Polymer brushes on flat and curved surfaces: How computer ...

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Polymer Brushes on Flat and Curved Surfaces: How Computer Simulations Can Help to Test Theories and to Interpret Experiments K. Binder,1 A. Milchev1,2 1

Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany

2

Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria

Correspondence to: A. Milchev (E-mail: [email protected]) Received 20 June 2012; accepted 6 August 2012; published online 26 September 2012 DOI: 10.1002/polb.23168

ABSTRACT: 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 50: 1515–1555, 2012

INTRODUCTION Polymer brushes are obtained by covalent

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 interest 67–76 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).

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

KEYWORDS: coatings; computer modeling; films; graft polymer;

interfaces; nanocomposites; statiscal thermodynamics; structure

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, 98–124 ). 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

© 2012 Wiley Periodicals, Inc.

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JOURNAL OF POLYMER SCIENCE PART B: POLYMER PHYSICS 2012, 50, 1515–1555

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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 the University 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.

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.

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 sidechains 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

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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

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)

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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 = 0.125,163 (upper-right panel) Polymer brush with chain lengt Nch = 100 at = 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 −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 (z) with increasing distance z from the grafting density. The blob diameter (z) increases then according to (z) ∝ [(z)]−/(d −1) ≈ [(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 = (h − df ), where h is the height of the brush, also is indicated. After Wittmer et al.46

of size a, such that under melt conditions the end-to-end distance would simply be Re = aN 1/2 , whereas under good solvent conditions in the dilute limit both Re and the gyration radius Rg scale as

between grafting sites is a second characteristic length that needs to be compared to Rgx . In fact, for > ∗ , defined by

Rg ∝ Re = aN .

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 −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 −1/2 = ag , in analogy to eq 1, therefore g = (a2 )−1/2 . Of course, this result makes only sense if g 1, that is, the brush must be “semidilute”.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,

(1)

Note that prefactors of order unity in this section are ignored throughout. The “Flory exponent”160 is = 3/4 in d = 2 dimensions and ≈ 0.588162 in d = 3 dimensions (in fact, simple “Flory-type arguments”160 would imply = 3/5, but this is not quite accurate). As is well known, > 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” 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 Rgx ∝ Rgz ∝ Rg(bulk) ∝ N . 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 −1/2 WWW.MATERIALSVIEWS.COM

∗ = (aN )−2 ,

h = −1/2 n = a(a2 ) 2 − 2 N ≈ a(a2 )1/3 N, 1

1

(2)

(3)

where in the very last step the Flory approximation160, 165 = 3/5 was used. From this “Alexander picture”26 of a polymer brush, one can immediately conclude that Re ∝ Rgz ∝ h. However, the consideration of Rgx is more subtle. Although naively one might


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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 (a1 )−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 (r ) in x -direction, proportional to r /f in y-direction, and f 1−1 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

conclude that Rgx = −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 −1/2 excluded volume interactions are screened,160 these lateral excursions should add up in a random walk-like fashion, and hence √ √ √ Rgx = Rgy ∝ −1/2 n = a(a2 )1/4−1/2 N ≈ a(a2 )−1/12 N (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 (h − z), while the density of chain ends e (z) is a delta function, (z) ∝ (a2 ) 2 − 2 (h − z) , e (z) ∝ (z − h). 3

1

(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 = 4 R2 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 4 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

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volume by blobs that touch each other, we clearly must have for the blob radius (r) (there are f blobs along the circle of radius r in d = 2, which has the circumference 2 r, whereas in d = 3 the f blobs share a surface area 4 r 2 for a sphere of radius r) (r) ∝ r/f (d = 2) ,

(r) ∝ r/f 1/2 (d = 3).

(6)

Because each blob contains g(r) ∝ [(r)]1/ monomers, we conclude that the density profile decays according to a power law for r < h, the brush height: (r) = g(r)/[(r)]d ∝ [(r)]1/−d ∝ (r/f )1/−2 (d = 2) which means (r) ∝ (r/f )−2/3 , while (r) ∝ (r/ f )1/−3 ≈ (r/ f )−4/3 (d = 3)

(7)

(8)

(Here in the last expression on the right hand side was put equal to the Flory value = 3/5.) 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 2 N= f

h (r)rdr

(d = 2),

4 N= f

0

h (r)r 2 dr

(d = 3)

0

(9) Using eqs 7 and 8, we readily obtain h ∝ f 1/4 N 3/4 ,

(d = 2)

(10)

and h ∝ f (1−)/2 N ≈ f 1/5 N 3/5 (d = 3)

(11)

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Analogous results follow then for the end-to-end distance Re and the squared gyration radius of the chains Re ∝ f 1/4 N 3/4 , Rg2 ∝ f 1/2 N 3/2

(d = 2)

(12)

Re ∝ f (1−)/2 N , Rg2 ∝ f 1− N 2

(d = 3)

(13)

and

Of course, for a spherical brush where chains are grafted on a sphere of radius Rc , the above relations, eq 10–13 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 Rc Re , the curvature of the substrate should have no effect). One can make a scaling assumption to describe this crossover as follows (in d = 3 F( ) is some crossover scaling function that we shall discuss below) h/h0 = F(h0 /Rc )

(14)

where h0 is the brush height at a planar surface, as given by eq 3. Noting that 4 Rc2 = f when f chains are grafted at a sphere of radius Rc , we see that eq 11 can be rewritten then as h ∝ (1−)/2 Rc1− N . Consequently, eq 14 becomes h/a ∝ (a )

2 1/(2)−1/2

2 1/(2)− 12

NF{a(a )

N/Rc },

(15)

therefore, for → ∞,

blobs, of cross-sectional area [(r)]2 each, provided they are spherical. Hence (omitting factors of order unity), p2 = Lr and, therefore, (r) ∝ (r/fa1 )1/2 . The blob volume then is of the order of V (r) = 3 (r) = (r/f 1 )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 g(r) ∝ [(r)/a]1/ ∝ a−1/ [r/(f 1 )]1/2 ,

(17)

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

g(r) ∝ a−1/ 3 (r)

r f 1

1−3 2 (18)

Using = 0.588162 this yields (r) ∝ r −0.65 while the Flory value = 3/5 yields (r) ∝ r −2/3 . The average height of the brush is then estimated by requiring that a total of a1 fN 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: h a1 fN = a

h (r)rdr ∝ a

1−1/

(1 f )

(3−1)/2

0

r (1−)/2 dr (19) 0

and hence 1−

2

h/a ∝ (1 fa) 1+ N 1+ F( ) ∝

−1

(16)

(1−)/2 Rc1− N

in order that the result h ∝ is reproduced in this limit. In the opposite limit, F( 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 1 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 a1 grafting sites occur along the axis whereby f chains per site are grafted, then in the z-direction the blob linear dimension is simply 1/1 , while in the tangential direction in the xy-plane [cf. Fig. 2(c)] the linear dimension must be of the order 2 r/f . However, in the standard theory 77, 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 (r), depending on the radial distance r from the axis. One considers a segment of the cylinder of length L containing p = Lf 1 polymers (as mentioned above, for infinitely thin rods 1 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 2 rL, there should then be p


(20)

