Lattice Self-consistent-field Theory And Self-assembly Of Coil-Rod Block Copolymers

In this thesis for doctorate that consists of four parts, we develop a novelself-consistent field lattice model and propose a general method to solve theself-consistent field equations. Applying this approach to block copolymers,we studied the self-assembly of coil-rod diblock copolymers and phasebehaviors of linear coil-rod multiblock copolymers.I: Lattice self-consistent field theory In this part, we begin with the exact partition function and obtain thefree-energy function, the end segment distribution functions, the single chainpartition function and self-consistent field theory (SCFT) equations. Thisprocess resembles usual continuous SCFT model and differ from previousSCFT lattice model, which begins with segment distribution functions. Inorder to treat the polymer that contains semiflexible or rigid blocks, weemploy the segments coupled with bond orientation to describe the chain. In the first description of this model, we consider a polymer melt systemconsisting of n AB diblock copolymer chains, each with N A segments oftype A monomer and N B segments of type B monomer distributed over alattice;the degree of polymerization of the chain is N ( = NA +NB)and thetotal number of lattice sites is N L. The A and B segments have the samesize and each segment occupies one lattice site ,and thus NL = nN. Forconvenience, we use the segment size of the block copolymer as the lengthunit. The partition function for this system is given by:{ }Z= {r j s∑}{ js}????NL z∏j n= ∏s N= rjjs s?? r j′s′j?s??????U,,,,1,,1, 1 211expαααλ (1)Where r j, sand α j,sdenote the position and bond orientation of the sthsegment of the j th copolymer, respectively. Here r′ denotes thenearest-neighboring site of r . α can be any of the allowed bondorientations depending on the lattice model used. For the simplest case , i.e.,cubic lattice ,there are six bond orientations. The summation is over all thepossible configurations of the system, i.e.,∑{ }{ } =∑{ }{ }∑{ }{ }∑{ }{ }r j, s, α j,sr1 ,1, α 1,1r1 ,2, α 1,2rn,N,αn,N , Here z is thecoordination number of the lattice .The transfer matrix lambda depends onlyon the chain model used. For a random-walk chain without direct backfolding,′′ ++??= ??? ? =?′+1(1)otherwise,1,0 ,,1,1,zjsjsrrjsjsjsjsλααα α (2)And for a rigid chain ,??′′= ??? =′?′??0otherwise,,11 ,,1,,1jsjsrrjsjsjsjsλααα α (3)U is the nonbonded monomers interaction potential .In this work, weconsider the potential contributions from the nearest neighboring interactionof A segment and B segment .The potential U is expressed as∑∑′=′r rU 12 χφ ?A (r)φ?B(r) (4)Where χ is the Flory-Huggins interaction parameter, which measures theincompatibility between A and B monomers.∑rmeans the summation overall the lattice sites. φ? A (r) and φ? B (r)are the volume fractions of the A and Bsegments at site r ,respectively. { } ∑ ∑A = Ajs=nj = sN =A rrjsr rr, 1 1 ,,φ? ()φ?(,)δ,andφ? B (r) is given by a similar expression .Considering the localincompressibility of the system, the volume fractions φ? A (r)and φ? B (r)satisfy φ? A (r )+ φ?B(r)?1=0 at every site r of the lattice. By employing aHubbard-Stratonovich transformation ,the partition function Z is convertedtoZ = N ∫ Dφ A DφBDωADωBexp{? F} (5)where N is a normalization constant, φ kis the volume fraction field of blockspecies k ,which is independent of individual polymer configurations,and ω k is the chemical potential field conjugated to φ k. ξ si thepressure potential field that ensures the incompressibility of the system, andD? (r)( ? = φA, φB,ωA,ωB,ξ)is the abbreviation ofD? ( r1 )D?(r2)D?(rNL).The free-energy function of F is defined by ,( ) ( ) ( ) ( )[( )]z(r ) (r ) nQrrrFrrrrr rABrABrBBrAA1'ln()1()()'+????=??∑∑∑∑∑χφρξφφωφωφ(6)whereQ = {r s ∑ }{α s}NL z∏s N= λαrs s? ? rαs′?s′????? ?∑sN =AωArs?s =∑NNB+ωBrs????, 21111exp()()11 (7)Is the single chain partition function.. Because all polymer chains are identical,we drop the subscript j , which indexes different polymers. Following thescheme of Leermakers and Scheutjens, 17 the end segment distributionfunction G(r ,s|1)α s that gives statistical weight of all possible walks startingfrom segment 1,which may be located anywhere in the lattice, and ending atsegment s at siter r, is evaluated from the following recursive relation :( ) ( )∑ ∑( )? ???