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Langmuir 2007, 23, 8252-8256
Two Mechanisms of Spontaneous Curvature of Strongly Adsorbed (2D) Double Comblike Copolymers Konstantin I. Popov† and Igor I. Potemkin*,†,‡ Physics Department, Moscow State UniVersity, Moscow 119992, Russian Federation, and Department of Polymer Science, UniVersity of Ulm, 89069 Ulm, Germany ReceiVed January 6, 2007. In Final Form: April 20, 2007 We propose a theory of spontaneous curvature of 2D comblike macromolecules with incompatible side chains of types A and B. It is expected that the side chains of both types are able to change their positions with respect to the backbone. We predict two mechanisms of curvature. In the case of strong incompatibility of the side chains, their complete segregation with respect to the backbone is responsible for the formation of the so-called “energetic” curvature that is a result of the difference in the length (or in the number) of the A and B chains. In the case of moderate incompatibility, partial mixing of the A and B side chains on the convex side of the molecule (a flip of shorter chains to the side of longer chains) can be entropically favorable. In this case, the stretching of the side chains decreases. The radius of the entropic curvature is determined by the length of the longer side chains.
1. Introduction The generation of a chemically heterogeneous surface pattern by the controlled adsorption and self-assembly of block copolymers in microdomains represents a topical challenge for functional substrates and templates in various branches of science and technology.1-3 One way of designing heterogeneous surfaces is based on the preparation of relatively thin films comprising a few periods of the bulk microstructure. The key parameters, which govern the orientation of the microdomains, are found to be the interfacial energies of the boundaries,4-11 the film thickness, and the molecular weight of the copolymers.12,13 For example, in the case of compositionally symmetric diblock copolymers, the perpendicular orientation of lamellae toward the substrate can thermodynamically be stable if the spreading (interaction) parameters of both blocks are similar14 or if the overall molecular weight of the copolymer is high enough.12,13 * To whom correspondence should be addressed. E-mail: igor@ polly.phys.msu.ru. † Moscow State University. ‡ University of Ulm. (1) Thurn-Albrecht, T.; Steiner, R.; DeRouchey, J.; Stafford, C. M.; Huang, E.; Bal, M.; Tuominen, M.; Hawker, C. J.; Russell, T. P. AdV. Mater. 2000, 12, 787. (2) Morkved, T. L.; Lu, M.; Urbas, A. M.; Ehrichs, E. E.; Jaeger, H. M.; Mansky, P.; Russell, T. P. Science 1996, 273, 931. (3) Thurn-Albrecht, T.; Schotter, J.; Ka¨stle, G. A.; Emley, N.; Shibauchi, T.; Krusin-Elbaum, L.; Guarini, K.; Black, C. T.; Tuominen, M. T.; Russell, T. P. Science 2000, 290, 2126. (4) Anastasiadis, S. H.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. Phys. ReV. Lett. 1989, 62, 1852. (5) Coulon, G.; Ausserre, D.; Russell, T. P. J. Phys. (Paris) 1990, 51, 777. (6) Menelle, A.; Russell, T. P.; Anastasiadis, S. H.; Satija, S. K.; Majkrzak, C. F. Phys. ReV. Lett. 1992, 68, 67. (7) Mayes, A. M.; Russell, T. P.; Bassereau, P.; Baker, S. M.; Smith, G. S. Macromolecules 1994, 27, 749. (8) Kellogg, G. J.; Walton, D. G.; Mayes, A. M.; Lambooy, P.; Russell, T. P.; Gallagher, P. D.; Satija, S. K. Phys. ReV. Lett. 1996, 76, 2503. (9) Huang, E.; Mansky, P.; Russell, T. P.; Harrison, C.; Chaikin, P. M.; Register, R. A.; Hawker, C. J.; Mays, J. Macromolecules 2000, 33, 80. (10) Mansky, P.; Russell, T. P.; Hawker, C. J.; Pitsikalis, M.; Mays, J. Macromolecules 1997, 30, 6810. (11) Huang, E.; Rockford, L.; Russell, T. P.; Hawker, C. J. Nature (London) 1998, 395, 757. (12) Busch, P.; Posselt, D.; Smilgies, D. M.; Rheinlander, B.; Kremer, F.; Papadakis, C. M. Macromolecules 2003, 36, 8717. (13) Potemkin, I. I.; Busch, P.; Smilgies, D. M.; Posselt, D.; Papadakis, C. M. Macromol. Rapid Commun. 2007, 28, 579. (14) Potemkin, I. I. Macromolecules 2004, 37, 3505.
