REFRACTIVE INDEX SELECTION FOR POWDER MIXTURES

Transcription

1 REFRACTIVE INDEX SELECTION FOR POWDER MIXTURES Laser dffracton s one of the most wdely used methods for partcle sze analyss of mcron and submcron sze powders and dspersons. It s quck and easy and provdes very consstent, repeatable nformaton. Snce these nstruments use scattered lght ntensty to determne partcle sze, knowledge of the optcal propertes, specfcally the complex refractve ndex of the materal beng analyzed s requred. Whle the optcal propertes of the ndvdual components of a mxture are readly avalable, what happens when we want to determne the sze dstrbuton of a mxture? Ths note brefly descrbes an approach that allows determnaton of a sutable ndex of a typcal ceramc mxture usng knowledge of the materals and statstcal goodness-of-ft parameters provded by the analyzer software. Background Determnng an accurate partcle sze dstrbuton of a mxture of materals usng laser dffracton partcle sze analyss can be a challengng task. Each ndvdual materal has a unque partcle sze dstrbuton, as well as dfferent partcle shapes, denstes and optcal propertes. Although many ndustres face ths ssue, t s partcularly common n the ceramcs ndustry. A typcal ceramc applcaton contans multple oxdes and carbonates of varous metals. Partcle sze dstrbutons and morphologes can vary wdely n these materals, and there are typcally grndng processes nvolved that further alter the partcle shapes and szes. Laser dffracton partcle sze nstrumentaton s based on the measurement of the ntensty of radaton scattered from the partcles as a functon of scatterng angle. Scattered ntensty and scatterng angle are both related to partcle sze. Smaller partcle szes scatter at hgher angles and wth lower ntensty than larger partcles. Detectors at varous angles measure the radaton ntensty and a lght flux pattern s generated. Ths pattern s the raw scattered Intensty Measured Theoretcal Channel Number Fgure ntensty data as a functon of detector, as shown n Fgure. The mathematcal theory, developed by Gustave Me [] allows very precse calculaton of the flux pattern for a partcle sze dstrbuton usng a gven set of optcal propertes []. However, t does not allow a partcle sze dstrbuton to be drectly calculated from the flux pattern. Therefore, an teratve approach s used to produce a scatterng pattern that matches the measured data. The number of teratons s usually fxed n the software for the Horba LA seres nstruments at one of two levels a lower number for wder dstrbutons and a hgher number for narrow dstrbutons. More nformaton about the affects of teraton number can be found n Ref.

2 []. The LA-950 software also allows selecton of any teraton value up to 00. The theoretcal raw data for the reported partcle sze dstrbuton as determned by Me theory s also plotted n Fgure. As shown, there are small dfferences between measured and theoretcal patterns. By changng the optcal propertes used by the theory to determne the theoretcal curve, the match between measured and theoretcal can be altered. Ideally, a good match between measured and theoretcal curves ndcates that the optcal propertes chosen were approprate for the measured dstrbuton. Goodness of Ft A value of Ch Square s calculated for each data set and provdes a quanttatve measure of the goodnessof-ft (GoF) between the measured and theoretcal dstrbutons. Ch square (χ ) s calculated as follows: χ = Where: y = Actual scatterng measurement y(x ) = σ [ y y( x )] Theoretcal scatterng measurement (based on nput optcal propertes) σ = Standard devaton of scatterng data Ch Square s a commonly used statstc for comparng dstrbutons of dscrete data, such as partcle sze data []. The closer ths value s to zero, the more closely the two dstrbutons match. Therefore, we can use ths value to help determne f the correct optcal propertes have been chosen. A relatvely new and hghly useful feature of the LA-950 software allows users to quckly and easly create new refractve ndces and recalculate the dstrbuton. By fndng the condtons where Ch Square s at or near a mnmum, we can home n on a sutable refractve ndex. Real Component Determnaton As descrbed n Horba TN8 [3], the complex refractve ndex requred by Me theory s made up of two parts the real part, whch s a physcal property of a materal, and an magnary part, whch accounts for the absorpton of lght durng t s nteracton wth the partcle. The complex refractve ndex, n*, s gven by: n* = n κ Here, n s the real component and κ s the extncton coeffcent. The real component of many ndvdual materals s readly avalable n the lterature. However, when confronted wth a mxture, whch value should be chosen? When the components of a mxture are of smlar partcle sze, t s commonly recommended that the mxture rule or weghted average be used: n mx = z = nv Where n mx s the refractve ndex of the mxture, n s the refractve ndex of the th component, V s the volume fracton of that component, and z s the total number of components. The real components of many materals are now avalable for selecton and use n the LA-950 software. Note that f any components are small (below ~ µm) and represent a sgnfcant fracton of the mxture, the results wll be more senstve to the refractve ndex used for analyss.

