Brushes of Cycled Macromolecules: Structure and Lubricating

Article pubs.acs.org/Macromolecules

Brushes of Cycled Macromolecules: Structure and Lubricating Properties Ekaterina B. Zhulina,†,‡ Frans A. M. Leermakers,§ and Oleg V. Borisov*,†,‡,∥ †

Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg, Russia St. Petersburg National University of Informational Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia § Physical Chemistry and Soft Matter, Wageningen University, 6703 NB Wageningen, The Netherlands ∥ Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254, CNRS, UPPA, Pau, France ‡

S Supporting Information *

ABSTRACT: We present a theory describing structural properties of brushes formed by cycled macromolecules tethered to planar surface. Diverse architectures of macromolecules that differ with respect to position and number of subchains in macrocycle are considered. We use the self-consistent field analytical approach based on strong stretching approximation and assumption of Gaussian elasticity of brush-forming molecules. Predictions of the analytical theory are systematically compared to the results of numerical SFSCF modeling. Similarity of the effects of cyclization and branching on largescale properties of polymer brush is demonstrated. It is also predicted that weaker interpenetration of sliding brushes formed by cyclic macromolecules might enhance their performance as lubricants. polymer architectures and studied neutral13−18 as well as charged19 single- and multicomponent16,18 brushes formed by treelike branched macromolecules with diverse topologies: starlike polymers (Ψ-shaped macromolecules), dendrons with number of generations up to 3, and short comblike macromolecules tethered to the surface through the stem segment. By using both analytical and numerical self-consistent field (scf) methods, we have analyzed how branching of these macromolecules mediates the brush thickness, polymer density profile, and distribution of the free ends. The aim of the present paper is to extend the analytical scf model of polymer brushes to a novel class of topologically complex macromolecules that contain some macrocycles. Intramolecular cyclization occurs in synthetic molecules20−22 and natural biopolymers.23 For example, the extended linear structure of the polysaccharide i-carrageenan in a salt-free solution changes upon additions of salt, demonstrating the ability to form cyclic structures with circumferences of different lengths depending on molecular weight of the polymer.23 Polysaccharides constitute a major component in extracellular brushlike layers that govern cell adhesive and interactive properties.24−26 Therefore, understanding how variation in the polymer topology (e.g., intramolecular macrocyclization) affects the brush structure and elasticity could help to unravel structure−function relations in these systems.

1. INTRODUCTION Tethering of macromolecules to solid−liquid interface is considered as one of the most promising approaches for surface functionalizations.1−3 Indeed, chemical linkage of branched molecules to a surface makes it possible to expose multiple functional groups, that are linked to the ends of the arms, to the environment. In order to reach high performance, multiple functional molecules have to be attached to such surfaces with sufficiently high surface densities, thus forming polymer brushes.4,5 Structural and dynamic properties of brushes formed by linear macromolecules studied both theoretically and experimentally in the past few decades are currently well comprehended. The effects arising from topological complexity (e.g., chain branching) of the brush-forming macromolecules have been addressed only recently, and work toward understanding these is still in progress. Meanwhile, it is believed6 that mere change of chain topology in macromolecular building blocks could significantly modify properties of the materials used in e.g. biomedicine. Studies on structural properties of brushes formed by regular dendritically branched macromolecules attached to planar surfaces have intensified in the past decade.7−12 It was demonstrated that the physical properties of such layers and the availability of terminal functional groups can efficiently be tuned by varying the molecular architecture, that is, by changing the number of dendron generations, the length of the stem and spacers, and the strength of intermolecular interactions. In our previous studies we have extended the spectrum of considered © XXXX American Chemical Society

Received: June 14, 2016 Revised: October 15, 2016

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DOI: 10.1021/acs.macromol.6b01275 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Self-assembled cycled structures might also arise in brushes of ssDNA probes in diagnostic assays or DNA chips27,28 upon hybridization with partially complementary ssDNA targets from the solution. Relevant studies29−31 focused mostly on minor probe/target mismatches for which hybridizaton did not essentially affect the conformation of formed dsDNA. Hybridization of ssDNAs with longer mismatched sequences could lead to the formation of loops and more complex structures in the dsDNA strands. Therefore, a theoretical approach presented in this paper could be also applied to model hybridization between relatively long tethered ssDNA probes and targets with perfectly matched peripheral sequences but a significantly mismatched longer segment in the central part. By using both analytical and numerical scf approaches, we investigate how brush properties are affected by the size and position of macrocycle in the tethered polymer chains. We note that the details of intermolecular interactions and structural organization in concentrated solutions and melts of cyclic macromolecules are still a subject of debate in the literature.32 In our approach we use the analytical model based on the strong stretching approximation suggested by Semenov33 for tethered polymers. It provides asymptotic analytical expressions for the brush thickness and polymer density distribution in the brush. Additional insights into the brush structure are obtained with the numerical SF-SCF model.34 The latter accounts for nonlinear elasticity of the strongly stretched macromolecules and gives the most probable distributions of all monomers, including the chain end-points. The SF-SCF model allows also for detection of the so-called “dead zones” depleted of the chain free ends,35 and we use this model throughout the paper to check/validate applicability of the analytical theory. In this study we focus on the simplest case of a macromolecule composed of two linear fragments separated by a cycled fragment (such polymers are referred below as “cycled” macromolecules). The remainder of the paper is organized as follows. In section 2 we formulate the model of a cycled macromolecule and briefly review the scf analytical formalism for brushes of nonlinear polymers. In section 3 we derive an equation for the topological coefficient in a brush of cycled macromolecules with arbitrary lengths of linear and cycled fragments and analyze a few particular cases of chain architectures. In sections 4 and 5 we discuss the results of the analytical and numerical scf models and present the conclusions. The details of numerical SF-SCF model are summarized in the Supporting Information.

Figure 1. Schematics of a cycled macromolecule in the brush (a) and its different realizations (b−f).

