Hee Hyun Lee et al- Orientational Proliferation and Successive Twinning from Thermoreversible...

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Orientational Proliferation and Successive Twinning from Thermoreversible Hexagonal-Body-Center ed Cubic Tra nsitions Hee Hyun Lee, Woon-Yong J eong, and Jin Kon Kim*  Department of Chemical Engineering and Polymer Research Institute, Electronic and Computer   Engineering Divisions, Pohang University of Science and Technology, Kyungbuk 790-784, Korea Kyo Jin Ihn  Department of Chemical Engineering, Kangwon National University, Chunchon 200-701, Korea Julia A. Kornfield and Zhen-Gang Wang  Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Shuyan Qi  Department of Chemical Engineering, University of California, Berkeley, California 94710  Rec eive d Ma y 31, 2001; Revised Man uscript R ec eive d S eptem ber 7, 2001 ABSTRACT: The determ inistic proliferat ion of the orientation of hexagonally packed cylinders (HEX) from the twinned body-centered cubic (BCC) phase is investigated by synchrotron small-angle X-ray scattering (SAXS) and rheo- optic al methods for di- and triblock copol ymers of styrene a nd isoprene (SI and SIS). Repeated heating and coo ling cycles star ting from a n initially aligned HEX produce a proli feration of cyli nder orientat ions from successivel y twinned BCC sta tes. The evolution of the orient ation distribution of HEX cylinders produces a decrease in t he birefringence an d increase in t he modulus with each successive generation. The cylinder axes of the degenerate HEX states coincide with the 11 1directions of the twinned BCC due to the epitaxial growth of cylinders from the twinned BCC. The distribution of the cyl inder axes of the degenerate H EX states a mong the 111 directio ns of the twinn ed BCC is fo und to be affected by memory of the prior HEX stat e, which decays with a nnea ling time in t he BCC state. 1. Introduction Controlling nanostructure in materials has attracted intense interest because of its relevance to technologies ran ging from bio medicine to photonics. Macrosco pic alignment of self-assembl ed nan ostructures in soft materials can be induced by applied fields, especially flow. Here we demonstrate that an initially flow-induced orientation can provide a platform for generating dis- tinct orientation distributions by thermal treatments that take the material back and forth across an order- order transition. The orientation distributions accessed in this way cannot be generated directly by any other methods. Spec if ically, we show that , starting from a single orientation of the hexagonal cylinder (HEX) phase of  an AB or ABA block copolymer, repeated heating and co oling cycl es across the tra nsition bounda ry between the HEX and the body-centered-cubic (BCC) sphere phases lead to continued proli feration of the HEX orienta tion an d su cc essive twinning of the BCC stru c- ture. To our knowledge, the phenomenon we report is the first example of deterministic proliferation of ori- enta tion of an ordered str uctur e in either soft-c ondensed materials or crystalline solids. This phenomenon re- quires a special crystallographic r elationship between two ordered ph ases connected by a n order-order transi- tion. The two ordered phases must each produce, in a deterministic manner, more than one orientation from a single orientat ion u pon the t ran sitio n from one to the other, and these multiple orientations must not form a closed set through repeated heating/cooling cycles across the transition boundary. Nondeterministic proliferation oc curs wh en one of the order -order tra nsitions pr oduces a co ntinu ous distribution of orienta tions, for example, the lamellae (LAM) to HEX t ran sition in block co- pol ymers: a single orientation of LAM produces HEX orientations tha t are completely ran dom in t he azimuth angle about the LAM normal. A lack of proliferation oc curs when t he orienta tions probed by the order-order transition form a closed set; for example the transition between the cubic and tetragonal phases in crystalline solids invo lves three orientations of the tetragonal pha se, all of which ma p back onto the sin gle o rient at ion of the original cubic phase, so repeated heating/cooling across the transition does not lead to any new orienta- tions. The crystallographic relation between two ordered pha ses of bloc k copolymers connected by an order -order transition is known as epitaxy. 1-4 In th e case of the HEX fBCC tran sition, epitaxy dictates th at one of the 11 1directions of the BCC phase co inci des with tha t of the cylinder axis of the original HEX, while the other three * To whom co rrespondence should be addressed: e-mail  jkkim@ postech.ac.kr. Present address: Advanced Materials Research Institute, LG Chemic al Ltd., Research Park, 104-1, Yuseong Science Town, Taejon, South Korea, 305-380. 78 5  Macromolecules 2002, 35 , 78 5-79 4 10.1021/ ma010951c CCC: $22.00 © 2002 A merican Chem ical Soc iety Published on Web 01/03/2002

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Orientational Proliferation and Successive Twinning from

Thermoreversible Hexagonal-Body-Center ed Cu bic Tra nsitions

H e e H y u n L e e ,† Wo o n - Yo n g J e o n g , a n d J i n K o n K im *

  Department of Chemical Engineering and Polymer Research Institute, Electronic and Computer   Engineering Divisions, Pohang University of Science and Technology, Kyungbuk 790-784, Korea

K y o J i n I h n

  Department of Chemical Engineering, Kangwon National University, Chunchon 200-701, Korea

Jul i a A . K ornfi el d and Z hen-G ang Wang

  Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125

S h u y a n Q i

  Department of Chemical Engineering, University of California, Berkeley, California 94710

  Received Ma y 31, 2001; Revised Man uscript R eceived S eptem ber 7, 2001

ABSTRACT: The determ inistic proliferat ion of the orientation of hexagonally pa cked cylinders (HEX)from the twinned body-centered cubic (BCC) phase is investigated by synchrotron small-angle X-rayscattering (SAXS) an d r heo-optical m ethods for di- and triblock copolymers of styrene a nd isoprene (SIand SIS). Repeated heating and cooling cycles star ting from a n initially aligned HEX produce aproliferation of cylinder orientat ions from successively twinned BCC sta tes. The evolution of the orient ationdistribution of HEX cylinders produces a decrease in t he birefringence an d increase in t he modulus witheach successive generation. The cylinder axes of the degenerate HEX states coincide with the ⟨11 1⟩directions of the twinned BCC due to the epitaxial growth of cylinders from the twinned BCC. Thedistribution of the cylinder axes of the degenerate H EX states a mong the ⟨111 ⟩ directions of the twinn edBCC is found to be affected by memory of the prior HEX stat e, which decays with a nnea ling time in t heBCC state.

