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Kinetics of Langmuirian Adsorption onto Planar, Spherical, and Cylindrical Surfaces Jouko Kankare* and Igor A. Vinokurov† Department of Chemistry, University of Turku, FIN-20014 Turku, Finland Received November 24, 1998. In Final Form: April 30, 1999 The kinetics of adsorption processes in solution onto adsorbents having different geometric shapes have been explored theoretically. The basic assumptions were (i) the diffusion of adsorbate in quiescent homogeneous solution with no convection, (ii) simple Langmuir-type adsorption kinetics and isotherm, (iii) the bulk concentration of the adsorbate in solution being sufficiently high to stay constant during the adsorption process, and (iv) the geometric shape of the adsorption surface being planar, spherical, or cylindrical. The diffusion equation with the time-dependent boundary conditions implied by the adsorption process was shown to lead to a nonlinear Volterra-type integral equation which is common for the three adsorbent geometries with a single definitive parameter, geometric factor, varying between 0 and 1. A numerical method was developed for solving this equation, and approximate analytical solutions were derived for the very beginning and the very end of the adsorption process. Implications of the results for the analytical methods based on the use of microparticles, such as various immunoassays, are discussed.
Introduction Adsorption processes from a liquid phase to a solid surface are of immense importance in diverse fields of everyday life, science, and technology. Although the equilibrium properties of different adsorption systems are well covered in the literature, the rate of the processes have attained less attention. Being heterogeneous, the kinetics of these processes depends on numerous different parameters. In solution the rate depends on the diffusion coefficient of the adsorbate molecules, convection, and the geometry of the adsorbent surface. At the interface the rate depends on the interaction forces between the adsorbate molecules and the surface. The simplest adsorption process is the Langmuirian adsorption process in which no interactive forces between the adsorbate molecules are assumed and only monolayers are assumed to be formed. Despite these simplifications, surprisingly many processes obey the Langmuirian kinetics. This work was initiated in the course of some analytical problems. In fact, adsorption plays an important role in various analytical methods. Traditionally in the voltammetric analytical methods where adsorption may have a significant role, its influence has been taken into account in deriving theoretical equations for the current-voltage curves. Another field which has grown enormously in practical importance during the recent years is different heterogeneous binding assays which are being used in various fields of diagnostic analysis. Generally in these methods one of the compounds participating in the assay is immobilized onto the surface and the other compound initially freely diffusing in the solution is captured by this surface-confined reactant. These processes include, e.g., different immunoassay procedures as well as DNA hybridization methods. The rate, selectivity, and affinity of these processes are of crucial importance for their practical utilization. Selectivity and affinity of the binding process are mainly governed by the choice of the reactants and these factors * To whom correspondence should be addressed: E-mail:
[email protected]. † On leave from Department of Chemistry, St. Petersburg University, 198904 St. Petersburg, Russia.
are not discussed in this treatment. The rate is strongly, often solely, dependent on the diffusion. If the geometrical features of the adsorbent are dimensionally of the same scale as the thickness of the diffusion layer, the rate becomes dependent on the shape of the diffusion front and thus on the geometry. By a suitable choice of the physical system, it is then possible to control the rate of the adsorption process. One of the goals of this study is to derive equations which enable us to compare the features and advantages of different adsorbate geometries. The most commonly used shapes of the adsorbent surfaces are either planar or spherical. Immunoassays are commonly done in microstrip wells in which antibody is immobilized onto the walls of the wells. Despite the curvature, the walls are effectively planar surfaces in the scale of diffusion. Spherical microparticles of the size of a few micrometers in diameter are becoming more and more popular in immunoassays. In this case the size of particles is even smaller than the thickness of a diffusion layer, and the spherical geometry in the diffusion equations should be taken into account. In some cases the highaffinity regions on the planar substrate are micrometersize spots which can be reasonably well approximated by a hemispherical diffusion front. A third type of geometry, which has certain advantages and may well become the basis of some assays, is the shape of a cylinder. A typical example is a biosensor based on optical fibers and the absorption of an evanescent wave on the surface of a fiber 50-200 µm in diameter. Also the biosensor may be a narrow active strip on a planar surface. In this case the diffusion geometry is effectively hemicylindrical if the width of the active area approaches the thickness of the diffusion layer. The spherical and cylindrical diffusion have been treated by electrochemists working with ultramicroelectrodes.1,2 However, although solutions for various geometries have been presented, the boundary conditions have generally pertained to common electrochemical cases. The adsorp(1) Fleischmann, M.; Pons, S. in Ultramicroelectrodes; Fleischmann, M., Pons, S., Rolison, D. R., Schmidt, P. P., Eds.; Datatech Systems, Inc.: North Carolina, 1987; p 17. (2) Aoki, K; Honda, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1985, 182, 267.
