Engineering physics/ matlab

Interaction of light with suspensions of particles

What is light?

• Light is: – A stream of photons (elementary quanta of electromagnetic

radiation traveling with the speed of light, c = 2.9979 x 108 m/s) – An electromagnetic wave (an electric wave and a magnetic wave

that generate each other as they propagate with the speed of light)

x

y

z

𝐸𝐸 𝑥𝑥, 𝑡𝑡 = 𝐸𝐸0 cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥 + 𝜑𝜑

• 𝐸𝐸0 is amplitude of electric field • 𝜔𝜔 = 2𝜋𝜋𝜋𝜋 is cyclic frequency • 𝑘𝑘 = 2𝜋𝜋𝜋𝜋

𝜆𝜆0 is wavenumber

• 𝜑𝜑 is phase • 𝑛𝑛 is refractive index of material • 𝜆𝜆0 =

𝑐𝑐 𝜈𝜈

is wavelength in vacuum • 𝜖𝜖0 is vacuum permittivity

𝐼𝐼0 𝑥𝑥 = 𝜖𝜖0𝑐𝑐 𝐸𝐸2 𝑥𝑥, 𝑡𝑡 = 𝜖𝜖0𝑐𝑐𝐸𝐸0 2 cos2 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥 + 𝜑𝜑

𝐼𝐼0 𝑥𝑥 = 1 2 𝜖𝜖0𝑐𝑐𝐸𝐸0

2time average

?

What is light?

What light?

Types of interaction of light with particles:

• Particles dispersed in solution, random and dynamically changing arrangement (due to the Brownian motion)

• Scattering: – Photons change their direction, energy of light is re-irradiated by the

particles (usually with different intensity and direction)

– Elastic scattering (same wavelength / energy) – Inelastic scattering (different wavelength / energy) – [reflection, refraction, diffraction – also scattering, but on highly ordered

particles]

• Absorption: – Photons disappear, energy of light is transformed into other forms of

energy (heat, chemical reactions)

– Light may be re-emitted at different wavelength (fluorescence, phosphorescence)

• [Change in polarization of light]

Types of interaction of light with particles:

𝐼𝐼 = 𝐼𝐼0 � 𝑒𝑒−𝜏𝜏𝜏𝜏

𝐼𝐼 = 𝐼𝐼0 � 10−𝛼𝛼𝜏𝜏

absorption coefficient

• Empirical observation: – Intensity of light is reduced by either absorption or scattering (or both)

exponentially with the thickness of the material

Scattering:

Absorption:

turbidity

Measurement of scattering:

• mean intensity 𝐼𝐼 reflects molecular weight (or size) of the particles

• characteristic time 𝜏𝜏 of intensity fluctuations reflects diffusion coefficient of particles

circular cuvette

• Scattered light preserves information about the particles (e.g. size, shape)

Measurement of scattering:

• Nephelometers are used to detect light that is scattered at various angles

• Turbidimetry measures a reduction in light transmission due to particle formation

Example: • measuring turbidity of Hb solubility

assay for detecting presence (and measuring concentration) of HbS

• monitoring bacterial growth in suspensions

• timing clot formation in cuvettes

Example: • measuring concentration of

antibody-antigen complexes

Nephelometry:

• Three types of light scattering in particulate solutions:

– 𝜆𝜆 ≫ 𝑑𝑑, light scatters symmetrically around the particle; minimal scatter at 90° (Rayleigh theory)

– 𝜆𝜆 ≈ 𝑑𝑑, more light scatters in the forward direction than in other directions (Rayleigh–Debye theory)

– 𝜆𝜆 ≪ 𝑑𝑑, light scatters forward (Mie theory)

• Example – measurement of antigen–antibody reactions:

– Typical diameter of antigen–antibody complexes 𝑑𝑑 = 250–1500 nm

– Wavelength of light used for measurements 𝜆𝜆 = 320–650 nm

– Forward light scattering (Rayleigh–Debye) nephelometry

Why do particles scatter light?

