Inna O Lebedeva

Ekaterina B Zhulina

Frans A M Leermakers

Sergei S Sheiko

Oleg V Borisov
email: oleg.borisov@uiv-pau.fr

The parabolic approximation for self-consistent molecular potential is widely used for theoretical analysis of conformational and thermodynamic properties of polymer brushes formed by linear or branched macromolecules. The architecture-dependent parameter of the potential (topological coefficient) can be calculated for arbitrary branched polymer architecture from the condition of elastic stress balance in all the branching points. However, the calculation routine for the topological coefficient does not allow unambigously identifying the range of applibility and the accuracy of the parabolic approximation. Here the limits of applicability of parabolic approximation are explored by means of numerical self-consistent field method for brushes formed by Y-shaped and comb-like polymers. We demonstrate that violation of the potential parabolic shape can be evidenced by appearance of multimodal distribution of the end monomer unit in the longest elastic path of the macromolecule. The asymmetry of branching of Y-shaped polymers does not disturb the parabolic shape of the potential as long as the degree of polymerization of the root segment remains sufficiently large. The same applies to comb-shape polymers with sufficiently long main chain and large number of branching points. For short comblike polymers multiple modes in the distribution of the end monomer unit of the main chain are observed and related to deviation from the parabolic shape of the potential.

Introduction

Polymer brushes are formed by linear or branched macromolecules attached to an interface at high surface coverage assuring dominance of inter-molecular interactions. Solvated or solvent-free (dry) polymer brushes are used in technology and medicine for modification of adhesive, tribological, and biointeractive surface properties. Brush-like layers of biological macromolecules are found inside living species, e.g., cilia in lung airways and cell membranes.

Theoretical understanding of structure and properties of polymer brushes stem from pioneering works of Alexander 1 and de Gennes, [START_REF] De Gennes | Conformations of Polymers Attached to an Interface[END_REF] based on scaling approach which capture the most essencial structure-property relationships for brushes formed by linear chain polymers. More recent theoretical activity in this field amply exploited the strong-stretching (or quasi-classical) selfconsistent field approximation suggested by Semenov [START_REF] Semenov | Contribution to the theory of microphase layering in block copolymer melts[END_REF] and further elaborated for solvated polymer brushes in refs [START_REF] Zhulina | Structure and Conformational Transitions in Grafted Polymer Chains Layers: New Theory[END_REF][START_REF] Milner | Theory of the grafted polymer brush[END_REF] (see also a comprehensive review [START_REF] Birshtein | Polymer brushes[END_REF] on theory of polymer brushes).

According to the Semenov's approach 3 each chain is assimilated to the trajectory of a forward motion of a classical particle in the self-consistent potential field. The particle velocity is mapped to the local chain stretching. The non-uniform distribution of the local stretching along the contour of the chain assures minimum of its free energy in the potential field. Irrespectively of the chain free end position (starting point of any trajectory) the terminal segment of the chain reaches the grafting surface. As long as the chains exhibit linear (Gaussian) elasticity this condition of equal paths length (or eaqul "flight times") can be satisfied only if the potential has a parabolic shape. The condition of self-consistency requires that polymer density distribution dictated by the parabolic potential can be formed by an ensemble of non-equally and non-uniformly stretched chains without dead zones depleted of the chains free ends.

The equivalent arguments were further applied for brushes of tree-like macromolecules (dendrons) by Pickett who suggested that self-consistent molecular potential acting in brushes of tethered regular dendrons has also a parabolic shape. [START_REF] Pickett | Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest[END_REF][START_REF] Zook | Hollow-Core Dendrimers Revised[END_REF] These findings paved the way for generalization of the analytical self-consistent field approach to brushes of other architectures. [START_REF] Polotsky | On the Two-Population Structure of Brushes Made of Arm-Grafted Polymer Stars[END_REF][START_REF] Zhulina | Theory of brushes formed by Ψ-shaped macromolecules at solid-liquid interfaces[END_REF][START_REF] Zhulina | Ideal mixing in multicomponent brushes of branched macromolecules[END_REF][START_REF] Lebedeva | Dendron and Hyperbranched Polymer Brushes in Good and Poor Solvents[END_REF][START_REF] Zhulina | Brushes of Cycled Macromolecules: Structure and Lubricating Properties[END_REF] The parameters of the parabolic self-consistent potential can be calculated for an arbitrary tree-like or even cyclic macromolecules. However, this does not guarantee that a real self-consistent field potential has indeed the parabolic shape.

Notably, if the brush is formed by linear chain macromolecules, the selfconsistent potential is parabolic only for stricktly monodisperse brushes. Even weak polydispersity of the brush-forming chains distores the parabolic shape of the potential, 14,[START_REF] Qi | Polydisperse Polymer Brushes: Internal Structure, Critical Behaviour and Interaction with Flow[END_REF] which is not always the case in brushes formed by branched macromolecules. [START_REF] Zhulina | Ideal mixing in multicomponent brushes of branched macromolecules[END_REF] The applicability of the parabolic potential approximation, i.e. the shape of the true self-consistent potential for any specific branched architecture of the brush-forming molecules can be verified by numerical self-consistent methods, that are free of any assumption about the stretching mode and do account for finite extensibility of the polymer chains.

The aim of the present paper is to explore the ranges of validity of the parabolic potential approximation for mono-component brushes formed by macromolecules with various branched architectures. In particular, we explore the simplest case of branched macromolecules, i.e. root-tethered Yshaped macromolecules and further extend the analysis for comb-shape polymers tethered to planar surface. To establish applicability limits of this approach for various macromolecular architectures,we use the Scheutjens-Fleer numerical self-consistent field (SF-SCF) method. [START_REF] Fleer | Polymers at Interfaces[END_REF] Although quantitative predictions of the lattice-based SF-SCF theory concerning chain elasticity in the non-linear stretching regime do not exactly match results obtained within continuum models (see, e.g., discussion in recent review article [START_REF] Li | The structure of brushes made of dendrimers: Recent advances[END_REF] ), this approach provides a robust tool for exploring both the shape of the self-consistent potential and the end-point distribution in brushes formed by branched macromolecules with varied architecture.

Notably, brushlike structures emerge as an outcome of self-assembly of copolymers comprising comb-shape blocks into diverse bulk morphologies. Such microphase segregated materials are being actively studied because of possibility to controll their optical and mechanical properties by tuning atchtectural parameters of constituent branched polymer blocks. [START_REF]Rzayev Synthethis of polystyrene-polylactide bottlebrush block copolymers and their melt self-assembly into large domain nanostructures[END_REF][START_REF] Liberman-Martin | Application of Bottlebrush Block Copolymers as Photonic Crystals[END_REF][START_REF] Song | Hierarchical Photonic Pigments via the Confined Self-Assembly of Bottlebrush Block Copolymers[END_REF][START_REF] Vatankhah-Varnosfaderani | Chameleon-like elastomers with molecularly encodded strain-adaptive stiffening and coloration[END_REF][START_REF] Bolton | Large pore size nanoporous materials from self-assembly of asymmetric bottlebrush block copolymers[END_REF][START_REF] Gai | Polystyrene-blockpoly(ethylene oxide) Bottlebrush Block CopolymerMorphology Transitions: Influence of Side Chain Length and Volume Fraction[END_REF][START_REF] Runge | Synthesis of High Molecular Weight Comb Block Copolymers and Their Assembly into Ordered Morphologies in the Solid State[END_REF][START_REF] Runge | Investigation of the Assembly of Comb Block Copolymers in the Solid State[END_REF][START_REF] Fei | Influence of Molecular Architecture and Chain Flexibility on the Phase Map of Polystyrene-block-poly(dimethylsiloxane) Brush Block Copolymers[END_REF][START_REF] Sunday | Self-Assembly of ABC Bottlebrush Triblock Terpolymers with Evidence for Looped Backbone Conformations[END_REF][START_REF] Bolton | Synthesis and Melt Self-Assembly of PS-PMMA-PLA Triblock Bottlebrush Copolymers[END_REF] The paper is organized as follows. In section 2 we present the general parabolic potential framework for arbitrary polymer architecture. In section 3 we employ numerical self-consistent field method to check range of applicability of the parabolic approximation to brushes of asymmetric Y-shaped polymers. In section 4 we extend our numerical analysis to brushes of comblike polymers. In section 5 formulate the conclusions. In the Appendix we demonstrate how the parabolic potential can be derived explicitly for the brush formed by Y-shaped macromolecules. 2 SS-SCF formalism.

s n 1 z 1 (a) (b) s m 1 q=3 z 1 z 2 z P m n n 2 n 3
According to previous studies, [START_REF] Pickett | Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest[END_REF][START_REF] Zook | Hollow-Core Dendrimers Revised[END_REF][START_REF] Polotsky | On the Two-Population Structure of Brushes Made of Arm-Grafted Polymer Stars[END_REF][START_REF] Zhulina | Theory of brushes formed by Ψ-shaped macromolecules at solid-liquid interfaces[END_REF][START_REF] Zhulina | Ideal mixing in multicomponent brushes of branched macromolecules[END_REF][START_REF] Lebedeva | Dendron and Hyperbranched Polymer Brushes in Good and Poor Solvents[END_REF][START_REF] Zhulina | Brushes of Cycled Macromolecules: Structure and Lubricating Properties[END_REF] the molecular potential U (z) in the brush can be presented as

U (z) k B T = 3 2a 2 κ 2 (H 2 -z 2 ) (1)
where k B T is the thermal energy, H is the brush thickness and κ is the topological coefficient. The latter has to be specified for each particular macromolecular architecture yet is remarkably independent of the grafting density and solvent quality. Eq 1 presumes the Gaussian (linear) elasticity of the tethered macromolecules on all length scales but can be generalized [START_REF] Birshtein | Polymer brushes[END_REF] for accounting finite chain extensibility. Crowding of the side chains near the branching points is not taken into account.

