O cálculo da densidade de elétrons do plasma foi desenvolvido utilizando a equação de Saha, dada pela Eq. (2.24). Foram escolhidas as combinações entre as linhas de Fe I: 293, 684 nm, 393, 288 nm, 396, 793 nm e 422, 544 nm e Fe II: 243, 542 nm, 252, 862 nm, 279, 102 nm e 330, 192 nm. Os parâmetros degenerescência e coeficiente de Einstein em-pregados foram obtidos da data-base do NIST, bem como a energia de ionização do Fe I, Eion = 7, 902 eV. A temperatura usada foi a obtida pelo gráfico de Boltztmann para as linhas de Fe II, T = 13000 K. Obtivemos no total, 16 valores para a densidade eletrônica (metade da combinação 2 a 2 dos 8 picos), aos quais foi aplicado o método estatístico IQR, já descrito na seção 4.2.
Após levarmos em consideração o critério dos limites de Tukey, foi possível obter uma densidade eletrônica média ne = 1, 684 × 1022 cm−3 com desvio-padrão σ = 2 × 1021cm−3. Uma explicação para o alto valor encontrado da densidade de elétrons quando comparado à literatura (geralmente da ordem de 1017- 1018) está no alto valor da temperatura encontrado (22). Isso é possível em virtude do curto delay time usado para o início da aquisição dos sinais. Na literatura encontramos temperaturas na faixa de (1 − 2)104K para o delay time da mesma ordem (22, 24). Em períodos longos de espera para a aquisição ( 2−4µs de delay), o plasma já se resfriou, resultando em temperaturas da ordem de (0, 6 − 1)104K. Este tempo também é suficiente para que processos de recombinação ocorram, levando à diminuição da densidade eletrônica.
Como já foi mencionado na seção (2.3.1), o critério de Mc Whirter estabelece a con-dição mínima em que o plasma gerado por LIBS é considerado em LTE, de modo que a
temperatura calculada possa, com boa aproximação, ser utilizada para descrevê-lo. Neste caso, como a densidade eletrônica encontrada foi maior que o limite estabelecido, estamos no regime de LTE e podemos caracterizar o plasma por meio da temperatura determinada na seção anterior.
Capítulo 5
Conclusões
Este trabalho permitiu validar a técnica LIBS como método de análise e possibilitou aumentar os campos de emprego, estendo-os à investigação de fármacos. Desta maneira, po-demos constatar sua versatilidade para estudo de diferentes matrizes. Notamos que, apesar da ampla aplicação, ela encontra-se sujeita a problemas como efeitos das matrizes escolhi-das, geometria da amostra e instrumentação, os quais limitam ainda sua reprodutibilidade e acarretam imprecisões na caracterização do plasma gerado e em análises quantitativas dos elementos identificados.
Como primeiro estudo, comprimidos do analgésico Dorflex foram submetidos à es-pectroscopia LIBS com o objetivo de estudarmos a uniformidade de distribuição dos com-ponentes em seu volume e identificarmos seus comcom-ponentes: o IFA e os excipientes (lubri-ficante e desintegrante). Ela foi verificada através da presença das linhas de emissão corres-pondentes aos elementos representativos do IFA, do lubrificante e do desintegrante, sendo C I, Na I e Mg I, respectivamente. Um estudo estatístico entre os quatro quadrantes indicou alta correlação entre seus espectros dois a dois, com coeficientes de correlação próximos de 1. Isto nos permitiu validar a homogeneidade do comprimido e a técnica LIBS como mecanismo de análise qualitativa.
Nosso segundo estudo consistiu na caracterização física do plasma gerado. Deter-minamos a temperatura a partir dos gráficos de Boltzmann, com as linhas de Fe II e Na I, resultando em T = 13000 ± 1000) K e T = (13000 ± 3000) K, respectivamente. A densidade
eletrônica foi calculada a partir da equação de Saha, sendo a média ne = 1, 684 × 1022cm−3 com desvio-padrão σ = 2 × 1021 cm−3. Em virtude desta alta densidade de elétrons, foi possível atestar o LTE do plasma comparando-a ao critério de Mc Whirter.
