-2 Cm 

1

Dynamique de précipitation de particules minérales en solution :

approche théorique et modélisation

C. Noguera C. Noguera

Institut des

Institut des NanosciencesNanosciences de Paris (France)de Paris (France)

Action soutenue par ANR PNANO

GDR Agrégats Atelier «Nucléation » Toulouse, 3-4 Novembre 2010

2

100 nm

20 nm

Hématite -Fe2O3

TiO2 rutile

Brookite TiO2

Soft Chemistry Size tailoring of oxide nano-particles

by precipitation in aqueous medium J. P. Jolivet et al.

2 3 4 6 7 pH

5 7 9 11

Mean diameter (nm)

0.0

0.1

 Cm-2

0.2

(a)

(b)

3

Evaporation

Atmospheric Pollution

Diagenesis Uses

Rejections Rains

Run off

Wells

Hydrothermalism

T

Deep

Infiltrations Superficial Infiltrations

Springs

Wells

Sea Water Salt

Lakes

Geochimie:

Cycle de l’eau

Contenusen sels minéraux très divers

des solutions aqueuses dans le milieu naturel - rain water : about 5 to 10 mg/l - spring waters : 10s to 100s of mg/l - sea water : 35 000 mg/l

- salt lakes : hundreds of g/l

4

Water-rock interaction:

- out-of-equilibrium contact between rocks and natural waters

- dissolution of primary minerals; precipitation of secondary minerals - evolution of the solution composition

- transport of the fluid Among secondary phases: clays:

- large surface/volume ratio - variability of composition - specific properties: swelling,

hydration, ion exchange

Geochemical codes

- thermodynamic and kinetic codes for water-clay rock interactions - solid-solution models to account for the variability of composition - BUT no good account of finite size effects

Challenge: improve the description of the first steps of nucleation and growth in solution

SiO₄ tetrahedra

(Si⁴⁺replaced by Al³⁺, Fe ³⁺…)

Al(OH)₂O₄ octahedra (Al³⁺replaced by Mg²⁺, Fe ²⁺…

Interfoliar cations

5 C. Noguera et al J. Cryst. Growth 297 (2006) 180

C. Noguera et al J. Cryst. Growth 297 (2006) 187 B. Fritz et al. GCA 73 (2009) 1340

B. Fritz and C. Noguera Rev. Mineralogy Geochemistry vol 70 (2009) 371

I: Precipitation of minerals of fixed composition - formalism

- similarity with other 1st order phase transitions - some applications

II: Precipitation of minerals of variable composition - formalism

- some applications

6

Mineral/aqueous solution interaction:

i i

iE M ^

 ^

Solubility product

eq eq i i

M K E

i

] [

]

[ ^

 

Ion activity product ]

[

] [ M Q E

i

i i

 



K I  Q

Saturation state

I <1

dissolution I =1

Thermodynamic equilibrium

I >1 precipitation

Out-of equilibrium Out-of equilibrium

7

Nucleation

I > 1 Small nuclei

Large surface/volume ratio ^{ }

 3 3 / 4

4

3

2 





4 ln 





 G nk

T I

Gibb’s energy variation to form a nucleus of radius 

3

3 4

nv 

Bulk term

negative if I >1 Surface term always positive

Competition between surface and volume terms Accretion of growth units:

Statistical approach

or classical nucleation theory AS prepared in a closed system

or results from dissolution of primary minerals

n = number of growth units = « size » of the nucleus Classical nucleation theory

8

Nucleation of non-spherical particles

Equilibrium shapes: from the knowledge of surface energies and adhesion energies, one can reconstruct the crystal habit

(Wulff and Wulff-Kaishev theorems) ⁱ

hi

 ^{is constant}

Boehmite in vacuum and in water

(H. Toulhoat et al.)

Hypothesis that 3D growth shape is identical to 3D equilibrium shape

lat adh bas

W l

e



  / 2



2D limit:

e = cst (→0)

_bas=W_adh/2

_i

_bas-W_adh

9

Nucleation



² 4 ln 





G nk_BT I The maximum of G defines

the size of critical nucleus n*

and the nucleation barrier G*



^*2

* 4

3

 1

G ₃

2 3

*

) ln (

3 32

I T k n v

B

 

Stationary nucleation rate _



 



 

 k T

F G F

B

* 0exp

I_c

Metastable zone

1 < I < Ic : nucleation is thermodynamically allowed but nucleation rate is exponentially small

At I_c, F=1

cZ F

 

Z= Zeldovich factor

Related to the curvature of G=f() (cf transition state theory)

F₀= highly unknown quantity

No data bank for  for most minerals

!

