Fundamentals of Polymorphism: The Phase Rule and Thermodynamic Relations

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Fundamentals of Polymorphism The Phase Rule and
Thermodynamic Relations Lian Yu University of
Wisconsin Madison, School of Pharmacy (608)
263 2263 lyu_at_pharmacy.wisc.edu
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Gibbs Findlay Westrum and McCullough McCrone Burg
er
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This Erice course will provide

• the theoretical basis for the existence of these
  diverse structural forms,
• the methodology to control the form, from the
  nucleation to macroscopic growth,
• the techniques used the characterize the variety
  of products obtained,
• the advantages resulting by this way of surveying
  structure/property relations for the design and
  preparation of new materials.

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• 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

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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)
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• c. the techniques used the characterize the
  variety of products obtained
• Calorimetry and thermal analysis are key
  techniques of polymorph characterization

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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
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Polymorphs are different solid phases of the same
component(s)
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An Example of Polymorphism in One-Component System
ON P21/c mp 114.8oC q 52.6
OP P21/c mp 112.7 oC q 46.1
YN P-1 q 104.1
ROY
Y P21/c mp 109.8 oC q 104.7
R P-1 mp 106.2 oC q 21.7
ORP Pbca q 39.4
J. Am. Chem. Soc. 2000, 122, 585
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An Example of Polymorphism in Two-Component System
Henck, J.-O. et al. J. Am. Chem. Soc. 2001, 123,
1834
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Two-Component Polymorphs of Racemic Compounds
x
R-tazofelone
S-tazofelone
Racemic Compound Space Group mp, ºC Form I
P21/c 156.6 Form II Pbca 154.7
Reutzel, S. Russell, V. Yu, L. J. Chem. Soc.
Perkin Trans 2 2000, 913
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Two-Component Polymorphs Racemic Compounds and
Conglomerates
R
S
R
RSRSRSRSRSRSRSRS SRSRSRSRSRSRSRSR RSRSRSRSRSRSRSRS
SRSRSRSRSRSRSRSR
R
S
R
S
R
R
R
S
R
S
R
S
R
S
R
R
R
R
R
R
R
S
R
S
S
S
S
S
racemic compound (single phase)
racemic liquid
polymorphs ?
RRRRRRR RRRRRRR RRRRRRR RRRRRRR
SSSSSSS SSSSSSS SSSSSSS SSSSSSS
conglomerate (two phases)

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The Phase Rule

• F C P 2
• P the number of phases
• C the number of components
• F the degree of freedom

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The Gibbs Free EnergyG 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
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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).

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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.
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Stability Relation between Two Polymorphs (Constan
t Pressure)
Enantiotropy
Monotropy
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LT-to-HT transition is endothermicHT-to-LT
transition is exothermic
LT low-temp. stable phase HT high-temp. stable
phase
This result leads to HTR (Heat of Transition
Rule) and HFT (Heat of Fusion Rule) see Henck
and Griesser
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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

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H and G of 1-Heptene Polymorphs
Data from McCullough, J. P. et al. J. Phys.
Chem. 1957, 61, 289
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Solubility

• Gi Gj RTln(xi/xj)
• xi and xj solubility of i and j in mole
  fraction
• T temperature in K

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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)/Td(1/T) (Hi Hj)
• If (Gi Gj) and (Hi Hj) are known at one
  temperature, (Gi Gj) at nearby temperatures can
  be estimated

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Melting Data

• Widely available for organic polymorphs because
  of their sluggish solid-solid transitions
• Easily measured by DSC

Tm,A
Tm,B
DHm,A
DHm,B
Heat flow
T
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The Heat of Fusion Rule
G - T curves
DSC data
enantiotropy
A
Tt
B
A
B
monotropy
A
B
A
B
Burger, A. Ramberger, R. Mikrochimica Acta
Wien 1979 II, 259-271 and 273-316.
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Quantitative Analysis of Melting Data
DG
extrapolation
Tm,A
Tm,B
Tt
B
T
A
slope
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
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Solubility vs. Melting Data Sulfathiazole
369 K
(HI - HIII) d(GI - GIII)/T/d(1/T) 7.1
kJ/mol
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Solubility, Heat of Solution and Melting Data
Solubility data (37oC)
Form B
Heat of solution data (25oC) provide the slope
DG (kJ/mole)
Auranofin1
Melting data
Form A
Form B
T, K
Reinterpretation of data of Lindenbaum, S. et
al. Int. J. Pharmaceutics 1985, 26, 123-132.
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Eutectic Melting Data

• Measured below pure melting points Te lt Tm
• Te changes with additive
• Standard technique of chemical microscopy

McCrone, W. C. Fusion Methods in Chemical
Microscopy Interscience Publishers, Inc. New
York, 1957.
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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
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Eutectic Melting Measured by DSC
ROY
Yu, L. et al. J. Am. Chem. Soc. 2000, 122, 585.
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xe2(G1-G2)(Te1) DHme2(Te2-Te1)/Te2
RTe1xe2ln(xe1/xe2) (1-xe2)ln(1-xe1)/(1-
xe2) DCp term xe1(G1-G2)(Te2) -
DHme1(Te1-Te2)/Te1-RTe2xe1ln(xe2/xe1)
(1-xe1)ln(1-xe2)/(1- xe1) DCp term
DG
x
x
Tm,A
Tm,B
T
Te1
Te2
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
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Relative Thermodynamic Stability of ROY Polymorphs
YN
L-sc
R
G-GY ,kJ/mol
ON
OP
Y
Y
OP
ON
L
T, oC
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Melting/Eutectic Melting Method Applied to Pairs
of Racemic Compounds and Conglomerates
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R Racemic CompoundC Conglomerate
Jacques, J. Collet, A. Wilen, S. H.
Enantiomers, Racemates, and Resolutions Krieger
Publishing Company Malabar, Florida, 1991.
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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
  non-thermodynamic explanations
• Combining thermodynamic, kinetic, and structural
  studies is necessary for understanding and
  controlling polymorphism

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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