Geoscience Reference

In-Depth Information

Tabl e 3. 1
The values of
"
,

.
"
/
(Eq.
3.125
),
u
"
,and

2

"

=q

"

.
"
/

u
"

2

.
u
"
/

2

1

1.615

8.994

3.902

1.737

.
u
"
/ (the set of Eqs.
3.158

at different dielectric

permeability ") and the ratio

of particle-to-ion

charges Q=q

3

1

1.777

6.447

2.402

1.959

5

1

1.926

5.136

0.1.684

2.174

1

1

2.22

3.8

1.002

2.615

2

2

1.427

16.351

8.661

1.492

3

2

1.535

11.56

5.415

1.633

5

2

1.633

9.107

3.846

1.767

1

2

1.821

6.617

2.331

2.037

2

1/2

1.894

5.05

1.718

2.117

3

1/2

2.139

3.673

1.041

2.47

5

1/2

2.369

2.96

0.721

2.817

1

1/2

2.833

2.229

0.421

3.54

Now we come to the result similar to that for metallic particles (Eq.
3.150
):

s
ˇq
2
e
2

2a

"
C
2

2"
C
3
C

"
1

"
C
2
:

".a/
2

(3.153)

3.5.5.2

Opposite Polarities

No noticeable simplifications comes up at "
D1
(metallic particles).

The analytical results for arbitrary " can be found for small particles. We again

return to Eq.
3.115
, rewritten in the dimensionless variables:

Z

e
u

2

".a/
D
.
"
/
C

.
u
/d
u
;

(3.154)

0

where
"
is the zero of
u
./ at arbitrary ". The first term is

2

"
C
S.
"
/:

.
"
/
D

(3.155)

At large , the integral on the right-hand side of Eq.
3.154
can be easily

estimated, for again, small
u
/
1= contributes to the integral. We have

".a/
.
"
/
C
"
:::

(3.156)

The values of
"
and .
"
/ are collected in Table
3.1
for a number of values of

the dielectric permeability ".