Using self-consistent field theory, we attempt to elucidate the links between microscopically determined properties, such as the bridging fraction of chains, and mechanical properties of multiblock copolymer materials. We determine morphological aspects such as period and interfacial width and calculate the bridging fractions, and compare with experimental data 

Kober, E. M. (Edward M.)

Lookman, T. (Turab)

Rasmussen, K. O. (Kim O.)

Saxena, A. B. (Avadh B.)

UNT Digital Library

LA-U R-b 3 -bg ( lApproved for public release;distribution is unlimited.Title:Author(s) :Submitted to:Bridging Properties of Multiblock CopolymersKim O . Rasmussen , 4ward M . Kober , Turab Lookman , andAvadh Saxen aMaterials - Research Society ' s Proceed ings Vol . 7341sAIamosNATIONA L LA B O R ATOR YLo s A la mos Nati on a l La boratory, an aff irmative action/equal opportunity employe r, is opera ted by the U niversit y o f Ca lifo rnia fo r t he U. S .Depa rtment o f Energ y u nde r contract W-7405 -ENG -3 6. By acceptance of this articl e, th e p ublishe r reco gnize s th at the U. S . Gove r nme ntre ta ins a n onexclusive, royalty -f ree license to pu blis h or rep ro duce the published form of this cont ribution , o r to a ll ow others to do so, for U .S .Gove rnment p urposes. Los A lamos Na tion al Labo rato ry req u est s that t h e publis he r ide ntify t his article as work performed unde r th ea us pic es of th e U . S. Depa rtme nt o f E ne r g y. Los A lamos N ational Labo ra tory strongly sup ports acade mic freedo m an d a researche r's rig ht top u b lish ; as an institution, however, the Labo rato ry does not endo rse the viewpoint of a publicat i on o r gu ara ntee its t ec h nica l c o rrectness .Form 836 (8/00)Bridging Properties of Multiblock CopolymersK.O. Rasmussen, E .M. Kober, T. Lookman, and A . SaxenaTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, U .S .A.ABSTRACTUsing self-consistent field theory, we attempt to elucidate the links between microscopicallydetermined properties, such as the bridging fraction of chains, and mechanical properties ofmultiblock copolymer materials . We determine morphological aspects such as period andinterfacial width and calculate the bridging fractions, and compare with experimental data .INTRODUCTIONBlock copolymers exhibit a fascinating variety of structured mesoscale phases when thedifferent blocks are sufficiently incompatible . Most work [1] has focused on diblock copolymersAnBm. The chemical dissimilarity between the A and B species (quantified through the segment-segment Flory-Huggins interaction parameter, X) produces periodic mesoscale morphologies [2-4]with symmetries determined by the overall volume fraction, f, of the A block and by the degree ofimmiscibility, XN , (where N=n+m is the total number of segments) . The lowest degree ofimmiscibility (xN ;:z 10.5) leading to mesoscale morphologies occurs for symmetric systems (f0.5 or nz m) where a lamellar structure emerges . Other equilibrium structures [2] established todate include spheres of A (B) arranged on a body-centered cubic lattice in a B (A) matrix, cylindersof A (B) arranged on a hexagonal lattice in a B (A) matrix, and bicontinuous gyroid channels .Multiblock copolymers [(AnB .)P or (A„Bm)pAn], on the other hand, have received considerablyless attention, although many commercial applications requiring mechanical strength rely on thesesystems [5] . It has been established that the mesoscopic phases of the multiblock systems areindistinguishable from those formed by diblocks . The onset of microsegregation, i .e . the disorder-order transition, however, occurs at increasingly higher x N as the diblock unit is repeated (larger p)[6, 7, 8) . The period of the segregated structures has similarly been found to decrease with p .The most comprehensive theoretical study of linear multiblock copolymers has been publishedby Matsen and Schick [9] in the many-block (p » 1) limit using the self-consistent mean-fieldtheory (SCFT) initially developed by Helfand [10] . Matsen and Schick theoretically established themorphological phase diagram and found that the periodicity D _ ,r116N213 and the interfacial widthw _ x-112 scale with the same exponents as for diblock systems . The principal feature that makesthe multiblock copolymers more attractive for commercial applications is the double tetheredcentral portion of the molecule . In contrast to diblock copolymers, this tethering allows blocks inthe central portions of a multiblock copolymer to bridge their respective domains . Bridging blocksare thought to provide multiblock copolymers with superior mechanical properties, relative to theirdiblock analogs, as they provide a physically cross-linked