Brushes of Dendritically Branched Polyelectrolytes - ACS Publications

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Brushes of Dendritically Branched Polyelectrolytes O. V. Borisov*,†,‡,§ and E. B. Zhulina‡,§ †

CNRS, UMR 5254, Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, Université de Pau et des Pays de l’Adour, Pau 64053, France ‡ Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg 199004, Russia § St. Petersburg National University of Informational Technologies, Mechanics and Optics, St. Petersburg 197101, Russia ABSTRACT: We present an analytical theoretical framework to describe a polymer brush formed by dendritically branched polyions root-tethered to a planar inert surface. First, a mean-field boxlike model is used to obtain the average thickness of the dendron brush with arbitrary number of generations in branched macromolecules. Here, finite extensibility of dendrons with high degree of ionization is taken into account. Second, a more refined selfconsistent field Poisson−Boltzmann formalism is developed to analyze the internal structure of the brushes formed by branched polyions with Gaussian elasticity. Brushes of first-generation ionic dendrons (arm-grafted starlike polymers) are considered in detail. Diagrams of states for such brushes are constructed in salt-free and salt-added solutions. The analytical dependences for the overall brush thickness in various regimes, the profiles of electrostatic potential, monomer units, and mobile ions are derived explicitly as a function of grafting density, branching functionality of macromolecules, and ionic strength in solution.

1. INTRODUCTION Brushes formed by dendritically branched or hyperbranched macromolecules that are root-tethered to planar substrates or surfaces of colloidal particles have attracted growing attention during the past decade. In addition to being ultrathin protective layers preventing protein adsorption and particle aggregation, these brushes could also introduce specific functions to the interface and tune its bioresponsive properties.1−5 It has been also predicted that dendron shells could improve tribological properties of the haired colloidal particles compared to those stabilized by linear macromolecules.6 The theory of nonionic dendron brushes has been developed in a number of papers7−14 and recently reviewed in ref 15. It was demonstrated that the scaling approach proposed by Alexander16 and de Gennes17 for brushes of linear chains can be extended to describe the structural properties of brushes formed by root-tethered dendrons. This extension is based on a reliable guess about strain distribution in the stretched dendrons.15 A more refined self-consistent field model of dendron brushes was originally formulated by Pickett7 and further extended by Polotsky et al.11 The latter model provided explicit analytical expressions for the molecular potential and the polymer density profiles in a particular case of the brush formed by the first-generation dendrons (arm-grafted stars). The predictions of the theory were found in a good agreement with the results of numerical modeling.11 The structure and properties of ionic dendron brushes are understood less. Up to now it remains unclear how the architecture of charged branched macromolecules mediates the thickness and the elastic response of ionic dendron brushes. At the same time, cationic dendrimers and dendrigrafts formed by © XXXX American Chemical Society

cationic (e.g., poly-L-lysine-based) dendrons root-tethered to linear chains or small colloidal particles are widely explored in diagnostics, drug, and gene delivery as nonviral gene vectors. They are also probed as nanopharmaceuticals capable of preventing formation of amyloid fibrils. Furthermore, the extracellular layers of branched ionic biomacromolecules (polysaccharides and glycoproteins) which can be assimilated to dendritic polyelectrolyte brushes play an important role in cell interaction and adhesion. In this paper we analyze theoretically how the interplay between branched architecture of root-tethered macromolecules and the electrostatic interactions between the monomers governs the equilibrium structure of a planar ionic dendron brush. To the best of our knowledge, this is a first analytical theoretical development in the direction of ionic dendron brushes. The article is organized as follows. First, we consider brushes formed by ionic dendrons with arbitrary number of generations and arbitrary functionality of the branching points using the boxlike model. As it was established in the earlier studies, the conformations of tethered linear polyelectrolytes (PEs) are controlled primarily by the spacial distribution of the mobile counterions. The counterions are distributed outside of the layer of loosely grafted polyions but condense inside the volume of PE brush upon an increase in the grafting density of the chains.18−20 Here, we analyze the impact of branching of the PE dendrons on the counterion localization and discuss the diagram of states of the ionic Received: October 22, 2014 Revised: January 16, 2015

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DOI: 10.1021/ma502157r Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules dendron brush. The effect of finite extensibility of the dendrons is also addressed. Second, we apply the self-consistent field (SCF) analytical model to get insights into the internal structure of PE dendron brushes. This can be achieved by linking the self-consistent molecular potential acting in the dendron brush7,11 with the generalized Poisson−Boltzmann approach formulated earlier23 for linear end-grafted PE chains. The SCF analytical model implies the Gaussian elasticity of the dendron branches on all length scales. We demonstrate that end-grafting of such macromolecules to a planar surface induces nonuniform normal stretching of the dendrons with widely spread distribution of their free ends. We focus further specifically on the case of firstgeneration PE dendron brush (which is equivalent to the brush of arm-tethered PE stars). The solution properties of such PE stars have been extensively studied during the past decade24−28 and are discussed in detail in the recent review.30 The selfconsistent field approach allows us to obtain explicit analytical expressions for the structural properties of the brush (polymer and counterions density profiles, distribution of the electrostatic potential, etc.) as a function of the number of arms in the brush-forming PE stars. Finally, we summarize the conclusions and discuss future extensions of this study.

