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thesis entitled

KINETIC EQUATIONS FOR THE PRECIPITATION OF CARBONATES
WITHIN THE THERMODYNAMIC STABILITY FIELD OF DOLOMITE

presented by

Robert E. Dedoes

has been accepted towards fulfillment
of the requirements for

Master of Science degree in Geological Sciences

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KINETIC EQUATIONS FOR THE PRECIPITATION OF CARBONATES
WITHIN THE THERMODYNAMIC STABILITY FIELD OF DOLOMITE

By

Robert E. Dedoes

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Geological Sciences

1987

Copyright by

ROBERT E. DEDOES

1987

9/2.;n:25”7’

ABSTRACT
KINETIC EQUATIONS FOR THE PRECIPITATION OF CARBONATES
WITHIN THE THERMODYNAMIC STABILITY FIELD OF DOLOMITE

By

Robert E. Dedoes

Equations which can potentially describe the precipitation of carbonate
phases within the thermodynamic stability field of dolomite are derived.
The equations are applicable only when all precipitating phases are
nucleating onto the same finite number of nucleation sites. The
equations appear consistent with published experimental data concerning
the precipitation of carbonates within the thermodynamic stability field
of dolomite. The equations allow several phases to compete with each
other for the same nucleation sites and by doing such,they account for
both Ostwald's step rule occurring during carbonate substrate to dolomite
transformations and for the distribution of carbonates in marine waters.
The equations also suggest that the number of nucleation sites per

unit volume of material being transformed is an important variable in
the rate of transformation of carbonate sediments to dolomite. The
equations can also be potentially used to predict porosity in dolomitic

rocks.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

I INTRODUCTION

II DERIVATION OF THE EQUATIONS

A. Time-dependent heterogeneous nucleation and
time-independent growth

B. Time-independent heterogeneous nucleation and
time-dependent growth

ii

iii

11

III THE VALIDITY OF THE DERIVED EQUATIONS AS COMPARED TO EXPERIMENTS 11

IV APPLICATION OF THE EQUATIONS TO EXPERIMENTAL AND NATURAL SITUATIONS

A. Proposed explanation for sequences of metastable
phases prior to the precipitation of the stable
phases (Ostwald's steps)

B. PrOposed explanation for the distribution of
carbonates from marine-type waters

C. Proposed explanation for the effect of substrate
grain size on the transformation rate

D. Proposed equation describing the porosity and
permeability of dolomitic rocks

V CONCLUSION
APPENDIX
A. Experimental procedure
B. Results
C. Interpretation of results

REFERENCES

15

19

22

23

24

25
26
26

33

Table 1:

Table 2:

LIST OF TABLES

EXPERIMENTAL RESULTS FROM X-RAY DIFFRACTION

DISTRIBUTION OF MARINE CARBONATES

ii

27

20

Figure 1:
Figure 2:
Figure 3:

Figure 4:

LIST OF FIGURES

Reaction progress versus time curves
Calcite to dolomite transformation
Nucleation rate per site versus supersaturation

Proposed curves for coarse and fine material

iii

12

16

23

I INTRODUCTION

 

Numerous studies have been concerned with the precipitation of
dolomite from aqueous solutions onto substrates under natural conditions
(see Fairbridge 1957, Ingerson 1962, Friedman and
Sanders 1967, Zenger 1972, Zenger and Dunham 1980, Morrow 19823 and
1982b). Two conclusions can be drawn from the current studies of
dolomite precipitation under natural conditions. First, dolomite occurs
in many different types of geologic environments yet it is absent from
many others where it is thermodynamically stable. Second, many types of
metastable carbonate phases precipitate within the thermodynamic
stability field of dolomite. These two conclusions indicate that dolo-
mite can precipitate under earth surface conditions and there is a
kinetic control of the precipitation of carbonates within the thermo-
dynamic stability field of dolomite.

Several experimental investigations have been undertaken to identify
the kinetic control(s) of carbonate precipitation within the thermo-
dynamic stability field of dolomite. Katz and Matthews (1977) found
that the dissolution-reprecipitation transformation of aragonite and
and calcite to dolomite occurred via a sequence of metastable carbonate
phases (Ostwald's step rule). Baker and Kastner (1981) found that
sulfate inhibits the transformation of calcite to dolomite. Gaines
(1980) found that in an aqueous solution both magnesium calcite and
aragonite underwent a faster transformation to dolomite than did low
magnesium calcite. He also found that the Mg/Ca ratio, the addition
of dolomite crystals, and the addition of certain additives all influ-
ence the transformation of aragonite to calcite.