When we use = 0.588,162 we find h/a ∝ (1 fa)0.259 N 0.74 while taking the Flory value = 3/4 yields h/a ∝ (1 fa)1/4 N 3/4 . This latter result happens to be identical to the estimate that one would obtain when one partitions the cylindrical brush into disks of width 1/1 , each disk containing a twodimensional 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 (r) ∝ (r/f 1 )1/2 happens to be the geometric mean of the characteristic length 1−1 in the z-direction and the length r/f in tangential direction, we conclude that the blob volume is of the same order, namely 3 (r) ∝ (r/f 1 )3/2 , irrespective whether we take it as spherical or of anisotropic shape with three different linear dimensions, namely (r/f 1 )1/2 , r/f , and 1−1 , 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 1−1 , because the chain can make random excursions (by ⫾1−1 ) 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

Rgy ∝ a(1 a)− 2+2 N 2+2 , Rgz ∝ fa(1 a)− 2+2 N 2+2 2−1

3

1+2

(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


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then the product f 1 /a in eq 20 needs to be reinterpreted as a grafting density per area. For Rcyl h we still have the result from eq 20, namely, h/a ∝ (a2 )(1−)/(1+) N 2/(1+) , while in the opposite limit Rcyl h we must recover the result for the flat brush, h/a ∝ (a2 )1/2−1/2 N (Eq. 3). The crossover scaling assumption analog to eq 15 then reads h/a ∝ (a2 )1/(2)−1/2 N Fcyl {a(a2 )1/(2) N/Rc }. 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 z − x plane (cross section of the cylinder, orienting now the cylinder axis along the y -axis) (a), with the size of the outermost blob (blob ) being delimited by the grafting density. In the z −y plane containing the cylinder axis, all blobs have a linear dimension blob = −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

and could not be present in the star polymer case. Finally, we emphasize that eqs 17–21 apply only if the grafting density 1 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 aN , the corresponding crossover scaling assumption for the height of the cylindrical brush is74 : h = aN Fmu (1 aN )

(22)

where Fmu ( ) is a crossover scaling function which for large arguments must reproduce eq 20 and hence 1−

Fmu ( ) ∝ 1+ ,

1.

(23)

Another interesting crossover displays the behavior of chains grafted to a thick cylinder of radius Rcyl a. First, we note that

(24)

The crossover scaling function Fcyl ( 1) = 1 while in the opposite limit it has a power law behavior Fcyl ( ) ∝ − 1+ , 1. 1−

(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 hence many different scaling regimes may occur (Fig. 5).78, 79 Note that there is a simple geometric relation between the grafting density of the anchor points at the cylinder wall and the average volume fraction of monomers contained in the cylinder. As the area A of the cylinder surface

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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We next consider the regime of wide tubes (D > aN ) where first (at = ∗ = a−2 N −2 ≈ a−2 N −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 ∗∗ , 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 > ∗∗ . Using h = 1/3 N (putting a = 1 and = 3/5 for simplicity, see eq 3), the condition h = D/2 leads to 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 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 ( = 3/5) has been made throughout, and a ≡ 1 was chosen. For wide tubes (D > N 3/5 ) with increasing only two crossovers are encountered: at = ∗ = N −6/5 (eq 2) from mushrooms to an (essentially) flat brush, and then at ∗∗ = (D/N)3 to a compressed brush. The latter regime ends at 4 = D/N, which means that the tube is densely filled with monomers (fraction of occupied sites in a lattice model = 1). For narrow tubes, D < N 3/5 , 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

is A = DL, the total number of monomers N contained in the cylinder is N = NA = N DL .

(26)

As the cylinder volume (for a cylinder of length L) is V = LD2 /4, we conclude that = N /V = 4N/D (note that we now choose a ≡ 1 for simplicity). The constraint < 1, i.e., < D/(4N), yields the rightmost straight line in Figure 5 (the region > 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 = g = g3/5 , one readily gets Rgy = DNblob = DN/g = ND−2/3 .

(27)

For D = N 3/5 a smooth crossover to Rgy = N 3/5 occurs, as it must be.


∗∗ = (D/N)3 ,

(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 of monomers is approximately uniform throughout the tube. Then, the blob size is = −3/4 , as for a semidilute solution in the bulk, and Rgx = Rgy = Nblob = −3/4 Nblob , 1/2

1/2

(29)

with Nblob being the number of blobs per chain. As there are g = 1/ = 5/3 = −5/4 monomers per blob, Nblob = N/g = N5/4 , and hence Rgx = Rgy = −1/8 N 1/2 = −1/8 D1/8 N 3/8 ,

(30)

where in the last step the relation = 4N/D was used. At the boundary = ∗∗ eqs 28 and 30 yield Rgx = Rgy = D−1/4 N 3/4 , which coincides with the prediction of eq 4, Rgz = Rgy = −1/12 N 1/2 = (D/N)−1/4 N 1/2 . 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


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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 multicomponent) 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 N 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 ⎡ ⎡ ⎤⎤ 2 N N 3 d r ⎣ D r (·) exp ⎣− ⎦⎦ = ds 2 2a ds =1 0 w × exp − (31) d r ˆ 2 ( r ) , 2 where the coordinate s runs along the contour of the chain, ˆ and the monomer density (r) in the system is defined in terms of the coordinates r (s) along all contours as ˆ r) = (

N N

ds[ r − r (s)].

(z) = −

ln Q[ ] = w(z),

(35)

with (z) being the average monomer density at distance z. The mean field free energy per polymer, in units of kB T, is 1 F = −w

dz

2 (z) − ln Q . 2

(36)

0

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 r (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 E(z, zo ) = −

ds(z, z0 ) dz

−1 (37)

so that the single-chain partition function becomes ∞ Q=

dz0

DE(·, ·) exp[−I(z0 )],

0 z m (z0 )

I(z0 ) =

dz 0

w(z) 3 |E(z, z0 )| + 2a2 |E(z, z0 )|

(38)

(32)

=1 0

ˆ and using the Inserting the identity 1 = D ( − ) ˆ integral representation of the delta function, ( − ) = ˆ r , one can carry out the Gaussian inteD exp [ − ]d gration over . This yields (the grafting density = N /A, of course) ∝ D exp[AF[]], (33) where is the external field and F[], the corresponding free energy functional, is given by (34) F[] = − dz2 (z)/(2w) − ln Q[],

1522

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


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 (z) follows by functional differentiation, (z) = − ln Q/(w). 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

REVIEW


path for a given z0 are eliminated, and hence the selfconsistent equation which determines the density becomes in the classical limit 1 ∞ dz0 (39) exp[−I(z0 )] (z) = Q z0 (z) |e(z, z0 )| Note that z0 (zm ) is the inverse function of zm (z0 ), and the end-point distribution e (z0 ) then is found as e (z0 ) =

exp[−I(z0 )] Q

(40)

Using the normalization dz0 e (z0 ) = , which expresses the fact that for each grafted chain the free end must be at some distance z0 , one can derive Q to find ∞ − ln Q =

e (z0 ) I(z0 ) +

dz0 0

∞ dz0 0

e (z0 ) e (z0 ) ln

To discuss this result, it is useful to introduce a rescaling in terms of the following dimensionless variables ˜ z ) ≡ (z) zˆ , ˜ e (˜z0 ) ≡ e (z0 ) z˜ , (˜ N zˆ ≡ N

2wa2 3

(42)

1/3

˜ z ) = 1, d z˜ (˜

0

(43)

1 2

d z˜ 0 ˜ e (˜z0 ) = 1.

