Kuan Fang REN Email: fang.ren@coria.fr tel: 02 32 95 37 43

Metrology of particles

Part 3: measurement techniques

Kuan Fang REN

Email: fang.ren@coria.fr tel: 02 32 95 37 43 UMR 6614/ CORIA

CNRS - INSA & University of Rouen

Measurement techniques

➢ Measurement of individual particles

▪ LDV and PDA

▪ Rainbow refractometry

▪ Imaging

➢ Measurement of cloud particles

▪ Extinction spectrometry

▪ Refractometry (Malvern)

▪ Global rainbow

Measurement techniques

Detector Particles Measured parameters

Points CCD

Cloud V D C T/m x, y, z Time

LDV X X X ? X y

PDA X X X X difficulty possible X y

Rainbow X X X X ? ?

Global rainbow

X X X X ? ?

Extinction spectroscopy

X X X X ? Y

PIV X X X Y

V : velocity D : diameter

C : concentration

T/m : temperature or refraction index x,y,z : positions of particles

Time : temporal evolution

Techniques and measured parameters

LDV - Laser Doppler Velocimetry

Configuration

particles

LDV - Laser Doppler Velocimetry

Incidents beam

Interference fringes

f

 v 

 = =

times

Measurement of velocity:

T → f

Principle of measurement

LDV - Laser Doppler Velocimetry

Demonstration of the fringe :

1

2

1 0 0

2 0 0

z x

z x

ik z ik x ik r

ik z ik x ik r

E E e E e

E E e E e

+



−



= =

= =

1 2 0

(

) 2

cos(

) E = E + E = E e

+ e

= E e k x

2 2

4

cos (

) I = E k x

The intensity is maximum:

x

sin

k x = k  x = p 

sin 2sin

p

p p

x k

 

 

= =

f

x x  v 

 = −

= =

Principle of measurement

PDA - Phase Doppler Anemometry

Incidents beams

Principle of measurement

particles

PDA – Model of fringes

f(d,n) < F(D,n)

Df

Df

Principle of measurement

PDA – Model of geometrical optics

Relation phase difference - diameter:

2 1

2 1

( )

refl refl refl refl

refr rerl refr refr

C D

C m D

  

  

D = − =

D = − =

Relation phase-diameter:

reflection

refraction reference

D m

Principle of measurement

PDA – model of waves

1

( , , , )

( , , , ) E

= E d m   + E d m  

 

0

1 cos(2 )

I = I + V  t + f

t I ₀

Rigorous simulation

Principle of measurement

PDA – signal processing

m=1.33, wl=0.633 µm, 2 alpha=5°, offaxis 30°,

-360 -300 -240 -180 -120 -60 0 60 120 180 240 300 360

0 50 100 150 200

Différence de phase [deg.]

Delda Phi=6°, Reflection Delda Phi=6° Réfraction

DF

PDA signal

filter—FFT

CSD

Principle of measurement

PDA – trajectory effect

In te ns it é (log )

perpendiculaire

parallèle 100° 40°

Reflection dominant

or refraction dominant

refl

C

D



D =

refr

C

( ) m D



D =

?

p=0

p=1 p=2

p=3

p=0

p=1 p=2

p=3

Effect of scattering mode

PDA – trajectory effect

refraction dominant

Independent of position

Plane wave or d < w

Reflection dominant

Refraction dominant

Dependent on position

Gaussian beam d ~ w

or d > w

trajectory effect

PDA – trajectory effect

Reflection dominant

refraction dominant

reflection

refraction

trajectory effect

ADL and PDA

6

For small particles

(Rayleigh scattering regime):

log( ) 6log( )

I  D → I  D

For big particles:

log( ) log( )

I  D → I  D

Big particules

Dynamics of measurement

Rainbow

Natural phenomenon

R

m

Incident rays

R

Rainbow – geometrical optics

1

cos

2 cos 2 2

go p

p i

p i q

 =

^  m ^  −  

 

0

2 , i

2( i i ')

 = −   = − −

2 2 ' 2 ' 2 ( 1)

p

p pi i p

 =  −  = − − − 

2 2

avec sin

1 p m

i p

= −

−

2 ' 2 0

d

di

di p di

 = − =

2 2 2

2 2 2 2

1 ( 1)

2 arctan 2 arctan 2

go p

m p m

p q

p m p m





GO model of rainbow Geometrical optics:

i

i’ i’

i’

i  ₂

 ₁ R ₀

R ₁ R ₂

m

I _A

A



i-i’

i

i i-i’

Rainbow – geometrical optics

 n 



486.1 1.3371 138.5° 128.0°

589.3 1.3330 137.9° 129.1°

656.3 1.3311 137.6° 129.6°

R ₂

m

Incident rays

R ₃

GO model of rainbow

Rainbow – in the nature

Can we observe personally a whole rainbow?

Why ?

How?

Rainbow – Airy theory

v

u 

➢ Phase difference is:

➢ Amplitude is constant for all emergent rays

➢ Amplitude of scattered field in  direction:

• The version of van de Hulst (1957) permits to predict the profile,

• But the absolute intensity was corrected later.

