Quantitative Phase Analysis of Ceramic Raw Materials Using a Nonnegative Least Squares Routine

Transcription

1 Citation & Copyright (to be inserted by the publisher) Quantitative Phase Analysis of Ceramic Raw Materials Using a Nonnegative Least Squares Routine C. Coelho 1,2, N. Roqueiro 3, D. Hotza 2,3 coelho@unisul.br 1 University of the South of Santa Catarina (UNISUL), Campus Pedra Branca, Palhoça, SC, Brazil hotza@materiais.ufsc.br 2 Graduation Program in Materials Science and Engineering (PGMAT), Federal University of Santa Catarina (UFSC), P.O. Box 476, Florianópolis, SC, Brazil nestor@enq.ufsc.br 3 Department of Chemical Engineering (EQA), Federal University of Santa Catarina (UFSC), P.O. Box 476, Florianópolis, SC, Brazil Keywords: Ceramic raw materials, phase composition, quantitative mineralogical analysis, rational analysis, nonnegative least squares routine Abstract. Combining quantitative chemical composition and qualitative mineralogical analysis, information enough can be obtained to solve the problem of quantitative determination of mineralogical phases. This concept is usually known as Rational Mineralogical Analysis. In this paper, a method using the nonnegative least squares routine for solving a system of linear equations is proposed, as a fast and reliable alternative to the quantitative analysis by X-ray diffraction. Introduction Reformulation techniques for ceramic bodies have been developed to permit varying the body composition to change physical properties, substitute raw materials, or to reduce batch cost. The linear programming can be used to reformulate an existing body rather than to develop a totally new body, reducing the amount of experimentation required by conventional methods [1,2]. The quality of ceramic bodies can be improved if the relationships between the characterizing parameters of raw materials and the physical properties of products are known. Two important but not exclusive parameters in order to describe either a constituent material or a ceramic body made up of a number of materials are chemical analysis (weight or mole percent) and mineral phase (weight percent). For using such techniques, which can be computadorized [3], quantitative input is necessary. This is no problem, for example to chemical analysis or particle mean size, but can be critical when no quantitative phase analysis is available. In the analysis of a multicomponent material by chemical and spectroscopic methods elemental composition can be obtained, but usually great difficulties are faced in distinguishing the chemical identity of the various components-phases present in the material and in determining the amounts of particular phases. X-Ray powder Diffraction (XRD) seems to be a good technique for the analysis of a multicomponent material. Each crystalline component of the material gives its characteristic diffraction pattern independently of the others, making it possible to identify the components of interest. The intensities of diffraction lines of each component are proportional to its amount, except for the absorption correction, so that an appropriate quantitative analysis can be performed.

2 2 Title of Publication (to be inserted by the publisher) The success of any quantitative X-Ray Diffractometry depends upon good control of particle size in the sample, elimination of preferred orientation of particles, and, in the case of the reference intensity method, the measurement of high-quality reference intensity ratios. This is a complicated and slow procedure, requiring attention to many details [4,]. UFSC Method The UFSC method proposes the simultaneous resolution of linear equations, as an alternative method to quantitative X-Ray Diffractometry. For a m constituent with n present mineralogical phases, a system with m equations can be built [6]. The calculation of each mineralogical phase can be then performed as follows: each mineralogical phase identified in the qualitative analysis should have a theoretical formula expressed in the form of oxides; the fraction in mass is calculated for each oxide in the theoretical formula from the atomic masses; a matrix is built where a line correspond to the amount of each oxide, except for the last one that represents the sum of each phase in the sample; a matrix column correspond to each phase, except for the last one that represents the amount of each oxide in the sample; there is usually a certain amount of elements that remain during the calculations, that is, the amount requested by the theoretical formulae is lower than that determined by the chemical analysis; this amount is added to a hypothetical phase denominated accessories to complete 100%. Almost any overdetermined (more lines than columns) or underdetermined (or more columns than lines) problem calls for a type of approximation method to a solution. Most frequently least squares is the approximation criterion chosen. The basic linear least squares problem can be stated as follows: given a real m n matrix A, a real m-vector B, find a real n-vector X minimizing AX B [7]. Lawson [8] describes a subroutine to computes a solution vector X, for the constrained least squares problem: solve AX B subject to X 0 (Nonnegative Least Squares NNLS). To solve the systems a algorithm is used. This algorithm assures that all the resulting values are positive. An example of rational mineralogical analysis follows. In a raw material, the present minerals are albite, kaolinite, and quartz. Non traceable phases are labeled as accessories. The theoretical formula of albite is Na 2 O.Al 2 O 3.6SiO 2, of kaolinite is Al 2 O 3. 2SiO 2. 2H 2, and of quartz is SiO 2. The chemical analysis is 34.22% Al 2 O 3, 0.2% CaO, 0.07% MgO, 49.81% SiO 2, 2.10% Na 2 O, 0.17% K 2 O, 1.12% TiO 2, 0.2% Fe 2 O 3, 11.74% ignition loss. The following system of linear equations can be set up, which correspond to Al 2 O 3, SiO 2, Na 2 O, and the sum row, respectively: 0.04 A K Q Z = 3.22 (1) 0.10 A K Q Z = (2) 0.06 A K Q Z = 2.1 (3) A K Q Z = (4) where A, K, Q, and Z correspond to albite, kaolinite, quartz and accessories.

