Ideal Mixing in Multicomponent Brushes of Branched Polymers

Article pubs.acs.org/Macromolecules

Ideal Mixing in Multicomponent Brushes of Branched Polymers Ekaterina B. Zhulina,†,‡ Frans A. M. Leermakers,§ and Oleg V. Borisov*,†,‡,∥ †

Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg 199004, Russia St. Petersburg National University of Informational Technologies, Mechanics and Optics, St. Petersburg 197101, Russia § Laboratory of Physical Chemistry and Colloid Science, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ∥ Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, CNRS, Université de Pau et des Pays de l’Adour UMR 5254, 64053 Pau, France ‡

ABSTRACT: We predict theoretically that in contrast to composite brushes of end-tethered linear polymers that exhibit vertical stratification, chemically identical branched macromolecules with different molecular weights and selected architectures can distribute their free ends all over the volume of composite brush and produce a unified polymer density profile. As a result of this, we expect the presence of terminal end-groups of all constituent macromolecular species in the vicinity of the external boundary of such composite brush. The predictions of the analytical theory are supported by numerical selfconsistent field calculations. Unified brushes offer new possibilities in the design of polymer-decorated surfaces and might improve our understanding of natural biointerfaces.

1. INTRODUCTION Chemically identical but diverse in molecular mass or topology macromolecules mix ideally in the solution. The effects of polydispersity on experimentally measurable solution properties of linear or branched macromolecules are well understood on the basis of existing theories.1,2 The situation is spectacularly different when macromolecules of various architectures or molecular masses are tethered to a surface giving rise to a mixed polymer brush. Modification of physical/chemical or biointeractive properties of surfaces by ultrathin (monomacromolecular) layers of polymers chemically attached to interfaces is widely exploited in advanced technological3,4 and biomedical applications.5 The requirement of robustness in fabrication of such polymeric coatings almost inevitably leads to polydispersity and architectural heterogeneity of the brush-forming macromolecules. Therefore, understanding of the effects of polydispersity/ heterogeneity of constituent macromolecules on the properties of manufactured brushes is of key importance. Binary brushes formed by linear polymers with significantly different molecular masses exhibit a pronounced vertical segregation (stratification) though remain laterally homogeneous.6,7 More specifically, shorter chains distribute all their segments (including the end-points) in the sublayer proximal to the surface. On the contrary, the end-segments of longer chains are distributed solely in the peripheral sublayer (with no free ends in the proximal sublayer). Similar stratification occurs in the brushes formed by multiple linear chain species.7 This vertical stratification has important consequences in terms of, e.g., availability of functionalized end-segments of the chains in biomedical applications of polymer brushes. Polymer coatings designed by surface-attached branched macromolecules have recently attracted considerable attention.8 © XXXX American Chemical Society

Branching of the tethered macromolecules provides potentially high surface density of terminal functional groups9 and affects thermodynamic stability of colloidal dispersions.10 In nature, extended layers of biopolymers decorating endothelial cells11 and bacterial surfaces12−14 comprise multiple types of branched and linear polysaccharides. The latter are believed to mediate cell−cell interactions and adhesion.15 Recent theoretical study16 pointed also at potentially enhanced lubrication properties of the branched polymer brushes compared to linear ones. Therefore, comprehensive analysis of the structure and properties of brushes formed by branched polydisperse polymers with diverse architectures is interesting from both experimental and theoretical viewpoints. The existing up-to-date theoretical studies focused mostly on brushes formed by monodisperse regularly branched polymers with equal length of all spacers and branches (referred below as symmetric dendrons).16−25 The analytical theoretical model was recently extended to incorporate brushes of asymmetric starlike (Ψ-shaped) macromolecules,26 and the dendrons of higher generations.27 It has been demonstrated26 that an increase in asymmetry of Ψ-shaped macromolecule (i.e., an increase in number of monomers in a free branch compared to that in a stem) leads to redistribution of stress in the tethered chains, and to the change in scaling behavior of the brush thickness as a function of the branching parameter. It has been also predicted analytically and confirmed by the self-consistent field numerical calculations that asymmetry of Ψ-shaped macromolecules can overrule the difference in their molecular Received: August 3, 2015 Revised: October 1, 2015

A

DOI: 10.1021/acs.macromol.5b01722 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematic illustration of the structure and typical conformation of (a) regularly branched symmetric dendron with the total number of generations g = 3, functionality of all the branching points q = 3, and the same number of monomer units in each spacer n; (b) asymmetric dendron of the second generation; n0, n1, n2 are the number of monomer units in the stem and in a spacer in the first and second generations, respectively; q1 = 3 and q2 = 2 are branching functionalities in the first and in the second generations, respectively; (c) comblike macromolecule with p = 2 repeat units and q = 3 side chains emanating from each branching point, in a polymer brush (other molecules are omitted for illustration purposes). The distance between the macromolecules equals to s , where s is the area per molecule (the inverse of the grafting density σ = a2/s). Chemically the molecules are homopolymers as all segments have the same interactions with the solvent, and thermodynamically the solvent quality is good.

weights, and mixed brushes of Ψ-shaped macromolecules can exhibit a unified polymer density distribution. In this study we argue, from a theoretical perspective, that specifically tailored dendritic homopolymers with various architectures and significantly different molecular weights could form a mixed brush-like coating with a unified polymer density profile. In such a brush, the free ends of the dendrons are distributed throughout the whole coating (i.e., they mix vertically), irrespective of the content of the constituent macromolecules. The unified density profile in mixed brushes of the dendrons with selected topologies should be contrasted to a layered structure in polydisperse brushes of linear chains.7 Recently, a layered structure of a mixed brush formed by starlike polymers and linear chains was found and examined by means of the numerical self-consistent field (SF−SCF) modeling.28 We argue that the homogeneous mixing of the free ends is feasible because the selected dendrons form one-component brushes with the same self-consistent molecular potentials. Because of this similarity, the elastic response of a unified coating to compression is controlled by only two parameters: the grafting density (number of tethered dendrons per unit area) and the brush thickness in an undeformed state. Therefore, brush layers with quite different compositions and architectures of dendritically branched macromolecules could demonstrate the same mechanical properties, i.e., give rise to identical force−distance profiles, if the grafting densities and thicknesses of these brushes in unperturbed states are equal. Below we focus on chemically identical branched macromolecules with varied architectures (symmetric and asymmetric dendrons, and comblike polymers) that give rise to mixed (composite) brushes with a unified polymer density distribution. The rest of the paper is organized as follows. In section 2, we formulate an analytical self-consistent field model of a planar brush formed by the tethered dendrons with various architectures. In section 3 we consider mixed brushes comprising dendrons of different architectures, while in section 4 we analyze the limits of applicability of the theory built under the assumption of Gaussian elasticity of tethered macromolecules. In section 5 we supplement the predictions of the analytical

theory by the results of numerical SF-SCF calculations. Finally, we briefly summarize the main results and formulate the conclusions