10.1021/la981642r CCC: $18.00 © 1999 American Chemical Society Published on Web 07/14/1999
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tion to the surface, especially if followed for longer times, causes special time-dependent boundary conditions to the diffusion equation. These boundary conditions lead to a nonlinear integral equation which in the case of the Langmuir or Frumkin adsorption isotherms can be solved only numerically. The planar case had been tackled for the first time 50 years ago by Ward and Tordai.3 The resulting integral equation has been solved by using analogue computation by Holub and Nemec4 assuming both planar and spherical geometry for the adsorbing surface and Langmuir and Frumkin isotherms. The first numerical solution obtained by using a digital computer seems to be due to Rampazzo.5 She assumed a planar geometry and Frumkin isotherm. The numerical solution for the diffusion-controlled Langmuirian adsorption on the planar surface has been given by Miller and Lunkenheimer.6 The solvability of the integral equations stemming from the diffusion-controlled adsorption to the planar and spherical surfaces has been treated by Vogel in the case of Langmuirian7 and Temkin8 isotherms. The effect of Langmuirian adsorption on the electrode reactions has been thoroughly treated by Guidelli9 and Reinmuth and Balasubramanian.10-11 A general method for treating the problems of diffusion-controlled adsorption kinetics was reported by Mysels12 and further elaborated by Frisch and Mysels.13 In a series of papers a Spanish group presents their theoretical and experimental results concerning the influence of adsorption on electroanalytical methods, taking into account the diffusion-controlled Langmuirian or Frumkinian adsorption on planar or spherical electrode surfaces.14,15 The present authors could not find any previous work on the heterogeneous rate process on a cylindrical surface simultaneously controlled by diffusion and interfacial reaction rate. A common feature of the previous reports dealing with the problems of diffusion-controlled adsorption kinetics is that the adsorption equilibrium is assumed to prevail during the process in the solution-solid interface; i.e., the adsorption process is assumed to be very fast compared with the transport rate by diffusion. This is certainly a viable assumption when we are dealing with small symmetric adsorbate molecules with no preferential adsorption site and/or the fast molecular rotation. The situation is different when dealing with macromolecules which are inherently anisotropic, proteins being a good example. Also if covalent bonding is occurring in the adsorption, the rate may be rather slow. A good recent example is the self-assembly techniques where e.g. thiols are adsorbed on a gold surface or silane derivatives are adsorbed on oxide surfaces.16 The analytical methods based on various binding processes of biological molecules form a very important field where the reaction yielding a
surface-confined complex may be the rate-limiting step depending on the volume concentration of the mobile reactant and surface concentration of the immobilized reactant. A typical example is one variant of the immunoassay methods where the antibody is immobilized on the surface and the antigen to be quantified is captured from the solution to form the immunocomplex with the antibody. In this case the rate of the interfacial reaction in addition to the diffusion process has been taken account by Sadana and Sii.17 However, they were interested only in the initial binding kinetics ignoring the influence of increasing coverage and desorption. It is important to be able to estimate the time needed for the quantitative adsorption, i.e., the long time limit of the process. It is also important to assess whether the process is diffusion controlled or controlled by the rate of the interfacial reaction. These are problems which have not been thoroughly treated in previous studies, and consequently a fresh approach is needed to derive equations suitable for the numerical computations and for the derivation of approximate formulas. The equation for the time dependence of coverage resulting from the combination of the diffusion equation and the rate law of adsorption is a nonlinear Volterra integral equation. It is formally highly complicated and it is impossible to see the influence of different parameters without incurring numerical computations. For the very beginning of the adsorption process the mathematical formulas can be linearized and reasonably simple series expansions can be derived. However, these series have proved to be often slowly converging, leading to rather complicated expressions which are useless for the very purpose they have been derived. On the other hand, in many cases a more useful approximation would be the asymptotic behavior of the adsorption process at the long time limit. Rather simple formulas of this kind have been derived in the present work for the different geometries. It is important to realize that this is still an idealized case where the solution is assumed to be quiescent and no convection is allowed. The realistic treatment should include the influence of stirring on the rate of adsorption. However, it is very difficult to assign a similar type of stirring to different shapes which would be necessary if any conclusions should be drawn on the effectiveness of convection. One of the objectives of this paper is to compare different geometries of the adsorbent in the adsorption processes, and only a stagnant solution provides a suitable reference condition.
(3) Ward, A. F.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (4) (a) Holub, K.; Nemec, L. J. Electroanal. Chem. 1966, 11, 1. (b) J. Electroanal. Chem. 1968, 18, 209. (5) Rampazzo, L. Electrochim. Acta 1969, 14, 733. (6) Miller, R.; Lunkenheimer, K. Z. Phys. Chem. (Leipzig) 1978, 259, 863. (7) Vogel, J. J. Collect. Czech. Chem. Commun. 1973, 38, 979. (8) Vogel, J. J. Collect. Czech. Chem. Commun. 1973, 38, 976. (9) Guidelli, R. J. Phys. Chem. 1970, 74, 95. (10) Reinmuth, W. H.; Balasubramanian, K. J. Electroanal. Chem. 1972, 38, 79. (11) Reinmuth, W. H.; Balasubramanian, K. J. Electroanal. Chem. 1972, 38, 271. (12) Mysels, K. J. Phys. Chem. 1982, 86, 4648. (13) Frisch, H. L.; Mysels, K. J. J. Phys. Chem. 1983, 87, 3988. (14) Mas, F.; Puy, J.; Sanz, F.; Virgili, J. J. Electroanal. Chem. 1983, 158, 217. (15) Puy, J.; Mas, F.; Sanz, F.; Virgili, J. J. Electroanal. Chem. 1983, 158, 231. (16) Ullman, A. Chem. Rev. 1996, 96, 1533.