Electric dipole moment: 𝜇𝜇 = 𝑑𝑑𝑄𝑄

Particles become induced dipoles under influence of external electric field:

𝜇𝜇 = 𝛼𝛼𝜖𝜖0𝐸𝐸 𝛼𝛼 is polarizability of the particle 𝜖𝜖0 is vacuum permittivity

(if 𝐸𝐸 is not very strong)

�⃗�𝜇 = �𝑟𝑟𝜌𝜌 𝑟𝑟 𝑑𝑑𝑑𝑑

Induced electric dipole moment for particles suspended in a medium:

𝜇𝜇 = 𝛼𝛼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑐𝑐𝑝𝑝𝑝𝑝 − 𝛼𝛼𝑚𝑚𝑝𝑝𝑚𝑚𝑝𝑝𝑚𝑚𝑚𝑚 𝜖𝜖0𝐸𝐸

(vacuum)

(medium)

x

y

z

Why do particles scatter light?

𝐸𝐸 𝑥𝑥, 𝑡𝑡 = 𝐸𝐸0 cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥

Particle becomes oscillating dipole…

𝜇𝜇 = 𝜇𝜇 𝑡𝑡 = 𝛼𝛼𝐸𝐸 = 𝛼𝛼𝐸𝐸0 cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥

…and oscillating dipole emits electromagnetic radiation in all directions!

(position of the particle)

(light polarized in XZ-plane)

Why do particles scatter light?

𝐸𝐸𝑠𝑠,1 𝑟𝑟 = 𝐸𝐸0 𝜋𝜋𝛼𝛼 sin 𝜙𝜙 𝑟𝑟𝜆𝜆2

cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑟𝑟 𝐸𝐸 𝑥𝑥, 𝑡𝑡 = 𝐸𝐸0 cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥

𝜇𝜇 𝑡𝑡 = 𝛼𝛼𝐸𝐸0 cos 𝜔𝜔𝑡𝑡 − 𝑘𝑘𝑥𝑥

𝐼𝐼𝑠𝑠,1 𝑟𝑟 = 𝜖𝜖0𝑐𝑐 𝐸𝐸𝑠𝑠,1 2 𝑟𝑟 =

1 2 𝜖𝜖0𝑐𝑐𝐸𝐸0

2 𝜋𝜋 2𝛼𝛼2 sin2 𝜙𝜙 𝑟𝑟2𝜆𝜆4

= 𝐼𝐼0 𝜋𝜋2𝛼𝛼2 sin2 𝜙𝜙

𝑟𝑟2𝜆𝜆4

Intensity of scattered light Intensity of incoming light

𝜙𝜙 = 90° (XY-plane) 𝐼𝐼𝑠𝑠,1 𝑟𝑟 = 𝐼𝐼0

𝜋𝜋2𝛼𝛼2

𝑟𝑟2𝜆𝜆4

Why do particles scatter light?

(incident light no longer polarized)

𝐼𝐼𝑠𝑠,1 𝑟𝑟 = 𝐼𝐼0 𝜋𝜋2𝛼𝛼2

𝑟𝑟2𝜆𝜆4 1 + cos2 𝜃𝜃

2

Average over all polarizations in YZ-plane:

Riding into the Sunset:

Intensity of scattered light for blue (𝜆𝜆 = 480nm) >4 times larger than for red (𝜆𝜆 = 690nm)

𝐼𝐼𝑠𝑠~ = 𝐼𝐼0 𝜆𝜆4

Additional Reading:

• https://www.olympus-lifescience.com/en/microscope- resource/primer/lightandcolor/

Homework assignment:

• Mandatory (last assignment):

— Homework #6 – due on Wednesday, 12/5, 11:59pm

• Extra credit:

— Super Project “Cells in Wells” – due on Monday, 12/10, at 11:59pm