We introduce also the curvilinear distance along the dendron from the root of the dendron to any of its terminal points (the longest elastic path). This path comprises 5 = n(g + 1) monomer units. The ratio N /5 = (q g + 1 − 1)/[(q − 1)(g + 1)] ≥ 1

is used following ref 11 as a measure of the degree of branching in the dendrons. For a linear chain (g = 0 or q = 1), N/5 = 1. In the particular case of g = 1, the tethered first-generation dendrons are equivalent to starlike polymers with f = q + 1 flexible branches (arms), with one branch (steam) attached to the surface by the end-point, and q = f − 1 unattached branches (see Figure 1b). Each branch has the degree of polymerization n ≫ 1. Total number of monomer units in the star is N = nf = n(1 + q). The PE dendron brush is in contact with bulk solution of monovalent salt with concentrations c− = c+ = cs of counter- and co-ions that specify the Debye screening length κ−1 = (8πlBcs)−1/2. The degree of ionization of the polyion α is assumed to be independent of the pH and salinity cs in the solution. The thermodynamic quality of the solvent for noncharged backbone of the macromolecule is assumed to be close to theta conditions which assures predominance of ionic interactions in the brush over the excluded-volume interactions. We introduce also the “bare” Gouy−Chapman length Λ related to charge density per unit area arising due to tethered polyions as s Λ= 2πlBαN (1)

2. THE MODEL We consider a planar brush of charged dendrons (shown schematically in Figure 1a) tethered by the root segment to an impermeable planar surface and immersed into a polar solvent (e.g., water).

It depends on the degree of ionization, α, and on polymer surface coverage, N/s, but is independent of branching.

3. PLANAR BRUSH OF CHARGED DENDRONS: BOXLIKE MODEL In a dendron brush the intermolecular interactions between tethered macromolecules dominate over the intramolecular ones. Within the boxlike model the brush is characterized by the average (or total) thickness H, and all the gradients in polymer density and electrostatic field distributions are neglected. The free energy (per dendron) in the brush with thickness H can be presented as F(H ) = Fion(H ) + Fex.vol(H ) + Fconf (H )

Figure 1. Schematic presentation of a root-tethered charged dendron of the second generation, g = 2 with functionality q = 2 (a) and of the arm-tethered to the surface polyelectrolyte star (first generation dendron, g = 1 with functionality q = 4) (b).

(2)

where Fion(H) and Fex.vol(H) account for the respective contributions of ionic and the short-range repulsive (excluded volume) interactions between monomer units, and Fconf(H) accounts for the conformational entropy losses in the stretched dendrons. In the linear elasticity regime the latter (entropic) contribution to the free energy is presented as

Each dendron with the number of generations g = 0, 1, 2, ... and the functionality of the branching points q = 1, 2, 3, ... is comprised of N = n(qg+1 − 1)/(q − 1) monomer units. Here n ≫ 1 is the number of monomer units per spacer, dendrons with g = 0 or with q = 1 are linear chains. The spacers in the dendrons are assumed to be intrinsically flexible (the Kuhn segment length on the order of monomer size a), and a ≃ lB, where lB = e2/εkBT is the Bjerrum length. The fraction α of monomer residues in the dendrons are permanently charged. In experimentally relevant situations α ≥ 1/n. Dendrons are tethered with area s per dendron (or, equivalently, with dimensionless grafting density σ = a 2 /s). When the intermolecular interactions dominate over the intramolecular ones, the tethered dendrons form a brush.

β Fconf (H ) 3 H2 ⎛ N ⎞ ⎜ ⎟ = kBT 2 a 2N ⎝ 5 ⎠

(3)

where the values of β = 1 and β = 2 correspond to the lower and upper estimates for the entropic penalty to stretch the dendrons in the brush.15 In scaling terms, the value of exponent β = 1 corresponds to the stretching of the longest elastic path in the dendron or, equivalently, to the decrease in the average degree of stretching of spacers as a function of the generation ranking number in compliance with the force balance condition B


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Macromolecules in each branching point. The analysis performed in ref 15 supports the value of β = 1 for nonionic dendron brushes in the linear elasticity regime. The value of β = 2 (used, e.g., in ref 8) corresponds to a uniform stretching of all the dendron branches. For the purpose of generality, in this section we keep 1 ≤ β ≤ 2 as an independent parameter. Making use of the results for PE brushes of linear chains,18−20 the ionic contribution to the free energy can be formulated as ⎧ αN[ln(αN /sH ) − 1], osmotic brush ⎪ Fion(H ) ⎪ charged brush ≅ ⎨−α 2lBN 2H /s , kBT ⎪ 2 2 ⎪ α (N /2c Hs), salted brush ⎩ s

In the particular case of N/5 = 1, eqs 8−10 are reduced to the boundaries between the regimes in the phase diagram of brush formed by linear PE chains. Equations 6, 8, 9, and 10 allow us to predict the following trends in the behavior of an ionic dendron brush at any value of β ≥ 1. Equation 8 indicates that for strongly branched dendrons a higher grafting density is required to entrap counterions inside the brush. As follows from eq 9, upon an increase in the branching parameter (N/5 ) the region of dominance of steric repulsions (quasi-neutral regime) extends toward smaller grafting densities. It follows from eq 10 that for strongly branched dendrons a higher salt concentration is required to provoke the salt-induced contraction of the brush. We analyze these trends more specifically for the case of tethered starlike polyions (see below). As demonstrated by eq 6, the strongest swelling of the brush caused by osmotic pressure of the confined counterions occurs in the osmotic regime. In this regime using of Gaussian approximation for the conformational free energy, eq 3, leads to underestimation of the restoring elastic force and, therefore, to overestimation of the brush thickness (“overstretching”). In order to take into account nonlinear elasticity effects (finite extensibility of the chains), we, following the lines in refs 21 and 22, introduce instead of eq 3 the nonlinear elastic potential

