Mechanisms and Rate Laws in Electrolyte Crystal Growth from

29 Mechanisms and Rate Laws in Electrolyte Crystal Growth from Aqueous Solution

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Arne E. Nielsen Medicinsk-Kemisk Institut, Panum Institute, University of Copenhagen, Blegdamsvej 3, DK-2200 Copenhagen N, Denmark When electrolyte crystals grow in an aqueous solution with a surface controlled rate following a parabolic or an exponential rate law, the rate-determining step is the integration of the cations at kinks in surface steps. The integration rate constant, or frequency, is about one-thousandth of the dehydration frequency of the cations. The factor, 10 , is assumed to be due to diffusion activation energy. Both the rate laws and the absolute rates observed can be accounted for by calculating the kink density by classical methods, and estimating the adsorption equilibrium constants by means of the ion pair stability constants. The calculated and the observed rates mostly agree within one order of magnitude. The rate-determining mechanism for crystal growth may change between several surface processes and transport processes (diffusion and convection in the liquid phase) when the concentration or the particle size is varied. -3

In g e o c h e m i s t r y , a s i n c h e m i s t r y i n g e n e r a l , a phenomenon i s n o t considered as completely understood u n t i l the e s s e n t i a l e m p i r i c a l f e a t u r e s o f t h e phenomenon (such a s , f o r i n s t a n c e , i t s k i n e t i c s ) a r e a c c o u n t e d f o r i n terms o f a r e a s o n a b l e m o l e c u l a r mechanism, c o n v i n c i n g l y v e r i f i e d by e x p e r i m e n t a l t e s t s . G e o c h e m i s t r y d e a l s p r i m a r i l y w i t h c r y s t a l l i n e b o d i e s , many o f which a r e e l e c t r o l y t e s t h a t have c r y s t a l l i z e d f r o m aqueous s o l u t i o n . The m o l e c u l a r mechanisms o f t h e s e c r y s t a l l i z a t i o n p r o c e s s e s a r e t h e r e f o r e o f g r e a t importance f o r t h e understanding o f geochemical processes t a k i n g p l a c e i n nature. When a c r y s t a l i s growing i n a s o l u t i o n two groups o f p r o c e s s e s a r e always t a k i n g p l a c e , t r a n s p o r t p r o c e s s e s b r i n g i n g t h e d i s s o l v e d growth u n i t s ( i o n s o r m o l e c u l e s ) from b u l k o f t h e s o l u t i o n up t o t h e c r y s t a l s u r f a c e , and s u r f a c e p r o c e s s e s t r a n s f e r r i n g t h e a r r i v i n g growth u n i t s t o t h e l a t t i c e p o s i t i o n s ( 1 - 3 ) . W i t h some s i m p l i f i c a t i o n s we may d e s c r i b e t h e s i t u a t i o n i n t h e way t h a t t h e c o n c e n t r a t i o n o f t h e growth u n i t s i s c i n t h e b u l k s o l u t i o n and c ' i n t h e s o l u t i o n j u s t o u t s i d e t h e c r y s t a l s u r f a c e and any a d s o r p t i o n l a y e r . The t o t a l

0097-6156/86/0323-O600$06.00/0 © 1986 American Chemical Society

In Geochemical Processes at Mineral Surfaces; Davis, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

29.

NIELSEN

Mechanisms

and Rate Laws in Electrolyte

Crystal

Growth

601

d r i v i n g f o r c e f o r c r y s t a l l i z a t i o n , ( c - c ) , where c i s t h e s o l u b i l i t y , i s d i v i d e d i n t h e d r i v i n g f o r c e (c-c') f o r t h e t r a n s p o r t p r o c e s s e s and ( c ' - c ) f o r t h e s u r f a c e p r o c e s s e s ( 1 - 3 ) . The s i m p l i f i c a t i o n s made a r e 1 ) , n e g l e c t i n g t h a t t h e d i f f e r e n t i o n s may have d i f f e r e n t , p e r haps even n o n - e q u i v a l e n t c o n c e n t r a t i o n s i n t h e s o l u t i o n , and 2 ) , n e g l e c t i n g t h a t t h e c o n c e n t r a t i o n v e r y c l o s e t o a growing c r y s t a l may v a r y a l o n g t h e s u r f a c e . s

s

s

Rate C o n t r o l I f c « c '

same

composed

general

although

control

time, and of

features

the

of

treatments

kinetic equation

for

the

(10)