Kinetic processes of crystallization cannot be

2
directly interpreted from experimental results without the aid of
theoretical kinetic equations. Currently there have been no studies
that investigate the theoretical aspects of the kinetics of precipi-
tation of carbonate phases within the thermodynamic stability field of
dolomite. The purpose of this study is to develop basic theoretical
kinetic equations for dolomite precipitation. The results of this study
are the derivation of equations which describe the simultaneous pre-
cipitation of several crystalline phases from solution onto a substrate.
These equations are an extension of the works of Johnson and Mehl (1939)
and Avrami (1939, 1940, 1941) to multiphase systems. The study also
shows how the equations can be used to interpret existing data on
carbonate precipitation within the thermodynamic stability field of

dolomite.

II DERIVATION OF THE EQUATIONS

 

The kinetic equations for the precipitation of carbonates within
the thermodynamic stability field of dolomite are based on two kinetic
processes of precipitation: nucleation and the growth of crystals from
those nuclei. It will be assumed that nucleation of carbonates within
the thermodynamic stability field of dolomite occurs only on a finite
number of nucleation sites and that all precipitating phases can nucleate
on the same nucleation sites. The general derivation of the equations
is based on the works of Johnson and Mehl (1939) and Avrami (1939, 1940,
1941).

Consider a volume of carbonate sediments bathed by an aqueous
solution whose volume is uV. The volume remains as uV until a time
t=t when the first nucleus of a transforming carbonate phase nucleates

. u
somewhere in V.

3

During the time interval I-+ dT the change in the number of nuclei

is given by:

dN = “vvrdr (1)

where vI is the nucleation rate for one type of nucleation site per unit
volume. A single type of nucleation site is any place or places on a
substrate where the AC of nucleation for a phase is the same for that
phase. Nucleation sites can be defects, corners, kinks, etc. on a sub-
strate. TV is the volume of uV left after the precipitation of nuclei
during the interval Y + dT. It will be assumed to be equal to uV.

The number of nuclei formed at any time since the onset of nucleation
is found by integrating Equation (1) as follows:

I

N(t) = uv VIdr (2)

an:

'1'

Assuming that the nucleus of a carbonate phase grows such that

G G , Gtz are the growth vectors along the x, y, and z axes res-

tx’ ty
pectively, then the amount of growth per unit time for three dimensions
3
18 Gtxdt X thdt X Gtzdz = R(tht) ; for two dimen51onal growth,
1
X a o ' ' z
Gtxdt thdt R(tht) , for one dimen51ona1 growth, Gtxdt R(tht)
and the general form for any dimensional growth is R(tht)n where R
is a geometric shape factor. The small t in Gt indicates that the
growth vector G can vary in magnitude with time.

The instantaneous change in volume of a growing crystal is given

by:

av = R(G(t)dt)n (3)

The volume change for a single crystal since the onset of nucleation

is:

t
V(t) = R I (c at)“ (4)
t=T (t)

The total volume change since the onset of nucleation of the car-

bonate phase is the product of Equations (2) and (4):

t t
A
V(t) = uv I vI [R I (cat)"] dr (S)
T=0 t=r
Consider many phases precipitating into the same nucleation sites.

Each phase will have a volume at time t given by Equation (5). The total

volume at time t since the onset of nucleation for all phases is:
N
V(t)tOtal = AV(t) + BV(t) + ... + V(t) (6)

N
where AV(t), BV(t), ... , V(t) are the volumes at time t since the
onset of nucleation of phases A, B, ..., N.
Substituting the V(t) for each phase as given in Equation (5) into

Equation (6) gives:

‘

N t V t n
V Z I I [R f (C(t)dt) ldT (7)
L 13A T=O t=T a

vtotal = u

Up to this point, the covering up of nucleation sites by growing

5
crystals and the mutual interference of growing crystals has not been
considered. These two processes are important in most precipitation
systems and therefore should be considered. This will be done using the
treatment of Avrami (1939, 1940). This treatment is valid only under
the following conditions: nucleation occurs randomly throughout the
volume, and impinging crystals meet with a common interface, but no
growth occurs along this interface.

To mathematically describe the covering up of nucleation sites by
growing crystals and the mutual interference of growing crystals, an
extended volume must first be defined. The extended volume is the vol-
ume of material which could have nucleated and grown if there was no
interference by crystals during growth and no covering up of nucleation
sites by crystals during their growth. In an extended volume,nuc1e-
ation is occurring within crystals as well as between them. The growth
of the crystals is in free space and unimpeded through each other.

