Dendron and Hyperbranched Polymer Brushes in Good and Poor

Article pubs.acs.org/Langmuir

Dendron and Hyperbranched Polymer Brushes in Good and Poor Solvents Inna O. Lebedeva,†,‡ Ekaterina B. Zhulina,§,∥ Frans A. M. Leermakers,⊥ and Oleg V. Borisov*,†,§,∥ †

Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS UPPA, Pau, France The Peter the Great St.Petersburg Polytechnic University, 195251, St. Petersburg, Russia § Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004, St. Petersburg, Russia ∥ National Research University of Information Technologies, Mechanics and Optics, 197101, St. Petersburg, Russia ⊥ Physical Chemistry and Soft Matter, Wageningen University, 6703 HB Wageningen, The Netherlands ‡

S Supporting Information *

ABSTRACT: We present a theory of conformational transition triggered by inferior solvent strength in brushes formed by dendritically branched macromolecules tethered to planar, concave, or convex cylindrical and spherical surfaces. In the regime of linear elasticity for brush-forming dendrons, an analytical strong stretching self-consistent field (SS-SCF) approach provides brush conformational properties as a function of solvent strength. A boxlike model is applied to describe the collapse transition in brushes formed by macromolecules with arbitrary treelike topology, including hyperbranched polymers. We demonstrate that an increase in the degree of branching, that is, an increase in the number of generations or/and functionality of branching points in tethered macromolecules, makes the swelling-to-collapse transition less sharp. A decrease in surface curvature has a similar effect. The numerical Scheutjens−Fleer selfconsistent field approach is used to analyze the collapse transition in dendron brushes in the nonlinear stretching regime. It is demonstrated that inferior solvent strength suppresses stratification that is exhibited under good solvent conditions by densely grafted dendron brushes.

1. INTRODUCTION Dendronized surfaces obtained by the covalent linkage of treelike branched macromoelcules to solid substrates have recently attracted a lot of attention. A significant impact to adhesive, rheological, and biointeractive properties of the surfaces can be achieved at sufficiently high degrees of surface functionalization (e.g., via the formation of dendron brushes or “forests”).1−4 Brushlike layers of dendritically branched macromolecules are used to stabilize nanoparticles in an aqueous environment. It is presumed that such layers create a sharper steric repulsive barrier capable of preventing particle aggregation caused by van der Waals attractive forces.5 It is also anticipated that dendron brushes might outperform brushes of linear polymers in the reduction of the friction force between sliding polymer-bearing surfaces. The self-assembly of linear−dendritic block copolymers in selective solvents (e.g., in water) gives rise to diverse nanostructures with interfaces decorated by brushlike layers of dendron blocks.6−9 The tuned solubility of the dendron blocks could lead to the change in the morphology of the nanostructure that can be used in the delivery of biologically active molecules. Hybrid nanoassemblies with cationic dendron coronas have the potential to be nonviral transfectants for genetic material.10−12 The experimental activities have motivated a significant theoretical effort13−25 aimed at understanding basic relationships between the architecture of branched macromolecules and © XXXX American Chemical Society

structural and interactive properties of the polymer coatings. The most recent developments in the analytical theory and computer simulations of dendron brushes can be found in the reviews.26,27 The conformations of anchored macromolecules and physicochemical properties of the dendronized surface can be efficiently tuned by varying environmental conditions such as the ionic strength and pH of the solution in the case of ionic dendrons and temperature in the case of thermoresponsive macromolecules. This opens an opportunity to design smart interfaces responsive to one or multiple external stimuli. In particular, ultrathin coatings made of water-soluble thermoresponsive branched macromolecules are promising in applications in microfluidics and biomedicine, including drug delivery, magnetic resonance imaging, and biolubrication because of their ability to change the degree of swelling and hydration dramatically as a response to the temperature variation. Thermoresponsive PEG-based dendron coatings demonstrate pronounced modulation of their protective properties caused by the temperature-induced swelling-to-collapse transition that leads to the reversible aggregation of dendron-stabilized nanoparticles.5 Understanding the conformational transitions experienced Received: December 6, 2016 Revised: January 4, 2017

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DOI: 10.1021/acs.langmuir.6b04285 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. Schematics of convex (a), planar (b), and concave (c) dendron brushes.

refs 19, 21, and 22 at high grafting density under good solvent conditions, the brush-forming dendrons may reach the limit of their extensibility. As a consequence, the elastic response of tethered dendrons becomes nonlinear, and the brush exhibits a pronounced stratification with populations of weakly and strongly stretched macromolecules, thus minimizing the overall conformational entropy losses. Inferior solvent strength leads to the decrease in dendron stretching, and one can expect a concomitant transition from a stratified to uniform structure of the brush to occur. To examine the evolution in the structure of very dense brushes upon inferior solvent strength, we use the numerical Scheutjens−Fleer method that does not involve any preassumption of the elastic response of dendrons and accounts explicitely for their finite extensibility. The rest of the article is organized as follows. In Section 2, we introduce a model of a nonionic dendron brush grafted onto an impermeable surface of different morphologies and immersed in a solvent of arbitrary thermodynamic quality. In Sections 3−5, we analyze the swelling-to-collapse transition in concave, convex, and planar brushes using the SS-SCF formalism. In Section 6, we present the results of numerical modeling of the collapse transition in dense dendron brushes using the Scheutjens−Fleer approach with the aim of unravelling the effect of solvent strength on brush stratification. In Section 7, we discuss a boxlike model of the swelling-to-collapse transition in the dendron and hyperbranched polymer brushes grafted to a planar surface or a surface of spherical and cylindrical particles with a vanishing radius of curvature. Finally, in Section 8 we present a discussion of our results and conclusions.