10 2D

growth

Spiral growth On flat face

Spiral growth

Growth

Growth mechanisms:

2D nucleation On flat face 3D growth

on rough surface

General scenario

as a function of saturation index value I

nucleation

t

3D growth

limited by diffusion in AS or

limited by interface reaction

11

 0 dt d 

Growth

3D growth on rough surfaces may be limited by:

• Diffusion of ions in the AS

• Interface reactions

) 1 ( 

 I dt

d 

) 1 ( 

 I dt

d





 _{Results from} the existence of a concentration gradient

Size independent law is valid for

large particles

For small particles, growth rate depends on size

G







 G

resorption (positive) growth

 0 dt d 

2 )) exp(

( 

 



T k I v

dt d

B





I >1 d / dt > 0 I < 1 d / dt < 0

2 ) exp( _*



 T k I v

B



12

) ln (

2 _*

I I T k

v

B

 



ln

2 _*

I I T k

v

B

 



 0 dt d 

 0 dt d 

Small nuclei dissolve Large nuclei grow

SOLUTION

Indirect transfer of matter from small to large nuclei through solution Especially important in CLOSED systems

Growth Ostwald ripening

(Coarsening)

13

) ) ( ln ) (

3 exp( 16 )

(

1 2 3

2 3 0

1

t I T

k F v

dt t dN

B

g

_{ } 

Analysis in terms of population of particles

' ) )

' , ( exp 2

) ' ( ( ) ' , 4 (

)) ( ln (

3 ) 32 , (

1 3

/ 2 3 1

1 2 3 1

1

t dt t T k t v

I t

t v n

t I T k t v

t n

B t

B t 

 



 



 ^  ^ _^

nucleation (algebraic) growth

Nucleation rate:

Total amount of precipitate:

Feed-back effect on the solution (closed system)

1 1

0

1) ( ( , ) 1) 1 (

)

( n t t dt

dt t dN t N

Q

t g Avog







Overall model t=0 t

Nucleation of new particles (if F≠0)

Size evolution of all particles nucleated at times t₁<t Size of particles:

Numerical resolution of these highly non-linear equations

14

The NANOKIN code

Reactant minerals Aqueous solution Secondary minerals

« infinite » size Can only dissolve d/dt = f

Closed system Speciation model

I(t) or I(x,t)

Nucleation growth

At each time we know:

- saturation of AS with respect to primary minerals - activity of all ions in solution

- particle population for each secondary mineral (time of nucleation, size, composition) Potentialities:

- homogeneous or heterogeneous nucleation - two growth modes

- various particle shapes (growth shape assumed equivalent to equilibrium shapes)

15

Similarity with other first order phase transitions

Condensation of droplets in the vapor phase

Cristallization of particles from a melt

P Laplace pressure in a droplet

C. Noguera et al J. Cryst. Growth 297 (2006) 180

A small object is more soluble than a big one

 =P/P₀ excess of pressure wrt gaz-liquid transition pressure P₀

4

ln 



B

T N

Nk

G    



Pressure _effwith which a particle of radius  is in equilibrium

0

) ) (

exp( 2

P P T

k v

B eff





 





Precipitation in an aqueous solution

I=Q/K saturation state of the aqueous solution Saturation state of a solution in equilibrium

with a particle of radius 

K K (  )







4 ln 





 G nk

T I

16 t₁ t

t Full description of particle population

0 10 20 30 40 50 60 Evolution of classes

n(ti,ti)

0 20000 40000 60000 80000 100000 120000 t

0 20000 40000 60000 80000 100000 120000

t

t

t1 Particle size

Saturation Index

Precipitation of a single mineral

(eg SiO2 from an AS saturated with H₄SiO₄)

17

Two scenarios of precipitation

From a supersaturated solution Via dissolution of granite Kaolinite precipitation as a function of initial conditions

Al₂ Si₂ O₅ (OH)₄

18

Precipitation in response to a rock alteration : granite dissolution

(3x10⁷s ~1 year)

19

I: Precipitation of minerals of fixed composition - formalism

-

similarity with other 1st order phase transitions - some applications

II: Precipitation of minerals of variable composition - formalism

- some applications

C. Noguera et al Chem Geol. 269 (2010) 89

B. Fritz and C. Noguera Rev. Mineralogy Geochemistry vol 70 (2009) 371

20

Mineral/water interaction:

Mineral with variable composition







_xB_xC   x A^q  xB^q  C^q A₁ (1 )

AC AC end-end-membermember (x=0)(x=0) BC endBC end--membermember (x=1)(x=1)



 

 A^q C^q

AC BC  B^q C^q



 



 

 k T

K G

B AC

AC exp 



 



 

 k T

K G

B BC BC exp





T

k_B   



BC

AC x G

G x

x

G     

 ( ) (1 )

Mixing enthalpy (zero if ideal SS) Mixing entropy

Change in Gibbs free energy for the formation of the compound

Mechanical mixture ....

) 1

(  

 Ax x Solid solution approach

21

Mineral/water interaction:

Mineral with variable composition



 



 

 k T

K G

B AC AC exp



 



 

 k T

K G

B BC BC exp





T

k_B   

BC 

AC x G

G x

x

G     

 ( ) (1 )

x x

x BC x AC B

x x

K T K

k x x G

K   ^  ^

 



 

 ( ) ¹ (1 )¹

exp )

(

x BC x

AC

x I x

x I

I 



 



 



 





 

 1

) 1 (

Thermodynamic equilibrium:

(dI/dx=0 and I(x)=1) I_BC=x₀ and I_AC=1-x₀

I(x₀)=I_AC+I_BC=1

Change in Gibbs free energy for the formation of the compound Ideal solid solution

Solubility product of the SS

Saturation state of the AS wrt SS

Roozeboom plot: x₀=f(X_aq) X_aq=[B]/ [A]+[B]=I_BCK_BC/I_ACK_AC

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

x

Xaq (Ba,Sr)CO3 Roosenbloom diagram

X_aq=0.5 x₀=0.85

22

Nucleation

Mineral of variable composition A_1-xB_xC

Gibb’s energy to form a nucleus of radius and composition x )

( 4

) ( ln )

,

( x nk T I x ² x

G



 



 

 



 

 

 





 

x n I

x n I

x I

n

ln

ln 1 )

( ln

The critical nucleus is at a saddle point of G(,x):

maximum wrt  minimum wrt x

Composition of the critical nucleus: Size of critical nucleus:

Nucleation barrier and rate Usually x*≠x₀

(1-x) n x n

v(x) = (1-x) v_AC+x v_BC

3

3 ) 4

(x  

nv

NB: if s functionof x, excessquantitiesat the surface

!

23

Growth:

Composition x of the deposited layers

Flux conservation of ions at the particle surface Local equilibrium at the particle-solution interface

Size variation of the particles x depends on time

and on the particle radius

accounts for Ostwald ripening this generates composition profiles



 



 



 

  2 ( ) ( ) ln ( , )

) 1 (

* ln1

ln * I x t

T k

x v v x

x v v x

x v x

B AC

BC BC

AC 



24

Feed-back effect on the aqueous solution (closed system)

Initial conditions:

composition of AS (speciation model)

At time t:

I(t,x) Nucleation of new particles:

their size *(t) their composition x*(t)

Growth of supercritical particles nucleated at t₁<t

composition of deposited layer x(t₁,t) function of I and size

or

Resorption of older subcritical particles:

nucleated at t1<t

Composition of removed layer x(t₁,) Change in the aqueous solution composition

25

Application :

precipitation of (Ba,Sr)CO₃ and (Ba,Sr)SO₄

Center of particles rich in Sr Ba_1-xSr_xCO₃ x=0

witherite

x=1 strontiannite K_BaCO3 ~ K_SrCO3

_BaCO3 ~ _SrCO3

Ba_1-xSr_xSO₄ x=0

barite

x=1 celestite K_BaSO4 << K_SrSO4

_BaSO4 >> _SrSO4

Core-shell particle

Variations of x* on a Roozeboomplot

26

Summary

- formalism for treating precipitation of multiple minerals in aqueous solutions (closed system)

- based on sets of coupled integro-differential equations solved by discrete steps (applied mathematics thesis of Y. Amal)

- extension to precipitation of minerals with variable composition (1 substitution; ideal solid solution)

- already some applications of increasing difficulty

Future: - non-ideality

- multiple substitutions (clays) - reactive transport

Acknowledgements: B. Fritz, A. B. Fritz, A. ClCléémentment (LHYGES Strasbourg)(LHYGES Strasbourg)