Estimation of the Interaction Parameter Using Melting Point Depression

The presence of a polymer changes the melting behavior of a drug in pharmaceutical systems if the drug is miscible with the polymer. Thus, a reduction in the melting temperature of the crystalline drug is observed, as, for example, for the piroxicam–

PVP system (Tantishaiyakul et al.1999). This is due to the fact that there is a negative free energy of mixing associated with the spontaneous mixing of the polymer with the liquid phase of the drug and the chemical potential of the drug in the mixture is decreased compared to that of the pure liquid drug. Thus, the melting process becomes thermodynamically more favorable which leads to melting point depression (Baird and Taylor2012). If there is an amorphous proportion of the drug present, it further lowers the initial chemical potential of the drug and thus lowers the melting point additionally. When a polymer is above its T_g, it can act as a solvent for the drug and the end point of the melting endotherm is the temperature at which all the drug has dissolved in the polymer. Accordingly, if the temperature scanning rate of the experiment is slow enough to allow the equilibrium to be reached, the solubility of the crystalline drug in the supercooled polymer can be obtained, as will be discussed in more detail later. The melting point depression is larger in the case of strongly exothermic mixing, less for weakly endothermal or athermal mixing, and if the drug and polymer are immiscible with each other, melting point depression does not occur (Marsac et al.2009). This phenomenon has been utilized for studying miscibility of polymer–polymer and polymer–solvent systems. Reduction of melting temperature of the crystalline phase as a function of composition and polymer–

polymer interaction has been analyzed using the Nishi–Wang equation (Nishi and Wang1975) based on the F–H theory:

T_m^pure− T_m^mix= −Tm^pureBV₂ϕ²₁

H_fus , (2.45)

where Tm^pure is the melting temperature of the pure crystalline component, T_m^mix is the melting temperature of the crystalline component of the mixture, B is the interaction energy density between blend components, V2 is the molar volume of the repeating unit of the crystalline component, ϕ1 is the volume fraction of the

2 Theoretical Considerations in Developing Amorphous Solid Dispersions 63 amorphous component in the blend, and H_fusis the heat (enthalpy) of fusion of the crystalline component per mole of the repeating unit. The value of B is independent of composition, and it represents the intensity of molecular interaction during mixing (Paudel et al. 2010). The more negative the value of B, the larger the interaction between the components.

The Nishi–Wang equation was found to predict T_m^mix, miscibility, and the existence of a specific interaction between the components in the molten state when investigat-ing the miscibility between 17β-estradiol and Eudragit RS (Wiranidchapong et al.

2008). The B value obtained from curve fitting was− 0.28 ± 0.0094 J/g cm³, indicat-ing some degree of interaction between 17β-estradiol and Eudragit RS in the system.

The interaction-related B value was found to depend on the molecular weights of the mixing components. Paudel et al. (2010) obtained B values of− 89.17, − 118.03, and

− 68.05 for naproxen mixed with PVP K12, PVP K25, and PVP K90, respectively.

This indicated that the interaction potential with the particular drug was highest with PVP K25. By extending the equations for polymer–solvent systems (Hoei et al.

1992), the melting point depression data from DSC measurements was related to the F–H interaction parameter for drug–polymer systems by Eq. 2.46 (Marsac et al.

2006b):

where T_m^mixis the melting temperature of a drug in the presence of a polymer, Tm^pureis the melting temperature of the pure drug without a polymer, H_fusis the heat of fusion of the pure drug, ϕ_d,pare the volume fractions of the drug or polymer, respectively, and m is the ratio of the volume of the polymer to the volume of the lattice site. Estimation of the interaction parameters from melting point depression data for nifedipine and felodipine mixed with PVP K12 was performed by rearranging Eq. 2.46 in order to establish a linear relationship as a function of ϕ²_p from which χ was determined as a slope of the curve. In addition, the visualization of the melting point depression by DSC required a reduced scanning rate and controlled particle size, due to slow mixing kinetics. For a polymer volume fraction up to 0.25, the curve was linear for both felodipine and nifedipine, but at higher concentrations, linearity was lost.

This was explained to originate from the composition dependence of the interaction parameter and the increasingly unfavorable kinetics of drug–polymer interaction as the melting point is depressed closer to the Tgof the polymer. Thus, only the linear part for low polymer concentrations was used to obtain χ values of− 3.8 and − 4.2 for nifedipine–PVP and felodipine–PVP, respectively. When compared to the values obtained by the solubility parameter approach (2.0 and 0.5, respectively), it could be concluded that the melting point depression method resulted in values that were better in agreement with the experimental observations (Marsac et al.2006a). In another study, the interaction parameters for felodipine with Soluplus and hydroxypropyl methyl cellulose acetate succinate (HPMCAS) were calculated using the solubility parameter and melting point depression methods. Similar values were obtained, indicating that either method is suitable in this case. The resulting values were in

64 R. Laitinen et al.

the range between 2.8 and 7.5 MPa^1/2which predicts the felodipine—Soluplus and felodipine—HPMCAS systems to show limited miscibility (Tian et al.2013).