2. MODEL AND FORMALISM Consider a planar layer formed by branched macromolecules with degree of polymerization N ≫ 1. The architecture of macromolecule specifies the relationship between its total degree of polymerization N, and the number and type of constituent spacers and free branches. We consider here two main architectures: the dendrons with number of generations g ≤ 3 (linear polymers correspond to dendrons of the zeroth, g = 0, generation), and comblike polymers with number of repeat units p ≤ 2. All the linear segments (spacers, branches) are assumed to be flexible with ratio of Kuhn segment length b to the monomer size a, b/a ≳ 1. The branched macromolecules are end-tethered to the surface by terminal monomer of the stem with dimensionless grafting density σ (the surface area per molecule s = a2/σ). The architectures of considered here branched polymers are depicted in Figure 1. 2.1. Polymer Architectures. 2.1.1. Symmetric (Regular) Dendrons. A symmetric dendron is characterized by the total number of generations g, number of spacers (or branches) q, each with degree of polymerization n ≫ 1, that emanate from each branching point of the macromolecule. For this type of macromolecules the degree of polymerization of the spacer, n, and that of the stem, n0, are equal, n0 = n. The value of g = 1 corresponds to end-tethered starlike polymers with total number of branches (q + 1) and number of free branches q, and in this case N = n(q + 1). The value of g = 2 corresponds to a regularly branched dendron with N = n(1 + q + q2) monomers while a symmetric dendron with g = 3 has degree of polymerization N = n(1 + q + q2 + q3). 2.1.2. Asymmetric Dendrons. In asymmetric dendrons branching activity q and degree of spacer polymerization n could vary within the same and/or different generations. That is, an asymmetric dendron of the first generation (g = 1) could have q free branches with degrees of polymerization ni (with i = 1, 2, ..., q). We refer to such macromolecules as Ψ-shaped B


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For low volume fraction of monomers, φ(z) ≪ 1, the free energy density f{φ(z)} can be presented as a virial expansion in polymer volume fraction,

polymers. The total degree of polymerization of Ψ-shaped macromolecule is N = ∑i=q i=0ni, where n0 is the degree of polymerization of the stem. For an asymmetric dendron with monodisperse branches (i.e., with ni ≡ n1), the total number of monomers is N = n0 + qn1 = n0(1 + qu), and the asymmetry of macromolecule is maintained when the asymmetry parameter u ≡ n1/n0 ≠ 1. An asymmetric dendron of the second generation (g = 2), could have q1 spacers with degree of polymerization n1 in the first generation, and q2 spacers with degree of polymerization n2 in the second generation. In this case, the total degree of polymerization N = n0 + q1(n1 + q2n2). In the case of equal spacers and stem lengths, n0 = n1 = n2 = n, the total number of monomers in the dendron is N = n(1 + q1 + q1q2), and the asymmetry of macromolecule is maintained due to the condition q1 ≠ q2. 2.1.3. Comblike Polymers. In the case of comblike architecture of the polymer molecule, we consider only a symmetric macromolecule that has p ≤ 2 repeat units, each composed of a backbone with n monomers and q free branches with degree of polymerization n each, emanating from the end monomer of the corresponding backbone section. The last backbone section has no free branches, and the first end-tethered backbone section serves as a stem with degree of polymerization n0 = n. For p = 1, the macromolecule constitutes a symmetric starlike polymer with number of free branches (q + 1) and total degree of polymerization N = n(q + 2). For p = 2, the total degree of polymerization of comblike polymer is N = n(2q + 3). 2.2. Self-Consistent Molecular Potential. In this study, we focus on moderate grafting densities σ that ensure a strong overlap between neighboring dendrons and a significant elongation of individual dendrons normally to the surface, but at the same time allow the stretched dendrons to remain in the linear (Gaussian) elasticity regime. In the analytical self-consistent field (SCF) model of regularly branched dendron brushes,17,19 the molecular potential U(z) is expressed as U (z ) 3 2 2 k (H − z 2 ) = 2ab kBT

a3f {φ(z)} = vφ 2(z) + wφ3(z) + ... kBT

with ν = 1/2 − χ, and w = 1/6 as dimensionless second and third virial coefficients of monomer−monomer interactions. In scaling terms, the mean-field approximation for f{φ(z)} in eq 3 as well as the Gaussian elasticity of the macromolecules in the solution are retained if the second virial coefficient of monomer−monomer interactions v < φ(b/a) 3, that is, remains moderately low.29 For higher values of v > φ(b/a)3, the scaling corrections due to the local chain swelling become important. In the analytical self-consistent field model of polymer brush the scaling corrections could be implemented along the lines in ref 30. Practically useful approximation for the polymer density profile and distribution function of the free ends in the scaling regime of good solvent have been also introduced in the recent study.31 Below we retain only the first term in eq 3 (that is, focus on the case of moderately good solvent). Under these conditions, the polymer density profile is parabolic: φ (z ) =

3 k2 2 (H − z 2 ) 2ab 2v

(4)

with brush thickness H and free energy F(H) per macromolecule expressed as16 ⎛ 2bσvN ⎞1/3 ⎟ H /a = ⎜⎜ 2 ⎟ ⎝ ak ⎠

and

⎛ a ⎞1/3 F (H ) 9 k 4H 5 ⎜ ⎟ (2vkσ )2/3 N 5/3 = = ⎝b⎠ kBT 20 vb2a3σ

(5)