1 . I n t r o d u c t i o n

Controlling nanostructure in materials has attracted

intense interest because of its relevance to technologiesran ging from biomedicine to photonics. Macroscopicalignment of self-assembled nan ostructures in softmaterials can be induced by applied fields, especiallyflow. Here we demonstrate that an initially flow-inducedorientation can provide a platform for generating dis-t inct orientation distr ibutions by thermal treatmentsthat take the material back and forth across an order -order transition. The orientation distributions accessedin this way cannot be generated directly by any othermethods.

Specifically, we show that , s tart ing from a singleorientation of the hexagonal cylinder (HEX) phase of an AB or ABA block copolymer, repeated heating andcooling cycles across the tra nsition bounda ry between

the HEX and the body-centered-cubic (BCC) spherephas es l ead t o cont i nued pr ol ifer at ion of t he H E Xorienta tion an d su ccessive twinning of the BCC stru c-ture. To our knowledge, the phenomenon we report isthe first example of deterministic proliferation of ori-enta tion of an ordered str uctur e in either soft-condensed

materials or crystalline solids. This phenomenon re-quires a special crystallographic r elationship betweentwo ordered ph ases conn ected by a n order -order transi-

tion. The two ordered phases must each produce, in adeterministic manner, more than one orientation froma single orientat ion u pon the t ran sition from one to theother, and these multiple orientations must not form aclosed set through repeated heating/cooling cycles acrossthe transition boundary. Nondeterministic proliferationoccurs wh en one of the order -order tra nsitions pr oducesa continu ous distribution of orienta tions, for example,the lamellae (LAM) to HEX t ran sit ion in block co-polymers: a single orientat ion of LAM produces HEXorientations tha t a re completely ran dom in t he azimuthangle about the LAM normal. A lack of proliferationoccurs when t he orienta tions probed by th e order -ordertransition form a closed set; for example the transition

between the cubic and tetragonal phases in crystallinesolids involves three orientations of t he tetragonalpha se, all of which ma p back onto the sin gle orient at ionof the original cubic phase, so repeated heating/coolingacross the transition does not lead to any new orienta-tions.

The crystallographic relation between two orderedpha ses of block copolymers connected by an order -ordertra nsition is known as epitaxy.1-4 In th e case of the H EXf BCC tran sition, epitaxy dictat es th at one of the ⟨111⟩directions of the BCC phase coincides with tha t of thecylinder axis of the original HEX, while the other three

* T o w h o m c or r e s p on d e n ce s h ou ld b e a d d r e s s ed : e -m a il [email protected].

† Present address: Advanced Materials Research Institute, LGChemical Ltd. , Research Park, 104-1, Yuseong Science Town,Taejon, South Korea, 305-380.

78 5 Macromolecules 2002, 35 , 78 5-79 4

10.1021/ma010951c CCC: $22.00 © 2002 American Chem ical SocietyPublished on Web 01/03/2002

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⟨111⟩ directions form th e skeletons of a t etra hedron withthe first, with th eir azimuth orienta tion conforming tothe hexagonal symmetry of the cylinders. Furt hermore,it was both predicted theoretically5 and demonstrated

experimentally6,7

tha t th e tra nsition from HE X to BCCshould yield two BCC structures connected by twinningas opposed to a single BCC. Upon t he r everse tra nsition,each of the ⟨111⟩ directions of the BCC can become thecylinder axis in the HEX phase. Since there are fourequivalent ⟨11 1⟩ directions in each of the BCC twins,with one of the four (along t he initial cylinder orienta -tion) shared by both, seven distinct HEX orientationsare to be expected after one HEX f twinned BCC fHEX cycle (Figure 1). However, t he ant icipated multi-plicity of the HEX orienta tions du ring th e BCC to HEXtra nsition wa s not reported in earlier studies by Koppie t a l.1 for a poly(ethylene-alt -propylene)-block -poly-(ethylethylene) (PEP -PEE ) diblock copolymer or byKimishima et al.8 for a polystyrene-block -polyisoprene

(SI) diblock copolymer.

In this paper, we provide clear small-angle X-rayscatt ering (SAXS) evidence showing the generation of seven cylinder directions upon cooling th e twinn ed BCCphas e back i nt o H E X . W e f ur t her dem ons t r at e t hatrepeated heating and cooling cycles lead to continuedproliferation of HEX orientations and successive twin-ning a s a result of the m ultiplicity of HEX orienta tionsproduced during each HEX f twinned BCC f H E Xcycle. This continued proliferation requires that eachcooling cycle produces n ew cylinder orienta tions notalready present in the previous generation. If the HEXto BCC transition yielded only one BCC structure, thenonly the first reverse t ra nsition would give new cylinder

orienta tions; repeated heat ing/cooling would not yieldany new cylinder orienta tions since t he four directionsform a closed set . Therefore, the fact that the HEXproduces a twinned BCC is essential for the subsequentprolifera tion of cylinder directions an d su ccessive twin-ning of the BCC dur ing repea ted h eat ing/cooling cycles.Since th e epitaxial relation between th e H EX and thegyroid phases is similar to that between HEX and BCC,w e ant i ci pat e a s i m i l ar phenom enon i n t he H E X T

gyroid t ran sition.

Our stu dy shows tha t th ere is a strong memory of theHEX orient at ion from one gener at ion t o the next, whichdecays with annealing t ime in the BCC phase. Thismemory effect may explain the apparent discrepancy

between th e present findings and those of the previousstudies.1,8

2 . E x p e r i m e n t a l S e c t i o n

Materials. We use a comm ercial-gra de SIS tr iblock copoly-mer , Vector 4111 (Dexco Polymer s Co.), whose ph ase beha viorhas been stu died previously.4 An SI diblock copolymer, SI-C1,was synthesized by anionic polymerization in cyclohexane(Aldrich Chemical Co.) with s-BuLi as an initiator at 40 °Cunder purified argon.9 The diblock is matched to almost half of the t riblock, and the t wo have nearly the sa me T OOT for theHE X-BCC transition (Table 1). We employed both SIS triblock and SI diblock copolymers in order to compar e with previousstudies of diblocks of PEP -P E E 1 an d SI.8 We found that thequalitat ive featu res of orienta tion proliferation are similar inthe diblock and triblock systems; we primarily show results