=

k

-F(S)-exp[-K

=

2av. (c y ) in s,ad m

e

or

can

"historically"

controlled

linear

at

on

the

nucleus)

to

in

for

surface

of

differ

only

the units

perfect of

the

growth

but

of

above right

the

typical

nuclei.

nucleation

face

growing

a

concentration

On s m a l l

observed,

it

than energy

critical

form before

model of

growth

free

the

growth

face

adsorbed

(a

surface

i t covers the

nucleus

on

stable

grow.

crystal

nucleation

energy

the

605

Growth

perfect

of

free

called

just

each

on a

groups

Depending is

only

surface

form

higher be

nuclei will

surface

silver

crystal led

surface

supersaturation,

to

Crystal

the

the

small a

formally

size

new

(e.g.

presents

may

group.

of

(1,15).

with

the

will a

face

1949

was

assumed

of

larger

degree

Before

literature

walls

certain

any


in

n u c l e i were

(4,12-14).

due


/(InS)]

e

(7)

where k

K

e =

e

π

F(S) a

2

=

=

S

7

mean =

c

Ύ /3^Τ

^

=

/

Spiral

Step

supersaturation suggest

that

dislocation, ing

to

the

according version

steps and

that

to

this

the

=

γ

is

γ

=

a c. 2

the of

In

(free)

the

steps

parabolic

,V '

layer,

grow

faster

This

lead

the

is

presence

not We

The

shall

small

of

destroyed

infinitely. (7).

at

Frank

(17) a

by

rate

use

the

energy (11)

a

per

tension one

to

screw spread law following

(10)

c s

frequency)

adsorption

7.

from

continues

n

(or

d

(S-1)

2

=

m

k ( S - l )

(11)

2

(y/kT)exp(y/kT)

surface

Equation

i n the

crystals

kind of

i

(10)

6

12

originate

is

/

constant

Equation

equation

9

formality

rate

Many

but

1

diameter

by

O.lav.

where

( l n S )

3

Equation

this

theory

v

/

concentration

edge,

kinetic

2

integration

may a l s o

crystal

of

( S - l )

allowed

(8) (9)

Control.

than

exp(-γ/kT)

2

ionic

see Surface

6

4/3

of

growth σ

in

the

unit

the

in

a

vertical

factors

(S-1)

step. step comes

Using

the

surface, from


the

606

G E O C H E M I C A L P R O C E S S E S AT M I N E R A L S U R F A C E S

net

flux

(growth u n i t s

linearly The

with

other

which

is

rates

we

to

the


that

rate

both

three

the

the

the

Adsorption

the

concentration of

But

it

c

Layer

been

,, ad

γ '

and

Y are

crystal.

the The

5000

that

ing

of

the

ways

of

as

the

As

average

an

the

of

them. of

Equation

be

are

1 where

"competing"

controlled not

growth

known

from

rates

we

find

macroscopic

is

not in

may b e

possible

the

to

measure

adsorption

calculated

by

layer.

means

of

the

values

c

These

may b e

(12)

estimated

from

a

the

the

anion,

respectively,

discussion

is

of

the

that (14)

T

1,1

electrolytes, for

3

for

2,2 all

i n the

between and

the

two

ions

situation, one

Κ

10V_K_^c«l) m ad

I

Q-factors

the

for

K^c ad

constant

and of

equal

ion

(«

K^.,

electrolyte

The

adsorption

Î

sum

approximately

(13)

mol/m

force

adsorbed site.

respectively.

QK

assumption

arranging

in

cases

parallel

the

= Κ "

for

200

between

adsorption

extreme

to

rate-controlling.