(44)

0

∞

d z˜ ˜ 2 (˜z )

0

∞

z˜ z0 ) m (˜

d z˜ 0 ˜ e (˜z0 )

+ 0

+

1

0

˜ z) (˜ d z˜ |˜e(˜z , z˜ 0 |) + |˜e(˜z , z˜ 0 )|

∞ d z˜ 0 z˜ e ˜ e (˜z0 ) ln[˜ e (˜z0 )]

(45)

0

where the parameter is proportional to the square of the ratio of the typical brush height zˆ to that of the unperturbed √ radius of gyration a N,

≡N

d z˜ = 1, |˜e(˜z , z˜ 0 )|

(47)

˜ z ) − (˜ ˜ z0 ) + (z which yields the condition e˜ 2 (˜z , z˜ 0 ) = (˜ ˜ 0 ), where (˜ ˜ z0 ) 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 ˜ and ˜ e require numerical work.51, 52 Figure 6 gives a typical example for several values of the parameter .51 Only in the SSL (included as thick broken lines in Figure 6) explicit simple formulas result, namely z˜ 2 ˜ z ) = (0) ˜ , (˜ 1− h˜ 2

3 ˜ , (0) = 2h˜

˜ z > h) ˜ = 0, (˜

(48)

3 z˜ 2 ˜ ˜ e (˜z0 > h) ˜ = 0. ˜ e (˜z0 ) = z˜ 0 1 − 0 , z˜ 0 < h, h˜ h˜ 2

(49)

∞

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

z0 ) z˜ m (˜

and ,

which satisfy the normalizations ∞

˜ z ) and the end-monomer disTo obtain the density profile (˜ tribution ˜ e (˜z ) 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 e˜ (˜z , z˜ 0 ), subject to the equal-length constraint

0

(41)

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

z z˜ ≡ , zˆ

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 → ∞, this term is neglected.29, 30

3w 2 2 2a2

1/3


.

(46)

In the SSL, there are no monomers beyond the rescaled brush height h˜ (note that h˜ differs from zˆ by a numerical factor), while the full SCFT shows a Gaussian tail even if 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 ∝ 1/3 N (eqs 3, 43) and show that indeed the monomer profile is neither the step function of the Alexander-de Gennes theory 26, 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 e (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


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

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. 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 (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/a3 a1/2 ). The validity of the classical approximation requires strong stretching, > 1 (thus, the curves for

≤ 1 in Figure 6 are not expected to be quantitatively reliable). On the other hand, a2 < 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 [2 a4 N 3 ]−1 < w 2 /a6 a2 < 1

(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

FIGURE 7 (a) Plot of rescaled brush height hD 2/3 , 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 (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 h at the Theta temperature , as a function of the dimensionless ‘‘solvent quality,’’ = (T − )/T . These data were obtained from the analysis of neutrons reflectivity measurements by Karim et al.189

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2 2 FIGURE 8 (a) Log-log plot of the squared radius of gyration component Rgz in the z-direction and in the xy -direction (Rgxy ) as a function of chain length N, for a bead spring model of polymers at a (normalized) grafting density = 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 R = /2 are included. (b) Total monomer density (z) and end-monomer density e (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

compressed brush as a function of brush height h and the associated osmotic pressure osm (h) = −dF(h)/dh, namely

2 h∗ h 1 h 5 F(h) = C + − , h h∗ 5 h∗ 2 2 2 1/3 12w 1/3 w h∗ = N, C =N 96 2 4 C h h h∗ 2 osm = ∗ −2 ∗ + , h ≤ h∗ h h h h∗

(51)

Here, h∗ is the height of the uncompressed brush (corresponding to h˜ in eq 48) and obviously osm = 0 for h ≥ h∗ . Writing h = h∗ (1 − ) with a small parameter, one finds that osm ∝ 2 while the Alexander brush picture would imply osm ∝ for small . 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 taken by the monomers can be reinterpreted as the total density in the system, and then osm 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 of mixed diagram ˆ r )2 in eq 31 polymer brushes. The energy term (w/2) d r ( then needs to be replaced by E(ˆ A , ˆ B )/kB T w wBB 2 AA 2 = d r A ( r ) + B ( r ) + wAB A ( r )B ( r ) , 2 2 WWW.MATERIALSVIEWS.COM

(52)

with ˆ A ( r ), ˆ B ( r )—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 -parameter ˜ = (2wAB − wAA − wBB )/2w denotes the mutual attraction (repulsion) between unlike monomers. It is related to the commonly used Flory-Huggins parameter160, 165 FH = w ˜ . By increasing ˜ > 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 = U − TS, 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 (z)). We now can write F = U − T(SA + SB ), where in SCFT

SA = kB {ln QA rG + d r A ( r )A rG ( r )}, (53) A rG

and similarly for species B. Here, QA rG means the single-chain partition function of an A chain that is grafted at the site r G at the grafting surface, A rG ( r ) the corresponding monomer density and A ( r ) 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 r G of all A chains. The density ˆ A rG ( r ) satisfies then the selfconsistency equation


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FIGURE 9 (a) Phase diagram for a symmetric polymer brush (NA = NB = N, average volume fractions A = B = 1/2). Under good solvent conditions (˜ < 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 ˜ > 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 > 0.23, while broken parts of the curves for < 0.23 are tentative extrapolations. The parameter is the inverse of the stretching parameter defined in eq 46. (b) Phase diagram of a symmetric binary polymer brush for = 0.5 as a function of the composition = A and the incompatibility ˜ . 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 Re = a N 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

A =

U =w kB Tˆ A r G

2 − ˜ 2 + ˜ A + B , 2 2

(ln QA rG ) ˆ A rG = − , A

(54)

For numerical solutions of these equations, it is useful to define the end segment distributions qA rG ( r , s) and q˜ A ( r , s). The first quantity is the probability that the end of a chain segment of length Ns is at r when this chain is grafted at r G , while q˜ A ( r , 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, R2 ∂qA = e qA − A qA (56) ∂s 6N These equations are solved subject to the initial conditions qA rG ( r , 0) = ( r − rG ) and q˜ A ( r , 0) = 1 for grafted and free chains, respectively. The density A rG ( r ) and the (normalized) chain partition function are then computed as N 1 ds qA rG (r, s)˜qA ( r , N − s), (57) A rG ( r ) = QA rG 0

1526

d r qA rG ( r , s)˜qA ( r , N − s).

N

(55)

and similar expressions hold for B and ˆ B rG . For simplicity, arguments r have been omitted in eqs 54 and 55. It is noteworthy that the self-consistent fields A , B do not depend on the grafting site r G .

QA rG =

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

(58)


A ( r ) = where qA ≡

ds qA ( r , s)˜qA ( r , N − s)

(59)

0

qA rG /QA rG .