for the SIS triblock copolymer in this paper.The samples for rheology, birefringence, and SAXS experi-

ments were prepared by solvent casting from toluene (10 wt%) in th e presen ce of an an tioxidan t (Irga nox 1010, Ciba-GeigyGroup), followed by annealing at 130 °C for 48 h in a vacuumoven. Although toluene is a slightly better solvent for PS thanfor PI, the HEX ordered structure after an nealing shows noevidence of a solvent effect. This sample is referred to as anas-cast sample. Well-aligned HE X cylinders were obtainedfrom t he a s-cast sam ple by imposing a large-amplitude oscil-latory shear (LAOS) with ω ) 1 r a d /s a n d 1 00 % s t r a inamp l it u d e a t 1 70 °C, a t emp erat u re b et w een t h e T g of thestyrene-rich domains and the T OOT of the H EX-BCC transi-tion. The LAOS was done with 50 mm diameter parallel platesin a Rheometrics RDS-II with nitrogen purge for 2.5 h untilthe birefringence and storage modulus, G′, reached steady-

state values (Figure 2). The sample was rapidly cooled (reach-ing ambient temperat ure in less tha n 1 m in) immediately afterLAOS to quench the well-aligned state.

Synchrotron Small-Angle X-ray Scattering. The SAXSmeasurements with synchrotron radiation were conducted atthe 3C2 and 4C1 beam lines at the Pohang Light Source (PLS),Korea.10 The wavelength of the primary beam (0.1598 nm,photon energy of 7.76 keV) was selected by use of a mono-chromator consisting of two Si(111) single crystals, and thenwas focused on a CCD detector plane by a bent cylindricalmi rror. T h e rect an gu l ar b ars are cu t from each q u en ch edspecimen at a position n ear the edge of the 50 m m disk. Thesecut sam ples are mounted in a temperatur e-controlled cell ,oriented such that xy , xz , an d yz planes a re probed, respec-tively. The x-, y-, a n d z-axes correspond to the flow, sheargradient, and vorticity directions, respectively. Synchrotron

F i g u r e 1 . Schematic of the tr ansition: HEX-0 f twinnedBCC-I f HEX-I.

T a b l e 1 . M o l e c u l a r C h a r a c t e r i s t i c s a n d TOOT o f S a m p l e s

U s e d i n T h i s S t u d y

sample code total MW,a g /m ol P S ,b wt % Mw /Mnc TOOT (°C)

Vect or 4111 143 000 18.3 1.11 185 ( 1SI-C1 67 200 18.5 1.02 182 ( 1

a By LALLS (low-angle laser light scattering). b By 1H NMR.c By GPC calibrated with PS standard.

F i g u r e 2 . Change of the birefringence and storage moduluswith time during LAOS with ω ) 1 rad/s and γ0 ) 100%.

78 6 Lee et al. Macromolecules, V ol. 35, N o. 3, 2002

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S AXS m e a su r e m en t s a r e p er for m e d in s it u in t h e t e m -p erat u re cel l as t h e s y s t em i s h eat ed an d co o l ed t h ro u g ht h e O OT as i n di cat ed i n Fi gu re 3 . T h e t w o-d imen s ion alpatterns are presented with a l inear intensity scale (Figures4 and 6).

R h e o - o p t i c a l Me a s u r e m e n t s . Both the birefringence an dthe storage modulus, G′, of a specimen during heating andcooling through the HEX-BCC t ran s i t ion w ere meas u redsimultaneously with a rheo-optical device.11 Here, the bire-

fringence (∆n xz ) i n t h e xz p l an e w as meas u red b y u s e o f apolarization-modulation optical train, which enables rapidmeasurement of the magnitude and orientation of birefrin-gence. The gap thickness used in the shear sandwich geometrywas t ypically 0.3 mm , corresponding to an optical pat h lengthof 0 .6 mm. For t h i s op t ical p at h l en g t h , n o i n di cat i on of  depolar ization of light wa s observed for t he in itial, polydomainstat e (for which it would be m ost severe). Therefore, depolar-ization wa s neglected in the ana lysis of the dat a. After LAOSwith ω ) 1 rad/s an d 100% stra in am plitude at 170 °C for 2.5h until the birefringence and G′ both reached steady-statev alu es , rep et i t iv e h eat i n g an d cool in g exp erimen t s w erecondu cted with sm all-amplitude, low-frequency dynamic shea r(1% strain amplitude and 0.2 r ad/s) to probe the mechanicalproperties of the sample while minimizing any potential effectof shear dur ing HEX-BCC and BCC-HEX tr ansitions. The

heating and cooling rate was 0.33 °C/min.T r a n s m i s s i o n E l e c t r o n M i c r o s c o p y . The microdomain

structures of well-aligned HEX (point a in Figure 3) a nd of the first degenerated H EX (point d in Figure 3) were inves-tigated by TEM after the sa mple was quenched with ice water.Cryogenic ultrasectioning was performed with a n RMC (MT-700) cryomicrotome at -1 00 °C. A H i t ach i t ran s mi ss ionelectron microscope operated at 75 kV was used to obtainmicrographs of the specimens sta ined with OsO4 for 7 min,which selectively stained the PI matrix black.

3 . R e s u l t s a n d D i s c u s s i o n

A li gnme nt of th e H E X Cyl i nders. Large-amplitudeoscillatory shear was effective in inducing alignment of t he H E X phas e. T he as -cas t s am ple has near ly zer obirefringence, indicating that there is no preferentialHEX orientation. But within 100 s of shearing at 170°C w i t h a r i m s t r ai n am pli t ude γR ) 1 0 0 % a n d afrequency ω ) 1 rad/s, the birefringence increasedr api dl y and G′ decreased a s the init ially r andomlyoriented HEX grains of the as-cast sample of Vector4111 became aligned along the shear direction.3 After100 s, both the birefringence and G′ changed slowly(Figure 2). The orientation was very strong. The ob-served birefringence was close to th eoretical pr edictionsfor a monodomain:12 ∼90% of the value based on formbirefringence alone (see Appendix). The SAXS peaksobserved in the xy a nd xz planes ar e very shar p (Figure4); as a mean s to qua ntify the degree of orienta tion, we

evaluate an orientation factor in both the xy a nd xzplanes 13

where

θ is as indicat ed in Figure 4b (the peak int ensity is alongπ   /2 and 3π   /2, which leads to the factor of  -2 i n eq 1relative to the more familiar expression for nematicorder oriented along 0 and π ), and I (θ) is evaluated atthe first-order peak in the xy plane for the {10 0} planeand at the second-order peak in the xz plane for the{11 0} plane, respectively. In the xy plane F  ) 0.90, an din the xz plane F  ) 0.88. (If the finite smear ing due tothe instr umen t were taken into account , a higher valuewould be obtained.) TEM images also confirm that ahigh degree of alignment ha s been induced (Figure 4).We refer to this initially aligned state as HEX-0.

