ABSTRACT

COMPARISON OF LOW-TEMPERATURE WITH
HIGH-TEMPERATURE DIFFUSION OF
SODIUM IN ALBITE

BY

Alan Bailey

These experiments were carried out in part, to test
the defect model for diffusion at lower temperatures.
More specifically, this study was designed to compare low-
temperature with high-temperature diffusion to determine
if these high-diffusivity paths are present in albite.

Diffusion coefficients and activation energies were
determined for sodium in albite at low temperatures by
means of exchange experiments and at high temperatures by
means of the sectioning technique.

From the exchange experiments, the following apparent

diffusion coefficients were determined:

25°C 6.18 X 10-24cm2/sec
45 8.05
75 12.1

The activation energy for the process was less than 5000

cal/mole Na.

Alan Bailey

From the sectioning experiments, a lattice diffusion
coefficient of about 8 x 10-l3cm2/sec was determined at
595°C. Using this and diffusion coefficients determined by
other workers at higher temperatures, an activation energy
of about 45 kcal/mole Na was determined for lattice dif-
fusion. Using the lattice diffusion coefficient determined
in the study and the activation energy for lattice dif-

fusion, diffusion coefficients were calculated for lattice

diffusion at the temperatures of the exchange experiments:

25°C 6 x 10-35cm2/sec
45 8 x 10-33
75 4 x 10'30

From these results and the behavior of the section-
ing curves, it was concluded that low-temperature release
of Na is controlled by diffusion along high-diffusivity
paths in the solid. Strained incipient cleavages, dis-
locations and twin planes are possible high-diffusivity
paths existing in the solid. The results of the study
indicate that these high-diffusivity paths become signifi-
cant at lower temperatures and should be incorporated in
models for alteration in weathering and low-rank meta-

morphism.

COMPARISON OF LOW-TEMPERATURE WITH
HIGH-TEMPERATURE DIFFUSION OF
SODIUM IN ALBITE

By

Alan Bailey

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Geology

1970

@4Q7;)QLI

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation
to:

Dr. T. A. Vogel for assistance in all areas through-
out the course of the study and constant examination of
the problem.

Dr. M. M. Mortland for indispensable advice on the
theoretical aspects of the study.

Dr. B. G. Ellis for advice on the theoretical aspects
of the study and the use of counters in the Soil Science
Department.

Dr. H. B. Stonehouse for advice and evaluation of
the study.

Dr. C. Spooner for reading the manuscript and sug-
gestions offered.

Dr. B. W. Wilkinson, Michigan State University Re-

actor Laboratory, for advice and use of facilities.

ii

TABLE OF CONTENTS

Chapter

I. INTRODUCTION . . . .

Statement of the Problem . . .

Method of Solving. .
Summary . . . . .

II. PREVIOUS WORK. . . .

Introduction . .

Mineral Dissolution Studies . .

Feldspar Studies .
Biotite Studies. .

Diffusion Studies. .

Studies in Which Theoretical

are Derived . . .
Summary . . . . .

III. THEORY . . . . . .

General Statement. .

Theory for Exchange Experiments .

Theory for the Sectioning Experiments

Heterogeneous Diffusion. . . .

Diffusion Coefficient
Effect of Variables on

IV. SUMMARY OF PROCEDURE .
Introduction . . .

Exchange Study. . .
Sectioning Study . .

iii

Diffusion.

Page

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10

l3
l4

l6

l6
16
20
22
29
31

33
33

33
35

Chapter
V. RESULTS . . . . . . . . . .
Treatment of Exchange Data . . .
Concentration-Time Plots . . .

Plots of Concentration vs /—. .
Plot of Log Slope vs l/T . . .

Diffusion Coefficients from Exchange

Process . . . . . . . .
Errors in Exchange Experiments .

Treatment of Sectioning Data . .

y vs x Plots for Sectioning . .
Sectioning Results for 300°C. .

Sectioning Results for Higher
Temperatures . . . . . . .
Errors in Sectioning Experiment .

Activation Energy for Lattice Diffusion.

Lattice Diffusion Coefficients at
Lower Temperatures . . . . .

VI. CONCLUSIONS. . . . . . . . .
Model Suggested by the Study . .

Exchange Evidence . . . . .
Sectioning Evidence. . . . .

Evidence for Heterogeneous Diffusion

Temperature Variation of the Process.

Time Variation of the Process . .

Nature of the High-Diffusivity Paths.

Application to Natural Processes .
VII. RECOMMENDATIONS FOR FUTURE STUDIES .
REFERENCES CITED . . . . . . . . .
APPENDICES
Appendix

A. Derivation of Diffusion Equation for
Exchange. . . . . . . . . .

B. Detailed Description of Procedure .

iv

Page
40
40
40
40
43

44
47

47
47
48

50
54
55
55
58
58
58
58
59
6O
62
64
65
67

69

73

75

LIST OF TABLES

Table Page
1. Summary of Previous Work . . . . . . . l5

2. Temperature Slope Data from Exchange

Experiments. . . . . . . . . . . 43
3. Diffusion Coefficients from Exchange
Experiments. . . . . . . . . . . 46

4. Lattice Diffusion Coefficients at Lower
Temperatures . . . . . . . . . . 57

LIST OF FIGURES

Figure Page

1. Gamma-Ray Spectrum of Irradiated Albite
(Taken 24 Hours After Irradiation) . . . 34

2. Schematic of Equipment Used During Dis-
solution Experiments. . . . . . . . 36

3. Grinder Unassembled and Assembled . . . . 38

4. Plot of PPM Sodium Released vs Minutes for
Albite in .lN Sodium Chloride Solution--

5 Grams Solid/500 ml Solution. . . . . 41
5. Plots of PPM NA Released vs VMins . . . . 42

6. Plot of Log 810 e vs l/T--Slope is in Units
of PPM NA/ M1ns X 10"2 . . . . . . . 45

7. x vs y Plots for Albite at 300°C--Diffusion
Time 24 Hours . . . . . . . . . . 49

8. x vs y Plots for Albite at 500°C--Diffusion
Time 69 Hours . . . . . . . . . . 51

9. x vs y Plots for Albite at 500°C—-Diffusion
Time 154 Hours. . . . . . . . . . 52

10. x vs y Plots for Albite at 595°C--Diffusion

Time 107 Hours. . . . . . . . . . 53

ll. Plot of Log D1 vs l/T--D1 is in Units of
10‘13 cm /sec . . . . . . . . . . 56

12. Plots of Log D1 and Log Db vs Temperature . 61

vi

CHAPTER I

INTRODUCTION

Statement of the Problem

 

This study is designed to determine how feldspars
alter in hydrous environments at temperatures of weather-
ing and low-grade metamorphism. A rate law and an acti-
vation energy are determined for the exchange process in
a simplified laboratory system consisting of an activated
albite in a salt solution. Diffusion in feldspar is also
examined at higher temperatures by the sectioning tech-
nique and diffusion coefficients and activation energies
determined. The results of both parts of the study are
combined to produce a model for alteration in weathering
and metamorphism.

At high temperatures, work by Sippel and others
indicates that diffusion in feldspars at these tempera-
tures is primarily lattice diffusion. Work by physicists
(Lidiard and Tharmlingham, 1959) and chemists (Harrison,
1961) indicate that, at lower temperatures, high-diffusivity
paths may become important in ionic solids.

Models developed for natural weathering include the

incongruent dissolution model of Correns, the model of

DeVore involving ionic groups and straight ionic exchange
models such as that suggested by Fredrickson. These models
do not incorporate the mechanical heterogeneity present in
real crystals. In light of the defect model suggested by
physicists and chemists for low-temperature diffusion, it
might be said that this study is designed to examine the
defect model as applied to feldspars.

Studies of the time and temperature dependence of
geologic reactions are needed to explain nonequilibrium
assemblages such as those found in weathering (feldspar+
kaolinite halloysite or sericite). More important is the
determination of how an assemblage, equilibrium or non-
equilibrium, formed. For an equilibrium assemblage, it
is implied that a certain temperature must have been
attained (perhaps greater than that predicted by equi-
librium thermodynamics--see Fyfe and Verhoogen, 1958)
and, probably that water or other volatiles must have been
present to catalyze the reaction. Obviously, equilibrium
thermodynamics will give us little information on cata-
lysts. It may not predict correct temperatures and will
say nothing about the time-dependence of the reaction or
the manner in which it proceeds. A study of the exchange
reactions involves diffusion and rates and activation
energies are fundamental data needed for any understand-
ing of diffusion in feldspars. Although only albite was
studied, this work should serve as an indication of rates

and energies needed for most feldspars.

Method of Solving

 

Much work has been done on equilibrium thermodynamics
but little on the reactions involved in the systems studied.
For this study the laboratory approach seemed most advis-
able because of the complexity of the heterogeneous re-
actions involved and lack of previous investigation. If
any fundamental information is to be gained, simplified
laboratory systems (such as feldspar—solution systems)
must be examined first.

A large number of such systems, however, could be
chosen for the study. The reaction chosen for this study
was that between albite and 0.1N NaCl solution in the low-
temperature range. This system was chosen because it
represents a simplified analog of a situation found in
nature. The low-temperature range was chosen because it
is experimentally easier to work with and because time can
be expected to play a more important part at low temper-
ature. Further, this exchange system provides a convenient
way of determining diffusion coefficients at low temper-
atures.

In addition to the feldspar-NaCl solution system,
dry diffusion in albite was studied. This was done be-
cause earlier similar work by metallurgists (Gulbrasen,
1943), chemists (Delmon, 1961) and geochemists (Correns,
1961) indicated that solid diffusion might play a major
role. There are, however, many ways in which material may

move through solids. The sectioning technique provides a

convenient way of studying diffusion at higher temper-
atures.

The exchange study involved exchange of non-
radioactive sodium for radioactive sodium (Na24) and, in
addition, the sectioning experiments also made use of
radioactive sodium Na22 so that self diffusion could be
examined. The driving force in such monoelement reactions
is thermal and is not complicated by chemical interaction.

(Data from such systems is more fundamental and can be

interpreted more definitely.)

Summary

In summary, an understanding of the way in which
minerals alter requires information on the time and temper-
ature variation of the reactions. The role of diffusion in
such heterogeneous reactions necessitates a knowledge of
solid diffusion and its variation with time and tempera-
ture. Any reaction, whether low- or high-temperature, must
involve diffusion and, therefore, rates and activation
energies play a major role. Because of the complexity of
heterogeneous reactions and lack of data at this stage,
simplified laboratory studies offer the best possibility

for gaining fundamental information on such systems.

