Impact of Macromolecular Architecture on Bending Rigidity of

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Impact of Macromolecular Architecture on Bending Rigidity of Dendronized Surfaces Ivan V. Mikhailov,† Frans A. M. Leermakers,‡ Oleg V. Borisov,*,†,§,∥ Ekaterina B. Zhulina,†,§ Anatoly A. Darinskii,†,§ and Tatiana M. Birshtein†,⊥ †

Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004 St. Petersburg, Russia Physical Chemistry and Soft Matter, Wageningen University, Stippeneng 4, 6708 WE Wageningen, The Netherlands § St. Petersburg National University of Informational Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia ∥ Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS UPPA, Pau, France ⊥ Department of Physics, St. Petersburg State University, 198904 St. Petersburg, Russia ‡

S Supporting Information *

ABSTRACT: Nanomechanical properties of natural and artificial nanomembranes can be strongly affected by anchored or tethered macromolecules. The intermolecular interactions in polymeric layers give rise to so-called induced bending rigidity which complements the bare rigidity of the membrane. Using analytical mean-field theory, we explore how macromolecular architecture of tethered polymers affects the bending rigidities of the polymer-decorated membranes. The developed theory enables us to consider explicitly various polymer architectures including regular dendrons, Ψ-shaped, star- and comblike macromolecules as well as macrocycles. Numerical self-consistent field computations for selected (regular dendritic) topology complement the analytical theory and support its predictions. We consider both cases of (i) quenched symmetric distribution of tethered molecules on both sides of the membrane and (ii) annealing distribution in which the tethered polymers can relocate from the concave to the convex side of the membrane upon bending. We demonstrate that at a given surface coverage an increase in the degree of branching or cyclization leads to the decrease in the induced bending rigidity. Relocation of the tethered molecules from concave to convex surfaces leads to the additional decrease in polymer contribution to the membrane bending rigidity. In the latter case, a decrease in configurational entropy due to this redistributions substantially contributes to the bending rigidity.

1. INTRODUCTION Membrane-like materials comprised of synthetic polymers and hydrogels are widely used to create stimuli-responsive interfaces1−3 in the emerging field of artificial organs4 and wound-dressing applications.5 Recent advances in synthesis of nanomembranes (free-standing two-dimensional sheets of cross-linked molecules) promoted intensive research in physical/chemical properties of these materials due to emerging applications in optics and electronics.6,7 Pairs of two-dimensional leaflets of amphiphilic molecules play a key role in biology as e.g. cell membranes. Such structures function as semipermeable barriers that regulate © XXXX American Chemical Society

transport phenomena and cell adhesive properties. Artificial double layers of amphiphiles often termed as bilayer membranes find diverse applications in technology. For example, liposomes composed of naturally occurring and/or synthetic phospholipids are used as drug delivery systems for carrying hydrophilic, lipophilic, and amphiphilic drugs.8 Crosslinking and immobilization of phospholipid bilayers on solid substrates gives rise to tethered membranes that paved the way Received: November 11, 2017 Revised: April 8, 2018

A

DOI: 10.1021/acs.macromol.7b02400 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

molecules attached to a thin membrane. In section 3 we introduce a unified analytical SCF approach to describe brushes of regularly branched or cycle-containing polymers. This approach is used to evaluate bending moduli of polymer brushes with a quenched symmetric distribution of tethered macromolecules between concave and convex surfaces. Next, the effect of annealing (i.e., equilibrium partitioning of tethered macromolecules between the two sides of the membrane) is considered. In section 4 we apply scaling arguments to incorporate the effects of solvent quality and corroborate the analytical theory. In section 5 we compare the analytical theory predictions to the results obtained with the SF-SCF numerical modeling for regular dendron brushes. The details of the SFSCF computational scheme are presented in the Appendix of the Supporting Information. We demonstrate that in most cases a good agreement between analytical and numerical approaches is found, and deviations between the results are rationalized in terms of approximations made in the analytical route. In section 5.3, we present the data for the bending rigidity of dendronized surfaces at high density of brushes investigated by SF-SCF numerical modeling. Finally, in section 6 we formulate the conclusions.

to biotechnology applications in healthcare, environmental monitoring, energy storage, etc.9 Decoration of natural and synthetic membranes with tethered macromolecules is an established route to mediate properties of underlying substrates. Anchoring of (water-)soluble macromolecules at soft interfaces has a major impact not only on their adhesive, tribological, and (bio)interactive properties but also on their nanomechanical characteristics. While the impact of end-grafted nonionic linear polymers on the bending rigidity of underlying substrate is relatively well understood (see, for example, recent review10 and references therein), less is known about the induced bending rigidity of polymer brushes formed by branched or cyclic macromolecules. It is reasonable to assume that the induced rigidity increases with the amount of polymer end-attached to the membrane. Intuition, however, does not help much to figure out the role of topology of the tethered macromolecules. On one hand, linear chains can bring segments further away from the surface than branched analogues, with similar molar mass, do. On the other hand, the density of segments near the surface is relatively low for linear compared to branched macromolecules. The first effect might increase the resistance against bending; the second effect has arguably the opposite effect. The major goal of the present work is to unravel the effect of polymer architecture on the induced rigidity of dendronized surfaces mimicking thin (nano)membranes. To reach this goal, we use two complementary methods: (i) the analytical strongstretching self-consistent field (SS-SCF) approach established originally for brushes of linear polymers11−13 and later extended to brushes of branched14−17 and cyclic18 macromolecules and (ii) the numerical self-consistent field (SF-SCF) model of Scheutjens and Fleer.19 The latter has been previously used by us to study dendronized surfaces;20,21 it spans extended parameter space and fills the gap between analytical, semianalytical, and scaling approaches and computer simulations.22−29 This combined analytical and numerical approach was used in our previous work30 where we specifically focused on bending rigidity of brushes formed by regular dendrons tethered to a surface with quenched grafting density. This limitation is relaxed in the present study: Here we allow tethered macromolecules to reallocate from concave to convex side of the bent membrane, thus reducing the free energy penalty for bending. Furthermore, in the present work we implement the rigorous SS-SCF analytical approach for calculating conformational entropy of branched (or cyclic) macromolecules in concave and convex brushes, in contrast to a simplifying approximation which was used in ref 30 and led to incorrect numerical values for the bending moduli. The SS-SCF analytical theory provides a unified framework for quantitative analysis of structural and nanomechanical properties of the interfacial layers formed by tethered macromolecules with diverse architectures, including regularly branched dendrons, star- and comblike macromolecules, and cycle-containing polymers. In the numerical SF-SCF modeling we focus exclusively on brushes formed by dendrons end-grated by their root segments to substrates of varied curvatures. This combined approach enables us to examine how increasingly complex architecture of the tethered macromolecules affects the polymer-induced bending rigidity. The rest of the paper is organized as follows. In section 2 we introduce the model of bilayer formed by branched macro-

2. MODEL Consider a thin membrane decorated by branched macromolecules (e.g., dendrons) terminally attached to both sides of the membrane surface (Figure 1). All the linear segments

Figure 1. A 3D schematic illustration of fragments of a bilateral polymer brush. The grafting surface has a planar configuration (in center, i = 1) or some bended configuration: cylindrical (left, i = 2) and spherical (right, i = 3) one. Hense, the geometry is specified by the index i = 1, 2, 3. The curvature radius R is indicated.

