Theoretical study of polymeric mixtures with different ... papers/67.pdf · help compatibilize...

JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 3 15 JANUARY 2000

Theoretical study of polymeric mixtures with different sequence statistics.I. Ising class: Linear random copolymers with different statisticalsequences and ternary blends of linear random copolymerswith homopolymers

Shuyan Qi and Arup K. ChakrabortyDepartment of Chemical Engineering, Department of Chemistry, and Materials Science Division,Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720

~Received 30 August 1999; accepted 19 October 1999!

We derive a Landau free energy functional for polymeric mixtures containing components withdifferent sequence statistics. We then apply this general field theory to two mixtures that belong tothe Ising universality class: mixtures of two different linear random copolymers, and ternarysystems of linear random copolymers and two homopolymers. We discuss the instability conditionsfor the homogeneous state of these mixtures, and calculate the structure factors for differentcomponents in the homogeneous state. The structure factors show interesting features which candirectly be compared with scattering experiments carried out with selectively deuterated samples.We also work out the eigenmodes representing the least stable concentration fluctuations for thesemixtures. The nature of these concentration fluctuations provides information regarding the orderedphases and the kinetic pathways that lead to them. We find various demixing modes for differentcharacteristics of the two mixtures~e.g., average compositions, statistical correlation lengths, andvolume fractions!. © 2000 American Institute of Physics.@S0021-9606~00!50303-1#

na

eno

ero

-

nrnm

o

ntererell

stohe

e

folic

tico-ear

t ofect

theBy

freee orilib-

nd

an-the

Onetua-

n to-itioneation-

I. INTRODUCTION

The ability of copolymers to reduce surface tension ahelp compatibilize polymer blends has led to their usesurfactants, detergents, solubilizers, and dispersion agThus, the behavior of mixtures of copolymers and hmopolymers and copolymer solutions is of significant intest. The physics of mixtures of homopolymers, mixturescopolymers with ordered sequences~e.g., diblock copoly-mers or DCPs! and mixtures of such copolymers with homopolymers has been studied extensively~e.g., see Refs. 1and references therein!. The behavior of such copolymers isolution has also been studied, and much has been leaabout the formation of micelles, membranes and othercroscopic structures in such systems~e.g., Refs. 2–4!. On theother hand, there are fewer studies of mixtures of randcopolymers~e.g., Refs. 5–19!.

Mixtures of random copolymers are practically and fudamentally important, however. This is because of the potial for random copolymers to be even more efficient atducing surface tension compared to copolymers with ordesequences,8,9 and because random copolymers are typicaless expensive to manufacture. Another reason for interethe physics of random copolymers is that they serve as gmodel systems to study the effects of frustrating quencdisorder embodied in their sequence distributions~e.g., Refs.11 and 12!. In addition, random copolymers share somcommon features with biopolymers~e.g., Ref. 13!. Sincemany biopolymers function in environments comprised omixture of species, a better understanding of random copmers in mixtures may also provide insights into the physof important biological processes.

1580021-9606/2000/112(3)/1585/13/$17.00

Downloaded 14 Jan 2003 to 128.32.198.56. Redistribution subject to A

dsts.--f

edi-

m

-n--d

yin

odd

ay-s

In this series of two papers, we develop a field-theoredescription of polymeric mixtures that include random cpolymers, and discuss its applications to mixtures of linrandom copolymers~LRCs! with other types of LRCs, withtwo homopolymers, and with diblock copolymers~DCPs!.Special attention is devoted to the behavior at the poininstability of the disordered phase. We also discuss dirconnections between our results and experiments.

As mentioned earlier, theorists have consideredphase behavior of random copolymers and their mixtures.using a Flory–Huggins free energy,14–16 ternary mixtures ofLRCs and two homopolymers have been studied. Theenergies of homogeneous phases containing either thretwo species are compared in order to determine the equrium state for a given set of conditions.

By using a field-theoretic treatment, Shakhnovich aco-workers18 and Fredrickson and co-workers19 have studiedmicrophase ordering in LRCs. Different types of phase trsitions have been predicted depending on the nature ofcorrelations that describe the sequence fluctuations.measure of the range of correlations describing these fluctions isl, which is defined asl5PAA1PBB21. HerePAA

and PBB are the probabilities of finding anA segment fol-lowing anA segment and finding aB segment following aBsegment, respectively. Macroscopic phase segregatioform two liquid phases is predicted whenl exceeds a threshold value. This segregation has been ascribed to composfluctuations. For smaller value ofl, this macroscopic phassegregation continuously changes to microphase segregthrough a Lifshitz point. Mixtures of two random copolymers have also been studied by Balazset al.20 andSchweitzer.21 Using a Flory–Huggins theory20 and a coarse-

5 © 2000 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

re

tu

tusrsf

ssonth

sttioonfehaaidexen

ur

veof

bity tors:td in

esal-exts. Inht

eg-

a

1586 J. Chem. Phys., Vol. 112, No. 3, 15 January 2000 S. Qi and A. K. Chakraborty

grained PRISM approach,21 they find that the conditions fodemixing of the homogeneous phase depend on the sequstatistics of the LRCs that comprise the mixture. These sies extend an earlier work by Scott.22

In this paper, we develop a field-theoretic approachstudy polymeric mixtures which contain LRCs. We focus odiscussion on the phase behavior of a mixture of two typeLRCs and a ternary mixture of LRCs in two homopolymeSimilar approaches have been used to study mixtures o~orpolydisperse! diblock copolymers by Hong and Noolandi,23

Benoit and Hadziioannou,24 Fredrickson and Leibler,25

Erukhimovich and Dobrynin26 and Balsaraet al.27 We de-scribe the instability of the homogeneous state and the aciated unstable concentration fluctuations of mixtures ctaining random copolymers in detail. The dependence ofinstability condition and fluctuation modes on sequencetistics ~average compositions and sequence correlalength! and volume fractions is studied. Detailed informatiregarding the dependence of the structure factors for difent components and the macroscopic and microscopic psegregation modes on the LRC sequence statistics isprovided. Furthermore, our theory can be applied to consordered microstructures in the weak segregation limit bypanding the free energy functional to a high order in conctration fluctuations~generally to quartic order!. Our fieldtheory does not account for compressibility or local structon scales shorter than the statistical segment length.


nced-

orof.

o--e

a-n

r-se

lsoer--

e

This paper is organized as follows: In Sec. II, we deria field-theoretic framework for studying the segregationpolymeric mixtures which include components that exhiquenched sequence disorder. We then apply the theorstudy two types of mixtures containing random copolymeIn Sec. III, a mixture of two types of LRCs with differencompositions and sequence fluctuations is studied, anSec. IV, a ternary mixture of LRCs~formed by type-A and -Bsegments! andA andB homopolymers is investigated. Thessystems may be considered to belong to the Ising univerity class. We discuss mixtures of LRCs and DCPs in the npaper as it is a system that belongs to the Brazovskii clasSec. V, we offer some concluding remarks and highligpertinent experiments discussed in preceding sections.

