4552
J. Phys. Chem. B 2008, 112, 4552-4560
Density Functional Approach to Adsorption of Simple Fluids on Surfaces Modified with a Brush-like Chain Structure A. Patrykiejew and S. Sokołowski Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska UniVersity, 20-031 Lublin, Poland
R. Tscheliessnig and J. Fischer BOKU UniVersity of Natural Resources and Applied Life Sciences, Institute for Chemical and Energy Engineering, Muthgasse 107, A-1190 Vienna, Austria
O. Pizio* Instituto de Quı´mica de la UNAM, Coyoaca´ n 04510, Me´ xico D.F., Me´ xico ReceiVed: NoVember 18, 2007; In Final Form: February 13, 2008
A density functional theory to describe adsorption of a simple fluid from a gas phase on a surface modified with pre-adsorbed chains is proposed. The chains are bonded to the surface by one of their ends, so they can form a brush-like structure. Two models are investigated. According to the first model all but the terminating segment of a chain can change the configuration during the adsorption of fluid species. The second model assumes that the chains remain “frozen”, and the system is considered as a nonuniform quenched-annealed mixture. We apply simple form of interactions to study adsorption phenomena, microscopic structure, and layering transitions. Our principal findings show that new layering phase transitions can occur because of a chemical modification of the substrate under certain conditions, in comparison with nonmodified surfaces. However, opposite trends, that is, smoothing the adsorption isotherms, can also be observed, depending on the surface density of the grafted chains.
I. Introduction Thermodynamic and structural properties of chemically modified adsorbents obtained by covering solid surfaces with a thin layer of pre-adsorbed molecules have been the subject of numerous studies.1-4 These investigations pointed to several applications of such adsorbents, connected with the stabilization of colloidal suspensions, change of adhesive properties, lubrication, and so forth. The surfaces covered by chemisorbed molecules are also used as the stationary phases in gas chromatography and in reversed-phase liquid chromatography.5 Among different types of modified adsorbents an important group of adsorbents is obtained by chemical attachment of chain molecules to the solid surface. The properties of the systems with end-grafted chains depend on several parameters, such as the length of chains, the chain stiffness, and the surface grafting density, as well as on the solvent. The end-grafted chains may form mushroom-like or brush-like structures depending on the grafting density. For low grafting densities, the chains are isolated and form coils or mushrooms at the surface. In contrast, if the grafting density increases, the interchain repulsion causes the polymers to stretch perpendicular to the surface. Then, the resulting structure is called the brush. One of the first theoretical treatments of grafted polymers was developed by Alexander6 and de Gennes.7 The next step toward a detailed understanding of these systems was made by Milner et al.8-10 who obtained an analytic solution of the classical self-consistent field equations. The self-consistent field * Corresponding author.
approach was then employed by several authors10-19 to describe the structure of modified adsorbents. An alternative theoretical approach to describe grafted chains is the so-called single-chain mean-field method, according to which the statistical mechanical problem of a single chain is solved exactly, and the interaction with the other chains is described using a mean-field treatment.20 More recently, McCoy and co-workers proposed a version of the density functional theory to study chains, tethered to a surface, both in the presence of a solvent and in a continuum-solvent approximation.21-25 Using as an input the expression for the free energy from the Flory-Huggins theory, they also showed that the proposed density functional approach reduces to a self-consistent field theory. Computer simulations have also been performed to study brush structures and parameters that affect them.26-36 One of the important objectives of the simulations was to test the validity of the self-consistent field approaches. Computer simulations were also used to investigate various problems related to specific applications of the adsorbents modified with polymer brushes. A challenging problem for some applications is the occurrence of wetting transitions in such systems.37,38 Another interesting phenomenon is the retention of solutes in chromatographic separations by column packings with the chemically bonded phases. This problem was investigated using molecular dynamics simulations.39-41 Quite recently, Boro´wko et al.42 applied a version of the density functional approach43-46 to describe adsorption of hardsphere mixtures on a surface covered with a brush. Monte Carlo