In the extended volume, during the interval T + dT , there are
vIquT nuclei formed in the open nucleation sites of one type and
vITVd‘r nuclei formed in the nucleation sites of that type now covered
up by crystals. The change in the number of nuclei since the onset of

nucleation is given by:

dN = vI (TV + uV) dr (8)

where TV and uV are the volumes of the transformed and untransformed

regions respectively.

The number of nuclei formed during a time interval since the onset

of nucleation is:

I dr (9)

Tv = uv.

TV is negligible for the small instance I + dt thus uV +
Multiplying Equations (9) and (4) gives the extended volume of

carbonate phases precipitated since the onset of nucleation:

t
T (C(t)

t ‘1'
eV = uv I vI [R dt)n]dr (10)

r=0 t

IIHII

When there are many phases precipitating into the same nucleation
sites of one type, each phase will have an extended volume given by
Equation (10). The total extended volume for all phases since the onset

of nucleation is:
ev(t)total = eV(t)A + eV(t)B + ... + eV(t)N (11)
Substituting the eV value for each phase as given in Equation (10)

into Equation (11) gives:

N t t
Vtotal = 11V X f VI[R f (G
i=A T=O t=r

e n
(t)dt) ] dr (12)

The extended volume is governed only by the kinetics of nucleation
and crystal growth. The real volume is governed by the kinetics of
nucleation and growth, by the effects of mutual interference of growing
crystals, and by the covering up of nucleation sites by growing crystals.

If the relationship between the transformed volume (TV) and the

e . . . .
extended volume ( V) 18 defined, then an equation can be written

7
relating the processes of nucleation, crystal growth, mutual interference
of growing crystals, and the covering up of nucleation sites by growing
crystals in terms of nucleation and crystal growth only. This relation-
ship is found as follows.

Consider a small, random volume, V. A fraction of this volume
remains without new crystals at time t. After a time interval dt, the
real and extended volumes expand by dTV and deV respectively. Next
consider many expansive occurrences; on an average, a fraction (1 - I!)
(where TV is the volume of new crystals) of deV will lie in an uv

untransformed region and thus contribute to dTV. This allows dTV to be

related to deV as follows:

dTV = (1 — Ty) deV (13)
uv
Upon integration, Equation (13) gives:
e u T
V=-Vln(1-_V_) (14)
uv
Substituting Equation (12) into Equation (14) gives:
T 11 N t V t n
- uv 1n (1 - _‘D = v I I[R I (C(t)dt) ] d1 (15)
uv i=A T=O t=T
Dividing both sides by uV and raising both sides to the exponential
gives:
T N t v t n
1-_\L= exp [ - Z I IlR I (C(t)dt) ] dT ] (16)
u i=A r=0 t=r

V

8
uV can be any volume of solid carbonates, packed carbonate particles,
or carbonate particles and solution, etc. TV is always the total solid
volume of the transforming crystals.

It should be mentioned that in certain cases TV will never reach
uV. This may occur for example, in dissolution-reprecipitation trans-
formations of a volume of carbonate grains and water where only the
volume of the grains is transformed. The result is the same as if a
transformation had been quenched before completion.

Graphically, the result of a transformation where TV will never
equal uV (or of a quenched transformation before uV equals TV) is a

truncation of the sigmoidal time I!_curve predicted by Equation (16)

(see Figure 1).

 

Time

Figure 1: Reaction progress versus Itime curves
. u
Curve A = reaction where V eventually reaches V. Curve B =

. T u
reaction where V never reaches V.

Equation (16) is one of the general equations for the precipitation
of carbonates within the thermodynamic stability field of dolomite when
random nucleation per unit volume is occurring. Equation (16) allows
for the precipitation of more than one phase. It takes into account
mutual interference of growing crystals and the covering up of nucle-
ation sites during growth. The equation can be arranged to model
time-dependent heterogeneous nucleation and time-independent growth, and

time-independent heterogeneous nucleation and time-dependent crystal

growth.

A. Time-dependent heterogeneous nucleation and time-independent
growth

 

For this case Equation (16) becomes:

N

X It
i=A 1 t

N t
1 - Iy.= exp - N z I J exp [R I (Gdt)n] dr (17)
uV =A T=0 t=r

where NO is the number of nucleation sites of one type and J1 is the

nucleation rate of phase 1 per site for one type of nucleation site.

