INTRODUCTION TO AMORPHOUS SOLID DISPERSIONS - Pharmaceutical Amorphous Solid Dispersions - page 23

where S mix are the enthalpy and entropy of mixing, respectively. Generally,

the mixing of different molecules leads to an increase in entropy, so that the positive
entropic term in Equation 1.21 leads to a loss of free energy and the favoring of
miscibility. On the other hand, if we consider the molecular mixing of two components A
and B,

Δ H mix and Δ

H mix can be either negative or positive depending on the relative strength of the
interactions between A:B and between A:A and B:B. Stronger A:B interactions lead to a
loss in enthalpy, while stronger A:A and B:B interactions lead to an increase in enthalpy.
Thus, the former will contribute to a loss in free energy, while the latter will contribute to
an increase in free energy. We can generalize further by saying that two organic
molecules having similar levels of polarity and, therefore, capable of undergoing
signi

Δ

cant intermolecular interaction, for example, hydrogen bonding, would be
expected to produce miscible amorphous mixtures, while chemical incompatibility
could lead to phase separation of the two or more amorphous forms if not offset by
positive entropy changes. An example of such a system is that of amorphous citric acid
(quite polar) and indomethacin (relatively hydrophobic) that phase separate beyond a
certain concentration [33]. To gain a more quantitative understanding of those factors that
control molecular mixing in the amorphous state, it would be desirable to have a theoretical
framework that describes the free energy of mixing of liquids at a fundamental level in
terms of measurable parameters. The Flory - Huggins lattice theory [34], presented in the

Flory Huggins equation, and originally developed for polymer solutions, describes the

Gibbs free energy of mixing per mole for a polymer and a

-

“ solvent ” as

Δ G mix RT n A ln ϕ A n B ln ϕ B n A ϕ B χ AB ; (1.22)

where R is the gas constant, is the interaction

parameter that describes the tendency for interaction between the components A and B,
and n is the number of moles of each component. The

ϕ is the volume fraction of each component, χ

first two terms in the parenthesis
represent the entropic contributions, while the third term represents the enthalpic
contributions from intermolecular interactions. For the purposes of this discussion,
component A will refer to the

and component B to the polymer. The smaller
or more negative the interaction parameter, the stronger the intermolecular interaction
between A and B. From a conceptual perspective, the Flory

“ solvent ”

Huggins equation is very
useful in demonstrating the importance of molecular size in affectingmiscibility through a
reduction in the entropy of mixing. For example, in Equation 1.23, the entropy of mixing
per mole can be described as

-

S mix R n A ln ϕ A n B ln ϕ B : (1.23)

If we now express the number of moles of each component in terms of the mass m and
molecular weight M , we can rewrite Equation 1.23 as

Δ S mix R m A = M A ln ϕ A m B = M B ln ϕ B : (1.24)

The important conclusion from this equation is that the greater themolecular weight of any
component, the less the contribution of entropy to the free energy of mixing, and thus the