′ ′??′′=? ′′?1 1111,|1,,1|1s sssssssG αrsGrsr α αr rαGαrsλ (8)For all the values of α 1,the initial condition is ( ,1|1) (,1)1G α r=Gr. G ( r,s)isthe free segment weighting factor and ,G (r , s) = ??? eexxpp {{?? ωω BA ((rr ss))}}N1A≤+1S<≤SN≤ANAnother end segment distribution function G(r sN)sα ,| is evaluated from thefollowing recursive relation :( ) ( )∑ ∑( )+ +++′ ′+′?′=′ ?′+1 1111,|,,1|1s sssssssG αrsNGrsr α αr rαGαrsλ (9)with the initial condition G(r , N|N) G(r,N)Nα =,for all the values ofα N .The expression for the single chain partition function Q finallybecomes :=∑∑( )N NNLrGr,N|Q N1 1zα α 1 (10)Minimizing the free-energy function F with φ A,φ B,ω A,ω B and ξ leads to thefollowing SCFT equations:( ) 1( ') ()'rrω A r = χz∑r φB+ξ (11)( ) 1( ') ()'rrω B r = χz∑r φA+ξ (12)φ A ( r )+ φB(r)?1=0 (13)∑∑( ) ( )==AsNssLA GrsGr,s|NGr,s|Qnφ ( r )N11zs 1α α( ,)α1 (14)∑ ∑( ) ( )=+= NLNBA sssGrsGr,s|NGr,s|Qnφ ( r )N11zs 1α α( ,α)1 (15)Here ∑r ′means the summation over the nearest-neighboring sites of r .The expression ( )∑1 z r ′ φ A (r′) represents the fraction of contacts the Bsegment experiences with its nearest-neighboring segments of type A at siter . ( )∑1 z r ′ φ B (r′) has the similar sense. The last two equations identifyφ A(r) and φ B(r) as the average numbers of A and B segments at r ascalculated in an ensemble of noninteraction polymers subjected to thefields , ω A(r) and ω B(r), which act on A and B segments, respectively.Ⅱ:A two-stage relaxation procedure to solve the SCFT equationsThe first stage relaxation procedureTo evolve the system to the free energy minimum as rapidly as possible, atwo-stage relaxation procedure is employed. In the first stage, ( )∑1 z r ′ φ k (r′),the nonlocal effects of the k component (contributions from the neighboringlattice sites of r ), is replaced with φ k(r), the "local" effects (contributionsdue to site r only). The Eqs.(6), (11), and (12) are rewritten asnQF r ArBrArArBrBrrArBrln'{()()()()()()}()[1()()]?= ∑ χφ φ?ωφ?ωφ?ξ?φ?φ(16)ω A ( r )= χφB(r)+ξ(r) (17)ω B ( r )= χφA(r)+ξ(r) (18)These rewritten self-consistent equations (i.e. Eqs. (13)-(15), (17) and (18))resemble the usual SCFT equations, except that the end segment distributionfunction is evaluated from the recursive relations (i.e., Eqs. (8) and (9)) ratherthan the modified diffusion equation. In order to prevent bias of the resultingmorphology, our calculations are initiated with randomly generated fields toscreen stable or metastable. The relaxation scheme for calculation thesaddle-point values for the fields is implemented through the following steps:1. random initial values are set for ω A and ω B,2. the end segment distribution equations are solved numerically tocalculate ( ,1|1) (,1)1G α r=Gr and G(r sN)sα ,|.3. these functions are substituted into Eqs. (14) and (15) to obtain φ Aand φ B,4. the chemical potential field are updated as follows:1 ( )+ 1()=()+()+1[()+()?1]++nAr nBrnArnBrnArnBrω ωωωλ φφ (19)ω nB+ 1 ( r )? ωnA+1(r)=ωnB(r)?ωnA(r)+λ 2???φ nA(r)?φnB(r)?(ω nB(r)?ωnA(r)) χ??? (20)5 the free energy is obtained by Eq. (16),6 return to step 2 until an assigned accuracy of the free energy F′ arereached.In the first stage, the local interaction is employed, which is convenient toinitiate with randomly generated conditions to study self-assembly of polymerand make the calculations more stable.The second stage relaxation procedureIn the second stage relaxation procedure, our calculations are employedunder nonlocal interaction conditions, i.e. Eqs(11)-(15). Here we usedrelaxation equations analogous to that in the first stage:1 ( )+ 1()=()+()+1[1∑(′)+1∑(′)?1]′′++rnrBnAnBnAnBnω A r ωrωrωrλ zφrzφr (21)? =?+??? ∑′?∑′?( ?) ???′′ω nB+1 ( r )ωnA+1(r)ωnB(r)ωnA(r)λ 21zr φnA(r)1zr φnB(r)ωnB(r)ωnA(r)χ(22)Using the fields obtained inthe first stage relaxationprocedure as the initialvalues, the scheme is thesame as that in the firststage. The calculation stopswhen the free energy Fchange at each interaction isreduced to 10?8 . Thestructures so obtainedcorrespond to either stable or metastable equilibrium. By comparing the freeenergy for different phasesobtained, we can obtain thephase diagram.This approach was firstlyused to study the phase diagramof coil-coil diblock copolymers(figure 1). The results areconsistent with theMatsen-Schick phase diagram.We show four stable structuresin Figure 2.