These tablet were prepared via direct compression using a compaction simulator (HB50, Huxley-Bertram Engineering, Cambridge, UK). The samples had a diameter of 10 mm and a thickness around 2 mm. The diameter and thickness were kept constant, while the filling weight of the powder material was adjusted to vary the tablet porosity. The powder was filled manually into the die of the compaction simulator to achieve precise powder filling. The compaction process was performed using a sinusoidal compaction profile, with an average speed of 0.026 m/s. The formulations of the tablets, the compression pressure, and the porosity values are given in Table 3.

The liquid penetration and tablet swelling were measured using a commercial terahertz-pulsed imaging system (TPI, TeraPulse 4000, Teraview Ltd., Cambridge, UK) in combination with a bespoke flow cell [33]. The TPI was set up with a fibre-based reflection probe equipped with an 18 mm focal-length silicon lens. The probe head was on a linear scale for the ease of spatial adjustment. The beam resulting from the THz optics had a beam waist of around 1 mm at the focus with an incident angle of 13°. The TPI setup with the flow cell (Figure 1a) measures the change of the back face of the tablet, which reflects the swelling of both the back face and the front face, where the liquid uptake occurs. Since the flow cell was not included in the DEM simulation (Figure 1b), only the front face swelling was recorded in the simulation. More details about the flow cell and its design can be foundin [33].

The compaction process was simulated by compressing the particles, which were assumed to be spherical under gravity, to a loose, random packing in a cylindrical compression die with a height of 3 mm and a radius of 1 mm. To decrease the computational cost, the simulated tablet was scaled down to a diameter of 2 mm and a thickness of 0.8–1 mm while using the experimental particle size distribution (Table 2) in the DEM. The concentration of each material, the compression force, and the punch speed were set according to the experimental setup described in Section 2.2.1 and Table 3. The number of particles simulated for each formulation is given in Table 4. The compression force and porosity were recorded during the simulation.

The Luding elasto-plastic contact model [22] was applied for particle–particle and particle–wall interactions; the model is given in Equations (1)–(3). The force in the normal direction, Fn, is given as (1)Fn=k1δnifk2(δn−δ0)≥k1δnk2(δn−δ0)ifk1δn≥k2(δn−δ0)≥−kcδn−kcδnif−kcδn≥k2(δn−δ0). with k1 as the loading stiffness, k2 as the plastic unloading stiffness, kc as the adhesion stiffness, δn as the normal overlap, and δ0 as the plastic contact deformation overlap. k2 is defined as: (2)k2=kpifδmax/δlim≥1k1+(kp−k1)δmaxδlimifδmax/δlim<1. where kp is the limit plastic unloading stiffness. δmax and δlim (see Equation (3)) are the maximum compression overlap and the plastic limit, respectively. (3)δlim=kpkp−k1ϕfR*, where ϕf is the dimensionless plasticity depth and R* is an equivalent radius. The values of the parameters used in this study are given in Table 5.

Reference [14] showed that the parameters k1 (loading stiffness) and kp (limit plastic unloading stiffness) of the Luding elasto-plastic model and the particle density impacted the compression profile mostly and the adjustment of these three parameters could cover most of the variations of the compression profile. As [14] increased the particle size in the model compared to the experiments, they accounted for this in the model by calibrating the particle density. In this work, the particle size in the model was the same as in the experiments, and hence, the measured particle (true) density was used for the simulations.

