JChemPhys_134_054906

THE JOURNAL OF CHEMICAL PHYSICS 134, 054906 (2011) Mesophase formation in two-component cylindrical bottlebrush polymers Igor Erukhimovich,1,a) Panagiotis E. Theodorakis,2 Wolfgang Paul,3 and Kurt Binder2 1A.N. Nesmeyanov Institute of Organoelement Compound, RAS and Moscow State University, Moscow 119992, Russia 2Institut für Physik, Johannes Gutenberg-Universität Staudinger Weg 7, 55099 Mainz, Germany 3Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, von Seckendorff-Platz 1, 06120 Halle, Germany (Received 3 November 2010; accepted 21 December 2010; published online 7 February 2011) When two types of side chains (A,B) are densely grafted to a (stiff) backbone and the resulting bottlebrush polymer is in a solution under poor solvent conditions, an incompatibility between A and B leads to microphase separation in the resulting cylindrical brush. The possible types of ordering are reminiscent of the ordering of block copolymers in cylindrical confinement. Starting from this analogy, Leibler’s theory of microphase separation in block copolymer melts is generalized to derive a description of the system in the weak segregation limit. Also molecular dynamics simulation results of a corresponding coarse-grained bead-spring model are presented. Using side chain lengths up to N = 50 effective monomers, the ratio of the Lennard-Jones energy parameter between unlike monomers (�AB) and monomers of the same kind (�AA = �B B) is varied. Various correlation functions are analyzed to study the conditions when (local) Janus-cylinder-type ordering and when (local) microphase separation in the direction along the cylinder axis occurs. Both the analytical theory and the simulations give evidence for short-range order due to a tendency toward microphase separation in the axial direction, with a wavelength proportional to the side chain gyration radius, irrespective of temperature and grafting density, for a wide range of these parameters. © 2011 American Institute of Physics. [doi:10.1063/1.3537978] I. INTRODUCTION Enabled by progress in chemical synthesis (see Refs. 1 and 2 for reviews), macromolecules with “bottlebrush” ar- chitecture, where flexible side chains are densely grafted to a backbone, have found much recent interest (e.g., Refs. 1– 13). Such molecules may be useful for various applications (such as sensors, actuators, building blocks in supramolecu- lar structures 7–9), since these systems are stimuli-responsive polymers, exhibiting large conformational changes when ex- ternal conditions vary. Apart from these synthetic bottlebrush polymers, also biopolymers with a related architecture are abundant in nature, e.g., proteoglycans.14 These brushlike polymers contain a protein backbone with carbohydrate side chains, and are held responsible for a large variety of biolog- ical functions (cell signaling, cell surface protection, joint lu- brication, etc.15–17). At the same time, the interplay between entropic effects and various enthalpic forces in these “soft” objects makes the understanding of structure–property rela- tionship of such bottlebrush polymers a challenging problem of statistical thermodynamics.18–55 In the present paper, we focus on the interplay between chain conformations and local order in binary A,B bottlebrush polymers, from the point of view of both theoretical argu- ments and molecular dynamics (MD) simulations. Previous work39, 41 has focused on the possibility of microphase sep- aration in the form of “Janus cylinders,”8 i.e., the cylinder splits, as shown in Fig. 1 in two halves (in the most sym- metric case, where the lengths NA, NB of both types of side a)Electronic mail: ierukhs@polly.phys.msu.ru. chains and their grafting densities σA, σB are equal), such that the A–B interface contains the cylinder axis (taken to be along the z-axis henceforth). Then it was pointed out46 that due to the quasi-one-dimensional character of this