#### Essentials of the Self-consistent Field Approach to Polymer Brushes

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

We begin with the continuum formulation of the theory, and write down the partition function for 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

- (31)

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

- (32)

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

- (33)

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

- (34)

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

- (35)

with ϕ(*z*) being the average monomer density at distance *z*. The mean field free energy per polymer, in units of *k*_{B}*T*, is

- (36)

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

- (37)

so that the single-chain partition function becomes

- (38)

Here, the prime on the functional integral over all stretching functions *E* means that only those are selected which satisfy the constraint that all polymer paths have the same length (*N*). Anticipating that paths that start out at a value *z*_{0} very near the grafting surface may first move away from the surface before returning to it, *z*_{m}(*z*_{0}) has been defined as the largest value of *z* reached by a path that starts at *z*_{0}.

From eq 38 the self-consistent equation for the density ϕ(*z*) follows by functional differentiation, ϕ(*z*) = −σδln*Q*/δ(*w*ϕ). 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*,*z*_{0}) which extremizes the mean field-free energy *F*. Thus, in the path integral all paths except the most probable path for a given *z*_{0} are eliminated, and hence the self-consistent equation which determines the density becomes in the classical limit

- (39)

Note that *z*_{0}(*z*_{m}) is the inverse function of *z*_{m}(*z*_{0}), and the end-point distribution ρ_{e}(*z*_{0}) then is found as

- (40)

Using the normalization ∫*dz*_{0}ρ_{e}(*z*_{0}) = σ, which expresses the fact that for each grafted chain the free end must be at some distance *z*_{0}, one can derive *Q* to find

- (41)

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

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

- (42)

- (43)

which satisfy the normalizations

- (44)

In terms of the rescaled stretching function *ẽ*(*z̃*, *z̃* = 0) ≡ *e*(*z*,*z*_{0})*N*/*ẑ*, the free energy eq 36 then becomes

- (45)

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

- (46)

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

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

- (48)

and

- (49)

In the SSL, there are no monomers beyond the rescaled brush height *h̃* (note that *h̃* differs from *ẑ* by a numerical factor), while the full SCFT shows a Gaussian tail even if β is very large. This smooth behavior near the brush height is well established from both experiment and simulation (Figs. 7 and 8).188, 189 These data are compatible with the predicted scaling behavior for the brush height *h* ∝ σ^{1/3}*N* (eqs 3, 43) and show that indeed the monomer profile is neither the step function of the Alexander-de Gennes theory26, 27 nor strictly parabolic, as implied by the SSL-SCFT,29, 30 while the data are (at least qualitatively) compatible with the classical approximation to the SCFT.51, 52 Note that Figure 7(b) includes already data on variable solvent quality, but we defer a discussion of this aspect to a later section (Section Varying the Solvent Quality: Theta Solvents and Poor Solvents). Unfortunately, we are not aware of any direct experimental measurements of ρ_{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 simulations can also study subtle effects due to topological interactions which appear when one grafts noncatenated rings (of length *N*_{R} = 2*N*) 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 ϕ(*z*), so local fluctuations in the density are ignored, and in addition, that one ignores all but the most probable polymer configuration (the “classical path” is assumed to dominate the functional integral, eq 31). Implicit in this treatment is the Gaussian chain statistics, which has been built in from the start (eq 31). The excluded volume interaction must be sufficiently weak, so that inside of the blobs (considered in Section Scaling Concepts for Brushes under Good Solvent Conditions) the chain conformation indeed would be Gaussian still (which requires *w*/*a*^{3} ≪ *a*σ^{1/2}). The validity of the classical approximation requires strong stretching, β > 1 (thus, the curves for β ≤ 1 in Figure 6 are not expected to be quantitatively reliable). On the other hand, σ*a*^{2} < 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

- (50)

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

- (51)

Here, *h*^{*} is the height of the uncompressed brush (corresponding to *h̃* in eq 48) and obviously Π_{osm} = 0 for *h* ≥ *h*^{*}. Writing *h* = *h*^{*}(1 − Δ) with Δ a small parameter, one finds that Π_{osm} ∝ Δ^{2} while the Alexander brush picture would imply Π_{osm} ∝ Δ for small Δ. Of course, eq 51 implies that *F*(*h* > *h*^{*}) ≡ 0 because there can be strictly no force between two brushes that are more than a distance 2*h*^{*} apart. Finally, we note that for “dry” brushes (no solvent being present) the volume fraction ϕ taken by the monomers can be reinterpreted as the total density in the system, and then Π_{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 diagram of mixed polymer brushes. The energy term (*w*/2) ∫ *d* )^{2} in eq 31 then needs to be replaced by

