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

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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: mattkohn@boisestate.edu 

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 

0016-7037/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.gca.2008.05.045

Trace element uptake in fossils? 3759 

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., 

2 

Trueman and Tuross, 2002) and loss of CO 3 

(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 

5–10 orders of magnitude higher than its F-rich, CO 3 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 f O2 ). 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 

(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 

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- 

A B C 

DA 

DR 

DMD 

Solid + fluid 

C 

x 

C 

Reacted 

Solid 

Unreacted 

Solid 

x 

C 

DA 

profile 

DMD 

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.


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 (t 0 ) and 

reduced distance squared (X 02 ). For this paper, t 0 = D eff t/l 2 

and X 0 = X/l, where D eff is the effective diffusion coefficient, 

t is time, l is the half-width of the tissue considered, and X is 

distance. Some workers prefer t 0 =[D eff t/l 2 ] 1/2 . The effective 

diffusion coefficient (D eff ) is the ratio of the tracer diffusion 

rate (D) in a fluid and the partition coefficient between biogenic 

apatite and diagenetic or pedogenic fluids (K d ), corrected 

for porosity (r) and tortuosity (s), i.e., D eff = Dr/ 

K d s 2 . Note that in applications discussed below D eff is a 

measured value, so it automatically accounts for contributions 

from porosity and tortuosity, and that K d , r, and s 

are presumed constant, so solutions to the diffusion equation 

scale equivalently with either D or D eff . 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 (t 0 is small) then a semi-infinite 

solution is used, and X 0 is expressed as the distance from 

that surface or boundary (where X 0 = 0). In contrast, if 

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 X 0 is expressed 

as the distance from the midpoint between the 

two boundaries: X 0 = 0 is the center of the sheet, and 

X 0 =1, 1 are the boundaries. 

Most models assume a fixed boundary condition, i.e., 

that the concentration at any bounding surface (C o ) 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, 

(C 0 , where C 0 = C/C o ), as a function of reduced distance 

(X 0 ) and reduced time (t 0 ). 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 t 0 Dep is used to describe the 

time since initiation of the boundary condition and is equivalent 

to the total duration, or t 0 , of a model. In practice, 

t 0 Dep is proportional to the true depositional age, with a 

proportionality constant of D eff /l 2 . The term t 0 app is used 

to describe the apparent age at position X 0 within a sample. 

In practice, t 0 app is proportional to the measured age with 

the same proportionality constant, D eff /l 2 . 

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 

C/Co 

1.0 

0.8 

0.6 

0.4 

0.2 

X’=0.9 

X’=0.7 

X’=0.5 

X’=0.3 

X’=0.1 

t' app = 

t' True 

∫ 

[ C/C o ]dt' 

t'=0 

t' Dep 

t'app 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0.0 

0.5 

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 

0.0 0.2 0.4 0.6 0.8 1.0 

t' Dep t' Dep Edge Distance 

X’=1.0 

X’=0.9 

X’=0.7 

X’=0.5 

X’=0.3 

X’=0.1 

Figure 

2C 

A B C 

t'app 

1.0 

0.9 

0.8 

0.7 

0.6 

1-X' 

1-X' 2 

t' Dep = 1.0 

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 X 0 = 1, and has constant composition C o throughout. Apparent age is determined by integrating concentration curve (in terms 

of C/C o ) at a particular value of X 0 over duration t 0 Dep , and ratioing to t0 Dep . (B) Apparent age ðt0 app Þ vs. depositional age ðt0 Dep 

, or age of initial 

establishment of fixed boundary condition). At boundary (X 0 = 1) apparent and depositional ages coincide, so the slope is 1. For long times, 

the slope of t 0 app 

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) t 0 app 

vs. distance, showing quadratic form with respect to distance, and linearity with respect to 

distance squared. Note that any function that is linear in X 02 is also linear in 1 X 02 . 

Core


X 0 were regressed assuming linearity with respect to X 02 . 

error-function to the observed data. For U-series data, 

ically calculated values of t 0 app for different values of t0 Dep and This result applies both to dimensionless time or to actual 

however, t 0 app is determined by integrating U concentration 

at that point over the duration of the model (t 0 Dep; Fig. 2A 

and B). Note that if normalized concentrations are used, C 0 

This model is widely applied to interpret U uptake in fossil 

bone (Millard and Hedges, 1996), and closely approximates 

the expression: 

at the boundary is 1.0, and the integrated area at X 0 =1is 

identically t 0 Dep. These t 0 app ages require that U is immobilized 

t 0 app t0 Dep þ 0:5ðe 2:25t0 Dep 

1:0Þð1:0 X 02 Þ ð2Þ 

upon uptake, either by recrystallization (DR), or or equivalently for t (measured age) and X (measured 

adsorption onto apatite crystallites (DA and DMD). In essence, 

distance): 

any U that accumulates at position X 0 may no longer 

diffuse but instead is subject to decay. 

