APPENDIX B: SCALING THE RIGIDICE MODEL
The Rigidice model of frost heave is a set of
1
ΓIW ≡
γ IW
γ IA
differential equations expressing mass, energy and
force balances in the frozen fringe for air-free, col-
and
(B3)
loid-free and solute-free soil (Miller 1978). A set
1
ΓAW ≡
γ AW .
of reduced or scaled variables for this model were
γ IA
later presented by Miller (1990). To avoid redun-
dancy, this later work did not contain derivations
To simplify, the remainder of this appendix will
but referred to the earlier classic works for ice-
address icewater systems. A similar set of equa-
free unsaturated soil (Miller and Miller 1955, Miller
tions for waterair and iceair systems are ob-
1980a). For completeness of this paper, this ap-
tained by direct analogy to the icewater equa-
pendix presents a brief derivation of the reduced
tion and are listed by Miller (1990).
variables presented by Miller for the liquid wa-
The distinguishing trait exhibited by water and
ter and solid ice phases. It also serves as an exam-
ice in a porous medium that does not exist in un-
ple of how the seemingly subjective process of
restricted bulk is the existence of curved inter-
scaling equations is performed. All scaled vari-
faces. The force balance across a curved interface
ables will be given upper-case letters to distin-
separating two phases in equilibrium is given by
guish them from the nonscaled dimensional vari-
the Laplace surface tension equation, which for
ables that are in lower case.
an icewater interface is
γ
pI - pW = 2 IW
(B4)
rIW
Air
where the absolute pressure is p (N/m2) and the
Water
mean radius of curvature is r. In the case of fro-
Ice
zen soil, this equation expresses the behavior of
water and ice contained within the pores. This
suggests a length scale on the order of the pore
γWA
size, so we introduce a microscopic length λ. Equa-
tion B4 contains three variables, pI, pW and r, and
one constant, γIW. Equation B3 shows how to re-
α
Figure B1. Equilibrium
γIA
γIW
duce the surface tension. At this point we need
configuration.
to decide how to reduce the remaining variables.
The simplest approach is to acknowledge that the
radius of curvature is a pore length dimension,
We start with Young's equation relating the
so it is reduced by dividing by λ. The reduced
contact angle α and the surface tensions for all
curvature is therefore
three phases of water:
1
γ IA = γ IW + γ WA cos α
RIW ≡
(B1)
r .
(B5)
λ IW
in which γ is the surface tension and the subscripts
Substituting B3 and B5 into the right hand side
I, W and A are for ice, water and air, respectively.
of B4 determines the reduced pressure by dimen-
Figure B1 is a schematic of the equilibrium con-
sional equivalence:
figuration and force diagram about the point of
λ
mutual contact for all three interfaces. Since cos
P≡
(B6)
p
γ IA
α is already dimensionless, eq B1 can only be made
dimensionless by dividing through by one of the
giving the reduced form of the Laplace surface
surface tensions. A reasonable choice is γIA with
tension equation:
the largest magnitude, which results in the re-
ΓIW
duced Young's equation
PI - PW = 2
.
(B7)
RIW
1 = ΓIW + ΓWA cos α
(B2)
At thermodynamic equilibrium the chemical
where the reduced surface tensions are
potentials of ice and water are equal and are ex-
14
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