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Self-consistent field theory for the distal ordering of adsorbed polymer: comparison with the Scheutjens-Fleer model

In a previous article [E.M. Blokhuis, K.I. Skau, and J.B. Avalos, J. Chem. Phys. 119, 3483 (2003)], a self-consistent field formalism was derived for weakly adsorbing polymers, valid for any chain length. It was shown that the presence of a solid wall induces an ordering of the polymers on the scale of the radius of gyration far away from the surface (the distal region). These oscillations in the polymer concentration profile were first noted in work by Semenov et al., and later observed in numerical solutions of the Scheutjens and Fleer self-consistent field model. In the present paper, we compare the weak adsorption model in more detail with the numerical results from the Scheutjens and Fleer model. Quantitative agreement is obtained for the polymer segment density profile in a good (athermal) solvent. For theta solvent and poor solvent conditions, it is necessary to extend the weak adsorption model to take the non-local character of the polymer-solvent interaction into account. Again, quantitative agreement is obtained for the polymer segment density profile, in particular for the transition from an oscillatory decaying profile to a monotonically decaying profile when the bulk polymer density is below a certain threshold value.


Introduction
The adsorption of polymer onto a solid surface has received considerable attention both from an experimental and a theoretical point of view. Interest in these systems is driven by practical applications, but also as a testing ground for various theoretical approaches [1][2][3].
Theoretically, surfaces with enhanced polymer adsorption were first studied by de Gennes [1,4] in the context of the Edwards self-consistent field theory [5].
In the de Gennes free energy functional, the so-called ground state dominance approximation [1,5,6] is made in which the polymer chain length is essentially set to infinity. Various extensions of the de Gennes model have been formulated. In work by de Gennes [4] and Rossi and Pincus [7] the correct scaling behavior was incorporated into the free energy functional, whereas Semenov and coworkers have extended the de Gennes model to determine finite chain length corrections [8][9][10][11][12].
The latter calculations are of particular interest since the polymer chain length is an important parameter in experiments [13] and computer simulations [14][15][16].
Variation of the chain length thus provides a more stringent testing of theoretical models [17].
A versatile model for the theoretical description of polymer adsorption is the Scheutjens and Fleer [18,19] self-consistent field model. The Scheutjens-Fleer model (SF) is a generalization of the Flory-Huggins mean-field lattice model [13] extended to describe inhomogeneous polymer systems. In the SF model, the Edwards [5] self-consistent field equations for the polymer's Green function are solved numerically on a lattice. The capability and flexibility of the SF model has been demonstrated with its use to calculate properties of inhomogeneous polymer systems containing homopolymers, copolymers and polyelectrolytes and also surfactant systems like micelles and membranes [2]. In recent years, the SF algorithm has been made available in the multipurpose computer program SFBOX [20].
Recently, a self-consistent field formalism was derived for weakly adsorbed polymers, valid for any chain length [21]. One notable difference from the classical ground state dominance result of de Gennes [1] is the distal ordering of the polymers; the finite chain length gives rise to oscillations on the scale of the radius of gyration away from the surface. These oscillations in the polymer concentration profile were first noted in work by Semenov et al. [8,11], and later observed [22] in the numerical solutions of the Scheutjens and Fleer self-consistent field model.
The onset of these oscillations has also appeared in the context of analytical work and computer simulations [15,21,23].
In the present paper, we compare the free energy functional formalism for weak adsorption [21] with the numerical results from the Scheutjens and Fleer model. In particular we address the observation made by van der Gucht et al. [22] that in a theta solvent, the oscillatory decaying function becomes a monotonically decaying function when the bulk polymer density is below a certain threshold value, similar to the Fisher-Widom transition [24,25] in the damped oscillations found in a simple liquid near a surface. Our comparison will therefore consider some general properties of the distal ordering in both good and theta solvent conditions.
The outline of this article is as follows: In the next two sections we discuss the self-consistent field theory for weak adsorption, in the context of which our calculations are made, and the Scheutjens and Fleer self-consistent field model. In Section 4, we make the comparison between our analytical results and numerical calculations carried out with the help of the computer program SFBOX. We end with a discussion of results.