Following the scf approach developed in refs 7 and 14 for polymer brushes of branched molecules, the self-consistent molecular potential U(z) in the brush of cycled macromolecules is presented in units of kBT as U (z ) 3 = 2 k 2(H2 − z 2) kBT 2a

(1)

with yet unknown topological coefficient k and where kB is the Boltzmann constant and T is the temperature. The topological coefficient k depends on the molecular weights of the fragments (n1, n3, n2) and the number q of linkers in the cycle. The expression in eq 1 is expected to hold when the tethered macromolecules exhibit Gaussian (linear) elasticity on all length scales and are noticeably stretched with respect to the Gaussian size in the direction normal to the grafting surface. The latter so-called strong stretching approximation allows for the implementation of “trajectories” to describe the conformations of the tethered macromolecules.33 The self-consistent potential in eq 1 assures a distribution of free ends throughout the brush that minimizes the overall conformational entropy penalty for the extension of the brush-forming chains. Therefore, a necessary condition of applicability of eq 1 is the absence of dead zones depleted of the chain end segments. In addition to the topological coefficient k, we introduce the topological ratio η = k/klin ≥ 1, where klin = π/(2N) is the topological coefficient for linear chain with number N of monomer units. In the case of nonionic monomers, the molecular potential in eq 1 is related to the polymer volume fraction profile ϕ(z) in the brush via a general equation

2. MODEL AND FORMALISM Consider a flexible macromolecule consisting of two linear fragments with numbers n1 and n3 of monomer units separated by a cycled fragment (see Figure 1a). The cycled fragment contains q ≥ 2 linkers, with number of monomer units n2 each, connecting the end-points of the linear fragments. The total number of monomer units (each with size a) in one macromolecule is N = n1 + n3 + qn2. The macromolecules are tethered with area s per chain to an impermeable planar surface by the terminal monomer of the first fragment (that is, the one with n1 monomer units). The grafting chain density 1/s is sufficiently high such that intermolecular interactions dominate over intramolecular ones, and the tethered macromolecules are found in the brush regime. In the brush regime macromolecules are extended in the direction normal to the grafting surface where the cutoff of the polymer density profile (brush thickness) occurs at a distance H from the surface.

a3

δf {ϕ(z)} = U (z ) δϕ(z)

(2)

where f{ϕ(z)} is the free energy density of monomer− monomer interactions.36 At low volume fractions ϕ(z) ≪ 1, the free energy density f{ϕ(z)} can be presented as a virial expansion a3f {ϕ(z)}/kBT = υϕ2(z) + wϕ3(z) + ...

(3)

with υa and wa as the second and third virial coefficients of monomer−monomer interactions. Below we focus on good solvent conditions under which the interaction free energy is dominated by binary contacts between monomers (described by the first term in eq 3). The generalization to arbitrary solvent strength is straightfor3

B

6


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3 k2 2 (H − z 2 ) 2a 2 2υ

fragment adjacent to the surface with position z1 of the lower junction is given by E1(z1 , z) = k

(4)

with the brush thickness

z12 sin 2(kn1)

− z2 (8)

Finally, the stretching function E2 of each linker in the cycle yields

⎛ 2a 2υN ⎞1/3 H /a = ⎜ ⎟ ⎝ sk 2 ⎠

E 2 (λ 2 , z ) = k λ 2 2 − z 2

(5)

(9)

Therefore, as soon as the topological coefficient k is known, one can find the brush thickness H and the polymer density distribution ϕ(z) for a given architecture of the tethered macromolecules. In contrast to linear polymers, the full endpoint distribution g(z) of branched macromolecules can be extracted only numerically, and we rely here on the SF-SCF modeling to obtain g(z). Importantly, in the linear (Gaussian) regime of elasticity for the tethered chains, the value of k depends neither on the state of the brush (ionized or neutral, solvent-free or swollen in solvent with arbitrary thermodynamic quality) nor on the grafting density and is solely determined by the polymer architecture. In other words, k serves as a universal parameter characterizing the polymer topology.

The conservation of linker length n2 provides the relationship between constant λ2 in eq 9 and the positions z1 and z2 of the junctions:

3. TOPOLOGICAL COEFFICIENT k To find the topological coefficient k for cycled chains, we introduce the stretching functions Ei = dz/dn (i = 1, 2, 3) for three different parts of the macromolecule: linear fragments (i = 1 and 3) with respective number n1 and n3 of monomer units and each of the linkers with n2 monomer units (i = 2). In the strong stretching approximation, the stretching functions Ei specify the elastic tension in the chain “trajectories”, which determine the most probable position z(j) of a monomer unit with ranking number j. By balancing chain tensions at junction points of the tethered macromolecules, one can formulate an equation for topological coefficient k. This approach is demonstrated below for a cycled macromolecule with number q ≥ 2 of linkers. Note that the analytical scf model is onedimensional and does not account for the crowding of chain fragments near junction points. The latter effect would smooth the jumplike changes in the E-functions at the junction points of the cycled chains. The stretching function E for a chain segment with n ≫ 1 monomer units exposed to the self-consistent potential in eq 1 has the general form

A similar force balance condition at the lower junction point with height z1 above the surface, E1(z1,z1) = qE2(λ2,z1), leads to the relationship between z1 and λ2: q z1 = λ2 q2 + cot2(kn ) (12)

E = k λ2 − z 2

z2 z2 z z 2 1 − 1 2 − 1 1 − 2 2 = sin(kn2) = λ2 λ2 λ2 λ2

3

1

Vertical positions z1 and z2 of the two junctions are thereby related to the position z3 of the chain free end as z 2 = z 3 cos(kn3)

and

z1 = z 3 cos(kn3)

q2 + tan 2(kn3) q2 + cot2(kn1) (13)

By substituting eqs 12 and 11 in eq 10, one arrives to the final equation for the topological coefficient k: [1 − tan(kn1) tan(kn3)] 1 + tan 2(kn2) tan(kn2) [1 + q−2 tan 2(kn3)][1 + q2 tan 2(kn1)]

=1 (14)