Ⅲ: Self-assembly of coil-rod diblock copolymersIn this part, we studythe phase diagram,self-assembled structuresand possible orientationsof rods in this structuresfor coil-rod diblockcopolymers 。 Threedimensionalself-assembled structuresand the phase diagram areobtained firstly by SCFT.In addition, we propose that the possible orientations of rods can be examinedby calculating the stability of the morphology with the assigned rod directions.All of this differs from previous works. In the case of other theories andmethods, three dimensional molecular simulation of this system is recentlyreported in only one literature and our results are consistent with theirs.As shown in Figure 3, it is clear that the phase diagram has four regions,i.e., lamellar, strip, gyroid and micelle phases, and is highly asymmetric. Theonly structures observed are lamellar structures at higher volume fraction ofthe rodlike component, while the cylinder and micelle phases occur only athigher coil content. Further, the order-disorder transition (ODT) occursat χ N≈6.4, which is lower than the corresponding one of coil-coil diblock,where χN ≈10.5. The gyroid phase only occurs at the vicinity of the criticalpoint. From this figure, we can also see that the block stiffness greatly affectsthe phase behavior and breaks the symmetry of the phase diagram observedwhen both blocks are flexible. And that, the transition from the disorder to theordered phase at higher rod contents occurs at a lower χ when comparedwith the transition from a disordered to a ordered morphology at a lower rodcontent.Figure 4 show the structures have been reported experimentally. Exceptfour stable morphologies, we observed the perforated lamellar structure in thestrip phase region. In our calculations, given the same composition andinteraction parameters, the strip structure has lower free energy and isamenable to initial conditions. The interesting zigzag structure[Figure 2(f)] isobserved for f rod =0.6 and χ N<15.As shown in figure 5, ina few instances, we arrivedat some interestingstructures, which have neverbeen reported. Thesestructures are obtained byinitiating with randomlygenerated conditions, andtheir free energies haveslightly higher than thosestable structures. Theseresults can also be used toguide the experimental work.In the present model, the orientations of the rods in different structurescan not be determined in the above calculations. In order to examine the rodorientation for each ordered phase obtained, we carry out a further SCFTcalculation. In this calculation, the values of the fields corresponding to thestructures obtained above are used as the initial conditions, and the rods areassumed to be oriented along assigned direction, which satisfies the localincompressibility constraint. If the morphology with the assigned roddirections is stable or metastable, the free energy of this structure will eitherbe the local minimum or can be distribution. In the morphology with theassigned roddirections is unstable,it will evolve to a newfree energy minimum.The free energy of thenewly arrivedmorphology is usuallysignificantly higherthan that of theoriginal one.From the above calculations, the possible orientations of the rods indifferent structures have been verified. As shown in figure 6, in lamellar,perforated lamellar, and gyoid structures, the rods are arranged in the samefashion that all rods aligningalong a common direction.In the hexagonal-packedstrip structure, the rodswithin the strips areassembled intointerdigitated bilayerstructure [Figure 7].As shown in Figure 8, for the zigzag structure, when we assumed that therods align perpendicular to the zigzag direction, the profile of this structure"collapses" after only three to four iterative steps;when we assumed the rodsare aligned within the planes parallel to the zigzag direction, the zigzag profileevolves slowly to the lamellar phase after thousands of iterations. This meansthat the zigzag phase of this orientation might be a metastable or unstablephase.Notice that the above rod orientations in the respective structures are onlysimplest ones. The possible orientations of the rods in the micelle structure[figure 4(a)] have not been obtained in our calculations. The rods must beassembled in more complicated forms in this structure.