- (52)

with , —the densities of both species *A*, *B*, and *w*_{AA}, *w*_{BB}, and *w*_{AB} describing the strength of the excluded volume forces between the different types of monomer pairs. Although *w* = (*w*_{AA} + *w*_{BB} + 2*w*_{AB})/4 characterizes the average strength of excluded volume in the brush, the normalized Flory χ-parameter = (2*w*_{AB} − *w*_{AA} − *w*_{BB})/2*w* denotes the mutual attraction (repulsion) between unlike monomers. It is related to the commonly used Flory-Huggins parameter160, 165 χ_{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 *k*_{B} 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 ω(*z*)). We now can write *F* = *U* − *T*(*S*_{A} + *S*_{B}), where in SCFT

- (53)

and similarly for species *B*. Here, means the single-chain partition function of an *A* chain that is grafted at the site at the grafting surface, the corresponding monomer density and 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 of all *A* chains.

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 short-range order and no true long range order:20 All the theory presented in this section has ignored fluctuations in the local density of the grafting sites. Such fluctuations are expected due to the process by which polymer brushes are prepared,7 of course. Although for homopolymer brushes simulations36 have indicated that the observables of interest [density profile ϕ(*z*) and ρ_{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 in polymer brushes is the polydispersity in the distribution of the chain lengths; this problem will be addressed in the next section. Recent extensions of SCFT on mixed polymer brushes include also detailed studies on self-assembly in confinement.195

We conclude this section by emphasizing that the numerical solution of differential equations such as eqs 55, that is inherent in the field-theoretic formulation of SCFT which was presented so far, eqs 31–58, can be circumvented by the lattice formulation due to Cosgrove et al.28 based on the Scheutjens-Fleer theory.185, 186 One considers in this theory a polymer-solvent system at a substrate surface, and aims to take into account all possible conformations, each weighted by its probability as given by the Boltzmann factor, assuming that both monomers of the chain and solvent particles can occupy the sites of a regular lattice, multiple occupancy of lattice sites being forbidden. To study a brush, where the chains are terminally attached to the substrate, conformations are restricted by requesting that the first chain segment is in the layer adjacent to the substrate. One considers a lattice of size *L* in *z*-direction perpendicular to the substrate, and taking now the lattice spacing as unit of length, one labels the layers consecutively as *z* = 1,2,…, *L* (layer 1 is adjacent to the substrate, layer *L* is outside of the brush in the dilute solution. We denote the coordination number of the lattice as *q* (*q* = 6 in the simple cubic (*sc*) lattice, *q* = 12 in the face-centered (*fcc*) cubic lattice), with *q*_{0} being the number of nearest neighbors in the same layer (*q*_{0}^{sc} = 4, *q*_{0}^{fcc} = 6). The fraction of sites λ_{0} = *q*_{0}/*q* in the same layer then is λ_{0}^{sc} = 2/3 or λ_{0}^{fcc} = 1/2, while the fraction of sites in an adjacent layer is λ_{1} = (1 − λ_{0})/2 = λ_{1}^{sc} = 1/6 or λ_{1}^{fcc} = 1/4, respectively. Of course, the choice of the type of lattice is arbitrary, and for long enough chains the results on brush properties should not depend on this choice.

The statistical mechanics of such a model could be obtained “numerically exactly” (i.e., avoiding systematic errors apart from statistical errors) by Monte Carlo simulation,196–198 see Appendix. The lattice formulation of SCFT, on the other hand, is based on making a mean field approximation in each layer, so the quantity that matters is just the monomer density profile ϕ(*z*); the profile of the solvent density is ϕ_{s} = 1 − ϕ(*z*), every site being occupied by either a monomer or a solvent molecule. Nearest-neighbor interactions between monomers and solvent are described by the Flory-Huggins parameter χ_{FH}. So the potential energy *u*(*z*) of a monomer (relative to that of a monomer in a bulk solution at concentration ϕ_{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, 〈ϕ(*z*)〉 = λ_{1} ϕ(*z* − 1) + λ_{0} ϕ(*z*) + λ_{1} ϕ(*z* + 1)). The logarithmic terms account for the change in translational entropy of solvent molecules, which is thus included into the effective potential *u*(*z*), while the entropy of the polymer needs to be calculated considering its conformational statistics, as described below.

