Conditions for the crossover from the strong to the superstrong segregation regime are analyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energy between the components required to induce the crossover to the superstrong segregation regime is inversely proportional to mb = 1 + n/m, where n is the degree of polymerization of the side chain and m is the distance between successive grafting points. As a result, the superstrong segregation regime, being rather rare in the case of ordinary block copolymers, has a much better chance to be realized in the case of diblock copolymers with combs grafted to one of the blocks.

Bates F.

Bates F. S.

Birshtein T. M.

de Gennes P.-G.

Dobrynin A. V.

Flory P. J.

Fredrickson G. H.

Goldacker T.

Hamley I. W.

Khalatur P. G.

Leibler L.

Matsen M. W.

Milner S. T.

Muthukumar M.

Nyrkova L. A.

Ruokolainen J.

Semenov A. N.

English

Vasilevskaya, V.V.

Gusev, L.A.

Khokhlov, A.R.

Ikkala, O.

ten Brinke, G.

Ikkala, AO

ARTS repository - University of Groningen

   University of GroningenDomains in melts of comb-coil diblock copolymers: Superstrong segregation regimeVasilevskaya, V.V.; Gusev, L.A.; Khokhlov, A.R.; Ikkala, O.; ten Brinke, G.; Ikkala, AOPublished in:MacromoleculesDOI:10.1021/ma001621fIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2001Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vasilevskaya, V. V., Gusev, L. A., Khokhlov, A. R., Ikkala, O., ten Brinke, G., & Ikkala, AO. (2001).Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: SuperstrongSegregation Regime. Macromolecules, 34(14), 5019 - 5022. https://doi.org/10.1021/ma001621fCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-09-2023Domains in Melts of Comb-Coil Diblock Copolymers: SuperstrongSegregation RegimeV. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ andG. ten Brinke*,⊥Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,9747 AG Groningen, The NetherlandsReceived September 19, 2000; Revised Manuscript Received April 4, 2001ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime areanalyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energybetween the components required to induce the crossover to the superstrong segregation regime is inverselyproportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is thedistance between successive grafting points. As a result, the superstrong segregation regime, being ratherrare in the case of ordinary block copolymers, has a much better chance to be realized in the case ofdiblock copolymers with combs grafted to one of the blocks.1. IntroductionMicrophase separation in copolymers of immisciblepolymer components A and B has been an area ofintensive research over the past decades.1-6 Theorieshave been developed to deal with the different regimesthat can be distinguished, e.g., weak, strong, andintermediate segregation regime. Apart from these morecommon regimes, it was found that in the case ofextremely strong immiscibility between the componentsof a diblock copolymer with one block (e.g., A chains)being rather small, the behavior of the system changesqualitatively from that of the usual strong segregationregime: a superstrong segregation regime emerges.7 Inthis regime the A chains become practically completelyextended within the corresponding microdomain. Cal-culations and experimental studies show that thesuperstrong segregation regime is not characteristic forordinary block copolymers; however, it can be used forthe description of aggregates and multiplets in associat-ing polymers.8These results apply for linear block copolymers. Butrecently, interest has shifted to the nature of themicrophase-separated structures of block copolymers ofa more complex linear9,10 and nonlinear architec-ture.6,11,12 For this paper comb-coil diblock copolymersconsisting of a nonlinear comb copolymer block and alinear block will be of direct interest. A peculiar subsetof these kind of systems, where the side chains are notcovalently bonded but attached by strong physicalinteractions (hydrogen bonding and ionic bonding), havebeen studied in detail recently.13-15 In this paper weconsider the microphase formation in a melt of comb-coil diblock copolymers and study the conditions for thecrossover from the strong to superstrong segregationregime.2. Results and DiscussionWe will consider the microdomain structure formedby comb-coil diblock copolymers consisting of a comb-like A block and a linear B block (see Figure 1). Themain characteristic parameters of the A block are thedegree of polymerization of the backbone NA, the degreeof polymerization of the side chains n, and the numberof monomer units between the grafting points m. TheB block is characterized by a single parameter NB whichis the degree of polymerization of this block.In correspondence with the main objective of thispaper, let us assume that the monomer units of the Ablocks are incompatible with the monomer units of theB blocks and that the immiscibility between the A andB blocks is strong enough to make a narrow interfacebetween the A and B phases; i.e., the structure formedcorresponds to the strong segregation limit.Spherical Micelles. Let us first assume that NA ,NB and that the A blocks (as minority component) formspherical micelles with radius R and aggregation num-ber Q.In ref 2 it was shown that in the strong segregationregime of ordinary block copolymers with immisciblelinear A and B blocks both the A chains within thespherical micelles and the B chain parts in the regionclose to the micellar core are significantly stretched.This is the case for the system under consideration aswell.To estimate the aggregation number Q and the radiusof micelle R, let us consider the free energy F of thesystem. This free energy can be presented as a sum ofthree terms:where Fsurf is the free energy of the micellar interfaceand the contributions Fconf-A and Fconf-B are due to thelosses in conformational entropy of A and B chains.† Russian Academy of Sciences.‡ Moscow State University.§ Helsinki University of Technology.⊥ University of Groningen.F ) Fsurf + Fconf-A + Fconf-B (1)5019Macromolecules 2001, 34, 5019-502210.1021/ma001621f CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 06/06/2001The interfacial free energy Fsurf can be written aswhere N is the total number of chains in the systemand γ is the interface tension coefficient determiningthe degree of immiscibility of A and B blocks. Thecontribution Fconf-B describes the loss in conformationentropy of the linear B blocks. These losses are causedby the expansion of B chains in the interfacial regiondue to the high density of B chains in the neighborhoodof the surface of A micelles. As shown in ref 7, thecontribution Fconf-B can be written asHere lB, aB, and vB are the Kuhn segment length, thelength, and the volume of the monomer units of the Bchains, respectively.Finally, Fconf-A is the conformational free energy ofthe A blocks. Generally speaking, the term Fconf-Aincludes two contributions: a contribution from theconformational entropy of the backbone chain and onefrom the conformational entropy of the side chains,assuming, as we will, that the backbone monomer unitsand the monomer units of the side chains have noenergetic interaction with each other. Because of thewell-known Flory theorem,16,17 the excluded-volumeinteractions are screened within the A micelle. As aconsequence, the side chains have a conformation closeto that of an ideal chain, and the following analysisshows that their conformations remain close to idealeven in the case of dense grafting of side chains (up tom ∼ 1). To see this, we assume that the side chains forma dense tube with radius D around the backbone.Equating the volume per single side chain, which is atleast D2m1/2a, with the total volume of the monomerunits of a side chain, na3, gives D ∼ n1/2a/m1/4. Thislatter represents the minimum value of D required toaccommodate the side chains. Since this value is smallerthan n1/2a up to m ∼ 1, the conformation of the sidechains will remain close to ideal. It implies that for mg1 the contribution to the free energy due to elasticdeformation of the side chains is negligible in compari-son to the contribution of backbone of A chains.Thus, the free energy Fconf-A includes only the elasticfree energy contribution of the A