Transcript No Slide Title

Charged Interfaces
(see P.C. Hiemenz and R. Rajagopalan)
• Introduction
• origin and importance of
surface charge
• The electrical double layer
•
•
•
•
Parallel plate capacitor model
Debye-Hückel approximation
Gouy-Chapman theory
Stern model: the inner layer
• Electrokinetic Theory
• Zeta Potential
• mobility (various models)
Introduction
Importance of Surface Charge
Determines colloid stability
Proteins--determines structure
Environment--groundwater purification,
sewage treatment
Industry--food, pharmaceuticals, paint.
Rat cytochrome b5 protein
Origin of Surface Charge
•Preferential adsorption of an ion onto
a previously uncharged surface
•Ionization of a surface group
AgI(s)  Ag+(aq) + I-(aq)
Ksp= 7.5 x 10-17
[Ag+] = [I-] = 8.7 x 10-9
[Ag+]< 3.0 x 10-6
[Ag+]zp = 3.0 x 10-6
zero point Ag+ concentration
RT [ Ag  ]

ln
nF [ Ag  ]zp
 150mV
at [Ag+] = [I-] = 8.7 x 10-9
[Ag+]> 3.0 x 10-6
The Electrical Double
Layer
Parallel Plate Capacitor model
1 q1q2
F
4  r 2
   o r
 d  F
E  -

 dx  q
E 

1 q


4  r 2
q
q
 *  , E 
A
4r 2
q
 E 
A
*




An estimate of the value for 
 = 0.1 Vm-1
 = or = 80 x 8.85 x 10-12 C2J-1m-1
* = 1 ion/nm2 = 1.6 x 10-2 Cm-2
E = 2.26 x 107 V/m
Thus  = 4.4 nm -- not unreasonable!
Problems
•ions move in solution
•more than 1 counter ion type may be
present.
•limited to one geometry
The Debye-Hückel Model
In the general case, the Poisson
equation gives the charge distribution
  ( x, y , z ) 
2
  * (x,y,z)

2
2
2



where  2  2  2  2
x y z
Solve for a 1-D planar surface
d2   * (x)

2
dx

Assume a Boltzmann
distribution for the
ions in solution near
the charged surface.
z e (x)

ni
 e kT
ni
i
 *   z ieni
n
i 1
  * (x)   z ieni e
n
i 1

z i e (x)
kT
Valid at low
Hückel Approximation!
potentials only!
2
3
4
x
x
x
e  x  1  x    ...  1  x
2! 3! 4!
n
z ie(x) 

 * (x)   z ieni 1
i 1
kT 

2 2
n
n z e n  (x)
i
 * (x)   zieni   i
i 1
i 1
kT
2 2
n z e n  (x)
i
 * (x)    i
i 1
kT
d2  1 n z i2 e 2ni (x)
 
2
dx
 i 1
kT
d2  e 2 (x) n 2
 z i ni

2
i
dx
kT 1
The definition of ionic strength, I, is
1n 2
I   z i ni
2 i 1
d2  2Ie 2 (x)
 2 
dx
kT
2Ie 2
2
Let  
kT
d2 
 2   2 (x)
dx
Conditions ...(x)  o as x  0
(x)  0 as x  
Solution.. .
(x)  o e  x
Can apply the same approximations
to a spherical particle: we give only
the solution here.
R s  r R 
(r)  o e
r
s
o
Rs
r
Signifigance of the D-H Parameter 
-1 has units of length: it is a measure
of the “width” of the double layer
(x)  o e  x
-1 is the distance for  to reach 1/e of
its initial value, o
 
1
 kT
2Ie2
-1 decreases as I increases
(i.e. as concentration and valency
of the ions increase)
-1 increases as T increases
(as T increases, ions diffuse more
readily)
-1 increases as  increases
(polarizable solvents have large
double layers associated with them)
-1 has values ranging from about
2 - 10 nm, depending on conditions:
this could be of the same order as
colloid size in solution.
1.2
1
y/yo
0.8
0.001 M
0.6
0.4
0.01 M
0.2
0
0
0.1 M
5
10
15
10
15
x(nm)
1.2
1
y/yo
0.8
0.6
1:1
0.4
2:2
0.2
3:3
0
0
5
x(nm)
Gouy-Chapman Theory
D-H model is limited to low potentials
G-C theory (textbook) overcomes this
problem, but:
•not as easy to visualise
•limited to case of symmetric electrolyte
(z+=z-)
Start in the same way:
d2 
 * (x)


dx 2

 * (x)   zieni e
n

z i e (x)
kT
i 1
 * (x)  zen  (e

ze (x)
kT
e

ze (x)
kT
ze (x)
 * (x)  -2zen  sinh
kT
)
d2 
ze (x)

2zen
sinh

dx 2
kT
Solution to this differential equation is:
2kT  1 e  *x 
(x) 
ln

 *x
ze  1 e 
ze (x)
2kT
1
2e2 z 2n
where   ze (x)
and  * 
 kT
e 2kT  1
e
This will reduce to the D-H equation
for low values of 
What does this function look like?
Debye-Huckel
Gouy-Chapman
•at large distances, they look similar
•G-C drops off more quickly
• * is still an indication of layer thickness,
but if anything is an overestimate
Stern Model: the inner layer
Up to now, we have assumed ions are
point charges, but they have finite size
and can thus adsorb preferentially on
surfaces.
The inner layer (continued):
 Stern in 1924 proposed a model in
which the double layer is divided
into two parts separated by the
“Stern plane”.
 The Stern plane is located about a
hydrated ion radius away from the
particle surface.
 The potential changes from o at
the wall to d (the Stern potential)
to zero far away in the diffuse
double layer.
 The Stern potential can be
estimated
from
electrokinetic
measurements.
 The Zeta Potential (z) will be
marginally smaller than d.
The Surface Potential - yo
The surface potential can be related to
the surface charge density in the
following way:

 o     * dx
o


 ze (x)
 ze (x)
kT
kT )
 * (x)  zen  ( e
e
Thus
 ze  o 
 o  8 n   kT 1 / 2 sinh 

 2kT 
At low potentials this reduces to:
 o    o
The Surface Potential - yo
 The surface potential depends on
both the surface charge density and
the composition of the continuous
phase (through ).
 If electrolyte is added to an
electrostatically stabilized sol, the
double layer will be compressed
and either the surface charge
density must increase or the
surface potential must decrease or
both must occur.
 Values of the double layer thickness
(1/) range from 100 to 10 to 1 nm
for electrolyte concentrations of 10-5
to 10-3 to 10-1 molar.
The Surface Potential - yo
IN GENERAL:
Increasing the electrolyte concentration
has the important effect of decreasing the
double layer thickness. At the same time
both the Stern and zeta potentials are
lowered and in certain cases their sign
may even be reversed!
y
yd
z
c1<c2<c3
c1
c2
0
x
c3