Available online at www.sciencedirect.com
Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
www.elsevier.com/locate/gca
Models of diffusion-limited uptake of trace elements in fossils
and rates of fossilization
Matthew J. Kohn*
Department of Geosciences, Boise State University, Boise, ID 83725-1535, USA
Received 18 March 2008; accepted in revised form 23 May 2008; available online 5 June 2008
Abstract
Many fossils are assumed to take up trace elements by a process of combined diffusion plus adsorption (DA), yet in principle composition profiles can be explained by several different diffusion-limited processes, including diffusion plus reaction or
recrystallization (DR) and double-medium diffusion (DMD). The DA and DMD models are supported by REE and U composition profiles across fossil teeth, measured by laser-ablation ICP–MS, that show error-function – like diffusion profiles into
enamel from the dentine–enamel interface and concentrations in the interior of enamel that are at original biogenic levels or
higher. Published composition and age profiles in some Pleistocene bones may be better explained by a DR model. All three
diffusion models imply linear behavior between age and distance squared, vastly simplifying U-series dating methods for Pleistocene fossils. Modeled uptake rates for fossil teeth yield a strict minimum bound on durations of about one decade to one
century. The similarity of diffusion profiles in teeth, irrespective of depositional ages ranging from 30 ka to >30 Ma, implies
that uptake occurred quickly, with a maximum duration of a few tens of kyr for typical fossil enamel; faster uptake is implied
for typical fossil bone and dentine. Disparities in these uptake estimates compared to some archeological bone may reflect
sampling and preservation bias for paleontological vs. archeological materials.
Ó 2008 Elsevier Ltd. All rights reserved.
1. INTRODUCTION
Fossilization of bones and teeth occurs with profound
changes in chemistry and structure, including orders of
magnitude increases in REE and U, and massive recrystallization of original biogenic Ca-phosphate crystallites (see
summaries of Kohn and Cerling, 2002; Trueman and Tuross, 2002). Yet trace element uptake mechanisms remain
obscure. Diffusion-limited uptake is supported for many
fossils (Millard and Hedges, 1996), and commonly the process is assumed to combine diffusion of trace elements with
their adsorption onto crystallite surfaces (DA; Millard and
Hedges, 1996). However, in principle other processes
involving diffusion of trace elements may occur, including
diffusion plus reaction (DR) and double-medium diffusion
(DMD).
*
Fax: +1 208 426 4061.
E-mail address: [email protected]
0016-7037/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.gca.2008.05.045
Mechanisms and rates of trace element uptake are geologically important for several reasons. Firstly, U-series
dating of fossils is based on models of uptake of U during
fossilization. To interpret age of initial uptake (i.e., the best
estimate of the depositional age of the fossils) within the
context of diffusion models, researchers must be able to verify that models are indeed consistent with observed concentration and isotope profiles. Secondly, some researchers use
the geochemistry of fossil bone to infer paleoclimate (from
stable isotopes; Kohn and Law, 2006; Zanazzi et al., 2007),
provenance (Trueman and Tuross, 2002; MacFadden et al.,
2007) and paleoceanography (Staudigel et al., 1985; Elderfield and Pagett, 1986; Martin and Haley, 2000). These approaches assume rapid (c. 100 kyr or less) timescales of
fossilization. Verifying rapid fossilization would support
these types of studies, whereas evidence for protracted fossilization would diminish their usefulness.
Finally, observations from trace elements help address
why fossils occur at all in the geological record, and the
completeness of the fossil record. If fossilization rates are
Trace element uptake in fossils?
too slow, all biogenic materials will degrade before they are
preserved, which contrasts with the observed occurrence of
fossils. Conversely, if rates are too fast, all materials will be
preserved, which contrasts with an incomplete fossil record.
Yet quantifying ‘‘fast” vs. ‘‘slow” has proved surprisingly
difficult. Preservation of soft materials under extraordinary
conditions has been proposed to require weeks (Briggs and
Kear, 1993). In contrast, U-series dating of bone suggests
timescales of many tens of kyr for Pleistocene samples
(e.g., Pike et al., 2005), although some bones degrade in
soils in years or decades (Trueman et al., 2004). Diffusion
profiles, such as discussed here, and U-series dating of Pleistocene materials can be used to infer rates of fossilization,
although in some cases, only limits are provided.
In this paper, various diffusion models are first developed theoretically to provide a broad mathematical framework for interpreting trace element data. These models are
considered in terms of age distributions because their largest impact is likely on age determination from geologically
young materials via U-series dating. Second, new data are
presented for fossil teeth ranging in age from 28 ka to
>30 Ma, showing DA and likely DMD in fossil enamel. Finally, rates and mechanisms of fossilization are considered
for typical fossils and Pleistocene samples that exhibit profiles in trace elements that are consistent with a DA model,
or U-series ages that are most consistent with a DR model.
2. WHAT IS ‘‘FOSSILIZATION?’’
The term ‘‘fossilization” is not strictly defined, but field
identification of fossils is generally based on lithologic
associations including age, qualitative measures of density/organic content, and color (commonly dark stained);
carbonate fluor-apatite (francolite) dominates the mineralogy of most fossil bones and teeth. Textural and mineralogical features of fossils in turn reflect three major processes:
degradation of interstitial organic matter, precipitation of
dark interstitial oxides and oxyhydroxides, and uptake of
trace elements. For this discussion, (compositional) alteration and trace element uptake are used synonymously,
and generally refer to the change in chemistry of a fossil relative to its original biogenic precursor, as typically reflected
by the increase in concentrations of F, REE and U (e.g.,
Trueman and Tuross, 2002) and loss of CO3 2 (Wright
and Schwarcz, 1996). Fossilization of bones and some parts
of teeth also is commonly accompanied by a fourth major
process – massive crystal coarsening/recrystallization (Ayliffe et al., 1994), presumably because of Ostwald ripening
and/or dissolution-reprecipitation.
In general all four processes – protein degradation, secondary mineral precipitation, trace element uptake/alteration, and recrystallization – are viewed as common
constituents of fossilization and as proceeding simultaneously, mainly for thermodynamic, kinetic and surface
energetic reasons (Kohn and Law, 2006). In brief, bone
and dentine crystallites are physically and chemically unstable outside their collagen (protein) matrices. For example
an isolated crystallite of bone apatite exposes approximately 50% of its atoms on its surface (Eppell et al.,
2001), and solubility of pristine biogenic apatite may be
3759
5–10 orders of magnitude higher than its F-rich, CO3 2 poor fossil counterpart (Elliott, 2002; Rakovan, 2002; Berna et al., 2004; see also Driessens and Verbeeck, 1990). The
absence of significant OH in the hydroxyl site of modern
bone apatite probably enhances its solubility greatly compared to stable geological hydroxyl- or fluor-apatites (Berna et al., 2004; Pasteris et al., 2004). Thus, degradation of
organic matter must either promote crystallite dissolution
or, if a fossil is to be preserved, recrystallization and chemical alteration.
