The ternary phase diagram was predicted using the parameters from
Table 2,
Table 3,
Table 4 and
Table 5, which were validated against the binary subsystems RIT/PVPVA, PVPVA/water, and RIT/water (
Supplementary Information). This prediction then serves as a basis to explain the release mechanism of RIT from RIT/PVPVA ASDs.
4.1.1. Amorphous Phase Separation and Liquid–Liquid Phase Separation in the Ternary Phase Diagram
When a dry ASD is exposed to an aqueous dissolution medium, generally, two phenomena happen concurrently: (1) the ASD absorbs the dissolution medium (water) and (2) the ASD dissolves in the dissolution medium. For hydrophobic APIs, this might lead to (1) amorphous phase separation in the wet ASD and (2) liquid–liquid phase separation (nanodroplet formation) in the liquid phase surrounding the ASD. These phenomena will be considered in detail in the following, first using the schematic phase diagram of an API/polymer/water system. Many APIs (e.g., RIT) show a huge miscibility gap with water (
Figure 1), which still exists when a polymer, such as PVPVA, is added. Generally, the size of the miscibility gap depends on the kind of polymer, API, and temperature. Every initial composition of API/polymer/water located within this miscibility gap leads to the formation of two liquid/amorphous phases.
Starting from a dry and homogeneous ASD (lower side of the triangle in
Figure 1), there are two possible pathways into this miscibility gap. When an initially dry ASD is exposed to a dissolution medium of certain water activity (equals RH when in contact with a vapor phase), water is absorbed along the straight line in
Figure 1a) between the point denoting the composition of the dry ASD and the apex depicting pure water. Upon water sorption, and depending on the ASD composition, the resulting water-containing ASD might be located within the miscibility gap (point F in
Figure 1a)). In this case, demixing occurs along the corresponding tie line resulting in an API-poor phase L1 and an API-rich phase L2 in the ASD (
Figure 1c)). The latter is often named amorphous phase separation in the literature. An mDSC measurement of such a phase-separated ASD would typically result in two glass-transition temperatures corresponding to the two phases.
Again, the same miscibility gap applies when an ASD is surrounded by liquid water as the dissolution medium. The only difference is that water concentration in the dissolution medium is much higher than in the wet ASD (
Figure 1b)). When an initially dry ASD is dissolved in pure water as a dissolution medium, API and polymer dissolve along the same straight line as before, connecting pure water and the point denoting the composition of the dry ASD. Depending on the amount of ASD dissolved in water, the composition of the resulting dissolution medium containing both API and polymer might be located in the same miscibility gap as before but at very high water concentrations (point F in
Figure 1b)). In this case, demixing occurs along the corresponding tie line again, leading to an API-poor phase L1 and an API-rich phase L2, this time in the dissolution medium (
Figure 1d)). This liquid–liquid phase separation can be observed as droplet formation in the dissolution medium. It is worth noting that the API concentration in the API-poor phase corresponds to the solubility of the amorphous API in the aqueous polymer-containing medium.
Considering mass conservation results in the lever rule enables calculating the mass ratio of the API-rich phase L2 to the API-poor phase L1 at equilibrium for a given feed point:
where
and
are the total masses of the API-poor phase L1 and the API-rich phase L2, respectively.
,
, and
are the mass fractions of component
in the feed F, the API-poor phase L1, and API-rich phase L2, respectively.
4.1.2. Prediction of the Ternary Phase Diagram
Figure 2 shows the ternary phase diagram of RIT/PVPVA/water predicted by PC-SAFT using the parameters from
Table 4 and
Table 5. These parameters were determined via fitting the binary subsystems only and were verified for these binary systems (
Supplementary Information). Using only one binary parameter per binary system, all types of experimental data, i.e., solid solubilities, liquid–liquid miscibility gaps, as well as vapor-liquid equilibria of the binary subsystems, could be modeled in near quantitative agreement with the experimental data.
None of the parameters were fitted to the ternary system. As seen in
Figure 2, the predicted miscibility gap for the ternary system RIT/PVPVA/water is huge. The predicted tie lines result in phases very poor in RIT and phases that contain almost only RIT. ASDs with low water content are predicted to be homogeneous (below the miscibility gap) and glassy at 25 °C (green region). Mixtures with compositions in the overlapping area of the miscibility gap and the glassy region are prone to demixing but are kinetically stabilized, which can hinder or delay demixing.
