Fundamentals of Polymorphism:
The Phase Rule and
Thermodynamic Relations
Lian Yu
University of Wisconsin – Madison,
School of Pharmacy
(608) 263 2263
lyu@pharmacy.wisc.edu
Gibbs
Findlay
Westrum and McCullough
McCrone
Burger
…
[This Erice course] will provide
a. the theoretical basis for the existence of these
diverse structural forms,
b. the methodology to control the form, from the
nucleation to macroscopic growth,
c. the techniques used the characterize the variety
of products obtained,
d. the advantages resulting by this way of surveying
structure/property relations for the design and
preparation of new materials.
a.the theoretical basis for the existence of these
diverse structural forms,
The stability of a polymorph is determined by
G = H - TS, not just H or S.
Energy-entropy compensation is important
b. the methodology to control the form, from the
nucleation to macroscopic growth
Thermodynamics tells us the direction and driving
force of transformations that yield the desired
form (but not the rate)
c. the techniques used the characterize the
variety of products obtained
Calorimetry and thermal analysis are key
techniques of polymorph characterization
d. the advantages resulting by this way of
surveying structure/property relations for the
design and preparation of new materials
Property = stability, solubility
Structure/stability relations:
The Close Packing Principle
The Density Rule
The greater stability of racemic compounds
over conglomerates
Polymorphs are different solid phases
of the same component(s)
An Example of Polymorphism in OneComponent System
ON P21/c
mp 114.8oC
q = 52.6°
OP P21/c
mp 112.7 oC
q = 46.1°
O
N
O
N
YN P-1
q = 104.1°
H
N
C
q
S
ROY
ORP Pbca
q = 39.4°
CH3
Y P21/c
mp 109.8 oC
q = 104.7°
R P-1
mp 106.2 oC
q = 21.7°
J. Am. Chem. Soc. 2000, 122, 585
An Example of Polymorphism in TwoComponent System
Henck, J.-O. et al. J. Am. Chem. Soc. 2001, 123, 1834
Two-Component Polymorphs of Racemic
Compounds
x
R-tazofelone
Racemic Compound
Form I
Form II
S-tazofelone
Space Group
P21/c
Pbca
mp, ºC
156.6
154.7
Reutzel, S.; Russell, V.; Yu, L. J. Chem. Soc. Perkin Trans 2 2000, 913
Two-Component Polymorphs: Racemic
Compounds and Conglomerates
S
S
R
R
S
R
racemic liquid
RRRRRRR
RRRRRRR
RRRRRRR
RRRRRRR
+
SSSSSSS
SSSSSSS
SSSSSSS
SSSSSSS
RSRSRSRSRSRSRSRS
SRSRSRSRSRSRSRSR
RSRSRSRSRSRSRSRS
SRSRSRSRSRSRSRSR
racemic compound
(single phase)
polymorphs ?
conglomerate
(two phases)
The Phase Rule
F =C–P+2
P = the number of phases
C = the number of components
F = the degree of freedom
The Gibbs Free Energy
G = H – TS
H = enthalpy  energy
S = entropy
G determines the stability of a phase at
constant pressure
The relative stability of two polymorphs
depends on their enthalpy difference and
entropy difference
For a one-component system at constant
pressure, the transition temperature Tt
between two polymorphs is unique
C = 1 (one component)
P = 2 (two polymorphs)
F=C–P+2=1
The condition of constant p removes one
more degree of freedom, making the
system invariant (F = 0).
Can two polymorphs have more than
one transition temperature?
Buerger, M. J. Chapter 6. Crystallographic Aspects of Phase Transitions. In Phase
Transitions in Solids; Smoluchowski, R. ; Mayer, J. E.; Weyl, W. A., Eds.; John Wiley &
Sons Inc.: New York, 1951.
Stability Relation between Two Polymorphs
(Constant Pressure)
Enantiotropy
Monotropy
G
G
B
A
transition
point Tt
virtual
transition
point Ttv
A
A
B
B
A stable B stable
L
L stable
TmA TmB
T
B stable
L
L stable
TmATmB
T
LT-to-HT transition is endothermic
HT-to-LT transition is exothermic
G
(GHT-GLT)
>0
=0
<0
LT: low-temp. stable phase
HT: high-temp. stable phase
HT
LT
Tt
LT
This result leads to
HTR (Heat of Transition
Rule) and HFT (Heat of
Fusion Rule): see
Henck and Griesser
HT
T
Quantitative Determination of DH, DS,
and DG at Constant Pressure
•
•
•
•
Low-temperature calorimetry
Solubility
Heat of solution and heat of transition
Melting and eutectic melting data
H and G of 1-Heptene Polymorphs
2000
T

S =
H=
H or G, cal/mole
1500
0K
T
1000
0K
500
Cpdt
Cpdlnt
HI
HII
G = H - TS
Form Tm, K
I
154.3
II
153.9
GI
0
GII
-500
Tt
Tm
-1000
-1500
T, K
0
20
40
60
80
100
120
140
160
Data from McCullough, J. P. et al. J. Phys. Chem. 1957, 61, 289
Solubility
Gi – Gj = RTln(xi/xj)
xi and xj = solubility of i and j in mole fraction
T = temperature in K
Heat of Solution
Heat of Transition
These measurements yield the enthalpy
difference between polymorphs (Hi – Hj), which
gives the temperature slope of their free-energy
difference:
d[(Gi – Gj)/T]d(1/T) = (Hi – Hj)
If (Gi – Gj) and (Hi – Hj) are known at one
temperature, (Gi – Gj) at nearby temperatures
can be estimated
Melting Data
• Widely available for organic polymorphs
because of their sluggish solid-solid transitions
• Easily measured by DSC
Heat flow
Tm,A Tm,B
DHm,A
DHm,B
T
The Heat of Fusion Rule
G - T curves
DSC data
enantiotropy
A
B
Tt
A
B
monotropy
A
A
B
B
Burger, A.; Ramberger, R. Mikrochimica Acta [Wien] 1979 II, 259-271 and 273-316.