CHAPTER II

PREVIOUS WORK

Introduction

 

A large number of studies have been done on the
alteration of feldspars in the laboratory. Quantitative
studies of the time and temperature variation of the
alteration process are much less numerous. Included below
are laboratory studies on the alteration of feldspar and
biotite which were directed toward time and temperature
variations and the mechanism of alteration (e.g., kinetics).
Biotite is also included because there have been a number
of quantitative rate studies done with this mineral. The
same general principles apply to both k-feldspar and bio-
tite; however, there are basic differences in the specific
mechanisms by which the two mineral groups alter.

Few studies on diffusion in feldspars have been
published and summaries of all are included below. One
study on diffusion in clays is also included, and finally,
included are two summaries of articles which develop theo-
retical models for the alteration of feldspar. The sum-

maries in each section are arranged chronologically.

Mineral Dissolution Studies

 

Feldspar Studies

 

Correns and Engelhardt (1938) studied the release of
A1203, 8102 and K20 from adularia in open systems. The
grain size was generally less than 1 micron and both
dialysis and ultrafiltration were used to separate the
solid from the solution. Temperature and pH were varied.
Using a variation of Fick's lst law, a diffusion coef-
ficient for K through the residual layer was determined

-190m2/sec.

which was of the order of magnitude of 10

Fredrickson and Cox (1954) examined the breakdown of
albite crystals suspended in a bomb for 100 hours. Pure
water and temperatures of 200 to 350°C were used. They
found that the crystal broke into small fragments and gel
and zeolites were formed. The mechanism suggested is a
weakening and expansion of the structure followed by break-
down to fragments, ions and gel. This study represents an
extreme in the studies given here in the sense that there
was complete breakdown of the mineral.

Nash and Marshall (1956) studied the surface chem-
istry of a K-feldspar and plagioclases in closed systems.
They found that the surface properties varied with the
mineral. They say that this is incompatible with theories
that postulate the formation of a completely amorphous

product layer. The release of Na from albite by HCl was

studied as a function of time and an initially rapid

release of Na determined was followed by a decreasing rate
of release. The time span was about 80 hours but only 3
points were used for each curve. Another interesting re-
sult of Nash and Marshall's study, which will be given
here for later use, is a set of curves giving the Na
released from albite as a function of the concentration
of various salts. Above a concentration of about .01
meq/ml the amount of Na released was fairly independent
of the salt concentration.

Correns (1961) summarizes the work done in his
laboratory on the experimental weathering of silicates.
In general, open systems were used. The effects of vari-
ation in flow-rate, pH and temperature are discussed and
some consideration is given to the composition of the
solution used to alter the minerals. The general process
arrived at is as follows: metals are first released
preferentially from the mineral leaving an amorphous layer
of Al 0 and SiO

2 3 2
rate. The further removal of metals is hindered by this

which also dissolves, but at a slower

residual layer more and more as it increases in thickness.
Eventually, a state is arrived at in which the A1203-8i02
removal rate becomes fixed relative to the rate of re-
lease of metals. The residual layer then maintains a
constant thickness. In addition to the above, Correns
also compares his work (dilute solutions, 20°C) with that
of Morey and Chen (loo-200°C, more concentrated solutions)

and concludes that more extreme conditions and closed

systems result in the formation of pseudomorphs rather
than only dissolution. He also argues against weathering
by exchange of alkalis by hydrogen or the hydronium ion
and postulates that the aluminum carries the residual
charge. Part of the evidence cited for this is the mi-
gration of particles of solid in electrodialysis. It is
argued that exchange does not really take place since the
framework is decomposed in the process.

Correns (1962) studied the release of materials from
adularia in a dialysis apparatus with a still attached to
provide a continuous supply of distilled water. Release
of material from albite powder was also studied. Results
are similar to those obtained in the above studies by
Correns and co-workers.

Wollast (1965) studied the release of Si and A1 from
orthoclase over periods of 150-300 hours under varying pH.
The release of Al was also studied over shorter periods
of time. The author reaches the conclusion that, initially,
release of Si and Al is consistent with diffusion from an
altered layer. In a closed system, the Al reaches the
equilibrium value for Al(OH)3 under the given conditions.
The Si is then said to reach a value which is determined
by diffusion into solution and reaction with Al(OH)3 to
form a dehydrated Al-silicate. No work was done with the
release of K or Na. Presumably, the work was done at room

temperature.

Manus (1968) studied the long-term (300 days) release
of constituents from 200-300 mesh perthitic feldspars in
columns at 40°C. Release of alkalis was retarded by a
residual layer. Phyllosilicates were identified in the
products by x-ray diffraction.

Parham (1969) leached potassium feldspars and
plagioclase in an apparatus that recycled the water. The
study was done at 78°C over a period of about 140 days.

By means of the electron micrOSCOpe he observed material
with the morphology of halloysite forming on the potassium
feldspar and possible plates of boehmite on the plagioclase.
The distribution of the product seemed to be controlled by

crystallographic features.

Biotite Studies

 

Mortland (1958) studied the release of K from biotite
in the presence of NaCl solutions. Work was done using
both open and closed systems and rates examined as a
function of a leaching rate, concentration of NaCl, temper-
ature and time. It is concluded that the release rate
is essentially independent of the amount of K left in the
biotite for a large percentage of the removal because the
rate depended on the concentration of K at reactive sites.
The concentration at reactive sites is maintained at a
constant level over a large part of the removal.

Mortland and Ellis (1959) studied the release of

fixed K from biotite that had been artificially weathered

10

to various degrees using 0.1N NaCl and then treated with
KCl to reintroduce the K. The rate of release of K was
found to be dependent on the amount of weathering. Loss
of charge, resulting in a possible change of diffusion
geometry, is suggested as the cause for this dependence.

Ellis and Mortland (1959) studied the release of
fixed K from vermiculite using 0.1N NaCl in an open system.
They found the process to be controlled by film diffusion
with an activation energy of 3550 cal/mole.

Mortland and Lawton (1961) studied the relationship
between K release from biotite using 0.1N NaCl and the
size of the grains. Initially, the small particles lost
a larger percentage of their K. After 50% of the K was
removed, however, the larger particles lost as much as
the fine. The Fez/Fe3 ratios varied with the size and

the alteration.

Diffusion Studies

 

Rosenquist (1949) measured diffusion coefficients
for Pb and Ra in albite and microperthite. The tracer
method was used with a thin layer of radioactive material
in the form of a glass being deposited on the surfaces of
several fragments. The samples were held at a fixed
temperature for a measured length of time. The glass was
then removed with a chisel and the activity of the surface
measured. Activation energies and diffusion coefficients

were determined for three directions in the albite for Ra

11

and one direction for Pb. For Ra at 823°C the following

results are given:

10

2.53 x 10‘ cmz/sec

1(001), D

i<01o), D = 1.33

 

#[100], D = 6.35 *1
Activation energies are 24.3, 24.0 and 29.2 kcal/mole, 3
respectively. For Pb at 873°C and 1(001), D = 4.53 x 10'11 .

cmZ/sec and the activation energy was found to be 43.2
kcal/mole. From the data for Ra, it can be seen that,
while some anisotropy is indicated, it is within an order
of magnitude.

Jensen (1952) measured diffusion coefficients of

12 11

10- cmz/sec and 10- cmZ/sec at 550°C for Na22 in micro-

cline perthite. The thin-layer method was used with a
slip coating of Na2CO3 with Na22 used as the initial layer.
No direction is specified and results are complicated to
some extent by the perthitic nature of the material used.

Sippel (1963) measured diffusion coefficients and
activation energies for the self-diffusion of Na22 in a
number of Na-bearing minerals. Again, the thin-layer was
used. In this case, however, the thin-layer was produced
by deuteron bombardment and was about 20p thick. The

material was annealed for 15-40 hours at fairly high

temperatures and then sectioned by grinding off layers

12

with silicon carbide paper. Most of these sectioning plots
of concentration vs x2 displayed curvature. Sippel attri-
butes this curvature to a combination of diffusion along
internal surface and lattice diffusion. From previous
work, Sippel (1959), using an extremely high resolution
proton scattering technique, concluded that the initial
slope of the sectioning plots gives an estimate of the
lattice diffusion coefficient even though the plots may
show curvature. This assumption was tested by measuring
the initial slope of the c vs x2 plot for polycrystalline
NaCl and fairly good agreement was found with that deter-
mined for single crystals. Using these methods, Sippel

-11

obtained diffusion coefficients of 8.0 x 10 cmz/sec at

-10cm2/sec at 940°C for polycrystalline

850°C and 2.8 X 10
samples of albite. The last value was determined above an
inversion point.

Jensen (1964) summarizes work done by metallurgists
and physicists in a discussion of the role of diffusion in
geologic situations and the factors affecting it. ‘The
effects of structural transformations with temperature,
chemical potential and pressure are emphasized. He also
considers the theoretical derivation of the diffusion coef-
ficient from an atomic model and mechanisms for diffusion.
He arrives at the conclusion that the primary mechanism
for lattice diffusion in the silicates is interstitial

network diffusion. In nature, material is said to move

long distances by movement into crystals along flaws and

l3

breaks. Ultimately, however, the process is controlled by
lattice diffusion.

Lai and Mortland (1968) studied the diffusion of Na
and Cs in plugs of Na-vermiculite and plugs of K-vermicu-
lite. The thin-layer initial condition and sectioning
technique were used to determine diffusion coefficients.
It was found that heterogeneous diffusion occurred in the
Na-vermiculite (which contains internal surfaces). The
heterogeneous nature of the diffusion is indicated by the
curvature of the sectioning plots and the time variation of
the diffusion coefficients. In the K-vermiculite, which
contains little internal surface, homogeneous diffusion
was observed. The results from the Na-vermiculite are
compared with theoretical models for heterogeneous dif-
fusion.

Studies in Which Theoretical
Models Are Derived

 

 

Fredrickson (1951) derives a model for the weathering
of albite in which a layer of crystalline water is said to
form on the surface of the feldspar. This is followed by
interdiffusion of H and Na resulting in the breakdown of
the mineral. The formation of structured water adsorbed
to the surface has been questioned by a number of workers.