Como perspectivas para trabalhos futuros, esperamos estudar e testar diferentes méto-dos e desenvolver algoritmos capazes de corrigir imprecisões que levam a grandes incertezas nos cálculos de intensidade, interferindo consequentemente no estudo das características do plasma e na análise quantitativa das amostras. No estudo da uniformidade, queremos em-pregar métodos quimiométricos, como a Análise do Componente Principal (PCA), afim de comprovar quantitativamente a uniformidade dos comprimidos.
Referências Bibliográficas
1 CREMERS, D. A.; RADZIEMSKI, L. J. History and fundamentals of LIBS. In: LIBS Fundamentals and Applications. [S.l.: s.n.], 2006. cap. 1, p. 1.
2 HADDAD, J. E.; CANIONI, L.; BOUSQUET, B. Spectrochimica Acta Part B Good practices in LIBS analysis : Review and advices. Spectrochimica Acta Part B: Atomic Spectroscopy, Elsevier B.V., v. 101, p. 171–182, 2014. ISSN 0584-8547. Disponível em: <http://dx.doi.org/10.1016/j.sab.2014.08.039>.
3 ST-ONGE, L. et al. Rapid analysis of liquid formulations containing sodium chloride using laser-induced breakdown spectroscopy. Journal of Pharmaceutical and Biomedical Analysis, v. 36, n. 2, p. 277–284, 2004. ISSN 07317085.
4 MANSOUR, S. A. Fast Quantitative Analysis of Pharmaceutical Products Using LIBS-Technique. IUG Journal of Natural and Engineering Studies, v. 23, n. 1, p. 15–25, 2015.
5 FORTES, F. J. et al. Laser-Induced Breakdown Spectroscopy. 2013.
6 ANZANO, J. et al. Rapid characterization of analgesic pills by laser-induced breakdown spectroscopy (LIBS). Medicinal Chemistry Research, v. 18, n. 8, p. 656–664, 2009. ISSN 10542523.
7 BÉCHARD, S.; MOUGET, Y. LIBS for the analysis of pharmaceutical materials. In: LIBS Fundamentals and Applications. [S.l.: s.n.], 2006. cap. 8, p. 314.
8 Food and Drug Administration. Guidance for Industry ANDAs: Impurities in Drug Products. n. November, 2010.
9 AGENCY, E. M. Guideline on setting specifications for related impurities in antibiotics. v. 44, n. June, 2012.
10 HARMONISATION, I. C. on. Impurities In New Drug Products Q3B (R2). n. June, 2006.
11 BRASIL. Resolução RDC no17, de 16 de abril de 2010. Dispõe sobre as Boas Práticas de Fabricação de Medicamentos. [S.l.]: Órgão emissor- ANVISA (Agência Nacional de Vigilância Sanitária), 2010.
12 ANVISA (Agência Nacional de Vigilância Sanitária). Farmacopeia Brasileira. v. 1, n. 5a, 2010.
13 United Nations Interregional Crime and Justice Research Institute. Couterfeiting: a global spread, a global threat. Trends in Organized Crime, v. 12, n. 1, p. 59–77, 2009. 14 LACHIVER, E. D. et al. Agglomeration tendency in dry pharmaceutical granular systems. European Journal of Pharmaceutics and Biopharmaceutics, v. 64, n. 2, p. 193–199, 2006. ISSN 09396411.
15 OTTINO, J. M.; KHAKHAR, D. V. Fundamental research in heaping, mixing, and segregation of granular materials: Challenges and perspectives. 2001. 117–122 p. 16 CARTILIER, L. H.; MOËS, A. J. Effect of Drug Agglomerates Upon the Kinetics of Mixing of Low Dosage Cohesive Powder Mixtures. Drug Development and Industrial Pharmacy, v. 15, n. 12, p. 1911–1931, 1989. Disponível em: <http: //dx.doi.org/10.3109/03639048909052510>.