Another strategy of reliable preparation of the chemically heterogeneous surface pattern is based on the preparation of ultrathin films whose thickness is much less than the equilibrium period of the bulk morphology. Selective adsorption of copolymers on the surface from dilute solution and their self-organization provide a heterogeneous surface structure. For example, the adsorption of PS-PVP copolymers on mica leads to the formation of micelles with (hemi)spherical or wormlike cores (PS blocks) and strongly adsorbed coronas (PVP blocks).15-20 The form of the micelles is determined by the composition of the diblock copolymer: the wormlike micelles are formed only by copolymers with long enough PS blocks.18,19 In contrast to the micelles in the bulk whose shape is controlled only by the composition of the copolymer, the structure of the surface micelles is sensitive to interactions with the surface. It has been shown that the variation of the “stickiness” of one of the blocks to the surface can result in the structural reorganization of the surface-induced nanopatterns.21 For the case of the single molecule, periodic variation of the stickiness can result in directional motion (reptation) of the copolymers on a solid surface.22 The next level of complexity of self-organization can be attributed to systems where there is an interplay of intra- and intermolecular self-organization. AB miktoarm stars,23 januslike spherical24-26 or cylindrical27-29 micelles, and comblike co(15) Spatz, J. P.; Ro¨scher, A.; Sheiko, S.; Krausch, G.; Mo¨ller, M. AdV. Mater. 1995, 8, 731. (16) Spatz, J. P.; Mo¨ller, M.; No¨ske, M.; Behm, R. J.; Pietralla, M. Macromolecules 1997, 30, 3874. (17) Kramarenko, E. Yu.; Potemkin, I. I.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P. Macromolecules 1999, 32, 3495. (18) Potemkin, I. I.; Kramarenko, E. Yu.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P.; Eibeck, P.; Spatz, J. P.; Mo¨ller, M. Langmuir 1999, 15, 7290. (19) Eibeck, P.; Spatz, J. P.; Potemkin, I. I.; Kramarenko, E. Yu.; Khokhlov, A. R.; Mo¨ller, M. Polym. Prepr. (Am. Chem. Soc., DiV. Polym. Chem.) 1999, 40, 990. (20) Spatz, J. P.; Eibeck, P.; Mo¨ssmer, S.; Mo¨ller, M.; Kramarenko, E. Yu.; Khalatur, P. G.; Potemkin, I. I.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P. Macromolecules 2000, 33, 150. (21) Potemkin, I. I.; Mo¨ller, M. Macromolecules 2005, 38, 2999. (22) Perelstein, O. E.; Ivanov, V. A.; Velichko, Yu. S.; Khalatur, P. G.; Khokhlov, A. R.; Potemkin, I. I. Macromol. Rapid Commun. 2007, 28, 977. (23) Zhu, Y.; Gido, S. P.; Moshakou, M.; Iatrou, H.; Hadjichristidis, N.; Park, S.; Chang, T. Macromolecules 2003, 36, 5719. (24) Erhardt, R.; Zhang, M.; Bo¨ker, A.; Zettl, H.; Abetz, C.; Frederik, P.; Krausch, G.; Abetz, V.; Mu¨ller, A. H. E. J. Am. Chem. Soc. 2003, 125, 3260.