The two unknown parameters in the model, k1 and kp, were determined through an optimisation procedure that minimises the error between the porosity values calculated from the DEM model and the experimental data. For the pure MCC tablets, this was performed for the medium compression pressure (15% porosity tablets) and was validated using the data from the experiments with low- and high-compression pressure. The calibration method was based on the work of [14]. As k1 only affects the loading stage of the compaction process, the process was simulated for various different values of k1, ranging from 500 to 20,000 N/m, simultaneously using the batch simulation mode in Yade. The root-mean-squared error (RMSE) between the experimental tablet porosity and simulated tablet porosity during loading for various k1 values was minimised to identify the optimal value. Reference [19] highlighted that the difference between initial experimental porosity and that of a simulated DEM tablet is primarily caused by the deviation of the real particle shape from the assumed spherical shape in the DEM. Other factors affecting the initial packing of the powder such as intra-particle porosity and surface asperities were also not considered in the DEM. This difference in initial porosity caused an error in the estimation of k1. To minimise the effect of this initial difference in the packing of the powder, the porosity in the loading process was scaled to a value between 0 (minimum observed porosity) and 1 (maximum observed porosity) and the calculated RMSE values were used to determine the optimal k1 value. This scaling significantly improves the estimation of k1 as this parameter primarily controls the curvature of the loading profile. kp was calibrated using the optimised k1 value, and we simulated the process again at different kp to reach the final desired tablet porosity.

The simulations of the swelling of the tablets modelled in Section 2.3.1 was set up to closely mimic the experimental work described. The time step (Δt) was set at 10−6 s/step to ensure simulation stability. The radius, mass, and inertia of individual particles were updated according to a single-particle swelling model (Equations (4)–(12)). Reference [29] originally derived this model to describe the swelling process of SAPs. The model assumes that the swelling of a particle is driven by the difference in the chemical potential between the particle and water (liquid medium) [36]. The single-particle swelling model is given as follows: (4)drp,idt=fwDϱsrp,iϱwQmax−QiabsQiabs, (5)Qiabs=miw+mismis=rp,i3ρwrp,03ϱs−ϱwϱs+1. where rp,0 is the initial particle radius, rp is the particle radius at time t, ϱs is the density of a dry particle, and ϱw is the density of the liquid (deionised water in this study). D (μm2/s) is the diffusion coefficient for water molecules in the particle, which was assumed to be constant. Qmax is the maximum absorption ratio. The values of D (μm2/s) and Qmax for the various materials used in this study were taken from [32] and are given in Table 1.

This model assumes that the entire particle is exposed to the liquid, which is not valid when the particle is part of a compact tablet. In a compact tablet, particles form bonds with neighbouring particles across a contact area. This reduces the effective (available) surface area of the particle that is exposed to the absorbing liquid. This is accounted for by introducing the factor fw in Equation (4) that describes the fraction of the available surface (Aactual) to the total particle surface area (Ap=4πrp). (6)fw=AactualAp.

Aactual is defined as the particle surface area subtracted by the sum of the overlapping area with neighbouring particles: (7)Aactual=Ap−∑inAcap,i,

Acap is the surface area of the normal displaced volume between two neighbouring particles, particles i and j, and defined as a function of the particle, centroid i, xi, yi, zi, and the coordinate of the contact point between particles i and j (xj, yj, zj). n is the number of neighbouring particles. Acap is given as (8)Acap=2πrph, with h as the height of the overlapping cap, defined as (9)h=rp−(xj−xi)2+(yj−yi)2+(zj−zi)2.

The position of the liquid was also updated every 100,000th time step using the experimental data. As the time instances of the experimental liquid penetration data did not match the simulation time points, a power law (y=a·tb) was fit to the experimental results from Section 2.2.2, enabling the calculation of the liquid front position in the tablet at the simulation time points. The fitting parameters for all formulations are shown in Table 6. The simulation of a single-particle swelling assumes that a particle starts to swell once the liquid reaches the particle centre; the model than considers the available wetted surface area to be Aactual. The particle size change was implemented in the DEM by defining a growth factor (f): (10)f=rp(t+MΔt)rp(t)=1+MΔtrp(t)drpdt.

To accelerate the simulation, the particle radius, mass, and inertia were updated only every 100,000th time step (M = 100,000). rp(t+MΔt) is the radius at time t+MΔt. Due to the absorption of the liquid by the particle, the mass (m) and inertia (J) of a particle were also updated: (11)m(t+MΔt)=m(t)·f3,(12)J(t+MΔt)=J(t)·f5.

The entire workflow to simulate the tablet swelling and break-up is presented in Figure 3.