backbone chains. Inthe simplest way this term can be written as the freeenergy of expansion of a backbone chain with NA unitsmultiplied by the total number N of polymer chains inthe system:where lA and aA are the Kuhn segment length andmonomer unit length of A chain, respectively.Since in the case under consideration the excluded-volume interactions are screened, the persistence lengthof the backbone is not expected to increase with increas-ing density of side chain grafting up to m ∼ 1 (note thatan increase in stiffness of the backbone of comb copoly-mers immersed in a solvent is caused by excluded-volume interactions between side chains).3,18,19By taking into account the fact that different A chainsare expanded differently within the spherical micelle,one can rewrite the contribution (4) as (see ref 7)Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that maybe formed by such macromolecules (B).Fsurf ) 4πγR2NQ(2)Fconf-B )38πkTR2vBlBaBNQ (3)Fconf-A ∼ kTN R2NAaAlA(4)Fconf-A )3π280kTN R2NAaAlA(5)5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001The value R of the radius of the A micelle is directlyconnected to Q by the space-filling condition. For thecase where the volume of the monomer units of thebackbone vA is equal to the volume of the monomer unitsof the side chains, the condition of space filling takesthe formwhere mb is the number of monomer units of the combblock per monomer unit of the comb backbone chain:Using eqs 2-6 together with the notations si ) vi/aiand Li ) Niai (i ) A, B) (si is the cross section; Li is thetotal contour length of corresponding blocks), the totalfree energy F can be rewritten aswhere θ ) 1 + (40/3π2)(sAlB/sBlA)(1/mb). For the case sA∼ sB, lA ∼ lB, and mb . 1, the value of parameter θ isclose to unity.Minimization of the free energy F, eq 8, with respectto Q gives the equilibrium value for the aggregationnumber Q:From eqs 6 and 9 we find for the equilibrium value ofthe micelle size, RFrom eq 9 it is clear that for large values of mb theaggregation number Q is almost independent of thestructural parameters of the coil block B. In this casethe parameter θ, which in general depends on lB andsB, is with high accuracy close to unity. Such a resultseems quite natural. As with an increase of mb theradius of the A micelle R increases (see eq 10), thedensity of B blocks in the neighborhood of the A micelleand thus the degree of expansion of these blocks drop.As a result, at high enough values of mb, the contribu-tion Fconf-B becomes negligible in comparison with thefree energy Fconf-A of the A blocks.With an increase in the degree of immiscibility γ, boththe aggregation number Q and the radius R of themicelle increase up to the crossover to the superstrongsegregation regime where the size of the micelle Rbecomes comparable with the length LA of a fullystretched backbone of the A block:7A simple calculation gives the following estimate for thecritical value (γ/kT)SS at the crossover point to thesuperstrong segregation regime:At γ/kT > (γ/kT)SS (within the superstrong segrega-tion regime) the size of the micelle RSS is a linearfunction of the contour length LA of the A blocks. Also,the aggregation number QSS no longer increases with adecrease of temperature:Equations 12 and 13 demonstrate that both the criticalvalue (γ/kT)SS and the aggregation number QSS in thesuperstrong segregation regime decrease as a functionof mb ) 1 + n/m.The characteristic sizes of the structure R and of thecritical value (γ/kT)SS for crossover to the superstrongsegregation regime in the case of lamellar and cylindri-cal structures have the same scaling dependencies onLA and mb as in eqs 11-13. To show this, we nowconsider the formation of lamellar structure.Lamellar Structure. In the case of a lamellarstructure the free energy F is given (by ignoring allirrelevant numerical factors) asHere σ is surface area of interface between A and Bblocks per chain.The self-consistent space-filling conditions areMinimization of free energy (14)-(16) with respect toσ give us the equilibrium value of characteristic size oflamellar structure RA and RB:Let us consider case NAmbvA = NBvB, where thevolume fractions of monomer units in the linear andcomb blocks are equal to each other. In this case, thethicknesses of the layers of the comb and the linearblocks are equal to each other as well: RA ) RB ) R.The period of the structure R is now given bywhere θl ) 1 + lAsB/mblBsA. For the case of mB . 1, theQNAmbvA4π3R3) 1 (6)mb ) 1 +nm(7)F ) ( 36πsALA)1/3mb2/3[ γkT LAsAQ1/3 + 3π320 Q2/3 sAlA θ] (8)Q ) γkT1603πLAlAθ(9)R ) ( γkT 40π2 LA2 lAsAθmb)1/3 (10)R ∼ LA (11)( γkT)SS ∼ π240LAθmblAsA(12)RSS ) LAQSS )4π3LA2mbsA(13)FkT)RA2NAlAaA+RB2NBlBaB+ γσ (14)RAσ ) NAmbvA (15)RBσ ) NBvB (16)RA ∼ ( γkT)1/3 NAmbvA(NAmb2vA2lAaA + NBvB2lBaB )1/3RB ∼ ( γkT)1/3 NBvB(NAmb2vA2lAaA + NBvB2lBaB )1/3(17)R ∼ ( γkT LA2 lAsAθl mb)1/3(18)Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021value of θl is close to unity and the period of thestructure R is determined by the structural parametersof the comb block only. Analysis shows that in thisparticular case, as in a case of spherical micelles, thefree energy F of elastic deformation of the linear B blockis negligible in comparison with elastic free energy ofthe A block due to the much higher stretching of thecomb A block.The crossover from the strong segregation regime (R∼ LA2/3) to the superstrong segregation regime (RSS ∼LA) proceeds atIn the superstrong segregation regime the character-istic period R is determined by the length LA of thebackbone of the comb block, which is much shorter (inthe case of high values of mb, see eqs 16-18) than thelength LB of coil block. In this regime period R does notincrease with further increase of parameter of im-miscibility γ.The crossover to the superstrong segregation regimeproceeds when the comb block is fully stretched, and inthe general case the crossover point depends on the ratiof ) NAmbvA/NBvB of the volume of monomer units in thecomb and the linear blocks:At f >1 the conditions for crossover coincides withcondition (19), whereas with decrease of parameter fbelow unity the critical value (γ/kT)SS increases.Note that the eq 20 gives for a wide range ofparameter values the same scaling dependence of criti-cal value (γ/kT)SS of the crossover to the supercriticalregime as eq 12. An increase in the degree of polymer-ization of the side chain n or a decrease in the distancem between the grafting points leads to softening of theconditions for the crossover from the strong to thesuperstrong segregation regime. In general, the periodof the structure significantly increases with an increasein the degree of polymerization of the side chains ortheir grafting density.In conclusion, we see that the possibility to reach thesuperstrong segregation regime is strongly enhanced fordiblock copolymers involving comb blocks.Acknowledgment. This work was supported by theNWO program for Dutch-Russian scientific cooperationand by the Program “University of Russia-FundamentalResearch” and by the Russian Foundation for BasicResearch (Grant 01-03-32672).References and Notes(1) Leibler, L. Macromolecules 1980, 13, 1602.(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,41, 525.(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,277, 1225.(6) Hamley, I. W. The Physics of Block Copolymers; OxfordUniversity Press: Oxford, 1998.(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules1993, 26, 3601.(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,A. N. Macromol. Theory Simul. 1996, 5, 713, 749.(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;Leibler, L. Nature 1999, 398, 137.(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,26, 276.(12) Milner, S. T. Macromolecules 1994, 27, 2333.(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;Thomas, E. L. Macromolecules 1998, 31, 3532.(14) Ruokolainen, J.; Mäkinen, R.; Torkkeli, M.; Mäkelä, T.;Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,557.(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,11, 777.(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-sity Press: Ithaca, NY, 1953.(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.MA001621F( γkT)SS ∼LAθlmblAsA(19)( γkT)SS ∼ LAmblAsA(1 + 1f lAsBmblBsA) (20)5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001

Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime:Superstrong Segregation Regime

University of Groningen

   University of GroningenDomains in melts of comb-coil diblock copolymers: Superstrong segregation regimeVasilevskaya, V.V.; Gusev, L.A.; Khokhlov, A.R.; Ikkala, O.; ten Brinke, G.; Ikkala, AOPublished in:MacromoleculesDOI:10.1021/ma001621fIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2001Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vasilevskaya, V. V., Gusev, L. A., Khokhlov, A. R., Ikkala, O., ten Brinke, G., & Ikkala, AO. (2001).Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: SuperstrongSegregation Regime. Macromolecules, 34(14), 5019 - 5022. https://doi.org/10.1021/ma001621fCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-04-2023Domains in Melts of Comb-Coil Diblock Copolymers: SuperstrongSegregation RegimeV. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ andG. ten Brinke*,⊥Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,9747 AG Groningen, The NetherlandsReceived September 19, 2000; Revised Manuscript Received April 4, 2001ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime areanalyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energybetween the components required to induce the crossover to the superstrong segregation regime is inverselyproportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is thedistance between successive grafting points. As a result, the superstrong segregation regime, being ratherrare in the case of ordinary block copolymers, has a much better chance to be realized in the case ofdiblock copolymers with combs grafted to one of the blocks.1. IntroductionMicrophase separation in copolymers of immisciblepolymer components A and B has been an area ofintensive research over the past decades.1-6 Theorieshave been developed to deal with the different regimesthat can be distinguished, e.g., weak, strong, andintermediate segregation regime. Apart from these morecommon regimes, it was found that in the case ofextremely strong immiscibility between the componentsof a diblock copolymer with one block (e.g., A chains)being rather small, the behavior of the system changesqualitatively from that of the usual strong segregationregime: a superstrong segregation regime emerges.7 Inthis regime the A chains become practically completelyextended within the corresponding microdomain. Cal-culations and experimental studies show that thesuperstrong segregation regime is not characteristic forordinary block copolymers; however, it can be used forthe description of aggregates and multiplets in associat-ing polymers.8These results apply for linear block copolymers. Butrecently, interest has shifted to the nature of themicrophase-separated structures of block copolymers ofa more complex linear9,10 and nonlinear architec-ture.6,11,12 For this paper comb-coil diblock copolymersconsisting of a nonlinear comb copolymer block and alinear block will be of direct interest. A peculiar subsetof these kind of systems, where the side chains are notcovalently bonded but attached by strong physicalinteractions (hydrogen bonding and ionic bonding), havebeen studied in detail recently.13-15 In this paper weconsider the microphase formation in a melt of comb-coil diblock copolymers and study the conditions for thecrossover from the strong to superstrong segregationregime.2. Results and DiscussionWe will consider the microdomain structure formedby comb-coil diblock copolymers consisting of a comb-like A block and a linear B block (see Figure 1). Themain characteristic parameters of the A block are thedegree of polymerization of the backbone NA, the degreeof polymerization of the side chains n, and the numberof monomer units between the grafting points m. TheB block is characterized by a single parameter NB whichis the degree of polymerization of this block.In correspondence with the main objective of thispaper, let us assume that the monomer units of the Ablocks are incompatible with the monomer units of theB blocks and that the immiscibility between the A andB blocks is strong enough to make a narrow interfacebetween the A and B phases; i.e., the structure formedcorresponds to the strong segregation limit.Spherical Micelles. Let us first assume that NA ,NB and that the A blocks (as minority component) formspherical micelles with radius R and aggregation num-ber Q.In ref 2 it was shown that in the strong segregationregime of ordinary block copolymers with immisciblelinear A and B blocks both the A chains within thespherical micelles and the B chain parts in the regionclose to the micellar core are significantly stretched.This is the case for the system under consideration aswell.To estimate the aggregation number Q and the radiusof micelle R, let us consider the free energy F of thesystem. This free energy can be presented as a sum ofthree terms:where Fsurf is the free energy of the micellar interfaceand the contributions Fconf-A and Fconf-B are due to thelosses in conformational entropy of A and B chains.† Russian Academy of Sciences.‡ Moscow State University.§ Helsinki University of Technology.⊥ University of Groningen.F ) Fsurf + Fconf-A + Fconf-B (1)5019Macromolecules 2001, 34, 5019-502210.1021/ma001621f CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 06/06/2001The interfacial free energy Fsurf can be written aswhere N is the total number of chains in the systemand γ is the interface tension coefficient determiningthe degree of immiscibility of A and B blocks. Thecontribution Fconf-B describes the loss in conformationentropy of the linear B blocks. These losses are causedby the expansion of B chains in the interfacial regiondue to the high density of B chains in the neighborhoodof the surface of A micelles. As shown in ref 7, thecontribution Fconf-B can be written asHere lB, aB, and vB are the Kuhn segment length, thelength, and the volume of the monomer units of the Bchains, respectively.Finally, Fconf-A is the conformational free energy ofthe A blocks. Generally speaking, the term Fconf-Aincludes two contributions: a contribution from theconformational entropy of the backbone chain and onefrom the conformational entropy of the side chains,assuming, as we will, that the backbone monomer unitsand the monomer units of the side chains have noenergetic interaction with each other. Because of thewell-known Flory theorem,16,17 the excluded-volumeinteractions are screened within the A micelle. As aconsequence, the side chains have a conformation closeto that of an ideal chain, and the following analysisshows that their conformations remain close to idealeven in the case of dense grafting of side chains (up tom ∼ 1). To see this, we assume that the side chains forma dense tube with radius D around the backbone.Equating the volume per single side chain, which is atleast D2m1/2a, with the total volume of the monomerunits of a side chain, na3, gives D ∼ n1/2a/m1/4. Thislatter represents the minimum value of D required toaccommodate the side chains. Since this value is smallerthan n1/2a up to m ∼ 1, the conformation of the sidechains will remain close to ideal. It implies that for mg1 the contribution to the free energy due to elasticdeformation of the side chains is negligible in compari-son to the contribution of backbone of A chains.Thus, the free energy Fconf-A includes only the elasticfree energy contribution of the A backbone chains. Inthe simplest way this term can be written as the freeenergy of expansion of a backbone chain with NA unitsmultiplied by the total number N of polymer chains inthe system:where lA and aA are the Kuhn segment length andmonomer unit length of A chain, respectively.Since in the case under consideration the excluded-volume interactions are screened, the persistence lengthof the backbone is not expected to increase with increas-ing density of side chain grafting up to m ∼ 1 (note thatan increase in stiffness of the backbone of comb copoly-mers immersed in a solvent is caused by excluded-volume interactions between side chains).3,18,19By taking into account the fact that different A chainsare expanded differently within the spherical micelle,one can rewrite the contribution (4) as (see ref 7)Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that maybe formed by such macromolecules (B).Fsurf ) 4πγR2NQ(2)Fconf-B )38πkTR2vBlBaBNQ (3)Fconf-A ∼ kTN R2NAaAlA(4)Fconf-A )3π280kTN R2NAaAlA(5)5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001The value R of the radius of the A micelle is directlyconnected to Q by the space-filling condition. For thecase where the volume of the monomer units of thebackbone vA is equal to the volume of the monomer unitsof the side chains, the condition of space filling takesthe formwhere mb is the number of monomer units of the combblock per monomer unit of the comb backbone chain:Using eqs 2-6 together with the notations si ) vi/aiand Li ) Niai (i ) A, B) (si is the cross section; Li is thetotal contour length of corresponding blocks), the totalfree energy F can be rewritten aswhere θ ) 1 + (40/3π2)(sAlB/sBlA)(1/mb). For the case sA∼ sB, lA ∼ lB, and mb . 1, the value of parameter θ isclose to unity.Minimization of the free energy F, eq 8, with respectto Q gives the equilibrium value for the aggregationnumber Q:From eqs 6 and 9 we find for the equilibrium value ofthe micelle size, RFrom eq 9 it is clear that for large values of mb theaggregation number Q is almost independent of thestructural parameters of the coil block B. In this casethe parameter θ, which in general depends on lB andsB, is with high accuracy close to unity. Such a resultseems quite natural. As with an increase of mb theradius of the A micelle R increases (see eq 10), thedensity of B blocks in the neighborhood of the A micelleand thus the degree of expansion of these blocks drop.As a result, at high enough values of mb, the contribu-tion Fconf-B becomes negligible in comparison with thefree energy Fconf-A of the A blocks.With an increase in the degree of immiscibility γ, boththe aggregation number Q and the radius R of themicelle increase up to the crossover to the superstrongsegregation regime where the size of the micelle Rbecomes comparable with the length LA of a fullystretched backbone of the A block:7A simple calculation gives the following estimate for thecritical value (γ/kT)SS at the crossover point to thesuperstrong segregation regime:At γ/kT > (γ/kT)SS (within the superstrong segrega-tion regime) the size of the micelle RSS is a linearfunction of the contour length LA of the A blocks. Also,the aggregation number QSS no longer increases with adecrease of temperature:Equations 12 and 13 demonstrate that both the criticalvalue (γ/kT)SS and the aggregation number QSS in thesuperstrong segregation regime decrease as a functionof mb ) 1 + n/m.The characteristic sizes of the structure R and of thecritical value (γ/kT)SS for crossover to the superstrongsegregation regime in the case of lamellar and cylindri-cal structures have the same scaling dependencies onLA and mb as in eqs 11-13. To show this, we nowconsider the formation of lamellar structure.Lamellar Structure. In the case of a lamellarstructure the free energy F is given (by ignoring allirrelevant numerical factors) asHere σ is surface area of interface between A and Bblocks per chain.The self-consistent space-filling conditions areMinimization of free energy (14)-(16) with respect toσ give us the equilibrium value of characteristic size oflamellar structure RA and RB:Let us consider case NAmbvA = NBvB, where thevolume fractions of monomer units in the linear andcomb blocks are equal to each other. In this case, thethicknesses of the layers of the comb and the linearblocks are equal to each other as well: RA ) RB ) R.The period of the structure R is now given bywhere θl ) 1 + lAsB/mblBsA. For the case of mB . 1, theQNAmbvA4π3R3) 1 (6)mb ) 1 +nm(7)F ) ( 36πsALA)1/3mb2/3[ γkT LAsAQ1/3 + 3π320 Q2/3 sAlA θ] (8)Q ) γkT1603πLAlAθ(9)R ) ( γkT 40π2 LA2 lAsAθmb)1/3 (10)R ∼ LA (11)( γkT)SS ∼ π240LAθmblAsA(12)RSS ) LAQSS )4π3LA2mbsA(13)FkT)RA2NAlAaA+RB2NBlBaB+ γσ (14)RAσ ) NAmbvA (15)RBσ ) NBvB (16)RA ∼ ( γkT)1/3 NAmbvA(NAmb2vA2lAaA + NBvB2lBaB )1/3RB ∼ ( γkT)1/3 NBvB(NAmb2vA2lAaA + NBvB2lBaB )1/3(17)R ∼ ( γkT LA2 lAsAθl mb)1/3(18)Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021value of θl is close to unity and the period of thestructure R is determined by the structural parametersof the comb block only. Analysis shows that in thisparticular case, as in a case of spherical micelles, thefree energy F of elastic deformation of the linear B blockis negligible in comparison with elastic free energy ofthe A block due to the much higher stretching of thecomb A block.The crossover from the strong segregation regime (R∼ LA2/3) to the superstrong segregation regime (RSS ∼LA) proceeds atIn the superstrong segregation regime the character-istic period R is determined by the length LA of thebackbone of the comb block, which is much shorter (inthe case of high values of mb, see eqs 16-18) than thelength LB of coil block. In this regime period R does notincrease with further increase of parameter of im-miscibility γ.The crossover to the superstrong segregation regimeproceeds when the comb block is fully stretched, and inthe general case the crossover point depends on the ratiof ) NAmbvA/NBvB of the volume of monomer units in thecomb and the linear blocks:At f >1 the conditions for crossover coincides withcondition (19), whereas with decrease of parameter fbelow unity the critical value (γ/kT)SS increases.Note that the eq 20 gives for a wide range ofparameter values the same scaling dependence of criti-cal value (γ/kT)SS of the crossover to the supercriticalregime as eq 12. An increase in the degree of polymer-ization of the side chain n or a decrease in the distancem between the grafting points leads to softening of theconditions for the crossover from the strong to thesuperstrong segregation regime. In general, the periodof the structure significantly increases with an increasein the degree of polymerization of the side chains ortheir grafting density.In conclusion, we see that the possibility to reach thesuperstrong segregation regime is strongly enhanced fordiblock copolymers involving comb blocks.Acknowledgment. This work was supported by theNWO program for Dutch-Russian scientific cooperationand by the Program “University of Russia-FundamentalResearch” and by the Russian Foundation for BasicResearch (Grant 01-03-32672).References and Notes(1) Leibler, L. Macromolecules 1980, 13, 1602.(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,41, 525.(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,277, 1225.(6) Hamley, I. W. The Physics of Block Copolymers; OxfordUniversity Press: Oxford, 1998.(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules1993, 26, 3601.(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,A. N. Macromol. Theory Simul. 1996, 5, 713, 749.(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;Leibler, L. Nature 1999, 398, 137.(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,26, 276.(12) Milner, S. T. Macromolecules 1994, 27, 2333.(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;Thomas, E. L. Macromolecules 1998, 31, 3532.(14) Ruokolainen, J.; Mäkinen, R.; Torkkeli, M.; Mäkelä, T.;Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,557.(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,11, 777.(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-sity Press: Ithaca, NY, 1953.(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.MA001621F( γkT)SS ∼LAθlmblAsA(19)( γkT)SS ∼ LAmblAsA(1 + 1f lAsBmblBsA) (20)5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001