ordering, for finite cylinder radius R, no true long-range order of Janus cylinder type should be expected: rather there exists a finite correlation length over which the orientation of the interface plane decorrelates. In addition, it was speculated47 that also the ratio �AB/� between the interaction strength �AB of unlike monomers and monomers of the same kind (�AA = �B B = �) matters: if the formation of A–B interfaces is energetically much more unfavorable rather than the formation of polymer– solvent interfaces, a phase separation along the axial (z) direc- tion of the cylinder could occur also in the form of a dou- ble cylinder (the A-rich polymers form a separate cylinder from the B-rich ones, both cylinders touch each other along the z-axis).47 Also intermediate cases (“Janus dumbbell” mor- phologies shown in Fig. 1 having crosslike sections of the bottlebrush) were suggested47 and some indications of such structures were observed.55 However, again only finite corre- lation lengths of such orderings along the z-axis can be ex- pected. In addition, one needs to consider that for not very large grafting densities the bottlebrush polymer under poor solvent conditions could form inhomogeneities in the form of a “pearl-necklace” structure43, 51, 56 instead of the homoge- neous (along the direction of the backbone) Janus cylinder (for simplicity we restrict attention to rigid backbones only). Note that in terms of the weak segregation theory we present further such a pearl-necklace structure could be considered as a lamellar-like one (see Fig. 1). 0021-9606/2011/134(5)/054906/22/$30.00 © 2011 American Institute of Physics134, 054906-1 Downloaded 05 May 2011 to 202.113.231.136. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054906-2 Erukhimovich et al. J. Chem. Phys. 134, 054906 (2011) FIG. 1. Janus cylinder (left), Janus dumbbell (middle), and lamellarlike (right) morphologies. The red and blue domains are filled by the A and B monomers. The problem which we investigate in the following sec- tions is whether it can be favorable to form a microphase- separated structure that is inhomogeneous along the z-axis. Following the information from experiment that one side chain can be grafted per backbone monomer,10 we assume an alternating grafting of the two types (A and B) of side chains.55 Of course, this situation is essentially equivalent to the case where symmetrical block copolymers57–61 would be anchored with their A–B junction to the backbone. It is then also interesting to ask how such a situation compares to the case where such block copolymers fill a cylinder (at the same density) without constraint on the locations of the A–B junc- tions. Of course, the latter problem has already been studied in other contexts.62–76 Here we generalize the treatment of Ref. 75 to take this constraint on the location of the A–B junctions into account to formulate a weak segregation theory77, 78 of or- dering in binary bottlebrushes (Sec. II). Then we present data from molecular dynamics simulations, extending the study of Ref. 55 by considering now the variation of properties with �AB/� (Sec. III) and discuss their results in Sec. IV. We sum- marize the main findings of our work in the concluding sec- tion (Sec. V). II. WEAK SEGREGATION THEORY OF MICROPHASE SEPARATION IN TWO-COMPONENT BOTTLEBRUSHES A. The condensed state of the two-component bottlebrush The general idea of our theoretical approach is as fol- lows. We consider the case when incompatibility of both sorts of blocks with the solvent is high enough to form a con- densed state (globule), which in the case of a rigid backbone would acquire the form of a cylinder uniformly filled by the monomers of the side chains. The height H of the cylinder would be just the length of the rigid backbone and the radius R would be related to the equilibrium monomer density ρ0 inside the cylinder. Such a condensed state could be consid- ered