It should be mentioned that frames 3–5 in Figure 14.5b are two-
dimensional and the real (three-dimensional) situations will generally be
different, with two principal radii of curvature; see Section 10.5.1. However,
trends will often be as mentioned.
Question
To what temperature above 0 8 C should ice be heated before water starts to form at
its surface? The surface tension of water at low temperature equals about
75 mN?m^1 andg(water-ice)&25 mN?m^1 (Table 10.1). The surface tension of
ice is unknown, but you may assume a value of 95 mN?m^1.
Answer
Assume the nucleus to be a spherical water cap on the ice surface. From Eq. (41.11)
we calculate cosy¼(9525)/75¼0.93 (corresponding toy¼ 218 ). Inserting this
value into Eq. (14.12) yieldsfcat¼0.0036. In the situation envisaged, all nuclei would
be the same, i.e., have the same radius and contact angle. This would mean that we
may insertfcatinDGmax, i.e., in the second exponent of Eq. (14.10). Table 14.2 gives
Jhomfor ice nucleation, and the same results would apply for water nucleation since
the absolute values ofgandDHwill be the same. We thus obtain for the present case
Jhet¼ 1036 exp (82,000fcat/(DT)^2 ). Putting this equal to 10^14 , we calculate forDTa
value of 2.3 K. Actually, the preexponential term may be higher than 10^36 , say 10^41 ,
because the transition ice!water does not involve a decrease but an increase in
entropy, but this would only cause a very small difference in the value ofDT.
Altogether, water would form at about 2 8 C.
Subsequent Question
In nature it is commonly observed, however, that ice becomes wet at, say, 0.3 8 C. On
the other hand, it can stay dry at, say, 5 8 C. How is this possible?
Answer
In the first place, the result obtained may be wrong, because of uncertainties in the
theory and in thegvalues. Nevertheless, overheating by a few degrees would be
needed on an ideal flat ice surface. However, ‘‘natural’’ ice will always contain little
crevices, and water can readily form in these, as illustrated in Figure 14.5b.
Furthermore, it was implicitly assumed that the air above the ice is exactly
saturated with water vapor at 0 8 C. At significant supersaturation of the air, water
can condense on the ice (beginning in crevices). The air may also be much drier, i.e.,
its relative humidity can be well below 100%. In that case, water will be removed
from the ice by desublimation (especially when a wind is blowing).