   University of GroningenDomains in melts of comb-coil diblock copolymers: Superstrong segregation regimeVasilevskaya, V.V.; Gusev, L.A.; Khokhlov, A.R.; Ikkala, O.; ten Brinke, G.; Ikkala, AOPublished in:MacromoleculesDOI:10.1021/ma001621fIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2001Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vasilevskaya, V. V., Gusev, L. A., Khokhlov, A. R., Ikkala, O., ten Brinke, G., & Ikkala, AO. (2001).Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: SuperstrongSegregation Regime. Macromolecules, 34(14), 5019 - 5022. https://doi.org/10.1021/ma001621fCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 16-07-2023Domains in Melts of Comb-Coil Diblock Copolymers: SuperstrongSegregation RegimeV. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ andG. ten Brinke*,⊥Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,9747 AG Groningen, The NetherlandsReceived September 19, 2000; Revised Manuscript Received April 4, 2001ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime areanalyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energybetween the components required to induce the crossover to the superstrong segregation regime is inverselyproportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is thedistance between successive grafting points. As a result, the superstrong segregation regime, being ratherrare in the case of ordinary block copolymers, has a much better chance to be realized in the case ofdiblock copolymers with combs grafted to one of the blocks.1. IntroductionMicrophase separation in copolymers of immisciblepolymer components A and B has been an area ofintensive research over the past decades.1-6 Theorieshave been developed to deal with the different regimesthat can be distinguished, e.g., weak, strong, andintermediate segregation regime. Apart from these morecommon regimes, it was found that in the case ofextremely strong immiscibility between the componentsof a diblock copolymer with one block (e.g., A chains)being rather small, the behavior of the system changesqualitatively from that of the usual strong segregationregime: a superstrong segregation regime emerges.7 Inthis regime the A chains become practically completelyextended within the corresponding microdomain. Cal-culations and experimental studies show that thesuperstrong segregation regime is not characteristic forordinary block copolymers; however, it can be used forthe description of aggregates and multiplets in associat-ing polymers.8These results apply for linear block copolymers. Butrecently, interest has shifted to the nature of themicrophase-separated structures of block copolymers ofa more complex linear9,10 and nonlinear architec-ture.6,11,12 For this paper comb-coil diblock copolymersconsisting of a nonlinear comb copolymer block and alinear block will be of direct interest. A peculiar subsetof these kind of systems, where the side chains are notcovalently bonded but attached by strong physicalinteractions (hydrogen bonding and ionic bonding), havebeen studied in detail recently.13-15 In this paper weconsider the microphase formation in a melt of comb-coil diblock copolymers and study the conditions for thecrossover from the strong to superstrong segregationregime.2. Results and DiscussionWe will consider the microdomain structure formedby comb-coil diblock copolymers consisting of a comb-like A block and a linear B block (see Figure 1). Themain characteristic parameters of the A block are thedegree of polymerization of the backbone NA, the degreeof polymerization of the side chains n, and the numberof monomer units between the grafting points m. TheB block is characterized by a single parameter NB whichis the degree of polymerization of this block.In correspondence with the main objective of thispaper, let us assume that the monomer units of the Ablocks are incompatible with the monomer units of theB blocks and that the immiscibility between the A andB blocks is strong enough to make a narrow interfacebetween the A and B phases; i.e., the structure formedcorresponds to the strong segregation limit.Spherical Micelles. Let us first assume that NA ,NB and that the A blocks (as minority component) formspherical micelles with radius R and aggregation num-ber Q.In ref 2 it was shown that in the strong segregationregime of ordinary block copolymers with immisciblelinear A and B blocks both the A chains within thespherical micelles and the B chain parts in the regionclose to the micellar core are significantly stretched.This is the case for the system under consideration aswell.To estimate the aggregation number Q and the radiusof micelle R, let us consider the free energy F of thesystem. This free energy can be presented as a sum ofthree terms:where Fsurf is the free energy of the micellar interfaceand the contributions Fconf-A and Fconf-B are due to thelosses in conformational entropy of A and B chains.† Russian Academy of Sciences.‡ Moscow State University.§ Helsinki University of Technology.⊥ University of Groningen.F ) Fsurf + Fconf-A + Fconf-B (1)5019Macromolecules 2001, 34, 5019-502210.1021/ma001621f CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 06/06/2001The interfacial free energy Fsurf can be written aswhere N is the total number of chains in the systemand γ is the interface tension coefficient determiningthe degree of immiscibility of A and B blocks. Thecontribution Fconf-B describes the loss in conformationentropy of the linear B blocks. These losses are causedby the expansion of B chains in the interfacial regiondue to the high density of B chains in the neighborhoodof the surface of A micelles. As shown in ref 7, thecontribution Fconf-B can be written asHere lB, aB, and vB are the Kuhn segment length, thelength, and the volume of the monomer units of the Bchains, respectively.Finally, Fconf-A is the conformational free energy ofthe A blocks. Generally speaking, the term Fconf-Aincludes two contributions: a contribution from theconformational entropy of the backbone chain and onefrom the conformational entropy of the side chains,assuming, as we will, that the backbone monomer unitsand the monomer units of the side chains have noenergetic interaction with each other. Because of thewell-known Flory theorem,16,17 the excluded-volumeinteractions are screened within the A micelle. As aconsequence, the side chains have a conformation closeto that of an ideal chain, and the following analysisshows that their conformations remain close to idealeven in the case of dense grafting of side chains (up tom ∼ 1). To see this, we assume that the side chains forma dense tube with radius D around the backbone.Equating the volume per single side chain, which is atleast D2m1/2a, with the total volume of the monomerunits of a side chain, na3, gives D ∼ n1/2a/m1/4. Thislatter represents the minimum value of D required toaccommodate the side chains. Since this value is smallerthan n1/2a up to m ∼ 1, the conformation of the sidechains will remain close to ideal. It implies that for mg1 the contribution to the free energy due to elasticdeformation of the side chains is negligible in compari-son to the contribution of backbone of A chains.Thus, the free energy Fconf-A includes only the elasticfree energy contribution of the A backbone chains. Inthe simplest way this term can be written as the freeenergy of expansion of a backbone chain with NA unitsmultiplied by the total number N of polymer chains inthe system:where lA and aA are the Kuhn segment length andmonomer unit length of A chain, respectively.Since in the case under consideration the excluded-volume interactions are screened, the persistence lengthof the backbone is not expected to increase with increas-ing density of side chain grafting up to m ∼ 1 (note thatan increase in stiffness of the backbone of comb copoly-mers immersed in a solvent is caused by excluded-volume interactions between side chains).3,18,19By taking into account the fact that different A chainsare expanded differently within the spherical micelle,one can rewrite the contribution (4) as (see ref 7)Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that maybe formed by such macromolecules (B).Fsurf ) 4πγR2NQ(2)Fconf-B )38πkTR2vBlBaBNQ (3)Fconf-A ∼ kTN R2NAaAlA(4)Fconf-A )3π280kTN R2NAaAlA(5)5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001The value R of the radius of the A micelle is directlyconnected to Q by the space-filling condition. For thecase where the volume of the monomer units of thebackbone vA is equal to the volume of the monomer unitsof the side chains, the condition of space filling takesthe formwhere mb is the number of monomer units of the combblock per monomer unit of the comb backbone chain:Using eqs 2-6 together with the notations si ) vi/aiand Li ) Niai (i ) A, B) (si is the cross section; Li is thetotal contour length of corresponding blocks), the totalfree energy F can be rewritten aswhere θ ) 1 + (40/3π2)(sAlB/sBlA)(1/mb). For the case sA∼ sB, lA ∼ lB, and mb . 1, the value of parameter θ isclose to unity.Minimization of the free energy F, eq 8, with respectto Q gives the equilibrium value for the aggregationnumber Q:From eqs 6 and 9 we find for the equilibrium value ofthe micelle size, RFrom eq 9 it is clear that for large values of mb theaggregation number Q is almost independent of thestructural parameters of the coil block B. In this casethe parameter θ, which in general depends on lB andsB, is with high accuracy close to unity. Such a resultseems quite natural. As with an increase of mb theradius of the A micelle R increases (see eq 10), thedensity of B blocks in the neighborhood of the A micelleand thus the degree of expansion of these blocks drop.As a result, at high enough values of mb, the contribu-tion Fconf-B becomes negligible in comparison with thefree energy Fconf-A of the A blocks.With an increase in the degree of immiscibility γ, boththe aggregation number Q and the radius R of themicelle increase up to the crossover to the superstrongsegregation regime where the size of the micelle Rbecomes comparable with the length LA of a fullystretched backbone of the A block:7A simple calculation gives the following estimate for thecritical value (γ/kT)SS at the crossover point to thesuperstrong segregation regime:At γ/kT > (γ/kT)SS (within the superstrong segrega-tion regime) the size of the micelle RSS is a linearfunction of the contour length LA of the A blocks. Also,the aggregation number QSS no longer increases with adecrease of temperature:Equations 12 and 13 demonstrate that both the criticalvalue (γ/kT)SS and the aggregation number QSS in thesuperstrong segregation regime decrease as a functionof mb ) 1 + n/m.The characteristic sizes of the structure R and of thecritical value (γ/kT)SS for crossover to the superstrongsegregation regime in the case of lamellar and cylindri-cal structures have the same scaling dependencies onLA and mb as in eqs 11-13. To show this, we nowconsider the formation of lamellar structure.Lamellar Structure. In the case of a lamellarstructure the free energy F is given (by ignoring allirrelevant numerical factors) asHere σ is surface area of interface between A and Bblocks per chain.The self-consistent space-filling conditions areMinimization of free energy (14)-(16) with respect toσ give us the equilibrium value of characteristic size oflamellar structure RA and RB:Let us consider case NAmbvA = NBvB, where thevolume fractions of monomer units in the linear andcomb blocks are equal to each other. In this case, thethicknesses of the layers of the comb and the linearblocks are equal to each other as well: RA ) RB ) R.The period of the structure R is now given bywhere θl ) 1 + lAsB/mblBsA. For the case of mB . 1, theQNAmbvA4π3R3) 1 (6)mb ) 1 +nm(7)F ) ( 36πsALA)1/3mb2/3[ γkT LAsAQ1/3 + 3π320 Q2/3 sAlA θ] (8)Q ) γkT1603πLAlAθ(9)R ) ( γkT 40π2 LA2 lAsAθmb)1/3 (10)R ∼ LA (11)( γkT)SS ∼ π240LAθmblAsA(12)RSS ) LAQSS )4π3LA2mbsA(13)FkT)RA2NAlAaA+RB2NBlBaB+ γσ (14)RAσ ) NAmbvA (15)RBσ ) NBvB (16)RA ∼ ( γkT)1/3 NAmbvA(NAmb2vA2lAaA + NBvB2lBaB )1/3RB ∼ ( γkT)1/3 NBvB(NAmb2vA2lAaA + NBvB2lBaB )1/3(17)R ∼ ( γkT LA2 lAsAθl mb)1/3(18)Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021value of θl is close to unity and the period of thestructure R is determined by the structural parametersof the comb block only. Analysis shows that in thisparticular case, as in a case of spherical micelles, thefree energy F of elastic deformation of the linear B blockis negligible in comparison with elastic free energy ofthe A block due to the much higher stretching of thecomb A block.The crossover from the strong segregation regime (R∼ LA2/3) to the superstrong segregation regime (RSS ∼LA) proceeds atIn the superstrong segregation regime the character-istic period R is determined by the length LA of thebackbone of the comb block, which is much shorter (inthe case of high values of mb, see eqs 16-18) than thelength LB of coil block. In this regime period R does notincrease with further increase of parameter of im-miscibility γ.The crossover to the superstrong segregation regimeproceeds when the comb block is fully stretched, and inthe general case the crossover point depends on the ratiof ) NAmbvA/NBvB of the volume of monomer units in thecomb and the linear blocks:At f >1 the conditions for crossover coincides withcondition (19), whereas with decrease of parameter fbelow unity the critical value (γ/kT)SS increases.Note that the eq 20 gives for a wide range ofparameter values the same scaling dependence of criti-cal value (γ/kT)SS of the crossover to the supercriticalregime as eq 12. An increase in the degree of polymer-ization of the side chain n or a decrease in the distancem between the grafting points leads to softening of theconditions for the crossover from the strong to thesuperstrong segregation regime. In general, the periodof the structure significantly increases with an increasein the degree of polymerization of the side chains ortheir grafting density.In conclusion, we see that the possibility to reach thesuperstrong segregation regime is strongly enhanced fordiblock copolymers involving comb blocks.Acknowledgment. This work was supported by theNWO program for Dutch-Russian scientific cooperationand by the Program “University of Russia-FundamentalResearch” and by the Russian Foundation for BasicResearch (Grant 01-03-32672).References and Notes(1) Leibler, L. Macromolecules 1980, 13, 1602.(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,41, 525.(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,277, 1225.(6) Hamley, I. W. The Physics of Block Copolymers; OxfordUniversity Press: Oxford, 1998.(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules1993, 26, 3601.(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,A. N. Macromol. Theory Simul. 1996, 5, 713, 749.(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;Leibler, L. Nature 1999, 398, 137.(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,26, 276.(12) Milner, S. T. Macromolecules 1994, 27, 2333.(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;Thomas, E. L. Macromolecules 1998, 31, 3532.(14) Ruokolainen, J.; Mäkinen, R.; Torkkeli, M.; Mäkelä, T.;Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,557.(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,11, 777.(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-sity Press: Ithaca, NY, 1953.(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.MA001621F( γkT)SS ∼LAθlmblAsA(19)( γkT)SS ∼ LAmblAsA(1 + 1f lAsBmblBsA) (20)5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001

NARCIS 

Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: Superstrong Segregation Regime

