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Keywords:

  • coatings;
  • computer modeling;
  • films;
  • graft polymer;
  • interfaces;
  • nanocomposites;
  • statiscal thermodynamics;
  • structure

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Theoretical descriptions of static properties of polymer brushes are reviewed, with an emphasis on monodisperse macromolecules grafted to planar, cylindrical, or spherical substrates. Blob concepts and resulting scaling relations are outlined, and various versions of the self-consistent field theory are summarized: the classical approximation and the strong stretching limit, as well as the lattice formulation. The physical justification of various inherent assumptions is discussed, and computer simulation results addressing the test of the validity of these approximations are reviewed. Also, alternative theories, such as the single chain mean field theory and the density functional theory, are briefly mentioned, and the main facts about the models used in the computer simulations are summarized. Both molecular dynamics and Monte Carlo simulations are described, the latter including lattice models and bead-spring models in the continuum. Also extensions such as brush–brush interactions or nanoparticles inside of brushes as well as the solubility of free chains in brushes are briefly mentioned. Pertinent experimental results, though still somewhat scarce, are mentioned throughout and their consequences on the status of the theoretical understanding of polymer brushes is emphasized. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Polymer brushes are obtained by covalent anchoring of long flexible macromolecules on a (nonadsorbing) substrate surface.1–9 Depending on the molecular weight of these end-grafted polymers and the density of the grafting sites, these polymers are more or less stretched perpendicularly to the grafting surface. Also variations in the chemical architecture of these macromolecules are interesting to consider: for example, one obtains “loop brushes,”10 if both chain ends of a linear macromolecules carry special reactive groups that are anchored at the substrate; or one may consider (noncatenated) brushes of ring polymers that are grafted to the substrate by a special chemical group.11 Even polymer brushes formed from grafted star polymers12, 13 or grafted comb polymers14 have been discussed. A further variation to this theme are binary mixed polymer brushes, where two chemically distinct linear chains (which we symbolically denote as A and B) are grafted at the substrate: depending on the balance between the binary interactions of monomers of type A and type B, and between these monomeric units and the solvent molecules, interesting mesophase ordering in these grafted macromolecular layers can occur.15–20 A related mesophase formation is possible when brushes are formed from end-grafted block copolymers.21–25 Another important aspect concerns the geometry of the substrate surface on which the chains are anchoring: the most frequently studied cases are planar substrates1–9, 26–59 albeit it is also possible to graft these polymers on the surface of spherical colloids60–66 or nanoparticles. Also grafting at the surface of cylindrical rods is of interest67–76 as well as on the inner surface of hollow cylinders.77–86 Interesting changes in the structure and properties of polymer brushes occur when the grafted chains are polydisperse87–93 rather than monodisperse, in particular, if the brush is almost monodisperse (containing a small minority of chains which are either much shorter or much longer than the majority).

The polymer chains in such polymer brushes organize themselves in very soft polymeric layers coating the substrate surface, which are easily perturbed by external stimuli (solvent quality,4, 40–43, 45, 94–97 temperature, applied electrical field and, last but not least, by applied mechanical forces3, 5, 9, 97–124). In this context, also the interaction with nanoparticles,125–130 proteins,131 and free flexible polymers in solutions132–137 are of interest, as well as surfactants or brush–brush interactions or interactions with polymer melts, polymer networks, and so forth138–156 Also the effects of monomer–substrate interactions deserve attention.8

A consequence of these facts is that polymer brushes find a lot of interest in the context of various applications: colloid stabilization, tuning the adhesion and wetting properties of surfaces, protective coatings preventing protein adsorption (“nonfouling surfaces” important for applications in a biological environment), improvement of lubrication properties of surfaces, devices controlling the flow through microfluidic channels, and so forth. However, since applications of polymer brushes have been reviewed extensively elsewhere,7 such applications shall not all be considered here further. Similarly, due to lack of expertise, we shall completely disregard all aspects of chemical synthesis (about these topics one also can find extensive accounts in the recent literature7). We rather focus here exclusively on the theoretical modeling of polymer brushes, in particular by computer simulation methods, because the predictive detail of analytic theories of polymer brushes is somewhat limited. We restrict attention to grafted neutral polymers: “polyelectrolyte brushes” and their interaction with ions in the solution is a rich topic in itself but outside the scope of the present review. Since we have recently given a brief review of polymer brushes under flow and in other out-of-equilibrium conditions elsewhere,9 we focus here on the static structure of polymer brushes in thermal equilibrium, and pay attention to possible phase transitions of polymer brushes. Also “bottle-brush” polymers, where side-chains are grafted to a flexible backbone macromolecule, will not be covered, since several other recent reviews exist.157–159 The outline of this review (which has its main focus on computer simulation approaches) is as follows: to set the perspective of the theoretical concepts that will be tested, we briefly review the scaling concepts that have been formulated in terms of the “blob picture”1, 160, 161 for polymer brushes in various geometries (Section Scaling Concepts for Brushes under Good Solvent Conditions) and then mention the main results due to the self-consistent field approach28–30, 50–52 (Section Essentials of the Self-Consistent Field Approach to Polymer Brushes). Other theoretical concepts will be summarized in Section Other Theoretical Concepts, while Section Varying the Solvent Quality: Theta solvents and Poor solvents deals with effects due to variable solvent quality, and possible microphase separation. Pertinent simulations and experimental results will be mentioned whenever available. Section Dense Brushes and Brushes Interacting with Polymer Melts considers dense brushes and brushes interactiong with polymer melts. Section LIVING POLYMER BRUSHES briefly discusses “living” polymer brushes. Section INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS OR WITH NANOPARTICLES then deals with (normal) forces between planar brushes (Section Normal Forces between Two Polymers Brushes), and the potential of mean forces between spherical brushes (Section Interactions Between Two Spherical Brushes), and the interaction between polymer brushes and nanoparticles, free chains, and so forth (Section Interaction of Polymer Brushes with Nanoparticles). Section Conclusions then presents some conclusions. After this summary of theoretical concepts, a short overview of the main computer simulation methods used to elucidate the validity of the theoretical concepts will be described in the Appendix.

A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Scaling Concepts for Brushes under Good Solvent Conditions

Flat Substrate Grafting Surface

In this section, only completely flexible linear chain molecules are considered and described in a crude coarse-grained way: each chain consists of N segments (effective monomeric units) of size a, such that under melt conditions the end-to-end distance would simply be Re = aN1/2, whereas under good solvent conditions in the dilute limit both Re and the gyration radius Rg scale as

  • equation image(1)

Note that prefactors of order unity in this section are ignored throughout. The “Flory exponent”160 is ν = 3/4 in d = 2 dimensions and ν ≈ 0.588162 in d = 3 dimensions (in fact, simple “Flory-type arguments”160 would imply ν = 3/5, but this is not quite accurate). As is well known, ν > 1/2 under good solvent conditions because the random coil-like structure of a macromolecule is swollen due to the excluded volume interaction.

If such polymers are grafted with one chain end to a planar nonadsorbing surface, and the “grafting density” σ of the “anchor points” is sufficiently small, the polymer conformation is sometimes referred to as “mushroom”:1, 27, 55 distinguishing now between the linear dimension Rgz in the z-direction perpendicular to the surface, and the lateral dimensions Rgx = Rgy parallel to it, one still has RgxRgzRg(bulk)Nν. Thus, the anisotropy of the conformation concerns only the (disregarded) prefactors in these power laws. When the grafting density is not so small, however, the mean distance σ−1/2 between grafting sites is a second characteristic length that needs to be compared to Rgx. In fact, for σ > σ*, defined by

  • equation image(2)

a crossover from “mushrooms” to “brush” occurs. According to the blob picture of Alexander,26 the polymers can be viewed as a cigar-like string of blobs of diameter σ−1/2, such that inside a blob excluded volume interactions are still fully operative, leading to a swelling of the chain. We hence find the number g of monomeric units per blob from the condition σ−1/2 = agν, in analogy to eq 1, therefore g = (σa2)−1/2ν. Of course, this result makes only sense if g ≫ 1, that is, the brush must be “semi-dilute”.160 The number n of such blobs in a string (Fig. 1) is then simply n = N/g, since each blob contains g monomeric units. This simple argument thus implies for the height h of the brush,

  • equation image(3)

where in the very last step the Flory approximation160, 165 ν = 3/5 was used.

thumbnail image

Figure 1. (upper-left panel) Snapshots of a polymer brush of linear chains with chain length N = 32 under good solvent condition and grafting density σ = 0.125,163 (upper-right panel) Polymer brush with chain lengt Nch = 100 at σ = 0.185 under poor solvent conditions: (a) side view, (b) top view. From Dimitrov et al.59 (lower-left panel) Polymer brushes comprising 42 chains of length N = 60 on a sphere with radius Rc = 7.9, at different temperature/solvent quality: T = 3.0, 2.2, 1.6, LoVerso and Binder (unpublished). (lower-right panel) Blob pictures of a polymer brush for the regime of good solvent quality. Case (a) shows the model due to Alexander26 and de Gennes:27 the chains form linear “cigars” of blobs with uniform diameter σ−1/2. Similar to a semidilute polymer solution, each blob contains only monomers of a single chain, and excluded volume interactions are screened only over distances larger than the blob diameter. Case (b) shows a non-uniform blob picture due to Wittmer et al.,46 allowing for a smooth decrease of monomer density ϕ(z) with increasing distance z from the grafting density. The blob diameter ξ(z) increases then according to ξ(z) ∝ [ϕ(z)]−ν/(dν − 1) ≈ [ϕ(z)]−3/4,160 because it can simply be put proportional to the screening length of excluded volume interactions (part c). The diameter df of the final blob, estimated from the condition df = ξ(hdf), where h is the height of the brush, also is indicated. After Wittmer et al.46

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From this “Alexander picture”26 of a polymer brush, one can immediately conclude that ReRgzh. However, the consideration of Rgx is more subtle. Although naively one might conclude that Rgx = σ−1/2, the blob diameter, actually the chain is not constrained to strictly stay within a single “cigar” but actually the chain conformation may make random excursions from the “cigar” starting at the grafting site into blobs belonging to neighboring “cigars.” Due to such excursions, the chain gains configurational entropy. Because on scales larger than σ−1/2 excluded volume interactions are screened,160 these lateral excursions should add up in a random walk-like fashion, and hence

  • equation image(4)

As further consequences of the Alexander picture, all chain ends reside in the region of the outermost blobs of the brush, and the monomer density profile is essentially uniform for z < h and zero for z > h, and thus proportional to the Heaviside step function θ(hz), while the density of chain ends ρe(z) is a delta function,

  • equation image(5)

We defer a critique of these results to the next sections, and discuss instead the extension of this scaling description in terms of blobs to the case of curved substrate surfaces, which leads to blobs of nonuniform size.

Curved Substrate Grafting Surfaces

We first consider a spherical polymer brush where chains are grafted to a sphere of radius R.60–66 Note that in this case the total number f = 4πR2σ of grafted chain is finite, and if the size of a single chain Re exceeds R, the situation is reminiscent of a star polymer.166–169 If f ≫ 1, the situation often is described by the Daoud–Cotton167 picture, Figure 2(a): the solid angle 4π is divided up into f conical sectors, so each arm of the star (or each chain grafted on the sphere, respectively) has the same volume in which it can spread out. When we fill such a conical volume by blobs that touch each other, we clearly must have for the blob radius ξ(r) (there are f blobs along the circle of radius r in d = 2, which has the circumference 2πr, whereas in d = 3 the f blobs share a surface area 4πr2 for a sphere of radius r)

  • equation image(6)

Because each blob contains g(r) ∝ [ξ(r)]1/ν monomers, we conclude that the density profile decays according to a power law for r < h, the brush height:

  • equation image(7)

which means ϕ(r) ∝ (r/f)−2/3, while

  • equation image(8)

(Here in the last expression on the right hand side ν was put equal to the Flory value ν = 3/5.)

thumbnail image

Figure 2. (a) Schematic construction of the Daoud-Cotton167 blob picture for star polymers. At a point-like center (or a small sphere), a total of f chains are grafted. (b, c) Schematic construction of a blob picture for a cylindrical brush, assuming that along the backbone (oriented along the z-axis) f(aσ1)−1 grafting sites occur at a regular spacing, such that at each grafting site f chains containing N monomeric units each are anchored. Part (b) shows a cross section of the cylindrical brush in the xz-plane, whereas part (c) shows a cross section in the (xy) plane, perpendicular to the cylinder axis. Although for a star polymer the space can be filled completely by spherical blobs whose radius increases linearly with the radial distance r from the center, for the cylindrical brush the blobs are ellipsoids with axes ξ(r) in x-direction, proportional to r/f in y-direction, and fσ1−1 in z-direction. For a particular chain, the coordinate system is chosen such that the chain center of mass is on the x-axis. After Hsu et al.74

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According to the Daoud-Cotton picture, the blobs that correspond to one particular chain just fill one conical sector of the star polymer (or spherical brush, respectively) up to the brush height h. In such a conical sector, there occur precisely all N effective monomers of the chain, therefore

  • equation image(9)

Using eqs 7 and 8, we readily obtain

  • equation image(10)

and

  • equation image(11)

Analogous results follow then for the end-to-end distance Re and the squared gyration radius of the chains

  • equation image(12)

and

  • equation image(13)

Of course, for a spherical brush where chains are grafted on a sphere of radius Rc, the above relations, eq 1013 can only hold when the end-to-end distance Re of the grafted chains is much larger than the sphere radius. If this is not the case, one expects a crossover to the behavior of polymer brushes on flat planar substrates (when RcRe, the curvature of the substrate should have no effect). One can make a scaling assumption to describe this crossover as follows (in d = 3 equation image(ζ) is some crossover scaling function that we shall discuss below)

  • equation image(14)

where h0 is the brush height at a planar surface, as given by eq 3. Noting that 4πσRc2 = f when f chains are grafted at a sphere of radius Rc, we see that eq 11 can be rewritten then as h ∝ σ(1 − ν)/2Rc1 − νNν. Consequently, eq 14 becomes

  • equation image(15)

therefore, for ζ [RIGHTWARDS ARROW] ∞,

  • equation image(16)

in order that the result h ∝ σ (1 − ν)/2Rc1 − νNν is reproduced in this limit. In the opposite limit, equation image(ζ ≪ 1) = 1, of course. However, the detailed behavior of the crossover scaling function can only be deduced by a more detailed theory, such as the self-consistent field theory (SCFT),61 it cannot be derived by such simple scaling arguments alone.

Next, we consider the case when chains are grafted to the surface of a (thin) cylinder (or, in the extreme case analog to the star polymer limit discussed above, we deal with chains grafted to a rigid rod). The grafting density σ1 is defined here per unit length. As Figures 2(b,c) demonstrates, in cylindrical geometry, the uniform filling of space with blobs necessarily requires nonspherical blob shapes. Indeed, requesting that the space is divided again in sectors so that aσ1 grafting sites occur along the axis whereby f chains per site are grafted, then in the z-direction the blob linear dimension is simply 1/σ1, while in the tangential direction in the xy-plane [cf. Fig. 2(c)] the linear dimension must be of the order 2πr/f.