The values of κ for macromolecules with different architectures are calculated [START_REF] Zhulina | Ideal mixing in multicomponent brushes of branched macromolecules[END_REF][START_REF] Zhulina | Brushes of Cycled Macromolecules: Structure and Lubricating Properties[END_REF] by using the conditions of length conservation for the segments between branching units, and by balancing the elastic forces in all branching units of the macromolecule. For linear chains κ lin = π/2N . [START_REF] Semenov | Contribution to the theory of microphase layering in block copolymer melts[END_REF] The topological coefficient κ ensures conservation of number of monomer units in all linear segments of the molecule at any position of the free endpoints. However, it does not ensure the absence of dead zones depleted of the free ends. Curvature of the grafting surface is known to produce dead zones in spherical and cylindrical convex brushes formed by both linear and branched polymers. Branching of the brush-forming macromolecules makes the free-end distribution more even, and may lead to virtual disappearance of the curvature-induced dead zones. [START_REF] Zook | Hollow-Core Dendrimers Revised[END_REF][START_REF] Rud | Dendritic Spherical Polymer Brushes: Theory and Self-Consistent Field Modelling[END_REF] As demonstrated below, the dead zones, which are always present in the convex brushes, may appear even in planar brushes for some particular macromolecular architectures. In the case of the dead zone emergence, the parabolic potential, eq 1 is not a true self-consistent one.

The profile of free ends of the tethered macromolecules should be therefore checked for every specific chain architecture to identify the parameter space where the parabolic potential is valid. We use the numerical self-consistent method of Scheutjens and Fleer (SF-SCF) to examine distributions of the free ends. [START_REF] Fleer | Polymers at Interfaces[END_REF] The molecular potential in eq 1 is related to the polymer density (volume fraction) profile φ(z) in the brush by a general equation

a 3 δf {φ(z)} δφ(z) = U (z) (2)
where f {φ(z)} is the free energy density of monomer-monomer interactions.

For non-ionic polymers, at φ(z) 1, it can be presented as a virial expansion in powers of polymer volume fraction, that is

f {φ(z)} = υφ 2 (z) + wφ 3 (z) + ... (3)
where υa 3 and wa 6 are the second and the third virial coefficients of monomermonomer interactions, respectively. The former is related to the Flory-Huggins polymer-solvent interaction parameter χ as υ = 1 2 -χ. Under good (υ wφ) or theta (| υ | wφ) solvent conditions one can keep only the respective dominant term in the virial expansion. This leads to explicit expressions for the polymer density profile,

φ(z) ≈    (3κ 2 )/(4a 2 υ)(H 2 -z 2 ) υ wφ {(κ 2 /2a 2 w)(H 2 -z 2 )} 1/2 | υ | wφ -υ/(2w) υ ≤ 0, | υ | wφ (4)
Under good solvent conditions, according to eqs 2 and 3 there is a simple relation between self-consistent potential and polymer density profile, that is

U (z) k B T = 2vφ(z)
and both have a parabolic shape. Through normalization of the polymer density profile, s a 3

H 0 φ(z)dz = N (5)
one finds the brush thickness

H/a =    (2a 2 υN/sκ 2 ) 1/3 v wφ (2w) 1/4 (4N a 2 /πsκ) 1/2 | v | wφ -2N wa 2 /(sυ) υ ≤ 0, | υ | wφ (6)
It is instructive to introduce the topological ratio

η = κ κ linear = 2κN π (7)
for brushes of branched polymers with the same number N of monomer units as in a linear macromolecule ( k linear = π/(2N )). The topological ratio quantifies a relative increase in the elastic free energy penalty for stretching of a branched macromolecule as compared to that for a linear one in a brush. The reduced thickness H/H linear of the brush formed by branched macromolecules can be expressed as

H H linear =    η -1/3 v wφ η -1/2 | v | wφ η 0 υ ≤ 0, | υ | wφ (8)
Free energy per chain in the brush is then presented as 12

F F linear =    η 2/3 v wφ η | v | wφ η 2 υ ≤ 0, | υ | wφ (9)
where the last line in eq 9 describes only the elastic contribution to the total free energy per chain.

Brushes of Y-shaped polymers

The simplest case of a branched macromolecule is an Y-shaped polymer with the stem composed of n 1 monomer units and two branches with n 2 and n 3 (n 3 ≥ n 2 ) monomer units emanating from a single branching point. Such polymer can be envisioned as a comblike polymer with one branching point or, equivalently, as an asymmetric starlike macromolecule with q = 2 free branches, Figure 1a.

As it is demonstrated in the Appendix, the self-consistency of the molecular potential in eq 1 in this case is attained if the topological coefficient κ is given by the minimal root of the equation tan(κn 1 )[tan(κn 2 ) + tan(κn 3 )] = 1 (10) In a symmetric case, n 1 = n 2 = n 3 = n, the solution of eq 10 yields κ = n -1 a tan(1/ √ 2) to give η = 3a tan(1/ √ 2), a known result for a brush of end-tethered symmetric stars, η = (q + 1)a tan(1/ √ q), with q = 2 . Notably, the solution of eq 10 can be found numerically for arbitrary values of n 1 , n 2 and n 3 . However, as we demonstrate below, it does not ensure the absence of dead zones (regions depleted of the free chain ends) and thus applicability of the parabolic molecular potential even in a planar geometry.

Numerical SF-SCF calculations for brushes of Yshaped polymers.

The Scheutjens-Fleer self-consistent field method (SF-SCF) was used earlier to analyze the organization of brushes formed by star-like polymers. [START_REF] Polotsky | On the Two-Population Structure of Brushes Made of Arm-Grafted Polymer Stars[END_REF] The details about SF-SCF method can be found elsewhere. [START_REF] Fleer | Polymers at Interfaces[END_REF] Here we apply good (athermal) solvent conditions, χ = 0, under which the effect of repulsive monomer-monomer interactions in the brush of branched macromolecules is maximal. In the parabolic molecular potential U (z), the polymer density profile ϕ(z) in a good solvent is expected to be parabolic as well provided that ϕ(z) 1 (eq 4, first line).

Below, we focus on analyzing the shape of self-consistent potential (more particularly, its deviation from the expected parabolic shape) and distribution of free ends of both branches and branching points in brushes formed by Y-shaped polymers with systematically and independently varied DPs of the stem, n 1 and both free branches, n 2 and n 3 .

As a reference system we use the brush formed by symmetric Y-shaped polymers with n 2 = n 3 and N = n 1 + n 2 + n 3 = 300 grafted to a planar surface with the density θ = N a 2 /s = 5. By plotting in Figure 2, the self-consistent potential is plotted as a function of z 2 , we show that upon variation of the length of the stem n 1 = N -(n 2 + n 3 ) in the range of 100 ≤ n 1 ≤ 200, the profile of the self-consistent potential remains parabolic while the distributions of the free ends of both branches are unimodal and exhibit no dead zone proximal to the surface. For very short stem length, n 1 = 2, the branching points are located proximal to the grafting surface and the brush of symmetric Y-shaped polymers is equivalent to conventional monodisperse brush formed by linear chains with DP n 2 = n 3 . The parabolic approximation is exact for such brushes as well. In the intermediate case of n 1 = 40 both distributions of the branching and end points exhibit a plateau shape which points to the onset of bimodal distributions of branching and end points. Such bimodal distributions are typical for stratified dendron brushes where the stems are strongly stretched in the non-linear elasticity regime. [START_REF] Borisov | Dendron Brushes and Dendronized Polymers: A Theoretical Outlook[END_REF] Let us now consider the effect of the branch bidispesity (asymmetry of branching) in brushes of Y-shaped macromolecules.

As demonstrated in the upper line in Figure 2, the asymmetry of branches does not perturb the parabolic shape of the self-consistent potential and does not lead to appearance of dead zones (depleted of the ends of longer branches) as long as the DP of the stem is sufficiently large (i.e., n 1 = 200). The distributions of the ends of long and short branches are close to each other and both retain unimodal shape while the distribution of the branching points is virtually unaffected by assymetry of branches.

For Y-shaped polymer with a constant degree of polymerization, N , and a long stem of n 1 ≥ N/3 monomeric units, the self-consistent potential in the body of the brush is almost perfectly parabolic even if the branches become asymmetric, i.e., n 2 ≤ n 3 , and the cumulative end-points distribution splits into two modes corresponding to the shorter and to the longer branch, respetively.

The effect of branching asymmetry on the brush structure gets more pronounced upon a decrease in the DP n 1 of the stem (and concomitant increase in total DP of branches n 2 + n 3 ). At n 1 = 100 distributions of the ends of long and short branches only partially overlap though without pronounced dead zone. However, the apparent shoulder in the distribution of the longer branch ends indicates approaching onset of their depletion in the region proximal to the grafting surface and enrichment in the distal region of the brush. The branching point distribution remains unimodal, but its maximum is displaced towards grafting surface upon increasing branching asymmetry.

These trends become and even more pronounced for Y-polymers with shorter stem, n 1 = 40. The distribution of the ends of longer branch demonstrates pronounced depletion near the grafting surface, where the self-consistent potential strongly deviates from the parabolic shape. Notably, as discussed above, at n 1 = 40, the stem approaches the non-linear elasticity regime and the brush structure is affected by both asymmetry of branching and stratification. Here the parabolic shape of the self-consistent potential is strongly violated.

At vanishing stem length, the brush of asymmetric Y-shaped polymers becomes equivalent to a bidisperse brush of linear chains 14 with two different DPs, n 2 and n 3 , and virtually non-overlapping end-point distributions of short and long chains: the ends of the longer chains are depleted from the proximal to the grafting surface region where short chains and their end points are located.

Brushes of comb-shaped polymers 4.1 Architecture of comb-shaped polymers

A comb-shaped polymer consists of P repeat units each comprising the main chain segment (spacer) with m i monomer units (i = 1, 2, ...P ) and q i ≥ 1 side chains each with n ij monomers (j = 1, 2, ...q i ) linked to a single branching point on the main chain (backbone), see Figure 1b. If q i > 1, such macromolecules are usually refered to as "barbwire" bottlebrushes.

The first backbone segment attached to the surface (the stem) has m 1 monomer units. The backbone has

M = P i=1 m i (11)
monomer units whereas total number of monomer units in one macromolecule is

N = P i=1 m i + P i=1 q i j=1 n ij (12)
4.2 Topolgical coefficient κ for comb-shaped polymers.