Crossref

Domains in Melts of Comb−Coil Diblock Copolymers: Superstrong Segregation Regime

University of Groningen Research Database

  
 University of Groningen
Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime
Vasilevskaya, V.V.; Gusev, L.A.; Khokhlov, A.R.; Ikkala, O.; ten Brinke, G.; Ikkala, AO
Published in:
Macromolecules
DOI:
10.1021/ma001621f
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2001
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Vasilevskaya, V. V., Gusev, L. A., Khokhlov, A. R., Ikkala, O., ten Brinke, G., & Ikkala, A. O. (2001).
Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: Superstrong
Segregation Regime. Macromolecules, 34(14), 5019 - 5022. DOI: 10.1021/ma001621f
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 10-02-2018
Domains in Melts of Comb-Coil Diblock Copolymers: Superstrong
Segregation Regime
V. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ and
G. ten Brinke*,⊥
Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,
Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;
Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015
HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
Received September 19, 2000; Revised Manuscript Received April 4, 2001
ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime are
analyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energy
between the components required to induce the crossover to the superstrong segregation regime is inversely
proportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is the
distance between successive grafting points. As a result, the superstrong segregation regime, being rather
rare in the case of ordinary block copolymers, has a much better chance to be realized in the case of
diblock copolymers with combs grafted to one of the blocks.
1. Introduction
Microphase separation in copolymers of immiscible
polymer components A and B has been an area of
intensive research over the past decades.1-6 Theories
have been developed to deal with the different regimes
that can be distinguished, e.g., weak, strong, and
intermediate segregation regime. Apart from these more
common regimes, it was found that in the case of
extremely strong immiscibility between the components
of a diblock copolymer with one block (e.g., A chains)
being rather small, the behavior of the system changes
qualitatively from that of the usual strong segregation
regime: a superstrong segregation regime emerges.7 In
this regime the A chains become practically completely
extended within the corresponding microdomain. Cal-
culations and experimental studies show that the
superstrong segregation regime is not characteristic for
ordinary block copolymers; however, it can be used for
the description of aggregates and multiplets in associat-
ing polymers.8
These results apply for linear block copolymers. But
recently, interest has shifted to the nature of the
microphase-separated structures of block copolymers of
a more complex linear9,10 and nonlinear architec-
ture.6,11,12 For this paper comb-coil diblock copolymers
consisting of a nonlinear comb copolymer block and a
linear block will be of direct interest. A peculiar subset
of these kind of systems, where the side chains are not
covalently bonded but attached by strong physical
interactions (hydrogen bonding and ionic bonding), have
been studied in detail recently.13-15 In this paper we
consider the microphase formation in a melt of comb-
coil diblock copolymers and study the conditions for the
crossover from the strong to superstrong segregation
regime.
2. Results and Discussion
We will consider the microdomain structure formed
by comb-coil diblock copolymers consisting of a comb-
like A block and a linear B block (see Figure 1). The
main characteristic parameters of the A block are the
degree of polymerization of the backbone NA, the degree
of polymerization of the side chains n, and the number
of monomer units between the grafting points m. The
B block is characterized by a single parameter NB which
is the degree of polymerization of this block.
In correspondence with the main objective of this
paper, let us assume that the monomer units of the A
blocks are incompatible with the monomer units of the
B blocks and that the immiscibility between the A and
B blocks is strong enough to make a narrow interface
between the A and B phases; i.e., the structure formed
corresponds to the strong segregation limit.
Spherical Micelles. Let us first assume that NA ,
NB and that the A blocks (as minority component) form
spherical micelles with radius R and aggregation num-
ber Q.
In ref 2 it was shown that in the strong segregation
regime of ordinary block copolymers with immiscible
linear A and B blocks both the A chains within the
spherical micelles and the B chain parts in the region
close to the micellar core are significantly stretched.
This is the case for the system under consideration as
well.
To estimate the aggregation number Q and the radius
of micelle R, let us consider the free energy F of the
system. This free energy can be presented as a sum of
three terms:
where Fsurf is the free energy of the micellar interface
and the contributions Fconf-A and Fconf-B are due to the
losses in conformational entropy of A and B chains.
† Russian Academy of Sciences.
‡ Moscow State University.
§ Helsinki University of Technology.
⊥ University of Groningen.
F ) Fsurf + Fconf-A + Fconf-B (1)
5019Macromolecules 2001, 34, 5019-5022
10.1021/ma001621f CCC: $20.00 © 2001 American Chemical Society
Published on Web 06/06/2001
The interfacial free energy Fsurf can be written as
where N is the total number of chains in the system
and ç is the interface tension coefficient determining
the degree of immiscibility of A and B blocks. The
contribution Fconf-B describes the loss in conformation
entropy of the linear B blocks. These losses are caused
by the expansion of B chains in the interfacial region
due to the high density of B chains in the neighborhood
of the surface of A micelles. As shown in ref 7, the
contribution Fconf-B can be written as
Here lB, aB, and vB are the Kuhn segment length, the
length, and the volume of the monomer units of the B
chains, respectively.
Finally, Fconf-A is the conformational free energy of
the A blocks. Generally speaking, the term Fconf-A
includes two contributions: a contribution from the
conformational entropy of the backbone chain and one
from the conformational entropy of the side chains,
assuming, as we will, that the backbone monomer units
and the monomer units of the side chains have no
energetic interaction with each other. Because of the
well-known Flory theorem,16,17 the excluded-volume
interactions are screened within the A micelle. As a
consequence, the side chains have a conformation close
to that of an ideal chain, and the following analysis
shows that their conformations remain close to ideal
even in the case of dense grafting of side chains (up to
m  1). To see this, we assume that the side chains form
a dense tube with radius D around the backbone.
Equating the volume per single side chain, which is at
least D2m1/2a, with the total volume of the monomer
units of a side chain, na3, gives D  n1/2a/m1/4. This
latter represents the minimum value of D required to
accommodate the side chains. Since this value is smaller
than n1/2a up to m  1, the conformation of the side
chains will remain close to ideal. It implies that for m
g1 the contribution to the free energy due to elastic
deformation of the side chains is negligible in compari-
son to the contribution of backbone of A chains.
Thus, the free energy Fconf-A includes only the elastic
free energy contribution of the A backbone chains. In
the simplest way this term can be written as the free
energy of expansion of a backbone chain with NA units
multiplied by the total number N of polymer chains in
the system:
where lA and aA are the Kuhn segment length and
monomer unit length of A chain, respectively.
Since in the case under consideration the excluded-
volume interactions are screened, the persistence length
of the backbone is not expected to increase with increas-
ing density of side chain grafting up to m  1 (note that
an increase in stiffness of the backbone of comb copoly-
mers immersed in a solvent is caused by excluded-
volume interactions between side chains).3,18,19
By taking into account the fact that different A chains
are expanded differently within the spherical micelle,
one can rewrite the contribution (4) as (see ref 7)
Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that may
be formed by such macromolecules (B).
Fsurf ) 4ðç
R2N
Q
(2)
Fconf-B )
3
8ð
kT
R2
vB
lBaB
NQ (3)
Fconf-A  kTN R
2
NAaAlA
(4)
Fconf-A )
3ð2
80
kTN R
2
NAaAlA
(5)
5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001
The value R of the radius of the A micelle is directly
connected to Q by the space-filling condition. For the
case where the volume of the monomer units of the
backbone vA is equal to the volume of the monomer units
of the side chains, the condition of space filling takes
the form
where mb is the number of monomer units of the comb
block per monomer unit of the comb backbone chain:
Using eqs 2-6 together with the notations si ) vi/ai
and Li ) Niai (i ) A, B) (si is the cross section; Li is the
total contour length of corresponding blocks), the total
free energy F can be rewritten as
where ı ) 1 + (40/3ð2)(sAlB/sBlA)(1/mb). For the case sA
 sB, lA  lB, and mb . 1, the value of parameter ı is
close to unity.
Minimization of the free energy F, eq 8, with respect
to Q gives the equilibrium value for the aggregation
number Q:
From eqs 6 and 9 we find for the equilibrium value of
the micelle size, R
From eq 9 it is clear that for large values of mb the
aggregation number Q is almost independent of the
structural parameters of the coil block B. In this case
the parameter ı, which in general depends on lB and
sB, is with high accuracy close to unity. Such a result
seems quite natural. As with an increase of mb the
radius of the A micelle R increases (see eq 10), the
density of B blocks in the neighborhood of the A micelle
and thus the degree of expansion of these blocks drop.
As a result, at high enough values of mb, the contribu-
tion Fconf-B becomes negligible in comparison with the
free energy Fconf-A of the A blocks.
With an increase in the degree of immiscibility ç, both
the aggregation number Q and the radius R of the
micelle increase up to the crossover to the superstrong
segregation regime where the size of the micelle R
becomes comparable with the length LA of a fully
stretched backbone of the A block:7
A simple calculation gives the following estimate for the
critical value (ç/kT)SS at the crossover point to the
superstrong segregation regime:
At ç/kT > (ç/kT)SS (within the superstrong segrega-
tion regime) the size of the micelle RSS is a linear
function of the contour length LA of the A blocks. Also,
the aggregation number QSS no longer increases with a
decrease of temperature:
Equations 12 and 13 demonstrate that both the critical
value (ç/kT)SS and the aggregation number QSS in the
superstrong segregation regime decrease as a function
of mb ) 1 + n/m.
The characteristic sizes of the structure R and of the
critical value (ç/kT)SS for crossover to the superstrong
segregation regime in the case of lamellar and cylindri-
cal structures have the same scaling dependencies on
LA and mb as in eqs 11-13. To show this, we now
consider the formation of lamellar structure.
Lamellar Structure. In the case of a lamellar
structure the free energy F is given (by ignoring all
irrelevant numerical factors) as
Here ó is surface area of interface between A and B
blocks per chain.
The self-consistent space-filling conditions are
Minimization of free energy (14)-(16) with respect to
ó give us the equilibrium value of characteristic size of
lamellar structure RA and RB:
Let us consider case NAmbvA = NBvB, where the
volume fractions of monomer units in the linear and
comb blocks are equal to each other. In this case, the
thicknesses of the layers of the comb and the linear
blocks are equal to each other as well: RA ) RB ) R.
The period of the structure R is now given by
where ıl ) 1 + lAsB/mblBsA. For the case of mB . 1, the
QNAmbvA
4ð
3
R3
) 1 (6)
mb ) 1 +
n
m
(7)
F ) ( 36ðsALA)1/3mb2/3[ çkT LAsAQ1/3 + 3ð320 Q2/3 sAlA ı] (8)
Q ) ç
kT
160
3ð
LAlA
ı
(9)
R ) ( çkT 40ð2 LA2 lAsAı mb)1/3 (10)
R  LA (11)
( çkT)SS  ð
2
40
LAı
mblAsA
(12)
RSS ) LA
QSS )
4ð
3
LA
2
mbsA
(13)
F
kT
)
RA
2
NAlAaA
+
RB
2
NBlBaB
+ çó (14)
RAó ) NAmbvA (15)
RBó ) NBvB (16)
RA  ( çkT)
1/3 NAmbvA(NAmb2vA2lAaA + NBvB2lBaB )1/3
RB  ( çkT)
1/3 NBvB(NAmb2vA2lAaA + NBvB2lBaB )1/3 (17)
R  ( çkT LA2 lAsAıl mb)1/3 (18)
Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021
value of ıl is close to unity and the period of the
structure R is determined by the structural parameters
of the comb block only. Analysis shows that in this
particular case, as in a case of spherical micelles, the
free energy F of elastic deformation of the linear B block
is negligible in comparison with elastic free energy of
the A block due to the much higher stretching of the
comb A block.
The crossover from the strong segregation regime (R
 LA2/3) to the superstrong segregation regime (RSS 
LA) proceeds at
In the superstrong segregation regime the character-
istic period R is determined by the length LA of the
backbone of the comb block, which is much shorter (in
the case of high values of mb, see eqs 16-18) than the
length LB of coil block. In this regime period R does not
increase with further increase of parameter of im-
miscibility ç.
The crossover to the superstrong segregation regime
proceeds when the comb block is fully stretched, and in
the general case the crossover point depends on the ratio
f ) NAmbvA/NBvB of the volume of monomer units in the
comb and the linear blocks:
At f >1 the conditions for crossover coincides with
condition (19), whereas with decrease of parameter f
below unity the critical value (ç/kT)SS increases.
Note that the eq 20 gives for a wide range of
parameter values the same scaling dependence of criti-
cal value (ç/kT)SS of the crossover to the supercritical
regime as eq 12. An increase in the degree of polymer-
ization of the side chain n or a decrease in the distance
m between the grafting points leads to softening of the
conditions for the crossover from the strong to the
superstrong segregation regime. In general, the period
of the structure significantly increases with an increase
in the degree of polymerization of the side chains or
their grafting density.
In conclusion, we see that the possibility to reach the
superstrong segregation regime is strongly enhanced for
diblock copolymers involving comb blocks.
Acknowledgment. This work was supported by the
NWO program for Dutch-Russian scientific cooperation
and by the Program “University of Russia-Fundamental
Research” and by the Russian Foundation for Basic
Research (Grant 01-03-32672).
References and Notes
(1) Leibler, L. Macromolecules 1980, 13, 1602.
(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.
(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,
41, 525.
(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.
(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,
277, 1225.
(6) Hamley, I. W. The Physics of Block Copolymers; Oxford
University Press: Oxford, 1998.
(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules
1993, 26, 3601.
(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,
A. N. Macromol. Theory Simul. 1996, 5, 713, 749.
(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.
(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;
Leibler, L. Nature 1999, 398, 137.
(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,
26, 276.
(12) Milner, S. T. Macromolecules 1994, 27, 2333.
(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;
Thomas, E. L. Macromolecules 1998, 31, 3532.
(14) Ruokolainen, J.; Ma¨kinen, R.; Torkkeli, M.; Ma¨kela¨, T.;
Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,
557.
(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,
11, 777.
(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-
sity Press: Ithaca, NY, 1953.
(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell
University Press: Ithaca, NY, 1979.
(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.
R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.
(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.
MA001621F
( çkT)SS 
LAıl
mblAsA
(19)
( çkT)SS  LAmblAsA(1 + 1f lAsBmblBsA) (20)
5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001