DeVore (1957) considers tflua surface crystallography
of the feldspars and arrives at a theoretical alteration
scheme in which chains of tetrahedra are released from the

(100) and (010) surfaces. These chains then polymerize

14

to sheets which are used to construct the various phyllo-

silicates.

Summary

The essential features of these studies described
above are listed in Table 1. In the portion of the table
involving reactions between solid and liquid, the arrange?
ment is approximately according to the amount of dis-
ruption produced in the mineral. In the work of Fred-
rickson and Cox the crystals were partially destroyed and
new minerals with completely different structures pro-
duced. In the work of Nash and Marshall, only the sur-
ficial material was effected. With respect to this scale,
the present study lies approximately between Parham's
study and Mortland's study. The process is exchange as
in the biotite studies. The exchange, however, is a
different type because of the difference in crystal struc-
ture. With respect to the dry diffusion portion, the dry
diffusion experiments carried out in this study are most
similar to those of Sippel, but conducted at much lower
temperatures where different mechanisms may become im—

portant.

 

15

 

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CHAPTER III

THEORY

General Statement

 

This section includes an outline of the theory for
the two sets of eXperiments, a discussion of heterogeneous
diffusion, consideration of a simplified atomic model for
diffusion and a discussion of the major variables effecting

the diffusion coefficient.

Theory for Exchange Experiments

 

Exchange between a solid (feldspar) and a liquid
(salt solution) falls in the province of heterogeneous
kinetics which are notoriously complex. One of the reasons
for this complexity is that, generally, the actual reaction
in the heterogeneous case can occur only at the surface of
contact. The displacement of this surface governs the rate
of reaction and the concept of order used so often in homo-
geneous kinetics has little use here. In the solid-liquid
==§> solid-liquid reaction the extent of reaction is indi-
cated by a quantity, a, which is the fraction of the
original volume of the reacting solid destroyed. The

fundamental rate of reaction is the rate of advance of

16

17

the interface. Since the gross rate observed depends on
the area of the reaction interface, the change in the
geometry of the interface with time is important. An
approximation used in this study is that the area of the
"reaction interface" does not change significantly during
the period of time over which the exchange is examined.

The linear displacement of the interface may be governed

by the rate of nucleation of new solid or it may be governed
by the rate of transfer of matter to or from the interface.
The exact case depends on how thoroughly the solid product
covers the wolid reactant (Pannetier and Souchy, 1967).
Initially, when little product is present, nucleation can
be expected to govern in any reaction. In a situation

such as that arranged in these exchange experiments (iso-
topic exchange), nucleation cannot be expected to play a
strong limiting role since the nuclei are physically and
chemically the same as the reactant and do not require any
addition of energy or attainment of critical size to grow.
The product, however, in this experiment effectively covers
the reactant.

Harrison (in Bamford and Tipper, 1969, pp. 404—413)
has obtained an equation suited for the present exchange
studies by solving Fick's law for one-dimensional self-
diffusion into our out of a semi-infinite solid into a
well-stirred fluid phase. The initial conditional applied
in the solution of Fick's second law is C = C0, x > 0,

t = 0. The boundary condition is C = Cl' x = 0, t Z 0.

18

If the experimental conditions are such that diffusion is
examined for a short period of time and the concentration
of diffusing species is effectively constant, then the
semi-infinite model can be used with the above boundary
and initial conditions. The solution to Fick's 2nd law

(Crank, 1956, pp. 18-19, 30; outline given in Appendix A) is:

(C — Cl) - erf [: X
_ _ ———-1-
(Co C1) 2(Dt)3

 

where C = concentration diffusing species at distance x,
time t.
C1 = concentration diffusing species in the solution.
CO = concentration diffusing species in the solid.

The mass of material passing through a surface of unit area
is given by the integral of the flux with respect to time.

By Fick's lst law, the flux at x = 0 is given by

Letting the concentration in the solution in the initial

9.2
EX

mined. The formula given for the total amount of material

portion of the reaction be small, [ Jx = 0 can be deter-

per unit surface area diffused into or out of the solid by

time t is then given by

t ’5
_ .85.: - 21:.
Mt —J[o D[3x]x = 0 dt _ 2Co[11}

19

where Mt = total amount of material released per unit
area.

CO = initial concentration diffusing species in
solid.

t = time.

If the diffusion control does not begin at time t = 0, then
the integration should not be carried out over the whole

interval. The indefinite integral gives

cO/E/E
Mt = 2 ——————-+ C
.4?
For the total amount of material diffused out of a solid

both sides are multiplied by the surface area of the solid.

cos/B/E
M =2————+c

t,tota1 /T-r-
If the volume into which the diffusion takes place is con-
stant, then the above equation can be put in terms of con-

centration changes:

Ctzzcofl+c
v/F

This indicates that, if the concentration is plotted

against the square root of time, then the slope will equal

cs/E
O

2...____._.
VA?

20

Theory for the Sectioning Experiments

 

One method of determining for material independently
of the above dissolution experiment is by means of the

sectioning method. This method involves the solution of
at 8x 3x

for certain conditions. These conditions are as follows:

Fick's 2nd law, ] (Shewmon, 1963, pp. 6-11),

lxl > 0, C + 0 as t + 0, for x = 0, C + w as t + 0. The
mathematical steps in the solution and application of the
boundary conditions are given in Appendix II of Wahl and
Bonner (1951). Experimentally, these conditions can be
realized by placing an extremely thin layer of the material
to be diffused on a slab of the material in which diffusion
is to take place. For these boundary conditions, the

solution to Fick's 2nd law is given by

 

2
-.3L_
C _ l 4Dt
6-..- e
o 2/nDt
where C = concentration diffusing Species at distance x

after time t.
C = concentration at interface at time t = 0.
D = diffusion coefficient.
t = time.

x = distance into solid.
Taking the log of both sides and combining constants gives

-0.1086

109 C = [ Dt

)x2 + constant .

21

This shows that for a plot of log C vs x2 the slope will
be :L%%§§, and t will therefore determine D from such a
plot. The technique is particularly easy if one uses a
radioactive tracer. Since the concentration of tracer is
prOportional to the activity, one can determine the dif-
fusion coefficient by diffusing tracer into the solid and
measuring the activity of layers after known amounts have
been ground off the surface. Diffusion coefficients at
several temperatures enable determination of activation
energies.

In the method used in this study the concentration
(activity) of a slice at depth x was not determined. In-
stead, the activity of the mineral fragment after a layer

of known thickness was ground off was determined. The

equation given above must be modified to suit this method.

Essentially, the modification involves allowance for
contribution to the measured activity by active atoms
below the surface of the mineral (Wahl and Bonner, p. 74).
Using the exponential absorption law, a layer dx thick and
x below the surface will contribute de = C eml‘IX dx to the
activity measured at the surface. k is the proportion-
ality constant between the activity and concentration and
u is the linear absorption coefficient for y-radiation in
albite. Putting the original expression for concentration
into this and integrating from the surface to infinite

depth in the mineral gives

22

 

2
I. A = O ~f e 4Dt dx
2/nDt o

as the expression for the activity at the surface after
diffusion has occurred. If a layer x units thick is ground

off, the activity is given by

2

x
A New}
II. A = o J[ e 4Dt

X Z/nDt x

 

 

 

x Ao 2Dt m 2
If we let y = + u/DE , I becomes A = —— eU e y dy
Z/DE /? y
Ao 2Dt 0 O
= 7? ep (1 - erf yo) [where yo = + u/DE = u/DE].
2/D—t
Ao 2Dt
In the same fashion II becomes AX = if e11 (l — erf y).
A l - erf
Taking the ratio of I and II,‘---‘E = y . Since

 

A l - erf yO

A A
7% ” Xé-well below the surface and since erf yO z 0,
0 A
erf = l - Xi . This says that y can be determined by
0

measuring A0 and Ax' Going back to the defining equation

 

for); y = .x + u/Dt , it can be seen that the slope of
2/Dt

a plot of y vs x equals with an intercept u/D . This

 

ZVDt
holds for large x. At smaller x there will be a slight

deviation.

Heterogeneous Diffusion
The thin-film solution to Fick's 2nd law given above

assumes homogeneous diffusion. In practice, many solids

23

may have high-diffusivity paths along grain boundaries,
fractures and "free" surface. Metallurgists have found
that diffusion under such conditions tend to occur by
movement along the high-diffusivity paths followed by
diffusion into the lattice from the paths. Fisher (1951)
has developed a simplified model for such a process.
Consider a high-diffusivity slab of thickness 6 imbedded

in a low-diffusivity medium (see diagram).

 

 

w

 

 

The z-dimension will be unity through this treatment. The
high-diffusivity coefficient is designated by Db (b for
boundary) and the low-diffusivity coefficient by D1 (1 for
lattice). At time t equals 0 a thin layer of tracer is
placed on a surface which is perpendicular to the high-
diffusivity slab. The y-coordinate is taken as parallel
the high-diffusivity slab and the x-coordinate as parallel
to the surface. If one takes an element dy of the slab,

how will the concentration change in it with time? The

24

change in concentration in the volume element with time
will be given by the flux of material entering the element
(times the area through which it enters) minus the flux of
material leaving the element (times the area through which
it leaves) all divided by the volume of the element. In
the form of an equation: change of concentration/unit
time = l/volume [area end of element (flux into element--
flux out other end) - area side of element (2 x flux out

side of element)]. In symbols

 

8C 1 aJy
8t ldyé 16[Jy [Jy 3y dy] 2dy x
_ _ BJy _ E'Jx'
_ QY 6
JX and Jy are given by Fick's lst law as
_ _ 3C _ _ BC‘
Jx - Dl[§§] and Jy — Db[7§?]

Putting these in the above equation we get

III —°C' = D —°2C' + ——2 D1 °—C
' at. b ayz 6 8x x = 0

In the above C' is the concentration in the slab and C
the concentration in the lattice. In Fisher's treatment,
he assumes that the tracer in the lattice is primarily

from the nearby grain boundary. Numerical analysis of

the above equation indicated that the concentration in

25

the slab rises at a quickly decreasing rate and spends most

of its time at its current value. This leads to the approxi-

BC'
3t

can simplify the problem to diffusion into slices at right

mate condition that = 0. Using this approximation one
angles to the high-diffusivity slab containing a constant
concentration at a given y (C' not a function of time).

