Morphologies and Bridging Properties of Graft Copolymers

Morphologies and Bridging Properties of Graft Copolymers

Liangshun Zhang, Jiaping Lin,* and Shaoliang LinKey Laboratory for Ultrafine Materials of Ministry of Education, School of Materials Science and Engineering,East China UniVersity of Science and Technology, Shanghai 200237, P. R. China

ReceiVed: August 4, 2006; In Final Form: NoVember 2, 2006

Morphologies and bridging properties of graft copolymers in the bulk state were studied by using a real-space algorithm of self-consistent field theory in two dimensions. The phase transition from cylindrical tolamellar phase can be triggered by changing the position of graft points and the number of branches. Thefraction of bridged conformation,fbridge, shows a tendency to decrease with increasing the length of free endblocks,τ1, and the number of branches,m. The value offbridge has a discontinuous drop when the transitionfrom cylindrical to lamellar phase takes place. The relationship betweenm and the number of bridged chainsper unit area,nb, which is associated with the mechanical properties of copolymers, was also examined. Itwas found thatnb increases with increasingm in the cylindrical phase. However, in the lamellar phase,nb

decreases whenm increases. It is proposed that the position of graft points and the number of branches aretwo important parameters for material design.

Introduction

Molecular architecture has been recognized as an importantrole in determining morphologies, phase behavior, and materialproperties of copolymers. For the simplest AB diblock copoly-mer, the morphologies depend on the composition and theinteraction between two different blocks.1 Whereas, as thearchitecture of copolymer changes from diblock copolymer tograft copolymer, the morphological behavior depends not onlyon the composition and interaction energy, but also on themolecular architecture, like the position of junctions and thenumber of branches. Systematic studies of the relationshipbetween molecular architecture and morphology were experi-mentally limited due to the unavailability of model graftcopolymers with well-defined architecture. Gido et al. used theconstituting block copolymer hypothesis to predict and interpretthe morphological behavior of graft copolymers.2-5 The con-stituting block copolymer hypothesis proposed that the mor-phological behavior of copolymers with a complex architectureis governed by the behavior of smaller block copolymer units(the constituting block copolymers) associated with the molec-ular architecture. It was concluded that the behavior of graftcopolymers is dictated by the behavior of the smaller architec-tural subunits from which they are comprised. Other theoreticalefforts have also been made to understand the influence ofmolecular architecture on the morphology and phase behaviorof graft copolymers.6-9 For example, the work of Balazs andher co-workers indicated that it is theøN of the averageconstituting single graft copolymer that determined the proximityof order-disorder transition for the graft copolymers (ø is theFlory-Huggins parameter andN is the total number of chemicalsegments in the single graft copolymer).9 Recently, Ye et al.systematically studied the morphology and phase behavior ofπ-shaped copolymers via a combinatorial screening methodbased on self-consistent field theory (SCFT).10 The transitionof the order-order phase was investigated by varying the

position of graft points. They predicted a hexagon-hexagonmorphology that has not been reported for linear and starcopolymers in the bulk state.

Morphological formation of A-g-Bm type graft copolymershas been investigated, and it was found that their microdomainstructures were considered to be almost identical to A-s-B typestar copolymers.2-5 Despite the similarity of the domainstructures between the graft and star copolymers, the backboneconformations of the A-g-Bm graft copolymers are different fromthose of the A-s-B star copolymers when backbone blocks formthe continuous matrix. Star copolymers have only the free endblocks which adopt a dangled conformation. However, thebackbone blocks of graft copolymers take either loopedconformation whose junctions localize in the same domain orbridged conformation whose junctions are anchored on thedifferent interfaces. The backbone blocks provide bridged“chains” across the continuous matrix that separates the discretedomains, thereby creating a physically cross-linked networkbetween domains. When material is fractured, the bridged blocksmust be pulled out of the microstructures, leading to a greatdeal of energy dissipation. A correlation between bridged blocksand mechanical strength in graft copolymers was experimentallydemonstrated.11-16 For example, Zhu et al. investigated mor-phological characteristics and mechanical properties of a seriesof graft copolymers, which exhibit high stress and strain atbreak.15 It was proposed that the high strength of graftcopolymers is attributed to bridged conformation of backbonein the microstructures. The blocks of backbone bridge adjacentdomains, resulting in the enhanced mechanical properties. Thus,the fraction of bridged conformation is one of the fundamentalissues in molecular design of graft copolymers that providesoptimal mechanical properties.