Defining a monomer weighting factor *G*(*z*) = exp[−*u*(*z*)/*k*_{B}*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

- (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 ϕ(*z*,*s*) of the *s*'th monomer in layer *z* becomes

- (62)

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

- (63)

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

- (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

- (65)

The normalization constant *C*_{g} is then fixed by the condition that every one of the *n*_{g} grafted chains has to have its end somewhere,

- (66)

From this formulation, it is obvious that an extension of the formalism to systems containing both grafted and free chains is rather straightforward39; also an adsorption potential due to the grafting wall is readily included. When we sum eq 65 over *s*, to obtain the profile ϕ(*z*) = ∑_{s = 1}^{N} ϕ_{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 (∫*dz*_{0} … become 4π∫*r*_{0}^{2}*dr*_{0} … for spheres or 2π ∫ *r*_{0}*dr*_{0} … 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,…,*L*_{z}, but in each plane one then envisages circles at radii *r* = 1,2,…, *L*_{r}. One now needs to define a-priori step probabilities, which are determined by the fraction of sites in neighboring layers adjacent to a given site on a lattice. In the *r*-direction, these probabilities Λ(*r*|*r* − 1) to go inside, or Λ(*r*|*r* + 1) to go outside, follow from geometric considerations199

- (67)

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

#### Other Theoretical Concepts

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

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

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

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 *k*_{B}*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 chemical potential functional μ(*z*) with respect to position *z*(*s*) of monomer *s*,

- (68)

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

- (69)

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

- (70)

Note that the results of the SSL of SCFT, namely30

- (71)

imply a very similar scaling of this product *h*ϕ(0)/*R*_{g}, namely

- (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*/*R*_{g} is simply proportional to (σ*R*_{g}^{2})^{(1 − ν)/2ν} and that ϕ(0) is proportional to (σ*R*_{g}^{2})^{(1/2)(3 − 1/ν)} were found already for *N* ≤ 100 (Fig. 10), and eq 70 was confirmed.

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*^{ν}) fully holds on the scale of a blob [in Fig. 1(b)]. Although this is true for the case of the Bond Fluctuation Model with purely repulsive interactions on the lattice, for real polymers the excluded volume strength often is relatively weak, in particular if the temperature of the polymer solution is not very far from the Theta-point θ: then one still has Gaussian-like statistics on the scale of thermal blobs ξ_{T} ∝ | *T* − θ]^{−1},160 and if the size of the thermal blobs would exceed the size of the blobs resulting from the grafting, ξ_{blob} ≈ σ^{−1/2}, excluded volume on this scale is negligible and thus the standard SCFT theory (Section Essentials of the Self Consistent Field Approach to Polymer Brushes) should hold.

This consideration leads to the question of describing the crossover between these two extreme cases55 and is addressed in Figure 11: the brush height *h* can actually be considered as a function of two scaling variables, Figure 11, namely ζ = σ*R*_{g}^{2} and ζ^{′} = *v*_{ex}*N*^{2}/*R*_{g}^{3}. For small ζ, one has mushroom behavior, while for large enough ζ, one has brush behavior; good solvent behavior where excluded volume dominates requires that ζ^{′} is large enough, while the regime where *v*_{ex}*N*^{2}/*R*_{g}^{2} 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

- (73)

but the scaling function *H̃* is only known well inside the regions shown in Figure 11, i.e. not near the broken straight lines in Figure 11 that indicate where a smooth gradual crossover from one limiting behavior to the other occurs. Note also that the regime of “marginal solvents” does not include the Theta point itself: as is well known, one needs to include the third virial coefficient at *T* = θ where the second virial coefficient (which is proportional to *v*_{ex} or *w*) vanishes; the behavior at *T* ≤ θ 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 consideration of these effects in terms of the SSL of SCFT89, 207 is difficult and only for special assumptions on the molecular weight distribution (MWD) explicit solutions are possible.89 One general result is that in a polydisperse brush each free chain end has a well-defined distance from the grafting surface in the brush, unlike the monodisperse case where the free ends exhibit a distribution in their position, that extends throughout the brush (eq 49), and thus each chain undergoes in a sense “critical fluctuations.”88 Because all chains are equivalent, each chain end explores the full range of the distribution, eq 49, rather than staying fixed in a particular region of the brush. This fact has particular consequences on the dynamics of chains in brushes,88 but this aspect is beyond the scope of the present article.