The expression NOJ exp H'Jit is derived as follows (adapted from
i=A
Avrami 1939). If there are No sites per unit volume at t=0, then after

a small time interval the number of sites that disappear is:

N
dN = - N z Jdt (18)
i=A

Integration of Equation (18) gives the number of sites remaining

at any time t:

N t N N
I 9%=-I ZJdt=ln%=2Jt (19)
N=N 0 i=A o i=A

Taking the exponential of both sides and dividing both sides by No

gives:
N
2.1::
N(t) = No ei=A (20)

Equation (20) represents the number of nucleation sites left at any

time. The rate at which these sites disappear is the time-dependent

10

nucleation rate per unit volume:

v
d N v 1
df_- No X J exp (21)

Setting t=T Equation (21) can be substituted into Equation (16) to give

Equation (17). For i=A and a constant crystal growth rate in three
dimensions, Equation (17) reduces to Equation (20).of Avrami (1940),

If a nucleation induction period is to be considered, then using
the expression vIt = vI exp :%- (Frenkel 1946, see also KashchieV’ 1969).
the disappearance of nucleation sites with time for one type of active

site is given by:
N v
dN = N 2: J exp t (22)

Integrating by parts gives:

' N IE. 5. (25.)1
N = N exp. 2 J exp t + t Ei t (23)
(t) 0 K...
si-A
where Ei denotes the integral exponential function and k is the time
until steady state nucleation is attained.
The time-dependent rate of nucleation is the change in the number
of nucleation sites with time:
' N 1‘- 5 .(in
dNolexP .2: Jexp t + t E1 t J (24)
i=A
dt
Setting t=1 , Equation (24) can be substituted into Equation (17)

instead of Equation (21) when a nucleation induction period is to be

considered.

11

B. Time-independent heterogeneous nucleation and time-dependent

growth

 

1 - v = exp - z (R vI I (cat)n d1) (25)
i=A t=T
where vI is the time-independent heterogeneous nucleation rate per unit
volume for one type of site.

The derivation of Equations (21) and (29) only applies (in a quan-
titative sense) to random nucleation of the volume-transforming
carbonates throughout the volume being considered (UV).

The general form of an equation when nuclei are confined to the
surface of sedimentary particles and when the crystals from these
surfaces can impinge upon the crystals from adjacent surfaces as well

as impinge upon each other is:

1 - I!_= exp (fn) (26)
uv

where (fn) is a function of nucleation and growth rate similar to those

rates described previously. Simple examples of this function are

derived in Cahn (1956). .Derivations of other geologically meaningful

examples of this function will not be presented here.
III THE VALIDITY OF THE DERIVED EQUATIONS AS COMPARED TO EXPERIMENTS

The purpose of this section is to examine experimental evidence
which is supportive of the equations' applicability to carbonate

precipitation.

Figure 2 represents the dissolution-reprecipitation transformation

of calcite to dolomite via an intermediate high magnesium calcite.

12

Q1

 

 

Figure 2: Calcite to dolomite transformation (from Katz and Matthews
(1977).

Curves B and C are sigmoidal in shape indicating that the fraction of

material increases with time by an exponential function. The presence

of these sigmoidal curves during the dissolution-reprecipitation trans-

formation reactions in Figure 2 indicate that equations of the type

given by (17), (25), and (26) are applicable. Figure 2 and Table 1

(see Appendix) indicate that metastable carbonate phases can precipitate

abundantly within the thermodynamic stability field of dolomite.

The equations can account for the precipitation of metastable
phases as follows. Under experimental conditions, nucleation occurs
heterogeneously on a limited number of nucleation sites and therefore
Equation (17) can be used to explain this. Assuming that each pre-
cipitating phase has a constant growth rate which is equal in three

dimensions, Equation (17) becomes:

- N
2 Jt
TV N x t I."
1 -';- - exp--No 2 RG I Ji exp (t-T ) it (27)

V i-A r-O

13

This equation can then be integrated to give:

 

N
_ X Jt
T N ’ i=A N
1 - -X- = exp - N E 6RiGi J exp - 1 + Z Jt
UV 0 i=A N i=
4 (28)
i=A N a 2 3 3
- Z J t + Z J t
1=A 1=A
2 6

Consider a solution supersaturated with respect to dolomite such
that the solution is also supersaturated with respect to magnesium

calcite and protodolomite. Let XC =

 

 

N i A i A ___ 13A
2.) 2 6
i=A
N
where X Ji is the summed nucleation rate of all precipitating phases
i=A

for one type of nucleation site.
Similarly, let XD and Xd represent protodolomite and dolomite

respectively. Then substituting these values into Equation (28) gives:

*3

V d

J 3
1-— =exp-[N (ch°+pcxp+
V O

’ d
C X )] (29)
where CG , pG , and dG are the growth rates for calcite, protodolomite
and dolomite respectively.

