18 Advanced Ceramics Processing 2.2 Particle morphology In only a relatively few types of powders the particle shape is sufficiently regular to provide a sin- gle definitive dime nsion for the"a spherical particle this is the diameter.while for a cube edto the sphere sieill be dependent upon the specific measurement technique employed The i”然ra xamples of types o e techniques ●r.·equivalent yol1 me diameter gas adsorntion ●x.·equivalent surface diameter microscom .x:equivalent settling diameter >sedimentation .x:equivalent mesh diameter →s1 eving More definitions of particle sizes are given in Table 2-1 Table 2-1 Some definitions of particle size Symbol Name Definition Is Surface diameter Diameter of a sphere having the same surface area as the particle v Volume diameter Diameter of a sphere having the same volume as the XsV Surface volume diameter Diameter of a sphere having the same surface area to volume ratio as the particle Stokes diameter(or Diameter of a sphere having the same sedimentation rate as the particle for laminar fow in a liquid XPA Projected area diameter Diameter of a cirele having the same area as the projected area of the particle Perimeter diameter Diameter of a circle having the same Width of the minimum square aperture through which the particle will pass Sieve diameter Width of the minimum square aperture through which the particle will pass Feret's diameter Mean value of the distance be etween pairs of parallel tangents to the projected outline of the particle Martin's diameter Mean chord length of the projected outline of the particle From:Rahaman MN(2003)Ceramic proo In the discussion given above each particle is characterised by only one scalar number,which does not contain any information about the shape of the particle.A measure of the shape of a particle can be obtained
18 Advanced Ceramics Processing 2.2 Particle morphology In only a relatively few types of powders the particle shape is sufficiently regular to provide a single definitive dimension for the “size”. For a spherical particle this is the diameter, while for a cube the length of the edge is a unique parameter for the particle size. In general, particle size data are related to the diameter of a sphere, which is equivalent to the particle regarded. This apparent sphere size will be dependent upon the specific measurement technique employed. The ISO standard (ISO, 1995) recommends the symbol x to be used for the particle size and a subscript denoting the equivalent diameter used. Examples of types of particle sizes and the techniques used for analysis are: • xv: equivalent volume diameter → gas adsorption • xs: equivalent surface diameter → microscopy • xw: equivalent settling diameter → sedimentation • xn: equivalent mesh diameter → sieving. More definitions of particle sizes are given in Table 2-1 Table 2-1 Some definitions of particle size* * From: Rahaman MN (2003) Ceramic processing and sintering, 2nd edn. CRC Press, New York In the discussion given above each particle is characterised by only one scalar number, which does not contain any information about the shape of the particle. A measure of the shape of a particle can be obtained from the sphericity index ψ, which is defined as the ratio of the surface area of a sphere with the same volume as the test particle to the actual surface area of the particle. The sphericity of the powder also has an influence on the packing density
2 sphericity fa 2.1) c e classifications defined by the inter natonal Organization for Standardization in ISO Standard 2 are represented in the table below These general powder morphologies are shown in Figure Classification Shape acicular needle shape angular roughly polyhedral shape with sharp edges dendritic branched crystalline shape fibrous (irregularly thread-like shape flaky plate-like shape irregular lacking any symmetry nodular round irregular shape spherical nominally spherical shape 2.3 Particle size distribution When the size of an individual particle is defined and determined then the next problem is how to express the distribution of particle sizes within a powder sample.A set of particle sizes can be plot- ted as a frequency distribution.In order to chara 2.61 Mode is the value where the frequency distri- ution reaches its maximum value Median divides the 。Meai is the main po int”of the frea quency distribution mode 20 30 -mean ( particle size not sufficient for a co mplete definition For exam of the mode,median distribution is not known in this way