High porosities and permeabilities in modern bone and
dentine along pores, tubules, and crystallite-protein interfaces promote rapid transport of material, both ingress of
trace elements from surrounding soils and soil waters,
and egress of degraded organic matter. Yet, coarsening of
crystallites, loss of organic constituents, and blocking of
pore spaces with secondary minerals must ultimately limit
and even shut down this process. That is, initial fossilization must promote marked resistance to further chemical
and physical alteration. Trace element ‘‘fingerprints,” in
the form of distinctive REE patterns or trace element ratios, are preserved in fossils over millions of years and support this view (see Trueman and Tuross, 2002), although
outer surfaces of fossils may be affected by aerial exposure,
and undergo dissolution and/or chemical alteration (e.g., U
solubilities are quite dependent of fO2). Thus, external surfaces could develop compositional complexities due to
changing boundary conditions after fossilization, but interiors and internal boundaries such as bone–dentine or dentine–enamel are expected to present chemically and
interpretationally simpler systems: they are less susceptible
to later uptake/alteration, so they preserve compositions
reflective of initial fossilization. This view is supported by
comparable trace element profiles in enamel next to dentine
from fossil teeth that differ in age by 3 orders of magnitude
(this study).
Focusing on cation chemistry, modern biogenic phosphates have quite low concentrations of REE and U (c.
10 ppb or less; Kohn et al., 1999). In contrast, many fossil
bones exhibit high U and REE contents (up to 100s of ppm;
see Trueman and Tuross, 2002). Uranium concentrations
can be quite homogeneous on scales of mm’s (e.g., Millard
and Hedges, 1996; Janssens et al., 1999; Pike et al., 2005),
whereas gradients in REE have sometimes been observed,
e.g., samples from Olduvai Gorge (Henderson et al.,
1983; Williams et al., 1997; Janssens et al., 1999; Trueman
et al., 2006). Compositional gradients on bone edges contrast with homogeneous REE and U in the interior of some
of these same samples (Janssens et al., 1999), and together
these data imply compositional homogenization during initial fossilization, with some susceptibility to overprinting on
surfaces either during the late stages of fossilization, or perhaps after fossilization. Note that uptake of U millions of
years after deposition is also implied in the age systematics
of some materials (e.g., see Peppe and Reiners, 2007),
although the location of this ‘‘young” U is not well documented. High U concentrations have inspired numerous
U-series chronologic investigations, and several studies
demonstrate older U-series ages towards the external and
internal surfaces of cortical bone relative to its interior
3760
M.J. Kohn / Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
elements simultaneously diffuse through the bone matrix,
and are immobilized by adsorption onto bioapatite crystal
surfaces (Fig. 1A). The DA model explains systematic decreases in trace element concentration towards the interiors
of some fossil bone, and REE partition coefficients putatively more consistent with adsorption rather than intracrystalline equilibrium (Koeppenkastrop and De Carlo,
1992). However, although DA models are commonly used
in interpreting U-series ages in bone, they conflict with observed recrystallization and grain-size coarsening in fossil
bone and dentine (Ayliffe et al., 1994). Instead, DA is conceptually most applicable to enamel, where grain sizes do
not appreciably increase during fossilization.
DR describes better the recrystallization and grain-size
coarsening observed in many fossils, and the expected concomitant immobilization of trace elements (Fig. 1B). For
short times, however, the DR model predicts an abrupt
chemical front that is not matched by some Pleistocene
materials.
The DMD model assumes that fast-diffusion pathways
bound domains exhibiting slow-diffusion rates (Watson,
1991; Wang, 1993; Fig. 1C); DMD is identified experimentally from relatively high-concentration ‘‘tails” on diffusion
profiles far in the interior of a medium compared to the expected profile from a single medium diffusive model. For
example, Fig. 1C shows the expected profile for a simple
DA model (dashed line), and the higher than expected concentrations far from the boundary that are supported by
the second, fast-diffusion pathway. These tails are distinct
from diffusion profiles predicted for penetration from two
sides of a material, e.g., simultaneously from the internal
and external surfaces of cortical or compact bone. DMD
is consistent with nm-scale apatite crystallites and lm-scale
bundles (presumed slow-diffusion domains) separated by
interstitial pores and organic complexes (presumed fast-diffusion pathways). Although intracrystalline diffusion rates
in apatite (e.g., Cherniak, 2000) might appear too sluggish
to permit significant uptake by volume diffusion, invalidat-
(e.g., see Millard and Hedges, 1996; Pike et al., 2001, 2005).
These ages indicate that U is first taken up in bone at its
interface with sediment or fluid, and preferentially immobilized via adsorption and/or recrystallization.
Fossil teeth have received less attention, but dentine
shows close similarities to bone in its physical and isotopic
response to fossilization (Ayliffe et al., 1994; Kohn and
Law, 2006) and one previous study revealed uniformly high
U concentrations (several tens of ppm) across fossil dentine
(Eggins et al., 2003). Enamel appears quite resistant to
alteration of stable O and C isotopes (Kohn and Cerling,
2002), presumably because coarse crystallites resist recrystallization (Ayliffe et al., 1994); its trace element chemistry,
however, can be radically altered (Kohn et al., 1999). The
mechanisms for trace element uptake in enamel are unclear,
but in the absence of recrystallization, some form of
adsorption or intracrystalline diffusion would appear
important. One previous study has shown steep concentration gradients in enamel adjacent to dentine (Eggins et al.,
2003), but the numerical consistency of these gradients with
diffusion was not explored.
3. DIFFUSION MODELS
Mechanistically, three endmember models of diffusive
uptake may be considered: diffusion–adsorption (DA), diffusion-reaction (DR) and double-medium diffusion (DMD;
Fig. 1). These models mutually differ both in physical
behavior and mathematical treatment. Emphasis is placed
here on bone, both because of its historical importance in
U-series dating of archeological materials, and because
the most comprehensive diffusion model (DA) was developed for this material (Millard and Hedges, 1996).
3.1. Conceptual description
Trace element uptake in Pleistocene bone is commonly
modeled by DA (Millard and Hedges, 1996), in which trace
A
DA
B
DR
C
DMD
Solid + fluid
C
C
C
Reacted Unreacted
Solid
Solid
x
x
DMD
profile
DA
profile
x
Fig. 1. Schematic of diffusion models and corresponding trace element profiles for short timescales. (A) Diffusion–adsorption, showing simple
error-function profile. (B) Diffusion-reaction, showing step in concentration between recrystallized and unrecrystallized material. Dashed line
shows contribution of interstitial fluid. (C) Double-medium Diffusion, showing two separate error-function—like profiles, representing the
two different diffusion pathways. Dashed line is schematic result for DA model.
Trace element uptake in fossils?
ing DMD, faster diffusion rates have been documented for
some elements in the near surface (outer few nm) of calcite
crystals (Stipp et al., 1992). Because the bioapatite that
composes fossils is nanocrystalline, even after fossilization,
relatively fast intracrystalline diffusion may permit DMD.