I G.O.

Airy theory

Airy theory (1838)

Rainbow – Airy theory

 _Airy,K I

Peak positions according to Airy theory :

) , (

,ⁱ

m d



Measurement of m et d

2 1/3

,1 ( , ) 1.087376 2

Airy go p m 12



 



 

− =  

 

2 1/3 2

, 2

9 1

( , )

4 4

Airy K go

p m  K H

 



   

− =      +     

( )

( )

2 / 1

2 3 2 2 2

2 2

1 1













−

−

= −

m m p p

H p

A _i : constants

A’ A

Interference b/t the rays A et A’

Ref: Wang & van de Hulst

Appl. Opt. 30(1):106-117 (1991).

Rainbow – Lorenz-Mie theory

p=0

p=2

Lorenz-Mie theory and Debye theory

Scattering diagram

Intensity

1 700 000 1 600 000 1 500 000 1 400 000 1 300 000 1 200 000 1 100 000 1 000 000 900 000 800 000 700 000 600 000 500 000 400 000 300 000 200 000 100 000 0

p=2; internal reflection only

Scattering diagram

Intensity

2 100 000 2 000 000 1 900 000 1 800 000 1 700 000 1 600 000 1 500 000 1 400 000 1 300 000 1 200 000 1 100 000 1 000 000 900 000 800 000 700 000 600 000 500 000 400 000 300 000 200 000 100 000 0

p=2; 0+2

Internal reflection

Int+ext reflections

Rainbow – GLMT

Ripple:

B+C

Airy:

B

Ondes de surfaceA+Cet

B+D

A B

C D

B A D

C

m d

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 1.E+07 1.E+07

137 137.5 138 138.5 139 139.5 140

Angle de diffusion

Intensité

FFT

Debye theory/series:

Separation of all the modes - Diffraction

- Reflection

GLMT and Debye theory

Rainbow – metrology

57

d =  f

100 700 1300 1900 2500

diameter ( um)

0 9 18 27 36 45

f ri

p p le ( 1/ de g .)

m=1.3324

Relation

diameter- ripple frequency An example of simulation

Typical error: 600 mm ± 2mm

- Does the factor depend-on the configuration?

- Suppose that the number of lobes  , find the relation between f et d.

For you to reply:

Measurement of diameter

Rainbow – metrology

Measurement of diameter variation

Rainbow – metrology

CSD

Measurement of diameter variation

Rainbow – metrology

0 400 800 1200 1600 2000

10 30 50 70 90

Experiment signal

Experimental setup

Rainbow – metrology

0 6 12 18

frequency (1/deg.)

0 0.2 0.4 0.6 0.8 1

l Fourier Amplitude l e x pe rim e nt

Sim ula tion

Airy

Ripple

Surface wave

Comparison between measured and calculated signals

Experimental results

Rainbow – metrology

1 4 7 10 13 16 19

10³ 10⁴

Logarithmic amplitude (a.u.)

0 5 1 0 1 5 2 0 2 5

distance from orifice (mm)

510 520 530 540 550 560 570 580 590 600

diameter (mm)

experiment prediction of BRUN flat model

Arrival of water

flowmeter

Cont ainer

alimentation

Heating

Ripples toward to the Alexandre band

1

: 

2

: 

Experimental results

Rainbow – metrology

126.5 128.5 130.5 132.5 134.5 136.5 138.5 0E+0

2E+6 4E+6 6E+6 8E+6 1E+7 1E+7

sca tt er in g in ten si ty ( a. u .) m=1.3216 b=-3

b=0 (linear) b=6

m=1.3316 (20°C)

(80°C) (80 ° C)

0 0.2 0.4 0.6 0.8 1

relative radius(x=r/r ^p) 1.321

1.324 1.327 1.33 1.333

refractive indice

b= 6 b= 0

m= 1.3316

m= 1.3216

b= -3

Measurement of temperature gradient

Rainbow – metrology

0 100 200 300 400 500

45 50 55 60

Angle de diffusion

Intensité diffusée

Ripple frequency : Dd ~ 0,1 µm Phase difference: Dd < 0,01 µm

FFT & CSD















Variant of rainbow refractometry

Rainbow – metrology

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 1.E+07 1.E+07

137 137.5 138 138.5 139 139.5 140

Angle de diffusion

Intensité

m _D

f

→d

d ^{< 5 µm}

FFT

D f →Dd

< 10 nm CSD

➢Refraction index

➢Particle size

✓Gradient of refraction index or of temperature

✓Precise size and sphericity

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 1.E+07 1.E+07

137 137.5 138 138.5 139 139.5 140

Angle de diffusion

Intensité

m _D

Summary of rainbow refractometry

Rainbow – metrology

Profile of liquid jet

Experimental scattering patterns.

Rainbow of a real liquid jet

Rainbow – metrology

Rainbow of a real liquid jet

Scattering patterns simulated with VCRM:

- A real jet illuminated by an elliptical gaussian beam.

- Tilt ripple fringes depend on the

vertical curvature of the jet.