II. THEORY

The mixture we investigate containsP types of differentpolymers, each comprised of either one or two types of sments. For each type of polymer, there arent chains. Eachchain hasNt segments~with the Kuhn length beingl t!, ofwhich f tNt are segments of type-A, and (12 f t)Nt are seg-ments of type-B. Each type of polymer is associated withparticular value ofl(l t). For this mixture, the partitionfunction can be written as

f

ns.

Z5E P t51P P j t51

nt Pst51Nt DRW j t

t ~st!expH 2(t, j t

3

2l t2 (

st51

Nt21

@RW j t

t ~st11!2RW j t

t ~st!#2J

3E DrA~rW !dFrA~rW !2 (t, j t ,st

11u j t

t ~st!

2d~RW j t

t ~st!2rW !G E DrB~rW !dFrB~rW !2 (t, j t ,st

12u j t

t ~st!

2d~RW j t

t ~st!2rW !G3d@rA~rW !1rB~rW !2r0#expH 2E drW drW8 (

m,n5A,Bxmn~rW2rW8!rm~rW !rn~rW8!J , ~1!

whereRW j t

t (st) is the position of thestth segment on thej tth chain of type-t polymer.u j t

t (st) measures the chemical identity o

thestth segment of thej tth chain of type-t polymer;u j t

t (st)51 if the stth segment is of type-A, u j t

t (st)521 if the stth segment

is of type-B. rA(rW) andrB(rW) are local coarse-grained concentrations for the type-A and type-B segments, respectively~e.g.,rA(rW)5( t51

P rAt (rW)!. The incompressibility constraint,rA(rW)1rB(rW)5r05const, is imposed by a delta function in Eq.~1!.

xmn(rW2rW)(m,n5A,B) are standard Flory–Huggins parameters which reflect the nature of the intersegment interactioWe rewrite the partition function as functional integrals over the local coarse-grained concentrations of type-A and type-B

segments coming from speciest ~rAt andrB

t !. To do this, we introduce conjugate fieldsgAt ~to rA

t ! andgBt ~to rB

t !. From Eq.~1!:

Z5E P t51P @DrA

t ~rW !DrBt ~rW !DgA

t ~rW !DgBt ~rW !#expF2E drW drW8 (

m,n5A,Bxmn~rW2rW8!rm~rW !rn~rW8!G

3dF(t

rAt ~rW !1(

trB

t ~rW !2r0G H E P t51P P j t51

nt Pst51Nt DRW j t

t ~st!expF2(t, j t

3

2l t2 (

st51

Nt21

@RW j t

t ~st11!2RW j t

t ~st!#2

1 i(tE drW@gA

t ~rW !rAt ~rW !1gB

t ~rW !rBt ~rW !#2 i (

t, j t ,stFgA

t ~RW j t

t ~st!!11u t~st!

21gB

t ~RW j t

t ~st!!12u t~st!

2 G G J , ~2!


yd

ofiot

ern

oemthtioat

a

-

of

ssae

d

e

o.n

c-

ffor

hattheond-ainsficisthe

onsi-i-

tialthe

.gen-icu-par-on,

ean-f

1587J. Chem. Phys., Vol. 112, No. 3, 15 January 2000 Polymeric mixtures with different sequence statistics. I

where i 5A21. The term in curly brackets is the entropcorresponding to a given set of macroscopic fields. Wefine it to be exp@S($rA

t %,$rBt %,$gA

t %,$gBt %)#. From Eq.~2!, it is

clear that the entropy can be written as

S5(t51

P

ntSt@rA

t ,rBt ,gA

t ,gBt #, ~3!

whereSt is the entropy contribution from a single chaintype-t and only depends on the coarse-grained concentratof speciest. This is a reflection of the fact that differenchains are coupled via the term corresponding to the enfor given macroscopic fields and the incompressibility codition. ExpandingSt in powers of the conjugate fields,gA

t

andgBt , we obtain the entropy as an expansion in powers

the concentration fields and the conjugate fields. Then,ploying a saddle point approximation allows us to obtainentropy as an expansion in powers of the concentrafields. Taking Fourier transforms, the entropy, upto quadrorder, can then be written as

S5S01VE dkW dW rT~kW !M21~kW !dW r~2kW !1O~dr3!, ~4!

whereV is the volume of the system, andS0 is a constantwhich only depends on the average concentration of especies. The concentration fluctuation vector is defineddrW T5(drA

1,drB1,drA

2,drB2,...,drA

P ,drBP). Because each spe

cies contributes independently to the entropy,M has a block-diagonal form; i.e.,M5diag@J1,J2,...,JP#, with Jt ~t51 toP! being 232 matrices. All the other elements inM areequal to zero. TheseJt’s can be obtained for specific typespolymers.

The interchain interactions come from the incompreibility condition and the energetic contributions proportionto the Flory–Huggins parameters. Considering both, theergetic contribution is

E522xE dkW (t1 ,t2

drAt1~kW !drA

t2~2kW !1const, ~5!

where we have assumed the Flory–Huggins parameter,fined by

x~rW2rW8!5 12@2xAB~rW2rW8!2xAA~rW2rW8!2xBB~rW2rW8!#,

~6!

to be of short rangex(rW2rW)5xd(rW2rW). Combining thiswith the entropy, we obtain the quadratic term of the frenergy functional to be

F5F01E dkW dW rT~kW !G2~kW !dW r~2kW !1O~dr3!, ~7!

where

G2~kW !5M21~kW !22xD. ~8!

Here D is a matrix defined asD2t121,2t22151 ~t1 and t2

ranging from 1 toP!, and all the other elements are zerThis structure ofD reflects the incompressibility condition

The incompressibility condition also reduces the dimesionality of G2(k). Recall that this condition implies( t@drA

t (kW )1drBt (kW )#50. This changes the matrixG2 from a

2P32P matrix to a (2P21)3(2P21) matrix. In the fol-


e-

ns

gy-

f-

en

ic

chas

-ln-

e-

e

.

-

lowing, we choosedrBP as the reference concentration flu

tuation and we substitute drBP(kW )52( t51

P21@drAt (kW )

1drBt (kW )#2drA

P(kW ) into Eq. ~7!. This leads to

F5F01E dkW drW 8T~kW !a~kW !drW 8~2kW !1O~dr3!, ~9!

wheredrW 8T5(drA1,drB

1,drA2,drB

2,...,drAP) and

a i j 5@G2# i j 1@G2#2P,2P2@G2# i ,2P2@G2#2P, j . ~10!

Equation~10! allows us to obtain the spinodal point othe homogeneous state as well as the structure factorsparticular components. The concentration fluctuations tdrive the instability are also interesting as they determinepathways of phase segregation. The eigenvector corresping to the eigenvalue that equals zero at the spinodal contthis information. We apply the above theory to two specimixtures which belong to the Ising universality class in thpaper and to a case belonging to the Brazovskii class infollowing paper.

