Microhydrodynamic Analysis

The dynamics of the inter-particle collisions in a dense slurry flow is referred to as microhydrodynamics. Eskin et al. (2005a, b) developed a model to calculate the mean velocity of bead oscillations in well-mixed slurries using the kinetic theory of granular flows and fundamental granular energy balance (Gidaspow, 1994). Salient features of this microhydrodynamic model with slight modification (Afolabi et al., 2014) in view of Eskin and Miller (2008) are presented here, and readers are referred to aforementioned literature for the assumptions and derivations. The power applied per unit volume of slurry P_w inside a stirred mill dissipates through several mechanisms, which are mathematically expressed as follows:

ht

where ε_visc is the energy dissipation rate due to both the liquid–beads viscous friction and lubrication, εcoll is the energy dissipation rate due to partially inelastic bead–bead collisions, and εht is the power spent on shearing the equivalent liquid (milled drug suspension). In Eq. (3.3), L is the apparent shear viscosity of the equivalent liquid, c is the bead volumetric concentration (volume fraction),  is the granular temperature defined as the bead–equivalent liquid relative mean-square velocity, Rdiss is the effective drag (dissipation) coefficient, d_b is the median size of the beads, k is the

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restitution coefficient for the bead–bead collisions (0.76 from Tatsumi et al. (2009)), and b is the density of the zirconia beads (6000 kg/m³).

The equivalent liquid properties µL andL as well as the power applied per unit volume in the presence of the beads Pw were measured. The energy dissipation rate for shearing the equivalent liquid εht was found to be negligibly small (much smaller than P_w) due to the low viscosity of the suspensions. MATLAB’s fsolve function was used to solve Eq. (3.3) for the granular temperature  using P_w measured and Rdiss values calculated (refer to Eqs. (A.1)–(A.5) in Appendix A). From the calculated  the frequency of single-bead oscillations  and the average oscillation velocity of the beads u_b were determined as follows:



Eskin et al. (2005a) advanced the microhydrodynamic model (Eskin et al., 2005b) by considering the elastic contact deformation of the beads along with the elastic–

perfectly plastic deformation of the particles caught between the beads. While the beads frequently collide due to their fluctuating motions in a slurry, which are characterized by , ub, and , the beads capture and compress the drug particles to be milled. The maximum contact pressure at the center of the contact circle σbmax

of the two colliding beads is given by

2

where F_bⁿ and α_b are the average maximum normal force during the collision of two

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elastic beads and the radius of the contact circle formed at the contact of two beads respectively (refer to Eqs. (A.6) and (A.7), respectively). The average frequency of

It must be noted that only a small fraction of energy consumption is actually used for deforming the drug particles, which is explained by the energy dissipation rate resulting from the deformation of the particles per unit volume Π and expressed as (Eskin et al., 2005a): suspension, reduced elastic modulus of the bead–drug particle contact, elastic modulus of the drug particles, Poisson’s ratio of the beads, contact pressure in a drug particle captured when the fully plastic condition is reached, radius of the drug milling intensity factor F, similar to Afolabi et al. (2014), as follows:

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3.3.1 Effects of Wet Media Milling on the Drug Particle Size and Morphology Two concerns during WSMM of BCS Class II drugs are physical instability of the milled drug suspensions and possible solid-state changes (Kesisoglou et al., 2007;

Kumar and Burgess, 2014). To this end, Runs 4 and 14 suspensions were selected for SEM imaging and XRPD diffractograms because they led to the coarsest (slowest, least intense breakage) and finest (fastest, most intense breakage) drug particles after 256 min milling, respectively, among all different stirrer speed‒bead size combinations (see Table 3.2). Figure 3.1 shows the SEM images of GF particles before and after 256 min milling. The SEM images show that as-received, coarse GF particles (Figure 3.1a) were broken into primary particles in the range of 0.20‒2.00 µm for Run 4 (Figure 3.1b) and in the range of 0.05‒0.20 µm for Run 14 (Figure 3.1c), which qualitatively agree with the particle sizes obtained from laser diffraction (Table 3.2). Results from laser diffraction and SEM images suggest that the drug suspension even with the smallest nanoparticles (Run 14) did not exhibit severe aggregation at the time scale of the milling process, and the electrosteric stabilization mechanism imparted by the HPC–SDS combination (Bilgili and Afolabi, 2012;

