Modeling Nanoparticle Optics

Abstract

As we discussed in the preceding Chap. 4, the optimal range of nanoparticle sizes and optimal wavelength of radiation for effective activating/heating of nanoparticles in tumor cells can be calculated on the basis of the Lorenz–Mie diffraction theory at the single-scattering approximation. The Mie formalism requires the use of two dimensionless input parameters: the complex refractive index of the nanoparticle and the surrounding medium at the given wavelength. In this chapter we introduce a computer software and perform Mie simulations of absorption, scattering, and extinction efficiencies of different types of nanoparticles as a function of particle size and the wavelength of radiation in different surrounding biological media. Also, we calculate and then use the differences in optical properties of normal and cancer cell organelles in order to design new cancer detection and treatment methods.

Appendices

Appendix A: Mie Code

This appendix contains the MATLAB code designed to compute the Mie coefficients. The comments within the MATLAB code should explain the used variables, procedures, and functions.

Note: This code comes directly from the MATLAB file and is therefore formatted to word-wrap according to MATLAB standard character-per-line limits.

1 function result = Mie.abcd (m, x) 2 % Computes the Mie coefficients: a_n, b_n, c_n, d_n. 3 % 4 % A criterion for approximating the minimum number of terms required to 5 % reach convergence (rather than determining convergence iteratively) is 6 % approximated by "nmax = 2 + x + 4*xˆ(1/3)" as suggested on p. 477 of 7 % Bohren & Huffman, Absorption and Scattering of Light by Small Particles. 8 % 9 % Mie coefficients a_1 through a_nmax, b_1 through b_nmax, etc. are 10 % calculated using the Eqs. (4.2). 11 % 12 % The only inputs are the size parameter x and the index parameter m. 13 % x=k*r, where k is the wavenumber and r is the sphere radius. 14 % Because k=2pi/wavelength, x results in a dimensionless parameter 15 % provided r and lambda are in the same units (e.g. nm). 16 % m is the relative complex refractive index m sphere/m medium. 17 % 18 % Colin E.W. Rice and Renat R. Letfullin – Rose-Hulman Institute of Technology 19 20 nmax=round(2+x+4*xˆ(1/3)); %convergence criteria to determine number of terms 21 22 n=(1:nmax);            % list of order, n 23 n_half=(n+0.5);            % list of order, n+1/2 (needed in Bessel functions) 24 25 mx=m.*x;                %m*x, defined for easier typing/reading below 26 mm=m.*m;                % mˆ2, again defined for easier typing. 27 28 jx=besselj(n_half,x).*sqrt(0.5*pi./x);    %spherical Bessel function j_n(x) 29 jmx=besselj(n_half,mx).*sqrt(0.5*pi./mx);    %spherical Bessel function j_n(mx) 30 yx=bessely(n_half,x).*sqrt(0.5*pi./x);    %spherical Bessel function y_n(x) 31 hx=jx+1i*yx;                   %spherical Hankel function h_n(x) 32 33 % The formula for above coefficients j_n(x), j_n(mx) and h-n(x) 34 % use derivatives [x.j_n(x)]' and [mx.j_n(mx)]' and 35 % [x.h_n(x)]' which can be calculated from recurrence relations using 36 % the above in addition to j_[n-1](x), j_ [n-1](mx), and h_[n-1](x). The 37 % above jx, jmx, hx functions range from n=1..nmax so the (n-1) versions 38 % will instead range from n=0..(nmax-1). These can be made simply by 39 % inserting the 0th order term to the front (and dropping the last term). 40 % The above naming convention is kept, with suffix 0 to denote the addition 41 % of the above 0 ^th order in the [n-1]th order terms. 42 43 jx0=[sin(x)/x, jx(1:nmax-1)];        %spherical Bessel function j_[n-1](x) 44 jmx0=[sin(mx)/mx, jmx(1:nmax-1]; %spherical Bessel function j_[n-1](mx) 45 yx0=[-cos(x)/x, yx(1:nmax-1];        %spherical Bessel function y_[n-1](x) 46 hx0=jx0+1i*yx0;                                                     %spherical Hankel function h_[n-1](x) 47 48 % With both the (n)th and (n-1)th order terms defined, the recurrence 49 % relations for [x. j_n(x)]' and [mx. j_n(mx)]' and [x. h_n(x)]' can be used. 50 % The above naming convention is used with suffix p to denote the prime' 51 % status when taking the derivative. 52 53 jxp=x.* jx0-n.*jx;            % [x.j_n(x)]' = x.j_[n-1](x) - n.j_n(x) 54 jmxp=mx.*jmx0-n.*jmx;    % [mx.j_n(mx)]' = mx.j_[n-1](mx) - n.j_n(mx) 55 hxp=x.*hx0-n.*hx;             % [x.h_n(x)]' = x.h_[n-1](x) - n.h_n(x) 56 57 % Finally the coefficients can be calculated from all of the above 58 % Bessel and Hankel functions. 59 60 an = (mm.*jmx.*jxp-jx.*jmxp)./(mm.*jmx.*hxp-hx.*jmxp); 61 bn = (jmx.*jxp-jx.*jmxp)./(jmx.*hxp-hx.*jmxp); 62 cn = (jx.*hxp-hx.*jxp)./(jmx.*hxp-hx.*jmxp); 63 dn = (m.*jx.*hxp-m.*hx.*jxp)./(mm.*jmx.*hxp-hx.*jmxp); 64 65 % Return the resulting lists of coefficients. 66 result=[an; bn; cn; dn]; 67 end