17 DUBEY, A. et al. Analysis of pharmaceutical tablet coating uniformity by laser-induced breakdown spectroscopy (LIBS). Journal of Pharmaceutical Innovation, v. 6, n. 2, p. 77–87, 2011. ISSN 18725120.
18 KUMAR, A. et al. Talanta Laser-induced breakdown spectroscopy-based investigation and classification of pharmaceutical tablets using multivariate chemometric analysis. Talanta, Elsevier B.V., v. 87, p. 53–59, 2011. ISSN 0039-9140. Disponível em: <http://dx.doi.org/10.1016/j.talanta.2011.09.040>.
19 ARANTES DE CARVALHO, G. G. et al. Evaluation of laser induced breakdown spectrometry for the determination of macro and micronutrients in pharmaceutical tablets. Journal of Analytical Atomic Spectrometry, v. 25, n. 6, p. 803, 2010. ISSN 0267-9477. 20 CONTRERAS, U. et al. Quantitative analysis of metformin in antidiabetic tablets by laser-induced breakdown spectroscopy. Proceedings of SPIE - The International Society for Optical Engineering, v. 8011, p. 80112Q–80112Q–10, 2011. ISSN 0277786X. Disponível em: <http://www.scopus.com/inward/record.url?eid=2-s2.0-84858391993{\&}partnerID= 40{\&}md5=4767d230c8968904f3a3e20120>.
21 MOWERY, M. D. et al. Rapid at-line analysis of coating thickness and uniformity on tablets using laser induced breakdown spectroscopy. v. 28, p. 935–943, 2002.
22 THAKUR, S. N. Atomic Emission Spectroscopy. Laser-Induced Breakdown Spectroscopy, p. 23–48, 2007.
23 FOOT, C. J. Atomic Physics. 2. ed. [S.l.]: Oxford University Press, 2005. 346 p. 24 MIZIOLEK, A. W.; PALLESCHI, V.; SCHECHTER, I. LIBS Fundamentials and Applications. [S.l.: s.n.], 2006. 640 p. ISBN 9780521852746.
25 FRANCO, M. A. d. M. Efeitos de matriz nas propriedades do plasma LIBS para quantificação de carbono. 126 p. Tese (Dissertação (Mestrado em Ciências)) — Universidade de São Paulo, 2017.
26 THAKUR, S. N.; SINGH, J. P. Fundamentals of Laser Induced Breakdown Spectroscopy. Laser-Induced Breakdown Spectroscopy2, p. 3–20, 2007.
Apêndice A
No anexo a partir da página seguinte, encontra-se um trabalho enviado para publi-cação. Nele, propomos uma montagem experimental ("homemade" simples e barata) que permita realizarmos medidas LIBS. Para isso, utilizamos a óptica ondulatória no regime pa-raxial, bem como a descrição gaussiana do feixe laser.
CALCULATION OF AN OPTICAL SETUP FOR A LIBS SYSTEM
Thalena C. Zanetti and Jader S. Cabral
Instituto de Física, Universidade Feral de Uberlândia, P.O. Box 593, 38400-902, Uberlândia, MG, Brazil.
ABSTRACT
In this work we will represent the necessary calculation for the optical setup on a LIBS spectroscopic system, as well as the parameters for the beam characterization. On the realized simulation, the beam was deal with the wave optics theory and its propagation with the paraxial approximation. It was described as a beam of Gaussian intensity profile and its propagation was acquired through the ABCD’s law. Our results have showed the necessary lens to hold a LIBS sign for different kinds of materials, depending on their bond cleavage . Also, we have showed a table with typical values of focus lens, used in LIBS setups, and its respective irradiance.