Self-consistent field theory
The Green function G(r, r ′ , N) describes the statistical weight of a single polymer chain of length N with one end at r and the other end at r ′ . The Green function is determined by the Edwards equation [5]: where a is the polymer segment length and U(r) is an as yet unspecified external potential. As an initial condition to the Edwards equation, we have that In terms of the Green function, one can construct the average segment density: where N p is the total number of polymer chains. This prefactor determines the scale of the Green function; it is chosen such that the density is equal to the given, uniform bulk density, φ(r) = a 3 N p N/V ≡ φ b , for a homogeneous system. Here, φ(r) is the polymer segment number density made dimensionless by the factor a 3 and can thus be interpreted as the polymer segment volume fraction.
The statistical weight G(r, n) of a polymer chain with one end at position r is obtained by integrating the Green function over one of the chain's ends: For the case of polymer adsorption onto a planar, solid surface, the segment density profile and Green function only depend on the coordinate z so that φ(r) = φ(z) and G(r, n) = G(z, n). In the (isotropic) bulk region G(z, n) → G b (n), i.e. independent of z. By solving the Edwards equation (2.1) with the initial condition in Eq.(2.2), one may show that in the bulk region: It is convenient to redefine the statistical weight to absorb this trivial n-dependence: The Edwards equation (Eq.(2.1)) and segment density (Eq.(2.3)) are then given with the initial condition: In the self-consistent field model, the external potential U(z) is in turn expressed in terms of the polymer segment density with the result that the set of equations (2.6)-(2.7) becomes self-consistently closed. Various forms for the self-consistent external potential in terms of the segment density may be proposed. It should in some way describe the interaction between different polymer segments and of the polymer segments with the solid wall. A convenient form would be: The first two terms describe the polymer segment interaction in a virial expansion where v 0 , the so-called excluded volume parameter, is proportional to the second virial coefficient and w 2 0 is proportional to the third virial coefficient of segmentsegment interactions. The third term describes the non-local character of the interaction and reflects the non-homogeneous nature of the polymer segment distribution. Such a term is analogous to the squared-gradient term in the van der Waals theory for the liquid-vapor interface. The last term in Eq.(2.9) gives the interaction with the wall. In the following we will assume that it is short-ranged and that it can be approximated by a delta function located at the wall [1]: The parameter d > 0 is termed the extrapolation length [1]; its inverse is a measure of the surface interaction strength and leads to enhanced polymer adsorption, In the following, we will investigate the situation where 1/d is small [21] leading to weak adsorption.

Weak adsorption
The assumption of weak polymer adsorption implies that and δZ(z, n) ≡ Z(z, n) − 1 ≪ 1. Linearization of Eqs.(2.6) and (2.7) then leads to: with the initial condition: The general form for the external potential expanded around φ = φ b is given by 14) The dimensionless parameter v is a generalized excluded volume parameter. Since we have expanded around φ = φ b rather than around φ = 0, as is done in the virial expansion in Eq.(2.9), we have that v = v(φ b ) is not necessarily equal to the second virial coefficient (v = v 0 ) and thus may contain higher order interactions. For stability, we do require v > 0; to describe the situation v < 0, it would be necessary to include more terms in the expansion around φ b . The above expression for the external potential is used when we make the comparison with the SF calculations.
For the situation that m = 0, it was shown [21] that the linearized set of equations in Eqs.(2.11)-(2.14) may be reformulated into a free energy functional formalism with the free energy functional given by where the polymer's radius of gyration R g ≡ Na 2 /6 and where Minimization of the free energy in Eq.(2.15) gives for the segment density profile [21] δφ where we have introduced the parameter ε as the (square of) the bulk correlation , divided by the polymer's radius of gyration: The expression for the free energy can be generalized to the situation m = 0 The resulting segment density profile is then given by where we have introduced the parameter β as the (square of) the length defined by the ratio of m and v, ξ m ≡ m/v, divided by the polymer's radius of gyration: The length ξ m is the typical length scale connected to spatial inhomogeneities in the segment density caused by the interactions between segments excluding interactions due to the chain's connectivity (which are described by ξ b ). In a good solvent ξ b dominates over ξ m while ξ m gains in significance near the theta region.
The main goal in this article is to compare the segment density profiles given in Eqs. (2.17) and (2.20) to numerical results from the Scheutjens and Fleer lattice model.