2 Characteristics of powders and compacts 19 A way to derive the sphericity of a particle is from the ratio of any two equivalent diameters. The most common example is the Wadell sphericity factor: 2 v w s x x ψ = (2.1) The shape of the individual particles within a powder significantly influences its bulk properties, for example, packing (apparent density), flowability and compressibility. The qualitative particleshape classifications, defined by the International Organization for Standardization in ISO Standard 3252 are represented in the table below These general powder morphologies are shown in Figure 2-5 2.3 Particle size distribution When the size of an individual particle is defined and determined, then the next problem is how to express the distribution of particle sizes within a powder sample. A set of particle sizes can be plotted as a frequency distribution. In order to characterise such a distribution the following three statistical quantities are very often used (see Figure 2-6): • Mode is the value where the frequency distribution reaches its maximum value, • Median divides the frequency distribution curve in two parts with equal surface area, • Mean is the “main point” of the frequency distribution. These three values give some information on the particle size distribution; however, these data are not sufficient for a complete definition. For example the “broadness” (standard deviation) of the distribution is not known in this way. Figure 2-5: Particle morphologies of powders: a) acicular, (b) angular, (c) dendritic, (d) fibrous, (e) flaky, (f) granular, (g) irregular, (h) nodular, (i) spherical. Classification Shape acicular needle shape angular roughly polyhedral shape with sharp edges dendritic branched crystalline shape fibrous (ir)regularly thread-like shape flaky plate-like shape granular irregular but approximately equidimensional irregular lacking any symmetry nodular round irregular shape spherical nominally spherical shape Qualitative particle-shape classifications according to ISO Standard 3252. Figure 2-6: Graphic representation of the mode, median and mean values of a particle size distribution
20 Advanced Ceramics Processing The particle size distribution is usually expressed by a distribution functionA well-known distri- (x)= 2)expl- (Inx -In (2.2) 2G2 In this equation o is the standard deviation of the distribution and can be calculated form a cumu- lative distribution as will be shown later in this section.is the geometrical mean particle size nlog log=画 (23) In equation (2.3)n expresses the number of parti- cles in the interval with mean size x The sum where is the number of intervals nich 235680 20304050 Aoegmlpatcs size distrib tion is symmet tical and lie exactly at 50%of the distribution.In Figure2-7:Alog Figure 2-7and Figure 2-8,respectively,a fraction- al and a cumulativ log-normal particl curve is most commonly uscd beca use these curves can he more easily internolated and normalised The cumulative curves may be chosen either undersize or oversize.For a log-normal distribution a straight line in the distribution curve is obtained if the logarithm of the particle siz is plotted aeanth the oversi n particle size is at 50%.The standard deviation ()is defined in such a way that the size of 6%of the particles in the distribution is within the interval:[The geometrical standard deviation(can be calculated from log=logr-logr=logx”-logx6(2.4) 100: 且40 -8)eto I hus m and star dard 710 ems an 4 le Allen 6 a are given 125103050709099 cumulative percentage undersize
20 Advanced Ceramics Processing The particle size distribution is usually expressed by a distribution function φ. A well-known distribution function is the log-normal distribution function (see Figure 2-7) and the frequency by which a particle with size xi is present, is then defined by: ( ) _ 2 2 1 (ln ln ) exp[ ] (2 ) 2 i g i x x φ x σ π σ − = − (2.2) In this equation σ is the standard deviation of the distribution and can be calculated form a cumulative distribution as will be shown later in this section. _ xg is the geometrical mean particle size: _ 1 log log i I i i i g n x x N = = = ∑ (2.3) In equation (2.3) ni expresses the number of particles in the interval I with mean size xi. The summation is over I, where I is the number of intervals in which the particle size distribution is divided. The total number of particles analysed is given by N. A log-normal particle size distribution is symmetric and therefore mode, median and mean are identical and lie exactly at 50% of the distribution. In Figure 2-7 and Figure 2-8, respectively, a fractional and a cumulative log-normal particle size distribution are given [6]. A cumulative distribution curve is most commonly used because these curves can be more easily interpolated and normalised. The cumulative curves may be chosen either undersize or oversize. For a log-normal distribution a straight line in the distribution curve is obtained if the logarithm of the particle size is plotted against the cumulative percentage oversize or undersize on a probability scale. From the cumulative distribution the mean particle size and the standard deviation can easily be determined. The mean particle size is at 50%. The standard deviation (σ) is defined in such a way that the size of 68% of the particles in the distribution is within the interval: g [ -σ, x +σ] xg . The geometrical standard deviation (σg) can be calculated from: 84 50 50 16 log log log log log g σ =−=− xx xx (2.4) where x84 means the size at 84% (40 µm in Figure 2-8) etc. Thus mean and standard deviation are sufficient parameters to complete the identification of the whole log-normal particle size distribution. A detailed description of particle size measurements and other mathematical functions used to describe particle size data are given in the book of Allen [6]. Figure 2-7: A log-normal distribution plotted as a relative percentage distribution, using a logarithmic scale for the particle size Figure 2-8: A log-normal distribution plotted on logprobability paper