The minimal solution of eq 14 provides a dependence of the topological coefficient k on the molecular parameters q, n1, n2, and n3 of a cycled macromolecule. It should be emphasized that the solution of eq 14 does not automatically guarantee positive value for the end-point distribution g(z) everywhere in the brush, and therefore we systematically check the shape and sign of g(z) by complementary SF-SCF calculations. In a few special cases analytical solutions of eq 14 are available, and we discuss these below. 3.1. Macromolecules with n1 = 0 (No Stem, Cycle Is Tethered to the Surface). Referring to Figure 1b, in the case that there is no stem and the cycle is directly tethered to the surface, eq 14 reduces to

(6)

z 22 − z2 , cos2(kn3)

1 + tan 2(kn2) (10)

The force balance condition at the upper junction point with height z2 above the surface, E3(z2,z2) = qE2(λ2,z2), provides the relationship between z2 and λ2: q z2 = λ2 q2 + tan 2(kn ) (11)

with the constant λ determined by the segment end-point positions. Similarly to stretching functions of free branches of dendrons and comblike polymers,18 the stretching function E3 of the peripheral chain segment with height z3 of end-point (see Figure 1a) is given by E3(z 3 , z) = k z 32 − z 2 = k

tan(kn2)

z 2 ≤ z ≤ z3 (7)

tan(kn2) tan(kn3) = q

where the position z2 of upper junction is related to z3 via the condition of the conservation of segment length n3. Similarly to the stretching function of stem in dendrons and comblike polymers, the stretching function E1 of the linear

(15)

and the value of the topological parameter k for the brush of cycled macromolecules is equal to that for the brush of Ψshaped macromolecules end-attached to the surface with q C


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of eq 17 is provided by eq 16 with q → 1/q in the corresponding coefficients bj. 3.3. Macromolecule with n2 = 0 (No Cycle, Linear Chain with n1 + n3 Monomers). In this case, depicted in Figure 1d, by using the identity

arms, with n2 monomer units each, and one free arm with n3 monomer units. When ratio n3/n2 is equal to an integer value j (e.g., j = 1, 2, 3, 4), the solution of eq 15 yields 1 k = arctan bj n2 (16)

1 − tan(kn1) tan(kn3) =

with b1 = q, b2 = q/(2 + q), and b3 =

3[(1 + q) −

b4 =

(1 + q)2 − 4q/9 ] 2

2 + 3q − 4+q

one finds the solution of eq 14 as tan [k(n1 + n3)] = ∞, or equivalently π k= 2(n1 + n3) (18)

;

⎛ 2 + 3q ⎞2 q ⎜ ⎟ − 4+q ⎝ 4+q⎠

That is, eq 18 reduces to the known result for linear chain with N = n1 + n2 monomer units, k = klin = π/(2N). 3.4. Symmetric Cycled Macromolecule with n1 = n2 = n3 = n. In this case depicted in Figure 1e, eq 14 reduces to

If ratio n3/n2 is equal to an integer number, n2 is substituted by n3 in eq 16. Although the solution of eq 15 is available analytically, or numerically for arbitrary ratios of n3/n2, the distribution function g(z) of the free ends might develop a dead zone upon an increase in length n3 of the terminal linear segment. In Figure 2 we present end-point distribution g(z) in a brush of cycled chains with n1 = 0 (cycle is tethered to the surface),

⎛ ⎞ 1 tan 2(kn)⎜1 + + q⎟ = 1 q ⎝ ⎠

(19)

with the solution k=

q 1 arctan 2 n q +q+1

(20)

For cycled chains with many linkers, q ≫ 1, the topological coefficient of a chain with fixed total number N of monomers, k ≈ q−1/2/n ≈ q1/2/N, increases with q. Therefore, irrespective of the presence of a peripheral linear fragment with n monomer units, the brush behavior becomes similar to that of a brush of multiarm starlike polymers. An increase in the number q of linkers in the cycle at a fixed value of N leads, according to eqs 20 and 5, to a monotonous decrease in the brush thickness. No dead zones were found for symmetric cycled macromolecules.

Figure 2. Variation in end-point distribution g(z) with increasing terminal segment length n3 in brushes of chains with macrocycle (q = 2, n2 = 125) tethered to the surface (n1 = 0). Total number N of monomer units in the molecule increases as N = 2·125 + n3 while surface coverage a2N/s = 5 is kept constant. Inset demonstrates solution of eq 15.

4. DISCUSSION 4.1. Topological Coefficient k at Arbitrary Values of n1, n2, and n3. To analyze the effect of macromolecular architecture at arbitrary values of n1, n2, and n3, we introduce dimensionless parameters: x = kn1, u = n3/n1, and v = n2/n1. The variation in u at a fixed length of the stem (the linear fragment with n1 monomer units next to the surface) corresponds to the variation in length n3 of the peripheral linear fragment. Variation in v corresponds to changes in the linker length n2 compared to the length n1 of the stem. By using x = kn1 as a new variable, we rewrite eq 14 as

linker length n2 = 125, number of linkers q = 2, and length of terminal segment n3 increasing from n3 = 0 up to n3 = 250. The inset demonstrates the corresponding solution of eq 15, kn2 as a function of n3/n2. As it is clearly seen from Figure 2, when n3 is relatively short, g(z) is unimodal and positive at all z ≥ 0. An increase in n3 leads to the decrease in the end-point concentration near the grafting surface, and when n3 becomes comparable to the length n2 of the linker (n3 = 150 in Figure 2), g(z) develops a “plateau” which later transforms in a dead zone (see the curve for n3 = 250 in Figure 2). 3.2. Macromolecule with n3 = 0 (No Peripheral Linear Fragment). In this case, depicted in Figure 1c, eq 14 reduces to q tan(kn2) tan(kn1) = 1

tan(kn1) + tan(kn3) tan[k(n1 + n3)]

[1 − tan(x) tan(xu)] 1 + tan 2(xv) tan(xv) [1 + tan 2(xu)/q2][1 + q2 tan 2(x)]

=1 (21)

In Figure 3 we present the dependences of product x = kn1 of the topological coefficient k and number of monomers n1 in the stem for macromolecules with complex cycle (q = 6) as a function of ratio v = n2/n1 at different values of reduced length u = n3/n1 of the peripheral segment (shown near the curves) . Hence, as follows from eq 21 and Figure 2, an increase in either u = n3/n1 or v = n2/n1 at constant N leads to the decrease in k and to the concomitant increase in the brush thickness H. At large values of v ≫ 1, the data for different values of u asymptotically collapse on a single straight line x = π/(2v), corresponding to a linear chain with n2 monomer units.

