Probing Biointerfaces: Electrokinetics - Biointerfaces: Where Material Meets Biology - page 60

is characterized by its dissociation pK value, denoted as pK j . The parameter
e j ¼ 1, ..., M takes the value 1, with e j ¼ 1 and e j ¼þ 1 corresponding to
ionization of the groups due to deprotonation or protonation reactions as
detailed elsewhere. 32 The boundary condition at the position X ¼ 0 can be
expressed in terms of the surface potential or surface charge density of the
hard carrier supporting the film. 31 For systems where the charge of the
supporting surface does not significantly contribute to the streaming current
and surface conductivity, the zero electric field condition may be imposed at
X ¼ 0. 31 In line with the electroneutrality far from the film/solution interface,
the additional required boundary condition for the potential is simply
written, y(X

d n 3 r 4 n g | 5

1/2) ¼ 0, which is correct in absence of overlap of the electric
double layers developed in the vicinity of

-

the two samples in the

microslit cell.

3.2.1.3 Streaming Current and Streaming Potential
To obtain the streaming current, I str , the spatial integral over the hydro-
dynamic flow field, V(X), [eqn (3.1)] times the charge density stemming from
charged mobile carriers [first term in the right hand side of eqn (3.3)] must
be solved. Using eqn (3.1) and (3.3), this integral can be written in the fol-
lowing form: 31

1 = 2

Z

V(X) X

N

DP ¼ 2 ' FH 3

I str

z i c i e z i y(X) dX (3 : 4)

ZL 0

i ¼ 1

0

.

Under conditions of low potentials (Debye-H ¨ ckel approximation)
and homogeneous distribution of polymer segments in the soft layer,
explicit analytical expressions may be derived for I str /DP. 30 For hard
surfaces, the solution of eqn (3.4) is identical to the Smoluchowski equation
that relates the streaming current to the zeta potential, the electrical po-
tential at the 'plane of shear'. 18 Similar expressions can be obtained for
the streaming potential, U str , by converting the streaming current into a
potential via Ohm's law, 18 where terms for the conductivity of the electrolyte
and for the surface conductivity of the film are involved [see eqn (3.5)].
Measurement of the streaming current and potential is outlined in

Appendix A.1.

3.2.2 Surface Conductivity
The excess conductivity, K s , is caused by the accumulation of mobile
counter-ions inside and outside a soft surface layer and can be determined
from streaming current and streaming potential measurements carried out
at a single or varying channel heights using the following equation:

I str = DP
U str = DP

L 0

2 ' ¼ H 2 K B þ K s

(3 : 5)