Estimation of distance between particles of powder in well dispersed sample for particle size analysis Krasnokutsky Dionys deonisk@yahoo.com While performing particle size analysis of different powders on Particle Size Analyzers (by laser diffraction) the next question can arise how to make more or less correct estimation of concentration of sample and how to estimate average distance between particles in well dispersed sample (liquid) the data can be important in understanding of manufacturing process in pharmaceutical industry (for example in case of wet granulation). By doing math a little and using data, obtained by Particle size analyzers (by laser diffraction) the estimation can be done. Discussion and results At first let's imagine that in well dispersed sample(liquid) each particle is situated in box (cube) with cube edge length a, then the whole sample consists of n cubes, having length a (Figure 1), where n is the number of particles. After making the assumption that each Figure 1 particle is situated in the middle of the cube, it can be seen that the shortest distance between centers of two particles: p1 and p2, which are situated in the middle of the cubes S = a. a is actually the shortest distance between 2 particles (between centers of two particles: p1 and p2), let's call it the average distance between 2 particles in the sample (liquid). If all particles had the same size it was easy to perform the calculations, in reality particles size can differ strongly and there is a wide range of particle sizes, included in particle size distribution. Total volume of powder, located in sample is Vmaterial = Vi ; i = 1, 2,...,n ; where n is the number of particles. mmaterial = ρmaterial Vmaterial, where ρmaterial is the material (powder) density and mmaterial is the material weight. It can be very difficult to calculate number of particles that are present in "well dispersed" sample ( liquid ) and to know size of each particle exactly, then in order to simplify analysis the next estimation can be done : number of particles - n doesn't differ too much from number of Estimation of distance between particles by Krasnokutsky Dionys Page 1 of 6
particles, occupying the volume a³ and having the same size: D(0.5) (or X50 )(1)- median particle diameter-at 50% of the distribution by Volume, D(0.5) (or X50 ) value can be obtained by different particle size analyzers ( by laser diffraction ). Then in case of distribution by Volume, the radius of each particle r = D(0.5)/2. Vparticle = 4/3 π r ³, where Vparticle is the volume of particle. Or Vparticle = 4/3 π [D(0.5)/2]³ (1.1) ; Vparticle = π [D(0.5)]³ / 6 (1.2) - Particle thickness value should be known in order to calculate particle volume for plate (2) like particles in case if distribution by Surface model is used in particle size calculation. If particles are thin long fibers or elongated particles with length/width ratio of 7 or more, then volume of particles can't be estimated simply by equation 1.2, it generally requires combination of different techniques including optical microscopy to estimate length of particles. Volume can also generally be calculated with help of Voronoi polyhedra construction using computers. By dividing the total volume of material (powder) Vmaterial by Vparticle, number of particles n can be defined : n = Vmaterial / Vparticle. Now by returning to previous statement that each particle is situated in the cube, having cube edge length a, the total volume of sample Vsample can be defined in the next way: Vsample = n a³ (1.3) or Vsample = a³ Vmaterial /Vparticle= 6 Vmaterial a³ /(π [D(0.5)]³) (1.4) ; or Vsample = 6 mmaterial a³ /ρmaterial (π [D(0.5)]³) (1.5 ) ; a = ³ Vsample ρmaterial (π [D(0.5)]³) / ( 6 mmaterial ) (1.6) ; -The next equation can be written too: Vsample = msample ρsample(averaged) Density of the dispersant and material density can differ strongly, in addition to it density of sample ( powder in dispersant (liquid) ) will have different values in different points of volume,- according to number, volume, density and distribution of particles ( particles shouldn't be dissolved in the dispersant ). It is obvious that the closer the value of particle density (material density) to the density of the dispersant (liquid), the closer one to each other will be values of densities in different points of sample and