Chuck Yeung, An-Chang Shi, Jaan Noolandi and Rashmi C. Desai- Anisotropic fluctuations in ordered...

8/3/2019 Chuck Yeung, An-Chang Shi, Jaan Noolandi and Rashmi C. Desai- Anisotropic fluctuations in ordered copolymer pha

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Macrom ol. Theory Simul. 5 , 291-298 (1996) 291

Anisotropic fluctuations in ordered copolymer phasesChuck Yeungavb), n-Chang Shi* "1 Jaan Noo landi"),Rashmi C. Des aib)a) Division of Science, Pennsylvania State University at Erie, The Behrend College,Erie, PA 16536, USAb, Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7") Xerox Research Centre of Canada, 2660 Speakman Drive, Mississauga, Ontario,Canada L5K 2L1(Received: September 14, 1995)SUMMARY:

The random phase approximation is reformulated to investigate the anisotropicfluctuations about an ordered polymer phase. This very general method is applied to thelamellar phase of block copolymers. The calculated anisotropic scattering intensity capturesthe main features observed experimentally including the secondary peaks due tofluctuations with hexagonal symmetry. We also determined the limits of metastability of thelamellar phase as well as the bending and elastic moduli of the lamellae.

The random phase approximation (RPA) has proven to be a valuable tool inunderstanding polymer systems. Essentially the RPA is a self-consistent expansionaround the mean field state. The RPA can give the stability and the magnitude ofGaussian fluctuations in an equilibrium state. For example, with the RPA one canpredict the scattering function and phase boundary of homopolymer I ) and copolymermelts'). However, the usual formulation of the RPA assumes a spatially uniformmean field state','). If the translational symmetry is broken this assumption is nolonger valid. Instead, the RPA must be reformulated as an expansion around thebroken-symmetry mean field state. Previous efforts in this direction have beenrestricted to an expansion around a non-interacting state3)or have not included thefull effects of incompressibility4). However, the correct mean field solution is crucialfor the validity of the RPA, otherwise one cannot distinguish between approximationsto the zeroth-order solution and contributions arising from Gaussian fluctuations.In this letter we formulate the RPA as an expansion about the exact mean field solutionincluding the full effect of interactions and incompressibility. Applying this method toblock copolymers in the lamellar phase, we obtain the density-density correlationfunction CRPAr, J), characterizing the anisotropic composition fluctuations. Wereport on the results of three important applications of this technique: (1) The experi-mentally observed scattering intensity which is the Fourier transform of CRPAr,J). Ourmethod accounts for the anisotropic nature of the fluctuations and captures the mainfeatures found experimentally5).(2)The mean field solution is linearly unstable if anyeigenvalue of [CRPA]-'s negative. The boundary where an eigenvalue first becomeszero is the limit of metastability of a particular phase, i. e., the spinodal line for thatphase. The least stable mode at this point dominates the fluctuations. (3) The elasticmoduli of the ordered phase can be and are extracted from the behavior of the scatteringfunction near the Bragg peak. Significant corrections to the limiting strong stretchingtheory result are found for experimentally relevant parameters.0 996, Huthig & Wepf Verlag, Zug CCC 1022-1344/96/$05.00


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292 C. Yeung, A.-C. Shi, J. Noolandi, R. C . DesaiDue to their amphiphilic nature, A B diblock copolymers form a variety of ordered

micro phase^^.^). The order-disorder transition ( O m ) from the fully disorderedhomogeneous melt to the ordered state occurs roughly at xZ = 10 for f = 1/22),wherex is the Flory-Huggins interaction parameter, Z is the degree of polymerization,andfis the chemical composition (volume fraction of theA blocks). Changes inf ffectthe shape and packing symmetry of the ordered structure. Besides the well-understoodlamellar, cylindrical, and spherical phases '), more complicated structures such as thebicontinuous cubic phase9) and the hexagonally modulated and perforated layeredphases 5, have been identified recently. The important role of composition fluctuationsin diblock copolymers was first studied by Fredrickson and Helfand '1. However, thevalidity of this theory is restricted to the regions near the order-disorder boundariessince it uses the structure factor of the disordered state. The approach developed in thisletter allows a general theoretical treatment of anisotropic composition fluctuationsand can be used as the framework for the systematic study of fluctuations within eachphase. As a first step in this study, we apply the generalized RPA to the diblockcopolymer lamellar phase. The anisotropic nature of the fluctuations is revealed in thescattering functions, which can be obtained via small angle neutron scattering. For thelamellar phase, Hamley et al. have observed from shear oriented samples that: (1) Atlow temperature, the scattering pattern shows meridional Bragg peaks at a wavevectork, with higher order reflections at 2k0 and 3k,, indicative of very small layerundulations. ( 2 ) At higher temperatures, four weak peaks corresponding to the layerundulations are observed, which shows an approximate hexagonal order. Theperiodicity of the in-plane undulations is about 10% larger than the interlayer spacing.Our calculated scattering functions for the lamellae capture all these features.

