Ind. Eng. Chem. Res. 2007, 46, 5785-5793
5785
Imbibition and Dissolution of a Porous Medium Nelly Brielles,†,§ Florence Chantraine,‡,§ Maryle` ne Viana,‡ Dominique Chulia,‡ Paul Branlard,§ Gilles Rubinstenn,§ Franc¸ ois Lequeux,| Didier Lasseux,⊥ Marc Birot,# Didier Roux,†,+ and Olivier Mondain-Monval*,† Centre de Recherche Paul Pascal, CNRS UPR 8641, UniVersity Bordeaux 1, AV. Schweitzer, 33600 Pessac, France; GEFSOD, EA2631, Pharmacy Dept, 2 rue du Docteur Marcland, 87025 Limoges, France; EUROTAB Co., ZAC les Peyrardes, 42 170 Saint-Just-Saint-Rambert, France; PPMD, UMR 7615, ESPCI, 10 rue Vauquelin, 75 231 Paris Cedex 05, France; TREFLE, ENSAM, Esplanade des Arts et Me´ tiers, 33 405 Talence, France; ISM, UniVersity Bordeaux 1, 351 cours de la Libe´ ration, 33 405 Talence, France; and Saint-Gobain-Les Miroirs, 18 aVenue d’Alsace, F-92096 La De´ fense Cedex, France
The disintegration of porous tablets, obtained by powder compression, is being studied. Tablets dissolve into a liquid through a two-step process including (i) an imbibition process inside the porous medium and (ii) a dissolution process of the powder aggregates. First, we show that the imbibition process follows the classical laws of capillarity. Then, we evidence and explain the influence of the presence of trapped gas inside the tablet. Finally, we demonstrate the existence of two limit disintegration regimes appearing at low and high porosity values. 1. Introduction Porous tablets, resulting from powder compression, are encountered in an increasing number of industrial applications such as pharmaceutics, detergents, food and feed products, drug delivery devices, etc. In most cases, the tablets are to be introduced into a liquid and go from their initial form to a molecularly dissolved state. This process is the result of two steps that may or may not be coupled depending on the experimental conditions. As the tablet is dipped into the liquid, imbibition takes place, i.e., the liquid invades the porous medium. When it is also a solvent of the initial powder, a second step takes place, which is the dissolution of the matrix. Understanding and controlling this whole two-step process is of paramount interest for the purpose of industrial applications. In pharmaceutics, the active component may constitute only a tiny part of the tablets. Thus, the dissolution kinetic may be tuned by the excipients. The choice of physico-chemistry and functionality of the excipient but also the compression pressure determine both the kinetics of dissolution and the mechanical properties in the solid state. The situation is different in detergency, since the active volume fraction of the tablets has to be as large as possible and typically never less than 80% of the tablets. It is, thus, important to understand how the invasion of the porous medium that constitutes a tablet couples with the dissolution of its compounds and how this coupling controls the kinetics of dissolution. If the two independent processes are fairly well-understood,1,2 coupling brings some complexity to the phenomenon. In this paper, we focus on the following question: is it possible to precisely tune such property by adapting the porosity of the system both in terms of pore size and global air content? For this purpose, we investigate the * Corresponding author. E-mail:
[email protected]. Tel.: 33 (0)5 56 84 56 69. Fax: 33 (0)5 56 84 56 00. † Centre de Recherche Paul Pascal. ‡ GEFSOD. § EUROTAB Co. | PPMD. ⊥ TREFLE. # ISM. + Saint-Gobain-Les Miroirs.