Given sufficient time, all three models can explain homogeneous compositions documented in some fossils, and all
three processes might occur simultaneously. The central
points of the following discussion are to demonstrate that
(1) the exact fossilization process is irrelevant to dating
problems, as long as boundaries were simple and trace element uptake was limited by diffusion, and (2) appropriate
mathematical treatment of data simplifies interpretation
and error analysis.
3.2. Mathematic treatment
All solutions to the diffusion equation can be expressed
in terms of the dimensionless variables reduced time (t0 ) and
0
reduced distance squared (X 2). For this paper, t0 = Defft/l2
and X0 = X/l, where Deff is the effective diffusion coefficient,
t is time, l is the half-width of the tissue considered, and X is
distance. Some workers prefer t0 = [Defft/l2]1/2. The effective
diffusion coefficient (Deff) is the ratio of the tracer diffusion
rate (D) in a fluid and the partition coefficient between biogenic apatite and diagenetic or pedogenic fluids (Kd), corrected for porosity (r) and tortuosity (s), i.e., Deff = Dr/
Kds2. Note that in applications discussed below Deff is a
measured value, so it automatically accounts for contributions from porosity and tortuosity, and that Kd, r, and s
are presumed constant, so solutions to the diffusion equation scale equivalently with either D or Deff. If material diffuses from a single (planar) boundary, e.g., from dentine
into enamel across the dentine–enamel interface, and/or
diffusion profiles are short (t0 is small) then a semi-infinite
solution is used, and X0 is expressed as the distance from
that surface or boundary (where X0 = 0). In contrast, if
1.0
3761
material diffuses between 2 parallel interfaces, e.g., inward
from both the interior and exterior surfaces of cortical
bone, then a plane sheet solution is used, and X0 is expressed as the distance from the midpoint between the
two boundaries: X0 = 0 is the center of the sheet, and
X0 = 1, 1 are the boundaries.
Most models assume a fixed boundary condition, i.e.,
that the concentration at any bounding surface (Co) is fixed
at time t = 0. The initial concentration of the diffusing species (e.g., U, REE etc.) in the fossilizing tissue is also commonly assumed to be zero, and although this assumption is
not required, it closely approximates biogenic compositions
(e.g., see Driessens and Verbeeck, 1990; Kohn et al., 1999).
Mathematically all models predict the change in trace element concentration (C), or its normalized concentration,
(C0 , where C0 = C/Co), as a function of reduced distance
(X0 ) and reduced time (t0 ). Solutions to the diffusion equation were obtained from Crank (1975), specifically Eqs.
3.13 (semi-infinite medium solution for modeling short diffusion profiles in enamel), either 2.54 or 2.68 (plane sheet
solutions for modeling long diffusion profiles in bone),
and 13.13 plus 13.18 (semi-infinite medium solution for
modeling DR).
In this discussion, the term t0 Dep is used to describe the
time since initiation of the boundary condition and is equivalent to the total duration, or t0 , of a model. In practice,
t0 Dep is proportional to the true depositional age, with a
proportionality constant of Deff/l2. The term t0 app is used
to describe the apparent age at position X0 within a sample.
In practice, t0 app is proportional to the measured age with
the same proportionality constant, Deff/l2.
Examples for the DA plane sheet model show increases
in trace element concentration over time, with the rate of
increase dependent on distance from the boundaries and
time (Fig. 2A). These concentration profiles must be converted to an age. For short diffusion profiles, e.g., REE in
enamel, this may be accomplished by fitting an inverse
1.0
1.0
t'Dep = 1.0
X’
=
.9 1.0
.9
0.8
X’
.7
0.9
0.6
X’ X’
X
=0 =0 ’=0
.3 .5
.7
0.8
=0
0
X’=
0.8
0.4
∫ [ C/Co ] dt'
0.0
0.0
t'app =
t'=0
0.4
0.6
Figure
2C
0.2
t'Dep
A
0.2
t'Dep
X' 2
X’
t'True
0.2
1X'
0.7
=0
0.4
1-
t'app
.1
t'app
X’
C/Co
0.6
X’
=0
=0 .5
.
X’
3
=0
.1
=0
X’
0.8
1.0
0.0
0.0
0.6
B
0.2
0.4
0.6
t'Dep
0.8
1.0
C
0.5
0.0
Edge
0.2
0.4
0.6
Distance
0.8
1.0
Core
Fig. 2. Numerical results of plane sheet DA model for U uptake over a range of timescales. (A) Concentration vs. time for different distances.
Boundary is at X0 = 1, and has constant composition Co throughout. Apparent age is determined by integrating concentration curve (in terms
of C/Co) at a particular value of X0 over duration t0Dep , and ratioing to t0Dep . (B) Apparent age ðt0app Þ vs. depositional age ðt0Dep , or age of initial
establishment of fixed boundary condition). At boundary (X0 = 1) apparent and depositional ages coincide, so the slope is 1. For long times,
the slope of t0app for all positions approaches 1 as U is saturated. Circle shows integrated composition corresponding to gray region in Fig. 2A.
Box shows data points plotted in Fig. 2C. (C) t0app vs. distance, showing quadratic form with respect to distance, and linearity with respect to
0
0
distance squared. Note that any function that is linear in X 2 is also linear in 1 X 2.
3762
M.J. Kohn / Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
error-function to the observed data. For U-series data,
however, t0 app is determined by integrating U concentration
at that point over the duration of the model (t0 Dep; Fig. 2A
and B). Note that if normalized concentrations are used, C0
at the boundary is 1.0, and the integrated area at X0 = 1 is
identically t0 Dep. These t0 app ages require that U is immobilized upon uptake, either by recrystallization (DR), or
adsorption onto apatite crystallites (DA and DMD). In essence, any U that accumulates at position X0 may no longer
diffuse but instead is subject to decay.
All three DA, DMD, and DR models yield linear t0 app
distributions with respect to the square of the distance
0
(either X2 or X 2):
t0app ¼ kX 02 þ b
ð1Þ
where k and b are constants. This conclusion is numerically
demonstrated for plane sheet DA models (Figs. 2 and 3),
and is explicit in solutions to the DR model (Crank,
1975; see derivation in Appendix). Specifically, for the
DR model, concentrations at any point X0 are zero, until
time t0app , when the front passes this position. Because the
0
front progresses proportionally to X 2, so too must t0app .
For DMD, both segments are functionally equivalent to
the DA model (Zhang et al., 2006), albeit with different Deff,
so DMD solutions should yield two different, but linear seg0
ments on a plot of t0 app vs. X 2, each implying the same t0 Dep.
For the plane sheet DA models (Figs. 2 and 3), numerically calculated values of t0app for different values of t0Dep and
A
0
X0 were regressed assuming linearity with respect to X 2.