F o r m a t io n o f t h e T w i n n e d B C C. The tran sit ionfrom oriented HEX-0 to the first-generation twinnedBCC-I is ma nifested in the mechanical an d opticalproperties and the SAXS pattern. Upon first heatingfrom H EX-0, the st orage modulus of the aligned sa mpleof Vector 4111 first increases gradually between 170 and182 °C, then increases sharply from 182 to 190 °C,before leveling off (Figure 5). The 2D SAXS results(Figure 6a -c) indicate that the init ial increase of   G′coincides with t he growth of undu lations on th e cylin-ders. As un dulations grow, the intense 2 qm peaks of thew el l-al igned H E X (Fi gur e 4a) dis appear and w eak 

F i g u r e 3 . Schedule of the thermal history of the heating/  

cooling cycles for 2D-SAXS experiment. Images were taken atthe following positions: (a) after alignment in the HEX (HEX-0), (b) in the HEX-BCC transition region (undulating HEX),(c) in t he BCC phase (twinned BCC-I), (d) in t he HEX regionafter the first cooling (HEX-I), and (e) in the HEX region afterthe second cooling (HEX-II).

F i g u r e 4 . Strong alignment of the HEX cylinders in t he flowdirection after LAOS (HEX-0, position a in Figure 3) isconfirmed by the 2D SAXS pa tterns and TEM images takenin thr ee diffraction plan es: (a, A) xy , (b, B) xz , and (c, C) zyplan es. The 2-fold qm an d 2qm peaks in the q xq y plane in panela arise from the (100) plane seen in TEM (panel A), and the

strong  3qm p eak i n t h e q xq z plane in panel b reflects the(110) plane of HEX seen in pan el B, which implies that thecylinders are well aligned with the (100) plane of the HEXcylinders parallel to the xz plane [on the basis of the relativeintensity of the first-order peaks in the q xq y a n d q xq z planes,there is less than 10% of the (110) orientation]. The hexagonalstructure in the zy plane of panel C is in good agreem ent withthe SAXS pattern (panel c).

F  ) -[3⟨cos2θ⟩ - 1] (1)

⟨cos2θ⟩ )∫0

π  I (θ) cos

2θ si n θ dθ

∫0

π  I (θ) sin θ dθ

(2)

 Macromolecules, V ol. 35, N o. 3, 2002 Thermoreversible HEX-BCC Tran sitions 78 7

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 2qm peaks appear (Fi gur e 6a). T he H E X t o B CCtransit ion temperature (T OOT ) of Vector 4111 is 185 (1 °C, as determined from synchrotron SAXS experi-ments during heating at a rate of 1 °C/min for HEX-0.14 T hus , t he s t r ong i ncr eas e i n G′ ar ound 185 °Ccoincides with the complete spli t t ing of undulatingcylinders into spheres,15,16 for m i ng a t w inned B CC(Figures 6d-f and 12). These beha viors are consistentwith the results in ref 3. In th e same temperat ure ra nge,∆n xz decreased slowly with the formation of undulatingcylinders and then decreased precipitously to zero ast he undul at ing cyli nder s s pli t i nt o s pher es at T OOT

(Figure 5). The twinned BCC obtained from HEX-0 uponheat ing is den oted BCC-I.

Prol i ferati on of C yl i nder O ri entati ons from Su c-c e s s i v e l y Tw i n n e d B C C . A distinctly differen t orien-tat ion distribution (HEX-I) is observed upon r etur ning

to the HEX state from BCC-I compared to the init ialaligned state HEX-0. The transition from the twinnedBCC-I t o HEX-I du ring t he first cooling occurs a t ∼17 2°C (Figure 5), well below T OOT , consistent with priorstudies.6,7,14 This can be explained by the existence of akinetic barrier for mer ging the spheres.7,16 Once spheresmerge into cylinders during the cooling process, themodulus drops and the birefringence increases; inter-estingly,∆n xz reaches only one-third t ha t of th e initiallyaligned HEX sample, while G′ i n t he new H E X s t at ebecomes 1 order of magnitude h igher than that of theinitially a ligned H EX sample (Figure 5). These r esultssuggest t ha t some of the cylinders h ave changed orien-tation from the initial shear direction.

The 2D SAXS pa ttern s pr ovide more direct evidence

of t he emergence of n ew cylinder orientations. TheSAXS pat tern s in both q xq y and q xq z planes (Figure 6g,h)for HEX-I are completely different from those obtained

for HE X-0 (Figur e 4a ,b). The  3qm high order peaks inFigure 6g-i confirm that the structure is that of theHEX pha se. The positions of the qm peaks in the threereflection planes in Figure 6g-i are similar to those of t he t w i nned B C C - I at 200 °C i n Fi gur e 6d-f, even

t hough t he hi gher or der peaks at  2qm have disap-peared. The similarity in the first-order peaks indicatest hat t he or i ent at i on of t he {10 0} pl anes of t he H E Xlattice coincide with the directions of the {110} planesof t he B CC l at t ice for al l t hr ee r efl ect i on planes .Equivalently, t he spheres along the ⟨11 1⟩ direction of 

the BCC lattice have been connected to form the axesof the new HEX phase, in accord with the so-calledepitaxial relationship between BCC and HEX.1-3,5 -7,17

The TEM micrograph in Figure 7 in the xy plane afterth e first cooling also reveals cylinder orienta tions otherthan the original aligned cylinder direction indicatedb y t h e a r r ow . T h er e for e , w e in fe r t h a t t h e s ev endegenerate HE X states t hat can be created with th eircylinder axes a long the seven ⟨11 1⟩ directions of atwinned BCC (Figure 1) are indeed formed upon thetransition from BCC-I to HEX-I.