   University of GroningenDomains in melts of comb-coil diblock copolymers: Superstrong segregation regimeVasilevskaya, V.V.; Gusev, L.A.; Khokhlov, A.R.; Ikkala, O.; ten Brinke, G.; Ikkala, AOPublished in:MacromoleculesDOI:10.1021/ma001621fIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2001Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vasilevskaya, V. V., Gusev, L. A., Khokhlov, A. R., Ikkala, O., ten Brinke, G., & Ikkala, AO. (2001).Domains in melts of comb-coil diblock copolymers: Superstrong segregation regime: SuperstrongSegregation Regime. Macromolecules, 34(14), 5019 - 5022. https://doi.org/10.1021/ma001621fCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Domains in Melts of Comb-Coil Diblock Copolymers: SuperstrongSegregation RegimeV. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ andG. ten Brinke*,⊥Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,9747 AG Groningen, The NetherlandsReceived September 19, 2000; Revised Manuscript Received April 4, 2001ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime areanalyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energybetween the components required to induce the crossover to the superstrong segregation regime is inverselyproportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is thedistance between successive grafting points. As a result, the superstrong segregation regime, being ratherrare in the case of ordinary block copolymers, has a much better chance to be realized in the case ofdiblock copolymers with combs grafted to one of the blocks.1. IntroductionMicrophase separation in copolymers of immisciblepolymer components A and B has been an area ofintensive research over the past decades.1-6 Theorieshave been developed to deal with the different regimesthat can be distinguished, e.g., weak, strong, andintermediate segregation regime. Apart from these morecommon regimes, it was found that in the case ofextremely strong immiscibility between the componentsof a diblock copolymer with one block (e.g., A chains)being rather small, the behavior of the system changesqualitatively from that of the usual strong segregationregime: a superstrong segregation regime emerges.7 Inthis regime the A chains become practically completelyextended within the corresponding microdomain. Cal-culations and experimental studies show that thesuperstrong segregation regime is not characteristic forordinary block copolymers; however, it can be used forthe description of aggregates and multiplets in associat-ing polymers.8These results apply for linear block copolymers. Butrecently, interest has shifted to the nature of themicrophase-separated structures of block copolymers ofa more complex linear9,10 and nonlinear architec-ture.6,11,12 For this paper comb-coil diblock copolymersconsisting of a nonlinear comb copolymer block and alinear block will be of direct interest. A peculiar subsetof these kind of systems, where the side chains are notcovalently bonded but attached by strong physicalinteractions (hydrogen bonding and ionic bonding), havebeen studied in detail recently.13-15 In this paper weconsider the microphase formation in a melt of comb-coil diblock copolymers and study the conditions for thecrossover from the strong to superstrong segregationregime.2. Results and DiscussionWe will consider the microdomain structure formedby comb-coil diblock copolymers consisting of a comb-like A block and a linear B block (see Figure 1). Themain characteristic parameters of the A block are thedegree of polymerization of the backbone NA, the degreeof polymerization of the side chains n, and the numberof monomer units between the grafting points m. TheB block is characterized by a single parameter NB whichis the degree of polymerization of this block.In correspondence with the main objective of thispaper, let us assume that the monomer units of the Ablocks are incompatible with the monomer units of theB blocks and that the immiscibility between the A andB blocks is strong enough to make a narrow interfacebetween the A and B phases; i.e., the structure formedcorresponds to the strong segregation limit.Spherical Micelles. Let us first assume that NA ,NB and that the A blocks (as minority component) formspherical micelles with radius R and aggregation num-ber Q.In ref 2 it was shown that in the strong segregationregime of ordinary block copolymers with immisciblelinear A and B blocks both the A chains within thespherical micelles and the B chain parts in the regionclose to the micellar core are significantly stretched.This is the case for the system under consideration aswell.To estimate the aggregation number Q and the radiusof micelle R, let us consider the free energy F of thesystem. This free energy can be presented as a sum ofthree terms:where Fsurf is the free energy of the micellar interfaceand the contributions Fconf-A and Fconf-B are due to thelosses in conformational entropy of A and B chains.† Russian Academy of Sciences.‡ Moscow State University.§ Helsinki University of Technology.⊥ University of Groningen.F ) Fsurf + Fconf-A + Fconf-B (1)5019Macromolecules 2001, 34, 5019-502210.1021/ma001621f CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 06/06/2001The interfacial free energy Fsurf can be written aswhere N is the total number of chains in the systemand ç is the interface tension coefficient determiningthe degree of immiscibility of A and B blocks. Thecontribution Fconf-B describes the loss in conformationentropy of the linear B blocks. These losses are causedby the expansion of B chains in the interfacial regiondue to the high density of B chains in the neighborhoodof the surface of A micelles. As shown in ref 7, thecontribution Fconf-B can be written asHere lB, aB, and vB are the Kuhn segment length, thelength, and the volume of the monomer units of the Bchains, respectively.Finally, Fconf-A is the conformational free energy ofthe A blocks. Generally speaking, the term Fconf-Aincludes two contributions: a contribution from theconformational entropy of the backbone chain and onefrom the conformational entropy of the side chains,assuming, as we will, that the backbone monomer unitsand the monomer units of the side chains have noenergetic interaction with each other. Because of thewell-known Flory theorem,16,17 the excluded-volumeinteractions are screened within the A micelle. As aconsequence, the side chains have a conformation closeto that of an ideal chain, and the following analysisshows that their conformations remain close to idealeven in the case of dense grafting of side chains (up tom  1). To see this, we assume that the side chains forma dense tube with radius D around the backbone.Equating the volume per single side chain, which is atleast D2m1/2a, with the total volume of the monomerunits of a side chain, na3, gives D  n1/2a/m1/4. Thislatter represents the minimum value of D required toaccommodate the side chains. Since this value is smallerthan n1/2a up to m  1, the conformation of the sidechains will remain close to ideal. It implies that for mg1 the contribution to the free energy due to elasticdeformation of the side chains is negligible in compari-son to the contribution of backbone of A chains.Thus, the free energy Fconf-A includes only the elasticfree energy contribution of the A backbone chains. Inthe simplest way this term can be written as the freeenergy of expansion of a backbone chain with NA unitsmultiplied by the total number N of polymer chains inthe system:where lA and aA are the Kuhn segment length andmonomer unit length of A chain, respectively.Since in the case under consideration the excluded-volume interactions are screened, the persistence lengthof the backbone is not expected to increase with increas-ing density of side chain grafting up to m  1 (note thatan increase in stiffness of the backbone of comb copoly-mers immersed in a solvent is caused by excluded-volume interactions between side chains).3,18,19By taking into account the fact that different A chainsare expanded differently within the spherical micelle,one can rewrite the contribution (4) as (see ref 7)Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that maybe formed by such macromolecules (B).Fsurf ) 4ðçR2NQ(2)Fconf-B )38ðkTR2vBlBaBNQ (3)Fconf-A  kTN R2NAaAlA(4)Fconf-A )3ð280kTN R2NAaAlA(5)5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001The value R of the radius of the A micelle is directlyconnected to Q by the space-filling condition. For thecase where the volume of the monomer units of thebackbone vA is equal to the volume of the monomer unitsof the side chains, the condition of space filling takesthe formwhere mb is the number of monomer units of the combblock per monomer unit of the comb backbone chain:Using eqs 2-6 together with the notations si ) vi/aiand Li ) Niai (i ) A, B) (si is the cross section; Li is thetotal contour length of corresponding blocks), the totalfree energy F can be rewritten aswhere ı ) 1 + (40/3ð2)(sAlB/sBlA)(1/mb). For the case sA sB, lA  lB, and mb . 1, the value of parameter ı isclose to unity.Minimization of the free energy F, eq 8, with respectto Q gives the equilibrium value for the aggregationnumber Q:From eqs 6 and 9 we find for the equilibrium value ofthe micelle size, RFrom eq 9 it is clear that for large values of mb theaggregation number Q is almost independent of thestructural parameters of the coil block B. In this casethe parameter ı, which in general depends on lB andsB, is with high accuracy close to unity. Such a resultseems quite natural. As with an increase of mb theradius of the A micelle R increases (see eq 10), thedensity of B blocks in the neighborhood of the A micelleand thus the degree of expansion of these blocks drop.As a result, at high enough values of mb, the contribu-tion Fconf-B becomes negligible in comparison with thefree energy Fconf-A of the A blocks.With an increase in the degree of immiscibility ç, boththe aggregation number Q and the radius R of themicelle increase up to the crossover to the superstrongsegregation regime where the size of the micelle Rbecomes comparable with the length LA of a fullystretched backbone of the A block:7A simple calculation gives the following estimate for thecritical value (ç/kT)SS at the crossover point to thesuperstrong segregation regime:At ç/kT > (ç/kT)SS (within the superstrong segrega-tion regime) the size of the micelle RSS is a linearfunction of the contour length LA of the A blocks. Also,the aggregation number QSS no longer increases with adecrease of temperature:Equations 12 and 13 demonstrate that both the criticalvalue (ç/kT)SS and the aggregation number QSS in thesuperstrong segregation regime decrease as a functionof mb ) 1 + n/m.The characteristic sizes of the structure R and of thecritical value (ç/kT)SS for crossover to the superstrongsegregation regime in the case of lamellar and cylindri-cal structures have the same scaling dependencies onLA and mb as in eqs 11-13. To show this, we nowconsider the formation of lamellar structure.Lamellar Structure. In the case of a lamellarstructure the free energy F is given (by ignoring allirrelevant numerical factors) asHere ó is surface area of interface between A and Bblocks per chain.The self-consistent space-filling conditions areMinimization of free energy (14)-(16) with respect toó give us the equilibrium value of characteristic size oflamellar structure RA and RB:Let us consider case NAmbvA = NBvB, where thevolume fractions of monomer units in the linear andcomb blocks are equal to each other. In this case, thethicknesses of the layers of the comb and the linearblocks are equal to each other as well: RA ) RB ) R.The period of the structure R is now given bywhere ıl ) 1 + lAsB/mblBsA. For the case of mB . 1, theQNAmbvA4ð3R3) 1 (6)mb ) 1 +nm(7)F ) ( 36ðsALA)1/3mb2/3[ çkT LAsAQ1/3 + 3ð320 Q2/3 sAlA ı] (8)Q ) çkT1603ðLAlAı(9)R ) ( çkT 40ð2 LA2 lAsAı mb)1/3 (10)R  LA (11)( çkT)SS  ð240LAımblAsA(12)RSS ) LAQSS )4ð3LA2mbsA(13)FkT)RA2NAlAaA+RB2NBlBaB+ çó (14)RAó ) NAmbvA (15)RBó ) NBvB (16)RA  ( çkT)1/3 NAmbvA(NAmb2vA2lAaA + NBvB2lBaB )1/3RB  ( çkT)1/3 NBvB(NAmb2vA2lAaA + NBvB2lBaB )1/3 (17)R  ( çkT LA2 lAsAıl mb)1/3 (18)Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021value of ıl is close to unity and the period of thestructure R is determined by the structural parametersof the comb block only. Analysis shows that in thisparticular case, as in a case of spherical micelles, thefree energy F of elastic deformation of the linear B blockis negligible in comparison with elastic free energy ofthe A block due to the much higher stretching of thecomb A block.The crossover from the strong segregation regime (R LA2/3) to the superstrong segregation regime (RSS LA) proceeds atIn the superstrong segregation regime the character-istic period R is determined by the length LA of thebackbone of the comb block, which is much shorter (inthe case of high values of mb, see eqs 16-18) than thelength LB of coil block. In this regime period R does notincrease with further increase of parameter of im-miscibility ç.The crossover to the superstrong segregation regimeproceeds when the comb block is fully stretched, and inthe general case the crossover point depends on the ratiof ) NAmbvA/NBvB of the volume of monomer units in thecomb and the linear blocks:At f >1 the conditions for crossover coincides withcondition (19), whereas with decrease of parameter fbelow unity the critical value (ç/kT)SS increases.Note that the eq 20 gives for a wide range ofparameter values the same scaling dependence of criti-cal value (ç/kT)SS of the crossover to the supercriticalregime as eq 12. An increase in the degree of polymer-ization of the side chain n or a decrease in the distancem between the grafting points leads to softening of theconditions for the crossover from the strong to thesuperstrong segregation regime. In general, the periodof the structure significantly increases with an increasein the degree of polymerization of the side chains ortheir grafting density.In conclusion, we see that the possibility to reach thesuperstrong segregation regime is strongly enhanced fordiblock copolymers involving comb blocks.Acknowledgment. This work was supported by theNWO program for Dutch-Russian scientific cooperationand by the Program “University of Russia-FundamentalResearch” and by the Russian Foundation for BasicResearch (Grant 01-03-32672).References and Notes(1) Leibler, L. Macromolecules 1980, 13, 1602.(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,41, 525.(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,277, 1225.(6) Hamley, I. W. The Physics of Block Copolymers; OxfordUniversity Press: Oxford, 1998.(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules1993, 26, 3601.(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,A. N. Macromol. Theory Simul. 1996, 5, 713, 749.(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;Leibler, L. Nature 1999, 398, 137.(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,26, 276.(12) Milner, S. T. Macromolecules 1994, 27, 2333.(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;Thomas, E. L. Macromolecules 1998, 31, 3532.(14) Ruokolainen, J.; Ma¨kinen, R.; Torkkeli, M.; Ma¨kela¨, T.;Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,557.(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,11, 777.(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-sity Press: Ithaca, NY, 1953.(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.MA001621F( çkT)SS LAılmblAsA(19)( çkT)SS  LAmblAsA(1 + 1f lAsBmblBsA) (20)5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001