by branched macromolecules in confined geometries is important for applications in multifunctionalized analytical microfluidics chips28 as well as for the design of mesoporous materials functionalized by organic, organometallic, and biological functions.29 The swelling-to-collapse conformational transitions in polymer brushes made of linear neutral polymers have been initially studied with a mean-field boxlike model.30,31 Later, a more refined analytical self-consistent field (SCF) approach based on the strong stretching (SS) approximation32 has been developed to study the swelling-to-collapse transition in a planar brush.33 The analytical SCF model accounted for the gradients in polymer distribution in the brush and predicted the variations in the brush properites with inferior solvent strength. One of the main findings of the theory was that in contrast to the phase character of the coil-to-globule transition in a single linear chain, the temperature-induced collapse of a planar brush composed of such chains occurs as a nonphase smooth transition. The aim of the present study is to understand how the polymer architecture affects the swelling-to-collapse transition in brushes of dendritically branched macromolecules end-attached to surfaces of various geometries. We focus primarily on planar and convex spherical and cylindrical brushes, which are of interest for biomedicine (antifouling, antimicrobal surfaces, bioactivation, etc.) and nanoparticle stabilization. We investigate also the swelling-to-collapse transition in brushes of dendrons end-grafted to concave matrixes, that is, to inner surfaces of the cylindrical and spherical pores. We demonstrate in this study that the swelling-to-collapse transition in dendron brushes grafted to surfaces with positive, zero, or negative curvature (that is, to convex, planar, or concave surfaces) can be investigated with a unified SS-SCF approach. The latter presumes the overlap and noticeable stretching of the end-tethered dendrons normally to the surface with respect to their unperturbed state. Although for concave and planar brushes this approach is explicit, it also remains relatively accurate for dendron brushes on weakly curved convex surfaces. Compared to convex brushes of linear chains with pronounced zones depleted of the chain free ends (dead zones) near the grafting surface,34,35 dead zones in convex dendron brushes decrease in size, and such brushes can be reasonably described by the SS-SCF formalism for a wide range of conditions. The analytical SS-SCF approximation is applicable as long as the brush-forming dendrons exhibit linear (Gaussian) elasticity on all length scales. Under these conditions, the brush retains a uniform structure with smooth distributions of the end segments and branching points in the direction perpendicular to the grafting surface. However, as has been demonstrated in

2. MODEL We consider brushes formed by dendritically branched macromolecules attached through the root segment to a planar, cylindrical, or spherical surface or to the inner surface of spherical or cylindrical pore (Figure 1). Each dendron is characterized by the number of generations g = 0, 1, 2, ..., where g = 0 corresponds to linear chains. In a general case, the spacers and the terminal branches may have different lengths, and the branching points may have different functionalities. In a regular dendron with constant functionality q of all of the branching points and a constant number n of monomer units in all of the spacers and free branches, the total number of monomer units equals N = n(qg+1 − 1)/(q − 1). The solvent strength is characterized by the temperaturedependent second virial coefficient of the monomer−monomer interaction v, which is positive or negative under good or poor solvent conditions, respectively, and vanishes in the theta point. The ternary repulsive interactions between monomer units are B


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The topological coefficient k in eq 1 depends only on the total molecular weight of the macromolecules and their connectivity but is remarkably independent of the grafting density and interactions in the brush. For linear chains with degree of polymerization N, the topological coefficient equals klin = π/2N.33 The topological coefficient enters as a prefactor in the local stretching functions for spacers and free branches of the dendrons. It is calculated from the condition of the balance of elastic tension in all of the branching points together with the condition of conservation of the number of monomer units in each linear segment. The details of the calculations and analytical expressions for k can be found elsewhere.18,21 The analytical expressions for k for a number of selected macromolecular architectures are collected in Table SI1. We also define a topological ratio η = k/klin. Obviously, the topological ratio for a linear chain is η = 1, whereas for any branched macromolecule η ≥ 1 and may serve as a quantitative characteristic of the degree of branching. For the interactions’ free-energy density f{c(z)}, we use the virial expansion

accounted for through the third virial coefficient w, which is positive and assumed to be temperature-independent. All of the spacers and free branches of the dendrons are assumed to be intrinsically flexible, with a statistical segment length on the order of the monomer unit length a. The grafting density σ = 1/s is characterized by the number of dendrons per unit area of the grafting surface (s is the surface area per dendron). For convex or concave grafting surfaces with radius of curvature R, 1/h = 2πR/s is the number of molecules per unit length of the cylindrical surface (or cylindrical pore) and p = 4πR2/s is the total number of chains attached to a spherical particle (or inside a spherical pore).

3. STRONG STRETCHING SELF-CONSISTENT FIELD APPROACH The strong stretching self-consistent field (SS-SCF) approach enables us to consider conformational transitions in dendron brushes of arbitrary morphology without a preassumption of the distribution of elastic stress in the dendrons. This approach was first suggested for linear chain brushes32,33 and further extended to brushes made of regular dendrons in a good solvent.13,18,21 The main prerequisite of the theory is the Gaussian (linear) elasticity of all spacers and branches in the brush-forming dendrons. Although this assumption could be violated at high chain grafting density under good solvent conditions, it becomes progressively more accurate with inferior solvent strength and a concomitant decrease in the dendron stretching. Another requirement for the applicability of the SS-SCF approach is that the end segments of dendrons are distributed across the brush with no dead zone proximal to the grafting surface. This condition always holds in concave and planar brushes but is violated in convex (cylindrical and spherical) brushes formed by linear polymers or weakly branched dendrons. However, an increase in the degree of branching (e.g., by increasing the number of generations and functionality of the branching points) leads to a more uniform structure of the convex brush, with the virtual disappearance of the dead zone in convex dendron brushes.14 This theoretical prediction is also confirmed by extensive numerical SF-SCF modeling of spherical dendron brushes.20 Hence, when applying the SS-SCF scheme to dendron brushes of arbitrary morphology, we keep in mind that it is exact (within the linear elasticity regime) for concave and planar brushes and is approximately accurate for brushes of strongly branched macromolecules at convex geometries. As demonstrated in refs 13, 14, 17, 18, and 21 in the case in which all of the spacers within a generation have the same length (and thus the lengths of all of the paths leading from any of the terminal points to the root of the dendron are equal), the dendron brush is characterized by the universal topological parameter k. The latter is specified by the dendron architecture and controls the sharpness of the self-consistent molecular potential in the brush linear elasticity regime. The self-consistent field theory provides the chemical potential of a monomer unit in the brush a distance z from the grafting surface as ∂f {c(z)} 3 = kBT 2 k 2(Λ2 − z 2) ∂c(z) 2a

f {c(z)} = vc 2(z) + wc 3(z) kBT

(2)

and parameter Λ in eq 1 has to be determined from the condition of vanishing osmotic pressure at the edge of the (nonconfined) brush, ⎛ ⎞ ∂f {c(z)} Π = ⎜c(z) − f (c(z))⎟ =0 ∂c(z) ⎝ ⎠z = H