The porosity was measured using the voxel porosity method [34]. This approach divides the whole volume into a dense grid of voxels at a given resolution (resolution = 200 μm) and counts the voxels that fall inside any of the particles. The porosity, ϵ, is calculated as (13)ϵ=V−VvV. where V is volume of the tablet and Vv is the volume of voxels that fall inside any particles.

The pore sizes were determined using the triangulation and pore finite volume method described in [37] and subsequently [29]. First, a triangulation procedure was applied to the pore space of the tablet using solid particle centres as vertices for the tetrahedra. The tetrahedron spans across four neighbouring particles and defines the pore space. This is referred to as a pore unit. The size of each pore unit in the tablet was then calculated as the radius of inscribed circle of the tetrahedron.

Cumulative porosity maps were generated to analyse the pore space spatially. This method is described in detail in [38]. In brief, PoreSpy [39] was used to generate a 3D voxel image of the tablet based on particles’ position and radius. A cuboid subsection (1400 × 1400 × 800 μm3) of the voxel image at the centre of the tablet was selected for this analysis. The maps were obtained by dividing the sum of the number of voxels classified as voids along each dimension (x, y, and z) by the total number of voxels per dimension. The generated maps depict the void fraction at each position.

Figure 7a,c show the porosity change during the swelling of the tablets for both the overall tablet and only the wetted volume. The initial delay in the porosity measurement for the wetted volume was due to the analysed volume being too small to be accurately measured and representative. The porosity decreased initially during the swelling of the wetted volume (Figure 7a). This was observed for PH101 tablets with ϵ0=15% and ϵ0=22%, indicating that the pores were closing in the initial stages of the swelling process. After the initial decrease, the porosity increased for both the wetted volume and the whole tablet. This increase was primarily attributed to the fact that the tablets were eroding, i.e., individual particles were breaking away from the tablet. For PH101/CCS (Figure 7c), the porosity of the wetted volume increased rapidly in the first three seconds. This was attributed to the fast swelling of CCS particles, resulting in a rapid increase in porosity close to the surface of the tablets. The changes in porosity were caused by particles’ movement. The changed in porosity along with the swelling of individual particles caused an increase in the tablet volume.

Unsurprisingly, the interparticle forces calculated in the DEM were strongly affected by the swelling process (Figure 7b,d). The swelling exerted stress on the particles, which caused an increase in the interparticle forces. The stress will reach a maximum, which then would lead to the disintegration of the tablet. The interparticle forces rose faster with increasing porosity (Figure 7b). For PH101/CCS (Figure 7d), the tablets with the two lower CCS concentrations had the same force profile, while the tablet with cCCS = 8% experienced a slower increase of the interparticle force.

Figure 8 shows the pore size (pore body) distributions (PSDs) of the tablets at different time points during the swelling process. As expected, the initial average pore size was larger for tablets with a higher initial porosity, ϵ0, i.e., a lower compaction pressure resulted in bigger pores. The initial pore size of the PH101/CCS tablets was in the same range as the PH101 tablets. PH101/CCS with cCCS= 2% tablets had similar PSDs as tablets with ϵ0=15%, as both had approximately the same initial porosity, and the initial PSD was primarily driven by the PH101 as the major component in the tablet. As expected for the tablet with a higher CCS content, cCCS= 5% and 8%, the number of smaller pores (∼10 μm) was higher than for cCCS = 2%. For all tablets, the pore size increased over time, and the pores opened up shortly before the break-up of the tablet. At the maximum swelling capacity, there were pores >70 μm, which were not present at the two previous time points.

Cumulative porosity maps were generated to analyse the pore space spatially. This method is described in detail in [38]. In brief, PoreSpy [39] was used to generate a 3D voxel image of the tablet based on particles’ position and radius. A cuboid subsection (1400 × 1400 × 800 μm3) of the voxel image at the centre of the tablet was selected for this analysis. The maps were obtained by dividing the sum of the number of voxels classified as voids along each dimension (x, y, and z) by the total number of voxels per dimension. The generated maps depict the void fraction at each position. These maps were obtained for different time instances during the swelling process (Figure 9 and Figure 10).