Vasilevskaya, V.V.,

Gusev, L.A.,

Khokhlov, A.R.,

Ikkala, O.,

Brinke, G. ten,

University of Groningen Digital Archive

Domains in Melts of Comb-Coil Diblock Copolymers: Superstrong
Segregation Regime
V. V. Vasilevskaya,† L. A. Gusev,‡ A. R. Khokhlov,*,‡ O. Ikkala,§ and
G. ten Brinke*,^
Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28,
Moscow 117823, Russia; Physics Department, Moscow State University, Moscow 117234, Russia;
Department of Engineering Physics and Mathematics, Helsinki University of Technology, FIN-02015
HUT, Espoo, Finland; and Department of Polymer Chemistry, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
Received September 19, 2000; Revised Manuscript Received April 4, 2001
ABSTRACT: Conditions for the crossover from the strong to the superstrong segregation regime are
analyzed for the case of comb-coil diblock copolymers. It is shown that the critical interaction energy
between the components required to induce the crossover to the superstrong segregation regime is inversely
proportional to mb ) 1 + n/m, where n is the degree of polymerization of the side chain and m is the
distance between successive grafting points. As a result, the superstrong segregation regime, being rather
rare in the case of ordinary block copolymers, has a much better chance to be realized in the case of
diblock copolymers with combs grafted to one of the blocks.
1. Introduction
Microphase separation in copolymers of immiscible
polymer components A and B has been an area of
intensive research over the past decades.1-6 Theories
have been developed to deal with the different regimes
that can be distinguished, e.g., weak, strong, and
intermediate segregation regime. Apart from these more
common regimes, it was found that in the case of
extremely strong immiscibility between the components
of a diblock copolymer with one block (e.g., A chains)
being rather small, the behavior of the system changes
qualitatively from that of the usual strong segregation
regime: a superstrong segregation regime emerges.7 In
this regime the A chains become practically completely
extended within the corresponding microdomain. Cal-
culations and experimental studies show that the
superstrong segregation regime is not characteristic for
ordinary block copolymers; however, it can be used for
the description of aggregates and multiplets in associat-
ing polymers.8
These results apply for linear block copolymers. But
recently, interest has shifted to the nature of the
microphase-separated structures of block copolymers of
a more complex linear9,10 and nonlinear architec-
ture.6,11,12 For this paper comb-coil diblock copolymers
consisting of a nonlinear comb copolymer block and a
linear block will be of direct interest. A peculiar subset
of these kind of systems, where the side chains are not
covalently bonded but attached by strong physical
interactions (hydrogen bonding and ionic bonding), have
been studied in detail recently.13-15 In this paper we
consider the microphase formation in a melt of comb-
coil diblock copolymers and study the conditions for the
crossover from the strong to superstrong segregation
regime.
2. Results and Discussion
We will consider the microdomain structure formed
by comb-coil diblock copolymers consisting of a comb-
like A block and a linear B block (see Figure 1). The
main characteristic parameters of the A block are the
degree of polymerization of the backbone NA, the degree
of polymerization of the side chains n, and the number
of monomer units between the grafting points m. The
B block is characterized by a single parameter NB which
is the degree of polymerization of this block.
In correspondence with the main objective of this
paper, let us assume that the monomer units of the A
blocks are incompatible with the monomer units of the
B blocks and that the immiscibility between the A and
B blocks is strong enough to make a narrow interface
between the A and B phases; i.e., the structure formed
corresponds to the strong segregation limit.
Spherical Micelles. Let us first assume that NA ,
NB and that the A blocks (as minority component) form
spherical micelles with radius R and aggregation num-
ber Q.
In ref 2 it was shown that in the strong segregation
regime of ordinary block copolymers with immiscible
linear A and B blocks both the A chains within the
spherical micelles and the B chain parts in the region
close to the micellar core are significantly stretched.
This is the case for the system under consideration as
well.
To estimate the aggregation number Q and the radius
of micelle R, let us consider the free energy F of the
system. This free energy can be presented as a sum of
three terms:
where Fsurf is the free energy of the micellar interface
and the contributions Fconf-A and Fconf-B are due to the
losses in conformational entropy of A and B chains.
† Russian Academy of Sciences.
‡ Moscow State University.
§ Helsinki University of Technology.
^ University of Groningen.
F ) Fsurf + Fconf-A + Fconf-B (1)
5019 Macromolecules 2001, 34, 5019-5022
10.1021/ma001621f CCC: $20.00 © 2001 American Chemical Society
Published on Web 06/06/2001The interfacial free energy Fsurf can be written as
where N is the total number of chains in the system
and ç is the interface tension coefficient determining
the degree of immiscibility of A and B blocks. The
contribution Fconf-B describes the loss in conformation
entropy of the linear B blocks. These losses are caused
by the expansion of B chains in the interfacial region
due to the high density of B chains in the neighborhood
of the surface of A micelles. As shown in ref 7, the
contribution Fconf-B can be written as
Here lB, aB, and vB are the Kuhn segment length, the
length, and the volume of the monomer units of the B
chains, respectively.
Finally, Fconf-A is the conformational free energy of
the A blocks. Generally speaking, the term Fconf-A
includes two contributions: a contribution from the
conformational entropy of the backbone chain and one
from the conformational entropy of the side chains,
assuming, as we will, that the backbone monomer units
and the monomer units of the side chains have no
energetic interaction with each other. Because of the
well-known Flory theorem,16,17 the excluded-volume
interactions are screened within the A micelle. As a
consequence, the side chains have a conformation close
to that of an ideal chain, and the following analysis
shows that their conformations remain close to ideal
even in the case of dense grafting of side chains (up to
m  1). To see this, we assume that the side chains form
a dense tube with radius D around the backbone.
Equating the volume per single side chain, which is at
least D2m1/2a, with the total volume of the monomer
units of a side chain, na3, gives D  n1/2a/m1/4. This
latter represents the minimum value of D required to
accommodate the side chains. Since this value is smaller
than n1/2a up to m  1, the conformation of the side
chains will remain close to ideal. It implies that for m
g1 the contribution to the free energy due to elastic
deformation of the side chains is negligible in compari-
son to the contribution of backbone of A chains.
Thus, the free energy Fconf-A includes only the elastic
free energy contribution of the A backbone chains. In
the simplest way this term can be written as the free
energy of expansion of a backbone chain with NA units
multiplied by the total number N of polymer chains in
the system:
where lA and aA are the Kuhn segment length and
monomer unit length of A chain, respectively.
Since in the case under consideration the excluded-
volume interactions are screened, the persistence length
of the backbone is not expected to increase with increas-
ing density of side chain grafting up to m  1 (note that
an increase in stiffness of the backbone of comb copoly-
mers immersed in a solvent is caused by excluded-
volume interactions between side chains).3,18,19
By taking into account the fact that different A chains
are expanded differently within the spherical micelle,
one can rewrite the contribution (4) as (see ref 7)
Figure 1. Schematic presentation of comb-coil diblock copolymer (A) and the superstrong segregated microstructure that may
be formed by such macromolecules (B).
Fsurf ) 4ðç R
2N
Q
(2)
Fconf-B ) 3
8ð
kT
R
2
vB
lBaB
NQ (3)
Fconf-A  kTN R
2
NAaAlA
(4)
Fconf-A ) 3ð
2
80
kTN R
2
NAaAlA
(5)
5020 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001The value R of the radius of the A micelle is directly