A scheme to calculate the topological coefficient κ and topological ratio η = 2kN/π for a brush composed of comb-shaped macromolecules is described in details in ref [START_REF] Zhulina | Brushes and lamellar mesophases of comb-shaped (co)polymers: a self-consistent field theory[END_REF] Here, we present only basic equations. The topological coefficient κ can be found for abitrary set of parameters {m i ; n ij } and arbitrary P as a minimal root of the equation det A P = 0 (13) where matrix A P is defined as

A P =         -B 1 (κ) 1 0 0 0 0 D 1 (κ) -B 2 (κ) 1 0 0 0 0 D 2 (κ) -B 3 (κ) . . . 0 0 0 0 . . . . . . 1 0 0 0 0 D P -2 (κ) -B P -1 (κ) 1 0 0 0 0 -1 C P (κ)         (14)
and its elements are related to parameters {m i ; n ij } as

B i (κ) = cos(κm i+1 )+ sin(κm i+1 ) tan(κm i ) -sin(κm i+1 ) q i j=1 tan(κn ij ), i = 1, 2, .., P -1 (15)
D i-1 (κ) = sin(κm i+1 ) sin(κm i ) , i = 2, 3, ...P -1

C P = cos(κm P ) -sin(κm P )

q P j=1 tan(κn P j ) (17)
Importantly, as we shall demonstrate below, the existence of the solution of eq 13 does not guarantee the parabolic shape of the self-consistent molecular potential.

The outlined above formalism provides a route to calculate the topological coefficient κ for macromolecules with a variety of comb-like architectures. In ref [START_REF] Zhulina | Brushes and lamellar mesophases of comb-shaped (co)polymers: a self-consistent field theory[END_REF] we focused below on a representative case of macromolecules with equal lengths of side chains, n ij = n, and equal branching activity of all the branching points, q i = q. These macromolecules have

M = m 1 + (P -1)m ( 18
)
monomer units in the backbone, and total number of monomer units

N = M + P qn = (m 1 -m) + P (m + qn) (19)
The condition m 1 = m i , i = 2, 3, ...P is most essential for deviation of the molecular potential from the parabolic shape.

The topological coefficient κ can be related to the topological ratio η , as

κ = πη 2N = πη 2M • (1 + qn/m) -1 , m 1 = m [1 + qn/m + (m 1 /m -1)/P ] -1 , m 1 = m ( 20
)
The asymptotic dependence for the topological ratio for such comb-shaped macromolecule can be obtained by assuming that the elastic stress propagates only through the main chain (backbone) of the macromolecule and reads

η ≈ κ κ linear = 2κN π = (1 + qn/m) 1/2 ≈ (qn/m) 1/2 , qn/m 1 ( 21
)
which coincides with asymptotic numerical solution of eq 13, as discussed in ref [START_REF] Zhulina | Brushes and lamellar mesophases of comb-shaped (co)polymers: a self-consistent field theory[END_REF]

SF-SCF results for comb-shaped polymers

Below we present the results of SF-SCF calculations for comb-shaped macromoelcules with m i = n ij = n, q 1 = q 2 = ... = q P -1 = q and q P = q + 1. For this architecture the total degree of polymerization is given by

N = n(1 + P + qP )
and the topological ratio is

η ≈ 1 + qP P + 1 1/2
The macromolecules are tethered to an impermeable planar surface with area s per chain. The grafting density θ = N a 2 /s ≥ 1 ensures strong overlap of individual macromolecules and predominance of intermolecular interactions over intramolecular ones (the brush regime). The Flory-Huggins parameter for both main and side chains is set as χ = 0, that corresponds to good (athermal) solvent conditions. In contrast to Y-shaped polymer, combshaped macromolecule with relatively small number of periods, P , demonstrate surprising features .

In Figures 3a,b , we present the polymer density profiles ϕ(z) and ϕ(z 2 ) in planar brushes of comb-shaped macromolecules with m i = n ij = n = 100, q = 5, fixed surface coverage θ = N a 2 /s = (1 + P + P q)a 2 n/s = 20, and varying P from P = 1(starlike polymer) to P = 10 (values of P are indicated near the curves). Variations in P at fixed surface coverage θ correspond to variations in the grafting area per macromolecules as s/a 2 = (1+P +qP )n/θ. As expected, at values of ϕ(z) 0.15, polymer density profiles ϕ(z 2 ) constitute almost perfect straight lines with conventional deviations at the brush edge and near the grafting surface. At larger values of ϕ(z) > 0.15, the profiles ϕ(z 2 ) become slightly convex, which is can be explained by larger polymer concentration and contributions of higher order monomer-monomer interactions.

        z  p =           (a)            z   p =           (b) z   2 z 1 P  1 P 
In Figure 4 , we present the distributions g(z) of the backbone endpoints for the same values of the parameters as in Figure 3. Surprisingly, distributions g(z) demonstrate fine structure with numerous extrema in spite of remarkably simple density profiles in Figure 3. Starlike polymer (P = 1) with q + 1 = 6 free branches exhibits a smooth distribution of the branching monomer units (stem ends), pointing at a single population of moderately stretched stars. Segregation of starlike polymers within the brush into two populations with weakly and stongly stretched stems, respectively, is known to occur at higher grafting densities in the nonlinear elasticity regime. [START_REF] Polotsky | On the Two-Population Structure of Brushes Made of Arm-Grafted Polymer Stars[END_REF] In contrast, distribution g(z) for comb-shaped macromolecules with P = 2 demonstrates two distinctive peaks indicating segregation of these macromolecules in two populations. Notably, this segregation occurs in the regime of Gaussian elasticity. Further increase in P = 2, 3, 4 leads to an increase in the number of peaks in g(z) pointing at the presence of P distinctive populations of the tethered bottlebrushes. At P > 5 the fine structure of g(z) smears and gradually disappears. No dead zones are found near the grafting surface, however, the depletion minimum in g(z) in the vicinity of z 50 (i.e., at approximately half counter length of the stem) deepens with increasing P .

To examine the behavior of this depletion in more detail we increased the length n of the side chains with the simulteneous decrease in spacer length m, surface coverage θ, and increase in the number of periods P . In Figures 5,6, we present polymer density profiles ϕ(z) and backbone endpoint distributions g(z) for m i = 50, n ij = 200, q i = 5 + δ(P -i), and θ = 5. Comparison of Figures 5 and4 indicates that fine structure of g(z) and the trend in its evolution observed upon an increase in P are retained. However, the relative contribution of the depleted zone in the total endpoint distribution, g(z), decreases upon an increase in P , suggesting that for macromolecules with long backbone, m 1 m, and P 1, the depletion effect could be neglected.

In Figure 6, the number of periods P increases up to P = 80. The evolution of distributions g(z) upon an increase in P indicates further decrease of the depth of the minimum, its shift to smaller values of z, and final disappearance leading to perfectly smooth g(z) at values of P 60.

A decrease in length m 1 of the stem (first spacer) leads to strong devia- tions of polymer density profile ϕ(z) from the parabola. The shape of ϕ(z) resembles the polymer density distribution in a two-component brush of linear polymers in which side chains emanating from the first branching point constitute one component while the rest of the bottlebrush -the second component. In accord with this picture, the "dead zone" free of the backbone ends (i.e. with g(z) = 0) develops near the surface.

In Figure 7 we present the polymer density profiles ϕ(z) and distributions of the backbone end-point, g(z), for macromolecules without a stem, m 1 = 0, that illustrate these features: complex (nonparabolic) shape of ϕ(z) and dead zone with g(z) = 0 near the grafting surface .

However, the trends found in the case of m 1 = m hold for m 1 = 0 as well. That is, an increase in P leads to gradual transformation of the polymer density profile in the parabola, and disappearance of both depletion in the vicinity of z 50, and dead zone near the surface.

Discussion and conclusions

In this study we have examined the applicability of the parabolic self-consistent potential approximation for planar brushes formed by macromolecules with branched architectures different from that of linear chains or regular dendrons. In the latter two cases, the parabolic approximation is known to be exact in the limit of linear conformational elasticity of the brush-forming macromolecules. For branched macromolecules with less regular architecture, the topological parameter of the parabolic potential can be calculated within SS-SCF scheme as well though this route does not provide a proof of self-consistency of the parabolic potential. We applied numerical SF-SCF scheme to calculate true self-consistent potential which appears to deviate from parabolic shape for such simple archittectures as asymmetric Y-shaped and comb-shaped polymers with short stems. The deviation of the selfconsistent potential from the parabola is accompanied by peculiar patterns in the free ends distribution functions.

Hence, the results of our study demonstrates that although topological coefficient in the parabolic potential can be routinely calculated for any branched polymer architecture, this calculation does not provide itself an unambigious proof of applicability of the SS-SCF analytical scheme to brushes formed by arbitrary tree-like macromolecules. Appendix: self-consistent potential for brushes of Y-shaped macromolecules

        z  p =                     z g p =          
Below we demonstrate how a self-consistent parabolic molecular potential

U (z) = U (0) - 3 2a 2 κ 2 z 2 (22)
(measured in k B T units) with specifically tailored coefficient κ ensures independence of the free energy penalty Ω for insertion of a probe Y-shaped polymer (depicted in Figure 1). In the strong stretching approximation (SSA), the chain conformations in the self-consistent or external potential field are described via trajectories that specify the most probable position of each monomer unit at given position of the chain end-point. We introduce the trajectory and stretching functions of the stem, E 0 (z) = dz/dj, and corresponding branches, E 1 (z) and E 2 (z) for the Y-shaped polymer with specified positions z = y 1 and z = y 2 of the ends of the shorter and the longer free branches, respectively.

If tethered to the surface Y-shaped polymers is subjected to external potential U ex (z), then the free energy of such polymer (measured in k B T units) can be presented as Ω = where the first, the second and the third integrals correspond to contributions of the stem, short and long free branches, respectively. The first term in each integral accounts for the elastic (conformational) free energy, whereas the second term is the energy of the corresponding chain segment (stem or free branch) in the potential U ex (z).

The stretching functions E 0 , E 1 , E 2 of the chain segments, and the position z 1 of the branching point can be derived by minimization of the free energy given by eq 23 with the account of conservation of the numbers of monomer units in each segment of the Y-shaped macromolecule, that is,

z 1 0 dz E 0 (z) = n 0 ; y 1 z 1 dz E 1 (z) = n 1 ; y 2 z 1 dz E 2 (z) = n 2 (24)
which is reduced to the minimization of the functional Φ = The indefinite Lagrange multiplies λ 0 , λ 1 , and λ 2 , insure the concervation of the number of monomers in each chain segment. Variation of functional Φ with respect to the unknown functions E 0 , E 1 , E 2 leads to the set of equations:

E i (z) = 2a 2 3 U ex (z) + λ i , i = 0, 1, 2 (26)
Lagrange multiplies λ 1 , and λ 2 are found from the condition of vanishing tension at free ends of the branches,

λ i = -U ex (y i ), i = 1, 2 (27)
to give U ex (z) -U ex (y 1 )dz (31) The subsequent minimization of the functional Φ given by eq 31 with respect to z 1 leads to the equation, U ex (z 1 ) + λ 0 -U ex (z 1 ) -U ex (y 2 ) -U ex (z 1 ) -U ex (y 1 ) = 0 (32)

E 0 (z) = 2a 2 3 U ex (z) + λ 0 (28)
E 1 (z) = 2a 2 3 U ex (z) -U ex (y 1 ) (29)
Notably, eq 32 is equivalent to the balance of elastic forces,

E 0 (z 1 ) = E 1 (z 1 ) + E 2 (z 1 )
in the branching point.

