Langmuir 2006, 22, 871-876
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The Kinetics and Saturation of Reversible Adsorption on Patterned (Heterogeneous) Surfaces K. Komaee,† G. Friedman,‡ and N. Dan*,† Department of Chemical and Biological Engineering, and Department of Electrical and Computer Engineering, Drexel UniVersity, 3141 Chesnut Street, Philadelphia, PennsylVania 19104 ReceiVed July 20, 2005. In Final Form: NoVember 8, 2005 We analytically examine the time-dependent adsorption of analyte (solute) on a finite-sized adsorption region as a model for sensors utilizing patterned or heterogeneous surfaces. We account for both reversible adsorption (assuming first-order reaction) and saturation of the adsorption patch that may arise either from packing constraints (finite area) or because of a finite number of binding sites (ligands). Our main conclusions include the following: (1) Saturation effects, due to either finite patch size or finite number of binding sites, become significant at extremely short times. (2) Increasing the strength of binding between the analyte and the adsorption sites increases the adsorbed amount at short times, but, at long times, the mass adsorbed on a weakly binding patch is higher than that on a strongly binding one. (3) The sensitivity of detection, as defined by the adsorption of the minimal analyte mass required for signaling, over a fixed period of time, does not scale as 1/detection time. As a result, increasing the time over which adsorption occurs increases sensitivity, but not linearly. Sensitivity of detection also increases with increasing patch area and initial binding strength.
Patterned surfaces with submicron periodicities provide a route for the fabrication of highly ordered colloidal crystals, biologically active substrates, and optical and electronic devices.1-6 Recent advances in surface patterning techniques enable the formation of submicron surface patterns,6-10 so that a large number of different sensing moieties, or ligands, can be attached onto a single substrate for the rapid detection of numerous target molecules. Micron and submicron patterns are of special interest for biomedical screening because of the reduced volume of suspension required for evaluation. One essential, universal parameter for sensor design is the degree of sensitivity. A common way of defining sensor sensitivity is through the lowest analyte concentration at which detection occurs, a measure that implies infinite adsorption time. Since many applications require rapid turnover, an alternate definition of sensitivity may be the shortest time needed at a given analyte concentration for detection to take place: if the time allowed for adsorption is shorter than this critical time scale, the sensor will indicate a false negative (namely, no signal will be observed despite the presence of analyte molecules in the solution). Since this time scale is expected to depend on the (typically unknown) concentration of analyte, successful determination of the minimal * Corresponding author. Tel: (215) 895 6624. Fax: (215) 895 5837. E-mail:
[email protected]. † Department of Chemical and Biological Engineering. ‡ Department of Electrical and Computer Engineering. (1) See, for example, Aizenberg, J.; Braun P. V.; Wiltzius, P. Phys. ReV. Lett. 2000, 84, 2997-3000. (2) Chen, K. M.; Jiang, X.; Kimerling, L. C.; Hammond, P. T. Langmuir 2000, 16, 7825-7834. (3) Guo, Q.; Arnoux, C.; Palmer, R. E. Langmuir 2001, 17 (22), 7150-7155. (4) Kokkoli, E.; Zukoski, C. F. Langmuir 2001, 17, 369-376. (5) Ye, Y. H.; Badilescu, S.; Truong, V.; Rochon P.; Natansohn, A. Appl. Phys. Lett. 2001, 79, 872-874. (6) Lateef, S. S.; Boateng, S.; Ahluwalia, N.; Hartman, T. J.; Russell, B.; Hanley, L. J. Biomed. Mater. Res. 2005, 72A (4), 373-380. (7) Marksich, M.; Whitesides, G. M. Trends Biotechnol. 1995, B13 (6), 228235. (8) Clark, S. L.; Montague, M.; Hammond, P. T. Supramol. Sci. 1997, 4 (12), 141-146. (9) Hidber, P. C.; Helbig, W.; Kim, E.; Whitesides, G. M. Langmuir, 1996, 12 (5), 1375-1380. (10) Xia, Y. N.; Whitesides, G. M. Annu. ReV. Mater. Sci. 1998, 28, 153-184.