III. MIXTURES OF TWO DIFFERENT TYPES OF LRCS

This mixture has two species (P52). There aren1 LRCchains with the average composition ofA segments beingf 1

and the correlation length specifying sequence fluctuatiequal tol1 , andn2 LRC chains with the average compostion of A segments beingf 2 and the correlation length specfying sequence fluctuations equal tol2 . In the spirit of con-structing the simplest picture which captures the essenphysics, we assume the two types of LRC chains havesame Kuhn length~l, chosen as the unit of the length! and thesame chain length~N! with no polydispersity in chain lengthIt is easy to relax these assumptions, and study a moreeral case. It is important to note that each chain of a partlar species has a different sequence. All sequences of aticular species belong to the same statistical distributihowever. Thus, in the following, we useu t(s) to describe thesequence distribution of speciest.

In order to obtain the quadratic coefficient of LRCs, wemploy a method that we have used previously to study rdomly grafted copolymers.12 We obtain the components othe matrixM (kW ) for type-I LRCs to be

M11I ~kW !5n1E

0

N

ds1 ds2 K @11u I~s1!#

2

@11u I~s2!#

2 L3exp$2xus12s2u%

5n1N2f 1~12 f 1!g28@N~x2 lnul1u!#

1n1N2f 12g2~Nx!,

M12I ~kW !5M21

I ~kW !

5n1E0

N

ds1 ds2S @12u I~s1!#

2

@11u I~s2!#

2 L3exp$2xus12s2u%

52n1N2f 1~12 f 1!g28@N~x2 lnul1u!#

1n1N2f 1~12 f 1!g2~Nx!


s

theuse


M22I ~kW !5n1E

0

N

ds1 ds2K @12u I~s1!#

2

@12u I~s2!#

2 L3exp$2xus12s2u%

5n1N2f 1~12 f 1!g28@N~x2 lnul1u!#

1n1N2~12 f 1!2g2~Nx!, ~11!

wherex5k2l 2/6. The second equalities in Eq.~11! are ob-tained by carrying out the quenched average over the

e

se

r-i

nae

f ta

vthm

e

igrea


e-

quence fluctuations. For the statistics that characterizeLRC sequences under consideration, this implies makingof the following relations:

^u I~s!&52 f 121,~12!

^u I~s1!u I~s2!&54 f 1~12 f 1!l1us12s2u

1~2 f 121!2.

In Eq. ~11!, g2(x)52@exp(2x)1x21#/x2 is the Debye func-tion, andg28 is a function that defined by

g28@Nz#51

N2 H N@11exp~2z!#

12exp~2z!2

2 exp~2z!

@12exp~2z!#2 @12exp~2Nz!# if l1.0,

N@12exp~2z!#

11exp~2z!1

2 exp~2z!

@11exp~2z!#2 @12~21!N exp~2Nz!# if l1,0

, ~13!

s

isy-ix-s

-

-

m-

ethe

-

cesr

e

et

she

nd

where z5x2 lnul1 u. The components ofM (kW ) for type-IILRCs can be obtained similarly.

Now substituting in Eq.~8! obtainsG2 for a mixture ofn1 type-I LRC chains andn2 type-II LRC chains:

G2~kW !5F M11I ~k! M12

I ~k! 0 0

M21I ~k! M22

I ~k! 0 0

0 0 M11II ~k! M12

II ~k!

0 0 M21II ~k! M22

II ~k!

G21

2F 2x 0 2x 0

0 0 0 0

2x 0 2x 0

0 0 0 0

G , ~14!

Notice thatx is inversely proportional to temperature. Thcorresponding reduced 333 matrix a @Eq. ~10!# is obtainedtrivially by imposing the incompressibility condition. Thiprojection from the four-dimensional spac(drA

I ,drBI ,drA

II ,drBII ) to the three-dimensional planedrA

I

1drBI 1drA

II1drBII50 makes the projected axes nono

thogonal. Such a nonorthogonal system complicates theterpretation of the nature of the concentration fluctuatioTo overcome this, we make a linear transformation toorthogonal system as explained in the Appendix. In the northogonal system, the eigenvalues and eigenvectors o333 matrix are calculated. In the homogeneous state,concentration fluctuations are stable, and so all the eigenues of this matrix are positive. The temperature at whichlowest eigenvalue equals zero is the spinodal of the hogeneous state. The eigenvector which belongs to this loweigenvalue indicates the least stable fluctuation mode~or theleast stable direction for the kinetic evolution of the system!.Next, we present the results of our analysis which shed lon the nature of the instabilities characterizing this mixtuPredictions regarding measurable structure factors arediscussed.

n-s.nwhell

al-eo-st

ht.

lso

Consider first a mixture of two types of symmetric LRC( f 15 f 250.5), with l150.8 andl2520.8. So we have amixture of two ensembles of LRC sequences, one whichstatistically blocky (l150.8) and the other is statisticallalternating (l1520.8). Figure 1~a! shows the structure factor S11(k) corresponding to the scattering pattern of a mture with deuterated type-A segments in the type-I LRC~which are statistically blocky!. A peak atk50 is clearlyseen. Asx increases~temperature decreases!, the intensity ofthis peak grows. The instability~divergence of the peak intensity! occurs withk50 ~at x50.4405! because of the lackof a natural length scale in the LRC sequences. Figure 1~b!shows the structure factorS33(k) corresponding to the scattering pattern of a mixture with deuterated type-A segmentsin the type-II LRCs~which are statistically alternating!. Apeak atk50 is also seen in this case. However, as the teperature changes within the same range as that in Fig. 1~a!,the curves in Fig. 1~b! are almost unchanged. In addition, wfind that close to the instability of the homogeneous statepeak intensity forS11 is much larger than that ofS33. Whenthe instability occurs, the peak inS11 diverges while the peakin S33 does not. The features of structure factors for typeBsegments of type-I LRCs (S22(k)) and type-II LRCs(S44(k)) are similar to those ofS11(k) and S33(k), respec-tively.

Insight into the reasons that underlie these differenbetweenS11(k) and S33(k) is provided by the eigenvectoshown in Fig. 2~a!. The four curves labeledA1 , B1 , A2 , andB2 in Fig. 2~a! depict the variation of the amplitudes of thconcentration fluctuationsdrA

I , drBI , drA

II anddrBII ~we refer

to them as eigenmodes!. They are obtained directly from theigenvector corresponding to the smallest eigenvalue ak50. At all values ofx, the sum of the four amplitudes iequal to zero, indicating the incompressibility condition. Teigenmodes are normalized so that

~drAI !21~drB

I !21~drAII !21~drB

II !251. ~15!