3 Journal Title and Volume Number (to be inserted by the publisher) 3 Using the NNLS algorithm to solve the example, the answer is 17.7% albite, 77.9% kaolinite, 1.3% quartz and 3.1% accessories. Results and discussions For the validation of the method, a series of pure ceramic phases was used as hypothetical raw materials, Table 1. A total of aleatory mixtures were defined, Table 2, considered in this case as standard compositions (column Std.). Table 1. Pure ceramic phases used as hypothetical raw materials. Raw materials Theoretical formulas Albite Na 2 O.Al 2 O 3.6SiO 2 Calcite CaO.CO 2 Kaolinite Al 2 O 3.2SiO 2.2H 2 O Dolomite CaO.MgO.2CO 2 Illite 3K 2 O.11Al 2 O 3.26SiO 2.8H 2 O Orthoclase K 2 O.Al 2 O 3.6SiO 2 Quartz SiO 2 Talc MgO.4SiO 2.H 2 O Table 2. Standard and UFSC method calculated mineralogical composition s Phases * Std. Calc. Std. Calc. Std. Calc. Std. Calc. Std. Calc. Albite Calcite Kaolinite Dolomite Illite Orthoclase Quartz Talc Accessories * samples with some null phase (mathematically undetermined)

4 4 Title of Publication (to be inserted by the publisher) The chemical composition of each standard mixture is in Table 3. From the chemical composition and qualitative analysis of the each mixture, the UFSC method was applied (column Calc. in Table 2). Table 3. Chemical composition of the samples expressed in oxides. Oxides s SiO Al 2 O Na 2 O K 2 O MgO CaO Most of the calculated mixtures (1 to 4) are very close to the standard, however some of them (mixture ) presented at least a phase with null result, although this phase had been identified by the qualitative analysis. In these case, all the other calculated values were incorrect. To solve this problem, a new calculation was performed, at first not using the null phase and your chemical composition (Tables 4 and ). It was observed that the results, again, were very close to the standard, except by the accessories, which had included the previous null phase. Finally, a calculation only with the null phase and the accessories was made (Table 6), allowing the quantification of this phase. Table 4 Chemical composition of the samples without the null phases. Table Mineralogical composition of the samples without the null phases. Table 6 Mineralogical composition of the null phases. Oxide Phase Oxide SiO Std. Calc. Std. Calc. MgO 2.00 CaO 1.2 Dolomite.0.0 Quartz Talc Accessories* * including the null phases Calcite Accessories.0 4.9

5 Journal Title and Volume Number (to be inserted by the publisher) Conclusions The UFSC method presented results very close to the hypothetical standard compositions. In addition, the method guarantees: the resolution of overdetermined systems; the resolution of underdetermined systems; the elimination of solutions that satisfy mathematically but not physically (negative amount of a phase); The rational analysis method presents some sources of errors, common to other methods, such as: the improper identification of crystalline phases, with the subsequent alteration of the numeric matrix; the occurrence of amorphous or sparsely crystallized phases in the sample, which are not sufficiently evidenced in the XRD; the presence of mixed-layered clay minerals; the simplification of theoretical formulas of some complex phases. Eventually, the UFSC method may present specific errors due to: the presence of two (or more) phases with the same theoretical formula; the presence of large amount of impurities which have a chemical element present in some other identified phases. From this calculation approach a computational routine was developed, allowing anyone and not necessarily a specialized technician to quantify mineralogical phases of ceramic raw materials. The UFSC method can be then a fast and reliable alternative to the direct quantitative determination of phases. References [1] S.K. Roy: Am. Ceram. Soc. Bull. Vol. 9 No. 3 (1980), p [2] R.L. Lehman, J.G. Weinstein, G.W. Phelps and K.M. Adans: Am. Ceram. Soc. Bull. Vol. 63 No. 8 (1984), p [3] REFORMIX Software: in http//cati.comp.ufscar.br/projetos.html, acessed in 07/31/2001. [4] L.S. Zevin and G. Kimmel: Quantitative X-ray Diffractometry (Springer-Verlag, New York 199). [] D.M. Moore and R.C. Reynolds Jr: X-ray Diffraction and the Identification and Analysis of Clay Minerals 2nd ed. (Oxford University Press, New York 1997). [6] B. Fabri, C. Fiori and A. Ravaglioli: Materie Prime Ceramiche Vol. 3 (Faenza Editrice, Faenza 1986). [7] J.W. Demmel: Applied Numerical Linear Algebra (SIAM - Society for Industrial and Applied Mathematics, Philadelphia 1997). [8] C.L. Lawson and R.J. Hanson: Solving Least Squares Problems (SIAM - Society for Industrial and Applied Mathematics, Philadelphia 199).