time for adsorption requires detailed understanding of the adsorption process. Several theoretical studies11-20 have examined the diffusional flux to heterogeneous substrates, commonly described as adsorbing patches distributed on otherwise nonadsorbing substrates. They find that the flux of molecules to the patterned surface depends on the adsorbing patch characteristics (geometry, dimensions), the spacing between patches, and the concentration of the adsorbing molecule in solution. However, these studies tend to focus on the steady state,11-15 neglect the effects of molecular transport,16 and/or assume that the analyte adsorbs instantaneously and irreVersibly to the patch surface.17-20 To the best of our knowledge, the effects of reversible binding and possible substrate saturation were not considered in analytical models to date. In this letter, we examine the time-dependent flux of analyte molecules to an adsorbing sensor patch. We assume that each adsorption patch binds only a specific analyte particle or molecule. Thus, despite the heterogeneity of surface patterning, the sensor is composedsfor any specific analytesof a small, localized adsorbing region embedded in a large surface that is nonadsorbing. We consider reversible adsorption (which also includes, as a special limit, irreversible binding), where the rate of adsorption depends on the concentration of analyte in the vicinity of the adsorbing patch. We also incorporate the fact that any real adsorption patch can be saturated, that is, can bind only a finite amount of analyte molecules or particles because of either finite patch size and/or finite number of adsorption sites (or binding ligands), as shown in Figure 1. Our goal is to determine the (11) Berg, H. C.; Purcell, E. M. Biophys. J. 1977, 20, 193-219. (12) Shoup, D.; Szabo, A. Biophys. J. 1982, 40, 33-39. (13) Zwanzig, R. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 5856-5857. (14) Nitta, T.; Yamaguchi, A. Langmuir 1993, 9 (10), 2618-2623. (15) Tovbin, Y. K. Langmuir 1997, 13 (5), 979-989. (16) Ceyrolles, W. J.; Viot, P.; Talbot, J. Langmuir 2002, 18 (4), 1112-1118. (17) Adamczyk, Z.; Weronski, P.; Musial, E. J. Colloid Interface Sci. 2002, 248 (1), 67-75 (18) Adamczyk, Z.; Weronski, P.; Musial, E. Colloids Surf., A 2002, 208 (1-3), 29-40. (19) Adamczyk, Z.; Siwek, B.; Weronski, P.; Musial, E. Appl. Surf. Sci. 2002, 196 (1-4), 250-263. (20) Faraudo, J.; Bafaluy, J.; Senger, B.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 2003, 119 (21), 11420-11428.
10.1021/la0519608 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/31/2005
872 Langmuir, Vol. 22, No. 3, 2006
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Figure 1. A sketch of an adsorption patch with a finite number of adsorption sites (ligands).
characteristic time scales for analyte adsorption, since the rate of adsorption, or mass accumulation on the binding patch, determines the sensors sensitivity and functioning. The adsorption region is taken to be a single disklike patch of an adsorbing moiety of radius a that binds an analyte from solution. This patch is surrounded by an (infinite) nonadsorbing surface. This assumption is valid for patterned substrates if the distance between adsorbing patches is large,6-8 or if each analyte can be captured by a single patch. The concentration profile and the flux of analyte to the patch is set by a combination of diffusion and convection terms.21,22 In a system such as the one discussed here, where there is no bulk reaction, the time-dependent concentration profile is set by a combination of diffusive and convective fluxes:
∂C + ∇(D∇C + VjC) ) 0 ∂t
(1a)
Ns ) (-D∇C + VjC)|s
(1b)
where C is the concentration of the target analyte molecules in the solution (commonly given in mass per unit volume), t is time, D is the diffusion coefficient of the molecule in the suspension, and Vj is the velocity field. Ns is the flux (mass of analyte per unit area per time) to the adsorbing patch and is composed of two contributions: The diffusional flux J ) (D∇C), a function of the diffusion coefficient and the concentration gradient at the surface, and the convective flux (VjC). We focus here on systems where there is no flow (Vj ) 0), so that transport is dominated by diffusion, and the solution is considered to be a “stagnant” film. For simplicity, we focus on the limit where the radius of the adsorbing patch is small, so that the system may be approximated by a point “sink”. This assumption is appropriate when the area of the patch is much smaller than that of the sensor surface. In this case, a point sink implies hemispherical symmetry, and eq 1 may be simplified:
∂2C 2 ∂C ∂C ≈ D 2 + ∂t r ∂r ∂r
(2)
where r now defines the radial distance from the patch/sink center. The initial boundary condition requires that the concentration be uniform everywhere in the system, as defined by C0. We assume that the suspension volume is large when compared to the adsorption area, so that no appreciable depletion takes place far away from the adsorbing surface, and C(r f ∞) ) C0. Since the flux to everywhere on the surface except the patch is zero, eq 1.b implies ∂C/∂r ) 0 for r g a. (21) Crank, J. The Mathematics of Diffusion; Oxford University Press: Oxford, 1975; eq 3.35. (22) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; J. Wiley: New York, 2002.