(spacers and free branches) of the grafted macromolecules are assumed to be intrinsically flexible, with statistical segment length on the order of the monomer unit length a. The solvent strength for monomer units of the tethered macromolecules is characterized by temperature-dependent Flory−Huggins parameter χ or, equivalently, by the dimensionless second virial coefficient of monomer−monomer interactions, v = 1/2 − χ. The surface grafting density σ = a2/s specifies the number of macromolecules tethered per unit area of the surface (s is the surface area per molecule). It is assumed to be large enough to ensure overlap of the neighboring macromolecules so that they form brushes on both sides of the membrane. The employed in section 3 analytical SS-SCF approach enables us to consider within unified formalism brushes formed by tethered macromolecules of diverse architecture, including regular dendrons, star- and comblike polymers, and macromolecules containing symmetric cycles. Numerical modeling performed in section 5 deals exclusively with brushes formed by regular dendrons with the number of generations g = 0, 1, 2, 3 (where g = 0 corresponds to linear chains), equal lengths of all spacers and free branches, and functionality of branching points q = 2. B


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becomes progressively more accurate upon inferior solvent strength.35 The second assumption is always applicable in planar and concave brushes but does not apply to convex brushes of linear chains. Moreover, in the case of asymmetrically branched or cycle-containing polymers certain molecular architectures give rise to dead zones even in planar and concave geometries.18 However, for symmetric dendrons the dead zone is absent not only in planar and concave brushes but also in weakly curved convex geometry.14,15 To ensure applicability of the SS-SCF approach, we also always check for dead zones by using the numerical SF-SCF model. As demonstrated in refs 14−18 in the case that all the spacers in each particular generation have the same length (and thus the lengths of all the paths leading from any of the terminal points to the root of the dendron are equal) the dendron brush is characterized by the topological coefficient k that controls the sharpness of the self-consistent molecular potential acting in the brush in the linear elasticity regime. This potential is presented as

An isotropic brush-decorated membrane is characterized by the mean and Gaussian bending moduli, κ C and κG , respectively.31 The change in free energy per unit area of the membrane uniformly bent with curvature radii R1 and R2 can be expressed as ΔF(R1 , R 2) =

2 κ 1 ⎛1 1 2 ⎞ κ C⎜ + − ⎟ + G 2 ⎝ R1 R2 R0 ⎠ R1R 2

(1)

Here, a planar configuration of the membrane with 1/R1 = 1/R2 = 0 corresponds to the reference state, and 1/R0 is the spontaneous curvature. If brushes on both sides are identical (like in our case), 1/R0 = 0. Usually, κC ≥ 0, and it determines the amplitudes of thermal bending fluctuations. The values of κC and κG can be found from the analysis of the incremental increases in the free energy of the membrane upon bending into a cylindrical ΔF2(R1 = R, 1/R2 = 0) and into a spherical ΔF3(R1 = R2 = R) (Figure 1) geometry:32−34 ΔF2(R ) = F2(R ) − F1 =

1 1 κC 2 R2

1 ΔF3(R ) = F3(R ) − F1 = (2κC + κG) 2 R

δfint {φ(r )} δφ(r ) (2)

= kBT

3 2 2 k (H − r 2 ) 2a5

(3)

Here, r is distance from the grafting surface, φ(r) is the dimensionless concentration (volume fraction) of the monomer units, f int{φ(r)} is the free energy of monomer−monomer interactions per unit volume of the brush, and H is the brush thickness. The analytical expressions for topological coefficient k for a variety of macromolecular architectures including dendrons up to the third generation, Ψ-shaped macromolecules, short combs, and cycle-containing macromolecules are summarized in refs 17 and 18. Remarkably, the topological coefficient that is calculated for a specific macromolecular architecture may also correspond to multiple other architectures.17 To quantify the degree of branching of the brush-forming macromolecules, we use also the topological ratio

where F1 is free energy per unit area in a planar configuration of the membrane. Below we calculate polymer-induced contribution to bending moduli by analyzing changes in the free energy of the polymeric layer upon bending of the substrate into a cylinder or into a sphere with given curvature radii. In the following we distinguish two cases: (i) the number of dendrons per unit area on both sides of the membrane is quenched and equal to σ = a2/s, or (ii) the partial grafting density on each side of the membrane is annealing, while the total grafting density is kept constant. This means that contributions from both side of the membrane add up to a fixed value 2σ = σ + + σ − , whereas partial densities corresponding to different sides of the membrane σ+ and σ− are fluctuating subjected to thermal equilibration. For symmetry reasons, in the case of planar configuration of the membrane σ+ = σ− = σ. However, if a finite curvature of the membrane is imposed, one can expect a symmetry break to occur and repartitioning of the tethered macromolecules between concave and convex sides of the membrane to appear. In the latter case, the configurational entropy of partitioning of the macromolecules between concave and convex sides provides an additional contribution to the bending moduli.

η=

k 2kN = klin π

(4)

which depends on the topology (connectivity) but does not depend on the overall polymerization degree N of grafted macromolecules. Here klin = π/2N is the topological coefficient for the brush of linear macromolecules.11 Obviously, the topological ratio for linear chains ηlin = 1 and η ≥ 1 for any branched or cycle-containing polymer architectures. For brushes of regular dendrons η monotonically increases as a function of the number of generations g = 0, 1, 2, ... and functionality of the branching points q = 1, 2, .... Equation 3 enables one to derive an expression for the polymer density profile φ(r) in the brush as soon as the functional form of f int{φ(r)} is specified. The equation for the brush thickness H± on both concave and convex sides can be obtained from the normalization of the density profiles

3. SELF-CONSISTENT FIELD ANALYTICAL THEORY The analytical strong stretching self-consistent field (SS-SCF) approach was first formulated for brushes formed by linear polymer chains.11−13 It was later extended to brushes made of branched macromolecules including dendrons, star- and comblike polymers, and cycle-containing macromolecules.14−18 The main requirements for the applicability of this approach are (i) the Gaussian (linear) conformational elasticity of the macromolecules on any length scale and (ii) distribution of the free ends of the chains throughout the brush without depletion of the end segments from the region proximal to the grafting surface (absence of the so-called “dead zone”). The first assumption can be violated at high grafting densities σ leading to strong stretching of the brush-forming macromolecules, particularly under good solvent conditions, but

∫0

H±

φ±(r )s±(r ) dr = Na3

(5)

where “+” and “−” signs refer to convex and concave curvatures, respectively. The r-dependent surface area per molecule on both sides of the curved membrane can be expressed as C


Article

Macromolecules ⎛ R ± r ⎞i − 1 ⎟ s±(r ) = s±⎜ , ⎝ R ⎠

i = 1, 2, 3

on good solvent conditions; an extension to theta-solvent conditions is straightforward. The thickness of a planar brush (i = 1) is obtained by substituting eq 15 into eq 7 to give

(6)

where i = 1, 2, and 3 for planar, cylindrical, and spherical surfaces, respectively, and s± ≡ s±(r = 0). The thickness H1 of the planar brush is found from the condition s

∫0

H1

3

φ(r ) dr = Na ,

i=1

1/3 H1(s) ⎛ 2vNa 2 ⎞ =⎜ ⎟ a ⎝ sk 2 ⎠

whereas the free energy per molecule in a planar brush is

(7)