Although both SCFT and scaling theories consider the asymptotic behavior of very long chains in not too dense brushes on coarse-grained length scales, it should be noted that much of the experimental work,7 and also some of the simulations discussed in Section Conclusions, are concerned with rather dense brushes of short chains. Then, an aspect comes into play that escapes both SCFT and scaling, namely, the problem of properly “packing” the monomers near the (hard) wall to which the chains are grafted. These types of effects that happen on the scale of individual monomers, rather than on the scale of blobs, can be captured by another type of mean field theory, namely density functional theory (DFT).208–211 Considering, for instance, a system where a brush interacts with free chains, one is interested in their monomer density profiles ρ_{f}( ), ρ_{g}() 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, *N*_{f} = 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 ρ_{f}(*z*) and ρ_{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 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* ∝ σ^{(1 − ν)/2}*R*_{c}^{1 − ν} *N*^{ν} ≈ σ^{1/5}*R*_{c}^{2/5}*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

#### 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 *f*_{int}(ϕ) is [*k*_{B}*T* ≡ 1 here]

- (74)

The corresponding exchange chemical potential (for replacing an effective monomer by a solvent particle) is then μ(ϕ) = ∂*f*_{int}(ϕ)/∂ϕ. This chemical potential must be combined with the contribution due to chain stretching in the brush. In SCFT-SSL, this contribution results from a potential *V*(*Z*) = 3π^{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(π*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

- (75)

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

- (76)

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

- (77)

and the osmotic pressure is

- (78)

with Π(ϕ(*Z* = *H*)) = 0, the osmotic pressure is zero at the outer boundary *Z* = *H* of the polymer brush. For small ϕ, eq 78 becomes Π(ϕ) ≈ ϕ^{2}( − χ) + ϕ^{3} and hence for χ > 1/2 (where a bulk polymer solution for *N* ∞ exhibits phase separation at Π = 0), the polymer-rich phase of the polymer solution has a monomer volume fraction ϕ_{coex} ≈ χ − 1/2. Then, the profile ϕ(*Z*) of the monomer volume fraction in the brush decreases from its maximum value ϕ_{max} = ϕ(*z* = 0) monotonically and continuously down to ϕ (*Z* = *H*) = ϕ_{coex} at *Z* = *H*, where a discontinuous jump to zero occurs. This jump vanishes linearly with χ when the critical point χ_{crit} = 1/2 of the polymer solution is reached. Although ϕ(*Z*) vanishes linearly with *Z* for χ < χ_{crit} (there eqs 48 and 49 apply), for χ = χ_{crit} one finds a square root singularity37

- (79)

and one also finds that for χ = χ_{crit} (i.e., at the Theta-point of the polymer solution) a different scaling with grafting density σ occurs, namely (we restore here again physical units, *w*_{3} is essentially the third virial coefficient37)

- (80)

while in the poor solvent region one predicts37

- (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 (Δ*F*_{el} = *h*^{2}/*N*) and of interactions,

- (82)

where now *w* ∝ (χ_{crit} − χ) may change sign at χ = χ_{crit}. At χ < χ_{crit} the term proportional to *w*_{3} can be neglected for small grafting density σ, of course. Minimizing the free energy Δ*F*_{el} + Δ*F*_{int} with respect to *h*, ∂(Δ*F*)/∂*h* = 0, yields for χ > χ_{crit} *h* ∝ *N*(*w*σ)^{1/3}x, 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

- (83)

- (84)

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 ϕ(

*z*) is a smooth function of

*z*, irrespective of solvent condition, and the same is true for ρ

_{e}(

*z*), see Figure

15. Disregarding the structure of the profiles at small

*z* near the grafting plane, we see that the profile ϕ(

*z*) of the monomer density is nearly parabolic in the good solvent regime but develops an almost horizontal part [where no solvent occurs, see part (b) of the figure] for the poor solvent case, separated from the pure solvent by a rather narrow surface region of an essentially polymer melt-like structure. The width of this region is not expected to be the intrinsic width of such an interface, but presumably is broadened by capillary waves.

213,

214 When

*N* is increased, keeping the grafting density and solvent quality unchanged, just the width of the horizontal part increases, but the qualitative behavior stays the same. When one passes the region of the Theta-point, all profiles change completely gradually and no singular behavior occurs in agreement with experiment, Figure

7(b).