(4)

Here, the first line corresponds to the regime of confinement of the counterions inside the brush at low ionic strength (the osmotic regime), the second line corresponds to the regime of barely charged brush (released counterions), whereas the third line corresponds to the regime of partial screening of Coulomb interactions in the brush by the ions of added salt (the salt dominated brush regime). The boundaries between different regimes are discussed below. Finally, the contribution of the excluded-volume interactions to the free energy can be presented, in the virial approximation, as Fex.vol(H ) = N (υϕ + wϕ2) kBT

β− 1 ⎡ ⎛ H ⎞2 ⎤ Fconf (H ) 3 ⎛N⎞ ⎟ ⎥ ln⎢1 − ⎜ = − 5⎜ ⎟ ⎝ 5a ⎠ ⎦ kBT 2 ⎝5 ⎠ ⎣

which reduces to eq 3 at weak extension, H ≪ 5a. The corresponding restoring elastic force

(5)

where ϕ ≡ ϕ(H) = a N/sH is the average volume fraction of monomer units in the brush with thickness H. Minimization of the free energy of the dendron brush (defined by eqs 2−5)with respect to H leads to the following power law expressions for the brush thickness H in the regimes dominated by the ionic interactions between monomers: 3

⎧ α1/2N (5/N )β /2 , osmotic brush ⎪ ⎪ 2 3 β H /a ≅ ⎨(α lBN a /s)(5/N ) , charged brush ⎪ 2 1/3 2 1/3 /3 β ⎪ N (a /s) (α /2c ) (5/N ) , salted brush ⎩ s

⎛ N ⎞ β− 1 1 ∂Fconf (H ) H /5a = 3⎜ ⎟ ⎝ 5 ⎠ 1 − (H /5a)2 kBT ∂H

(6)

⎡ α ⎢ 3 H /a ≅ 5⎢ ⎢1 + ⎣

⎛ N ⎞2 − β α⎜ ⎟ ≪1 ⎝5 ⎠

(8)

(9)

The crossover from the osmotic to the salt dominated regime occurs at the threshold salt concentration cs ≃ (α1/2/as)(N /5)β /2

⎤1/2

(13)

(14)

and under these conditions eq 13 reduces to the second line of eq 6. However, an increase in the degree of ionization α or/and (at β ≤ 2) in the degree of branching N/5 leads at (α /3)(N /5)2 − β ≥ 1 to the full stretching of the longest elastic path and H approaches asymptotically the value of 5a. The nonlinear increase of the entropic elastic force leads to weaker stretching of the dendrons and, consequently, to higher concentration of the counterions entrapped in the brush in the osmotic regime. Therefore, account of nonlinear corrections

The ionic interactions dominate over the short-range repulsions under theta-solvent conditions as long as s ≥ α −1(N /5)β /2 w−1/2a 2

2−β

( 5N ) ⎥ ⎥ α N 2−β ⎥ ( ) ⎦ 3 5

As follows from eq 13, the dendrons in the brush are weakly stretched compared to the length of the longest elastic path, H ≪ 5a, as long as

(7)

The onset of localization of the counterions inside the brush (that is, the crossover between the osmotic and the charged brush regimes) occurs at s ≃ α 3/2lBaN 2(5/N )β /2

(12)

diverges at H → 5a, which assures limiting extensibility of the dendrons up to the contour length of the longest elastic path 5a at any value of the applied stretching force or swelling osmotic pressure. Using eqs 4 and 11, we obtain from the condition of minimum of the free energy a more general expression for the thickness of the dendron brush in the osmotic regime

At sufficiently high grafting densities of macromolecules providing the predominance of nonelectrostatic interactions between monomers (specified below), the brush thickness H is given by the balance of Fex.vol(H) and Fconf(H) as 2 1/3 1/3 ⎧ υ (5/N )β /3 , good solvent ⎪ N (a / s ) H /a ≅ ⎨ ⎪ 2 1/2 1/4 β /4 ⎩ N (a /s) w (5/N ) , theta solvent

(11)

(10) C


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which is proportional to α1/2. The characteristic length H0(k) depends also on the number of monomer units per dendron, N, number of generations, g, and branching functionality, q, via the parameter of molecular potential, k. The latter can be calculated numerically for arbitrary N, g, and q from the condition of elastic force balance in the branching points following the scheme proposed by Pickett.7 Remarkably, k is independent of the grafting density σ and of the details of intermolecular interactions in the brush. For linear polyelectrolyte chains comprising N monomer units, the electrostatic length defined by eq 16 coincides with the characteristic length

leads to widening of the osmotic regime in which the brush thickness remains virtually unaffected by addition of salt. Remarkably, if β = 2 applies in a dendron brush, the stretching of a dendron in the osmotic regime does not depend on N and is controlled only by the length of the longest elastic path 5 and degree of ionization α. On the contrary, if β = 1, the brush thickness H in the osmotic regime depends on the degree of dendron branching. Below we demonstrate that for the specific case of first-generation polyelectrolyte dendron brush the self-consistent field Poisson−Boltzmann analysis supports the value of β = 1.