5 F1̃(s) 9 sk 4 ⎛ H1(s) ⎞ = ⎜ ⎟ kBT 20 va 2 ⎝ a ⎠

that follows from eq 5 at R → ∞ (or by setting i = 1). The free energy of the curved brush (calculated per molecule, which is indicated below by tilde) can be expressed as (±)

F̃

=

(±) ̃ Fint

+

(±) ̃ Felastic

(±)

∫0

H±

(8)

fint {φ±(r )} ·s±(r ) dr

⎧ 3⎛ 3 H± ⎞ ⎟, i=2 ⎪ H± ⎜1 ± ⎝ 2⎪ 8 R⎠ s±k ⎨ N= 2 2va 2 ⎪ 3⎛ 3 H± 1 H± ⎞ ⎜ ⎟⎟ , i = 3 H 1 ± + ⎪ ±⎜ 4 R 5 R2 ⎠ ⎩ ⎝

(9)

accounts for the contribution of excluded volume interactions in the brush while the second term (±)

̃ Felastic =

∫0

H±

felastic (r ) ·s±(r ) dr =

1 2

∫0

H±

T (±)(r ) ·s±(r ) dr

⎧⎛ H ⎞−1 ⎪ ⎜1 ± 3 ± ⎟ , i=2 8 R⎠ ⎛ H± ⎞3 ⎪ ⎪⎝ ⎜ ⎟ =⎨ 2 −1 ⎝ H1(s±) ⎠ ⎪⎛ 3 H± 1 H± ⎞ + ⎪ ⎜⎜1 ± ⎟⎟ , i = 3 ⎪⎝ 4 R 5 R2 ⎠ ⎩

is the contribution due to the conformational entropy of extended macromolecules in the brush. Here 1 T (r ) 2

(11)

is the entropic contribution to the free energy per unit volume of the brush, T(r) is the tension per unit area at distance r from the surface, and eq 11 holds under the conditions of the Gaussian (linear) elasticity for brush-forming molecules. By assuming the relationship between T(r) and φ(r), similar to that in linear chain brushes, i.e. T (±)(r ) 3k 2 = 5 kBT a s±(r )

∫r

H±

r′φ±(r′)s±(r′) dr′

(±)

∫0

H±

φ±(r ) ·s±(r )r 2 dr

h± =

3 2 2 k (H± − r 2) 4a 2v

,

ρ=

H1(s) R

(20)

⎧ 3⎛ 3 ⎞ i=2 ⎪ h± ⎝⎜1 ± h±ρ⎠⎟ ⎪ 8 s =⎨ s± ⎪ 3⎛ 3 1 2 2⎞ ⎪ h± ⎝⎜1 ± 4 h±ρ + 5 h± ρ ⎠⎟ i = 3 ⎩

(13)

(21)

The concentrational part of the free energy of curved brushes (eqs 9, 14, and 15) is equal to ⎧⎛ 5 H± ⎞ ⎟, i=2 ⎪ ⎜1 ± (±) ⎝ ⎠ 4 5 R 16 ⎪ ̃ k s H Fint 3 ± ± ⎨ = 2 kBT 10 va 7 ⎪ ⎛ 5 H± 1 H± ⎞ ⎜ ⎟⎟ , i = 3 ± + 1 ⎪⎜ 8 R 7 R2 ⎠ ⎩⎝

(14)

(22)

The latter can be also presented as

The polymer density (volume fraction) profile in the brush is then given by eq 3 as φ±(r ) = φ(r ) =

H± H1(s)

and using eq 18 present eq 19 as

(12)

Equations 8−13 may be used to calculate the free energy of a planar or curved (cylindrical, spherical) polymer brush at any curvature radius and arbitrary solvent strength specified by the functional form of f int{φ(r)}. Under good solvent conditions and low polymer density, φ(r) ≪ 1, the density of the free energy of monomer−monomer interactions in the brush can be approximated36 as fint {φ(r )}/kBT ≈ vφ 2(r )

(19)

Obviously, at R → ∞, H± → H1(s), and both lines in eq 19 reduce to eq 16. We now introduce reduced variables

and by substituting eq 12 into eq 10, we obtain final expression for the elastic free energy as ̃ Felastic 3k 2 = 5 kBT 2a

(18)

Equations 18 can be also be presented (using eq 16) as

(10)

felastic (r ) =

(17)

Equations for the thicknesses H+ and H− of the brushes on the concave and convex surfaces with curvature radius R (obtained by integration of eq 5 with φ±(r) specified by eq 15) have the following form:

where the first term ̃ = Fint

(16)

⎧ 5⎛ ⎞ 5 h±ρ⎟ , i=2 ⎪ h± ⎝⎜1 ± 16 ⎠ 2 F1̃(s) ⎪ ⎨ = s±kBT 3 skBT ⎪ 5⎛ 5 1 2 2⎞ ⎪ h± ⎝⎜1 ± 8 h±ρ + 7 h± ρ ⎠⎟ , i = 3 ⎩ (±) ̃ Fint

(15)

and is valid in planar (i = 1) and curved (concave and convex) cylindrical (i = 2) and spherical (i = 3) brushes. Below we focus

(23) D


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Macromolecules The elastic part of the free energy is given by

Then, the total free energy (per molecule) in the curved brush

⎧⎛ 5 H± ⎞ ⎟, i=2 ⎪ ⎜1 ± (±) ⎝ 4 5 8 R⎠ ̃ Felastic 3 k s±H± ⎪ ⎨ = 2 kBT 20 va 7 ⎪ ⎛ 5 H± 3 H± ⎞ ⎜ ⎟⎟ , i = 3 ± + 1 ⎪⎜ 4 R 7 R2 ⎠ ⎩⎝

equals ⎧⎛ ⎞ 5 h±ρ⎟ i=2 ⎪ ⎝⎜1 ± (±) (±) (±) ̃ ̃ ⎪ 12 ⎠ Fint Felastic F1̃(s) F̃ 5 = + = s±h± ⎨ kBT kBT kBT skBT ⎪ ⎛⎜ 5 5 2 2 ⎞⎟ h±ρ , i = 3 ⎪ ⎝1 ± h±ρ + ⎠ ⎩ 6 21

(24)

that can be also presented as

(26)

⎧ 5⎛ 5 ⎞ i=2 ⎪ h± ⎝⎜1 ± h±ρ⎠⎟ , 8 1 F1̃(s) ⎪ ⎨ = s±kBT 3 skBT ⎪ 5⎛ 5 3 2 2⎞ ⎪ h± ⎝⎜1 ± 4 h±ρ + 7 h± ρ ⎠⎟ , i = 3 ⎩

The change ΔF in the total free energy per unit area of the

(±) ̃ Felastic

membrane upon bending (comprising contributions from both sides) is given by

(25)

⎧ 5 5 5 (h+6 − h−6)ρ , i=2 (+) (−) ⎪ h + h− − 2 + F ̃ (s+) 2F1̃(s) F1̃(s) ⎪ + 12 F ̃ (s−) ΔF = + − = ×⎨ 5 6 5 kBT s+kBT s−kBT skBT skBT ⎪ 5 5 6 (h+7 + h−7)ρ2 , i = 3 ⎪ h+ + h− − 2 + (h+ − h− )ρ + ⎩ 6 21 2 κC 9(2v 4)1/3 ⎛ a 2 ⎞ 5 F1̃(s)H1 (s) = = ⎜ ⎟ kBT s 16 16 ⎝ s ⎠