10.1021/la070035d CCC: $37.00 © 2007 American Chemical Society Published on Web 06/20/2007
Spontaneous CurVature of Double Comblike Copolymers
polymers with side chains of types A and B30-33 are examples of macromolecules revealing intra- and intermolecular selforganization. Intramolecular segregation of A and B species in such macromolecules makes them structural units of peculiar form, such as octopuslike (januslike stars in a selective solvent), coiled worms (januslike combs in a selective solvent), and so forth. Depending on the form of the structural units, the intermolecular aggregation can be either promoted or suppressed. In the present article, we analyze the intramolecular selforganization of comblike copolymers with side chains of types A and B strongly adsorbed on a flat surface. The intermolecular aggregation of such macromolecules on the surface will be reported in a forthcoming publication. Densely grafted comblike (brush) macromolecules are known to possess a large amount of stiffness induced by the interactions of the side chains.34-39 At first glance, such macromolecules could be used to make regular 1D-surface nanostructures. However, the breaking of local cylindrical symmetry of the molecule because of strong adsorption of the side chains on the surfaceleadstothespontaneouscurvatureofthemacromolecules.40-44 This curvature is a result of the locally uneven distribution of strongly adsorbed (2D) side chains with respect to the backbone. Such a distribution is thermodynamically stable and corresponds to the optimum stretching of the side chains. Despite a high bending modulus, equilibrium conformations of strongly adsorbed (2D) comblike macromolecules are curved.45 In the case of the strong adsorption of combs with side chains of different chemical structure of types A and B (double combs), one can expect a competition between (i) elastically favorable mixing of A and B side chains and (ii) energetically favorable segregation of them. Depending on the strength of the AB interactions, one can expect stability of different conformations of the brush. For example, two kinds of conformations, horseshoeand meanderlike, were observed in the case of adsorption of the brush with PVP and PMMA side chains.30 In the present article, we analyze the equilibrium curvature of strongly adsorbed (2D) double combs. It is expected that side chains of types A and B are able to change their positions with respect to the backbone. The mechanism responsible for the curvature is discussed. (25) Erhardt, R.; Bo¨ker, A.; Zettl, H.; Kaya, H.; Pyckhout-Hintzen, W.; Krausch, G.; Abetz, V.; Mu¨ller, A. H. E. Macromolecules 2001, 34, 1069. (26) Xu, H.; Erhardt, R.; Abetz, V.; Mu¨ller, A. H. E.; Goedel, W. A. Langmuir 2001, 17, 6787. (27) Zhang, M.; Mu¨ller, A. H. E. J. Polym. Sci., Part A: Polym. Chem. 2005, 43, 3461. (28) de Jong, J.; ten Brinke, G. Macromol. Theory Simul. 2004, 13, 318. (29) Stepanyan, R.; Subbotin, A.; ten Brinke, G. Macromolecules 2002, 35, 5640. (30) Stephan, T.; Muth, S.; Schmidt, M. Macromolecules 2002, 35, 9857. (31) Uhrig, D.; Mays, J. W. Macromolecules 2002, 35, 7182. (32) Snyder, J. F.; Hutchison, J. C.; Ratner, M. A.; Shriver, D. F. Chem. Mater. 2003, 15, 4223. (33) Palyulin, V. V.; Potemkin, I. I. Polym. Sci., Ser. A 2007, 49, 473. (34) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A. R.; Yurasova, T. A. Polym. Sci. U.S.S.R. 1987, 29, 1293. (35) Fredrickson, G. H. Macromolecules 1993, 26, 2825. (36) Rouault, Y.; Borisov, O. V. Macromolecules 1996, 29, 2605. (37) Saariaho, M.; Subbotin, A.; Szleifer, I.; Ikkala, O.; ten Brinke, G. Macromolecules 1999, 32, 4439. (38) Saariaho, M.; Ikkala, O.; ten Brinke, G. J. Chem. Phys. 1999, 110, 1180. (39) Potemkin, I. I. Macromolecules 2006, 39, 7178. (40) Potemkin, I. I.; Khokhlov, A. R.; Prokhorova, S.; Sheiko, S. S.; Mo¨ller, M.; Beers, K. L.; Matyjaszewski, K. Macromolecules 2004, 37, 3918. (41) Sheiko, S. S.; da Silva, M.; Shirvaniants, D. G.; Rodrigues, C. A.; Beers, K.; Matyjaszewski, K.; Potemkin, I. I.; Mo¨ller, M. Polym. Prepr. (Am. Chem. Soc., DiV. Polym. Chem.) 2003, 44, 544. (42) Potemkin, I. I. Eur. Phys. J. E 2003, 12, 207. (43) de Jong, J. R.; Subbotin, A.; ten Brinke, G. Macromolecules 2005, 38, 6718. (44) Subbotin, A.; de Jong, J.; ten Brinke, G. Eur. Phys. J. E 2006, 20, 99. (45) Potemkin, I. I. Macromolecules 2007, 40, 1238.
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Figure 1. Schematic representation of a small fragment of a 2D brush with complete segregation of the side chains.