There have already been a number of research efforts towardthe bridging properties of linear block copolymer. Experimentalefforts have been directed toward determining the bridgingfractionbydielectricaltechniquesandrheologicalmeasurements.17-20

A theoretical challenge to evaluate the bridging fraction wasmade by Matsen and Thompson.21 They used the self-consistent-

* Address correspondence to this author. Phone:+86-21-64253370.Fax: +86-21-64253539. E-mail: [email protected].

351J. Phys. Chem. B2007,111,351-357

10.1021/jp0650432 CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 12/21/2006

field theory to evaluate the bridging properties of triblockcopolymers with lamellar, cylindrical, and spherical morphol-ogies, and the bridging fractions are about 40-50%, 60-65%,and 75-80%, respectively, which agreed with the results ofexperimental determination. Their calculation results alsoindicated that the bridging fraction depends weakly on the degreeof segregation and the copolymer composition. Drolet andFredrickson developed a real-space self-consistent algorithm forevaluating the bridging fraction of internal blocks in ABABApentablock copolymers.22 They anticipated that the pentablockcopolymers with A-B-A-B-A composition of 10-15-50-15-10 in volume percentages should be optimally tough. Rasmussenet al. applied SCFT to calculate the bridging fraction of chainin (AB)p multiblock copolymer systems.23 It was found thatthe bridging fraction has a sharp raise when polymerizationindex p goes from 2 to 3 and followed a slight decrease forcopolymers withp more than 3. They also calculated thebridging fractions of the polymer chain that bridges consecutivelamellar domains. Daoulas et al. studied the bridging propertiesof styrene block copolymer self-adhesive materials by using self-consistent-field theory.24 The SCFT results were combined witha slip tube model of rubber elasticity to predict the elasticbehavior. Their predicted results were in good qualitativeagreement with the experimental observations. These studieshave focused on the limits of the linear block copolymers,whereas, as far as we know, a theoretical study of bridgingproperties for A-g-Bm type graft copolymers has not beenreported.

In this paper, we use a combination screening method basedon the real-space implementation of the SCFT, developed byFredricksion and co-workers,22,25-28 to study the equilibriummicrostructures assembled by graft copolymers in two-dimensional space. The average bridging fraction of backboneis evaluated by utilizing the equilibrium value of the mean fieldsobtained from the SCFT calculation in terms of the number ofbranches and the position of the first graft point. Our simulationresults are useful for understanding the morphology-propertyrelationship and optimizing the material performance.

Theory

We consider a system with volumeV, containingnG graftcopolymers. Each copolymer is comprised of a flexible ho-mopolymerA backbone along whichm flexible homopolymerB grafts are spaced. The degrees of polymerization of theAandB chains areNA andNB, respectively. The volume fractionof A-type monomer in the system is denoted byfA, and that ofB-type monomer isfB ) 1 - fA. A schematic representation ofthe architecture of graft copolymer is shown in Figure 1. Theith graft is located atτi given by

The length of blocks between neighbor junctions (in units ofNA) is denoted by∆τ ) (1 - 2τ1)/(m - 1).

In the mean-field theory the configuration of the singlecopolymer chain is determined by a set of effective chemicalpotential fieldsωK(r ) (K ) A, B), replacing actual interchaininteractions within the melt. These potential fields are conjugatedto the monomer density fieldsæK(r ). We invoke an incompress-ibility ( æA(r ) + æB(r ) ) 1) by introducing a Lagrange multiplierú(r ). For such an A-g-Bm melt, the free energy per chain (inunits of kBT) is given by

where the Flory-Huggins parameterøAB characterizes therepulsive interaction betweenA- andB-type monomers,NG )NA + mNB denotes the total degree of polymerization of thegraft copolymer, andQG ) ∫ dr qA(r ,1) is the partition functionfor a single noninteracting, grafted chain subject to the fieldsωA(r ) andωB(r ) in terms of the backbone propagatorqA(r ,s).The contour lengths increases continuously from 0 to 1 fromone end of the homopolymer chain to the other. The spatialcoordinater is in units of Rg, whereRg