We now give a brief outline of the derivation of the generalized RPA, leaving thedetails to a future publication. We begin with the many-chain Edwards Hamiltonianand introduce the monomer concentrations @,(I-)nd two auxiliary fields w,(r )(a = A, B ) following Helfand") and Hong and NoolandiI2). The partition functionof the system can be written as a functional integral over @ , ( I ) and w,(r ) with a freeenergy functional,

where V is the volume of the system, and Q, is the partition function of a singlediblock copolymer chain in the external fields, mu@) 2) . The mean field approxima-tion reduces the interacting chain system to a system of independent chains in the self-consistent fields of the other chains. Xchnically, this amounts to evaluating thefunctional integral over w and @ by the saddle-function method, i.e., replacing theintegral by the maximum value of the integrand 12 ) . This extremization yields a set ofmean field equations, with solution @ i o ) ( r )nd of)@).he mean field free energy isthen obtained by inserting the mean-field solution into the free energy expression.

To include fluctuations we expand @ o, round the mean field solution @ u ( r )@f)(r)+ & , ( r ) , w,( r ) = of)(r) + a m a @ ) .The single-chain partition function Q,can be then expanded in terms of 6w,(r)and it is easily shown


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Anisotrop ic fluctuations in ordered copolymer phases 293

( - 1 ,7 drl. . .dr,v / Z ) InQc = C t )_. . ,n ( r , , . ., r,)6cou,(rl). . 6wan(r,,)fl n . a,... n

where Ci:! , , , ,un(r l , . ., r , ) = (Ga,(r1). @un(r,,))$o)s the nth-order cumulantcorrelation function for a noninteracting diblock chain in the self-consistent externalfields w$"(r). These correlation functions can be obtained once the mean fieldsolution is known. The free energy can now be expanded as F = F(O)+ F( ' )+ F(') +. . . whereFco) s the mean field free energy,F( ' )= 0, and the higher-order terms canbe written in terms of the single-chain correlation functions. To this point theformulation of the theory is exact. Now we introduce the RPA, which amounts toreplacing the functional integral over the fields dco,(r) (for fixed by themaximum value of the integrand, i. e., the RPA is simply the mean field theory appliedto an ensemble in which is held fixed. The lowest-order, the RPA free energybecomes

12F = F(O)+- dr dr' CRPAI , r' ) 6@(r) @ (r ' )+ O(6G3)

and the second-order RPA correlation function CRPAs of thehere a@=form

-

where Zis the unit operator, and c(r, J ) s the correlation function for the independentincompressible chains

E(r, r') = C(O)(r,J ) - [d(O) , Z ( ~ ) ) - ~ L I ( ~ ) Ir, J )and the quantities C(O), Z ( O ) , andfunctions ch9$(r, r ' )

are combinations of the two-point correlation

C(O)(r,J) = G?(r, ) - 2G$(r, J ) + G j ( r ,J)Z(O)(r,r ' ) = C,? (r, J ) + 2cJ ( r , r') + C#(r, r ')A(O)(r,r') = c$'j (r, J ) - C$j(r, r')

Formally the equations above are identical to those for the high-temperature RPA.The essential difference is that the independent chain correlation functions c:)8(r, r')arise from the full mean field solution and hence include the anisotropic nature of thesystem. If q:b are replaced by the correlation function for Gaussian coils we recoverthe result of Leibler').

We now apply the theory to the lamellar phases. For fixed x , Z, f and lamellarperiodicity 1,we solve the mean field equations numerically. The free energy densityis then minimized as a function of I to obtain the equilibrium periodicity for that setof x,f nd Z. Due to the periodic structure the eigenmodes of the correlation functionwill be Bloch functions in the normal (z)direction and Fourier modes in the transverse( x ) directions. We therefore write all the correlation functions in terms of this basis,


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294 C. Yeung, A.-C. Shi, J . Noolandi, R. C. Desai

where K(n) = k,/n with k , = 2x /L CF(z, z) is obtained for a range of q and Kand the eigenvalues and eigenmodes of [CRPA]- re identified. If there are nonegative eigenvalues for any q and K the lamellar structure is linearly stable and weobtain the scattering function by Fourier transforming with respect to z andz.We thenincreasef nd repeat the procedure until the lamellae become unstable. This is repeatedfor different values ofx and2 to obtain the limits ofmetastability of the lamellar phase.