The rate of analyte adsorption may be described by a boundary condition at the adsorbing patch surface: for example, fast, irreversible adsorption onto a patch that does not saturate (namely, where there is no limit to the analyte uptake) is defined by C(r e a) ) 0, where C is the concentration of free analyte and a is the patch radius. Alternately, one may consider an equilibrium, or reVersible, rate of adsorption that is proportional to the local concentration of free analyte in solution. In this case, the flux to the patch, -D∂C/∂r is equal to the binding reaction rate kC for r e a, where the binding coefficient k accounts for the probability of binding (or strength of binding). In the limit where k is large, this condition reduces to the irreversible adsorption condition and effectively imposes that C ) 0 on the adsorbing surface. It should be noted that taking k to be a constant implies that the patch does not saturate, so that the rate of adsorption at any given time is independent of the patch history, sensitive only to the amount of analyte in the vicinity of the adsorption site. As shown by Crank,21 the interfacial concentration at the patch edge, and the diffusional flux to the patch (per unit area), for this type of system is given by
[
(
])
C0 xt(D + ak) 2 2 D + 2ak - aket(D+ak) /Da erfc D + ak axD (3)
C|r)a )
|
∂C ) ∂r a kC0 xt(D + ak) 2 2 D + aket(D+ak) /Da erfc D + ak axD
J ) -D
[
(
])
(4)
where erfc(x) ) 1 - 2/(xπ∫x0e-x2 dx) is the complementary error function. Since there is no flow in the system, the overall flux N is given by the diffusional flux J: Initially, this is given by
(
J|tf0 ) kC0 1 -
2kxt xπD
)
(5a)
while, at longer time, the flux reaches steady state, becoming independent of t:
J|tf∞ ) kDC0/(D + ak)
(5b)
The overall mass of analyte on the patch can be calculated by integrating the flux (multiplied by the patch area, πa2) over time, which yields
M(t) ) πa4DkC0(Dt/τ + ak{2xt/πτ - 1} + aket/τerfc[xt/τ]) (D + ak)3
(6)
where τ is a characteristic time scale, defined as
τ)
a2D (D + ak)2
(7)
In the limit of t f 0, the adsorbed mass is given by πta2kC0; namely, it is linearly proportional to the area of the patch, the strength of adsorption (as denoted by k), and the concentration of analyte in solution. The adsorbed amount is independent of the diffusion coefficient initially, since mass transport does not play a role at early times when the analyte concentration is uniform and given by C0. After a long period of time, the system reaches steady state, where the adsorbed mass increases as
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πta2kC0D/(D + ak). In this limit, the increase in the analyte mass on the surface is due to a balance between the rate of transport to the adsorbing patch (as defined via D, the diffusion coefficient) and the strength/rate of binding (as defined through k). The characteristic time scale for adsorption in this case is defined by τ (eq 7). We see that the time scale for adsorption is small either if diffusion is rapid when compared to the adsorption constant (D . k) or in the opposite case (D , k). The time scale for adsorption passes through a peak; that is, it is maximal when D ) ak. In the case of irreversible adsorption (not physically appropriate for many systems of interest) k is infinite; In this limit, the steady-state value for the adsorbed mass is equal to πtaC0D, in agreement with previous analysis for steady-state, irreversible adsorption.5 Any “real” adsorption patch contains a finite number of adsorption sites, and thus reaches saturation at some point. Saturation may be due to packing constraints on the surface, arising from the finite size of the analyte particles and the adsorption patch, or be the result of a finite number of adsorption sites, or binding ligands, on the surface. In the following, we will focus on the latter case, namely, a finite number of adsorption sites. However, the analysis generally applies to finite size effects as well (except for the necessary assumptions involved in the diffusion model itself). In a saturating system, the binding rate k depends on the number of free, or unbound, surface sites, a value that decreases with time as more analyte adsorbs. It is important to note that considering saturation does not necessarily imply irreversible adsorption; the rate of adsorption is defined by a self-consistent combination of the analyte concentration near the substrate, the number (or density) of free adsorption sites, and the strength of binding. In systems characterized by strong binding, saturation will occur quickly, while, in systems with reversible adsorption, it may take a long time. Assuming first-order reaction between the analyte and the binding ligand, we may define k, the coefficient of binding, as k0CL(t). Here k0 is a constant, and CL(t) is the number of available surface sites. The latter is linked to the flux to the surface via the relationship (assuming q analyte molecules can bind to one ligand, or site)