A rG

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. However, in reality one can expect at best extended shortrange 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 (z) and c (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

REVIEW


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 56, that is inherent in the field-theoretic formulation of SCFT which was presented so far, eqs 31–58, can be circumvented by the lattice formulation due to Cosgrove et al.29 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 fcc same layer (qsc 0 = 4, q0 = 6). The fraction of sites 0 = q0 /q in fcc the same layer then is 0sc = 2/3 or 0 = 1/2, while the fraction of sites in an adjacent layer is 1 = (1 − 0 )/2 = 1sc = 1/6 fcc or 1 = 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 (z); the profile of the solvent density is s = 1 − (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 FH . So the potential energy u(z) of a monomer (relative to that of a monomer in a bulk solution at concentration b ) becomes (following the formulation of Wijmans et al.)39

(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, (z) = 1 (z − 1) + 0 (z) + 1 (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


Defining a monomer weighting factor G(z) = exp[−u(z)/kB T], 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 G(z, s) = G(z, s − 1)G(z) ,

(61)

which starts with G(z, 1) = G(z) and is computed for all s ≤ N. 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 (z, s) of the s’th monomer in layer z becomes (z, s) = CG(z, s)G(z, N − s + 1)/G(z),

(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 (z) = (z, s) and hence C = b /N. This s

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 Gg (z, 1) = G(z)z1 ,

(63)

where we use a subscript g to clearly distinguish grafted from nongrafted chains. Therefore, the iteration eq 61 is replaced by Gg (z, s) = Gg (z, s − 1)G(z)

(64)

In the chain, monomers can be considered as the joint between a grafted chain of s monomers and a nongrafted free chain of N − s + 1 monomers. Eq 61 still holds for the latter part, and thus for grafted chains eq 62 is replaced by g (z, s) = Cg Gg (z, s)G(z, N − s + 1)/G(z)

u(z)/kB T = FH [1 − 2(z) − 1 + 2b ] − ln[1 − (z)] + ln[1 − b ]

to be calculated considering its conformational statistics, as described below.

(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,

Gg (z, N). (66) Cg = / s

From this formulation, it is obvious that an extension of the formalism to systems containing both grafted and free chains


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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 (z) = Ns=1 g (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 4 r02 dr0 . . . for spheres or 2 r0 dr0 . . . 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 (r|r − 1) to go inside, or (r|r + 1) to go outside, follow from geometric considerations199 (r|r − 1) = S(r − 1)/3L(r),

(r|r + 1) = S(r)/3L(r), (67)

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

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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 (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 = 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.

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.

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 , F = 3/5, rather than using its correct value,162 ≈ 0.588. Then, both SCFT and scaling theory in terms of blobs predict for the brush height a scaling h ∝ 1/3 N in the good solvent regime. However, the proximity of the actual estimate for to F 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.

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 (z) of

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 kB T, 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


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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 (R 2 )(1−)/2 , for chain lengths N = 20, 30, 40, 50, and 100, as indicated, and various grafting densities (denoted as g in the figure). A straight line fit in the region where the brush is strongly stretched is indicated. Note that for (R 2 )(1−)/2 < 1, a crossover to nonstretched mushrooms occurs. (left) Plot of the volume fraction at the grafting plane relative to the overlap volume fraction, (0)/∗ , plotted against the scaling variables (R 2 )(3−1)/2 . The straight line shows a fit by the scaling theory. From Wittmer et al.46

chemical potential functional (z) with respect to position z(s) of monomer s, (z) = kB T

N ds[A()(dz/ds)2 + U()].

(68)

Note that the results of the SSL of SCFT, namely 30 2 h2 3 12 2 N, h= wa (0) = 2 8wN 2

(71)

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

0

Here, the potential U() is the work required to insert a monomer at a distance z from the wall, and in the semidilute regime scales like U() ∝ 1/(3−1) . As depends on z, U() can be considered as a potential depending on z too. The semidilute regime requires that exceeds the overlap density = N/(4 Rg3 /3), where Rg is the gyration radius of a free chain in dilute solution. The prefactor of the gradient square term A() ∝ (2−1)/(3−1) (= 1/4 in the Flory approximation) would be simply a constant for Gaussian chains, and U() would be harmonic, U() ∝ 2 , cf. Section Scaling Concepts for Brushes under Good Solvent Conditions. Introducing then the coordinate Z defined by

h(0) 3Rg2 N = . Rg 2Rg3

(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 (Rg2 )(1−)/2 and that (0) is proportional to (Rg2 )(1/2)(3−1/) were found already for N ≤ 100 (Fig. 10), and eq 70 was confirmed.

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 (Z) explicitly.46 The final result for the brush height is of the form of eq 3, h/Rg ∝ (Rg2 )(1−)/2 , 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, (0), namely

The theoretical results, eqs 68–70, 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) ∝ s ) 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 : then one still has Gaussian-like statistics on the scale of thermal blobs T ∝ |T − ]−1 ,160 and if the size of the thermal blobs would exceed the size of the blobs resulting from the grafting, blob ≈ −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.

(Rg2 )N h(0) ≈ 6.09(Rg2 )∗ ≈ 1.454 Rg Rg3

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

dZ/dz = (/∗ )(−1/2)/(3−1) ,


(69)

(70)


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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. FIGURE 11 Schematic log-log plot showing in the plane of the scaling variables relating to excluded volume (vex ) and grafting density 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 vex N 2 /Rg3 ∝ (Rg2 )(2−3)/2 ], and to the right of the brush-mushroom crossover for marginal solvents [described by vex N 2 /Rg2 ∝ (Rg2 )−1 ]. The dependence of the scaling function H˜ of the brush height (eq 73) on the scaling variables is indicated in the various regimes. Note that SCFT implies h ∝ 1/3 , 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.

a function of two scaling variables, Figure 11, namely = Rg2 and = vex N 2 /Rg3 . For small , one has mushroom behavior, while for large enough , one has brush behavior; good solvent behavior where excluded volume dominates requires that is large enough, while the regime where vex N 2 /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 ˜ ), h/Rg = H( ,

(73)

but the scaling function H˜ 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 = where the second virial coefficient (which is proportional to vex or w) vanishes; the behavior at T ≤ will be discussed in the next section. 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

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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 f ( r ), g ( r ) 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 f (z) and g (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. 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

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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 → ∞ from the start.97 Then, the free energy density due to the local interactions fint () is [kB T ≡ 1 here] fint () = (1 − ) ln(1 − ) + (1 − )

(74)

The corresponding exchange chemical potential (for replacing an effective monomer by a solvent particle) is then () =

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, is the range parameter, LJ , of a Lennard-Jones potential, and the solvent density 3 in the bulk is chosen as bulk LJ = 0.32. The energy parameters refer to energies between pairs of particles of type , (see Section APPENDIX: COMPUTATIONAL METHODS for a precise definition of the model). Chain length is Ng = 64 and grafting density is = 0.32 (choosing LJ = 1, as the unit of length). From Ref. 211.

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). 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 → ∞. Recall, that we expect from eqs 15 and 16, a scaling h ∝ (1−)/2 Rc1− N ≈ 1/5 Rc2/5 N 3/5 in the limit N → ∞. Figure 14 shows, however, that the DFT results65 seem to converge towards a scaling h ∝ N 2/3 rather than h ∝ N 3/5 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


FIGURE 13 Log-log plot of the monomer density (r ), (upper part), and of the end-monomer density e (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 = 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, (r ) ∝ r 1/−3 ≈ r −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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and the osmotic pressure is () = () − fint () = − ln(1 − ) − 2 − ,

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 . MD results (open symbols) and DFT results (full symbols) are shown for three grafting densities: = 0.185, triangles; = 0.118, crosses; = 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

∂fint ()/∂. This chemical potential must be combined with the contribution due to chain stretching in the brush. In SCFTSSL, this contribution results from a potential V (Z) = 3 2 Z 2 /8, if Gaussian chain statistics is assumed, Z = z/Na. Amoskov and Birshtein97 used a finite stretching model instead, V (Z) = −3 ln[cos( Z/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 () − min = V (H) − V (Z),

(75)

where min = (min ) is the chemical potential at the outer brush boundary, which defines the (normalized) brush height, H = h/(Na). Note that min = 0 for good and Theta solvents ( = 1/2 for the latter160, 165, 191 ) but we shall find min > 0 for > 1/2: min 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, H (Z)dZ.