t 

All three DA, DMD, and DR models yield linear t 0 app t Dep þ 0:5ðe 2:25t Dep 1:0Þðl 2 X 2 Þ 

app 

D 

ð3Þ 

distributions with respect to the square of the distance 

Note that a function that is linear in X 02 must also be linear 

(either X 2 or X ): 

in (1 X 02 ), and that these relationships are least accurate 

t 0 app ¼ kX 02 þ b 

ð1Þ for short times, where the distribution of t 0 app vs. X02 is least 

linear (Fig. 3B). Analogous expressions apply to either of 

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, 

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): 

1975; see derivation in Appendix). Specifically, for the 

DR model, concentrations at any point X 0 are zero, until 

X 2 ¼½ðC 1 C x Þ=C x Š½8D eff =pŠt ð4Þ 

time t 0 app 

, when the front passes this position. Because the 

front progresses proportionally to X 02 , so too must t 0 app . 

For DMD, both segments are functionally equivalent to 

the DA model (Zhang et al., 2006), albeit with different D eff , 

so DMD solutions should yield two different, but linear segments 

on a plot of t 0 app vs. X02 , each implying the same t 0 Dep. 

For the plane sheet DA models (Figs. 2 and 3), numer- 

where C 1 and C x 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. 

1.00 

A 

t' Dep =1.75 

1.75 

B 

t' app ~ t' Dep + 0.5(e -2.25t' Dep - 1.0)(1-X' 2 ) 

0.80 

t' Dep =1.0 

1.50 

t' Dep =1.75 

t' Dep =0.4 

1.25 

C/Co 

0.60 

0.40 

0.20 

Near-edge 

U loss 

t' Dep =0.1 

= SWC3 [U] 

0.00 

0.0 0.2 0.4 0.6 0.8 1.0 

Edge 

1-X' 2 

Core 

t' app 

1.00 

0.75 

0.50 

0.25 

t' Dep =1.0 

67±9 ka 

t' Dep =0.4 

t' Dep =0.1 

SWC3 

U-series 

0.00 

0.0 0.2 0.4 0.6 0.8 1.0 

Edge 

1-X' 2 

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 t 0 Dep 

, concentrations become essentially constant across 

tissue; steep concentration profiles are realized only for low values of t 0 Dep 

. Homogeneous compositions from Pleistocene bone (excepting nearedge 

U loss) require t 0 Dep P 1. (B) t0 app . Slope steepens between t0 Dep 

¼ 0to1.0, as U is taken up and develops a chemical diffusion profile. 

For t 0 Dep 

> 1:0, U is saturated (concentration profiles are flat) and slope does not change (see Fig. 2B). Data for natural sample SWC3 

assume D eff =4 10 14 cm 2 /s; linearity is consistent with diffusive uptake, but DA model cannot simultaneously explain steep slope (low t 0 Dep ) 

and homogeneous U across bone (high t 0 Dep 

). 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.


measured ages because t 0 app 

involves only measured age 

(t app ) 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. 

4. DIFFUSION PROFILES IN ENAMEL 

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 14 C 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 mm 2 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/cm 2 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 

Ce, Yb (ppm) 

10000 

1000 

100 

10 

“in” profile 

“wash-in” 

Dentine 

“in” 

Enamel 

Cerium 

Ytterbium 

1 

Enamel 

Dentine 

Enamel 

0.1 

0 100 200 300 400 500 600 

Distance (m) 

“out” profile 

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. 

ICP–MS. Masses ranging from 43 to 238 were measured, 

most importantly 43 Ca (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 

“out” 

washout


40 wt% Ca (stoichiometric biogenic calcium phosphate), 

and apparent REE and U concentrations ranged from


Ce(ppm)/100, Yb(ppm)/10 

60 

50 

40 

30 

20 

10 

125 

Leptomeryx REE profiles U profiles 

A 

Cerium/100 

HAFOpp 

Mean Ce 

Ytterbium/10 

27691g 

100 

Leptomeryx 

78018i 

Mean Yb 

U (ppm) 

75 

50 

25 

B 

0 

0 25 50 75 100 125 

Distance (μm) 

0 

0 25 50 75 100 125 150 175 


erfc -1 (C/Co) 

2.5 

2.0 

1.5 

1.0 

Dt=4x10 -7 

Dt=2x10 -6 

Dt=1x10 -5 

C 

log(time, yrs) 

6 

5 

4 

3 

2 

1 

33 Ma 

3.2 Ma 

REE 

U 

D 

HAFOpp U 

0.5 

Leptomeryx Ce 

Leptomeryx Yb 

27691g U 

0.0 

0 25 50 75 100 125 


0 

-1 

-2 

Leptauch. 