3.1. Bending Moduli: Quenched Case. We start with the analysis of bending moduli of the polymer-decorated membrane in the case that the grafting areas on both sides of the membrane can not change upon bending; that is, they are quenched and equal: s+ = s− = s. The polymer contribution to the bending moduli of the polymer-decorated membrane (the so-called induced bending rigidity) arises due to extra crowding of tethered polymers that leads to an increase in the free energy of repulsive (under good or theta-solvent conditions) intermolecular interactions and concomitant decrease in the conformational entropy of the macromolecules due to their extra stretching on the concave side of the membrane. These losses in the free energy are partially compensated by a decrease in the free energy (per molecule) in the brush on the convex side. Then, eq 21 provides an expansion (up to the second order) of the brush thicknesses on both sides of the membrane in powers of the reduced curvature ρ = H1(s)/R as ⎧ 1 3 2 1∓ ρ+ ρ , i=2 ⎪ ⎪ 8 64 ⎨ H± ≈ H1(s) ⎪1 ∓ 1 ρ + 29 ρ2 , i = 3 ⎪ ⎩ 4 240

=

7/3

N7/3 k 2/3

9(2v 4)1/3 ⎛ πη ⎞−2/3 3 7/3 ⎜ ⎟ Nσ ⎝ 2 ⎠ 16

(30)

2 κG 12(2v 4)1/3 ⎛ a 2 ⎞ 4 F1̃(s)H1 (s) =− =− ⎜ ⎟ 35 kBT 21 s ⎝ s ⎠

=−

12(2v 4)1/3 ⎛ πη ⎞−2/3 3 7/3 ⎜ ⎟ Nσ ⎝ 2 ⎠ 35

7/3

N7/3 k 2/3 (31)

Here we remark that eqs 30 and 31 differ by numerical prefactors from similar ones, derived in ref 30 for a particular case of regular dendron brushes. As it was mentioned above, this difference is a result of incorrect calculation of the entropic contribution to the free energy of the bent brush in ref 30. As follows from eq 31, the bending moduli decrease as ∼η−2/3 upon an increase in the degree of branching (or cyclization) of the brush-forming macromolecules provided that N and σ are fixed. Remarkably, the ratio of bending moduli κG 64 =− κC 105

(32)

is independent of the brush architecture, that is, degree of polymerization N, grafting density σ, and topology of the brushforming macromolecules (value of η) and is the same as obtained for linear chains (which correspond to η = 1) in ref 32. 3.2. Bending Moduli: Annealing Case. Now we consider the case that upon bending a fraction ψ ≤ 1 of the brushforming molecules are redistributed from the concave to the convex side of the membrane to reduce the overcrowding of grafted molecules and minimize the free energy. Then, the grafting densities on the convex and concave sides are given by

(28)

By substituting eq 28 into eq 27 and retaining only terms up to the second order in the reduced curvature ρ = H1(s)/R ≪ 1, we obtain ⎧ 5 2 ⎪ ρ , i=2 F1̃(s) ⎪ 32 ΔF ⎨ = kBT skBT ⎪ 73 ρ2 , i = 3 ⎪ ⎩ 168

(27)

(29)

σ+ =

Subsequently, from eq 1 one obtains the bending moduli E

1+ψ a2 = a2 s+ s

(33) DOI: 10.1021/acs.macromol.7b02400 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

now includes also the configurational entropy change ΔS due

and σ− =

1−ψ a2 = a2 s− s

to the unequal partitioning of molecules between concave and convex sides of the membrane

(34)

so that σ+ + σ− = 2σ; i.e., the total number of molecules tethered on both side of the membrane is conserved. Equation 21 is now rewritten as ⎧ 3 3 h ± h± 4ρ , i=2 ⎪ ⎪± 8 1±ψ=⎨ ⎪ h 3 ± 3 h 4ρ + 1 h 5ρ2 , i = 3 ⎪± ± ± ⎩ 4 5

ΔS = − σ[(1 + ψ ) ln(1 + ψ ) + (1 − ψ ) ln(1 − ψ )] ≈ σψ 2 kB (37)

where the latter equality refers to the situation of weak bending, ψ ≪ 1. The superscript asterisk in eq 36 and below is used to (35)

indicate the free energy and bending moduli in the case of

The total change in the membrane free energy per unit area upon bending

annealing distribution of the tethered molecules between two

ΔF * ΔF − T ΔS = kBT kBT

sides of the membrane. At weak bending (ρ ≪ 1) the total free energy change per

(36)

unit area of the membrane is therefore given by

⎧ 5 5 h + h− 5 − 2 + (h+6 − h−6)ρ , i=2 ⎪ ⎪+ σF1̃ 12 a2ΔF * 2 = σψ + ×⎨ 5 6 5 kBT kBT ⎪ 5 5 6 (h+7 + h−7)ρ2 , i = 3 ⎪ h+ + h− − 2 + (h+ − h− )ρ + ⎩ 6 21

At a given value of the reduced curvature ρ = H1(s)/R, the reduced thicknesses h+ and h− in eq 35 as well as ΔF in eq 27 become functions of a single variable ψ. Minimization of ΔF* in eq 38 with respect to ψ leads to

By expanding eqs 41 and 35 at small curvatures ρ ≪ 1, we find the leading contribution in fraction ψ of relocated molecules

∂ ⎛ ΔF * ⎞ ∂ ⎛ ΔF ⎞ ∂h+ ∂ ⎛ ΔF ⎞ ∂h− + =0 ⎜ ⎟ = 2a−2σψ + ⎜ ⎟ ⎜ ⎟ ∂ψ ⎝ kBT ⎠ ∂h+ ⎝ kBT ⎠ ∂ψ ∂h− ⎝ kBT ⎠ ∂ψ (39)

⎧ ⎛ ⎞−1 ⎪ 3 ρ⎜1 + 9 kBT ⎟ , i = 2 ⎪ 10 F1̃(s) ⎠ ⎪8 ⎝ ψ (ρ ) ≈ ⎨ −1 ⎪3 ⎛ 9 kBT ⎞ ⎪ ρ ⎜1 + ⎟ , i=3 ⎪4 ⎝ 10 F1̃(s) ⎠ ⎩

According to eq 35 ⎧ ⎛ ⎞−1 3 ⎪±⎜3h±2 ± h±3ρ⎟ i=2 ⎠ 2 =⎨ ⎝ ∂ψ ⎪ 2 3 4 2 −1 ⎩±(3h± ± 3h± ρ + h± ρ ) i = 3

∂h±

5 F1̃(s) 2 (h+ − h−2) = 0, σ 3 kBT

i = 2, 3

(42)

Hence, the equilibrium fraction of molecules repartitioned from the concave to the convex side of the membrane increases proportionally to the reduced curvature and is more pronounced upon bending into spherical as compared to bending into cylindrical geometry. The reduced thicknesses of the brushes h± are given by

(40)

and eq 39 reduces to 2σψ +

(38)

(41)