However, in the standard theory77, 170, 171 the nonspherical character of the blob shape is not explicitly accounted for: rather, it is argued that one can characterize the blobs by a single effective radius ξ(r), depending on the radial distance r from the axis. One considers a segment of the cylinder of length L containing p = Lfσ1 polymers (as mentioned above, for infinitely thin rods σ1 means a grafting density per unit length and not per area). On a surface of a cylinder of radius r and length L, which has an area 2πrL, there should then be p blobs, of cross-sectional area [ξ(r)]2π each, provided they are spherical. Hence (omitting factors of order unity), pξ2 = Lr and, therefore, ξ(r) ∝ (r/fa σ1)1/2. The blob volume then is of the order of V(r) = ξ3(r) = (r/fσ1)3/2. Invoking again the principle that self-avoiding walk statistics holds inside a blob, one concludes that the number g(r) of effective monomers in a blob is

  • equation image(17)

so once again a power law for the density profile ϕ(r) results

  • equation image(18)

Using ν = 0.588162 this yields ϕ(r) ∝ r−0.65 while the Flory value ν = 3/5 yields ϕ(r) ∝ r−2/3. The average height of the brush is then estimated by requiring that a total of aσ1fN monomers per unit length in the z-direction along the axis of the cylindrical brush are found when we integrate from r = 0 to r = h:

  • equation image(19)

and hence

  • equation image(20)

When we use ν = 0.588,162 we find h/a ∝ (σ1fa)0.259N0.74 while taking the Flory value ν = 3/4 yields h/a ∝ (σ1fa)1/4N3/4. This latter result happens to be identical to the estimate that one would obtain when one partitions the cylindrical brush into disks of width 1/σ1, each disk containing a two-dimensional f-arm star polymer. However, eq 20 indicates that this idea is a misconception: the problem is entirely controlled by excluded volume effects in d = 3 and not in d = 2. Because the correlation length ξ(r) ∝ (r/fσ1)1/2 happens to be the geometric mean of the characteristic length σ1−1 in the z-direction and the length r/f in tangential direction, we conclude that the blob volume is of the same order, namely ξ3(r) ∝ (r/fσ1)3/2, irrespective whether we take it as spherical or of anisotropic shape with three different linear dimensions, namely (r/f σ1)1/2, r/f, and σ1−1, respectively.74 Although the anisotropic blob shape hence does not affect the results for the brush height, eq 20, it does have observable consequences74 for the shape of a grafted chain in a cylindrical brush: again the linear dimension of a chain in the z-direction is not given by σ1−1, because the chain can make random excursions (by ±σ1−1) when one moves outward in radial direction in Figure 2 from one shell to the next one. Adding up these excursions in a random walk-like fashion, Hsu et al.74 predicted

  • equation image(21)

while the chain extension in radial direction is of the same order as the brush height. The difference between Rgy and Rgz is a clear consequence of the anisotropic character of the blobs and could not be present in the star polymer case. Finally, we emphasize that eqs 1721 apply only if the grafting density σ1 is sufficiently large so that a significant stretching of the chains in radial direction away from the rod on which the chains are grafted does in fact occur. If this is not the case, a crossover toward “mushrooms” grafted along a rod takes place. Because the linear dimension of a mushroom is of order aNν, the corresponding crossover scaling assumption for the height of the cylindrical brush is74:

  • equation image(22)

where equation image(ζ) is a crossover scaling function which for large arguments must reproduce eq 20 and hence

  • equation image(23)

Another interesting crossover displays the behavior of chains grafted to a thick cylinder of radius Rcyla. First, we note that then the product fσ1/a in eq 20 needs to be reinterpreted as a grafting density σ per area. For Rcylh we still have the result from eq 20, namely, h/a ∝ (σa2)(1 − ν)/(1 + ν)N2ν/(1 + ν), while in the opposite limit Rcylh we must recover the result for the flat brush, h/a ∝ (σa2)1/2 ν − 1/2N (Eq. 3). The crossover scaling assumption analog to eq 15 then reads

  • equation image(24)

The crossover scaling function equation image(ζ ≪ 1) = 1 while in the opposite limit it has a power law behavior

  • equation image(25)

The last case which we shall discuss in detail are brushes grafted on the inner surface of a cylinder of diameter D (Fig. 3). Now the tendency of the chains to stretch away from the grafting surface (as in all cases considered so far) is counteracted by the fact that near the cylinder axis less volume is available. Thus, the validity of blob concepts becomes questionable,79, 80 when the end-to-end distance of the grafted chains becomes comparable to the cylinder radius D/2, as was assumed in Figure 3. One then could argue in favor of a picture where the monomer density in the cylinder is more or less uniform, and the blob diameter is then given in terms of this density, similar to the “concentration blobs”160, 161 in semidilute solutions; then one expects that the free chain ends can be located anywhere in the cylinder [Fig. 4(a)], rather than being confined within a conical sector [Fig. 3(a)]. Of course, when we consider again low grafting densities, now not only the crossover toward polymer mushroom-like conformations needs to be considered, but rather in very narrow tubes also cigar-like configurations (strings of blobs oriented along the cylinder axis) may occur [Fig. 4(b)]. Depending on chain length N, tube diameter D and grafting density σ hence many different scaling regimes may occur (Fig. 5).78, 79 Note that there is a simple geometric relation between the grafting density σ of the anchor points at the cylinder wall and the average volume fraction ϕ of monomers contained in the cylinder. As the area A of the cylinder surface is A = πDL, the total number of monomers equation image contained in the cylinder is

  • equation image(26)

As the cylinder volume (for a cylinder of length L) is V = LD2π/4, we conclude that ϕ = equation imageN/V = 4 Nσ/D (note that we now choose a ≡ 1 for simplicity). The constraint σ < 1, i.e., σ < D/(4N), yields the rightmost straight line in Figure 5 (the region σ > 1 is unphysical). The mushroom to brush crossover is independent of D for wide tubes (eq 2 then just represents the vertical straight line in the left part of Fig. 5), while for narrow tubes we have “cigars” rather than mushrooms in the dilute limit: one has Rgx = Rgz = D, while the longitudinal component is much larger. Assuming160, 172 that the cigar conformation can be treated as a string of Nblob = N/g blobs of diameter D, with g monomers per blob, D = gν = g3/5, one readily gets

  • equation image(27)

For D = N3/5 a smooth crossover to Rgy = N3/5 occurs, as it must be.

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Figure 3. Schematic illustration of the generalization of the Alexander-de Gennes picture for brushes to the case of grafting at the inner surface of a cylinder as proposed by Sevick.77 It is implied that each chain occupies a conical sector in the zx-plane (cross section of the cylinder, orienting now the cylinder axis along the y-axis) (a), with the size of the outermost blob (ℓblob) being delimited by the grafting density. In the zy plane containing the cylinder axis, all blobs have a linear dimension ℓblob = σ−1 along the y direction, but become progressively smaller than a−1 in the direction toward the cylinder axis (b). After Dimitrov et al.79

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thumbnail image

Figure 4. (a) Choice of coordinates in a cylindrical tube of diameter D, necessary for the analysis of the gyration radius components of polymer chains that are grafted to the walls of the tube. For each polymer, then a separate coordinate system is chosen, whose origin is at the grafting site, and the z-axis is perpendicular to the grafting surface at this site, while the y-axis is parallel to the cylinder axis. Note that z-coordinates larger than D/2 are possible when one does not assume that the chains are confined to a conical sector [cf. Fig. 3(a)]. (b) Schematic blob picture of a “cigar”-like polymer configuration of a long polymer chain grafted (at small grafting density) in a long narrow cylinder. After Dimitrov et al.79

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thumbnail image

Figure 5. Schematic diagram for the polymer linear dimensions Rgx, Rgy, Rgz of a grafted chain of length N for the case of polymer brushes grafted at the wall of a tube of diameter D and grafting density σ in good solvent. Due to the choice of logarithmic scales, crossovers between the various regimes (which for simplicity are assumed to be sharp, while actually they will be rather gradual) show up as straight lines in this diagram. For the exponents describing both the linear dimensions and the crossover laws, the Flory approximation (ν = 3/5) has been made throughout, and a ≡ 1 was chosen. For wide tubes (D > N3/5) with increasing σ only two crossovers are encountered: at σ = σ* = N−6/5 (eq 2) from mushrooms to an (essentially) flat brush, and then at σ** = (D/N)3 to a compressed brush. The latter regime ends at 4σ = D/N, which means that the tube is densely filled with monomers (fraction ϕ of occupied sites in a lattice model ϕ = 1). For narrow tubes, D < N3/5, instead of mushrooms one has cigars [Fig. 4(b)], elongated along the tube axes. One must distinguish swollen cigars from compressed and overlapping cigars. After Ref.79

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We next consider the regime of wide tubes (D > aNν) where first (at σ = σ* = a−2 N−2νa−2N−6/5) a crossover from mushrooms to expanded brushes takes place, as for planar brushes (see eq 2). But when the chains in the cylindrical brush stretch more and more, this would lead to too much crowding of monomers in the center of the cylinder: we may estimate that this happens when the brush height h (and hence also the gyration radius component Rgz in the direction normal to the grafting surface in the grafting point, cf. Fig. 4), become comparable to the tube diameter D.

This yields a second crossover grafting density σ**, separating a weakly compressed brush with properties not much different from an ordinary planar brush from a strongly compressed brush which has Rgz = h = D for all σ > σ**. Using h = σ1/3N (putting a = 1 and ν = 3/5 for simplicity, see eq 3), the condition h = D/2 leads to

  • equation image(28)

where prefactors of order unity were omitted. This crossover line is also included in Figure 5 (of course, all these crossovers between different types of power laws are gradual and smooth, there do not occur any abrupt changes of behavior, unlike in the case of a true phase diagram). The lateral linear dimensions of chains in such a compressed brush can be estimated from the fact that the density ϕ of monomers is approximately uniform throughout the tube. Then, the blob size is ξ = ϕ−3/4, as for a semidilute solution in the bulk, and

  • equation image(29)

with Nblob being the number of blobs per chain. As there are g = ξ1/ν = ξ5/3 = ϕ−5/4 monomers per blob, Nblob = N/g = Nϕ5/4, and hence

  • equation image(30)

where in the last step the relation ϕ = 4Nσ/D was used. At the boundary σ = σ** eqs 28 and 30 yield Rgx = Rgy = D−1/4N3/4, which coincides with the prediction of eq 4, Rgz = Rgy = σ−1/12N1/2 = (D/N)−1/4N1/2. Of course, all chain linear dimensions must exhibit smooth crossovers at any of the crossover lines shown in Figure 5.

These considerations can straightforwardly be carried out also for the crossover from (expanded) cigars to compressed cigars and the region of overlapping cigars (which is again a kind of homogeneous filling of the cylinder with blobs, as in a semidilute solution). We direct the reader to Ref.79 for full details on these problems. Similarly, we also do not discuss in any details the possibility of grafting chains on the inner surface of a sphere, for which similar scaling considerations in terms of blobs can be made.

Essentials of the Self-consistent Field Approach to Polymer Brushes

The SCFT is one of the most important and powerful approaches to compute properties of polymeric systems. It is not only applicable to polymer brushes but has originally been developed for interfaces in polymer blends173–176 and for the description of mesophase order in block copolymers in the strong segregation limit,177–181 including surface and confinement effects, for example.182, 183 We cannot attempt to cover the rich literature on this subject, and rather refer to a recent book,184 restricting ourselves instead to SCFT applications to polymer brushes. We note, however, that the theory exists in many different variants: formulations exist both in the continuum (e.g., Refs.19, 22, 29, 30, 33, 50–52, 173–184) and on the lattice (e.g., Refs.28, 39, 83, 85, 185, 186) We do not wish to describe either of these approaches in any technical details, but only characterize the starting point and then give the flavor of the approach. As a rule,the main idea is to formulate the statistical mechanics of a single macromolecule exposed to an effective field provided by the other macromolecules that needs to be computed self-consistently. We stress that the same idea is inherent in related yet somewhat different approaches as well.4, 48, 187 Moreover, the inherent mean-field approximation leads to errors that cannot always be controlled. Nonetheless, compared with scaling theories, one should note that the SCFT provides much greater detail (such as the overall monomer- and end-monomer density distribution functions in a brush); one can consider both variable solvent conditions and generalizations to binary (or multi-component) brushes as well as block copolymer brushes, and even polydispersity effects can be treated. Due to this broad applicability, SCFT is so widely used.

We begin with the continuum formulation of the theory, and write down the partition function for equation image Gaussian linear macromolecules of N units, end-grafted to an area A, interacting with each other via a quadratic repulsion with the strength w of the excluded volume interaction.51

  • equation image(31)

where the coordinate s runs along the contour of the chain, and the monomer density equation image(r) in the system is defined in terms of the coordinates equation image(s) along all contours as

  • equation image(32)

Inserting the identity 1 = ∫ equation imageϕδ(ϕ − equation image) and using the integral representation of the delta function, δ(ϕ − equation image) = ∫ equation imageΩexp [∫Ω[ϕ − equation image]dequation image], one can carry out the Gaussian integration over ϕ. This yields (the grafting density σ = equation image/A, of course)

  • equation image(33)

where Ω is the external field and F[Ω], the corresponding free energy functional, is given by

  • equation image(34)

with Q(Ω) the partition function of a single chain in an external field Ω. Note that in eq 34 it was explicitly assumed that the (effective) field Ω(z) depends on the z-coordinate in the direction normal to the planar grafting surface only. In the limit A [RIGHTWARDS ARROW] ∞, the free energy per polymer is then given by the minimum value of F[Ω]. This minimum value occurs for a self-consistent field Ω(z) = ω(z), with ω(z) given by the self-consistency equation

  • equation image(35)

with ϕ(z) being the average monomer density at distance z. The mean field free energy per polymer, in units of kBT, is

  • equation image(36)

In the lattice formulations of SCFT, one gets the single-chain partition function (disregarding excluded volume) exactly numerically, while in the analytic work often the classical approximation to the path integral is used. Therefore, for each position of the free end point (z = z0) of the polymer equation image(s = 0), only the most probable polymer configuration is used, disregarding fluctuations around this most probable path described by z(s,z0). To make this approximation clear, one may change the description of the polymer path from the variable z(s,z0) to the inverse function s(z,z0) with s(z0,z0) = 0 and s(0,z0) = N. One then needs to define the stretching function

  • equation image(37)

so that the single-chain partition function becomes

  • equation image(38)

Here, the prime on the functional integral over all stretching functions E means that only those are selected which satisfy the constraint that all polymer paths have the same length (N). Anticipating that paths that start out at a value z0 very near the grafting surface may first move away from the surface before returning to it, zm(z0) has been defined as the largest value of z reached by a path that starts at z0.

From eq 38 the self-consistent equation for the density ϕ(z) follows by functional differentiation, ϕ(z) = −σδlnQ/δ(wϕ). The classical approximation to this partition function Q then means that the functional integral over E in eq 38 is replaced by the result when one evaluates the integrand with the function e(z,z0) which extremizes the mean field-free energy F. Thus, in the path integral all paths except the most probable path for a given z0 are eliminated, and hence the self-consistent equation which determines the density becomes in the classical limit

  • equation image(39)

Note that z0(zm) is the inverse function of zm(z0), and the end-point distribution ρe(z0) then is found as

  • equation image(40)

Using the normalization ∫dz0ρe(z0) = σ, which expresses the fact that for each grafted chain the free end must be at some distance z0, one can derive Q to find

  • equation image(41)

Recalling eq 38, we recognize that this expression is just the free energy of a single polymer in the external field wϕ(z). Using this result in eq 36, we obtain the desired free energy (per chain) of the polymer brush.

To discuss this result, it is useful to introduce a rescaling in terms of the following dimensionless variables

  • equation image(42)
  • equation image(43)

which satisfy the normalizations

  • equation image(44)

In terms of the rescaled stretching function (, = 0) ≡ e(z,z0)N/, the free energy eq 36 then becomes

  • equation image(45)

where the parameter β is proportional to the square of the ratio of the typical brush height to that of the unperturbed radius of gyration a equation image,

  • equation image(46)

The right hand side of eq 45 contains three terms: the first term is the direct binary interaction of the monomers, the second term represents the free energy cost of stretching, and the third term is the entropy of distributing the free chain ends in the brush. In the Strong Segregation Limit (SSL) of SCFT, which means β [RIGHTWARDS ARROW] ∞, this term is neglected.29, 30

To obtain the density profile equation image() and the end-monomer distribution equation image() explicitly, one must minimize the free energy functional, eq 45, with respect to these two functions, subject to the normalization constraints, and with respect to the stretching function (, 0), subject to the equal-length constraint

  • equation image(47)

which yields the condition 2(, 0) = equation image() − equation image(0) + equation image(z0), where equation image(0) is a Lagrange multiplier whose physical meaning is the magnitude of the stretching at its free endpoint.

Although this analytical treatment has required several approximations, explicit solutions for equation image and equation image require numerical work.51, 51 Figure 6 gives a typical example for several values of the parameter β.51 Only in the SSL (included as thick broken lines in Figure 6) explicit simple formulas result, namely

  • equation image(48)

and

  • equation image(49)

In the SSL, there are no monomers beyond the rescaled brush height (note that differs from by a numerical factor), while the full SCFT shows a Gaussian tail even if β is very large. This smooth behavior near the brush height is well established from both experiment and simulation (Figs. 7 and 8).188, 189 These data are compatible with the predicted scaling behavior for the brush height h ∝ σ1/3N (eqs 3, 43) and show that indeed the monomer profile is neither the step function of the Alexander-de Gennes theory26, 27 nor strictly parabolic, as implied by the SSL-SCFT,29, 30 while the data are (at least qualitatively) compatible with the classical approximation to the SCFT.51, 52 Note that Figure 7(b) includes already data on variable solvent quality, but we defer a discussion of this aspect to a later section (Section Varying the Solvent Quality: Theta Solvents and Poor Solvents). Unfortunately, we are not aware of any direct experimental measurements of ρe(z) while in the simulation this information is readily accessible [Fig. 8(b)]. The simulations confirm also the scaling of the chain extensions in the xy-directions parallel to the surface (eq 4), which can be estimated from the blob model, but is outside the scope of the one-dimensional version of the SCFT, described above. As a side remark, we mention that the simulations can also study subtle effects due to topological interactions which appear when one grafts noncatenated rings (of length NR = 2N) instead of two linear chains of length N, as Figure 8 also demonstrates.

