- Email: [email protected]
Fundamentals of transmission fluctuation spectrometry with variable spatial averaging
CHINA PARTICUOLOGY Vol. 1, No. 6, 242-246, 2003
FUNDAMENTALS OF TRANSMISSION FLUCTUATION SPECTROMETRY WITH VARIABLE SPATIAL AVERAGING Jianqi Shen1,2,*, Ulrich Riebel2, Marcus Breitenstein2 and Udo Kräuter2 1
Institute of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China 2 Brandenburg Technical University of Cottbus, 03013 Cottbus, Germany *Author to whom correspondence should be addressed. Tel: 0086-21-65686483, E-mail: [email protected]
Abstract Transmission signal of radiation in suspension of particles performed with a high spatial and temporal resolution shows significant fluctuations, which are related to the physical properties of the particles and the process of spatial and temporal averaging. Exploiting this connection, it is possible to calculate the particle size distribution (PSD) and particle concentration. This paper provides an approach of transmission fluctuation spectrometry (TFS) with variable spatial averaging. The transmission fluctuations are expressed in terms of the expectancy of transmission square (ETS) and are obtained as a spectrum, which is a function of the variable beam diameter. The reversal point and the depth of the spectrum contain the information of particle size and particle concentration, respectively. Keywords
expectancy of transmission square, particle size analysis, transmission fluctuation spectrometry
1. Introduction Extinction measurement is traditionally performed in a wide beam, which is described by Bouger-Lambert-Beer’s law (BLBL). Based on a quasicontinuum approach, the BLBL does not account for the discrete nature of particles or their extension and arrangement. If an extinction measurement is performed with a high spatial and temporal resolution, the transmitted intensity signal shows significant fluctuations. The technique of transmission fluctuation was first established by Gergory (1985) and then by subsequent workers (Matsui et al., 1993; Ohto et al., 1993; Wessely et al., 1995) to determine the average particle size in suspensions or aerosol based on extinction measurements with a narrow beam of radiation. The average and the standard deviation of the extinction fluctuation were used. Unfortunately, this approach is suitable only for narrow size distributions, as only the average particle diameter can be obtained. Signals from broader size distributions cannot be interpreted properly. The technique of transmission fluctuation spectrometry (TFS) was later developed by Kräuter (1995), Breitenstein et al. (1999) and Breitenstein (2000). Radiated with an infinitesimally thin beam, one absorbent particle is sufficient to block the beam completely so that the transmission is a “binary” signal jumping between 0 and 1. The transmission signal T(t) originating from this arrangement is subjected to averaging with a variable time constant τ, leading to the averaged signal Tτ ( t ) . Instead of the average and the standard deviation, the transmission fluctuations are expressed in terms of the expectancy of the transmission square (ETS), e (Tτ2 ) , defined as:
( )
e Tτ2
1 = lim ts →∞ 2t s
+ ts
∫ Tτ ( t ) dt . 2
− ts
(1)
In the limiting case of the averaging time constant tending to 0, we have:
(
)
+ ts
1 t s →∞ 2t s
1 t s →∞ 2t s
2 ∫ T ( t ) dt = lim
e Tτ2→0 = lim
− ts
+ ts
∫ T ( t ) dt = e (T ) .
− ts
(2) The second equality exists due to the fact that transmission signal is equal to 0 or 1 for the case of an infinitesimally narrow beam. Another limiting case is τ → ∞ , whereby the ETS is found to be:
(
)
1 t s →∞ 2t s
e Tτ2→∞ = lim 1 lim ts →∞ 2t s
+ ts
∫ ⎡⎣e (T )⎤⎦
− ts
2
+ ts
∫ Tτ ( t ) dt = 2 →∞
− ts
.
dt = ⎡⎣e (T ) ⎤⎦
(3)
2
So, by increasing the averaging time interval from zero to infinity, a translation of the ETS from the average transmission e (T ) to its square ⎡⎣e (T )⎤⎦
2
must occur.
Based on a layer model, the 3-dimensional particle suspension was taken as a piling up of monolayers. Each monolayer is a 2-dimensional arrangement of particles moving through the beam path with a flow velocity v . For monodisperse particles of diameter x , the thickness of a monolayer is at least the particle diameter, i.e., Zml = ( P 1.5 ) x . Here, P is the structural parameter with a value larger than 1.5. So, for a suspension with a thickness of ΔZ , the number of layers is ΔZ 1.5 ΔZ Nml = = . (4) Zml P x Based on the assumptions of geometric ray propagation, completely absorbent particles and low concentration, the average transmission and the ETS are obtained as products of those through the single layers,
Shen, Riebel, Breitenstein & Kräuter: Fundamentals of Transmission Fluctuation Spectrometry Nml
e (Ttot ) = ∏ e (Tml,i ) = e (Tml )
Nml
i
(
Nml
) = ∏ e (T ) = e (T )
2 τ ,tot
2 τ ,ml,i
e T
2 τ ,ml
.
