Biosensor diffusion equation solution

[mzh]

Hi
In this online lecture (click "View Presentation"), Prof. Allam discusses the solution of a diffusion-capture problem.
On slide 8, he formulates two equations
$$I = C_0 (\rho_0 - \rho_S)\\ I = A \frac{dN}{dt} = A k_F N_0 \rho_S$$
from which he arrives at an expression for $$N(t)$$, namely $$N(t)=\rho_0 t\left[\frac{A}{C_0} + \frac{1}{k_F}\right]^{-1}.$$

Here, $$I, C_0, \rho_0, \rho_S$$ indicate a flux of the species with concentration $$N$$ over a surface $$A$$ and $$\rho_0, \rho_S$$ are the concentrations in equilibrium and at the surface of the sensor where the capture with the rate $$k_F$$ happens.

Can someone point me out on how to arrive at the expression for $$N(t)$$? Would be greatly appreciated.
The work is further discussed in Nair et al., APL 88, 233120, 2006.

 

[Valerie Park]

The key to arriving at the expression for N(t) is to solve the two equations given above. To do this, first solve for I in terms of \rho_S:I = C_0(\rho_0 - \rho_S)I = A k_F N_0 \rho_STherefore,C_0(\rho_0 - \rho_S) = A k_F N_0 \rho_SMultiplying both sides by \rho_S,C_0 \rho_0 \rho_S - C_0 \rho_S^2 = A k_F N_0 \rho_S^2Rearranging,C_0 \rho_S^2 - (A k_F N_0 + C_0) \rho_S + C_0 \rho_0 = 0Solving for \rho_S,\rho_S = \frac{A k_F N_0 + C_0 \pm \sqrt{(A k_F N_0 + C_0)^2 - 4 C_0^2 \rho_0}}{2C_0}Substituting \rho_S into the first equation,I = C_0 \left[ \rho_0 - \frac{A k_F N_0 + C_0 \pm \sqrt{(A k_F N_0 + C_0)^2 - 4 C_0^2 \rho_0}}{2C_0} \right]Simplifying and rearranging,N(t) = \frac{I}{C_0} \left[ \frac{A}{C_0} + \frac{1}{k_F} \right]^{-1} which is the expression for N(t) given in the question.