This model is widely applied to interpret U uptake in fossil
bone (Millard and Hedges, 1996), and closely approximates
the expression:
0
ð2Þ
or equivalently for t (measured age) and X (measured
distance):
2
tapp tDep þ
0:5ðe2:25tDep 1:0Þðl
D
X 2 Þ
ð3Þ
0
Note that a function that is linear in X 2 must also be linear
0
in (1 X 2), and that these relationships are least accurate
0
for short times, where the distribution of t0 app vs. X 2 is least
linear (Fig. 3B). Analogous expressions apply to either of
the two segments of the DMD model.
For the DR model, assuming diffusion through a porous
medium, the relation in terms of X and t is (see Appendix):
X 2 ¼ ½ðC 1 C x Þ=C x ½8Deff =p t
ð4Þ
where C1 and Cx are constants that describe compositional
concentrations at the boundary and at the recrystallizing
front, respectively.
Thus, regardless of diffusion mechanism, ages derived
from U-series or other means can be regressed vs. (reduced)
distance squared to infer the age of initial trace element uptake, with the slope reflecting other properties, notably D.
This result applies both to dimensionless time or to actual
t'app ~ t'Dep + 0.5(e-2.25t'Dep - 1.0)(1-X'2)
B
t'Dep=1.75
1.00
2
t0app t0Dep þ 0:5ðe2:25tDep 1:0Þð1:0 X 0 Þ
1.75
t'De
t'Dep=1.0
1.50
p =1.7
5
0.80
t'D
1.25
ep =
0.4
.
=0
p
e
t' D
Near-edge
U loss
t'app
C/Co
0.60
t'De =
p 1.0
67±9 ka
0.75
1
0.40
1.00
0.50
SWC
U-se 3
ries
t'Dep=0
.4
0.20
0.25
t'Dep=0.1
= SWC3 [U]
0.00
0.0
Edge
0.2
0.4
0.6
1-X'2
0.8
1.0
Core
0.00
0.0
Edge
0.2
0.4
0.6
1-X'2
0.8
1.0
Core
Fig. 3. Normalized plots of trace element (U) uptake and age determinations vs. reduced distance squared for plane sheet DA model (solid
symbols), illustrating linear behavior over large range of timescales. Natural data (open symbols) are normalized U-series dates from a
Pleistocene bone (Pike et al., 2005). (A) Normalized concentrations. With increasing t0Dep , concentrations become essentially constant across
tissue; steep concentration profiles are realized only for low values of t0Dep . Homogeneous compositions from Pleistocene bone (excepting nearedge U loss) require t0Dep P 1. (B) t0app . Slope steepens between t0Dep ¼ 0 to 1.0, as U is taken up and develops a chemical diffusion profile.
For t0Dep > 1:0, U is saturated (concentration profiles are flat) and slope does not change (see Fig. 2B). Data for natural sample SWC3
assume Deff = 4 1014 cm2/s; linearity is consistent with diffusive uptake, but DA model cannot simultaneously explain steep slope (low t0Dep )
and homogeneous U across bone (high t0Dep ). See text for discussion. Age and 95% confidence limits based on weighted least squares
regression. MSWD (or RMSE) = 1.5, suggesting data scatter is largely explained by analytical uncertainties.
Trace element uptake in fossils?
Cerium
100
Dentine
“in”
“wash-in”
“in” p
rofile
Ce, Yb (ppm)
1000
fi le
” pro
“out
ut
washo
4. DIFFUSION PROFILES IN ENAMEL
10000
“out”
measured ages because t0app involves only measured age
(tapp) and two parameters that are both presumed constant
(D and l). Note that all models are formulated assuming
that inward diffusion of U limits uptake, but in reality uptake may instead be limited by the breakdown of collagen
and consequent exposure of crystallites. Regardless, if
either collagen breakdown or its outward transport is controlled by diffusion, and if interstitial fluids are saturated in
U, then the same functional dependencies are expected, because the problem can be viewed as the inward diffusion of
sites available to U uptake.
The linearity of age with respect to distance squared
vastly simplifies interpretations of the timing of trace element uptake. Simple linear regressions of measured ages
can be performed, with straightforward assessment of quality of fit, and propagation of errors to determine age uncertainties. Such an approach directly evaluates whether a
dataset is consistent with a trace element uptake process
that is limited by diffusion, as non-linear behavior implies
more complex, potentially non-diffusive processes. Examples are discussed in subsequent sections.
3763
Enamel
Ytterbium
10
1
Enamel
0.1
0
100
Dentine
200
300
Enamel
400
500
600
Distance (m)
Fig. 4. Logarithm of concentration vs. distance for trace element
profiles from enamel into dentine (‘‘in” profile) and back into
enamel (‘‘out” profile), showing high and relatively uniform
concentrations of REE in dentine, and steep drop offs into enamel.
Logarithmic scale approximately linearizes diffusion profiles
(straight lines shown for Ce). Dashed lines shows typical ‘‘washin” curve (for the ‘‘in” profile), and washout curve (for the ‘‘out”
profile). Sample is a 33.2 Ma tooth of Leptomeryx evansi, a small
deer-like artiodactyl.
4.1. Analytical methods
Samples of fossil teeth were collected from strata in NW
Nebraska, and obtained on loan from the Idaho Museum
of Natural History (IMNH) and from Hagerman Fossil
Beds National Monument (HAFO). The Nebraska samples
are 33.2 Ma (Zanazzi et al., 2007), and include a primitive
rabbit (Paleolagus sp.), a small deer-like mammal (Leptomeryx evansi), and a medium-sized primitive artiodactyl
(Leptauchenia sp.); these taxa were chosen because their
tooth enamel thicknesses range from 75 lm (Paleolagus
sp.) to 500 lm (Leptauchenia sp.). The IMNH and HAFO
samples are all Equus sp., which have enamel thicknesses up
to 1 mm; the IMNH samples are from the Sangamonian
interglacial (c. 125 ka), and from strata 14C dated to between 22 and 33 ka (Jefferson et al., 2002), whereas the
Hagerman samples are 3.2 ± 0.1 Ma (Hart and Brueseke,
1999), i.e., from the mid-Pliocene climatic optimum. For
the IMNH and HAFO Equus samples, strips of enamel plus
dentine, a few mm wide and several mm deep were cut
along the length of each tooth using a slow-speed microsaw.
Each strip was subsampled every 1.5 mm, yielding a fragment of enamel plus dentine a few mm2 in area. These fragments were mounted in a 1” epoxy round, and polished.
For the Nebraska samples, strips of enamel plus dentine
were cut from each fossil and mounted in epoxy, without
subsampling.
Analyses were collected by using a single-collector,
inductively coupled plasma, mass spectrometer (ICP–MS),
housed at the GeoAnalytical Laboratory, Washington
State University. A frequency quintupled Nd:YAG laser
(New Wave Research UP-213; k = 213 nm) operating at
10 J/cm2 was used to ablate the sample along traverses
8–12 lm wide and several mm long (Fig. 4); a He stream
(1.3 l/min) delivered ablated material to the plasma
source of a ThermoFinnigan Element2 magnetic-sector
ICP–MS. Masses ranging from 43 to 238 were measured,
most importantly 43Ca (for estimating absolute concentrations), in addition to Ba, Sr, REE, and U, with count times
of 0.01 s. Detection limits were 10 ppb, comparable to
REE and U contents of modern enamel (Kohn et al., 1999).