network . Several theoretical studies havebeen directed towards predicting the bridging fraction in multiblock copolymer systems . Thesestudies have focused on the limiting cases of the either triblock (ABA) or the many-block (p >>1)systems. In contrast we will study theoretically the intermediate systems (1 < p<10) .Although there have been quite extensive attempts to calculate morphologies and bridgingfractions in triblock and many-block systems, there is still no real understanding of howmorphology and bridging characteristics relate to mechanical properties of the systems . However,recently Spontak and Smith [11] published experimental results from a careful study ofpoly(styrene-b-isoprene)p (S I)p (1 < p< 4) multiblock copolymers . Two series were considered :(1) a constant chain mass (CCM) series, and (2) a constant block mass (CBM) series . Here, weconcentrate on the CCM series where the molecular weight was kept at 120kg/mol resulting in theblock weight-the weight of a SI segment-varying as Mb=120/(2p) kg/mol . Since the system isalmost symmetric, a lamellar structure emerges as a result of the microphase segregation . Theperiod of the lamellar structures was found to be in the 5 nm-18 nm regime, with the followingdependence on p: D - p~0'8 . Further, the authors performed rheological measurements on the phasesegregated system and extracted the dependence of the tensile modulus, E, and the tensile yieldstress ay on the block repeat number, p . Unfortunately, no simple (power-law, exponential)functional relationship could be established in either of these cases, but both quantities increasedramatically as a function of p and this increase is intuitively ascribed to the increased bridgingability of the copolymer. This experimental study provides a unique opportunity for us to begin toelucidate the quite poorly understood morphology-property relationship and also to qualitativelyunderstand how bridging of the central parts of multiblock copolymers affects elastic properties .RESULTS AND DISCUSSIONThe self-consistent field theory applied to multiblock copolymers and the technique used tocalculate the bridging fractions, i .e. the number of molecules bridging a given domain has beendetailed elsewhere [ 12] and will not be repeated . Here, it will suffice to discuss the main results .As the experiments use equal overall mass fractions of the two polymer species, we candetermine the volume fractions required by the theory using the homopolymer densities 1 .00 and0.84 g/cm3 (from Ref. [13]) for polystyrene and polyisoprene, respectively to get fS=0 .54/p andfI=0.46/p. These almost equal amounts of styrene and isoprene lead to a lamellar morphologythroughout the study. Reliable information on the Flory-Huggins parameter x is not available, sowe elect to examine at the range xN=105, 100, 95, 90 .First, we have determined the period, D, of the equilibrium lamellar structures . We find thatthis quantity is virtually independent of xN . We therefore show in Fig. 1 only the results (circles)for XNt =100 . In Fig. 1, the solid line represents the functional relationship D/Rg=5 .9 p 0'81 whichhas been obtained as the least squares fit to the six data points . It is important to note that theradius of gyration is constant for all data points since the polymerization Nt is kept constant . Thisrelation is exactly what was found experimentally, however here it is based on six data points ratherthan four as in Ref. [11] . The diblock (p=1) value D=5 .8Rg is consistent with the value previouslyobtained by other authors [14] . In Fig. 1 we also show (diamond) the period for ABA triblocksystem which is intermediate to the diblock and the tetrablock . This data point is seen to fall belowthe curve for multiblock systems. Based on the results of Ref . [7] this is expected as that workpredicts a strong dependence on the type of terminal block. This effect is expected to diminish asthe number of blocks increases [15] . We finally note that taking Rg= 8 .5 nm makes the agreementbetween theoryD/R9Figure 1 . Lamellar period versus p for xN=100 (circles) . Solid line represents the fit toD/ Rg = 5.9p"' . The diamond shows the period of an N segment ABA triblock (taken from Ref.[15]) .and experiment qualitative as far as the lamellar period is concerned, and this value of the radius ofgyration is consistent with independent experiments [13, 16] .A morphological feature of some importance for the mechanical properties that theexperiments have not been able to extract is the interfacial width of the lamellar structures .However, in the context of the SCFT this particular feature is easy to extract . In the equilibriumconfiguration, the interface between an S-rich region and an I-rich region (see Fig . 