10.1021/jp710978t CCC: $40.75 © 2008 American Chemical Society Published on Web 03/27/2008
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simulation indicated that this theory precisely reproduced simulation data.42 In this work, we apply a similar theory to study the adsorption of simple fluid of spherical particles from a gaseous phase onto a surface modified by pre-adsorbed chain molecules. Our principal aim is to study how the pre-adsorbed layer influences the layering transitions, as a part of the scenario leading to the formation of a macroscopic film of an adsorbate on a chemically modified substrate. Two different models are used. In the first model, the model A, the configuration of the chain particles can change during adsorption of spherical species. The tools used to describe this model is an extended version of the approach given in ref 42. According to the second model, the model B, the pre-adsorbed chains are kept fixed. The approach appropriate to describe this model relies on a generalization of the theory of the so-called quenched-annealed systems, developed by Schmidt.47-49 The utility of the model B relies on a possibility of investigations of adsorption on “rough” surfaces. This type of substrate can be formed as a result of deposition of complex particles, not necessarily spherical on bare surfaces. II. Theory We begin with general aspects of the approach developed to describe adsorption of spherical particles S on a surface covered by a known amount of pre-adsorbed chain molecules, C. Each chain consists of M spherical segments of diameter σ(C) that are tangentially jointed. The chain connectivity is ensured by the binding potential acting between nearest-neighbor segments. The total binding potential Vb is given by:43 M-1
exp[-βVb(R)] )
[δ(|ri - ri+1|) - σ(C)]/4π(σ(C))2 ∏ i)1
(1)
In the above equation, R ) r1, r2, ..., rM is the vector of coordinates of all the segments. The external potential acting on all but the first segment is a hard-wall potential
V(C) i (z) )
{
∞, z e σ(C)/2, 0, otherwise
(2)
for i ) 2, 3, ..., M. The first segment is grafted at the surface plane z ) σ(C)/2; that is, it interacts with the surface via a very strong but short-ranged potential (C) exp[-βV(C) 1 (z)] ) Cδ(z - σ /2)
(3)
where C is a constant. We will see below that the precise value of the constant C is irrelevant if the total amount of grafted chains is fixed. The total external potential acting on a chain M particle is V(C)(R) ) ∑i)1 V(C) (zi). The spherical particles S i interact with the wall via (9,3) Lennard-Jones potential 9 3 V(S) (z) ) (S) gs [(z0/z) - (z0/z) ]
(4)
All of the segments of chains and spherical molecules interact via (12,6) Lennard-Jones potentials
U(ij)(r) ) 4(ij) [(σ(ij)/r)12 - (σ(ij)/r)6]
(5)
where i, j ) S, C, σ(ij) ) 0.5(σ(i) + σ(j)), and σ(S) is the diameter of spherical particles. The interactions between all spherical species (segments and spherical molecules) are treated in a perturbational manner; that
is, the potentials U(ij)(r) are divided into the repulsive (reference) and attractive (perturbation) parts. Then the repulsive interactions are approximated by hard-sphere potentials with the effective hard-sphere diameters d(ij). The division of U(ij)(r) is carried out according to the Weeks-Chandler-Andersen scheme;50 that is, the attractive part of the potential U(ij)(r) is defined by
u(ij) att (r) )
{
-(ij), r < 2(1/6)σ(ij) U(ij) (r), r g 2(1/6)σ(ij)
(6)
Although different approaches can be used to determine the effective hard-sphere diameters d(ij), in this work, we assume that d(ij) ) σ(ij) for the sake of simplicity. For both models in question (i.e., for the models A and B), the Helmholtz free energy of the system containing chains and spherical molecules results from the theory outlined in details in refs 42-46 and 51-53. We introduce the local densities of individual segments, F(C) si (z) and the total segment density, M (C) F(C) s (z) ) ∑1 Fsi (z) besides the local density of chain molecules, F(C)(R) and the local density of spherical particles, F(S)(z). The definitions of individual segment densities are given in refs 42-46 and 51-53, but we present them below for a better understanding for the reader M