The curve labeledj1 represents the smallest eigenvalue, a


uey

Itaat

th

ntuuati

tu

eer

lyte

ed

ix-

e


the curve labeledj2 represents the second lowest eigenvalWe have rescaledj1 andj2 by the same factor so that thefit in the figure. At low values ofx ~or high temperatures!, j1

is positive and thus the homogeneous state is stable.clear from this figure that when temperature is high, the lestable concentration fluctuation is the one that segregtype-I and -II LRCs~type-a concentration fluctuation!. Uponfurther increase inx, the system reaches a point wherej1

5j2 . At this point, the least stable mode changes tosegregation of type-A and type-B segments~type-b concen-tration fluctuation!. Thus, we see that the nature of the cocentration fluctuations changes significantly as temperais lowered. At high temperature, the concentration flucttions are such that chains with different sequence statisattempt to separate from each other. At lower tempera~i.e., temperature below that corresponding toj15j2! theimportant concentration fluctuations are ones where sments of different type attempt to separate from each othis this type-b segregation that occurs at the spinodal~i.e.,whenj150!

FIG. 1. Structure factors derived fromG2(k) for a mixture of two symmet-ric LRCs (f 15 f 250.5), with the first species of LRCs being statisticalblocky (l150.8) and the second species of LRCs being statistically alnating (l1520.8). Both LRCs have the same chain lengthN15N2

51000, and the number ratio isn1 : n251 : 1. In ~a!, the vertical axis rep-resents the intensity of the structure factor for deuterated type-A segments ofthe first species of LRCs (S11(k)), and in~b!, the vertical axis represents thintensity of the structure factor for deuterated type-A segments of the seconspecies of LRCs (S33(k)).


.

isstes

e

-re-csre

g-.It

r-

FIG. 2. The smallest two eigenvalues~labeledj1 andj2! and the normal-ized eigenvector~labeledA1 , B1 , A2 , andB2 which depict the variation ofthe amplitudes for the concentration fluctuationsdrA

I , drBI drA

II , anddrBII ,

respectively! of the smallest eigenvalue at different temperatures for a mture of two symmetric LRCs (f 15 f 250.5) with n1 : n251 : 1 and~a! l1

52l250.8 and~b! l152l250.2. j1 and j2 are rescaled by the samfactor to fit in the figure. In~c!, the values ofx at the spinodal (xs) areplotted for different values ofl1 and l2 . The chain lengths areN15N2

51000 in all the cases.


encolvodr

on-IIchsllyt-

thn-

isptn-

asastc-te

at-tis

to

s

ofn

m

n

taw

y

her

ua-

nt.hichre,-of

be-e-

yCsn-her-spires de-la-

of

estear., ofsi-r-

-

o

f theet-

m-


It is important to note that the eigenmodes forj1,0provide information pertinent to what happens under supcooling conditions. When considering a temperature queacross the spinodal boundary, the system starts to evfrom the homogeneous state, and the initial direction of elution is along this least stable mode. Thus, the eigenmoat j1,0 tell the initial direction of the kinetic pathway undedifferent temperature quench conditions.

In Fig. 2~a!, it is clear that the amplitudes for segregatiof A and B segments are different for type-I and typeLRCs. The fluctuation amplitudes of type-I LRCs are mularger than those of type-II. This is because the two typeLRCs are statistically different: type-I chains are statisticablocky (l150.8) and type-II chains are statistically alternaing (l2520.8). It is harder to segregate type-A and type-Bsegments in statistically alternating LRCs because ofhigher entropic penalty. Indeed, if we decrease the differein the values ofl characterizing the two types of LRC sequences, this difference in the amplitudes is reduced. Thshown in Fig. 2~b! where we have a similar mixture excethat l152l250.2. We also find that in this case the cocentration fluctuations become of type-b at a higher value ofx. The spinodal temperature also becomes lower. The refor the shift of the spinodal to lower temperatures is thdemixing of the homogeneous state is dominated by thetistically blocky LRCs~species I!, and as the sequence flutuations of the statistically blocky LRCs become correlaover shorter scales~smaller l1!, it becomes entropicallymore difficult to separate the type-A and type-B segments.

We demonstrate the dependence of the value ofx corre-sponding to the spinodal (xs) on l1 and l2 in Fig. 2~c!.These results show that the spinodal temperature decresharply asl1 and l2 become strongly statistically alternaing. This comes from the severe entropic penalty for statically alternating LRCs to exhibit type-b concentration fluc-tuations. A ridge occurs along the locusl15l2 . This isbecause, for fixedl11l2 , going from l15l2 to l1Þl2

necessarily increases the correlation length characterizingsequences of the statistically blocky of the two typesLRCs. This, in turn, makes type-b concentration fluctuationsmore facile because the segregation is dominated by thetistically blocky LRCs.

In Fig. 3, we show the eigenmodes and eigenvaluesmixture of two LRCs which are asymmetric in compositioThe two types of LRCs have the same compositionf 15 f 2

50.2, but different values ofl(l152l250.8). The eigen-modes of this mixture show complicated variations with teperature. At low values ofx ~or high temperature!, the leaststable fluctuation is again the segregation of type-I atype-II LRCs ~type-a concentration fluctuation!. As the twosmallest eigenvalues approach each other, the least sconcentration fluctuations smoothly pass through the folloing three segregation types: type-B segments of type-I LRCssegregate from all other segments~type-gB concentrationfluctuation! in a narrow temperature window, followed btype-A and type-B segments segregating~type-b concentra-tion fluctuation! in another narrow window, and finally


r-h

ve-

es

of

ece

is

onta-

d

ses

-

hef

ta-

a.

-

d

ble-

type-A segments of type-I LRCs segregate from all the otsegments~type-gA concentration fluctuation! at low tempera-tures. Notice that at the spinodal, the concentration flucttions are of type-b.

At high temperature, entropic penalties are dominaSo, the least stable concentration fluctuations are those wcost least entropy. Therefore, in Fig. 3, at high temperatuwe get type-a concentration fluctuations. At low temperatures, type-gA concentration fluctuations occur becausethe following reason. Statistically blocky LRCs withf 1

Þ0.5 exhibit concentration fluctuations with their type-A andtype-B segments segregated with relative ease. This iscause composition fluctuations leading to long runs of typAsegments become likely. It is difficult~entropically! to seg-regate the type-A and type-B segments of the statisticallalternating LRCs. So, since the statistically alternating LRare largely composed of type-B segments, energetic and etropic considerations lead to their aggregating with ttype-B segments of the statistically blocky LRCs. At intemediate temperatures, energetic and entropic factors conto cause the complicated changes in the unstable modepicted in Fig. 3. At the present time, we have no clear expnation for these complicated variations.

In Fig. 4, the eigenmodes and eigenvalue of a mixturetwo asymmetric LRCs with different fractions~f 150.8, f 2

50.2! and different values ofl(l152l250.8) are shown.The second lowest eigenvalue is much larger than the loweigenvalue, and it is not presented in the figure. It is clthat there is only one segregation mode in this case; i.etype-a. This is because the difference in average compotions of type-I and type-II LRCs is so large that the unfavoable interactions between type-A and type-B segments separates the type-I~almostA! and type-II~almostB! LRCs.