2. Model Let us consider a 2D model of the brush molecule with two types (A and B) of incompatible, flexible side chains. We also assume that the side chains of each type are incompatible with surrounding medium (i.e., attractive forces among A(B) monomer units prevail). We expect that attraction results in the dense packing of monomer units so that the side chains are examined as a 2D melt. It is assumed that a sequence of grafting points of A and B chains is regularly alternating. In our model, an exchange of side chains between two sides of the brush is possible. Let us denote by N, MA, and MB the number of segments in the backbone and A and B side chains, respectively, where N . MA . 1, and MB . 1. The linear size of each segment of the brush is assumed to be the same and equal to a. We study densely grafted double combs (i.e., the side chains are attached to each segment of the backbone and their number is equal to N). The fraction of side chains of A type is denoted by φ ) NA/N, where NA is the total number of side chains of type A. Notice that the applicability of the 2D model to the description of the adsorption of the copolymer is possible if spreading parameters SA and SB for side chains of both types are high enough. For example, a nongrafted chain attains the 2D conformation if its dimensionless spreading parameter is Sh ) Sa2/kBT g S* ≈ 4π2/3.21 One can expect that for the grafted chains the value of S* is a bit higher. It is known that many polymers, such as linear and grafted PBA, PVP, and PEO, practically form a monolayer on mica and can be modeled as 2D ones. 2.1. Completely Segregated Side Chains. In the case of incompatibility of A and B side chains with each other, their complete segregation with respect to the backbone is energetically favorable (Figure 1). If the lengths of the A and B chains are different (or if their fractions are different), then the brush has a curvature. The total free energy of the curved brush per unit of its length comprises the elastic free energy of the side chains and the backbone,40 the mixing entropy of the side chains, and the energy of interaction with the surrounding medium:
f0 ) fconv + fconc + fbb + fmix + fint el el
(1)
Let us assume that chains of type A form the convex side of the brush. Using the approach in ref 40, one can find
fconv ) el
φ a2
∫RR
A
E(r) dr
(2)
where kBT ≡ 1 and E(r) is the local stretching of the side chain that depends on radial coordinate r. This function is defined as
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a derivative of r over the contour length of the chain s, E(r) ) dr/ds. RA and R in eq 2 are the coordinates of the free ends of the side chains and the radius of curvature of the backbone, respectively. To calculate E(r), let us use a differential form for the dense packing condition of the monomer units in a circular sector of the angle, dφ (Figure 1):
r dr dφ ) a ds dNA
(3)
Taking into account that φ ) dNAa/(R dφ), we derive
E(r) )
dr φR ) ds r
(4)
Substitution of eq 4 into eq 2 gives
) fconv el
(
)
2aMAφ φ2R RA φ2R ) ln ln 1 + R R a2 2a2
(5)
Figure 2. Energetic curvature z as a function of the x ) MA/MB parameter at different values of the fraction of side chains of type A, φ: (a) 0.3, (b) 0.5, (c) 0.7, and (d) 0.8.
where RA is expressed through R via integration of eq 3 over r, R2A - R2 ) 2RaMAφ. It is convenient to rewrite eq 5 in the form
) fconv el
Mφ2 ln(1 + 2zxφ) a(x + 1)z
(6)
where we introduce the average number of segments in the side chain, M ) (MA + MB)/2, the ratio x ) MA/MB g 1, and the dimensionless curvature z ) 2aM/(x + 1)R. The expression for the elastic free energy of the side chains forming the concave side of the brush, f conc el , can be derived in a similar way:
)fconc el
M(1 - φ)2 ln(1 - 2z(1 - φ)) a(x + 1)z
(7)
Figure 3. Schematic picture of a small fragment of the 2D brush with partial mixing of the side chains.