F(C) s (z) )
∑ i)1
M
F(C) si (z) )
∫ dRδ(r - ri)F(C)(R) ∑ i)1
(7)
Let us proceed to the discussion of the free energy. The ideal contribution to the free energy, Fid, is given by an exact expression (cf. eq 8 of ref 43). The excess free energy terms due to hard-sphere contribution, Fhs, and due to the chain connectivity, Fch, are evaluated from the fundamental measure theory and from the first-order theory of Wertheim by using eqs 9 and 16-19 from ref 43. For the sake of brevity, we do not repeat them here. The free energy resulting from the attractive potential (eq 6), Fatt, is obtained within a mean-field approximation. Thus, we have
Fatt )
1 2
∫ dr1 dr2F(S)(r1)F(S)(r2)u(SS)(r12) + (SC) (r12) + ∫ dr1 dr2F(S)(r1)F(C) s (r2)u 1 (CC) (C) (r12) ∫ dr1 dr2F(C) s (r1)Fs (r2)u 2
(8)
The free energy functional is thus the sum of all the contributions
F(F(C)(R), F(S)(z)) ) Fid(F(C)(R), F(S)(z)) + Fhs(F(C)(R), F(S) (z)) + Fch(F(C)(R), F(S)(z)) + Fatt(F(C)(R), F(S)(z)) (9) First, we consider the case when the configuration of the pinned chain molecules can change during the adsorption of spherical particles (model A). It is worth mentioning that in this model the first segments of chains are pinned in the plane z ) σ(C)/2. All remaining segments of chains can move in space. However, their movement is restricted by the repulsive cores of molecules and by the binding potential. The equilibrium structure of the chains and of the spherical particles in this model is obtained by minimizing the thermodynamic potential42 in the ensemble in which the total number of particles of onecomponent (chains) is fixed. We have
Y ) F(F(C)(R), F(S)(z)) +
∫ drF(S)(z)[V(S)(z) - µ(S)]
(10)
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where µ(S) is the chemical potential of adsorbate species. The functional Y is minimized under the constraint of a constant number of tethered chain particles per unit surface area, that is,
RC )
∫ Fs1(C)(z) dz
(11)
Employing the method described in refs 42, 46, and 51-53, we arrive at the following equations for the density profiles
F(S)(z) ) exp[-βV(S)(z)] exp{βµ(S) - βλ(S)(z)}
(12)
and
F(C) si (z) ) (C) (L) (R) exp[-βV(C) i (z)] exp[-βλi (z)]Gi (z)GM+1-i(z) (C) (L) (R) ∫ dz exp[-βV(C) i (z)] exp[ - βλi (z)]Gi (z)GM+1-i(z)
(13)
From the above equation we see that the normalization condition makes irrelevant the value of the constant C in eq 3. In the above
λ(S)(z) )
δ(F - Fid) (C) δ(F - Fid) , λi (z) ) (S) δF (z) δF(C) s (z)
(14)
and the functions GLi and GRi are determined from the recurrence relations42,46,51-53
GLi (z) )
(C) (C) (z)] exp{-βλi-1 (z)} × ∫ dz′ exp[-βVi-1
θ(σ
- |z - z′|)
(C)
2σ
(C)
L Gi-1 (z′) (15)
(C) (C) (z)] exp{-βλM-i+2 (z)} × ∫ dz′ exp[- βVM-i+2
θ(σ(C) - |z - z′|) 2σ(C)
L Gi-1 (z′) (16)
for i ) 2, 3, ..., M and with GR1 (z) ) GL1 (z) ≡ 1. Alternatively, if the chain particles remain in a fixed configuration after the pre-adsorption step (model B), the grand potential of such quenched-annealed system is expressed as a functional, dependent on both the density distribution of the annealed component, F(S)(z), and that of the quenched component, F(C)(R). The latter function is evaluated for the system involving only chain molecules, according to the theory of Yu and Wu.43 Once F(C)(R) is known, then the density distribution of the adsorbate, F(S)(z), can be obtained by minimization of the grand potential47-49,54
Ω1(F (R), F (z)) ) F(F (R), F (z)) - F(F (R), (C)
(S)
(C)
F(S)(z) ≡ 0) +
(S)
(C)
∫ drF(S)(z)[V(S)(z) - µ(S)]
(17)
To obtain F(S)(z), one needs to solve the equation
δΩ1(F(C)(R), F(S)(z)) δF(S)(z)
)0
constant. The minimization of the grand potential Ω1 yields the density profile equation that is formally identical to eq 12. III. Results and Discussion
and
GRi (z) )
Figure 1. Excess adsorption isotherms for the model A, Γ* ) Γσ2, versus bulk density, F/b ) Fbσ3 at T* ) 0.75. The results are for M ) 6 and for different amounts of attached chains, RC. The inset shows details of the isotherm at RC ) 0.7 obtained at higher bulk densities. The arrows indicate the bulk densities that correspond to the profiles in Figure 2.
