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Fundamentals of Polymorphism: The Phase Rule and
Thermodynamic Relations
Lian YuUniversity of Wisconsin – Madison,
School of Pharmacy (608) 263 2263
GibbsFindlay Westrum and McCulloughMcCroneBurger …
[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 RuleThe greater stability of racemic compounds
over conglomerates
Polymorphs are different solid phases of the same component(s)
An Example of Polymorphism in One-Component System
R P-1mp 106.2 oC= 21.7°
ROY
ORP Pbca= 39.4°
OP P21/cmp 112.7 oC= 46.1°
ON P21/cmp 114.8oC= 52.6° YN P-1
= 104.1°
Y P21/cmp 109.8 oC= 104.7°
N
S
HNO O
C
N
CH3
J. Am. Chem. Soc. 2000, 122, 585
An Example of Polymorphism in Two-Component System
Henck, J.-O. et al. J. Am. Chem. Soc. 2001, 123, 1834
x
R-tazofelone S-tazofelone
Two-Component Polymorphs of Racemic Compounds
Racemic Compound Space Group mp, ºCForm I P21/c 156.6 Form II Pbca 154.7
Reutzel, S.; Russell, V.; Yu, L. J. Chem. Soc. Perkin Trans 2 2000, 913
Two-Component Polymorphs: Racemic Compounds and Conglomerates
R
S
R R
R
R
R
RS
R
S
R
R
R
S SS
S
RR
S
RSRSRSRSRSRSRSRSSRSRSRSRSRSRSRSRRSRSRSRSRSRSRSRSSRSRSRSRSRSRSRSR
RRRRRRRRRRRRRRRRRRRRRRRRRRRR
SSSSSSSSSSSSSSSSSSSSSSSSSSSS
+
racemic compound(single phase)racemic
liquid
conglomerate(two phases)
R
SR
S
S
RR
S
S
R
polymorphs ?
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 EnergyG = H – TS
H = enthalpy energyS = 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.
Enantiotropy
G
AA
B
B
L
TmA TmB
A stable B stable
transitionpoint Tt
L stable
T
G
A
B
L
TmA TmB
B stable
virtualtransitionpoint Ttv
L stable
T
Monotropy
Stability Relation between Two Polymorphs(Constant Pressure)
LT-to-HT transition is endothermicHT-to-LT transition is exothermic
LT
HT
Tt
HT
LT
T
G (GHT-GLT)> 0 = 0 <0
LT: low-temp. stable phaseHT: high-temp. stable phase
This result leads to HTR (Heat of Transition Rule) and HFT (Heat of Fusion Rule): see Henck and Griesser
Quantitative Determination of H, S, and G 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
Data from McCullough, J. P. et al. J. Phys. Chem. 1957, 61, 289
-1500
-1000
-500
0
500
1000
1500
2000
0 20 40 60 80 100 120 140 160
Form Tm, K I 154.3 II 153.9
H o
r G
, cal
/mol
e HI
HII
GI
GII
Tt
Tm
T, K
H = T
0KCpd
S = T
0KCpdln
G = H - TS
Solubility
Gi – Gj = RTln(xi/xj)
xi and xj = solubility of i and j in mole fraction
T = temperature in K
Heat of SolutionHeat 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
T
Hea
t flo
w
Tm,A Tm,B
Hm,A Hm,B
AB
A B
A
B
enantiotropy
A
B
monotropy
Tt
The Heat of Fusion Rule
DSC data G - T curves
Burger, A.; Ramberger, R. Mikrochimica Acta [Wien] 1979 II, 259-271 and 273-316.