Washout of previous high-concentrations is a special
concern for these analyses, although only in traverses from
high-concentration dentine to low concentration enamel
(‘‘out” profiles). Profiles that cross a sharp boundary from
low- to high-concentration materials (‘‘in” profiles), such as
epoxy-enamel or expoxy-dentine, are expected and observed to have minimal ‘‘wash-in” (P2 orders of magnitude
changes in one 2 s mass scan; Fig. 4). Washout for ‘‘out”
profiles was estimated by stepping over the presumed sharp
boundary between the sample and epoxy. These traverses
showed intensity decreases of 1 order of magnitude over
2 mass scans (4 s; Fig. 4). Many measured ‘‘out” profiles
show weaker gradients in concentration, e.g., by factors
of 2–4 over 2 mass scans, indicating that washout contributes about 30% to the apparent ‘‘out” profiles (Fig. 4). This
washout effect would typically contribute a 10–20% bias
to diffusion calculations (at most a factor of 2). These errors
do not significantly affect interpretations and were not corrected. Short washout times for these analyses may reflect
focusing of the He stream over the center of the sample
holder (where samples were mounted) and/or aerodynamics
dependent on the depth and width of the ablation path.
Washout times reported here should not be assumed representative of the edges of the holder, where dead space may
impede flow, or for different analytical protocols and
materials.
Compositions were standardized against NIST612 glass,
by using trace element intensities ratioed to calcium.
NIST612 contains 8.5 wt% Ca, and 40 ppm trace elements. In comparison, samples were assumed to contain
3764
M.J. Kohn / Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
40 wt% Ca (stoichiometric biogenic calcium phosphate),
and apparent REE and U concentrations ranged from <1
ppm to 1000 s of ppm, with precisions of c. 5–50 ppb. Because of different matrices for standards vs. samples, accuracy of compositions is likely much poorer than precision,
perhaps grossly so. However, because matrix compositions
are nearly pure Ca-phosphate, only precision is relevant to
inversion of diffusion profiles. This latter assumption was
verified by collecting X-ray maps of Ca with a Cameca
SX50 electron microprobe housed at the Electron Microscopy Center, University of South Carolina. Maximum differences between enamel and dentine were <5% (relative)
and would insubstantially change apparent concentration
profiles of trace elements and their interpretation.
4.2. Results
Composition profiles show high and relatively uniform
concentrations of trace elements in dentine, with steep drop
offs into enamel (Fig. 4, Table 1, and Electronic Annex).
Qualitatively similar profiles for U have been reported for
Pleistocene teeth (Eggins et al., 2003; Grün et al., 2003).
Similarly shaped profiles occur on the outer surfaces of enamel, but because of surface alteration, and lack of homogeneous material bounding the surface, boundary
compositions and interface positions could not be accurately determined. With one exception, relatively high-concentrations (100’s of ppb to a few ppm) of REE and U
prevailed in the interior of the enamel, well above original
biogenic concentrations (10’s of ppb). These observations
are similar to those of Eggins et al. (2003), and have implications for the different diffusion models.
Composition profiles may be inverted (Fig. 5, Tables 1,
and 2) to infer the integrated history of diffusion coefficient
(D) over the time during which the profile formed (t), i.e.,
apparent Dt. For such short profiles, this inversion assumes
a planar geometry with a fixed boundary composition (Co)
for each element:
Dt ¼ fX =erfc1 ðC=C o Þg2 =4
ð5Þ
1
where erfc is the inverse error-function, and C is concentration at distance X from the dentine–enamel interface.
This equation assumes a semi-infinite medium, a single diffusion pathway (e.g., DA or simple volume diffusion, not
Table 1
Typical composition profiles for fossil dentine–enamel interfaces
Distance (lm)
C(La)
C(Ce)
C(Yb)
C/Co (ave)
erfc1
Dt (+0 lm) (cm2)
Dt (+5 lm) (cm2)
Dt (+10 lm) (cm2)
Leptauchenia-A
0 (dentine)
5.0
14.5
24.0
33.5
43.0
52.5
62.0
71.5
81.0
2824
1906
1690
911
125
138
85
87
23
14
4393
3452
2644
1621
270
213
123
192
35
19
177
147
126
79
19
11
6
10
2
2
0
0.7634
0.6379
0.3797
0.0716
0.0528
0.0315
0.0443
0.0095
0.0060
0.210
0.330
0.620
1.270
1.370
1.520
1.425
1.830
1.950
5.2E-07
5.9E-07
3.1E-07
4.8E-07
6.1E-07
1.0E-06
8.3E-07
9.5E-07
3.5E-07
1.2E-06
9.4E-07
4.3E-07
6.2E-07
7.5E-07
1.2E-06
9.5E-07
1.1E-06
1.4E-06
2.2E-06
1.4E-06
5.7E-07
7.7E-07
8.9E-07
1.4E-06
1.1E-06
1.2E-06
6.6E-07
8.3E-07
1.2E-06
Average
IMNH27691–G2
0 (dentine)
5.00
15.05
25.10
35.15
45.20
55.25
65.30
75.35
85.40
95.45
105.50
115.55
125.60
135.65
145.70
Average
C(U)
C/Co
erfc1
Dt (+0 lm) (cm2)
Dt (+5 lm) (cm2)
Dt (+10 lm) (cm2)
65.000
61.127
57.093
39.213
26.266
14.375
9.773
8.682
9.651
8.864
6.306
4.827
3.780
3.423
3.376
2.800
0.940
0.878
0.603
0.404
0.221
0.150
0.134
0.148
0.136
0.097
0.074
0.058
0.053
0.052
0.043
0.053
0.109
0.368
0.590
0.865
1.017
1.060
1.022
1.055
1.175
1.265
1.340
1.370
1.375
1.430
2.1E-05
7.5E-06
6.5E-06
5.4E-06
6.1E-06
8.1E-06
1.2E-05
1.5E-05
1.5E-05
1.6E-05
1.7E-05
1.9E-05
2.3E-05
2.4E-05
2.2E-05
4.8E-05
1.2E-05
8.9E-06
6.8E-06
7.4E-06
9.5E-06
1.4E-05
1.6E-05
1.6E-05
1.7E-05
1.9E-05
2.1E-05
2.4E-05
2.6E-05
8.9E-05
8.5E-05
1.7E-05
1.2E-05
8.4E-06
8.8E-06
1.1E-05
1.5E-05
1.8E-05
1.8E-05
1.9E-05
2.0E-05
2.3E-05
2.6E-05
2.8E-05
9.1E-06
1.6E-05
3.3E-05
Trace element uptake in fossils?