Each of these seven, locally-aligned HEX statesobtained after the first cooling upon repeated heatingand cooling will produce a twinned BCC again withfurther proliferation of more HEX orientations in thesecond cooling. The process thu s ha s a treelike stru cture(Figure 8). The further decrease in ∆n xz and increasein G′ during the second cooling process (Figure 5)indicate this continued proliferation of HEX orienta-tions. The SAXS patterns shown in Figure 6j-l afterthe second cooling (at point e in Figure 3) are similar

F i g u r e 5 . Repeated temperature sweeps of G′ a n d ∆n xz withω ) 0.2 ra d/s an d γ0 ) 1% at a ramping rate of 0.33 °C/minstarting from the HEX-0 initial condition with no h oldingperiod at the upper and lower temperatures.

F i g u re 6 . (a -c) Two-dimensional SAXS patterns at theH E X-BCC transition temperature (undulating HEX, point bin Figure 3) show four new peaks indicated by the arrows inboth (a) q xq y an d (b ) q xq z planes due to the fluctuation of undulating cylinders.21-24 (The real space illustration of un-dulating cylinders is shown in ref 22.) (d -f) Two-dimensiona lSAXS patterns of BCC-I (point c in Figure 3) are characteristic

of a twinned BCC. The distinct appearance of four  2qm

peaks in (e) q xq z ar e from t he (200)a and (200)b planes (Figure12a) of the twinned BCC structure.25 The strong 2-fold qm

peaks along the q y axis in (d) q xq y are from the reflections of (110)f  and (110)g planes belonging to each twin (Figure 12b).26

The clear 6-fold patterns at qm an d  2qm in (f) q zq y indicatet h at t h e cy li n ders s p li t i n t o BCC s p h eres al on g t h e ⟨11 1⟩direction. (g-i) Two-dimensional SAXS pa tter ns of HEX-I sta teat 170 °C after t he first cooling (point d in Figur e 3). (j-l) Two-dimensional SAXS patterns of HEX-II at 170 °C after thesecond cooling (point e in Figure 3).

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to those after the first cooling, although t he scatt eringpeaks show weaker intensity than those of the f irstcooling. The positions of the SAXS peaks of the first and

second generation (HEX-I an d HEX-II, r espectively)agree well with the patterns expected for Bragg diffrac-tion from the corresponding ensembles of HEX orienta-tions (e.g., azimuthal plots of the first-order peaks,Figure 9a). The calculated SAXS patterns reveal thereason that HEX-I and HEX-II have similar scatteringpatt erns (Figure 9b): the seven orienta tions present inHEX-I give the predominant contribution to the scat-tering in the xy , xz , and yz planes for HEX-II as well.However, the intensity is lower for HEX-II since some(roughly 9 / 32) of the m aterial is distr ibuted am ong th e

other 18 orientations present in HEX-II (Figure 8). Theaccord between observed and calculated patterns sug-gests tha t th e same epitaxial relationship holds for thesecond cooling as well.

Repeated heat ing/cooling cycles lead to furt her pro-liferat ion of new cylinder orienta tions, wh ich is ma ni-fested in smearing of the SAXS profiles. As is clear fromFi gur e 9a, t he azim ut hal i nt ens it y dis t r ibut ion ap-proached a nearly isotropic state after nine heating/  cooling cycles.

Memory of the Prece di ng H E X State. The cylinderaxes of the seven degenerate HEX states in the f irstgeneration are determined from the corresponding ⟨111⟩directions in the twinned BCC; however, the relativeamounts of the distinct orientations in HEX-I appearnot t o be proportional to t he corresponding ⟨111⟩ orien-tations in BCC-I. The SAXS patterns show that theorientation corresponding to HEX-0 is overrepresentedin HEX-I compared to a calculation that gives each of the seven orientat ions a weighting in proportion t o thecorresponding ⟨11 1⟩ orientation in BCC-I (indicated inFigure 8): the r elative intensity of the peak s at 90° and270° is higher in t he observed SAXS pat tern (Figur e 9a,first generation) than in the calculated pattern (Figure

9b, first generation). A bias in favor of the orienta tionof HEX-0 would also explain why a nonzero birefrin-gence is observed for HEX-I: if the seven orienta tionsin HEX-I were present in proportion to the correspond-in g ⟨111⟩ dir ect i ons , t he m at er ial w oul d r et ain onaverage the cubic symmetry present in BCC-I, and thebirefringence would be zero for HEX-I and all subse-quent HEX generations. Indeed, we can use the ob-served birefringence in HEX-I to estimate the preferencefor HEX-0 over the other six orientations; and we canuse the birefringence of HEX-II to see whether this

F i g u r e 7 . TEM micrograph of HEX-I taken in the xy plane.The white ar row points at cylinders h aving the initial HEXcylinder direction.

F i g u r e 8 . Proliferat ion of HEX orientat ions u pon the first and second cooling (volume fra ctions indicated in par enth eses ar e forthe case that all the ⟨111 ⟩ orientations of each BCC are equally likely to produce corresponding HEX orientations). The underlinedHEX entr ies ar e six newly formed orientations from th e first cooling (HEX-I), an d th ose shown in boldface type a re 18 new onesformed upon th e second cooling (HEX-II). Upon heat ing thr ough th e HE X f BCC transition, each HEX orientation gives rise totwo BCC twins, one of which is the BCC t win tha t pr oduced the pa rt icular HEX orientat ion in th e preceding BCCfHEX tr ansition.Upon cooling through the BCC f HEX tr ansition, each BCC t win produces four HEX orientat ions, one of which is the HEX fromwhich t hat particular twin came in the preceding HEX f BCC.

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preference for the orientation of “the HEX from whichyou came” holds for subsequent generations, as well.

The birefringence of an ensemble of HEX orienta tionscan be com put ed given t he dir ect i on and r elat i veamounts of th e various orientations. In HEX-I, th eorienta tions a re t hose of the ⟨11 1⟩ directions in the twot w ins pr esent i n B CC -I . B ot h t w ins have a ⟨11 1⟩direction along the flow direction, denoted e A. In addi-t ion, each twin contributes three more orientat ions,denoted (e B, e C, e D) and (e B′, e C′, e D′) (Figur e 13). If the

unit vector e A is set to (1/  3)(111), the vector compo-

nents of the degenerate orientat ions of HEX-I are t hosegiven in Table 2 (calculated as described in the Ap-pendix). The six new cylinder orientations obtainedupon first cooling from twinned BCC-I are under linedin Figure 8.