Hence, as soon as the parameter k is specified, one can establish the dependences of brush thickness H and free energy F(H) on architectural parameters of the dendrons. 2.3. Force Balance. To find k, we consider force balance in each branching point of the tethered macromolecule. When the stem and the spacers are noticeably extended with respect to their Gaussian sizes, but are still far from the full extension, their conformations can be described through the so-called trajectories that specify the most probable position z(j) of monomer with ranking number j. The local elongation of any spacer is determined by its stretching function E = dz(j)/dj, and that of the stem, E0 = dz(j0)/dj0. In the parabolic molecular potential U(z) specified by eq 1, the stretching function E for a chain segment (spacer) with positions z1 and z2 > z1 of two end-points has functional form

(1)

where constant k depends on the architectural parameters of tethered dendrons, z is the distance from the surface, H is the brush thickness (cutoff of the polymer density profile), and kBT is thermal energy. The parabolic shape of the molecular potential U(z) is directly linked to the linear (Gaussian) elasticity of the tethered dendrons on all length scales. The molecular potential in eq 1 is related to the polymer density (volume fraction) profile φ(z) by a general equation a3δf {φ(z)}/δφ(z) = U (z)

(3)

(2)

where f{φ(z)} is the free energy density of interactions between monomers. In the case of nonionic macromolecules, f{φ(z)} could be expressed using the Flory−Huggins model of semidilute polymer solution,

E = k λ2 − z 2

(6)

where constant λ = λ (z1, z2, n) is specified by conservation of the total number of monomers, n, in the chain segment. Therefore, the stretching functions of all spacers in tethered branched macromolecules have functional form similar to that in eq 6. The full set of constants {λ} for macromolecule with given architecture is uniquely specified by the positions of its branching points. For a spacer in the last generation (free branch) of a dendron and/or a side chain in a comblike polymer, {λ}-set ensures zero tension (E = 0) at the chain free ends. For a spacer connecting two branching points (including the

a3f {φ(z)} = [1 − φ(z)] ln[1 − φ(z)] kBT + χφ(z)[1 − φ(z)]

Here, χ is the Flory interaction parameter that governs the thermodynamic quality of the solvent. C


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Table 1. Analytical Expressions for the Parameter k in the Self-Consistent Field Potential U (Eq 1) for Various Architectures of Branched Macromoleculesa

a

Notations are explained in the text.

to give the following equation for the parameter k,26

stem end-attached to the surface), {λ}-set ensures the conservation of the spacer length and force balance in the branching points. Here we present the scheme of derivation of λ-s for the main branched macromolecular architectures considered below, that is for the first (g = 1) and the third (g = 3) generation dendrons and short comb-like polymers. 2.3.1. Asymmetric Dendron of the First Generation, g=1. To illustrate the scheme of finding λ-s, we start with a brush formed by asymmetric dendrons (Ψ-shaped macromolecules) with arbitrary degrees of polymerization ni (i = 1, 2, ..., q) of the free branches. The expression for stretching function Ei = dz(j)/dj of the ith free branch, emanating from the branching point located at distance z0 above the surface is given by z02

Ei(z 0 , z) = k

2

cos (kni)

− z2 ,

z0 ≤ z ≤

q

tan(kn0)∑ tan(kni) = 1 i=1

The minimal solution of eq 10 relates the parameter k to the architectural parameters of Ψ-shaped macromolecule (ni, n0, and q). eq 10 for asymmetric Ψ-shaped macromolecule as well as similar equations for other polymer architectures are typically solved numerically. However, in some cases the analytical solutions are available. In particular eq 10 allows for the analytical solution (cf. Table 1) for star-like polymers with equal lengths of spacers,19 n0 = ni = n, i = 1, 2, 3, ..., and in some special cases for Ψ-shaped macromolecule with n 0 ≠ n i = n, i = 1, 2, 3, .... 2.3.2. Symmetric Dendron of the Third Generation, g = 3. Consider a brush of symmetric dendrons of the third generation with number of monomers n ≫ 1 per spacer, branching activity q, and z-coordinate of the free ends z3 (see Figure 1a). We introduce the stretching functions for the last generation of spacers (i = 3) as

z0 cos(kni) (7)

where zi = z0/cos(kni) is the position of the free end of the ith branch, that follows from the conservation condition ∫ zz10 E−1(z0,z) dz = ni. The stretching function E0 of the stem yields E 0 (z 0 , z ) = k

z02 sin 2(kn0)

2

−z ,

0 ≤ z ≤ z0

E3 = k z 3 2 − z 2

(8)

Ei = k λi 2 − z 2

i=1

(12)

where i = 0 corresponds to the stem that attaches the dendron to the surface, i = 1, 2 correspond to spacers in the first and second generations, respectively, and λi are unknown constants. eq 11 ensures vanishing tension t at all the free ends of dendron, t(z3) = 3kBTa−2E(z3) = 0. We express all the parameters λ and positions z3 of the free ends and of the branching points zi (i = 0,1), as a function of

q

∑ Ei(z 0 , z 0)

(11)

and stretching functions for spacers in generations i = 0, 1, and 2 as

In the linear elasticity regime, balance of normal forces is equivalent to balance of the stretching functions for all the spacers emanating from the branching point. That is, balance of forces at the branching point yields E0(z 0,z 0) =

(10)

(9) D


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Macromolecules the position z2 of the branching points of the last generation (i = 2) by the following steps.