dΓocc/dt ) k+csΓunocc - k_Γocc
Theory The first approximation to the rate law of the heterogeneous adsorption or more generally binding process is the Langmuir kinetic law
(1)
Here Γocc denotes the surface concentration of the bound species, Γunocc denotes the surface concentration of the vacant sites still available for the adsorption, and cs denotes the concentration of the adsorbate in solution in the immediate vicinity of the surface. We assume further that the rate coefficients k+ and k- do not depend on the surface concentration of the adsorbate and that
Γocc + Γunocc ) Γs ) constant
(2)
With θ, the fractional coverage (17) Sadana, A.; Sii, D. J. Colloid Interface Sci. 1992, 151, 166.
Langmuirian Adsorption onto Surfaces
θ ) Γocc/Γs
Langmuir, Vol. 15, No. 17, 1999 5593
(3)
we may write eq 1 to another form
dθ/dt ) k+cs(1 - θ) - k_θ
(4)
For later use this can still be written to the third form
θ K d ln(1 - θ) ) Kcs 1 - θ k+ dt
(7)
The sorption process causes a special boundary condition. Let J be the mass flux in the system. Then according to Fick’s first law
J ) -D∇c
(8)
and the rate of change of the coverage is directly proportional to the flux J
dθ ) -Jsurface ) D(∇c)surface Γs dt
(9)
According to Oldham and Spanier20 the diffusion eq 7 can be represented (by using a somewhat different notation) in the following general form in the case of the semiinfinite medium
∂2c 2GD ∂c ∂c )D 2+ ∂t r + R ∂r ∂r
(10)
where r is the spatial coordinate directed normal to the boundary and having its origin at the interface and R is the radius of the sphere or cylinder. The geometric factor G is 1 for a convex sphere, 1/2 for a convex cylinder, and 0 for a plane. With certain restrictions the equation is valid also for a concave sphere(G ) -1) and concave cylinder (G ) -1/2). Although the equation with these discrete values of G can be solved separately, we want to evaluate this equation in the general case speculating (18) Ekins, R.; Chu, F.; Biggart, E. Anal. Chim. Acta 1989, 227, 73. (19) Ekins, R.; Chu, F. Clin. Chem. 1991, 37, 1955. (20) Oldham, K. B.; Spanier, J. The Fractional Calculus; Academic Press: New York, 1974; p 198.
τ ) Dt/R2
F ) r/R + 1;
(11)
eq 10 becomes
∂γ ∂2γ 2G ∂γ ) + ∂τ ∂F2 F ∂F
(12)
Application of the Laplace transform gives
d2γ j j 2G dγ - sγ j)0 + 2 F dF dF
(6)
An additional assumption is that the pool of adsorbate is large enough so that the adsorbed amount on the surface has only a negligible influence on the total dissolved amount. With this assumption there is no need to define the outer boundaries of the vessel containing the solution. Although this seems to be a rather restrictive assumption as many assays are based on the quantitative depletion of the analyte from solution to surface, in fact, this does not restrict the comparison of different adsorbent shapes. On the other hand, this situation refers directly to the methods which are based on the principle of “fractional antibody occupancy”.18,19 In these methods the analyte and antibody concentrations are adjusted to give a fractional occupancy on the surface after the equilibration period. The partial differential equation to be solved in our problem is the diffusion equation or Fick’s second law
∂c/∂t ) D∇2c
γ ) c - c0;
(5)
where K stands for the equilibrium constantscommonly also called “affinity constant” in the biochemical literature
K ) k+/k-
that the geometric factor G may attain other values for the arbitrary shapes of the surface. With the substitutions
(13)
This equation is brought to the normal form by the transformation
γ j (s,F) ) u(s,F)F-G
(14)
where
[
]
G(1 - G) d2u + -s + u)0 2 dF F2
(15)
With a further substitution
1 z F) 2xs
(16)
this becomes
[
]
1 G(1 - G) d2u + - + u)0 2 4 dz z2
(17)
This is the normal form of Whittaker’s equation,21 which has its solutions as Whittaker functions M0,G-1/2(z) and W0,G-1/2(z). Because the function should remain finite as z f ∞, we chose the latter function multiplied by an undetermined function A(s) which does not depend on F. But we have21
1 1 W0,G-1/2(z) ) z1/2KG-1/2 z 2 xπ
( )
(18)
where KG-1/2 is the modified Bessel function of the second kind. Substituting back eq 16 and 14 we have finally
γ j (s,F) )
x2 A(s)s1/4F1/2-GKG-1/2(xsF) xπ
(19)
from which we may solve the function A(s)
A(s) )
xπ2 s
-1/4
KG-1/2-1(xs)γ j (s,1)
(20)
Using the rules of differentiation of Bessel functions we obtain
|
KG+1/2 (xs) ∂γ j γ j (s,1) F)1 ) -xs ∂F K (xs)
(21)
G-1/2
(21) Slater, L. J. In Handbook of Mathematical Functions, 9th printing, Abramowitz, M., Stegun, I. A., Ed.; Dover Publications: New York, 1970; p 503.