Homework Exercises

5.1.1 Section 5.1: Lorenz–Mie Simulations of Nanoparticle Optics

• 5.1. Describe the research task in modeling the radiation interaction with nanoparticles.

• 5.2. List three input parameters required for Lorenz–Mie calculations.

• 5.3. What are the output results from Lorenz–Mie simulations?

• 5.4. True or false? The complex refractive index of the nanoparticle could be engineered.

• 5.5. Why is a gold nanoparticle good for biological applications?

• 5.6. True or False? The index of refraction of materials doesn’t depend on the wavelength of the radiation.

• 5.7. How is the optimal range of the wavelengths determined for therapeutic applications of gold nanoparticles?

• 5.8. How is the optimal range of the wavelengths defined for the diagnostic applications of gold nanoparticles?

• 5.9. At what wavelength is a plasmon resonance absorption observed for gold nanoparticles?

• 5.10. True or false? The surrounding medium doesn’t affect the absorption properties of the nanoparticles.

• 5.11. True or false? The optical properties of the nanoparticles depend on the shape and internal structure of the particle.

• 5.12. Perform Mie simulations of absorption and scattering efficiencies for a silver nanosphere vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of silver nanospheres. Make conclusions about the scattering and absorption properties of silver nanospheres similar to the Practice Example #1.

• 5.13. Perform Mie simulations of absorption and scattering efficiencies for a silica–gold nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of silica–gold nanoparticles. Make conclusions about the scattering and absorption properties of silica–gold nanoparticles similar to the Practice Example #1.

• 5.14. Perform Mie simulations of absorption and scattering efficiencies for a silver nanocube vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal radius of particle for therapeutic and diagnostic applications of silver nanocubes. Make the conclusions about the scattering and absorption properties of silver nanocubes similar to the Practice Example #1.

• 5.15. Perform Mie simulations of absorption and scattering efficiencies for a silica–silver nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of silica–silver nanoparticles. Make conclusions about the scattering and absorption properties of silica–silver nanoparticles similar to the Practice Example #1.

• 5.16. Perform Mie simulations of absorption and scattering efficiencies for an aluminum nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of aluminum nanoparticles. Make conclusions about the scattering and absorption properties of aluminum nanoparticles similar to the Practice Example #1.

• 5.17. Perform Mie simulations of absorption and scattering efficiencies for a nickel nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of nickel nanoparticles. Make conclusions about the scattering and absorption properties of nickel nanoparticles similar to the Practice Example #1.

• 5.18. Perform Mie simulations of absorption and scattering efficiencies for a carbon nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of carbon nanoparticles. Make conclusions about the scattering and absorption properties of carbon nanoparticles similar to the Practice Example #1.

• 5.19. Perform Mie simulations of absorption and scattering efficiencies for a fullerene nanosphere vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of fullerene nanospheres. Make conclusions about the scattering and absorption properties of fullerene nanospheres similar to the Practice Example #1.

• 5.20. Perform Mie simulations of absorption and scattering efficiencies for a polystyrene nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of polystyrene nanoparticles. Make conclusions about the scattering and absorption properties of polystyrene nanoparticles similar to the Practice Example #1.

• 5.21. Perform Mie simulations of absorption and scattering efficiencies for a glass nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of glass nanoparticles. Make conclusions about the scattering and absorption properties of glass nanoparticles similar to the Practice Example #1.

• 5.22. Perform Mie simulations of absorption and scattering efficiencies for an aluminum oxide nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of aluminum oxide nanoparticles. Make conclusions about the scattering and absorption properties of aluminum oxide nanoparticles similar to the Practice Example #1.

