pubs.acs.org/Langmuir © 2009 American Chemical Society
Ternary Protein Adsorption onto Brushes: Strong versus Weak A. Halperin*,† and M. Kr€oger‡,§ †
Laboratoire de Spectrom etrie Physique, Universit e Joseph Fourier - CNRS, UMR 5588, BP 87, 38402 Saint Martin d’H eres, France, and ‡Polymer Physics, ETH Zurich, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland. §URL: www.complexfluids.ethz.ch. Received March 10, 2009. Revised Manuscript Received June 9, 2009
Attractive interactions between proteins and polyethylene glycol (PEG) give rise to ternary adsorption within PEG brushes. Experimental evidence suggests two ternary adsorption modes: (i) weak, due to nonspecific weak attraction between PEG monomers and the surface of the protein, as exemplified by serum albumin and (ii) strong, due to strong binding of PEG segments to specific protein sites as it occurs for PEG antibodies, which can involve the terminal adsorption of free chain ends or backbone adsorption due to binding to interior chain segments. Ternary adsorption affects the capacity of brushes to repress protein adsorption. The strong adsorption of antibodies can trigger an immune response that may affect the biocompatibility of the surface. Theoretical adsorption isotherms and protein concentration profiles of the three cases are compared for “parabolic” brushes, allowing for the grafting density, 1/Σ, and degree of polymerization of the PEG chains, N, as well as the volume and surface area of the proteins. The amount of adsorbed protein per unit area, Γ, exhibits a mode-specific maximum in all three cases. For backbone and weak adsorption, Γ ∼ N, whereas for terminal adsorption, Γ ∼ N0. In every case, the concentration profile of adsorbed proteins, ctern(z), exhibits a maximum at zmax>0 that shifts outward as Σ decreases; zmax =0 occurs only for weak and backbone adsorption at a high Σ value.
I. Introduction Surfaces displaying brushes of terminally anchored polyethylene glycol (PEG) chains are known for their resistance to protein adsorption.1-5 This effect is now utilized to prolong the circulation time of protein drugs and drug-loaded liposomes.6,7 It also motivates research aimed at the development of biocompatible surfaces incorporating PEG chains.8 Overall, the PEG chains are considered to be protein-repellent, having repulsive interaction with proteins. This view suggests that protein adsorption onto a surface displaying PEG brushes occurs because of primary adsorption at the surface or secondary adsorption at the outer edge of the polymer layer.9-12 The primary adsorption is due to short-range protein-surface attraction, and the secondary adsorption arises because of van der Waals protein-surface attraction at the outer edge of the polymer layer. Within this picture, proteins do not adsorb within the PEG brush itself because the insertion penalty incurred is not balanced by an attraction term. Importantly, the applicability of this view is limited because certain proteins experience attractive interactions to PEG. As a result, such proteins can undergo ternary adsorption *Corresponding author. E-mail:
[email protected]. (1) Goddard, J. M.; Hotchkiss, J. H. Prog. Polym. Sci. 2007, 32, 698–725. (2) Elbert, D. L.; Hubbell, J. A. Annu. Rev. Mater. Sci. 1996, 26, 365–394. (3) Lee, J. H.; Lee, H. B.; Andrade, J. D. Prog. Polym. Sci. 1995, 20, 1043–1079. (4) Harris, J. M. Poly(Ethylene Glycol) Chemistry: Biotechnical and Biomedical Applications; Plenum Press: New York, 1992. (5) Senaratne, W.; Andruzzi, L.; Ober, C. K. Biomacromolecules 2005, 6, 2427– 2448. (6) Janoff, A. S., Ed.; Liposomes Rational Design; Marcel Dekker: New York, 1998. (7) Harris, J. M.; Chess, R. B. Nature Rev. Drug Discovery 2003, 2, 214–221. (8) Kawakami, H. J. Artif. Organs 2008, 11, 177–181. (9) Jeon, S. I.; Lee, J. H.; Andrade, J. D.; de Gennes, P.-G. J. Colloid Interface Sci. 1991, 142, 149–158. (10) Szleifer, I. Biophys. J. 1997, 72, 595–612. (11) Halperin, A. Langmuir 1999, 15, 2525–2533. (12) Currie, E. P. K.; Norde, W.; Cohen Stuart, M. A. Adv. Colloid Sci. 2003, 100-102, 205–265.
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within PEG brushes. Experimental evidence suggest two ternary adsorption scenarios: one is due to weak and nonspecific monomer-protein attraction as illustrated by the case of serum albumin.13-19 The second involves strong polymer binding at specific protein sites as realized, for example, by PEG antibodies.20,21 This last scenario may involve either terminal binding of free ends22 or backbone binding of internal chain segments23 (Figure 1). The first, the weak adsorption case, gives rise to protein adsorption within the brush without measurable complexation of proteins by free polymers in the bulk. In the second, the strong adsorption case, both brushes and free chains in solution can bind proteins. Existing theoretical models consider the pure repulsion9-11 and the weak ternary adsorption13,14 scenarios. In the following, we present a theoretical model comparing the three ternary adsorption modes (i.e., strong terminal and backbone adsorption vs weak adsorption). (13) Currie, E. P. K.; Van der Gucht, J.; Borisov, O. V.; Cohen Stuart, M. A. Pure Appl. Chem. 1999, 71, 1227–1241. (14) Halperin, A.; Fragneto, G.; Schollier, A.; Sferrazza, M. Langmuir 2007, 23, 10603–10617. (15) Baskir, J. N.; Hatton, T. A.; Suter, U. W. Macromolecules 1987, 20, 1300– 1311. (16) Baskir, J. N.; Hatton, T. A.; Suter, U. W. J. Phys. Chem. 1989, 93, 2111– 2122. (17) Abbott, N. L.; Blankschtein, D.; Hatton, T. A. Macromolecules 1992, 24, 4334–4348. (18) Abbott, N. L.; Blankschtein, D.; Hatton, T. A. Macromolecules 1992, 25, 3917–3931. (19) Abbott, N. L.; Blankschtein, D.; Hatton, T. A. Macromolecules 1992, 25, 3932–3941. (20) Armstrong, J. K.; Hempel, G.; Koling, S.; Chan, L. S.; Fisher, T.; Meiselman, H. J.; Garratty, G. Cancer 2006, 110, 103–111. (21) Ishida, T.; Wang, X.; Shimizu, T.; Nawata, K.; Kiwada, H. J. Controlled Release 2007, 122, 349–355. (22) (a) PEG antibodies recognizing the terminal methoxy group are available at www.epitomics.com. (b) Terminal dextran antibodies are discussed in Cisar, J.; Kabat, E. A.; Dorner, M. M.; Liao, J. J. Exp. Med. 1975, 142, 435–459. (23) Cheng, T.-L.; Cheng, C. M.; Chen, B. M.; Tsao, D. A.; Chuang, K. H.; Hsiao, S. W.; Lin, Y. H.; Roffler, S. R. Bioconjugate Chem. 2005, 16, 1225–1231 as used in ref 14 correspond to our Σ + and Σ -, respectively .
Published on Web 08/12/2009
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Figure 1. (a) Terminal vs (b) backbone adsorption in the mushroom regime. In terminal adsorption, the protein binds only to free ends whereas in backbone adsorption it binds to interior chain segments.
