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UMIST
olloids
C
rystals &
C
Interfaces
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esearch
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NUCLEATION and CRYSTAL
GROWTH.
Roger J. Davey, Molecular
Materials Centre, UMIST,
Manchester, UK.
Roj’s Golden Triangle
Kinetics
Thermodynamics
Structure
‘From molecules to crystallizers’ Roger Davey and John
Garside, Oxford Chemistry Primers, OUP, 2000
Supersaturation
Nucleation of a crystal
Molecular clustering
Derivation of a rate equation and its extension to
a polymorphic system.
Simple experiments – induction times
Crystal Growth – a surface sensitive process
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supersaturated
Concentration, c
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b
a
metastable
xi
c eq (T)
undersaturated
Temperature,T
Quantifying supersaturation
eq = 0 + RTlnxeq
ss = 0 + RTlnxss
 =  /RT = ln(xss/xeq)
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Some equations (sorry!!) – the critical size
g z  zb g b  z s g s or
g z  ( zb  z s ) g b  ( g s  g b ) z s
define   ( g s  gb ) z s / A
because
Az
2/3
Equilibrium between
monomers and clusters
hence
g z  zgb  A
it follows that
g z  zb  z 2 / 3
zA  Az
G  ( z b  z 2 / 3 )  z(  o  kT ln x ss )
G  zkT ln( x ss / xeq )  z 2/ 3
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Free Energy G
(a)
0
zc(b)
zc(a)
Cluster size, z
(b)
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C o l lo i d s
C r y s t a ls &
I n te rfa c e s
R e s e a rc h
G r o u p
C
CI
Nu c l e a t i o n :a s s e m b l y
p r oce sse s
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experiments
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C o l lo i d s
C r y s t a ls &
I n te rfa c e s
R e s e a rc h
G r o u p
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CI
SAXS
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Continuous flow cell - WAXD
WAXS data recorded for 2,6 nitrobromoaniline
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UMIST
C o l lo i d s
C r y s t a ls &
I n te rfa c e s
R e s e a rc h
G r o u p
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CI
Su l p h a t h i a z o l e
(c) Form III
(a) Form I
(b) Form II
I
II
III
IV
(d) Form IV
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Solubility of small crystals
pb  p f  2 / r
b  eq   o  RT ln xeq (r )
b (T , pb )  b (T , p f )  ( pb  p f )vc
b (T , pb )  b (T , p f )  2vc r
ln xeq (r )  ln xeq ()  2vc / rRT
xeq (r ) / xeq ()  exp2vc / rRT
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Relative solubility, x(r)/x_
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1.08
1.07
1.06
20 mJ/m2
1.05
1.04
10 mJ/m2
1.03
1.02
1.01
1
0.01
0.1
Crystal size r, microns
1
The rate equation
zc A  Ac
K z  [ Ac ] /[ A]
[ Ac ]  [ A]zc exp(Gc / RT )
ln K z  Gc / RT
zc
Gc  (4rc3/3)Gb  4rc2
[ Ac ]  [ A]zc exp( 8rc2 / 3RT )
J  P[ Ac ]  P[ A]zc exp(16 3 c 2 / 3R3T 3 2 )

J  K J exp  BJ  3 / T 3 2

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Nucleation rate, J
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crit.
Supersaturation, 
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Induction times –
a simple measure
of J.
600
Induction time / hr
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400
200
0
0.1
0.2
Supersaturation
0.3
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Nucleation in polymorphic systems
xi
Phase I
Solubility
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xI
xB
Phase II
xII
Ti
Temperature
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

 i  xi  xII xII
 x  xI  xII  xII
J I  K J , I exp[BI /( i   x )2 ]
J II  K J , II exp[BII /  i 2 ]
with the two values of B being given by
the appropriate values of the term 16
3 2/3R3T 3. Both KJ,I and KJ,II are
functions of xeq and T
By defining the dimensionless variables:
a = x / BII1/2, b = (BI / BII) and c = [a / ln
(KJ,II /KJ,I)]1/3
a
If KJ,I > KJ,II, then above some value of
supersaturation, the metastable phase I has the
highest nucleation rate whereas below this value
phase II appears more rapidly.
b
If KJ,II > KJ,I and (1 – a/c)3 < b, the stable
phase II has the higher nucleation rate at all
supersaturations.
c
If KJ,II > KJ,I and (1 – a/c)3 > b, the metastable
phase has the higher nucleation rate only over the
intermediate range of supersaturations.
.
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AI

AII

AII
AI
J
J


b
a
AII

AI
J
c

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Solution chemistry: polymorphic
forms of 2,6,dihydroxybenzoic
acid
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Time = 15 hrs.
Time = 0 hrs.
0.2mm
Time = 30 hrs.
Time = 45 hrs.
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Polymorphs 1 and 2 of 2,6 DHB
1
(Toluene)
2
(Chloroform)
Chloroform
Ideal
solubility
of 2
Toluene
1&2
1&2
So lu b ility (m o l/m o l)
0 .0 1
0 .0 0 8
0 .0 0 6
0 .0 0 4
0 .0 0 2
0
20
30
40
50
60
70
80
o
T ( C)
Solubility of DHB
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1
Form II toluene
Form I toluene
From II chloroform
0.9
1/(Induction Time) (1/mins)
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0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.4
0.6
0.8
1
1.2
1.4
Supersaturation(lnx/xeqII)
1.6
1.8
2
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Toluene
1
Chloroform
2
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Crystal Growth
Kinked,
stepped and
flat faces
Growth in direction 1 > 2
• Flux of growth units to surface exceeds net loss.
• Integration into lattice dependent, amongst others (e.g. molecular
recognition) upon strength and number of interactions.
• Assume linear growth rate, v, of a face is proportional to total
binding energy of growth unit to surface:
vK > vS > vF
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Figure 1. The Morphology of  glycine
taken from Groth5. b = {010}, q = {011}, n
= {110}, m = {210}
Sucrose Morphology
+C
-A
-B
+B
+B
+A
+A
+C
-C
-B
-A
Saccharin
Morphology
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[010]
[010]
[100]
[100]
Form 1
(a)
(b)
Form 2
(b)
(c)
Crystal growth – the importance of mass transfer,
surface integration and temperature.
RG  k r (ci  ceq )
r
RG   cry v
RG  k d (c  ci )
RG  k r


c  ceq  RG k d

r
When r = 1
RG  k G (c  ceq )
with
1
1
1


kG kd kr
Concluding remarks –
You cannot define a crystallisation experiment
without a phase diagram (solubility curve).
In a polymorphic system the relative nucleation
rates of different structures are not predictable.
In a polymorphic system growth rates of different
structures will be different and appearance depends
on J.kG
A structural model for nucleation based on crystal
structure works well – see next lecture.
Finally …….. The bees in Roj’s bonnet
Solubility
Solubility
Polymorph I
Polymorph I
Ttransition
Polymorph II
Temperature
(a)
Polymorph II
Temperature
(b)
Concomitant polymorphs and energies
Disappearing polymorphs and control