(17)

Equation 17 is similar to the equation for the topological parameter k in a brush of Ψ-shaped macromolecules with q free branches of n2 monomer units each and the stem with n1 monomer units.16 Indeed, in the framework of the strong stretching approximation a terminal cycle composed of q linkers is equivalent to q free branches. The analytical solution D


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architecture. In some applications it is of interest to tune the tension force t(0) acting at the tethered monomer of the branched macromolecules.38 In nonionic swollen brushes tension force exerted per unit area of the grafting surface, t(0)/s, is balanced by the osmotic pressure, Π(0), that is, t(0) = sΠ(0). Under good solvent conditions, the osmotic pressure Π = kBTa−3υϕ2 in a nonionic brush with ϕ ≪ 1 arises due to the short-range repulsive binary contacts between monomers. In the case of a dense, solventfree brush39 t(0)/s = a−3U(0). These results are summarized in eq 22. The first line in eq 22 corresponds to a nonionic brush in a good solvent with υa3 ≃ 1, while the second line to a dry, solvent-free brush ⎧ 9s k 4 ⎪ H 4 ∼ a−1/3(kN )4/3 /s1/3 t(0) ⎪ 16a 7 v =⎨ kBT ⎪ 3 2 2 2 ⎪ 5 k H s ∼ a(kN ) /s ⎩ 2a

Figure 3. Dependence of reduced topological coefficient kn1 on parameter v = n2/n1 for different values of u = n3/n1 = 0.1, 1, 3, and 10 (indicated near the curves), q = 6. Horizontal lines indicate asymptotic dependence kn1 = π/[2(1 + n3/n1)]; dashed line with slope −1 indicates asymptotic dependence kn1 = πn1/(2n2).

In Figure 4 the topological ratio η = k/klin = 2kN/π is presented for cyclic macromolecules as a function of v = n2/n1

(22)

As it follows from eqs 22, cyclization-induced increase in tension depends on the solvent quality. To estimate magnitude of this effect, we consider symmetric cycled chains with q ≥ 2 linkers. By substituting kN ≈ q1/2 (eq 20) in eqs 22, we find that the value of exponent in the power law dependence of the tension t on the degree of chain branching q at a fixed number of monomers N and grafting area s increases from t(0) ∼ q2/3 in a swollen brush to t(0) ∼ q in a solvent-free dense brush. However, the ratio of the forces in dense and swollen brushes, (kN)2/3/(s/a2)2/3 ≃ q1/3/(s/a2)2/3 < 1, indicates a larger tension force in a good solvent. To get more insights into the dependence of the tension t(0) on the position of cycle within the chain, we present in Figure 5

Figure 4. Topological ratio η = k/klin = 2kN/π for cyclic macromolecules with the number of linkers q = 6 (closed symbols) and q = 2 (open symbols) as a function of the asymmetry parameter v = n2/n1 for values of u = n3/n1 = 0.1 (triangles), 1.0 (circles), and 10 (squares). Solid horizontal lines indicate asymptotes η = q. Insets depict cycled chains with q = 6 and q = 2.

for various values of u = n3/n1 and q = 2 (simple cycle) and q = 6. The value of η increases as a function of the relative cycle size v ∼ n2, and as a consequence the brush height H decreases. At small values of v ≪ 1, all the data approach the asymptote η = 1 while at large values of v ≫ 1, the dependences with different u but same value of q approach the asymptotic dependence η = q. The first asymptote (η = 1) corresponds to a linear chain with N = n1 + n3 monomer units, while the second asymptote (η = q) corresponds to a starlike macromolecule with q branches of N/q monomers each, tethered by its center to the surface. The effect of linear fragments encompassing intramolecular cycle is most noticeable for symmetric macromolecules with u ≃ v ≃ 1. Here, an increase in u = n3/n1 leads to a moderate increase in the topological ratio η. This effect becomes more pronounced for larger values of q. 4.2. Elastic Force in Tethered Macromolecules. In the brush regime, tethered macromolecules experience an elastic tension t(z), which is maximal at the grafting surface z = 0. It is influenced by the grafting density of macromolecules, the strength of intermolecular interactions, and the chain

Figure 5. Ratio of tensions t(0)/tlin(0) at grafting surface, z = 0, in brushes of cycled and linear chains with the same N and grafting densities as a function of number n1 = 250 − n3 of monomer units in the stem. (a) N = 1000, q = 6; (b) N = 500, q = 2. The linker length is n2 = 125 in both cases.

the analytically predicted ratio of the tension forces t(0)/tlin(0) = φ2(k, z = 0)/φ2(klin, z = 0) = [2kn2(q + 2)/π]4/3 exerted at the grafting surface (z = 0) by cycled and linear chains with the same mass N/s per unit area as a function of the stem length n1. Depending on the number q of linkers, the tension force t(0) in brushes of cycled macromolecules can be significantly larger than in brushes of linear chains with the same polymer mass per unit area. In the shadowed area of the parameters (located at relatively small values of n1), the numerical SF-SCF data indicate brush stratification and deviations in polymer density profile φ(z) from the parabola due to the onset of nonlinear chain elasticity. Because the monomer concentration φ(0) in nonlinear elasticity regime is larger than predicted by the E