If the product of growth rate and time-dependent nucleation rate

3
of calcite (CG x Xc) is considerably greater than the products of

time-dependent nucleation rate and growth rate of protodolomite

p ’ p . d ’ d p ’ p d ’ d
( G X X ) and dolomite ( G X X ), then ( G X X ) and ( G X X ) are
3
negligible compared to (CG x XC) and thus Equation (29) becomes:

14

3
TV “NCGX
1—— = exp 0 (30)
U
v
c 3 c ’ d 3 d
If ( c x x ) = (pc x xp) = ( c x x ), then:
TV c ’ c 3 d 3 d
1--u—=exp-[N0(Gx +pcxp+ (311)] (31)
v

and the final transformed volume will contain equal amounts of dolomite,
protodolomite and magnesium calcite.

All of the equations therefore can account for the precipitation
of metastable phases when the nucleation and growth rates relative to
other precipitating phases are significant.

In conclusion, supportive evidence for the equations in being able
to predict the dissolution-reprecipitation transformation of carbonates
is twofold. 1) The equations predict the sigmoidal-shaped curves for
the dissolution-reprecipitation of carbonates. These curves are
observed in experimental data (see Figure 2). 2) The equations can
account for the precipitation of one or more metastable phases and
predict that these phases should precipitate sigmoidally with time.

This is also observed experimentally.

IV APPLICATION OF THE EQUATIONS T0 EXPERIMENTAL AND NATURAL SITUATIONS

The purpose of this section is to qualitatively apply the equations
to four aspects of the precipitation of carbonates within the thermo-
dynamic stability field of dolomite. This will demonstrate how these

equations may be used to explain experimental and natural situations.

15

A. Proposed explanation for sequences of metastable phases prior

to the precipitation of the stablegphases (Ostwald's steps)

In this study and in the study of Katz and Matthews (1977), a
sequence of metastable phases have been observed during the dissolution-
reprecipitation transformation of 4MgC03°Mg(0H)2'4H20, aragonite and
calcite (see Appendix). The previously derived equations will provide
an explanation for these observed sequences after the following five
paragraphs are presented.

From the classical nucleation theory, a relationship for the
nucleation rate per site type can be derived as follows. Under steady
state conditions, the rate of nucleation onto a substrate is given by

Turnbull and Fisher (1949):

3 2

NSRT _ c _ 82 v
3—— —- I l * 2
Q h exp RT 3"" R T (1138—) (32)

where Q = the nucleation rate per second
NS = the number of ions in a boundary layer on the substrate surface

R = the gas constant per ion or molecule

T = temperature

h = Planck's constant

C = Gibbs function of activation for diffusion

B = nucleus shape factor

2 = total surface free energy of the nucleus

v = volume per molecule in the solid

%;.- supersaturation ratio (saturation is the minimal saturation

state necessary to form a nucleus on the substrate)
(Modified from Strickland-Constable 1968).

The nucleation rate per site for one type of site is given by:

J = Q (33)
total number of sites

 

Equation (33) can be simplified as follows. Noting that the term

16

N RT . .
8 exp :§_is essentially constant at constant temperature, then for a

h RT
certain substrate it can be set equal to K. Doing this gives:

Bz v
J a K exp - R3T3(1n a): (34)
s

 

Equation (34) can be used to construct a plot of saturation state
versus the nucleation rate per site type for dolomite and each metastable
phase. Unfortunately, the plot (see Figure 3) must be constructed with
relative values since values for z of carbonate phases for heterogenous
nucleation are not available. Discussions using Figure 3 therefore can

only be in relative or qualitative terms.

log
nucleation
rate per
site type

 

 

 

Figure 3: Nucleation rate per site versus supersaturation

Note: §:_is not fixed but depends on the curve being veiwed. For
3
example, when viewing the dolomite curve, §:_represents the dolomite
S a a
supersaturation ratio. Curves are constructed such that Bz v of

A < Bz3v1 of C < Bziv3 of P < Bz3v3 of D. Curve D is the thermo-
dynamically stable phase. A, C, P, D are metastable carbonate phases
composed of some combination of Ca + C0;, Mg + C0; or Ca + Mg + C03.
8, 81’ 82, x, y are explained in the text.