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Figure 6. (a) Rescaled density profile equation image() as a function of the rescaled distance from the grafting surface in a polymer brush, according to the classical limit of the SCFT, for four different values of the stretching parameter β: β = 0.1,1,10, and 100 (dotted, dash-dotted, dashed, and solid lines, respectively). The strong stretching limit (SSL) (β [RIGHTWARDS ARROW] ∞) is shown as a thick broken line (eq 48). (b) Rescaled end-point distribution function equation image(0) [denoted as in this figure] as a function of the rescaled end-point position0, for the same values of β as in (a). Again the limit obtained for β [RIGHTWARDS ARROW] ∞ (SSL) is included as a thick broken line (eq 49). From Netz and Schick.51

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Figure 7. (a) Plot of rescaled brush height hD2/3, where D is the mean distance between grafting points, for polydimethylsiloxane chains end-grafted on porous silica, using dichloromethane as a good solvent, versus the molecular weight M of the chains. These data were extracted from the analysis of small angle neutron scattering by Auroy et al.188 (b) Polymer volume fraction ϕ(z) as a function of distance z for end-grafted polystyrene (M = 105000) at polished silicon as a substrate, in cyclohexane as a solvent, varying the solvent quality by changing the temperature, as shown. Inset shows brush height h normalized by its value hθ at the Theta temperature θ, as a function of the dimensionless “solvent quality,” τ = (T − θ)/T. These data were obtained from the analysis of neutrons reflectivity measurements by Karim et al.189

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Figure 8. (a) Log-log plot of the squared radius of gyration component Rgz2 in the z-direction and in the xy-direction (Rgxy2) as a function of chain length N, for a bead spring model of polymers at a (normalized) grafting density σ = 0.125 (see Appendix for a characterization of the model). For comparison, also data for grafted, noncatenated ring polymers of ring length NR = 2N and grafting density σR = σ/2 are included. (b) Total monomer density ϕ(z) and end-monomer density ρe(z) plotted vs. distance z; for the same model as in part (a). Four values of the chain length N = 32, 64, 128, and 256 are included, as well as corresponding data for rings with NR = 2N. From Reith et al.11

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However, the classical SCFT of polymer brushes is even much more restricted in its validity, as the above review of its derivation clearly shows. The essential approximations are that the theory is formulated in terms of the mean density ϕ(z), so local fluctuations in the density are ignored, and in addition, that one ignores all but the most probable polymer configuration (the “classical path” is assumed to dominate the functional integral, eq 31). Implicit in this treatment is the Gaussian chain statistics, which has been built in from the start (eq 31). The excluded volume interaction must be sufficiently weak, so that inside of the blobs (considered in Section Scaling Concepts for Brushes under Good Solvent Conditions) the chain conformation indeed would be Gaussian still (which requires w/a3aσ1/2). The validity of the classical approximation requires strong stretching, β > 1 (thus, the curves for β ≤ 1 in Figure 6 are not expected to be quantitatively reliable). On the other hand, σa2 < 1 is necessary to avoid the regime of dense melts. The regime where the theory is supposed to hold is hence very restricted, namely given by the inequalities

  • equation image(50)

This restricted applicability of the theory is often ignored when even the SSL-SCFT is widely used to interpret experiments and simulations. The SSL-SCT is most widely used for describing polymer brushes, simply because it leads to simple explicit formulas, such as eqs 48 and 49. It also yields a useful expression for the free energy function F(h) of a compressed brush as a function of brush height h and the associated osmotic pressure Πosm(h) = −dF(h)/dh, namely

  • equation image(51)

Here, h* is the height of the uncompressed brush (corresponding to in eq 48) and obviously Πosm = 0 for hh*. Writing h = h*(1 − Δ) with Δ a small parameter, one finds that Πosm ∝ Δ2 while the Alexander brush picture would imply Πosm ∝ Δ for small Δ. Of course, eq 51 implies that F(h > h*) ≡ 0 because there can be strictly no force between two brushes that are more than a distance 2h* apart. Finally, we note that for “dry” brushes (no solvent being present) the volume fraction ϕ taken by the monomers can be reinterpreted as the total density in the system, and then Πosm is reinterpreted as the total pressure. However, the above equations are not expected to be accurate in the limit of dry brushes, see Section Dense Brushes and Brushes Interacting with Polymer Melts.

We also note that the SCFT can be easily extended to treat problems such as block copolymer brushes, mixed polymer brushes containing two types (A, B) of chains, and so forth. As an example, we briefly mention, following Müller,19 the formulation of SCFT used for computing the phase diagram of mixed polymer brushes. The energy term (w/2) ∫ d equation image)2 in eq 31 then needs to be replaced by

  • equation image(52)

with equation image, equation image—the densities of both species A, B, and wAA, wBB, and wAB describing the strength of the excluded volume forces between the different types of monomer pairs. Although w = (wAA + wBB + 2wAB)/4 characterizes the average strength of excluded volume in the brush, the normalized Flory χ-parameter equation image = (2wABwAAwBB)/2w denotes the mutual attraction (repulsion) between unlike monomers. It is related to the commonly used Flory-Huggins parameter160, 165 χFH = wequation image. By increasing equation image > 0, we increase the incompatibility between the two species, and this may lead to a microphase separation190 between the two species (due to the constraint of irreversible grafting of the chains, macroscopic phase separation like in polymer blends160, 165, 191 is impossible).

Just as in eqs 36 and 45, the free energy of a one-component brush was written in the form [in eq 52 and in the following we display again Boltzmann's constant kB and the absolute temperature explicitly] F = UTS, where U is the internal energy and S the total conformational entropy per chain (that includes the effect of stretching through the self-consistent field ω(z)). We now can write F = UT(SA + SB), where in SCFT

  • equation image(53)

and similarly for species B. Here, equation image means the single-chain partition function of an A chain that is grafted at the site equation image at the grafting surface, equation image the corresponding monomer density and equation image the self-consistent field. Of course, when we deal with lateral microphase separation in a brush, we should not reduce the problem to a one-dimensional problem that depends on the normal distance z from the grafting surface only, but one must keep the x, y, and z coordinates. Note that the sum in eq 53 runs over all grafting points equation image of all A chains.

The density equation image satisfies then the selfconsistency equation

  • equation image(54)
  • equation image(55)

and similar expressions hold for ωB and equation image. For simplicity, arguments equation image have been omitted in eqs 54 and 55. It is noteworthy that the self-consistent fields ωA, ωB do not depend on the grafting site equation image.

For numerical solutions of these equations, it is useful to define the end segment distributions equation image(equation image,s) and A(equation image,s). The first quantity is the probability that the end of a chain segment of length Ns is at equation image when this chain is grafted at equation image, while A(equation image,s) is the analogous quantity for a free (i.e., not grafted) chain.

The Gaussian statistics (cf. eq 31) comes into play by requesting that both end segment distributions satisfy diffusion equations, Re being the end-to-end distance of a free chain,

  • equation image(56)

These equations are solved subject to the initial conditions equation image(equation image,0) = δ(equation imageequation image) and A(equation image,0) = 1 for grafted and free chains, respectively.

The density equation image and the (normalized) chain partition function are then computed as

  • equation image(57)
  • equation image(58)

The total monomer density of species A in the brush then becomes

  • equation image(59)

where qAequation image.

Figure 9 shows typical predictions for the phase behavior.19 The stronger the stretching of the brush becomes, the larger the tendency for the development of mesophase long range order. The problem is highly nontrivial, because several distinct mesophases with different superstructures compete against each other.

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Figure 9. (a) Phase diagram for a symmetric polymer brush (NA = NB = N, average volume fractions ϕA = ϕB = 1/2). Under good solvent conditions ( equation image < 2), a transition occurs between a disordered phase and a “ripple” phase, where the two species cluster into a periodic arrangement of cylinders, every second one being rich in the A component. The transition to a layered structure (segregation occurring only in one dimension) is shown as a broken curve (because this transition is always preempted by the transition to the other phases, this layered structure is only metastable). For equation image > 2 also “dimple phases” occur, where the two species form clusters arranged on two-dimensional lattices: the “dimple S” phase corresponds to a square lattice, while “dimple A(B)” means A(B) clusters forming an hexagonal lattice, where the B(A) component is collapsed and fills the space between the A-rich (B-rich) clusters. Note that calculations were limited to δ > 0.23, while broken parts of the curves for δ < 0.23 are tentative extrapolations. The parameter δ is the inverse of the stretching parameter β defined in eq 46. (b) Phase diagram of a symmetric binary polymer brush for δ = 0.5 as a function of the composition ϕ = ϕA and the incompatibility equation image. See part (a) for an explanation of the various phases. The insert shows the lateral unit cell size L in units of the end-to-end distance Ra = aequation image of unperturbed chains along the phase boundary between the “dimple” and “ripple” phases. The solid line corresponds to the “ripple”, the broken line to the “dimple” phase. After Müller.19

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However, in reality one can expect at best extended short-range order and no true long range order:20 All the theory presented in this section has ignored fluctuations in the local density of the grafting sites. Such fluctuations are expected due to the process by which polymer brushes are prepared,7 of course. Although for homopolymer brushes simulations36 have indicated that the observables of interest [density profile ϕ(z) and ρc(z), linear dimensions of the chains in the brush, etc.] are essentially identical for a regular and a random arrangement of grafting sites, this is not the case for binary polymer brushes:20 the fluctuations in the relative concentration of grafting sites of A-chains (or B-chains, respectively) lead to some energetic preference for the locations of the A-rich (or B-rich) dimples.20 This effect is analog to the action of a “random field” on the “order parameter” of a phase transition: it is well known that random fields destroy long range order in two-dimensional systems, stabilizing ordered domains of large but finite size.192–194 Another important source of randomness in polymer brushes is the polydispersity in the distribution of the chain lengths; this problem will be addressed in the next section. Recent extensions of SCFT on mixed polymer brushes include also detailed studies on self-assembly in confinement.195

We conclude this section by emphasizing that the numerical solution of differential equations such as eqs 55, that is inherent in the field-theoretic formulation of SCFT which was presented so far, eqs 3158, can be circumvented by the lattice formulation due to Cosgrove et al.28 based on the Scheutjens-Fleer theory.185, 186 One considers in this theory a polymer-solvent system at a substrate surface, and aims to take into account all possible conformations, each weighted by its probability as given by the Boltzmann factor, assuming that both monomers of the chain and solvent particles can occupy the sites of a regular lattice, multiple occupancy of lattice sites being forbidden. To study a brush, where the chains are terminally attached to the substrate, conformations are restricted by requesting that the first chain segment is in the layer adjacent to the substrate. One considers a lattice of size L in z-direction perpendicular to the substrate, and taking now the lattice spacing as unit of length, one labels the layers consecutively as z = 1,2,…, L (layer 1 is adjacent to the substrate, layer L is outside of the brush in the dilute solution. We denote the coordination number of the lattice as q (q = 6 in the simple cubic (sc) lattice, q = 12 in the face-centered (fcc) cubic lattice), with q0 being the number of nearest neighbors in the same layer (q0sc = 4, q0fcc = 6). The fraction of sites λ0 = q0/q in the same layer then is λ0sc = 2/3 or λ0fcc = 1/2, while the fraction of sites in an adjacent layer is λ1 = (1 − λ0)/2 = λ1sc = 1/6 or λ1fcc = 1/4, respectively. Of course, the choice of the type of lattice is arbitrary, and for long enough chains the results on brush properties should not depend on this choice.

The statistical mechanics of such a model could be obtained “numerically exactly” (i.e., avoiding systematic errors apart from statistical errors) by Monte Carlo simulation,196–198 see Appendix. The lattice formulation of SCFT, on the other hand, is based on making a mean field approximation in each layer, so the quantity that matters is just the monomer density profile ϕ(z); the profile of the solvent density is ϕs = 1 − ϕ(z), every site being occupied by either a monomer or a solvent molecule. Nearest-neighbor interactions between monomers and solvent are described by the Flory-Huggins parameter χFH. So the potential energy u(z) of a monomer (relative to that of a monomer in a bulk solution at concentration ϕb) becomes (following the formulation of Wijmans et al.)39

  • equation image(60)

The angular brackets 〈…〉 denote a weighted average over three layers, to account for the fraction of contacts that a segment or solvent molecule has with its nearest neighbors in these layers (thus, 〈ϕ(z)〉 = λ1 ϕ(z − 1) + λ0 ϕ(z) + λ1 ϕ(z + 1)). The logarithmic terms account for the change in translational entropy of solvent molecules, which is thus included into the effective potential u(z), while the entropy of the polymer needs to be calculated considering its conformational statistics, as described below.

Defining a monomer weighting factor G(z) = exp[−u(z)/kBT], isolated monomers (being not part of a chain) would be distributed according to G(z), within the mean field approximation made in eq 60. To take chain connectivity into account, one defines a function G(z,s) that describes the average statistical weight of all conformations of an s-mer of which the last monomer is located in layer z, and the first monomer is located anywhere. The monomer with label s − 1 then must be in one of the layers z − 1, z or z + 1. This means that G(z,s) must be proportional to 〈G(z,s − 1)〉, the weighted average of statistical weights of (s − 1) mers, of which the last segment is in one of the layers z + 1, z or z + 1. As segment s in layer z contributes a factor G(z), one finds the recursion relation

  • equation image(61)

which starts with G(z,1) = G(z) and is computed for all sN. As the segment s of a (free) polymer of N segments can be considered simultaneously to be the end of an s-mer and of an (N + 1 − s)-mer, the total monomer volume fraction ϕ(z,s) of the s'th monomer in layer z becomes

  • equation image(62)

where the denominator accounts for the fact that monomer s is counted twice. The normalization constant C can be found in the bulk solution, as ϕ(z) = equation imageϕ(z,s) and hence C = ϕb/N. This volume fraction profile of (nongrafted) chains near a surface should be consistent with eq 60 for all L values of z, which means that one has a set of L coupled nonlinear equations, which requires a numerical iteration scheme for its solution (see e.g.).39 Turning now to grafted chains, the iteration analog to eq 61 must start with

  • equation image(63)

where we use a subscript g to clearly distinguish grafted from nongrafted chains. Therefore, the iteration eq 61 is replaced by

  • equation image(64)

In the chain, monomers can be considered as the joint between a grafted chain of s monomers and a nongrafted free chain of Ns + 1 monomers. Eq 61 still holds for the latter part, and thus for grafted chains eq 62 is replaced by

  • equation image(65)

The normalization constant Cg is then fixed by the condition that every one of the ng grafted chains has to have its end somewhere,

  • equation image(66)

From this formulation, it is obvious that an extension of the formalism to systems containing both grafted and free chains is rather straightforward39; also an adsorption potential due to the grafting wall is readily included. When we sum eq 65 over s, to obtain the profile ϕ(z) = ∑s = 1N ϕg(z,s), we immediately realize that this is just the lattice analog of the result eq 59 of the continuum theory, of course.

Although it is clear that the continuum formulation can be, in principle, generalized to curved substrate surfaces (∫dz0 … become 4π∫r02dr0 … for spheres or 2π ∫ r0dr0 … for cylinders, respectively), the extension of the lattice formulation to such geometries deserves a more detailed comment. As an example, we consider a cylinder, where one then introduces lattice planes perpendicular to the cylinder axis at z = 1,2,…,Lz, but in each plane one then envisages circles at radii r = 1,2,…, Lr. One now needs to define a-priori step probabilities, which are determined by the fraction of sites in neighboring layers adjacent to a given site on a lattice. In the r-direction, these probabilities Λ(r|r − 1) to go inside, or Λ(r|r + 1) to go outside, follow from geometric considerations199

  • equation image(67)

when S(r) = 2πr is the surface area between the two cylinders of unit height, and L(r) = π[(r + 1)2r2] is the difference between the volumes of these two cylinders. Λ(r|r) then is given by the sum rule, Λ (r|r) = 1 − Λ(r|r − 1) − Λ (r|r + 1). The transition probabilities along the z-direction are given by Λ(z,z) = 1/3 where z = z − 1,z, or z + 1. For each lattice site (z,r), one then obtains nine transition probabilities λ(z,r|zr) = Λ(z,z) Λ(r,r), where (z,r) = (z + α, r + β) with α, β = −1,0,1. These transition probabilities generalize the constants λ0, λ1 discussed above. Of course, no discretization of the angular coordinate is possible (and also not needed in this mean field theory).

Other Theoretical Concepts

One basic ingredient of SCFT is the input how a single polymer in the absence of the self-consistent field due to the other chains is described, namely as a Gaussian chain. This fact appears very explicitly in the continuum formulation (Eqs 31 and 56), but it is also true in the lattice formulation, where the constraint that only a single monomer can occupy a lattice site is not obeyed strictly but rather on the average only. Of course, it is clear that under good solvent conditions (and small grafting densities where excluded volume interactions are not screened160) this approximation is unsatisfying.

One interesting attempt to improve the situation is the single chain mean field theory4, 94, 187, 200–203 (SCMF). It is based on computing the properties of one “central” chain with all its interactions fully (which can be done for various models of polymer chains by Monte Carlo methods,196–198 for instance) while the interaction with both surrounding chains and solvent molecules is taken into account within a mean field approximation. The mean field interactions are functions of the average local monomer density, which depends on the distance z from the surface, because the profile ϕ(z) of the monomer density is inhomogeneous, of course. Because the probability of a given chain configuration will depend on both its intramolecular interactions and on this mean field, which in turn depends on the chain conformation via the profile ϕ(z), a self-consistency problem results which again requires iterative numerical solutions. Obviously, the spirit of this approach is very similar to SCFT, but differs from it because it is not restricted to the simplest form of Gaussian chain models for the statistical mechanics of the “central” single chain. If one uses these simple models, then SCMF and SCFT are equivalent,94 using, for example, the rotational isomeric state model204 which models a chain where each bond has three states (trans, gauche+, gauche−) with angles ϕ = 0, 120° and −120°, respectively). Effects of local chain stiffness can be incorporated in the description.94 By using torsional and bending potentials, this local stiffness can be systematically varied. The approach can also be used for branched chains, for chains tethered to curved surfaces, and so forth.4 Although this approach has clearly several advantages in comparison with SCFT, the existing applications seem to be restricted to relatively short chains (N ≤ 100).4 In this regime of chain lengths, however, it is possible to avoid mean-field approximations altogether, and simulate the whole brush, a multichain system (see Section Conclusions) rather than simulating a single chain in an effective field that is inhomogeneous and needs to be found by an iteration procedure, that also is numerically costly. Because the SCMF approach has been reviewed elsewhere in detail,4 we do not dwell on it here further.

Another problem of both SCFT and SCMF, however, is the correct treatment of the excluded volume interaction. Already in the section on the “blob theory” (Section Scaling Concepts for Brushes under Good Solvent Conditions), we have seen that for polymer brushes under good solvent conditions the polymer conformation is swollen in the semidilute regime, that is, inside of a blob in Figure 1 we have self-avoiding walk statistics, while SCFT implies on a small scale that Gaussian chain statistics holds. This discrepancy is somewhat hidden when one makes the Flory approximation for the exponent ν, νF = 3/5, rather than using its correct value,162 ν ≈ 0.588. Then, both SCFT and scaling theory in terms of blobs predict for the brush height a scaling h ∝ σ1/3N in the good solvent regime. However, the proximity of the actual estimate for ν to νF is a kind of numerical accident, and hence it is still worthwhile to explore what happens when one wants to include excluded volume effects in quantitatively accurate descriptions of polymer brushes.