(18)
and, obviously, the chain density field, F(C)(R), is an input quantity. Similarly as in the case of the model A, the calculations are carried out assuming that the surface density of chains is
The principal aim of the calculations presented was to study how the presence of chains, tethered at a solid surface, influence thermodynamic and structural properties of an adsorbed film, and how the selection of the model setup, model A or model B, influences the obtained results. We have assumed that all of the segments and spherical molecules are of the same size, σ(C) ) σ(S) ) σ and that the parameter z0, entering eq 4, is given by z0 ) σ(S) in all cases considered. Moreover, all of the Lennard-Jones energy parameters, (ij), i, j ) C, S, were assumed to be equal, (ij) ) , and the energy parameter of eq 4 was set to be gs/ ) 12. This value for the adsorption energy is half the strength of that for argon on graphite substrate. On the other hand, it is two times higher than the adsorption energy of argon on solid CO2. Under the chosen conditions, the system containing no pre-adsorbed chain molecules exhibits layering transitions at low temperatures. We define the reduced temperature as T* ) kT/ and the reduced density as F* ) Fσ3. We begin with the discussion of the behavior of excess isotherms, defined by the equation
Γ)
∫0∞ [F(S)(z) - Fb] dz
(19)
where Fb is the bulk density of one-component system containing spherical molecules at the same temperature T* and the chemical potential µ(S) as in the inhomogeneous system under study. Figure 1 shows isotherms Γ* ) Γσ2 evaluated according to the model A at T* ) 0.75 for different amounts of pre-adsorbed chains, RC. The number of segments in a chain equals M ) 6. In the case of adsorption on the bare surface, that is, the surface without attached chains, the isotherm exhibits the layering transition within the first layer adjacent to the wall. This transition is manifested by the hysteresis loop in Figure 1. Although the formation of the consecutive layers on the bare
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Figure 2. Density profiles of spherical particles at different bulk densities, listed in the figure, as a function of the distance from the wall. Results are given for different amounts of pre-adsorbed chain particles, RC ) 0, 0.01, 0.1, 0.5, 0.7. The calculations are for the model A (M ) 6, T* ) 0.75).
surface is continuous, the steps connected with their filling-up are visible. In the presence of a small amount of pre-adsorbed chains, RC e 0.1, the adsorption is higher compared with the bare surface, but the first layering transition is not present. An increase of adsorption is due to an increase of attraction in the system, but the attractive centers (segments) are distributed in space. Consequently, the layering transition disappears at this temperature. At higher bulk fluid densities, the adsorption isotherms evaluated for the bare surface and for RC ) 0.01 flow together. At low RC, the amount of pre-adsorbed chains is too low to create an attractive field that is able to influence the adsorption in remote layers. If RC grows to 0.1, the isotherm becomes smooth without any transitions within the investigated range of bulk fluid densities. For a very high amount of pre-adsorbed chains, RC ) 0.7, the initial part of the isotherm (up to F/b = 0.002) is completely smooth. At higher bulk fluid densities, layering-like transitions develop (see the inset to Figure 1). Under these conditions, the layer of pinned chains covering the surface is very dense and its outer part, that is, that exposed to the fluid of spherical particles, behaves as an adsorbing surface. It is the source of an external potential field, and the layering transitions occur at this new surface. In Figure 2, we show the density profiles of the adsorbate at three different bulk densities. They have been marked by arrows in Figure 1. In Figure 2a, at a low bulk density a small amount of pre-adsorbed chains yields a higher peak of the local density. At higher bulk densities, and for small RC, the adsorbate density profiles are almost identical to those obtained for a nonmodified surface (Figure 2b). In fact, for F/b ) 0.001 the profile for RC ) 0.01 is almost indistinguishable from the profile for the bare surface. On the other hand, high values of RC lead to a diminishing adsorption; compare with Figure 2a,c. We see that at F/b ) 0.0001 the height of the local density peak evaluated for RC ) 0.7 is much lower than the corresponding peak at the profile obtained for RC ) 0.1. A similar effect has been found at higher bulk fluid densities (Figure 2b,d), and its origin is quite obvious. The first layer is formed at the distance corresponding to the minimum of the adsorbing potential, but
Figure 3. (a) Total segment density profiles are given for three different bulk densities. The first peak has been cut in height. (b) Individual segment density profiles of chains for segments i ) 2 and 6, at different bulk densities F/b ) 0.00001, 0.001, 0.0024: lines, lines connecting open symbols, and lines decorated with filled symbols in left panel and at two bulk densities F/b ) 0.00001 and 0.0024 in right panel. The solid line and solid line with filled symbols almost coincide for i ) 2 in the right panel. Amounts of pre-adsorbed chain particles equal RC ) 0.01 for left panels and RC ) 0.7 for right panels. The calculations are for the model A (M ) 6, T* ) 0.75).
when RC is high enough, the layer built of chains is dense and volume exclusion effects prevent the adsorbate molecules from entering its interior. Therefore, at low RC, the effects due to attractive segment-adsorbate molecule forces prevail, whereas for high RC, the dominant role is played by the volume exclusion effects. The structure of adsorbed fluid is thus influenced by a competition between these two effects. According to the model A adsorption of fluid particles can change the structure of pre-adsorbed layer. In particular, for RC ) 0.01, an increase of Fb causes that the chains become more and more straight (left panels of Figure 3a,b). For Fb ) 0.00001, almost all segments lie within the region of z/σ < 3. An increase of Fb causes the total density profile of segments to exhibit a layered structure; see left panel in Figure 3a. This is also visible in the case of the outermost segments, i ) 6; compare with the left panel in Figure 3b. However, if RC is high, the adsorption of fluid species exhibits little influence on the structure of pre-adsorbed chains. In fact,
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Figure 4. Excess adsorption isotherms, Γ*, versus bulk density for the model A (M ) 6, T* ) 0.75). The values of RC are given in the figure.