(3)

By substituting eq 2 into eq 1, one finds that 2vc(z) + 3wc 2(z) =

3 2 2 k (Λ − z 2 ) 2a 2

(4)

and an explicit expression for the polymer density profile is derived as c(z) = −

v + 3w

⎛ v ⎞2 k2 ⎜ ⎟ + (Λ2 − z 2) ⎝ 3w ⎠ 2a 2w

(5)

(1) Figure 2. Polymer concentration profiles in the brushes formed by linear chains (blue lines) and by second-generations dendrons, η = 2.09; q = 3 (red lines) under the same good, θ, and poor solvent strength conditions, as indicated.

where c(z) is the concentration (number density) of monomer units and f{c(z)} is the free energy of interactions per unit volume of the brush. C


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Langmuir By combining eqs 1, 2, and 3, one finds the relation between parameter Λ and brush thickness H as ⎧ H2 , v≥0 ⎪ 2 Λ =⎨ v 2a 2 ⎪ H2 − , v≤0 ⎩ 6wk 2

where ⎛ H ⎞(i = 1) α=⎜ ⎟ ⎝ Hθ ⎠

and (6)

Hθ(i = 1) ⎛ 4 N =⎜ ⎝ π as a

Finally, the equation for the brush thickness is obtained from the normalization of the polymer density profile

∫0

H

⎛R ± z⎞ ⎟ dz c(z)s⎜ ⎝ R ⎠

=N

(7)

β=

c(z)s dz = N , i = 1

(8)

c(z)z i − 1 dz = Nσi

⎧ ⎛ ⎞2 ⎪ β2 α2 z + − ⎜⎜ (i = 1) ⎟⎟ , v ≥ 0 ⎪ −β η + η η η ⎝ Hθ , η = 1 ⎠ ⎪ c(z) ⎪ =⎨ c0 ⎪ ⎛ ⎞2 β2 α2 ⎪ ⎜ z ⎟ , v≤0 − + + − β η η ⎪ ⎜ (i = 1) ⎟ η 4η ⎪ ⎝ Hθ , η = 1 ⎠ ⎩

(10)

4. SWELLING-TO-COLLAPSE TRANSITION IN A PLANAR DENDRON BRUSH (i = 1) The final equation for the planar brush thickness can be presented in reduced variables as ⎞ ⎟, ⎟ ⎠

β2 4

(15)

where c0 is the concentration of monomers at the grafting surface for the linear chain brush under the θ-solvent condition: c0 = cθ(i,=η =1)1(z = 0) =

v≥0 ⎞ ⎟ ⎟, v ≤ 0 ⎟ ⎠

(14)

The polymer concentration profiles in a planar brush can be presented as

where the generalized grafting density σi is defined by the equation

⎧ ⎛ α ⎪− αβ + (α 2 + β 2)arcsin⎜ ⎜ ⎪ 2 + β2 α ⎝ ⎪ ⎪ π =⎨ ⎛ 2 ⎪ 2⎞ ⎛ ⎜ α ⎪− 5 αβ + ⎜α 2 + β ⎟arcsin⎜ ⎪ 2 4⎠ ⎝ 2 ⎜ α + ⎪ ⎝ ⎩

s 1 a1/2v ≡ s N 3 × 21/4 w 3/4η1/2

⎧ ⎛ 2 3 ⎞1/3 ⎪ ⎜ 8a vN ⎟ , β ≫ 1 2 2 ⎪ H = ⎨ ⎝ sπ η ⎠ ⎪ 2Nw ⎪ , β ≪ −1 ⎩ s| v |

(9)

⎧1/s , i=1 ⎪ σi = ⎨1/2πh , i = 2 ⎪ ⎩ p/4π , i = 3

π 1/2 a1/2v 3 × 23/4 w 3/4k1/2

is a reduced parameter that characterizes the solvent strength (deviation from the θ point). The asymtotic solutions of eq 11 in the limits of good (β ≫ 1) and poor (β ≪ −1) solvents lead to the following expressions for the brush thickness

H

H

(12)

(13)

In the opposite limit of a thin cylinder or small sphere, R → 0, the normalization condition in eq 7 can be presented as

∫0

1/2 2w ⎞ ⎟ η ⎠

is the brush thickness in the θ point and

i−1

where signs + and − refer to convex and concave surfaces, respectively. Here we ascribe index i = 1, 2, 3 to planar, cylindrical and spherical brushes, respectively. In the limit R → ∞ (or by setting i = 1), eq 7 reduces to the following form for a planar brush:

∫0

1/2 ⎛ 8 N2 2w ⎞ ⎟ ≡⎜ 2 k ⎠ ⎝ π as

klin 2(Hθ(i,=η =1)1)2 2a 2w

The dependence of the reduced polymer concentration c(z)/c0 on the distance z from the grafting surface under good, θ, and poor solvent conditions is presented in Figure 2 for two selected values of the branching ratio η corresponding to linear chains

(11)

Figure 3. Dependence of H/Hθ (a) and H/Hθ, η=1 (b) on reduced solvent strength parameter u ≡ βη1/2 ≃ vs1/2w−3/4 for the planar brush of linear chains η = 1 (curves 1) and dendrons of the second, η = 2.09 (curves 2), and third, η = 3.43 (curves 3), generations; q = 3. D


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5. CONVEX AND CONCAVE DENDRON BRUSHES The integration of the polymer density profile in eq 7 leads to the following equations for the reduced thickness of curved dendron brushes (where signs + and − refer to convex and concave surfaces, respectively). 5.1. Cylindrical Brush.