Bilgili et al., 2016d) was effective. Hence, the presence of coarser primary particles in

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Run 4 suspension can be attributed to the slower breakage in Run 4, which will be further elucidated.

Figure 3.1. SEM images showing GF particle size and morphology: (a) before milling and after 256 min milling at (b)  = 1000 rpm with db = 1500 µm beads (Run 4) and (c) 4000 rpm with db = 50 µm beads (Run 14). Before milling, the GF particles had d50 = 16.08 ± 0.05 m and d90 = 35.47 ± 0.55 m.

Short-term physical stability of the suspensions was studied for a storage period of 7 days at 8 °C (Table 3.2). The suspensions were physically stable and remained colloidal owing to the synergistic stabilizing action of the HPC–SDS combination. The slight size increase observed in most samples could have resulted from the temperature cycle during the sample preparation following the milling process, i.e., initial cooling to 8 °C and equilibration to room temperature after 7-day

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storage, and associated particle aggregation. These observations are in line with our recent work (Bilgili et al., 2016d; Li et al., 2016b), where nanoparticles of GF and other poorly water-soluble drugs such as azodicarbonamide, phenylbutazone, and indomethacin were adequately stabilized by HPC–SDS. Long-term physical stability of the suspensions was not investigated in the present study. Reproducibility was established by repeating Run 4 (the slowest, least intense breakage among all runs) and Run 14 (fastest, most intense breakage among all runs), which essentially captures the full range of breakage dynamics observed in all milling experiments. For both runs, the time-wise evolution of PSD was almost identical in the repeated runs, with slight deviations within experimental accuracy of the size measurements (see Figure B2 in Appendix B). Hence, the milling process is considered reproducible, which is in line with some previous investigations on WSMM (see Li et al., (2016a), and the references cited therein).

Another potential concern with wet media milling is potential solid-state changes to the drugs. Figure 3.2 presents the XRPD diffractograms of as-received GF, unmilled physical mixture of GF‒HPC‒SDS, as well as 256 min milled GF suspensions (Runs 4 and 14) after overnight drying. The characteristic peaks of GF appeared in all diffractograms without a broad halo after milling, despite the contribution of amorphous HPC in the samples. As compared to the as-received GF pattern, a slight reduction in the GF peak intensities in the unmilled physical mixture is observed due to dilution and surface coverage of GF particles by HPC (Hecq et al., 2005). On comparing Runs 4 and 14 patterns with the unmilled physical mixture, it is noted that the peak positions remained the same despite a slight reduction in peak

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heights after milling, which can be attributed to defect formation–accumulation during milling (Monteiro et al., 2013) besides more effective coverage of GF particles by HPC in dried Runs 4 and 14 samples than that in the physical mixture. While XRPD cannot detect minor amount of amorphous phase due to indirect inference and crystal orientation effects, the aforementioned XRPD results overall suffice to show that the crystalline state of GF was largely preserved after 256 min milling.

2degree)

10 20 30 40

Intensity (a.u.)

Run 4 Physical mixture

HPC As-received GF

Run 14

Figure 3.2 XRPD diffractograms of as-received GF, HPC, unmilled physical mixture of GF‒HPC‒SDS, and dried, milled suspensions of Runs 4 and 14. Run 4 refers to the drug suspension milled at  = 1000 rpm with db = 1500 µm beads and Run 14 refers to the drug suspension milled at  = 4000 rpm with db = 50 µm beads.