dissolution behavior of tablets of various porosities. Our system is composed of a chlorine provider (sodium dihydrate dichloroisocyanurate) used as a disinfectant in industrial and domestic applications as a more convenient substitute to hypochlorite. The initial powder is compressed at various pressures, which leads to tablets of various porosities. We study the dissolution kinetic of this system and, at the same time, measure the penetration of the liquid phase inside the tablet. By using two different liquids, a good (water) and a bad (silicone oil) solvent of the chlorine powder, we are able to decouple the processes of dissolution and imbibition. In a first part, we will present the experimental system before characterizing the tablet porosity by different techniques such as Hg porosimetry, permeametry, and tablet imbibition by silicone oil. Then, we will show how the silicone imbibition process is driven by capillarity and evidence the role of trapped air. Measurements of tablets internal pressure will confirm the importance of porosity distribution in the imbibition process. Finally, we will study the behavior of the tablets in the presence of water (a dissolving liquid), which results in the coupling of imbibition and dissolution. 2. Experimental Procedures 2.1. Tablet Composition. Industrial disinfectant tablets contain at least four basic components (chlorine provider, organic acid, and carbonate for effervescence and a disintegrant). A model system containing only the chlorine provider (DCCNa, sodium dihydrate dichloroisocyanurate, ACL 56, Oxychem, U.S.A.) has been chosen to study the tablet dissolution kinetics. The particle-size distribution of the powders was evaluated with a laser diffraction analyzer Mastersizer 2000 (Malvern Instruments Ltd., Worcestershire, United Kingdom) by suspending the particles in the air (dispersion pressure ) 1 bar; vibration rate ) 30%). The volume equivalent median diameter (D0.5, µm) was calculated from three measurements. The specific surface area, quantified by the extend of the powder/gas interface, was determined by nitrogen adsorption on the surface of the material using a Gemini 2360 analyzer (Micromeritics Instruments Inc., Norcross, GA).
10.1021/ie0701569 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/14/2007
5786
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007
Table 1. Physical Characteristics of DCCNa Powder product
D0.5 (µm)
specific surface area (m2‚g-1)
dpycno (g‚cm-3)
DCCNa
588
1.434 ( 0.148
2.001 ( 0.001
Finally, the pycnometric density of the powder (dpycno, g‚cm-3) was determined using a helium pycnometer (Accupyc 1330, Micromeritics Instrument Inc., Norcross, GA). The DCCNa powder characteristics are summed up in Table 1. 2.2. Tablet Manufacturing Process. The tablet compression was performed using a lab-scale alternative tabletting machine (Korsch, Type EK0, Korsch Maschinenfabrik, Germany) with 1 cm2 flat punches and a die volume of 1 cm3. The pressure was adjusted from 30 to 300 MPa to produce tablets with controlled porosity. 2.3. Description of the Experimental Setup. First, we aimed at characterizing the imbibition kinetics independently from the dissolution process. For this purpose, the tablets are immersed in a nondissolving liquid (low-viscosity silicone oil: Fluka silicone oil DC 200, of viscosity 10 mPa‚s) that invades the compacted powder. Then, we use two setups in order to follow the imbibition process in two different situations: (i) The silicone oil is penetrating the tablet from the bottom while the occluded gas in the tablet is free to evade by the top of the tablet (Figure 1a); (ii) The tablet is stuck between two glass slides before being dipped into silicone oil. In such case, the silicone oil penetrates inside the tablet from its whole free faces and the occluded gas is trapped inside the tablet (Figure 1b). In both cases, the progression of the silicone front is observed by backlighting the tablet. The part that is wetted by the silicone oil looks brighter than the dry part. In order to measure the impregnated length h as a function of time, some pictures of the tablets are taken during the imbibition process. The tablet dissolution kinetic is followed using conductimetry to study the whole imbibition/dissolution process. In fact, the dissolution and hydrolysis of DCCNa lead to an ionic solution whose concentration is proportional to the solution conductivity. During the analysis, the tablets are immersed in 300 mL of distilled water, with magnetic stirring to ensure a good