Ternary adsorption is significant because it constitutes a possible route to brush failure in vivo. This is especially the case for the strong adsorption of PEG antibodies. In the cases of PEGylated protein drugs and PEG bearing “stealth” liposomes, it diminishes their circulation time in the blood.20,21 For macroscopic surfaces, the strong adsorption of PEG antibodies can trigger an immune response that may affect their biocompatibility. Importantly, these effects may occur even though PEG antibodies are minority components in the blood. The in vivo consequences of weak nonspecific protein adsorption remain to be established, but it certainly diminishes the protein repellency of the brushes. The repression of ternary adsorption thus becomes an important issue in the design of PEG brushes for biomedical applications. With this in mind, we present a theoretical picture of the three ternary adsorption scenarios aiming at two observable properties: One is the concentration profile of adsorbed proteins, ctern(z), as a function of the altitude z. The second is the adsorption isotherm relating the amount of adsorbed protein per unit area, Γ, to the area per PEG chain, Σ, the concentration of the protein solution, cp, and the temperature T. We obtain analytical expressions for ctern(z) and Γ(Σ,cp) specifying their dependence on three sets of parameters: (i) Σ and the degree of polymerization, N, of the grafted PEG chains describe the brush; (ii) the proteins are characterized by their volume Vp and surface area Ap and (iii) the interaction energy between the protein and the PEG. The analytical results that we obtain pertain primarily to the conception and analysis of physical chemistry experiments using model proteins. Thus, we focus on small, univalent, recombinant antibody fragments such as scFv fragments24,25 rather than on intact antibodies. This avoids modeling difficulties due to the structural complexity, flexibility, and multivalency of intact immunoglobulins. With regard to weak nonspecific adsorption, we consider the relatively well studied case of serum albumin.13,19 With these reservations, our results also provide general insights concerning the design of PEG brushes for biomedical applications and their limitations. The advantages of analytical expressions relating multiple parameters are attained at the price of simplifications affecting the applicability of the theory. First, in common with preceding models, the theory assumes thermodynamic equilibrium.9-11,13,14 Second, the analysis concerns roughly spherical globular proteins with an approximate radius of Rp , H0 ≈ N(a2/Σ)1/3a, where H0 denotes the unperturbed brush height (Figure 2) and a is the monomer size. As we shall discuss in section II, this condition is fulfilled for the model systems considered. The applicability of the theory is further limited to brushes with relatively large N and small Σ when the self(24) Holliger, P.; Hudson, P. J. Nat. Biotechnol. 2005, 23, 1126–1136. (25) Kindt, T. J.; Goldsby, R. A.; Osborne, B. A. Kuby Immunology, 4th ed.; W. H. Freeman: New York, 2006.
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Figure 2. (a) Insertive vs (b) compressive modes of roughly spherical proteins. Small proteins with Rp , H0 enter the brush via the insertive mode, with only a local perturbation of the concentration profile. The compressive mode is realized by large proteins with Rp . H0 that can approach the grafting surface only by compressing the brush.
consistent field theory (SCF) is applicable and the concentration profile of the brush is parabolic.26,27 Because our primary interest is in the repression of ternary adsorption, we focus on the dilute regime, where protein-protein interactions within the brush are negligible. For brevity, we consider pure ternary adsorption and ignore contributions due to primary and secondary adsorption. The generalization to allow for the combined effect is straightforward because the different contributions to Γ are additive in the dilute brush regime.14 Our discussion focuses on the interplay of two free energies. Ternary adsorption is driven by an attraction term reflecting attractive interactions between the PEG brush and the proteins. In all its forms, it is countered by the insertion penalty incurred by proteins entering the brush. In the following we approximate the insertion free energy by the osmotic penalty. Strong adsorption and weak adsorption differ, however, with respect to the attraction term. Strong adsorption involves a specific site at the surface of the protein that binds a definite number of consecutive monomers. The protein binds to an individual PEG chain. The associated binding free energy in itself is not modified by the brush. In particular, it is independent of the protein surface area Ap and the monomer volume fraction φ. Antibody fragments of different size exhibit identical binding free energies24 because their binding sites are identical. The brush affects the PEG-protein binding equilibrium in two respects: (i) The mass action law reflects the concentration profile in the brush. (ii) The equilibrium constant is modified by the osmotic penalty. Whereas strong adsorption reflects a binary protein-PEG reaction, the weak adsorption emerges as a form of partitioning between two phases: a PEG-rich brush and the neat solvent. The protein does not bind to an individual PEG chain because the adsorption threshold is not attained. Rather, the ternary adsorption is driven by a large number of weak attractions incurred in the brush, interactions that involve many chains.14 The weak nonspecific adsorption onto the brush differs from the partitioning of proteins between two bulk polymer phases15-18 in that the grafted polymer chains cannot exchange with the solution. It is modeled assuming a uniform protein surface with weakly attractive monomer-protein contacts. A monomer in contact with the surface of the protein experiences an adsorption free energy of -ɛkBT with a positive ɛ , 1 in the range of 0.010 for all Σ whereas for the two other modes zmax=0 occurs in a certain Σ range. (iii) For weak adsorption and strong backbone adsorption, Γ ∼ N whereas for terminal adsorption Γ is independent of N, i.e., Γ ∼ N0. (iv) At high grafting densities, Γ decreases as Σ1/3 for weak adsorption and backbone adsorption whereas for terminal adsorption Γ approaches a constant. Altogether Γ due to weak and strong backbone adsorption can be lowered by using brushes with small N and Σ. In contrast, the contribution of the strong terminal adsorption does not vanish for low N and Σ and may thus survive in brushes comprising short chains at high grafting densities. Its repression may require a brush of loops with no free ends, as may be obtained upon adsorption of triblock copolymers comprising a middle nonadsorbing PEG block and two terminal adsorbing blocks. Ternary protein adsorption is of interest from a number of perspectives. One concerns drug delivery and the design of biocompatible interfaces as discussed earlier. The second point of view is that of polymer physics. Ternary adsorption provides an additional probe of brush properties, and the terminal adsorption of antibodies is of special interest because it is sensitive to the free ends distribution. The polymer science interest in this system is further illustrated by studies of protein adsorption onto polyelectrolyte brushes, where electrostatic interactions and counterion release play a dominant role.28-30 Finally, from a surface science point of view, ternary adsorption is a case of adsorption onto a soft, penetrable surface with tunable properties. From all points of view, it is important to note that ctern(z) due to ternary protein adsorption can be characterized experimentally by neutron reflectometry using deuterated proteins31 whereas standard techniques, such as ellipsometry, can probe only Γ(Σ). While our discussion focuses on PEG, similar effects also occur for other antigenic neutral water-soluble polymers (NWSP) such as polyvinylpyrrolidone (PVP)32 and dextran.33 The theory as presented applies to brushes of NWSP immersed in marginal solvent in general. Numerical results shown in our graphs are, however, obtained for parameters specific to PEG brushes. Furthermore, whereas our discussion of strong adsorption underlines the role of antibodies in NWSP, the scope of this effect is wider. In particular, one should note the existence of incidental strong binders,34 such as peptide deformylase,35 that bind PEG with no biological rationale. Finally, preceding discussions of ternary adsorption11,13 invoked the Alexander model assuming a steplike monomer concentration profile and uniform stretching of the chains. In the following text, we analyze the different ternary adsorption cases when the brush is described by SCF theory leading to a parabolic concentration profile. (28) Biesheuvel, P. M.; Leermakers, F. A. M.; Cohen Stuart, M. A. Phys. Rev. E 2006, 73, 011802. (29) de Vos, W. M.; Biesheuvel, P. M.; de Keizer, A.; Kleijn, J. M.; Cohen Stuart, M. A. Langmuir 2008, 24, 6575–6584. (30) de Vos, W. M.; Biesheuvel, P. M.; de Keizer, A.; Kleijn, J. M.; Cohen Stuart, M. A. Langmuir 2009, DOI:10.1021/la900791b. (31) For preliminary results, see Schollier, A.; Frangneto. G, ; Halperin. A.; Sferraza, M. ILL report 9-13-197 (2007) available at http://club.ill.fr/cv/servlet/ ReportFind. (32) Gill, T. J.; Kunz, H. W. Proc. Natl. Acad. Sci. U.S.A. 1968, 61, 490–496. (33) Kabat, E. A.; Berg, D. J. Immunol. 1953, 70, 514–532. (34) Hasek, J. Z. Kristallogr. Suppl. 2006, 23, 613–618. (35) Becker, A.; Schlichting, I.; Kabsch, W.; Schultz, S.; Wagner, A. F. V. J. Biol. Chem. 1998, 273, 11413–11416.