189 Although this smooth behavior is also obtained with numerical versions of the SCFT theory, we are not aware of any explicit analytic theory describing the rounding of the singularities that were derived above.

- (ii)
For not too long chains and small enough grafting density, one finds microphase separation in the brush, it laterally decomposes into monomer-rich clusters separated by regions that are almost free of monomers.

40,

42 Figure

1 (upper-right panel) and Figure

16 give characteristic examples. Of course, due to the constraint that for each chain one end is irreversibly grafted to the substrate, lateral motion of this end being forbidden, no monomer can get arbitrarily far from the grafting site of the chain to which the monomer belongs; thus, macroscopic lateral phase separation cannot occur.

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

In fact, in a polymer solution under poor solvent conditions it is known that phase separation occurs for *T* < *T*_{c}(*N*), with the critical temperature scaling as160, 165, 191 *T*_{θ} − *T*_{c}(*N*) ∝ *N*^{1/2}, and the critical monomer volume fraction ϕ_{c} scaling as ϕ_{c} ∝ *N*^{−1/2} as well. For *T* ≪ *T*_{c}(*N*) phase separation occurs between essentially pure solvent and a semidilute solution of volume fraction ϕ_{coex} ∝ T_{θ} − *T*(∝ χ − χ_{crit}, if the model eq 74 is used). From eq 80 we can conclude that at *T*_{θ}, or nearby, the volume fraction inside of a brush is ϕ ≈ *N*/(σ^{−1}*h*) since σ^{−1}*h* is (apart from a prefactor of order unity) the volume available per grafted chain, and hence ϕ ∝ σ^{−1}. On the other hand, at *T* = *T*_{θ} the crossover between mushroom behavior (where *h*_{m} ≈ ) and brush occurs for *h*/*h*_{m} ≈ 1, therefore σ^{1/2}*N*/ ≈ const, that is, σ^{*} ≈ 1/*N*.

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

Tang and Szleifer45 presented an interesting attempt to compute the collective structure factor of the grafted polymer layer, using the random phase approximation (RPA)160, 191 which relates the collective structure factor to the structure factor of an ideal (noninteracting) system of grafted chains, *S*_{0}( )

- (85)

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

- (86)

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

- (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*(σ*a*^{2}) ≫ 1 while the inhomogeneous structures occur for *q* of order unity (and the characteristic size of the “cluster” or “dimples” is of order *L*_{cluster} ≈ 2π/*k*^{*} ≈ 2 *a*, consistent with the simulations.40, 42 Again, as in the case of microphase separation in mixed binary polymer brushes, one may argue that the random fluctuations in the density of grafting sites destroy sharp phase transitions, so the transitions from collapsed mushrooms to “clusters” of collapsed chains (“dimples”) and then to a homogeneous layer are just gradual crossovers.

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 *R*_{cyl} ≫ *a* 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 ν = 3/5) one would predict that the swollen brush has a radius *R* ∝ *N*^{3/4}σ^{1/4}τ^{1/4}217 (τ = |(*T* − *T*_{θ})/*T*_{θ}|, cf. also eq 20), for τ = 0 (Theta conditions), one rather finds *R* ≈ *N*^{2/3}σ^{2/3}), while the collapsed brush scales as *R* ∝ (*N*σ/|τ|)^{1/2}. Sheiko et al.217 predict that the pearls rather have a radius *R*_{pearl} ∝ *N*^{1/2}(σ/|τ|)^{1/4} and an axial length *L*_{pearl} ∝ *N*^{1/2}(|τ|/σ)^{1/2} so each pearl contains *n*_{pearl} ∝ *N*^{1/2}|τ|^{1/4}σ^{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 Theta-solvent conditions, the monomer density profiles were found66 to qualitatively resemble those of planar brushes. Thus, for large enough grafting density σ and long enough chains (20 ≤ *N* ≤ 80 could be studied66), the density profile has an extended horizontal part (corresponding to almost melt density inside of the brush). Thus, the spherical core to which the chains are grafted is coated by a “shell” of thickness *h*, the brush height. Although it turned out not to be feasible to study the crossover from spherical to planar brushes with so short chains, a numerical SCFT calculation66 (using a lattice formulation adapted to spherical geometry) could show that the brush height *h*(*R*,*N*, σ) scales as in eq 15. Yet, as the height *h*_{0} ∝ σ^{1/2}*N* for *T* = *T*_{θ} (cf. eq 80) and the size of the star polymer scales as *R* ∝ *N*^{1/2} at the Theta-point, the scaling function *f*(*h*_{0}/*R*_{c}) in eq 15 scales like *F*(ζ) ∝ ζ^{−1/2}, and this scaling has been verified.65