⎧ ⎛ ⎞−1 ⎛ ⎞−1 ⎪1 ∓ 1 ⎜1 + 10kBT ⎟ ρ + 1 ⎜1 + 10kBT ⎟ ρ2 , i=2 ⎪ 8⎝ 9F1̃ ⎠ 16 ⎝ 9F1̃ ⎠ ⎪ H± =⎨ h±(ρ) = −1 −1 H1(s) ⎪ 10kBT ⎞ 10kBT ⎞ 2 1⎛ 1⎛ 1 2 ⎪1 ∓ ⎜1 + ρ + ⎜1 + ρ − ρ , i=3 ⎟ ⎟ ⎪ ̃ ̃ F F 4 9 4 9 15 ⎝ ⎠ ⎝ ⎠ ⎩ 1 1

(43)

By substituting ψ(ρ) from eq 42 and h±(ρ) from eq 43 in eq

36, we obtain F


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Macromolecules ⎛ ⎞ ⎛ H ⎞2 H F (i , ±) = F1⎜1 ± b1(i) 1 + b2(i)⎜ 1 ⎟ + ...⎟ , ⎝R⎠ R ⎝ ⎠

⎧ ⎪ 9 ρ2 ΔF * ⎪ 64s =⎨ kBT ⎪− 4 F1̃(s) ρ2 + 9 ρ2 ⎪ 21 sk T 16s B ⎩

i = 2, 3 (47)

⎧ ⎛ ⎞2 ⎪ 9 ⎜ H1(s) ⎟ , i=2 ⎪ 64s ⎝ R ⎠ =⎨ 2 ⎪ 4 F ̃ (s ) ⎛ H (s ) ⎞ 2 9 ⎛ H1(s) ⎞ 1 ⎪− ⎜ 1 ⎟ + ⎜ ⎟ , i=3 16s ⎝ R ⎠ ⎩ 21 skBT ⎝ R ⎠

where F1 and H1 are the free energy (per unit area) and the thickness of a planar brush (at R = ∞), and numerical (i) coefficients b(i) 1 and b2 may depend on solvent strength but are presumably independent of polymer architecture. A similar approach was used earlier for brushes of linear chains.33 The difference in the free energy (per unit area) of the bent symmetric bilateral brush with respect to its planar configuration is

(44)

ΔF (i) = 2b2(i)

Hence, within the accuracy of the second-order terms in the expansion in powers of reduced curvature, the free energy of bending into a cylindrical geometry is controlled by the entropy of repartitioning of the macromolecules from concave to convex side of the membrane. In the case of a spherically curved surface, the repartitioning entropy provides a (small) correction to the main negative term. Using eq 44, we can evaluate the bending moduli 2 κC* 9(2v 4)1/3 ⎛ πη ⎞−4/3 2 5/3 9 H1 (s) ⎜ ⎟ = = Nσ ⎝ 2 ⎠ kBT 16 32 s

2 FH 1 1

R2

,

i = 2, 3

(48)

and thus 2 κC = 4b2(2)FH 1 1

(49)

2 κG = (2b2(3) − 8b2(2))FH 1 1

(50)

The ratio of bending moduli is given by κG b (3) − 4b2(2) = 2 κC 2b2(2)

(45)

(51)

b(2) 2

2

4 1/3

κG* 12(2v ) 4 F1̃(s)H1 (s) =− =− 35 kBT 21 skBT

⎛ πη ⎞ ⎜ ⎟ ⎝ 2 ⎠

provided that does not vanish. Hence, to obtain scaling dependences of the bending moduli, the free energy and the thickness of unperturbed planar brush should be known. We keep using here the mean-field approximation and present the free energy of the brush (per molecule) as

−2/3

N3σ 7/3 (46)

Hence, the mean bending modulus κC* is controlled by the entropy of the molecule repartitioning and has a small positive value. On the contrary, the Gaussian modulus κ*G is negative and has the same value as the modulus κG for the membrane with quenched (and equal) number of molecules on both sides of the membrane. Both moduli are decreasing functions of the topological coefficient η, that is, κC*/kBT ∼ η−4/3, whereas κG*/kBT ∼ η−2/3. In contrast to the quenched case, eq 32, the ratio of the bending moduli κG/κC ∼ −Nσ2/3η2/3 in the annealing case is not a constant number but a function of the main brush architectural parameters, i.e., degree of polymerization N, grafting density σ, and topological ratio η.

̃ (H ) + Fint ̃ (H ) F (̃ H ) = Felastic

(52)

as a sum of conformational entropy contribution (F̃ elastic) and excluded volume interactions (F̃ int). The latter has the form independent of the polymer architecture 2 3 good solvent ̃ (H ) ⎧ ⎪ N a /sH, Fint ≅⎨ ⎪ 3 6 2 kBT ⎩ N a /(sH) , theta‐solvent

(53)

(for simplicity, we have omitted here dimensionless second and third virial coefficients, both on the order of unity) whereas the conformational entropy term F̃elastic(H) depends explicitly on polymer architecture. As it was discussed in ref 35 in brushes of regular dendrons or cycle-containing polymers F̃ elastic(H) can be presented using the topological ratio η as

4. SCALING ANALYSIS OF BENDING MODULI The presented above analysis is applicable if the shape of the self-consistent potential is parabolic and ensures absence of the dead zones. These conditions are fulfilled if decorating the membrane brushes are formed by regular dendrons (where within each generation the lengths of all spacers and functionalities of all branching points are the same) or symmetric cycle-containing polymers. In this case the analytical model can be used to calculate the free energies of brushes on concave and convex sides of the membrane. We might extend our results to brushes formed by polymers with arbitrary treelike branched architecture under good and theta solvent conditions by using the following heuristic arguments: The free energies (per unit area) of the weakly bent into cylinder or into a sphere brushes can be presented as an expansion in terms of reduced curvature

̃ (H ) Felastic H2 2 ≅ η kBT Na 2

(54)

Minimization of the free energy with the account of eqs 53, 54 leads to 3 2 7/3 −2/3 ⎧ , good solvent η ⎪ N (a / s ) ≅⎨ ⎪ 3 2 3 0 kBT theta‐solvent ⎩ N (a / s ) η ,

κC,G

(55)

That is in the case of a good solvent bending moduli are identical (with the accuracy of numerical coefficient) to eqs 30 and 31. On the contrary, under theta-solvent conditions the mean-field approach predicts independence of bending moduli of polymer architecture (κC,G ∼ η0). G


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Figure 2. Polymer density profiles φ(i)±(r) for cylindrically curved brushes (a) and for spherically curved brushes (b) of linear chains (g = 0) and dendrons of the third generation (g = 3) in cases when the redistribution of the polymers is not possible, i.e., for the geometrically “quenched” brush. In this plot, x⃗ and R⃗ are vectors emanating from a single point, r = |x⃗ − R⃗ | is distance from grafting surface, and negative and positive values of (x − R) correspond to concave and convex sides of the brush, respectively. In both cases, the radius of curvature is R = 200, the number of monomer units in the grafted molecules is N = 420, and the grafting density is σ = 0.02. The dashed-dotted lines corresponds to parabolic dendity profiles predicted by analytical SS-SCF theory. 3 2 7/3 ⎧ (N /5)−β /3 , good solvent ⎪ N (a / s ) ≅⎨ ⎪ 3 2 3 kBT theta‐solvent ⎩ N (a / s ) ,

A decrease in absolute values of the bending moduli upon replacement of linear chains by dendrons under good solvent conditions can be also interpreted as follows: The dependence of moduli κC and κG on the structural parameters of brushes can be represented as ⎛ πη ⎞−2/3 κC ∼ |κG| ∼ N3σ 7/3⎜ ⎟ ∼ (Nσ )2 H1 ⎝ 2 ⎠

κC,G

In the limit of q ≫ 1 the topological ratio for regular dendrons scales as η ∼ qg/2. Therefore, comparison of eqs 55 and 58 indicates that they asymptotically match each other if β = 1. This is not surprising since setting β = 1 assumes elastic force balance at any branched point; that is also an essential requisite of the analytical SCF approach. Hence, the scaling analysis points to the decrease in absolute values of the bending moduli of a dendronized membrane in good solvent upon an increase in the degree of branching, whatever model to account for the branched topology of decorating membrane macromolecules is used. In pronounced contrast to the case of a good solvent, the mean-field approximation predicts no dependence of the induced rigidity on polymer branching under theta-solvent conditions.