Scheutjens and Fleer self-consistent field model
The Scheutjens-Fleer model [18,19] gives an efficient way to solve, self-consistently, the discrete version of the Edwards equation (2.6) on a lattice. The lattice distance is taken to be equal to the polymer segment length a, and 1/λ is the lattice coordination number that we set equal to 1/λ = 6 corresponding to a cubic lattice.
The distance z away from the surface is here a discrete variable, z = 0, 1, . . . , M layers (See Figure 1). A hard surface is located at z = 0, and at z = M we have reflective boundary conditions (M should be chosen large enough for the system to reach bulk density).
The polymer is modeled as a one-dimensional walk perpendicular to the surface, where the walk can either move to the neighboring layer or stay in the layer it already is. Each step is weighted with the probability of going to a neighboring layer (λ) or staying in the same layer (1 − 2λ). This gives the following recurrence relation for the Green function G(z, s) (the discrete analog of Z(z, n)), which is the statistical weight of finding the end segment s of a chain in layer z, As a starting point for the recurrence relation we have that G(z, 1) = G(z), the statistical weight of a single polymer segment. The brackets < . . . > denote the neighbor-weighted average as defined above. In the first layer, z = 1, the neighborweighted average is defined as: In analogy with the continuous case, the segment density profile is obtained by summing over all possible paths of a polymer of length N passing through position z: where the factor G(z) in the denominator accounts for the double counting of the statistical weight at layer z. The statistical weight G(z) is written as the Boltzmann factor of the segment potential The segment potential U SF (z) accounts for the interactions between polymer segments and the surface-monomer interaction. The zero of the potential is at z = ∞ so that U SF (z = ∞) = 0 and G(z = ∞) = 1. In the Scheutjens and Fleer model the form for the segment potential is chosen such that it is consistent with the Flory-Huggins free energy expression for a homogeneous polymer solution [13]. It is given by [18]: (3.5) The first term is derived from the translational entropy of the "solvent molecules" (the sites not occupied by the polymer segments) in the Flory-Huggins theory.
The second term describes the effective monomer-monomer interaction energy through the Flory parameter χ [13]. (Notice that this term is non-local since the interactions between monomers in different layers is explicitly taken into account by the use of < φ(z) > instead of φ(z).) The final term describes the surfacemonomer interaction through the Silberberg parameter χ s [26]. This parameter corresponds to the energy gain for a polymer segment to replace a solvent molecule at the surface (z = 1).
As input parameters for the SF calculations we thus have: φ b , N, χ, and χ s .
Many extensions to this model are possible and many have been made [2,20].
One may, for example, introduce different polymer architectures, inhomogeneities in more than one dimension, and curved lattice geometries, but the underlying principles remain the same.