homogeneity of the solution. The solution conductivity is measured using a conductivity meter (CDM210, Radiometer Analytica, France) with a 2-pole conductivity cell (CDC641T, Radiometer Analytica, France). This cell design permits continuous measurements because of the easy flowing of solution through the two platinum plates. The final dissolution time is the one at which conductivity is stabilized, i.e., when all the components are dissolved. 3. Porosity Characterization A porous system is a complex system that is difficult to describe.3 The geometry is characterized by many variabless from the simplest to the most complex ones: volume fraction, size of the voids averaged with a given technique, pair distribution of the mattersobtained by X-ray or neutrons scattering, topological properties, etc., and none of them is sufficient to give a precise description of the porous system. Here we will measure some properties of the porous system, in terms of estimation of the pores’ diameters that depend on the technique used, as a function of the tablet global porosity. 3.1. Porosity. The porosity (%) is deduced from the pycnometric density of the powder (dpycno, g‚cm-3) and the apparent density of the tablets (dtablet, g‚cm-3), measured 24 h after manufacturing. For a compression pressure varying from 30 to 300 MPa, the porosity decreases from 30% to 10%. 3.2. Porous Distribution. Porosity measurements were performed using a mercury intrusion porosimeter (AutoPore IV 9500 Micromeritics Instruments Inc.). This method consists of measuring the volume of intruded mercury into the pores as a function of the applied pressure. From the total intruded mercury volume (VHg, cm3) and the apparent volume of the tablet (Vtablet, cm3), the porosity (φmeasured, %) is determined,
φmeasured )
VHg × 100 Vtablet
where Vtablet ) S × h with S being the section of the tablet (1 cm2) and h being its height (cm). Because the maximal pressure used for the high-pressure measurement is 207 MPa, the minimal pore size in which mercury can enter is ∼6 nm. It is clearly established that the calculation of the total porosity underestimates reality. However, we have checked that the values obtained from mercury intrusion and pycnometric densities are comparable. The mercury intrusion measurements permit one to have a description of the poresize distribution. In fact, each intruded volume corresponds to an applied pressure ∆P. Knowing the mercury surface tension (γHg ) 485 mN/m) and assuming that its contact angle θ at the surface of the tablet is equal to 130°, it is, thus, possible to deduce the characteristic pore radius RHg corresponding to each applied pressure ∆P from the Laplace equation:
RHg ) 2γHg cos θ/∆P
Figure 1. (a) Scheme of the setup used to measure the imbibition of a tablet from the bottom: gas is free to escape. (b) Scheme of the setup used to measure the imbibition length in the case of a total immersion of the tablet (gas may be trapped inside the tablet).
(1)
(2)
The pore size diameters are thus calculated, and the evolution of the differential volume of intrusion is plotted as a function of the pore characteristic size. A median pore diameter (volume) is calculated for each porous distribution. The “median diameter” on the volume basis is defined as the diameter of the 50% point on the volumetric cumulative distribution. 3.3. Permeametry. The tablet permeability is measured on a minipermeameter (Figure 2). Permeametry measurements consist of measuring the gas flow (N2) that takes place through
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 5787
Figure 2. Scheme of the minipermeameter used to determine Rv. Figure 4. Time evolution of the tablets imbibition square length when silicone oil is penetrating the tablet by the bottom (situation of Figure 1a). Table 2. Physical Characteristics of Silicone Oil and Water Used for Imbibition Experiments
silicone water
γ (mN/m)
cos θ (on tablets’ surface)
η (mPa‚s)
20 72
∼1 ∼1
10 1
Figure 3. Scheme of a porous material made of parallel capillaries.