(η = 1) and second-generation dendrons (η = 2.09, q = 3). As one can see from Figure 2, an increase in the degree of branching of the brush-forming macromolecules does not qualitatively change the shape of the concentration profiles but leads to a compacting of the brush. This compacting is more pronounced under good or θ solvent conditions whereas under poor solvent conditions the density profiles level off irrespectively of the branched architecture of the brush-forming chains. Under poor solvent conditions, all of the density profiles exhibit a discontinuity at the edge of the brush. The magnitude of the jump is independent of the polymer architecture and equals the density in a polymer globule, −v/2w. This feature of the density profiles is inherent in the analytical SS-SCF model, which specifies only the most probable state of the system. As we see below, the numerical SF-SCF model achieves better resolution of the polymer density profiles by accounting for the thermal fluctuations of the free branches. As a result, a jumpwise drop in the polymer density under poor solvent conditions is replaced by a continuous though sharp decrease in the polymer density profiles near the edge of the brush. In Figure 3, we present the solvent strength dependence of the dendron brush thickness normalized by the dendron brush thickness under θ solvent conditions H/Hθ (Figure 3a) or normalized by the thickness of the brush formed by linear chains under θ solvent conditions, H/Hθ,η=1 (Figure 3b). As follows from Figure 3a, an increase in the degree of branching (expressed as an increase in the topological ratio η at fixed {N,σ}) makes the collapse transition progressively less sharp. This trend can be explained by the increasingly important role of repulsive ternary interactions in the brushes formed by strongly branched macromolecules.

⎧ ⎛ ⎞ α ⎪− αβ + (β 2 + α 2)arcsin⎜ ⎟+ ⎜ 2 ⎪ 2 ⎟ β α + ⎝ ⎠ ⎪ ⎪ ⎪ 2 ⎛⎜− 3 α 2β + (α 2 + β 2)3/2 − β 3⎞⎟, v≥0 ⎠ ⎪ 3ρc ⎝ 2 ⎪ π =⎨ ⎛ ⎞ 2 ⎪ ⎛ β2 ⎞ ⎜ ⎟ α 5 2 ⎪− αβ + ⎜ + α ⎟arcsin⎜ ⎟+ 2 ⎪ 2 ⎝4 ⎠ ⎜ β + α2 ⎟ ⎪ ⎝ 4 ⎠ ⎪ ⎞ ⎪ 2 ⎛ 3 2 β β 2 3/2 3/2 ⎪ ⎝⎜− α β + (α + ) − ( ) ⎠⎟, v ≤ 0 2 4 4 ⎩ 3ρ

where ρc = ±

Rc Hθ(i = 1)

Under θ solvent conditions (β = 0), eqs 16 reduce to α2 +

4 3 α = 1, β = 0 3πρc

(17)

to give α ≅ 1 − /3πρc at ρc ≫1 and α ≅ (3πρc/4) opposite limit of ρc ≪1. 5.2. Spherical Brush. 2

1/3

⎧ ⎛ ⎞ α ⎪−αβ + (β 2 + α 2)arcsin⎜ ⎟ + 2 ⎜⎛ − 3 α 2β + (α 2 + β 2)3/2 − β 3⎟⎞ ⎜ 2 ⎪ ⎠ 2 ⎟ 3ρs ⎝ 2 ⎝ β +α ⎠ ⎪ ⎪ ⎛ ⎞⎞ ⎪ 1 ⎛ 8 3 α 3 2 2 2 2 2 ⎟⎟ , v ≥ 0 ⎪+ 2 ⎜⎜ − α β − 2αβ + αβ(α + β ) + (β + α ) arcsin⎜⎜ 2 2 ⎟⎟ 3 4 ρ ⎪ ⎝ β + α ⎠⎠ s ⎝ ⎪ ⎪ π ⎛ ⎞ =⎨ 3/2 ⎞ ⎛ β2 ⎜ ⎟ 2 ⎛ 2 ⎛ β ⎞3/2 ⎞ β⎞ 2 ⎛ 3 2 α ⎪ 5 2 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ arcsin − αβ + + α + − α β + α + − ⎟ ⎜ ⎜ 2 ⎟ ⎜ ⎪ 2 ⎝ ⎝ 4 ⎠ ⎟⎠ 4 3ρs ⎝ 2 4⎠ β ⎠ ⎝ 2 ⎟ ⎜ ⎪ ⎝ 4 +α ⎠ ⎪ ⎪ ⎛ ⎛ ⎞⎞ 2 ⎪ 3 2⎞ ⎜ 8 3 ⎞ ⎛ β2 ⎛ 2 ⎜ ⎟⎟ β β 1 α 2 ⎪+ 2 arcsin − α β − α + αβ α + + + α ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟⎟ , v ≤ 0 2 ⎪ 4ρs 2 ⎜ 3 4 4⎠ ⎝4 ⎠ ⎝ ⎜ β + α 2 ⎟⎟ ⎪ ⎝ 4 ⎠⎠ ⎝ ⎩

Rs Hθ(i = 1)

Under θ solvent conditions (β = 0), eqs 18 reduce to α2 +

4 3 1 4 α + α = 1, β = 0 3πρs 4ρs 2

in the

(18)

In the limit of |ρ| → ∞, eqs 16 and 18 cross over to eq 11 for the planar case. Note that for concave brushes, eqs 16 and 18 are applicable only at |ρ| ≥ 1. In the opposite limit of ρ ≪ 1 (only for convex brushes), eqs 16 and 18 assume simpler forms. 5.3. Cylindrical Brush (i = 2, ρc ≪ 1).

where ρs = ±

(16)

c

⎧(α 2 + β 2)3/2 − β 3 , v≥0 c c ⎪ c ⎪ 3 2 3/2 1 = − αc βc + ⎨ ⎛ ⎛ β 2 ⎞3/2 β2⎞ 2 ⎪ ⎜α 2 + c ⎟ − ⎜ c ⎟ , v ≤ 0 c ⎜ ⎟ ⎜ 4 ⎟ ⎪⎝ 4 ⎠ ⎝ ⎠ ⎩

(19)

leading to α ≅ 1 − 2/3πρs at ρs ≫1 and α ≅ (4ρs)1/4 in the opposite limit of ρs ≪1. E

(20)


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Figure 4. Dependence of H(i=3)/H(iθ = 1) (a) and (H/Hθ)(i=3) (b) on β in the case of a spherical grafting surface (eqs 18); the reduced radii of curvature of the surface in panel (a) are ρc = −5 (1), ∞(2), 2 (3), 0.2 (4); the reduced radii of curvature of the surface in panel (b) are ρc = −5 (1), ∞(2), 0.1 (3), 0.01 (4).