Under these conditions the concentration in each slice is

 

 

 

 

 

 

 

 

 

 

 

given by
IV. C = M(y) erfc (Shewmon, p. 14)
2/Dlt
where M(y) = C'. The problem becomes one of finding M(y).
Placing 2%%XL = 0 and C = M(y) erfc [ in III gives
2/Dlt
2 2 D
0 Db§_fli%)_+ 61 8M(y;:rfc __x__
8y 2|E1t x=0
where
F ) I 2“...
_ x
8M(y)erfc x = M(y) e 2/Dlt
3" z/Dt _ /D1rt
l X—O l L x
L. d __ ddx=0
M()

26

2 2D M(y) 2 2D
III becomes 0 = Dbfli¥+ —L——= wig—+-——l—M(y)
3y G/nDlt 3y Dbd/nDlt

A solution to this differential equation is

2Dl
M(y) = exp - -———————-y
Dbé/nDlt

 

Putting this expression for M(y) back into IV

 

 

2Dl x
C = exp - ————————-y erfc
DbéinDlt 2 Dlt
This gives the concentration as a function of x, y and t.
In order to find the concentration of tracer, x, y and t
must be determined. In order to find the quantity of
tracer in an infinitely wide layer dy high and 1 unit
thick, the above function is integrated over the volume

giving

 

 

2Dl +m x
dQ(y,t) = exp - ———————— y dyjf erf dx
Dbd/nDlt -m 2¢Dlt

Taking the log of both sides gives

2Dl
V. 1n Q(y,t) = - ———————- y + constant

DbélnDlt

This equation indicates that if the logarithm of the

quantity of tracer is plotted against the diffusion
2D1

Dbé/nDlt

coordinate, a straight line with slope -

27

results when dealing with heterogeneous diffusion as
described above. Among other conditions, this solution
will only be good in regions where there is no contribution
from the original surface. A more detailed treatment has
been carried out by Whipple (1954) with less restrictive
assumptions which allow for the contribution from the sur-
face through the lattice to a point x,y. In addition to
being more correct, it shows the time dependence of the
overall distribution of the diffusing species better.
Whipple's solution for the concentration as a function of

x, y and t is

 

 

" -212 11 1

C = erfc n + -3- exp ”4 erfc §-—§—-- BE dT
n o T B
where
1

_""2T _y _Db 5
c-———-. n- , 8—5——1 .

VDlt VDlt l 2/Dlt

It is difficult to get a physical feeling for this un-
wieldy solution. A better physical interpretation can be
achieved by use of a method in which the shape of contours
of the ratio c/cO in a grain and in the vicinity of a grain
boundary is examined. In particular, the tangent of the
angle at which a particular contour intersects the grain
boundary can be taken as a measure of the influence of the
grain boundary. On the next page is a diagram taken from

Whipple.

28

 

 

 

 

 

 

 

 

From this diagram one can see that, for smaller 8 (larger
times), the concentration contours are less distorted.
The diffusion coefficients from sectioning and ex-
change experiments in heterogeneous substances can be
confusing without clear definition of what is being mea-
sured. Harrison (1961) has subdivided heterogeneous
diffusion into three types. Type A is for long times and
corresponds to small thetas in Whipple's diagram. The
effect is that of homogeneous diffusion, but the bulk
diffusion coefficient will be a weighted average of D1
and Db. If DC is the macroscopic, measured coefficient,

then the relationship between Dc and D1 and Db is given by

D=fD

c b + (1 - f)Dl

where f is the fraction of volume that can be assigned to

high-diffusivity paths.

29

Type B corresponds to intermediate times and is best
visualized by means of Whipple's contours.

Type C is for very short times in which the diffusion
distance in the lattice is very small. Here the diffusion
into the bulk solid is primarily along high—diffusivity
paths. The macroscopic diffusion picture is similar to
that in homogeneous diffusion with the exception that
diffusion occurs only through a fraction, f, of the cross-
sectional area. If diffusion coefficients are calculated
by means of an exchange experiment using the relationship
Mt = 2CO[I—DT-T£]11 the D calculated will be an apparent coef-
ficient (Da). The relationship between Da and Db is given

2

by Da = f Db'

Diffusion Coefficient

 

The above discussion has been concerned with macro-
sc0pic diffusion only. However, more complete understand-
ing of the factors involved in diffusion, Fick's lst law
must be considered from an atomic aspect.

As an initial point, consider an interstitial atom
vibrating with frequency v in a crystal lattice (Girifalco,
1964, pp. 38-47). In order to move from one interstitial
position to the next, the atom will have to squeeze between
several neighboring atoms of the lattice. In doing so, it
will have to overcome an energy barrier. The number of
times an atom will do this is given by the probability an

atom receives sufficient thermal energy (this depends on

30

the height of the energy barrier and the temperature
through the Boltzmann equation e-E/RT times the number of
attempts it makes (the vibrational frequency). The jump
frequency, then, is given by F = ve-E/RT.
Such random jumps, however, will give no net move-
ment of matter. For net movement a concentration gradient
must be present. To see this, consider the movement of
matter across a plane midway between two interstitial
planes. The flux of matter from right to left will be

given by J = aFN(R) a = geometric factor, F = jump

R+L
frequency, N(R) = number. In the same way, the flux from

left to right is given by J = aPN(L). The net flux of

L+R
matter across the plane will be given by J = dF[N(L) - N(R)].
This indicates that, if there are no atoms on the left

and a certain number on the right, then there will be a
net movement from right to left until the concentration
gradient is eliminated. This can be accomplished merely
by thermal agitation without any chemical driving force.
If A is the spacing between the interstitial planes and

we are concerned with the movement through lcmz, the above

formula can be put in terms of concentrations

J = aI‘1[C(L) - C(R)]
LC _ 3.32 ' 1
A — 3X on a megascopic sca e
J = aFAZ £9

31

By comparison with Fick's lst law (J = D %%) we find that

e-E/RT' 2e-E/RT e-E/RT.

D = dFAZ. Since F = v D = d1 = DO

Effect of Variables on Diffusion

 

Diffusion-controlled reactions, homogeneous diffusion
and heterogeneous diffusion, and the diffusion coefficient
have been examined. The study involves the effect of
temperature. In general, an increase in temperature gives
an exponential increase in the diffusion coefficient. At
low temperatures only a few atoms have the thermal energy
to squeeze between adjacent ions into a nearby hole, how-
ever at higher temperatures, more atoms have the required
thermal energy and a greater amount of material diffuses.
In certain cases, with lattice diffusion, the quantity E
also contains factors allowing for the readjustment of the
structure after the ions have moved. With heterogeneous
diffusion, then the effects of diffusion become more com-
plex. Generally, activation energies are lower for grain
boundary diffusion than for lattice diffusion, and this
means that an increase in temperature will increase the
lattice diffusion coefficient more than the grain boundary
coefficient. The activation energies for grain boundary
diffusion in metals are about half that for the lattice
diffusion (Shewmon, 1963, p. 171). However, no activation
energies could be found for diffusion along high-diffusivity
paths in silicates. Depending on the type of disruption,

the activation energy could vary all the way down to that

32

for surface diffusion. Consider the following equation

 

D _
2 = Db,o €031 Eb)/RT
Dl Dl,o
(El - Eb)/RT
where El - Eb > 0. When T increases e becomes
smaller and D1 becomes larger relative to Db. At a high

enough temperature, lattice diffusion will become as im-
portant as grain boundary diffusion. Above this tempera-
ture only the effects of lattice diffusion will be seen
because, though both types of diffusion are occurring, the
lattice makes up a much larger volume of the solid than
the grain boundaries. The presence of grain boundaries
will no longer effect the diffusion coefficient. This
effect has been demonstrated with diffusion in single
crystals and polycrystalline samples of silver (LeClaire,
1953). Above 700°C the diffusion coefficients for both
the single crystals and the polycrystalline samples were
found to be the same.

Among the major variables not considered in this
study are pressure, impurities and chemical potential of
all species in a system. The interested reader should

consult Shewmon.

CHAPTER IV

SUMMARY OF PROCEDURE

Introduction

 

The study involved two parts: in one part the ex-
change of activated sodium in albite for nonactivated
sodium from .lN NaCl solution was studied, and in the
second part, self-diffusion was examined in albite and a
self-diffusion coefficient determined by means of the
sectioning technique. A more detailed description of the
procedure used in both parts is given in the Appendix.
The procedure for the exchange part will be summarized

first below.

Exchange Study

 

Thirty-five grams of almost pure albite from Amelia
County, Virginia were ground to a fine powder with a
specific surface area of 1.4 M2/gm and 5 gram portions
irradiated in the MSU reactor for 10 minutes at a flux of
1012 neutrons/cm2 sec. Standard 10 ppm Na solutions were
irradiated at the same time. After 24 hours only Na24

remained active (see Figure l). The 5 grams were then

placed in a semipermeable membrane (along with some NaCl

33

34

 

 

>92 ©5N

vm< z

 

 

>Ommzm

>92 mm._

cm<z

v_<wn_
wQ<Umm

 

 

 

AZOCEOdiE
”Eh“? meOI VN zmxdfiv
mtméq Ow.._.<_0<mm_
uO ZDGHUMQm >4ml<§§<®

 

Elva lNOOD

 

 

35

solution) and the membrane suspended in a reaction cell
containing more solution. In all runs, 5 grams of solid
were used per 500 ml of solution and the reaction cell was
placed in a controlled-temperature bath (see Figure 2).

A stirring bar agitated the solution inside the cell while
rotation of the membrane agitated the sample. To make the
runs, 2 ml of solution were collected from the cell out-
side the membrane, at approximately 1 hour time intervals
and the exact time recorded. Two ml of 0.1N NaCl solution
were used to replace that withdrawn. The samples and
standards were then counted on a y-spectrometer and the
concentration of the sodium released calculated to give
release of sodium as a function of time. Three runs were
made at 25, 45, and 75°C and runs with and without the
membrane were made at 65°C. The solid was centrifuged

out when no membrane was used (a much slower and less
complete procedure). Similar curves were obtained with
and without the membrane, indicating it had little effect

on the movement of sodium.