3 Another approach to fndng the real component of a mxture s that of Gladstone-Dale [5]. Here the refractve ndex s lnearly related to densty, ρ, by a constant, k, as follows: n = + ρk For a mxture, the relaton would change to: z n = + ρ m k mx mx = The term m s the molar fracton of the component n the mxture and k s ts Gladstone-Dale constant. These constants have been tabulated for many oxdes and carbonates []. Note that the densty, ρ, s for the mxture. Whle ths method was orgnally developed for mnerals, whch are essentally mxtures of oxdes, t has applcablty n other areas [7]. Usng ether one of these approaches wll gve the user a reasonable startng pont for the real refractve ndex. Imagnary Component As mentoned earler, the magnary component of the refractve ndex s assocated wth absorpton phenomena. Snce tabulatons of magnary values are not usually avalable, one must use some rules of thumb to fnd a sutable startng pont. Some rules to keep n mnd are: - For opaque materals, a hgh magnary s needed. - The smaller the partcles the greater the mportance of the magnary. - Value of 0.0 only vald for transparent spheres (ISO330- ) - Colored materals such as pgments may requre hgher magnary values. - Partcles wth a rough surface or rregular shape may requre hgher magnary values. These rules only provde general gudelnes. To get a more quanttatve dea of the sutablty of the value, we can mnmze the GoF parameter, Ch Square, provded n the software by changng the magnary component used n the calculatons. An Example The approach descrbed above was used to fnd the refractve ndex for a ZrO - Y O 3 -Al O 3 mxture. Applyng the mxture rule, the real component s found to be.35. The Gladstone-Dale approach gves us a value of.05. To determne an approprate magnary component, we start wth the rules of thumb. These are whte powders, wth a sgnfcant fracton below µm. Snce they were mlled together, they wll have rregular shapes. Wth these facts n mnd, t can be concluded that a nonzero magnary s requred and t may turn out to be farly hgh. Usng the two real ndces determned above, the magnary component was vared and changes n Ch Square montored. Fgures (a) and (b) show the change n Ch Square and medan partcle sze as a functon of magnary component. Both real components gve smlar results, ndcatng that ether value could be used n practce. At an magnary value of 3.0, Ch Square levels out and does not change sgnfcantly after that. Although 3.0 s not the locaton of the mnmum of Ch Square, a look at some tabulated magnary values shows that, typcally, only metals have magnares as hgh as.0 or 5.0. Therefore, 3.0 should be used n ths case.

4 Ch Square 8 Ch LA-950 Medan SEM Medan n= Sze (µm) Ch Square Ch LA-950 Medan SEM Medan n= Sze (µm) Imagnary Component Imagnary Component 0.00 (a) Fgure (b) Verfcaton Obvously, mnmzng Ch Square s only approprate f t provdes accurate nformaton about the actual partcle sze dstrbuton. Verfcaton usng a scannng electron mcroscope (SEM) mage s the preferred method for partcles of ths sze. Image analyss, was performed usng ImageJ, a freeware applcaton developed by The Natonal Insttute of Health. An example of the mages at varous stages of processng s shown n Fg. 3. The software calculated the area of each dentfed partcle and both number and volume statstcs were determned. To mnmze the error of these calculatons, at least 00 partcles should be analyzed. For the purposes of ths example, only ~00 were ncluded n the analyss of a total of sx SEM mages. Table, below, summarzes and compares the mage analyss results to the data from the LA-950. The hstogram n Fg. compares the volume based dstrbutons from the LA- 950 usng both the mxture rule ndex and the Gladstone-Dale approach to the SEM data. As shown, there s relatvely good agreement between the LA-950 dstrbutons and the SEM data. Wth ths level of correlaton between the mage based technque and the laser scatterng method, t s far to conclude that the optcal model that was chosen based on mnmzaton of Ch Square s Fgure 3

5 0 8 SEM Frequency (%) Sze (µm) Fgure sutable and should gve accurate partcle sze dstrbuton nformaton. Concluson Table Medan Sze (µm) Volume Number SEM Although partcle sze dstrbutons of mxtures can be a challenge to measure accurately, use of the statstcal nformaton provded by the LA-950 analyzer can be of great help. Usng a systematc approach to home n on sutable optcal propertes can provde data that s consstent, repeatable, and accurate. References [] Me, Gustave, Betrage zur Optk trubermoden, specel Kolladaler Metallosungen, Annalen der Physk, Vol. 5, No. 3,pp , 908 [] Algorthm Iteratons, TN9, Horba Instruments Techncal Note [3] Gude to Selectng Refractve Index, TN8, Horba Instruments Techncal Note [] NIST/SEMATECH e-handbook of Statstcal Methods, November, 007. [5] e-dale.shtml [] Larsen, E.S., Berman, H., The Mcroscopc Determnaton of the Non-Opaque Mnerals, Second Edton, Unted States Department of the Interor, Geologcal Survey Bulletn 88, 93, US Government Prntng Offce, Washngton, DC. [7] Mandarno, J. A., The Gladstone-Dale Relatonshp: Part III: Some General Applcatons, Dept. Mneral. Geol., Royal Ontaro Mus., Toronto, ON, Can. Canadan Mneralogst (979), 7[], 7-. Copyrght 008, HORIBA Instruments, Inc.