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The remainder of the article is organized as follows. In section II, we discuss the underlying ingredients of our analysis: assumptions, length scales, and free energies, focusing mainly on the brush regime. In particular, we summarize the relevant properties of antibodies, antibody fragments, and serum albumin. Ternary adsorption in the mushroom range is outlined in section III. This is helpful for clarifying the physical chemistry of ternary adsorption even though its relevance to the repression of protein adsorption is limited. Strong ternary adsorption in brushes is analyzed in detail in section IV focusing on the case of “intermediate” proteins incurring an osmotic penalty large enough to affect the binding equilibrium yet sufficiently small so as not to affect the brush structure. The weak adsorption case is analyzed in section V, having in mind the case of serum albumin. Mathematical details concerning the strong and weak adsorption regimes are delegated to Appendices A-D. Qualitative features of the strong adsorption of large proteins, incurring an osmotic penalty sufficient to perturb the brush structure, are briefly discussed in Appendix E. A summary and concluding discussion are provided in section VI.
II. Ternary Adsorption: Length Scales and Free Energies Our discussion of ternary adsorption mostly concerns the brush regime when the grafted chains crowd each other and the distance between two grafting sites, Σ1/2, is smaller than RF, the Flory radius of the polymer. For simplicity, we consider a monodisperse planar brush comprising chains that do not adsorb onto the surface. The crowded regime is the most relevant because our interest is in brushes deployed for the repression of protein adsorption. We will also briefly discuss ternary adsorption in the mushroom regime when the surface bearing the chains is nonadsorbing. Whereas this case is not significant for repressing protein adsorption, it merits attention because it allows us to discriminate easily between the three adsorption scenarios as well as to determine the binding constants for the strong adsorption. In this section, we summarize the assumptions, results and experimental information that will be invoked in our subsequent analysis, focusing primarily on the brush case. Ternary adsorption within a brush is strongly influenced by its monomer concentration profile, c(z). This determines the penalty incurred upon insertion of a protein into the brush. For weak adsorption, c(z) also determines the attraction term. Because we aim to distinguish between terminal and backbone adsorption, our model should allow for the concentration profiles of both monomers and free ends. The SCF theory of polymer brushes in marginal solvents,26,27 leading to a parabolic monomer concentration profile, ! z2 ð1Þ cðzÞ ¼ c0 1 - 2 H0 accordingly provides a suitable foundation. Within this picture, the volume fraction at z=0, φ0 =c0a3 is a2 φ0 ¼ B Σ
!2=3
3 π 2 a3 , B¼ 2 8pv
!1=3
and H0, the height of an unperturbed brush, is given by !1=3 H0 3 N a2 ¼ 2B Σ a
ð2Þ
ð3Þ
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v ≈ a3τ where τ=1 - Θ/T, a is the monomer size, Θ is the theta temperature, and Σ is the area per chain. Because in a typical experiment one varies Σ and N at a constant T for a single polymer species, it is helpful to group the numerical factors as well the polymer-specific p and v into a constant B, thus obtaining more compact expressions. The SCF theory yields the concentration profile of free ends or terminal groups, 3z ce ðzÞ ¼ ΣH0 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 1- 2 H0
ð4Þ
and the average free energy per chain in the unperturbed brush, 0 Fbrush 6v a2 ¼ 3 BN 5a kB T Σ
!2=3 ð5Þ
The parabolic brush picture is appropriate to the case of PEG brushes in aqueous media because of evidence that water is a marginal solvent for PEG.14,36,37 Comparison to computer simulations demonstrates that this approach accounts well for the leading features of the brush structure.38 Full consensus regarding the values of a, p, and τ for PEG remains to be established. In the following text, we will utilize a ≈ 5 A˚, p=2,39 and τ ≈ 0.25 (thus v ≈ 30 A˚3 and B ≈ 2). PEG brushes with a wide range of N and Σ can be realized using the Langmuir-Schaeffer technique and polystyrene(PS)-PEG diblock copolymers. 40 The reported ranges of N and Σ are respectively 48 e N e 770 and 170 A˚2 e Σ e 10 000 A˚2. The corresponding H0 range for N=770, as obtained from eq 3, is thus 390 A˚ e H0 e 1500 A˚. To estimate the insertion and attraction free energies, we assume that the insertion of protein at altitude z is similar to its insertion into a bulk solution of concentration c = c(z). As discussed below, this insertion assumption is justified when the proteins are small enough and when their concentration in the brush is sufficiently low. With regard to the protein size, it is useful to distinguish between two modes of inserting nonadsorbing, spherical particles into a brush. When Rp . H0, the particle can approach the grafting surface only by compressing the brush and the insertion assumption is not justified (Figure 2). It is, however, reasonable for relatively small particles, Rp , H0, because their insertion affects the monomer concentration only in their immediate vicinity.11 In the Rp , H0 case, the insertion assumption is also justified because the variation of the parabolic c(z) (cf. eq 1), over a distance Rp is small. Its performance is further enhanced because the sphere’s volume elements closer to the grafting surface experiences higher c(z) than the outer volume elements. As a result, the errors in estimating the insertion penalty partially cancel. These two regimes occur also for nonspherical particles, but the regime boundaries require a more elaborate analysis. For example, for disklike particles it is necessary to distinguish between edgewise insertion and insertion with the disk surface parallel to the grafting surface. For simplicity, we confine our analysis to roughly spherical particles with Rp , H0. Clearly, the insertion assumption is valid only when protein-protein interactions within the brush can be neglected. For simplicity, (36) Hansen, P. L.; Cohen, J. A.; Podgornik, R.; Parsigian, V. A. Biophys. J. 2003, 84, 350–355. (37) Marsh, D. Biophys. J. 2004, 86, 2630–2633. (38) Lai, P. Y.; Zhulina, E. B. J. Phys. II (France) 1992, 2, 547–560. (39) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: Oxford, England, 2003. (40) de Vos, W. M.; de Keizer, A.; Kleijn, M.; Cohen Stuart, M. A. Langmuir 2009, 25, 4490–4497.
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we focus on proteins at their isoelectric pH, when their net charge is zero. In this regime, electrostatic interactions do not play an important role, and hard core excluded volume interactions are dominant. We thus limit our discussion to the dilute brush range corresponding roughly to ΓkBT, thus (41) Louis, A. A.; Bolhuis, P. G.; Meijer, E. J.; Hansen, J. P. J. Chem. Phys. 2002, 116, 10547–10556. (42) de Gennes, P.-G. C.R. Acad. Sci. Paris, Ser. B 1979, 288, 359–361. (43) Odijk, T. Macromolecules 1996, 29, 1842–1843. (44) Hanke, A.; Eisenriegler, E.; Dietrich, S. Phys. Rev. E 1999, 59, 6853–6878. (45) Wang, S.; van Dijk, J. A. P. P.; Odijk, T.; Smit, J. A. M. Biomacromolecules 2001, 2, 1080–1088. (46) Grosberg, A.Yu.; Khokhlov, A. R. In Statistical Physics of Macromolecules; American Institute of Physics: New York, 1994. (47) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610– 1619.