A phenomenological extension of the SCFT to properly include excluded volume in the SSL, as formulated by Milner et al.,30 has been given by Wittmer et al.46 In the SSL, the dominant contribution to the chemical potential per chain is proportional to the number of monomers, and therefore, the entropy of the end points of the chains, which is of order kBT, cf. eq 45, can be neglected. As mentioned in Section Scaling Concepts for Brushes under Good Solvent Conditions, fluctuations around the most probable conformation of the chain are ignored. The conformation of each chain is obtained by minimization of the chemical potential functional μ(z) with respect to position z(s) of monomer s,

  • equation image(68)

Here, the potential U(ϕ) is the work required to insert a monomer at a distance z from the wall, and in the semidilute regime scales like U(ϕ) ∝ ϕ1/(3ν − 1). As ϕ depends on z, U(ϕ) can be considered as a potential depending on z too. The semidilute regime requires that ϕ exceeds the overlap density ϕ = N/(4πRg3/3), where Rg is the gyration radius of a free chain in dilute solution. The prefactor of the gradient square term A(ϕ) ∝ ϕ(2ν − 1)/(3ν − 1) (=ϕ1/4 in the Flory approximation) would be simply a constant for Gaussian chains, and U(ϕ) would be harmonic, U(ϕ) ∝ ϕ2, cf. Section Scaling Concepts for Brushes under Good Solvent Conditions. Introducing then the coordinate Z defined by

  • equation image(69)

the chain is essentially transformed to a Gaussian string of swollen blobs. In the actual minimization of this functional, eq 68, one needs to explicitly impose the constraint that in a monodisperse brush each chain reaches the wall after N steps: this “equal time”-requirement imposes a parabolic potential in Z, which allows to construct ϕ(Z) explicitly.46 The final result for the brush height is of the form of eq 3, h/Rg ∝ (σRg2)(1 − ν)/2ν, and clearly the prefactor is nonuniversal. However, the theory implies a nontrivial universal relation between brush height and the (coarse-grained) density at the wall, ϕ(0), namely

  • equation image(70)

Note that the results of the SSL of SCFT, namely30

  • equation image(71)

imply a very similar scaling of this product hϕ(0)/Rg, namely

  • equation image(72)

Computer simulations of the bond fluctuation model on the simple cubic lattice205, 206 have been performed to test eq 70,46 see Figure 10. As it is known that this model exhibits strong excluded volume effects already for rather small N, rather good scaling properties verifying that h/Rg is simply proportional to (σRg2)(1 − ν)/2ν and that ϕ(0) is proportional to (σRg2)(1/2)(3 − 1/ν) were found already for N ≤ 100 (Fig. 10), and eq 70 was confirmed.

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Figure 10. (right) The brush height h = (8/3)〈z〉 in units of the gyration radius Rg (denoted as R in the figure) of a free chain in a good solvent, plotted vs. the scaling variable (σR2)(1−ν)/2ν, for chain lengths N = 20, 30, 40, 50, and 100, as indicated, and various grafting densities σ (denoted as σg in the figure). A straight line fit in the region where the brush is strongly stretched is indicated. Note that for (σR2)(1−ν)/2ν < 1, a crossover to nonstretched mushrooms occurs. (left) Plot of the volume fraction at the grafting plane relative to the overlap volume fraction, ϕ(0)/ϕ*, plotted against the scaling variables (σR2)(3ν−1)/2ν. The straight line shows a fit by the scaling theory. From Wittmer et al.46

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The theoretical results, eqs 6870, apply to situations where the excluded volume interaction is so strong that excluded volume statistics (end-to-end distances r(s) of a piece of a chain containing s monomers scale as r(s) ∝ sν) fully holds on the scale of a blob [in Fig. 1(b)]. Although this is true for the case of the Bond Fluctuation Model with purely repulsive interactions on the lattice, for real polymers the excluded volume strength often is relatively weak, in particular if the temperature of the polymer solution is not very far from the Theta-point θ: then one still has Gaussian-like statistics on the scale of thermal blobs ξT ∝ | T − θ]−1,160 and if the size of the thermal blobs would exceed the size of the blobs resulting from the grafting, ξblob ≈ σ−1/2, excluded volume on this scale is negligible and thus the standard SCFT theory (Section Essentials of the Self Consistent Field Approach to Polymer Brushes) should hold.

This consideration leads to the question of describing the crossover between these two extreme cases55 and is addressed in Figure 11: the brush height h can actually be considered as a function of two scaling variables, Figure 11, namely ζ = σRg2 and ζ = vexN2/Rg3. For small ζ, one has mushroom behavior, while for large enough ζ, one has brush behavior; good solvent behavior where excluded volume dominates requires that ζ is large enough, while the regime where vexN2/Rg2 is not so large describes “marginal solvents”. So in general the brush height h is a function of both of the two scaling variables defined above

  • equation image(73)

but the scaling function is only known well inside the regions shown in Figure 11, i.e. not near the broken straight lines in Figure 11 that indicate where a smooth gradual crossover from one limiting behavior to the other occurs. Note also that the regime of “marginal solvents” does not include the Theta point itself: as is well known, one needs to include the third virial coefficient at T = θ where the second virial coefficient (which is proportional to vex or w) vanishes; the behavior at T ≤ θ will be discussed in the next section.

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Figure 11. Schematic log-log plot showing in the plane of the scaling variables relating to excluded volume (vex) and grafting density σ the regime of mushrooms (left) and brushes (right). Crossovers between the various regimes are indicated by broken or dash-dotted straight lines. SCFT holds below the crossover line to the regime of the scaling theory [described by vexN2/Rg3 ∝ (σRg2)(2−3ν)/2ν], and to the right of the brush-mushroom crossover for marginal solvents [described by vexN2/Rg2 ∝ (σRg2)−1]. The dependence of the scaling function of the brush height (eq 73) on the scaling variables is indicated in the various regimes. Note that SCFT implies h ∝ σ1/3, both for marginal and for good solvents, but the prefactors exhibit different dependences on the strength of the excluded volume interaction (recall that vex corresponds to the second virial coefficient). According to the scaling theory, however, the strength of the excluded volume does no longer matter. From Ref.9

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Unfortunately, apart from some numerical work,55 we are not aware of further studies of the problem posed by eq 73 and Figure 11. It should be noted, of course, that experimental work on the precise scaling properties of brushes is complicated by inevitable polydispersity effects. The theoretical consideration of these effects in terms of the SSL of SCFT89, 207 is difficult and only for special assumptions on the molecular weight distribution (MWD) explicit solutions are possible.89 One general result is that in a polydisperse brush each free chain end has a well-defined distance from the grafting surface in the brush, unlike the monodisperse case where the free ends exhibit a distribution in their position, that extends throughout the brush (eq 49), and thus each chain undergoes in a sense “critical fluctuations.”88 Because all chains are equivalent, each chain end explores the full range of the distribution, eq 49, rather than staying fixed in a particular region of the brush. This fact has particular consequences on the dynamics of chains in brushes,88 but this aspect is beyond the scope of the present article.

Although both SCFT and scaling theories consider the asymptotic behavior of very long chains in not too dense brushes on coarse-grained length scales, it should be noted that much of the experimental work,7 and also some of the simulations discussed in Section Conclusions, are concerned with rather dense brushes of short chains. Then, an aspect comes into play that escapes both SCFT and scaling, namely, the problem of properly “packing” the monomers near the (hard) wall to which the chains are grafted. These types of effects that happen on the scale of individual monomers, rather than on the scale of blobs, can be captured by another type of mean field theory, namely density functional theory (DFT).208–211 Considering, for instance, a system where a brush interacts with free chains, one is interested in their monomer density profiles ρf( equation image), ρg(equation image) which are found from a minimization of a free energy functional where one tries to account explicitly for repulsive and attractive interactions, starting, for example, from hard-sphere-like expressions for the repulsive part, while the attractive part is accounted for in a mean-field approximation. Figure 12 shows that for not too long chains results can be obtained that are in fair agreement with corresponding Molecular dynamics (MD) Simulations,59 where a solvent rather than free chains were used (formally, Nf = 1 in the theory). The clear advantage of both DFT and MD is that solvent molecules can be considered explicitly, and one sees a “layering” (i.e., density oscillations) near the wall both for ρf(z) and ρg(z), and for the chosen model in fact some enrichment of solvent particles at the wall occurs. Of course, neither scaling theory nor SCFT would yield any statement on this local behavior near the wall, and no information on the solvent density distribution whatsoever would be available.

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Figure 12. Density profiles of the monomers of grafted chains (open symbols and broken curves) and of solvent particles (full dots and full curves) at a planar wall, for three types of solvent conditions: good solvent (upper panel), Theta solvent (middle panel), and poor solvent (lower panel). Here, σ is the range parameter, σLJ, of a Lennard-Jones potential, and the solvent density in the bulk is chosen as ρbulkσLJ3 = 0.32. The energy parameters ϵαβ refer to energies between pairs of particles of type α, β (see Section APPENDIX: COMPUTATIONAL METHODS for a precise definition of the model). Chain length is Ng = 64 and grafting density is σ = 0.32 (choosing σLJ = 1, as the unit of length). From Ref.211

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The DFT can also be directly extended to spherical polymer brushes, and Lo Verso et al.65 have compared such DFT calculations to corresponding MD simulations (Fig. 13). The results shown here all refer to the good solvent regime. Again, one notices the characterizing layering effect (density oscillations in the first few “shells” of monomers around the sphere to which the chains are grafted), and this behavior is well captured by DFT also in this case. At somewhat larger distances, the results are nicely compatible with the power law decay predicted by Daoud and Cotton167 for star polymers. However, at this point we note the criticism212 where it was pointed out that this model does not correspond to the minimal free energy of such a spherical brush [even when one assumes that the end monomers are all on the outer edge of the brush, which clearly is not the case, as Fig. 13(b) demonstrates]. Zhulina et al.212 showed that the true solution of such a model involves a nonlocal free energy expression, and hence the polymer density profile does not follow a single-exponent power law. However, these deviations from the Daoud-Cotton scaling are numerically small, and hence were not resolved in Figure 13(a).

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Figure 13. Log-log plot of the monomer density ρ(r), (upper part), and of the end-monomer density ρe(r), (lower part), for spherical brushes where f = 162 chains are grafted to a sphere of radius Rc = 8.35 so that a grafting density σ = 0.185 is realized (all lengths are measured in units of the Lennard-Jones parameters of the Weeks-Chandler-Andersen potential acting between effective monomers, cf. Section APPENDIX: COMPUTATIONAL METHODS). Chain lengths N = 20, 40, 60, and 80 are included (from the left to right), MD results are shown by broken curves, DFT by full curves (including also much longer chains in the upper part, namely N = 250, 500, and 750, the three right-most curves). The thick straight line illustrates the Daoud-Cotton167 scaling, ρ(r) ∝ r1/ν−3r−4/3. Note that r is measured here from the center of the sphere to which the chains are grafted (the dash-dotted vertical line in the upper part indicates the sphere surface). From Lo Verso et al.65

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We also note that DFT and MD results almost coincide for N = 20 and N = 40, but with increasing chain length deviations become indicative of a breakdown of DFT, N [RIGHTWARDS ARROW] ∞. Recall, that we expect from eqs 15 and 16, a scaling h ∝ σ(1 − ν)/2Rc1 − ν Nν ≈ σ1/5Rc2/5N3/5 in the limit N [RIGHTWARDS ARROW] ∞. Figure 14 shows, however, that the DFT results65 seem to converge towards a scaling hN2/3 rather than hN3/5 in this limit. The reason for this failure of DFT is not yet known. By (numerical) SCFT approaches, it is possible to verify the correct structure of crossover scaling between planar and spherical brushes, however.61 In view of these problems, we do not give any detail of the approximations involved in DFT, but rather refer to the literature.65, 211

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Figure 14. Log-log plot of the normalized brush height hav/h0 against the normalized inverse radius h0/Rc of the sphere onto which f chains of the spherical brush are grafted. Here, h0 denotes the corresponding value of a planar brush at the same grafting density σ. MD results (open symbols) and DFT results (full symbols) are shown for three grafting densities: σ = 0.185, triangles; σ = 0.118, crosses; σ = 0.068, asterisks; (see Fig. 13 for more details of the model). The theoretical asymptotic slope (−2/5), predicted from scaling (Eqs 15 and 16) and confirmed by SCFT,61 is shown as a broken straight line while the DFT slope (−1/3) is shown as a full straight line. From Lo Verso et al.65

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Varying the Solvent Quality: Theta Solvents and Poor Solvents

In this section, we again return to SCFT in the SSL. Using as in eq 59, the Flory-Huggins theory of a polymer solution160, 165, 191 as a starting point, we consider the limit N [RIGHTWARDS ARROW] ∞ from the start.97 Then, the free energy density due to the local interactions fint(ϕ) is [kBT ≡ 1 here]

  • equation image(74)

The corresponding exchange chemical potential (for replacing an effective monomer by a solvent particle) is then μ(ϕ) = ∂fint(ϕ)/∂ϕ. This chemical potential must be combined with the contribution due to chain stretching in the brush. In SCFT-SSL, this contribution results from a potential V(Z) = 3π2Z2/8, if Gaussian chain statistics is assumed, Z = z/Na. Amoskov and Birshtein97 used a finite stretching model instead, V(Z) = −3 ln [cos(πZ/2)], where coordinates are chosen such that Z = 1 corresponds to maximum extension of the (infinitely long!) chains. Note that for small Z this potential reduces to the standard parabolic potential. Now the exchange chemical potential and the self-consistent field potential are connected via

  • equation image(75)

where μmin = μ(ϕmin) is the chemical potential at the outer brush boundary, which defines the (normalized) brush height, H = h/(Na). Note that ϕmin = 0 for good and Theta solvents (χ = 1/2 for the latter160, 165, 191) but we shall find ϕmin > 0 for χ > 1/2: ϕmin just turns out to be the volume fraction of the polymer rich phase coexisting with pure solvent, for a polymer solution that is described by eq 74. The brush height H is connected to the grafting density by the normalization condition,

  • equation image(76)

The profile is then described by the equation, resulting from eq 74 and 75,

  • equation image(77)

and the osmotic pressure is

  • equation image(78)

with Π(ϕ(Z = H)) = 0, the osmotic pressure is zero at the outer boundary Z = H of the polymer brush. For small ϕ, eq 78 becomes Π(ϕ) ≈ ϕ2( equation image − χ) + ϕ3 and hence for χ > 1/2 (where a bulk polymer solution for N [RIGHTWARDS ARROW] ∞ exhibits phase separation at Π = 0), the polymer-rich phase of the polymer solution has a monomer volume fraction ϕcoex ≈ χ − 1/2. Then, the profile ϕ(Z) of the monomer volume fraction in the brush decreases from its maximum value ϕmax = ϕ(z = 0) monotonically and continuously down to ϕ (Z = H) = ϕcoex at Z = H, where a discontinuous jump to zero occurs. This jump vanishes linearly with χ when the critical point χcrit = 1/2 of the polymer solution is reached. Although ϕ(Z) vanishes linearly with Z for χ < χcrit (there eqs 48 and 49 apply), for χ = χcrit one finds a square root singularity37

  • equation image(79)

and one also finds that for χ = χcrit (i.e., at the Theta-point of the polymer solution) a different scaling with grafting density σ occurs, namely (we restore here again physical units, w3 is essentially the third virial coefficient37)

  • equation image(80)

while in the poor solvent region one predicts37

  • equation image(81)

At this point, we note that qualitatively the results eqs 80 and 81 can be understood in terms of simple Flory arguments,26 when one does not care about prefactors. In fact, the Flory argument amounts to writing the chain free energy in a brush as a sum of free energies of elastic stretching (ΔFel = h2/N) and of interactions,

  • equation image(82)

where now w ∝ (χcrit − χ) may change sign at χ = χcrit. At χ < χcrit the term proportional to w3 can be neglected for small grafting density σ, of course. Minimizing the free energy ΔFel + ΔFint with respect to h, ∂(ΔF)/∂h = 0, yields for χ > χcrit hN(wσ)1/3x, that is, eqs 15 and 71, while for w = 0 eq 80 results, and for |w| [RIGHTWARDS ARROW] ∞, eq 71 results. However, the Flory theory does not give valid results on the distribution of chain ends, while SCFT-SSL predicts37

  • equation image(83)
  • equation image(84)

All results of this subsection, however, refer to the limit N [RIGHTWARDS ARROW] ∞ that the SCFT-SSL describes. Of course, for finite chain lengths two effects occur40, 42, 43, 45, 48, 59, 95:

  • (i)
    The density profile ϕ(z) is a smooth function of z, irrespective of solvent condition, and the same is true for ρe(z), see Figure 15. Disregarding the structure of the profiles at small z near the grafting plane, we see that the profile ϕ(z) of the monomer density is nearly parabolic in the good solvent regime but develops an almost horizontal part [where no solvent occurs, see part (b) of the figure] for the poor solvent case, separated from the pure solvent by a rather narrow surface region of an essentially polymer melt-like structure. The width of this region is not expected to be the intrinsic width of such an interface, but presumably is broadened by capillary waves.213, 214 When N is increased, keeping the grafting density and solvent quality unchanged, just the width of the horizontal part increases, but the qualitative behavior stays the same. When one passes the region of the Theta-point, all profiles change completely gradually and no singular behavior occurs in agreement with experiment, Figure 7(b).189 Although this smooth behavior is also obtained with numerical versions of the SCFT theory, we are not aware of any explicit analytic theory describing the rounding of the singularities that were derived above.
  • (ii)
    For not too long chains and small enough grafting density, one finds microphase separation in the brush, it laterally decomposes into monomer-rich clusters separated by regions that are almost free of monomers.40, 42 Figure 1 (upper-right panel) and Figure 16 give characteristic examples. Of course, due to the constraint that for each chain one end is irreversibly grafted to the substrate, lateral motion of this end being forbidden, no monomer can get arbitrarily far from the grafting site of the chain to which the monomer belongs; thus, macroscopic lateral phase separation cannot occur.
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Figure 15. Monomer (above) and solvent (below) density profiles, obtained from MD simulation of a bead-spring model (see APPENDIX: COMPUTATIONAL METHODS) of polymers in explicit solvent (described as point particles interacting with each other by a Weeks-Chandler-Andersen potential). Solvent quality for the polymers ranges from very good solvent (curve with the largest brush height) to poor solvent (curve with the smallest brush height). All data refer to N = 60 and σ = 0.185 (choosing Lennard-Jones diameter as length unit). From Dimitrov et al.59

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Figure 16. Snapshot pictures from the MD simulation of a bead-spring model of polymers of length N = 50 (a,c) and N = 100 (b), grafted on a flat repulsive surface, for a temperature below the Theta point (T/Tθ ≈ 0.67) and two choices of grafting densities: σ = 0.03 (a,b) and σ = 0.10 (c). All monomer positions are projected into the grafting plane (z = 0). Lengths are measured in units of the Lennard-Jones diameter of the Weeks-Chandler-Andersen potential between the beads. From Grest and Murat.42

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There were several theoretical treatments to interpret these results. Yeung et al.43 carried out numerical SCFT work that did not restrict spatial inhomogeneity from the start to the z-direction perpendicular to the grafting surface, as was described in Section Essentials of the Self-Consistent Field Approach to Polymer Brushes, but allowed also for lateral inhomogeneity. Carignano and Szleifer187 calculated the osmotic pressure Π(σ) as a function of grafting density σ from their SCMF theory (see Section Other Theoretical Concepts) and found that for σ less than some critical value the pressure is negative, and also the compressibility is negative. They interpreted this as an indication for microphase separation. To corroborate this result, they computed also the phase behavior for grafted chains that are laterally mobile along the grafting surface.4, 187, 201 They showed that for temperatures T sufficiently less than the Θ-temperature Tθ, the isotherms Π(σ) vs. 1/σ developed a loop rather reminiscent of a van der Waals loop in a simple fluid, confirming thus the speculative proposal of Lai and Binder40 that the microphase separation (first observed by them) in a polymer brush with fixed grafting sites under poor solvent conditions is a kind of “arrested” macroscopic phase separation.