The application of the melting point depression method for the evaluation of the miscibility for several drug–polymer systems showed that systems identified as miscible exhibited melting point depression, while systems identified as immiscible or only partially miscible showed only slight or no melting point depression (Marsac et al.2009). Applying the melting point data to Eq. 2.46, χ values for the systems could be obtained. Negative or close to zero values were estimated for all PVP systems that were miscible, and systems previously known to be immiscible gave large positive χ values. Although the theoretical interaction parameters have often been found to be in reasonable agreement with the experimental results, there are limitations associated with the melting point depression method. First, application of the method to pharmaceutical systems requires that the drug and polymer are stable over the temperature range of interest and that there are sufficient physical interactions between the components for the melting point depression to be observed. In addition, the melting point of the drug should be sufficiently high for the polymer to exist in the supercooled liquid state, allowing mixing and interaction with the drug. Thus, the method is best suited for systems where the polymer has a Tgsignificantly lower than the Tm of the drug. Furthermore, Eq. 2.46 is linear only at comparatively low polymer concentrations, which probably is due to the kinetics of mixing during the experiment. It should also be noted that the method does not provide a universal value for χ but an estimation close to the melting point of the drug.

Zhao et al. (2011) attempted to correlate χ with temperature and construct a temperature phase diagram for indomethacin–PVP-VA systems. This would allow an estimation of the miscibility behavior of a drug polymer pair at different tem-peratures and compositions. In general, phase diagrams are useful in describing the compatibility of binary mixtures (Lin and Huang2010). Phase diagrams give infor-mation on the temperature ranges in which the particular mixture would be miscible and/or unstable. An unstable binary mixture will separate into two phases, the com-positions of which can be seen from the phase diagram. Two different sets of χ values, χ₁(T₁) and χ₂(T₂), were calculated, the first set being obtained from melting point depression experiments at T_mand the other set from solubility parameter cal-culations at room temperature. This allowed the authors to obtain values for A and B in Eq. 2.38. By substituting Eq. 2.38 into Eq. 2.36 and taking into account the fact that the component volume fractions equal 1, they were able to obtain the free energy versus composition phase diagram for the indomethacin–PVP-VA system at different temperatures (Fig.2.9a). Subsequently, this relation was transformed to a temperature-composition phase diagram which summarizes the phase behavior of the mixture by showing regions of stability, instability, and metastability (Fig.2.9b). The first phase boundary was determined by the tangent of the free energy curve where the first derivative of the free energy of mixing with respect to volume fraction is set equal to zero (Fan et al.1992; Zhao et al.2011):

2 Theoretical Considerations in Developing Amorphous Solid Dispersions 65 Fig. 2.9 a Free energy versus

composition phase diagram for the indomethacin and PVP-VA system and b temperature versus composition phase diagram for the indomethacin and PVP-VA system showing the spinodal curve (the binodal curve was not determined in the study). (Adapted from Zhao et al.2011)

∂G_mix

∂ϕ_drug = ln ϕdrug+ 1− 1

m_poly− 1

m_polyln (1− ϕdrug)+ (1 − 2ϕdrug)χ_drug−poly=0, (2.47) where G_mix is free energy of mixing, ϕ_drug and ϕ_polyare the volume fractions of the drug and the polymer, respectively, m_polyis the degree of polymerization of the amorphous polymer, and χ_drug−polyis the F–H interaction parameter.

The numerical solution of Eq. 2.47 is possible by combining it with Eq. 2.36 for the interaction parameter at different temperatures. The phase boundary obtained corresponds to the boundary between the stable and metastable region and is known as the binodal curve. In addition, the spinodal curve that represents the boundary between the unstable and metastable region can be obtained by setting the second

66 R. Laitinen et al.

derivative of the free energy to zero:

∂²G_mix The highest or lowest point of the spinodal curve is the critical point (ϕ_c), a point where the spinodal and binodal curves meet. It can be calculated by setting the third derivative of the free energy to zero:

∂³G_mix The critical interaction parameter χcand the critical temperature Tccan be obtained from substituting back to Eqs. 2.36 and 2.48. Binodal and spinodal curves do not exist if the mixture has no critical point. This is usually due to χ being negative or very small over the entire temperature range, i.e., the system is miscible at all conditions.

From Fig.2.9, it can be observed that the Gmix curve is negative and convex at 100 ^◦Cƒ. At this temperature, thermodynamically stable single-phase mixtures are obtained at all indomethacin–PVP-VA compositions. The temperature at which

G_mix becomes positive depends on the volume fraction of the polymer, i.e., the bigger the polymer fraction, the lower the temperature. The critical temperature for the indomethacin–PVP-VA system was found to be 73^◦C (Fig.2.9). In addition, the critical point is at a very low polymer concentration, meaning that a high amount of polymer is required to prevent phase separation. The spinodal curve is the boundary between unstable and metastable regions, meaning that below this curve, the system would spontaneously phase separate into two phases with the compositionsϕα and ϕβ at the temperature in question with any given homogenously mixed state ϕ0

(Fig.2.9). After formation of the bicontinuous polymer-rich and drug-rich domains, phase separation would proceed through a coarsening process. Generally, the binodal curve (which was not studied in the publication in question) would be located above (and adjacent to) the spinodal curve. Together, these give boundaries for a region where the mixture is metastable, in the sense that the mixture will only start to phase separate after a sufficiently large fluctuation in concentration or temperature. For systems between the binodal and spinodal boundaries, amorphous phase separation would occur through nucleation and growth, i.e., the drug-rich domains first appear as small droplets which subsequently grow in size (Fan et al.1992; Lin and Huang 2010).