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Kankare and Vinokurov
Table 1. Parameters and Functions for the Planar, Spherical, and Cylindrical Geometry in Equations 25 and 26 planar D1/2/KΓ
L M
f(s) ) F(τ)
a
KG+1/2(xs)
xsKG-1/2(xs)
spherical
cylindrical
K/k+ ) 1/k-
R/KΓs KD/k+R2
R/KΓs KD/k+R2
1/s1/2
1/s1/2 + 1/s
K1(xs)/xsK0(xs)
1/xπτ
1/xπτ + 1
4 π2
s
∫
∞
0
exp(-τx2) x[J02(x) + Y02(x)]
dx
J0 and Y0 are the Bessel functions of the first and second kind, respectively.
where Taking the Laplace transform from eq 9 and applying eq 21 gives
-
KG+1/2 (xs) KG-1/2(xs)
xsγj (s,1) )
Γs sθ h R
(22)
Taking into account of the transform of variables given in eq 11 and taking the Laplace transform from the both sides of eq 5 gives
L
{1 -θ θ} - kKDR sL{ln(1 - θ)} ) Kγj(s,1) + +
2
Kc0 s
(23)
Elimination of γ j (s,1) from 22 and 23 gives finally
θ h)
[
θ R KG+1/2(xs) Kc0 -L + KΓs xsK s 1 θ (xs) G-1/2
{
}
]
KD sL{ln(1 - θ)} (24) k+R2 Integral Equation. Equation 24 has been derived for the spherical and cylindrical cases, but actually a completely analogous equation is obtained for the planar case by only ignoring the transformation of variables of eq. 11 Generally we may write
[
{
}
Kc0 θ -L + MsL{ln(1 - θ)} θ h ) Lf(s) s 1-θ
]
(25)
where the parameters L and M and the function f(s) are defined in Table 1. It should be noted that the ratio of the modified Bessel functions in eq 24 is simplified to a rational function of s1/2 when G attains integer values.22 Hence by Laplace inversion and using the convolution theorem we obtain
θ)L
[
θ(σ)
∫0τF(τ - σ) Kc0 - l - θ(σ) + M
]
d ln(1 - θ(σ)) dσ dσ (26)
where
F(τ) ) L-1{f(s)} ) 4 q(G) + 2 π
∫0
∞
1 2πi e
K
(xs)
c+i∞ sτ G+1/2 e ∫c-i∞ xsK
G-1/2(xs)
ds )
du (27) u[JG-1/22 (u) + YG-1/22(u)]
-τu2
(22) Antosiewicz, H. A. In Handbook of Mathematical Functions, 9th printing, Abramowitz, M., Stegun, I. A., Ed.; Dover Publications: New York, 1970; p 435.
q(G) )
{
2G - 1 if 1/2 e G e 1 0 if 0 e G e 1/2
}
The last step is derived in the Supporting Information. Equation 26 is a nonlinear Volterra integral equation which can be solved only by numerical methods. It should be noted that in the planar case τ is the real time whereas in the spherical and cylindrical cases it is the dimensionless time defined in eq 11. Previously reported considerations of this problem have been focused on the planar and spherical cases and the assumption on the local adsorption equilibrium at the interface. Equation 26 is more general, covering both the equilibrium and nonequilibrium cases and including also all three geometries. The equilibrium case corresponds to a completely diffusion-controlled reaction, meaning that the rate constant k+ f ∞. Consequently M f 0 and the integral eq 26 is somewhat simplified by leaving off one term. The numerical solution of the integral eq 26 was done by using a simple recurrence relation (see Supporting Information). The time interval was divided quadratically into N subintervals starting with a denser spacing. Corresponding values of coverage θ were calculated by using a rectangular approximation for the integrals and recurrence relation
θi+1 ) P(τk,θk;k)1,...,i)
i)1, ..., N - 1 (28)
where P stands for the numerical equivalent of the convolution integral. The method works well in most cases, but it is subject to instability at certain values of parameters. Numerical solution of eq 26 gives coverage as a function of time, but the influence of different parameters is more or less obscure and can be based only on a large number of numerical simulations at different values of parameters. A better way to visualize the functional relationship between the parameters and coverage is to derive simple approximate solutions for, e.g., the short time and long time limits. These approximations can be best derived starting from the Laplace plane solutions (25). For these derivations the complicated form of the ratio of two modified Bessel functions can be approximated between 0 < s < ∞ by a rather simple function of s1/2
KG+1/2(xs) KG-1/2(xs)
≈1+
G xs
(29)
In fact, this is not an approximation but an exact form in the planar (G ) 0) and spherical (G ) 1) cases. In the cylindrical case (G ) 1/2) eq 29 is an approximation and represents the first two terms of the asymptotic expansion as shown in the Supporting Information. Numerical
Langmuirian Adsorption onto Surfaces
Langmuir, Vol. 15, No. 17, 1999 5595
Table 2. Terms of the Series Expansions of the Surface Coverage for the Short Time Limit in Different Geometries ∞