(or, equivalently, q(q + 1)z 2 sin(kn) = λ12 − z12 ), leads (through eqs 21 and 20) to the relationship between z0 and z2

Step 1. Conservation of the number of monomers n in the free branch (third generation of the dendron, i = 3), z ∫ z32dz/E3 = n, specifies height z2 of the last branching point as kn = π/2− arcsin (z2/z3) to give z2 z3 = cos(kn) (13) The stretching function E3 of spacers in the last (i = 3) generation is therefore specified as E3 = k

z 0 = z1 cos(kn) − q(q + 1)z 2 sin 2(kn) = z 2 cos2(kn)[1 − q(q + 2) tan 2(kn)]

and specifies the value of λ1 as λ12 = z12 + q2(q + 1)2 z 2 2 sin 2(kn) = z 2 2 cos2(kn)[(1 − q tan 2(kn))2 + q2(1 + q)2 ]

z22 cos2(kn)

− z2

E0 = k

sin 2(kn)

−z

E0(z0) = qE1(z0), gives z 0 = q tan(kn) λ12 − z 02 or, equivalently, ⎛λ 2 − z 2⎞ q2 tan 2(kn)⎜ 1 2 0 ⎟ = 1 ⎝ z0 ⎠

Step 3. Conservation of spacer length in the second generation (i = 2), ∫ zz21 dz/E2 = n, provides the relationship between z1 to λ2 and z2, kn = arcsin(z2/λ2) − arcsin(z1/λ2), or equivalently,

q2 tan 2(kn)

qz 2 tan(kn) =

λ2 − z 2

2

(16)

k=

(18)

(19)

By substituting λ2 from eq 18 in eq 16 one finds the expression for z1 as a function of z2, z1 = z 2 cos(kn) − qz 2 tan(kn) sin(kn) 2

= z 2 cos(kn)[1 − q tan (kn)]

z02 q2 tan 2(kn)

(20)

k=

z02 cos2(kn)[1 − q tan 2(kn)]2

(27)

1 1 arctan n q(q + 2)

(28)

The expression in eq 28 constitutes a special case of a more general expression for the parameter k, derived for asymmetric dendron of the second generation.27 2.3.4. Symmetric Comblike Polymers with p = 2 Repeat Units. A symmetric comblike polymer is depicted in Figure 1c. The polymer molecule consists of p = 2 repeat units each

z 0 = λ1 sin[arcsin(z1/λ1) − kn] λ12 − z12 sin(kn)

= z12 =

with the solution

Step 5. Conservation of spacer length in the first generation (i = 1), ∫ zz10dz/E1 = n, gives kn = arcsin(z1/λ1) − arcsin(z0/λ1), or, equivalently,

= z1 cos(kn) −

1 2 arctan 2 n q[(q + 2q + 3) + (q2 + 2q + 3)2 − 4 ]

Note that it reduces to expected value, k = π/8n = π/2N, for q = 1 (linear chain of N = 4n monomers). The expression in eq 26 was presented previously27 as a conjecture without derivation. 2.3.3. Symmetric Dendron of the Second Generation, g = 2. A similar procedure is applicable to find the parameter k for symmetric regularly branched dendron of the second generation g = 2 (see Figure 1(b)). By substituting i → i − 1 for values of i = 3 and 2, and eliminating Step 6, we find the corresponding expressions for the set of parameters λ and zi. By using force balance condition at the branching point, qE1(z0) = E0(z0), one finds the following equation for the parameter k,

(17)

The stretching function E2 of spacers in the second generation (i = 2) is then expressed as E2 = k z 22[1 + q2 tan 2(kn)] − z 2

=1

(26)

which provides the expression for λ2 as λ 2 2 = z 2 2(1 + q2 tan 2(kn)]

[1 − q(q + 2) tan 2(kn)]2

The solution of eq 25 provides an analytical expression for the parameter k for symmetric regularly branched dendrons of the third generation (g = 3),

Step 4. Force balance at the branching point with coordinate z2, qE3(z2) = E2(z2), yields 2

tan 2(kn)[q tan 2(kn) − (1 + q + q2)]2

(25)

z1 = λ 2 sin[arcsin(z 2/λ 2) − kn] λ 2 2 − z 2 2 sin(kn)

(24)

By substituting λ1 and z0 from eqs 23 and 22 in eq 24, one finds (after some algebra) the final equation that specifies parameter k as a function of q and n,

2

(15)

= z 2 cos(kn) −

(23)

Step 7. Force balance at the first branching point (i = 0),

(14)

Step 2. Conservation of number n of the stem monoz mers (i = 0), ∫ 00 dz/E0 = n, specifies λ0 = z0/sin(kn) to determine the stretching function of the stem as z02

(22)

(21)

Step 6. Force balance at the branching point of the first generation (i = 1) with height z1, qE2(z1) = E1(z1) E


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Macromolecules comprising of a backbone segment with n monomers, and q side chains (each with number of monomers n), emanating from the terminal point of the backbone segment. A repeat unit consists of n(1 + q) monomers, the last backbone segment with n monomers has no side chains. This last backbone segment is equivalent to an additional free branch emanating from the second repeat unit. Positions of the branching points are z0 and z1 (see Figure 1c). The total number of monomers in the macromolecule is N = n(2q + 3). Similarly to the stretching functions of the free branches in dendrons, we introduce the stretching functions Esi for side

macromolecule is symmetric, and in this case the expression for k is presented in the first row of Table 1. The fifth row is for an asymmetric dendron with equal lengths of spacers (n) but different branching activities: q1 in the first, and q2 in the second generations, respectively. Finally, the last row in Table 1 corresponds to a symmetric comblike polymer with p = 2 repeat units, each composed of backbone segment and q side chains emanating from its terminal monomer, and one terminal backbone segment with no side chains.

3. MULTICOMPONENT (COMPOSITE) BRUSHES In brushes formed by any of the individual species considered above, the self-consistent molecular potential has the same (parabolic) shape and differs only by the value of parameter k and by additive constants which depend on N and σ. As a consequence, when branched macromolecules of different molecular weights and architectures, that have the same k parameter are mixed in arbitrary proportion to form a composite brush, they produce, and individually experiences, the same molecular potential U(z). As a result, such brush is characterized by a unified (parabolic in a good solvent) polymer density profile. Fixation of parameter k to a certain value specifies for a given molecular architecture the molecular weight of the dendron N(k) (see, Table 1 for particular architectures). Mixing different macromolecules with the same k parameter in a brush merely mediates its thickness H, which is governed by the chain grafting density σ, and the average molecular weight of constituent macromolecular species,

chains in the comblike polymer, Eis = k yi 2 − z 2 , where yi is the position end monomer of the side chain in the ith repeat y unit (i = 0,1). Conservation of the side chain length, ∫ zii dz/Esi = n, provides the relationship between yi and zi (i = 0, 1) as yi = zi/cos(kn). Stretching functions Ebi of the backbone segments (i = 0,1) are introduced as Eib = k λi 2 − z 2 with unknown constants λi to be determined. Similarly to the case of dendrons with g = 2, we aim to express all λ1, λ0, and z0 as a function of z1. The conservation of monomers in the first (i = 0) backbone z segment (that is, stem of the macromolecule), ∫ 00 dz/Eb0 = n, provides the relationship between λ0 and z0 as λ0 = z0/sin(kn). Similar condition of conservation of monomers in the second backbone segment (i = 1) leads to the relationship between z1, z0, and λ1, z 0 = z1 cos(kn) −