=

(76)

0

The profile is then described by the equation, resulting from eq 74 and 75, − 1 − 2 − ln(1 − ) − min = −3 ln cos( H/2) + 3 ln cos( Z/2) ,

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(77)


(78)

with ((Z = H)) = 0, the osmotic pressure is zero at the outer boundary Z = H of the polymer brush. For small , eq 78 becomes () ≈ 2 ( 12 − ) + 3 and hence for > 1/2 (where a bulk polymer solution for N → ∞ exhibits phase separation at = 0), the polymer-rich phase of the polymer solution has a monomer volume fraction coex ≈ − 1/2. Then, the profile (Z) of the monomer volume fraction in the brush decreases from its maximum value max = (z = 0) monotonically and continuously down to (Z = H) = coex at Z = H, where a discontinuous jump to zero occurs. This jump vanishes linearly with when the critical point crit = 1/2 of the polymer solution is reached. Although (Z) vanishes linearly with Z for < crit (there eqs 48 and 49 apply), for = crit one finds a square root singularity 37 Z2 (Z) ∝ 1 − 2 , (79) H and one also finds that for = crit (i.e., at the Theta-point of the polymer solution) a different scaling with grafting density occurs, namely (we restore here again physical units, w3 is essentially the third virial coefficient 37 ) h=

1/4 4 w3 (a2 )2 Na, 2

(80)

while in the poor solvent region one predicts37 h∝

w3 a2 Na, | − 1/2|

| − 1/2| 1.

(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 (Fel = h2 /N) and of interactions, N N 2 + Nw3 (82) Fint ∝ Nw h h where now w ∝ (crit − ) may change sign at = crit . At < crit the term proportional to w3 can be neglected for small grafting density , of course. Minimizing the free energy Fel + Fint with respect to h, ∂(F)/∂h = 0, yields for > crit h ∝ N(w)1/3 , that is, eqs 15 and 71, while for w = 0 eq 80 results, and for |w| → ∞, eq 71 results. However, the Flory theory does not give valid results on the distribution of chain ends, while SCFT-SSL predicts37 z e (z) = 2 , h

1 e (z) ∝ z 1−

= crit ,

, z2 h2

− crit 1.

(83)

(84)


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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.

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 = 0.185 (choosing Lennard-Jones diameter as length unit). From Dimitrov et al.59

All results of this subsection, however, refer to the limit N → ∞ that the SCFT-SSL describes. Of course, for finite chain lengths two effects occur40, 42, 43, 45, 48, 59, 95 : (i) The density profile (z) is a smooth function of z, irrespective of solvent condition, and the same is true for e (z), see Figure 15. Disregarding the structure of the profiles at small z near the grafting plane, we see that the profile (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

(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. 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 () as a function of grafting density from their SCMF theory (see Section Other Theoretical Concepts) and found that for 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 -temperature T , the isotherms () vs. 1/ developed a loop rather reminiscent of a van der

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 /T ≈ 0.67) and two choices of grafting densities: = 0.03 (a,b) and = 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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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 T − Tc (N) ∝ N 1/2 , and the critical monomer volume fraction c scaling as c ∝ N −1/2 as well. For T Tc (N) phase separation occurs between essentially pure solvent and a semidilute solution of volume fraction coex ∝ T − T (∝ − crit , if the model eq 74 is used). From eq 80 we can conclude that at T , or nearby, the volume fraction inside of a brush is ≈ N/(−1 h) since −1 h is (apart from a prefactor of order unity) the volume available per grafted = T the chain, and hence ∝ −1 . On the other hand, at T √ ≈ N) and crossover between mushroom behavior (where h m √ brush occurs for h/hm ≈ 1, therefore 1/2 N/ N ≈ const, that is, ∗ ≈ 1/N. Thus, we conclude that the microphase separation has its onset near the crossover from mushroom to brush behavior, because the volume fraction ∗ at the mushroom to brush crossover (∗ ∝ (∗ )1/2 ∝ N −1/2 ) scales in the same way as c (N) in a polymer solution does. Although in poor solvents for ∗ , we observe the collapse transition of isolated mushrooms, fully analogous to the collapse of isolated chains in a extremely dilute solution,160 for near ∗ 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 (k)

= [S0 (k)]

−1 + a3 ∂ 2 fint /∂2 , S −1 (k)

(85)

where fint () = a−3 (2 /2 + w3 3 /3), in analogy to eq 82, and stands for the distance from the Theta-point. One finds

−1 has a minimum at k = k ∗ with k ∗ Na2 /6 ≈ 1.893, that [S0 (k)] −1 with S0 (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 √ p = N/w3 , q = N w3 a2 (86) √ this stability limit can be written as ( = h/ Na2 ) pq/ + 2q2 /2 = 2.624

(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, N(a2 ) 1 while the inhomogeneous structures occur for q of order unity (and the characteristic size of√ the “cluster” or “dimples” is of order Lcluster ≈ 2 /k ∗ ≈ 2 Na, consistent with the simulations.40, 42 Again, as 1534


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 = N/w3 . The Theta-point occurs for = 0) √ and scaled grafting density q, with q = N(a 2 ) w3 . 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 T 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

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. 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 Rcyl a N 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. 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 = 3/5) one would predict that the swollen brush has a radius R ∝ N 3/4 1/4 1/4 217

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height h(R, N, ) scales as in eq 15. Yet, as the height h0 ∝ 1/2 N for T = T (cf. eq 80) and the size of the star polymer scales as R ∝ N 1/2 at the Theta-point, the scaling function f (h0 /Rc ) in eq 15 scales like F( ) ∝ −1/2 , and this scaling has been verified.65

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 = N 1/2 (appropriate in one dimension, when chains are grafted to a line at distance −1 between grafting points), and the ordinate variable is the scaled distance from the Theta-temperature (T = in this figure) z = N 1/2 |(T − )/|. The good solvent regime (T > ) 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

( = |(T − T )/T |, cf. also eq 20), for = 0 (Theta conditions), one rather finds R ≈ N 2/3 2/3 ), while the collapsed brush scales as R ∝ (N/||)1/2 . Sheiko et al.217 predict that the pearls rather have a radius Rpearl ∝ N 1/2 (/||)1/4 and an axial length Lpearl ∝ N 1/2 (||/)1/2 so each pearl contains npearl ∝ N 1/2 ||1/4 3/4 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 Thetasolvent conditions, the monomer density profiles were found66 to qualitatively resemble those of planar brushes. Thus, for large enough grafting density 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


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 ≥ , 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 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 → 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 F(h) = Fel + Fos ,

Fel /kB T = h2 /(a2 N)

(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 = a−1/2 the distance between the grafting sites, so the volume taken by a chain is hD2 ) Fos /kB T = ex N 2 /(hD2 ) + a6 N 3 /(hD)2 .