4. BRUSH OF CHARGED DENDRONS: THE SELF-CONSISTENT FIELD APPROACH 4.1. General Formalism. To study the brush of ionic dendrons in more details, we use the analytical self-consistent field (SCF) approach combined with the Poisson−Boltzmann framework to account for the electrostatic interactions between charged species. These interactions are described in terms of the self-consistent electrostatic potential Ψ(z) originating due to the charged monomers of the tethered dendrons and thermally equilibrated mobile salt ions. In the case of dendritically branched strong PE with quenched (positive) fractional charge α per monomer (independent of the pH and salinity cs in the solution), the molecular potential U(z) acting in the brush (chemical potential of a monomer unit) coincides with the electrostatic energy αeΨin per monomer. That is

H0 =

d2ψin(z) dz 2

ψin(z) ≈

ρ(z) = αc(z) + c+(z) − c −(z) =

Q ( k ) = ρ (z )H =

H0(k) =

(15)

Λ̃(k) =

1/2

2a α 3 k

⎡ ̃ ⎛ (κ Λ̃)2 + 1 − 1 ⎞ ⎟ + 2 ln⎢ κ Λ + ψout(z) = 2 ln⎜⎜ ⎟ ⎢⎣ κ Λ̃ + κ Λ̃ ⎝ ⎠

(19)

H 2πlBH0 2(k)

(20)

H 2(k) 1 = 0 2πlBQ (k) H

(21)

associated with the electrostatic potential ψout(z) above the brush edge, i.e., at z > H

(16)

(κ Λ̃)2 + 1 − 1 + (κ Λ̃ − (κ Λ̃)2 + 1 − 1 − (κ Λ̃ −

⎤ (κ Λ̃)2 + 1 + 1) exp[−κ(z − H )] ⎥ (κ Λ̃)2 + 1 + 1) exp[−κ(z − H )] ⎥⎦ (22)

⎡ H2 − z 2 ⎤ ⎥ c±(z) = c±(H ) exp⎢ ∓ ⎣ H0(k)2 ⎦

The mobile ions are distributed according to the Boltzmann law as c±(z) = c±(H ) exp[∓ψ (z)]

1 2πlBH0 2(k)

The latter determines the Gouy−Chapman length

with the characteristic length 2

(18)

is independent of distance z from the surface. Therefore, the uncompensated number of elementary charges Q per unit area of the brush

2

U (z ) 3k H −z (H 2 − z 2 ) ≡ ≈ 2 αkBT 2αa H0 2(k)

= −4πlBρ(z)

one finds that net number charge density ρ(z) inside the brush

The reduced (dimensionless) electrostatic potential ψin(z) = eΨin(z)/kBT (measured in kBT units) at distance z from the surface is therefore specified as 2

(17)

introduced previously.23 The expression for electrostatic potential ψin(z) in eq 15 allows us to obtain the charge density profile ρ(z) in the brush. By using the Poisson equation

αe Ψin(z) U (z ) ≈ = αψin(z) kBT kBT

2

8 aα1/2N 3π 2

with the corresponding concentrations c±(H) at the brush boundary z = H specified as

(23)

with

c±(H ) = cs exp{± [ψout(z = ∞) − ψout(H )]} ⎛ (κ Λ̃)2 + 1 − 1 ⎞±2 ⎛ (κ Λ̃)2 + 1 ∓ 1 ⎞2 ⎜ ⎟ ⎜ ⎟ = cs⎜ ⎟ = cs⎜ ⎟ κ Λ̃ κ Λ̃ ⎝ ⎠ ⎝ ⎠

⎧ 0 α−1/2(lB/a)−1, the counterions condense in the intramolecular volume. Here, the size of starlike polyion is specified by the first line in eq 37. The onset of the regimes of individual polyions (ISo and ISc) is determined as

Figure 2. Dependence of the reduced thickness H( f)/H0 of the brush formed by arm-tethered starlike polyelectrolytes with the same total number of monomer units N, degree of ionization α, and grafting density σ and varied number of branches f = 2 (linear chains), f = 4, and f = 8 as a function of the reduced salt concentration 2πlBH02cs ≡ (κH0/2)2. Here H0 is the characteristic length for a brush linear PE chains given by eq 17, and H0/Λ = 15.

H0 specified in eq 17) calculated according to eqs 31 and 33 for arm-tethered polyelectrolyte stars with the same total degree of polymerization, N, the same grafting density, σ, but different number of arms, f. By using these coordinates, we can compare the features of PE dendron brushes with different degrees of branching. As one can see from Figure 2, the effect of branching on the brush thickness is more pronounced in the low-salt (osmotic) regime. Here, the decrease in the brush thickness H ∼ h is stronger than in the salt dominated regime located at larger salt concentrations, cs. Moreover, the transition to the salt dominated regime is shifted to larger values of cs. This is because in a more compact PE dendron brush the concentration of counterions is larger than in an equivalent PE brush of linear chains, and therefore a larger concentration of salt ions, cs, is required to reach the onset of the salt dominated regime. Equation 35 can be further simplified in the case of large number of arms, f ≫ 1, using eq 34 to give

s ≃ R star 2

At this crossover grafting area of starlike macromolecules, the intermolecular and intramolecular interactions become of the same order of magnitude. At more dense grafting of the polyions with s < Rstar2, the intermolecular interactions significantly perturb the structure of individual macromolecules, F


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brush of starlike macromolecules. The current conjecture (used in construction of diagrams in Figures 3 and 4) is that the

Figure 3. Diagram of states of the arm-tethered starlike polyelectrolytes in a salt-free solution. f is the , and s is the surface area per macromolecule. The ratio of Bjerrum length lB to monomer size a is set to unity. The diagram comprises the regimes of individual armtethered stars that either keep the counterions in the intramolecular volume (regime ISo) or release the counterions to the solution (regime ISc), the charged brush regime PB, the osmotic brush regime OB, the regime QnB of a quasi-neutral brush with predominance of nonelectrostatic interactions, and the regime SSB of the brush of stars with fully stretched longest elastic path.