Sectioning Study

 

For solid diffusion studies, the sectioning method
was used with the thin-layer initial condition. Cleavage
fragments of the same albite as used for the dissolution
experiments were selected and ground (using a wet diamond
lap) on a side opposite a major cleavage until that side

was parallel to the cleavage. The cleavage face was then

36

 

 

mmmdfim
2.512042

mhzmémmdxm 20:540on
@2330 0mm: ._.Zm_2n:30m .10 U_._.<§mIom

$051506. 5

 

 

m musmflm

 

37

ground slightly so that it was exactly parallel to the

back side (as indicated by measurement in several places
with a micrometer). The surface was abraded slightly with
dry carborundum paper. A layer of Na22 was produced on the
prepared surface by clamping it against a very slightly
dampened Millipore filter that had been saturated with a

Na22

Cl solution until an activity of about 700,000 cpm
was obtained and then dryed. After about 1 hour, contact
produced an activity on the surface of lO-20,000 cpm. The
fragment was placed in an oven, the temperature raised
immediately to 300°C and, at a rate of 25°/30 min., from
there to the diffusion temperature. The sample was kept

at the required temperature for a recorded length of time
and removed after decreasing the temperature in the same
way it was increased. The sample was mounted on a brass
plug (for later grinding), the thickness of the sample plus
plug measured with a micrometer, and the activity deter-
mined in a controlled-geometry counter with a NaI detector.
The counts taken before diffusion were taken with the same
counting geometry throughout. After counting, the sample
(mounted on the plug) was placed in a grinder which was
constructed so as to grind off small amounts of material in
layers parallel to the surface of the fragment (see

Figure 3 and the section on grinding in the Appendix).

A layer approximately .1 mil (.0001") was ground off, the
sample cleaned, the thickness of the sample plus plug

measured with a micrometer, and the activity measured in

Figure 3

brass

  
 

  

3 , Sdhplev

GRINDER UNASSEMBLED

 

GRINDER ASSEM BLED

39

the controlled-geometry counter. Since Na22 has a half
life of 2.6 years, no correction was made for radioactive
decay in the solid diffusion studies. The counting-
grinding operation was repeated 4 to 8 times for each
sample and the data used to construct plots of y vs x.
Four diffusion runs were made: one run at 300°C with a
time of 24 hours, two runs at 500°C with times of 69 and
154 hours and one run at 595°C with a time of 107 hours.
In each run three fragments, cut parallel the major
cleavages (010, 001 and 110), were used. The designation
of the cleavages was verified by optical examination of
thin sections of fragments on a universal stage. In the
500° runs the 001 fragments were lost through breakage.

At 595° the 110 fragment was lost.

CHAPTER V

RESULTS

Treatment of Exchange Data

Concentration-Time Plots

 

From the exchange experiments, curves of concen-
tration of radioactive sodium in the solution vs elapsed
time were obtained at 25, 45 and 75°C. These are shown
in Figure 4. The general features include a rapid rise
during the first two or so hours to about 7, 10 and 13 ppm.
This is followed by a decreasing rate of increase over the
remaining 20 or so hours to concentrations of 10, 14 and

18 ppm.

Plots of Concentration vs /t

 

In order to test the results against the exchange
equation given in the theory section, the concentrations
were plotted against the square root of time. Plots for
25, 45 and 75°C are shown in Figure 5. The curves are
fairly linear after the initial portion. Slopes deter-
mined by the least-squares method using values of /E from

17 to about 40 are given in Table 2.

4O

41

 

 

'3. 1|"

 

 

 

 

009 89 m2: 0% 00

a 4 A

D O O D O D
oomm. . a a n a o . ,o_
Dom? o o o o o o d _i _
i
m
i
q q 4 a q a q q . <2 Ema i
HOE . 4 a .

zopiomnsooonom mzqqomll ‘ M
6m

ZOEDJQm Hemoulurééoom Z: Z_mkfi4< dOu
mmejzi w> Qmm<m4mm EDEOm Edd do FOBQ

£111? L

 

 

v mudmflm

 

42

 

ow om .mwem\» Om

m2: \:w> Qmm<mjmd <2 Edd .10 mHOIE

 

00mm

<2 Edd

 

 

0mm

0

 

m musmflm

 

 

43

TABLE 2.--Temperature slope data from exchange experiments.

 

 

Temperature Slope
25°C 6.78 x 10‘8ppm//min
45 7.74
75 9.48

 

It can be seen that the slopes increase with temperature.

Plot of Log Slope vs 1
T
From the theory section, the slopes of the plots

 

described in the last section are equal to
C s/B

_2____

v/F

2

The primary temperature dependence of this term comes from

D. D varies with temperature according to the equation

where D0 is fairly independent of temperature, E is the
activation energy, R the gas constant and T the tempera-
ture in degrees kelvin. The slope can then be written

.2.
2RT

c s/D
o o_
v/F

2 e

44

A plot of the common logarithm of the slope vs % will

therefore equal

 

4.6R '

Such a plot is given in Figure 6. From this an activation
energy of about 2730 cal/mole Na was determined. It must
be emphasized that the exact value obtained for the
activation energy depends upon the model used to derive
the formula for the slope. By looking at the concen-
tration-time curves, however, it can be seen that the
temperature dependence is not strong.

Diffusion Coefficients from
Exchange Process

 

 

In addition to activation energies, the diffusion—
control equation given in the theory section and the
slopes enable calculation of diffusion coefficients.

From the theory section, the slope of concentration
vs the square root of time is equal to
c s/E
_2____

V/T—T

2

As an example of the calculation of D, consider the case

at 25°C:

45

Figure 6

 

 

PLOT OF LOG SLOPE vs V}

“SLOPE IS IN UNITS OF

PPM NA -2
,4MN53X'O

- IOOO

LOG SLOPE

-IOO

 

2£9 30 GM 32 2&5 54
I/% Q4(XIO3 )

 

 

“Jul... 1... .1

46

 

 

 

C S 2 x .223 Jflg x 7 x 104cm2
2 o _ cm
v ” 3
500 cm
= 6.24 x 103 53%
cm
gm
3 ~—
6.78 x 10 3 cm = 6.24 x 103 J£%- %
7min cm

1.085 x 10’11 cm = / 9

 

IT
mln
-22 c_m3 .. 9.
1.18 x 10 min - W
cm2
6.18 x 10'24 sec = D

In like manner, the apparent diffusion coefficients were
calculated for 45 and 75°C. The coefficients are given

in the table below.

TABLE 3.--Diffusion coefficients from exchange experiments.

 

 

Temperature Coefficient
25°C 6.18 x 10'24cm2/sec
45 8.05

75 12.1

47

Errors in Exchange Experiments

With respect to errors in the dissolution part of
the experiments, there are several possible sources within
each concentration-time curve (neglecting errors due to
original weighing and volume determinations). The errors
occur in concentration determination and time determination.
The time error is within a few minutes and is not felt to
be significant. The concentration determination involves
both volume determination and activity determination.
Since the total counts ranged from about 100,000 initially
to about 40,000 after 24 hours, the errors in activity can
be expected to be in the neighborhood of 1%. Again, this
is not significant for the purposes of this experiment.
The major source of errors is felt to be the volume deter-
minations. Even though equal volumes of sample and stand-
ard were collected at the same temperature and the pipet
rinsed after each sample, it is difficult to measure 2 ml
of 0.1 N NaCl solution. This is particularly true at 75°C.
The volume determinations are therefore felt to be the

major source of error.

Treatment of Sectioning Data

y vs x Plots for Sectioning
For the examination of diffusion in solid individual
fragments y was plotted against x for several directions

in the lattice and several different temperatures. Some

48

runs were also made in which time was varied. The effects
of direction were found to be unimportant at the level of
accuracy reached in this part of the study and the curves
in the three directions (001, 010 and 110) are treated as
if they are equivalent. Justification for this approxi-
mation can be found in the work of Rosenquist. In working
in the range of 800-1000°C, Rosenquist found diffusion
coefficients that varied in different directions. The
variation, however, was within an order of magnitude. As
indicated in the section on the discussion of errors, the
diffusion coefficients determined at lower temperature
(595°C) by the sectioning method are felt to be good only
to a little less than an order of magnitude. Therefore the

differences due to direction are ignored.

Sectioning Results for 300°C

The differences that occur when the temperature is
changed, however, are marked and felt to be significant.
At 300°C (see Figure 7) the plots consist of slightly
curved lines with shallow slopes and y-intercepts of .5-.8.
According to the theory outlined for homogeneous diffusion,
such plots of y vs x should give straight lines with
intercepts of u/DE where u is the linear absorption coef-

24 y-radiation (1.3 Mev) in albite, D is the

ficient for Na
diffusion coefficient and t is time. u can be calculated
from mass absorption coefficients for the elements for

1.3 Mev radiation (Chappel, 1956) and the formula for

49

Figure 7

 

 

x v5 Y PLOTS FOR ALBITE AT 300°C——-DIFFUSION

TIME 24 HOURS

 

 

 

 

 

.9I' [:3 E
:2
Y
J. (0 IO)
00 50
08’"
Y
_I_ (OOI)
00 3b
I.I -
Cal—5::
[:23
[:3
y _I_ (HO)
O L...— - .-_ ._.-.._ L _.__.____
O x IXIO“CMI 3O

”7

 

50

albite. When this is done it is found that u 3 .145/cm.
Assuming a diffusion coefficient of lO-llcmz/sec (probably

much too large) and a time of 105 seconds we get

 

 

 

2
Yo = .145 »//10_11 X 105 EE— sec
cm sec
= .145 /10'6
= .145 x 10‘3

This is essentially 0. The initial slope of the 300°C
curves must be quite steep for the curves to pass through
the point 0,0. But the theory for homogeneous diffusion
predicts straight lines. From these considerations it

was deduced that, at 300°C, the curves must represent the
heterogeneous diffusion. Material is moving through the
solid along high-diffusivity paths faster than through the

lattice of the crystal.