the density profiles in the case of RC ) 0.7 and evaluated at F/b ) 0.001, 0.0024 are almost identical; see right panels of Figure 3a,b. The shape of adsorption isotherms changes if the temperature is lowered. At T* ) 0.7, the adsorption on the bare surface shows a sequence of layering transitions in the first, second, third, fifth, sixth, and seventh (and presumably higher) adsorbed layers; see Figure 4. There is no transition within the fourth layer; that is, the critical temperature for this transition is lower than 0.7. For RC ) 0.01, the layering transitions occur in the first two layers adjacent to the surface, and also within higher layers, except for layers 3 and 4. For sufficiently high values of RC, we observe layering transitions within layers located remote from the wall, but not in the layers in the vicinity of the substrate. The formation of consecutive layers during the adsorption process is accompanied by the changes in the structure of the adsorbed film and of the structure of pre-adsorbed chains. If the adsorbed film becomes thicker, the observed structural changes are illustrated in Figure 5. The calculations were done for RC ) 0.01 (part a) and for RC ) 0.7 (part b). Left panels in both parts of Figure 5 display the density profiles of the adsorbed fluid, whereas right panels show the total segment density profiles. Similar to the case of a higher temperature, T* ) 0.75 (see Figures 2 and 3), if RC is low enough, the adsorbed film enforces substantial changes in the structure of the attached chains. However, for RC ) 0.7, the transition connected with the filling of the seventh adsorbed layer (Figure 5b) causes only marginal changes in the structure of the total segment profile. Indeed, these changes are only weakly seen within the tail of the density distribution (see the inset to the right panel of Figure 5b). The summary of our results obtained for the model A and for the chains composed of M ) 6 segments are given in Figure 6. We show here the phase diagrams, in the chemical potentialtemperature plane, obtained for the nonmodified surface and for two modified surfaces, characterized by RC ) 0.01 and RC ) 0.5. Each branch of the diagram corresponds to the layering transition within a given layer. We have restricted our study to the transitions within the first seven layers. Also, we did not perform calculations at very low temperatures, because the
Figure 5. Left panels give density profiles of fluid molecules, while right panels show the total segment density profiles of the attached chains. First peak in right panels has been cut in height. Part a is for RC ) 0.01, and part b is for RC ) 0.7. Bulk densities are given in the figure. The inset magnifies the tail of the total segment density profile. The calculations are for the model A (M ) 6, T* ) 0.7).
validity of the approach employed here is limited to the range of temperatures above the triple point temperature. The presence of the attached chain particles at the RC ) 0.01 shifts the layering transitions toward lower chemical potentials and lowers the corresponding critical temperatures. The differences are significant up to the transition within the fourth layer. The branches representing the layering transitions within higher layers (n g 5), on both bare and modified surfaces are not distinguishable. For a high amount of attached chains, RC ) 0.5, only the transitions within the layers n g 5 are present in the investigated range of temperatures. Their critical temperatures are higher than in the case of two remaining surfaces (bare and characterized by RC ) 0.01). In all cases, we observe non-monotonic changes of the critical temperatures with the layer number. Note that, according to a phenomenological theory, described by Pandit et al.,55 the critical temperatures for layering transition tend to the roughening temperature with n f ∞. Because the evaluation of the phase diagrams becomes more and more tedious as the chemical potential approaches the bulk coexistence, we did not carry out the relevant calculations and therefore are not able to discuss the behavior of the critical temperatures for higher layers.
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Figure 6. Phase diagrams in the chemical potential - temperature plane for the model A. Curves from left to right correspond to consecutive layering transitions and include the transitions within the first seven layers. Open circles correspond to the nonmodified surface, dotted lines with crosses are obtained for RC ) 0.01 and solid lines for RC ) 0.5. In the latter case the diagrams are for the layers from 5 to 7.
Figure 8. (a) Right panel gives the total segment density profiles and the left panel shows the fluid density profiles at bulk densities given in the figure. The chains lengths are M ) 6 and 10. The first peak has been cut in the height in the right panel. (b) Segment density profiles of particular segments for chains of length M ) 6 (dashed lines), and M ) 10 (solid and dotted lines). The calculations are for the model A.
Figure 7. Excess adsorption isotherms, Γ* vs F/b, obtained for three chain lengths M ) 2, 6, 10 (dashed lines, solid lines, and dashed-dotted lines, respectively). RC ) 0.01 in the left panel, while the right panel contains results for RC ) 0.1, 0.7: lines and lines decorated with circles, respectively. The calculations are for the model A.