θ)C
∑a τ
i/2
i
i)1
equilibrium C a1
2/xπ
a2
-
k+c0 0
D1/2 G 1 - KΓs KΓs R
(
3xπK -
1
)
4D
a3 a4
kinetic control
D1/2c0/Γs
2
Γs2
(1 - 2KΓ GR)
(
)
D3/2 G G2 1 - 3KΓs + K2Γs2 2 3 3 R 2K Γs R
comparison with the more exact approximation described in the Supporting Information shows that in this case G should be taken as an adjustable parameter which in practical cases is between 0.1 and 0.2. Short Time Limit. As discussed previously, the adsorption processes of many important adsorbates, such as proteins, are complicated, and the simple Langmuirian adsorption law may not be valid, especially at later stages. The analytical methods which are based on the equilibrium coverage give doubtful results unless the conditions have been adjusted for a very fast process, e.g., by using very small spherical or cylindrical adsorption regions. It would be then practical to have simple equations for describing the first moments of the adsorption process which may allow the estimation of the bulk concentration of the adsorbate or its heterogeneous rate constant k+. In the present case the solution of integral eq 26 can be obtained for the very beginning of the process when the coverage is low allowing the linearization of the expressions. It is feasible to start the derivation from the expression (25) in the Laplace domain. The solution is obtained in a rather complicated form containing, e.g., roots of the third-order polynomial (see eq S29 in the Supporting Information). However, a series expansion can be derived, although with a quite narrow domain of convergence. The full derivation for the series expansions and the recursive relations for the coefficients are shown in the Supporting Information. The first four terms of the series expansions are shown in Table 2 where also the dimensionless time τ has been changed to real time. Long Time Limit for the Spherical and Cylindrical Adsorbent Geometries. Among of the most important information needed from an analytical method based on adsorption is the time needed to accomplish the process up to the required accuracy. If the adsorption isotherm and diffusion coefficients are known, this time can be calculated from, e.g., the numerical approximation of the integral equation. However, to be able to visualize the influence of different parameters, one needs an approximate and reasonably simple formula for estimating the adsorption time. A considerable simplification is introduced by making assumptions on the magnitudes of parameters M and L. For instance, when dealing with the modern immunoassays made in clinical chemistry by using antibody-coated spherules, the diameter of these spherules is a few micrometers. The diffusion coefficient of the macromolecular analytes is 10-7 to 10-6 cm2 s-1, and the affinity constant K may be of the order of 1012 cm3 mol-1. Assuming that the rate constant k+ is large enough, e.g., 108 to 109 cm3 mol-1 s-1, we see that the dimensionless parameter
4k+Γs
-
s
3xπD1/2
k+2Γs2 1 G - k- + k+Γs 2D 2 2R
M, which plays a significant role in the equations, is of the order of 104 to 106. We can estimate the time th needed to reach the point where the coverage of the surface is within a factor h from the final value θeq (see the complete derivation in the Supporting Information)
[
]
Γ sR 1 K + × 1 + Kc0 k+ GD(1 + Kc0) G D(1 + Kc0) ln (1 + Kc0) 1+ R k+ΓsKc0 -ln h + ln G D(1 + Kc0) 1+ R k+Γs
{ [
t ) th ≈
]}
(30)
This equation allows investigation of the influence of different parameters on the rate of the adsorption process. For instance, the assumption of a local equilibrium at the interface means that k+ f ∞ keeping K at the same time constant. This gives
lim th ≈ -
k+f∞
RKΓs ln h
(31)
DG(1 + Kc0)2
Also we obtain for low values of concentration c0
(
lim th ) -K
c0f0
)
ΓsR 1 + ln h k+ GD
(32)
Long Time Limit for the Planar Adsorbent Geometry. For many practical applications we may derive a reasonably accurate approximation (see Supporting Information)
t ) th ) K2Γs2
[
1+
πDh2(1 + Kc0)4
]
D ln(1 + Kc0)(1 + Kc0)3 k+Γs2K2c02 1-
D(1 + Kc0)4 k+KΓs2
2
(33)
In most cases of any practical interest with the fast interfacial kinetics, the term in brackets is close to unity and we can write
th =
K2Γs2 πDh2(1 + Kc0)4
(34)
5596 Langmuir, Vol. 15, No. 17, 1999
Figure 1. Calculated dependence of coverage θ on time for small spherules of radius 2 µm with various rates of adsorption. The other parameters are D ) 5 × 10-7 cm2 s-1, c0 ) 10-11 mol cm-3, K ) 1011 cm3 mol-1, and Γs ) 10-11 mol cm-2.