The last three terms of eq 1 can be neglected in comparison with the elastic free energy of the side chains. The small values of fbb and fmix are demonstrated in ref 40. Let us show that the conformation-dependent part of fint is also small. This term has two contributions coming from (i) the interactions of the monolayer with the substrate and the air and from (ii) the line tension. The first contribution is constant. (The total area of the side chains forming the 2D melt is constant.) We denote by γA and γB the line (surface) tension coefficients of chains A and B, respectively. Then fint takes the form (Figure 1) of
2.2. Partial Mixing of the Side Chains. Relying on the existence of the spontaneous curvature of 2D brushes with homopolymer side chains,40-44 one can expect that in the case of double combs complete segregation of the side chains is not the only state corresponding to the minimum in the free energy and that partial mixing of the side chains can also be favorable. Thus, let us assume that some fraction of shorter (B) side chains can coexist with the longer ones on the convex side of the brush (Figure 3). Our previous calculations showed that such coexistence on the concave side is unfavorable, and this situation is excluded from the analysis. The main contributions to the total free energy of the brush, f, come from the elastic free energy of the side chains and from the energy of interactions between A and B side chains:
fint )
γARA + γBRB + const ) γAx1 + 2zφx + R γBx1 - 2z(1 - φ) + const (8)
Relying on the estimates for the line tension coefficients, γAa ≈ γBa e 1, one can see that f conc and f conv are M times larger el el than the line tension contribution of fint. The equilibrium curvature of the brush segment with completely segregated side chains is found by the minimization of f0 ≈ f conv + f conc over z. Taking into account that complete el el segregation is driven by the repulsion of A and B side chains, we call this kind of the curvature “energetic”. Figure 2 shows how the energetic curvature depends on the x parameter at different values of fraction φ. The curvature is defined as positive if side chains of type A form the convex side of the brush and vice versa. For all values of fraction φ, curvature z is an increasing function of x (i.e., the higher the asymmetry of chain lengths MA and MB, the larger the curvature). The curvature becomes negative if the fraction of shorter (B) chains, 1 - φ, is high enough.
f)
M(φ + )2 ln(1 + 2z(φ + )) + a(x + 1)z 1 + 2z(xφ + ) M(1 - φ - )2 Mφ2 ln × a(x + 1)z 1 + 2z(φ + ) a(x + 1)z 4MγAB (x1 + 2z(φ + ) - 1) ln(1 - 2z(1 - φ - )) + (x + 1)z (9)
The first two terms of eq 9 correspond to the elastic free energy of the side chains on the convex side. The side is divided in two layers with widths of R′B - R and RA - R′B (Figure 3). It is assumed that in the first layer the B chains and MB segments of the A chains have equal stretching. Thus, the total elastic free energy of this layer (the first term of eq 9) is equivalent to that
Spontaneous CurVature of Double Comblike Copolymers
Figure 4. Interfaces between A and B chains approximated by a contour of the circular segment.
of monodisperse chains each having MB segments and whose fraction is equal to φ + . The elastic free energy of the second layer formed by the A chain fragments (each fragment contains MA - MB segments, and the fraction of the fragments is φ) is contained in the second term of eq 9. The first two terms of eq 9 reproduce eq 6 by setting ) 0. The third term of eq 9 is the elastic free energy of the side chains forming the concave side of the brush. It is calculated by eq 7 where 1 - φ is substituted by 1 - φ - . Finally, the last term of eq 9 is the main contribution to the line tension of the AB interfaces. We assume that the interpenetration of the A and B chains is unfavorable because of their high stretching. Therefore, the interfaces have a welldefined shape that is approximated by a contour of the circular segment (Figure 4). The interfacial energy per chain takes the form of γAB(2(R′B - R) + R′Bdθ) ≈ 2γAB(R′B - R) (Figures 3 and 4), and the energy of all interfaces per unit length of the brush is given by the fourth term of eq 9. Here, γAB is the line tension coefficient of the AB interface. The equilibrium value of the free energy f is found by the minimization of eq 9 over and z. Taking into account that in the case of partial mixing of the A and B side chains the spontaneous curvature is promoted by the elasticity of the side chains, let us call this kind of the curvature “entropic”.
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Figure 5. Difference in free energies ∆f as a function of curvature z at different values of the surface tension coefficient, γ ) γABa: (a) 0.25, (b) 0.2, (c) 0.18, (d) 0.17, and (e) 0.15. The ratio is x ) MA/MB ) 7/5, and the fraction of side chains of type A is φ ) 0.6. The dotted line corresponds to the maximal possible curvature.
Figure 6. Radius of the entropic curvature having a minimal possible value of RA - R, which is determined from the condition of the contact of the side chains forming the convex side of the brush.
