Fundamentals of Polymorphism: The Phase Rule and

Thermodynamic Relations

Lian YuUniversity of Wisconsin – Madison,

School of Pharmacy (608) 263 2263

lyu@pharmacy.wisc.edu

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