Results and Discussion The algorithms described in detail in the Supporting Information are able to compute the time dependence of coverage in all three geometric cases. The common assumption in the previously published studies has been very fast interfacial kinetics compared to the rate of diffusive transport. This assumption leads to the simplification of mathematics as discussed in this work. Although very often this assumption is in reasonable conformity with the experimental results, there is a need to estimate the influence of the finite adsorption rate on the functional time dependence of coverage in order to see at least the order of magnitude of the interfacial rate constants which starts to have a non-negligible influence. To illustrate the influence of different factors, the values of parameters used for simulations should be carefully chosen. Various binding assays are presently of great practical importance. In these measurements the heterogeneous reaction occurs between macromolecules of biological origin and surfaces coated with specific, selective compounds. A typical value of diffusion constant for these macromolecules is 5 × 10-7 cm2 s-1. The site density on the surface capable of binding these macromolecules might be Γs = 10-11 mol cm-2. The equilibrium constant for the adsorption processes varies in wide limits, but a typical value for immunoassays is K ) 1012 cm3 mol-1. The rate constant for the adsorption process is typically around 109 cm3 mol-1 s-1. The diameter of the spherules commonly used in various binding assays is around 4 µm. In Figure 1 we see the influence of the variation of the rate constant on the shape of the curve describing coverage vs time. In the same figure the curve calculated with k+ ) 1011 cm3 mol-1 s-1 is indistinguishable from a curve describing coverage calculated by assuming the local equilibrium at the interface (k+ ) ∞). As one can see, the differences are significant, at least within the range from k+)109 to 1011. Usually the problems in the methods based on the adsorption concern the very beginning or the very end of the process. The rate of the process in the beginning, as the adsorbate and adsorbent have been brought to the mutual contact, is of concern for the analyst who is developing analytical methods based on kinetics. For example, the use of biosensors has been strongly advocated
Kankare and Vinokurov
to be used in the kinetic mode.23 There are obvious advantages to do assays by using kinetic methods. In addition to that, the assay is certainly faster and there is no need to know the mechanism in detail; it is enough to know that the initial rate of the reaction depends on the concentration of the analyte. The solution should be quiescent in order to keep reproducible conditions. We have then the conditions which are the same as those assumed to prevail when deriving the expressions in this work. The measurement of the initial rate is usually done graphically by plotting the measured quantitysusually coverage or a physical quantity proportional to itsas the function of time or automatically by using some procedure to extract kinetic data from experimental results. In both cases better accuracy is assured if the rate law is known. Table 2 shows the terms in the series expansions in different cases. The first thing to note is that the rate laws are quite different depending on whether the local equilibrium at the interface is assumed to prevail or there is a kinetic control for the adsorption process. In the former case the first term depends on the square root of time whereas in the latter case the term depends linearly on time. The geometrical shape of the adsorbent surface does not have any influence on the form of the first term. Even the second terms in the kinetically controlled cases do not depend on the geometry. In the equilibrium case already the second term depends on the geometry, sometimes even quite strongly because the dimensionless parameter kΓs/R may be in many cases considerably greater than unity. The first inspection of experimental adsorption data is commonly done by using some graphical procedure. If nothing else but concentration is needed in the analytical procedure, the way of plotting data is not too important as long as the procedure is reproducible. But as soon as one would like to determine physicochemical parameters such as rate coefficients, diffusion constants, limiting coverage, etc., the knowledge of the rate law is necessary. On inspecting again the terms of the series expansions in Table 2, one is tempted to infer that by plotting θ/t1/2 vs t1/2 it is possible decide whether the adsorption process at the interface is at equilibrium or kinetically controlled. If the plot passes through the origin, i.e., the intercept of the line is zero, the process is kinetically controlled, whereas if the intercept has a positive nonzero value, the process is at equilibrium. Naturally the success of this procedure depends on the relative magnitude of the parameters and the precision of the measurements. One should be also aware that the series do not converge very rapidly or even they may diverge with the realistic parameter values, especially in the case of spherical and cylindrical geometries with kinetic control. Consequently the linearization of the series by including only the first two terms may be reasonably valid only at the time range which is not experimentally attainable. The range of validity of the series expansion can be seen in an example shown in Figure 2 where a complete numerical solution of a spherical geometry based on the integral eq 26 is compared to the corresponding series expansion with 200 terms. It is clearly seen that at least in this case the series expansion is a valid approximation only within such a range that it has no relevance in practical applications. On the other hand, in the same figure the approximation of eq S29 follows quite closely the numerical solution up to ca. θ ≈ 0.2. The drawback of this approximation is its extremely complicated form due to the necessity of solving (23) Place, J. F.; Sutherland, R. M.; Riley, A.; Mangan, C. In Biosensors with Fiberoptics; Wise, D. L., Wingard, L.B., Jr., Eds.; Humana Press, Inc.: Clifton, NJ, 1991; p 253.