Figure 1 :

 1 Figure 1: Schematic of Y-shape macromolecule (a) and comb-like polymer (b) tethered to the surface by the root segment of the main chain.

Figure 2 :

 2 Figure 2: Self-consistent potential (left). free ends distributions (middle) and branching points distribution (right) in brushes of Y-shaped polymers, calculated numerically by SF-SCF method. The DP of the Y-shaped polymers N = 300 is kept constant, the DP of the stem n 1 is varied from 200 to 2, n 2 ≤ n 3 are the DPs of branches.

Figure 3 :

 3 Figure 3: Polymer density profiles ϕ as a function of z (a) and z 2 (b) in planar brush of comb-shaped macromolecules with m 0 = m = n and total number of monomer units N = n + P (q + 1)n at fixed surface coverage N a 2 /s = 20. Other parameters are m = n = 100, q = 5, number of periods P = 1, 2, 3, , 10 indicated at the curves. The Flory-Huggins parameter χ = 0.

Figure 4 :Figure 5 :Figure 6 :

 456 Figure4: Distribution of the free ends of the backbone g(z) in planar brush of comb-shaped macromolecules with m 0 = m = n and total number of monomer units N = n + P (q + 1)n at fixed surface coverage N a 2 /s = 20. Other parameters are m = n = 100, q = 5, number of periods P = 1, 2, 3, , 10 indicated at the curves. The Flory-Huggins parameter χ = 0.

Figure 7 :

 7 Figure7: Polymer density profiles φ(z) and distribution of the free ends of the backbone g(z) in planar brush of comb-shaped macromolecules with stem length m 0 = 0 with total number of monomer units N = P (q + 1)n at fixed surface coverage θ = N a 2 /s = 20. Other parameters are m = n = 100, q = 5, number of periods P = 1, , 10 indicated at the curves. The Flory-Huggins parameter χ = 0.

2 1 z 1 3 2a 2 E 1 y 2 z 1 3 2a 2 E 2

 211212122 E 0 (z) + U ex (z) + λ 0 E 0 (z) dz+ y (z) + U ex (z) + λ 2 E 1 (z) dz + (z) + U ex (z) + λ 1 E 2 (z) dz(25)

E 2 (z) = 2a 2 3 U√ 6 = z 1 0U 2 z 1 U 1 z 1

 23612111 ex (z) -U ex (y 2 ) (30) With the account of eqs 28, 29, 30, functional Φ reduces to Φ a ex (z) + λ 0 dz + y ex (z) -U ex (y 2 )dz + y

Acknowledgment

This work was supported by Russian Science Foundation grant 20-13-00270

The application of normalization conditions given by eq 24 provides three equations,

By substituting eqs 28 -30, and 33 -35 in eq 23, one finds the penalty for insertion of an Y-shaped polymer into the external field U ex (z)

We now show that if U ex (z) is given by eq 22 with the topological coefficient κ given by the minimal solution of equation tan(κn 0 )[tan(κn 2 ) + tan(κn 1 )] = 1, (37) then the insertion free energy penalty Ω in eq 36 is independent of the position y 2 of the free end of the longest branch (or, equivalently, on positions y 1 or z 1 of the shorter branch or the branching point, respectively) of the Y-shaped polymer.

Substituting U (z) = U ex (z) given by eq 22 into eqs 28 -30 leads to

whereas substituting eq 22 into eqs 33 -35 leads to

where eq 41 can be presented as

Eq 32 reduces to

and, finally, with the account of eqs 42 -44

Substitution of eqs 42 -44 in eq 46 leads to eq 37. The relative positions of the free end of the shorter branch and of the branching point yield

while the stretching function E 0 (z) is given by

By using eqs 38, 39,40 and eq 44 into eq 36 the insertion free energy penalty of probe chain can be calculated as

Hence, Ω does not depend on position y 2 of the free end of the inserted Y-shaped polymer. If the molecular potential acting on each monomer unit inside the brush is given by eq 22, then chain insertion free energy Ω given by eq 49 coincides with the chain chemical potential.

Whether for any given set of {n 1 , n 2 , n 3 } the potential given by eq 22 is really self-consistent can be proven only by numerical calculations which indicate absence of presence of the dead zone and provide the trueshape of the self-consistent potential. As demonstrated above, in spite of existing solution of eq 37, not for any set of {n 1 , n 2 , n 3 } the parabolic potential given by eq eq 22 is a self-consistent one.


Mikhail Y Laktionov

Oleg V Shavykin

Frans A M Leermakers

Ekaterina B Zhulina

Oleg V Borisov
email: oleg.borisov@uiv-pau.fr

Colloidal particle interacting with a polymer brush: a self-consistent field theory

Interaction of colloidal particle with planar polymer brush immersed into a solvent of variable thermodynamic quality is studied by numerical self-consistent field method combined with analytical mean-field theory. The effect of embedded particle on the distribution of polymer density in the brush is analyzed and the particle insertion free energy profiles are calculated for variable size and shape of the particle and sets of polymer-particle and polymer-solvent interaction parameters. In particular, both cases of repulsive and attractive interactions between particle and brush-forming chains are considered. It is demonstrated that for large particles the insertion free energy is dominated by repulsive (osmotic) contribution and is approximately proportional to the particle volume in accordance with earlier theoretical predictions [Halperin, A. et al, Macromolecules 2011, 44,3622].

1

For the particles of smaller size or/and large shape asymmetry the adsorption or depletion of polymer from the particle surface essentially contributes to the insertion free energy balance. As a result, depending on the set of the polymer-solvent and polymer-particle interaction parameters and brush grafting density the insertion free energy profile may exhibit complex patterns, i.e., from pure repulsive effective potential barrier to attractive well. The results of our study allow for predicting equilibrium partitioning and controlling diffusive transport of (bio)nanocolloids across (bio)polymer brushes of arbitrary geometry including polymer-modified membranes or nanopores.

Introduction

Understanding transport and partitioning of nanocolloids and large molecules in polymeric media, e.g., concentrated solutions, melts, gels, promotes comprehension of processes occurring in technological applications (separation, purification) and living tissues. The principals of selective transport and partitioning of macromolecules adopted from nature can be successfully exploited in bioinspired nanotechnology.

For instance, the membranes of bacterial and endothelial cells are decorated by anchored biopolymers (lipopolysaccharides, glycoproteins) that form brushlike interfacial layers. [START_REF] Klein | Polymers in living systems: From biological lubrication to tissue engineering and biomedical devices[END_REF][START_REF] Camesano | Heterogeneity in bacterial surface polysaccharides, probed on a single-molecule basis[END_REF][START_REF] Vu | Bacterial extracellular polysaccharides involved in biofilm formation[END_REF][START_REF] Avila-Calderón | Outer Membrane Vesicles of Gram-Negative Bacteria: An Outlook on Biogenesis[END_REF] Similar structural motif for intrinsically disordered proteins is found inside nucleopores perforating membrane of nucleus. These brushlike arrays of biopolymers constitute steric penetration barriers providing selective permeability of globular proteins and macromolecules. [START_REF] Suntharalingam | Peering through the pore: Nuclear pore complex structure, assembly, and function[END_REF][START_REF] Kubitscheck | Nuclear transport of single molecules: Dwell times at the nuclear pore complex[END_REF][START_REF] Tu | Large cargo transport by nuclear pores: Implications for the spatial organization of FGnucleoporins[END_REF][START_REF] Fu | Investigating molecular crowding within nuclear pores using polarization-PALM[END_REF][START_REF] Denning | Disorder in the nuclear pore complex: The FG repeat regions of nucleoporins are natively unfolded[END_REF][START_REF] Yang | Nuclear import time and transport efficiency depend on importin β concentration[END_REF] In the realm of medicine, polymer brushes formed by biocompatible hydrophilic polymers, e.g. PEG, PMOXA etc. are used for fabrication of biocompatible non-fouling surfaces [START_REF]Practical approach series[END_REF][START_REF] Chen | 50th Anniversary Perspective: Polymer Brushes: Novel Surfaces for Future Materials[END_REF][START_REF] Halperin | Polymer brushes that resist adsorption of model proteins: Design parameters[END_REF][START_REF] Ayres | Polymer brushes: Applications in biomaterials and nanotechnology[END_REF][START_REF] Krishnamoorthy | Surfaceinitiated polymer brushes in the biomedical field: Applications in membrane science, biosensing, cell culture, regenerative medicine and antibacterial coatings[END_REF][START_REF] Doberenz | Thermoresponsive polymers and their biomedical application in tissue engineering-A review[END_REF][START_REF]Responsive Polymer Materials: Design and Applications[END_REF] of artificial implants and medical devices, as well as providing stealth properties for nanoscale drug delivery systems. [START_REF] Nishiyama | Nanostructured devices based on block copolymer assemblies for drug delivery: Designing structures for enhanced drug function[END_REF][START_REF] Schacher | Functional block copolymers: Nanostructured materials with emerging applications[END_REF][START_REF] Sakai-Kato | General considerations regarding the in vitro and in vivo properties of block copolymer micelle products and their evaluation[END_REF][START_REF] Betageri | Liposome Drug Delivery Systems[END_REF][START_REF] Stealth Liposomes; Lasic | [END_REF] In spite of broad use, the mechanisms of interaction (selective penetration, accumulation or expulsion) of (bio)nanocolloidal particles, first of all globular proteins, in polymer brushes are still not fully understood.

In particular, balance of steric repulsive and short-range attractive interactions between brush-forming polymers and particles may have an impact on dependencies of the partition coefficients on the size of the particles and brush grafting density. There are indications from experiments that some proteins, e.g. BSA, may exhibit weak affinity to PEG chains and therefore accumulate in PEG brushes. [START_REF] Norde | Interaction of bovine serum albumin and human blood plasma with PEO-tethered surfaces: Influence of PEO chain length, grafting density, and temperature[END_REF][START_REF] Bosker | BSA adsorption on bimodal PEO brushes[END_REF] Furthermore, solubility of partially hydrophobic or thermosensitive brush-forming chains may provide additional control parameters for manipulating partitioning and transport of bionanocolloids through polymer brushes.