(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


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forces, and this term can be understood from a virial expansion of the Flory- Huggins-equation of state of a polymer solution160, 165, 191 ln + (1 − ) ln(1 − ) + (1 − ) N 1 1 ≈ ln + (1 − 2)2 + 3 . (90) N 2 6

Fmix /kB T =

Equation (90) is the free energy of mixing per site in a lattice model, 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 , we now can conclude that the second virial coefficient ex is related to the Flory -Huggings parameter as ex = a3 (1 − 2), 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 ex which becomes negative below the Theta temperature (for moderately good solvents 0 < < 1/2 and hence ex < a3 ). This consideration justifies the choice of coefficients in eq (89). Note that a3 N/(hD2 ) is the volume fraction of monomers in the volume taken by a chain, as usual, and the total enthalpy due to pair interactions is put (hD2 )2 vex 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 ∗ < < 1 = (ex /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 1/3 1/3 N(a/D)2/3 = ex N1/3 , that is eqs (3), (43), unity) that h = ex and (51). However, for > 1 it is the term due to the third virial coefficient which dominates, and hence one obtains instead from ∂F(h)/∂h = 0 that h = (a2 /D)N = a1/2 N,

> 1 = (ex /a3 )2 .

(91)

Note that there occurs a smooth crossover between eqs (51) and eq (91) at = 1 , with h = Nex /a2 there. We also observe that the average volume fraction of monomers in the brush in the regime described by eq (91) is = Na3 /(hD)2 = 1/2 ; so only for → 1 one also obtains = 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 > 1 predicts an elliptic rather than parabolic volume fraction profile44 (z) = (0)[1 − (z/h)2 ]1/2 ,

√ h = (2/ )Na2 /D,

(92)

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

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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 P N. 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 N , P = 1 − N 153, 165, 191 Fmix /kB T =

N 1 − N ln N + ln(1 − N ) N P

(93)

which in the limit of small N leads to Fmix /kB T ≈

N 1 2 1 3 ln N + + + ··· N 2P N 6P N

(94)

and hence one can conclude that ex 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 = 0). Writing then, in the limit where pair interactions dominate, in analogy to Eq. 89 F(h)/kB T = h2 /(a2 N) + (a3 /P)N 2 /(hD2 ),

(95)

and the condition ∂F(h)/∂h = 0 yields (neglecting the term from the third virial coefficient) h = aNP −1/3 1/3 ,

< 1 = P −1/2 .

(96)

The monomer volume fraction N = Na/h = P 1/3 2/3 . Note that for = 1 , we find that N = 1, which means that the mobile chains for > 1 are already completely expelled! For that reason, for > 1 one simply has h = Na, > 1 ,

N = 1 (dry brush).

(97)

Actually one finds that inside the moderate coverage regime, there occurs another crossover, at = 1 = P −2 ; while for < 1 the picture of a chain is a string of subunits of size D for > 1 it is a string of subunits of size = aP 1/3 −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 1 , however. These results can be summarized in a schematic (P, ) diagram, Figure 19, where for completeness also the crossovers to the mushroom regime are included.154 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

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

LP Fbrush =

∞

c(N)[ln c(N)] + 1 N + Fchain [(z), c(N), N] (99)

N=1

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, h ≈ aN 1/2 (mushrooms for which excluded volume is screened out); region 2, h ≈ aN 3/5 P −1/5 (mushrooms for which excluded volume is effective); region 3, h ≈ aNP −1/3 1/3 (good solvent region of the stretched brush); region 4, h ≈ aN 1/2 (layer of overlapping chains that are not stretched); region 5, h = aN (dry brush). From Aubouy et al.154

a recent experimental review on nanocomposites formed from brush-coated nanoparticles, see Ref. 237.

‘‘LIVING POLYMER’’ POLYDISPERSE BRUSHES

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: c(N) = (1 −

pr )pN−1 r

M0 = MN

M0 1− MN

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

c(N) ∝ exp(−1 N − Fchain (N))

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

,

z

s(z) =

(98)

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,

(100)

In the simplest case of dilute nonoverlapping living polymers (mushrooms) tethered to a plane in good solvent, one expects Fchain (N) = ln(N) with the exponent = 1 − s where the universal “surface” exponent s ≈ 0.65 < 1. Hence, one expects to find a weakly singular behavior c(N) ∝ N − exp(−1 N). In a weakly stretched polymer brush at low grafting density , 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 (z) ∝ z the monomer density would scale as (z) ∝ z − , = (3−1)/. The chain-end density, like the blob density, e (z) ∝ 1/(z)3 ∝ z − with = 3. Because c(N)dN = e (z)dz, one readily finds c(N) ∝ N − with = 1 + 2 ≈ 11/5.

N−1

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.


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 selfconsistent 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.

0

e (z0 ) dz0 , (z0 )

(101)

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

H

e (z )dz =

NH

c(N )dN ∝ 1/ .

(102)

N

with = 2/(3 − 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 (z) ∝ z − , e (z) ∝ z − , c(N) ∝ N − , and z(s) ∝ s ⊥ , as JOURNAL OF POLYMER SCIENCE PART B: POLYMER PHYSICS 2012, 50, 1515–1555

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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 − with ≈ 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

in the case of the weak stretching limit. Equations 101 and 102 then yield244 (3 − 1)( − 1) , 1 + − (1 − ) (3 − 1) − + 1

= + 1= , 1 + − (1 − ) 1 + − (1 − ) ⊥ = . 2 =

(103) (104) (105)

where three relations, eqs 103–105, relate the four exponents. One can, therefore, either impose the condition ⊥ = , 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 103–105, correspond to a living polymer brush, grown by diffusion-limited aggregation (DLA), which yields = 2, thereby fixing uniquely the set of exponents as = 7/4, ⊥ = 3/4, and = 2/3.245 Indeed, one can verify that these theoretical predictions agree well with the MC simulation results (Figure 20).244 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

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compression. As shown by Milner et al.,30 the height of a polydisperse brush within the SSL-SCFT is


h[˜c (N)] =

12 2

Nmax

dN 3 − c˜ (N ),

(106)

0

where c˜ (N) are the number of chains per unit area of length less than N. Therefore, one can estimate that the effective ¯ increases as a square root of height h() = h0 (1 + /(2N)) the variance , where MW /MN = 1 + 2 /(3N¯ 2 ). 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

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

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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) 3/4 ∗ 9/4 D 2h T k B − , 2h∗ < D (107) f˜ (D) ∝ 3 −3/2 a D 2h∗ Note that for small relative compression = 1 − D/2h∗ , the force scales linear in . 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 f˜ (D) ∝ 2 for small D. However, because in reality chains are never infinitely long, there is some interpenetration between brushes taking place, and likewise, the force f˜ (D) is not strictly zero for D > 2h∗ , due to the “tail” of the monomer density profile (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 Grest 138 introduced a quantity I(D), defined as D I(D) = D/2

D 1 (z)dz /

1 (z)dz,

(108)