Fig. 1 shows the calculated scattering intensities Sk,q or f = 0.65, xZ = 25.6 andX Z = 15.0, corresponding to the lamellar system used by/Hamley et aL5)at low andhigh temperatures. The anisotropic nature of the fluctuations are clearly shown in thesescattering functions. There are two Bragg peaks at k = f ,, q = 0 corresponding tothe lamellar order. The higher order Bragg peaks at k = k2k,, q = 0 are apparent,indicative of strong lamellar ordering. We also observe two wings extending fromeach Bragg peak. These wings increase in amplitude as X Z s decreased eventuallyleading to four clear secondary peaks at k = +k,/2 and q = k0.87k0.As discussedbelow these peaks are due to in-plane fluctuations with apprximately hexagonal order.The calculated scattering patterns correspond to the experimentally observed onesremarkably well. In particular, the main features of the experiments, the appearanceof the four weak peaks due to in-plane fluctuations, the approximate hexagonal orderof secondary peaks5) and the 10% difference in the periodicities in q and k , are allreproduced in the theoretical scattering functions.

XZ= 25.60

-2k2kO

XZ 15.00

-2k0

Fig. 1 . Scattering functions forf = 0.65. Dark regions corres-pond to large scattering. The la-mellar phase becomes linearly un-stable at X Z = 14.8. The spreadof the wings around the prima-ry Bragg peaks at X Z = 25.6 isdue to scattering from a mode ofhexagonal symmetry. AtX Z = 15.0 this leads to clear se-condary peaks at k = *k , / 2 ,q = k0.87 k,


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Anisotropic fluctuations n ordered copolymer phases 295

AFig. 2. The leastunstable modes atk = d,l = k , / 2 . Inmode A the minorityphase bulges and pin-ches while in mode Bthe majority phase bul-ges and pinches. ModeA becomes unstablefirst with increasingf

Fig. 2 shows the primary modes contributing to the scattering k = k o / 2 (A) themode in which the minority phase pinches and bulges while the majority phase hasconstant thickness and (B ) the mode in which the majority phase pinches and bulgeswhile the minority phase has constant thickness. The energies of the two modes dependon the transverse wavenumber q. At smaller q the B mode is less stable than the A modewhile the A mode becomes less stable at larger q. In the weak segregation limit (WSL)the scattering is dominated by mode A. In this regime the energyof mode A is minimumat q = 0.87 k , which indicates that the dominant fluctuations have hexagonalsymmetry (qhm= p k , / 2 = 0.8600 ko) .This is consistent with the experiments ofHamley et al. who also identified mode A as being the lamellar deformation mode5).

The lamellae become linearly unstable with increasing f a t fixedXZ or equivalentlydecreasingXZat fixedf). The first unstable mode is always mode A. However, the ratioq*/k, whereq* is the transverse wavenumber of the unstable mode becomes somewhatsmaller with increasing xZ. Hence the instability is no longer towards a hexagonalsymmetry in the strong % 10) segregation limit (SSL). An additional qualitativedifference between strong and weak segregation is the relative stabilities of modes Aand B. For large xZ, mode B is the least stable nearf = 0.5. It is only near the stabilityboundary that the mode A becomes less stable. This is because, nearf = 0.5, a defor-mation of the majority phase will result in smaller strain. However, near the instabilityline, the minority phase thickness is of order of the radius of gyration, R , = b mwhereb is the Kuhn length. Collective behavior of chains is possible on this length scaleand leads to clumping of the minority phase. In the WSL mode A is always the leaststable since the thickness of the minority phase is of order R , for allf.