Comparison of results
To make a comparison between the density profiles from the analytical theory and the Scheutjens-Fleer self-consistent field model, one needs to relate the parameters in the analytical theory to those in the lattice model. In particular, one needs to: To make the correspondence between χ and the parameters v and m, one therefore needs to expand the expression for U SF (z) in Eq.(3.5) around φ = φ b , using that < φ >→ φ + (a 2 /6) φ ′′ in the continuous limit (cf. Appendix A), and compare the result with δU(z) in Eq.(2.14). One then finds that v ←→ It is less straightforward to arrive at a relation between the extrapolation length d and the surface interaction parameter χ s . It should be realized that the presence of a solid surface is treated fundamentally different in the two models.
In the weak adsorption model, the treatment of the presence of the solid surface is derived from the de Gennes model for polymer adsorption [1]. In the de Gennes model, the free energy functional is strictly defined for z > 0; an interaction energy is added located at z = 0 (Eq.(2.10)), but such a term merely results in a boundary condition to the differential equation obtained from minimizing the free energy functional. As a result, the polymer segment density does not necessarily go to zero at the wall, φ(0) = 0.
In the SF model, it is implicitly assumed that the polymer segment density at the wall is zero. This can be read off from the definition of the neighbor-weighted average in the first layer (see Eq.(3.2)), which for the segment density reads: i.e. φ(0) = 0. It turns out that a consequence of this condition is that enhanced adsorption only occurs in the SF model above some threshold value, χ s > χ sc . The situation χ s = χ sc then corresponds to the condition 1/d = 0, which gives φ(z) = φ b for all z in the de Gennes model. The threshold value as a function of χ may therefore be determined by using that at the threshold one should have φ(z) ≈ φ b for all z. For very long chains one finds: χ sc = −λ χ − ln(1 − λ) [27]. Next, by comparing the expressions for the wall interaction terms in the two models, Eqs.(2.14) and (3.5), one arrives at the following identification [9,27] where we used that δ z,1 ←→ a (1 − λ) δ(z) [27].
We now have the necessary ingredients to compare the analytic expressions for the segment density profiles in Eqs. Only when one wants to make a quantitative comparison with the SF model, it is necessary to use the above relation between 1/d and χ s .
Next, we first consider the case of an athermal solvent (χ = 0) and compare the SF model results to Eq.(2.17). In particular, we will investigate the distal oscillations for weak and strong polymer adsorption. Second, we consider a theta solvent (χ = 1/2), as was previously done by van der Gucht et al. [22], who showed that when the bulk density is below some threshold value, the oscillations in the distal profile disappear similar to the Fisher-Widom transition [24,25] in the damped oscillations for a simple liquid near a surface. A quantitative comparison is made with the analytical expression for the segment density profile in Eq. (2.20).
Finally, we consider the location of the Fisher-Widom transition with respect to the complete Flory-Huggins phase diagram [13] and also make a comparison with MC simulations [15].

Athermal solvent
For an athermal solvent we have that χ = 0 so that: here, and in later plots, the absolute value has been taken. Also shown as the solid line is the ground state dominance segment density profile [1]: with the integration constant x 0 = (1/2) arcsinh(2d/ξ b ). In the comparison of The profiles corresponding to the two chain lengths are close to each other and close to the ground state dominance profile in this distance range. At very large distances, however, markedly different behavior can be observed; in Figure   2b, the same SF model results as in Figure 2a

Theta solvent
We next consider the case of a Θ-solvent (χ = 1/2) where we have that It is good to realize that theta solvent conditions do not correspond to a vanishing generalized excluded volume parameter (v = 0).
In Ref. [22] the observation was made that the oscillations, prominently present in Figures 2 and 3, disappear in a Θ-solvent when the density is below a certain threshold bulk density, similar to the Fisher-Widom transition [24] for a simple liquid. In Figure 4, we show the characteristic signature of the Fisher-Widom or to an exponentially decaying sinusoid: . To compare the analytically determined Fisher-Widom transition in Figure 5b with the results from the SF model, we located the Fisher-Widom transition by determining the bulk density where the first oscillation in the numerical density profiles disappears. This is the point where the first minimum (located closest to the wall) of the oscillating density profile crosses the value of the bulk density.
This procedure gives only an approximation to the location of the real Fisher-