will be closer to ρsample (averaged). Now let's examine equation 1.6. From particle size analysis of Caffeine (Anhydrous caffeine) powder (in well dispersed sample) next results were obtained: D(0.1) = 20 µm = 2 10 ³ cm ; D(0.5) = 61 µm = 6.1 10 ³ cm ; D(0.9) = 226 µm = 22.6 10 ³ cm. The data was obtained using Particle Size Analyzer - Mastersizer 2000, Hydro 2000S wet dispersion unit (Malvern Instruments) - size range: 0.02 µm 2000 µm. Caffeine density is about 1.2 g/cm³ (was found in internet), powder (about 40 mg) mmaterial = 0.04 g was dispersed in 20 ml of dispersant (silicone oil), Vsample 20 cm³, - 1 cm³ is equal to 1 ml, 1 cm³ =1 ml, 1 liter = 1000 cm³. Using equation 1.6 the averaged distance a between centers of particles can be found: a = ³ [ 20 1.2 3.14 (6.1 10 ³ )³ / 6 0.04] 41.5 10 ³ cm or 415 µm Another example is Theophylline Sodium Glycinate. It had density of about 1.23 g/cm³ Estimation of distance between particles by Krasnokutsky Dionys Page 2 of 6
( was calculated ), D(0.5) = 29 µm = 2.9 10 ³ cm the data was obtained using Particle Size Analyzer - Mastersizer 2000, Hydro 2000S wet dispersion unit ( Malvern Instruments ), mmaterial = 0.03 g was dispersed in 15 ml of dispersant ( silicone oil ), Vsample 15 cm³. a = ³ [15 1.23 3.14 (2.9 10 ³)³ / 6 0.03] 19.8 10 ³ cm or 198 µm Material (powder) density values can be found by displacement of a liquid to find volume, described in different printed sources and in different sources in Internet or/and using special laboratory equipment /devices. Results, obtained for Theophylline Sodium Glycinate and Caffeine(Anhydrous caffeine) samples give a good enough estimation of distance between particles - in average, see figures 2 and 3 -pictures have been made by WAT-202D digital color camera (Watec), connected to the microscope - x40 magnification, using Metric 8.01 software. Several values of distance between particles are represented in figures 2 and 3 - the measured distances appear in the status line (bar) under / near lines, showing distances between particles, reference scale is displayed up vertically. Figure 2. 30 mg of Theophylline Sodium Glycinate in 15 ml of silicone oil Estimation of distance between particles by Krasnokutsky Dionys Page 3 of 6
Figure 3. Caffeine : 40 mg of powder in 20 ml of silicone oil To estimate sample concentration at known (or ''required'') distance a between the particles and by using equations 1.1-1.5 the next equation will be obtained for concentration C: C (%) = [mmaterial /msample] 100 (%) = [ρmaterial π [D(0.5)]³/( ρsample (6 a³ ) )] 100 (%) (1.7), where mmaterial and msample are in the same measurement units. Or C (%) [0.52 ρmaterial D(0.5) ³/(ρsample a³) ] 100 (%) (1.8), where ρmaterial, and ρsample are in g/cc (g/cm³) or in g/ml measurement units and Vsample in ml. Equations (1.5) and (1.6) can't give very accurate results, but can help to estimate the concentration of sample. While dealing with coarse particles, for example in case if D(0.5) 300 µm, the adhesion forces ( attraction forces ) are much lower than in case of micronized powders, and by assuming that a should be about 3 D(0.5) ; a = 3 D(0.5) to overcome adhesion forces (attraction forces) and using equation 1.6, the next equation will be obtained: C (%) 0.058 ρmaterial / ρsample 100 (%) (1.8). In case of micronized powders (less than 30 µm), by assuming that a should be about 10 D(0.5) to overcome adhesion forces (attraction forces), which in case of micronized powders can be huge and by looking at equations 1.6,1.7 next equation will be obtained: C (%) 0.00052 ρmaterial /ρsample 100 (%) (1.9). Estimation of distance between particles by Krasnokutsky Dionys Page 4 of 6
In case if D(0.5) is between 20 µm and 100 µm and assuming that in this case a = 5 D(0.5) to overcome attraction forces between particles and using equations 1.6 and 1.7: C (%) 0.0042 ρmaterial / ρsample 100 (%) (1.10) In case if ρsample(averaged) 1 g/cm³ (g/ml) the equations 1.8, 1.9 and 1.10 will depend only on value of ρmaterial density of material (powder). Let's now find how a depends on densities of material (particle) and liquid (dispersant). In case if all particles have the same size ρaveraged (ρ sample ) is the averaged density of cube ( and sample too ), hanving lenght a, and ρsample = ( mparticle + mliquid ) /a³ (1.11), where mparticle and mliquid are masses