In fact, in a polymer solution under poor solvent conditions it is known that phase separation occurs for T < Tc(N), with the critical temperature scaling as160, 165, 191 TθTc(N) ∝ N1/2, and the critical monomer volume fraction ϕc scaling as ϕcN−1/2 as well. For TTc(N) phase separation occurs between essentially pure solvent and a semidilute solution of volume fraction ϕcoex ∝ TθT(∝ χ − χcrit, if the model eq 74 is used). From eq 80 we can conclude that at Tθ, or nearby, the volume fraction inside of a brush is ϕ ≈ N/(σ−1h) since σ−1h is (apart from a prefactor of order unity) the volume available per grafted chain, and hence ϕ ∝ σ−1. On the other hand, at T = Tθ the crossover between mushroom behavior (where hmequation image) and brush occurs for h/hm ≈ 1, therefore σ1/2N/equation image ≈ const, that is, σ* ≈ 1/N.

Thus, we conclude that the microphase separation has its onset near the crossover from mushroom to brush behavior, because the volume fraction ϕ* at the mushroom to brush crossover (ϕ* ∝ (σ*)1/2N−1/2) scales in the same way as ϕc(N) in a polymer solution does. Although in poor solvents for σ ≪ σ*, we observe the collapse transition of isolated mushrooms, fully analogous to the collapse of isolated chains in a extremely dilute solution,160 for σ near σ* clusters are formed where several grafted chains form together a relatively dense “dimple.”

Tang and Szleifer45 presented an interesting attempt to compute the collective structure factor of the grafted polymer layer, using the random phase approximation (RPA)160, 191 which relates the collective structure factor to the structure factor of an ideal (noninteracting) system of grafted chains, S0( equation image)

  • equation image(85)

where fint(ϕ) = a−3(τϕ2/2 + w3ϕ3/3), in analogy to eq 82, and τ stands for the distance from the Theta-point. One finds that [S0( equation image)]−1 has a minimum at k = k* with k*Na2/6 ≈ 1.893, with S0−1(k = k*) = 2.624/N (note that in the ideal system of grafted, noninteracting chains Gaussian statistics applies, of course). The treatment hence is reminiscent of the RPA for block copolymers,191, 215 and putting S−1(k = k*) = 0, one finds a stability limit (“ordering spinodal”216). In terms of variables

  • equation image(86)

this stability limit can be written as (α = h/ equation image)

  • equation image(87)

and is shown in Figure 17. One can see that laterally homogeneous brushes (denoted as “layers” in Fig. 17) are always stable for large q, that is, Na2) ≫ 1 while the inhomogeneous structures occur for q of order unity (and the characteristic size of the “cluster” or “dimples” is of order Lcluster ≈ 2π/k* ≈ 2 equation imagea, consistent with the simulations.40, 42 Again, as in the case of microphase separation in mixed binary polymer brushes, one may argue that the random fluctuations in the density of grafting sites destroy sharp phase transitions, so the transitions from collapsed mushrooms to “clusters” of collapsed chains (“dimples”) and then to a homogeneous layer are just gradual crossovers.

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Figure 17. “Phase diagram” of polymers grafted to a plane for the case of poor solvents, in terms of the scaled distance p from the Theta-point (p = τ equation image. The Theta-point occurs for τ = 0) and scaled grafting density q, with q = Na2)equation image. The laterally homogeneous brush is denoted as “layer.” The (smooth) crossover between mushrooms (in the left part of the figure) and clusters or layers is shown by a broken line (this line is vertical near Tθ yet bends to the left for very poor solvents, corresponding to large negative values of p). The stability limit, eq 87, is shown as full curve (it is physically meaningful only on the right side of the brush to mushroom crossover). From Tang and Szleifer.45

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This latter conclusion also applies when one considers polymer brushes where chains are grafted to cylinders76, 217–223 or spheres66 under poor solvent conditions. Particularly delicate is also the extension to binary polymer brushes in cylindrical geometry, where the quasi-one-dimensional character of the system provides additional fluctuation mechanisms to destroy the long-range mesophase order that mean field-type treatments predict.219–223 However, the latter problem is too specialized to be discussed at length here.

Restricting attention to one-component brushes, we show in Figure 18 the “phase diagram” of polymers grafted to an (infinitely thin) cylinder, so the crossover between cylindrical brushes and flat brushes when the cylinder radius Rcyla equation image is out of question here. The “pearl-necklace” structure in Figure 18 is the analog of the “clusters” (Figs. 16 and 17) discussed above for the planar brush.

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Figure 18. Schematic phase diagram of a cylindrical polymer brush (for cylinder radius Rc comparable to the monomer length a) under poor solvent conditions. The abscissa variable is the scaled grafting density x = N1/2σ (appropriate in one dimension, when chains are grafted to a line at distance σ−1 between grafting points), and the ordinate variable is the scaled distance from the Theta-temperature (Tθ = θ in this figure) z = N1/2|(T − θ)/ θ|. The good solvent regime (T > θ) is not included here. The crossover lines shown in the diagram have been first proposed by Sheiko et al.,217 and should be understood as qualitative estimates for smooth crossovers. Representative simulation snapshots are shown to illustrate the existence of the proposed states, where the grafted chains form either Gaussian (z < 1) or collapsed (x > 1) isolated mushrooms (on the left side of the diagram), or a stretched cylindrical brush (upper right region), collapsed cylindrical brush (middle right region) or pearl-necklace structure (lower right region). From Theodorakis et al.218

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The various crossover lines in Figure 18 were estimated217 from a free energy minimization, motivated by the Flory theory,160 amended by scaling arguments. It is interesting to note that the “pearls” of the pearl-necklace structure are not predicted to be spherical objects, but rather somewhat elongated along the cylinder axis. Although in the scaling limit and good solvents (using ν = 3/5) one would predict that the swollen brush has a radius RN3/4σ1/4τ1/4217 (τ = |(TTθ)/Tθ|, cf. also eq 20), for τ = 0 (Theta conditions), one rather finds RN2/3σ2/3), while the collapsed brush scales as R ∝ (Nσ/|τ|)1/2. Sheiko et al.217 predict that the pearls rather have a radius RpearlN1/2(σ/|τ|)1/4 and an axial length LpearlN1/2(|τ|/σ)1/2 so each pearl contains npearlN1/2|τ|1/4σ3/4 chains. Although the simulations76, 218 gave some qualitative evidence for the diagram of states reproduced in Figure 18, these scaling predictions could not yet be tested, simply because for poor solvent conditions only rather short chains (N ≤ 50) could be equilibrated.

For spherical polymer brushes under poor solvent and Theta-solvent conditions, the monomer density profiles were found66 to qualitatively resemble those of planar brushes. Thus, for large enough grafting density σ and long enough chains (20 ≤ N ≤ 80 could be studied66), the density profile has an extended horizontal part (corresponding to almost melt density inside of the brush). Thus, the spherical core to which the chains are grafted is coated by a “shell” of thickness h, the brush height. Although it turned out not to be feasible to study the crossover from spherical to planar brushes with so short chains, a numerical SCFT calculation66 (using a lattice formulation adapted to spherical geometry) could show that the brush height h(R,N, σ) scales as in eq 15. Yet, as the height h0 ∝ σ1/2N for T = Tθ (cf. eq 80) and the size of the star polymer scales as RN1/2 at the Theta-point, the scaling function f(h0/Rc) in eq 15 scales like F(ζ) ∝ ζ−1/2, and this scaling has been verified.65

Note that in the poor solvent regime and at small grafting densities and chain lengths, one observes again a crossover from collapsed mushrooms to collapsed clusters and finally to a dense layer, similar as for planar brushes. Although configuration snapshots65 have given qualitative evidence for this crossover, it has not been analyzed in detail yet. For dense brushes under variable solvent conditions, an interesting issue is the distribution of end monomers: while near the sphere at which the chains are grafted the end monomer density is very small for T ≥ θ, in the poor solvent regime end-monomers are more or less randomly distributed in the brush, as expected for a dense melt where no strong long range correlations in the single-chain structure exist.

Dense Brushes and Brushes Interacting with Polymer Melts

Although in the previous section we have given a (qualitative) discussion of the phenomena in poor solvents, where high monomer density in a brush can result from the collapse of the grafted chains, but still the grafting density σ was assumed to be small (emphasizing even the crossover to the collapsed mushroom regime), we consider now different regimes where high monomer density in a brush matters: such a case occurs also for good solvent (and Theta solvent) conditions when one considers the limit σ [RIGHTWARDS ARROW] 1, as well as for the case of polymer brushes interacting with (dense) polymer melts. These cases are not at all a straightforward extension of the treatments of sections Scaling Concepts for Brushes Under Good Solvent Conditions and Essentials of the Self-Consistent Field Approach to Polymer Brushes, since there the treatment of monomer-monomer interaction was basically reduced to consider the second virial coefficient (cf. eq 31, Fig. 11, etc.). However, this is not at all appropriate in the present case, as pointed out already by de Gennes27 and thoroughly elaborated and extended by Raphael et al.44 and Aubouy et al.154 We follow Refs.44, 154 by restricting the level of discussion to the Alexander-type picture of the brush (disregarding the actual monomer density profile) and Flory-type arguments. Thus, we write the free energy of a chain in a brush as

  • equation image(88)

where Fel is the elastic contribution resulting from stretching the chain to the brush height h, and the osmotic free energy is given as (denoting here D = aσ−1/2 the distance between the grafting sites, so the volume taken by a chain is hD2)

  • equation image(89)

Here, the first term corresponds to the effect of the two-body interactions (as previously, this represents excluded volume forces), whereas the second term corresponds to three body forces, and this term can be understood from a virial expansion of the Flory- Huggins-equation of state of a polymer solution160, 165, 191

  • equation image(90)

Equation (90) is the free energy of mixing per site in a lattice model, ϕ being the fraction of sites taken by polymers, the second and third terms on the right hand side of this equation were already encountered in eq (74). From the expansion of eq (90) for small ϕ, we now can conclude that the second virial coefficient υex is related to the Flory -Huggings parameter χ as υex = a3 (1 − 2χ), remembering that a is the lattice spacing of the Flory-Huggins model, a single lattice site hence is associated with a volume a3. The third virial coefficient is a6, and is temperature-independent, unlike υex which becomes negative below the Theta temperature (for moderately good solvents 0 < χ < 1/2 and hence υex < a3). This consideration justifies the choice of coefficients in eq (89). Note that a3N/(hD2) is the volume fraction of monomers ϕ in the volume taken by a chain, as usual, and the total enthalpy due to pair interactions is put (hD22vex in eq (89). Factors of order unity are ignored throughout.

Now two cases must be considered, depending on which of the two terms in eq (89) dominate when we minimize F with respect to h. One easily finds that for σ* < σ < σ1 = (υex/a3)2, it is the excluded volume interaction term that dominates; neglecting hence the term with the third virial coefficient the condition ∂F(h)/∂h = 0 yields (ignoring prefactors of order unity) that h = υex1/3N(a/D)2/3 = υex1/3 Nσ1/3, that is eqs (3), (43), and (51). However, for σ > σ1 it is the term due to the third virial coefficient which dominates, and hence one obtains instead from ∂F(h)/∂h = 0 that

  • equation image(91)

Note that there occurs a smooth crossover between eqs (51) and eq (91) at σ = σ1, with h = N υex/a2 there. We also observe that the average volume fraction of monomers in the brush in the regime described by eq (91) is ϕ = Na3/(hD)2 = σ1/2; so only for σ [RIGHTWARDS ARROW] 1 one also obtains ϕ = 1. Eq (91) was already quoted in eq (80) for brushes at the Theta temperature, but it is important to note that it also holds in the good solvent regime at high enough grafting densities. We also note that an approximate extension of SCFT to this regime σ > σ1 predicts an elliptic rather than parabolic volume fraction profile44

  • equation image(92)

which suggests that now the Alexander picture is closer to the actual behavior of real brushes for σ near unity than for small σ. However, also in this case the profile ϕ(z) does not exhibit for finite N a sharp vanishing at z = h, as simulations show.224, 225 Of course, the limit σ [RIGHTWARDS ARROW] 1 is delicate, because then crystallization of the brush may need to be considered.226

We now discuss the extension of the theory to the case where a polymer brush interacts with a polymeric matrix as a solvent, rather than a small molecule fluid. We here restrict attention to the case that the polymer melt that a polymer brush interacts with has a degree of polymerization PN. A free chain of length N dissolved in a melt of shorter, chemically identical chains is swollen, but the effective strength of the excluded volume interaction is reduced.227, 228 One can easily see this from the Flory-Huggins entropy of mixing of chains of lengths N, P and volume fractions ϕN, ϕP = 1 − ϕN153, 165, 191

  • equation image(93)

which in the limit of small ϕN leads to

  • equation image(94)

and hence one can conclude that υex in the previous treatment needs to be replaced by a3/P, and the third virial coefficient becomes a6/P (note that we consider only a brush in a melt of identical chains, so χ = 0). Writing then, in the limit where pair interactions dominate, in analogy to Eq. 89

  • equation image(95)

and the condition ∂F(h)/∂h = 0 yields (neglecting the term from the third virial coefficient)

  • equation image(96)

The monomer volume fraction ϕN = Naσ/h = P1/3σ2/3. Note that for σ = σ1, we find that ϕN = 1, which means that the mobile chains for σ > σ1 are already completely expelled! For that reason, for σ > σ1 one simply has

  • equation image(97)

Actually one finds that inside the moderate coverage regime, there occurs another crossover, at σ = σ′1 = P−2; while for σ < σ′1 the picture of a chain is a string of subunits of size D for σ > σ′1 it is a string of subunits of size Λ = aP1/3σ−1/3 > D, so one has no longer a picture of dense packing of spherical subunits which do not overlap (as in the original Alexander-de Gennes picture), but now the spherical subunits necessarily overlap. The expression for h does not change when one crosses σ′1, however.

These results can be summarized in a schematic (P, σ) diagram, Figure 19, where for completeness also the crossovers to the mushroom regime are included.154

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Figure 19. Schematic diagram for the states of a polymer brush exposed to a chemically identical high-molecular-weight solvent which has degree of polymerization P while the brush has degree of polymerization N(N > P). The brush height in the different regions scales as follows: region 1, haN1/2 (mushrooms for which excluded volume is screened out); region 2, haN3/5P−1/5 (mushrooms for which excluded volume is effective); region 3, haNP−1/3σ1/3 (good solvent region of the stretched brush); region 4, haN1/2 (layer of overlapping chains that are not stretched); region 5, h = aNσ (dry brush). From Aubouy et al.154

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Some of these predictions have been successfully tested by experiments (e.g., Ref.229) and simulations (e.g., Ref.230). Very recently, very much interest has been devoted to the problem of embedding brush-coated nanoparticles (i.e., spherical polymer brushes) into polymer melts under various conditions.231–236 Of particular interest then is the interaction between two brush-coated nanoparticles. Some aspects of this problem will be discussed further in Section INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, ON WITH NANOPARTICLES of the present article. For a recent experimental review on nanocomposites formed from brush-coated nanoparticles, see Ref.237

“LIVING POLYMER” POLYDISPERSE BRUSHES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Although the overwhelming part of theoretical studies on polymer brushes deals with strictly monodisperse systems in which the process of polymerization is terminated, while in fact, polymer brushes are usually created by means of surface-initiated polymerization238 and are characterized by non negligible polydispersity.239 A large variety of synthetic routes for the generation of polymer brushes includes, for example, ionic-,240 ring-opening-,241 atom transfer radical-,242 and reversible addition-fragmentation chain transfer-243 polymerization. Most frequently, for example, one observes a Flory-Schulz Molecular Weight Distribution (MWD) of chain lengths inside the polymer brush:

  • equation image(98)

where c(N) is the fraction of chains with length N, pr ≤ 1 is the probability that a monomer has reacted, M0 is the molecular weight of a monomer, and MN is the number-averaged molecular weight.