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polymers,14 bimodal g(z) indicated vertical segregation of the tethered chains in two populations: one with strongly extended stem and free branches and another with weakly stretched macromolecules. A similar brush stratification is anticipated for cycled chains. Upon an increase in the stem length n1 the shape of polymer density profile approaches a parabola, and the corresponding end-point and branching-point distributions acquire a smooth unimodal character. In Figure 8 we present a selection of polymer density profiles φ(z) together with corresponding end-point distributions g(z) and distributions of the first and second junctions (denoted as B1 and B2) for n1 = 25, 75, 125, and 200. The bimodal character of the distribution functions for relatively short stem with n1 = 25 and 75 is indicative of brush stratification into populations of stronger and weaker stretched chains. According to the analytical model with molecular potential U(z) specified by eq 1, end-point distribution, g(z), and distributions of junctions, B1(z) and B2(z), should be related as

parabolic potential, the analytical curves in Figure 5 provide a lower boundary estimate for tension t(0). The actual tension would be higher. In Figures 6 and 7 we present polymer density profiles φ(z2) and the corresponding end-point distributions g(z) at different positions of macrocycle with varying n1 and n3 = 250 − n1.

Figure 6. Calculated by SF-SCF method polymer density profiles ϕ(z2) in brushes of chains with different positions of macrocycle (varying stem length n1 at fixed length of linker n2 = 125 and length of terminal segment n3 = 250 − n1). (a) q = 2, N = 500, a2N/s = 0.01; (b) q = 6, N = 1000, a2N/s = 0.005.

B1(z) =

z 3 ⎛ z1 ⎞ g ⎜z ⎟ z1 ⎝ z 3 ⎠

and

B 2 (z ) =

z3 ⎛ z 2 ⎞ g ⎜z ⎟ z 2 ⎝ z3 ⎠

(23)

with ratios z1/z3 and z2/z3 determined by eq 13. In Figure 9, the distribution functions of junctions B1(z) and B2(z) presented in Figure 8 are compared with renormalized distributions B1,r(z) and B2,r(z) derived from the end-point distribution g(z) in Figure 8 according to eqs 23. As it is seen from Figure 9, at relatively small stem lengths, n1 = 25 and 75, the differences between actual and renormalized distributions are strong, consistent with deviations of the molecular potential U(z) from the parabolic shape. At larger stem lengths, n1 = 125 and 200, the two sets of distributions become almost indistinguishable, confirming the applicability of the molecular potential U(z) in eq 1 to these systems. 4.3. Effect of Macrocycle Size and Asymmetry. We first consider how the brush thickness H is affected by an increase in linker length n2 at fixed constant values of n1 and n3. In Figure 10, we present the dependence H(n2) calculated with the analytical (square symbols) and numerical (circles) scf models for brushes of chains with a simple macrocycle, q = 2 and n1 = n3. An almost perfect linear dependence is found in

Figure 7. End-point distributions g(z) for the same values of parameters as in Figure 6 (varied length n1 of the stem, linker length n2 = 125, and length of the terminal segment n3 = 250 − n1). (a) q = 2, N = 500, a2/s = 0.01; (b) q = 6, N = 1000, a2/s = 0.005.

As it is seen from Figures 6 and 7, at small stem lengths n1 and long terminal segments n3, polymer density profiles φ(z2) deviate from straight lines predicted by the parabolic molecular potential, while the end-point distributions g(z) either demonstrate a dead zone or exhibit bimodal behavior. As it was demonstrated earlier for the brush of neutral starlike

Figure 8. SF-SCF data for distributions of junction points B1(z) and B2(z) (dotted lines). The end-point distributions g(z) and polymer density profiles φ(z) are shown by red and black solid lines, respectively. N = 500, q = 2, a2/s = 0.01. F


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Figure 9. Distributions of junction points calculated by SF-SCF method: actual distributions B1(z) and B2(z) (black lines) and those obtained by renormalization of end-point distributions g(z) according to eq 23, B1,r and B2,r (red lines). N = 500, q = 2, a2/s = 0.01.

In the first scenario topological coefficient k is specified by eq 20, and for q = 2 it gives k1n = arctan( 2/7 ) ≈ 0.49. The corresponding brush thickness H1 ∼ k1−2/3 (see eq 5). In the second scenario a cycled macromolecule is equivalent to a symmetric tethered star with q + 1 = 3 free branches. The corresponding expression14 for k for a symmetric starlike polymer with q + 1 free branches is kn = arctan(1/ q + 1 ) which gives k2n = arctan(1/ 3 ) ≈ 0.52 for q = 2. The corresponding brush thickness H2 decreases only slightly, with ratio H2/H1 = (k1/k2)2/3 ≈ 0.96. In Figure 11, we present the dependence of brush thickness Ha as a function of cycle asymmetry x (i.e., for an asymmetric

Figure 10. Brush thickness H as a function of linker length n2 in symmetric cycle with q = 2, N = 250 + 2n2, a2N/s = 5. Inset demonstrates end-point distributions g(z) for different values of n2 = 125, 150, 175, 200, 225, and 250 (indicated near the curves). Circles and squares correspond to the numerical SF-SCF data and to analytical predictions, respectively. Lines are a guide for the eye.

both cases; the correspondence between the analytical and numerical values is also good. The end-point distributions g(z) shown in the inset in Figure 10 remain unimodal upon an increase in n2. The developed analytical model is not applicable to the chains with asymmetric macrocycle composed of linkers with different lengths. However, in some cases the effect of cycle asymmetry on the brush properties can be estimated. Consider, for example, symmetric macromolecules (n1 = n2 = n3 = n) with a simple cycle composed of q = 2 linkers and compare the values of topological coefficient k in two limiting cases: (i) symmetric cycle with equal lengths n of the two linkers and (ii) asymmetric cycle with zero length of one linker and length 2n in the second linker. That is, in the second case a loop of 2n monomers is attached to the junction point between two linear fragments with n monomers each (see Figure 1f). In both cases the total number of monomer units in macromolecule is N = (q + 2)n = 4n.