In the next generation (HEX-I to BCC-II to HEX-II),each of the seven vectors, e A, e B, e C, e D, e B′, e C′, a n de D′, w hich ar e t he cyli nder or i ent at i ons of H E X-I ,produces an other p air of BCC twins. For example, if weconsider e D′ [HEX-I(D′) in Figure 8], one of the twins[BCC-I(A′) in Figure 8], has ⟨111⟩ directions of th e twinin BCC-I th at gave rise to the H EX along e D′ [i.e., BCC-I(A′)], and the other twin [BCC-II(D ′) in Figure 8]sanew BCCsintroduces three new ⟨111⟩ orientations, e A′D′,

e B′D′, and e C′D′ (Figur e 13). (The n ew orienta tion vectorsfor th e second cycle a re denoted e ij, indicating the it horientation proliferated from e j.)

In this wa y, 18 new cylinder orienta tions in H EX-IIare obtained from e B, e C, e D, e B′, e C′, and e D′ after thesecond cooling. The 18 newly prolifera ted HE X orient a-tions upon the second cooling are shown in boldface typein Figure 8 (vectors are listed in Table 2). Indeed, noneof these 18 vectors coincide with each other. Using thesecylinder orienta tions, we can calculate th e relative valueof t he bir efr i ngence i n t he xz plane, ∆n xz , o f t h edegenerate HEX states relative to the birefringence of the initial well-aligned HEX-0, ∆n xz

ma x (see Appendix):

where v i is the volume fraction of HEX-i, and θi a n d φi

are angles of the vector of the cylinder orienta tion i.I f a l l t h e ⟨111⟩ orientations are equally l ikely to

produce HE X states with cylinders along t heir orienta-tion, th en the volume fractions of each orienta tion arethose indicated in paren theses in Figure 8. In th is case,∆

n xz

eff 

becomes zero for the first an d a ll subsequent H EXstates, as mentioned above. Instead, the experimentaldata show considerable birefringence in HEX-I andHEX-II (Figure 5), reflecting the preference for theinitial HEX orientation (HEX-A in Figure 1) evident inthe SAXS pattern (Figure 9). The birefringence observedin HEX-I (Figure 5) is consistent with a 3-4-fold grea terpreference for the orientation of the HEX from which agiven BCC came relative to the other HEX orienta tionsformed in the BCC f HEX tra nsition. For example, thebirefringence of HEX-I is captured by eq 3 with νi (i )

 A) ≈ 3.3ν j ( j *  A). Further, the birefringence of HEX-IIrelative to HEX-I is also captured by a 3 -4-fold biasfactor when it is applied for each of the HEX orienta-tions propagating forward from HEX-I through BCC-II

to HEX-II.To observe how the memory of th e prior HE X orienta -

t ion d eca ys w it h t im e in t h e B CC s t a te , w e h a vemeasured the birefringence of HEX-I as a function of the ann ealing time in th e twinned BCC-I. The birefrin-gence of HEX-I decreases with increasing an nealingtime in the twinned BCC-I, both for an SI diblock copolymer having a HEX-BCC transition (Figure 10)an d for t he SIS tr iblock. The two types of systems s howsimilar HEX-I structure in both SAXS and rheo-opticalexperiments as functions of annealing time in BCC-I.The bias factor observed for the case of no annealing inthe BCC phase was similar for th e diblock (3.8 for SI-C1 in Figure 10) and triblock (3.3 for Vector 4111 inFigure 5).

Since the relative change in birefringence is used toi nfer t he bias fact or , i t r epr es ent s a fai r ly r obus testimate. The analysis is not affected by the lack of aquantitative model for the value of the birefringence of a HEX monodomain.18 Sma ll imperfections of the initialalignment of HEX-0 have little effect on the estimate,since a small but finite breadth of the initial orientationdistribution cancels out of the birefringence rat io. Wedo not detect depolarization of light by our polydomainsamples; if it were significant, the actual bias factorcould be larger than inferred by this method.

Wh y H a s H E X P r o l i f e r a t i o n N o t B e e n R e p o r t e dB e f o r e ? We now contrast our f indings with those of  Koppi et al.1 and Kimishima et al.8 Koppi et al.1 studied

F i g u r e 9 . Change in the scattering intensity of first-orderpeaks with proliferation of new HEX directions in the q xq y

plane: (a) SAXS experiment and (b) calculated a zimutha l

intensity distribution. The calculation assumes Gaussianpeaks whose width is determined from the SAXS data forHEX-0.

∆n xzeff  ) ∆n xz

ma x ∑i) A

G

v i(cos2θi - si n

2θi cos

2φi) (3)

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t he H E X t o B C C t o H E X t r ans i t i on i n a PE P -P E Ediblock copolymer with PEP cylinder domains ( f PE P )0.25) using dynamic shear (2% strain amplitude and 0.5rad/s) during hea ting an d cooling at a rat e of 1 °C/min,with no ann ealing in t he BCC state. Upon the reversetransition from BCC to HEX, the modulus of the newHEX essentially returned to that of the initial alignedHEX. In contr ast to Figure 5, this lack of change in t hemodulus of th e HEX pha se from th e initial aligned HEXindicates t hat the orientat ion distribution in th eir first-generation HEX with a short dwell t ime in the BCCphase (no annealing) is s imilar to the init ial al ignedHEX orienta tion distribution. Kimishima et al.,8 usinga polydomain, ordered SI diblock copolymer, also ob-served that the original cylinder orientation was recov-ered in each grain following the HEX to BCC to HEXtra nsitions, even when t he sam ple was an nealed for anhour in the BCC sta te. As we ha ve described, there is amemory effect of the pr ior H EX orienta tion th at persists

for some time in the BCC phase. We infer that in bothprior studies, the residence time in the BCC state wass hor t com par ed t o t he m em or y decay t i m e of t hei rmaterials.