A “living” polymer brush has the unique feature that the chains are dynamic objects with constantly fluctuating lengths. Subject to external perturbation, they are able to respond dynamically via polymerization – depolymerization reactions allowing new thermodynamic equilibrium to be attained. Here, we consider a typical case of living ionic polymerization whereby chains grow from initiators, fixed on a grafting plane, by end- monomer attachment – detachment while the total number of chains remains constant.

The free energy FbrushLP (per unit surface) can be written in Flory-Huggins-like manner as244

  • equation image(99)

where the first term allows for the entropy of mixing of the grafted chains, the second one entails the Lagrange multiplier for the conserved number of polymer segments, Mg, and the last term is the free energy of a reference chain in the self-consistent density profiles, created by the neighboring chains. The mean-field eq 99 assumes that the length of a given reference chain is not correlated with those of its neighboring chains.

On minimization of eq 99 via functional derivation FbrushLP[c(N)] with regards to c(N), one obtains the equilibrium MWD

  • equation image(100)

In the simplest case of dilute nonoverlapping living polymers (mushrooms) tethered to a plane in good solvent, one expects Fchain(N) = τln(N) with the exponent τ = 1 − γs where the universal “surface” exponent γs ≈ 0.65 < 1. Hence, one expects to find a weakly singular behavior c(N) ∝ N−τ exp(−μ1N). In a weakly stretched polymer brush at low grafting density σ, where the living polymers do not overlap strongly, the excluded volume interactions are expected to favor longer chains which can explore more dilute regions of the living polymer brush. This would lead to a power law MWD (plus exponential cutoff). Therefore, for a self-similar mushroom structure of the living polymer brush with blob size ξ(z) ∝ z the monomer density would scale as ϕ(z) ∝ z−α, α = (3 ν − 1)/ν. The chain-end density, like the blob density, ρe(z) ∝ 1/ξ(z)3z−β with β = 3. Because c(N)dN = ρe(z)dz, one readily finds c(N) ∝ N−τ with τ = 1 + 2ν ≈ 11/5.

Within the Alexander-de Gennes picture of a SSL brush, assuming that the polymer brush is described as a compact layer of concentration blobs ξ(z) ∝ ϕ(z)− ν/(3 ν − 1), the chains are described again as classical trajectories,

  • equation image(101)

that are strongly stretched at distances, larger than ξ. Hence, the number of chains per unit surface at z is given by

  • equation image(102)

with ε = 2ν/(3ν − 1), H denoting the upper edge of the pile of blobs, and NH = s(z = H) being the upper cutoff of the layer. If scaling holds, one may then use power law functions to express ϕ(z) ∝ z−α, ρe(z) ∝ z−β, c(N) ∝ N−τ, and z(s) ∝ smath image, as in the case of the weak stretching limit. Equations 101 and 102 then yield244

  • equation image(103)
  • equation image(104)
  • equation image(105)

where three relations, eqs 103105, relate the four exponents. One can, therefore, either impose the condition ν = ν, implying the existence of smooth crossover from the SSL limit to the afore mentioned case of weakly stretching, or assume that the three SSL equations, eqs 103105, correspond to a living polymer brush, grown by diffusion-limited aggregation (DLA), which yields β = 2, thereby fixing uniquely the set of exponents as τ = 7/4, ν = 3/4, and α = 2/3.245 Indeed, one can verify that these theoretical predictions agree well with the MC simulation results (Figure 20).244

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Figure 20. (upper left) A snapshot of a living polymer brush with mean length 〈N〉 ≈ 18, indicating the strong polydispersity of the grafted chains. Neither the ambient free monomers nor the initiators are shown. (upper right) MWD of a dense “living” brush for different overall concentration ∝ Lz−1 showing a power law c(N) ∝ N−τ with τ ≈ 7/4. (lower row) Density profiles of grafted monomers vs. z at different concentration (left) and the same for end-monomers (right). Monte Carlo (MC) data from Milchev et al.244

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It is also interesting to examine the effect of polydispersity on the effective mean height of a polymer brush, and the ensuing force exerted by the polymer brush on a plane upon compression. As shown by Milner et al.,30 the height of a polydisperse brush within the SSL-SCFT is

  • equation image(106)

where (N) are the number of chains per unit area of length less than N. Therefore, one can estimate that the effective height h(Δ) = h0 (1 + Δ/(2N)) increases as a square root of the variance Δ, where MW/MN = 1 + Δ2/(3N2). Accordingly, the repulsive force of the polydisperse brush increases - cf. Figure 21 (right panel)—and is always larger that that of a monodisperse polymer brush with chains of length equal to the mean length of the polydisperse brush N.

INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Normal Forces Between Two Polymer Brushes

The fact that two brushes (under good solvent conditions) repel each other, when they come into close contact, is considered as a basic mechanism for colloid stabilization.246–249 The colloidal particles have radii in the micrometer range, while the heights of the polymer brushes h used in this context are in the range between 10 and 100 nm. As a consequence, one can approximate the problem as an interaction between two (equivalent) planar brushes a distance D apart. This problem was first considered using the Alexander-de Gennes26, 27 model, where one assumes that the two brushes cannot interpenetrate at all when they come into contact. So the force is strictly zero for 2h* < D, while for 2h* > D the problem is equivalent to the compression of a single brush to a height h = D/2 which is smaller than the equilibrium height h* (corresponding to zero osmotic pressure at the outer end of the brush).

Using a Flory-type argument, one estimates a force-distance relation (the first term represents the osmotic part of the free energy and the second term the elastic part, see Ref.44)

  • equation image(107)

Note that for small relative compression Δ = 1 − D/2h*, the force scales linear in Δ. In contrast, the SCFT-SSL treatment also implies250 that there is negligible interpenetration between the two brushes, as is generally expected in the limit of brushes with infinitely long chains,251 but using h = D/2 in eq 51 yields (D) ∝ Δ2 for small D. However, because in reality chains are never infinitely long, there is some interpenetration between brushes taking place, and likewise, the force (D) is not strictly zero for D > 2h*, due to the “tail” of the monomer density profile ϕ(z) extending beyond z = h (see section Scaling Concepts for Brushes under Good Solvent Conditions). Thus, it is not surprising that experimental data252, 253 can be fitted both by eq 107, where the prefactor and h* are two adjustable parameters, and the corresponding expression, eq 51, from SCFT-SSL. As a consequence, eq 107 has remained the basis for experimentalists who measure force vs. distance curves in brushes (e.g., Refs.254–256) although one should keep in mind the limitation of this approach. More detailed SCFT calculations can also be found in Zhulina et al.257, 258

Computer simulations138, 140–145 have focused in particular on the study of the degree of interpretation between two polymer brushes pressed against each other, because on the one hand this quantity is not directly accessible to experiment, and on the other hand, it seems to be of central importance for the interpretation of the excellent lubrication properties when two brushes are sheared against one another (see e.g., the reviews3, 5, 9). There are several ways to measure the degree of interpenetration. Murat and Grest138 introduced a quantity I(D), defined as

  • equation image(108)

where ϕ1 is the monomer density of brush 1, and suggested that

  • equation image(109)

using scaling arguments of Witten et al.139 The MD results138 were roughly compatible with eq 109, and as h* ∝ σ1/3N, the scaling of I(D) as I(D) ∝ N−2/3 σ−4/9 shows that I(D) for strong compression (Δ of order unity) vanishes rather slowly. Although interpenetration hence is quite important, there is nevertheless no contradiction with the experimental finding that f(D) becomes very small when Δ becomes small, see Figure 21. As a caveat, we mention that for a quantitative account of the experimental data in Figure 21 also polydispersity should be accounted for.250

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Figure 21. (left) Interaction energy between two planar brushes, normalized per unit area, as a function of the normalized distance D/2h*, according to the MD simulation of a bead spring model.138 Cases include N = 100, σ = 0.03 (full dots), N = 50, σ = 0.03 (full squares); N = 50, σ = 0.1 (open triangles); N = 50, σ = 0.2 (open circles). The insert shows a comparison between the experimental results of Taunton et al.252 (data points) and the MD results for N = 100, σ = 0.03 (full line, vertically shifted upward by an arbitrary amount). The experimental points include results from measurements with polymers of various molecular weight at corresponding grafting densities. From Murat and Grest.138 (right) Force vs. separation curve for the Alexander - de Gennes ansatz26, 27(dotted line), for a monodisperse polymer brush according to SCFT30(dashed line), and for a polydisperse brush with MW/MN = 1.02 (solid line). Circles denote experimental data.252 Data from Ref.250

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A different measure of interpenetration was defined by Chakrabarti et al.143

  • equation image(110)

which is useful also in the case when the two brushes 1 and 2 are asymmetric in chain length and/or grafting density (I(D) was also used in SCFT work259). Again the conclusion did emerge that for Δ not too small, considerable interpenetration occurs. Nonetheless, the forces between the brushes are reasonably well compatible142 with the SCFT-SSL prediction, eq 51, which ignores interpenetration.

Still another measure of interpenetration is the interpenetration length L, defined by Witten et al.139 and used also in the study of sheared brushes.121–124 It is defined as the distance over which in eq 110 the product ϕ1(z2(z) is essentially different from zero. For very dense brushes (melt conditions), one has139 LN2/3a4/3D−1/3, while for the semidilute brushes under strong compression121 one needs to replace a by the size ξ∝aa3)ν/(1 − 3ν) of the “concentration blobs,” assuming that in two strongly compressed brushes the monomer concentration is constant, ϕ ∝ Nσ/D. In this rescaling of the expression139 for L one needs to replace N by the number n = N/g of blobs per chain, with g ∝ (ξ/a)1/ν the interpenetration length becomes (using ν ≈ 0.588162)

  • equation image(111)

In MD simulations of a bead-spring model, Spirin et al.122 obtained rough agreement with this prediction using rather short chains (30 ≤ N ≤ 120).

Recent experiments (e.g., Refs.260–262) measure forces between a planar polymer brush and a brush-coated colloidal probe attached to the arm of a cantilever. According to the Derjaguin approximation,263 the force between a flat surface and a curved surface of radius R can be related to the interaction energy per unit area U(D) of two flat surfaces via

  • equation image(112)

for DR. Using colloidal probes where R = 3μm262 and measuring forces in the range from a few nm to about 200 nm, the Derjaguin approximation was found to hold for these experiments. Using either eq 107 or eq 51, it is found that both theories give a reasonable account of the data, again pointing to the problem that for a stringent experimental test of the theory one should work with a well-characterized monodisperse brush, for which the density profile is determined independently (and then the brush height h* is no longer an adjustable fit parameter).

Interactions between Two Spherical Brushes

Here, we are not concerned with the interactions between brush-coated colloidal particles with colloid diameters in the μm range and hence much longer than typical brush heights, but rather with brush-coated nanoparticles, so the particle diameter is comparable to the brush height. Thus, already for the single-spherical brush, one has a nontrivial crossover from the star polymer limit to that of a planar brush, see section Essentials of the Self-consistent Field Approach to Polymer Brushes. Compared to the problem of two planar brushes that are compressed against each other at distance D, the problem of spherical brushes at distance D has some new features, as chains can avoid interpenetration of the two brushes by “escaping” in the direction perpendicular to the “axis” connecting the centers of the two spheres151 (Fig. 22).

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Figure 22. Two-dimensional density distribution ρ(z,r) = const of two spherical polymer brushes under good solvent conditions, as obtained from MD simulation. Here, the z-axis is oriented such that it connects the centers of the spherical particles, on which f chains are grafted, and which have the radii Rc = 7 (a,d), 7.9 (b,e) and 8.35 (c,f) Lennard Jones diameters, corresponding to choices f = 42 (a,d), f = 92 (b,e), and f = 162 (c,f). A chain length N = 20 is chosen throughout, and two choices for the distance between the particle centers r have been used, r = 30 (left side) and r = 20 (right side). Here, r is a radial coordinate in the plane perpendicular to the z-axis. Colors indicate the density ϕ, as shown by the scale given at the bars. From Lo Verso et al.151

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The effective forces between two spherical polymer brushes have again been studied by a variety of methods, including SCFT,146 Monte Carlo,147 and MD151 simulation, as well as DFT methods.151 Witten and Pincus247 were the first to consider this problem, approximating the spherical brush as a star polymer. They assumed that when two particles with f arms each are brought together to a very close distance, the system can be approximated as a single particle but now with 2f arms. From this argument, they concluded that the interaction potential between the two brushes depends logarithmically on the separation r, VWP(r) ∝ f3/2 ln[Rc/(r − 2Rc)], and hence the force is

  • equation image(113)

However, this expression is useful at best for small distances. Cerda et al.147 proposed a Flory-type treatment for the large distance behavior of the force, to conclude that

  • equation image(114)

where 〈Re2〉 is the mean square end-to-end distance of a grafted chain. Cerda et al147 argue that their Monte Carlo results are compatible with eqs 113 and 114 in the respective regimes. Lo Verso et al.151 obtain the potential of mean force

  • equation image(115)

for two interacting brushes from extensive MD simulations (Fig. 23). These data show a divergence of w(r) for r [RIGHTWARDS ARROW] 2Rc, but the data imply that it is somewhat stronger than predicted by eq 113. The large distance behavior of w(r) can be fitted by a Flory-like expression, analog to eq 114, but the precise meaning of the adjusted parameters remains somewhat doubtful.151 Lo Verso et al.151 also compare their results to DFT predictions. It was found that DFT describes the observed trends qualitatively but not quantitatively, in view of the limitations found in the DFT description of isolated spherical brushes, this lack of quantitative agreement has been expected, however.

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Figure 23. Potential of mean force W(r) of two interacting spherical polymer brushes under good solvent conditions, for the same model as used for Figure 22, on a log-linear plot vs. the square of brush-brush separation r. Note that w(r) diverges at direct contact between the two spherical cores. Three choices of f (f = 42, 92, and 162) and several chain lengths are included, as indicated. From Lo Verso et al.151

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Very recently, the work of Lo Verso et al. has also been extended to variable solvent conditions, and complemented by SCFT calculations.164 It was found that for T < Tθ the potential of mean force exhibits a minimum, implying an attractive force between two spherical brushes. The same conclusion was also reached by Marla and Meredith149 for a similar model, but much less complete data. The SCFT results164 show that the minimum of w(r) gets shallower, and ultimately disappears as T reaches Tθ.

Although the work described in this section deals exclusively with spherical polymer brushes embedded in low ‘molecular’ weight solvent, there has also been a lot of interest in the interaction between spherical nanoparticles embedded in polymer melts.150, 152 However, for a discussion of this problem, it is useful to better understand the interaction between polymer brushes and free chains in a solution or melt. This is the problem to which we turn next.

Interaction of Polymer Brushes with Nanoparticles

It has been mentioned in the Introduction that polymer brushes are expected to play a most prominent role in the design and applications of smart, multiscale, interfacial materials. For instance, versatile adaptive surfaces are capable of responding to changes of temperature, solvent polarity, pH, and other stimuli, most frequently by reversible swelling.264 In many aspects, this is based or achieved by combination of a polymer brush with small particles of size in the nm range, comparable to the brush height.265–274 Small amounts of such additives like, e.g., metal nanocrystals273, 274 for molecular photovoltaic semiconductors, metal (Au, Ag) nanoparticles immobilized on end-functionalized pH-responsive polymer brush,265, 275 protein globules,276, 277 and fullerines278 play a key role in the behavior of such hybrid interfaces. As a rule, they are created by first chemical grafting of a specific polymer brush, followed by the attachment of pre/in-situ formed nanoparticles (e.g., by binding of preformed functionalized nanoparticles or in situ formation of nanoparticles at active centers in the brush). Therefore, from the point of view of both basic science as well as regarding potential applications, a particularly challenging and intensively studied problem in polymer science at present is the behavior of such hybrid systems composed of polymer brushes and nanoparticles.265–272, 279 A key question of principal importance124 concerns thereby the organization and structuring of nanoparticles in a polymer brush. In general, one needs to understand how are nanoparticles spatially distributed and what is the free energy cost for the polymer brush to accommodate them, for example, regarding the size of the nanoparticles.

The behavior of nanoparticles dispersed in a polymer brush has been considered125–127 within the framework of SCFT, using an early SSL version suggested by Semenov179 for a pure non-solvated brush. As in section Essentials of the Self-consistent Field Approach to Polymer Brushes, one starts with an expression for the free energy of the brush, similar to eq 41, yet taking the presence of nanoparticles of fraction ϕnano(r) into account. An important difference to the approach, presented in section Essentials of the Self-consistent Field Approach to Polymer Brushes, however, is the assumed noncompressibility of the polymer brush.126 One imposes the condition that monomers plus nanoparticles are space filling, that is, the monomer volume fraction ϕ(r) = 1 − ϕnano(r). The latter means that the sum of monomer density contributions from all chains passing through z whose ends z0 lie between z and h, ∫zh dz0 ρe(z0)/|∂z(z0,s)/∂ s| = ϕ/a3, which yields Semenov's expression for the chain end distribution, ρe = equation image:

  • equation image(116)

(One should note, however, that due to incompressibility the brush height h increases as h ∝ σN, in contrast to the generally established eqs 3 and 43). With incompressibility imposed, the distribution of nanoparticles ϕnano(r) in the polymer brush is determined from the following brush free energy (in terms of the chain configurations z(s,z0)125:

  • equation image(117)

which yields Semenov's result for the pure nonsolvated brush179 without nanoinclusions for ϕnano(r) ≡ 0.