Leptomeryx 

Paleolagus 

HAFO 

78018 

2347 

Sample 

27691 

26768 

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 cm 2 ), 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 C o . 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


Table 2 

Composition and modeling results for fossil dentine–enamel interfaces 

Sample Element Dt (cm 2 ) t(yr) Sample Element Dt (cm 2 ) t(yr) 

Leptauchenia-A La,Ce,Yb 4 10 6 100 Leptomeryx-A1 U 4 10 6 7 

Leptomeryx-A1 La, Ce, Yb 4 10 6 100 Leptomeryx-A2 U 1 10 5 20 

Leptomeryx-A2 La, Ce, Yb 3 10 6 90 Leptomeryx-B1 U 5 10 6 8 

Leptomeryx-B1 La, Ce, Yb 2 10 6 50 Leptomeryx-B2 U 7 10 6 10 

Leptomeryx-B2 La, Ce, Yb 3 10 6 100 Paleolagus-A1 U 2 10 6 3 

Leptomeryx-B3 La, Ce, Yb 2 10 6 70 HAFO-pp1 U 1 10 5 20 

Paleolagus-A1 La, Ce, Yb 7 10 6 200 HAFO-pp2 U 4 10 7 1 

Paleolagus-A2 La, Ce, Yb 3 10 6 100 IMNH78018-I1 U 6 10 6 9 

Paleolagus-A3 La, Ce, Yb 3 10 6 80 IMNH78018-I2 U 4 10 6 7 

Paleolagus-A4 La, Ce, Yb 2 10 6 60 IMNH2347-aa1 U 2 10 5 40 

Paleolagus-A5 La, Ce, Yb 5 10 6 200 IMNH2347-aa2 U 1 10 5 20 

Paleolagus-A6 La, Ce, Yb 2 10 6 60 IMNH27691-G1 U 1 10 5 20 

Paleolagus-A7 La, Ce, Yb 8 10 7 30 IMNH27691-G2 U 2 10 5 30 

IMNH27691-A-3a U 3 10 6 5 

IMNH27691-A-3b U 5 10 6 7 

IMNH26768-ss U 4 10 6 7 

Average 96 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 erfc 1 

(C/C o ) 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 (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 D eff 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 D eff , which in turn depends strongly on 

the partition coefficient between biogenic apatite and diagenetic 

or pedogenic fluids (K d ). Mono- and di-valent cations 

diffuse through modern enamel at uniform D’s of 

1 10 8 cm 2 /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.


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., 10 8 cm 2 /s. The phosphate-fluid 

partition coefficient (K d ) is believed to be at least 

5 10 5 for U (Millard and Hedges, 1996) and 1 10 6 

for REE (Koeppenkastrop and De Carlo, 1992), with 

uncertainties of about one order of magnitude. Thus D eff 

is 62 10 14 cm 2 /s for U and 61 10 14 cm 2 /s for REE 

in modern enamel, with uncertainties of about 1 1 2 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 

D eff in enamel reflect an adsorption rather than bulk equilibrium 

distribution coefficient, and both K d 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 10 14 cm 2 /s for U and 

1 10 14 cm 2 /s for REE) is warranted. 

Substituting these values of D eff 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 

D eff could reconcile the REE- and U-derived estimates. Generally 

shorter penetration distances for U compared to REE 

suggest lower D eff for U than for REE, in contrast to published 

K d ’s that suggest the opposite. 

Calculated durations are strict minima for two reasons. 

First, as described previously, K d may be underestimated, 

resulting in too large an assumed D eff . Second, preservation 

of the diffusion profiles requires some mechanism for 

retarding D eff 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


The U-series ages were then converted to reduced apparent 

age ðt 0 app Þ using a preferred value of D eff of 4 10 14 cm 2 /s 

(Millard and Hedges, 1996), and plotted vs. 1 X 02 for direct 

comparison to DA predictions in Fig. 3. This value of D eff implies 

t 0 Dep 

0.8, which balances DA model misfit between the U 

concentration data (Fig. 3A) and the chronologic data (Fig. 

3B). Increasing D eff to P5 10 14 cm 2 /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 D eff to 

61 10 14 cm 2 /s maximizes the quality of fit for the chronologic 

data (Fig. 3B), but the resulting t 0 Dep 

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 t 0 Dep 

. This conclusion 

presumes that t 0 Dep 

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 t 1=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 X 2 ). The following derivation 

for DR shows that k is proportional to D 1/2 , yielding 

linearity between X 2 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.


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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Associate editor: James McManus

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

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