• 5.23. Perform Mie simulations of absorption and scattering efficiencies for a chromium nanoparticle vs. the particle radius (r ₀ = 1–1000 nm) and radiation wavelength (λ = 200–1000 nm) in cytoplasm, blood, and collagen by using optical data from Tables 5.1 and 5.2. Find the optimal wavelength of radiation and optimal particle radius for therapeutic and diagnostic applications of chromium nanoparticles. Make conclusions about the scattering and absorption properties of chromium nanoparticles similar to the Practice Example #1.

• 5.24. Perform Mie simulations for gold nanoparticles and silica–gold nanospheres in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

• 5.25. Perform Mie simulations for silver nanocubes and silica–silver nanospheres in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

• 5.26. Perform Mie simulations for aluminum nanoparticles and nickel nanoparticles vs. the particle radius and radiation wavelength in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

• 5.27. Perform Mie simulations for carbon nanoparticles and fullerene nanospheres vs. the particle radius and radiation wavelength in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

• 5.28. Perform Mie simulations for polystyrene and glass nanoparticles vs. the particle radius and radiation wavelength in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

• 5.29. Perform Mie simulations for aluminum oxide and chromium nanoparticles vs. the particle radius and radiation wavelength in water, blood, and cell membrane by using optical data from Tables 5.1 and 5.2. Make a comparative analysis of the scattering and absorption efficiencies for those particles similar to the Practice Example #1.

5.1.2 Section 5.2: Simulations of Optical Properties of Biological Particles

• 5.30. What are the differences in the morphological properties of normal and cancer cells?

• 5.31. How can the differences in the morphological properties of normal and cancer cells be used for diagnostic and therapeutic applications?

• 5.32. Describe the research task in modeling the radiation interaction with biological particles.

• 5.33. Describe how to create a rix file for the MiePlot software.

• 5.34. True or false? Cancerous ribosomes are considerably more efficient in the scattering and absorption of light than normal ribosomes in the optical range of spectrum.

• 5.35. In what wavelength range of the spectrum are cancer ribosomes strong scatters of light?

• 5.36. In what wavelength range of the spectrum are cancer ribosomes strong absorbers of light?

• 5.37. What is the optimal wavelength for cancerous ribosome detection?

• 5.38. What is the optimal wavelength for cancerous ribosome therapy?

• 5.39. True or false? About 70 % of total bone material is formed mostly from calcium phosphate in a crystalline structure known as hydroxyapatite.

• 5.40. Describe the experimental setup shown in Fig. 5.20.

• 5.41. True or false? Bone tissue is an effective absorber of laser radiation over a wide range of the optical spectrum.

• 5.42. What are the wavelengths of two windows of transparency of bone tissue corresponding to the minimum absorption efficiency?

• 5.43. What are the differences in optical properties (absorption and scattering) of bone particles and tooth material?

• 5.44. Perform Mie calculations and a comparative analysis of the absorption and scattering efficiencies for normal and cancerous cell mitochondria surrounded by cytoplasm in the optical range of spectrum (λ = 400 nm to 1 μm) by using optical data from Tables 5.2 and 5.6.

• 5.45. Perform Mie calculations and comparative analysis of the absorption and scattering efficiencies for normal and cancerous cell nuclei surrounded by cytoplasm in the optical range of the spectrum (λ = 400 nm to 1 μm) by using optical data from Tables 5.2 and 5.6.

• 5.46. Perform Mie calculations and comparative analysis of the absorption and scattering efficiencies for normal and cancerous cell lysosome surrounded by cytoplasm in the optical range of the spectrum (λ = 400 nm to 1 μm) by using optical data from Tables 5.2 and 5.6.

• 5.47. Perform Mie calculations and comparative analysis of the absorption and scattering efficiencies for normal and cancerous cell cytoskeleton surrounded by cytoplasm in the optical range of the spectrum (λ = 400 nm to 1 μm) by using optical data from Tables 5.2 and 5.6.

• 5.48. Perform Mie simulations of the scattering efficiency for a bone particle vs. the particle radius (r ₀ = 10–1000 μm) and radiation wavelength (λ = 200–1000 nm) in water, blood, and air by using optical data from Tables 5.2 and 5.9. Find the optimal wavelength of radiation and optimal particle radius for diagnostic applications. Make conclusions about the scattering properties of bone particles similar to the Practice Example #3.

• 5.49. Perform Mie simulations of the scattering efficiency for a tooth particle vs. the particle radius (r ₀ = 10–1000 μm) and radiation wavelength (λ = 200–1000 nm) in water, blood, and air by using optical data from Tables 5.2 and 5.9. Find the optimal wavelength of radiation and optimal particle radius for diagnostic applications. Make conclusions about the scattering properties of tooth particles similar to the Practice Example #3.