Theories of interactions of nanocolloids with planar polymer brushes were developed in the past decades mostly on the basis of strong stretching self-consistent field approximation. In solvent-free (dry) brushes, a penalty for nanoparticle insertion was described in terms of the pressure field acting at nano-object. [START_REF] Kim | Morphology Selection of Nanoparticle Dispersions by Polymer Media[END_REF] Polymer-soluble nanoparticles smaller than a brushdetermined threshold were shown to disperse in the film to a depth scaling inversely with particle volume. In solvated brushes penetration of nanoobjects, e.g., small proteins [START_REF] Halperin | Polymer brushes that resist adsorption of model proteins: Design parameters[END_REF][START_REF] Halperin | Primary versus ternary adsorption of proteins onto PEG brushes[END_REF] and ultrathin AFM tips [START_REF] Halperin | Atomic force microscopy of polymer brushes: colloidal versus sharp tips[END_REF] invoked short ranged perturbation of the brush concentration profile, in agreement with the simulation results. [START_REF] Murat | Molecular Dynamics Simulations of the Force between a Polymer Brush and an AFM Tip[END_REF][START_REF] Ermilov | Colloids in brushes: The insertion free energy via monte carlo simulation with umbrella sampling[END_REF] In the seminal paper by Halperin et al 31 a fundamental relation between particle insertion free energy and osmotic pressure inside a planar polymer brush was established. This theory accounted for the thermodynamic quality of the solvent to quantify the osmotic contribution to the insertion free energy of nano-object. The latter was proportional to the particle volume, depended on the particle-solvent interactions, and served as a good approximation for sufficiently large particles. However, for particles of smaller size (comparable to polymer density correlation length) such approximation becomes inaccurate. Furthermore, the details of short-range interactions between monomer units of the brush-forming chains and particle surface are crucially important and may strongly affect the shape of the insertion free energy profiles and thus particle transport through the brush.

The aim of the present paper is to study interaction of nanocolloidal particle with non-ionic polymer brush and calculate the position-dependent insertion free energy with the account of most relevant interactions, particle size and shape and brush grafting density. The effect of embedded particle on the density distribution of polymer in the brush was analyzed as well. Knowledge of the shape of the free energy versus particle position curves does not only allow predicting equilibrium distribution of particles between the brush and solution, but also paths the way to study diffusion of particles across polymer-modified membranes. Because of the high complexity of the system, the problem cannot be adequately treated neither analytically, nor with the aid of direct computer simulations, which would require long computational times for exploring relevant range of control (thermodynamic solvent quality for polymers, polymer-particle interaction) and architectural (polymer grafting density, particle size and shape) parameters. Therefore, we employ molecular realistic self-consistent field modelling which is the most appropriate technique. Specifically, we use the Scheutjens-Fleer self-consistent field (SF-SCF) lattice method [START_REF] Fleer | Polymers at Interfaces[END_REF] in its two-gradient version. The SF-SCF numerical method allows obtaining structural and thermodynamic properties of complex polymer systems with the accuracy close to that provided by coarsegrained molecular simulation (Brownian Dynamics, Monte Carlo) techniques, but with orders of magnitude higher computational efficiency, that is crucial in sdudies of many-chains systems (e.g. polymer brushes). On the other hand, this method is free of many approximations (e.g., strong stretching of polymer chains) unavoidably used by analytical theory of polymer brushes. The comparison of the obtained numerical results to those derived within simplified analytical model enables us to rationalize the main features in the insertion free energy patterns and, at the same time, to estimate accuracy and identify limits of applicability of the mean-field analytical approach.

The rest of the paper is organized as follows:

We start in Section 2 with brief reviewing principles of analytical the-ory of polymer brushes in solvent of arbitrary quality based on analytical strong-stretching self consistent field (SS-SCF) approximation and formulate analytical approach for calculating insertion free energy of a particle into the brush in Section 3. In Section 4 we present results of our numerical calculations for variable polymer-particle interaction parameter and solvent quality for the brush-forming chains and confront them to the results derived using analytical theory. Finally, in Section 5 we summarize our conclusions.

2 Analytical theory of polymer brushes: polymer concentration profile and osmotic pressure

We consider a brush formed by long flexible polymer chains with degree of polymerization (DP) N and one unit length Kuhn segment grafted by one end to an impermeable planar surface and immersed into a solvent. The grafting density is characterized by the area per chain s (or the inverse value σ). The solvent quality is characterized by the Flory-Huggins parameter χ P S where subscript PS refers to polymer-solvent interactions. Theory of structural properties of planar polymer brushes formed by nonionic polymer chains in solvent of arbitrary thermodynamic quality was developed in original papers of the authors [START_REF] Zhulina | Structure and conformational transitions in grafted polymer chain layers. A new theory[END_REF][START_REF] Zhulina | Coil-Globule Type Transitions in Polymers. 1. Collapse of Layers of Grafted Polymer Chains[END_REF][START_REF] Lebedeva | Dendron and Hyperbranched Polymer Brushes in Good and Poor Solvents[END_REF] where a virial approximation valid at small volume fraction of monomer units in the brush, φ 1, was employed. The virial approximation accounts only for binary and ternary monomer-monomer interactions in the brush and allows for exploring properties of relatively sparsely grafted brushes under good or close to theta-solvent conditions. Below we formulate a more general scheme which is operative at arbitrary polymer concentration. This allows us to explore the effects of solvent strength and polymer-particle interactions in a wide range of parameters, including collapsed in poor solvent brushes with polymer volume fraction approaching unity, φ 1.

Within the analytical strong-stretching self-consistent field (SS-SCF) approximation the self-consistent molecular potential inside the brush exhibits a parabolic dependence on the distance z from the grafting surface.

∂f {φ(z)} ∂φ(z) = 3 2 κ 2 (Λ 2 -z 2 ) ( 1
)
where φ(z) is the profile of volume fraction of monomer units in the brush, f {φ(z)} is the free energy of interactions in the brush per unit volume, κ is a coefficient dependent on the DP and topology of the brush-forming chains (for linear chains κ = π/2N ) and parameter Λ is specified below.

Eq. 1 presumes Gaussian (linear) conformational elasticity of the brushforming chains on all the length scales and absence of dead zones proximal to the grafting surface and depleted of the chain ends. Remarkably eq 1 is applicable irrespectively of the specific type of interactions (functional form of f {φ(z)}) in the brush. Here we apply the mean field Flory-Huggins approximation

f {φ(z)} k B T = (1 -φ(z)) ln(1 -φ(z)) + χ P S φ(z)(1 -φ(z)) + φ(z)(1 -χ P S ) (2)
Combining eqs 1 and 2 we obtain implicit dependence of the polymer volume fraction in the brush on z as

-ln(1 -φ(z)) -2χ P S φ(z) = 3 2 κ 2 (Λ 2 -z 2 ) (3)
The cut-off length for polymer volume fraction profile (the brush thickness) D has to be found from the normalization condition

D 0 φ(z)dz = N s (4)
The osmotic pressure inside the brush is given by

Π(z) = φ(z) ∂f {φ(z)} ∂φ(z) -f {φ(z)} = k B T [-ln(1 -φ(z)) -φ(z) -χ P S φ 2 (z)] ≈ k B T [( 1 2 -χ P S )φ 2 (z) + 1 3 φ 3 (z) + ...] (5)
where the last equality corresponds to the virial expansion in terms of φ valid at φ 1. If we define the polymer volume fraction at the edge of the brush, z = D as φ(D) ≡ φ D , then, from the condition of vanishing osmotic pressure at the edge of the brush, Π(z = D) = 0, we find an equation for φ D as a function of

χ P S -ln(1 -φ D ) -φ D -χ P S φ 2 D = 0 (6)
By substituting z = D into eq 3 we find that

Λ 2 = D 2 - 2 3κ 2 [ln(1 -φ D ) + 2χ P S φ D ] (7)
Figure 1: Cylindrical colloidal particle embedded in a polymer brush, schematic representation of SF-SCF 2-gradient version implementation with cylindrical symmetry. Concentric circles on the left side represents grafting surface (z = 0), the particle center position is denoted as z c , particle diameter and height are w and h, respectively.

As follows from eq 6, φ D = 0 at χ P S ≤ 1/2, (i.e., under good or thetasolvent conditions) and φ D ≥ 0 at χ P S ≥ 1/2 (poor solvent conditions). Evidently, under poor solvent conditions φ D defined by eq 6 coincides with the polymer volume fraction in a large globule (precipitate polymer phase). Then, as follows from eq 7, Λ 2 = D 2 at χ P S ≤ 1/2 and Λ 2 ≤ D 2 at χ P S ≤ 1/2.

3 Free energy of the particle insertion into the brush: an analytical approach Without major loss of generality and having in mind implementation of SF-SCF numerical self-consistent field scheme in its 2-gradient version, we consider the brush with embedded particle which possesses cylindrical symmetry with respect to the axis perpendicular to the grafting surface (Figure 1). Let the z-coordinate of the center of the particle be z c . The radius of the particle is described by the function r c (z ), where z ∈ [z c -h/2, z c + h/2] and h is the dimension of the particle in the z direction.

If one can neglect gradient in polymer density inside the brush in the zdirection on the length scale of the order of the particle size, h • |dφ(z)/dz| φ(z), then the insertion free energy (with the reference state of the particle outside the brush) can be approximated as

∆F (z c ) =∆F osm (z c ) + ∆F surf (z c ) = Π(z c ) • V + γ{φ(z c )} • A (8)
where

V = π zc+h/2 zc-h/2 r 2 c (z )dz (9)
and

A = 2π zc+h/2 zc-h/2 [r c (z )+ 1 2 δ(z -(z c -h/2))r 2 c (z )+ 1 2 δ(z -(z c + h/2))r 2 c (z )]dz (10)
are the volume and the surface area of the particle, respectively and γ{φ(z c )} is the excess free energy per unit area of the particle in contact with polymer solution of concentration φ(z c ).