2 ) NGa2/6 (a is thestatistical segment length). The backbone propagator is dividedinto m + 1 segments

where each segment satisfies the modified diffusion equation

and is subject to the following initial conditions:

Here, qB(r ,s) is a propagator forB graft that satisfies thefollowing modified diffusion equation

and is subject to the initial conditionqB(r ,0) ) 1 for the freeend of the graft ats ) 0. We also define a back-propagator ofthe jth B chain,qBj

+ (r ,s). It satisfies eq 6 and starts on the endof theB chain tethered to the backbone. It is therefore subjectto the initial condition

In terms of these propagators, the monomer densitiesæA(r )andæB(r ) become

Figure 1. Molecular architecture of the graft copolymer.

τi ) τ1 +(i - 1)(1 - 2τ1)

m - 11 e i e m (1)

F ) -lnQG

V- 1

V∫ dr [ωAæA + ωBæB - øABNGæAæB +

ú(1 - æA - æB)] (2)

qA(r ,s) ) qA(j)(r ,s) (3)

for τj e s < τj+1; j ) 0, 1, ...,m; τ0≡ 0, τm+1 ≡ 1

NG

NA

∂qA(j)(r ,s)

∂s) ∇2qA

(j)(r ,s) - ωA(r )qA(j)(r ,s) (4)

qA(j)(r ,τj) ) qA

(j)1)(r ,τj)qB(r,1); j ) 1, 2, ...,m; qA(0)(r ,0) ) 1

(5)

NG

NB

∂qB(r ,s)

∂s) ∇2qB(r ,s) - ωB(r )qB(r ,s) (6)

qBj+ (r ,0) )

qA(j)(r ,τj)qA

(j)(r ,1-τj)

qB2(r ,1)

(7)

æA(r ) ) ∑i)1

m+1

æAi (r ) )

VfA

QG∑i)1

m+1 ∫τi-1

τi ds qA(r ,s)qA(r ,1-s)

(8)

æB(r ) )VfB

mQG∑j)1

m ∫0

1ds qB(r ,s)qBj

+ (r ,1-s) (9)

352 J. Phys. Chem. B, Vol. 111, No. 2, 2007 Zhang et al.

where,æAi (r )is the density coming from blocks betweenτi-1

and τi. Finally, the minimization of the free energy,F, withrespect toæA, æB, andú is achieved by satisfying the mean-field equations:

To solve the SCFT equations, we use a variant of thealgorithm developed by Fredrickson and co-workers.22,25-27 Westart from a general random initial state. For the solution of thediffusion equations eq 4 and eq 6, we employed the Baker-Hausdorff operator splitting formula proposed by Rasmussenet al.29,30 The density fieldsæI(r ) of species I, conjugated thechemical potential fieldsωI(r ), are evaluated based on eqs 8and 9 and eqs 10 and 11. The chemical potential fieldsωI(r )can be updated by using a two-step Anderson mixing scheme.31

Next, we briefly describe the calculation that determines theequilibrium properties of the backbone of the graft copolymerexisting in bridged and looped conformations. Here, we considerA-g-Bm graft copolymer melt in a region of the parameter spacewhere theA blocks form the continuous matrix and theB blocksproduce the cylindrical domains. The fraction ofA blocks thatbridge the neighborB cylinders can be evaluated by utilizingthe equilibrium configuration of fieldsωA andωB, obtained fromthe algorithm as previously described. The first step is to performa Voronoi tessellation with respect to the center of a cylindricaldomain. Next, we focus on a single Voronoi cellD1. Followingthe approach of Matsen and Thompson,21 the function is definedas

constraining theith graft point that belongs toD1. Thedistribution can now be propagated by solving eq 4 as the initialcondition ats ) τi up to s ) τi+1.32 The fraction,fi, of loopedconformation, where the blocks between theith graft point andthe (i+1)-th graft point have both ends in the cellD1, is givenby24

The average looped fraction,floop, for the cell D1 is thenobtained as

The fraction of bridged conformation can be given byfbridge

) 1 - floop. An average value offbridge can be evaluated byrepeating the calculation for all cells and performing anarithmetic average.

The different step of the evaluating procedure for the bridgingfraction between the lamellar phase and the cylindrical phaseis the Voronoi tessellation. It is preformed with respect to theinterfaces of backbone-rich lamellar phase.