a porous material when a well-defined pressure difference is applied between the input and the output. The permeability K, homogeneous to a surface, can be deduced from the Darcy law applied to compressible gases,
K)
2µQ1P1 aG0(P1 - P2 ) 2
2
(3)
where µ is the gas dynamic viscosity, Q1 is the nitrogen flow rate, P1 is the input pressure (typically from 1 to 6 bar), P2 is the output atmospheric pressure, a is the nozzle radius, and G0 is the tablet shape factor (G0 ) 5.5 in our case). The calculated mean free path of nitrogen is ∼30 nm at Patm and ∼6 nm when P1 ) 6 bar, which is lower than the calculated pore diameter. It is, thus, possible to deduce an equivalent hydraulic radius Rv from the tablet permeability thanks to the Bear relation,4
Rv )
x8TK Φ
(4)
in which the porosity (φ) of tablets is modeled as a group of parallel capillaries of radius Rv and tortuosity T ) 1 (Figure 3). 3.4. Imbibition of a Porous System by Silicone Oil. The laws that govern the imbibition of a porous material are clearly established as long as the gas or the fluid that is initially present inside the porosity is free to escape. If we consider a fluid traveling through a capillary and in a laminar flow, the loss of pressure ∆P due to the viscous energy loss is described by the Poiseuille law,
∆P )
8ηh dh Rv2 dt
(5)
where ∆P is the pressure drop inside the capillary, η is the fluid dynamic viscosity, h is the length filled by the fluid, t is the time, and Rv is the hydraulic radius. Besides, the Laplace law gives another expression of this pressure drop that is due to the curvature of the meniscus that is formed inside the capillary tube,
∆P )
2γ cos θ RL
where ∆P is the pressure drop, γ is the surface tension, θ is the contact angle, and RL is the capillary radius. From eqs 5 and 6, and neglecting the contribution of gravity, one can deduce what is referred to as the Washburn law (through a time integration between 0 and t),5
(6)
h2 )
RWγ cos θ t ) A wt 2η
(7)
where
Rv2 RW ) RL
(8)
RW is called the Washburn radius, and Aw has a dimension of a diffusion coefficient. Of course, in this approach, the porous materials are considered as composed of parallel capillaries, which obviously is a rough approximation. Yet, surprisingly, earlier studies2,6 have shown that this law can give a good description for the imbibition of porous materials of different types with porous structures very different from parallel capillaries. We have measured the evolution of the length h as a function of time for tablets with various porosities (9, 15, and 20%). On Figure 4, we plot the square length h2 as a function of time, which clearly reveals a good agreement with eq 7. Knowing the values of γ, θ, and η for silicone (see Table 2), we will deduce the various values of RW from the slopes of the different curves. 3.5. Discussion on the Results of Porosity Characterization. We have plotted the three typical lengths deduced from, respectively, porosimetry, permeability, and imbibition, as a function of porosity (Figure 5). It is interesting to notice that these lengths are proportional and follow the logarithmic relation
log10(Ri)) Aφ + Bi
(9)
where Ri is the length of type i (i refers to the technique used to measure R), φ is the porosity, A is a constant between 6 and 8, and Bi is a constant different for each technique. We have no explanation for this relationship, but it evidences that the ratios between the different characteristic lengths remain
5788
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007
Figure 5. Evolution of the lengths measured by the imbibition (RW), mercury porosimetry (RHg), and permeametry (Rv) experiments as a function of porosity.
Figure 6. Evolution of the hydraulic radius Rv as a function of porosity. The empty squares and triangles are the calculation of Rv using eq 11 and two different values for RS.
constant over the whole compression range. This means that the porous structure of the material remains similar upon compression. Such behavior is reminiscent of what can be expected in the particular case of compacted spheres of radius RS. In that situation, the relation between the permeability K and the material porosity φ is given by the Ergun equation:7
K≈
φ3RS2 45(1 - φ)2
(10)
It is, thus, possible to relate the hydraulic radius Rv ) x8K/Φ, to the sphere radius
Rv )
x
8 RSφ ‚ 45 1 - φ
(11)
Thus, though the initial powder constituting our tablets is very different from spheres, we tried to fit our data to eq 11 (solid line on Figure 6). As one can see on Figure 6, a good agreement is obtained between our data and the prediction in the range of porosity φ < 15% and with RS ) 10 µm. This obtained value for RS is much smaller than the average radius of the particles constituting the initial powder (〈R〉 ≈ 400 µm). However, the study of powder deformation under pressure8,9 evidences that the DCCNa powder has a brittle behavior under compression. As a consequence, the grains composing the DCCNa powder might be smaller at smaller porosities. In order to get a good agreement between the experimental data and eq
Figure 7. Time evolution of the wetted length of tablets when totally immersed in silicone oil (situation of Figure 1b).