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having a significant effect on the ternary adsorption. This requirement implies Fin ≈ Fosm ≈ Π(φ(z))Vp ∼ φ2(z). Note, however, that Fin ≈ Fosm>kBT is never realized throughout the brush because the parabolic profile φ(z) =φ0(1 - z2/H02) implies that Fin e kBT at its outer edge where Fin/kBT ≈ Rp/ξid ∼ φ(z). The applicability range of Fin ≈ Fosm>kBT can be estimated from the condition Fosm(z)/kBT=Π(z)Vp=Π0Vp(1 - z2/H02)2 ≈ 1, where !2 Fosm ðzÞ Σosm 4=3 z2 1- 2 ¼ kB T Σ H0
ð9Þ
Here the osmotic pressure at z=0 is Π0a3/kBT ≈ τφ20, and Σosm, defined as Σosm τB2 Vp 2 a a3
!3=4 ð10Þ
corresponds to the condition Π0Vp = kBT or τB2(Vp/a3) (a2/Σosm)4/3 = 1. This last definition concerns a hypothetical situation with the protein center located at z=0, a situation that cannot be realized because the protein cannot attain z > Σ osm > < Γ 0:68 ≈ for R ¼ 1 or Σ ¼ Σosm > c p K0 Σ osm > > > > : 1 for R , 1 or Σ . Σosm Σ
ð27Þ
In the brush regime, Γ decreases monotonically as Σ increases (Figure 4). Importantly, within our simplified model Γ does not vanish for Σ , Σosm. It is also helpful to compare Γ to (54) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; National Bureau of Standards: Washington, DC, 1967.
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Figure 3. Protein volume fraction profiles of terminally adsorbed protein, φp(z) = ctern(z)Vp, multiplied by N/cpK0, vs the reduced brush height z/H0 for Π0Vp/kBT , 1, Π0Vp/kBT ≈ 1, and Π0Vp/ kBT . 1 as obtained for Σ = 0.2Σosm (---), Σ = Σosm (-), and Σ = 5Σosm ( 3 3 3 ). The PEG parameters are a = 5 A˚, τ = 0.25, and p = 2, hence B = 2.03.
Γ(Vp=0)=cpK0/Σ. As expected, Γ ≈ Γ(Vp=0) for small proteins incurring a weak insertion penalty (R e 1) whereas in the opposite limit Γ/Γ(Vp =0) ≈ Σ/Σosm , 1. The scaling behavior of Γ at large and small R, as given by eq 27, can be obtained using a simple argument. This invokes the approximation Γ ≈ ctern(zmax)Δz where Δz is the characteristic width of ctern(z). For R , 1, the width is roughly Δz ≈ H0 and ctern (zmax) ≈ cpK0/ΣH0, corresponding to proteins bound at the maximal concentration of unperturbed free ends. Γ ≈ ctern (zmax)Δz thus leads to Γ ≈ cpK0/Σ. In the opposite limit, R . 1, the maximal ctern(zmax), according to eq 23, is lowered by a factor of R-1/4. The corresponding Δz is narrowed by a factor of R-1/2 to Δz ≈ H0 - zmax where zmax ≈ [1 - 4/4R1/2]H0. This yields Δz ≈ H0/4R1/2 and hence Γ ≈ ctern(zmax)Δz ≈ cpK0/ΣR3/4= cpK0/Σosm. Within our model, Γ(Σ) of terminal adsorption in the brush regime decreases monotonically with Σ but does not vanish when Σ f 0. As we shall see, this is in contrast to the behavior of weak ternary adsorption and strong backbone adsorption. This brings up the question of a possible artifact due to the underestimation of Fin by Fosm. However, as discussed in Appendix B, this conclusion is independent of the form of Fin. B. Backbone Adsorption in the Linear Regime. The concentration profile of backbone adsorbed proteins is
ctern ðzÞ ¼ cp K0 c0
! 2 !2 3 2 z2 z 1 - 2 exp4 -R 1 - 2 5 H0 H0
ð28Þ
where c0 φ0/a3 is discussed in the beginning of this section, and R, the reduced the osmotic penalty at z=0, is given by eq 22 or eq 26. The pre-exponential factor ctern(z, Vp=0)=cpK0c(z) is the concentration profile of ideal pointlike proteins incurring no insertion penalty. ctern(z, Vp = 0) attains its maximal value at the wall, z=0, where the concentration of binding monomers is highest. When R , 1, the protein concentration profile is proportional to c(z). As the osmotic penalty grows, ctern(z) decreases in amplitude and exhibits a maximum at zmax > 0 given by zmax 2 1 1 Σ 2=3 ¼ 1 - pffiffiffiffiffiffi ¼ 1 - pffiffiffi H0 2R 2 Σosm 11628 DOI: 10.1021/la9008569
ð29Þ
Figure 4. Reduced surface fraction of terminally adsorbed protein ΓR2p/cpK0 vs Σ/Σosm calculated for scFv using Rp ≈ VscFv1/3.
√ for R> 1/2 or equally Σ (4 8)Σosm, the maximum occurs at z = 0 (Figure 5). The corresponding
Γ ¼ c p K0
φ0 a3
Z 0
H0
1-
2
z H0 2
!2 3 z exp4 -R 1 - 2 5 dz H0
!
2
2
ð31Þ
evaluates as c p K0 N Γ¼ 2 F2 Σ
!
3 5 7 1, , , , -R 2 4 4
ð32Þ
where 2F2 is a special pFq generalized hypergeometric function (Appendix C). The amount of adsorbed protein per unit area Γ exhibits a maximum around R ≈ 1 or Σ ≈ Σosm and tends to zero for both R f ¥ or Σ f 0 and for R f 0 or Σ f ¥ (Figure 6): 8 > 3 1 Σ 1=3 > > for R . 1 or Σ , Σosm > > > cp K0 N > Σ osm > > > 1 > : for R , 1 or Σ . Σosm Σ
ð33Þ
It is helpful to compare Γ to Γ(Vp = 0) = cpK0N/Σ of pointlike proteins. As expected, Γ ≈ Γ(Vp=0) for small proteins incurring a weak insertion penalty, R e 1, whereas in the opposite limit Γ , Γ(Vp =0). As for the case of terminal adsorption, we can obtain the scaling behavior of Γ at large and small R, as seen in eq 33, using the Γ ≈ ctern(zmax)Δz approximation. For R , 1, the maximum in ctern occurs at ctern(z=0)=cpK0c0 and Δz ≈ H0, leading to Γ ≈ cpK0c0H0 ≈ cpK0N/Σ. In the opposite limit, R . 1, the maximum is located at zmax/H0 ≈ 1 - 8-1/2(Σ/Σosm)2/3, and the width of the concentration profile ctern(z) is Δz = H0 - zmax ≈ H0/(8R)1/2. Because ctern(zmax) ≈ cpK0c0(Σ/Σosm)2/3, we obtain Γ ≈ 1/3 ctern (zmax)Δz ≈ cpK0Σ-1 osmN(Σ/Σosm) . C. Terminal vs Backbone Adsorption and the Comparison of scFv and Fab Fragments. For terminal and backbone adsorption in the linear regime, ctern(z) = η(z)x(z) and x(z) ≈ exp{-R(1 - [z/H0]2)2}. The features distinguishing the two modes Langmuir 2009, 25(19), 11621–11634
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Σ , Σosm range, as obtained from eq 27, is 3=4 ΓscFv ðΣÞ VFab 3=4 55 ≈ ≈ ¼ 1:66 ΓFab ðΣÞ 28 VscFv
ð35Þ
As before, we assumed that VFab/VscFv is well approximated by the ratio of their molecular weights. Similarly, eq 33 for backbone adsorption in the Σ , Σosm regime leads to ΓscFv ðΣÞ VFab 55 ¼ 1:96 ð36Þ ≈ ≈ ΓFab ðΣÞ 28 VscFv In both cases, ΓscFv/ΓFab f 1 as Σ increases toward the overlap threshold Σ ≈ RF2 where the insertion penalty is negligible.