A more general expression for the insertion free energy which we use in our numerical calculations is

∆F (z c ) = π zc+h/2 zc-h/2 Π(z )r 2 c (z )dz + 2π zc+h/2 zc-h/2 γ{φ(z )}[r c (z ) + 1 2 δ(z -(z c -h/2))r 2 c (z )+ 1 2 δ(z -(z c + h))r 2 c (z )]dz (11)
Eqs 8, 11 without the second (proportional to the particle surface) term are equivalent to those derived in ref. [START_REF] Halperin | Colloid-brush interactions: The effect of solvent quality[END_REF] Importantly, the particle is treated as a probe, that is, upon evaluating ∆F we use unperturbed profiles of polymer concentration φ(z) and osmotic pressure Π(z) in the brush (in the absence of a particle). Moreover, splitting of the insertion free energy into volume and surface terms is justified only if the particle size is much larger than the width of the zone of perturbed by the particle polymer density distribution (the correlation length or adsorbtion layer thickness). As demonstrated below, a significant deviation between approximate analytical and exact numerical results are found for small particles.

The insertion free energy ∆F (z c ) is directly related to the partition coefficient K(z c ) = exp(-∆F (z c )/k B T ). The particles can be either expelled from the brush, K ≤ 1 (depletion regime) or accumulated in the brush, K ≥ 1 (absorption regime). The concentration of particles inside the brush is either small or larger than in the bulk of the solution at K ≤ 1 or K ≥ 1, respectively.

While the volume contribution to the insertion free energy (the first term in eq. 8 or 11) is always positive, since Π(φ(z)) ≥ 0, the sign of particle surface contribution (the second term in eq 8 or 11 proportional to γ) depends on polymer-particle interactions and can be either positive or negative depending on the set of interaction parameters, {χ P S , χ P C , χ CS }, and local concentration of polymer φ(z), i.e γ(z) = γ{χ P S , χ P C , χ CS ; φ(z)}. Here χ P S , χ P C , χ CS describe direct polymer-solvent (PS), polymer-colloid (PC) and colloid-solvent (CS) interactions, respectively, within employed below discrete lattice model. Without loss of generality we set χ CS = 0. Then physically relevant control parameter describing polymer-colloid interaction is

χ ads = χ P C -χ P S (1 -φ) (12)
(free energy change upon replacement of a contact of monomer unit with solvent by a contact with the surface).

If the surface of the particle is repulsive (χ ads ≥ 0) or even weakly attractive (χ (crit) ads ≤ χ ads < 0) for polymers, then, because of imposed by the impermeable surface steric constraint on the available set of polymer conformations, polymer is depleted from the particle surface and γ ≥ 0. At χ ads = χ (crit) ads conformational entropy losses due to presence of the surface and free energy gain upon monomer-surface contact cancel each other and γ vanishes. [START_REF] Fleer | Polymers at Interfaces[END_REF][START_REF] Birshtein | Adsorption of polypeptides on solid surfaces. I. Effect of chain stiffness[END_REF][START_REF] Birshtein | Conformations of polymer chains grafted to an impermeable plane surface[END_REF][START_REF] Eisenriegler | Adsorption of polymer chains at surfaces: Scaling and Monte Carlo analyses[END_REF] The value of χ (crit) ads depends on the local properties of the lattice where polymer chain performs its random walk. For the simple cubic lattice χ (crit) ads = 6ln(5/6). Finally, at χ ads ≤ χ (crit) ads polymer is sufficiently strongly attracted to the surface of the particle and forms enriched by polymer adsorbed layer entrapping the particle, and γ ≤ 0.

The difference χ ads -χ

(crit)
ads serves as a universal parameter characterizing switch between adsorption and depletion regimes at vanishing polymer concentration. However, as it will be confirmed below by SF-SCF calculations, γ(z) varies throughout the brush and exhibits more complex dependence on φ(z) than it follows from eq 12. Furthermore, for small particles χ

Impact of an embedded particle on the brush structure

For numerical SF-SCF calculations of the particle insertion a polymer brush with a degree of polymerization N = 1000 and the grafting density of σ = 0.02 has been selected. The embedded particle has a cylindrical shape, with the height h and the diameter of the base w. The symmetry axis of the cylindrical particle is directed perpendicular to the grafting surface and coincides with the z-axis of our cylindrical coordinate system. (Figure 1). As shown in Figures 2a and2b the profiles of polymer density φ(z) and the profiles of osmotic pressure Π(z) calculated analytically (SS-SCF) and numerically (SF-SCF) for particle-free brush at different values of χ P S ranging from good (χ P S = 0) to theta-(χ P S = 1/2) and poor (χ P S ≥ 1/2) solvents almost perfectly match each other. A smoother decay in the numerically calculated density profiles near the edge of the brush are due to non-stretched terminal segments of the chains that exhibit Gaussian fluctu-ations not accounted for in the SS-SCF scheme. For plotting "numerical" osmotic pressure profiles Π(z) presented in Figure 2b we used analytical eq 5 where numerically calculated by SF-SCF density profiles φ(z) were substituted. A contraction of the brush caused by decreasing solvent strength is accompanied by an increase in polymer density inside the brush and concomitant decrease in the osmotic pressure. A zone of negative osmotic pressure appearing under poor solvent conditions (χ P S ≥ 1/2) close to the edge of the brush is a manifestation of positive surface tension at the interface between collapsed polymer brush and poor solvent. [START_REF] Grosberg | Statistical Physics of Macromolecules[END_REF] Two-gradient SF-SCF calculations of a polymer brush with an embedded particle allows obtaining detailed information about spatial polymer density distribution in the brush. In Figures 3,4 spatial polymer density diagrams and profiles are presented for the brush with an embedded particle.

An embedded particle causes perturbation of the polymer density distribution both in the directions perpendicular (z -axis) and parallel (x -axis) to the grafting surface, as one can see it as a halo of different color around the particle in Figures 3 or wells in Figure 4.

Let us take a closer look on the polymer density depletion and enrichment regions. To examine disturbances in polymer density a volume (highlighted with a red borders rectangle in upper right frame of Figure 3a) proximal to an embedded particle replotted in Figure 3b in a different scale. The sequence which diagrams are presented in corresponds to the ones on Figure 3a The diagrams of Figure 4 show the difference between an unperturbed particle-free polymer brush and a polymer brush with an embedded particle φ -φ * , where φ * is the polymer concentration of the corresponding unperturbed polymer brush. The difference φ -φ * is color-coded with diverging color scale, where blue color means polymer depletion, red color means an excess of polymer, while white means that the polymer concentration remains unchanged.

In Figure 4 the polymer concentration profiles in z-(Figure 4a, 4b) and in x-directions (Figure 4c) are plotted. (The x-axis perpendicular to the z-axis z has its origin in the particle center z c ). Figure 4a illustrates polymer concentration of an unperturbed brush φ * (z) and the ones with an embedded particle of the same size and position but different affinity to the polymer (χ P C = {-0.5, -0.75, -1.0, -1.25}). To make the profiles in the vicinity of the particle easier to trace we present Figure 4b as a zoomed version.

Note that the polymer concentration profiles in the lateral x-direction in unperturbed brush (Figure 4c) has the shape of a straight horizontal line, meaning no disturbances in the brush when no particle is introduced.

In the case of weakly attractive particle χ P C = -0. 3a,3b and blue trace in Figure 4), there is pronounced depletion of the polymer near the embedded particle, which can be seen as the blue halo in Figure 3b. In contrast, around strongly attractive particles χ P C = -1.0, -1.25 (the frames in the second row in Figures 3a,3b and the green and red traces in Figure 4), polymer density is increased in the vicinity of the embedded particle. Note the red halo around the particle in the corresponding frames of Figure 3b.

The behavior of polymer density in the vicinity of the particle in case of χ P C = -0.75 is more complex (see the diagram with the red borders rectangle in Figure 3a and the diagram with red axis spines Figure 3b or the orange color trace in Figure 4). Since χ P S = 0.5 is chosen for these calculations, the value of χ P C = -0.75 at z c = 40 corresponds approximately to critical adsorption conditions, i.e., χ ads ≈ χ crit , so that γ(z c ) ≈ 0. Despite of the vanishing surface term in free energy change ∆F surf ∼ γ ≈ 0, the weak depletion and enrichment regions near the particle are observed.

Insertion free energy profiles: good solvent conditions, varied polymer-particle interactions

In Figure 5 we present particle insertion free energy profiles ∆F (z c ) normalized by the particle volume V = πhw 2 /4, for particles with variable dimensions w = h = {4, 6, 8, 10, 12}, and different strengths of polymer-particle interaction, χ P C = {-0.5, -1.1, -1.5}. The solvent is good (athermal) for polymers, χ P S = 0. The case χ P C ≥ 0 would corresponds to pure repulsive polymer-particle interactions, whereas at χ P C ≤ 0 a short-range attraction operates between monomer units of the brush-forming chains and particle surface. As a reference line, the osmotic pressure Π(z c ) in the particle-free brush is also plotted in Figure 5. It is expected that for large particles or/and vanishing interfacial tension at the particle-polymer solution interface, γ ≈ 0, the normalized insertion free energy profiles ∆F (z c )/V should match Π(z c ) curves (see eq 8).

As we can see from Figure 5 in the case of weakly attractive, χ P C = -0.75 (or, similarly, repulsive) particle, with χ ads ≥ χ (crit) ads , the normalized insertion free energy ∆F (z c )/V curves go systematically beyond Π(z c ) because of excess positive free energy contribution due to depletion layer formed by the polymer around the particle, leading to γ ≥ 0. Hence, even without polymer adsorption on the particle, the proportionality of the insertion free energy to the particle volume is violated for sufficiently small particles.

In the case of strongly attractive particle, χ P C = -1.5, (χ ads < χ the ∆F (z c )/V curves go beneath Π(z c ), since γ ≤ 0 and the surface contribution to the insertion free energy is negative. Remarkably, for small particles regions with negative total insertion free energy appears, as the surface contribution outbalances the volume contribution. Which is corresponds to spontaneous particle absorption into such regions of the brush and increased concentration of particles compared to the bulk of the solution, as the partition coefficient becomes K(z c ) ≥ 1.

As expected, the larger the particle, the smaller its surface-to-volume, A/V ratio, the closer the ∆F (z c )/V curves to the osmotic pressure profile Π(z c ) are.

The middle drawing of Figure 5 corresponds to the case when χ ads ≈ χ P C ≈ χ (crit) ads (we remind that here χ P S = 0) and the surface contribution to free energy fairly vanishes, i.e. γ ≈ 0. In this case the normalized free energy ∆F (z c )/V curves are close to osmotic pressure profile regardless of the particle size. Indeed, as follows from eq 12 at χ P S = 0, χ ads -χ (crit) ads is independent of polymer concentration and can be tuned to zero throughout the brush. For any other arbitrary selected value of χ P S = 0 there is no single value of χ P C which would assure that χ ads -χ (crit) ads = 0 and γ(z c ) = 0 ∀z c ∈ [0, D] and ∆F (z c )/V and Π(z c ) profiles match each other in the whole z c range.