Figure 11. Total brush thickness Ha and the first moment of polymer density distribution H1 as a function of the cycle asymmetry x = n2 − 125, N = 500, q = 2, a2/s = 0.01. The values of x = 0 and x = 125 correspond to symmetric cycle with linker length n2 = 125 and loop with total length 2n2 = 250, respectively. Inset demonstrates end-point distributions g(z) for the values of x = 0, 25, 50, 75, 100, and 125 (indicated near the curves).

cycle for which the length of the longer linker n + x = 125 + x monomers and the length of the shorter linker 125 − x monomers) obtained from the SF-SCF calculations. A symmetric cycle with n2 = n = 125 corresponds to x = 0, while a loop of 250 monomers to x = 125. The brush thickness Ha2 was obtained by extrapolating the linear part of the polymer density profile φ(z2) to zero. As is seen from Figure 11, the G


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Macromolecules

25. The values of Lring/Llin obtained from the MD simulations varied in the range 0.76−0.83 depending on the molecular weight N of polymer and distance D between plates. The corresponding ratios mring/mlin varied in the range 1.6−1.75 and were consistent with the expectation of larger number m of monomers in the interpenetration zone for ring polymers compared to linear ones. However, the numerical values of mring/mlin were higher than predicted by the theoretical model. This discrepancy could be attributed to an oversimplified treatment of a loop in the interpenetration zone as a coil with unrestricted position of the second end. In agreement with the theoretical prediction on branchinginduced decrease in friction between brushes with interpenetration length L independent of V (that is, in linear regime of friction),16 the MD simulations37 confirmed that at the same polymer mass per unit area and small sliding velocities the tethered rings indeed produce smaller friction than linear macromolecules. 4.5. Gaussian Elasticity Threshold. The presented analytical model is applicable when tethered macromolecules exhibit linear (Gaussian) elasticity on all length scales. The onset of nonlinear elasticity can be estimated from the condition that maximal normalized tension in the mostly stretched chains with z3 ≃ H is equal to unity. That is, t(0)a/kBT = 3E1(z1, z = 0)/a ≅ 1. By using the second relationship in eq 13, this condition can be formulated as

brush thickness Ha monotonously decreases with increasing cycle asymmetry x. The ratio of the two values Ha(x=0)/Ha(x=125) ≈ 0.95 is in perfect agreement with the analytical prediction H2/H1 ≈ 0.96. The unimodal character of g(z) in brushes with asymmetric cycles is verified by the corresponding SF-SCF data in the inset in Figure 11. Therefore, an influence of cycle asymmetry (that is, the unequal lengths of linkers in a given macrocycle) on the topological coefficient k and brush thickness H is relatively small. It is about 6% for symmetric macromolecules with q = 2 and expected to decrease even more for larger values of q. 4.4. Interpenetration of Brushes Formed by Cycled Chains. In a general case of nonionic cycled chains with an extended peripheral fragment, the interpenetration of weakly compressed brushes follows the same power law dependence as for treelike polymers with independently fluctuating peripheral branches.16 That is, the interpenetration length L≃

a 4/3 k D

2/3 1/3

≃ L linη−2/3

(24)

of opposing brushes coincides with the last unstretched segment of the chain (last elastic blob) and decreases upon an increase in the topological ratio η = 2Nk/π. However, if the peripheral segment is short or absent (n3 = 0), eq 24 for the interpenetration length L changes. To find L in the case of tethered rings, we estimate the free energy penalty δF to insert qδn ≃ qL2/a2 monomer units from one brush into its opposing counterpart. In the compressed brushes separated by distance D, the molecular potential U(z) varies at distances z = (D/2 − L) ≫ L in the middle of the gap between surfaces as ΔU /kBT = − [U (D/2) − U (D/2 − L)]/kBT ≃

3

a 4/3 q1/3k 2/3D1/3

(26)

For example, in the case of symmetric macromolecules with n1 = n2 = n3 = n, and q ≫ 1, sin(kn) ≈ tan(kn) ≈ q−1/2, cos(kn) ≈ 1, the topological coefficient k ≈ N−1q1/2, and the brush thickness H in a good solvent with 2v = 1 is estimated as

k2 DL a2

The free energy penalty δF/kBT ≃ (qδn)(k2DL/a2) ≃ (L/a)3(qk2D/a) ≃ 1 for the transferring qδn monomer units from the center of the gap, z = D/2, to z ≅ D/2 − L specifies the interpenetration length L of the two brushes as L≃

(q2 − 1) cos2(kn3) + 1 H k =1 a (q2 − 1) sin 2(kn1) + 1

H /a =

a 2N sk 2

1/3

( )

≃ N (s /a 2)−1/3 q−1/3.

The condition (26) gives H/Na ≃ q−1 to specify the threshold grafting area s*/a2 ≃ 3q2 below which Gaussian elasticity of the stem becomes significantly violated. However, for q ≳ 1, a more accurate estimate must be obtained from eq 26. For example, for q = 2, n1 = n2 = n3 = 125, eq 20 specifies kn = arctan √2/7 ≈ 0.491, and eq 26 gives s*/a2 ≈ 150. According to this estimate of s*, the value of grafting chain density a2/s = 0.01 used in SF-SCF calculations was slightly above the Gaussian elasticity threshold a2/s* ≈ 0.007. However, because a relatively small fraction of the chains have z3 ≃ H, major parts of the chains were stretched less, and at the chosen value of a2/s = 0.01 deviations in the polymer density profile φ(z) from the parabola were negligible. Similarly to starlike polymers,14 cycled chains with s ≪ s* and strong nonlinear elasticity are expected to segregate vertically into populations with strongly and weakly stretched macromolecules.