V. Weak Ternary Adsorption of Intermediate Proteins Figure 5. Protein volume fraction profiles of backbone adsorbed protein, φtern(z) = ctern(z)Vp, divided by cpK0, vs the reduced brush height z/H0 for R = Π0Vp/kBT , 1, R ≈ 1, and R . 1 as obtained (cf. eq 26) for Σ = 0.2Σosm (---), Σ = Σosm (-), and Σ = 5Σosm ( 3 3 3 ).
Figure 6. Reduced surface fraction of backbone adsorbed protein ΓRp2/NcpK0 vs Σ/Σosm calculated for scFv using Rp ≈ VscFv1/3.
are traceable to differences between η(z)=c(z) and η(z)=ce(z). As a result, ctern(z) of terminal adsorption exhibits always a maximum at zmax>0 whereas for backbone adsorption zmax=0 occurs for Σ>Σosm. In the brush regime, as in the mushroom range, Γ ≈ N for backbone adsorption while Γ ≈ N0 for terminal adsorption. Within our discussion of the mushroom case, this reflects the entropy of the adsorbed proteins whereas in the brush regime it arises because of the mass action law. Γ of backbone adsorption exhibits a maximum within the brush regime, around Σ ≈ Σosm. For terminal adsorption, Γ decreases monotonically with Σ in the brush range. However, because Γ ≈ K0cp/Σ in the mushroom regime, the combination of the two trends suggests that the terminal adsorption Γ exhibits a maximum around the overlap threshold, when Σ ≈ RF2. The above considerations lead to predictions for experiments comparing the ternary adsorption of univalent scFv and Fab fragments with identical K0. For both terminal and backbone adsorption, the ratio of the concentration profiles in the linear regime is 2 !2 3 2 cscFv ðzÞ Π ΔV z 0 p ð34Þ ¼ exp4 1- 2 5 cFab ðzÞ kB T H0 Here, ΔVp = VFab - VscFv, where VFab and VscFv denote, respectively, the volumes of the Fab and scFv fragments. The corresponding ratio of Γ for terminal adsorption in the R . 1 or Langmuir 2009, 25(19), 11621–11634
The free energy of weakly adsorbed proteins in a brush, Ftern= Fbind + Fosm, differs from that of strongly adsorbed proteins in the nature of the attraction term. Fbind(z)/kBT ≈ -c(z)Apaɛ (eq 12) reflects attractive (ɛ > 0) interactions of strength ɛkBT experienced by a monomer in grazing contact with the protein surface. As opposed to the strong adsorption case, the attractive interactions do not involve a specific surface site and, in our simplified description, the protein surface is considered to be uniform. The chemical potential of the protein within the brush is μtern ≈ Ftern(z) + kBT ln ctern(z), where ctern(z) is the dimensionless number concentration of ternary adsorbed proteins per unit area at altitude z. The adsorption isotherm is obtained by equating μtern and the chemical potential of a free protein in the bulk solution, μbulk ≈ kBT ln cp, whose dimensionless number concentration is denoted by cp. The adsorption isotherm is Ftern ðzÞ ð37Þ ctern ðzÞ ¼ cp Ktern ðzÞ ¼ cp exp kB T where the local equilibrium constant is 2 ! !2 3 2 2 Ap z Vp z Ktern ðzÞ ¼ exp4 2 Eφ0 1 - 2 - 3 τφ20 1 - 2 5 ð38Þ a a H0 H0 Ktern(z) determines the form of the protein concentration profile due to weak ternary adsorption, ctern(z), and the total amount adsorbed per unit area, Z H0 Ktern ðzÞ dz ð39Þ Γ ¼ cp 0
To obtain Γ in the number of adsorbed proteins per unit area, it is necessary to express cp in eq 39 as a number concentration. This does not violate our earlier discussion because ctern(z)=cpKtern(z), as derived in the paragraph before eq 37 for dimensionless cp, remains valid for the dimensional form of cp. The onset of ternary adsorption at z=0 occurs roughly when Fbind/kBT ≈ (Ap/a2)ɛφ0 ≈ 1. This occurs at φ=φ+ a2/Apɛ or equivalently at Σ = Σ+ where Σ+ is the characteristic area Σ+ corresponding to φ+ as defined by55 Σþ BAp E 3=2 ð40Þ a2 a2 The ternary adsorption at z = 0 begins to be repressed when Fbind ≈ Π0Vp. This crossover occurs at (Ap/a2)ɛφ0=(Vp/a3)τφ02 or φ-=aApɛ/τVp, thus defining another characteristic area Σ-,55 !3=2 ΣBτVp ð41Þ a2 aAp E (55) Σad and Σco as used in ref 14 correspond to our Σþ and Σ-, respectively.
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Figure 7. Reduced concentration profiles of weakly adsorbed pro-
teins, ctern(z)/cp=Ktern(z), for Σ+=5Σ- depicting Σ=0.3Σ- (thick continuous line), Σ=Σ- (thick continuous line), Σ=81/2Σ- (---), and Σ= 10Σ- ( 3 3 3 ). Plots are qualitatively similar for other choices of Σ+/Σ- .
Notice that Σ- and Σ+ are related via Σ- Σ+ =Σosm2. Utilizing Σ- and Σ+ allows us to express Ftern(z) in the form !2 ! Ftern ðzÞ Σosm 4=3 z2 Σ þ 2=3 z2 ¼ 1- 2 1 - 2 ð42Þ kB T Σ Σ H0 H0 leading to " Ktern ðzÞ ¼ exp
Σþ Σ-
#
2=3
ðQ - Q Þ 2
ð43Þ
where Q (Σ-/Σ)2/3(1 - z2/H02). In this form, it is immediately evident that the maximum in Ktern(z) and thus in ctern(z) is located at Q(zmax)= 1/2, hence, zmax 2 1 Σ 2=3 ¼12 ΣH0 2
ð44Þ
Accordingly, a maximum in ctern(z) occurs when Σ < 81/2Σwhereas for Σ g 81/2Σ- the maximal value occurs at the wall (Figure 7), at z=0. When Σ+ , Σ-, the adsorption is negligible because the osmotic penalty overwhelms the adsorption free energy before Fbind ≈ kBT is attained. Significant adsorption occurs for Σ+ . Σ- when Γ exhibits a maximum atxA ΣMAX ≈ 81/2Σ-. Inserting eq 43 into eq 39 yields
Γ≈
8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > :
2 Σ þ 2=3 Σ þ 1=3 cp H0 ðΣ þ Þ 1 þ Σ 3 Σ for Σ . Σ - and Σ . Σ þ pffiffiffi Σ þ 2=3 π c H0 ðΣ þ Þexp Σ 2 p for Σ þ . Σ . Σ 9 Σ - 1=6 1 Σ þ 2=3 c p H0 ðΣ - Þ exp 10 4 ΣΣþ pffiffiffi for Σ ¼ ΣMAX ≈ 8Σ pffiffiffi Σ 1=3 1 Σ þ 2=3 π exp c p H0 ðΣ - Þ Σþ 4 Σ4 for Σ , Σ - , Σ þ
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ð45Þ
ð45Þ
Figure 8. Reduced surface fraction of weakly adsorbed protein, ΓRp2/Nφp, where φp = cpVp is the volume fraction of the bulk protein, vs Σ/Σ- for Σ+ = 5Σ- (-), Σ+ = Σ- (---), and Σ+ = 0.1Σ(3 3 3)
The detailed derivation of eq 45 is discussed in Appendix D1, and the graphs of Γ(Σ) are shown in Figure 8. Expressions for the maximal amount of adsorbed protein, ΓMAX, and the corresponding area per chain, ΣMAX, are delegated to Appendix D2. As in the strong adsorption scenarios, it is possible to recover the leading features of the adsorption behavior using the Γ ≈ ctern(zmax)Δz approximation. In the weak adsorption case, where ctern(z)=cpKtern(z), its application invokes Ktern(zmax) (Appendix D3). The overall picture is as follows: There is essentially no ternary adsorption when Σ+ < Σ- and the osmotic penalty dominates for all Σ. Significant adsorption occurs when Σ+ . Σ-. When Σ+ . Σ . Σ-, the maximum in ctern occurs at z=0 where Ktern(0) ≈ exp[(Σ+/Σ)2/3] and the width Δz ≈ H0(Σ+) is roughly constant. In this range, Γ increases as Σ decreases because Ktern(0) grows with the adsorption free energy ∼ (Σ+/Σ2/3). Γ attains a maximum around Σ ≈ Σ- where both Δz ≈ H0(Σ) ≈ H0(Σ-) and Ktern(0) ≈ exp[(Σ+/Σ-)2/3/4] reach their maximum values. Further lowering of Σ causes a decrease of Γ. In this range, Ktern(zmax) ≈ exp[(Σ+/Σ-)2/3/4] does not depend on Σ whereas Δz decreases with Σ because the large osmotic penalty near the grafting surface gives rise to a protein exclusion region.