It is reasonable to hypothesize that the decay times

of t he pr ior m at er ial s coul d be m uch l onger t hanobserved in the present materials (several hours, Figure10). In the case of Koppi et al . , 1 the high degree of  entanglement ( N PE P /  N ePE P of 12 an d N PE E /  N ePE E of 6.6)coul d r es ult i n s low er r elaxat ion t han t he pr esentsystem (SI diblock with N PS /  N ePS of 0.7 an d N PI /  N ePI of 8.3).19 In the case of Kimishima et al., 8 proximity of th eexperimental temperatures (T OOT of 116 °C, T anneal of 120 °C) to th e T g of the PS-rich domains (∼80 °C) couldresult in slower relaxation than our material, SI diblock (T OOT of 182 °C, T anneal of 190 °C).14

The nature of the memory effect and, consequently,the mechanism for i ts relaxation are not known. Ki-mishima et al.8 speculate t hat ther e is some an isotropyin the BCC structure reflecting the HEX orientationfrom which it came. If such an anisotropy is present, itis below the detection limit of SAXS. Also, since theanisotropy they suggest is on nanoscopic length scales,it would be expected to relax rapidly (on the time scalefor redistribution of S-I junctions over the surface of an individual spherical domain). Alternatively, Matsen 17

suggested that the poor correlation of spheres originat-ing from different rupt uring cylinders in both t ime andspace cause the memory effect. However, we did notobserve a sharpening or an increase in intensity of theBCC diffraction peaks with time in the BCC phase. 14

Further studies of the transient structure in the BCCstate are needed to determine the physical basis of themem ory effect.

T a b l e 2 . N e w C y l i n d e r D i r e c t i o n s f r o m ⟨11 1⟩ D i r e c t i o n s o f F i r s t a n d S e c o n d T w i n n e d B C C s a

a Each eij of HEX-II is proliferated from ei of HEX-I. The θ a n d φ are t he an gles defined in Figure 1, taking eA a n d e z in Figure 13 tobe the unit vectors of  x a n d z, respectively.

F i g u r e 1 0 . Effect of annealing in the twinned BCC state on∆n xz of HEX-I for SI-C1 diblock copolymer.

 Macromolecules, V ol. 35, N o. 3, 2002 Thermoreversible HEX-BCC Tran sitions 79 1

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4 . C o n c l u s i o n s

I n t hi s w or k, t he t her m al l y r ever s i bl e H E X-BC Ctransition was examined through several repeated heat-ing/cooling cycles. A key finding of our study is t heorientational proliferation of the HEX phase and suc-cessive t winning of the BCC dur ing t hese cycles. Thisunique phenomenon is a consequence of two crucialmechanisms in the HEX-BCC tran sition, the epitaxialrelationship and the generation of a twinned, rather

than a single, BCC from the HEX phase. These twom echani sm s dict at e t hat , s t ar t ing fr om an i nit i all yaligned HE X, seven cylinder orienta tions (including th einitial one) are created after the first heating/coolingcycle and 25 cylinder orientations (including the sevenpresent in HEX-I) are generated after the second cycle.Two-dimensional SAXS, rheo-optics, and TEM resultsare fully consistent with th ese mechanisms. The n onzerobirefringence of HEX-I suggests a preference for thecylinder orientation in the initially shear-aligned direc-tion, which indicates some memory of t he init iallyaligned HE X state. This memory effect is shown todecrease with annealing time in the twinned BCC state.Differences in the memory effects between our systeman d th ose of previous st udies ma y be responsible for the

fact that previous studies did not observe the orienta-tional prolifera tion an d successive twinning r eported inour work.

A c k n o w l e d g m e n t . We a cknowledge h elpful discus-sions with Professors T. Hashimoto, T. Lodge, S. Oka-moto, C. Y. Ryu, S. Sakurai and A. C. Shi. This work was supported by KOSEF (97-05-02-03-01-3), by theApplied Rheology Center governed by KOSEF (2000G0202), by TND projects supported by th e Ministry of Science an d Technology (MOST) an d by NSF (CTS-9729443) and AFOSR (LC MURI). Synchrotron SAXSat t he PLS (3C2 and 4C1 beam lines) was supported byMOST an d Poha ng Ir on an d St eel Co. (POSCO). Vector4111 was kindly provided by Professor C. D. Han at theUniversity of Akron.

Ap p e n d i x : D e r i v a ti o n o f  ∆ n xz f r o m t h eL o r e n t z-L o r e n z E q u a t i o n

For a single-crystal-l ike HEX cylindrical micro-domain, the form birefringence in the plane of the flowand vorticity directions, ∆n xz , was estimated by12

where v i is the volume fraction of the i block, n a2 )

vPS

nPS

2 + vPI

nPI

2, and ni

is the bulk refractive indices of species i (n PS ) 1.59 and n PI ) 1.52). This equationassumes strong segregation, leading one to expect theact ual for m bir efr i ngence t o be s m all er t han t hi sestimate. On the other h and, th is equat ion n eglects th eintrinsic birefringence, which is expected to increase t hebirefringence in the case of PS cylinders in a PI m atr ix(PS more stretched tha n P I),18 leading one to expect t heactu al birefringen ce (form + intrinsic) to be larger tha nthe form estima te alone. It appea rs th at t hese two errorsapproximately cancel out, since the agreement betweenthe estimate from eq A1 and experimentally observedvalues for well-aligned st yrene-diene cylinders wasfound to be surprisingly good (calculated value was 6%higher than the observed value).12 Fr om eq A1, with th e

volume fraction of each block  vPS ) 0.16 an d v PI ) 0.84at 170 °C , ∆n xz of perfectly oriented Vector 4111 ispredicted to be 4.3 × 10 -4. A t heor et ical at t em pt t oaccount for the intrinsic birefringence of a relatedstyrene-diene HE X18 led to an unphysical result (theo-retical prediction for the intrinsic contribution was muchlarger than the observed ∆n ); thus, it is not possible forus to compare to a theoretical result that accounts forboth form a nd intrinsic contributions.