λ12 − z 0 2 sin(kn)

(29)

while force balance at the second branching point (i = 1), located at height z1 above the surface, Eb1(z1) = (q + 1) Es1(z1) gives λ12 = z12[1 + (q + 1)2 tan 2(kn)]

N̅ (k) =

i

Here, f i is the number fraction of macromolecular species of type i with molecular weight Ni(k), and the summation runs over all specimen present in the brush. In terms of the average molecular weight N (which depends on the parameter k through dependences Ni = Ni(k) for individual species, and the brush composition {f i}), the thickness H of a unified brush is given by

(30)

By substituting λ1 in eq 30 in eq 29, one finds z 0 = z1 cos(kn)[1 − (q + 1) tan 2(kn)]

∑ fi Ni(k)

(31)

Finally, force balance in the first branching point (i = 0), Eb0(z0) = Eb1(z0) + qEs0(z0), provides (after some algebra) an equation for the parameter k,

H /a =

(q + 1)q tan 4(kn) − 3(q + 1) tan 2(kn) + 1 = 0

⎛ 2bσv ⎞1/3 ⎜ ⎟ N̅ (k)1/3. ⎝ ak 2 ⎠

(33)

The free ends of all constituent specimen are distributed throughout the whole brush volume. The absence of segregation of the free ends allows one to express the free energy per macromolecule in a unified brush as

with solution ⎡ ⎤ 2 1 ⎢ 3(q + 1) − 9(q + 1) − 4q(q + 1) ⎥ k = ·arctan⎢ ⎥ n 2q(q + 1) ⎢⎣ ⎥⎦

1/3 F (H ) 9 k 4H 5 9 ⎛⎜ a ⎞⎟ (2vkσ )2/3 N̅ (k)5/3 = = 20 vb2a3σ 10 ⎝ b ⎠ kBT

(32)

In the case of q = 0, eq 32 reduces to expected result for linear chain with N = 3n monomers, k = n−1 arctan(1/ 3 ) = π /(6n) = π /(2N ). In Table 1 we collect the analytical expressions for parameter k obtained for polymer architectures described above. The first three rows in Table 1 correspond to symmetric dendrons with different number of generations (g = 1, 2, and 3). The forth row is for an asymmetric Ψ-shaped macromolecule (g = 1) comprising q monodisperse free branches (ni = n1) and the stem with n0 monomers. The analytical expression for k is available when the asymmetry parameter u = n1/n0 adopts integer and/or inverse integer values. If u = 1, the Ψ-shaped

(34)

As follows from eqs 33 and 34, the thickness H and the free energy F(H) of a unified polydisperse composite brush are obtained from the corresponding properties of a monodisperse dendron brush by substitution N → N. The elastic deformation of two unified brushes brought at distance 2D < 2H between the grafting surfaces, leads to the increase in free energy F(D) per chain as F(D) = F

⎛ D2 5 1 D5 H⎞ F (H ) ⎜ 2 − + ⎟ 9 5 H5 D⎠ ⎝H

(35)


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Macromolecules while the corresponding disjoining pressure increases as Π(D) = −

⎞2 ∂F(D)σ 5 F (H )σ ⎛ H 3 2 = − D ⎜ ⎟ 9 a 2H 5 ⎝ D ⎠ a 2∂D

(36)

with thickness H and free energy F(H) of the undeformed brush given by eqs 33 and 34, respectively. Therefore, as far as the brush thickness H and the grafting density σ of the tethered dendrons are known and F(H) is determined from eq 34, the elastic properties of the composite brush eqs 35, 36 are fully specified. Using eq 10, one can select the architectural parameters of Ψ-shaped macromolecules that would ideally mix in a composite brush without vertical segregation and form a unified brush with a parabolic polymer density profile. Consider, for simplicity, brush of asymmetric Ψ-shaped macromolecules with equal length of all the free branches, ni = n1 ≠ n0. In this case, eq 10 reduces to q tan(kn0) tan(kn1) = 1

Figure 3. Combinations of arm (and stem) length n and number of free arms q which produce a given value of the k parameter (as indicated) for monodisperse brushes, that is for brushes of starlike dendrons of the first generation with stem and the arms that have the same length n ≡ n0 = n1. The lines are to guide the eye. The points are for integer values of q and n.

(37)

4. THRESHOLD OF THE GAUSSIAN ELASTICITY The assumption of linear (Gaussian) elasticity in the elongated dendrons imposes restrictions on the grafting density σ of the tethered macromolecules. Because the stem stretching is maximal near the grafting surface (z = 0), the threshold of linear chain elasticity could be specified by the condition kz 0 a−1E0(z = 0) = ≃1 a sin(kn0) (38)

In Figure 2, we present the solution of eq 37 for a fixed selected value of k(q, n1, n0) = 0.002. This solution constitutes

where n0 is the number of monomers in the stem, and z0 is vertical position of the first branching point in a dendron with its end points located at height ≃ H above the grafting surface. The relationship between z0 and brush thickness H, z0 = z0(H), depends on the dendron architecture, and the condition 38 must be formulated for each chain architecture separately. To illustrate the scheme of finding the grafting density σ* corresponding to threshold of the dendron Gaussian elasticity, consider the simplest case of symmetric stars (g = 1) with equal lengths of the stem and branches (n0 = ni = n). In this case, position of the branching point z0 in the most strongly stretched dendrons with z1 = H is related to the brush height H as z0 = H cos (kn). Therefore, the Gaussian elasticity in the brush of starlike dendrons is maintained if