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

#### Dense Brushes and Brushes Interacting with Polymer Melts

Although in the previous section we have given a (qualitative) discussion of the phenomena in poor solvents, where high monomer density in a brush can result from the collapse of the grafted chains, but still the grafting density σ was assumed to be small (emphasizing even the crossover to the collapsed mushroom regime), we consider now different regimes where high monomer density in a brush matters: such a case occurs also for good solvent (and Theta solvent) conditions when one considers the limit σ 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

- (88)

where *F*_{el} 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*σ^{−1/2} the distance between the grafting sites, so the volume taken by a chain is *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 forces, and this term can be understood from a virial expansion of the Flory- Huggins-equation of state of a polymer solution160, 165, 191

- (90)

Equation (90) is the free energy of mixing per site in a lattice model, ϕ being the fraction of sites taken by polymers, the second and third terms on the right hand side of this equation were already encountered in eq (74). From the expansion of eq (90) for small ϕ, we now can conclude that the second virial coefficient υ_{ex} is related to the Flory -Huggings parameter χ as υ_{ex} = *a*^{3} (1 − 2χ), remembering that *a* is the lattice spacing of the Flory-Huggins model, a single lattice site hence is associated with a volume *a*^{3}. The third virial coefficient is *a*^{6}, and is temperature-independent, unlike υ_{ex} which becomes negative below the Theta temperature (for moderately good solvents 0 < χ < 1/2 and hence υ_{ex} < *a*^{3}). This consideration justifies the choice of coefficients in eq (89). Note that *a*^{3}*N*/(*hD*^{2}) is the volume fraction of monomers ϕ in the volume taken by a chain, as usual, and the total enthalpy due to pair interactions is put (*hD*^{2})ϕ^{2}*v*_{ex} in eq (89). Factors of order unity are ignored throughout.

Now two cases must be considered, depending on which of the two terms in eq (89) dominate when we minimize *F* with respect to *h*. One easily finds that for σ* < σ < σ_{1} = (υ_{ex}/*a*^{3})^{2}, it is the excluded volume interaction term that dominates; neglecting hence the term with the third virial coefficient the condition ∂*F*(*h*)/∂*h* = 0 yields (ignoring prefactors of order unity) that *h* = υ_{ex}^{1/3}*N*(*a*/*D*)^{2/3} = υ_{ex}^{1/3} *N*σ^{1/3}, that is eqs (3), (43), and (51). However, for σ > σ_{1} it is the term due to the third virial coefficient which dominates, and hence one obtains instead from ∂*F*(*h*)/∂*h* = 0 that

- (91)

Note that there occurs a smooth crossover between eqs (51) and eq (91) at σ = σ_{1}, with *h* = *N* υ^{ex}/*a*^{2} there. We also observe that the average volume fraction of monomers in the brush in the regime described by eq (91) is ϕ = *Na*^{3}/(*hD*)^{2} = σ^{1/2}; so only for σ 1 one also obtains ϕ = 1. Eq (91) was already quoted in eq (80) for brushes at the Theta temperature, but it is important to note that it also holds in the good solvent regime at high enough grafting densities. We also note that an approximate extension of SCFT to this regime σ > σ_{1} predicts an elliptic rather than parabolic volume fraction profile44

- (92)

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

We now discuss the extension of the theory to the case where a polymer brush interacts with a polymeric matrix as a solvent, rather than a small molecule fluid. We here restrict attention to the case that the polymer melt that a polymer brush interacts with has a degree of polymerization *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 ϕ_{N}, ϕ_{P} = 1 − ϕ_{N}153, 165, 191

- (93)

which in the limit of small ϕ_{N} leads to

- (94)

and hence one can conclude that υ_{ex} in the previous treatment needs to be replaced by *a*^{3}/*P*, and the third virial coefficient becomes *a*^{6}/*P* (note that we consider only a brush in a melt of identical chains, so χ = 0). Writing then, in the limit where pair interactions dominate, in analogy to Eq. 89

- (95)

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

- (96)

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

- (97)

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

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

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