All simulations were carried out in two dimensions on a 64× 64 lattice with periodic boundary conditions. Contour step

sizes for theA backbone andB grafts were set at 0.01,respectively. The numerical simulations were carried up to aconvergence of 10-6 in free energy and the achievement of theincompressibility condition. In the simulations the box wasoptimized for each system to minimize the free energy. Thebox size corresponding to the lowest free energy upon conver-gence was chosen as the most appropriate one.33

Results and Discussion

In this work, we investigate the molecular architecture effecton the morphologies and bridging properties of graft copolymers.We focus on two factors: the number of branches and the blocklength between the neighbor junctions. With the constraint ofthe graft pointsτi (eq 1), the block length between the neighborjunctions can be determined only by the first junctionτ1. Forexample, ifτ1 ) 0, the first and last graft points are located atdifferent ends of the backbone. Ifτ1 ) 0.5, the graft copolymerbecomes an A2-s-Bm star copolymer and the bridging fractionbecomes zero. We restrict attention to graft copolymers with70.0% backbone content. At this composition, the equilibriummorphology is predominantly a hexagonal array of cylindersin a continuous matrix rich in backbone blocks, while lamellarstructure can also form under certain conditions. According tothe work of Balazs and her co-workers,9 we fix the interactionstrength of the average constituting single graft copolymerh )øABNG/m at a value of 20.0, sufficient to produce microphaseseparation in the melt.

A typical total density fieldæA(r ) for graft copolymers withm ) 2 andτ1 ) 0.05 is shown in Figure 2a. The black (gray)areas represent the regions of lower (higher) density ofA-typemonomers. The graft blocks self-assemble into cylindersdispersed in a matrix rich in backbone blocks. According tothe calculations, the cylindrical phase is observed over the range0 e τ1 e 0.50 withm ) 2, 0 e τ1 e 0.38 withm ) 3, and 0e τ1 e 0.33 with m ) 4. The cylindrical to lamellar phasetransition takes place when the number of branches and positionof the first graft point are changed. For example, the lamellarphase forms whenm ) 3 andτ1 ) 0.40. The total density fieldæA(r ) of such a lamellar phase is shown in Figure 2b. The gray(black) color is assigned to the higher (lower) density ofbackbone blocks. The lamellar phase is produced over the range0.38< τ1 e 0.50 withm ) 3, and 0.33< τ1 e 0.50 withm )

Figure 2. Total density fieldæA(r ) for a graft copolymer melt withfA) 0.70 andh ) øABNG/m ) 20.0: (a)m ) 2, τ1 ) 0.05, box size is8.20Rg × 14.20Rg; (b) m ) 3, τ1 ) 0.4, box size is 6.70Rg × 6.70Rg.

ωA(r ) ) øABNGæB(r ) + ú(r ) (10)

ωB(r ) ) øABNGæA(r ) + ú(r ) (11)

æA(r ) + æB(r ) ) 1 (12)

qjA(r ,s)τi) ) {qA(r ,s)τi) r ∈ D1

0 r ∉ D1} (13)

fi )∫D1

dr qjA(r ,τi+1)qA(r ,1-τi+1)

∫Vdr qjA(r ,τi+1)qA(r ,1-τi+1)

(14)

floop )1

m - 1∑i)1

m-1

fi (15)

Morphologies and Bridging Properties of Graft Copolymers J. Phys. Chem. B, Vol. 111, No. 2, 2007353

4 in our calculations. Regarding the mechanism of the phasetransition between the lamellar and the cylindrical phase whenτ1 andm are changed, it can be explained as follows. In the flatstructure of a lamellar phase, the junctions of graft copolymerspredominately localize along the same interface. When∆τ(length of blocks between neighbor junctions) increases, cor-responding to a decrease inτ1 or m, blocks between junctionsbecome overcrowded. This results in more blocks stretchingnormal to the interface on the side of the backbone. Theinterfacial curvature could increase to alleviate this effect causedby the increase in∆τ. As a result, the transition from the flatstructure of a lamellar phase to the curved structure of acylindrical phase occurs to lower the entropic contribution ofthe inner blocks to the overall free energy when∆τ increases.