Allowing for the symmetry of ϕnano(r) in x, y-direction, and using eq 116 for the end-monomer distribution, ρe, which is the appropriate expression for dry brushes that have ϕ(zh) = const., the 1d expression for F along z then reads:

  • equation image(118)

where the first term is the chain stretching energy and the second one constrains the chain grafting density to σ. The third term in eq 118 expresses the space filling condition. The pressure field Π(z) depends here only on z and the Lagrange multiplier η is the chemical potential of a chain. As shown by Semenov,180 the self-consistent pressure field, representing the effect of all the other chains on a given chain, has quadratic form in SSL,

  • equation image(119)

The height h in eq 119 is increased by the presence of nanoparticles in the brush relative to the inclusion-free value h0, and the volume conservation leads to

  • equation image(120)

where equation image is the overall nanoparticles fraction. For the minimization of the the total free energy, eq 118, each chain path must satisfy the condition

  • equation image(121)

where the second term denotes the work needed to insert a chain into the polymer brush against the pressure while the explicit value of η is obtained by considering a chain at z0 = 0 (i.e., with no stretching energy) at pressure Π0Na3. Using eq 121, and changing integration variable from s to z, one can obtain directly the total stretching energy of the polymer brush (i.e., the first term in eq 118) as

  • equation image(122)

For small inclusion density, equation image ≪ 1, the dependence on equation image in the first two terms in eq 122 vanishes to first order in equation image so that the change in Fstretch due to nanoparticles is simply

  • equation image(123)

indicating that inserting a nanoparticle with volume Vnano = b3 at height z generates energy penalty Π(z)Vnano. Therefore, Finc is the free energy of inclusion profile ϕnano(z) compared to the free energy, provided all nanoinclusions were above h. Because Π(z) decreases with z (note that Π(h) = 0), the nanoparticle experiences upward buoyancy trying to expel it from the polymer brush.

Because at moderate volume fractions, the nanoinclusion free energy Fnano is dominated by translational entropy, it should be taken into account too, and one gets

  • equation image(124)

in which interactions between nanoparticles of order ϕnano(z)2 and higher have been neglected, according to the equation image ≪ 1 condition, eq 123, while the last term enforces the mean nanoparticles density.

Minimizing Fnano, one then derives125, 126

  • equation image(125)

where the pressure Π(z) remains quadratic albeit substrate pressure and height have increased. Thus, the insertion of a nanoparticle into the polymer brush creates a repulsive pressure, which decays exponentially away from the nanoparticle location.280–282

If nanoparticles strongly attract to functional groups of tethered chains, the enthalpy of interaction can be strong enough to overcome the deformation-induced repulsion by the polymer brush and the nanoparticle are “dissolved” in the brush. One may, therefore, distinguish two types of nanoparticles: soluble or insoluble, depending on the crucial role of interactions with the surrounding medium. As argued by Kim and O'Shaughnessy,125, 126 small (soluble) particles disperse freely within the polymer brush while brush-“insoluble” nanoparticles tend to aggregate at the brush interface with the surrounding medium—Figure 24. Formation of such aggregates of insoluble nanoparticles (on varying the strength of nanoparticle—polymer brush interactions and the concentration of nanoparticle) in the distal brush region has been observed129 in MD simulations. At low concentration, the nanoparticle assemble into a cylindric-shaped nanodroplet which is vertically oriented, as predicted.125 At higher concentration such aggregates reshape into horizontally – oriented baguette-like clusters at the rim of the polymer brush.129

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Figure 24. (left) Distribution of nanoparticles depends strongly on their size: (a) nanoparticles smaller than b* completely mix. (b) nanoparticles in the size range b* < b < bmax partially mix and remain near the soft-free brush surface. The penetration depth is δ = equation image. (c) If mean nanoparticle density exceeds the saturation density ϕsat = equation image, excess nanoparticles are expelled from the polymer brush and separate. (d) nanoparticles larger than bmax do not penetrate the polymer brush. (middle) Predicted phase diagram describing polymer brush - nanoparticle mixtures: complete mixing b < b*, partial mixing b* < b < bmax, exclusion b > bmax. The typical blob size and surface blob size are ξblob and ξsurf, respectively. (right) insoluble nanoparticles aggregate at the polymer brush - air interface of baguette shape in y-direction while its cross section comprises two half-lenses above and below the polymer brush surface. After Kim and O'Shaughnessy.125, 126

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As Π(h) = 0 and Π ≈ 2Π0 (1 − z/h) for small depths, one obtains ϕnano(z) ∝ exp[−(hz)/δ] where δ ≡ h(b*/b)3 and b* ≈ (N/h)2/3 = σ−2/3. Therefore, above a threshold size b*, equilibrium particle penetration is limited to a depth δ ≪ h, and the polymer brush has a loading capacity ϕmax whereby both δ and ϕmax scale inversely with nanoparticle volume b3. In the language of blobs, soluble inclusions with size b smaller than the blob size ξ, do not significantly interfere with the unperturbed path of the grafted chains while larger inclusions induce strong lateral stretch. In the latter case, chain trajectories are forced to make lateral displacements larger than ξ so the ensuing path can only be determined, provided Π(r) is known.126

Beyond a second threshold of nanoinclusions' size b > bmax ≈ (N/σ)1/4, nanoparticles cannot penetrate the polymer brush. For bb*, the penetration depth, δ ≈ 1/b3, is much less than the height h until δ = b defines the upper threshold size b = bmax ≈ (Ng)1/4 which can infiltrate the brush. In this partial penetration regime, b* < b < bmax, the maximum nanoparticle density ϕmax = δ/h ≈ 1/b3.

Evidently, the brush excess free energy, Finc, due to inclusions, must be known to determine their organization in a polymer brush—nanoparticle composite. However, the computation of Finc as with all entropy-related quantities is not straightforward in a simulation. Using an efficient method for direct determination of Finc by means of gradual inflation (as a special kind of thermodynamic perturbation method), Milchev et al.283 used a Monte Carlo procedure to get the insertion free energy of a spherical nanoparticle of radius R placed at depth D in a dense polymer brush. It was shown that, indeed, FincR3 for deeply placed nanoparticles, whereas FincR2 for shallow particles at the distal region of the polymer brush. In Figure 25(a), we show the ensuing buoyant force f, the Potential of Mean Force W = ∫0Rf(r)dr (that is, Finc itself, and the monomer density profile ϕ against nanoparticle position z. It turns out that f depends essentially on the distance from the brush surface and even goes through a maximum at the inflection point of ϕ(z) so that deep inside the brush f nearly dwindles due to canceling of its symmetric lateral components.

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Figure 25. (a) Buoyant force f exerted on a nanoparticle by the polymer brush. Full lines denote a best fit with a second-order polynomial. The inset shows f(R = 3.0) for different distances z = D′ from the grafting plane. The polymer brush density profile ϕ(z) is also given (appropriately normalized for better visibility). Vertical bars denote the work W (reduced for visibility by a factor of 10), which is necessary to place this nanoinclusion at depth D′ between the sphere surface and the grafting plane. (b) Angular distribution of the buoyant force f exerted on a nanoparticle of radius R at distance D′ = 14.0, close to the polymer brush surface. The brush distortion by the smaller particle, R = 1.0, does not reach the brush rim, hence, the force is laterally symmetric with no component normal to the grafting plane. The distortion field around the bigger nanoparticle, R = 3.0, reaches the free space above the brush end and the resultant force exerted by the brush acts upwards. From Milchev et al.283

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These findings deviate somewhat from the SCFT predictions125, 126 yet have been confirmed by Halperin and coworkers284 who also found no evidence for the theoretically predicted FincR4/3 De Gennes-behavior. One should note, in addition, that at equal volume ΔFinc depends essentially also on the specific geometric shape of the nanoinclusion125, 126 in close analogy between grafted chain paths next to inclusion and hydrodynamic streamlines around an obstacle. This analogy was used by Kim and O'Shaughnessy to demonstrate that, for example, the energy cost for a diskoidal inclusion essentially depends on its orientation with respect to the polymer chains in the brush, or a rod-like inclusion, perpendicular to the grafting plane, is significantly less than when the rod is placed parallel.

Another interesting and little explored aspect of nanoinclusion behavior in a polymer brush are the effective brush-mediated interactions between neighboring nanoparticles. Even noninteracting spherical nanoparticles at constant distance between each other may experience effective repulsion or attraction, depending on their mutual (dipole) position in the polymer brush.126, 286 Generally, the complex organization of nanoparticle in a polymer brush has been studied by a variety of computational methods, most notably MD,128, 129 Monte Carlo,285 and DFT or SCFT.131, 287, 288 By varying the length of the chains N, the grafting density σ, and the size of the nanoparticles, b, Yaneva et al.128 determined the equilibrium particle penetration depth δ and the average concentration of nanoparticles equation image in a polymer brush with explicit solvent, using a constant pressure (NPT) ensemble and Dissipative Particle Dynamics (DPD)—thermostat. They found that for nonadsorbing spherical nanoparticles of size bb* the thickness of the infiltration layer δ ∝ h(b/b*)3 and b* ∝ σ−2/3, in agreement with the predictions of Kim and O'. Shaughnessy125—Figure 26.

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Figure 26. (a)-left: Interpenetrating density profiles of monomers,128 ϕp(z), and of nanoparticles, ϕnano(z), for N = 60, σ = 0.185, and varying nanoparticle size b given as parameter. The nanoparticle density has been magnified by a factor of 100 for better visibility. (a)-right: Density profiles ln ϕnano(z) ∝ (zh)/δ of nanoparticles of size b. In the inset, one observes δ ∝ 1/b2 rather than the predicted 1/b3 behavior, probably due to the small depth of nanoparticle infiltration. (b) The mean concentration of inclusions equation image(b) against reduced size b/b* where equation image(b) = ∫ϕp(znano(z)dz/∫ ϕp(z)dz, and the master curve Γ ∝ (b*/b)3 (full line). From Yaneva et al.128

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It was found that the mean density of nanoparticles, equation image(b), scales as predicted, equation image(b) ∝ (b/b*)3 within the whole range of parameter variation. For too large nanoparticles, b > bmax, no infiltration into the polymer brush was observed. It was also found that the anisotropy of the polymer brush affects the nanoinclusion dynamics so that the mobility perpendicular to the grafting plane is about 20% higher than the lateral mobility. The respective diffusion coefficients, Dnano and Dnano were observed to vary with nanoparticle size b in agreement with theoretical expectations.289 A nice recent example for experimental work on nanoparticles in a brush can be found in.290

An especially important aspect of the incorporation of nanoparticle into a polymer brush pertains to the adsorption of oligomers and linear macromolecules into a homopolymer-,287, 291 or mixed brush.266, 292 One is strongly interested, for example, in reducing undesirable protein adsorption on poly(ethylene glycol) (PEG) brushes in order to prevent clotting in blood-contacting devices, fouling of contact lenses, or diminished circulation of therapeutic proteins and drug-bearing lyposomes.293–297

The problem has been investigated theoretically by means of SCFT already two decades ago,257, 258, 298 and also experimentally,299–301 concerning surfactant adsorption on a polymer brush, or the grafting of free chains into a preexisting polymer brush. Theoretical insight has been provided by the SCF treatment of Zhulina and collaborators.257, 258 The change in free energy ΔF, due to the presence of free chains in it, consists again of two contributions due to elastic stretching of the grafted chains and volume interactions with the free chains inside the polymer brush: ΔF = ΔFstretch + Finc. As before, ΔFstretch is given by the first term in eq 118, namely ΔFstretch = equation image0h ρe(z0)dz00math image dzE(z,z0). The normalization condition ∫0h dzE(z,z0) = N with E - the local stretching, cf. eq (37). The second contribution (given that grafted chains do not contribute to translational entropy) reads

  • equation image(126)

where L denotes the length of (oligomer) chains (LN) in the polymer brush per grafted chain, ϕo is the (free) oligomer density, and ϕ(z) = ∫zh g(z)|E(z,z)|−1 dz. Here χmath image are the Flory-Huggins interaction parameters. Making use of the fact that the local chain stretching E(z,z) does not depend on the particular interactions in the polymer brush, and is uniquely determined by the Gaussian type of local stretching, E(z,z0) = equation image, then in the simplest case of chemically identical grafted and free chains, χmath image = 0, the enthalpy term in eq (126) is dropped while for z > h one has ϕ = 0 so that in the bulk the free energy density is simply F/a3 = (1 − ϕob) ln (1 − ϕob) + L−1ϕob ln ϕob. The total free energy of the infiltrated polymer brush is then described by the equations258:

  • equation image(127)
  • equation image(128)

where z is the distance from the surface and K2 = 3π2/8a2N. The height of the polymer brush is then determined from ∫0hϕ(z) dz = σNa3 as

  • equation image(129)

where D(x) = exp(−x2)∫0x exp(t2)dt is the Dawson integral. Eq. (128) and (129) then yield the volume fraction profiles ϕ(z), ϕo(z), and the brush thickness h as functions of N, L, and σ for given ϕob in the SSL as long as the oligomer length LN. To extend predictions to low grafting densities and arbitrary lengths of the free polymer chains, one has to use scaling considerations, or resort to the lattice SCF model.39

For different compatibility χmath image ≠ 0 between a polymer brush and free chains in the solution, a comprehensive investigation by means of MC and SCFT291 has revealed that the dissolved species get progressively more and more ejected out of the polymer brush with growing length L at χ = 0, whereas the compatible ones (for χ < 0) undergo a sharp crossover from weak to strong adsorption (the adsorbed amount Γ(L) ≈ 100%), discriminating between oligomers (1 ≤ L ≤ 8) and longer chains Figure 27.

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Figure 27. Density profiles of the polymer brush, ϕp(z), (shaded area) and of free chains, ϕo(z), of length L, (given as parameter) at two grafting densities: σ = 0.25: (a, b), and σ = 1.00 (c, d). (a) and (c) illustrate good compatibility between brush and free chains, ϵpo = 2.0, while (b, d) demonstrates a case of bad compatibility, ϵpo = 0.04. Thin solid lines in (a) and (b) denote results from the DFT calculation, thick lines - MC data.291 The densities in (a) are normalized so as to reproduce the correct ratio of brush to free chains concentrations ϕp and ϕo (the absolute particle concentration ci is indicated in the alternative y-axis. For the sake of better visibility, in (b) and (c) the density of all species is normalized to unit area.

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This entropic effect agrees with scaling theory,27 SCFT,133 and MD simulations156 results. In Figure 27c one can even observe the onset of the so called wetting autophobicity effect302 when the penetration of free chains into a sufficiently dense polymer brush vanishes.

Although the conformations of adsorbed free chains in a polymer brush retain largely their shape of Self Avoiding Walks,291 an interesting aspect concerns the structure of free chains in concave brush-coated nanocylinders.83, 85 Nanotubes, coated inside by a polymer brush are rather interesting from technological point of view because they can be used as smart valves or size-selective filter membranes.267 Moreover, it has been shown recently that the density profile of a diblock AB-copolymer brush exhibits a typical “cart wheel” configuration whereby the A-blocks (placed in the distal region of the brush) collapse upon deterioration of the solvent quality with respect to A-segments while the solvent remains “good” regarding the B-segments that are grafted to the tube wall.83 An example for the complex behavior of adsorbed free chains was recently demonstrated by a combination of MC and SCF computations whereby a free chain confined in such a brush-coated pore non-monotonously changes its size when the tube radius R is systematically varied83, 85—cf. Figure 28. This somewhat unexpected result reflects a delicate interplay between a possible interpenetration of the free macromolecule and the brush chains and an axial stretching as the tube radius gets narrow and can be interpreted in terms of confinement-induced blobs of the free chain, ξ(R), and grafting-density blobs ξg(σ) in the brush coating.85

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Figure 28. (a) SCFT results for the end-to-end distance 〈Rez〉 of a free chain of chain length L along the cylinder axis plotted vs. the cylinder radius R, for several grafting densities (as indicated) and N = 16, L = 100, and L = 200. A single intersection point is observed at Ris ≈ 7.74R. (b) Change in polymer conformations on decreasing nanopore radius R): In the polymer brush coating, the blob size ξg is determined by the grafting density of the brush while the size of the free chain blob, ξ, depends on the clearance width above the brush surface along the pore axis. When ξ ≈ ξg, the free macromolecule penetrates into the polymer brush. From Ref.85

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A very interesting other system where cylindrical symmetry matters are systems of nanorods with grafted chains embedded in melts. This problem has very recently been considered by both DFT, MC, and experiment.303

The present brief overview pertaining to the properties and behavior of nanoparticles in polymer brushes does not permit to go deeper into related important yet scarcely explored problems, as, for example, the kinetics of free chain adsorption or penetration into a polymer brush, the impact of flow on nanoparticles distribution in a polymer brush, or the transport of nanoparticle (e.g., nutrients, etc.) in brush-coated nanocapillaries. Here, we should only mention that recent MD studies130 indicated the existence of strong and efficient screening by the polymer brush of the nanoinclusions even at strong shear flow. On the other hand, theoretically predicted transport properties of tracer particles, for example, time-dependent concentration profiles are found to be in good agreement with those observed in the computer experiment.81

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

In this review, static properties of coarse-grained molecular models for polymer brushes and their investigation by Monte Carlo and MD computer simulation methods were discussed. The emphasis of the work that was covered was on general theoretical concepts and their validity; it is one particular strength of computer simulations that information is available in molecular detail and any desired spatial resolution. As is well known, polymer brushes have an inhomogeneous structure in the direction perpendicular to the substrate on which the chains are grafted, but in some cases (homopolymer brushes in poor solvent conditions, or binary polymer brushes, or grafted block copolymers, etc.) also nontrivial inhomogeneous structures form in directions parallel to the substrate surface. The information that one can extract from experimental studies of such systems is much more limited, of course, and hence computer simulations of polymer brushes provide an information that is not directly available from experiments, and hence complements them and helps their interpretation.