Orientation distr ibutions other than monodomainorder result in lower values of the birefringence. Theserelative changes can be treated in a robust way thatgives the approximate ratio of the birefringence for anew orientation distribution to that of a monodomain.These predicted ratios can be compared to observedrat ios to obtain a result t hat is insensitive to imperfectorienta tion of the a ctual HE X-0 sta te a nd t o cancel outinaccuracies of the existing m odels for t he birefringenceof t he H E X phas e. W e can es t i m at e ∆n xz from th epolarizability tensor for an anisotropic material usingthe Lorentz-Lorenz equation.20 In Figure 11, the coor-dinate system for an anisotropic cylinder is depicted.The induced dipole p is calculated from the following

empirical relationship:

where E is the in cident electr ic field. For an an isotr opicmat erial, the polarizability is a t ensor tha t depends onthe relative orientation of the principal axes of thematerial and the incident electric field. If the polariz-ability is biaxial a nd of the form

each polarizability component can be expressed in term sof the laboratory frame, e x, e y, e z in Figure 11.

Therefore

∆n xz ) n x - n z )vPS vPI (n PS

2 - n PI2)

2

2n a[(vPS + 1)n PI2 + vPI n PS

2]

(A1)

F i g u r e 1 1 . Coordinat e system of an anisotropic cylinder a ndpolarizability components.

 p ) R E (A2)

R ) (R3 0 0

0 R2 0

0 0 R1) (A3)

R1 ) (cos φ si n θ)e z + (sin φ sin θ)e y + (cos θ)e x (A4)

R2 ) (cos φ cos θ)e z + (sin φ cos θ)e y - (sin θ)e x (A5)

R3 ) (sin φ)e z - (cos φ)e y (A6)

(R3

R2

R1) ) (

si n φ -cos φ 0

cos θ cos φ cos θ si n φ -si n θ

si n θ cos φ si n θ si n φ cos θ)(

e z

e y

e x) (A7)

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Applying rotation matrixes defined in eq A7 to R an dthe uniaxial symmetry of cylinders (R2 ) R3)

Therefore

From the Lorentz-Lorenz equation:

The birefringence ∆n xz and the a verage refractive indexnj are defined as follows:

Then

Subtraction of eq A12 from eq A11 yields

Therefore

The orientation angles θ a n d φ of the distinct HEXorientations are determined from the geometry of theBCC twins. There are four ⟨11 1⟩ directions in each twinpr es ent i n B C C - I , but t he one al ong t he x direction

(initial shear direction) is shared by both (Figure 12b).The un it vector of the ⟨111⟩ direction that coincides withth e cylinder a xes of the origina l aligned HE X is denotede A. The other t hree ⟨11 1⟩ vectors in one of the t wins (e B,e C, e D) correspond to the bonds of a diamond tetra-hedron skeleton with th e angle between the bonds being109.47° (Figure 13). The other three orientations, e B′,e C′, a n d e D′, comprising the other BCC twin can beobtained by respectively inverting e B, e C, a n d e D a n dr efl ect i ng w i t h r es pect t o t he plane nor m al t o e A.Mathematically:

as illustrated in Figure 13.

R e f e re n c e s a n d N o t e s

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R(θ,φ) ) (si n φ cos θ cos φ si n θ cos φ

-cos φ cos θ si n φ si n θ si n φ

0 -si n θ cos θ)

(R2 0 0

0 R2 0

0 0 R1)(

si n φ -cos φ 0

cos θ cos φ cos θ si n φ -si n θ

si n θ cos φ si n θ sin φ cos θ))

R2 I + (R1 - R2)

(cos2

φ si n 2θ cos φ si n φ si n 2

θ cos φ si n θ cos θ

cos φ si n φ si n 2θ si n 2

φ si n 2θ si n φ si n θ cos θ

cos φ si n θ cos θ si n φ si n θ cos θ cos2θ

)(A8)

R zz ) R2 + (R1 - R2)sin2θ cos

2φ (A9)

R xx ) R2 + (R1 - R2)cos2θ (A10)

n xx2 - 1

n xx2 + 2

)4π 

3R xx (A11)

n zz2 - 1

n zz2 + 2

)4π 

3R zz (A12)

∆n xz ) n xx - n zz (A13)

nj ) 1 / 2(n xx + n zz ) (A14)

n xx ) nj + (∆n xz  /2) (A15)

n zz ) nj - (∆n xz  /2) (A16)

n xx2 - 1

n xx2 + 2

-n zz

2 - 1

n zz2 + 2

)6nj∆n xz

(n xx2 + 2)(n zz

2 + 2)≈

6nj∆n xz

(nj2 + 2)2 )

4π 

3 (R1 - R2)(cos2

θ - si n2

θ cos2

φ) (A17)

∆n xzeff ≈

2π (nj2 + 2)2

9nj(R1 - R2)(cos

2θ - si n

2θ cos

2φ) ≈

∆n xzma x

(cos2θ - si n

2θ cos

2φ) (A18)

Figure 12. Reflection pla nes a nd ⟨111 ⟩ directions of a t winnedBCC shown in (a) x- z plane and (b) x- y plane.

F i g u r e 1 3 . Determination of the cylinder orientations fromt he ⟨111 ⟩ directions of the twinned BCC. The e z is the projectedvector of  eD on the plane normal to eA.

e i′ ) - e i + 2(e i·e A)e A (i ) B , C , D ) (A1 9)

 Macromolecules, V ol. 35, N o. 3, 2002 Thermoreversible HEX-BCC Tran sitions 79 3

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(24) Qi, S.; Wang, Z.-G. Macromolecules 1997, 30 , 4491.

(25) Almdal, K.; Koppi, K. A.; Bates, F. S. Macromolecules 1993,26 , 4058.

(26) The four wea k diagonal peaks of qm at aroun d 30°, 150°, 210°,and 330° in the q xq y plane in Figure 6a perhaps result fromthe overlap of residual int ensity of the Ga ussian Bragg peaksof the twinned BCC. These an gles are slightly different fromthose (35°, 145°, 215°, and 325°) in the q xq z plane (Figure6b), which are the positions where two rings intersect eachother (i.e.,(K B3 a n d (QB3 in Figure 3a of ref 5). When Braggpeaks are not Delta function but Gaussian type with a finitewidth, the intensity from the overlap of the residual tails of Bragg peaks from (K B3 a n d (QB3 become a maximum at thepositions at 30°, 150°, 210°, and 330° on q xq y. Alternatively,these weak peaks may reflect a small amount of orientedHEX with [110] orientation. We note tha t these four weak peaks in the q xq y plane were not reported in the SANS studiesby Almdal et al.25

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