(56)

Because for given N and σ the brush thickness H1 decreases with increasing degree of branching, the brushes of linear chains show higher rigidities than brushes of dendrons. While the dependences of the induced moduli on the brush parameters N and σ remain universal, the type of chain branching manifests itself through the dependence of the brush thickness H1 on the topological ratio η. On the contrary, under theta-solvent conditions Fint ∼ H−2, and as a result, the bending moduli which are proportional to FH2 are independent of the brush height and thus are independent of the architecture of brush-forming macromolecules. For arbitrary branched treelike polymer (“irregular dendron”) the conformational entropy losses in stretched dendrons can be presented20 as β Felastic H2 ⎛ N ⎞ ≅ 2 ⎜ ⎟ kBT a N ⎝5 ⎠

5. NUMERICAL SELF-CONSISTENT FIELD MODELING To check and complement the analytical theory predictions, we use the SF-SCF numerical model. The SF-SCF formalism rigorously solves for the system partition function and hence enables one to obtain the thermodynamic functions for polymer brushes in various geometries on a freely jointed chain level and subject to the Bragg−Williams mean-field approximation.36 The method remains accurate even when parts of the macromolecules are stretched beyond the Gaussian elasticity regime or when ternary and higher order monomer− monomer interactions come into play. Therefore, this method can be used to evaluate the bending moduli in a wider region of grafting densities than the analytical theory would allow for. By doing so, we can also determine the range of parameters for which the analytical approximation of section 3 is applicable. The details of its computation scheme are summarized in the Supporting Information. In order to single out the effect of branching of tethered macromolecules on the bending rigidity of brushes, we have considered brushes of regular dendrons with fixed branching activity q = 2 and different number of generations (g = 0, 1, 2, 3) keeping the same total number of monomer units N and the same grafting density σ.

(57)

where 5 is the number of monomer units in the longest elastic (chemical) path from the root (grafting point) to any of terminal points of the branched polymer and the exponents β = 1 and β = 2 correspond to minimal and maximal estimates for the conformational entropy losses, respectively.20 The ratio N/5 ≥ 1 can be also used as a quantitative measure of the degree of branching. For a regular dendron of generation g (with all the spacers of the same length n and functionalities q of all the branching points) 5 = n(g + 1), N = n

q g+1 − 1 q−1

(58)

and

thus N /5 = (q g + 1 − 1)/(q − 1)(g + 1). Using eq 57, we obtain the following scaling relations for bending moduli: H


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Figure 3. Change in free energy (in kBT units) per unit area of the brush upon cylindrical (a) and upon spherical (b) bending as a function of the square of the surface curvature 1/R2.

Figure 4. (a) Dependence of the induced mean bending modulus κC (in kBT units) on the combination of the structural parameters (polymerization degree N, grafting density σ, and topological ratio η) in the quenched case in double-logarithmic coordinates (b) The negative of the induced Gaussian (−κG) plotted versus the induced mean (κC) bending modulus (both in kBT units) in double-logarithmic coordinates.

Figure 5. Polymer density profiles φ±(r) for planar brushes (dashed lines) and for cylindrically (a) and spherically (b) curved brushes (solid lines) of linear chains (g = 0) and dendrons of the third generation (g = 3) for the “annealing” case. In this plot, x⃗ and R⃗ are vectors emanating from a single point, r = |x⃗ − R⃗ | is the distance from the grafting surface, and negative and positive values of (x − R) correspond to the concave and convex sides of the brush, respectively. In cases of curved brushes, the radius of curvature is R ≈ H1, the number of monomer units per grafted molecule is N = 420, and the grafting density is σ = 0.02.

For considered here case q = 2, the values of η are 1.1755, 1.5144, and 2.0140 for dendrons of generations g = 1, 2, and 3, respectively.17 For higher number of generations (up to 6) the topological coefficients were calculated numerically in the pioneering publication of Pickett.14 To ensure that the condition of a constant number of monomer units N in dendrons is fulfilled, N should be multiple of 105. Therefore, the values of N = 105, 210, 420, and 840 were chosen for calculations. The grafting density was set in the interval from σ = 0.01 to σ = 0.08. These values of σ suffice for grafted macromolecules to occur in the brush regime.

5.1. Bending Moduli. Quenched Case. Let us first consider the quenched case, i.e., when tethered molecules can not translocate from the concave to the convex side of a membrane upon bending. Thus, the density of the monomer units increases upon bending on the concave side and decreases on the convex side. According to the analytical theory, the volume fraction profiles preserve the parabolic form on both sides. This is confirmed by SF-SCF modeling for both cylindrically and spherically curved brushes at low graft densities (see Figure 2). I


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Figure 6. SF-SCF results for the fraction of the redistributed molecules ψ as a function of curvature radius R at cylindrical (a) and at spherical (b) bending of brushes. Dashed lines show theoretical slopes at ⟨H1⟩/R ≪ 1.

Figure 7. Change in free energy (in kBT units) per unit area upon cylindrical (a) and upon spherical (b) bending in annealed case as a function of the square of the membrane curvature 1/R2.

5.2. Bending Moduli. Annealing Case. In the case when repartitioning of the brush-forming molecules from concave to convex side of the membrane is possible, the system uses this additional degree of freedom to minimize its free energy. Equivalently, chemical potentials of tethered molecules on the concave and convex sides of the brush upon bending are equalized. This results in similar segment density distributions on either side of the grafting plane (Figure 5a) upon bending into a cylinder. Upon spherical bending such conservation of volume fraction profiles is impossible due to the system geometry (Figure 5b). The condition for normalizing the volume fraction profiles

At small curvatures, the change in free energy upon bending of the membrane into cylindrical (ΔF2) and into spherical (ΔF3) geometry are proportional to the square of curvature 1/R2. As an example, Figure 3 shows the dependences of the change in the free energy ΔF2 and ΔF3 obtained by the SF-SCF method for brushes of linear chains and dendrons up to third generation that have a total number of segments N = 420 and grafting density σ = 0.01. As can be seen in this figure, the slope of ΔF2 versus 1/R2 dependences decreases as a function of degree of branching of the brush-forming molecules at the same parameters of N and σ. The values of the induced bending moduli κC and κG may be estimated from the slopes of ΔF2(R) and ΔF3(R) in the limit 1/R → 0. The full set of data obtained by SF-SCF computations for the bending modulus κC is shown in Figure 4. The calculated values of the mean bending moduli are in good agreement with the

2

+

πη −2/3 3 7/3 Nσ 2

( )

predicted analytical dependence κC = 9/32

∫0

(eq 30). However, some deviations from the analytical predictions become apparent upon an increase in the grafting density and degree of branching of grafted dendrons (i.e., with increasing g at the same N and σ). These deviations can be attributed to increasingly important contribution of ternary and higher order monomer−monomer interactions in the brush and nonlinear elasticity of strongly stretched relatively short spacers. Figure 4b presents the ratio between induced moduli κG and κC. It is seen that the relation between the Gaussian modulus κG and the mean bending modulus κC corresponds to the theoretical dependence κG = −64/105κC (eq 32).