As the liquid moves in one direction from the top to the bottom, resulting in the swelling of the particles from top to bottom, it is expected that the porosity will also mostly be affected along the z direction. However, it can be observed that the porosity changed across the entire tablet, though it had not been fully wetted.

The cumulative porosity analysis highlighted that the porosity was higher on the edges during the swelling process, which indicates that the tablet began to break up from the edges. For the PH101 tablets with ϵ0=10% and ϵ0=15%, the trend in the porosity distribution was similar across the entire tablet, while for PH101, ϵ0=22%, the porosity was significantly higher on the edges compared to the tablet centre.

The cumulative porosity maps for the PH101/CCS tablets were similar at the initial and the halfway times. The porosity was slightly higher at the edges for PH101/CCS, cCCS = 8%, at the halfway point. At the final time, where the tablets reached their maximum swelling capacity, the porosity at areas close to the surface increased with increasing CCS content. This was in contrast to the overall swelling and disintegration of these tablets, where cCCS = 8% was the tablet with the slowest swelling, i.e., the time to reach the maximum normalised swelling capacity. The disintegration of the cCCS = 8% tablet was, therefore, not controlled by the swelling of the tablet, but the liquid uptake process, which is further discussed below. This slower liquid uptake means that a smaller number of particles were swelling for the cCCS = 8% tablet, which in turn resulted in a slower change of the interparticle forces for this case compared to the lower CCS concentrations.

Both experiments and simulations showed a slower swelling rate for the tablet with the highest CCS content (cCCS = 8%) (see Figure 6). These tablets also had the slowest liquid penetration rate. A similar trend was observed by [41,42], who showed that the disintegration time is prolonged with higher CCS content. As the liquid penetration rate is directly linked to the pore size, it is crucial to understand how the pore size changes in the wetted domain, as well as in the dry domain of the tablet.

The average pore size of the wetted domain of PH101/CCS, cCCS = 8%, decreased during the initial stages of the swelling (Figure 11a). The pore size of PH101/CCS, cCCS = 5%, slightly decreased initially, whereas PH101/CCS cCCS = 2% was constantly increasing. Reference [30] explained that the reason for the decrease in the pore size (porosity) was due to the swelling of particles being much faster than that particle contacts could dissipate their potential energy. Therefore, the particle movement would be limited and the particle packing would clog (i.e., the porosity would tend to decrease). The decrease in average pore size is an indication that the pores were closing, which reduces the liquid flow. The decrease in the pore size behind the liquid front could be observed across all formulations, but it was most significant (>30% reduction in pore size) for the tablets with cCCS = 8%. Interestingly, the pore size reduction had a nonlinear dependence on the liquid penetration depth. CCS swelled significantly faster compared to PH101, causing the closing of the pore space, which reduced the liquid flow into the tablet.

As the particle swelling of the material studied was omnidirectional, it can affect the dry volume of the tablet. A change in the pore space of the dry volume impacts the liquid penetration rate, which is mostly driven by the capillary action. The capillary pressure depends on the pore size at the liquid front, i.e., at the interface of the wetted and the dry volume.

The pore space of the wetted and dry volume can be analysed through the pore size ratio (PS/PS0) across the tablet height at different liquid front positions (Figure 12). The focus was on the smaller pores (<30 μm), as smaller pores have a greater impact on liquid flow compared to large pores.

The number of pores decreasing in size at the surface of the tablet was largest for PH101/CCS, cCCS = 8%. At three seconds, the small pores of the cCCS = 5% and cCCS = 2% tablets were opening up more significantly compared to cCCS = 8%. It can also be observed that the dry pores changed in size substantially during the swelling of the wetted volume with a decrease in the size of many small pores. Again, this phenomenon was most significant for the PH101/CCS tablet with cCCS = 8%. The reduction in the pore size of the dry pore and wetted pores slowed down the water uptake of these tablets, which resulted in a slower swelling and ultimately delayed disintegration of the entire tablet.