Sectioning Results for Higher Temperatures

 

Runs were made at 500 and 595°C. Looking at Figures
8 and 9 we see that, at 500°C, there is still a marked
curvature. The curves, however, have a definite slope
toward the origin, even though strongly curved. Curves
from longer runs at 500°C seem to flatten out even more
(Figures 8 and 9). At 595°C (Figure 10) there are fairly
straight lines which pass through the origin. These
represent fairly homogeneous diffusion. According to

Sippel (1963), estimates of the crystal diffusion

51

 

 

 

 

 

e 368 x 60 e 39.03 x

L L
m
5 e: 66:.
1 y n“
I ”U
1
i,
_ HHHHU

nUIIIU
H H
U
U .JNJ

1 9130: me we:
_ zomfiuaI oooom Z Em? moi $36 > m> x

 

m:

 

m mudmflm

 

52

 

Om

3 ob x x 0 ON 39.9 x x o

 

 

 

 

 

_ O 1 0
Elm > 6514 x
U H. ”U
H.
IO._ IO._
U
U
mdDOI 69 m2:
ZO_mDnE_QIUoOOm .2 m._._m4< doll mHOIE > w> x

 

m omzmflm

 

53

 

 

e sob;
4 1 x co m._ 599x x

so Ifi 6611

I NJ U

 

mEDOI no. 92;
anm3 IIIIIIII
“7:0 Oommm F4 mtmnz mod mHOIE > m> x

1 m._

 

 

3 653m

 

54

coefficient can be obtained from initial slopes of such
plots as the ones obtained at 595°C. From such initial
slopes (lines through 0,0 and the first two points) values

13

of D = 5 X 10- cmz/sec for the direction (010) and

12cmz/sec for the direction (001) were ob-

D = l x 10-
tained. These were calculated to slide rule accuracy,
but are probably only good to half an order of magnitude.
An average of the two values was taken as the value for
the diffusion coefficient at 595°C and was found to be

equal to about 8 X lO-l3cm2/sec.

Errors in Sectioning Experiment

The errors in the solid diffusion curves result from
two sources. These are counting errors and errors due to
the measurement of thickness. The counting errors are on
the order of a percent. Errors in thickness (determination
of x) are fairly large. Thickness was estimated to 0.1 mil
using a Lufkin micrometer. While curves can be found in
which measurements were made to 0.1 mil (.0001 inches)
(Turnbull and Hoffman, 1954) this is an estimate which is

4cm).

felt to be good only to about 1.05 mil (11.27 X 10-
Even with such errors, however, the average diffusion coef-
ficient is felt to be good under an order of magnitude, as

indicated by its compatibility with Sippel's data. This is,

however, working at the limit of the sectioning method

under these conditions.

55

Activation Energy for Lattice Diffusion
Sippel (1963) has determined diffusion coefficient

11 10 2
cm /sec

of 8.00 X 10- cmZ/sec at 850°C and 2.8 X 10-
at 940°C for Na in albite at higher temperatures. The
value at 940°C is said by Sippel to occur above an in-
version point. This inversion, however, is from triclinic
to monclinic albite and, since the triclinic albite is
almost monoclinic, one cannot expect the inversion to
alter the diffusion coefficient much. The logs of the
three values of the diffusion coefficient were plotted
against the reciprocal of temperature (absolute) (Figure
11). From the slope of this plot an activation energy of
about 45.8 k cal/mole Na was calculated. This is similar
to other values obtained for the diffusion of more mobile
constituents in the lattice of silicates.
Lattice Diffusion Coefficients at
Lower Temperatures

Using the activation energy and diffusion coefficient

determined in this study at 595°C, lattice diffusion coef-

ficients were calculated at 25, 45 and 75°C. As a sample

of the calculations consider the case at 25°C:

 

—E/RT
D D e 595 3
1,595 = o _ eE/R(3.36 - 1.15) x 10
D -EET ‘
1,25 D e 25
O
3 3

 

e 45.8 x 10 x 2.2 x 10'

= 850.9
1.99

 

 

 

56

Figure 11

 

7..-..1” 1-.

LIOOO

LOG DL

LIOO

___- _. .___l___._-- ..

PLOT OF LOO DL VS I/T
——-DL IS IN UNITS OF
IO"'3 CMZ/SEC

IL- . l -__.____I_-____

I/T XIO 3/K

-I__ ___._,- .

Io?)

L-

 

 

 

,2__._ ..1_.__.'_——d

57

_ D1,595 _ 7.94 x 10"13 —35 cm2

--—————— — = 6.3 x 10 .
1,25 e50.9 1.26 x 1022 sec

D

 

Below is a table giving crystal diffusion coefficients

at 25, 45 and 75°C.

TABLE 4.--Lattice diffusion coefficients at lower temper-

 

 

atures.
Temperature . Coefficient
25°C 6.30 x 10-35cm2/sec
45 7.94 x 10’33
-30

75 3.97 X 10

 

CHAPTER VI

CONCLUSIONS

Model Suggested by the Study
The results of the study are compatible with a model
for low-temperature alteration in which release of alkalis
is controlled by movement along high-diffusivity paths not
accessible to liquids in the normal sense. The evidence
for this comes from both the exchange data and the dif-

fusion studies.

Exchange Evidence

 

Considering the exchange data first, beyond the very
initial portion, the data for the low-temperature release
of sodium fit a model in which the rate of release is
controlled by diffusion. This is indicated by the para-
bolic rate law (predicted by diffusion-control theory) and
the low activation energy obtained for the process (if the
process were controlled by chemical reaction, a much

stronger temperature dependence would be expected).

Sectioninngvidence
When the high-temperature data for solid lattice

diffusion are considered, however, a high activation

58

59

energy (reasonable for lattice diffusion in most silicates)
is obtained. This stronger temperature dependence means
that, when the Arrhenius equation is used to extrapolate
to the temperatures of the dissolution experiments, dif-
fusion coefficients are calculated which are much smaller
(10 orders of magnitude) than the diffusion coefficients
obtained from the dissolution data. The large difference
between the diffusion coefficients and the activation
energies determined by the exchange experiments and those
lattice coefficients determined by the sectioning method
means that, though exchange is controlled by diffusion,
it is not controlled by lattice diffusion. Control by
diffusion through an adsorbed film of solution is ruled
out by the small values of the coefficients.

Evidence for Heterogeneous
Diffusion

 

 

There are, however, several ways material may move
through a solid. The series of solid diffusion curves
offers evidence as to the processes involved. At 300 and
500°C plots of y vs x gave curved lines, whereas the
theory for homogeneous diffusion predicts straight lines
passing through the origin. These curved lines indicate
heterogeneous diffusion and that high-diffusivity paths
must be important in addition to lattice diffusion. This
is verified by the fairly straight curves passing through
the origin obtained at 595°C. Since the higher activation

energy for lattice diffusion causes it to be more effected

60

by a temperature increase than diffusion along high-
diffusivity paths, a point is reached at which lattice
diffusion and movement along high-diffusivity paths occur
with equal ease. Since the volume of the lattice is much
greater than the volume in high diffusivity paths, the

diffusion becomes homogeneous above this temperature.

Temperature Variation of the Process

The dissolution diffusion coefficients are inter-
preted to be diffusion coefficients where high-diffusivity
paths are dominant and the coefficients obtained at 595°C
to be diffusion coefficients for lattice diffusion. In
Figure 12, the logs of the lattice diffusion coefficients
and coefficients for high-diffusivity paths are plotted as
functions of temperature. To construct the curves, the
activation energies and diffusion coefficients determined
in this study were used with an equation of the type

:r.=_[_1___1_]
D =DeRT Tx

T T
x

At this point it is stated again that the coefficients
from the exchange coefficients are apparent coefficients.
If high-diffusivity paths are operative, then the apparent

coefficients must be modified.

61

Figure 12

 

 

PLOTS OF LOG 0L AND

LOG DB VS

TEMPERATURE

 

l
o 300 T m 600

 

 

 

62

The real high-diffusivity coefficients must be calculated

using the equation

Da = f Db
or

Da

__ _—_ D

f2 b

where f is the fraction of the crystal perpendicular to

the diffusion direction which is taken up by the high-
diffusivity paths. If f is, say .01, then Db will be in-
creased by four orders of magnitude. The line representing

Db therefore represents a minimum.

Time Variation of the Process
The absolute values of the diffusion coefficients,
however, are not sufficient alone to obtain a good picture
of the process. The overall distribution of diffusing
species in the solid can be visualized by examining con-

tours of the quantity c/co in a cross-section of the solid.

 

 

 

 

63

This type of distribution (not predicted by Fisher's treat-
ment) comes about because material will move through the
lattice to a point (x,y) from the surface in addition to
moving from the grain boundary. The angle theta is a
measure of the relative importance of movement along the
high-diffusivity path. The shape of a particular contour
varies with time as well as the ratio Dl/Db and this vari-
ation is indicated by a diagram, taken from Whipple's
paper, in which the tangent of theta is plotted as a

D
function of B = [—2 - l] 6

 

D1

tane

(after
Whipple, 1954).

 

 

 

Since tane is inversely proportional to /E longer times
will produce smaller thetas. The contours will not be as

distorted by the high-diffusivity path. tane, however, is
D
directly proportional to [-2 - l]—i—. For very small D

D1 .51
there will be large thetas and

l
and relatively large Db’

the concentration contours will be greatly distorted.

64

Looking at the plots of D1 and Db vs temperature, the
hypothesis can be made that, as lower and lower tempera-
tures are approached, it will take more and more time to

overcome the increasing difference between the diffusion

coefficients.

Nature of the High-Diffusivity Paths

The low-temperature diffusion control, heterogeneous
nature of the diffusion and its dependence on temperature
and time have been discussed. What can be said about the
nature of the high-diffusivity paths? The most obvious
high-diffusivity paths are incipient cleavages (010, 001
and 110). Even without any disruption, these directions
would have higher diffusivity for alkali ions due to the
higher densities of these ions on these planes. Exami-
nation of thin sections of the material, however, shows
the presence of many incipient cleavages. Some disruption
is therefore present. It seems reasonable that this type
of disruption is also present in the grains of the ground
sample used for the dissolution experiments. Along these
partially disrupted cleavage planes activation energies
would be lowered by separation of the neighboring ions and
general disruption. Dislocations and twin planes are
other possible high-diffusivity paths. Strain, which
would be localized along these defects may also play a
role. These factors may be enhanced in the exchange

experiments by the grinding.

65

Application to Natural Processes

The results of the study indicate that relatively
rapid movement of ions along high-diffusivity paths (not
accessible to solutions in the normal sense) is possible.
With this model the extensive reaction of inert silicates
in diffusion-controlled situations can be explained.