Figure 4. Diagram of states of the arm-tethered starlike polyelectrolytes in a salt-added solution. cs is salt concentration, and s is the surface area per macromolecule. The ratio of Bjerrum length lB to monomer size a is set to unity. The diagram comprises the regimes of individual osmotic, ISo, or salt-dominated, ISs, arm-tethered stars, the osmotic brush regime OB, the salted brush regime SB, the regime QnB of a quasi-neutral brush, and the regime SSB of the brush of stars with fully stretched longest elastic path.

leading to preferential elongation of the branches normally to the surface (the brush regimes). As demonstrated above, in all the brush regimes for the firstgeneration dendrons (starlike polyions) located to the left of the boundary s ≃ Rstar2 in Figure 3, the expressions for the brush thickness given by eq 36 are consistent with the general equation, eq 6, if the value of exponent β = 1. In the regime CB the brush is almost net charged. In such a charged brush (also referred in the literature as Pincus brush), counterions are distributed above the surface within the Gouy−Chapman length Λ (eq 1) that exceeds by far the brush thickness H (i.e., Λ ≫ H). The brush thickness H in this regime is given by the second line in eq 6. In the osmotic brush regime OB, the counterions condense into the volume of the brush, and its thickness H is given by the first line in eq 6. The boundary between the charged (CB) and the osmotic (OB) brush regimes is specified by eq 8 with β = 1, and N /5 = f /2. At larger grafting densities, the nonelectrostatic repulsions between monomers (ternary contacts under theta-solvent conditions) dominate over the electrostatic interactions, and the brush of starlike polyions demonstrates quasi-neutral behavior. The onset of quasi-neutral (QnB) brush regime is specified by eq 9 with β = 1, and N /5 = f /2 . The brush thickness H in this regime is given by the second line in eq 7. Finally, a wide regime SSB corresponds to starlike polyions with almost fully stretched longest path composed of 5 = 2n monomers, and the brush thickness here is given by H ≈ 2an. While crossovers between the brush regimes (all with β = 1) are smooth, the boundary s ≃ Rstar2 (shaded region in Figure 3) does not ensure smooth crossover between the brush regimes (located at s ≪ Rstar2) and the regimes of individual polyions (located at s ≫ Rstar2). The lack of smooth crossover at the boundary s ≃ Rstar2 is attributed to the change in exponent β from β = 2 (for individual starlike polyions) to β = 1 (for starlike macromolecules in the brush regimes). Here, the threedimensional spherical symmetry of a single star is substituted by the one-dimensional symmetry of a laterally homogeneous

transition into brush regimes upon crossing the boundary s ≃ Rstar2, and the concomitant increase in the vertical size of the tethered stars from Rstar to H ≫ Rstar takes place in a narrow interval of grafting densities that is not expressed in scaling terms. This interval of grafting densities needs a more accurate consideration. 4.3.2. Salt-Added Solution. In Figure 4 we demonstrate how additions of salt ions in the solution modify the diagram of states presented in Figure 3. We focus here on the case of starlike polyions with relatively large number of branches, α−1 > f > α−1/2(lB/a)−1, that entrap the mobile counterions inside the volume of macromolecule (typical value of f is indicated by the dashed line in Figure 3). To construct the diagram of states for these conditions, we use the volume fraction of salt ions (csa3) and the area per chain (sa−2) as variables. At zero salt concentration, (i.e., along X-axis, cs = 0) one finds the brush regimes SSB, QnB, OB, and the regime ISo of individual starlike polyions with entrapped counterions. All these regimes have been discussed earlier (see also Figure 3). Addition of salt in the solution leads to partial substitution of the osmotic brush regime (OB) by the salt dominated brush regime (SB) and extension of the regime of quasi-neutral brush behavior (QnB) to larger values of grafting area, s. At loose grafting of the macromolecules with s > Rstar2 (where Rstar is given by the first line in eq 37) an increase in salt concentration cs leads to transition in the salt-dominated regime of individual polyion (ISs) and the decrease in size of tethered starlike macromolecule as30 R star /a ≃ (α 2/a3cs)1/5 f 1/5 n3/5

(38)

The boundaries between different regimes in the diagrams of states in Figures 3 and 4 are collected in Table 1.

5. DISCUSSION In this study we explore theoretically the equilibrium structure of a planar polyelectrolyte dendron brush with fixed degree of G


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Table 1. Power Law Expressions for Boundaries between Different Regimes in the Diagrams of States in Figures 3 and 4a SSB−QnB OB−SB OB−ISs, ISo ISs−ISo OB−SBB a

s/a2 ≃ f 3/2 csa3 = f1/2α1/2a2s−1 s/a2 ≃ αn2 csa3 = fα−1/2n−2 f ≃ α−1

QnB−OB SB−QnB SB − ISs, ISo ISo−ISc OB−PB, ISc

s/a2 ≃ f1/2α−1 csa3 = f−1/4α2a−1s1/2 s/a2 ≃ αn2 f ≃ α−1/2 s/a2 ≃ f4/3α4/3n2

Here, the ratio lB/a is set equal to unity.