In Figure 4a,b, we present the dependences of H/Hθ(ρ = ∞) and H/Hθ(ρ) on solvent strength expressed through parameter β ≈ v for different values of reduced curvature radius ρ for the case of a spherical surface (the trends demonstrated by dendron brushes grafted onto cylindrical surface are qualitatively the same) calculated according to eqs 18. As one can see from Figure 4a, a decrease in the curvature of the grafting surface (in both positive and negative ranges) leads to a progressive increase in the swelling of the dendron brush at any value of the solvent strength. Figure 4b demonstrates that the sharpness of the swelling-tocollase transition increases with an increase in the curvature of the surface. For any finite positive value of the curvature radius, the width of the transition range |Δvtr| → 0 at N → ∞. That is, in the limit of infinitely long chains the transition recovers the character of the second-order phase transition, similar to that in the convex brushes formed by linear polymer chains. This follows from the analysis of eqs 16 and 18 because in the N → ∞ limit the l.h.s. is dominated by the highest order in the 1/ρ term. Clearly, this does not apply in the case of the concave grafting surface.

where αc =

H Hθ(i = 2)

⎞1/3 ⎞1/3 ⎛ 3 N 2 Hθ(i = 2) ⎛ 3 N ⎟ 2 w 2 w =⎜ ≡ ⎟ ⎜ ⎝ 2πa 2 kh ⎠ a ⎠ ⎝ π 2a 2 ηh

and βc =

1/3 1/3 ⎛ 2 ⎞4/3 a 2/3 v ⎛ hN ⎞ 22/3π 1/3a 2/3 v ⎛ h ⎞ ⎜ ⎟ ⎜ ⎟ ≡ ⎜ ⎟ 2 ⎝ 3 ⎠ π 1/3 w 2/3 ⎝ η2 ⎠ 34/3 w 2/3 ⎝ k N ⎠ (21)

Spherical Brush (i = 3, ρs ≪ 1). ⎧− 2α β 3 + α β (α 2 + β 2)+ s s s s s s ⎪ ⎪ ⎛ ⎞ ⎪ 2 αs 2 2 ⎜ ⎟, v ≥ 0 ⎪(αs + βs ) arcsin⎜ 2 2 ⎟ α β + ⎪ ⎝ s s ⎠ ⎪ 2 ⎪ ⎛ 2 βs ⎞ 8 π = − αs 3βs + ⎨ 2α |β |3 αs|βs|⎜⎝αs + 4 ⎟⎠ s s 2 3 ⎪− + + ⎪ 4 2 ⎪ ⎞ ⎛ ⎪ 2 ⎟ ⎜ βs 2 ⎞ ⎪⎛ 2 αs ⎟⎟ arcsin⎜ ⎟, v ≤ 0 ⎪ ⎜⎜αs + 2 4⎠ βs ⎟ ⎜ 2 ⎪⎝ α + ⎝ s 4 ⎠ ⎩

6. STRATIFICATION IN A DENDRON BRUSH: EFFECT OF THE SOLVENT STRENGTH The SS-SCF approach and the obtained results are based on the assumption of the Gaussian (linear) elasticity of the brushforming dendrons on all length scales. This approximation would be first violated in dense brushes under good solvent conditions. Here, strong repulsions between the dendrons could cause their stretching up to the limit of extensibility. As has been demonstrated in refs 19−21, the stretching of the dendrons in planar or convex brushes up to the limit of extensibility gives rise to the intrabrush stratification. This effect is most pronounced in brushes formed by the firstgeneration dendrons, i.e., arm-tethered polymer stars. At high grafting density or/and a large number of arms per star, the brush exhibits pronounced two-layer structure. The layer proximal to the surface is formed by weakly stretched stars with stems and arms remaining in the linear elasticity regime. It is also composed of almost fully stretched stems of the stars whose branches form the distal layer of the brush. The free arms of the upper population of stars are found exclusively in the distal layer. As a result, the distribution function of the end segments of the free arms exhibits a strongly bimodal shape. The bimodal distribution of

(22)

where αs =

H Hθ(i = 3)

⎞1/4 ⎞1/4 ⎛ 8 pN 2 Hθ(i = 3) ⎛ 4 pN 2w ⎟ ≡ ⎜ 3 3 2w ⎟ =⎜ 2 3 ⎝π a k ⎠ a ⎠ ⎝π a η

and βs =

1/4 25/8a3/4 v ⎛ N 2 ⎞ π 1/2a3/4 v 3 −1/4 ⎟ ⎜ ( k pN) ≡ 3 3 × 21/8 w 5/8 3π 1/4 w 5/8 ⎝ η p ⎠

(23) F


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Figure 5. Monomer density profiles (left), the end-point distribution function (middle), and the branching point distribution (right) in a brush of starlike dendrons with different numbers q of free arms, as indicated in the panels. The total number of monomers in a star is kept constant and equal to N = 900; the grafting density is σ = 0.1. The values of the χ parameter corresponding to varied solvent strength are indicated near the curves.

the end segments has been first observed in the numerical SCF calculations19,20 and later analyzed on the basis of the generalized SS-SCF approach that accounts for finite extensibility effects.21

Here we use the same numerical SF-SCF method as in refs 19 and 21 in order to analyze the collapse transition in the firstgeneration dendron brush that exhibits a two-layered structure G