Figure 3 illustrates that the nucleation rate of a metastable

phase can increase much faster than the nucleation rate of the

17
thermodynamically stable phase with an increase in the saturation state
of the stable phase when the following two conditions are met. First,
the starting saturation ratio s: in terms of the saturation state of the
stable phase must be appropriate (i.e. it must occur before the expo-
nential upsweep of the stable phase curve). Second, an increase in the
saturation state of the stable phase must provide an appropriate
increase in the saturation state of the metastable phase. For example,
consider metastable phase A represented by curve A (see Figure 3). It
has the appropriate starting saturation ratio (given as s on curve D
and 0 on curve A) and has the product Bz,v3 appropriately lower than
that product for curve D. When the saturation state of phase D is
raised by the amount X for the stable phase D, producing an increase in
Y of the saturation state for the metastable phase A, the nucleation
rate of phase A increases by several orders of magnitude over the stable
phase.

From this theoretically derived concept, the Ostwald's steps
obtained during the transformation of 4MgC0:'Mg(OH);°4H20, aragonite

and calcite to dolomite in this study (see Table 1, Appendix) can be
explained as follows.

Consider a substrate (4MgC03°Mg(0H)a'4H20) in contact with a
dolomite supersaturated solution in a closed system. On this substrate
there exists a number of nucleation sites of one type. Let a nucleus
of aragonite have a nucleation rate versus the saturation state on
these nucleation sites be represented by line Alin Figure 3. Let the
nucleation rate versus supersaturation of calcite, protodolomite and
dolomite be represented by lines C, P, and D respectively in Figure 3.

If the solution in contact with the substrate has a dolomite

supersaturation state of 81 (which has a corresponding nucleation rate

18
R1 for all phases indicated in Figure 3), then the nucleation rate of
aragonite will be several orders of magnitude greater than all the other
phases and recalling Equation (28) gives:
1 — g! = exp - [NOAG’XA] (35)

V
If the growth rate of aragonite is competitive with the growth rate of
the other phases, then the transformation will proceed to all or virtu-
ally all aragonite.

In a closed system, the transformation to aragonite has lowered the
saturation state of dolomite within that system (see Interpretation of
Results section in Appendix for reasoning). This lower dolomite satu-
ration state coupled with the new aragonite surface provides a range of
AGS of nucleation for all phases which are some combination of Ca2+,

2+ 3_
Mg , C0: .

If the nucleation and growth rates of certain metastable phases are
much faster than the nucleation and growth rates for the thermodynamically
stable phase, then this transformation will also convert to a metastable
phase or phases in a manner exactly as explained previously for the
transformation to aragonite.v For example, a drop in the dolomite
saturation state from s1 to 32 (see Figure 3) lowers the relative rate
of nucleation of aragonite relative to protodolomite, low magnesium
calcite and dolomite. If the rates of nucleation of protodolomite and
low magnesium calcite are still much greater than the rate of nucleation
of dolomite (a drop from R1 to R2 in Figure 3), then the aragonite will
be transformed into these two phases instead of dolomite.

This process may be repeated several times before the precipi-

tation of the thermodynamically stable phase thus giving rise to the

19
the sequences illustrated in Table 1 (see Appendix). The basic con-
cepts used here may explain sequences of metastable phases (Ostwald's

step rule) in other systems as well.

B. Proposed explanation for the distribution of carbonates from
marine-type waters

 

 

The most notable features of Table 2 are the virtual absence of the
thermodynamically stable phase, dolomite and the widespread precipi-
tation of various types of metastable carbonate phases. Equilibrium
thermodynamics does not provide an easily attainable explanation for
this distribution and thus a kinetic explanation must be sought.

Using the explanation of Ostwald's steps given in the previous
section, the distribution of carbonates in Table 2 can be explained by
the equations in the following manner. The nucleation and/or growth
rates of aragonite and magnesium calcite are substantially faster than
the rates of nucleation and/or growth of protodolomite in shallow
marine solutions supersaturated with respect to aragonite and high
magnesium calcite. This allows these phases to transform pore space
into aragonite and magnesium calcite preferentially over protodolomite
during the time the pores are bathed in this solution. This is equi-

3 3 A

3 J
valent to the situation where CG X = G X

>> pc x, and then
substituting these values into Equation (29).