γ(φ(z)): Impact of polymer-particle interface

Obviously, γ = γ(z) essentially depends not only on the set of χ-parameters, but also on local polymer concentration φ(z). The value of γ{χ P S , χ P C ; φ(z)} can be evaluated from numerical SF-SCF data by subtracting the volume contribution from the calculated insertion free energy, using eq. 11

γ num (z c ) = ∆F (z c ) -∆F osm (z c ) A = ∆F (z c ) A - π A zc+h/2 zc-h/2
Π(z )r 2 c (z )dz (13) where A = π 2 (w 2 + 2wh) is the particle surface area. We remind that calculated according to eq 13 value of γ{χ P S , χ P C ; φ(z)} should be considered as an apparent one because the width of the interface between the particle and surrounding polymer solution (the brush interior) in our calculations is comparable to the particle dimensions.

In Figure 6 we present γ as a function of χ P C calculated for selected values of χ P S and different particle position z c inside the brush using SF-SCF numerical data and eq. 13. The observed trends are consistent with expected dependence γ ∼ (χ ads -χ ads (marked with the red circle in Figure 6). For χ P S ≥ 0 the value of χ P C corresponding to inversion of sign of γ gets systematically smaller as z c decreases (i.e. φ(z c ) increases). This observation illustrates the fact that even if critical adsorption conditions are met at some distance z c from the surface, they are violated for other particle locations in the brush. Notably, the slope of γ(χ P C , z c ) curves in Figure 6 slightly increases upon a decrease in z c , that is, upon an increase of local polymer concentration φ(z c ) inside the brush.

(crit) ads ) ≡ χ P C -χ P S (1 -φ(z c )) -χ (crit) ads : All
In Figure 7 we present apparent (calculated by eq 13 from numerically evaluated insertion free energy) values of γ at different solvent strength (cor-responding to different color of the points), different values of the parameter χ P C of polymer-particle interactions and different particle position z c in the brush. In the 3D plot γ is plotted as a function of χ ads = χ P C -χ P S (1-φ(z c ) and local polymer concentration φ(z c ).

Columns of points with the same {χ ads , φ} correspond to different particle size. Closer inspection indicates that the points get closer to each other upon an increase in the particle size, that is, weakening of the effect of the particle curvature and approaching of the apparent values of γ calculated from eq. 13 to real interfacial tension between polymer solution with concentration φ and solid particle.

The set of obtained data points was fitted by the analytical function

γ(χ ads , φ) = (χ ads -χ (crit) ads )(a 1 φ + a 2 φ 2 ) ( 14
)
where χ ads = φ P C -φ P S (1-φ) and a 1 , a 2 are numerical fitting parameters and we neglected higher order terms in φ in the last bracket in eq 14. The rough estimate of a 1 and a 2 are 0.16 and 0.08, respectively.

As follows from eq. 12 and one can see in Figure 7, at sufficiently low polymer concentration in the brush, φ 1, the value of γ is controlled by the parameter χ ads (φ = 0) = χ P C -χ P S . Hence, a decrease in χ P C or/and an increase in χ P S result in stronger polymer affinity to the particle surface and smaller net free energy penalty for the particle insertion into the brush. In the first approximation, the sign of γ is controlled by the difference χ ads -χ

(crit) ads ≈ χ ads (φ = 0) -χ (crit) ads ≈ χ P C -χ P S -χ (crit)
ads . Note that, dγ dχ ads = (a 1 φ + a 2 φ 2 ) so the slope of γ(χ ads ) increases with polymer concentration, which can be seen as a counter-clockwise rotation of the grey surface around the χ ads = χ (crit) ads line. Thus the sign of γ remains the same for χ ads (φ = 0) and χ ads at arbitrary φ or z. Since χ ads is a function of φ too -the aforesaid approximation is invalid for arbitrary φ especially in the poor solvent regime with χ P S > 0.5. With higher values of χ P S and φ the approximation gets off, underestimating γ.

The region with φ(z) 1 in poor solvent regime is either absent or has a very narrow interval of z, φ(z) profile according to ASCF predictions is not continuous and has a sharp edge, numerical results confirms this prediction exhibiting very steep change in polymer concentration at the polymer brush's edge. In Figure 7 this caused the absence of numerical results for χ P S = 0.6 in low concentration range.

Having γ(χ ads , φ) calculated, we can evaluate insertion free energy as

where the osmotic pressure Π(φ(z)) is given by eq 5 and we use the polymer density profile φ(z) calculated by SF-SCF method for the particle-free (unperturbed) brush.

In Figure 8 we compare theoretically calculated (by eq 15) insertion free energy (red trace) with SC-SCF numeric results (blue scatter). The analytically calculated osmotic term ∆F osm (z c ) and interfacial term ∆F surf (z c ) are presented by grey dashed line and grey dotted line, respectively. The results are presented for selected values of χ P S = {0.0, 0.3, 0.6} and χ P C = {-0.5, -1.0, -1.5} for a cylindrical particle with dimensions h = w = 10.

The solvent quality is decreasing from the left (athermal solvent, χ P S = 0) to the right (down to moderately poor solvent, χ P S = 0.6).

According to ASCF, for the poor solvent regime χ P S > 0.5 the polymer concentration profile became a step-like function, with φ(z = 0) = 0. Using ASCF approximation, polymer concentration for a brush with N = 1000 and σ = 0.02 at χ P S = 0.6 has been evaluated to be φ(z = 0) ≈ 0.325, φ(z = D) ≈ 0.245 where D ≈ 66.2, which can also be seen in the Figure 2. In Figure 8 the right panel correspond to the poor solvent regime, thus covers only the cases where φ / ∈ [0, φ(z = 0)]. Note that χ ads ≈ χ ads (φ = 0) can not deliver right result and more rigorous eq. 12 has to be used.

The polymer-particle contact attraction energy is increasing from up to down. The adsorption strength |χ ads | = |χ P C -χ P S (1 -φ)| for a given φ is thus increasing from upper left to lower right corner.

In the upper line χ P C = -0.5 and at any solvent quality χ ads ≥ χ

(crit)
ads , that is, there is a depletion layer around the particle. For χ P S = 0.6 for ∀φ ∈ [φ(z = D), φ(z = 0)] γ > 0. As a result, γ ≥ 0 and both, the osmotic and the surface contributions to ∆F zc are positive though decreasing in magnitude upon a decrease in solvent strength.

The middle line corresponds to moderately strong polymer colloid attraction, χ P C = -1.0. At χ P S = 0 and χ P S = 0.3 nearly critical adsorption conditions, χ ads ≈ χ (crit) ads are fulfilled and surface contribution ∆F surf ∼ γ to the free energy fairly vanishes. The net free energy is dominated by the osmotic term. Under poor solvent conditions , χ P S = 0.6 adsorption of polymer on the particle takes place, because

χ ads = χ P C -χ P S (1 -φ) ≤ χ (crit)
ads , ∀φ ∈ [φ(z = D), φ(z = 0)], γ < 0 and, as a result, negative surface free energy overcompensate weak positive osmotic contribution: the free energy balance upon particle embedding into the brush is negative.

The lower line in Figure 8 corresponds to strong polymer-particle attraction, χ P C = -1.5, that is, polymer is adsorbed onto the particle surface, χ ads ≤ χ (crit) ads irrespectively of the solvent strength. Under good solvent conditions, χ P S , the left panel, the insertion free energy is dominated by osmotic repulsion and is positive at any particle position, except the very edge of the brush, where polymer concentration is low and osmotic repulsion is insufficient to overcome gain the free energy due to adsorption of terminal segments of the brush forming chains onto the particle: As a result, a shallow potential well emerge at the periphery of the brush. Upon a decrease in the solvent strength (an increase in χ P S ) the adsorption strength increases whereas the magnitude of osmotic repulsion decreases. As a result, the minimum in the insertion free energy becomes more pronounced and extends deeper into the brush. Only in the pre-surface region with higher polymer concentration the osmotic repulsion dominates over attractive surface term. Eventually, under poor solvent conditions (lower right corner of the Figure ), osmotic contribution is small whereas adsorption strength is large so that ∆F (z c ) is negative throughout the brush, which implies spontaneous accumulation of particles in the brush due to high polymer-particle affinity; the osmotic repulsive term ∆F osm ≥ 0 under these conditions is negligible ∀z ∈ [0, D].

Altogether, Figure 8 demonstrates good quantitative agreement between insertion free energy profiles ∆F (z c ) obtained by exact numerical SF-SCF calculations and by approximate analytical approach.

Effects of particle size and shape

As discussed above, in the cases of inert or weakly attractive for polymer segments particle, both terms in the insertion free energy are positive and ∆F (z c ) monotonously grows upon a decrease in z c (deeper embedding of the particle into the brush). If χ ads ≤ χ (crit) ads and γ ≤ 0, then the shortrange attraction operating between polymer segments and the particle leads to (partial) cancellation of the osmotic penalty for the particle insertion into the brush. However, for large particle the osmotic contribution, which is proportional to the particle volume, remains dominant even if the brushforming chains do adsorb onto the particle surface, γ ≤ 0.

In Figure 9 we plot ∆F/V as a function of χ ads at constant z c for varied particle volume V (the case of d = h) and varied solvent strength.

The osmotic pressure Π(φ(z c ) at given distance z = z c is controlled by grafting density σ, degree of polymerization N and the solvent strength χ P S . Hence, for a given particle position the osmotic pressure is a function of χ P S , which is depicted by horizontal lines on Figure 9. The normalized by the particle volume insertion free energy then can be approximated as ∆F (z c , χ P S , χ P C ) V ≈ Π(z c , χ P S ) + γ(z c , χ P S , χ P C ) A V (16) in the limit of large particles A V → 0, thus ∆F (z c , χ P S )

V ≈ Π(z c , χ P S ) (17)
As the particles get bigger the contribution related to the particle surface have less impact on the overall apparent insertion free energy. In Figure 9 we see that the slope of SF-SCF numerical results decreases with an increasing particle volume. For sufficiently big particles the curves are expected to collapse to horizontal lines Π(z c , χ P S ).

The ∆F (z c ) curves may acquire more complex shape for adsorbing polymer particles, γ ≤ 0, with smaller volume-to-surface ratio, that is for sufficiently small and asymmetric (h d or d h) particles. In Figure 10 we present ∆F (z c ) curves for adsorbing polymer particles with constant volume and varied ratio d/h for γ ≤ 0.

