koff - LEPA

The dimensionless parameter ψ is useful to define the adsorption process. If ψ is much smaller
than unity, then we have an adsorption from a dilute solution where the surface coverage is small
and directly proportional to the bulk concentration; we then speak of the linearized Langmuir
isotherm. If ψ is much greater than unity, then the coverage tends to a full monolayer. When ψ
is equal to unity, the equilibrium coverage is half of the full coverage. In the case of biosensing,
we deal generally with dilute solutions and therefore the equilibrium surface concentration is
directly proportional to the bulk concentration.
! eq = ! max Kc bulk (3)
Exercise : For all the calculations, we shall take K=10 9 M –1 , k on =10 6 M –1 ·s –1 , a bulk analyte
concentration of 0.1 pM unless specified otherwise, and ! max " 10 #9 mol·m #2 .
Calculate the area of an adsorption site, the equilibrium surface coverage and ψ .
1.2 Kinetic aspects
The rate law for a binding reaction is just given by the difference between the adsorption process
considered as a second order reaction between the reactant and a free site and the desorption
process simply considered as a first order reaction, and we can write
d!
dt
= k on c bulk (1!!) ! k off! (4)
This differential equation can be re-arranged as
d!
!
"
dt +! k onc bulk + k off
and the solution is then given by
! =
k on c bulk
k on c bulk + k off
#
$ = konc bulk (5)
1! exp ! konc bulk " " ( + k
#
off )t $ $
#&
% %' =
where τ is the time constant for the adsorption defined as
! =
1
k on c bulk + k off
=
t d
1+"
"
"# 1! exp [ !t /# ] $
1+"
% (6)
where td is the desorption time ( td = 1/ koff ) . This equation clearly shows that the adsorption
kinetics from dilute solutions is controlled by the desorption time.
The adsorption kinetics is illustrated schematically in Figure 2 for different values of ψ
Fig. 2. Time dependence of the surface coverage
(7)
2