ρ
surface charge
D
d
κ1
screening charge
1
κ1
2
Figure 8. Schematic dependence of the space charge density ρ on the dis-
Lsc
tance to the ice surface x (after Petrenko and Colbeck 1994). Scales are arbi-
trary. d is the thickness of liquid-like layer, D is the thickness of the ice layer
rubbed off by a slider from the ice surface, and κ1 1 and κ2 1 are the length of
screening by Bjerrum defects and ions correspondingly.
1
-
e2 (n10 + n20 )
ice surface comes from the high adsorption coefficient
2
1
≅
of ice at temperatures above 40C (Ocampo and
(7)
κ 2 kBTε 0εs
Klinger 1983), and the results of computer simulations
carried out by Bush and Devlin (1991). At the same
where ε0 =
dielectric permittivity of vacuum
time the polarization of water molecules in the bulk is
ε∞ =
r
r
3.2, the high-frequency permittivity of ice
εs ≈ 100 =
the surface (x = 0) to 0 (x = ∞). This nonuniform polar-
static dielectric permittivity of ice
kB =
Boltzmann constant
ization results in a surface charge of a high density
e3 = 0.38e =
charge of Bjerrum defects
∞
concentrations of H3O+ ions
n10 =
P
∫
λs = -
dx = P0 .
(5)
concentrations of OH ions
n20 =
x
0
n30 =
concentrations of D-defects
An estimation using eq 5 gives λs ≈ 2 101 C/m2.
n40 =
concentrations of L-defects.
Other possible reasons accounting for the formation of
-
At the screening length κ1 1, which is determined by the
λs could be adsorption of ions from the air and the
concentration of majority carriers (n30 + n40) and coin-
presence of the surface electronic states.
cides with the well-known expression for the Debye
-
screening length κ1 1, the field drops down from the
be neutralized by the opposite screening charge λsc,
original value λs/ε0ε∞ to λs/ε0εS. As we have seen, εS ≈
equal to the former in magnitude, which is distributed,
102, that is, κ1 1 is a characteristic length at which εS
-
however, in a much thicker screening layer. The
attains a steady-state value. At larger x the field drops
screening of the surface charge in ice was considered
down to zero because of screening by minority charge
in Petrenko and Ryzhkin (1984a), and in Petrenko and
carriers, i.e., ions with a larger characteristic screening
Maeno (1987). The screening charge is composed of
-
length κ 21 determined by ion concentrations (n10 + n20).
protonic charge carriers (Bjerrum D- and L-defects
Substitution of charge carrier concentrations in pure ice
and H3O+ and OH ions). There are two screening
(Petrenko 1993b) into eq 7 and 8 gives at T = 5C
-
-
lengths in ice κ1 1 and κ 21
1
≅ 3.4 10 -8 m
1
(8)
-
1 e3 (n30 + n40 )
κ1
2
2
≅
(6)
κ1 ε 0ε ∞kBT
1
≅ 3 10 -5 m.
(9)
κ2
6
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