CL(r,t) ) C0L -
1 qm
t ∫t′)0 (-D∂C∂r )r,t′ dt′
(8)
where C0L is the initial number of adsorbing ligands (or sites) per unit area, and m is the molecular weight (MW) of the analyte (when C is given in units of mass/vol. If C is a molar concentration, m is unity). Since CL is given as the number of ligands per unit area, the units of k0 are those of Volume per unit time (rather than the length/time, the units of k). The boundary condition on the patch is given, then, for r e a
(-D∂C∂r )
r,t
) k0C(r,t)CL(r,t) )
{
k0C(r,t) C0L -
1 qm
t ∫t′)0 (-D∂C∂r )r,t′ dt′}
(9)
namely, the rate of adsorption at any given time t depends on the number of previously adsorbed analyte molecules, that is, patch history. The diffusion eq 2, subject to the boundary condition defined by eq 9, is too complex to solve analytically. We use a pseudosteady-state approach where the rate of change in the number of available ligands is taken to be slow when compared to the
change in C. The differential equation is solved assuming that the flux is approximately constant at time t (namely, that CL is fixed) and then the new adsorption rate is calculated using eq 9 to define CL(t). In the limit of low k0, we find the effective binding coefficient to be given by
k0C0L
keff ≈ 1+
k 0t (πa3DC0 + DmqC0L + ak0mq{C0L}2) aDmq
(10)
Initially, all adsorption sites are available, and k ) k0C0L, namely, the value of k if no adsorption has taken place and all surface sites are available. At long times, most or all sites are occupied, and keff approaches zero. The time-dependent flux is then given by eq 4 where keff (eq 10) replaces k. In the limit where k0 is small, this leads to
|
( {
aC0L tC0L πa2C0t ∂C + + + ≈ k0C0C0L 1 - k0 ∂r a D a mq aC0L Dt/a2 xDt e erfc (11) D a
J ) -D
[ ]})
In Figure 2 we plot the flux to the adsorbing patch as a function of time. As may be expected, the flux decreases rapidly as time progresses, in a manner somewhat similar to that of the flux to a patch that does not saturate. However, while the flux in the nonsaturating case approaches a finite value in the limit of t f ∞ (steady state), on the saturating substrate, the limit value of the flux is zero. We find that deviations between the saturating and nonsaturating cases develop at relatively short times, much shorter than the characteristic time scale at which steady state (or saturation) is achieved. Examining the effect of different parameters on the flux, we see (Figure 2) that increasing the number of analyte molecules or particles that can bind to each site, q, increases the value of J (note that J0 ) C0C0Lk0 and is independent of q) at any given time. The flux is more sensitive to the concentration of binding sites than to the concentration of analyte in solution. The flux, J, as defined by eq 11 may be integrated to obtain the adsorbed mass at any given time.23 In Figure 3, we plot the adsorbed mass as a function of time for a system with a finite capacity for adsorption, namely, an adsorption patch that can saturate. We see that, once the adsorbed mass reaches a significant fraction of the overall amount (in this case, approximately 70%), the rate of adsorption decreases significantly, and a regime of (23) Analytical integration of the flux to the saturating surface is not possible. The results presented in Figures 3 and 4 were obtained through numerical integration (using Mathematica software). We developed an approximate analytical expression for the adsorbed mass as a function of time that captures the main features of the system; however, this expression deviates somewhat from the full integrated result at both low and high t: M(t)/πa2 ≈
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Figure 3. The adsorbed mass on a saturating patch, as a function of time, obtained by numerically integrating the flux (eq 11). All parameters are as defined for Figure 2A. M∞ is given by the maximal adsorbed amount, namely, mqπa2C0L. Time is given in units of D/( C0Lk0)2. We see that the adsorbed mass increases rapidly until reaching ∼70% of the total amount (at t ≈ 2), after which it increases very slowly. Examining the initial behavior (inset), we see a very short induction period where little adsorption takes place, followed by a region of rapid uptake. The induction period is due to the reversibility of the binding.
over which the effect of saturation on the adsorbed mass is negligible is when t is smaller than D/(k0C0L)2.