VI. Summary and Discussion Our study of ternary protein adsorption onto PEG brushes considered three modes differing in the nature of the PEGprotein attraction, in particular, weak nonspecific attraction and strong binding at specific protein sites. In the last category, we distinguish between sites binding to free ends and sites that bind to the polymer backbone. The focus on these three scenarios is based on current experimental evidence and does not exclude the possibility of other adsorption modes. In every case, the protein adsorption is opposed by a free-energy penalty incurred upon inserting the impenetrable protein into the brush. In practical terms, this leads to significant effects when the proteins are large enough and the insertion gives rise to an osmotic penalty. The three modes are characterized in terms of two observables: the protein concentration profile, ctern(z), within a brush and the total amount of adsorbed protein, Γ. For the case of dilute protein solutions, the equilibrium adsorption isotherms lead to analytical results. In particular, ctern(z) for strong terminal adsorption, strong backbone adsorption and weak nonspecific adsorption is given, respectively, by eqs 21, 28, and 38. The corresponding Γ is specified by eqs 24, 32, and 45. These equations relate the observables to the leading tuning parameters of the system: N, Langmuir 2009, 25(19), 11621–11634
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Σ, Vp, and Ap. Our analytical results for Γ have been tested against the numerical integration of ctern(z) and visualized for cases of scFv fragments and serum albumin. An online facility56 allows numerical calculations for different parameters so as to facilitate quantitative comparison with other systems. The assumptions underlying our rough analysis limits its applicability to brushes such that (i) brush height H0 is large compared to the span of the protein, (ii) Σ is sufficiently small to ensure Π0Vp/kBT>1, or, equivalently, Σ0 regimes. This can be achieved by the use of the Pincus model because the chain ends’ distribution does not affect the weak adsorption behavior. In contrast, a discussion of strong ternary adsorption must incorporate SCF-based approaches that can correctly distinguish between terminal and backbone adsorption. The physical chemistry approach to protein adsorption, be it experimental or theoretical, often involves surfaces in contact with a solution of single proteins. The predictive value of this approach has been criticized on the grounds that it cannot capture the complicated processes that occur in vivo.59 As our discussion underlines, this difficulty is partially traceable to the choice of protein studied. Much work in this area focused on majority species, such as serum albumin in the blood and lysozyme in the tear fluid. This choice is reasonable because the majority species are expected to dominate the adsorption on kinetics grounds, at least in an early stage. This choice is also dictated by practical considerations because such proteins are well characterized and easily available. Less effort was directed at the study of minority species such as PEG antibodies partially because their commercial availability is only recent. However, the ternary adsorption of PEG antibodies as well as other NWSP antibodies merits study because it may trigger an immune cascade that can affect the in vivo performance of the brush. Thus far, the immune response to PEG and NWSP was studied for free chains32,33 and for nanoparticles decorated by PEG, such as PEGylated proteins20 and stealth liposomes.21 However, it is not obvious that the immune response to macroscopic surfaces displaying brushes is identical if only because their residence time in the body is longer. Acknowledgment. We thank Katia Zhulina and Efim Katz for helpful discussions. This project has been supported through EUNSF contract NMP3-CT-2005-016375 and FP6-2004-NMP-TI-4 STRP 033339 of the European Community.
Appendix A: Mushroom Regime We first discuss the case of terminal adsorption and then comment on the modification required for the description of backbone adsorption. The adsorption isotherm of proteins onto terminally adsorbing mushrooms is specified by the equilibrium condition μad =μnb + μp, where μad is the chemical potential of chains with a bound protein, μnb is the chemical potential of nonbinding chains, and μp is the chemical potential of the free proteins. In the first step, we obtain the exchange chemical potential μex = μad - μnb by minimizing the free energy per grafting site, γsite(x), with respect to x, the fraction of chains with bound proteins. When each chain can bind at most a single protein, γsite is γsite ¼ kB T½x ln x þ ð1 -xÞlnð1 -xÞ þ xμ0ad þ ð1 -xÞμ0nb
ðA1Þ
The first term allows for the mixing entropy of binding and nonbinding chains. μ0ad is the standard chemical potential of a chain with a bound protein, and μ0nb is the standard chemical potential of a bare chain with no adsorbed protein. The exchange chemical potential is thus μex ¼
Dγsite x þ μ0ad - μ0nb ¼ kB T ln 1 -x Dx
(59) Gorbet, M. B.; Sefton, M. V. Biomaterials 2004, 25, 5681–5703.
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and the adsorption isotherm is specified by μex=μp. The chemical potential of the free proteins is μp =μ0p + kBT ln ap, where μ0p is their standard chemical potential and ap is their dimensionless activity ap =γpcp, where cp is the protein number concentration and γp is the activity coefficient such that γp f 1 when cp f 0.60 Because our discussion focuses on dilute protein solutions, we will for simplicity use μp=μ0p + kBT ln cp, where cp is the dimensionless number concentration. With this in mind, the equilibrium condition μex = μp yields the terminal adsorption isotherm for dilute protein solutions (eq 14). A similar argument may be used to obtain the backbone adsorption isotherm when cp is sufficiently small so as to ensure that on average no chain adsorbs more than one protein, implying x , 1. In this case, all binding sites along the homopolymer backbone are equivalent. When the binding site incorporates m consecutive monomers, the protein cannot adsorb within m/2 monomers from the chain ends. For a chain comprising N monomers, the protein can bind at N - m positions,61 and the standard chemical potential of a chain carrying a single bound protein is μ0ad - kBT ln(N - m), where μ0ad is the standard chemical potential of a protein bound to a chain segment of length m. The equilibrium condition μex=μp yields eq 15. Notice that our discussion of backbone adsorption is not valid for higher cp such that each chain adsorbs on average more than one protein. This regime requires an alternative description, replacing the ideal mixing entropy per chain Smix = k[x ln x + (1 - x) ln (1 - x)] by a mixing entropy allowing for multiple chain occupancy by impenetrable proteins that cannot reside on the same chain segment.13 For an average fraction of monomers binding proteins θ=mn /N and an average number of adsorbed proteins per chain n > 1, this can be approximated by Smix = kB(N/m)(θ ln{θ/[m(1 - θ) + θ]}) + m(1 - θ)ln{m(1 - θ)/[m(1 - θ) + θ]}. This last form is, however, qualitatively misleading for the n < 1 case, which we consider, because it assigns uniform occupancy to all chains whereas in reality some chains bind a single protein and others are bare.