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Langmuir

Hence, the results obtained within the boxlike model should be applicable to brushes made of macromolecules with arbitrary treelike architecture. In this section, we assume that the radius of curvature R of spherical or cylindrical surfaces is much smaller than the characteristic thickness of the brush H. The free energy of the brush is presented as

under good solvent conditions. The details of the method can be found in refs 21 and 42. As has been demonstrated in ref 17, the onset of stratification in the brush formed by arm-tethered stars occurs when the most stretched tethers reach the limit of their extensibility close to the grafting surface. For the stars with q arms in a good solvent, this corresponds to grafting density σ* ≡ a2/s* ≅ [q(q+ 1)]−1. In Figure 5, we demonstrate how an increase in the number q of star free arms at fixed values of N = 900 and σ = 0.1 triggers a stratification of the brush in a good solvent while preserving a uniform brush structure in a poor solvent. The set of polymer density profiles with increasing q from top (q = 2) to bottom (q = 5) is presented in the left column in Figure 5. The values of the Flory polymer−solvent interaction parameter are indicated at the curves. The corresponding end-point and branching-point distributions are presented in the right and middle columns, respectively. For a star with a small number of free arms q = 2, the stratification threshold is estimated to be σ* = 1/6 = 0.17, and the chosen value of σ = 0.1 is below σ*. Under these conditions, the brush structure is uniform in a good solvent, which is indicated by smooth distributions of both branching points and free ends. An inferior solvent strength modifies these distibutions: the endpoint distribution develops a maximum at the brush edge, and the distribution of branching points exhibits the corresponding maximim in the middle of the brush. The development of these maxima reflects the expected divergency in the distibution of free ends, similar to that in a condensed brush of linear chains. The shapes of the end-point and branching-point distributions in a poor solvent do not change with an increase in q, and the presense of the boundary maximum in the end-point distribution does not indicate brush stratification. An increase in q of up to q = 4 leads to the development of sharp maxima in both end-point and branching-point distributions in a good solvent whereas the corresponding distributions stay almost intact in a poor solvent. For q = 4, the threshold grafting density σ* = 1/20 = 0.05 is below the chosen value of σ = 0.1, and the brush is expected to undergo stratification in a good solvent. The stratification is indicated by progressively increasing maxima in the end-point and branchingpoint distributions upon further increase in q. The data presented in Figure 5 proves that inferior solvent strength could suppress the stratification in dendron brushes.

F = Fint + Fconf

(24)

Here the first term Fint ≅ kBTN (vc + wc 2)

(25)

accounts for the interactions between monomer units of the dendrons as a function of the average concentration

σiN Hi of the monomer units in the brush The second term accounts for the conformational entropy losses in stretched dendrons and can be presented in the general form as c≅

̃

β Fconf H2 ⎛ N ⎞ ≅ 2 ⎜ ⎟ kBT a N ⎝5 ⎠ (26) where exponents β̃ = 1 and 2 correspond to minimal and maximal estimates of the conformational entropy losses, respectively,19 and 5 is the number of monomer units in the longest elastic (chemical) path from the root (grafting point) to any of the terminal points of the branched polymer. For a regular dendron (with all the spacers the same length) 5 = n(g + 1) and thus N /5 = (q g + 1 − 1)/(q − 1)(g + 1). If we assume that for a hyperbranched polymer, similar to a randomly branched one, 5 ≈ N3/4 , then N /5 ≈ N1/4 . Minimization of the free energy of the brush, eq 24, with respect to H (or to c) leads to the following equation for the brush thickness as a function of v v i α 2i + 2 − α − 1 = 0, i = 1, 2, 3 (27) v*

where we have introduced a reduced thickness of the brush ⎛ H ⎞(i) α=⎜ ⎟ ⎝ Hθ ⎠

7. COLLAPSE OF DENDRON AND HYPERBRANCHED POLYMER BRUSHES: A BOXLIKE MODEL With the aim of unravelling the effects of branched topology on the characteristics of the swelling-to-collapse transition in planar and convex brushes formed by macromolcules with arbitrary treelike topology (e.g., dendron or hyperbranched ones), we employ a simplified boxlike mean-field model similar to that proposed in ref 31 for a description of the corresponding transition in conventional linear chain brushes. The boxlike model neglects radial gradients in the polymer density and the free ends distribution but enables us to obtain a general equation for the swelling coefficient of a brush of arbitrary topology as a function of solvent strength that captures qualitatively combined effects of both polymer branching and surface topology on the swelling-tocollapse transition. The boxlike model enables us to characterize the brush-forming dendrons by only two parameters: the total number of monomer units N in a dendron and the number of monomer units 5 in the longest elastic path on the dendron. The ratio N/5 ≥ 1 may serve as an alternative to the introduced topological ratio η as a quantitative measure of the degree of branching. (For linear chains, it obviously equals unity.)

Here ̃ 1/(2i + 2) Hθ(i) ⎡⎢ 2i 2 4⎛ N ⎞−β ⎤⎥ ⎜ ⎟ = σ wN 2i i ⎝ 5 ⎠ ⎥⎦ a ⎣⎢ 3a

(28)

is the thickness of the brush at the θ point (v = 0) and (i + 2)/2(i + 1)⎡

3 ⎛ 2i ⎞ v* = ⎜ ⎟ i⎝3⎠

iβ ̃ /2

⎢σi⎛⎜ N ⎟⎞ ⎢⎣ ⎝ 5 ⎠

⎤1/(i + 1) N1 − iw(i + 2)/2a−(4i + 3)⎥ ⎥⎦

(29)

In the limits of good (v ≫ v*) and poor (v ≤ 0, |v| ≫v*) solvents, the brush thickness is given by the following asymptotic expressions

H (i) a

H

⎧⎡ ̃ 1/(i + 2) ⎛ N ⎞ β ⎤⎥ ⎪⎢ i 3⎜ ⎟ v ≫ v* , ⎪ ⎢ i σivN ⎝ ⎠ ⎥ 5 ⎦ ⎪ ⎣ 3a =⎨ ⎪⎡ −i ⎤1/ i ⎪ ⎢ 2σiwa N ⎥ , v ≤ 0, |v| ≫ v* ⎪ ⎣ |v| ⎦ ⎩

(30)


Article

Langmuir At N/5 = 1 eqs 27 and 29 are reduced to those obtained in ref 31 for brushes formed by linear chains. The value of v* determines the width of the transition from swollen to collapsed states of the brush. For a planar brush, i = 1, v*≈ N0, whereas for i = 2, 3 the width of the transition vanishes with an increase in N.

