Transitions to periodic structures in block copolymer melts.

\Vc analyzc ncarly continuous transitions to pcriodic structures expanding the dcnsity of thc periodic strllcture in multiple harmonics. A thrce dimensional hexagonal structure identified as a "Lamellar-Catenoid" is the equilillrium structurc near thc transition. It consists of alternating lamellar domains connectcd by an orthogonal array of hexagonally packed cylinders. \Ve obtain the periodicity, the density distribution inside the unit cell and the equilibrium morphology for diblock copolymers melts of various molecular compositions. RESUMEN. Analizamos transiciones a estructuras periódica. .• expandiendo la densidad. de la estruc. tura en armónicos. La estructura de equilibrio encontrada es una lamela.catenoide, una estructura hexl,gonal en tres dimensiones. Consiste en dominios lamelares conectados ortogonalmente por cilindros formando un arreglo hexagonal. Obtenernos la periodicidad, la densidad en la celda unitaria y la morfología de equilibrio en bi.copolímeros de varia') composiciones moleculares. PAes: 64.70.Dv; 05.20.-y; 36.20.Ey The possibility of nearly continuous transitions from an isotropie state to a periodie strlletllre has led to numerolls stlldies of the eqllilibrillm strueture at the transition point. in systems ranging from weak erystallization in Iiqllids [1-4J to self-assembly systems, sueh as block eopolymers [5-8]. The eonventional Landall mean field approaeh [1,2J (LMFA) prediets a bee st.ruetllre. When the densit.y fluetuations are self-eonsistent.ly inclllded in the analysis [4J a lamellar ,t.rllet.ure is obtained. By expanding the density in t he periodie struet.lIre in mlllt.iple harmonies, we show here that a three dimensional hexagonal strlleture is more favorable. In block eopolymers, this ,tmeture is ident.ified as a "Iamellareatenoid" [81; it eonsists of alternating lamellar domain, eonneeted by an ort.hogonal array of hexagonally paeked eylinders. The transition is analyzed expanding t.he free energy of the isotropie state in power series of the density fllletllations Óp(r) arollud the mean density Po [1), 6F(pd _ ~ ~ p-k kT 2! L-Pk So(k) + ,}. I¿ f3(k, k', k")PkPk,pk"Ó(k + k' + k") ,l. + ,} I¿f.,(k, k', k", k"')Pkpk'Pk"Pk",Ó(k + k' + k" + k"') (1) 4. 206 MÓNICA OLVERA DE LA CRUZ where Pk are the Fomier eomponents of bp(r). In most systems Sül(lkl) has a minimum at a single waveveetor magnitude k'. In the LMFA, a eontinuous transition oeeurs at Te when the symmetry is sueh that f3 = O. At Te. So(lkl = k') diverges leading to an instability of the isotropie state to density fiuetuatians of wavevector magnitude equal to k'. At the transition bp has the symmetry of the periodie struetme. It can therefore be expanded in plane waves whose veetors are the reeiproeal lat tiee veetors of the periodie struetme, the eompositional harmonies [1]. The largest eontribution to the free energy in nearly eontl:lUOUS transitions (f3 ~ O) is from the first harmonie veetors [21 11\';1)1 = k', r = 1, ... 1 ni, whcrc nI is the number oC ncarcst-neighhors (n.n.) in tite feciprocailattice; "1 = I for a lamellar (Jam), "1 = 3 for hexagonally paeked eylinders (hpe), "1 = G for a bee, etc. The distanee between n.n. planes in the strueture is d = 2rr/ k'. Sinee higher harmonie veetors (i > 1) ha\"(' larger magnitudes (1/\;;)1 = nik', r = 1, ... , "i, with ni > 1), their eontribution is often negleeted. There are struetures, however, whose first ami seeond harmonie veetors have equal magnitudes. Consider a hexagonal (hexe(a) struetme with indiees (h/.-ie), e = 2n [i = _ (h +k )1, amI distanee between planes d( hkf) = {4(h 2+ k2 + hk )/3a + e2 / e2} -1(2, where a is the lattiee spacing in the hexagonal plane, and e is the length of the perpendicular axis. The first harmonie veetors are {1\;1)}r=I.2.3 = {(IojO),(0IjO),(fl00)}, and the seeond is I\¡') = (0002). In a hexagonal structure the ratio e/a is not fixed. In a c1ose-paeked struetme e/a = j873 and l\¡'i = LOG07k'. \Vhen e/a = v'3, however, I/\;I)¡ = Wn = k'. To analyze the hexc(a struetme. higher arder harmonies have lo be eonsidered in bp. \Ve expanded bp up to the m-th harmonie for whieh 1/\;mll = 21,' for various struetures (for f3 ~ O the higher order harmonies i > m are negligible near lhe transition), m ni bp(r) = ¿ ai ¿(exp{i(rl\;;) + 'P~II)}+ e.e.)