K ,c m ad

cation

3

roughly

are

same

equal

will

c ^

constant

, « ad

mol/m

fundamental force

mechanism are be

one

constituent

result

constants

the

.

density

ν

equilibrium

and

is

kink

in

1+10V

electrolytes,

the

the

processes

It

stability

Κ

The

varies

equilibrium)

surface

ions

[XY]

Q =

from

discussion

the

[X][Y]

where

which

(10,18,19)

adsorption

ion pair

X and

in

among

normally

Κ

growing

come

will

Concentration.

ad

where

kink

(solubility

and

surface

are

suggested

equation

the

and the

slower

for

namely

The

the

1

spiral

rate

one

with

transport

which 2

where

surface

growth

faster

the

equations

measurements,

Langmuir

each

=

Parameters

parameters

has

S

denominator

mechanisms,

when

into

for

(S-l)/[(γ/kT)exp(γ/kT)]. and

compared

control,

of

the

of be

time)

zero

the

to

two

rate

should

Estimates In

these

the

unit

and

Consequently

concluded

for

(S-l)

nucleation

of

This

per

^ and becomes

proportional

mechanisms. equal

a (

factor

Surface the

c

estimates

of

crystal

and ions

two

of

the

the

the

an of

0.06,

for

Q and Kj

give

for

1,1

electrolytes

1,2

and

200

for

2,2

electrolytes

(as

consist

(as

surface

at

BaSO

the

molecules

situation and

it

possible (_18*J_9) .

1 m /mol, 3

finally

(18)

KCl)

electrolytes

type.

that to

h y d r a t i o n water ion pair

= 0.002,

for

is,

different

Kj

10

charge

ion pair

crystal

the

same

adsorbed

the

their

2,1 stability

an

for

in

and pair

ion

the

and

1,2 Ion

Q-factors

in

30

2,1

ion pairs

theory

ion

for

account

and

may t a k e

500

electrolytes.

"1 (as

Κ SO

and

BaCl

)


)Ml5) J

29.

NIELSEN

Mechanisms

These v a l u e s o f


Crystal

607

Growth

a r e assumed t o be good w i t h i n f a c t o r s 0.2

to

5.

I n t e r f a c i a l Tension. The i n t e r f a c i a l energy σ between a c r y s t a l and an aqueous s o l u t i o n cannot ( a t l e a s t i n g e n e r a l ) be measured by m a c r o s c o p i c methods. But i t may be deduced from homogeneous n u c l e a t i o n d a t a (20-24). F o r t h e p u r p o s e o f d e t e r m i n i n g t h e edge energy γ = a o one may e i t h e r t a k e the i n d i v i d u a l v a l u e d e t e r m i n e d on t h e a c t u a l s u b s t a n c e ( i f i t i s determined) o r use t h e g e n e r a l c o r r e l a t i o n w i t h t h e s o l u b i l i t y c , e x p r e s s e d f o r i n s t a n c e by (10,18) 2

g

c

Ύ


kT

* , 3 - 0.272 l n mol /mΊ

2.82

(16)

The a c c u r a c y o f t h i s e q u a t i o n was t e s t e d on 37 s u b s t a n c e s and was found t o be b e t t e r than ί 20% f o r 57 p e r c e n t o f t h e 37 s u b s t a n c e s . ( T h i s t e s t has n o t been p u b l i s h e d b e f o r e , b u t t h e d a t a used were p u b l i s h e d as F i g u r e 4 i n R e f e r e n c e ( 10) ).. The I n t e g r a t i o n Rate C o n s t a n t . R e i c h and K a h l w e i t s u g g e s t e d t h a t v-j^n i s e q u a l t o the r a t e c o n s t a n t (or f r e q u e n c y ) v ^ o f d i s s o c i a t i o n o f a water m o l e c u l e from t h e i n n e r c o o r d i n a t i o n s h e l l o f the h y d r a t e d c a t i o n (25,26). They found o n l y p a r t i a l v e r i f i c a t i o n o f t h i s h y p o t h e s i s because o f t o o few d a t a . I t has been shown s u b s e q u e n t l y t h a t i f a l l t h e e s t i m a t e s mentioned above a r e made, a c o n s i d e r a b l e amount o f c r y s t a l growth d a t a may be e x p l a i n e d (18,19) by e s t i m a t i n g the i n t e g r a t i o n f r e q u e n c y as n