(5)
Nml
i
Therefore, the transmission signal can be studied for a monolayer. For low particle concentrations, the expression of the ETS through a monolayer e (Tτ2,ml ) was found to be given by:
(
)
e Tτ2,ml = 1 − β H (Θ ) ,
(6)
243
effect of temporal averaging. So an additional measurement on the flow velocity is needed and the results on particle size distribution are affected by both the fluctuation of the flow velocity and the contour of the flow velocity in the measuring zone. In this paper, the transmission fluctuation spectrometry is presented with a variable spatial averaging by changing the beam diameter.
2. Mathematical Formulation
where β = PCV is the fraction of the monolayer area cov-
2.1 Expectancy of transmission square
ered with particles, called the monolayer density. Here, CV
In the first step, we consider a monolayer of monodisperse spherical particles, which is illuminated by a circular beam (as shown in Fig. 1). Based on the assumptions of
is the particle volume concentration. In Eq. (6), H (Θ ) is a function obtained as:
⎧ ⎪ ⎪Θ < 1: ⎪⎪ H (Θ ) = ⎨ ⎪ ⎪ ⎪Θ ≥ 1: ⎩⎪
9 ⎞ ⎛ 2Θ 2 −⎜ − ⎟ 1− Θ ⎝ π 2πΘ ⎠ 2⎞ 8 ⎛ 1 ⎜ 2πΘ 2 + π ⎟ arcsinΘ − 3πΘ ⎝ ⎠
1−
2−
(
7 1− Θ 2 3πΘ
1⎛ 8
−
)
3 2
(
)
1 2
+ Beam
,
1 ⎞
Θ ⎝⎜ 3π 4Θ ⎠⎟
(7) where Θ is the dimensionless temporal averaging parameter, defined as: vτ , (8) Θ= x and H (Θ ) has values between 1 in the limiting case of Θ → 0 and 2 in the limiting case of Θ → ∞ . Therefore, by changing the averaging time interval τ (or Θ ), the ETS is obtained as a spectrum. For a monodisperse system, Eq. (5) can be approximately rewritten as the logarithm of the ETS: 1.5ΔZ ⎛ vτ ⎞ ln e Tτ2,tot = Nml ln e Tτ2,ml ≈ − CV H ⎜ ⎟ . (9) x ⎝ x ⎠ This expression can be extended to a polydisperse particle suspension:
(
)
(
)
CV ( x j )
⎛ vτ ⎞ ≈ −1.5ΔZ ∑ ln e T (10) H⎜ ⎟, ⎜x ⎟ x j j ⎝ j⎠ where, x j is the mean particle diameter of the jth fraction
(
2 τ ,tot
)
of particle size distribution. Eq. (9) shows that the logarithm of the ETS is linear with the particle concentration. Furthermore, it is shown in Eq. (10) that, the effects from different particle size fractions superimpose linearly to form the resulting ln(ETS). This leads to the conclusion that particle size distribution (PSD) and particle concentration can be deconvoluted from a simple measurement of transmission fluctuations. The approach needs a very narrow beam especially for the measurements of small particles, which is unfortunately unavailable in practice due to light diffraction. Furthermore, it is necessary to know the flow velocity to evaluate the
Particle
Fig.1
Transparency function of a monolayer of monodisperse particles.
geometric ray propagation of radiation and completely K K absorbent particles, the local transmission Tml ( r ) at r may be described by the transparency function: N K ⎛x K K ⎞ Tml ( r ) = 1 − ∑ Heav ⎜ − r − rk ⎟ , ⎝2 ⎠ k =1
(11)
K
where x is the particle diameter, rk are the particle center coordinates and Heav is the Heaviside function defined as: ⎧u ≥ 0 : 1 . (12) Heav ( u ) = ⎨ ⎩u < 0 : 0 For a circular beam of diameter D crossing the K monolayer at r0 (the coordinate of the beam center), the
transmission is the result of spatial averaging over the beam cross section: +∞ +∞ K K K K Tml,D ( r0 ) = ∫ ∫ Tml ( r ) BD ( r , r0 ) drx dry ,
K K
(13)
−∞ −∞
where, BD ( r , r0 ) is the normalized beam intensity distribution. For example, the profile of a Gaussian beam can be expressed as: K K 8 − 8 ( rK0 − rK )2 D2 , (14) BD ( r , r0 ) = e πD 2
244
CHINA PARTICUOLOGY Vol. 1, No. 6, 2003
and for a circular beam of uniform intensity is: K K ⎧⎪ r0 − r ≤ D 2 : 4 πD 2 K K . BD ( r , r0 ) = ⎨ K K ⎪⎩ r0 − r > D 2 : 0
(
can be written as a convolution: K Tml,D ( r0 ) = (Tml ⊗ BD )rK .
j
(15)
Eqs. (14) and (15) mean that the beam intensity function K K K K BD ( r , r0 ) can also be expressed as BD ( r0 − r ) . So, Eq. (13) (16)
0
Therefore, the Fourier transform of the transmission K signal Tml,D ( r0 ) is a product of the Fourier transforms of
F {Tml,D } = F {Tml } ⋅ F {BD } .