as an equivalent of a bottlebrush compressed up to the monomer density ρ0 via squeezing it into a tube of the radius R. To consider the composition fluctuations in such a con- densed (squeezed) bottlebrush we employ some new ideas, which were put forward in Refs. 71 and 75 to generalize the seminal Leibler theory77 (see also Refs. 57–61) to confined polymer systems. To begin with, we find, based on the theory of polymer globules by Lifshitz et al.,79 the value of ρ0, which is to provide the minimum of the two-component bottlebrush free energy given the values of the χ -parameters χi j (a sim- ilar approach has been employed by Sheiko et al.43 for one- component bottlebrushes in a poor solvent). To determine the equilibrium value of a two-component bottlebrush forming a condensed state due to its overall incompatibility with the solvent we notice that, according to Lifshitz et al.,79 the free energy of any flexible polymer system is (in a zeroth approx- imation) just the sum of a structural (entropic) contribution, which allows for the effect of connectivity (“linear memory” in Lifshitz terms) and an energetic contribution, which allows for the van der Waals attraction and the excluded volume re- pulsion effects: F = Fstr({ρi (r)}) + F∗({ρi (r)}). (1) The precise form of the structural free energy Fstr({ρi (r)}) as a functional of the spatially nonuniform dis- tributions {ρi (r)} of the local number densities of the repeat units of the i th sort (i = A, B) is determined by the actual microscopic structure of the bottlebrush. We analyze it some- what later when studying the bottlebrush stability with respect to its longitudinal, angular, or helical ordering. But now, when we focus on the basically uniform cylindrical globule, the structural term could be simply disregarded as compared to the energetic contribution to the free energy: F = F∗({ρi (r)}) = T V f ∗({ρi }). (2) Here V = π R2 H = 2Mπ R2a is the volume of the cylindri- cal globule (where 2M and a are the total number of both A and B side chains and the distance between the neighboring A and B chains), ρi is the value of the local number density of the repeat units of the i th sort, averaged over the whole vol- ume V of the bottlebrush, and the specific (per unit volume) free energy f ∗({ρi }) depends on the type of the interactions between the solvent molecules and repeat units (monomers) forming the bottlebrush. For simplicity, we assume in this pa- per that these interactions correspond to the conventional in- compressible Flory–Huggins lattice model, in which case v0 f ∗ = (1 − φ) ln(1 − φ) + φ + χABφAφB + χASφAφS +χBSφBφS, (3) Downloaded 05 May 2011 to 202.113.231.136. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054906-3 Mesophases in two-component bottlebrushes J. Chem. Phys. 134, 054906 (2011) where v0 is the excluded volume assumed, for simplic- ity, to be the same for both sorts of the repeat units and solvent molecules, φi (r) = v0ρi (r) is the local volume frac- tion of the particles of the i th sort within the globular bot- tlebrush, φ = φA + φB is the total polymer volume fraction (volume fraction of all monomers), φS = 1 − φ is the volume fraction of the solvent molecules, and χi j are the conventional Flory–Huggins energetic parameters. (The linear addendum φ is irrelevant for phase equilibrium; we introduced it to elim- inate the linear term in the expansion of the function f ∗ in powers of φ.) For the homogeneous bottlebrush we are considering here, φi (r) = φi = fiφ = v0ρi , where fi is the fraction of the monomers of the i th sort, and, thus, the function f ∗({ρi }) can be rewritten in the form v0 f ∗({ρi }) = v0 f ∗(φ) = (1−φ) ln(1−φ) + φ−χ˜ φ2 + χφ, (4) where χ˜ = χAS f A + χBS fB − χAB f A fB is the effective χ - parameter describing the overall bottlebrush-solvent incom- patibility and the parameter χ = χAS f A + χBS fB does not af- fect the equilibrium value of φ. Indeed, taking