Widom transition. A closer investigation reveals that the density oscillations do
not disappear all at once at the same polymer bulk density; the first oscillation disappears before the second oscillation with decreasing bulk density, and presumably before all oscillations beyond that. Still, since the bulk densities at which subsequent oscillations vanish differ only slightly, we believe this procedure to give a fairly accurate approximation.
In Figure 6, the polymer segment density profile is shown for a Θ-solvent with N = 10000 for various bulk densities just below and above the Fisher-Widom Good quantitative agreement is obtained.
In Figure 7, we have located the Fisher-Widom transition in the SF model by determining the bulk density where the first oscillation disappears for different chain lengths (circles). The surface interaction strength χ s = −1/12, as in Ref. [22].
The numerical data are consistent with a scaling of the bulk density φ * b ∼ 1/N. The solid line is the Fisher-Widom transition shown in Figure 5b, where we have used that N = ε/(2β 2 ) and φ b = 2β/ε to transform from (ε, β) to (N, φ b ) as variables. Good agreement is obtained with the SF calculations.
Also shown in Figure 7, as the squares, are the SF calculations results for N = 200 and N = 1000 reported in Ref. [22], which differ somewhat from our results.
Since these SF calculations are done with exactly the same set of parameters used to determine our results represented by circles, this difference must be attributed to the different procedure used in Ref. [22] to locate the Fisher-Widom transition.
In the analysis in Ref. [22] the segment density profile is taken to be of the following form: Then, by locating, as a function of φ b , the position where ξ b,num /2 equals R g /A num , the Fisher-Widom transition is determined.
Even though the analysis in Ref. [22] is done in a more judicious manner than simply fitting the numerical segment density profile to Eq.  Figure 4 for φ b = 0.01 lies in the purely exponentially decaying-region, which seems unlikely.
Nevertheless, it is expected that this fitting procedure is more reliable in the region where the different length scales are well-separated, e.g. when φ b ≫ φ FW b . In this region, the fit parameters were numerically determined as [22] A num ≃ 1 0.19 These values can be compared to the analytical density profile in the corresponding regime: ε ≪ 1 and β ≪ 1. For β = 0 and expanding in ε, it was shown that the segment density profile is then given by [21] δφ(z) Thus, we find in the region φ b ≫ φ FW b good agreement between the analytical results [21] and the fitting parameters in Eq.(4.10).

Phase diagram
Since the Fisher-Widom transition found in a Θ-solvent seems to be absent in an athermal solvent, it is worthwhile to locate the Fisher-Widom transition as a function of solvent quality.
In Figure 8,

Comparison with Monte Carlo simulations
Recently, de Joannis et al. [15] performed Monte Carlo (MC) simulations of polymer chains at surfaces in an athermal solvent. In Figure 9a, the reduced segment density δφ(z)/φ b is shown as a function of z/R g (Figure 9b shows in the polymer concentration profile can be observed, even though it is not pos-sible from the numerical data (and it is hard to get more accurate profiles) to observe more than the first oscillation.