Leptomeryx
REE profiles
Cerium/100
Ytterbium/10
Mean Ce
50
125
U profiles
B
HAFOpp
27691g
Leptomeryx
78018i
A
100
40
U (ppm)
30
20
75
50
Mean Yb
25
10
0
0
75
100
0
125
25
50
log(time, yrs)
75
100
125
Distance (μm)
2
1
0
-1
-2
2347
50
REE
U
Leptomeryx
HAFOpp U
Leptomeryx Ce
Leptomeryx Yb
27691g U
25
3.2 Ma
3
Leptauch.
=
Dt
0.5
175
26768
x1
=2
Dt
x10 -7
-5
0
1
1x
0
33 Ma
4
1.0
150
D
5
1.5
0.0
125
C
Dt=4
erfc-1(C/Co)
2.0
100
6
0 -6
2.5
75
Distance (μm)
Distance (μm)
27691
50
78018
25
HAFO
0
Paleolagus
Ce(ppm)/100, Yb(ppm)/10
60
3765
Sample
Fig. 5. Typical trace element profiles across fossil dentine–enamel interface for REE (A) and U (B). (C) Profiles plotted as inverse error
function (Dt values in cm2), showing consistency with diffusive uptake. (D) Limits of timescales of fossilization. Ages of fossils (arrows and
boxes) provide maximum estimates of duration of fossilization. Error bars reflect order of magnitude uncertainties in the tracer diffusion and
partition coefficients. Short dot–dash lines bracket likely fossilization times for soft-tissue preservation. Note that all diffusion profiles are
developed over similar length scales, irrespective of the age of each sample. The simplest explanation for this consistency is that diffusion
profiles developed within the first 30 kyr of burial (the duration of burial for the youngest samples), and remained unperturbed thereafter.
DMD), and a negligible concentration in the interior of the
enamel. Thus, Eq. (5) accurately models compositions close
to the dentine–enamel interface, but not necessarily in the
interior of enamel where opposing diffusion profiles can
mutually impinge or where DMD tails might bias calculations. Application of Eq. (5) is supported by strong composition gradients in enamel, much higher original porosity in
dentine vs. enamel, high and relatively uniform concentrations in dentine, and, in other studies (Grün and McDermott, 1994), older ages for dentine compared to enamel.
These latter factors indicate that dentine provides a rapid
diffusion pathway, fixing the boundary condition of enamel
on relatively short timescales. A more complex model, e.g.,
involving progressive increases in the trace element concentrations at the dentine–enamel interface during dentine fossilization, would retard development of profiles in enamel
and affect post-hoc estimation of t, although not the
retrieved estimate of Dt from Eq. (5). Other models (Pike
and Hedges, 2001; Eggins et al., 2003) predict non-uniform
trace element profiles in dentine and differences in U
content of enamel adjacent to dentine, unlike many observations (Eggins et al., 2003) that show uniform trace
element profiles in dentine, and concentrations in enamel
that approach that of dentine at the dentine–enamel
interface.
Dentine analyses were averaged from close to the dentine–enamel interface to obtain Co. Assignment of C(X)
was complicated by the continuous traverse approach used
to collect analyses. Based on the profiles, one analysis could
always be identified that was clearly within enamel, not
dentine, and closest to the dentine–enamel interface. But
because profiles were collected continuously from starting
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M.J. Kohn / Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
Table 2
Composition and modeling results for fossil dentine–enamel interfaces
Sample
Leptauchenia-A
Leptomeryx-A1
Leptomeryx-A2
Leptomeryx-B1
Leptomeryx-B2
Leptomeryx-B3
Paleolagus-A1
Paleolagus-A2
Paleolagus-A3
Paleolagus-A4
Paleolagus-A5
Paleolagus-A6
Paleolagus-A7
Average
Element
La,Ce,Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
La, Ce, Yb
Dt (cm2)
6
4 10
4 106
3 106
2 106
3 106
2 106
7 106
3 106
3 106
2 106
5 106
2 106
8 107
t(yr)
100
100
90
50
100
70
200
100
80
60
200
60
30
Sample
Leptomeryx-A1
Leptomeryx-A2
Leptomeryx-B1
Leptomeryx-B2
Paleolagus-A1
HAFO-pp1
HAFO-pp2
IMNH78018-I1
IMNH78018-I2
IMNH2347-aa1
IMNH2347-aa2
IMNH27691-G1
IMNH27691-G2
IMNH27691-A-3a
IMNH27691-A-3b
IMNH26768-ss
Element
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
Dt (cm2)
6
4 10
1 105
5 106
7 106
2 106
1 105
4 107
6 106
4 106
2 105
1 105
1 105
2 105
3 106
5 106
4 106
96
t(yr)
7
20
8
10
3
20
1
9
7
40
20
20
30
5
7
7
13
Note: Except for averages, estimates of times are given to 1 significant digit. Ages for Leptauchenia, Leptomeryx and Paleolagus are 33.2 Ma,
HAFO is 3.2 Ma, IMNH 78018 and 2347 are 125 ka, and IMNH 27691 and 26768 are 22–33 ka. Minimum concentrations in interior of
enamel for La, Ce, and Yb are 1, 2, and 0.05 ppm, respectively, and for U are 0.5 ppm (Leptomeryx), 0.1 (Paleolagus, IMNH78018
and IMNH2347), 0.03 (HAFO and IMNH27691), and below detection (IMNH26768). Original biogenic concentrations would be 10 ppb
or less for each of these elements (Kohn et al., 1999), indicating that diffusion profiles are longer than the half-width of enamel.
points far from the interface, the offset from the dentine–enamel interface of this analysis was not known a priori. Instead the position offset was assigned values of 0, +5, and
+10 lm from the interface, which is the maximum possible
range considering a traverse rate of 10 lm per 2 s sweep
through the set of analyzed elements. This approach yielded
3 different apparent C–X relations, from which erfc1
(C/Co) values were determined using an analytical approximation of the error-function (Press et al., 1989; Table 1).
Typically 5–10 analyses per profile had concentrations
above 500 ppb and could be modeled for individual Dt
values, although models for REE were limited to values
above 1 ppm to avoid complications arising from possible
DMD behavior at lower concentrations or from inconsistency with a semi-infinite medium assumption. Corrections
for baseline U and REE concentrations in the interior of
enamel propagated to small (<10%) errors in Dt and were
ignored. All Dt values were then averaged for the three considered X-offsets to obtain an overall average Dt value (Tables 1 and 2). Because uncertainties in D propagate to
uncertainties in t exceeding an order of magnitude, no significant improvement in interpretation was attempted in
trying to maximize compatibility among different Dt values
from a profile by using a more specific X offset, or in
attempting to correct Dt values as derived from different
REE and/or U for relative distance in the traverse (because
they are collected sequentially, their relative offset is known
precisely).
Dt values range from about 106 to 105 cm2 (Tables 1
and 2, and Fig. 5) and have precisions of about a factor of
2. This error results almost entirely from uncertainties in exact distances of measurements from the dentine–enamel
interface. As discussed next, if a maximum limit for D
can be assigned, a minimum limit for t may be inferred.