Imaging technique – principle

TLMG/TLM Phase function

of the lens Huygens-Fresnel integration

Optical imaging system:

lens

Theoretical principle

Imaging technique – principle

Imaging (SDV) Digital holography

lens

particle Incident

beam Image

out of focus

Imaging in different stages

Imaging technique

Image of a hole by TLMG:

Effect of diffraction

Imaging technique

Out of focus Imaging

Measurement techniques - 2

➢ Measurement of individual particles

▪ LDV and PDA

▪ Rainbow

▪ Imaging technique

➢ Measurement of cloud particles

▪ Extinction spectrometry

▪ Refractometer (Malvern)

▪ Global rainbow

Extinction spectroscopy - Principle

Balance of energy:

Beer-Lambert law: ^{I = I exp } _{ } ^{− L} 

^{C ( , ) ( ) r  n r dr } _{ }

( ) _ext ( , ) dI = − I n r C r  dx

I ₀ I

dx

Beer-Lambert law

Extinction spectroscopy

4 2

, d 

 ^ − ^

 

Extinction factor behaviors

m=1.33

Extinction spectroscopy

Mono-dispersed particles :

) exp(

0 LC N

I

I = − ^ext LC ^ext N = ln( I ⁰ / I )

LC

I N = ln( I

/ )

Poly-dispersed particles :

 

 



= 

 ^{C ( r , ) N ( r )} _L ^{1 ln} _{I (} ^I

_{j )}

i

i j

i

ext

 

Inversion problem :

) ( )

( ) ,

0 A ( r  n r dr = s 

 ^

Inversion problem

Extinction spectroscopy

reference:

measured:

I

( )  I ( ) 

lumière blanche

spectrometer

 

 −

= I 0 exp L  0 ^ C ( r , ) N ( r ) dr

I _ext  RINVERSE: N(r)

Inversion problem

Extinction spectroscopy

0 0.5 1 1.5 2

0.5 0.6 0.7 0.8 0.9

Longueur d'onde

Indice de réfraction

partie réelle

partie imaginaire avec ligne chauffée

Radius (µm)

0.01 0.1 1.0

Relative number distribution (%)

7

6

5

4

3

2

1

0

Measured Verified

Ln(I0/I)

0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08

Measurement of soot emitted by a motor (CERTAM)

Extinction spectroscopy

1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

10 100 1000 10000

diam ètre [nm ]

dN/d(log(D)) [part/cm3]

SMPS ELPI Exinction spect.

D moyen (nm) 86 111 101

Dmodal (nm) 68 71 101

f _v 1.07E-8 1.62E-7 6.07E-8

Measurement of soot emitted by a motor (CERTAM)

Extinction spectroscopy

I ₀ I

L Ld

N(r) d

• N(r)L must not be too small (measurement noise) neither too big (multiple broadcast),

• d/Ld must be small enough.

Caution for a dense medium

Refractometry

Mono-dispersed particles :

2 1

0

sin

) sin ) (

( 



 



=   

 ka

ka I J

I

Poly-dispersed particles :

Inversion problem :

) ( )

( ) ,

0 A ( r  n r dr = I 

 ^

 ^ _ ^{ } _ ^

=

i i

i

i

kr

kr r J

N I

I

2 1

)

0

sin

) sin ( (

)

(   

General inversion problem

Refractometry

Principle of refractometer (Malvern)

Global rainbow

Mono-dispersed Particles :

) , , ( m r 

I

Poly-dispersed Particles

Inversion problem:

) ( )

( ) ,

0 I ( r  n r dr = I ^t 

 ^



=

i

i i

t

I N r I r

I (  )

( ) ( ,  )

f _ ~ r, Df~D r,  _airy ~ m

Principle of global rainbow

Inversion problem

Fredholm equation of the first kind ) ( )

( ) ,

0

K ( x y f y dy = g x





= +

i

i j i j i

i w K f

g  ^,

Discretization:

 _i – measurement error, w _i – discretization weight.

Usually this is an ill conditioned equation.

Regularization:

B g

Af

M = − ² +

Algorithm of inversion problem

Inversion problem

Phillips-Twomey :

( A A ^T +  H f ) = A g ^T

( ) 0

( 0) 0

( ) 0

'( 0) 0 n r

f r f r

f r



= =

→  =

= =

Physical constraints:

Algorithm NNLS (Lawson, SVD, …)

Algorithm of inversion problem

Rainbow

Fig (a) presents the scattering diagram of a water droplet illuminated by a plane wave and Fig (b) a schema of rays around the rainbow angles.

1. Explain the formation of the ripple structure (high frequency) and the Airy structure in Fig. (a) with help of the Fig. (b).

2. Estimate the radius of the particle from the Fig. (a).

1. The ripple structure (high frequency) is formed by the interference of rays external reflected rays B and the internal reflected rays and A’. The Airy structure is formed by the rays around A, i. e. rays of types represented by solid and dashed red lines.

We have 35 ripple peaks over 5°, i.e. the peak number over 180° is N=35*180/5=1260 ~ . So the