1. Introduction
The LIBS (Laser Induced Breakdown Spectroscopy) system is widely used in several research fields for atomic analysis of many materials, including alloys, soils, fertilizers, grains, architectonic objects, semiconductors and medicines, among others. The works (1-3) illustrate this great applicability. A short period of time have passed since the first experiments, that came thereupon the laser creation in the 1960s, and the technic progress is still far away to stagnate. The fact of LIBS system owns a great potential as an analytical tool originated a kind interest by the scientific community in the last decades. As a result, the number of international publications related with the technic has grown. In 2011, a LIBS system was coupled to the Curiosity robot and then it was sent to Mars, showing its maturity and utility (4,5).
The LIBS technic is a type of atomic emission spectrometry that uses the plasma generation by high-power short pulses of a laser. The laser’s pulse duration is typically of nanoseconds (ns), however there are experiments with lasers of picoseconds (ps) and femtoseconds (fs) (2,6-8). The high-power laser focusing in the sample (~GW/cm²) causes ablation of a small portion of matter, in order of Nano grams, which generates a plasma plumb with a temperature over 50.000K. In this temperature, the material dissociates into ions and excite atoms, emitting a continuous spectrum of radiation, which is not useful to characterize materials. Due to the high velocity of plasma elements, occurs a supersonic and adiabatic expansion, cooling the plasma between 5.000K and 15.000K in approximately 1 to 2 μs, when it is possible to measure the atomic/ionic emission lines and identify the elements present in the sample. The requisite time between the ablation pulse and the radiation acquirement can vary in the range of 1 to 10 μs.
Owing the small amount of material which is wasting in the analysis (ng and fg), LIBS is considered a non-destructive technic for several applications, since with an average power density less than 1 W, the sample don’t heat out of ablation region. Because of its nature, this technic enables in loco analysis, which deals out complex sample preparation processes, thus eliminating production of chemistry residues (9) or dry combustion products in DNA analysis (10). Other possibility is to use it in gaseous, liquid or solid samples, in conductor and non-conductor materials and also on that hardly dissolution (11).
In this work we present the calculation achieved for an optical LIBS system setup supposing the Gaussian profile of the laser beam. To do so, we applied the Maxwell’s Equations resolution in cylindrical coordinates and the Gaussian beam propagation method in optical systems. Thereby, we have found the best optical setup which will allow us the suitable power density (Gw/cm²) to reach the breakdown of different sample matrixes.
2. Gaussian beam and its propagation
From Maxwell’s four equations, those describe wave light behavior, we found the wave equation for electric field 𝐸⃗ , given by eq. (1):
∇2𝐸⃗ + 𝑘2(𝑟 )𝐸⃗ = 0 (1) where 𝑘 = 𝜇𝜖𝜔2 is the wave vector, 𝜇 is the magnetic permeability of the medium, 𝜖 is the electric permittivity and 𝜔 is the wave frequency. For simply, we’ll treat the case in which the medium is homogeneous and non-magnetic, i. e., the vector 𝑘⃗ is a constant.
Writing the laplacian operator in cylindrical coordinates and separating it into transverse portion to beam propagation (𝑟, 𝜃) and parallel to beam propagation (z), we achieved the wave equation resolution e we found the electric field expression (12,13).
As the wave intensity is proportional to the square of electric field magnitude, thus the light intensity 𝐼(𝑟, 𝑧), resultant from 𝐸⃗ wave equation, is given by eq. (2) and exhibits the Gaussian form, with maximum intensity 𝐼0(𝑤0
𝑤 (𝑧))2 at radial coordinate 𝑟 = 0.
𝐼(𝑟, 𝑧) = 𝐼0(𝑤0
𝑤 (𝑧))2𝑒−2𝑤 2(𝑧)𝑟2 (2)
For 𝑟 = 𝑤 (𝑧) , the beam intensity decreases 1/𝑒2 from its maximum value and this distance is called beam ray. Then the parameter 𝑤0 is called waist and its relation with 𝑤 (𝑧) is given by eq. (3). Note that, in the origin of beam propagation 𝑧 = 0, the ray reaches its minimum value 𝑤0.