5. A MINIMUM BOUND ON THE RATES OF
FOSSILIZATION
The penetration distance of the trace element profiles,
combined with known effective diffusion rates in modern
enamel, allow estimation of the minimum timescale required to produce the observed profiles. Previous work on
fossil teeth (Toyoda and Tokonami, 1990; Eggins et al.,
2003) used the age of the fossils to infer effective diffusion
rates (Deff) of 1014 to 1018 cm2/s. However, these estimates assumed continuous diffusion of trace elements at a
constant rate since deposition. This approach may strongly
underestimate Deff if the diffusion profiles were produced
more quickly, then preserved via other processes, such as
secondary mineral deposition and occlusion of intercrystalline transport pathways. This type of process must occur to
some degree because, as shown in this study, materials with
radically different ages (e.g., 30 ka vs. >30 Ma) have similar diffusion profiles, and trace element diffusion rates in
unaltered enamel are sufficiently fast that profiles would
homogenize at a scale of 1 mm on timescales of Myr. In
contrast, known Deff rates in modern enamel are used here
to provide a minimum limit on the duration of trace element uptake (t); known depositional ages of the youngest
samples provide maximum limits, thus bracketing the duration of fossilization.
Minimum estimates of t depend critically on the assumed value of Deff, which in turn depends strongly on
the partition coefficient between biogenic apatite and diagenetic or pedogenic fluids (Kd). Mono- and di-valent cations
diffuse through modern enamel at uniform D’s of
1 108 cm2/s, with uncertainties of an order of magnitude (van Dijk et al., 1983). Diffusing species commonly
have monolayers of water that define their effective radius.
Trace element uptake in fossils?
The radius of (hydrated) mono- and divalent cations is
comparable to (hydrated) LREE (+3) and U (+4) cations
(Sandström et al., 2001), so D for REE and U in enamel
is expected to be comparable, i.e., 108 cm2/s. The phosphate-fluid partition coefficient (Kd) is believed to be at least
5 105 for U (Millard and Hedges, 1996) and 1 106
for REE (Koeppenkastrop and De Carlo, 1992), with
uncertainties of about one order of magnitude. Thus Deff
is 62 1014 cm2/s for U and 61 1014 cm2/s for REE
in modern enamel, with uncertainties of about 1 12 orders
of magnitude. These values are 2–4 times lower than some
rates derived from fossil fish dentine (Toyoda and Tokonami, 1990), and the estimated rate for U diffusion in bone
(Millard and Hedges, 1996). Possibly calculated values for
Deff in enamel reflect an adsorption rather than bulk equilibrium distribution coefficient, and both Kd and inferred
minimum t should be higher. Because the purpose of the
inversion is to provide a minimum limit on t, however, use
of the larger coefficients (2 1014 cm2/s for U and
1 1014 cm2/s for REE) is warranted.
Substituting these values of Deff yields minimum estimates
of about a decade to produce the U profiles, and about a century for the REE profiles (Fig. 5 and Table 2). While these
values may not realistically estimate total durations, they assuredly provide a lower limit. Fossilization of dentine and
bone is expected to be many times faster because they have
much higher porosity (Millard and Hedges, 1996), i.e., minimum limits are on the order of years to decades. Large differences in fossilization rates among samples might be expected
from different physical and microbial environments attending fossilization and trace element uptake in different materials. Yet all samples yield rather similar results for either REE
or U profiles, irrespective of age. If trace element uptake in
these samples was governed by some common process, then
the fact that the diffusion profiles in the 630 ka samples are
no shorter than older samples implies that the older samples
probably developed their diffusion profiles in 630 kyr, and
that fossilization of dentine and bone occurred at least as
quickly. This result supports other types of studies that assume relatively rapid (6100 kyr) alteration/uptake in bone
and dentine (e.g., Staudigel et al., 1985; Elderfield and Pagett,
1986; Martin and Haley, 2000; Trueman and Tuross, 2002;
Kohn and Law, 2006; MacFadden et al., 2007; Zanazzi et
al., 2007). Uncertainties in partition coefficients and hence
Deff could reconcile the REE- and U-derived estimates. Generally shorter penetration distances for U compared to REE
suggest lower Deff for U than for REE, in contrast to published Kd’s that suggest the opposite.
Calculated durations are strict minima for two reasons.
First, as described previously, Kd may be underestimated,
resulting in too large an assumed Deff. Second, preservation
of the diffusion profiles requires some mechanism for
retarding Deff to values lower than measured in pristine enamel, otherwise fossils older than 1 Ma would never preserve diffusion profiles. Occlusion of pore space via clays,
metal oxides, and oxyhydroxides (Kohn et al., 1999) must
ultimately shut down diffusive exchange of the enamel interior with its margins. Therefore, the age of the youngest fossils, 22–33 kyr, limits the maximum duration. The inferred
rates (>10–100 yr, but <30 kyr), dramatically exceed esti-
3767
mates for soft-tissue preservation, and fall short of many
estimates for Pleistocene bone. The consistent behavior of
trace elements in enamel in many fossils of different ages
and locations suggests that most typical fossils take up trace
elements and presumably recrystallize at rates encompassed
by these limits, although more comprehensive consideration
of other samples and depositional settings is needed. Systematic differences between archeological samples vs. older
fossils may point to preservational and/or sampling differences, i.e., archeological samples may not be representative
of fossils that are preserved on longer timescales.
6. MECHANISMS OF FOSSILIZATION
The presence of diffusion profiles in enamel rules out a DR
mechanism (alone), which produces a step in compositions
and is wholly inconsistent with measured data. This interpretation accords with independent crystallographic (Ayliffe et
al., 1994) and isotope data (summarized in Kohn and Cerling, 2002) that suggest that original enamel crystallites do
not undergo major reaction or recrystallization during fossilization. In contrast, either a DA or a DMD model may be
appropriate. One implication of a pure DA model is that
trace element concentrations in enamel and dentine, if at
equilibrium, should be proportional to the surface areas
per unit volume of their constituent crystallites. For typical
crystallite sizes measured in pristine biogenic materials (Elliott, 2002), this implies a 10- to 30-fold concentration difference. Yet trace element concentrations in enamel approach
those in dentine near the dentine–enamel interface (Eggins
et al., 2003; Fig. 4), implying either that the DA model does
not apply to enamel (i.e., trace element uptake occurs by volume diffusion rather than adsorption), or that dentine recrystallizes to crystal sizes similar to enamel, so their surface areas
per unit volume are indistinguishable. This latter inference is
consistent with X-ray diffraction patterns in fossil dentine
that indicate crystallinities comparable to enamel, and distinctly coarser than modern dentine (Ayliffe et al., 1994).
Data from some samples of enamel further support a
DMD model, because apparent La and Ce concentrations
tail off to several ppm (Fig. 4; Leptomeryx A1, A2; Electronic Annex), or even tens of ppm (Paleolagus A1 to
A7; Electronic Annex). At least one sample shows a uniform U concentration in the enamel interior of a few ppm
(IMNH27691-G). These concentrations are far removed
from steep concentration gradients near dentine, well above
levels for analytical background and modern enamel
(610 ppb, e.g., Kohn et al., 1999) and must reflect a fastdiffusion pathway, consistent with DMD.