In this study, the situation has been one of pure
exchange in a short period of time with the mineral main-
taining its structural integrity. In nature, diffusion
and chemical attack, such as that described by Correns,
may occur. Even in this situation, diffusion-controlled
reactions occur. Diffusion may occur before much dis-
truction of the lattice occurs and the completely open
system of Correns is not correct because residual material
in the alteration of a feldspar may form an inert product
(kaolinite, gibbsite, illite, feldspar) which may cover
the reacting material. These situations give a diffusion-
controlled situation similar to that in this study. If
the product is another silicate similar to albite (another
feldspar), then the situation will be very similar to the
one in this study. Diffusion controlled reactions can
also be envisioned in the formation of oriented products
seen scattered throughout grains in thin sections of rocks.
Part of this orientation is due to the location of favor-
able nucleation sites, but a diffusion problem still

exists.

66

In conclusion, natural application can be found for
the principles demonstrated by this study. They clarify
the reaction and the effects of temperature and time on
such situations. They help explain extensive alteration
of inert silicates under mild conditions and are of use in
considering diagenesis in sediments and the formation of

metamorphic rocks.

CHAPTER VII

RECOMMENDATIONS FOR FUTURE STUDIES

Several areas for further study are indicated by the
present work. One is a more detailed examination of im-
perfections in silicates (dislocations, grain boundaries,
microscopic and submicroscopic cleavages). Autoradio-
graphy, electron microscopy and etching are possible tech-
niques that could be used.

Another is the quantitative examination of strain
induced by grinding in the exchange process. This might
be studied by examining exchange rates after annealing to
remove the strains or by examining rates of release as a
function of grinding time. Determination of the specific
surface would be necessary to remove the effect of in-
creased surface area in the second method.

The quantitative examination of interdiffusion is
a more complex extension of the present study. This could
be carried out by changing the composition of the solution
in the exchange experiments. Exchange in .l N KCl is an

example.

67

68

A study of the rate of redistribution of radioactive
tracer between two solids in aqueous solution is still

more complex, but closer to the natural situation.

REFERENCES CITED

REFERENCES CITED

Chappell, D. G. (1956). Gamma ray attenuation. Nucleonic,
vol. 14, pp. 40-41.

Correns, C. W. and W. Engelhardt. (1938). Neue Unter-
suchungen ifixn: die Verwitterung des Kalifeldspates.
Chemie der Grde. Bd 12, pp. 1-22.

Correns, C. W. (1963). Experiments on the decomposition
of silicates and discussion of chemical weathering.
Clays and Clay Minerals, Tenth Conf. pp. 443-459.

Correns, C. W. (1962). Uber die Chemische Verwitterung
von Feldspaten. Norsk Geol. Tidsskr. Bd 42 (part 2),
pp. 272-282.

Crank, J. (1956). Mathematics of Diffusion. Oxford Uni-
versity Press.

Delmon, B. (1961). L'Etude des réactions limitées par
1es processus d'interface. Revue de L'Institut
Francais du Pétrole. Vol. 16, pp. 1-57.

DeVore, G. W. (1951). Surface chemistry of feldspars as
an influence on their decomposition products. Clays
and Clay Minerals. Vol. 6, pp. 26-41.

Ellis, B. G. and M. M. Mortland. (1959). Rate of
potassium release from fixed and native forms. Soil
Sci. Soc. Amer. Proc. Vol. 23, pp. 451-453.

Fisher, J. C. (1951). Calculation of diffusion pene-
tration curves for surface and grain boundary
diffusion. Journal of Applied Physics. Vol. 22,
pp. 74-77.

Fredrickson, A. F. (1951). Mechanism of Weathering.
Bull. GSA. Vol. 62, pp. 221-232.

69

70

Fredrickson, A. F. and J. E. Cox. (1954). "Solubility"
of albite in hydrothermal solutions. Am. Miner.
Vol. 35, pp. 738-749.

Fyfe, W. S., F. J. Turner and J. Verhoogen. (1958).
Metamorphic reactions and metamorphic facies. Geol.
Soc. America Memoir 73, pp. 1-259.

Girifalco, L. A. (1964). Atomic Migration in Crystals.
Blaisdell Publishing Company.

Harrison, L. G. (1961). Influence of dislocations on
diffusion kinetics in solids with particular refer-
ence to the alkali halides. Trans. Faraday Society.
Vol. 57 (part 2), pp. 1191-1199.

Harrison, L. G. (1969). The theory of solid phase
kinetics. In Comprehensive Chemical Kinetics, edited
by C. H. Bamford and C. F. H. Tipper, Vol. 2, pp.
377-462, Elsevier Publishing Co.

Hoffman, R. E. and D. J. Turnbull. (1951). Lattice and
grain boundary self-diffusion in silver. Journal
of Applied Physics. Vol. 22, pp. 634-639.

Jensen, M. L. (1952). Solid diffusion of radioactive
sodium in perthite. Am. J. Sci. Vol. 250, pp.
808-821.

Jensen, M. L. (1964). The rational and geological aspects
of solid diffusion. Canadian Mineralogist. Vol. 8,
pp. 271-290.

Lai, T. M. and M. M. Mortland. (1968). Cationic diffusion
in clay minerals: I. Soil Sci. Soc. Amer. Proc.
Vol. 32, pp. 56-61.

Lidiard, A. B. and K. Tharmalingam. (1959). Diffusion
processes at low temperatures. Disc. Faraday Soc.
No. 28, pp. 64-86.

Manus, R. (1968). Experimental chemical weathering of
two alkali feldspars. Ph.D., University of
Cincinnati. pp. 1-100.

Mortland, M. M. (1958). Kinetics of potassium release
from biotite. Soil Sci. Soc. Amer. Proc. Vol. 22,
pp. 503-508.

Mortland, M. M. (1959). Release of fixed potassium as a
diffusion controlled process. Soil Sci. Soc. Amer.
Proc. Vol. 23, pp. 363-364.

71

Mortland, M. M. and K. Lawton. (1961). Relationships be-
tween particle size and potassium release from bio-
tite and its analogues. Soil Sci. Soc. Amer. Proc.
Vol. 25, pp. 473-476.

Nash, V. E. and C. B. Marshall. (1956). The surface
reactions of silicate minerals, Part I. The reaction
of feldspar surfaces with acidic solutions. Univ.
Missouri Coll. Agr. Res. Bull. 613, pp. 1-36.

Nash, V. E. and C. E. Marshall. (1956). The surface
reactions of silicate minerals, Part II. Reactions
of feldspar surfaces with salt solutions. Univ.
Missouri Coll. Agr. Res. Bull. 614, pp. 1-36.

Pannetier, G. and P. Souchay. (1967). Chemical Kinetics.
Elsevier Publishing Co.

Parham, W. (1969). Formation of halloysite from feld-
spar: low temperature artificial weathering versus
natural weathering. Clays and Clay Minerals.

Vol. 17. pp. 13-22.

Rosenquist, I. T. (1949). Some investigations in the
crystal chemistry of silicates. I. Diffusion of
Pb and Ra in feldspars. Acta Chemica Scandinavica.
Vol. 3, PP. 569-583.

Shewmon, P. G. (1963). Diffusion in Solids. McGraw-Hill
Book Company.

Sippel, R. F. (1959). Diffusion measurements in the
system Cu-Au by elastic scattering. Phys. Rev.
Vol. 115, pp. 1441-1445.

Sippel, R. F. (1963). Sodium self diffusion in natural
minerals. Geochim. Cosmochim. Acta. Vol. 27,
pp. 107-120.

Wahl, A. C. and N. A. Bonner. (1951). Radioactivity
Applied to Chemistry. John Wiley and Sons.

Wang, J. H. (1951). Radioactivity applied to self-
diffusion studies. In Radioactivity Applied to
Chemistry, edited by Wahl and Bonner, pp. 62-81,
John Wiley and Sons.

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Vol. 45. pp. 1225-1236.

72

Wollast, R. (1967). Kinetics of the alteration of k-
feldspar in buffered solutions at low temperature.
Geochim. Cosmochim. Acta. Vol. 31, pp. 635-648.

APPENDICES

APPENDIX A

DERIVATION OF DIFFUSION EQUATION

FOR EXCHANGE

APPENDIX A

DERIVATION OF DIFFUSION EQUATION

FOR EXCHANGE

Consider Fick's 2nd law for the case described in

the text and the following initial and boundary conditions:

ll
0

C = C , x > 0, t

x = 0, t Z 0

o e f(t)dt] on both

Using the Laplace transform [f(p) =Jf

sides of Fick's 2nd law gives

00 2 co
I‘f e-Pt3_£dt_1fept§£dt=o
2 D at
0 3x 0

Since the order of differentiation and integration can be

interchanged, the first integral becomes

2 m 2—
II. —3—2- ceptdt=-3—§-
3x 0 8x

Integrating by parts, the second integral becomes

73

A general solution to this equation is

//P J/P
—-x - —'x C
D D "E

o
kle + k2e + 3' --
k1 = 0 since CVf+ w as x + m
C - C C
k = l 0 since 5': —£ at x = 0
2 p p
P
c — c -//: x c
then C = [ l ] e D + —9
P P

this corresponds to

 

c = (cl - co) erf C ‘x + cO
2/Dt

in a table of Laplace transforms and can be rearranged to

 

APPENDIX B

DETAILED DESCRIPTION OF PROCEDURE

APPENDIX B
DETAILED DESCRIPTION OF PROCEDURE

Exchange Diffusion Study

Sample Preparation

Approximately 35 grams of unaltered albite from
Amelia County, Virginia were found in a tungsten carbide
mortor in portions of a few grams. The resulting powder
was placed in a plastic jar and shaken with a tungsten
carbide ball until thoroughly mixed. The specific surface
area of a portion was then determined by nitrogen ad-
sorption and use of the BET equation and found to be about
1.4 Mz/gram. The composition of this albite was essen-
tially that of pure albite (NaAlSi 08) with about .25% K.O

3 2

and .20% CaO as determined by x-ray fluorescence.

Solution Preparation

 

Approximately 6 liters of 0.1N NaCl were prepared by
placing 5.8448 grams NaCl in a 1 liter volumetric flask
and bringing to volume with distilled, deionized water.
This operation was repeated six times to give six liters

which were all placed in a five-gallon polyethylene bottle.

75

76

The pH as measured using a Photovolt Digicord pH meter

with a D 3204 glass electrode was 5.57 at 23°C.