The limit of metastability of the lamellae is shown in Fig. 3 along with the mean fieldphase boundaries 3 ) . The linear instability occurs at larger values off than the meanfield lamellar-to-cylindrical (L-to-C) and lamellar-to-gyroid (Lto-G) transitions. Thepresence of a small amplitude phase should be reflected in the linear stability. Ourresults show that in the WSL the stability boundary and phase boundary are very close,


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296 C . Yeung, A.-C. Shi, J. Noolandi, R. C . Desai

Nx

fFig. 3. The limit of m etastability (i.e., the spinodal line) for the lamellar pha se (dashedline) alo ng with the m ean field phase d iagram (solid lines) with L = lamellar, G = gyroid,C = cylindrical an d S being the spherical phase. Varying Z at fixed x Z has little effect onthe stability. Th e instability is always at larg erfth an the L to -G transition bu t the two almostcoincide in the weak segregation regime

thu s f luctuat ions may be able to stabil ize a small ampl itude phase. O n the other han d,in the SSL the stabili ty bo un dar y deviates strongly f rom th e G to -C transi tion. In thisregime a small amplitude hexagonal phase is not possible even if f luctuations areincluded. A free energy comparison including fluctuation effects using our RPAanalysis is straightforward b ut requires t he RPA arou nd al l the ordered phases whichwe leave to a future study.

We now turn to the calculat ion of t he elastic an d bending m oduli of th e system. T heelastic an d bend ing mo duli o f diblock lamellae have been previously calculated in theW SL j 4 , Is ) and SSL 1 6 ) . However, the availability of the scattering function s permitsa general calculation of these elastic moduli for the entire range of xZ. For smalldistor t ions of the per iodic lamellae we can write @ ( r )= @o(z )- u (z , x)d@, /dz +O(u2) here @o(z) s the lamellar solut ion an d u(z, x ) is the deviation of the lamellarpositio n f rom th at o f perfect ordering. Hence, the scattering intensity is given by

1.( @ k , + k , q @ - ( k o + k ) , - q ) = @? ('k,q u - k , - 4 ) where @ I = 1 I - ' 1 dz @O(z ) ik.z 1 .

0Th e elast ic modulus B and bending modulus K ca n be extracted f rom th e scatteringnear th e Bragg peak since, from general distortion theory, (uk, u - k , - q ) = k, T(Bk2+ K q 4 ) - ' .A s a test of this meth od we obtained th e elastic mod uli B f rom both thek dependence of S,, nd the L dependence of the free energy density. We find verygo od agreement using these two techniques. Since there is n o alternative meth od forobtaining the bending m odulus we focus on K . Th e inser t of Fig. 4 shows tha t Sk o , ,behaves as 4-4 t small q as expected. T he K obtained from th e analysis of the small


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Anisotrop ic fluctuations in ordered copolymer phases 297

Fig. 4. The bendingmodulus K (in unitsof k , T/b)vs. XZ orZ = 1280,f = 0.5.Th e solid line is theprediction of strongsegregation theory. AtX Z = 44.8, K isapproximately 1 /2that of the SSL result.The insert shows theq -4 behavior of S , ,4(in units of b3k, T?near the Bragg peakfor XZ = 12.8

lo-k4

1o-210 20 30 40 50xz

q behavior is show n in Fig. 4 alo ng with the result f rom strong segregation theory Is).A s expected K vanishes as XZ pproaches the ODT a t xZ = 10.5. At larger values ofxZ, he behavior of K i s consistent with the strong segregation scaling al tho ug h a largerrange of XZ s required to clear ly test the scal ing. Th e imp orta nce of cor rec t ions to thestrong segregation app roximation is demonstrated in our result. For example, for XZ= 44.8 (of ten considered t o be i n th e strong segregation regime) th e K obta ined f ro mthe ful l mea n f ield RP A is approximately 1/2 tha t o f the strong segregation result .

We tha nk P rof. R S. Bates for a useful discussion, an d Dr. M. Matsen for providing themean field phase boundaries used in Fig. 3. R. C. D. and C . Y. are grateful for sup port fromthe Natural Sciences and Engineering Research Council of Canada.

) P.-G. De Gennes, Scaling Concepts in Polymer Physics: Cornell University Press,2, L. Leibler, Macromolecules 13, 1602 (1980)3, J. Marko, T. Witten, Phys. Rev. Lett. 66, 1541 (1991)4, C. Yeung, A. C . Balazs, D. Jasnow, Macromolecules 26, 1914 (1993)5 , I. W. Hamley, K . A. Koppi, J. H . Rosedale, F. S. Bates, K. Almdal, K. Mortensen,6, F. S. Bates, Sciences (Washington,D.C.) 151, 898 (1991);F. S. Bates, G. H. Fredrickson,

Ithaca, NY 1979

Macromolecules 26, 5959 (1993)Annu. Rev. Phys. Chem. 41, 525 (1990)


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Chuck Yeung, An-Chang Shi, Jaan Noolandi and Rashmi C. Desai- Anisotropic fluctuations in ordered...

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