Quantitative Analysis of Melting Data
DG
extrapolation
B
Tt
slope
Tm,A
Tm,B
T
A
dDG0/dT = -DS0 =
-DHm,A/Tm,A + DHm,B/Tm,B + DCp term
value
DG0 = DHm,B (Tm,A/Tm,B - 1)+ DCp term
Yu, L. J. Pharm. Sci., 1995, 84, 966
Solubility vs. Melting Data: Sulfathiazole
3
N
GI-GIII (kJ/mole)
N
N
solubility
2
S
S
O
O
sulfathiazole
1
0
Tt
= 369 K
-1
melting
-2
(HI - HIII) = d[(GI - GIII)/T]/d(1/T) = 7.1 kJ/mol
-3
270
320
370
T (K)
420
470
Solubility, Heat of Solution and Melting Data
4
DG (kJ/mole)
3
Form B
Solubility
data (37oC)
Au
O
O
P
S
O
O
Heat of
solution data
(25oC) provide
the slope
2
O
O
O
O
O
Auranofin1
1
Melting
data
0
Form A
Form B
-1
250
300
350
400
T, K
450
Reinterpretation of data of Lindenbaum, S. et al.
Int. J. Pharmaceutics 1985, 26, 123-132.
Eutectic Melting Data
Tmi
• Measured below pure
melting points: Te < Tm
• Te changes with additive
i
Tmj
j
Tma
a
• Standard technique of
chemical microscopy
Tej
Tei
0
xej xei
x
1
McCrone, W. C. Fusion Methods in Chemical Microscopy; Interscience Publishers,
Inc.: New York, 1957.
HMX Polymorphs Studied through Eutectic Melting
“Free energy-temperature diagram for HMX. The
intersection temperatures are measured points,
but the actual slopes are unknown.”`
Teetsov, A. S.; McCrone, W. C.
Microscope & Crystal Front 1965, 5, 13
Haleblian, J.; McCrone, W. C. J. Pharm. Sci.
1969, 58, 911
0.6
Eutectic Melting Measured
by DSC
0.3
ON
H
N
Y
-0.3
-0.6
ON
Y
ON
Tm ON
L
pure forms
Y
Y
Y
ON
80
ON
melting
eutectic melting
60
q
ROY
ON
ON
Y
40
C
S
Y
0
DSC Signal
N
Tm Y
Tt
N
O
L-sc
+azobenzene
GON-GY, kJ/mol
0.9
O
100
Yu, L. et al. J. Am. Chem. Soc. 2000, 122, 585.
120
T, oC
CH3
DG
xe2(G1-G2)(Te1)= DHme2(Te2-Te1)/Te2+ RTe1{xe2ln(xe1/xe2)
+ (1-xe2)ln[(1-xe1)/(1- xe2)]} + DCp term
xe1(G1-G2)(Te2)= - DHme1(Te1-Te2)/Te1-RTe2{xe1ln(xe2/xe1)
+ (1-xe1)ln[(1-xe2)/(1- xe1)]} + DCp term
xx
Tm,A Tm,B
Te1 Te2
T
slope
d DG0/dT = -DS0 =
-DHm,A/Tm,A + DHm,B/Tm,B + DCp term
value
DG0 = DHm,B (Tm,A/Tm,B - 1)+ DCp term
Relative Thermodynamic Stability of ROY Polymorphs
1.2
YN
G-GY ,kJ/mol
0.8
L-sc
0.4
R
ON
OP
0
Y
Y
OP
-0.4
ON
L
30
50
70
90
110
T, oC
130
Melting/Eutectic Melting Method Applied to Pairs of
Racemic Compounds and Conglomerates
S
Tg
10
N
O
G-GRII, kJ/mole
8
O
tazofelone
B
6
C, A
4
2
TmC
A
RI, RII: racemic compounds
A = enantiomorph (+ or -)
C = conglomerate
0
-2
300
TmA
LA
TmRI
RI
RII
Tt TmRII
350
400
LR
T, K
450
R = Racemic Compound
C = Conglomerate
(GC-GR) TmA = DHmR(TmR - TmA)/TmR + TmARln2 + DCpmR[TmA-TmR-TmAln(TmA/TmR)]
8
(SC-SR) TmA = DHmR/TmR - DHmA/TmA - Rln2 + DCpmRln(TmA/TmR)
9
where TmA and TmR are the melting temperatures of A and R, respectively; DHmA and DHmR the
corresponding latent heats; and DCpmR the heat capacity change upon melting R. The subscript
TmA signifies that the properties are calculated at TmA.
Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates, and Resolutions; Krieger
Publishing Company: Malabar, Florida, 1991.
Summary
Thermodynamic studies provides
the relative stability of polymorphs
driving forces of crystallization and polymorph
conversion
the basis for structure-stability studies
Thermodynamics does not address kinetic and
structural aspects of polymorphism. Many
behaviors of polymorphic systems require nonthermodynamic explanations
Combining thermodynamic, kinetic, and structural
studies is necessary for understanding and
controlling polymorphism
The fascination of a growing science lies in
the work of the pioneers at the very
borderland of the unknown, but to reach this
frontier one must pass over well traveled
roads; of these one of the safest and surest
is the broad highway of thermodynamics.
G. N. Lewis and M. Randall, 1923