𝑤2(𝑧) = 𝑤02{1 + (𝑧𝑧
0)2} (3)
The parameter 𝑧0 =𝑘𝑤𝑜2
2 =𝜋𝑛𝑤02
𝜆 is called Rayleigh’s length and represents the beam focusing region, where 𝑤0< 𝑤 (𝑧) < √2𝑤0. For small 𝑧0, there is a short focalization and for great 𝑧0, there is a long focalization. Figure 1 illustrates the Gaussian profile of a laser beam.
Figure 1: Perfil Gaussiano de intensidade um feixe de luz laser.
To find the beam ray 𝑤 (𝑧), the waist 𝑤0 and the curvature ray of the wavefront 𝑅(𝑧) as the beam propagates and interacts with the optical elements (mirrors and lens), we need to apply the ABCD’s law, widely used in geometric optics and that, with the correct construction of parameters which describe the beam, can be utilized in the Gaussian wave (13). To do so, it is defined the parameter 𝑞(𝑧), given by eq. (4), which is a complex number. Note that its real part gives information about the beam wavefront 𝑅(𝑧) and its imaginary part depends on the beam ray 𝑤 (𝑧).
1
𝑞(𝑧)=𝑅(𝑧)1 −𝜋𝑛𝑤𝑖𝜆2(𝑧) (4)
where 𝑛 is the refraction index of the propagation medium and 𝜆 is the wavelength in the vacuum.
In accord with the matrix optics, each optical element can be represented by a unitary matrix 2𝑥2 and its elements are called A, B, C and D in the form (𝐴 𝐵
𝐶 𝐷). When the beam strikes an optical element, the new parameter 𝑞2(𝑧) is given by eq. (5) and thus it is possible to find a new beam ray 𝑤 (𝑧) and the new wavefront 𝑅(𝑧).
𝑞2(𝑧) =𝐴𝑞 (𝑧)+𝐵𝐶𝑞(𝑧)+𝐷 (5)
This way, in an optical system composed of various elements, we only need to apply this routine in each element or to calculate the total matrix of the optical system, given by the multiplication of individual matrixes, to characterize the Gaussian beam, i. e., to find its ray 𝑤 (𝑧) and the ray of the wavefront 𝑅(𝑧).
To determine the best optical setup of a LIBS system, we have solved the following problem: Assuming the beam arrives collimated at the focusing lens of the LIBS system, which is the beam waist 𝑤0 after the lens? Besides, we need to answer if the joined 𝑤0 is enough to presents a power density (irradiance) able to cause the material ablation and to create the LIBS signal. Figure 2 illustrates the calculated situation.
Figure 2: Optical setup of the LIBS assembly.
As the beam is collimated when strikes the lens, its wavefront ray 𝑅1 → ∞. Thus, by eq. (4), we have 𝑞1
1= −𝑖𝑛𝜋𝑤𝜆
12. Using ABCD’s law, eq.(5), and a thin lens matrix that is ℳ𝐿𝐸𝑁𝑇𝐸 = (−1/𝑓 1) , we obtain that the parameter 𝑞1 0 2, immediately after the lens, satisfies the following relation: 1 𝑞2= − 1 𝑓+𝑖 𝜆 𝑛𝜋 𝑤 12 (−𝑓1)2+( 𝜆 𝑛𝜋 𝑤 12)
2 , where 𝑓 is the lens focusing distance.
After the lens, we apply the ABCD’s law once more, eq. (5), for a translation with distance 𝑙, whose matrix is ℳ𝑇𝑅𝐴𝑁𝑆𝐿 = (1 𝑙
0 1) and we obtain that the parameter 𝑞3 satisfies the relation given by eq. (6).