11 for the lowest porosities, the grains should have a diameter around 20 µm. As a result, we believe that RS varies with the porosity and cannot be considered as constant over the whole range of porosity investigated in this study. In our case, the observed exponential law thus seems to correspond to a “modified” Ergun relation because of the decrease of the sphere radius as the porosity φ decreases (plastic deformation and fragmentation). Furthermore, we notice some significant discrepancies between the characteristic size values obtained from the different methods. The values obtained with Hg porosimetry and permeability are very similar and about ten times larger than the ones deduced from the imbibition measurements (RW). As expressed in eq 8, for a simple situation, RW is a function of the two lengths, which are Rv and RL. These two averaged radiuses are different in the case of real capillary with nonuniform pores.10 Whereas the resistance to capillary rise is dominated by the smallest diameter (Rv) along the capillary, the driving force is dominated by the largest diameter (RL). Consequently, the apparent radius RW is smaller than the values obtained with Hg porosimetry or permeability. This phenomenon has already been quantified by Bico,6 who also observed a ratio of 10 between the real microscopic radii and the Washburn ones. 4. Tablet Imbibition/Dissolution Kinetics 4.1. Tablet Imbibition from All Sides. First, we consider the situation that occurs when a tablet is entirely plunged into the invading liquid and when the gas that is present is trapped inside the porous medium. For this purpose, we used the experimental setup that is described in Figure 1b. At first, we can notice that the Washburn law still gives a good description of the imbibition curve when some gas is trapped inside the system (Figure 7). From the obtained curves, we deduce the values of RW and plot them as a function of porosity together with the ones deduced from the curves presented in the previous paragraph (Figure 8). The same profile of logarithmic relationship between RW and the porosity is observed. This means that the imbibition mechanisms must be very similar in both cases, i.e., with or without gas entrapment. However, a quantitative difference is observed and the imbibition process is much slower when the tablet is totally immersed in silicone oil. From the slope of the curve, one can deduce a Washburn radius RW that is approximately twice smaller than the value obtained in the first case. We believe that the mechanism proposed by Nogami et al.11 can account for these results. In their paper, the authors calculate the speed of progression of a wetting liquid inside a capillary tube of radius r that is connected on its other side to a capillary
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 5789
Figure 8. Comparison between the Washburn radii measured in situations of bottom imbibition (Figure 1a) and total imbibition (Figure 1b).
Figure 10. Evolution of the different measured lengths as a function of porosity.
Figure 9. Scheme of the imbibition of a nonuniform capillary.
tube of radius R (R > r) (Figure 9). In the first step of the imbibition process, the liquid penetrates more rapidly inside the large tube than on the other thin end of the capillary (as expected from eq 7). However, as time goes by, the liquid penetrates the thinner tube, which gives rise to a capillary pressure P1 ) 2γ/r in this part of the tube. Since this pressure is larger than the one endured at the other end, P ) 2γ/R, the liquid is pushed away from the larger tube and the trapped gas can escape under the form of bubbles of radius connected to R. This is very likely what happens in our system. When some trapped gas is present inside the tablets, the imbibition process occurs by the smallest pores, which gives rise to an inner capillary pressure that overcomes the capillary pressure inside the biggest capillaries. This is consistent with the observation of gas bubbles escaping by one or two pores during the tablet imbibition. One can easily have a rough estimate of the surface capillary diameters (2R) by measuring the average diameters of the bubbles (2RB) that escape from the tablet. The relation between these two quantities is given by the equilibrium between gravity and surface tension when an air bubble forms at the surface of the tablet in a capillary of radius R:
R ) (2RB3FLg)/3γ cos θ
(12)
We have measured, on blown-up pictures, the average sizes of the air bubbles that escape from tablets having different porosities. From this, we deduce a surface capillary radius RS using eq 12 (Figure 10). Though this value of RS is subjected to many uncertainties since RB is very hard to precisely determine, RS evolves with the porosity as the other length. This surface radius seems to be slightly bigger than the equivalent radius measured by porosimetry and permeametry. It is consistent with the previous assumption as we expect bubbles to go out by the biggest pores. This radius may well be underestimated, as bubbles may detach from the surface before reaching their equilibrium weight because of the flow of water entering the tablet and pushing the air outside. To definitely conclude on this issue, we have measured the inner pressure that exists inside the tablets when the imbibition
Figure 11. Time evolution of the measured tablet internal pressure in case of total immersion (Figure 1b).