Appendix B: Terminal Adsorption Rewriting eq 21 upon changing the variable from z to y z2/ H02 yields cp K0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ctern ðyÞ ¼ 3 ð1 -yÞy exp½ -Rð1 - yÞ2 ΣH0
Γ¼
3 cp K0 1 4 Σ R3=4
1
t -1=4 e -t dt ¼
0
3 cp K0 γð3=4, RÞ 4 Σ R3=4
ðB1Þ
ðB2Þ
(60) Moore, W. J. Physical Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1972. (61) The number of binding sites may be smaller because steric hindernce may prevent the protein from binding monomers in the immediate vicinity of the grafting site.
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8 > < 4 R3=4 - 4 R7=4 þ 2 R11=4 for R , 1 3 7 11 γð , RÞ ≈ 3 0:907 for R ¼ 1 > 4 : -1=4 -R Γð3=4Þ -R e for R . 1
ðB3Þ
Combining eqs B2 and B3 yields eq 27. For the hypothetical case of ideal proteins with Vp=0 and thus R=0, the protein concentration profile is ctern(z)=cpK0ce(z) yielding Γ(Vp =0)=cpK0/Σ. In the limit of large R, the difference (zmax/H0)2 - (zosm/H0)2 ≈ (Σ/Σosm)2/3/2 f 0 as Σ f 0 and the adsorption takes place mostly within a region where Fin > Fosm is operative. This raises the possibility that the adsorption at the outer edge is repressed by the higher Fin. To conduct a rough investigation of this option, we replace R(1 - y2)2 in eq 24 with Fin/kBT = const0 (Rp/a)B(a2/Σ)2/3 (1 - y2) as given by eq 6. In the √ limit of small Σ, this again leads to a nonvanishing Γ ≈ cpK0(3( π)/4)(a/Rp)3/2, albeit of a slightly different form.62
Appendix C: Backbone Adsorption As before, it is convenient to change the variable from z to y z2/H02, yielding ctern(y)=cpK0c0(1 - y) exp[- R(1 - y)2]. Because dy/dz=2z/H02 g 0, the extremum condition dctern(y)/dy=0 leads to 2R(1 - y)2 =1 and eq 29. In terms of y, eq 31 is rewritten as Γ¼
cp K0 N Σ
Z 0
1
3 ð1 -yÞ 2 pffiffiffi exp½ -Rð1 - yÞ dy 4 y
ðC1Þ
where the integral in eq C1 is identical to the definition of the generalized hypergeometric function 2F2({1, 3/2}, {5/4, 7/4}, -R),54 which evaluates to 0.527 for R=1 and behaves as 1 - 24R/35 + 64R/2312 for R , 1 and as 3/8R + 3π1/2/32R3/2 + 9/64R2 for R . 1, thus leading to eq 33.
Appendix D: Weak Ternary Adsorption R 1. Effect of Area per Chain. To calculate Γ=cp H0 0 Ktern(z) dz with Ktern given by eq 43, we change the variable from z to w 1 - z2/H20 to obtain Z
Because dy/dz=2z/H20, the extremum condition dctern(y)/dy=0 leads to a nonlinear equation, 1 - 2R(1 - y)2=1/2y. For R . 1, when δ 1- y ≈ 0, this reduces to 1-2Rδ2 ≈ 1 or y ≈ 1 - 1/2R1/2. In the opposite limit, R , 1, the extremum condition yields y ≈ 1/2. The simplest expression reducing to these two limits, as given by eq 23, provides an excellent approximation for the exact numerical solution for all R values (maximum deviation of 1.6% at R=2). Rewriting eq 24 upon changing the variable from z to t R(1 - [z/H0]2)2, such that 3z dz=-3H02/4(Rt)1/2, leads to Z
where γ (3/4, R) is the lower incomplete gamma function γ(3/4, R) R R -1/4 exp(-t) dt with the properties54 0t
Γ ¼ cp H0 ðΣÞ 0
1
2=3
Σþ dw exp ðQ -Q2 Þ pffiffiffiffiffiffiffiffiffiffiffi Σ2 1 -w
ðD1Þ
Here Q, as used in eq 43, is Q (Σ-/Σ)2/3w. We will separately consider several limiting cases of eq D1 where Σ is either large or small compared with Σ- and/or Σ+. (i) When Σ . Σ-, the termR Q2 in eq√D1 is negligible. The integral is thus√ of the form (1/2) 10 dw etw/ (1 - w)=1F1(1, 3/2, t) (π1/2/2)erf ( t), where 1F1(1, 3/2, t) is the Kummer confluent hypergeometric function54 and erf is the error function. With H0(Σ)=H0(Σ+)(Σ+/Σ)1/3, this leads to " # " # pffiffiffi Σ þ 2=3 Σ þ 1=3 π erf exp Γ ¼ cp H0 ðΣ þ Þ Σ Σ 2
ðD2Þ
In the Σ . Σ- case, it is also possible to distinguish between two limits: When Σ . Σ+, we utilize the series expansions R (62) In this case, Γ=cpK0Σ-1 0H0(3z/H02)(1 - z2/H20)1/2 exp[-M(1 - z2/H20)2] 2 2/3 dz with M = const0 (R √ p/a)φ0 = const0 (Rp/a)B(a /Σ) , which √ evaluates to Γ = (3π1/2/4) cpK0 Σ-1[erf( M)/M3/2 - 3e-M/2M. Because erf ( M) approaches unity in the limit of large M, the first term dominates, and Γ thus approaches Γ f (3π1/2/4)cpK0Σ-1M-3/2=(3π1/2/4)cpK0(const0 BRp/a)-3/2.