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estimate, but the present theoretical results provide the worst-case limits obtained in quiescent solutions. Equation 30 provides a useful approximation for estimating the time period needed for the adsorption process in the spherical and cylindrical cases. This equation comprises two parts where the first part depends on the various parameters of the diffusion process and surface characteristics but the second part depends only on the dissociation rate and coverage. Their mutual magnitude determines whether the process is kinetically or diffusion controlled. We see that as concentration c0 increases and c0 . K-1, the second term decreases more slowly than the first term. Apparently at a certain concentration level the process becomes predominantly kinetically controlled. We see also that as the second term of eq 30 depends on the radius of the sphere or cylinder, the process becomes faster as the radius is decreasing, but only up to a certain limit because the first term does not depend on the radius. This limit can be estimated by equating the terms and solving for R. At a low coverage (θeq ≈ 0) we obtain Figure 2. Comparison of different approximations for solving the coverage vs time at the initial stages of the adsorption process on a spherical particle. The insert shows the very beginning of the process. Curve 1: Full numerical solution of eq 26. Curve 2: Approximation by eq S29. Curve 3: Series expansion as in Table 2 with 200 terms calculated by using the recursive relation described in the Supporting Information. Parameters: D ) 5 × 10-7cm2 s-1, K ) 1012 cm3 mol-1, Γs ) 10-11 mol cm-2, c0 ) 10-12 mol cm-3, R ) 2 µm, k+ ) 1010 cm3 mol-1 s-1.
the roots of the third-order equation. Even then it is more convenient for simulations than the full numerical solution due to the tendency of the numerical solution for instability at the low values of coverage. The geometric factor shows up in the series expansions in an interesting way. We see in Table 2 that the geometric factor G and radius R are present in equations always as the ratio G/R. In fact, the same ratio can be seen also in eq 30 for the long-time limit. Hence we are led to the conclusion that the mathematical forms of the kinetic laws for the spherical and cylindrical geometries are equivalent, because any change in G can be always compensated by the corresponding relative change in R. Actually this is not strictly valid because the approximation (29) is not very accurate in the cylindrical case, but for all practical purposes we can say that the time required to reach a near-equilibrium value of the adsorption coverage onto the surface of a cylinder is the same as the adsorption time onto a sphere with a 5 to 10 times greater radius. It is quite probable, although not easily proved, that adsorption onto any kind object of whatever shape follows approximately the same rate law but with different values of a single parameter, G/R. The other end of the adsorption process, close to the equilibrium state, is important for various analytical methods. For instance, the on-line monitoring of a chemical process which might be based on the use of biosensors needs a reasonably short response time. The response time depends on the rate of the interfacial process and the transport rate of the analyte. Other examples are the multitude of different binding assays used in clinical diagnostics and other fields. These heterogeneous processes are mostly followed up close to the equilibrium state. Rapid assay is in these cases essential or at least one should be able to predict approximately how long it takes until the process is sufficiently quantitative. Influence of convection processes is in most cases impossible to
Rlimit ≈ GD/k+Γs
(35)
Hence we come to an interesting conclusion which may have practical implications: Although it is common knowledge to use particles as small as possible in binding assays in order to speed up the process, there is a limiting radius below which no significant improvement is achieved. Using the parameter values represented above, we can calculate the limiting value in our “typical” case to be Rlimit ) 0.5 µm, not very far from the commonly used particle sizes. The same result is valid for cylindrical sensors. For example, in the case of fiberoptic biosensors it has been shown that the smaller radius results in a higher sensitivity.24 According to eq 30 also the response time diminishes by decreasing the radius. Again, eq 35 sets the limit also in this case, although the mechanical limitations may come earlier. Equation 33 gives the adsorption time in the planar geometry. An important difference with eq 30 is in the functional dependence on h. In the planar case the diffusion term depends inversely on the square of h whereas in the spherical and cylindrical cases the dependence is on the logarithm of h. Another difference is the fourth power dependence on the concentration. This causes a very fast decrease in the diffusion term as concentration increases. There is one reported case where the adsorption time has been calculated by the numerical solution of the integral equation for the planar case.6 The results of Miller and Lunkenheimer together with the function th from eq 34 are shown in Figure 3. The agreement is nearly perfect. The numerical solution of the integral eq 26 by using the recurrence relation (28) serves mainly as checking the approximations and their range of validity. The numerical method, which is based on the finite differences and described in Supporting Information, is rather slow even on a 200 MHz personal computer, especially in the cylindrical case. Also if the calculations are extended to reach the near-equilibrium state, the full numerical solution of the integral equation requires a very large number of points in order to achieve any level of accuracy, meaning computing time of several tens of minutes on an up-to-date computer. In this case it is much more (24) Love, W. F.; Button, L. J.; Slovacek, R. E. In Biosensors with Fiberoptics; Wise, D. L., Wingard, L.B., Jr., Eds.; Humana Press Inc., Clifton, NJ, 1991; p 139.