Of course, the adsorption and the structure of adsorbed fluid depend on the length of the chains attached to the surface. Figure 7 illustrates changes in the adsorption isotherms at T* ) 0.7 with the change of the chain length. The results presented are for M ) 2, 6, and 10 and for three values of RC: RC ) 0.01 in the left panel, RC ) 0.1, and RC ) 0.7 in the right panel of Figure 7. Only at low values of RC, the isotherms are not very sensitive to the length of the chains. The isotherms displayed in the left panel of Figure 7 demonstrate that in the case of chains of M ) 6 and 10 only the condensation of the first two layers occurs. For M ) 10, the critical temperature for the first layering transition is close to 0.7, whereas for M ) 6, the critical temperature for this transition is ≈0.74 (cf. Figure 6), and for M ) 2 it is even higher. However, all of the isotherms obtained
for M ) 2, 6, and 10 exhibit layering transitions within the sixth and seventh layer. The increase of RC up to 0.1 leads to more significant changes in the behavior of adsorption isotherms for systems which differ by the length of attached chains. In the case of the surface covered with dimers, the layering transitions occur within layers numbers 5, 6, and 7. For M ) 6, the transition takes place in the layer no. 7, whereas for M ) 10, no layering transitions are observed within the investigated range of the bulk density. For RC ) 0.7, the isotherm for M ) 2 essentially differs from the two remaining isotherms, obtained for M ) 6 and 10. In fact, for M ) 6 and M ) 10, the adsorption of a fluid takes place mainly on an external surface of the attached chains, whose structure corresponds to a dense layered liquid. It has already been demonstrated that, if RC is high enough, the adsorbed film has a small impact on the structure of attached chains. For a low number of the attached chains, however, the adsorbate may significantly influence the structure of the preadsorbed layer. For longer chains, this effect is strong; see Figure 8. This figure illustrates the influence of the bulk density on the total segment density profiles (right panel of part a) and on
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Figure 9. Consecutive isotherms from left to the right are for the temperatures T* ) 0.67, 0.68, 0.69, 0.70, and 0.71, and for RC ) 0.1. Two models are compared. Model A is dashed lines, whereas model B is solid lines. The inset, from left to right, gives phase diagrams for transitions within the 5th, 6th, and 7th layers (model A) and within the 6th and 7th layers (model B).
the individual segment density profiles (part b). The left panel of Figure 8a shows the density profiles of fluid species. The changes occurring in the system with chains of M ) 10 are compared with the behavior observed for the shorter chains of M ) 6. In both cases, the parameter RC was equal to 0.1. An interesting observation is that the changes in the structure of the chains with adsorption of the fluid seem to be more pronounced for longer chains and that, instantaneously, the distributions of the most inner (of the second and, in part, of the fourth segment) are almost independent of the chain length. However, even for the bulk fluid density close to the bulk coexistence the ends of the chains composed of 10 segments are not far from the surface. Indeed, the z coordinates of a great majority of the end segments (i ) 10) do not exceed 7σ. These trends are observed because of energetic aspects of adsorption. At higher temperatures (we do not show the relevant figure), longer chains are more erected. In contrast to the model A, the model B assumes that the chain configuration remains fixed after the pre-adsorption step. The density profiles for pre-adsorbed chains are calculated at the same temperature as the adsorption of fluid species. One can expect that the two models yield similar results when the amount of pre-adsorbed chains is either very low or very high. For very low values of RC, RC f 0,both models must give adsorption on a bare surface. If RC is very high, then the possibility of movement of the segments in the model A is low, and thus, the pre-adsorbed layer is quite similar for the two models in question. Figure 9 compares adsorption isotherms obtained for the two models in question, whereas Figure 10 shows how the structure of the adsorbed fluid changes with the bulk density. The amount of pre-adsorbed layers equals RC ) 0.1. Note that the values of Γ are plotted in Figure 9 by using logarithmic scale. At low bulk fluid densities, the model A predicts significantly higher adsorption than the model B does. This is due to blocking the surface by the pre-adsorbed chains in the model B. Indeed, the total segment density profile evaluated for this model shows that the chains are almost parallel or remain bent with respect to the surface (Figure 10a). Therefore, at low bulk densities a
Figure 10. Part a shows the total segment densities, whereas part b shows local densities of fluid molecules. In part a, the first peak has been cut in the height. Results for model A are indicated by lines; results for the model B are indicated by lines connecting filled circles. In both cases, RC ) 0.1 and T* ) 0.7. Bulk fluid densities are shown in the figure.