Not until a marine solution has a lower rate of precipitation
of aragonite and magnesium calcite will protodolomite and/or possibly
low magnesium calcite be able to precipitate (or transform the substrate
if the solution is undersaturated with respect to aragonite and mangesium
calcite) during the time these substrates are in contact with that

solution.

20

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21

There are two principal ways in which the product (growth rate X
nucleation rate) for aragonite and high magnesium calcite can be lowered
relative to this product for protodolomite and/or low magnesium calcite.
First, a change in the dolomite saturation state such that the activity
of Ca3+ and/or C031- is lowered. This can reduce (as suggested in the
previous section) the rates of nucleation of these two phases by an
amount greater than it reduces the rate of nucleation of protodolomite.

Second, the addition or removal of nucleation and/or growth
inhibiting substances which many inhibit the nucleation and/or growth
rate of one carbonate phase more than another may also lower the product
of growth rate X nucleation rate. There is some evidence suggesting
inhibition of carbonate precipitation in marine-type waters (Baker and
Kastner 1981, Berner et a1. 1978, Chave and Seuss 1967, Seuss 1970),
but whether there are any selective inhibitors remains to be demon-
strated.

The placement of ordered dolomite within this scheme is not as
simple because there is not definite evidence which suggests that
ordered dolomite is able to nucleate onto aragonite or magnesium calcite
substrates at the dolomite saturation states found in seawater or in
modified seawater. There are therefore, two possible reasons for the
present distribution of ordered dolomite in marine-type waters. First,
ordered dolomite is outcompeted by faster nucleating and growing meta-
stable phases on aragonitic, calcitic, and protodolomitic substrates
except when the dolomite saturation state is such that it is close to
or below protodolomite saturation. The potentially low dolomite satu-
ration state will thus produce 1ow nucleation and growth rates of
dolomite. These rates may be slow even in terms of geologic time

thereby preventing dolomite from precipitating to any large extent

22
during the time a substrate is in contact with such a solution. Second,
the surface free energy to form a dolomite is so much higher on aragonite
and magnesium calcite substrates that its nucleation rate on those
substrates is zero at the dolomite saturation state in marine-type
waters. It would therefore only be able to nucleate on a specific
substrate such as a protodolomite.

C. Proposed explanation for the effect of substrate grain size

on the transformation rate

If the number of initial nucleation sites on a substrate are
directly proportional to the surface area, then a fine grained sediment
will have a greater surface area per unit volume and thus a greater
number of nucleation sites per unit volume than a coarse grained
sediment.

The greater number of initial nucleation sites per unit volume in
the fine grained sediment with respect to the coarse grained sediment
results in a larger number of nuclei per unit volume in the fine grained
sediment compared to the coarse grained sediment. Examination of
Equation (17) indicates that at any time t, the larger the value of the
combined No, the closer I - IV_ is to zero or the closer the trans-
formation is to completion “V (see Figure 4). Thus a fine grained
sediment can undergo transformation at a faster rate than a coarse

grained sediment.

23

fine
coarse

 

 

Time

Figure 4: Proposed curves for coarse and fine material

Bartlett (1984) experimentally examined the effect of carbonate
substrate size on the rate of transformation of that substrate to dolo-
mite. His results point out the importance of the number of nucleation
sites per unit volume on the transformation rate of carbonate sediments
to dolomite. His curves differ somewhat from those in Figure 4. The
reason for this discrepancy may be that other factors besides the number
of nucleation sites per unit volume were important during his trans-
formation reactions. It is not within the scope of this study to
discuss potential candidates for these factors.

D. Proposed equation describing the porosity and permeabilityiof
dolomitic rocks

 

Equations describing the generation of porosity and permeability
in carbonate sediments as a function of time do not exist. The
porosity and permeability of dolomitic rocks are dependent on the density
per unit volume of dolomite crystals and the degree of mutual inter-
ference between these crystals. Therefore, porosity and permeability
can be explained quantitatively in terms of the equations proposed in
this study.

For example, consider a volume of carbonate sediments being

24
transformed into a dolomitic rock. If the untransformed portion of
the carbonate is considered to be porosity (it gets removed through
later diagenesis), then the porosity of the dolomitic rock will be

equal to E!_. From Equation (17):
u
V

TV N t -Jt ‘1 n
— = 1 - exp [- N Z < I J exp [R (Gdt) ldT)] (36)
uv ° A r=0 t=T

Permeability will depend on porosity and the degree of mutual
interference between dolomite crystals and therefore will vary directly
with Equation (36):

T

permeability = fcn-E% (37)

V CONCLUSION

 

Equations have been derived for the precipitation of carbonate
phases within the thermodynamic stability field of dolomite in terms
of the two kinetic processes: nucleation and crystal growth. The
equations can be used quantitatively and qualitatively in natural and
experimental cases where transformation of a phase proceeds by random
nucleation within that volume.