Apparatus

 

The sample cell consisted of a 1 liter polyethylene
bottle with the upper portion cut off at the shoulder and
plugged with a number 14 rubber stOpper in which two large
holes were drilled. During the runs the bottle was placed
in a water bath, the temperature of which was controlled
by a Lauda/Brinkmann model K-2/R circulator. A teflon-
coated stirring bar was placed inside the cell and a
thermometer in a water-filled bottle set in the bath to
determine the temperature. The sample itself was contained
(with some solution) in a cellophane semipermeable membrane
which was suspended through one of the holes from a stirrer.
The sack was used to keep the solid particles out of the
solution collected. One run was made without the membrane,
indicating it had little effect on the movement of Na.

When no membrane was used the solid was centrifuged out,

as indicated in a later description of the experimental
runs. The material inside the membrane was agitated by
rotation of the membrane, while the stirring bar agitated
the solution outside the membrane. Samples of the solution
were collected by inserting a 2 ml pipet through the other
hole in the stopper. A diagram of the apparatus is shown

in Figure 2.

77

Experimental Runs

 

Five grams of albite and several 7 m1 portions of
10 ppm NaCl solution were placed in polyvials and irradi-
ated for 10 minutes in the MSU reactor at a flux of 1012
neutrons/cmzsec. Previous examination of the spectrum of
the irradiated albite using a high-resolution [Ge(Li)]
detector indicated that, while Si and Al were activated,
only sodium remained activated 24 hours after the irradi-
ation (Figure 2). About 24 hours after the irradiation
the solid and approximately 200 ml of 0.1N NaCl solution
were placed in a presoaked cellophane membrane and the
membrane suspended in the reaction cell containing about
400 m1 of the 0.1N NaCl solution. In all runs 5 grams
of albite were used per 500 ml of NaCl solution. The
magnetic stirrer and stirrer from which the sample was
suspended were then started. The polyvials containing the
irradiated NaCl solution were all emptied into a 100 ml
volumetric flask and the flask suspended in the water
bath. A 100 ml flask of unirradiated NaCl solution was
also placed in the bath. To collect a sample of the
solution, both stirrers were stopped and 2 m1 of the
solution withdrawn from the cell, outside the membrane,
using a 2 ml transfer pipet inserted through the hole in
the stopper. The sample was placed in a labeled plastic
counting tube, the tube capped and the time recorded.
Two ml of unirradiated solution were used to replace the

solution withdrawn. The position of the surface of the

78

solution in the bottle with respect to a reference line
drawn on the side of the bottle was noted to detect loss
due to evaporation. If any decrease was noted, deionized
water was added to maintain the volume. Two m1 of the
irradiated 10 ppm NaCl solution were also placed in the
counting tubes. After 10 or so samples had been collected,
the samples, standards and background were counted on a
Packard automatic y-ray spectrometer. Each sample was
counted for 10 minutes using a voltage of 8.1, a gain of
9.5% and the counter windows open (since only the activated
sodium was present). The counting sequence was: empty
tube--sample--standard--empty tube. The concentration of

released Na was calculated using the following formula:

concentrated = (activipypsample--background) concentration
sample (activity standard--background) standard
Runs were made at 25, 45 and 75°C.

As mentioned above, one run was made without the
membrane in order to ascertain possible effects of the
membrane. This was carried out at 65°C using a centrifuge
to remove the solids. The same experimental setup except
that 5 grams of albite was placed loose in 5000 ml of
solution in the cell. The suspension was agitated with
both a magnetic stirring bar and with a stirrer with a
paddle. To collect a sample, both stirrers were stopped,
the time recorded and 5 ml of the suspension withdrawn.

The stirrers were then started again and the 5 m1

79

centrifuged in a bench centrifuge at about 1275 rpm for
10 minutes. The centrifuge tube was then carefully removed
and the top 2 ml of solution pipeted off. The 2 ml was
clear and no turbidity could be seen. The 2 ml samples
and standards were then counted as described above. The
remaining liquid in the centrifuge tube was poured off
the solid in the bottom and 5 ml of fresh 0.1N NaCl
solution at the bath temperature added. The solid was
brought back into suspension using a teflon—coated stirring
rod and the suspension poured back into the sample cell to
restore the lost solid and solution.

With the membrane, samples were collected every hour
for 24 hours. Without the membrane, samples were collected
every two hours for 24 hours. The time-concentration and

temperature-rate relationships were then examined.

Sectioning Diffusion Study
The procedure used in the study of diffusion in the
solid will be described here. To determine diffusion coef—
ficients for the albite, the method outlined in the theory

section for homogeneous diffusion was attempted.

Sample Preparation

Small fragments with the largest cleavage surface
parallel one of the major cleavages (010, 001, and 110)
were selected so as to be as free of fractures as possible.

The fragments were prepared by first grinding the side

80

opposite the cleavage on a diamond lap in water until it
was approximately parallel the cleavage surface on the
front side. Since the cleavage surfaces were not perfect,
it was necessary to grind a small portion off the cleavage
and as parallel to it as possible. The thickness of the
resulting fragment was then measured in several places
with a micrometer (1.0001 inches) to make sure the back
side was parallel the front side. The front side was then
abraded slightly with 400 mesh carborundum paper dry and
wiped with a Kimwipe to remove the dust. The resulting

2 and about

fragments had flat surfaces about 5 x 10 mm
parallel the major cleavages. At this stage the sample
was ready for a layer of tracer to be placed on it. The
transfer of a layer of Na22 was made in the following way.
Rectangles (10 X 20 mmz) of Sn Millipore filter paper were
cut and taped to standard petrographic slides. The filter
paper was then saturated repeatedly with a weak solution
of NaClZZ. Ten drops were placed on each slide over a
period of 24 hours. The final activity of the slide was
about 700,000 counts/minute. The slides were allowed to
dry completely. The transfer was then made by clamping
the mineral fragment between the d0ped filter paper on

the slide and another petrographic slide in a small c-
clamp. This was done in such a way that the dryly abraded
surface was pressed firmly against the active Millipore

filter. The Na22

23

in the filter was then exchanged with

the Na in the surficial layer of the albite. It was

81

found that little transfer could be made when the filter
was completely dry. The transfer was attempted again after
dampening the paper by spraying a fine mist of distilled
water in the air over the paper using a plastic atomizer.
The paper still had a dry appearance, but the transfer was
as efficient as when the paper was moistened by placing a

small drop of the Na22

Cl solution on the paper with a
dropper. The transfer time was an hour to two hours and
the activity of the mineral surface resulting from such a
treatment was in the range of lO-20,000 counts/minute
(background was about 1200 counts/minute). The best
transfer seemed to occur when the mineral was clamped most
squarely against the filter paper as indicated by the in-
dentation in the paper after removal. By grinding off
several layers of the mineral before much diffusion had
occurred (right after doping and mounting) it was found
that the activity was primarily in the top several microns
of the mineral. After transfer and before measuring the
activity, the surface of the mineral was cleaned with a
dry Kimwipe to remove any NaCl crystals. Measurement of

the activity before and after such cleaning indicated

little decrease in activity.

Experimental Runs and Apparatus
After the above preparation, the sample was placed
in an oven and the temperature increased slowly to the

temperature at which diffusion was to be examined. It

82

was found that the temperature of the oven could be in-
creased rapidly to 300°C. Above this temperature, however,
rapid increase of the temperature caused the fragments to
break. A rate of increase of 25°C per 30 minutes was found
to be a reasonable compromise between a slow rate of temper-
ature increase and a short period of variable temperature
relative to the period under which diffusion in the mineral
was to be examined. Fragments parallel the 001 cleavage
still broke at 500°C however.

After diffusion for a given period of time, the
temperature was reduced to 300°C at a rate equal to the
rate it was increased at and the sample was removed from
the oven. As mentioned in the section on homogeneous dif-
fusion, one needs activity as a function of depth in the
mineral. In order to do this, layers of the mineral had
to be ground off parallel the doped surface. A device
was machined from brass and aluminum to attain this end.
In general, it consisted of a brass well which contained
a disc of 400 mesh carborundum paper mounted in the bottom
and an aluminum plug which fitted in the well. The bottom
of the well could be removed to change the carborundum
paper. A hole was machined in the end of the aluminum
plug (offset from the axis of the plug) in which a small
brass with mounted sample could be placed. A picture of
the device is given in Figure 3.

The mineral fragment was mounted on the brass plug

using epoxy and the epoxy cured in a low-temperature oven

83

at 60°C for 45 minutes. After mounting, the activity was
measured using a Nuclear-Chicago Model 8725 Analyzer/Sealer
with a standard NaI scintillation crystal. The sample on
the plug was placed under the crystal in a plastic stand
in such a way to maintain the same geometry with respect
to the crystal each time. The stand was located in a

lead "house" to lower the background. The activities
measured before diffusion were measured in exactly this
way with the exception that the fragments were not epoxied
to the brass plug, but merely set on it. New brass plugs
had to be machined whenever a fragment was epoxied to the
plug.

After measuring the activity, a layer was ground off
in the device described above by placing the brass plug in
the aluminum plug and then placing the aluminum plug in
the well. The face of the mineral fragment then rested
against the carborundum paper. When the plug was turned,
material was ground off the mineral fragment. All grinding
was done with methanol in the well to prevent radioactive
dust from being thrown into the air. One or two turns
were generally sufficient to remove .0001-.0002 inches of
material. This was determined by measuring the change in
thickness of the sample and plug with a micrometer after
grinding. The thickness could be estimated to .0001
inches and, by taking several measurements, the parallel
nature of the surface could be verified. Before measuring

the activity and after grinding, the fragment was cleaned

84

carefully with a Kimwipe and methanol to remove dust. All
tissues and abrasive were carefully discarded in radio-
active waste containers while wet so as to not allow
radioactive dust into the air.

After the above grinding, the activity and background
were measured again. From such activity-thickness measure-
ments, plots of x vs y were obtained as outlined in the
section on homogeneous diffusion. The first three runs
were made using 010, 001 and 110 fragments at 300°C for
24 hours. Runs were then made with 010, 001 and 110 frag-
ments at 500°C for about three days and at 500°C for about
six days. During these runs the 001 fragments broke. Four
fragments were sectioned (two after three days, two after
six days) and plots made of x vs y. Runs were then made
at 595°C using 010, 001 and 110 fragments. These runs were
about five days in length and the 110 fragment broke.
Again, the fragments were sectioned and plots made of x

vs y.