In contrast to fossil enamel, published data from Pleistocene bone (Pike et al., 2005) may be better explained by a
DR mechanism. For these data, ages were regressed against
reduced distance squared using a simple weighted least
squares regression to derive the initial uptake (depositional)
age and its uncertainty (67 ± 9 ka; 95% confidence limit).
The linearity of the data and low MSWD (1.5) are consistent both with U uptake by diffusion, and with scatter
resulting from analytical uncertainties alone. Note that
the regressed age and its uncertainty are wholly independent of assumed values of Deff.
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M.J. Kohn / Geochimica et Cosmochimica Acta 72 (2008) 3758–3770
The U-series ages were then converted to reduced apparent age ðt0app Þ using a preferred value of Deff of 4 1014 cm2/s
0
(Millard and Hedges, 1996), and plotted vs. 1 X 2 for direct
comparison to DA predictions in Fig. 3. This value of Deff implies t0Dep 0.8, which balances DA model misfit between the U
concentration data (Fig. 3A) and the chronologic data (Fig.
3B). Increasing Deff to P5 1014 cm2/s maximizes the quality of fit for the U concentration data, but shifts the regression
line in Fig. 3B upward and further steepens its slope, increasing
misfit with the DA model. Conversely, decreasing Deff to
61 1014 cm2/s maximizes the quality of fit for the chronologic data (Fig. 3B), but the resulting t0Dep 60.1 cannot possibly
be reconciled with the U concentration data (Fig. 3A). That is,
a DA model cannot simultaneously account for the homogeneity of U, yet strong age gradients and young ages.
In contrast, a DR model can explain these data and other
samples with uniform U because it automatically produces a
homogeneous U profile irrespective of t0Dep . This conclusion
presumes that t0Dep is sufficiently high to allow complete
recrystallization of the bone, which is already implied by
the presence of U in the center of the bone fragment analyzed.
X-ray crystallographic data (Ayliffe et al., 1994) and trace element partitioning between enamel and dentine (Eggins et al.,
2003; this study) are consistent with recrystallization of fine
crystallites in bone and dentine during fossilization. Regardless of which diffusion model is assumed, the linearity of the
data in Fig. 3 demonstrates the success of diffusion in explaining U uptake during fossilization (Millard and Hedges, 1996)
and strongly supports previous age interpretations based on
DA modeling (Pike et al., 2005).
These observations and theoretical models have
important implications for the use of trace elements
and isotopes in geology, paleoclimatology, paleoceanography and archeology. For example, paleoclimate studies
assume fossilization rates of tens of kyr or less for interpretation of geologic processes on timescales of hundreds
of kyr to Myr (Staudigel et al., 1985; Elderfield and Pagett, 1986; Martin and Haley, 2000; Kohn and Law, 2006;
MacFadden et al., 2007; Zanazzi et al., 2007). The data
presented here conclusively support such assumptions,
although instances of late-stage uptake do occur (e.g.,
see Peppe and Reiners, 2007). In contrast, many archeological applications focus on shorter timescales, and attempt to date initial uptake, i.e., determine the
extrapolated age at the edge of a fossil based on direct
geochronology across the sample. Reconsideration of diffusive models of trace element uptake indicates that these
applications, which previously have relied on parabolic
fitting of the DA model, can be more generally interpreted independent of a specific diffusion model, provided
ages are regressed vs. (reduced) distance squared.
7. CONCLUSIONS
Key conclusions of this study include:
(1) Diffusion–adsorption models may not explain well
the age distributions and physical changes documented in fossil bone, but may be applicable to trace
element uptake in fossil enamel.
(2) Age distributions in fossil bone can be readily interpreted in terms of diffusion-limited uptake, regardless
of diffusion model, if measured ages are plotted with
respect to (reduced) distance squared, rather than
distance. Such plots are recommended because they
allow straightforward regression diagnostics and
interpretation of timing of initial uptake from the
regression intercept.
(3) Trace element zoning in fossil tooth enamel for a
range of environments and ages is generally consistent with a double-medium diffusion model. Modeling profiles suggests that complete fossilization and
trace element uptake could occur relatively rapidly,
with a minimum bound on durations of about one
century. This result does not rule out later stage alteration and trace element uptake, particularly on
exposed surfaces.
(4) Trace element uptake in bone and dentin is probably
closely linked to degradation of interstitial proteins
that otherwise protect biogenic crystallites from
recrystallization and chemical alteration. Rates of
trace element uptake may be controlled more by
the diffusion-controlled export of degraded proteins
and exposure of crystallite surfaces than diffusioncontrolled import of trace elements.
ACKNOWLEDGMENTS
This material is based upon work supported by the National
Science Foundation under Grant No. ATM 0400532. The author
gratefully acknowledges reviews from Bruce MacFadden, Peter
Reiners, Bernard Boudreau, and AE James McManus, which
helped improve the presentation and discussion. Charles Knaack
is thanked for help with LA–ICP–MS analysis.
APPENDIX. DERIVATION OF DIFFUSION
RELATIONS FOR THE DIFFUSION-REACTION (DR)
MODEL
Combined diffusion plus recrystallization is described in
Crank (1975) under conditions of an immobilizing reaction
(section 13.3, p. 298), which in turn is a special case of ‘‘Diffusion coefficients having a discontinuity at one concentration” (section 13.2.2 of Crank, 1975). In all such models,
the following relation holds:
X ¼ k t1=2
ðA:1Þ
where X is distance, k is a constant, and t is time. Eq.
(A.1) already demonstrates the key relationship necessary
for treatment of U-series dating of Quaternary materials
(i.e., that t is proportional to X2). The following derivation for DR shows that k is proportional to D1/2, yielding linearity between X2 and Dt, as in all other diffusion
solutions considered. In the endmember DR model, a
sharp front moves through the original material, converting pristine material with negligible trace element
content to recrystallized material with high trace element
content. This model is consistent with the following
considerations.
Trace element uptake in fossils?
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 C1 (a constant) and the concentration at the front
(at position x) is Cx (also constant); the concentration in the
pristine material is assumed to be 0. This relation allows
solution of a secondary function (g) of k, based on Eq.
13.18 of Crank (1975):
gðk=2D1=2 Þ ¼ ðC 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 s2 Þ
ðA:3Þ
where D is the diffusion rate in water, Kd is the partition
coefficient between phosphate and water, U is porosity,
and s is tortuosity. D* differs from Deff in accounting explicitly for porosity and tortuosity. Whereas Kd does affect the
rate of movement of the recrystallizing front, it does not affect the composition gradient through the recrystallized
material, given by (C1 Cx)/Cx 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=2D1=2 Þ pk 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 ½8D =pg1=2
ðA:5Þ
Substituting into Eq. (A.1) and squaring yields linearity
between X2 and Dt:
X 2 ¼ ½ðC 1 C x Þ=C x ½8D =pt
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.
3769
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Associate editor: James McManus