3. Results and Discussion To find conditions of stability for the energetic and the entropic curvatures, we compare the free energies of the corresponding states of the brush. Let us introduce dimensionless function ∆f, which is the difference in the dimensionless free energies ∆f ) hf(z) - hf0. Here, hf(z) is found by the minimization of eq 9 over , hf(z) ) min{af/M}, and hf0 is the equilibrium value of f0, which is hf0 ) minz{af0/M}. Characteristic behavior of ∆f is shown in Figure 5. This function is defined in the limited range of z values. The maximal possible value zmax (dotted line in Figure 5) is related to the excluded volume of the brush and is determined from the condition of the contact of the side chains forming the convex side of the brush (Figure 6). At relatively high values of γ, the minimum of ∆f is equal to zero (Figure 5a). This means that f and f0 coincide (i.e., the A and B side chains are completely segregated ( ) 0) and the energetic curvature is the only possible structure of the brush). With the decrease in γ, the minimum of ∆f gradually becomes negative (Figure 5b,c). In this case, partial mixing of the side chains driven by a gain in their elastic free energy is possible. Taking into account that the minimum in this regime corresponds to z < zmax, one can conclude that the AB interactions stabilize the curvature of the brush. At relatively
Figure 7. Optimum curvature of the brush z as a function of dimensionless line tension coefficient γ ) γABa. x ) 7/5 and φ ) 0.6.
low values of γ (Figure 5d,e), the minimum in ∆f is attained at zmax (i.e., the entropic curvature is stabilized by the excluded volume of the brush fragments). The curvature of the brush as a function of the dimensionless line (surface) tension coefficient γ ) γABa is shown in Figure 7. Here, curve a corresponds to the entropic curvature stabilized by the excluded volume interactions. It slightly increases with γ. Such behavior can be explained by the smaller radial stretching
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(larger lateral size) of the B chains on the convex side of the brush because of the tendency to reduce the AB line tension. The decreasing branch of curve b is the entropic curvature stabilized by AB interfacial interactions whereas the constant part of curve b corresponds to the energetic curvature. The higher value of the entropic curvature in comparison with the energetic curvature is related to the higher asymmetry in the left-right distribution of the side chains. Our prediction of the existence of two curvatures can be useful for explaning two kinds of conformations (horseshoe- and meanderlike) observed in the case of brushes having PVP and PMMA side chains.30 It was shown that quaternized PVP-coPMMA-QEB50 spin-cast from chloroform solution onto mica revealed a horseshoelike conformation whereas macromolecules of higher quaternization, PVP-co-PMMA-QEB70, spin-cast from aqueous solution onto mica had a meanderlike conformation. Quaternization in chloroform does not give a pronounced electrolyte effect (dissociation of counterions) because of the low dielectric constant of chloroform (around 5). Thus, one can expect that the complete segregation of PVP and PMMA side chains is realized (horseshoelike conformation). However, quaternization in water is accompanied by a strong polyelectrolyte effect, and the PVP side chains accrue a high positive charge. In this case, complete segregation is energetically unfavorable because the location of all similarly charged PVP chains on one side would give a high increase in the electrostatic energy. Electrostatic repulsion between PVP chains leads to their partial mixing with PMMA chains. In terms of our model, where longrange interactions are not taken into account, switching on Coulomb repulsions between the PVP chains can be considered
PopoV and Potemkin
to be a reduction of the interaction parameter between PVP and PMMA chains. Partial mixing reduces the radius of curvature in comparison with the radius under complete segregation, and a meanderlike conformation similar to that of Figure 6 is formed.
4. Conclusions We propose a theory of strongly adsorbed (2D) comblike macromolecules (brushes) with side chains of different chemical structure of types A and B (double combs). It is expected that the side chains of both types are able to change their positions with respect to the backbone. Depending on the strength of the AB interactions, we predict two mechanisms of spontaneous curvature of the brush. In the case of the strong incompatibility of A and B side chains, their complete segregation is responsible for the so-called energetic curvature. The radius of this curvature is determined by the ratio of the chains lengths and by the relative fractions of the A and B side chains. If the incompatibility is less pronounced, then partial mixing of the A and B chains on the convex side of the brush can arise. Such mixing is favorable because of the decrease in stretching of the side chains, and the curvature, which is called entropic, is characterized by a lower radius in comparison with that of the energetic curvature. This radius is determined by the length of the longest side chains. Acknowledgment. The financial support of the Deutsche Forschungsgemeinschaft within SFB 569, the VolkswagenStiftung (Germany), and the Russian Foundation for Basic Research is gratefully acknowledged. LA070035D