(25)

That is, simultaneous interpenetration of q chain segments into the opposing brush occurs on length scales by a factor of q1/3 smaller than predicted by eq 24. It is instructive to compare the theoretical predictions of the analytical model with the results of recent MD simulations37 on compressed sliding brushes composed of linear and ring polymers. In the theoretical model, a flexible ring polymer is characterized by the following values of the parameters: q = 2, n1 = n3 = 0, and eq 14 reduces to sin(kn2) = sin(kN/2) = 1 to give kring = π/Nring. In MD simulations,37 the length N of linear polymers was kept twice shorter than Nring, that is, N = Nring/2. For this choice of system parameters klin = π/(2N) = π/Nring = kring. Therefore, eq 25 predicts that at the same distance D between the plates Lring/Llin = q−1/3 = 2−1/3 ≈ 0.79. The ratio of the average numbers m of monomers per chain in the interpenetration zone, mring/mlin, is then given by qLring2/Llin2 = 21/3 ≈ 1.26. That is a ring polymer delegates more of its monomer units in the interpenetration zone than a linear chain. The results of MD simulations at zero and small sliding velocities V demonstrated excellent agreement on the shrinkage of the interpenetration zone (the decrease in L) predicted by eq

5. CONCLUSIONS In this study we have developed an analytical scf model for a novel class of polymer brushes. The latter are formed by macromolecules comprising an intramolecular macrocycle with an arbitrary number of equally long linkers. We focused here on the macromolecules with a single macrocycle. However, the developed formalism allows for extension of the model to polymers containing more than one symmetric cycle with varying numbers of linkers. Although the analytical scf approach H


Article

Macromolecules Notes

is applicable to macromolecules with symmetric cycles only, we have demonstrated that in symmetric macromolecules with equal lengths of linkers and linear segments the cycle asymmetry leads to a difference on the order of few percent in structural properties of the brush. By using the numerical SF-SCF calculations, we outlined the range of parameters in which the molecular potential U(z) is well approximated by the parabola. Outside of this range, the end-point distribution g(z) loses its unimodality or exhibits a dead zone. In particular, the placement of a macrocycle closer to the chain grafting point (a decrease in n1 ≲ n2 at fixed values of n2 and of the total molecular weight N = n1 + 2n2 + n3) leads to noticeable deviations in the polymer density profile φ(z) from the expected parabola and to brush stratification (Figure 6). On the other hand, when the terminal chain segment is absent (n3 = 0) or relatively short, a parabolic shape of the molecular potential is well maintained for chains with tethered macrocycle (Figure 2). We have shown that in the strong stretching limit the effect of cyclization on the brush thickness is quantitatively close to the chain branching. That is, the topological coefficients for a cycled chain with q linkers and a starlike macromolecule with q free branches differ by only few percent. The maximal difference between the topological coefficients is found for symmetric macromolecules with simple cycle (q = 2). In this case, opening a macrocycle (that is, detaching one linker from the second junction and transforming the cycled macromolecule in an asymmetric tethered star with one free branch of n monomer units and the second one of 2n monomer units) decreases kn by several percent. The chain cyclization might have stronger impact on dynamic and tribological properties of the brushes. It has been predicted in our earlier study16 that chain branching leads to the decrease in the interpenetration length L of opposing brushes. As a result, a decrease in friction between sliding brushes is expected at least in the linear friction regime in which the interpenetration length L is independent of the sliding velocity. The cyclization-induced decrease in the interpenetration length L in the linear friction regime is now confirmed by MD simulations37 of brushes formed by ring polymers (cycled macromolecules with no linear segments and number of linkers q = 2). One might expect that topological repulsions40 between brush-forming cyclic macromolecules could enhance this effect, thus making cyclic brushes even better lubricants. Our findings may be also useful to understand the features of biomacromolecular brushes in which intramolecular complexation with guest species is driven by hydrogen bonding or nucleotide paring.

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by Russian Science Foundation grant 16-13-10485.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01275. Details of how brushes of macromolecules that contain a macrocycle can be accounted for in the self-consistent field approach using the discretization scheme of Scheutjens and Fleer (SF-SCF) (PDF)

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REFERENCES

(1) Gillich, T.; Benetti, E. M.; Rakhmatullina, E.; Konradi, R.; Li, W.; Zhang, A.; Schlüter, A. D.; Textor, M. J. Self-Assembly of Focal Point Oligo-catechol Ethylene Glycol Dendrons on Titanium Oxide Surfaces: Adsorption Kinetics, Surface Characterization, and Nonfouling Properties. J. Am. Chem. Soc. 2011, 133, 10940−10950. (2) Schüll, C.; Frey, H. Grafting of Hyperbranched Polymers: From Unusual Complex Polymer Topologies to Multivalent Surface Funtionalization. Polymer 2013, 54, 5443−5455. (3) Minko, S. Responsive Polymer Brushes. J. Macromol. Sci., Polym. Rev. 2006, 46, 397−420. (4) Alexander, S. Adsorption of chain molecules with a polar head: a scaling description. J. Phys. (Paris) 1977, 38, 983−987. (5) de Gennes, P. G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069−1075. (6) Vitelli, V.; Irvine, W. The geometry and topology of soft materials. Soft Matter 2013, 9, 8086−8087. (7) Pickett, G. T. Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest. Macromolecules 2001, 34, 8784−8791. (8) Merlitz, H.; Wu, C.-X.; Sommer, J.-U. Starlike Polymer Brushes. Macromolecules 2011, 44, 7043−7049. (9) Merlitz, H.; Cui, W.; Wu, C.-X.; Sommer, J.-U. Semianalytical Mean-Field Model for Starlike Polymer Brushes in Good Solvent. Macromolecules 2013, 46, 1248−1252. (10) Cui, W.; Su, C.-F.; Merlitz, H.; Wu, C.-X.; Sommer, J.-U. Structure of Dendrimer Brushes: Mean-Field Theory and MD Simulations. Macromolecules 2014, 47 (11), 3645−3653. (11) Kröger, M.; Peleg, O.; Halperin, A. From Dendrimers to Dendronized Polymers and Forests: Scaling Theory and its Limitations. Macromolecules 2010, 43, 6213−6224. (12) Gergidis, L. N.; Kalogirou, A.; Vlahos, C. Dendritic Brushes under Good Solvent Conditions: A Simulation Study. Langmuir 2012, 28, 17176−17185. (13) Polotsky, A. A.; Gillich, T.; Borisov, O. V.; Leermakers, F. A. M.; Textor, M.; Birshtein, T. M. Dendritic versus Linear Polymer Brushes: Self-Consistent Field Modelling, Scaling Theory, and Experiment. Macromolecules 2010, 43, 9555. (14) Polotsky, A. A.; Leermakers, F. A. M.; Zhulina, E. B.; Birshtein, T. M. On the two-population structure of brushes made of arm-grafted polymer stars. Macromolecules 2012, 45, 7260−7273. (15) Borisov, O. V.; Polotsky, A. A.; Rud, O. V.; Zhulina, E. B.; Leermakers, F. A. M.; Birshtein, T. M. Dendron Brushes and Dendronized Polymers: A Theoretical Outlook. Soft Matter 2014, 10, 2093−2101. (16) Zhulina, E. B.; Leermakers, F. A. M.; Borisov, O. V. Theory of brushes formed by Ψ-shaped macromolecules at solid-liquid interfaces. Langmuir 2015, 31 (23), 6514−6522. (17) Zhulina, E. B.; Leermakers, F. A. M.; Borisov, O. V. Selfconsistent field model of brushes formed by root-tethered dendrons. Scientific and Technological Journal of Information Technologies, Mechanics and Optics 2015, 15 (3), 493−499. (18) Zhulina, E. B.; Leermakers, F. A. M.; Borisov, O. V. Ideal mixing in multicomponent brushes of branched polymers. Macromolecules 2015, 48, 8025−8035. (19) Borisov, O. V.; Zhulina, E. B. Brushes of dendritically-branched polyelectrolytes. Macromolecules 2015, 48, 1499−1508. (20) Johnson, J. A.; Finn, M. G.; Koberstein, J. T.; Turro, N. J. Construction of Linear Polymers, Dendrimers, Networks, and Other Polymeric Architectures by Copper-Catalyzed Azide-Alkyne Cycloaddition “Click” Chemistry. Macromol. Rapid Commun. 2008, 29, 1052−1072.