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curves shows clearly what has been generally observed empirically: The use of small particles speeds the adsorption processes enormously. However, what is not so evident beforehand is the behavior of the cylindrical geometry. Apparently the use of thin strings as adsorbent is inferior to the spherules but still only by a small constant factor. Conclusions
Figure 3. Time required to reach 99% of the equilibrium value in the adsorption process to a planar surface described by Miller and Lunkenheimer:6 solid curve eq 34; circles, data from ref 6. Parameters: D ) 10-6 cm2 s-1, K ) 2 × 105 cm3 mol-1, Γs ) 5 × 10-10 mol cm-2.
Figure 4. Comparison of different adsorbent geometries for the same adsorption process: curve 1, sphere with radius of 2 µm; curve 2, cylinder of radius 2 µm; curve 3, plane. Parameters: D ) 5 × 10-7 cm2 s-1, K ) 1011 cm3 mol-1, Γs ) 10-11 mol cm-2, c0 ) 10-11 mol cm-3, k+ ) 1011 cm3 mol-1 s-1.
convenient to use the approximate formulas (30) and (33). In practice, eq 30 gives estimates which are 20-30% lower than the estimates from the full numerical solutions, but for any relevant applications this accuracy is sufficient. Equation 33 gives considerably more accurate estimates for the planar case. There is no doubt that optimization would result in a considerably faster algorithm for the numerical solution, but this task was not pursued further. A further drawback of the numerical solution is the frequent problem with the stability which, however, can be circumvented in most cases by recalculation with different time intervals. One of the goals of this paper is to create mathematical methods for comparison of the spherical, cylindrical, and planar geometries in the adsorption kinetics. Numerical comparison is now possible by using the approximate formulas (30) and (33), but one should remember the limitations of these formulas. If the various parameters of the adsorption processes can be estimated a priori, it is possible to use the uncompromising numerical solution of the integral equation (26). Figure 4 shows the coverage θ as the function of time for these three geometrical cases with one typical set of parameters. Comparison of the
The rate equations for the adsorption onto planar, spherical, and cylindrical surfaces have been derived and approximations deduced for the first time. The most important results are the series expansions (Table 2) for the beginning of the adsorption processes and the expressions (30) and (33) for the adsorption time. The very beginning of the adsorption process is different depending upon whether the process at the interface is controlled by heterogeneous kinetics or the process is very fast and the local equilibrium or “steady state” is maintained at the interface. A remarkable observation is that the same expansion is valid for the planar, spherical, and cylindrical geometries by changing only a single parameter, geometric factor G, between 0 and 1. It can be speculated that any geometric shape of the adsorptive particle can be handled by the same expansion by taking the ratio G/R as the single variable parameter. The equations have been derived on the assumption that there is no convection in the system. Even a small convection induced by thermal gradients may disturb the system. Any convection would speed the adsorption rate, and thus the theory gives the results for the worst case. Another restriction in the use of the results presented in this report is the assumption of a “sufficient” amount of adsorbate in the solution. The assumption that the bulk concentration of adsorbate stays constant during the adsorption process is relevant in the case of biosensors but not in the case of many practical binding assays where the analyte is quantitatively depleted from the solution to the surface of the adsorbent. The real power of the theory is in providing a reference framework within which one is able to compare the adsorption rates of planar, spherical, and cylindrical geometries in a relative scale. The simple formulas derived for the long-time limit allow one to estimate the influence of different variables on the adsorption rate. Glossary c c0 cs γ D ∆ Γ(z) Γ Γs Γocc Γunocc G h
concentration (variable) bulk concentration of adsorbate concentration of adsorbate in solution in the immediate vicinity of the interface ) c - c0 diffusion coefficient of the adsorbate in solution deviation from the equilibrium coverage ()θeq - θ) Euler gamma function of z surface concentration of adsorbate on the adsorbent maximum surface concentration of adsorbate at full coverage surface concentration of the sites occupied by adsorbate on the adsorbent surface concentration of unoccupied sites on the adsorbent geometric factor (G ) 0 for plane,G ) 1 for sphere,G ) 1/2 for cylinder) relative deviation from the equilibrium coverage ()(θeq - θ)/θeq)
Langmuirian Adsorption onto Surfaces Jν K k+ kKν L{F(t)} r R F t th τ
Bessel function of the first kind of order ν equilibrium constant for the heterogeneous adsorption equilibrium rate constant for adsorption rate constant for desorption modified Bessel function of the second kind of order ν Laplace transform of function F(t) distance to the adsorbing surface radius of sphere or cylinder dimensionless distance in the spherical coordinates time time needed to reach the relative deviation h from the equilibrium coverage dimensionless time
Langmuir, Vol. 15, No. 17, 1999 5599 θ θeq Yν
surface coverage surface coverage at equilibrium ()Kc0/(1 + Kc0) Bessel function of the second kind of order ν
Acknowledgment. The financial support from the Academy of Finland (Grant 30579) is gratefully acknowledged. Supporting Information Available: Detailed mathematical derivation of eqs 27 and 29 and various equations in the sections Short Time Limit, Long Time Limit for the Spherical and Cylindrical Geometries, Long Time Limit for the Planar Geometry, and Numerical Solution of Integral equation (26). This material is available free of charge via the Internet at http://pubs.acs.org. LA981642R
