4.3.2. Ryshkewitch-Duckworth Analysis

The compactibility of the material is described by Ryshkewitch-Duckworth Equation (Equation (6) [45]. Tensile strength at zero porosity (σ0) can be determined using this mathematical model. Since the tensile strength of a material is normalized by the bonding area at zero porosity; σ0 represents interparticulate bonding strength (BS) of the material undergoing compaction. The effect of bonding area was minimal as the similar particle shape and PSD were used for CTC profiling.

$$
\sigma = \sigma\_0 \,\, \mathfrak{e}^{-a\varepsilon} \tag{6}
$$

where σ is tensile strength, α is a constant and ε is porosity. In its logarithmic form, a linear relationship between porosity and the log of the tensile strength was obtained. The value of σ0 was 2.75, 2.58, and 3.21 MPa for MAL, RIV, and RIV-MAL Co, respectively. The higher σ0 value indicates a greater BS for the cocrystal over the API and coformer. At the same time, the BS of MAL was higher than that of RIV. Thus, Ryshkewitch analysis supports the findings of the compactibility plot and confirmed the greater BS of the cocrystal.

The higher tabletability of RIV-MAL Co was the outcome of both higher BA and BS, as demonstrated by its low yield pressure (Py 83 MPa) and high σ0 (3.21 MPa) (Figure 5). Comparable tabletability profiles were observed for RIV and MAL, despite the low plasticity of MAL. The tabletability of RIV was predominantly contributed by the BA, as indicated by the low yield pressure (Py of 133 MPa). BS predominantly governs the tabletability of MAL, as supported by the observed σ0. The high Py of MAL can be correlated to its brittle nature. Based on the compactibility plots and relative tensile strength at zero porosity (σ0), the order of BS in the three solids can be ranked as RIV < MAL < RIV-MAL Co. The Ryshkewitch–Duckworth analysis confirmed the higher BS for MAL (<sup>σ</sup>0 = 2.75 MPa) than RIV (<sup>σ</sup>0 = 2.58 MPa), while the highest BS was observed for RIV-MAL Co (<sup>σ</sup>0 = 3.21 MPa). In other words, the higher work of adhesion was observed at significantly lower compaction pressure in RIV-MAL Co as compared to RIV and MAL.

### *4.4. Particle Level Deformation: Quantifying Crystal Deformation by Nanoindentation*

Crystals with smooth surfaces were subjected to nanoindentation to decipher the particle-level deformation behaviour. The direction of the applied stress was perpendicular to the slip plane predicted based on the visualization and attachment energy calculations (applicable for RIV and RIV-MAL Co). Nanoindentation parameters, i.e., elastic modulus (E), mechanical hardness H, and 1/E values for crystal samples, are presented in Table 2.


**Table 2.** Elastic modulus (E), mechanical hardness H, and 1/E values for crystal samples.

a Average values are presented, while standard deviations are shown in parentheses (n = ≥10).

Indentation hardness (mechanical hardness) H denotes the resistance offered by the material to plastic deformation [29]. A low value of H is indicative of lower resistance offered by the material to undergo irreversible (plastic) deformation. However, organic molecular solids show initial elastic deformation followed by plastic deformation. Before plastic deformation takes place, the elastic limit has to be exceeded by the applied stress.

E is a measure of the resistance to elastic deformation and is a function of the energy of the interaction between molecules and their distances of separation [29]. The 1/E (compliance) can be correlated with elastic recovery (ER). A high ER indicates dominance of elastic deformation, which adversely contributes to plasticity (BA) and thus to the tabletability of the material.[46]

The lower the value of E, the larger is the 1/E and hence higher will be the elastic recovery. Amongst the three molecular solids, RIV possesses the lowest E (03.41 GPa), indicating the highest elastic recovery. The high elastic recovery of RIV crystals was also verified by the higher (*hmax-hp*) value (178 GPa) (Figure 6), which is the elastic recovery determined from the (*p-h*) loading–unloading curve. The value of (*hmax-hp*) was significantly higher for RIV (178 GPa) as compared to MAL (74 GPa) and RIV-MAL (75 GPa). The high E value (i.e., low 1/E) and low (*hmax-hp*) for MAL and RIV-MAL Co were also indicative of lower ER of the coformer and cocrystal as compared to the API.

**Figure 6.** Nanoindentation parameters for crystal deformation representing (**a**) The (*hmax-hp*) values depicting elastic recovery for crystals of RIV, MAL, and RIV-MAL Co; (**b**) Inverse correlation of H/E ratio (primary axis) and interparticulate bonding strength, <sup>σ</sup>0 (secondary axis).

The high H of MAL (0.71 GPa) depicts a higher intermolecular interaction between MAL molecules, which can be directly correlated to the crystallographic features of MAL. Careful evaluation of the crystal structure of MAL showed hydrogen bonding interactions along all three axes (3D H-bonding), which provided the greater hardness to MAL (this is thoroughly discussed in Section 4.5.3). The comparatively higher hardness may hinder plastic deformation of MAL crystals when subjected to bulk compaction. RIV-MAL Co showed an "intermediate" value of H when compared with the H values of MAL and RIV (Table 2). At the same time, the cocrystal exhibited low elastic recovery as evidenced by the low values of 1/E and (*hmax-hp*) (Figure 6). Thus, the crystals of cocrystal possess a dominance of plasticity over elasticity.

Wendy and Hewitt in 1989 found that the H/E ratio can be used to predict bulk deformation (compaction) behaviour of materials based on particle-level deformation study using the microindentation technique [31]. In this study, acetaminophen possessed the largest ratio of H/E and exhibited the poorest compaction, i.e., tablets capped and delaminated extensively during decompression and ejection from the die. Adipic acid compacts with a relatively large H/E ratio also underwent delamination during wear testing. Conversely, the materials with a lower H/E ratio could form tablets free from the above defects. This means Hewitt"s work only "qualitatively" correlated H/E ratio with compaction behaviour because the compaction behaviour of the materials was qualitatively described as "good" (for materials whose compacts were free from of capping and lamination) or "poor" (for materials whose compacts showed capping or lamination). Interestingly, the present work provided a correlation of H/E ratio with a "quantitative" bulk deformation parameter, i.e., interparticulate bonding strength at zero porosity (σ0). In this work, the H/E was found to inversely correlate to σ0 (Figure 6b). The cocrystal had the lowest H/E (0.029) and exhibited the highest σ0 (3.21 MPa), while RIV had the highest H/E (0.58) and showed the lowest σ0 (2.58 MPa). The value of H/E ratio was "intermediate" for MAL (0.040) and hence MAL exhibited the "intermediate" σ0 (2.75 MPa) among the three molecular solids.