ionization α. We use two approaches: (i) a boxlike model that ignores the details of polymer and ion distributions and (ii) an analytical self-consistent field (SCF) model that specifies the internal organization of an ionic dendron brush (i.e., polymer density profile and spacial distributions of mobile ions). To the best of our knowledge, our study presents the first analytical theory of the planar brush formed by ionic dendritically branched macromolecules. The first (boxlike) model provides the power law dependences for the average brush thickness H in a wide range of conditions, including strong (non-Gaussian) stretching of the macromolecules. It does not account explicitly for the force balance in the branching points of ionic macromolecule but incorporates the parameter 1 ≤ β ≤ 2 which governs the elasticity of the tethered dendrons. The value of β = 2 corresponds to equal stretching of all the branches (the upper estimate for the elastic free energy of the macromolecule in eq 3) while β = 1 corresponds to the strong stretching of the longest path with the elastic contribution of all other branches on the same order of magnitude (the lower estimate for the elastic free energy of the macromolecule in eq 3). The second (SCF) model assumes the linear (Gaussian) elasticity of the branches of dendron on all length scales and is therefore limited to the range of relatively low values of branching parameter N/5 and degrees of ionization α (see discussion below). It however accounts for the force balance in all the branching points of the dendron and for the distribution of the free ends of the macromolecules all over the volume of the brush. The explicit analytical expressions for the brush thickness H are available for ionic dendrons of the first generation (g = 1) that constitute polyelectrolyte starlike macromolecules with number of branches f ≥ 2, end-tethered by the terminal monomer of one of the branches. On the basis of the results of both models, we construct the scaling type diagrams of states for a starlike PE brush under salt-free (Figure 3) and salt-added (Figure 4) conditions. We demonstrate that similarly to the case of a neutral brush of starlike dendrons,11 the results of the SCF analytical model are consistent with the value of exponent β = 1 in the scaling expression for the elastic free energy in eq 3. This is the direct consequence of (i) lateral homogeneity of the brush with the polymer density gradient in the direction normal to the surface and (ii) local balance of the elastic forces in all the branching points of the tethered dendron. Because of this force balance, the elastic free energy of the stem (i.e., the branch connecting the center of the star with the surface) and the elastic contribution of all other free branches become on the same order of magnitude. The value of β = 1 is retained in all the brush regimes with linear (Gaussian) elasticity of the branches (regimes OB, PB, SB, and QnB in Figures 3 and 4). As follows from the results of the analytical SCF model, in the equations for brush properties (e.g., eqs 30 and 31) all the details of the dendron architecture are incorporated in the parameter of molecular potential, k (eqs 15 and 16). Therefore,

the polymer density profile, c(z), the distributions of mobile ions, c±(z), and the electrostatic potential inside and outside of the ionic dendron brush, Ψin(z) and Ψout(z), can be obtained from the corresponding properties of the brush of linear polyions23 by renormalization of the variables according to eqs 26−29. The situation is more complex for the distribution of free ends of the dendrons. In contrast to the case of linear polymer brushes in which the relationship between polymer density profile, c(z), stretching function, E(z 1,z), and distribution function of the free ends, g(z1), is provided by the Abel integral equation,29 in dendron brushes this relationship is more complex due to different stretching functions E for branches in different generations g ≥ 1 of the dendron (see, e.g., the Appendix for starlike dendrons with g = 1). Up to now, distribution of the free ends of tethered dendrons, g(z1), in the whole range of distances 0 ≤ z1 ≤ H can be obtained only numerically. We emphasize that the analytical (SCF) model developed in this study is based on the assumption of the linear (Gaussian) elasticity of the dendron branches on all length scales. While this assumption is justified under salt-free conditions in the theta solvent, it could be violated upon addition of salt ions (in the upper part of regime SB in the diagram of states in Figure 3). As has been discussed recently,31 the intramolecular repulsion between the changed monomers of a linear polyion imposes intrinsic tension in its backbone. As a result, the Gaussian elasticity of a linear polyion in the brush is modified when the Debye screening length κ−1 becomes smaller than distance √s between tethered macromolecules but exceeds the size ξel of the electrostatic blob that governs intrinsic tension ∼kBT/ξel in the backbone. Electrostatically induced modification of polyion elasticity leads to the variations in exponent −1/3 in the salt dependence of brush thickness H(cs). That is, to appearance of a plateau (with exponent 0) at intermediate salt concentrations with subsequent more rapid decrease (with exponent −3/7) in brush thickness H upon further increase in cs (see ref 31 for details). A similar effect could be expected for branched polyions as well. The boxlike and SCF models developed in this paper do not distinguish between the “own” charges of dendron and the charges that belong to neighboring macromolecules. Incorporation of intrinsic tension due to own charges in the branched polyion would divide the salt dominated brush regime (SB) with single exponent −1/3 (third line in eq 32) in several subregimes with correspondingly modified exponents. The regime of individual starlike polyion ISs (with exponent −1/5, eq 38) would be modified as well. The analytical SCF model developed in this paper presumes that the brush thickness H remains noticeably smaller than the contour length of the longest path, a5 , in all the brush regimes. As follows from eqs 32 and 35, an increase in the number of branches of constant length n (i.e., a decrease in k in a general case or an increase in f in the particular case of starlike dendrons) leads to rapid increase in the brush thickness H. As a result, the dendrons approach the threshold of linear elasticity H