Figure 2. Combination of stem lengths n(0) ≡ n0, arm lengths n(1) ≡ n1 and number of arms q, to give a fixed value of the parameter k = 0.002 as given in eq 37. Only the points on the surface where the three parameters n0, n1, q have integer values are relevant (the remainder of the surface is to guide the eye).

tan(kn) H < a k

By substituting the brush thickness H from eq 5 in inequality 39, one finds the threshold grafting density σ* above which the Gaussian elasticity of starlike macromolecules becomes violated,

the surface in the (q, n1, n0) coordinates. The points on this surface with integer values of q, n1, and n0 are the parameters of Ψ-shaped macromolecules that are predicted to ideally mix vertically in a composite brush (with k = 0.002). In Figure 3 we present the spacer length n(q) = k arctan(1/ q ) of symmetric starlike polymers corresponding to three different values of the parameter k (indicated near the curves). In the case of k = 0.002, the values of n(q) correspond to the intersection of the surface in Figure 2 by the plane passing through the diagonal n0 = n1. Another instructive example is a binary brush formed by symmetric Ψ-shaped macromolecules with equal lengths of the stem and free arms, n0 = n1 = n (star-like polymers) and linear chains of length N. In this case a unified density profile (without vertical stratification) is expected for the length of linear chains N=

(39)

σ* =

2va tan 3(kn) 2va 1 = b kN b q3/2 arctan 1 (q + 1) q

( )

2va 1 , ≈ b q2

when q ≫ 1

At grafting densities σ < σ*, the linear elasticity of starlike dendrons is retained. In the case of asymmetric stars with the stem length n0 and different lenghs of branches ni (among which the maximal length is nmax), position of the branching point z0 in mostly stretched dendrons with zmax = H is given by z0 = H cos(knmax), and the threshold grafting density is specified as

−1 πn ⎛ 1 ⎞ ⎜⎜arctan ⎟⎟ 2 ⎝ q⎠

σ* = G

sin 3(kn0) 2va b kN cos3(knmax ) DOI: 10.1021/acs.macromol.5b01722 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules Here, the value of parameter k is obtained numerically by solving eq 10. A similar scheme applies to all other architectures considered in this paper, and also to mixed (composite) brushes with fixed constant value of the parameter k. In the latter case, the grafting density threshold σ* is determined by the brush component with mostly stretched stem.

5. SF−SCF NUMERICAL MODEL To check the predictions of the analytical theory, we used the method of Scheutjens and Fleer (SF−SCF), who mapped the SCF equations onto a lattice where the lattice site is chosen to fit the segment size a. The chains are modeled as strings of amorphous beads where neighboring segments along the chain sit on adjacent lattice sites. Using an efficient propagator formalism, all possible and allowed freely jointed chain (FJC) conformations are included to evaluate the partition function33 and the density profiles. The FJC model has a finite extensibility and this makes the numerical SCF also applicable for the cases where the chains or some chain-parts become stretched comparable to the contour length. The numerical SF−SCF approach deals (on the mean field level) with all interactions between segments, i.e., on top of the binary interactions also ternary and higher order interactions are accounted for as in the full Flory−Huggins equation of state. The SF−SCF method requires an iterative procedure to optimize the free energy of the system and after this optimization thermodynamic and structural information is typically obtained with an accuracy of 7 significant digits.34 5.1. Composite Brush of Ψ-Shaped Macromolecules. Below we report key SF−SCF predictions for unified composite brushes that consist of 10 specimen of Ψ-shaped macromolecules. We limit ourselves to those specimen for which the analytical theory predicts that the molecular fields of the individual brushes are the same, that is they all have the same k-value. Here we select the value k = 0.002 and we refer to Figure 2 for the values of the stem lengths n0, the free arm lengths n1 and the number of arms q, which produces the specified k-value. We have fixed the individual grafting densities σ = 0.001 for each component in the composite brush, so that the overall grafting density of each of the composite brushes is given by σ = 0.01. An athermal solvent (with χ = 1 − 2v = 0) is assumed in all the SCF calculations, and also all interactions with the substrate onto which the chains are grafted are set to the athermal value. In Figure 4, we present results for the overall volume fraction profile φ(z) as found by the SF−SCF method for three composite brushes, each having 10 specimen of Ψ-shaped macromolecules in equal proportion (f i = 0.1). In the curve labeled with q = 5, we have selected a composite brush which consists of molecules which all have 5 free arms. We varied the stem length n0 and adjusted the n1 values as specified in the legend, such that the k-value for each specimen is given by k = 0.002. The curve labeled n0 = 300, represents a composite brush formed by a selection of 10 species which have in common that the stem length is 300 segments long. The number of arms q is changed from 1 to 10 and the corresponding value of the arm length that results in k = 0.002 is specified in the legend. Finally, in the third composite brush we focused on a mixture of 10 specimen with symmetrical arms, that is for a brush that consists of Ψ-shaped macromolecules for which the stem and arm length are the same n ≡ n0 = n1. The value of number of

Figure 4. Overall volume fraction of polymer segments of three composite brushes which are composed of 10 specimen of Ψ-shaped macromolecules each with partial grafting density σ = 0.001 (so total grafting density is σ = 0.01) plotted as a function of z2. Each specimen on itself forms a brush with a fixed value of the k parameters, namely k = 0.002, i.e, the parameters n0, n1, q are chosen to be on the surface in the parameter space presented in Figure 2. The curve with label q = 5 corresponds to the brush consisting of Ψ-shaped macromolecules with stem and arm lengths (n0,n1) = (40,595), (60,514), (100,389), (140,304), (210,211), (300,142), (400,96), (500,64), (600,39), (700,17); The curve with label n0 = 300 (stem lengths fixed) corresponds to the brush consisting of Ψ-shaped macromolecules with the number q and the lengths of the arms are selected as (q,n1) = (1,485), (2,316), (3,227), (4,175), (5,142), (6,119), (7,102), (8,90), (9,81), (10,73); The curve with label n0 = n1 (≡ n) consists of Ψ-shaped macromolecules with the number of arms q and the length of the arm/stem n selected as (q,n) = (1,392), (2,308), (3,262), (4,232), (5,211), (6,194), (7,181), (8,170), (9,161), (10,153).