Figure 12. Evolution of the different measured lengths as a function of porosity (data from Figures 10 and 11).
process takes place. A membrane captor is hermetically stuck at the surface of the tablet while it is immersed into the invading liquid. The recorded pressure is plotted as a function of time for different porosities (Figure 11). In all cases, we observe an increase in the pressure to reach a plateau value. Such excess pressure value ∆P can be used to calculate the average radii of the largest capillaries R ) 2γ cos θ/∆P. This “largest” radius is plotted as a function of tablet porosity in Figure 12. As expected, this radius is significantly bigger than those measured previously. Besides, it is particularly interesting to notice that these radii are consistent with the porogram obtained from mercury porosimetry (Figure 13). The internal pressure is fixed by the largest pores when the tablet is totally immersed as gas is escaping through the “easiest” way. Such a behavior
5790
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007
Figure 13. Mercury porosimetry porograms of tablets with different porosities: 6, 13, and 18%. The “largest diameter” deduced from internal pressure measurement is pointed in each case.
Figure 14. Evolution of the Washburn radii as a function of porosities for tablets immersed by one side in water and silicone. On the average, there is a difference of ∼10 nm between the RW’s measured for water and for silicone oil.
also explains why the average RW found in the situation of totally immersed tablets is smaller than the one obtained in the first case. Indeed, since the gas escapes from the largest pores, they no longer can participate in the imbibition process. As a consequence, the imbibition process is then mostly driven by the smallest capillaries. 4.2. Water Imbibition/Dissolution Kinetics. 4.2.1. Water Imbibition from One Side. The tablets imbibition was further studied with a dissolving liquid, water in this particular case. In this situation, the liquid front always remains homogeneous, whatever the porosity. As a consequence, the imbibition speed is the same everywhere inside the tablet. We never observe the formation of channels as described in the literature when the invading liquid is injected into the porous material at various speeds.12,13 It is, thus, possible to unambiguously define a wetted length characteristic of the imbibition. Once again, eq 7 is well-appropriated to fit our imbibition curves for tablets partially immersed, and we can deduce the Washburn radii from imbibition experiments. Whereas we could expect faster imbibition kinetics resulting from pore enlargement due to dissolution, we obtain smaller RW’s at equal porosities. Everything happens as if the pore average radius had decreased by ∼10 nm (Figure 14). We believe that this might be the result of a nonhomothetic swelling of powder seeds as soon as they are wetted by water, resulting in a reduction of the intergrain spacing, or of a viscous layer of water containing DCCNa just below its solubility limit. 4.2.2. Water Imbibition from All “Free” Sides. Let us now consider imbibition kinetics when tablets are totally immersed in water (Figure 15). Two cases can be distinguished. On the one hand, for tablets with low porosity (φ < 10%), it is no
Figure 15. Time evolution of the square wetted length for tablets totally immersed in water.
Figure 16. Evolution of the different measured Washburn radii as a function of porosities for experiments in water and silicone and in situations of partial and total immersion.