0

where 1 is the monomer density of brush 1, and suggested that ∗ 43 2 2h D 3 3 N 1− , (109) I(D) ∝ h∗4 D 2h∗


using scaling arguments of Witten et al.139 The MD results138 were roughly compatible with eq 109, and as h∗ ∝ 1/3 N, the scaling of I(D) as I(D) ∝ N −2/3 −4/9 shows that I(D) for strong compression ( 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 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 A different measure of interpenetration was defined by Chakrabarti et al.143 ⎡ ⎤ D D D I (D) = D 1 (z)2 (z)dz ⎣ 1 (z)dz 2 (z)dz ⎦ , (110) 0

0

0

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 work 259 ). Again the conclusion did emerge that for 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 1 (z)2 (z) is essentially different from zero. For very dense brushes (melt conditions), one has139 L ∝ N 2/3 a4/3 D−1/3 , while for the semidilute brushes under strong compression121 one needs to replace a by the size ∝ a(a3 )/(1−3) of the “concentration blobs,” assuming that in two strongly compressed brushes the monomer concentration is constant, ∝ N/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 ∝ (/a)1/ the interpenetration length becomes (using ≈ 0.588162 ) 1 a 1− 9−3 ≈ aN 0.51 (a2 )−0.15 (D/a)−0.18 L ∝ N 2 (a2 )2(1−) D (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 f˜ (D) = 2 R U(D),

(112)

for D R. Using colloidal probes where R = 3 m262 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


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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, = 0.03 (full dots), N = 50, = 0.03 (full squares); N = 50, = 0.1 (open triangles); N = 50, = 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, = 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

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 m 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). 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

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arms. From this argument, they concluded that the interaction potential between the two brushes depends logarithmically on the separation r, VWP (r) ∝ f 3/2 ln[Rc /(r − 2Rc )], and hence the force is f˜WP (r) ∝

f 3/2 r − 2Rc

(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 r2 f˜ (r) ∝ r exp −const 2 , (114) Re 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 ∞ W (r) =

dr F(r )

(115)

r

for two interacting brushes from extensive MD simulations (Fig. 23). These data show a divergence of w(r) for r → 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


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FIGURE 22 Two-dimensional density distribution (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 , as shown by the scale given at the bars. From Lo Verso et al.151

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. 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 < T 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 T . 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.


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distribution of nanoparticles nano (r) in the polymer brush is determined from the following brush free energy (in terms of the chain configurations z(s, z0 )125 : N 2 ∂z 3 ds dz0 e (z0 ) F= 2 ∂s 0 + dr(r)[(r) + nano (r) − 1] + Fnano [nano ] (117) which yields Semenov’s result for the pure nonsolvated brush179 without nanoinclusions for nano (r) ≡ 0. Allowing for the symmetry of nano (r) in x, y-direction, and using eq 116 for the end-monomer distribution, e , which is the appropriate expression for dry brushes that have (z ≤ h) = const., the 1d expression for F along z then reads: N h 3 ∂z(s, z0 ) 2 dz0 e (z0 ) 2 ds F= 2a 0 ∂s 0 h − dz0 e (z0 ) − 0

h

+ 0

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

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 Selfconsistent 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 nano (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 (r) = 1 − nano (r). The latter means that the sum of monomer density contributions from all chains h passing through z whose ends z0 lie between z and h, z dz0 e (z0 )/|∂z(z0 , s)/∂s| = /a3 , which yields Semenov’s expression for the chain end distribution, e = h √ z20 2 : h −z0

2z0 e (z0 ) =

f (h)

− h2 − z02

h z0

f (x)

dx √ h2 − x 2

with f (x) =

1 − nano (x) 2Na3

(116)

(One should note, however, that due to incompressibility the brush height h increases as h ∝ N, in contrast to the generally established eqs 3 and 43). With incompressibility imposed, the

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h

dz(z)a3

dz0 z

1 − nano (z) e (z0 ) − |∂z(z0 , s)| a3 (118)

where the first term is the chain stretching energy and the second one constrains the chain grafting density to . The third term in eq 118 expresses the space filling condition. The pressure field (z) depends here only on z and the Lagrange multiplier 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, (z) = 0 (1 − z 2 /h2 ),

0 ≡ 3 2 h2 /8N 2 a5 .

(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 h h0 ¯ , nano ≡ dznano (z)/h, (120) h= 1 − ¯ nano 0 where ¯ nano is the overall nanoparticles fraction. For the minimization of the the total free energy, eq 118, each chain path must satisfy the condition N N ∂z(s, z0 ) 2 3 ds + ds(z(z0 , s))a3 = = 0 Na3 , 2a2 0 ∂s 0 (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 is obtained by considering a chain at z0 = 0 (i.e., with no stretching energy) at pressure 0 Na3 . 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 h stretch dz[0 − (z)][1 − nano (z)] F 0

h

= 0 h/3 − 0 nano (z) +

(z)nano (z) 0

(122)


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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 ∗ 3 ∗ 3 = h2 bb . (c) If mean nanoparticle density exceeds the saturation density sat = 12 bb , 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 blob and surf , 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

For small inclusion density, ¯ nano 1, the dependence on ¯ nano in the first two terms in eq 122 vanishes to first order in ¯ nano so that the change in F stretch due to nanoparticles is simply h F inc = F stretch = (z)nano (z) + O(¯ 2nano ), (123) 0

indicating that inserting a nanoparticle with volume Vnano = b3 at height z generates energy penalty (z)Vnano . Therefore, F inc is the free energy of inclusion profile nano (z) compared to the free energy, provided all nanoinclusions were above h. Because (z) decreases with z (note that (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 h h nano (z) Fnano = dz(z)nano (z) + dz ln nano (z) Vnano 0 0 h + h¯ nano − dznano (z) . (124) 0

in which interactions between nanoparticles of order nano (z)2 and higher have been neglected, according to the ¯ nano 1 condition, eq 123, while the last term enforces the mean nanoparticles density. Minimizing Fnano , one then derives125, 126 nano (z) = nano (h)e−(z)b

3

(125)

where the pressure (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 As (h) = 0 and ≈ 20 (1 − z/h) for small depths, one obtains nano (z) ∝ exp[−(h − z)/] where ≡ h(b∗ /b)3 and b∗ ≈ (N/h)2/3 = −2/3 . Therefore, above a threshold size b∗ , equilibrium particle penetration is limited to a depth h, and the polymer brush has a loading capacity max whereby both and max scale inversely with nanoparticle volume b3 . In the language of blobs, soluble inclusions with size b smaller than the blob size , 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 so the ensuing path can only be determined, provided (r) is known.126 Beyond a second threshold of nanoinclusions’ size b > bmax ≈ (N/)1/4 , nanoparticles cannot penetrate the polymer brush. For b b∗ , the penetration depth, ≈ 1/b3 , is much less than the height h until = b defines the upper threshold size b = bmax ≈ (N/g )1/4 which can infiltrate the brush. In this partial penetration regime, b∗ < b < bmax , the maximum nanoparticle density max = /h ≈ 1/b3 .