arms q = 1,...,10 and the lengths n of these symmetric specimen in the composite brush are presented in the legend. Completely in a line with the analytical predictions we find in Figure 4 that the overall volume fractions profiles are almost perfectly parabolic and that the composite brush has a single k-value as anticipated which is equal to the k-value of the individual monodisperse brushes (not shown). By presenting the volume fraction profiles φ as a function of z2, and getting the straight lines we prove that the overall profile in the body of the brush is indeed parabolic. Slight deviations from the parabola are attributed to the approach of the Gaussian elasticity threshold by some of the specimen. The slope of each profile leads to the value of k which in turn characterizes the molecular field. All three composite brushes have the same slope and thus all three composite brushes experience the same molecular field. In these composite brushes the average chain lengths N are not the same: the brush with the highest average has also the highest overall density and largest height. According to eq 33, at fixed values of k = 0.002 and σ = 0.01, the ratio of thicknesses (H) for mixed brushes with q = 5 (N = 1490) and n0 = 300 (N = 981.5) is estimated as (1490/981.5)1/3 ≈ 1.15 (see legend to Figure 4 for sets of molecular weights for the corresponding systems). An independent estimate for this ratio can be obtained from the SF−SCF calculations by extrapolating φ(z2) to zero as shown by the dashed lines in Figure 4, and by specifying the brush thickness H as a square root of the x-coordinate of intersection with x-axis. This procedure gives the respective values of H/a ≈ 160 and 138 with ratio 160/138 ≈ 1.16, which is in excellent agreement with the prediction of the analytical model. In passing we note that when a composite brush is formed by two or more specimen that do not have the same k-value, two or more distinct slopes are found in the plot of φ(z2) (not shown). All three examples of the composite brushes in Figure 4 H


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Figure 5. Volume fraction profiles φ(z) (left panels) and corresponding end-point distributions normalized by the number of arms g(z)/q (right panels). Each composite brush consists of 10 Ψ-shaped molecules as specified in Figure 4. The composite brush with n0 = 300 stem lengths (top row) are labeled by by the number of arms q. The composite brush with q = 5 (middle row) is labeled by the stem length n0. The brush composed of specimen with corresponding stem and arm lengths n0 = n1 are labeled by the number of arms q.

have a perfect parabolic profile and that is why we refer to such brush as being unified. In Figure 5 we present the polymer density profiles φ(z) of individual specimen of the composite brushes (left panels) and the corresponding distribution functions g(z) of their free ends (right panels). As it is seen from the plots, there is no stratification in these composite brushes, all exhibiting the same fixed value of k = 0.002. 5.2. Interactions between Composite Brushes. To check further the predictions of analytical model, we performed the SCF numerical calculations for compressed composite brushes of Ψ-shaped macromolecules. For simplicity, we focused here on the so-called “mirror” dendrons26 that have q = 4 free branches and the same longest path of n0 + n1 = 2000 monomer units, and are mixed at various ratios 0 ≤ f ≤ 1 within each brush. The grafting density of the tethered dendrons is σ = 0.005 per chain. The first component has stem length n0 = 1700 (and free branches with n1 = 300 monomer units each) and the total degree of polymerization NI = n0 + qn1 = 2900. Its mirror counterpart has stem length n0 = 300 (and free branches with n1 = 1700 monomer units each) and the total degree of polymerization NII = n0 + qn1 = 7100. The number fraction f of the second component in each brush was varied from f = 0 (pure component I) to f = 1 (pure component II) with composition increment Δf = 0.1. As indicated by eq 37, the mirror dendrons have the same value of parameter k, and thereby the free energy increase due to compression of the two apposing brushes should obey eqs 35, 34. In Figure 6 we demonstrate the increase in the free energy per unit area, σΔF(D)/a2 = σ (F(D) − F(H))/a2 as a function

Figure 6. Free energy per unit area, σF(D)/a2, as a function of half distance D/2 between the grafting surfaces at various compositions f of the mirror dendrons in the composite brushes.

of half distance D/2 between the grafting surfaces at various compositions f of the mirror dendrons in composite brushes. The theoretical thickness H ≈ D* of an unperturbed brush with a given composition f was evaluated from the value of polymer volume fraction in the middle of the gap φ(D*/2) = 0.005. Although such evaluation of the thickness is approximate due to the presence of fluctuation-induced tail at the brush edge, it still gives a reasonable estimate for H, which could be used to collapse the data on a master curve. In Figure 7 we present the dependence of D* as a function of the brush composition f. Because the average molecular weight in a mixed brush of mirror dendrons, N = (1 − f) NI + f NII = N1 + f(NII − NI) is proportional to the brush composition f, the brush thickness H should increase linearly with f. The numerical estimate for the brush thickness H ≈ D* in Figure 7 is consistent with this expectation. I