2, left panel) isfitted with a functional form [ 14] from which the interfacial width w is found .w/R~, .. . .9PFigure 2 . Left : Profile of S-monomers in the lamellar phase for XN =100 and p=4, the period of thelamellar structure D and the interfacial width w are indicated . Thick solid line shows the leastsquares fit to the functional form used to extract the interfacial width (for details see Ref . [12]) .Right: Interfacial width w versus p for several xN's : XN =105 (diamonds), XN =100 (circles), XN=95 (squares), and XN =90 (stars) . For XN =90 the p=6 point is missing as the segregation is veryweak at this point .In contrast to the period D, the interfacial width has noticeable dependence on XN . Thereforewe plot, in the right panel of Fig.2, w versus p for all four different values of XN The interfacialwidth is seen to depend rather strongly on p, particularly as p becomes larger than p=4 . This ismainly due to the fact that the system approaches the weak segregation limit . In this system, thedisorder-order transition occurs [8] at XN z 15 .CO~c~XfReNC~E0C02 .Figure 3. (Color) Illus tration .of the junction distribution leading to the bridging fractiondetermination . Thick solid line shows the S monomer density profile (right axis) in a system thathas segregated into four lamellae . Thin solid line shows the distribution of the first S-I junctionpoint related to the central lamella and blue line shows the distribution of the second junction pointunder the cons traint that the first junction was within the central lamella . Similarly, red and greenlines show the dis tribution of the third and fourth junctions, respectively .Having determined that SCFT can reproduce the experimentally observed morphologicalfeatures we now investigate the looping and bridging behavior of the chains in the system . First, weillustrate in Fig. 3 how the junctions between polymer species S and I are distributed within a givenlamellar (the density of S monomers is shown with a thick solid line referring to the right axis) . Notsurprisingly the junctions tend to reside in the interfacial regions as this clearly helps minimize thefree energy of the system . Figure 3 also illustrates how the bridging and looping occurs : the bluecurve shows the spatial distribution of the S-I junction in the polymer chain under the constraintthat the first S-I junction belongs to the central lamella . Clearly, in this particular case a little lessthan 50% of the chains bridge to reach the adjacent lamellae . As it has been argued before [14],based on the symmetry and the assumed diffusive character of the chains, the maximum bridgingfraction for a lamellar configuration will be 50% at least in the context of SCFT . Further, the redline shows the distribution of the junctions connecting the first I-block with the second S-block .From this distribution we can extract the fraction of polymer chains that bridge an S and an Idomain. Similarly, the green curve shows the distribution of the junction between the second I-block and the second S-block from which we can calculate the fraction of polymer chains bridgingthree consecutive lamellar domains .In Table I we list the bridging fractions ( XN = 100) resulting from a complete study of thebehavior of SI multiblock (p=l, 2, . . ., 6) copolymers . The empty slots in the table indicate that theparticular bridging fraction is not defined for that multiblock copolymer structure .First, we see that the single domain bridging fraction, vl is relatively insensitive to the totalextent of the molecule. Although there is a sharp rise form 0 .30 to 0 .44 in this bridging fractionwhen going from tetrablock copolymers to hexablock copolymers, there is a slight decrease for thehigher order systems. This decrease can be ascribed to the fact that the system becomes lessstrongly segregated as p is increased [15] . Further, we see that this bridging fraction has noapparent correlation to the measured mechanical properties (the tensile modulus and yield stress arelisted in Table I for comparison) . There is only a moderate enhancement of the mechanicalproperties for the hexablock in comparison to tetrablock and diblock while there is an increase inthe single domain bridging fraction vl. On the other hand the mechanical properties aresignificantly enhanced for the octablock in contrast to the slight decrease in the single domainbridging fraction .Table I. All possible bridging fractions (subscripts denote the number of consecutive domains thatis bridged by a single molecule) and measured tensile modulus E and yield stress 6y taken from Ref .[11] .P 1 2 3 4 5 6v 1 0.30 0.44 0.43 0.42 0.4022 0.15 0.22 0.24 0.23 0.2 11)3 0.16 0.19 0.20 0.1 824 0.14 , 0.15 0.17 0.1 5vs 0.15 0.16 0.1 326 0.08 0.14 0.1 2v 7 0.09 0.1 1v g 0 .05 0 . 