Gido et al. used the constituting block copolymer hypothesis(CBCH) to predict and interpret the phase behavior of graftcopolymers.2-5 The phase behavior of graft copolymers isdetermined by the behavior of the constituting star copolymersassociated with the architecture of graft copolymers. ExistingMilner’s theory is used to predict the phase behavior of theconstituting star copolymers. In Milner theory, the phasebehavior of the constituting star copolymers relies not only onthe volume fraction of respective blocks, but also on themolecular asymmetry parameter.6 Phase transition of the graftcopolymers is predicted according to the variation of themolecular asymmetric parameter and the volume fraction ofrespective blocks.

The CBCH can predict the cylindrical to lamellar phasetransition for the graft copolymers modeled in the present workunder certain conditions. For example, when∆τ is asymptoti-cally zero, the graft copolymer can be regarded as a starcopolymer. The molecular asymmetry parameter of a symmetricstar copolymer is given byε ) (nB/nA)(lB/lA)1/2, whereni (i )A, B) is the arm number of thei-type block andli is the ratioof the segment volume to the square of the statistical segmentlength for thei block.6 The A-g-B2, A-g-B3, and A-g-B4 typegraft copolymers can be regarded as A2-s-B2, A2-s-B3, and A2-s-B4 type star copolymers, respectively. The molecular asym-metry parameters of A2-s-B2, A2-s-B3, and A2-s-B4 copolymersare 1.0, 1.5, and 2.0, respectively. The CBCH predicts acylindrical phase for A2-s-B2 and a lamellar phase for A2-s-B3

and A2-s-B4 with fA ) 0.70. In our simulations, the morphologiesof A-g-B2, A-g-B3, and A-g-B4 type graft copolymers arecylindrical, lamellar, and lamellar phase, respectively. The phasetransitions predicted by CBCH upon changing the molecularstructures agree well with our simulation results.

We subsequently investigate the conformation behavior ofbackbone in the bulk state. The looped, bridged, and dangledconformations of the backbone of graft copolymers in acylindrical phase withB domains in anA matrix are shown inFigure 3a. The Voronoi tessellation is performed with respectto the centers of the domains. Similarly, the conformations ofbackbone in a lamellar phase are illustrated in Figure 3b. TheVoronoi tessellation is preformed with respect to the interfacesof the A-rich lamellar phase. The free end blocks of thebackbone dangle at the interfaces. The backbone blocks betweenthe neighbor junctions are localized in the same Voronoi celland form the looped conformation, while they are localized inthe neighbor Voronoi cells and adopt the bridged conformation.

The bridging fractionfbridge as a function of the position ofthe first graft pointτ1 for graft copolymers with number ofbranchesm ) 2, 3, and 4 is shown in Figure 4a. Whenm ) 2,fbridge versusτ1 has a slight raise over the range 0.05e τ1 e0.10, a slow drop over the interval 0.10e τ1 e 0.33, and a

rapid drop asτ1 increases beyond 0.33, whereas, whenm ) 3and 4,fbridge versusτ1 only includes two regions: a slow dropand a rapid drop. For a givenτ1, fbridge shows a decrease asmincreases. To have a further insight into the bridging behavior,we also calculated the average distance between centers of thenearest-neighbor domains (shown in Figure 3a) as a functionof τ1, as shown in Figure 4b. There is a drop, a plateau, and araise in the intercylinder spacing. Whenτ1 is smaller, theintercylinder spacing is controlled by the∆τ, the block lengthbetween neighbor junctions. With an increase inτ1, the lengthof ∆τ has shrunk and the distanceDcyc has dropped. Althoughthe length of blocks, bridging the neighbor cylinders, hasdecreased, the bridging fraction has only a slow drop due tothe intercylinder spacing being close. Whenτ1 is larger, thefree end blocks are the longest ones that control the distancebetween the neighbor cylinders. With further increase inτ1, the

Figure 3. Typical conformations of A-g-Bm graft copolymers in (a)cylindrical phase withB domains in anA matrix and (b) lamellar phase.The looped, bridged, and dangled conformations are shown.