Intuitively it is expected that an increase in A/V ratio would cause surface term in free energy ∆F surf to grow in magnitude proportional to the particle surface A, while osmotic term proportional to the particle volume remains constant. So the curves should go systematically beneath the osmotic pressure profile in a manner, that the higher value of A is, the lower the corresponding curve goes, since γ ≤ 0. Thus we should have observed the next order in which all the curves lie in Figure 10: the particle with h = 8, w = 8 would be the closest to Π(z)/k B T profile, having the smallest surface; the particle with h = 2, w = 16, on the other hand, would be the lowest one.

As we can see Figure 10 SF-SCF numerical results evidence more complex behavior of the ∆F zc as a function of the particle shape. As expected, the insertion free energy for "cigar-like" particle (h = 32, w = 4) is systematically lower than for a symmetric (h = 8, w = 8) particle with the same volume because of larger surface area A for the former one: the minimum in ∆F (z c ) is shifted towards center of the brush and gets deeper. However, the profile of the insertion free energy ∆F (z c ) for the disk-like particle (h = 2, w = 16) is going beyond that for the symmetric particle, particularly in the inner denser part of the brush in spite of larger surface area of the disk. This effect can be attributed to stronger perturbation of conformations of the brush-forming chains upon embedding of the disk-like particle in quanched orientation with the base parallel to the grafting surface. A proper account of rotational

Effects of grafting density

Evidently, an increase in grafting density leads to both an increase in the brush thickness and (under good and theta-solvent conditions) to an increase in the polymer concentration in the brush. As soon as polymer density φ(z c ) affects in a different ways the osmotic and to the surface parts of the insertion free energy, we may expect that increasing grafting density may affect not only the extension and the magnitude, but also the shape of the insertion free energy ∆F (z c ) profiles.

The result of SF-SCF numerical calculations for varied grafting density σ compared to the predictions of the analytical model presented in Figure 11.

As expected, for non-adsorbing polymer particles, the magnitude of ∆F (z c ) progressively increases and the onset of the brush-particle repulsion is shifted to larger z c (due to increasing brush thickness) upon an increase in grafting density.

On the contrary, in the case of polymer-adsorbing (χ ads ≤ χ (crit)

ads particles, at large grafting densities pronounced peripheral minimum and proximal to the grafting surface maximum are observed. A decrease in the grafting density leads to a significant decrease in the maximum magnitude (and its eventual disappearance) and simultaneous deepening and widening of the the peripheral minimum in the insertion free energy profiles.

Discussion and conclusions

We have used self-consistent field SF-SCF numerical method to study insertion of a (nano)colloidal particle into a polymer brush. The effect of embedded particle on the polymer density distribution in the brush and the free energy penalty (or gain) upon particle insertion into the brush were analyzed. The latter controls equilibrium partitioning of particles between the brush and surrounding solution.

The position-dependent insertion free energy of the particle in the brush was analyzed as a function of (i) thermodynamic quality of the solvent for the polymer; (ii) short-range polymer-particle attraction/repulsion; (iii) particle size and shape; (iv) grafting density of polymer chains. The effect of embedded particle on the density distribution of polymer in the brush was analyzed as well.

The obtained by numerical SF-SCF method insertion free energy profiles were systematically compared to those calculated within approximate analytical scheme. The latter was developed using strong stretching self-consistent field theory of polymer brushes and is based on the assumption that the insertion free energy can be decoupled into two contributions proportional to the partial volume and partial surface, respectively. This decoupling scheme is asymptotically exact for large particles embedded in quasi-infinite reservoir of polymer solution. However, it can serve as a quantitatively reasonable approximation even for small particles inserted into finite-size brush considered here.

The free energy term ∆F osm proportional to the product of particle volume and local osmotic pressure is always positive and provides repulsive contribution to the net free energy balance. It grows upon increasing solvent strength and grafting density. For sufficiently large particles this term is always the dominant one. However, for the particles of small size and thus smaller surface-to-volume ratio the surface-related contribution ∆F surf may be comparable and even outperforming the volume term. As a result, in the case of short-range attractive interactions between polymer chains and the particle surface the net insertion free energy may become negative. We remind that for small particles splitting of the free energy into volume and surface controlled terms is ambiguous and can serve only as an approximation.

Depending on the sign of ∆F and shape of insertion free energy profile, different scenario of particle exclusion from or accumulation in the brush may take place.

If the particles are "inert" or weakly adsorbing for polymer chains, the insertion free energy profile has a shape of a potential barrier with the height increasing towards grafting surface. Remarkably, both the width (extension) and the magnitude of the barrier can be efficiently controlled by solvent strength for the brush-forming chains and decrease upon a decrease in the solvent strength. At given solvent strength the insertion free energy grows upon an increase in the grafting density, but is relatively insensitive to the DP of the brush-forming chains (the latter parameter affects primarily the width of the barrier). Hence, an increase in either grafting density or/and in solvent strength lead to exclusion of particles from the brush. This is a target in application of water-soluble polymer brushes in prevention of non-specific protein adsorption and design of antifouling surfaces.

The short-range attractive interactions between the particle and polymer chains may not only reduce the free energy penalty for particle insertion into the brush and lower the potential barrier, but also qualitatively change the insertion potential profile. The latter may acquire non-monotonic shape and exhibit local minima at the periphery of the brush followed by a repulsive barrier close to the grafting surface. The height of this barrier decreases upon increasing polymer-particle attraction energy or/and decreasing grafting density. In this case instead of exclusion, the particles will be accumulated inside the brush. Hence, our theory explains maximum in the dependence of absorbed by the brush number of nanoparticles on the brush grafting density, as observed in experiments. [START_REF] Gage | Adsorption of Nanocolloidal SiO2 Particles on PEO Brushes[END_REF][START_REF] Mora-Sierra | Stabilization and Kinetics of an Adsorbed Protein Depends on the Poly(N-isopropylacrylamide) Grafting Density[END_REF] As demonstrated above, the shape of the ∆F (z) curves can be tuned not only by solvent strength or polymer-particle affinity, but also by changing surface-to-volume ratio which depends on the size and shape asymmetry of the particles. This allows, for example, separating particles with respect to their shape.

We remind that within our theory we disregard either direct or indirect (by perturbation of the structure of the brush) interactions between the particles localized inside the brush. Hence, this theory is applicable at sufficiently low particle concentration inside the brush which is usually the case for pure repulsive shape of ∆F , though may be violated upon accumulation of particles in the brush.

If the brush is tethered to a (semi) permeable membrane, tuning of the width and shape of the barrier by changing solvent strength may allow controlling permeability of the polymer-modified membrane for particles. Obtained in this work results allow us to study quantitatively diffusion of particles across membrane-attached polymer brushes that will be in the scope of the forthcoming publication. Furthermore, because the developed in the present paper scheme enables calculating the insertion free energy as a function of local polymer concentration, solvent strength and polymer-particle interaction parameter, it can be further applied to analysis of partitioning and diffusion of nanoparticles in polymer brushes of more complex architectures, e.g., in polymer-modified nanopores.

Figure 2 :

 2 Figure 2: Polymer density (a) and osmotic pressure profile (b).Strong stretching self-consistent approximation (ASCF) and SF-SCF numerical results at χ P S = {0, 0.5, 0.6, 0.8} for N = 1000, σ = 0.02

  and follows the next order χ P C = [[-0.5, -0.75][-1.0, -1.25]].

23 <Figure 3 :Figure 4 :

 2334 Figure 3: Perturbation of the polymer density in a brush caused by an embedded particle for z c = 40, h = w = 10, χ P S = 0.5, N = 1000, σ = 0.02, at different polymer -colloid interaction parameters χ P C = {-0.5, -0.75, -1.0, -1.25} (a) -general view of the polymer density distribution. Polymer density is mapped with blue to yellow palette. Black contour lines trace selected polymer density values. Red viewport highlights space proximal to the embedded particle. (b) -proximal polymer density change (φ -φ * ), where φ * -polymer density of an unperturbed brush.)

Figure 5 :

 5 Figure 5: Insertion free energy change ∆F (z c ) normalized by the particle volume V at different particle center positions z c , particle sizes h = w = {4, 6, 8, 10, 12}, particle affinity χ P C = {-0.75, -1.1, -1.5} compared to osmotic pressure profile Π(z) for N = 1000, σ = 0.02 in an athermal solvent χ P S = 0

Figure 6 :

 6 Figure 6: Excess of surface free energy per unit particle area γ(z c ) as a function of χ P C at different particle center positions z c , at different solvent regimes χ P S = {0.0, 0.3, 0.5} for particle size h = w = 10, N = 1000, σ = 0.02. Positions z c are shown in the legend. Red circe highlights common intersection point.

Figure 7 :

 7 Figure 7: The dependence of γ and γ num on {χ ads , φ(z)} at different solvent regimes χ P S ∈ [0.0, 0.6] and particle affinity to the polymer χ P C ∈ [-1.5, -0.5] for N = 1000, σ = 0.02 SF-SCF numerical results (scatter) and corresponding analytical model surface. Green line corresponds to γ = 0

Figure 8 :Figure 9 :

 89 Figure 8: Apparent insertion free energy ∆F on z c at χ P S = {0.0, 0.3, 0.6} and χ P C = {-0.5, -1.0, -1.5} for N = 1000, σ = 0.02, h = w = 10. SF-SCF numerical results (scatter) and analytical model (red line). By gray dashed and dotted lines the osmotic, ∆F osm (z c ) and the surface, ∆F surf (z c ), contributions to the analytically calculated free energy are depicted.

Figure 10 :

 10 Figure10: The dependence of ∆F/V on particle geometry (h, w) = {(2, 16),(8, 8), (32, 4)} with fixed particle volume for χ P S = 0, χ P C = -1.5, N = 1000 and σ = 0.02 compared to the osmotic pressure profile Π(z)/k B T of corresponding unperturbed brush

Figure 11 :

 11 Figure 11: Apparent insertion free energy change ∆F on z c at σ = {0.01, 0.015, 0.02, 0.025, 0.03} and χ P C = {-0.5, -1.5} for w = h = 10, χ P S = 0.3 and N = 1000. SF-SCF numerical results (connected dots) and analytical model (dashed lines). Weak adsorbing particles (χ P C = -0.5) on the left and strong adsorbing particles (χ P C = -1.5) on the right

Acknowledgments

This work was supported by Russian Foundation for Basic Research, grant 21-53-10005.