1
𝑞3=(𝑎2+𝑏[𝑙(𝑎22)[𝑙(𝑎+𝑏22)−𝑎]+𝑏22)−𝑎−𝑖𝑏]+𝑏2 (6)
where we introduce the parameters 𝑎 =1
𝑓, 𝑏 = 𝜆
𝑛𝜋𝑤12= 1
𝑧01 to simplify the equation.
As Figure 2 shows, at 𝑞3 plan the wavefront ray is 𝑅3→ ∞, therefore the real part of eq. (6) is null. This information takes us to find the translation distance 𝑙 as a function of the lens focusing distance 𝑓 and the Rayleigh’s length 𝑧01, eq. (7).
𝑙 = 𝑓
1+(𝑓
𝑧01)
Note that, if 𝑧01≫ 𝑓, we have 𝑙 = 𝑓, i. e., we are at geometric optics regime. Matching the imaginary part of eq. (6) with imaginary part of eq. (4), we have:
(𝑎2+𝑏2)𝑏
[𝑙(𝑎2+𝑏2)−𝑎]2+𝑏2= 𝜆
𝜋𝑛𝑤02 (8)
Substituting the values of 𝑎 and 𝑏 in the equation above, we achieved the relation between the beam waist 𝑤0 and its initial ray 𝑤1, given by eq. (9).
𝑤0= 𝑓
𝑧01√1 + ( 𝑓𝑧
01)2 ∙ 𝑤1
Note that, if the Rayleigh’s length 𝑧01 is larger than the lens focusing distance 𝑓, eq. (9) will assume the simple form 𝑤0=𝑛𝜋𝑤𝑓𝜆
1.
To represent a LIBS system, we use a pulsed Nd:YAG laser, with pulse width of 10 nanoseconds, in fundamental harmonics (λ = 1064 nm) and energy about 50 mJ per pulse. Figure 3 shows the variation of the lens focusing distance value and the calculation of light irradiance at 𝑤0, i. e., where the analyzed sample was placed. The irradiance calculation was achieved dividing the beam power (50 𝑚𝐽
10 𝑛𝑠) by the lighted area 𝜋𝑤02 in the sample.
Figure 3: Irradiance and 𝑤0 versus Focus Distance plot for a LIBS assembly. The 𝑤1 utilized on simulation was 1mm, according to actual lasers Nd:YAG.
For sample matrixes utilized in LIBS spectroscopy, bond cleavage following material ablation happened in the range of 5 to 10 GW/cm². In this case, we can overcome this values with the initial use of a lens with focusing distance of to 20 cm. If in the experiment it was necessary a higher or lesser irradiance in the sample, we would change it easily according to the calculation realized and the Figure 3. In Table 1, we show the values of 𝑤0 and irradiance for some 𝑓 values usually used in LIBS assemblies.
Table 1. 𝑤0 and Irradiance values for focus distances usually used in LIBS assemblies. Focus Distance 𝑓 (cm) 𝑤0(𝜇𝑚) Irradiance (GW/cm2)
5,0 16,9 5,55 x 102 7,5 25,4 2,46 x 102 10,0 33,8 1,39 x 102 12,5 42,3 8,90 x 101 15,0 50,7 6,18 x 101 20,0 67,6 3,48 x 101 4. Conclusion
Utilizing the Gaussian beam formalism to treat the laser used in Laser Induced Breakdown Spectroscopy (LIBS) and the ABCD’s law to model its propagation in the optical system, it was possible to find the values of focusing distance more suitable for LIBS setup. The simulation was reliable enough to change the optical assembly in the laboratory without compromise its validity. With a lens of 𝑓 = 7,5 cm we achieved an irradiance in the sample of approximately 246 GW/cm², which is enough to cause material ablation and consequently to obtain a LIBS signal.
The realized simulation in this work was necessary to know the beam parameters before and after it strikes the sample. Thus, the optical assembly of the LIBS system has all of its parameters known, that permits more agility and efficiency in the measurements and sample changes.
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