pronounced adsorption of fluid particles within the first two layers is not possible. In contrast, the model A permits mutual accommodation of pre-adsorbed chains and fluid species. Compared with the model A the filling of the first adsorbed layer in the model B is lower even at high bulk densities. If the bulk fluid density becomes high enough, differences in the structure of remote layers, predicted by both models, become smaller (Figure 10b). Consequently, the adsorption isotherms (Figure 9) almost flow together for higher bulk fluid densities. The inset to Figure 9 shows phase diagrams for the layering transitions that occur within the fifth, sixth, and seventh layers. In the model A, higher critical temperatures and lower values of the chemical potential at the transition points are observed comparing with the model B. That means that the stability of the particular layered structures is higher in model A than in model B. IV. Summary In this work, we have proposed and tested theoretical tools to study adsorption on surfaces modified by brush-like structures formed by pre-adsorbed chains. The proposed approach seems to open new perspectives to study adsorption and wetting of simple and complex fluids on chemically modified surfaces. The theory discussed here is based on the development of Yu and Wu43-46 for nonuniform chain fluids. It has already been demonstrated42 that in the case of adsorption of hard spheres and their mixtures on surfaces modified with hard-sphere chains
Adsorption of Simple Fluids on Surfaces this type of approach leads to a good agreement with computer simulation data. Both models used in this study assume a random distribution of pinned segments of chains. Moreover, particular cases studied within this paper are simple, and we have not explored all of the possibilities of the proposed approach so far. In particular, the effects of segment wall attraction have not been investigated. Nevertheless, we have accomplished our principal objective to investigate the formation of adsorbed layers on a substrate and the corresponding layering phase transitions. We have chosen intermediate attractive surface and observed either new layering phase transitions due to chemical modification or smoother adsorption isotherms that describe a lower number of layering phase transitions in comparison witho the bare substrate. These observations are relevant to future studies of the mechanism of wetting phenomena on chemically modified surfaces and location of the layering and wetting lines with respect to the bulk phase diagram. Moreover, we have realized that the important factors influencing the phase behavior of this type of adsorption system are the energetic contrast between the bare surface and the segments of pre-adsorbed chains and entropic type effects influencing configuration of grafted chains, as well as the parameters of the model such as the number of segments in chains, grafted density, and temperature. These factors permit us to tune the phase behavior at the level of the formation of consecutive layers, but undoubtedly they will play a similar role in the case of wetting phenomena. It is important to mention that the energetic effects of possible segment-wall attraction could contribute to the formation of adsorbed layers in a nontrivial manner. In such case, the density of chain molecules can play a more important role compared with the models of the present work. Our theoretical developments are well-suited to study various combinations of energetic parameters. However, we have restricted our attention to simple situations in which a substrate affects chain segments only entropically via geometrical restriction. More complex models will be described in a future work now in progress in our laboratory. At present, our results were obtained for a single value of the gas-solid interaction energy, that is, for gs/ ) 12. We expect similar trends for higher values of this parameter.56 For lower values, however, a crossover between nonwetting and wetting behavior is expected. This issue requires further investigations. On the other hand, it would be interesting to get an insight into the effect of the parameters determining the structure of a quenched layer in the model B, specifically of the quenching temperature. It is known that quenching conditions significantly influence adsorption properties in the case of quenched-annealed charged fluids.57 We should also mention some limitations of the model and theory. We deal with tangentially jointed spheres; that is, our model is not appropriate for the description of chains formed by overlapping segments. Moreover, the applied model does not allow us to treat chains as built of monomers that are bonded together via elastic springs.58,59 Nevertheless, there exist a possibility to consider more complex structures of molecules used as surface modificators.60 It is possible to derive a modified version of the model and the proposed approach assuming that the chains form a “crystal-like” structure, that is, that the pinned segments are periodically distributed on a lattice. Moreover, one can think about a possible “combination” of both models A and B considered here, to put a combined model closer to experimental setup. It is known that the synthetic route to form brushlike structures often involves the surface initiated polymerization.61