The equations are not restricted to carbonate precipitation but
can be applied to the transformation of a volume by precipitation of one
or more phases in igneous and metamorphic as well as sedimentary environ-
ments if the following condition. is met. In the case to be considered

all precipitating phases can nucleate on all available nucleation sites.

APPENDIX

25

APPENDIX

A. Experimental procedure

 

The purpose of these experiments was to transform various carbonate
substrates to dolomite. Particular attention was given to the solubility
of that substrate relative to the other substrates and the sequence
of metastable carbonate phases precipitating prior to the precipitation
od dolomite.

All experiments were performed inside 10 ml stainless steel bombs
and Parr model #4749 Teflon-lined bombs with a 23 ml capacity.

Temperature was kept at 210°C for all experiments inside an auto-
matic temperature-controlled Linberg oven.

The reaction solutions were prepared by dissolving analytical
grade CaCla'ZHaO (Fischer) and MgCla'6HaO (Mallinckrodt) into twice-
distilled water in separate erlenmyer flasks to a concentration of 1M.

The reactants used were 4MgC03'Mg(OH)a'4H20, reagent grade calcite
and ground natural aragonite.

Experimental charges consisted of .2000 grams of a reactant, 5.4 m1
of CaCl. and 3.6 ml of MgCl; (only one mineral per charge at any one
time). Charges were allowed to react for a certain length of time and
then quenched with cold water. The contents of each charge were
emptied into Whatman #1 11 cm filter paper and then rinsed with distilled
water. The charge contents and filter paper were then dried together
at 180°C until thoroughly dry. The charge contents were then placed in
a small plastic container with a snap-on cap. The container was then
marked with the type of reactant and the length of time elapsed between
placing the charge into the oven and the quenching of that charge.

Phase identification of the solid charge contents was made by

26

X-ray powder diffraction using Ni filtered CuK radiation. Scanning was
done from 14°C to 43° C 29 at a rate of .5°/minute. Samples were
scanned as a powder smear on a recessed portion of a glass slide. The

internal standard used was fluorite.

B. Results

See Table 1: Experimental Results from X-ray Diffraction

C. Interpretation of Results

 

By extrapolating Rosenberg's et a1. (1967) experimental boundaries
for the stability field of dolomite in chloride solutions to 210°C, the
solutions within the charges are found to lie within the thermodynamic
stability field of dolomite.

Table 1 indicates that dolomite, regardless of the initial reactant,
precipitates after the precipitation of a series of one ore more meta-
stable phases. Based on Katz and Matthews (1977) work it is assumed
that one metastable phase replaces another through dissolution-repre-
cipitation in these experiments.

'The dissolution-reprecipitation sequence is interpreted to
represent a decreasing dolomite saturation state within the charge as
time progresses. The following explanation is given in support of this.
For one phase to undergo replacement of another phase through a dis-
solution-reprecipitation transformation, that phase must be able to
dissolve. Consider the series of observed phases when aragonite was
used as the initial reactant:

1) At t=0, the charge is saturated with aragonite

3

+ 2 2
2) As protodolomite precipitates, the product of aCa )<a C0:

must decrease in order for aragonite to dissolve and eventually be

27

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32

replaced. This occurs because the AG of dissolution of aragonite is
positive at aragonite supersaturation, zero at aragonite saturation
and negative at aragonite undersaturation. Thus the energy input and
direction of flow necessary to drive the kinetic processes of dis-
solution on the aragonite surface can occur only at aragonite under-
3+ 2+ 2 3-
saturation. Therefore the aCa X aMg X a C0: is lower after
protodolomite precipitates than at t=0.
2+ 2+ 3 2

3) As dolomite replaces protodolomite, the aCa x aMg x a C03
must be lower than step 2 above. Thus there is a general decrease in
the dolomite saturation state with time.

A theoretical explanation for the observed sequences of metastable

phases precipitating prior to dolomite precipitation is given in section

IV of this paper.

REFERENCES

33

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, 1940, Kinetics of phase change, II. Transformation-time
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