G0 = Hm,B (Tm,A/Tm,B - 1)+ Cp term
T
G
Tm,A Tm,B
dG0/dT = -S0 =-Hm,A/Tm,A + Hm,B/Tm,B Cp term
slope
value
Tt
Quantitative Analysis of Melting Data
Yu, L. J. Pharm. Sci., 1995, 84, 966
B
A
extrapolation
-3
-2
-1
0
1
2
3
270 320 370 420 470
GI-G
III(k
J/m
ole)
Tt
T (K)
melting
solubility S
N
OO
N
S
N
sulfathiazole
S
N
OO
N
S
N
sulfathiazole
= 369 K
(HI - HIII) = d[(GI - GIII)/T]/d(1/T) = 7.1 kJ/mol
Solubility vs. Melting Data: Sulfathiazole
Solubility, Heat of Solution and Melting Data
-1
0
1
2
3
4
250 300 350 400 450
Meltingdata
Solubilitydata (37oC)
Heat ofsolution data(25oC) providethe slope
G (
kJ/m
ole)
Form A
Form B
Form BT, K
Reinterpretation of data of Lindenbaum, S. et al. Int. J. Pharmaceutics 1985, 26, 123-132.
O
OO
O S
O
Au
OO
O
PO
Auranofin1
Eutectic Melting Data
McCrone, W. C. Fusion Methods in Chemical Microscopy; Interscience Publishers, Inc.: New York, 1957.
x0 1
Tmi
xei
Tei
Tmj
xej
Tej
Tma a
i
j
• Measured below pure melting points: Te < Tm
• Te changes with additive
• Standard technique of chemical microscopy
Teetsov, A. S.; McCrone, W. C. Microscope & Crystal Front 1965, 5, 13
Haleblian, J.; McCrone, W. C. J. Pharm. Sci. 1969, 58, 911
HMX Polymorphs Studied through Eutectic Melting
“Free energy-temperature diagram for HMX. The intersection temperatures are measured points, but the actual slopes are unknown.”`
40 60 80 100 120
-0.6
-0.3
0
0.3
0.6
0.9
DS
C S
igna
l
+th
ymol
+az
oben
zene
+be
nzil
+ac
etan
ilide
pure forms
YYON
YY
ONONY
ONON
GO
N-G
Y, k
J/m
ol
meltingeutectic melting
Tt
T, oC
Tm Y
Tm ON
L
L-sc
YY
ON
ON
Eutectic Melting Measured by DSC
Yu, L. et al. J. Am. Chem. Soc. 2000, 122, 585.
N
S
HNO O
C
N
CH3
ROY
G0 = Hm,B (Tm,A/Tm,B - 1)+ Cp term
T
G
Tm,A Tm,B
d G0/dT = -S0 =-Hm,A/Tm,A + Hm,B/Tm,B Cp term
slope
value
xe2(G1-G2)(Te1)=Hme2(Te2-Te1)/Te2+ RTe1{xe2ln(xe1/xe2)+ (1-xe2)ln[(1-xe1)/(1- xe2)]} Cp termxe1(G1-G2)(Te2)=Hme1(Te1-Te2)/Te1-RTe2{xe1ln(xe2/xe1)+ (1-xe1)ln[(1-xe2)/(1- xe1)]} Cp term
Te1 Te2
x x
-0.4
0
0.4
0.8
1.2
30 50 70 90 110 130
G-G
Y ,k
J/m
ol
R
Y
OP
ON
ON
OP
L
L-sc
YN
Y
T, oC
Relative Thermodynamic Stability of ROY Polymorphs
Melting/Eutectic Melting Method Applied to Pairs of Racemic Compounds and Conglomerates
-2
0
2
4
6
8
10
300 350 400 450
B
RI, RII: racemic compoundsA = enantiomorph (+ or -)C = conglomerate
T, K
G-G
RII, k
J/m
ole
RII
C, A
TmRII
LR
LA
TmA
Tg
TmC
A
RI TmRI
Tt
O
S
N
O
tazofelone
R = Racemic CompoundC = Conglomerate
(GC-GR) TmA = HmR(TmR - TmA)/TmR + TmARln2 + CpmR[TmA-TmR-TmAln(TmA/TmR)] 8 (SC-SR) TmA = HmR/TmR - HmA/TmA - Rln2 + CpmRln(TmA/TmR) 9
where TmA and TmR are the melting temperatures of A and R, respectively; HmA and HmR the corresponding latent heats; and CpmR 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.
SummaryThermodynamic studies provides
the relative stability of polymorphsdriving forces of crystallization and polymorph
conversionthe basis for structure-stability studies
Thermodynamics does not address kinetic and structural aspects of polymorphism. Many behaviors of polymorphic systems require non-thermodynamic 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