longer possible to fit the curve to eq 7 since the imbibition is very slow compared with dissolution. On the other hand, for tablets with high porosity (φ > 10%), it is still possible to use the Washburn law to fit the imbibition kinetics. In that case, we observe a smaller RW when compared to the values obtained for tablets in silicone. As explained above, this slowdown of the imbibition is likely related to the dissolution phenomena of DCCNa (Figure 16). However, the evolutions of the RW’s with porosities, measured in each case, are no longer comparable. We believe that this is due to the following reasons: (1) The whole process becomes much slower, and the dynamic of the wetting couples with the one of the swelling. (2) For the most compacted tablets, the imbibition is so slow that it couples with the dissolution of the powder. Only a small area around the tablet is impregnated by water, and the wetted length is nearly constant during all the dissolution process (about 1 or 2 mm). In that case, we can assume that the imbibition is the limiting step of the disintegration process. We will further examine these mechanisms in the last section of this paper. 4.2.3. Disintegration Kinetics for the Totally Immersed Tablets. In order to evaluate the influence of imbibition on the whole disintegration process, tablets dissolution kinetics are recorded for increasing porosities. Three different kinetic regimes are observed with decreasing values of the tablet porosity (Figure 17): fast, intermediate, and low disintegration regimes. We believe that the process of dissolution of our tablet involves different processes with associated characteristic times. This is schematized on Figure 18: (1) t1 and t2 are the characteristic times for the product and reactant transports from the bulk to the solid surface. As our
Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 5791
Figure 17. Dissolution kinetics of tablets with different porosities. Dashed line fits the “low porosity” curve to the “Hixson-Crowell” eq 15 in the text.
Figure 19. Dissolution kinetic of a low-porosity tablet (φ ) 10%) at various temperatures.
In our particular case, d ≈ 11 mm and h ≈ 5-6 mm. Thus, we can roughly consider that d ≈ 2h and, consequently, that S ≈ 4πh2 and V ≈ πh3. We are, thus, very close to the case of a spherical tablet of radius R (S ) 4πR2 and V ) 4/3πR3). So, V ) K1h(t)3 and dV/dt ) K1h(t)2 dh/dt. Besides,
dV/dt ) -KS ) K2h(t)2
Figure 18. Schematic representation of a tablet while dissolving. The different characteristic times are associated to the dissolution process (t1 and t2, product and reactant transport; t3, dissolution; t4, imbibition; and t5, molecular diffusion).
tablets are immersed in strongly stirred solutions (600 rpm), these characteristic times can be considered as much shorter than all the other ones. (2) t3 is the characteristic time for the reaction of the chlorine provider with the water (the dissolution reaction). (3) t4 is the characteristic time for imbibition. This time can be measured using the procedure described previously and strongly depends on the tablet porosity. (4) t5 is the molecular diffusion characteristic time. This time can be evaluated using the Stoke-Einstein equation for the diffusion coefficient D ) kT/6πηr where kT is the thermal energy and r is a characteristic molecular length (typically 1 nm, i.e., much smaller than the characteristic pore size value). For 1 mm, t5 ≈ 2500 s. We first consider the case of the low-porosity tablets (φ < 10%, low disintegration regime). In that case, a very slow imbibition of the tablet is observed (Aw ≈ 10-9 m2/s) and the wetted length is very small (typically 20%) disintegrate a few seconds after their immersion in water. Therefore, their behavior is very close to the one of powder grains. The dissolution is nearly linear with time. This is probably due to a case II type of invasion of water in the individual grains.15 We have seen the importance of imbibition speed, which strongly depends on the pore average size. We believe that this is the critical parameter that is driving the transition from one regime to the other. Thus, in principle, it should be possible to
Porous tablets made of compressed powder of polydispersed grains are intrinsically very inhomogeneous materials, resulting in the presence of many pore characteristic lengths in the system. Despite this aspect, our study shows that the imbibition of such a system may obey very simple laws as far as observations are performed at the macroscopic scale. Moreover, the porous structure of the system is found to be similar upon compression when the grains composing the tablet have a fragmentary behavior.8 As a consequence, the different characteristic lengths of the system all obey the same exponential variation law with the porosity. The imbibition of the porous medium is shown to be sensitive to the presence of gas trapped inside the tablet. An excess pressure inside the tablet slows down the imbibition process. However, the gas evacuation is possible and we evidence that it simply occurs through the largest pores of the system, as expected from the Laplace law on capillary pressure. When the tablets are immersed in a dissolving liquid, the powder dissolution couples with the material imbibition, leading to three different regimes depending on the porosity of the system: (i) At high porosity (>20%), the whole disintegration process is driven by the reaction of dissolution of the powder as the tablet rapidly breaks apart in different fragments as soon as it is introduced in the liquid phase (very fast imbibition process). (2) At very low porosity (