The average transmission can be obtained with Eq. (21): K K 1 e (Tml,D ) = lim ∫∫ Tml,D ( r0 ) dr0 . (27) A →∞ A A The solution for a monodisperse monolayer is found to be independent of the beam diameter: e (Tml,D ) = e (Tml ) = 1 − β = 1 − PCV .
(17)
The Fourier transform of the local transmission Tml is K K πx ⎛ x ω ⎞ N − iωK ⋅rKk F {Tml } = 4π2δ (ω ) − K J1 ⎜⎜ . (18) ⎟∑e ω ⎝ 2 ⎟⎠ k =1 The Fourier transform of the normalized beam profile is
{ }=e
K2 − ω D 2 32
K
{ }
)
where A is the area of the integration. Using Parseval’s theorem of the Fourier transform, we have:
)
(22)
(
)
xj
.
(29)
is the average transmission through a
monolayer and e (Ttot ) is that of a 3-dimensional suspension.
Comparing Eq. (24) with Eq. (6), we find that 2 − χ ( Λ ) is somewhat equivalent to H (Θ ) . Fig. 2 shows the numerical results. The dimensionless spatial averaging parameter Λ and the dimensionless temporal averaging parameter Θ are plotted on a common abscissa and the transition functions of 2 − χ (U) ( Λ ) and H (Θ ) are plotted 2.0 1.9
22 −− χχ( Λ(Λ) )
1.7
H ((ΘΘ))
1.5
+∞ +∞
2 1 F {TML } ⋅ F {BD } dω x dω y . = lim A →∞ 4 π2 A ∫ ∫ −∞ −∞
(23)
)
2 e Tml, Λ = 1− β ⎡ ⎣2 − χ ( Λ ) ⎤⎦
,
1.4 1.3
Combining Eq. (23) with Eqs. (18~20), the final solution of the ETS through a monolayer of particles at low concentration is found to be
(
Here, e (Tml )
1.6
or further, e T
CV ( x j )
j
1.8
+∞ +∞
2 1 F {Tml,D } dω x dω y , A →∞ 4 π2 A ∫ ∫ −∞ −∞
2 e Tml, D = lim
2 ml,D
ln e (Ttot ) ≈ −1.5ΔZ ∑
(19)
πD ⎛ D ω ⎞ U F BD( ) = K J1 ⎜⎜ (20) ⎟ ω ⎝ 2 ⎟⎠ for a circular uniform beam. In the second step, the expectancy of transmission square is obtained with: K K 1 2 2 (21) e Tml, Tml, D = lim D ( r0 ) dr0 , A →∞ A ∫∫ A
(
(28)
3. Numerical Calculation and Discussion
for a Gaussian beam and
(
(26)
Therefore, for a 3-dimensional polydisperse suspension, we have:
profile BD :
F BD
CV ( x j ) ⎡ ⎛ D ⎞⎤ ⎢2 − χ ⎜ ⎟ ⎥ . ⎜ x ⎟⎥ x j ⎣⎢ ⎝ j ⎠⎦
2.2 Average transmission
the local transmission Tml and of the normalized beam
( G)
)
ln e TD2,tot ≈ −1.5ΔZ ∑
(24)
1.2 1.1 1.0 0.01
0.1
1
Θ ((Θ Λ )) H
10
100
∞ ⎧ (G) 2J12 ( u ) − u 4Λ Fig.2 Transition functions of the ETS with variable temporal averaging e du for a Gaussian beam ⎪χ ( Λ ) = ∫ u (dashed line) and with variable spatial averaging (solid line). ⎪ 0 . χ (Λ) = ⎨ 2 2 ∞ ⎡ ⎤ 2 J u 2 J u Λ ) du for a uniform beam on the vertical ordinate. The value of 2 − χ (U) ( Λ ) increases ⎪ (U) 1 ( ) 1( ⎢ ⎥ ⎪χ ( Λ ) = ∫ u ⎣ uΛ ⎦ 0 ⎩ 2 2
(25)
Here, χ ( Λ ) is called the transition function and Λ = D x
is the beam-to-particle diameter ratio, which is a dimensionless spatial averaging parameter. Similar to Eq. (10), the logarithm of the ETS through a 3-dimensional suspension can be obtained as:
from 1 to 2 monotonously and it has the highest slope in the vicinity of Λ = 1, i.e., the reversal point. This gives an estimate of the particle size. Evidently, 2 − χ (U) ( Λ ) and H (Θ ) are quite similar, which means that averaging in
space and averaging in time give similar results. The limiting values of 2 − χ (U) ( Λ ) for Λ → 0 and Λ →∞ corre-
Shen, Riebel, Breitenstein & Kräuter: Fundamentals of Transmission Fluctuation Spectrometry spond to the limiting values of H (Θ ) for Θ → 0 and
Θ →∞ , respectively. However, the slope for 2 − χ ( Λ ) is ( U)