In Fig. 5, we show the dependence of the value ofx atthe spinodal (xs) on the average compositions of the twtypes of LRCs~with l152l250.8!. A ridge appears atf 1

5 f 2 with f 15 f 250.5 being a saddle point. Whenu f 12 f 2uincreases,xs decreases sharply. The dependence ofxs on f 1

FIG. 3. The smallest two eigenvalues and the normalized eigenvector osmallest eigenvalue at different temperatures for a mixture of two asymmric LRCs (f 15 f 250.2) with n1 : n251 : 1 andl152l250.8. The chainlengths areN15N251000. The labels for the concentration fluctuation aplitudes are the same as in Fig. 2.


d

s-

ucs

a-e

asen

pes

ins

rmpo-

of

an-

e

f tme

a-

t

s ofale


and f 2 is clearly different from Scott’s mean-fielprediction22 that xs only depends onu f 12 f 2u. This differ-ence is the largest whenf 15 f 2 , where the sequence statitics plays an important role. Whenu f 12 f 2u is large, Scott’sresult becomes more accurate.

For a better understanding of these findings, it is instrtive to consider the types of the concentration fluctuationatthe spinodal~we shall refer to them as segregation types!. InFigs. 6~a! and 6~b!, we depict the different types of segregtion using a gray scale. The two figures differ in the volumfractions of the two types of LRCs. Both figures show thwhen u f 12 f 2u is large, type-a segregation happens becauthe two types of LRCs are so different in composition. Othe contrary, whenu f 12 f 2u is very small, type-b segregationhappens.28 This is because of two reasons. Whenf 15 f 2 ,

FIG. 4. The smallest two eigenvalues and the normalized eigenvector osmallest eigenvalue at different temperatures for a mixture of two asymric LRCs ~f 150.8, f 250.2! with n1 : n251 : 1 andl152l250.8. Thechain lengths areN15N251000. The labels for the concentration fluctution amplitudes are the same as in Fig. 2.

FIG. 5. The values ofx at the spinodal (xs) are displayed against differenaverage compositions of the two LRCs. The chain lengths areN15N2

51000 andn1 : n251 : 1.


-

t

there is no significant energetic advantage for the two tyof LRCs to segregate~i.e., type-a segregation!. Type-b seg-regation, however, does lead to significant energetic gabecause the unfavorable interactions between type-A andtype-B segments are minimized. However, type-b segrega-tion is entropically expensive. The entropic costs to foA-rich andB-rich domains depend upon the average comsitions and the composition fluctuations. For givenl1 andl2 , equal average compositions makes the two typesLRCs most compatible for formingA andB microdomains.Our results indicate that this along with the energetic advtage leads to type-b segregation whenf 1' f 2 .

In the intermediate region between type-a and type-bsegregation, type-gA and type-gB segregation take place. Wfind that in both type-gA and type-gB segregation, the type-Aand type-B segments of the LRCs with the smaller value ofl

het-

FIG. 6. Segregation types for different average compositions of two typeLRCs with l152l250.8. From the lightest to the darkest, the gray screpresents the regions in whicha, b, gA , and gB types of concentrationfluctuations occur at the spinodal. The number ratio is~a! n1 : n2510 and~b! n1 : n2520. The chain lengths areN15N251000 in both cases.


psenite

ns

. A

so

hee

eo

-

imin

en-

o-mem-the

anni-

ofalera-us

ofthe

ena

viz.,

tionld-opics

ng

ndcon-arealso

n

fn


do not demix. This is because the entropic penalty to serate type-A and type-B segments is larger than that for LRCwith larger values ofl. The reason the system chooses typgA or type-gB segregation at different average compositioof LRCs lies in interaction energy changes. Compared wtype-b segregation, the energy gain associated with conctration fluctuations of type-gA is proportional to x@(12 f 1) f 2#, while the energy gain of concentration fluctuatioof type-gB is proportional tox@ f 1(12 f 2)#. Thus whenf 1

. f 2 , type-gA segregation is favored on energetic groundsdecrease inxs is observed as we go from type-b to type-asegregation region~see Fig. 5!. This is because type-b seg-regation is associated with the largest entropic loss, andhappens at the lowest temperature; type-a segregation leadsto the lowest entropic loss, and thus it happens at the higtemperature. Type-gA and type-gB happen at intermediattemperatures.

Comparing Figs. 6~a! and 6~b!, we see that the regionwhere type-a segregation occurs decreases while the arfor the type-g segregation increase as the volume fractionstatistically blocky LRCs increases.28 This is because ahigher volume fraction of statistically blocky LRCs decreases the total entropic penalty of separating type-A andtype-B segments in the mixture. This demonstrates theportance of the volume fractions of the two types of LRCsthe mixture in determining the demixing conditions.

In Figure 7 we show the dependence ofxs on the vol-ume fraction of LRCs of type-I (f1). For f 15 f 2 , increasingf1 results in a monotonic decrease in the value ofxs . How-

FIG. 7. The values ofx at the spinodal (xs) are displayed as a function othe volume fraction of the first species of LRCs in the mixture for differeaverage compositions of the two species withl152l250.8. The chainlengths areN15N251000.


a-

-shn-

it

st

asf

-

ever, slightly away fromf 15 f 2 , xs exhibits a minimum at aspecial value off1 . As u f 12 f 2u increases, this special valuapproachesf150.5, andxs decreases progressively. To uderstand this, let us first consider the limit of largeu f 1

2 f 2u. In this case, the segregation is of type-a because themixture is close to a blend of two different types of hmopolymers. As for homopolymer blends, when the volufractions of each component are equal, the transition teperature to a two-phase state is the highest. This is whyminimum of xs approachesf150.5 whenu f 12 f 2u is large.On the other hand, whenf 15 f 2 , we find the segregation isof type-b. When f1 increases~more statistically blockyLRCs!, it is easier for type-b segregation to occur. So,xs

decreases monotonically. Between the two limits, we findintermediate region which exhibits a minimum and the mimum approachesf150.5 with increasingu f 12 f 2u.

The variation of xs with volume fraction discussedabove suggests that, for a given choice of the two typesLRCs, a particular volume fraction will allow experimentstudy of the demixing phenomenon at the highest tempture. This particular choice of the volume fraction may thbe the most convenient system for experimental studies.

IV. TERNARY BLENDS OF LRCS IN TWOHOMOPOLYMERS

In this case, LRCs consisting of type-A and type-B seg-ments are in a mixture with type-A and type-B homopoly-mers. We study the demixing of the homogeneous statethis mixture and its dependence on the characteristics ofthree types of polymers.