3

v. = 10~ u in dh

(17)

where v ^ i s the d e h y d r a t i o n f r e q u e n c y o f t h e c a t i o n s (27-29). The f a c t o r 10~ may be i n t e r p r e t e d as the A r r h e n i u s f a c t o r exp(-E*/kT) f o r an i o n making a d i f f u s i o n jump. V a l u e s o f t h e d e h y d r a t i o n f r e q u e n c i e s o f a l l m e t a l i o n s o f i n t e r e s t have been measured i n v a r i o u s ways (10). n

3

P r e d i c t i o n o f Growth Rates From t h e above s t a t e m e n t s i t f o l l o w s t h a t i t s h o u l d be p o s s i b l e t o d e r i v e t h e growth k i n e t i c s and c a l c u l a t e the growth r a t e o f u n c o n t a m i n a t e d e l e c t r o l y t e c r y s t a l s when the f o l l o w i n g p a r a m e t e r s a r e known: molecular weight, d e n s i t y , s o l u b i l i t y , c a t i o n d e h y d r a t i o n frequency, i o n p a i r s t a b i l i t y c o e f f i c i e n t , and the b u l k c o n c e n t r a t i o n o f t h e s o l u t i o n (or t h e s a t u r a t i o n r a t i o ) . I f t h e growth r a t e i s t r a n s p o r t c o n t r o l l e d , one s h a l l a l s o need the p a r t i c l e s i z e . In t a b l e I we have made t h e s e c a l c u l a t i o n s f o r 14 e l e c t r o l y t e s o f common i n t e r e s t . F o r t h e s a t u r a t i o n r a t i o and p a r t i c l e s i z e we have chosen v a l u e s t y p i c a l f o r t h e range where k i n e t i c e x p e r i m e n t s have been p e r f o r m e d (29,30). The e m p i r i c a l r a t e s a r e g i v e n f o r comparison. F o r t h e c a l c u l a t i o n we used V k

m D

6

= M/p = DV

;

a =

c /r ; m s

(V / V L ) m k

T

= r f o r r < 10 ym

= DV and

1 / 3

(18/19)

c / 10


ym

608

GEOCHEMICAL PROCESSES AT MINERAL SURFACES

en

ω

4J

ο

ω

ω

4->

x

S

α


ω

\

CM

4->

S

α

r—ι

•

EH

ο ο ro CM

^

•

ΚΩ

•

οτ-Η

Ο CM

ΚΩ

LO

ro Ο

ro

LD

CO

Ο

Ο Γ-

Γ ΟΟ Ο Ο

Γ

• ro

Ο en ο CM

ΚΩ

Ο Ο

m ο "

ι—Ι

^

CM • ro

CM ro • Ο

•

Ο m CM ro

Ο

κΩ

Γ ΟΟ • CM

Ο Ο en Γ-

,

H

ο•

ζ

en ro • 00

ΚΩ ι-Ι

r

• ι-Η

Ο

r—1

Ο

•

If)

Ο

CO

ι—ι

Ο Ο in

τ—I

s

Ο Ο m

rH 0

ro

αί Ο

H

e

ro

Ο

s

rH αϊ 4-»

ω >ι

ro

>1

ω

ε

Ο

•

ω m

ο

en r-

Ο

00 m m

ο CM

ro 00

ο

ro

CM

ο ro

Ο ro

Φ

ιο

I

τ-Η

CM

en • CM

ro • ro ro

in ο •

CM

ro

ο

CM

0 •Η

τ-Η

ro • ro

ro •

τ-Η

r—I r—I

•

m

rH

4->

rQ (d EH

ϋ Φ rH

ω

κΩ

r-

CM • en

m en • ο

ο ω

ι—1

:

CM

CM

Η

• 00

ζ

• m CM

00 Ο

1—1 ο

Ο

ro rH U CP

S

0

constant

constant

mass

("molecular

number

of

weight")

mass

transfer

constant 1/3

r

defined

radius

5

saturation

t

time

Τ

temperature

of

ratio

particle =

=

(3ν/4π)

c/c s

ν

particle

ν^

growth

volume

V

molar

γ

edge

6

thickness

rate

«

volume energy

dr/dt =

=

diffusion solution

η

viscosity

of

number

ions

V

in

i

n

t

e

9

r

a

t

i

o

σ

of

ν

of

M/p a

n

in

a

formula

rate

constant, constant

dehydration

rate

ρ

density;

= density

σ

surface

Δρ

layer

("unstirred

layer")

unit or (or

frequency frequency)

difference

(SI for

unit,

s

cation

(SI

crystal-liquid

tension


unit,

s

29.

NIELSEN

Mechanisms


Crystal

Growth

613

Indices, e t c . ad


dh

adsorption

layer

dehydration

D

diffusion

control

e,E

exponential

r a t e law ( p o l y n u c l e a r

in

i n t e g r a t i n g jump

L

l i n e a r r a t e law

m

molar

Ρ

p a r a b o l i c r a t e law

s

solubility

equilibrium,

surface

nucleation)

saturated

S

surface

Τ

transport

control

2

p a r a b o l i c r a t e law ( s u r f a c e s p i r a l

control step)

Acknowledgments T h i s work h a s been s u p p o r t e d b y t r a v e l g r a n t s f r o m t h e D a n i s h N a t u r a l S c i e n c e R e s e a r c h C o u n c i l , and by t h e P e t r o l e u m R e s e a r c h Fund o f t h e American C h e m i c a l S o c i e t y .

Literature Cited 1. Nielsen, A. E. "Kinetics of Precipitation"; Pergamon Press: Oxford etc., 1964. 2. Nielsen, Α. Ε. Croatica Chem. Acta 1980, 53, 255-79. 3. Nielsen, A. E. In "Treatise on Analytical Chemistry"; Kolthoff, I. M.; Elving, P. J., Eds.; Wiley: New York, 1983; Part 1, Vol. 3, Chap. 27. 4. Volmer, M. "Kinetik der Phasenbildung"; Steinkopff: Dresden and Leipzig, 1939. 5. Nielsen, A. E. J. Phys. Chem. 1961, 65, 46-9. 6. Nernst, W. Z. Physik. Chem. 1904, 47, 52-5. 7. Burton, W. K.; Cabrera, N.; Frank, F. C. Phil. Trans. Roy. Soc. London 1951, A243, 299-358. 8. Nielsen, A. E. Acta Chem. Scand. 1959, 13, 1680-6. 9. Mullin, J. W. "Crystallisation"; Butterworths: London, 1972. 10. Nielsen, A. E. J. Crystal Growth 1984, 67, 289-310. 11. Nielsen, A. E. In "Industrial Crystallization 78"; de Jong, E. J . ; Jančić, S. J., Eds. North Holland Publishing Company: Amsterdam, 1979, pp. 159-68. 12. Stranski, I. Ν. Z. physik. Chemie 1928, 136, 259-78. 13. Kaischev, R.; Stranski, I. Ν. Z. physik. Chem. 1934, B26, 317-26. 14. Becker, R.; Döring, W. Ann. Physik 1935, 24, 719-52. 15. Cabrera, N.; Burton, W. K. Disc. Faraday Soc. 1949, 5, 40-8. 16. Bostanov, V. J. Crystal Growth 1977, 42, 194-200. 17. Frank, F.C. Disc. Faraday Soc. 1949, 5, 48-54. 18. Nielsen, A. E. Pure and Appl. Chem. 1981, 53, 2025-39.



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G E O C H E M I C A L PROCESSES AT MINERAL SURFACES

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Mechanisms and Rate Laws in Electrolyte Crystal Growth from

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