The employed analytical formalism is essentially based on the assumption of the Gaussian (linear) elasticity of the dendrons extended in the brush. This condition might be violated under good solvent conditions/high grafting density when (some of) the linear segments within the dendrons reach the limit of extensibility.19,21 However, upon a decrease in solvent strength and deswelling of the brush as a whole, the Gaussian elasticity approximation becomes progressively more and more reliable. We anticipate, therefore, that stratification exhibited by dendron brushes highly swollen in a good solvent disappears with brush deswelling caused by a decrease in the solvent strength. This prediction is confirmed by the numerical self-consistent field modeling based on the Scheutjens−Fleer approach presented above: we considered the brush formed by the first-generation dendrons that is equivalent to the brush of arm-grafted stars. When the grafting density and/or branching-point functionality is chosen to be sufficiently high, then two prononced maxima in the end-segments distribution point to the stratification of the brush into two sublayers under good solvent conditions. Our results demonstrate that these two maxima merge into a single one with a decrease in the solvent strength that causes a decrease in the extension of the brush-forming dendrons. Because the stretching of the chains in the brush increases with a decrease in curvature, we anticipate that the effect of stratification is less (more) prononced in convex (concave) brushes. As a result, sufficiently dense concave brushes may retain a two-layer structure even under sufficiently poor solvent strength conditions. On the contrary, if the brush is formed by ionic (polyelecrolyte) dendrons, then a decrease in the solvent strength may trigger brush collapse with the concomitant appearance of stratification even when the brush was homogeneous under good solvent conditions. A similar effect has been predicted earlier for brushes formed by linear polyelectrolytes.36 We aim to consider this effect in detail in a forthcoming publication. Within SS-SCF theory, the branched architecture of the brushforming dendrons is accounted for though the topological coefficient k or, equivalently, though the topological ratio η. At η = 1, the behavior of the linear chain planar or convex brushes is recovered. On the other hand, a closer inspection of eqs 12 and 30 enables us to reconstruct for a dendron brush the classical Alexander−de Gennes boxlike model37−39 developed initially for brushes of linear chains. This model is based on the minimization of the free energy of the brush (per chain). In the case of the dendron brush, the free energy can be expressed as a function of a single large-scale parameter such as the overall brush thickness H as

̃

For dendritic brushes, however, v* ≈ (N /5)iβ /2(i + 1). Hence, the increase in the degree of branching leads, as expected, to a widening of the transition and its shift further below the θ point due to increasingly important higher-order (ternary) repulsions. It is important to keep in mind that the derivation presented above assumes that the mode of stretching of the branched macromolecules (specified by parameter β̃) is not changing upon contraction of the brush. As demonstrated in ref 17, for regular dendrons the results of the boxlike model match those obtained within the more accurate SS-SCF approach if β̃ is set to unity, which could be considered to be a direct consequence of the force balance in branching points implemented in both schemes. On the other hand, for strongly asymmetric Ψ-shaped macromolecules with a relatively short stem and long free arms, the crossover to the β̃ = 2 stretching mode, which assumes fairly equal stretching of all of the branches, was predicted.17 Furthermore, we assumed that the brush remains laterally homogeneous in course of collapse, which is justified for sufficiently dense brushes under moderately poor conditions. A deep collapse of the brush may lead to the appearance of lateral undulations and, eventually, to a splitting of the uniform planar or cylindrical dendron brush into separated collapsed clusters, which is not considered here.

8. DISCUSSION AND CONCLUSIONS Using the strong stretching self-consistent field approximation, we have developed a theoretical model of brushes formed by dendritically branched macromolecules grafted to planar, convex, and concave surfaces of different morphologies and immersed in solvents of arbitrary strength. The most important finding of the work is that the interplay of branching and surface curvature affects the sharpness of the swelling-to-collapse transition triggered by a decrease in the solvent strength: an increase in the degree of branching (e.g., by increasing the number of generations or the functionality of the branching points) leads to a smoothing of the transition. A similar effect is caused by an increase in the grafting density or by a decrease in the surface curvature. This trend persists for convex as well as concave surfaces. For convex surfaces, the transition acquires the character of the second-order phase transition (within the vanishing transition-temperature range) in the thermodynamic limit of N → ∞. The latter observation should have the most important implications for the molecular design of smart coatings made of stimuli-responsive brushes used in microfluidics: an increase in the degree of branching may lead to a decrease in the sensitivity of the brush to the temperature variations. Remarkably, as has been shown in ref 18, multicomponent brushes formed by branched macromolecules with the same topological parameter k exhibit (within Gaussian elasticity limits) unified density profiles and a fairly uniform distribution of the end segments of all of the brush-forming species across the brush irrespective of the grafting density and solvent strength. Hence, we expect that this unified density profile (given by eq 5) will be maintained for any composition of the multicomponent brush undergoing the collapse transition.

F (H ) F (H ) F (H ) = conf + int kBT kBT kBT

(31)

where the first term Fconf (H ) 3 H2 2 = η kBT 2 Na 2

(32)

describes the conformational entropy contribution to the free energy, whereas Fint(H ) N3 N2 =v + w 2 2 + ... kBT sH sH

(33)

accounts (in the virial approximation) for binary and ternary short-range intermonomer interactions. Renormalization of the first term in eq 31 by a factor of η2 enables us to properely I