0929 0.07210 0.05E [MPa] 100 100 125Fay [MPa] 2.3 3.0 3. 6From Table I it is clear that for p>2 the (SI)P molecule is able to bridge multiple domains as a largeincrease in v2 and v3 is seen. This ability results in a material that is physically cross-linked overrelatively large distances . The increase in the multi-domain bridging correlates well with theenhancement of mechanical properties for p>2 . It therefore appears that the bridging fractionsdescribing longer range bridging behavior are more appropriate descriptors of the mechanicalproperties than the single domain bridging fractions . The ability of the molecules to bridge multipledomains creates a strong physical cross linking of the material, in contrast to materials with a lownumber p=1 or 2 of blocks where the material strength to a large degree relies on entropicinteractions between molecules. However, it is worth noting that although the single domainbridging fraction is not a strong function of p the actual number, nb, of single domain bridges perunit area of interface formed is : nb =2v1(p-1) . That is, the actual number of bridges formed isapproximately a linearly increasing function of p . Nevertheless, this is still not sufficient to accountfor the enhancement of the mechanical properties . While these predictions of the self-consistentfield theory provide useful insight into the key features determining mechanical properties ofmultiblock copolymer systems, it should be stressed that these are based on equilibriumconsiderations alone. Non-equilibrium considerations would, however, be expected to play a role inthe development of nanostructures and desirable properties of copolymer materials . In our case nonequilibrium effects will most strongly affect the bridging fractions near the middle of the chain .CONCLUSIO NWe have theoretically investigated the morphological and property characteristics ofmultiblock copolymer blends in the few (p < 6) blocks limit. Using self-consistent field theory wehave found that the equilibrium period of the lamellar morphology scales as p-0'g , in accordanc ewith experimental findings on (SI)P i .e . styrene-isoprene, systems with p< 4 . Moreover, for the firs ttime we have investigated the single and multidomain bridging behavior of multiblock copolymer sand found that significant amounts (>25%) of the polymer molecules bridge multiple domains i nthe phase segregated morphology . Finally, we have argued that the multidomain bridgin gphenomenon plays a dominant role in determining the mechanical properties of multibloc kcopolymer materials .ACKNOWLEDGEMENTSWe gratefully acknowledge discussions with Drs . Russell Thompson and Thomas Sewell . Thisresearch was supported at Los Alamos National Laboratory by the U .S. Department of Energyunder contract W-7405-ENG-36 .REFRENCES[1] I.W. Hamley, The Physics of Block Copolymers (Oxford University Press, New York,1998).[2] A.K. Khandpur, S . Forster, F.S. Bates, I.W. Hamley, A .J. Ryan, W. Bras, K . Almdal, and K .Mortensen, Macromolecules 28, 8796 (1995) .[3] E.L. Thomas and R .L. Lescanec, Philos . Trans . R. Soc. London Ser . A 348, 149 (1994).[4] M.W. Matsen and F .S. Bates, J . Chem. Phys. 106, 2436 (1997) .[5] G . Holden, N .R. Legge, R. Quirk, and H .E. Schroeder, Thermoplastic Elastomers (HanserPublishers, New York, 1996) .[6] H . Benoit and G. Hadziioannou, Macromolecules 21, 1449 (1988) .[7] T.A. Kavassalis and M .D . Whitmore, Macromolecules 24, 5340 (1991) .[8] K.O. Rasmussen, T.D. Sewell, T. Lookman, and A. Saxena in Filled and NanocopositePolymer Materials, edited by A.I. Nakatani, R.P. Hjelm, M. Gerspacher, and R .Krishnamoorti, (Mater . Res. Soc . Proc. 661, Warrendale, PA, 2001) pp . KK4.6.1-KK4.6.6 .[9] M.W. Matsen and M. Schick, Macromolecules 27, 7157 (1994) .[10] E. Helfand J. Chem. Phys. 62, 999 (1975) ; E. Helfand Macromolecules 8, 552 (1975) .[11] R.J. Spontak, and S .D. Smith, J . Poly. Sci . B: Polym. Phys . 39, 947 (2001) .[12] K.O. Rasmussen, E .M. Kober, T. Lookman, and A . Saxena J . Poly. Sci . B : Poly. Phys. 41,xxxx (2003) .[13] D. A . Hajduk, P.E. Harper, S .M. Gruner, C .C. Honeker, G. Kim, E.L. Thomas, and L.J .Fetters, Macromolecules 27, 4063 (1994) .[14] M.W. Matsen and M. Schick, Macromolecules 27, 187 (1994) .[15] M.W. Matsen and R .B . Thompson, J . Chem. Phys . 111, 7139 (1999) .[16] The molecular weight of the diblock used in Ref. 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Bridging properties of multiblock copolymers.

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Bridging properties of multiblock copolymers.

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