Figure 4. (a) bridging fractionfbridge and (b) average distanceDcyc (inunits ofRg) between the centers of the nearest-neighbor domains as afunction of the position of the first graft pointτ1 for the graft copolymerswith m ) 2, 3, and 4. The inserts showfbridge andDlam as a function ofτ1 for graft copolymers withm ) 3, 4.


distanceDcyc has a raise and∆τ becomes shorter. As a result,the bridging fraction has a rapid drop. Whenτ1 is intermediate,Dcyc is deep in the state with a big plateau of almost constantvalue due to the competition between the lengthτ1 and∆τ. Theinserts of Figure 4 show the bridging fractionfbridge and theaverage distanceDlam (shown in Figure 3b) as a function ofτ1

over the range 0.40e τ1 e 0.45 withm ) 3, and 0.35e τ1 e0.44 with m ) 4, where the morphology transits from thecylindrical to the lamellar phase. There is a discontinuous changein the bridging fraction and the average distance at the phasetransition. The value offbridgehas a rapid drop and is asymptoti-cally zero, butDlam has a raise whenτ1 increases. These trendscan be ascribed to the fact that free end blocks favoring loopedconformation become long asτ1 increases.

To achieve further understanding of the behavior of loopedand bridged conformations, density fieldsφA

1(r ) of free endblocks andφA

2 (r ) of the backbone blocks between the junctionτ1 andτ2 have also been studied. Figure 5 shows a result form) 2 andτ1 ) 0.05, corresponding to the longer middle blocks.The total density field ofA blocks is shown in Figure 2a. Theblack regions in Figure 5a,b represent the higherA volumefraction contributed from the free end blocks and middle blocksbetweenτ1 andτ2, respectively. The free end blocks are localizedalong the interfaces of cylinders, while the middle blocks areconcentrated in the continuous matrix rich inA-blocks. Thecalculation shows that 72.0% of the blocks adopt bridgedconformation. Whenτ1 increases to 0.43, corresponding to thelonger free end blocks, theA-rich domain is predominantly thefree end blocks and the interface regions contain the middleblocks, as shown in Figure 6, parts a and b, respectively. Thefree end blocks fill in the gap between the cylinders. The middleblocks betweenτ1 andτ2 reside in the interfacial regions, withonly 43.8% of these blocks forming bridged conformation asrevealed by the calculations. Thus, the shorter middle blocksare more favorable to adopt the looped conformation, while thelonger middle blocks are more favorable to form the bridgedconformation. As previously described, the lamellar phase isproduced whenm ) 3 andτ1 ) 0.40. The bridging fraction is13.5%, which is lower than that in cylindrical morphology. Thedensity fieldsφA

1(r ) and φA2(r ) are shown in Figure 7. The

black areas represent the regions of higher densities of free endblocks (Figure 7a) and middle blocks betweenτ1 andτ2 (Figure7b). Blocks betweenτ1 andτ2 reside in the interfacial area andbroaden the interface.

The bridging behavior can be further viewed in Figure 8a inwhich fbridge is plotted against the number of branchesm for thegraft copolymers with the position of the first graft pointτ1 )0.10, 0.15, and 0.20. With increasing the value ofm, the bridgingfraction sharply decreases and then remains almost constantwhenτ1 ) 0.10, whereas there is always a decrease ascribed tothe fact that the block length between neighbor junctionsbecomes shorter forτ1 ) 0.15 and 0.20. As stated in theIntroduction, the bridged chains of graft copolymers create aphysically cross-linked network between neighbor domains,resulting in the enhanced mechanical properties. According torefs 19, 23, and 24, the tensile at the break is proportional tothe number of bridged “chains” per unit area,nb ) (m - 1)-fbridge, if the strength of the bridged “chains” is the same and

Figure 5. Density fields (a)æA1(r ) of free end blocks and (b)æA

2(r ) ofthe backbone blocks between the junctionsτ1 and τ2 for the graftcopolymer withm ) 2, τ1 ) 0.05. Box size is 8.20Rg × 14.20Rg.

Figure 6. Density fields (a)φA1(r )of free end blocks, and (b)φA

2(r ) ofthe backbone blocks between the junctionsτ1 and τ2 for the graftcopolymer withm ) 2, τ1 ) 0.43. Box size is 7.60Rg×13.16Rg.