At this point we want to emphasize differences betweour field-theoretic treatment and previous work usingFlory–Huggins free energy approach.14–17 In these studies,only three independent concentrations are considered;the concentrations of type-A homopolymers, of type-B ho-mopolymers and of LRCs. Because of this, the segregaof A andB segments of LRCs is not discussed. In our fietheoretic treatment, besides a careful account of the entrcontribution, we explicitly study concentration fluctuationof all four independent components in the mixture, includiconcentration fluctuations of type-A and type-B segments ofLRCs. In addition, unlike the previous studies, we obtain adescribe the eigenmodes which determine the unstablecentration fluctuations. These concentration fluctuationsannouncements of the impending phase transition, andtell us about the pertinent kinetic pathways.

Following Eq.~8!, we obtain the quadratic coefficient ithe free energy functional for this mixtures:

t

G2~kW !5F M11I ~k! M12

I ~k! 0 0

M21I ~k! M22

I ~k! 0 0

0 0 n2N22g2~N2x! 0

0 0 0 n3N32g2~N3x!

G 21

2F 2x 0 2x 0

0 0 0 0

2x 0 2x 0

0 0 0 0

G , ~16!


d

3

w

-m

--

d

l-

ith

-la-sen-

ga-e

ra-

ingCsthis

to avi-ture-

ithndig.

ra-is

kes-

pe

rr

ers


whereMi jI (k)( i , j 51,2) is defined in Eq.~11!, x5k2l 2/6, n2

andn3 are the number of homopolymer chains of type-A andtype-B, andN2 and N3 are the chain lengths of type-A andtype-B homopolymers.~In the following, subscripts 1, 2, an3 represent LRCs, type-A homopolymers and type-B ho-mopolymers, respectively.! The corresponding reduced33 matrix is obtained from Eq.~10!. Again, to reveal theessential physics of the demixing process of this mixture,study the case ofN15N25N351000 andl 15 l 25 l 35 l ~andwe choosel as the unit of length!.

In Fig. 8~a!, the structure factors from deuteratedA seg-ments of LRCs (S11(k)) at different Flory–Huggins parameters are presented. These curves show little change frox50.002 tox50.0029 andS11(k) does not exhibit any divergences. In the same range ofx, the structure factors of deuterated type-A homopolymers (S33(k)) show a large change@see Fig. 8~b!#. In this figure, as the temperature is lowerethe peak intensity atk50 grows. We find thatS33(k) di-verges atk50 upon lowering temperature. Thus, in the folowing, we only show the eigenmodes fork50. The featuresof structure factors for type-B segments of LRCs (S22(k))

FIG. 8. Structure factors derived fromG2(k) for a ternary mixture of sym-metric (f 150.5), statistically blocky (l150.8) LRCs in two homopoly-mers. LRCs and two homopolymers have the same chain lengthN15N2

5N351000, and the number ratio isn1 : n2 : n351 : 1 : 1. In ~a!, the ver-tical axis represents the intensity of the structure factor for deuterated tyAsegments of the first species of LRCs (S11(k)), and in~b!, the vertical axisrepresents the intensity of the structure factor for deuterated type-A ho-mopolymers (S33(k)).


e

,

and type-B homopolymers (S44(k)) are similar to those ofS11(k) andS33(k), respectively.

The least stable eigenmode of a mixture of LRCs wsymmetric composition (f 150.5) andl150.8 in a mixtureof n1 : n2 : n351 : 1 : 1 is shown in Fig. 9. The curve labeledj1 represents the smallest eigenvalue. The curvesbeledA1 , B1 , A, andB represent the normalized amplitudefor the concentration fluctuations corresponding to the eigvalue j1 for type-A segments of LRCs, type-B segments ofLRCs, type-A homopolymers and type-B homopolymers, re-spectively. Concentration fluctuations leading to a segretion of the two types of homopolymers is obvious in thtemperature window shown in Fig. 9. The segregation ofA1

andB1 occurs to only a small extent even when the tempeture is lowered below the spinodal value~defined byj150!.This segregation is even smaller for statistically alternatLRCs. The small extent of segregation of segments of LRis because a large entropic penalty is associated withtype of separation. This penalty is larger whenl1 is smaller,and hence, the statistically alternating LRCs segregatelesser extent. By using a similar terminology as in the preous section, we can say that, through the whole temperawindow shown in Fig. 9, the eigenvector indicates a typebsegregation.

The least stable eigenmodes of this ternary mixture wcompositionally asymmetric LRCs is shown in Fig. 10, ais in stark contrast to the results displayed in Fig. 9. In F10, the statistically blocky LRCs (l150.8) have less type-Asegments (f A50.2). We find that the least stable concenttion fluctuation through the whole temperature windowsuch that the LRCs segregate from type-A homopolymers~type-gA concentration fluctuation!. This is because type-B isthe major component of the LRC segments, which maLRCs favor type-B homopolymers. Similar results are obtained for statistically alternating LRCs.

In Fig. 11 we show the dependence ofxs on the se-quence statistics of the LRCs~measured byl1 and f 1!. We

-

FIG. 9. The smallest eigenvalue~labeledj1! and the normalized eigenvecto~labeledA1 , B1 , A, andB which depict the variation of the amplitudes fothe concentration fluctuationsdrA

I and drBI for LRCs, drA for type-A ho-

mopolymers anddrB for type-B homopolymers, respectively! of the small-est eigenvalue at different temperatures for a mixture of two homopolymand a symmetric LRCs (f 150.5) with n1 : n2 : n351 : 1 : 1 andl150.8.The chain lengths areN15N25N351000.


don

tp

da

--on

re-oci-ho-ared,

e

thentPS

eriners.

pi-

nts

I in

ix-

e-it

f ttw

ig.

tio


see that changes inl1 have a small effect onxs . In contrast,f 1 has a dramatic effect onxs . A mixture containing sym-metric LRCs (f 150.5) exhibits the largest value ofxs .Thus, they are the most efficient compatibilizers for blenof different homopolymers. For a fixed average compositiwe show the dependence ofxs on l1 for a mixture contain-ing symmetric LRCs in Fig. 12. It is clear that decreasingl1

leads to an increase inxs . This is consistent with the facthat statistically alternating LRCs experience larger entropenalties when separating type-A and type-B segments.

By putting PS-r-PVP into a thin film containing PS anPVP homopolymers~PS and PVP are immiscible and formsharp interface!, Dai et al.8 found that the maximum interfacial reinforcement occurs atf PS50.5. Although these experiments were performed under strong segregation conditi

FIG. 10. The smallest eigenvalue and the normalized eigenvector osmallest eigenvalue at different temperatures for a ternary mixture ofhomopolymers and asymmetric LRCs (f 150.2) with n1 : n2 : n3

51 : 1 : 1 andl150.8. The chain lengths areN15N25N351000. Thelabels for the concentration fluctuation amplitudes are the same as in F

FIG. 11. The values ofx at the spinodal (xs) as a function ofl1 and f 1 inthe ternary mixture of LRCs in two homopolymers with a number rabeing n1 : n2 : n351 : 1 : 1 and the chain lengths beingN15N25N3

51000.


s,

ic

s,

and our study of the ternary mixture is for the weak seggation limit, demixing of the homogeneous phase is assated with an increase of surface tension between twomopolymers. In this sense, our results shown in Fig. 11consistent with the findings of Ref. 8. On the other hanKulasekereet al.9 found that, by putting PS-r-PMMA into athin film containing PS and PMMA homopolymers, thmaximum interfacial reinforcement occurs atf PS50.68. Theauthors of Ref. 9 explained this difference by consideringFlory–Huggins parameterx to be concentration-dependefor PS and PMMA, but concentration-independent forand PVP.