Discussion
In this article, we have compared the analytic expressions for the segment density profiles obtained in the context of the weak adsorption model [21] to the numerical results of the Scheutjens-Fleer model [18,19]. The analytic expressions are expected to be valid when the polymer adsorbs only weakly to the solid surface or, for strong adsorption, when the distance to the surface is large. Indeed, excellent quantitative agreement for the distal, oscillatory density profile was obtained for very good (athermal) solvent conditions, for both weak and strong polymer adsorption (Figures 2 and 3).
We have further investigated the previously observed Fisher-Widom transition in SF model calculations by van der Gucht et al. [22] for a Θ-solvent. We demonstrated that the location of the Fisher-Widom transition can be described by the weak adsorption model when the non-local character of the monomer-solvent interaction is taken into account. The quantitative agreement that is then achieved (see Figures 6,7 and 8), gives confidence in identifying the non-local character of the interactions in the SF model as the source of the Fisher-Widom transition observed [22]. To further substantiate this identification, we also carried out separate SF calculations in which the non-local interactions were removed by replacing < φ(z) > by φ(z) in Eq. (3.5). In this way, we could explicitly verify [29] that without the non-local interactions present, the oscillations always remain with decreasing bulk polymer concentration while maintaining the quantitative agreement with the segment density profiles of the weak adsorption model.
Although it is shown that the introduction of a non-local interaction term is  Figure 8).
We therefore expect our mean-field results to be most relevant to describe the occurrence and location of the Fisher-Widom transition in experimental polymer systems in the theta region.
It may seem somewhat surprising that the non-local interactions in the SF model, whose range is only one lattice distance a, may interfere with the oscillatory behavior on the scale of the polymer's radius of gyration. However, the length scale connected with the non-local character is rather ξ m ≡ m/v which diverges (similar to the bulk correlation length near the critical point in an ordinary liquid) when v is small, i.e. when the solvent is poor enough.
The Fisher-Widom transition in a simple liquid occurs when the attractive part of the interaction potential between the molecules dominates over the repulsive part [24,25]. For high densities the hard core interaction gives damped oscillations away from the surface, while these oscillations disappear when the inter-molecular attraction balances the repulsion. The similarity with the transition observed for polymers is striking; the Θ-point defines the system conditions where the monomer attraction cancels the monomer repulsion. For a good solvent the effective monomer interaction is repulsive and oscillations are predicted for all densities [21]. There are, however, nontrivial differences between monomers and liquid molecules; the most important difference, of course, is the polymer chain connectivity. The chain connectivity gives rise to oscillations with a wavelength of the size of polymer coil, while in a simple liquid the wavelength is given by the molecular radius. It is tempting to think of the polymer solution as a "liquid" of polymer coils close to the surface. But this intuitive idea fails to consider the strong interdigitation of the coils at semi-dilute concentrations and that the coil changes its conformation and size close to the surface (while a liquid molecule keeps its hard core shape). One should therefore be careful to make too much out of such analogies.
In the comparison of the Scheutjens and Fleer model [18,19] to analytical theories such as the de Gennes model [1,4], and extensions thereof such as the weak adsorption model [21], a number of differences can be distinguished. A clear difference is the fact that the SF model is a lattice model with the obvious consequence that the parameter describing the distance to the substrate is discrete; z = 1, 2, 3, . . .. The discreteness of space is most consequential in the very vicinity of the wall, where the monomer length scale becomes important.
Another important difference between the SF model and the de Gennes-like models concerns the boundary condition at the wall. In these latter models, the free energy functional is only strictly defined for z > 0. An interaction energy is defined located at z = 0, but the addition of such a term merely serves as a boundary condition to the differential equation which results from the minimization of the free energy functional. The result is that the polymer segment density not necessarily goes to zero at the wall, φ(0) = 0. Even though in real polymer systems, the polymer segment density should become zero (or exponentially close to it) at some point, it is expected that a microscopic shift in the precise location of the wall-this location is not well-defined within a microscopic distance anyway-should restore the validity of the de Gennes-like models at least for enhanced polymer adsorption [4].
In the SF model, the free energy functional is already defined for whole space from the start. The reason that the density profile is zero when z ≤ 0 is due to the explicit addition of an external potential which gives an infinite energy penalty for z ≤ 0. Apart from this external potential there may be an energy contribution to the first layer at z = 1 which will generally be attractive. The result of the infinite repulsion at z ≤ 0 is that the polymer density is zero at the wall φ(0) = 0. The presence of this extra boundary condition makes the SF model essentially different from the de Gennes-like models, but similar to the single chain mean-field theory by Szleifer and others [30,31] and computer simulations [14][15][16]. This boundary condition (but without the presence of any additional attraction) is also used in the calculations by Eisenriegler and coworkers [3,32,33]. Still, it seems, as we have done here, that a meaningful comparison between the SF model and the de Gennes-like models can still be made when the adsorption energy in the first layer is above some given threshold value.

A Discrete and continuous Edwards equation
In this appendix, we show [2] how the recurrence relation for G(z, s) in Eq. (3.1) in continuous form reduces to the Edwards equation (2.6). First, we rewrite the recurrence relation for G(z, s) as: where we used the expression for G(z) in Eq. and, keeping in mind that the distance z was rescaled by a, that When G(z, s) is a slowly varying function, it implies that the potential U SF (z) is close to its asymptotic value U SF (z → ∞) = 0, so that we can expand

B Asymptotic behavior of the density profile
In this appendix, we investigate the asymptotic behavior of the polymer segment density profile given in Eq. (2.20). We write Eq.(2.20) as: The integral in Eq.(B.1) can be solved [34] ∞ 0 dt cos(xt) where R ′ ≡ sum of residues of Q(z) e ixz in the upper half plane.
The poles are determined by the zero's of g(z) in the complex plane (see also [25]).
It turns out that there are an infinite number of poles in the upper half plane.
Luckily, to describe the asymptotic behavior of Eq.(B.1), it is sufficient to locate the pole with the lowest imaginary part. Depending on the value of ε and β, this pole is either purely imaginary-let's denote it as z 0 = iA 0 -or it is complex and then it comes in pairs: z 1 = ±B 1 + iA 1 , where A 0 , A 1 , and B 1 are positive, real numbers. Taking only these three competing poles into account, we can determine the residues with the result: where and where z 1 is taken to be the pole in the upper right quadrant.
As a final result, we thus have for the asymptotic density profile: