l - fix11 x

(2 exp(-fx) + x exp(-fx) - 2 + x ) / x 4 (C-13a)(1 - exp(-fx) - 2 x 2 exp(-fx)/2 - x exp(-fx))/x4(C -13b)

f 3 ( f , h )={h- 1 - h exp(-fx) +exp(-hfx)j/h(h - l ) x 2for h =0, 1, 3, 4

f 3 ( f , 0) =glv, x ) / 2

(C-14)

(C-14a)f3(f, 1) =(1 - exp(-fx) - x exp(-fx)j/x2 (C-14b)In order to obtain GzZz1 (hl,t 2 ) =Gzzlz(hl, 2)=GZlz2(hl,h2)=Glzzz(hl, 2 ) nd Gzzzz ne should change f to 1 -in eq C-8 and C-7, respectively.References and Notes

(1 ) (a ) J. Polym. Sci. , Part C 1969, 26. (b) Aggarwal, S., Ed .Block Copolymers; Plenum P ress: New York, 1970. ( c )

Theory of Microphase Separation in Block Copolymers 1617Burke, J. J., Weiss, V., Eds. Block and Graft Copolymers;Syracuse University Press: Syracuse, NY, 1973. (d ) Polym.Eng. Sci. 1977, 17 ( 8 ) .(2 ) Skoulios, A; Finaz, G; Parro d, J. C. R. Hebd. Seances A cad.Sci. 1960. 251. 739.(3 ) Molau, G. E. In ref I b, p 79 .(4 ) Kiimpf, G.; Kromer, H.; Hoffmann, M. J. Macromol. Sci.,Phys: 1972, B6, 167.(5 ) Skoulios, A. In ref IC,p 121.(6 ) Folkes, M. J.; Keller, A. In ref IC, 87 .(7 ) Folkes, M. J.; Keller, A. In The Physica of Glassy Polymers;Howard, R. N., Ed.; Halsted Press: New York, 1973.(8) Skoulios, A. Adu. Lip. Cryst. 1975, 1, 169.(9 ) Kiimpf, G.; Hoffmann, M.; Kromer, H. Ber. Bunsenges. Phys .Chem. 1970, 74, 851.(10) Folkes, M. J. ; Keller, A,; Scalisi, F. P. Kolloid Z. 2. olym.1973, 251, 1.(11) Terrisse J. Ph.D. Thesis, University of Strasbourg, 1973.(12) Mathis, A.; Hadziioann ou, G.; Skoulios,A. Polym. Prepr. , Am.Chem. SOC. ,Diu. Polym. Chem. 1977, 18,282.(13) Math is, A.; Hadziioannou, G.; Skoulios, A. Polym. Eng. Sci.1977, 17, 570.(14) Gervais, M.; Gallot, B. Makromol. Chem. 1973, 171, 157.(15) Cervais, M.; Gallot, B. Makromol. Chem. 1973, 174, 193.(16) Dunn , D. J.; Krause, S.J.Polym. Sc i ., Polym. Le t t . Ed. 1974,12. 591.(17) Toporowski, P. M.; Roovers, J. E. L. J. Polym. Sci. , Polym.Chem. Ed. 1976, 14, 2233.(18) Krause, S.; Dunn, D. J.; Seyed-Mozzaffari, A.; Biswas, A. M.Macromolecules 1977, 10, 786.(19) Hadziioannou, G. Th6se d e 36me Cycle, University of Stra s-bourg, 1978, unpublished.(20) Widmaier, J. M.; Meyer, G. C. Polymer 1978, 19, 398.(21) Meier, D. J. J.Polym. Sci. , Part C 1969, 26, 81 .(22) Inoue, T.; Soen, T.; Hashimoto, T .; Kawai, H. J.Polym. Sci. ,Par t A-2 1969, 7, 1283.(23) Marker, L. Polym. Prepr. , Am . Chem. SOC., iu. Polym. Chem.1969, 10, 524.(24) Learv. D. F.: W illiam . M. C. J.Polvm. Sci.. Part B 1970.8.335.(25) Kirgbaum, W. R. ; Y h g a n , S.;T d b e r t , W. R. J.Polyh.Sci. ,Polym. Ed. 1973, 11, 511.(26) Meier, D. J. In The Solid State of Polymers; Geil, P. H.,Baer, E., Wad a, Y., Eds.; Marcel Dekker: New York, 1974.(27) Helfand. E. Macromolecules 1975.8, 552.(28) Helfand, E.; Wasserman, Z. R. M&romolecules 1976, 9, 879.(29) Helfand, E.; W asserman, Z. R. Macromolecules 1978,11,961.(30) Meier. D. J. In ref IC.D 105.

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H e l f h d , E. Acc . Che k. Res. 1975, 8 , 295.Helfand, E. ;Wasserman. Z. R. Polvm. Enp. Sci. 1977. 17. 582.. .Krause S . In ref IC,p 141.de Gennes, P. G. J.Phys. (Paris) 1970, 31, 235.de Gennes, P. G. Scaling Concepts in Polymer Physics;Cornel1 University Press: Ithaca, NY, 1979.Joanny, J. F. Th6se de 3Bme Cycle, University of Pa ris VI,1978.Brazovskii, S. A. Zh. Eksp. Teor. Fir . 1975,68, 175.Alexander, S. ; Mc Tague, J. Phys . Rev . Le t t . 1978, 41, 702.Helfand, E. Macromolecules 1978, 11, 683.Flory, P. J. Principles of Polymer Ch emistry; Cornell Uni-versity Press: Ithaca, NY, 1953; Chapter 13.Krause, S. J.Macromol. Sci., Rev. Macromol. Ch em. 1972, C7,251.de Gennes, P. G. J.Phys . Le t t . (Par is ) 1977, 38, L441.BrBzin, E.; Le Guillou, J. C.; Zinn-Justin , J. In Phase Tran-sitions and Critical Phenomena ; Domb, C., Green, M. s., de.;Academic Press: New York, 1976; Vol. 6.Peliti, L. Ph.D. Thesis, Queens Mary College, University ofLondon, 1976.Wo hrer, M. These de 3Bme Cycle, Univ ersity of Pa ris VI, 1976.BouB, F.; Daoud, M.; Nierlic h, M.; William s, C.; C otto n, J. P. ;Farnoux, B.; Ja nni nk , G.; Benoit, H.; Dupplessix, R.; Pico t, C.Neutron Inelast ic Scat ter ing , Ro c . Symp. , 1977 1978,1 , 563.Landau, L. D. Phys. Z. Sou. 1937,11, 26,545.Landa u, L. D.; Lifshitz, E. M. Statistichesk aia Fizika, Part1; Nauka: Moscow, 1976; Chapter XIII.Hoffmann, M.; Kampf, G.; Kramer, H.; Pampus, G. Adu.Chem. Ser. 1971, N o . 99, 351.Grosiud, P.; G allot, Y.; Skoulios, A. Mol. Chem. 1970,132,35.

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Theory of Microphase Separation in Block Copolymers