Figure 13~a! shows the dependence ofxs on n2 andn3 .The surface defined byxs is symmetric with respect ton2

andn3 . The surface also has a valley whenn25n3 . This issimilar to the phase behavior of a binary homopolymblend. The similarity is due to the fact that the demixingour case always requires the demixing of the homopolymFor small volume fraction of the homopolymers, Fig. 13~a!shows thatxs increases rapidly. This is because it is entrocally more costly to segregate theA and B segments whenthere are more LRCs.

Figure 13~b! shows the types of segregation for differevalues ofn2 and n3 for a ternary mixture containing LRCwith symmetric composition. We find that aroundn25n3 ,type-b segregation occurs. Forn2.n3 , we find type-gA seg-regation, and forn2,n3 , we find type-gB segregation. Thereason why the type-gA and type-gB regions are locatedabove and below the linen25n3 is explained by the energychange arguments similar with those we made in Sec. IIthe context of LRC mixtures@with f 2 replaced byn2N2 /(n2N21n3N3)#.

Figure 14 shows the segregation types for a ternary mture with asymmetric LRCs. We find that the type-b segre-gation region tilts to the region whereinn2,n3 , and the arearepresenting type-gA segregation becomes larger. This is bcause the LRCs have more type-B segments, and thereforeis easier for them to aggregate with type-B homopolymers.

In the experiments of Kulasekereet al.9 for PS-r-PMMA

heo

9.

FIG. 12. The dependence of the values ofx at the spinodal (xs) on l1 in theternary mixture of symmetric LRCs (f 150.5) in two homopolymers with anumber ratio ofn1 : n2 : n351 : 1 : 1. The chain lengths areN15N25N3

51000.


, i

f

egel

thly

e

-meera-me

iste

.

rialceg

r ofac-oodhed

ns,r oftheex-e

achvialdes

t

ym

ret

at

ho-tricy


in a thin film composed of homopolymer PSs and PMMAsis observed that PS-r-PMMAs are miscible~inmiscible! inPSs forf PS.0.78(, 0.78), and are miscible~inmiscible! inPMMAs for f PS,0.6(. 0.6). In the intermediate range oLRC compositions, PS-r-PMMAs locate at the PS/PMMAinterface. Although our theory is applicable in the weak sregation limit, these experimental findings are qualitativmanifested in our results~see Figs. 13 and 14!. When thefraction f 1 decreases, for a fixed ratio ofn2 : n3 , the segre-gation types change from type-gB to type-b and eventuallyto type-gA . In between, the segregation is of type-b, whichindicates a kinetic pathway to a final phase whereinLRCs locate at the interface of the two types of homopomers.

By adding random copolymers to binary homopolym

FIG. 13. ~a! The values ofx at the spinodal (xs) as a function of thenumbers of chains of the two homopolymers in ternary mixtures of smetric LRCs (f 150.5) with l150.8 in two homopolymers. The chainlengths areN15N25N351000.~b! Segregation types for the same mixtuas in ~a!. From the lightest to the darkest, the gray scale representsregions in whichb, gA , andgB types of concentration fluctuations occurthe spinodal.


t

-y

e-

r

blends, Rigbyet al.15 find a maximum in the spinodal temperature when the two homopolymers have different volufractions, and a monotonic decrease of the spinodal tempture when the two homopolymers have the same volufractions~see Fig. 1 in Ref. 15!. This qualitative behavior isclearly shown in Fig. 13~a!. Away from n25n3 addition ofLRCs is equivalent to decreasing the value ofn2 and n3 ,keepingn2 : n3 fixed. Generally, a nonmonotonic variationseen whenevern2 : n3Þ1 : 1. The value ofxs decreases firsand then increases beyond a threshold. The point whern2

5n3 gives the minimum value ofxs . In contrast, along then25n3 line, adding LRCs makes the values ofxs increasemonotonically. These findings are consistent with Ref. 15

V. CONCLUDING REMARKS

Random copolymers with linear architectures~LRCs!are manufactured in large amounts. The resulting mateoften contains a mixture of LRCs with different sequenstatistics. LRCs also offer much potential for compatibilizinhomopolymer blends. Understanding the phase behaviomixtures of LRCs and other polymers is thus of some prtical importance. LRCs have also been shown to be gmodel systems to study the effects of frustrating quencrandomness~embodied in the sequence distribution!.

Motivated by these fundamental and practical concerwe have developed a field theory to describe the behaviomixtures of LRCs and other polymers. The theory treatsquenched disorder carried by the sequence distributionplicitly. Unlike Flory–Huggins approaches employed in thpast, the concentrations of each type of monomer in etype of chain are the order parameters. This is a nontripoint as our results show the existence of segregation mo~e.g., of the type-b! which cannot be found without explici

-

he

FIG. 14. Segregation types for different numbers of chains of twomopolymers in ternary mixtures of two homopolymers and asymme( f 150.4) LRCs withl150.8. From the lightest to the darkest, the grascale represents the regions in whichb, gA , andgB types of concentrationfluctuations occur at the spinodal. The chain lengths areN15N25N3

51000.


toretiu

abesenthle

heaysedietuons

bplo-apd

f tu

ki-tiomg

ck

e

en

de

ebe

huc

ath

or-

al

m

us

di-

he

one

s

e

di-

mn


treatment of each type of entity. Our theory allows usconsider the entire phase diagram including the ordephases. However, in this paper, we have restricted attento the conditions which make the homogeneous statestable. Special attention is given to the nature of the unstconcentration fluctuations at the spinodal and below it. Thconcentration fluctuations are announcements of the imping phase transition, and provide information concerningmost facile kinetic pathways and the behavior of supercoomixtures. We also predict pertinent scattering profiles.

For LRC mixtures, we find that, depending upon ttype of mixture, four distinct types of concentration fluctutions lead to the spinodal instability. The instability alwaoccurs withk* 50, and the scattering profiles are predictto be dramatically different depending upon which moietare deuterated in experiments. For LRCs in a ternary mixwith two homopolymers, only three types of concentratifluctuations occur at the spinodal. Our results seem content with some existing experiments.

The results that we describe can be tested directlyscattering experiments with selectively deuterated sam~e.g., Ref. 5!. For example, the equilibrium scattering prfiles can be directly compared to our predictions. Perheven more interestingly, experiments that measure thenamic modes could be compared to our considerations ounstable concentration fluctuations. Such experiments woserve as motivation for further theoretical work on thenetic pathways for the phase transitions under considera

In this paper, we have used our theory to study systethat belong to the Ising universality class. In the followinpaper, we examine mixtures of LRCs and diblocopolymers—a system in the Brazovskii class.