slightly steeper, so that better solution might be expected when the transmission fluctuation spectra with variation of the beam diameter are deconvoluted to particle size distributions. In order to see the sensitivity of the transmission fluctuations to the beam profile, the relative deviation of the transition functions between a Gaussian beam and a circular uniform beam is given in Fig. 3. The transition function for a Gaussian beam is larger than that for a circular uniform beam due to the contour of the beam, which implies that a Gaussian beam gives a weaker spatial averaging of the transmission fluctuations than a circular uniform beam. Furthermore, the relative deviation is related to the beam-to-particle diameter ratio. It reaches a maximum in the vicinity of Λ = 2.51 and becomes smaller toward the limiting cases of Λ → 0 and Λ →∞ . This is not difficult to understand. When the beam-to-particle diameter ratio tends to zero, the beam diameter is very small compared with the particle size so that the effect of beam profile should cancel out and hence χ ( Λ ) tends to 1:
⎧ lim χ ( G) ( Λ ) = 1 ⎪Λ →0 lim χ ( Λ ) = ⎨ . Λ →0 χ ( U) ( Λ ) = 1 ⎪⎩ Λlim →0
suspension of particles passes through the cuvette with a pathlength of ΔZ = 16.5 mm . Transmitted light is collected on a photodiode by a convex lens. The signals may be sent to a personal computer and then processed. Alternatively, they can be treated by a true RMS chip and then sent to the computer. With this setup, the transmission fluctuation spectrum is acquired for a monodisperse system ( x = 2.2 mm ) at a concentration of 1.4% in volume. The measured results are shown in Fig. 5 in closed dots. Simulated results (in open dots) and the theoretical curve (in line) are also given in the figure so as to make a comparison with the measurements.
Fig. 4
Scheme of the optical setup.
0.1
-0.05
ln(ETS)
-0.25
(31)
0.08
( ) ( )]
χχ(G( ) (ΛΛ ) − χχ( (U) () ΛΛ) 1 1 ⎡ χ((GG)) ( Λ ) + χ (U(U) ()Λ )⎤ Λ + χ Λ⎦ 2 ⎣χ 2
[
U
Model
-0.2
-0.3
0.12
( ) ( )
Measurement
-0.15
-0.35
G
10
Simulation
-0.1
⎧ lim χ ( G) ( Λ ) = 0 ⎪ Λ →∞ lim χ ( Λ ) = ⎨ . Λ →∞ χ ( U) ( Λ ) = 0 ⎪⎩ Λlim →∞
0.10
1
0
(30)
On the other hand, as the beam diameter is much larger than the particle diameter, the transmission fluctuations can be completely smoothened so that χ ( Λ ) tends to 0:
245
Fig. 5
0.06
Λ
Comparison between experimental results, simulations and theoretical curve. Experiments and simulations are performed with parameters as follows: the pathlength is 16.5 mm; the particle concentration is 1.4% in volume and the particle diameter is 2.2 mm; the diameter of the uniform beam varies in a range of 0.22 mm to 8 mm. Theoretical curve is calculated with Eq. (26).
0.04
5. Conclusions 0.02 0.00 0.01
Fig.3
0.1
1
Λ Λ
10
100
The relative deviation of the transition functions for a Gaussian beam and a circular uniform beam.
4. Experimental Evidence Measurement is performed with a simple optical setup portrayed in Fig. 4 (Kräuter, 1995). The He−Ne laser is expanded to a circular beam with variable diameters. The
By changing the beam diameter, the spatial resolution of the transmission signals varies accordingly, whereby the transmission fluctuation spectrum is obtained. For a monodisperse suspension, the reversal point of the spectrum is in the vicinity of Λ = D x = 1 , which can be used to evaluate the particle size. From Eqs. (26) and (29), and Fig. 5, we can find that the extreme values of ln(ETS) at Λ → 0 and Λ → ∞ are the logarithm of the average transmission and its double respectively, from which the particle concentration may be calculated. As to a polydisperse suspension, the transmission fluc-
CHINA PARTICUOLOGY Vol. 1, No. 6, 2003
246
tuation spectrum is given in Eq. (26), whereby the effects from different particle size fractions superimpose linearly to form the ln(ETS), making it possible to reduce the PSDcalculation to a linear inversion problem solved by known methods. It is proven that, compared to the TFS with variable temporal averaging, the transmission fluctuation spectrometry with variable spatial averaging has a high resolution for particle size analysis and is independent of flow velocity. Due to the simplicity of the optical arrangement, this method is predestined for low-cost in-situ particle size measurements.