H1

φ (r ) d r =

∫0

H(i) −

φ(i) −

∫0

H(i) +

φ(i) +

(R + r )i − 1 dr Ri − 1

(R − r )i − 1 dr = 2Nσ Ri − 1

(59)

can be satisfied with φ(r) = φ(i)− = φ(i)+ for cylindrical bending (i = 2) and is not feasible for spherical bending (i = 3). The fraction of redistributed molecules ψ calculated by the SF-SCF method is shown in Figure 6 as a function of the reduced curvature ⟨H⟩/R, where ⟨H⟩ is the first moment of the segment distribution: ⟨H ⟩ =

∑r φ(r )r ∑r φ(r )

(60)

3

that is equal to 8 H1 in the case of a parabolic profile (eq 15). J


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Figure 8. Dependence of (a) the induced mean bending modulus κC* and (b) the induced Gaussian bending modulus κG* (both in kBT units) on a combination of the structural parameters (i.e., of the degree of polymerization N, the grafting density σ, and the topological coefficient η) in the annealing case.

The dependence of ψ on (⟨H⟩/R) tends to ψ ≈ ⟨H⟩/R upon cylindrical bending and close to ψ ≈ 2⟨H⟩/R upon spherical bending at large radii of curvature R ≫ ⟨H⟩ (see Figure 6). This behavior of ψ upon bending agrees with the predictions of the analytical theory (eq 42). The reported redistribution of grafted molecules greatly affects the change in free energy upon bending of brushes. In the case of weak cylindrical bending (Figure 7a), the change in free energy ΔF2* in the annealing case turns out to be significantly smaller than ΔF2 in the absence of the redistributions (Figure 3a). In the case of spherical bending, the change in free energy ΔF3* in the annealing case is negative (Figure 7b). Thus, the spherical bending is thermodynamically advantageous in the annealed case, in contrast to the quenched case, when ΔF3 > 0 (Figure 3b). Remarkably, this trend is independent of the architecture of the brush-forming macromolecules. The induced bending moduli κC* and κG* obtained by the numerical SF-SCF method fall on master curves (Figure 8) that correspond to the prediction of the analytical theory. Small deviations of the order of the numerical coefficient in the equations arise due to inaccuracies of the free energy calculation that are associated with the discreteness of the (numerical) brush model. Altogether, the dependencies of the induced bending moduli on the structural parameters of the brush obtained by SCF modeling are in good agreement with the analytical predictions, even in the region of high grafting densities that do not correspond to the parabolic density profile. 5.3. Bending Moduli of Brushes with High Density. In sections 5.1 and 5.2 we have demonstrated the quantitative agreement between predictions of the analytical theory and the results of numerical SF-SCF method at moderate values of the grafting density (σ ≲ 0.1). The analytical approach assumes a number of approximations, and thereby it is valid in a certain parameter range. The SF-SCF numerical method is free from the limitations of the analytical theory. Therefore, a comparison of the results of the theory and numerical calculations allows us not only to confirm the validity of the analytical model but also to estimate the limits of its applicability and to study the behavior of the system in a wider parameter range. Figure 10 presents the SF-SCF data on the mean and Gaussian moduli of quenched and annealing brushes composed of linear chains (g = 0) and dendrons (g = 1 ÷ 3) under good solvent conditions in a wide range of σ up to σmax.

Figure 9. Negative of the induced Gaussian bending modulus in quenched (κG) versus the negative of the annealing (κ*G ) bending modulus in double-logarithmic coordinates.

The maximal grafting density σmax is determined as the ratio of the number of monomer units 5 in the longest path of the grafted macromolecule (the latter determines maximum possible thickness of the brush and was defined in section 4) and the total number N of monomer units in the macromolecule, g+1 5 σmax = = g+1 (61) N 2 −1 for the branching points functionality q = 2. In the case of quenched brushes the dependence of the mean bending modulus κC on the grafting density σ (Figure 10a) changes abruptly upon transition from small values of σ to values close to σmax. This change is related to the increase of mean brush density ⟨φ⟩ caused by an increase in σ (see Figure11). When σ → σmax, the value of ⟨φ⟩ → 1. In Figure 12a, the data in Figures 10a and 11 are presented as dependence of ratio κC(g)/κC(g = 0) on the mean density ⟨φ(g)⟩ for fixed value of σ. At small ⟨φ(g)⟩ up to ≈0.3÷0.4, a good quantitative agreement between ratio κC(g)/κC(g = 0) and the value of η−2/3 (predicted in eq 30) is observed. Deviations from the theoretical prediction are observed at larger ⟨φ(g)⟩ > 0.4. This is natural since developed above analytical theory accounts for only bunary repulsive interactions between monomr units and thus is valid only under good solvent conditions and at relatively low monomer volume fraction in the brush. At ⟨φ⟩ > 0.4 ternary and higher order repulsive interactions taken into account within SF-SCF K


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Figure 10. Dependencies of the normalized mean bending moduli κC and κ*C (both in kBT units) on the grafting density σ in the quenched (a) and annealing (b) cases, respectively. The dependencies are shown in redused coordinates κC/N3 and κC*/N2 for a better comparison of data at the different polymerization degrees. (c) Ratio of −κG/κC as a function of the grafting density for polymer brushes consisting of dendrons with different polymerization degree N and with different generation number g. (d) Dependence of negative values of Gaussian moduli in quenched (κG) and annealing (κ*G ) cases. In (a), (b), and (c) the crosses indicate the maximum possible values of grafting density σmax(g ) = 5/N , where 5 is the contour length of path from end point to grafting point in grafted chain.

dendrons in the brush. Up to the “transitional” value of ⟨φ(g)⟩ the dendron brushes remain more flexible than brushes consisting of linear chains with the same N and σ. At larger polymer volume fraction the brushes become very stiff due to the sharp increase of the free energy. This stiffening occurs at different σ for dendron brushes and those composed of linear chains. The value of σ ≈ σmax (i.e., ⟨φ(g)⟩ ≈ 1) for the dendron brushes corresponds to smaller density ⟨φ⟩ for brushes consisting of linear chains. Therefore, these brushes continue to be flexible at such σ. This leads to a sharp increase in the ratio κC(g)/κC(g = 0). Thus, quenched dendron brushes at mean densities ⟨φ(g)⟩ ≳ 0.6 (if brush exist at such densities) are hardly bendable. Figures 10c and 12c show the dependencies of the ratio of the Gaussian modulus (negative in magnitude) to the mean bending modulus on the grafting density and on the mean polymer volume fraction in the brush, respectively. This ratio decreases (in absolute values) with an increase polymer volume fraction in the brush. The effect of polymer concentration in the brush on the Gaussian modulus is even stronger than on the mean bending modulus. Consider now annealing brushes. According to the analytical theory, at low grafting densities the mean modules κ*C is determined only by loss of the configurational entropy due to the redistribution of grafted macromolecules, and therefore values of κ*C are relatively small (much less than value of κC for analogous quenched brushes). Figure 10b shows that the relatively small values of κC* are realized practically in the entire range of variation of the grafting density σ and, correspond-