Figure 7. Density fields (a)φA1(r ) of free end blocks and (b)φA

2(r ) ofthe backbone blocks between the junctionsτ1 and τ2 for the graftcopolymer withm ) 3, τ1 ) 0.40. Box size is 6.70Rg × 6.70Rg.


the “chains” are simultaneously pulled out. The higher valueof nb could lead to better mechanical properties. As shown inFigure 8a, thenb increases with increasing the number ofbranches, and graft copolymers with smallerτ1 values producelargernb. Therefore, the molecular architecture with largermand smallerτ1 should have optimal mechanical propertiesaccording to the above arguments. Weidisch et al. haveexperimentally investigated the tensile properties of graftcopolymers comprised of polyisoprene (PI) backbone andpolystyrene (PS) grafts.14 It was found that strain and tensilestrength at the break increase with increasing number ofbranches. The strain at the break is about 2300% for the graftcopolymer sample with ten branches. Recently, Zhu et al. haveinvestigated the effect of chain architecture on the tensileproperties of a series of PI-g-PS graft copolymers and foundthe fact that tensile strength increases linearly with the num-ber of junction points.16 Therefore, our simulation resultsagree well with the general features of these experimentalobservations.

Next, we focus onfbridge andnb at the higher value ofτ1 )0.40, where the transition from the cylindrical to the lamellarphase takes place. The dependence offbridgeandnb onm is shownin Figure 8b (note that the graft copolymers form the cylindricalphase whenm ) 2; the bridging fraction of 0.52 is for thecylindrical phase). The bridging fraction becomes 0.13 whenthe cylindrical phase transforms to the lamellar phase (m ) 3).A drop in the number of bridged chains per unit area also occursdue to the transition from the cylindrical to the lamellar phase.The values offbridge and nb show a decrease asm furtherincreases from 3 to 5. In such a case, the mechanical propertiesbecome worse whenm increases. The above calculation resultssuggest that the position of graft points and the number ofbranches are two important parameters for controlling themechanical properties of graft copolymers. On the other hand,although these predictions of self-consistent-field theory provideuseful insight into the key features determining mechanicalproperties of graft copolymers, the connectivity betweenfbridge

and mechanical properties needs to build a more sophisticated

model and consider more detailed information such as theinterlocked entanglement.

Conclusions

Morphologies and bridging properties of graft copolymersin the bulk state were investigated by using the two-dimensionalself-consistent-field theory. The graft copolymers exhibit cy-lindrical and lamellar mesostructures depending on the positionof graft points and the number of branches. Studies on the chainconformation behavior of the backbone in the cylindrical phaserevealed thatfbridgeshows a tendency to decrease with increasingτ1. When morphology transforms from the cylindrical to thelamellar phase, the value offbridgehas a discontinuous drop anddecreases rapidly asτ1 increases.fbridge andnb as a function ofm were also examined. It was found thatfbridge decreases withincreasingm in both cylindrical and lamellar phases.nb, whichis associated with the mechanical properties of copolymers,increases with increasingm in the cylindrical phase. However,in the lamellar structure,nb decreases whenm increases.

Acknowledgment. This work was supported by the NationalNatural Science Foundation of China (20574018, 50673026).Support from the Doctoral Foundation of Education Ministryof China (Grant No. 20050251008), the Program for NewCentury Excellent Talents in University in China (NCET-04-0410), and the Project of Science and Technology Commissionof Shanghai Municipality (05DJ14005) is also appreciated.

References and Notes

(1) Bates, F. S.; Fredrickson, G. H.Phys. Today1999, 52, 32.(2) Gido, S. P.; Lee, C.; Pochan, D. J.; Pispas, S.; Mays, J. W.;

Hadjichristidis, N.Macromolecules1996, 29, 7022.(3) Xenidou, M.; Beyer, F. L.; Hadjichristidis, N.; Gido, S. P.; Tan,

N. B. Macromolecules1998, 31, 7659.(4) Lee, C.; Gido, S. P.; Poulos, Y.; Hadjichristidis, N.; Tan, N. B.;

Trevino, S. F.; Mays, J. W.Polymer1998, 39, 4631.(5) Beyer, F. L.; Gido, S. P.; Buschl, C.; Iatrou, H.; Uhrig, D.; Mays,