Ⅳ: Phase behaviors of coil-rod multiblock copolymersIn this part, we first investigate self-assembly of a coil-rod-coil triblockcopolymer, the obvious difference between this copolymer chain and acoil-rod diblock one is that the rod is grafted by two coils at both ends. In thisarticle, we discuss the effects of temperature on self-assembly behaviors ofcoil-rod-coil triblock copolymers, and therefore, two coils in a copolymerchain keep the same length. Although we minish the parameter range,varieties of self-assembled structures and the complexity of phase behaviorare more complicated than diblock copolymer.As shown in Figure 9, we fist present the phase diagram for acoil-rod-coil triblock copolymer under the weak segregation limitation.Different from the phase diagram for coil-rod diblock copolymer, in which thelamellar morphology occupies most of the phase diagram, the regions of thecylinder morphology is bigger than others. We also observe that varieties ofphase behaviors for a given component parameter with change of interactionsparameter χ N between coil and rod, when a rod is grafted by two coils atboth ends. Figure 10 shows the examples of morphology for coil-rod-coiltriblock copolymers. Among these, there is a novel structure between thecylinder phase and the lamellar phase, i.e., cylinder-lamellar (CL) structure. Inaddition, we also observe a interesting phase behavior that the perforatedlamellar phase region is clipped by two cylinder phase regions, and the gyriodphase region has the same plight. The reason for these may be the competitionbetween the interfacial energy and the stretching energy of the coils.We extend our study to linear coil-rod pentablock copolymers. As shownin figure 11, comparing with coil-rod triblock copolymers, the pentablock onepresents more complex thermotropic phase behaviors. In addition, we alsoobserved a new net structure, which resembles cylinder-assembled structuresand is called "CS", and we also observe the transition of this structure tocylinder one [figure 11]. We show the assembled fashion of cylinders in thisstructure in figure 12.We also study self-assembly of a linear coil-rod heptablock copolymerand find that its phase behaviors resemble the pentsblock ones. For instance,using the same parameters in figure 11(c) (i.e., n c=4 and n rod=6), we findthat coil-rod heptablock copolymers present the phase transitions of gyriod tolamella to perforated lamella and that each phase region is increased.Furthermore, for χ N=70, the perforated lamella structure is still stable.Therefore, we think that a linear coil-rod pentablock copolymer can presentmost phase behaviors of linear coil-rod multiblock copolymers.Finally, we discuss that the effect of the length of the coil between the rodblocks on the microphase separation for a linear coil-rod pentablockcopolymer, and find that the phase behaviors becomes same with coil-rod-coiltriblock copolymers.In conclusion, we have developed a novel self-consistent field latticemodel by combining the advantages of previous SCFT and propose a generalmethod to solve the self-consistent field equations. Our method can beexpediently and rapidly applies to polymer system in three dimensions.Applying our method to coil-rod diblock copolymers, three dimensionalself-assembled structures are first obtained by SCFT and the orientations ofthe rods is these structures can be examined by further SCFT calculations. Inaddition, our results also reveal new morphologies not yet observed inexperiment and may guide experimental work. We investigate thermotropicphase behaviors of coil-rod mutilblock copolymers, i.e., main chain liquidcrystal polymer, which has never been studied by any simulation method, andour results help guide experimental design of new materials.