Models of diffusion-limited uptake of trace elements in fossils and ...

Trace element uptake in fossils? 3769
Once a fossil has recrystallized and coarsened, trace
elements may be immobile on Myr timescales (Trueman
and Tuross, 2002), so intracrystalline trace element transport
through the recrystallized apatite must be quite slow.
Rather, transport of a trace element from the surface to
the recrystallization front probably occurs through an
interstitial aqueous medium, taking advantage of a fossil’s
porosity, even after recrystallization. The concentration
of trace elements in the fluid is small compared to
the recrystallized material, and diffusion rates of aqueous
species are quite high. Even accounting for the lowest
likely porosities of a few percent, the diffusivity of the
trace element, D, is much larger than the rate of movement
of the recrystallization front. These considerations
yield a small composition gradient through the recrystallized
material (plus interstial fluids), and a sharp composition
front coinciding with the recrystallization front
(Fig. 1).
The trace element concentration at the outer surface of
the material (solid plus interstitial fluid) can be assigned a
value of C 1 (a constant) and the concentration at the front
(at position x) is C x (also constant); the concentration in the
pristine material is assumed to be 0. This relation allows
solution of a secondary function (g) ofk, based on Eq.
13.18 of Crank (1975):
gðk=2D 1=2 Þ¼ðC 1 C x Þ=C x ðA:2Þ
where D * is the effective diffusion coefficient of the trace element
through the recrystallized material. Assuming diffusion
through a porous medium, D * is given by:
D ¼ D U=ðK d s 2 Þ
ðA:3Þ
where D is the diffusion rate in water, K d is the partition
coefficient between phosphate and water, U is porosity,
and s is tortuosity. D * differs from D eff in accounting explicitly
for porosity and tortuosity. Whereas K d does affect the
rate of movement of the recrystallizing front, it does not affect
the composition gradient through the recrystallized
material, given by (C 1 C x )/C x and controlled by the diffusion
rate in the fluid (D).
Because D is large in the fluid, composition gradients are
small and g(k/2D * 1/2 ) approaches 0, implying that k/2D * 1/2
also approaches 0 (Fig. 13.6 of Crank, 1975). In the limit,
Eq. 13.13 from Crank (1975) reduces to:
gðk=2D 1=2 Þpk 2 =8D
ðA:4Þ
Thus, for expected conditions during fossilization, combination
of Eqs. (A.2) and (A.4) gives:
k f½ðC 1 C x Þ=C x Š½8D =pŠg 1=2 ðA:5Þ
Substituting into Eq. (A.1) and squaring yields linearity
between X 2 and Dt:
X 2 ¼½ðC 1 C x Þ=C x Š½8D =pŠt ðA:6Þ
APPENDIX A. SUPPLEMENTARY DATA
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/j.gca.2008.
05.045.
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