Figure 2. (A) The flux J of adsorbing analyte to a nonsaturating (eq 4) and a saturating (eq 11) adsorption patch, as a function of time. J is given in units of C0k and C0C0Lk0 for the nonsaturating and saturating cases, respectively. Time is given in units of D/k2 and D/ (C0Lk0)2, respectively. All parameters (m, q, D, a, and C0) are unity, as are k, k0, and C0L. We see that deviations between the saturating and nonsaturating cases develop when the adsorption time exceeds ∼0.01. Both cases reach a more or less constant value of J at times that are longer than ∼5 (inset); however, in the saturating case, this “steady-state” value is approximately zero, while, in the nonsaturating case, it is on the order of 0.5. (B) The effect of system parameters on the flux to a saturating adsorption patch. The solid line (standard) applies to the case described in panel A, and J0 ) C0C0Lk0. We see that the flux is most sensitive to changes in the density, or number of adsorbing sites.
very slow adsorption develops. Initially, the adsorbed mass is given by
(
M|tf0 ≈ πa2C0C0Lk0t 1 -
)
C0Lk0t1/2 D1/2
(12)
so that, at very short times, it increases as πta2k0C0LC0, that is, it is identical to the initial, linear rate of adsorption in the nonsaturating case (as indeed may be expected). However, once t is on the order of D/(k0C0L)2, the rate of mass accumulation decreases when compared to this initial rate; thus, the time range
Our analysis is based on reversible adsorption, thereby allowing us to estimate the role of the binding strength (as defined by k0) on the adsorbed mass. One would expect that the adsorbed mass will be higher, at any given time, for strongly binding sites (e.g., hydrogen or covalent binding) when compared to weakly binding ones (van der Waals, screened electrostatics). In Figure 4, we compare mass adsorbed from identical analyte solutions (same m, C0, and D) on two identical, saturating patches (same a, q, and C0L) but with different binding strengths (k0). As may be expected, initially the amount of adsorbed mass is smaller for the weakly binding adsorption patch. However, at some (very long) time the mass accumulated on the weakly binding patch increases aboVe that of the more strongly binding one. This surprising result may be understood when examining the effective adsorption coefficient, keff (eq 10), in the limit of short and long times:
keff|tf0 f k0C0L keff|tf∞ f
C0LDmq t(πa3DC0 + DmqC0L + ak0mq{C0L}2)
(13a) (13b)
so that, if all other parameters are equal, the effective binding coefficient is lower for lower binding strength initially. At extremely long times, however, the effective binding coefficient for the more strongly binding system is lower. The time scale required to obtain a fixed amount of analyte on the substrate, M*, for example, the mass required to initiate signaling, is of interest for sensor design. In the limit where M*
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Langmuir, Vol. 22, No. 3, 2006 875
time of adsorption by a given factor will not reduce the detectable concentration by that same ratio. Can we estimate the various parameters calculated here, such as the rate of analyte accumulation on a characteristic patch or the time scales at which saturation becomes important (eq 12)? Some quantities are simple to estimate: for example, the patch size (typically on the order of 1-100 µm) or the diffusion coefficient (∼10-5 m2/s for small analyte molecules). The value of m is known for any analyte (or is unity if the concentration is given in molar units), and q is known for any ligand type. Determination of k0 and C0L is more complex, however. To evaluate these, we can use the results obtained from adsorption isotherms, which describe the adsorbed amount as a function of solute concentration in the limit of long time. For low k0, we find that the long time limit of the adsorbed mass is given by
M*|tf∞ ≈ Figure 4. The effect of binding strength, as defined by k0, on the mass adsorbed on a saturating patch, calculated by numerically integrating eq 11. All parameters, except for k0, are as defined for Figure 2A. The maximal adsorbed amount, mqπa2C0L is therefore equal to π. We see that, over a wide range of time, the adsorbed mass for the weaker binding patch (k0 ) 0.1) is lower than that for the strongly binding one (k0 ) 1). However, at very long times, the two reverse, as may be seen from eq 13.