Acknowledgement The work presented here is performed under DFG Grant Ri 533/7-1 and Ri 533/7-2, which is gratefully acknowledged.
ts
averaging time interval in statistics
u v x Zml
integration variable particle flow velocity particle diameter thickness of the monolayer
β
monolayer density
δ(
χ(
ω
area of monolayer beam intensity distribution particle volume concentration
A B CV
D e(
beam diameter expectancy
)
e (Tml2 )
ETS F{ }
expectancy of transmission square through a monolayer expectancy of transmission square of the total suspension expectancy of transmission square Fourier transform
H(
transition function of the transmission spectrum
( )
e Ttot2
)
Heav (
)
Heaviside function
i j
layer number number of particle fraction in PSD
J1
1 order Bessel function
ml Nml
monolayer number of monolayers
P PSD K r K r0 K rk
structure parameter of monolayer particle size distribution coordinates coordinate of beam center
st
)
Dirac delta function
)
pathlength dimensionless averaging time beam-to-particle diameter ratio averaging time transition function
ΔZ Θ Λ τ K
Nomenclature
time transmission transmission fluctuation spectrometry transmission of monolayer
t T TFS Tml
space reciprocal
References Breitenstein, M. (2000). Grundlagenuntersuchung zur statistischen Partikelgerößenspektrometrie mittels Auswertung der Transmissionsfluktuation von Licht in diepersen Systemen (pp.33-92). Doctoral thesis, Brandenberg Technical University of Cottbus, Germany. Breitenstein, M., Kräuter, U. & Riebel, U. (1999). The fundamentals of particle size analysis by transmission fluctuation spectrometry, part 1: a theory of temporal transmission fluctuations in dilute suspensions. Part. Part. Syst. Char., 16, 249-256. Gergory, J. (1985). Turbidity fluctuations in flowing suspensions. J. Colloid Interface Sci., 105(2), 357-371. Kräuter, U. (1995). Grundlagen zur in-situ Partikelgrößenanalyse mit Licht und Ultraschall in konzentrierten Partikelsystemen (pp.155-162). Doctoral thesis, Karlsruhe. Matsui, Y., Tambo, N., Ohto, T. & Zaitsu, Y. (1993). Dual wavelength photometric dispersion analysis of coagulation and flocculation. Water Sci. Technol., 27, 153-165. Ohto, T., Zaitsu, Y. & Tambo, N. (1993). Advances in photometric dispersion analyzer with dual wavelength light. Water Sci. Technol., 27, 257-260. Wessely, B., Stinz, M. & Ripperger, S. (1995). In-line-fähiges Meßverfahren zur Überwachen von Flockungsprozessen in einem weiten Konzentrations-bereich. Chem. Ing. Tech., 67, 754-757.
th
coordinate of the k particle center Manuscript received August 19, 2003 and accepted October 27, 2003.