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Macromolecules (21) Choi, S.-J.; Kwon, S.; Kim, T.-H.; Lim, Y. Synthesis and conformational analysis of macrocyclic peptides consisting of both 3b1-helix and polyproline helix segments. Biopolymers 2014, 101, 279−286. (22) O’Bryan, G.; Ningnuek, N.; Braslau, R. Cyclization of 3b1, 3c9 heterotelechelic polystyrene prepared by nitroxide-mediated radical polymerization. Polymer 2008, 49, 5241−5248. (23) Stokke, B. T.; Elgsaeter, A.; Kitamura, S. Macrocyclization of polysaccharides visualized by electron microscopy. Int. J. Biol. Macromol. 1993, 15 (1), 63−68. (24) Park, B.-J.; Abu-Lail, N. I. Variations in thenanomechanical properties of virulent and avirulent Listeria monocytogenes. Soft Matter 2010, 6, 3898−3909. (25) Button, B.; Cai, L.-H.; Ehre, C.; Kesimer, M.; Hill, D. B.; Sheehan, J. K.; Boucher, R. C.; Rubinstein, M. Periciliary Brush Promotes the Lung Health by Separating the Mucus Layer from Airway Epithelia. Science 2012, 337, 937−941. (26) Vu, B.; Chen, M.; Crawford, R. J.; Ivanova, E. P. Bacterial Extracellular Polysaccharides Involved in Biofilm Formation. Molecules 2009, 14, 2535−2554. (27) Marshall, A.; Hodgson, J. DNA chips: an array of possibilities. Nat. Biotechnol. 1998, 16, 27−31. (28) Heller, M. J. DNA microarray technology: devices, systems, and applications. Annu. Rev. Biomed. Eng. 2002, 4, 129−153. (29) Peterson, A. W.; Wolf, L. K.; Georgiadis, R. M. Hybridization of Mismatched or Partially Matched DNA at Surfaces. J. Am. Chem. Soc. 2002, 124, 14601−14607. (30) SantaLucia, J.; Hicks, D. The thermodynamics of DNA structuralmotifs. Annu. Rev. Biophys. Biomol. Struct. 2004, 33, 415−440. (31) Halperin, A.; Buhot, A.; Zhulina, E. B. On the Hybridization Isotherms of DNA Microarrays: The Langmuir Model and its Extensions. J. Phys.: Condens. Matter 2006, 18, S463−S490. (32) Obukhov, S.; Johner, A.; Baschnagel, J.; Meyer, H.; Wittmer, J. P. Melt of polymer rings: The decorated loop model. Europhys. Lett. 2014, 105, 48005. (33) Semenov, A. N. Contribution to the theory of microphase layering in block copolymer melts. Sov. Phys. JETP 1986, 61, 733−742. (34) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993. (35) Wijmans, C. M.; Zhulina, E. B. Polymer Brushes at Curved Surfaces. Macromolecules 1993, 26, 7214−7224. (36) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. Theory of Steric Stabilization of Colloid Dispersions by Grafted Polymers. J. Colloid Interface Sci. 1990, 137, 495−511. (37) Erbas, A.; Paturej, J. Friction between ring polymer brushes. Soft Matter 2015, 11, 3139−3148. (38) Sheiko, S. S.; Panyukov, S.; Rubinstein, M. Bond Tension in Tethered Macromolecules. Macromolecules 2011, 44, 4520−4529. (39) Kim, J. U.; O’Shaughnessy, B. Morphology Selection of Nanoparticle Dispersions by Polymer Media. Phys. Rev. Lett. 2002, 89, 23801. (40) Vologodskii, A. V.; Frank-Kamenetskii, M. D. Topological aspects of the physics of polymers: The theory and its biophysical applications. Usp. Fiz. Nauk 1981, 134, 641−673.

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Brushes of Cycled Macromolecules: Structure and Lubricating

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