is much smaller than the maximal adsorbed amount, we find
t(M*)|M*f0 ) M*{1 + 3C0Lk0M*/2πa2DC0 - xC0Lk0M*/πa2DC0} πa2k0C0C0L
(14)
As may be expected, the time required to obtain a critical amount of analyte on the patch increases with the required mass (M*) and decreases with the analyte concentration in solution, the density of binding ligands, the adsorption coefficient (k0), and the patch area. All of our results so far, regarding the mass accumulation of the adsorption patch, the flux to it, and the characteristic time scales, depend on the concentration of analyte in solution, a quantity that is usually unknown. However, we can determine the minimal concentration of analyte that may be detected over a given period of time for adsorption: eq 14 yields the time required to obtain M* on the substrate (the critical amount required for detection) as a function of the analyte concentration. However, if we define t* as the contact time allowed for adsorption, we find
Cmin 0 |t*f0
)
M*{1 + C0Lk0xt*/D} πa2k0t*C0L
(15)
assuming that t* is relatively short. As may be expected, the minimal concentration that may be detected increases with M*, the amount of material required for detection/signaling, and decreases with increasing adsorption patch area, or strength of (initial) adsorption, k0. Thus, systems that require a large amount of adsorbed material for detection will detect only high analyte concentrations. Allowing adsorption to take place over longer periods of time will enable the detection of anlaytes that may be present in trace amounts. However, the time dependence of the detectable concentration is not linear; thus, increasing the
amqC0C0L(D - 2ak0C0L) 2D(πa3C0 + mqC0L)
(16a)
or, written as the fraction of maximal mass,
Θ)
πa3C0/mqC0L M* ) M*|C0f∞ 1 + πa3C /mqC0 0
(16b)
L
which is identical to the Langmuir isotherm. Thus, we can identify the coefficient of the Langmuir isotherm (commonly denoted by KL) and the equilibrium adsorbed mass with some of our parameters. To examine our predictions, we can take, as an example, the adsorption of n-propanol at air/water interfaces: studies show24 that M*(tf∞) is 7 × 10-5 mol/m2, and the Langmuir isotherm coefficient, KL, is 5.5 × 10-3 m3/mol. The diffusion coefficient is on the order of 10-5 m2/s. Since there are no specific binding ligands (saturation is due to excluded volume at the interface) and the units are molar, q and m are unity. To estimate k0C0L, however, we need a characteristic size scale for the experiment at which the adsorbed mass was measured, since M*(tf∞) varies with a (see eq 17). Assuming that the area used was on the order of 10-1 m, we find a value of 8.7 × 10-5 for k0 and 0.57 for C0L, so that k0C0L ) 5 × 10-5 m/s. Now we can apply these numbers to estimate the characteristic adsorption parameters. We find that, for a 1 µm patch, taking the characteristic time scale for detection to be 10 s, the minimal detectable concentration is approximately (eq 15) (7 × 1014)M*, where M* is the amount of analyte (in moles) required for signaling. Thus, if one molecule is required for signaling, the patch can detect concentrations on the order of 10-8 M, but a patch that requires 106 molecules will be able to detect in that time period only concentrations higher than 10-3 M. One of the more surprising predictions of our model is that, at some time, the mass accumulated on a weakly binding patch may increase aboVe that of a more strongly binding one. To examine this point, we compare n-pentanol to n-octanol, which has the same maximal adsorbed mass 24 but a Lanmguir isotherm coefficient24 that is 2.5 m3/mol. The k0 value of n-octanol comes out as 0.04, which is 3 orders of magnitude higher than that of n-pentanol. Yet, both end up with the same maximal adsorbed amount. Examining the effective rates, we find that keff for pentanol is equal to 0.00003/(1 + 0.002t), while that of octanol is given by 0.000048/(1 + 1.25t). Thus, initially, the rate constant for octanol adsorption is more rapid, as expected, but, after a (24) Chang, C. H.; Franses, E. I. Colloids Surf., A 1995, 100, 1-45
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period of time (in this case, a relatively short time of approximately 0.55 s due to the large difference in k0), the rates reverse, and pentanol has the higher effective adsorption constant. In conclusion, we examine here the time-dependent adsorption of analytes on a finite-sized patch, as a model for patterned or heterogeneous surfaces. We account for both reversible adsorption (assuming first-order reaction) and saturation of the adsorption patch. Our somewhat surprising, main conclusions include the following: (1) Saturation effects, either due to finite patch size or to finite number of binding sites, become significant at extremely short times. As a result, assuming that saturation effects are negligible may lead to large overestimates of the flux and the adsorbed amount. (2) Increasing the strength of binding (k0) increases the adsorbed amount at short times. However, at (very)
Letters
long times, the mass adsorbed on a weakly binding patch is higher than that adsorbed on a strongly binding one. (3) The sensitivity of detection, as defined by the adsorption of a minimal analyte mass required for signaling, in a fixed period of time, does not scale as 1/detection time. As a result, increasing the time over which adsorption occurs increases sensitivity, but not linearly. Sensitivity of detection also increases with increasing patch area and initial binding strength. These results may be used in the design of sensors based on analyte binding to patches on the substrate. Acknowledgment. The support of NSF-NIRT Grant No. 0304453 is acknowledged. LA0519608