2 ders and compacts 2 2.4 Chemical and physical characterisation methods .Particle size.Several methods for determining the particle size distribution are available,e.g o Sieving Dynamic light scattering Crystallite size determination by X-ray line broadening .Thermal analysis(DTA.DSC.TGA) .Chemical analysis(X-ray fluorescence.) 2.4.1 Dynamic light scattering Dynamic light scattering DLS)measures the Brownian motion of a particle and relates this to its s the rar om movement of particles due to the bombardment by solvent lecule wer the d(H)=3xnD kT (2.5) With d(H)the hydrodynamic diameter.D translational diffusion cocfficient.k Boltzmann's con- stant,T absolute temperature and n the viscosity.The diameter obtained with this technique is the ient as the particle.This means size also the Su o tio ac and typ the Brownian motion. cosity of the solvent and sity ha tion in the sus n will cause pon-random movement that will ruin a interpretation.For the surface morphology any change of the surface of a particle that will influ ence the dffu spe will also in ce th apparent partic th size.An absorbed polymer laye nd the surfac lons in the solution and the rotal ionic effect the elec double layer around the particle(see for detail chapter 3)and thereby,the diffusion speed.A low ded double layer of ions around the particle,reducing the
2 Characteristics of powders and compacts 21 2.4 Chemical and physical characterisation methods • Particle size. Several methods for determining the particle size distribution are available, e.g.: ◊ Sieving ◊ Dynamic light scattering ◊ Crystallite size determination by X-ray line broadening ◊ (Electron) microscopy • Specific surface area: This is defined as the accessible area of solid surface per unit mass of material (m2 /g) • Thermal analysis (DTA, DSC, TGA) • Chemical analysis (X-ray fluorescence,.) 2.4.1 Dynamic light scattering Dynamic light scattering (DLS) measures the Brownian motion of a particle and relates this to its size. Brownian motion is the random movement of particles due to the bombardment by solvent molecules. The larger the particle, the slower the Brownian motion will be. The velocity of the Brownian motion is defined by the translational diffusion coefficient (D). From this coefficient the hydrodynamic diameter of particles is calculated using the Stokes Einstein equation: ( ) 3 kT d H πηD = (2.5) With d(H) the hydrodynamic diameter, D translational diffusion coefficient, k Boltzmann’s constant, T absolute temperature and η the viscosity. The diameter obtained with this technique is the diameter of a sphere that has the same translational diffusion coefficient as the particle. This means that DLS measurements do not give any indication on the actual shape of a particle. Besides the size also the temperature, surface morphology, concentration and type of ions in solution influences the Brownian motion. A constant temperature is important because it determines viscosity of the solvent and viscosity has a direct relationship with the Brownian motion. The temperature also needs to be constant, because otherwise convection in the suspension will cause non-random movement that will ruin a correct interpretation. For the surface morphology any change of the surface of a particle that will influence the diffusion speed will also influence the apparent particle size. An absorbed polymer layer stacking out of the particle surface will reduce the diffusion speed more than a polymer that is nicely absorbed around the surface. Ions in the solution and the total ionic strength effect the electric double layer around the particle (see for detail chapter 3) and thereby, the diffusion speed. A low conducting medium will produce an extended double layer of ions around the particle, reducing the diffusion speed and resulting in a lager hydrodynamic diameter