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√ √ erf( t) ≈ 2(t1/2 - 1/3t3/2 + ...)/ π and exp(t) ≈ 1 + t + ... to obtain the√first line of eq 45.√ In the opposite case, for Σ , Σ+ , when erf( t) ≈ 1 - (exp(-t)/ (πt)) + o(t-3/2) ≈ 1, we obtain the second line of eq 45. (ii) When Σ , Σ- and Q is large, it is useful to express Ftern(z) in the form Ftern(z)/kBT = (Σ+/Σ-)2/3[(Q - 1/2)2 - 1/4]. Because (Q - 1/2)2 ≈ Q2 and (Σ+/Σ- )2/3(Σ-/Σ)4/3 =(Σosm/Σ)4/3 =R, this leads to " #Z 1 1 Σ þ 2=3 dw 2 Γ ¼ cp H0 ðΣÞexp e -Rw pffiffiffiffiffiffiffiffiffiffiffi ðD3Þ 4 Σ2 1 -w 0 where the integral in eq D3 matches the definition of the hypergeometric function54 2F2({1/2, 1}, {3/4, 5/4}, -R), which evaluates to 0.623 for R = 1 and is approximated by the first terms of a series, 1 - 8R/15 + 64R2/315 for R , 1, and by (π/R)1/2/4 + 1/8R + 3(π/R3)1/2/64 + 5/64R2 for R . 1. The latter feature is used to obtain Γ in the fourth line of eq 45. In the Σ , Σ- case, there is a regime where expressions simplify dramatically: When Σ . Σosm or R=(Σosm/Σ)4/3=(Σ+Σ-/Σ2)2/3 , 1, we obtain Σ+ , Σ-. Thus, the exponential can be expanded, as exp[(Σ+/Σ-)2/3/4] ≈ 1 + (Σ+/Σ-)2/3/4. Combining these relationships leads to Γ ≈ cpH0(Σ+)(Σ+/Σ)1/3[1 + [1 + (Σ+/Σ-)2/3/4] [1 - 8(Σosm/Σ)4/3/15] with a leading term Γ ≈ cpH0(Σ) in the range of Σosm , Σ , Σ-. 2. Maximal Adsorption. Numerically we observe a maximum in Γ around ΣMAX ≈ 81/2Σ-. With this insight, it is possible to obtain analytical results for both the Σ- , Σ+ and Σ- . Σ+ regimes. In turn, these lead to a useful numerical approximation for the maximal Γ over a wide range of Σ-/Σ. To this end, it is convenient to introduce χ 2(Σ-/Σ)2/3, such 1/2 2/3 to that χ=1 at ΣMAX √ = (8 )Σ- and β (Σ+/Σ-) /4 leading H0(Σ) = H0(Σ-) R (χ/2) and Ftern(w)/kBT = β(χw - 1)2 - β. Altogether Γ=cp H 0K √tern(z) dz becomes the product of a prefactor, cpH0(Σ-)exp(β)/ 2 , and an integral Iβ(χ), Z 1 dw pffiffiffi ðD4Þ Iβ ðχÞ χ exp½ -βðχw -1Þ2 pffiffiffiffiffiffiffiffiffiffiffi 2 1 -w 0 which is Σ-dependent via χ. To investigate the β . 1 or Σ+ . Σregime, when χ ≈ 1 or Σ ≈ 81/2Σ-, we expand Iβ(χ) in powers of χ around χ=1, 0
Iβ ðχÞ ¼ Iβ ð1Þ þ ðχ -1ÞI β ð1Þ þ
2
ðχ -1Þ 00 I β ð1Þ 2
ðD5Þ
where the prime denotes a derivative with respect to χ. To proceed,R we need some properties of the kernel of the integral √ Iβ(χ) = 10h(w, β, χ)[dw/(2 (1 - w))], where h is given by eq D4 and we denote its value at χ=1 by h1 h(w, β, χ=1) = exp[-β(w - 1)2]. The first and second derivatives of h with respect to χ evaluate as h0 (χ=1)=[1/2 + 2βw(1 - w)]h1 and h00 (χ=1)=-{1/4 2βw + 4βw2[1 -β(1 - w)2]}h1, respectively. With these derivatives at hand, we need only basic integrals to evaluate eq D5 as Iβ ðχÞ ¼
γð1=4, βÞ 4β1=4 -
þ
χ -1 -β ½e þ β1=4 γð3=4, βÞ 2
ðχ -1Þ2 -β ½e þ β3=4 γð1=4, βÞ 8
ðD6Þ
The integral Iβ(χ) attains its maximal value at χ=χMAX specified by Iβ0 (χ)=Iβ0 (1) + ( χ - 1)Iβ00 (1)=0 or 0
χMAX ¼ 1 -
2 þ 2β1=4 expðβÞγð34 , βÞ I β ð1Þ ¼1 þ 00 I β ð1Þ 1 þ β3=4 expðβÞγð14 , βÞ
Langmuir 2009, 25(19), 11621–11634
ðD7Þ
We next simplify this expression in the limits of β . 1 and β , 1 corresponding to Σ+ . Σ- and Σ+ , Σ-. (i) For β . 1, we use γ(3/4, β) ≈ Γ(3/4) - exp(-β)β-1/4 and 1 γ( /4, β) ≈ Γ(1/4) - exp(-β)β-3/4 to obtain χMAX =1 + 2Γ(3/4)/ Γ(1/4)β1/2 ≈ 1 + 1.35 (Σ-/Σ+)1/3. Recalling the definition χ= (Σ-/Σ)2/3, the maximum corresponds to an area per chain " 1=3 # pffiffiffi ΣΣMAX ≈ 8Σ - 1 -2:03 ðD8Þ Σþ Because this expression was obtained from an expansion to second order around χ= 1, it is valid only so long as |χMAX 1| , 1. Accordingly, the maximum Iβ(χMAX) is well approximated by Iβ(1) when Σ- , Σ+ and χMAX ≈ 1. In this β . 1 limit when γ(1/4, β) ≈ Γ(1/4), one has I(β, 1) ≈ Γ(1/4)/4β1/4 with Γ(1/4)/4 ≈ 0.9064, leading to Γ(χMAX) stated in eq 45. (ii) For β , 1, a different approach is required because χMAX given in eq D8 √ no longer satisfies |χMAX - 1| , 1. To this end, we expand h= χ exp[-β(χw - 1)2] around β=0 in powers of β (i.e., h ≈ h(β = 0) + β(dh/dβ)|β=0 + o(β2)), leading √ to h ≈ √ χ [1 - β + 2βχw - βχ2w2] and thus to Iβ(χ) = χ[1 - β + 4βχ(5 - 2χ)/15]. Now, χMAX is specified by the condition Iβ0 (χMAX) = 0. Because Iβ0 (χ) 3 - 3β - 4βχ(3 - 2χ) = 0, we obtain χMAX=3(1 + [1/3 + 2/3β]1/2)/4 ≈ (3/8β)1/2=(3/2)1/2(Σ-/ Σ+)1/3 or equivalently ðD9Þ ΣMAX ≈ 2:09 ðΣ þ Σ - Þ1=2 ¼ 2:09 Σosm √ By substituting χMAX=3[1+ (1/3 + 2/3β)]/4 into eq D6, we obtain Iβ ðχMAX Þ ¼
ffiffiffi 1 2p 5β1=4 5 4 6 1=4 þ pffiffiffi - β3=4 5 2 6 4 β
ðD10Þ
and the leading term of ΓMAX Γ(χMAX) for Σ+ , Σ- is 1=6 " 2=3 # ffiffiffi 2p Σ1 Σþ 4 6cp H0 ðΣ - Þ exp ΓMAX ¼ 5 4 ΣΣþ
ðD11Þ
In this range, we can further neglect the exponential correction, thus leading to Γ(χMAX) ≈ 0.623 cpH0(Σ-)(Σ- /Σ+)1/6 where (Σ-/Σ+)1/6 ≈ 1 for any feasible Σ-/Σ+. The presented limiting expressions of χMAX suggest a numerical fit of the form χMAX = c1 + (c2 + c3/β)1/2 and indeed χMAX = 0.669 + (0.235 + 0.533/β)1/2 or equivalently 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 -3=2 2=3 pffiffiffi Σ5 ΣMAX ¼ 8Σ - 40:669 þ 0:235 þ 2:132 Σþ ðD12Þ reproduces the numerical solution to within 1% in the range of 0 e β (Σ+/Σ-)2/3/4 e 10.87. Similarly, eq D10 suggests a numerical fit of Iβ(χMAX) in the form of Aβ-1/4 + Bβ1/4 - Dβ3/4, which indeed leads to a useful approximation, when A=0.76, B= 0.33, and D=0.09 such that ΓMAX
" # " 1=6 1 Σ þ 2=3 Σþ 0:23 ≈ cp H0 ðΣ - Þexp 4 ΣΣ 1=2 1=6 # Σþ ΣðD13Þ - 0:03 þ 1:08 ΣΣþ DOI: 10.1021/la9008569
11633
Article
Halperin and Kr€ oger
is a useful approximation (