Article

Langmuir

Adsorption Kinetics Surface Characterization, and Nonfouling Properties. J. Am. Chem. Soc. 2011, 133, 10940−10950. (2) Yeh, P. Y. J.; Kainthan, R. K.; Zou, Y.; Chiao, M.; Kizhakkedathu, J. N. Self-assembled monothiol-terminated hyperbranched polyglycerols on a gold surface: a comparative study on the structure, morphology, and protein adsorption characteristics with linear poly(ethylene glycol). Langmuir 2008, 24, 4907−4916. (3) Hyperbranched Polymers: Synthesis, Properties and Applications; Frey, H., Cao, C., Yan, D., Eds.; John Wiley, 2011. (4) Schüll, C.; Frey, H. Grafting of hyperbranched polymers:From unusual complex polymer topologies to multivalent surface functionalization. Polymer 2013, 54, 5443−5455. (5) Gillich, T.; Acikgöz, C.; Isa, L.; Schlüter, A. D.; Spencer, N. D.; Textor, M. PEG-Stabilized Core-Shell Nanoparticles: Impact of Linear versus Dendritic Polymer Shell Architecture on Colloidal Properties and the Reversibility of Temperature-Induced Aggregation. ACS Nano 2013, 7, 316−329. (6) Wurm, F.; Frey, H. Linear-dendritic block copolymers: the state of the art and exciting perspectives. Prog. Polym. Sci. 2011, 36, 1−52. (7) Blasco, E.; Pinol, M.; Oriol, L. Responsive Linear-Dendritic Block Copolymers. Macromol. Rapid Commun. 2014, 35 (12), 1090−1115. (8) Garcia-Juan, H.; Nogales, A.; Blasco, E.; Martinez, J. C.; Sics, I.; Ezquerra, T. A.; Pinol, M.; Oriol, L. Self-assembly of thermo and light responsive amphiphilic linear dendritic block copolymers. Eur. Polym. J. 2016, 81, 621−633. (9) Mirsharghi, S.; Knudsen, K. D.; Bagherifam, S.; Niström, B.; Boas, U. Preparation and self-assembly of amphiphilic polylysine dendrons. New J. Chem. 2016, 40, 3597−3611. (10) Yu, T.; Liu, X.; Bolcato-Bellemin, A.-L.; Wang, Y.; Liu, C.; Erbacher, P.; Qu, F.; Rocchi, P.; Behr, J.-P.; Peng, L. An amphiphilic dendrimer for effective delivery of small interfering RNA and gene silencing in vitro and in vivo. Angew. Chem. 2012, 124, 8606−8612. (11) Liu, X.; Zhou, J.; Yu, T.; Chen, C.; Cheng, Q.; Sengupta, K.; Huang, Y.; Li, H.; Liu, C.; Wang, Y.; Pososso, P.; Wang, M.; Cui, Q.; Giorgio, S.; Fermeglia, M.; Qu, F.; Pricl, S.; Shi, Y.; Liang, Z.; Rocchi, P.; Rossi, J. J.; Peng, L. Adaptive Amphiphilic Dendrimer?Based Nanoassemblies as Robust and Versatile siRNA Delivery Systems. Angew. Chem. 2014, 126, 12016−12021. (12) Liu, X.; Liu, C.; Zhou, J.; Chen, C.; Qu, F.; Rossi, J. J.; Rocchi, P.; Peng, L. Promoting siRNA delivery via enhanced cellular uptake using an arginine-decorated amphiphilic dendrimer. Nanoscale 2015, 7, 3867−3875. (13) Pickett, G. T. Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest. Macromolecules 2001, 34, 8784−8791. (14) Zook, T. C.; Pickett, G. T. Hollow-Core Dendrimers Revised. Phys. Rev. Lett. 2003, 90 (1), 015502. (15) Kröger, M.; Peleg, O.; Halperin, A. From Dendrimers to Dendronized Polymers and Forests: Scaling Theory and its Limitations. Macromolecules 2010, 43, 6213−6224. (16) Gergidis, L. N.; Kalogirou, A.; Vlahos, C. Dendritic Brushes under Good Solvent Conditions: A Simulation Study. Langmuir 2012, 28, 17176−17185. (17) 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. (18) Zhulina, E. B.; Leermakers, F. A. M.; Borisov, O. V. Ideal mixing in multicomponent brushes of branched polymers. Macromolecules 2015, 48 (21), 8025−8035. (19) 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, 9555−9566. (20) Rud, O. V.; Polotsky, A. A.; Gillich, T.; Borisov, O. V.; Leermakers, F. A. M.; Textor, M.; Birshtein, T. M. Dendritic Spherical Polymer Brushes: Theory and Self-Consistent Field Modelling. Macromolecules 2013, 46, 4651−4662. (21) 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.

account for the impact of branching on the conformational entropy losses due to the extension of the brush-forming chains. Obviously, at η = 1 the classical Alexander−de Gennes result for a linear chain brush is recovered. The most important feature of eq 32 is that it enables the correct prediction of the scaling dependence of the large-scale properties (i.e., the overall brush thickness) on the degree of branching irrespectively of the particular nature of interactions in the brush. For instance, it can be readily applied for constructing a boxlike model of convex dendron brushes, similar to those developed in refs 40 and 41 for linear chain convex brushes. The power law dependences obtained from eq 31 in the asymptotic limit of good, θ, or poor solvents match (with the accuracy of numerical factors) those given by eqs 11 and 16. At this point, we remind the reader that the universal analytical SS-SCF description and account of the topology of the brushforming chains through parameter η apply only to (mono- or multicomponent) brushes of regular dendrons. However, the predicted trends should be valid for arbitrary branched architectures, including hyperbranched polymer brushes as well. In practice, the dendronization of the surface aimed at the modification of its physicochemical and interactive properties is performed via robust surface-induced radical polymerization, leading to brushes of irregularly (hyper)branched chains. In this situation, the above-described approach is not strictly applicable, though it might provide rough estimates of the effect of branching. Instead, a more coarse-grained approach, as described in Section 7 and based on the ad hoc preassumption of the dominant stretching mode, can be used. Remarkably, qualitative predictions of the analytical SS-SCF theory concerning the effect of branching on the features of the conformational collapse-toswelling transitions remain valid for brushes of hyperbranched chains as well. Moreover, for regular dendrons in the limit of q ≫ 1, both models lead to the same dependence of the brush thickness on q and g if the longest path stretching mode (β̃ = 1) is assumed.

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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.langmuir.6b04285. Analytical expressions for the topological coefficient k for a number of selected macromolecular architectures (PDF)

■

AUTHOR INFORMATION

ORCID

Oleg V. Borisov: 0000-0002-9281-9093 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship (PIRSES-GA-2013612562-POLION) within the Seventh European Community Framework Programme and by the Government of the Russian Federation, grant 074-U01.

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REFERENCES

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Dendron and Hyperbranched Polymer Brushes in Good and Poor

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