into account that the polymer volume frac- tion within the uniform bottlebrush is φ = v0 N/(2π R2a), (5) where N = NA + NB is the total average number of all monomers A and B per distance 2a of the backbone, one can reduce the total free energy (2) to the form F = T M N 1 φ [(1 − φ) ln(1 − φ) + φ − χ˜ φ2 + χ φ]. (6) Minimization of the free energy (6) with respect to φ results, finally, in the desired equation (it corresponds to the so-called volume approximation of the polymer globule theory79) for the equilibrium polymer volume fraction of the two-component bottlebrush: v0 p∗/T = v0 p/T − φ = − ln(1 − φ) − φ−χ˜φ2 = 0, (7) where p∗(φ) is the pressure of the semidilute polymer mixture with infinitely long polymer chains given the composition f A and the total polymer volume fraction φ. Obviously, the finite equilibrium polymer volume frac- tion of the two-component bottlebrush φ0, which is the so- lution of Eq. (7), depends on the value of χ˜ only and has a physically meaningful positive value only for τ = χ˜ − 1/2 > 0. (8) The corresponding asymptotics read{ φ = 3τ − (27/4)τ 2 + . . . , τ � 1 φ = 1 − exp(1 − χ˜ ), χ˜ � 1. (9) It is worth noticing that in so-called Hildebrand approximation,80 in which the value of the χ -parameter for the i th and j th species is proportional to the squared differ- ence of their solubility parameters: χi j = v0(δi − δ j )2/(2T ), the effective χ -parameter is always positive: χ˜ = (2 f − 1 + x) 2 4 χAB, where x = (2δS − δA − δB)/(δB − δA) is the selectivity parameter.81 If the A and B monomers are compatible (δA = δB) then χ˜ = χAS = χAB . To conclude this subsection we discuss the conditions of validity of our consideration. First, it is worth noticing that the bottlebrushes under consideration are quasi-one-dimensional systems. Therefore, their equilibrium volume fraction stays finite even in the -solvent, where R2 ∼ Nl2 (here l is the statistical segment length supposed, for simplicity, to be the same for both sorts of the side chains). Indeed, it follows from Eq. (5) that in this case φ ∼ Li/(π a˜) , (10) where we introduced the reduced distance a˜ = a/ l between the neighboring side chains and the Lifshitz number Li = v0/ l3, which characterizes the chains’ flexibility. Compar- ing Eq. (10) and the first of Eq. (9) and taking into account that the condensed bottlebrush should be much denser than that in -solvent we conclude that the first of the validity conditions reads κ = φ /φ = R2/(Nl2) ≈ Li/(3π a˜τ ) � 1 . (11) On the other hand, the volume approximation (7) is known79 to be valid when R � rc, (12) where rc is the correlation radius within the globule. For dense globules (1 − φ � 1 or τ � 1) the condition holds always but for semidilute globule (β/3 � τ � 1) the correlation ra- dius is known79, 82 to increase with decrease of τ : rc ∼ l/τ . (13) It follows from Eqs. (5), (9), and (13) that close to the -temperature the condition (11) is satisfied only when the side chains are long enough: N � 1/(φ τ ). (14) Condition (14) is rather severe since both quantities φ and τ are small. Thus, the fact that the monomer density profile usually observed in computer simulation of bottle- brushes in poor solvents close to the -temperature is not steplike, which is expected for a condensed (globular) state, but rather smooth51, 55 is explained by insufficiently high de- grees of polymerization of the side chains, which do not sat- isfy condition (14). B. The random phase approximation for confined polymer systems 1. General theory To consider the fluctuations within the two-component condensed bottlebrush we are to expand the total free energy (1) in powers of the density fluctuations counted from the Downloaded 05 May 2011 to 202.113.231.136. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054906-4 Erukhimovich et al. J. Chem. Phys. 134, 054906 (2011) uniform globular state i (r) = ρi (r) − ρi : F = F0 + �F2 + �F3 + �F4 + . . . , (15) where F0 is defined by Eqs. (2) and (6), the quadratic term of the expansion reads �F2 = T2 ∫ �i j (r1, r2) i (r1) j (r2) dV1 dV2 (16) [in Eq. (16) and thereafter we employ the rule of summation over the repeating indices] and the next terms of the expansion (15) are defined similarly.77 Our purpose in this subsection is to find the explicit form of the kernel �(r1, r2) and conditions ensuring that the quadratic form �F2 is positive definite, which implies that the fluctuations are small. It follows from Eqs. (1)–(3) and (16) that �αβ(r1, r2) = T −1 δ2 F({ρi (r)}) δρα(r1) δρβ(r2) = γαβ(r1, r2) − cαβ(r1, r2). (17) Here we introduced new designations γαβ(r1, r2) = T −1 δ2 Fstr({ρi (r)}) δρα(r1) δρβ(r2) , (18) cαβ (r1, r2) = −T −1 δ2 F∗({ρi (r)}) δρα(r1) δρβ(r2) = δ(r1 − r2) Cαβ, (19) where the number matrix C = ‖Cαβ‖ reads C = ∣∣∣∣T −1 ∂2 f ∗∂ρα ∂ρβ ∣∣∣∣ = v0 ( 2χAS − φ−1S k − φ−1S k − φ−1S 2χBS − φ−1S ) (20) and k = χAS + χBS − χAB . Thus, our problem is reduced to calculating the second derivative (18) of the structural free energy functional. For this purpose we consider an auxiliary thermodynamic potential �str({ϕi (r), T }) = −T ln Z ({ϕi (r), T }), (21) which has meaning of the free energy of an ideal polymer system with a specified architecture (in our case it is the two- component bottlebrush) affected by a set ϕi (r) of external fields applied to the particles of the i th sort. The thermody- namic potential �str({ϕi (r), T }) could be readily calculated (see below). On the other hand, it is directly related to the desired thermodynamic potential Fstr(ρi (r)). Indeed, the par- tition function Z appearing in Eq. (21) is the integral over all possible nonuniform density distributions ρi (r) of the parti- cles of the i th sort in the volume of the system: Z = − ∫ ∏ i δρi (r) exp(− F˜str({ρi (r)}, {ϕi (r)})/T ) (22) with F˜str({ρi (r)}, {ϕi (r)}) = Fstr{ρi (r)} + ∫ ρi (r)ϕi (r) dV . Definition and calculation of the density integral Z is, gener- ally, rather cumbersome but it becomes trivial in the saddle- point approximation: �str({ϕi (r)}) = min{ρi (r)} F˜str({ρi (r)}, {ϕi (r)}). (23) It follows from (23) that if the thermodynamic poten- tial Fstr({ρi (r)}) is known then the thermodynamic potential �str({ϕi (r)}) is parametrically defined as follows: ϕα(r) = − δFstr({ρi (r)})/δρα(r), (24) �str({ϕi (r)}) = Fstr({ρi (r)}) − ∫ ρi (r) δFstr({ρi (r)}) δρα(r) dV . (25) Equations (24) and (25) imply that the thermodynamic poten- tial �, which is a functional of all external fields ϕi (r) applied to the particles of the system, is the Legendre transform of the thermodynamic potential F , which is a functional of all num- ber densities ρi (r) of these particles. Therefore, the reciprocal relationships hold ρα(r) = δ�str({ϕi (r)})/δϕα(r), (26) Fstr({ρi (r)}) = �str({ϕi (r)}) − ∫ ϕi (r) δ�({ϕi (r)}) δϕα(r) dV . (27) It follows from Eqs. (16) and (24)–(27) that γαβ(r1, r2)= δ 2 Fstr({ρi (r)}) δρα(r1) δρβ(r2) ∣∣∣∣ ρi (r)=const = −δϕα(r1) δρβ(r2) ∣∣∣∣ ρi (r)=const . (28) Furthermore, if we define the structural matrix gαβ(r1, r2) = − δ 2�str({ϕi (r)}) δϕα(r1) δϕβ(r2) ∣∣∣∣ ϕ(r)=0 = − δρα(r1) δϕβ((r2)) ∣∣∣∣ ϕ(r)=0 , (29) we obtain∫ γαβ(r1, r) gβγ (r, r2)dr = δαγ δ(r1 − r2) . (30) 2. Calculation of the structural matrix for copolymers in bulk The derivation of the explicit expression for the struc- tural matrix g of copolymer systems under confinement is a straightforward extension of that in bulk. So, we remind the reader how the structural matrix is derived in bulk. Instead of calculating the density integral (22) we con- sider an equivalent expression for the partition function Z ({ϕi (r)}) of an ideal polydisperse n-component system of linear macromolecules affected by the external fields {ϕi (r)} applied to the monomers of the i th sort. The partition function Z ({ϕi (r)}) can be written directly in terms of the correspond- ing discrete microscopic model: Z ({ϕi (r)}) = ∏ S Z MSS ({