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4559 The density functional approach used in this work is based on the fundamental measure theory for hard spheres, on the perturbational theory for chain connectivity, and on the meanfield approximation. In the case of nonuniform fluid of tangentially bonded hard-spheres, this approach performs well with respect to computer simulations.62,63 In the case of systems involving pinned chains and hard-sphere component, the predicted structure of nonuniform fluid reproduces well computer simulation data.42,64,65 It is known that the mean-field approximation for the attractive interactions is not perfect. A possible alternative is the first-order mean-spherical approximation developed by Tang.66 However, in our opinion, in the case of model calculations, accuracy at a qualitative level suffices and the mean-field approximation is good enough to get insight into the system behavior; compare with refs 67-70. Acknowledgment. This work was partially supported by EU under the TOK Contract No. 509249. References and Notes (1) Netz, R. R.; Andelman, D. Phys. Rep. 2003, 380, 1. (2) Liakos, I. L.; Newman, R. C.; McAlpine, E.; Alexander, M. R. Surf. Interface Anal. 2004, 34, 347. (3) Weaver, J. F.; Carlsson, A. F.; Madix, R. J. Surf. Sci. Rep. 2003, 50, 107. (4) Carlsson, A. F.; Madix, R. J. J. Phys. Chem. B 2001, 105, 8155. (5) Dorsey, J. G.; Dill, K. A. Chem. ReV. 1989, 89, 331. (6) Alexander, S. J. Phys. (France) 1977, 38, 983. (7) de Gennes, P. G. Macromolecules 1980, 13, 1069. (8) Milner, S. T.; Witten, T. A.; Cates, M. E. Europhys. Lett. 1988, 5, 413. (9) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (10) Milner, S. T. Science 1991, 251, 905. (11) Zhulina, E. B.; Borisov, O. V.; Pryanitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140. (12) Shim, D. F. K.; Cates, M. E. J. Phys. (France) 1989, 50, 3535. (13) Whitmore, M. D.; Noolandi, J. Macromolecules 1990, 23, 3321. (14) Pattanayek, S. K.; Pham, T. T.; Pereira, G. G. J. Chem. Phys. 2005, 122, 214908. (15) Pang, P.; Koska, J.; Coad, B. R.; Brooks, D. E.; Haynes, C. A. Biotechnol. Bioeng. 2005, 90, 1. (16) Manciu, M.; Ruckenstein, E. Langmuir 2004, 20, 6490. (17) Stoyanov, S. D.; Paunov, V. N.; Rehage, H.; Kuhn, H. Phys. Chem. Chem. Phys. 2004, 6, 596. (18) Leermakers, F. A. M.; Philipsen, H. J. A.; Klumperman, B. J. Chromatogr. A 2002, 959, 37. (19) Patra, C. N.; Yethiraj, A.; Curro, J. G. J. Chem. Phys. 1999, 111, 1608. (20) Szleifer I.; Carignano, M. A. AdV. Chem. Phys. 1996, XCIV, 165. (21) McCoy, J. D.; Ye, Y.; Curro, J. G. J. Chem. Phys. 2002, 117, 2975. (22) Ye, Y.; McCoy, J. D.; Curro, J. G. J. Chem. Phys. 2003, 119 555. (23) Hooper, J. B.; McCoy, J. D.; Curro, J. G.; van Swol, F. J. Chem. Phys. 2000, 113, 2021. (24) McCoy, J. D.; Teixeira, M. A.; Curro, J. G. J. Chem. Phys. 2001, 113, 4289. (25) Frischknecht, A. L.; Weinhold, J. D.; Salinger, A. G.; Curro, J. G.; Frink, L. J. D.; McCoy, J. D. J. Chem. Phys. 2002, 117, 10385. (26) Grest, G. S.; Murat, M. Macromolecules 1993, 26, 3108. (27) Lai, P. Y.; Binder, K. J. Chem. Phys. 1991, 95, 9288. (28) Lai, P. Y.; Binder, K. J. Chem. Phys. 1992, 97, 586. (29) Lai, P. Y. J. Chem. Phys. 1993, 98, 669. (30) Weinhold, J. D.; Kumar, S. K. J. Chem. Phys. 1994, 101, 4312. (31) Grest, G. S. J. Chem. Phys. 1996, 105, 5532. (32) Pastorino, C.; Binder, K.; Kreer, T.; Mu¨ller, M. J. Chem. Phys. 2006, 124, 064902. (33) Ohno, K.; Sakamoto, T.; Minagawa, T.; Okabe, Y. Macromolecules 2007, 40, 723. (34) Descas, R.; Sommer, J. U.; Blumen, A. J. Chem. Phys. 2006, 125, 214702. (35) Wongkoblap, A.; Do, D. D. J. Colloid Interface Sci. 2006, 297, 1. (36) Patra, M.; Linse, P. Nano Lett. 2006, 6, 133. (37) Mu¨ller, M.; MacDowell, L. G. J. Phys. Condens. Matter 2003, 15, R609. (38) MacDowell, L. G.; Mu¨ller, M. J. Chem. Phys. 2006, 124, 084907. (39) Klatte, S. J.; Beck, T. L. J. Chem. Phys. 1993, 97, 5727. (40) Klatte, S. J.; Beck, T. L. J. Chem. Phys. 1995, 99, 16024.
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