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Macromolecules

the grafting surface. The macromolecules are stretched nonuniformly with the degree of local chain extension E = (dz/dm) decreasing to zero at the end of each trajectory, m = 2n. When the maximal chain extension Emax = (dz/dm)m=0 remains noticeably below unity (that is, at small grafting densities σ and/or for weakly branched stars with f ≳ 1), tethered macromolecules exhibit the Gaussian (linear) elasticity on all length scales. We refer to this range of conditions as linear elasticity regime of tethered stars. The self-consistent field analytical theory of a planar neutral dendron brush was originally formulated by Pickett.7 He developed a model for dendrons with flexible spacers (comprising of n ≫ 1 monomers each), functionality q = 2, and arbitrary number of generations g ≥ 1. The model assumes elastic force balance in each branching point in combination with quadratic self-consistent field potential acting on monomer units of the brush-forming dendrons. In the g = 0 limit dendrons are equivalent to linear chains, and the wellknown results of the self-consistent field theory of the linear chain brushes are recovered. In ref 11 this model was applied for a brush formed by the first-generation dendrons (g = 1) equivalent to the brush of arm-grafted stars with arbitrary number f of arms. Following the lines of ref 11, we demonstrate below how the explicit analytical equation for the self-consistent molecular potential acting in the brush formed by arm-grafted stars can be derived. In terms of chain trajectories, the end points of all the free branches of the star (m = 2n) are located at the same height above the surface. Let z1 and z0 be the respective positions of the end-points and of the branching point (the star center) while Efreeand Estem be the stretching functions of the free branches and of the steam, respectively. Following refs 7 and 11, we introduce the self-consistent potential U(z) acting at the monomers at distance z from the surface

regime. For starlike dendrons this threshold can be quantified by monitoring the extension of stem near the grafting surface (z = 0) in the most strongly stretched stars (i.e., the stars with end-points of the branches located at z1 = H). The requirement Estem(z1 = H , z = 0) ≲ 1

or equivalently H ≲ an

1 (f − 1) arccos

f−1 f

(39)

specifies the conditions of the linear elasticity regime for tethered ionic stars (see Appendix for details). By substituting the corresponding expressions for H collected in eq 35, one finds the threshold of linear elasticity regime in various regions of the diagram of states. For example, in the regime of osmotic brush (OB) inequality (39) reads α≲

3 2(f − 1) ln[2ζ(f )/ π ]

(40)

with ζ( f) ≃ α3/2f 3/2n2alB/s. Thereby smaller fractions α of charged monomers are required to ensure linear elasticity of highly branched tethered stars with f ≫ 1. Remarkably, with the accuracy of a logarithmic factor, eq 40 follows from eq 14 derived above for an arbitrary dendron brush, provided that N /5 ∼ f and β is set to unity. In the two other brush regimes, the charged brush (CB) and salt-dominated brush (SB), the extension of dendrons is less than in the osmotic brush regime (OB). Therefore, inequality (40) provides an estimate for the threshold of linear elasticity regime in the whole diagram of states of a PE dendron brush. The experimental realization of the dendron polyelectrolyte brushes would require grafting of ionic dendrons or PE stars to planar substrates or to the surfaces of colloidal particles. Both approaches are experimentally feasible and would provide excellent model systems to verify our theoretical predictions. We anticipate that in addition to planar brushes of charged dendrons the presented theory is applicable also to spherical or cylindrical colloidal dendron PE brushes when the curvature radius of the particles is larger than the characteristic thickness of counterion cloud (given by eq 21 in salt-free solution) or larger than the brush thickness (in salt-added solution). However, at the moment, we are not aware of any systematic experiments on such systems. Finally, we would like to mention few possible developments of the analytical SCF model of the dendron PE brushes formulated in this paper. The model can be extended to address (i) the effect of dendron polydispersity (branched macromolecules with different lengths of the branches), (ii) the effect of finite extensibility of branched macromolecules, and (iii) the brush of dendrons with pH-sensitive ionization. We aim to address these issues in our future studies.

U (z ) 3 = 2 k 2(H2 − z 2) kBT 2a

(41)

Here, H is the brush thickness and k is an unknown constant to be determined later. In the linear elasticity regime the stretching function of free branches Efree(z1,z) with end-points located at distance z1 from the surface is specified as Efree(z1 , z) =

dz = k z12 − z 2 dm

z 0 ≤ z ≤ z1

(42)

It ensures vanishing tension ∼ Efree(z1,z1) = 0 at the end-point of any free branch. By using the conservation condition

∫z

z1 0

dz =n Efree(z1 , z)

(43)

one finds position z0 of the branching point in a star with endpoints of free branches located at height z1

■

z 0 = z1 cos kn

(44)

The stretching function of the stem is introduced as

APPENDIX. SELF-CONSISTENT MOLECULAR POTENTIAL U(z) To study the brush of starlike dendrons, we use the analytical self-consistent field (SCF) model that assumes strong stretching of the tethered macromolecules with respect to their Gaussian size. The strong stretching approximation allows for introduction of the so-called chain “trajectories” that specify position z(m) of each star monomer with ranking number m (calculated from the stem grafting monomer with m = 0) above

Estem(z 0 , z) = k λ − z 2

(45)

with indefinite constant λ to be specified later. The balance of elastic forces acting at the branching point (mechanical equilibrium at the star center) yields Estem(z 0 , z 0) = qEfree(z1 , z 0)

(46)

The condition of stem length conservation is formulated as I


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Macromolecules

∫0

z0

dz =n Estem(z 0 , z)

(6) Borisov, O. V.; Zhulina, E. B.; Polotsky, A. A.; Leermakers, F. A. M.; Birshtein, T. M. Macromolecules 2014, 47, 6932−6945. (7) Pickett, G. T. Macromolecules 2001, 34, 8784. (8) Kröger, M.; Peleg, O.; Halperin, A. Macromolecules 2010, 43, 6213. (9) Merlitz, H.; Wu, C.-X.; Sommer, J.-U. Macromolecules 2011, 44, 7043−7049. (10) Merlitz, H.; Cui, W.; Wu, C.-X.; Sommer, J.-U. Macromolecules 2013, 46, 1248−1252. (11) Polotsky, A. A.; Leermakers, F. A. M.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 2012, 45, 7260. (12) Gergidis, L. N.; Kalogirou, A.; Vlahos, C. Langmuir 2012, 28, 17176−17185. (13) Guo, Y.; van Beek, J. D.; Zhang, B.; Colussi, M.; Walde, P.; Zhang, A.; Kröger, M.; Halperin, A.; Schlüter, A. D. J. Am. Chem. Soc. 2009, 131, 11841. (14) Gergidis, L. N.; Kalogirou, A.; Charalambopoulos, A.; Vlahos, C. J. Chem. Phys. 2013, 139, 044913. (15) Borisov, O. V.; Polotsky, A. A.; Rud, O. V.; Zhulina, E. B.; Leermakers, F. A. M.; Birshtein, T. M. Soft Matter 2014, 10, 2093. (16) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (17) de Gennes, P.-G. Macromolecules 1980, 13, 1069. (18) Pincus, P. A. Macromolecules 1991, 24, 2912. (19) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. J. Phys. II 1991, 1, 521. (20) Wittmer, J.; Joanny, J. F. Macromolecules 1993, 26, 2691. (21) Chen, L.; Merlitz, H.; He, S.-Z.; Wu, C.-X.; Sommer, J.-U. Macromolecules 2011, 44, 3109−3116. (22) Attili, S.; Borisov, O. V.; Richter, R. P. Biomacromolecules 2012, 13, 1466−1477. (23) Zhulina, E. B.; Klein Wolterink, J.; Borisov, O. V. Macromolecules 2000, 33, 4945−4953. (24) Mays, J. W. Polym. Commun. 1990, 31, 170. (25) Heinrich, M.; Rawiso, M.; Zilliox, J. G.; Lesieur, P.; Simon, J. P. Eur. Phys. J. E 2000, 131−142. (26) Karaky, K.; Reynaud, S.; Billon, L.; Francois, J.; Chreim, Y. J. Polym. Sci., Part A: Polym. Chem. 2005, 43, 5186−5194. (27) Plamper, F. A.; Becker, H.; Lanzendörfer, M.; Patel, M.; Wittemann, A.; Ballauff, M.; Müller, A. H. E. Macromol. Chem. Phys. 2005, 206, 1813−1825. (28) Plamper, F. A.; Walther, A.; Müller, A. H. E.; Ballauff, M. Nano Lett. 2007, 7, 167−170. (29) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733. (30) Borisov, O. V.; Zhulina, E. B.; Leermakers, F. A. M.; Ballauff, M.; Müller, A. H. E. Adv. Polym. Sci. 2011, 241, 1. (31) Zhulina, E. B.; Rubinstein, M. Soft Matter 2012, 8, 9376−9383.

(47)

Solution of eqs 46 and 47 together with eq 44 allows us to determine λ and Estem as

λ = z12q = z12(f − 1)

(48)

and Estem(z 0 , z) = k (f − 1)z12 − z 2 0 ≤ z ≤ z 0 ; z 0 ≤ z1 ≤ H

with k=

q 1 1 arccos = arccos n q+1 n

f−1 f

(49)

In the linear elasticity regime positions z0 and z1 of the branching point and of the free ends of the starlike dendrons are therefore related as

z0 = z1

f−1 f

(50)

and the self-consistent molecular potential U(z) in the starlike dendron brush is finally specified as U (z ) 3 ⎛ = 2 2 ⎜⎜arccos kBT 2a n ⎝

2 f − 1⎞ ⎟⎟ (H2 − z 2) f ⎠

(51)

Equation 51 is valid for both neutral and charged dendron brushes in salt-free and salt-added solutions. For neutral dendrons, the molecular potential U(z) is governed by the short-range (excluded volume) interactions between monomers.11 When, however, the electrostatic interactions dominate over the nonelectrostatic interactions, the molecular potential U(z) in eq 51 is of essentially electrostatic origin and is directly linked to the electrostatic potential Ψin(z) in the brush.

■

AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship (PIRSES-GA2013-612562-POLION) within the seventh European Community Framework Programme and partially supported by the Russian Foundation for Basic Research (Grants 14-03-00372a and 14-03-91338), by the Department of Chemistry and Material Science of the Russian Academy of Sciences, and by Government of Russian Federation, Grant 074-U01.

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REFERENCES

(1) Gillich, T.; Benetti, E. M.; Rakhmatullina, E.; Konradi, R.; Li, W.; Zhang, A.; Schlüter, A. D.; Textor, M. J. Am. Chem. Soc. 2011, 133, 10940. (2) Gillich, T.; Acikgöz, C.; Isa, L.; Schlüter, A. D.; Spencer, N. D.; Textor, M. ACS Nano 2013, 7, 316−329. (3) Schlüter, A. D. Top.Curr.Chem. 2005, 245, 151−191. (4) Teertstra, S. J.; Gauthier, M. Prog. Polym. Sci. 2004, 29, 277−327. (5) Schüll, C.; Frey, H. Polymer 2013, 54, 5443−5455. J


Brushes of Dendritically Branched Polyelectrolytes - ACS Publications

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