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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 (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

Evidently, the brush excess free energy, F inc , due to inclusions, must be known to determine their organization in a polymer brush—nanoparticle composite. However, the computation of F inc as with all entropy-related quantities is not straightforward in a simulation. Using an efficient method for direct determination of F inc 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, F inc ∝ R3 for deeply placed nanoparticles, whereas F inc ∝ R2 for shallow particles at the distal region of the polymer brush. In Figure 25(a), we show the ensuing buoyant force f , the R Potential of Mean Force W = 0 f (r)dr (that is, F inc itself, and the monomer density profile 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 (z) so that deep inside the brush f nearly dwindles due to canceling of its symmetric lateral components. 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 F inc ∝ R4/3 De Gennes-behavior. One should note, in addition, that at equal volume F inc 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

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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 , and the size of the nanoparticles, b, Yaneva et al.128 determined the equilibrium particle penetration depth and the average concentration of ¯ in a polymer brush with explicit solvent, nanoparticles nano using a constant pressure (NPT) ensemble and Dissipative Particle Dynamics (DPD)—thermostat. They found that for nonadsorbing spherical nanoparticles of size b ≥ b∗ the thickness of the infiltration layer ∝ h(b/b∗ )3 and b∗ ∝ −2/3 , in agreement with the predictions of Kim and O’. Shaughnessy 125 —Figure 26. It was found that the mean density of nanoparticles, ¯ nano (b), scales as predicted, ¯ nano (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 mobilnano and D||nano were ity. The respective diffusion coefficients, D⊥ 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

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FIGURE 26 (a)-left: Interpenetrating density profiles of monomers,128 p (z), and of nanoparticles, nano (z), for N = 60, = 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 nano (z) ∝ (z − h)/ of nanoparticles of size b. In the inset, one observes ∝ 1/b 2 rather than the predicted 1/b 3 behavior, probably due to the small depth of nanoparticle infiltration. (b) The mean concentration of inclusions ¯ nano (b) against reduced size b/b ∗ where ¯ nano (b) = p (z)nano (z)dz/ p (z)dz, and the master curve ∝ (b ∗ /b)3 (full line). From Yaneva et al.128

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 drugbearing 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 F, 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: F = F stretch + F inc . As before, F stretch is given by the z first term in eq 118, h namely F stretch = 2a32 0 e (z0 )dz0 0 0 dzE(z, z0 ). The normalh ization condition 0 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 F inc =

a3

h

(1 − (z) − o ) ln(1 − (z) − o ) o 1 + ln o + k,k k,k k (z)k (z) dz L 2 0

(126)

where L denotes the length of (oligomer) chains (L N) in the polymer brush per grafted chain, o is the (free) h oligomer density, and (z) = z g(z )|E(z, z )|−1 dz . Here k,k 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 z02 − z 2 , then in the simplest case stretching, E(z, z0 ) = 2N of chemically identical grafted and free chains, k,k = 0, the enthalpy term in eq (126) is dropped while for z > h one


has = 0 so that in the bulk the free energy density is simply F/a3 = (1 − bo ) ln(1 − bo ) + L−1 bo ln bo . The total free energy of the infiltrated polymer brush is then described by the equations258 : 2 2 2 2 2 2 (z) = (1 − bo ) 1 − e−K (h −z ) + bo 1 − e−K (h −z )L (127) o (z) = bo e

−K 2 (h2 −z 2 )L

(128)

where z is the distance from the surface and K 2 = 3 2 /8a2 N. The polymer brush is then determined from h height of the 3 (z)dz = Na as 0 √ √ Kh = NKa + (1 − bo )D(Kh) + b D(K Lh)/ L

(129)

x where D(x) = exp (−x 2 ) 0 exp (t 2 )dt is the Dawson integral. Eq. (128) and (129) then yield the volume fraction profiles (z), o (z), and the brush thickness h as functions of N, L, and for given bo in the SSL as long as the oligomer length L N. 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 k,k = 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 = 0, whereas the compatible ones (for < 0) undergo a sharp crossover from weak to strong adsorption (the adsorbed amount (L) ≈ 100%), discriminating between oligomers (1 ≤ L ≤ 8) and longer chains Figure 27. 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 effect 302 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


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FIGURE 27 Density profiles of the polymer brush, p (z), (shaded area) and of free chains, o (z), of length L, (given as parameter) at two grafting densities: = 0.25: (a, b), and = 1.00 (c, d). (a) and (c) illustrate good compatibility between brush and free chains, po = 2.0, while (b, d) demonstrates a case of bad compatibility, po = 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 p and o (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.

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 ABcopolymer 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, (R), and grafting-density blobs g () in the brush coating.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

In this review, static properties of coarse-grained molecular models for polymer brushes and their investigation by Monte


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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 R is ≈ 7.74R. (b) Change in polymer conformations on decreasing nanopore radius R): In the polymer brush coating, the blob size g is determined by the grafting density of the brush while the size of the free chain blob, , depends on the clearance width above the brush surface along the pore axis. When ≈ g , the free macromolecule penetrates into the polymer brush. From Ref. 85.

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 ScheutjensFleer 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


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of simulation results to theoretical predictions; as well as to experiments whenever appropriate.

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

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.

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

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. ACKNOWLEDGMENTS

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. APPENDIX: COMPUTATIONAL METHODS

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 Utot = ULJ + UFENE + UW

(A1)

The first term ULJ stands for the pair interactions, a LennardJones potential with a cut-off at its minimum and a shift such that only the repulsive branch acts 12 6 1 , r < rc − + (A2) ULJ (r) = 4 r r 4

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UFENE (r) = −30

R0

2

r2 log 1 − 2 , R0 = 1.5. R0

(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, distinct from rc ≈ 1.12. 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 -temperature, which for this model, generally referred to as that of and Kremer Grest,306 has been established as302 kB T/ ≈ 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 (pp , ps ), 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 12 6 − + 1/4 − , r ≤ · 21/6 , ULJcos (r) = 4LJ r r (A4) where { } 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 pp = 1 and LJ = 1 (the Boltzmann’s constant kB = 1). For r ≥ ab 21/6 , we choose the cosine potential as follows ⎧ 2 ⎨ 1 r cos a + b − 1 , 21/6 ≤ r/ ≤ 1.5 cos 2 ULJ (r) = ⎩ 0, r ≥ 1.5 . (A5) Here, a and b are determined as the solution of the two equations, 21/3 a + b = , 2.25a + b = 2 ,

(A6)


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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) ⇔ 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

which yield a = 3.1730728678 and b = −0.85622864544. As desired, both ULJcos (r) and dULJcos (r)/dr are continuous at the potential minimum r/ = 21/6 and at the cut-off r/ = 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 2 9 3 UW (z) = wall − (A7) 15 z z 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 monomermonomer 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 ) = bend (1 − cos ) where describes the polar angle between successive bonds.163 The equation of motion of each particle is given by the Langevin equation, m

drj ∂Utot d 2 rj = + + Rj dt 2 dt ∂t

(A8)

where m is the monomer mass, ri is the position of the ith monomer, is the friction coefficient. and Ri is a white

noise random force with the correlation function Ri Rj = 2kB Tij with , 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 2 / value m = 1, the characteristic time = mpp LJ becomes unity. A single - integration step is usually chosen in the interval [10−3 , 10−2 ] 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 N 2+1 ≈ N 2.2 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 l 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 l = ⫾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


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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. 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 = lmax − l0 , r = l − l0 , 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 is measured in units of thermal energy kB T. The nonbonded interactions are described by the Morse potential,

whereas = 0.25 would correspond to a rather concentrated polymer solution. For the chain model, M /kB T = 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 kB /M = 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 coll = 0.1. a much shorter range, rmin 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

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UM (r) = exp (−2(r − rmin )) − 2 exp (−(r − rmin )) , M

(A9)

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