of particle and liquid, that are situauted in the same cube, or using (1.2) ρsample = [ ρmaterial π [D(0.5)]³ / 6 + Vrest ρliquid ] / a³ (1.11), where Vrest is the volume of liquid in the cube, containing both particle and liquid. From (1.11) it can be found that Vrest = ( 6 a³ ρsample - ρmaterial π [D(0.5)]³ ) / 6 ρ liquid (1.12 ) or Vrest = k a³ - ρmaterial π [D(0.5)]³ ) / 6 ρ liquid (1.13), where k = ρsample /ρliquid ; ρsample = msample / Vsample Relation Vrest / Vcube = ( k a³ - ρmaterial π [D(0.5)]³ ) / 6 ρ liquid ) / a³ (1.13 ) or Vrest / Vcube = k-(ρmaterial π/6 ρ liquid) (D(0.5)/a)³ (1.14) From the other side it is known that that Vrest = Vcube Vparticle = a³- π [D(0.5)]³ /6 (1.15) or ( a³ - π [D(0.5)]³ /6 ) / a³ = k-(ρmaterial π/6 ρ liquid ) (D(0.5)/a)³ (1.16) from 1.16 the next equation will be obtained: k-1 = π [D(0.5)]³ ( ρmaterial - ρliquid ) / 6 ρliquid a³ (1.17) From (1.17) value of a the averaged distance between centers of 2 particles can be defined: a = ³ π [D(0.5)]³ (ρmaterial - ρ liquid) /(6 ( k-1) ρliquid ) (1.18) where ρmaterial,ρ liquid and ρsample can be in g/cc (or in g/ml) units, ρmaterial,ρ liquid and ρsample can be calculated using graduated cylinder and analytical weights ( ρmaterial, by adding the powder into the dispersant ( liquid ), and by performing simple calculation ρmaterial = mmaterial / Vmaterial, where mmaterial is the weight of powder, added to the liquid (dispersant ) and Vmaterial is the difference between final volume ( after addition of powder into the dispersant ) and initial volume ( before addition of powder ).If using dispersant (liquid) other than water, the density of the liquid should be accounted too. ρsample = msample / Vsample, where msample is the weight of sample (dispersant with powder) and Vsample is total volume (powder and dispersant ). Results, obtained by 1.6 and 1.18 can give different values, it reflects the fact that equation 1.18 includes density of both ρliquid and ρmaterial and doesn't take into account differences in particle size of different particles, in reality particles of pharmaceutical powders are generally doesn't have the same size and there is a wide range of particle sizes. Equation 1.18 would give accurate enough results in case if all particles had the same size. To get accurate results, additional factors should be taken into account: particle shape, particle pore size, etc., - accuracy of results also depends on symmetry of the distribution and calculation of averaged distance between particles can become much more complicated. For bimodal distribution another mathematical model should be developed. Estimation of distance between particles by Krasnokutsky Dionys Page 5 of 6
Summary Equations (1.6) and (1.7) can not give very accurate results ( although in some cases results, obtained with help of these equations can be accurate enough ), but with help of these equations distance between centers of particles and concentration of powder in liquid (dispersant) can be estimated this estimation can help a lot to prepare the representative sample for particle size analysis and it can play an important role in understanding of manufacturing process of medications and it can help to estimate Van der Waals forces of attraction between particles. Acknowledgements Special thanks to my lovely son Edwin, my wife, my sister, my parents and to all people, who inspired me for writing this article. I also would like to thank all of QC Laboratory employees and my friends for their help and support. References 1. Basic principles of particle size analysis. Malvern Application Note: MRK034 (www.malvern.co.uk) ; ISO 13320:1996(E), page 3. Particle size analysis Guidance on laser diffraction methods ; 2. "Optical Microscopy", General test <776>, USP 29 (The United States Pharmacopeial Convention, Rockville, MD, 2006), page 2717. Recommended articles for reading: - Witold Dzwinel and David A. Yuen, Mesoscopic Dispersion of Colloidal Agglomerate in a Complex Fluid Modelled by a Hybrid Fluid-Particle Model, Journal of Colloidal and Interface science 247, 463--480 (2002) - Tadeusz GLUBA, Andrzej OBRANIAK, Estera GAWOT-MLYNARCZYK, The Effect of Granulation Conditions on Bulk Density of a Product, Physicochemical Problems of Mineral Processing, 38 (2004) 177-186 Estimation of distance between particles by Krasnokutsky Dionys Page 6 of 6