FUNDAMENTALS OF TRANSMISSION FLUCTUATION SPECTROMETRY WITH VARIABLE SPATIAL AVERAGING Jianqi Shen1,2,*, Ulrich Riebel2, Marcus Breitenstein2 and Udo Kräuter2 1
Institute of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China 2 Brandenburg Technical University of Cottbus, 03013 Cottbus, Germany *Author to whom correspondence should be addressed. Tel: 0086-21-65686483, E-mail: [email protected]
Abstract Transmission signal of radiation in suspension of particles performed with a high spatial and temporal resolution shows significant fluctuations, which are related to the physical properties of the particles and the process of spatial and temporal averaging. Exploiting this connection, it is possible to calculate the particle size distribution (PSD) and particle concentration. This paper provides an approach of transmission fluctuation spectrometry (TFS) with variable spatial averaging. The transmission fluctuations are expressed in terms of the expectancy of transmission square (ETS) and are obtained as a spectrum, which is a function of the variable beam diameter. The reversal point and the depth of the spectrum contain the information of particle size and particle concentration, respectively. Keywords
expectancy of transmission square, particle size analysis, transmission fluctuation spectrometry
1. Introduction Extinction measurement is traditionally performed in a wide beam, which is described by Bouger-Lambert-Beer’s law (BLBL). Based on a quasicontinuum approach, the BLBL does not account for the discrete nature of particles or their extension and arrangement. If an extinction measurement is performed with a high spatial and temporal resolution, the transmitted intensity signal shows significant fluctuations. The technique of transmission fluctuation was first established by Gergory (1985) and then by subsequent workers (Matsui et al., 1993; Ohto et al., 1993; Wessely et al., 1995) to determine the average particle size in suspensions or aerosol based on extinction measurements with a narrow beam of radiation. The average and the standard deviation of the extinction fluctuation were used. Unfortunately, this approach is suitable only for narrow size distributions, as only the average particle diameter can be obtained. Signals from broader size distributions cannot be interpreted properly. The technique of transmission fluctuation spectrometry (TFS) was later developed by Kräuter (1995), Breitenstein et al. (1999) and Breitenstein (2000). Radiated with an infinitesimally thin beam, one absorbent particle is sufficient to block the beam completely so that the transmission is a “binary” signal jumping between 0 and 1. The transmission signal T(t) originating from this arrangement is subjected to averaging with a variable time constant τ, leading to the averaged signal Tτ ( t ) . Instead of the average and the standard deviation, the transmission fluctuations are expressed in terms of the expectancy of the transmission square (ETS), e (Tτ2 ) , defined as:
( )
e Tτ2
1 = lim ts →∞ 2t s
+ ts
∫ Tτ ( t ) dt . 2
− ts
(1)
In the limiting case of the averaging time constant tending to 0, we have:
(
)
+ ts
1 t s →∞ 2t s
1 t s →∞ 2t s
2 ∫ T ( t ) dt = lim
e Tτ2→0 = lim
− ts
+ ts
∫ T ( t ) dt = e (T ) .
− ts
(2) The second equality exists due to the fact that transmission signal is equal to 0 or 1 for the case of an infinitesimally narrow beam. Another limiting case is τ → ∞ , whereby the ETS is found to be:
(
)
1 t s →∞ 2t s
e Tτ2→∞ = lim 1 lim ts →∞ 2t s
+ ts
∫ ⎡⎣e (T )⎤⎦
− ts
2
+ ts
∫ Tτ ( t ) dt = 2 →∞
− ts
.
dt = ⎡⎣e (T ) ⎤⎦
(3)
2
So, by increasing the averaging time interval from zero to infinity, a translation of the ETS from the average transmission e (T ) to its square ⎡⎣e (T )⎤⎦
2
must occur.
Based on a layer model, the 3-dimensional particle suspension was taken as a piling up of monolayers. Each monolayer is a 2-dimensional arrangement of particles moving through the beam path with a flow velocity v . For monodisperse particles of diameter x , the thickness of a monolayer is at least the particle diameter, i.e., Zml = ( P 1.5 ) x . Here, P is the structural parameter with a value larger than 1.5. So, for a suspension with a thickness of ΔZ , the number of layers is ΔZ 1.5 ΔZ Nml = = . (4) Zml P x Based on the assumptions of geometric ray propagation, completely absorbent particles and low concentration, the average transmission and the ETS are obtained as products of those through the single layers,
Shen, Riebel, Breitenstein & Kräuter: Fundamentals of Transmission Fluctuation Spectrometry Nml
e (Ttot ) = ∏ e (Tml,i ) = e (Tml )
Nml
i
(
Nml
) = ∏ e (T ) = e (T )
2 τ ,tot
2 τ ,ml,i
e T
2 τ ,ml
.
(5)
Nml
i
Therefore, the transmission signal can be studied for a monolayer. For low particle concentrations, the expression of the ETS through a monolayer e (Tτ2,ml ) was found to be given by:
(
)
e Tτ2,ml = 1 − β H (Θ ) ,
(6)
243
effect of temporal averaging. So an additional measurement on the flow velocity is needed and the results on particle size distribution are affected by both the fluctuation of the flow velocity and the contour of the flow velocity in the measuring zone. In this paper, the transmission fluctuation spectrometry is presented with a variable spatial averaging by changing the beam diameter.