J. W.; Chang, M. Y.; Garetz, B. A.; Balsara, N. P.; Tan, N. B.;Hadjichristidis, N.Macromolecules2000, 33, 2039.

(6) Milner, S. T.Macromolecules1994, 27, 2333.(7) Olvera de la Cruz, M.; Sanchez, I. C.Macromolecules1986, 19,

2501.(8) Dobrynin, A. V.; Erukhimovich, I. Y.Macromolecules1993, 26,

276.(9) Shinozaki, A.; Jasnow, D.; Balazs, A. C.Macromolecules1994,

27, 2496.(10) Ye, X.; Shi, T.; Lu, Z.; Zhang, C.; Sun, Z.; An, L.Macromolecules

2005, 38, 8853.(11) Kennedy, J. P.; Delvaux, J. M.AdV. Polym. Sci.1991, 38, 141.(12) Kennedy, J. P. InThermoplastic Elastomers; Legge, N. R., Holden,

G., Quirk, R., Schroeder, H. E., Eds.; Hanser: Munich, Germany, 1996.(13) Peiffer, D. G.; Rabeony, M.J. Appl. Polym. Sci.1994, 51, 1283.(14) Weidisch, W.; Gido, S. P.; Uhrig, D.; Iatrou, H.; Mays, J. W.;

Hadjichristidis, N.Macromolecules2001, 34, 6333.(15) Zhu, Y.; Weidisch, R.; Gido, S. P.; Velis, G.; Hadjichristidis, N.

Macromolecules2002, 35, 5903.(16) Zhu, Y.; Burgaz, E.; Gido, S. P.; Staudinger, U.; Weidisch, R.;

Uhrig, D.; Mays, J. W.Macromolecules2006, 39, 4428.(17) Watanabe, H.Macromolecules1995, 28, 5006.(18) Watanabe, H.; Sato, T.; Osaki, K.; Yao, M.; Yamagishi, A.

Macromolecules1997, 30, 5877.(19) Takano, A.; Kamaya, I.; Takahashi, Y.; Matsushita, Y.Macromol-

ecules2005, 38, 9718.(20) Takahashi, Y.; Song, Y.; Nemoto, N.; Takano, A.; Akazawa, Y.;

Matsushita, Y.Macromolecules2005, 38, 9724.(21) Matsen, M. W.; Thompson, R. B.J. Chem. Phys.1999, 111, 7139.(22) Drolet, F.; Fredrickson, G. H.Macromolecules2001, 34, 5317.(23) Rasmussen, R. Ø.; Kober, E. M.; Lookman, T.; Saxena, A.J. Polym.

Sci. Part B: Polym. Phys.2003, 41, 104.(24) Daoulas, K. Ch.; Theodorou, D. N.; Roos, A.; Creton, C.

Macromolecules2004, 37, 5093.(25) Drolet, F.; Fredrickson, G. H.Phys. ReV. Lett. 1999, 83, 4317.(26) Ganesen, V.; Fredrickson, G. H.Europhys. Lett.2001, 55, 814.

Figure 8. Bridging fraction (fbridge) and number of bridged chains perunit area (nb) as a function of the number of branchesm with theposition of the first graft point (a)τ1 ) 0.10, 0.15, and 0.20 for thecylindrical phase and (b)τ1 ) 0.4 for lamellar phase and cylindricalphase.


(27) Fredrickson, G. H.; Ganesan, V.; Drolet, F.Macromolecules2002,35, 16.

(28) Patel, D. M.; Fredrickson, G. H.Phys. ReV. E 2003, 68, 051802.(29) Tzeremes, G.; Rasmussen, R. Ø.; Lookman, L.; Saxena, A.Phys.

ReV. E 2002, 65, 041806.(30) Rasmussen, R. Ø.; Kalosakas, G.J. Polym. Sci. Part B: Polym.

Phys.2002, 40, 1777.

(31) Eyert, V.J. Comput. Phys.1996, 124, 271.(32) The boundary conditions were not used for the calculation of the

bridging fraction due to the performance of the Voronoi tessellation.Therefore, the propagationqjA(r ,s) was not accomplished with the Baker-Hausdorff operator splitting formula. It was calculated by the Crank-Nicolson algorithm.

(33) Bohbot-Raviv, Y.; Wang, Z.Phys. ReV. Lett. 2000, 85, 3428.


Documents

Morphologies and Bridging Properties of Graft Copolymers