22 Advanced Ceramics Processing Analysis method c radiation)in with an intensity dependant on particle size.Fol thc in ered light is tion between particle size and intensity there is a is equal in all direction.This theory is valid for particles that are much compared to the wave length of the laser s thar particles the Rayleigh a imation is n anymore,the intensity of the scattered light will from ent on the particle size but also on the refractive For analysis a sample is illuminated with a lase beam with well- intensity of th scattered light fluctuates usinga Figure:Particle si analysis by light scatterin suitable optical a can be determi amplc wit are where the phase additions of the scattered light are mutually destructive The bright blobs of light in the pattern are spots where the light scattered from particles with the same phase interfere. erved where the the moving particles is constantly evolving and forming new pattems.The rate at which these in- tensity fluctuationsoc ur will depend on the size of the particl s.Small particles cause the intensity to fluctuate more rap large ones.It is poss le to mea so.A better way is be using a device called a digital auto correlator.In the correlator signals at dif. ferent time are compared with each other.The time interval is normally very small,nanoseconds or intensity of a ce signal at time t will be com with the i slowly and the correlation will r for time.If the rapidly)then correlation will reduce faster.The time at which the correlation starts to significantly decay is an ind the powd er.The steeper the line,the more mon odisperse the sample is.Size can be obtained from the correlation function by using various algorithms Further details can a/o be found in the paper:"Particle size determination:An undergraduate lab in Mie scattering"by Weiner et al
22 Advanced Ceramics Processing Analysis method When light (electromagnetic radiation) interacts with a particle, this particle will scatter the light with an intensity dependant on particle size. Following the Rayleigh approximation of scattering the intensity of the scattered light is I = d6 with d the particle diameter. Because of the strong relation between particle size and intensity there is a danger that a little pollution of large particle will overrule the scatter light of the smaller ones. In Rayleigh scattering the intensity of scattered light is equal in all direction. This theory is valid for particles that are much compared to the wave length of the laser used and at least be less than one tenth the wavelength λ (d < λ/10). For larger particles the Rayleigh approximation is not valid anymore, the intensity of the scattered light will vary with respect to the angle it is scattered from the particle. It must also be mentioned that the degree of Rayleigh scattering is not only dependent on the particle size but also on the refractive index of the material studied. When the particle size is in the order of the wave length of the laser the Mie theory is used. For analysis a sample is illuminated with a laser beam with well-defined wave length. When the laser interacts with the particles, the light will be scattered and by measuring the rate at which the intensity of the scattered light fluctuates using a suitable optical arrangement the size of the particles can be determined. When light is scattered by a sample with stationary particles a classical speckle pattern would be observed. This pattern will be stationary both in speckle size and position because the whole system is stationary. Dark spaces are where the phase additions of the scattered light are mutually destructive. The bright blobs of light in the pattern are spots where the light scattered from particles with the same phase interfere. For a system of particles undergoing Brownian motion, a speckle pattern is observed where the position of each speckle is seen to be in constant motion. This is because the phase addition from the moving particles is constantly evolving and forming new patterns. The rate at which these intensity fluctuations occur will depend on the size of the particles. Small particles cause the intensity to fluctuate more rapidly than the large ones. It is possible to directly measure the spectrum of frequencies caused by intensity fluctuations arising from Brownian motion, but it is inefficient to do so. A better way is be using a device called a digital auto correlator. In the correlator signals at different time are compared with each other. The time interval is normally very small, nanoseconds or micro seconds. The intensity of a certain signal at time t will be compared with the intensity at t = t+δt The correlation in intensity will decrease in time. For large particles the signal will change slowly and the correlation will persist for a long time. If the particles are small (so moving more rapidly) then correlation will reduce faster. The time at which the correlation starts to significantly decay is an indication of the mean size of the powder. The steeper the line, the more monodisperse the sample is. Size can be obtained from the correlation function by using various algorithms. Further details can a/o be found in the paper: “Particle size determination: An undergraduate lab in Mie scattering” by Weiner et al.11 _ Figure 2-9: Particle size analysis by light scattering