2. Mathematical Formulation
where β = PCV is the fraction of the monolayer area cov-
2.1 Expectancy of transmission square
ered with particles, called the monolayer density. Here, CV
In the first step, we consider a monolayer of monodisperse spherical particles, which is illuminated by a circular beam (as shown in Fig. 1). Based on the assumptions of
is the particle volume concentration. In Eq. (6), H (Θ ) is a function obtained as:
⎧ ⎪ ⎪Θ < 1: ⎪⎪ H (Θ ) = ⎨ ⎪ ⎪ ⎪Θ ≥ 1: ⎩⎪
9 ⎞ ⎛ 2Θ 2 −⎜ − ⎟ 1− Θ ⎝ π 2πΘ ⎠ 2⎞ 8 ⎛ 1 ⎜ 2πΘ 2 + π ⎟ arcsinΘ − 3πΘ ⎝ ⎠
1−
2−
(
7 1− Θ 2 3πΘ
1⎛ 8
−
)
3 2
(
)
1 2
+ Beam
,
1 ⎞
Θ ⎝⎜ 3π 4Θ ⎠⎟
(7) where Θ is the dimensionless temporal averaging parameter, defined as: vτ , (8) Θ= x and H (Θ ) has values between 1 in the limiting case of Θ → 0 and 2 in the limiting case of Θ → ∞ . Therefore, by changing the averaging time interval τ (or Θ ), the ETS is obtained as a spectrum. For a monodisperse system, Eq. (5) can be approximately rewritten as the logarithm of the ETS: 1.5ΔZ ⎛ vτ ⎞ ln e Tτ2,tot = Nml ln e Tτ2,ml ≈ − CV H ⎜ ⎟ . (9) x ⎝ x ⎠ This expression can be extended to a polydisperse particle suspension:
(
)
(
)
CV ( x j )
⎛ vτ ⎞ ≈ −1.5ΔZ ∑ ln e T (10) H⎜ ⎟, ⎜x ⎟ x j j ⎝ j⎠ where, x j is the mean particle diameter of the jth fraction
(
2 τ ,tot
)
of particle size distribution. Eq. (9) shows that the logarithm of the ETS is linear with the particle concentration. Furthermore, it is shown in Eq. (10) that, the effects from different particle size fractions superimpose linearly to form the resulting ln(ETS). This leads to the conclusion that particle size distribution (PSD) and particle concentration can be deconvoluted from a simple measurement of transmission fluctuations. The approach needs a very narrow beam especially for the measurements of small particles, which is unfortunately unavailable in practice due to light diffraction. Furthermore, it is necessary to know the flow velocity to evaluate the
Particle
Fig.1
Transparency function of a monolayer of monodisperse particles.
geometric ray propagation of radiation and completely K K absorbent particles, the local transmission Tml ( r ) at r may be described by the transparency function: N K ⎛x K K ⎞ Tml ( r ) = 1 − ∑ Heav ⎜ − r − rk ⎟ , ⎝2 ⎠ k =1
(11)
K
where x is the particle diameter, rk are the particle center coordinates and Heav is the Heaviside function defined as: ⎧u ≥ 0 : 1 . (12) Heav ( u ) = ⎨ ⎩u < 0 : 0 For a circular beam of diameter D crossing the K monolayer at r0 (the coordinate of the beam center), the
transmission is the result of spatial averaging over the beam cross section: +∞ +∞ K K K K Tml,D ( r0 ) = ∫ ∫ Tml ( r ) BD ( r , r0 ) drx dry ,
K K
(13)
−∞ −∞
where, BD ( r , r0 ) is the normalized beam intensity distribution. For example, the profile of a Gaussian beam can be expressed as: K K 8 − 8 ( rK0 − rK )2 D2 , (14) BD ( r , r0 ) = e πD 2
244
CHINA PARTICUOLOGY Vol. 1, No. 6, 2003
and for a circular beam of uniform intensity is: K K ⎧⎪ r0 − r ≤ D 2 : 4 πD 2 K K . BD ( r , r0 ) = ⎨ K K ⎪⎩ r0 − r > D 2 : 0
(
can be written as a convolution: K Tml,D ( r0 ) = (Tml ⊗ BD )rK .
j
(15)
Eqs. (14) and (15) mean that the beam intensity function K K K K BD ( r , r0 ) can also be expressed as BD ( r0 − r ) . So, Eq. (13) (16)
0
Therefore, the Fourier transform of the transmission K signal Tml,D ( r0 ) is a product of the Fourier transforms of
F {Tml,D } = F {Tml } ⋅ F {BD } .
The average transmission can be obtained with Eq. (21): K K 1 e (Tml,D ) = lim ∫∫ Tml,D ( r0 ) dr0 . (27) A →∞ A A The solution for a monodisperse monolayer is found to be independent of the beam diameter: e (Tml,D ) = e (Tml ) = 1 − β = 1 − PCV .
(17)
The Fourier transform of the local transmission Tml is K K πx ⎛ x ω ⎞ N − iωK ⋅rKk F {Tml } = 4π2δ (ω ) − K J1 ⎜⎜ . (18) ⎟∑e ω ⎝ 2 ⎟⎠ k =1 The Fourier transform of the normalized beam profile is
{ }=e
K2 − ω D 2 32
K
{ }
)
where A is the area of the integration. Using Parseval’s theorem of the Fourier transform, we have:
)
(22)
(
)
xj
.