ACKNOWLEDGMENT

Financial support for this work was provided by thU.S.D.O.E.~Office of Basic Energy Sciences!.

APPENDIX

The incompressibility condition imposes the requiremS t51

P drAt (kW )1drB

t (kW )50. Including this condition into thefree energy functional decreases the number of indepenorder parameters by one. Accordingly, the 2P32P matrixG2(kW ) becomes a (2P21)3(2P21) matrix. This proce-dure is equivalent to a projection from 2P dimensions ontothe 2P21 dimension plane~1, 1,..., 1, 1!. Here the 2Pdimensions are in a coordinate system (drA1

,drB1

,...,drAP,drBP

). However, the resulting matrix in th2P21 dimension plane is not in an orthogonal systemcause the projection of an orthogonal coordinate systemnot necessarily an orthogonal coordinate system itself. Twe cannot obtain the eigenvalues and eigenvectors direfrom matrix a @see Eq.~10!#.

To overcome this, we perform a rotational transformtion on the matrix. There are many ways of doing this. Infollowing, we chooseP52 ~four-dimensional space! to il-lustrate such a rotational transformation.


donn-leed-ed

-

sre

is-

yes

sy-held

n.s

t

nt

-iss,tly

-e

In the four-dimensional space, we choose the new codinate system such thate1851/2)(3,21,21,21), e2851/&(0,1,21,0), e3851/A6(0,1,1,22), and e4851/2(1,1,1,1). The transformation matrix from the origincoordinate system e15(1,0,0,0), e25(0,1,0,0), e3

5(0,0,1,0), ande45(0,0,0,1) to this new coordinate systeis

U5F )/2 0 0 1/2

2)/6 &/2 A6/6 1/2

2)/6 2&/2 A6/6 1/2

2)/6 0 2A6/3 1/2

G . ~A1!

The 333 matrix in an orthogonal coordinate system is thobtained by

a85UTaU, ~A2!

where the matrix product is taken through the first threemensions, anda is the matrix defined in Eq.~10!. The eigen-values of a8 are then obtained in the standard way. Teigenvectors in the original coordinate system~such that theyrepresent the ratio of concentration fluctuatidrA1

: drB1: drA2

: drB2! are obtained through a revers

transformation:vW i5UvW i8 , where i 51, 2, 3, andvW i8 are thethree eigenvectors ofa8 plus a fourth component which iset to zero.

In the case of a 333 matrix, the eigenvalues can bobtained analytically. The results are

j1522AQ cos~f/3!2a/3,

j222AQ cos~~f12p!/3!2a/3, ~A3!

j3522AQ cos~~f22p!/3!2a/3,

where

a52~a118 1a228 1a338 !,

b5a118 a228 1a118 a338 1a228 a338 2a128 a128 2a138 a138 2a238 a238 ,~A4!

c52~a118 a228 a338 12a128 a138 a238 2a138 a138 a228 2a128 a128 a338

2a238 a238 a118 !,

and Q5(a223b)/9, R5(2a329ab127c)/54, f5arccos(R/AQ3).

The structure factors for specific components can berectly obtained from

S5Ua821UT. ~A5!

For example,S11 is the structure factor for type-A segmentsof species 1. In this case, the elements of the fourth coluand the fourth row of matrixa821 are equal to zero.

1I. W. Hamley, The Physics of Block Copolymers~Oxford Univ. Press,Oxford, 1998!.

2Z. Tuzar and P. Kratochvil, inSurface and Colloid Scienceedited by E.Matijevic ~Plenum, New York, 1993!, Vol. 15.

3J. Noolandi and K. M. Hong, Macromolecules16, 1443~1982!.4L. Leibler, H. Orland, and J. C. Wheeler, J. Chem. Phys.79, 3550~1983!.5N. P. Balsaraet al., Macromolecules25, 6137~1992!.


a

.


6W. W. Graessley, R. Krishnamoorti, N. P. Balsara, and D. J. Lohse, Mromolecules26, 1137~1993!.

7J. Rhee and B. Crist, J. Chem. Phys.98, 4174~1994!; B. Crist and J. Rhee,J. Polym. Sci.32, 159 ~1994!.

8C.-A. Dai et al., Phys. Rev. Lett.73, 2472~1994!.9R. Kulasekereet al., Macromolecules29, 5493~1996!.

10C. Yeung, A. C. Balazs, and D. Jasnow, Macromolecules25, 1357~1992!.11D. Bratko, A. K. Chakraborty, and E. I. Shakhnovich, Phys. Rev. Lett.76,

1844 ~1996!; S. Srebnik, A. K. Chakraborty, and E. I. Shakhnovichibid.77, 3157~1996!.

12S. Qi et al., Phys. Rev. Lett.82, 2896~1999!.13A. Sali, E. I. Shakhnovich, and M. Karplus, Nature~London! 369, 248

~1994!; H. Frauenfelder and P. G. Wolynes, Phys. Today47, 57 ~1994!.14L. Leibler, Makromol. Chem., Rapid Commun.2, 393 ~1981!.15D. Rigby, J. L. Lin, and R. J. Roe, Macromolecules18, 2269~1985!.16R. Koningsveld and L. A. Kleintjens, Macromolecules18, 243 ~1985!.17D. Broseta and G. H. Fredrickson, J. Chem. Phys.93, 2927~1990!.18E. I. Shakhnovich and A. M. Gutin, J. Phys.~France! 50, 1843~1989!; C.


c- D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E51, 4727~1995!.

19G. H. Fredrickson and S. T. Milner, Phys. Rev. Lett.67, 835~1991!; G. H.Fredrickson, S. T. Milner, and L. Leibler, Macromolecules25, 6341~1992!.

20A. C. Balazset al., Macromolecules19, 2188~1985!.21K. S. Schweizer, Macromolecules26, 6050~1993!.22R. L. Scott, J. Polym. Sci.9, 423 ~1952!.23K. M. Hong and J. Noolandi, Macromolecules14, 727 ~1981!.24H. Benoit and G. Hadziioannou, Macromolecules21, 1449~1988!.25G. H. Fredrickson and L. Leibler, Macromolecules22, 1238~1989!.26I. Drukhimovich and A. V. Dobrynin, Macromol. Symp.81, 253 ~1994!.27H. S. Jeon, J. H. Lee, and N. P. Balsara, Macromolecules31, 3328~1998!;

31, 3340~1998!.28In Figs. 6~a! and 6~b!, type-b segregation happens withinu f 12 f 2u

,0.01, with the area in Fig. 6~a! being slightly larger than that in Fig6~b!.


Theoretical study of polymeric mixtures with different ... papers/67.pdf · help compatibilize...

Documents

Transcript of Theoretical study of polymeric mixtures with different ... papers/67.pdf · help compatibilize...