Figure 11. Mean volume fraction of monomer units ⟨φ⟩ = ∑rφ2(r)/ ∑rφ(r) for planar brushes as a function of grafting density.

method contribute substantially to the free energy of the brush and therefore calculated properties of the brush deviate from those predicted by analytical theory. This result is in a line with the scaling theory prediction (section 4) concerning independence of bending moduli on polymer architecture under theta-solvent conditions (i.e., under dominance of ternary repulsive interactions). However, the bending moduli of the dendron brushes remain less than the bending moduli of the brushes composed of linear chains (for the same N and σ) as long as the density of the dendron brushes is less than ⟨φ⟩ ≈ 0.5÷0.6. This “transitional” value of ⟨φ(g)⟩ increases with the generation number g of L


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Figure 12. Dependensies of κC (a) and κ*C (both in kBT units) (b) on the mean volume fraction of monomer units in dendron brushes at different g ≥ 1 at the same grafting density normalized by the values of the same moduli for brushes of linear chains (g = 0). On the ordinate axis, the crosses indicate the values of the topological coefficients η−2/3(g) and η−4/3(g) in the cases of quenched and annealing brushes, respectively. (c) Ratio of −κG/κC as a function of the mean volume fraction of monomer units in brushes. (d) The negative Gaussian modulus normalized by N3 as a function of ⟨φ⟩.

such brushes. Furthermore, scaling analysis has enabled us to generalize our predictions for brushes formed by irregularly branched treelike macromolecules. One of the key predictions of the theory is that at constant surface coverage the polymer-induced bending rigidity is a decreasing function of the degree of branching. That is at constant grafting density and degree of polymerization of the tethered macromolecules the polymer contributions to the mean and Gaussian membrane bending moduli decrease upon polymer branching. This prediction has a general character and applies to many architectures of branched macromolecules (e.g., regular dendrons, hyperbranched macromolecules, polymers containing macrocycles, etc.). Moreover, the prediction applies not only to monodisperse brushes of branched polymers but also to mixed brushes of macromolecules with different topologies provided that constituent architectures demonstrate the same topological coefficient and absence of dead zones. This trend has been observed by the numerical SCF analysis performed in our previous work30 on brushes formed by regular dendrons with quenched grafting densities under good solvent conditions. However, the analytical model used in ref 30 was based on a simplifying assumption of independence of ratio between equilibrium values of Fint and Felastic of the brush curvature. Although this approximation provides correct values of the exponents in power law dependencies of bending moduli, it leads to incorrect numerical prefactors. This inaccuracy is eliminated in the present study where an adequate dependence of ratio Fint/Felastic on the brush curvature is implemented.

ingly, in the whole range of variation of polymer volume fraction ⟨φ(g)⟩ in dendron brushes. As can be seen in Figure 12b, at low polymer concentration in the brushes the ratios κC*(g)/κC*(g = 0) are equal to the theoretical values of η−4/3 and are virtually independent of ⟨φ⟩. At larger ⟨φ⟩ the change of these ratios vary nonmonotonically. but the inequality κ*C (g)/κ*C (g = 0) < 1 is valid up to maximal values of the polymer density in the brush. It should be noted that the Gaussian moduli of equivalent annealing and quenched brushes coincide in the entire interval of variation in the grafting density (κG* = κG, ∀σ, see Figure 10d). Therefore, at high polymer concentration in the brush, the dependence of the Gaussian modulus on σ becomes more pronounced (Figure 12d). Because the dependence of the mean modulus on σ changes weakly in the annealing case, the instability of a planar brush with respect to spherical bending increases sharply as the density increases.

6. DISCUSSION AND CONCLUSIONS Using the strong stretching self-consistent field method, we have developed a theory that enables us to predict how the induced bending rigidity of polymer-decorated interfaces depends on architecture of the tethered macromolecules. The analysis is operational when the topological coefficient k that controls the shape of the self-consistent potential is known; that is the case for regular dendrons (including Ψ-shaped and starlike macromolecules), comb-shaped and cycle-containing macromolecules, and when the analytical SCF approach can be used to calculate the free energies of volume interactions in M


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In the case of regular dendrons the power law dependences that follow from the coarse-grained scaling analysis are justified by the analytical SCF theory and well confirmed by the results of numerical SF-SCF modeling. We demonstrate excellent agreement between the results of numerical SF-SCF method and the predictions of SS-SCF analytical theory for both quenched and annealing brushes. It should be noted that the model underlying the analytical theory assumes an absence of the “dead zone” for free ends of grafted macromolecules. As it was discussed above, this restriction is not fulfilled on the convex side of the brush, especially for brushes consisting of linear chains. Nevertheless, even in this case, the complete quantitative agreement is observed between the values of the modules predicted by the analytical theory developed in this work and the results of the numerical SF-SCF analysis (see also works30,34). The weak effect of “dead zones” on the bending rigidity of brushes can be explained by the fact, that the bending moduli are calculated from the free energy change at very small curvatures of grafting surface. Similar to the case of quenched brushes formed by linear chains, the induced moduli κC and κG of dendron brushes are determined by the change in conformational free energy and proportional to each other. Their ratio is independent of the grafting density, degree of polymerization, and architecture of the brush-forming molecules. In the case of an annealing brush with grafted molecules repartitioning from the concave to the convex side of the bent membrane a concomitant decrease in combinatorial entropy of the system contributes substantially to the induced bending modulus κ*C . In total, the repartitioning leads to a significant decrease in the mean bending modulus as compared to that in the quenched system but does not affect the Gaussian modulus. Importantly, repartitioning can change the scaling behavior of the induced mean bending modulus. In particular, the induced mean bending modulus (manifested in the membrane elastic response upon bending into a cylinder) is controlled by the configurational entropy penalty for repartitioning of the brushforming molecules from concave to convex side of the membrane, while the variations in the free energy of excluded volume interactions and conformational entropy on both sides of the membrane cancel each other. A similar effect of the chains repartitioning was observed in SF-SCF calculations performed in ref 34 for curved brushes of linear chains. Interestingly, in the annealing case, the mean bending modulus is determined only by the thickness of the brush κC* ∼ H12 whereas the ratio of the moduli depends on all the abovementioned parameters of the brush. The absolute value of the ratio |κC*/κG*| ∼ F̃1−1 is small and decreases with an increase in the degree of branching. The Gaussian moduli κG and κ*G are the same for both quenched and annealing cases. All these regularities take place at not too high densities of the dendron brushes. At high densities, the stability of the planar configuration of the dendronized membrane increases dramatically in the case of a quenched brush and decreases in the annealing case. Our finding may have important implication for molecular design of artificial polymer-modified membranes as well as for understanding of the relations between supramolecular organization and nanomechanical properties of biological membranes decorated by “forests” of branched polysaccharides.37

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02400. Details of the SF-SCF computational scheme; Figures S1−S4 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (O.V.B.). ORCID

Frans A. M. Leermakers: 0000-0001-5895-2539 Oleg V. Borisov: 0000-0002-9281-9093 Notes

The authors declare no competing financial interest.

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

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REFERENCES

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