Of course, one particular strength of computer simulation that becomes increasingly important recently is that one can study chemically realistic models where polymer chains are considered in full atomistic detail. Such models have the particular merit that quantitative comparisons with corresponding experimental data become possible, without the need to rescale any parameters. However, such work is not reviewed in this article: first of all, it is still somewhat scarce, and often there are uncertainties about the precise properties of the substrates and the polymer–substrate interactions; such simulations clearly are more demanding than chemically realistic simulations of bulk polymer solutions and melts. We do not cover any work on chemically realistic models of polymer brushes, not because we consider it as not important, but because our lack of expertise on such questions, and the different scope of this article.

A subject of great recent interest that also is not covered in our review are the so-called “hairy polymer” nanoobjects304 where two types of chains are grafted to spheres, cylinders, or both sides of thin plates or membranes, which can form “Janus particles,” “sandwich structures,” or other special shapes.

We also do not discuss the dynamical aspects of polymer brushes, despite the fact that from an application perspective (e.g., lubrication properties of polymer brushes exposed to shear deformations and flow) this is a very important aspect. We note, however, that we have presented a brief review of such phenomena recently elsewhere. Furthermore, we do not cover another very important aspect of polymer brushes, namely their use to control the wettability of surfaces. As is well known, when a surface is exposed to a fluid that may exhibit coexistence between two phases (e.g., vapor and liquid for one-component fluids, or A-rich and B-rich phases for binary AB-mixtures) it is of great interest to know whether a droplet of the minority phase in equilibrium would meet the surface under a nonzero contact angle, or rather spread out. Coating a surface with a polymer brush is a powerful means to control wetting properties, in particular for complex fluids, but this also is a subject that is not covered.

Rather, this review focuses on elementary theoretical concepts, such as the Alexander de Gennes blob picture of polymer brushes and its extensions from flat to curved substrate surfaces, and the SCFT in each various versions: field-theoretic classical formulation and its SSL, as well as the Scheutjens-Fleer lattice formulations. Some complementary theories, such as SCMF and DFT are mentioned only very briefly, and others not at all. This focus simply is due to the fact that an extraordinarily large number of original papers is based on the use of the Alexander de Gennes model or the SCFT theory (thus we could not attempt an exhaustive discussion of all papers that can be found in the literature), and the aim of our review in fact is to provide the reader with a guide to this large body of work, and help him in making a judgment on the reliability of these studies. Thus, we have emphasized in our presentation the underlying model assumptions of all these approaches, and the limitations of their validity resulting from these assumptions. These limitations sometimes are not strongly emphasized in the original papers, while our review focuses on the lessons learned from the comparison of simulation results to theoretical predictions; as well as to experiments whenever appropriate.

We also have not covered the use of polymer brushes in the context of responsive polymer interfaces, but rather refer to Cohen-Stuart et al.305 for a recent authorative review.

Of course, one must remember that although computer simulations in principle yield the exact statistical thermodynamics of the considered model system, they do have their own problems (the need for sufficient equilibrium prevents the use of very long chains, often an exhaustive coverage of the desirable parameter range in chain length, grafting density, chi-parameter and so forth. is not possible; the study of lateral mesophase structures is hampered by finite size effects; etc.). However, we feel that this review shows that a wealth of useful insight nevertheless has been gained.

This review pays little attention to a problem that is very relevant experimentally, namely the polydispersity of chains. Apart from a short section on “living polymer brushes,” strictly monodisperse brushes have been assumed throughout. Also interesting effects when the brushes are almost monodisperse, containing a small fraction of significantly shorter chains (which then are collapsed), or longer chains (which then take a “stem plus flower”-configuration) are not dealt with. Also the section of interactions between polymer brushes and interactions between brushes and free chains and nanoparticles and so forth is intended to “wet the appetite” of the reader, rather than presenting an exhaustive coverage. We note also, that a lot of work on this subject is still ongoing. Thus, we hope that this review will be useful for many colleagues in the field, and stimulate further work to close the gaps that still exist in our theoretical understanding of polymer brushes.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

Partial support by the Deutsche Forschungsgemeinschaft (grants No. BI314/23, SFB 625/A3) is acknowledged. We wish to thank many colleagues for their fruitful collaboration on the original work on which this article is based, in particular D.I. Dimitrov, J. Yaneva, S.A. Egorov, H.-P. Hsu, J.-F. Joanny, A. Johner, J. Klein, L. Klushin, T. Kreer, P.-Y. Lai, F. LoVerso, M. Müller, C. Pastorino, W. Paul, D. Reith, P. Virnau, R. Wang, and J. Wittmer.

REFERENCES AND NOTES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information

APPENDIX: COMPUTATIONAL METHODS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information
Appendix A: Molecular Dynamics

One of the most broadly used models for simulation of polymer brushes is that of MD. As a rule, one uses a coarse-grained bead-spring model where the total potential acting on a monomer can be described as a sum

  • equation image(A1)

The first term ULJ stands for the pair interactions, a Lennard-Jones potential with a cut-off at its minimum and a shift such that only the repulsive branch acts

  • equation image(A2)

where rc = 21/6σ. Here ε characterizes the strength and σ − the diameter of a chain segment.

In addition, cf. eq A1, monomers bonded to each other as nearest neighbors along a polymer chain interact with a finitely extensible nonlinear elastic (FENE) potential306

  • equation image(A3)

The choice of these parameters ensures that the minimum of the total potential between two bonded monomers along the chain occurs for307, 308 r ≈ 0.97 σ, distinct from rc ≈ 1.12 σ. This misfit between the two distances ensures that there is no tendency of monomers to form a simple crystal structure even if the density is very high and/or the temperature is rather low. Because in most simulations of polymer brushes the solvent is treated as a continuum with no explicit solvent particles, this choice corresponds to the so called “good” solvent conditions (simulation data in Figures 14 and 21 were derived within the framework of this model, for instance).

Whenever the properties of polymer brushes are examined at poor solvent conditions, one deals usually with the full Lennard-Jones potential with an attractive branch, rather than with eq ( A1) - see, for example, Figure 16 for a simulation where this model was used. Thereby the temperature T must be kept below the corresponding Θ-temperature, which for this model, generally referred to as that of and Kremer Grest,306 has been established as302 kBT/ϵ ≈ 3.3.

Alternatively, the nonbonded interactions between monomers (including, in the case of explicit solvent, monomer–solvent interactions) may be modeled by truncating the Lennard-Jones potential in its minimum, shifting it to some desired depth (ϵpp, ϵps), and then continuing from its minimum to zero with a cosine potential having a cosine form.59 This has the advantage that one deals with a potential that is both continuous and has a continuous derivative at the cut-off too. Thus, our potential is

  • equation image(A4)

where {αβ} stands for the different types of pairs, polymer – polymer, polymer – solvent, pp, ps, and solvent – solvent ss, respectively. Scales for length and energy (and temperature) are chosen such that both σpp = 1 and ϵLJ = 1 (the Boltzmann's constant kB = 1). For r ≥ σαβ21/6, we choose the cosine potential as follows

  • equation image(A5)

Here, a and b are determined as the solution of the two equations,

  • equation image(A6)

which yield a = 3.1730728678 and b = −0.85622864544. As desired, both ULJcos(r) and dULJcos (r)/dr are continuous at the potential minimum rαβ = 21/6 and at the cut-off rαβ = 1.5.

The last term in eq A1 denotes the potential of the grafting wall located at z = 0, which is frequently taken as a structureless repulsive potential,53

  • equation image(A7)

acting in z-direction. Frequently, however, the grafting wall is represented by atoms forming a densely-packed triangular lattice. The potential with which effective monomers and wall atoms interact is chosen to be the same as the monomer-monomer potential, eq A2. Also, the wall atoms are fixed in the rigid positions of an ideal crystal lattice. This guarantees that no monomer can cross the wall.

One should note that in those cases when the properties of semiflexible polymer brushes are of interest, one usually adds to the pair potentials in eq A1 an additional bending potential Ubend(ri − 1,ri,ri + 1) = ϵbend (1 − cos θ) where θ describes the polar angle between successive bonds.163

The equation of motion of each particle is given by the Langevin equation,

  • equation image(A8)

where m is the monomer mass, ri is the position of the i-th monomer, γ is the friction coefficient. and Ri is a white noise random force with the correlation function 〈RiαRjβ〉 = 2kBTγδijδαβ with α, β denoting spatial components of the force. Most frequently in these standard MD methods, one uses the velocity Verlet algorithm309 for advancing the trajectories of the beads in time. Assigning for the monomer mass m, the value m = 1, the characteristic time τ = equation image becomes unity. A single - integration step is usually chosen in the interval [10−3τ, 10−2τ] and one needs as a rule about 106 such time steps for equilibration, bearing in mind that the typical Rouse relaxation time for a linear polymer chain under good solvent conditions scales as N2ν + 1N2.2 with chain length N. Usually, runs of about 107 ÷ 108 time steps are performed to collect the necessary statistics.

Appendix B: Monte Carlo Method

During the last two decades, the bond fluctuation method (BFM)205, 206 has proved to be a very efficient computational method for Monte Carlo simulations of large numbers of linear polymers. This is a coarse-grained model of polymer chains, in which an ”effective monomer” consists of an elementary cube which occupies 2 × 2 × 2 sites on a hypothetical cubic lattice as shown in Figure 29 (left panel) so that these eight sites are blocked for further occupation. A polymer chain is made of “effective monomers” joined by bonds. A bond corresponds to the end-to-end distance of a group of 3 ÷ 5 successive chemical bonds and can fluctuate in some range. It is represented by vectors equation image of the set P(2,0,0), P(2,1,0), P(2,1,1), P(3,0,0), and P(3,1,0), which guarantee that intersections of the polymer chain with other chains, or with itself, are virtually impossible. All lengths are measured in units of the lattice spacing and the symbol P(a,b,c) stands for all permutations and sign combinations (e.g., 108 bonds in 3d) of the Cartesian coordinates equation image = ±a, ±b, ±c. Monomer – monomer interactions may extend over a certain range in the lattice and the semiflexibility of the chains may be accounted for by ascribing some additional energy to certain bond angles. The algorithm displays Rouse behavior for all spatial dimensions and combines typical advantages of the lattice MC methods with those of the continuous Brownian dynamics algorithm. Thus, one may increase the density of the polymer brush up to that of a melt and still keep the system mobile.

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Figure 29. (left) Schematic plot of the BFM with two chains of length N = 7, placed in a single plane for better visibility. (right) (a) Scheme for scission and recombination of bonds in ”living” polymer brushes. Each monomer has two (saturated or unsaturated) bonds. Chains consist of coupled bonds, each pointing to a counterpart: jbond = pointer(ibond) [LEFT RIGHT DOUBLE ARROW] bond = pointer(jbond). The pointers of unsaturated bonds point to 0. (b) Recombination of two initially unsaturated bonds, ibond = 2 and jbond = 5 connect monomers imon = 2 and jmon = 5. (c) Scission of a saturated bond implies resetting of the pointers of the coupled bonds ibond and jbond = pointer(ibond) = 0.310

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For a number of problems, however, as, for example, the study of polymer brushes grafted on cylindrical or spherical substrates, and so forth, lattice MC models turn inappropriate and one resorts rather to off-lattice model.11, 85, 163, 311 The non-bonded interactions between monomers are then usually expressed by means of the Lennard-Jones potential, eq (A2), whereas the spring potential between two neighboring monomers is taken as a FENE interaction, eq (A3). The interaction between the grafted wall and the effective monomer is taken either as a long-range-, eq (A7), or as a (modified) short-range attraction,85 eq (A2). A very efficient MC model,11, 163, 311 that combines the advantages of an off-lattice MC code with the productivity of a lattice-based algorithm was developed as a hyper-fine 216 × 216 × 216 grid whereby discretization effects are virtually absent. The high performance of the code is based on expressing the monomer coordinates as long integers that can be handled by fast bit-wise operations. Thus, (i) the most heavily involved modulo operations which provide periodicity of coordinates are accomplished by single use of the &-operator (bitwise AND), and (ii) the minimum image condition for taking distances between interacting particles is simply reduced to type conversion (or, casting) of all variables, standing for the coordinates, from long into short integers, which is probably one of the fastest computational operations.312

In this MC model, the polymer brush consists of linear chains of length N grafted at one end to a flat structureless surface. The effective bonded interaction is described by the FENE potential, eq A3, where R0 = lmaxl0, r = ll0, and the elastic constant K = 20. Here lmax = 1, l0 = 0.7, lmin = 0.4 between nearest neighbor monomers is l0 = 0.7. The maximal extension of the bonds, lmax = 1, is used as a unit length while the potential strength ϵ is measured in units of thermal energy kBT.

The nonbonded interactions are described by the Morse potential,

  • equation image(A9)

with α = 24, rmin = 0.8, ϵM/kBT = 1.

The polymer chains are tethered to grafting sites which constitute a square, checkered, or triangular periodic lattice on the substrate whereby the closest distance between grafting sites is l0. Thus, the largest grafting density σ = 1.0, if the polymer chains are anchored at distance l0, and σ = 0.25, if the “lattice constant,” that is the distance between adjacent head monomers on the surface is equal to 2l0. Note that Σ = 1.0 corresponds to a simulation where the monomer density in the brush near the wall is close to the density of a polymer melt, whereas σ = 0.25 would correspond to a rather concentrated polymer solution.

For the chain model, ϵM/kBT = 1 corresponds to good solvent conditions since the Theta-temperature for a (dilute) solution of polymers described by the model, eqs A3–A9 has been estimated311 as kBΘ/ϵM = 0.62. As usual, solvent molecules are not explicitly included311 but work which includes solvent explicitly54 would yield very similar results. The data shown in Figure 8 were obtained with this model.

Nanoinclusion, if considered, are taken as spherical (nanocolloid) particles of radius R at distance D away from the grafting surface.283 We consider both attractive and repulsive interactions of the nanocolloid with the monomers of the brush and the colloid potential is again modeled as a Morse potential, eq A9, “smeared” at the surface of the spherical particle, with a much shorter range, rminLPcoll = 0.1.

Eventually, we end this section with the more complex case of a living polymer brush, which has also been modeled by this MC algorithm.310 One of the main challenges in a living polymer brush is the constant scission and recombination of bonds in the state of dynamic equilibrium whereby both the length of individual tethered chains in the ensuing polydisperse brush as well as the identity of monomers (end- or middle-monomers, free building units, etc.) change instantaneously after each MC step. The necessary bookkeeping in such algorithm is rather involved and would normally require huge resources of operational memory. Therefore, in Figure 29 (right panel), we sketch one efficient scheme of allowing for the unceasing breakage and creation of bonds (chains) based on attaching two (saturated or not) bonds to each building unit and the redirecting of coupled bonds by means of pointers.312

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information
Thumbnail image of

Andrey Milchev

Andrey Milchev studied physics at the University in Sankt-Peterburg, Russia, where he got his Masters Degree in theoretical physics in 1970. In 1977 he graduated in solid state physics at the University of Leipzig, East Germany. Since 1979 he is in the Institute of Physical Chemistry at the Bulgarian Academy of Sciences in Sofia where now he is a full professor and head of the Group of Computer Modeling in the Department of Amorphous Materials. From 1984 to 1986 Andrey Milchev was an Alexander von Humboldt fellow at theUniversity of Mainz, Germany, in the group of Prof. Kurt Binder, with whom a long-standing research partnership and collaboration since then exist. Andrey Milchev's interests cover computer modeling of soft condensed matter (polymers, micelles, membranes), microfluidics, diffusion and phase transitions.

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. A BRIEF REVIEW OF THEORETICAL CONCEPTS ON POLYMER BRUSHES IN THERMAL EQUILIBRIUM
  5. “LIVING POLYMER” POLYDISPERSE BRUSHES
  6. INTERACTIONS OF A POLYMER BRUSH WITH ANOTHER BRUSH, WITH FREE CHAINS, OR WITH NANOPARTICLES
  7. CONCLUSIONS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. APPENDIX: COMPUTATIONAL METHODS
  11. Biographical Information
  12. Biographical Information
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Kurt Binder

Kurt Binder studied Technical Physics at the Technical University Vienna, Austria, where he got his PhD in “Technical Sciences” in 1969. He was an IBM postdoctoral fellow at IBM Zürich Research Laboratory/Switzerland in 1972 - 1973. In the years 1977 - 1983 he was a full Professor at the University of Cologne, and Director of the Institute of Theory II, Inst. Solid State Research, Research Center Jülich/Germany. Since October 1983 he is a full Professor for Theoretical Physics at the Johannes Gutenberg-University in Mainz. Kurt Binder is member of numerous Scientific Councils and Academies of Sciences. Among the many awards one should mention Berni J. Alder CECAM prize (European Physical Society, 2001), Staudinger Durrer Medal (ETH Zürich, 2003) and the Boltzmann medal, 2007.