connected to Q by the space-filling condition. For the
case where the volume of the monomer units of the
backbone vA is equal to the volume of the monomer units
of the side chains, the condition of space filling takes
the form
where mb is the number of monomer units of the comb
block per monomer unit of the comb backbone chain:
Using eqs 2-6 together with the notations si ) vi/ai
and Li ) Niai (i ) A, B) (si is the cross section; Li is the
total contour length of corresponding blocks), the total
free energy F can be rewritten as
where õ ) 1 + (40/3ð2)(sAlB/sBlA)(1/mb). For the case sA
 sB, lA  lB, and mb . 1, the value of parameter õ is
close to unity.
Minimization of the free energy F, eq 8, with respect
to Q gives the equilibrium value for the aggregation
number Q:
From eqs 6 and 9 we find for the equilibrium value of
the micelle size, R
From eq 9 it is clear that for large values of mb the
aggregation number Q is almost independent of the
structural parameters of the coil block B. In this case
the parameter õ, which in general depends on lB and
sB, is with high accuracy close to unity. Such a result
seems quite natural. As with an increase of mb the
radius of the A micelle R increases (see eq 10), the
density of B blocks in the neighborhood of the A micelle
and thus the degree of expansion of these blocks drop.
As a result, at high enough values of mb, the contribu-
tion Fconf-B becomes negligible in comparison with the
free energy Fconf-A of the A blocks.
With an increase in the degree of immiscibility ç, both
the aggregation number Q and the radius R of the
micelle increase up to the crossover to the superstrong
segregation regime where the size of the micelle R
becomes comparable with the length LA of a fully
stretched backbone of the A block:7
A simple calculation gives the following estimate for the
critical value (ç/kT)SS at the crossover point to the
superstrong segregation regime:
At ç/kT > (ç/kT)SS (within the superstrong segrega-
tion regime) the size of the micelle RSS is a linear
function of the contour length LA of the A blocks. Also,
the aggregation number QSS no longer increases with a
decrease of temperature:
Equations 12 and 13 demonstrate that both the critical
value (ç/kT)SS and the aggregation number QSS in the
superstrong segregation regime decrease as a function
of mb ) 1 + n/m.
The characteristic sizes of the structure R and of the
critical value (ç/kT)SS for crossover to the superstrong
segregation regime in the case of lamellar and cylindri-
cal structures have the same scaling dependencies on
LA and mb as in eqs 11-13. To show this, we now
consider the formation of lamellar structure.
Lamellar Structure. In the case of a lamellar
structure the free energy F is given (by ignoring all
irrelevant numerical factors) as
Here ó is surface area of interface between A and B
blocks per chain.
The self-consistent space-filling conditions are
Minimization of free energy (14)-(16) with respect to
ó give us the equilibrium value of characteristic size of
lamellar structure RA and RB:
Let us consider case NAmbvA = NBvB, where the
volume fractions of monomer units in the linear and
comb blocks are equal to each other. In this case, the
thicknesses of the layers of the comb and the linear
blocks are equal to each other as well: RA ) RB ) R.
The period of the structure R is now given by
where õl ) 1 + lAsB/mblBsA. For the case of mB . 1, the
QNAmbvA
4ð
3
R
3
) 1 (6)
mb ) 1 + n
m
(7)
F ) (
36ð
sALA)
1/3
mb
2/3[
ç
kT
LAsA
Q
1/3 + 3ð
320
Q
2/3sA
lA
õ] (8)
Q ) ç
kT
160
3ð
LAlA
õ
(9)
R ) (
ç
kT
40
ð
2
LA
2lAsA
õ
mb)
1/3
(10)
R  LA (11)
(
ç
kT)SS
 ð
2
40
LAõ
mblAsA
(12)
RSS ) LA
QSS ) 4ð
3
LA
2
mbsA
(13)
F
kT
)
RA
2
NAlAaA
+
RB
2
NBlBaB
+ çó (14)
RAó ) NAmbvA (15)
RBó ) NBvB (16)
RA  (
ç
kT)
1/3 NAmbvA
(
NAmb
2vA
2
lAaA
+
NBvB
2
lBaB)
1/3
RB  (
ç
kT)
1/3 NBvB
(
NAmb
2vA
2
lAaA
+
NBvB
2
lBaB)
1/3 (17)
R  (
ç
kT
LA
2lAsA
õl
mb)
1/3
(18)
Macromolecules, Vol. 34, No. 14, 2001 Comb-Coil Diblock Copolymers 5021value of õl is close to unity and the period of the
structure R is determined by the structural parameters
of the comb block only. Analysis shows that in this
particular case, as in a case of spherical micelles, the
free energy F of elastic deformation of the linear B block
is negligible in comparison with elastic free energy of
the A block due to the much higher stretching of the
comb A block.
The crossover from the strong segregation regime (R
 LA
2/3) to the superstrong segregation regime (RSS 
LA) proceeds at
In the superstrong segregation regime the character-
istic period R is determined by the length LA of the
backbone of the comb block, which is much shorter (in
the case of high values of mb, see eqs 16-18) than the
length LB of coil block. In this regime period R does not
increase with further increase of parameter of im-
miscibility ç.
The crossover to the superstrong segregation regime
proceeds when the comb block is fully stretched, and in
the general case the crossover point depends on the ratio
f ) NAmbvA/NBvB of the volume of monomer units in the
comb and the linear blocks:
At f >1 the conditions for crossover coincides with
condition (19), whereas with decrease of parameter f
below unity the critical value (ç/kT)SS increases.
Note that the eq 20 gives for a wide range of
parameter values the same scaling dependence of criti-
cal value (ç/kT)SS of the crossover to the supercritical
regime as eq 12. An increase in the degree of polymer-
ization of the side chain n or a decrease in the distance
m between the grafting points leads to softening of the
conditions for the crossover from the strong to the
superstrong segregation regime. In general, the period
of the structure significantly increases with an increase
in the degree of polymerization of the side chains or
their grafting density.
In conclusion, we see that the possibility to reach the
superstrong segregation regime is strongly enhanced for
diblock copolymers involving comb blocks.
Acknowledgment. This work was supported by the
NWO program for Dutch-Russian scientific cooperation
and by the Program “University of Russia-Fundamental
Research” and by the Russian Foundation for Basic
Research (Grant 01-03-32672).
References and Notes
(1) Leibler, L. Macromolecules 1980, 13, 1602.
(2) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.
(3) Bates, F.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990,
41, 525.
(4) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091.
(5) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997,
277, 1225.
(6) Hamley, I. W. The Physics of Block Copolymers; Oxford
University Press: Oxford, 1998.
(7) Nyrkova, L. A.; Khokhlov, A. R.; Doi, M. Macromolecules
1993, 26, 3601.
(8) Khalatur, P. G.; Khokhlov, A. R.; Nyrkova, I. A.; Semenov,
A. N. Macromol. Theory Simul. 1996, 5, 713, 749.
(9) Bates, F. S.; Fredrickson, G. H. Phys. Today 1999, 52, 32.
(10) Goldacker, T.; Abetz, V.; Stadler, R.; Erukhimovich, I. Ya.;
Leibler, L. Nature 1999, 398, 137.
(11) Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993,
26, 276.
(12) Milner, S. T. Macromolecules 1994, 27, 2333.
(13) Ruokolainen, J.; Tanner, J.; Ikkala, O.; ten Brinke, G.;
Thomas, E. L. Macromolecules 1998, 31, 3532.
(14) Ruokolainen, J.; Ma ¨kinen, R.; Torkkeli, M.; Ma ¨kela ¨, T.;
Serimaa, R.; ten Brinke, G.; Ikkala, O. Science 1998, 280,
557.
(15) Ruokolainen, J.; ten Brinke, G.; Ikkala, O. Adv. Mater. 1999,
11, 777.
(16) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-
sity Press: Ithaca, NY, 1953.
(17) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell
University Press: Ithaca, NY, 1979.
(18) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A.
R.; Yurasova, T. A. Vysokomolekul. Soed. 1987, 25A, 1169.
(19) Fredrickson, G. H. Macromolecules 1993, 26, 2825.
MA001621F
(
ç
kT)SS

LAõl
mblAsA
(19)
(
ç
kT)
SS

LA
mblAsA(1 + 1
f
lAsB
mblBsA) (20)
5022 Vasilevskaya et al. Macromolecules, Vol. 34, No. 14, 2001

Principles of Polymer Chemistry;

Scaling Concepts in Polymer Physics;

The Physics of Block Copolymers;

Theory Simul.

Today

Domains in Melts of Comb-Coil Diblock Copolymers: Superstrong Segregation Regime

Domains in Melts of Comb-Coil Diblock Copolymers: Superstrong Segregation Regime

Abstract

Similar works

Full text

Available Versions

ARTS repository - University of Groningen

University of Groningen

ARTS repository - University of Groningen

NARCIS

Crossref

University of Groningen Research Database

University of Groningen

University of Groningen Digital Archive