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the free ends of the constituent macromolecules are vertically mixed and distributed all over the brush volume. The physical reason for the vertical mixing of the free ends is that the macromolecules experience the same, parabolic molecular potential U(z), irrespective of their molecular architecture. In such a coating, the free ends of a dendron with a relatively low molecular weight may be still found at the brush periphery, and hence can be available for interactions with targets in the solution. A unified density profile, rather than a layered one, may have potentially interesting applications in surface modifications. The proposed analytical model specifies the architectural parameters of linear and branched macromolecules that could provide a unified polymer distribution with an unrestricted distribution of the free ends in mixed brushes (e.g., through eq 10 for Ψ-shaped macromolecules). The model assumes the Gaussian (linear) elasticity of the tethered macromolecules that limits the interval of grafting densities σ, and/or number of branches q to moderately high values. A violation of the linear elasticity of the tethered macromolecules modifies the molecular potential U(z),32 and induces a vertical brush segregation similar to that described earlier in brushes of symmetric starlike polymers,18,19,21 and mixed brushes of linear and starlike macromolecules.28 The predictions of the analytical model are supported by the SF−SCF numerical calculations. The unique organization of a unified brush governs its elastic properties. The increases in the free energy F(D) eq 35 and disjoining pressure Π(D) eq 36 upon brush compression are formulated in terms of only two parameters - thickness H of a unified brush in an unperturbed state eq 33, and grafting density σ of the macromolecules (or, equivalently, the brush free energy in an unperturbed state F(H) ∼ H5/σ, eq 34). We have checked this analytical prediction by performing the numerical SF−SCF analysis of the compressed binary composite brushes that are composed of mixture of mirror dendrons. The mirror dendrons could have quite different molecular weights but nonetheless exhibit the same value of the parameter k, and thereby form brushes with unified density profiles and exhibit the corresponding elastic properties. It is tempting to speculate that brush-like coatings of dendritically branched biomacromolecules (e.g., extracellular polysaccharides on the bacterial surfaces) could possibly adopt a unified brush structure and thereby exhibit the nanomechanical properties reminiscent of those of linear chain brushes. This might explain why the scaling models developed for linear chain brushes are effective in interpretation of the experimental data for (bio)systems incorporating brush-like coatings of polydisperse branched macromolecules. Although in this study we focused only on planar brushes of nonionic dendrons in good solvents, equations that specify the parameter k for various architectures (e.g., eqs 10, 26, 32, and all other expressions collected in Table 1) are applicable in a much wider range of conditions. In particular, they hold for chemically identical nonionic polymers under Θ and poor solvent conditions, including polymer melts with φ(z) = 1. They are also directly applicable to concave (inwardly curved) dendron brushes and could be practical in approximate description of convex (outwardly curved) dendron brushes. In the latter case, the parabolic molecular potential becomes more rigorous with increasing number of generations in the branched macromolecule.25,36

Figure 7. Dependence of unperturbed brush thickness D* as a function of the brush composition f.

Figure 8. Increase in the free energy σΔF(D)/a2 upon compression of the brush in the reduced variables that follow from eqs 35 and 34

In Figure 8, we present the increase in the free energy σΔF(D)/a2 = σ [ F(D) − F(H) ]/a2 in the reduced variables that follow from eqs 35, 34. Because the free energy F(H) of an unperturbed brush with the fixed values of parameter k and grafting density σ is proportional to H5 eq 34, we plot F(D)/F(H) ∼ F(D)/H5 ≈ F(D)/(D*) 5 as a function of the reduced distance between plates, D/D*. In full agreement with eq 35, the numerical SCF data obtained for different values of the brush composition f collapse on a master curve in the whole range of distances D between surfaces. We emphasize that eq 35 is valid at relatively small compressions D/D*≲ 1 providing the predominance of binary contacts between monomers. However, addition of monomer−monomer interactions of the higher orders at D/D* ≪ 1, does not prevent collapse of the data on the same master curve. This is because in this range of brush compressions, the elastic contribution to the free energy is negligible, and F(D) is controlled by the average volume fraction φ(D) ∼ D−1 of monomers in the gap between surfaces.

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DISCUSSION AND CONCLUSIONS In this study we have formulated an analytical SCF model of brushes formed by regularly branched macromolecules of various architectures root-tethered to a planar inert surface. The types of considered macromolecules include: symmetric dendrons of g ≤ 3 generations, asymmetric dendrons of g ≤ 2 generations, and short symmetric comblike polymers. Some features of the selected architectures have been addressed in our earlier studies,26,27 and the available analytical expressions for k parameter are collected in Table 1. Recall that the parameter k determines the self-consistent field potential U(z) eq 1, which specifies the equilibrium brush properties. The developed analytical model predicts that topologically tailored sets of chemically identical branched macromolecules with significantly different molecular weights could form a brush with a unified polymer density distribution. In such a brush, J


Macromolecules

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Moreover, the presented expressions for parameter k remain valid also for chemically identical ionic macromolecules (provided that the stretched polyions remain in the linear elasticity regime). In particular, the molecular potential U(z) in a strong polyelectrolyte brush (with degree of ionization α independent of the external conditions) is linked to the electrostatic potential Ψin(z) in such a brush as U(z) = αeΨ in(z) where αe is the fractional charge per monomer unit, and e is elementary charge. Therefore, general results for brushes of strong polyelectrolyte dendrons35 can be specified for polyions with any particular architecture (considered in Table 1) by substitution of an appropriate expression for the parameter k. For tethered weak polyelectrolytes (whose degree of ionization is pH-sensitive), the molecular potential U(z) is linked to the profile of ionization α (z) of the monomers in the brush as ln[1 − α (z) ]= Λ + U(z) where constant Λ is related the degree of monomer ionization at the external brush boundary, z = H. A link to brushes of both strong and weak polyelectrolyte dendrons in the salt dominated brush regime is established if the second virial coefficient v of the short-range binary monomer− monomer interactions is substituted by the effective salt-controlled second virial coefficient veff = α2/(2a3cs) where cs is the total concentration of salt ions in the surrounding solution. Such substitution provides the scaling dependences for monolayers of charged dendrons in the salt dominated regime. In conclusion, the developed analytical theory describes planar polymer brushes formed by branched macromolecules with diverse architectures, including regularly branched dendrons up to the third generation and short comblike polymers. It assumes the Gaussian elasticity of the tethered dendrons on all length scale, and operates in terms of the self-consistent field potential acting on the monomers of brush-forming macromolecules. For neutral (nonionized) macromolecules the model is applicable in solvents of various strength (moderately good, Θ, and poor solvents, including melted solvent-free brushes) with polymer density profiles adjusted accordingly eq 2. For charged dendrons, the main regimes of polyelectrolyte brushes (osmotic and salt dominated) could be now recovered with the account of specifics of polyion architecture (Table 1). An interesting prediction that follows from the analytical model and finds support from the numerical SF−SCF calculations is that tailored polydispersity in molecular weights and architectures of the tethered dendrons can overrule vertical stratification, and such layers of branched macromolecules exhibit a unified polymer density profile with free chain ends distributed all over the volume of the brush.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship (PIRSES-GA2013-612562 - POLION) within the seventh European Community Framework Programme and partially supported by the Russian Foundation for Basic Research (Grant 14-0300372a), by the Department of Chemistry and Material Science of the Russian Academy of Sciences, and by Government of Russian Federation, Grant 074-U01. K


Ideal Mixing in Multicomponent Brushes of Branched Polymers

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