4.1.3. Experimental Validation of the Ternary Phase Diagram
To validate the predicted miscibility gap (
Figure 2), RIT/PVPVA ASDs with different DLs and various mass fractions of water were analyzed via turbidity inspection and mDSC. Different water concentrations in the samples were realized via storing these samples for at least 2 days at different RHs. Afterward, the water concentrations were determined gravimetrically.
First, the ASDs were stored at 97% RH and visually inspected. As shown in
Figure 3, after 2 days of storage, the turbidity and occurrence of a second amorphous phase were visible for ASDs with 5–40 wt% DL. This validates the existence of the amorphous phase separation predicted by PC-SAFT at 97% RH (
Figure 2). In samples with DLs higher than 25 wt% (
Figure 3g–i), both phases were macroscopically distinguishable. Our results are in accordance with DSC measurements from Purohit et al. [
34], which showed two glass-transition temperatures, depicting two phases, for RIT/PVPVA ASDs with DLs between 10 wt% and 50 wt% after exposure to 97% RH.
Second, for the ASDs stored at 23% and 53% RH for at least 10 days and monitored by DSC, only one glass-transition temperature and no melting peaks were detected (see
SI,
Figure S4). Thus, these samples did not show amorphous phase separation because they are not located within the miscibility gap, as predicted by PC-SAFT (
Figure 2).
DSC scans of RIT/PVPVA ASDs with DLs of at least 10 wt% stored at 94% RH for a minimum of 7 days showed two glass transitions and no melting peak (
SI,
Figure S5 and
Table 6). The two glass-transition temperatures indicate a phase split into an RIT-rich phase and an RIT-poor phase, which is again in qualitative agreement with the miscibility gap predicted in
Figure 2. The glass-transition temperatures of the RIT-rich and RIT-poor phases corresponded to the glass-transition temperatures of pure RIT and pure PVPVA, respectively, at the same RH. This means that the RIT-rich phase contained almost pure RIT, whereas the RIT-poor phase contained almost pure PVPVA confirming the huge miscibility gap predicted using PC-SAFT.
We used the two measured
s per ASD stored at 94% RH (
Table 6) to determine the compositions of two evolving amorphous phases. For that purpose, the deviations between the experimental glass-transition temperatures
and
and the glass-transition temperatures
and
calculated using the Kwei equation (Equation (4)), and the parameters from
Table 2 and
Table 3 were minimized simultaneously for the two phases, L1 and L2 (Equation (14)).
This optimization was performed with respect to the masses
and
of every component
(polymer, API, and water) in the two phases L1 and L2, considering the constraints given by the mass conservation (Equation (15)) as well as the lower and upper bounds for the masses of the components in phases L1 and L2 (Equations (16) and (17)). Total masses
of API, polymer, and water were known from the gravimetric measurements of the dry ASDs and of the ASDs after storage at a fixed RH. The mass fractions
and
of the two phases L1 and L2 evolving from the phase separation upon water sorption were then obtained from the component masses according to:
The resulting equilibrium concentrations are shown in
Figure 2 and
Table 6 for all four samples with varying DLs. Regardless of the DL, all equilibrium concentrations determined by the novel approach are almost identical and are in very good agreement with the predictions using PC-SAFT (see
Figure 2). Thus, the results not only qualitatively verify the existence of the miscibility gap but, moreover, show that both the length and the slope of the predicted tie lines are in quantitative agreement with the experimental data. To the best of our knowledge, this is the first time ever that a quantitative phase diagram for the system RIT/PVPVA/water is provided. It has been predicted via PC-SAFT solely based on pure-component parameters and experimental data of the binary subsystems and was experimentally validated against various types of experimental data in this work. Based on the same modeling,
Figure 4 shows a plot of the predicted phase mass ratios
calculated (Equation (13)) by PC-SAFT, which are in excellent agreement with the experimental data (
Figure 4). It is worth mentioning that
Figure 4 illustrates the lever rule: the smaller the DL, the smaller the total amount of evolving RIT-rich phase L2.
Figure 2 and
Figure 4 will be subsequently used to understand and explain the release behavior of RIT/PVPVA ASDs in water.