(29)
is the average transmission through a
monolayer and e (Ttot ) is that of a 3-dimensional suspension.
Comparing Eq. (24) with Eq. (6), we find that 2 − χ ( Λ ) is somewhat equivalent to H (Θ ) . Fig. 2 shows the numerical results. The dimensionless spatial averaging parameter Λ and the dimensionless temporal averaging parameter Θ are plotted on a common abscissa and the transition functions of 2 − χ (U) ( Λ ) and H (Θ ) are plotted 2.0 1.9
22 −− χχ( Λ(Λ) )
1.7
H ((ΘΘ))
1.5
+∞ +∞
2 1 F {TML } ⋅ F {BD } dω x dω y . = lim A →∞ 4 π2 A ∫ ∫ −∞ −∞
(23)
)
2 e Tml, Λ = 1− β ⎡ ⎣2 − χ ( Λ ) ⎤⎦
,
1.4 1.3
Combining Eq. (23) with Eqs. (18~20), the final solution of the ETS through a monolayer of particles at low concentration is found to be
(
Here, e (Tml )
1.6
or further, e T
CV ( x j )
j
1.8
+∞ +∞
2 1 F {Tml,D } dω x dω y , A →∞ 4 π2 A ∫ ∫ −∞ −∞
2 e Tml, D = lim
2 ml,D
ln e (Ttot ) ≈ −1.5ΔZ ∑
(19)
πD ⎛ D ω ⎞ U F BD( ) = K J1 ⎜⎜ (20) ⎟ ω ⎝ 2 ⎟⎠ for a circular uniform beam. In the second step, the expectancy of transmission square is obtained with: K K 1 2 2 (21) e Tml, Tml, D = lim D ( r0 ) dr0 , A →∞ A ∫∫ A
(
(28)
3. Numerical Calculation and Discussion
for a Gaussian beam and
(
(26)
Therefore, for a 3-dimensional polydisperse suspension, we have:
profile BD :
F BD
CV ( x j ) ⎡ ⎛ D ⎞⎤ ⎢2 − χ ⎜ ⎟ ⎥ . ⎜ x ⎟⎥ x j ⎣⎢ ⎝ j ⎠⎦
2.2 Average transmission
the local transmission Tml and of the normalized beam
( G)
)
ln e TD2,tot ≈ −1.5ΔZ ∑
(24)
1.2 1.1 1.0 0.01
0.1
1
Θ ((Θ Λ )) H
10
100
∞ ⎧ (G) 2J12 ( u ) − u 4Λ Fig.2 Transition functions of the ETS with variable temporal averaging e du for a Gaussian beam ⎪χ ( Λ ) = ∫ u (dashed line) and with variable spatial averaging (solid line). ⎪ 0 . χ (Λ) = ⎨ 2 2 ∞ ⎡ ⎤ 2 J u 2 J u Λ ) du for a uniform beam on the vertical ordinate. The value of 2 − χ (U) ( Λ ) increases ⎪ (U) 1 ( ) 1( ⎢ ⎥ ⎪χ ( Λ ) = ∫ u ⎣ uΛ ⎦ 0 ⎩ 2 2
(25)
Here, χ ( Λ ) is called the transition function and Λ = D x
is the beam-to-particle diameter ratio, which is a dimensionless spatial averaging parameter. Similar to Eq. (10), the logarithm of the ETS through a 3-dimensional suspension can be obtained as:
from 1 to 2 monotonously and it has the highest slope in the vicinity of Λ = 1, i.e., the reversal point. This gives an estimate of the particle size. Evidently, 2 − χ (U) ( Λ ) and H (Θ ) are quite similar, which means that averaging in
space and averaging in time give similar results. The limiting values of 2 − χ (U) ( Λ ) for Λ → 0 and Λ →∞ corre-
Shen, Riebel, Breitenstein & Kräuter: Fundamentals of Transmission Fluctuation Spectrometry spond to the limiting values of H (Θ ) for Θ → 0 and
Θ →∞ , respectively. However, the slope for 2 − χ ( Λ ) is ( U)
slightly steeper, so that better solution might be expected when the transmission fluctuation spectra with variation of the beam diameter are deconvoluted to particle size distributions. In order to see the sensitivity of the transmission fluctuations to the beam profile, the relative deviation of the transition functions between a Gaussian beam and a circular uniform beam is given in Fig. 3. The transition function for a Gaussian beam is larger than that for a circular uniform beam due to the contour of the beam, which implies that a Gaussian beam gives a weaker spatial averaging of the transmission fluctuations than a circular uniform beam. Furthermore, the relative deviation is related to the beam-to-particle diameter ratio. It reaches a maximum in the vicinity of Λ = 2.51 and becomes smaller toward the limiting cases of Λ → 0 and Λ →∞ . This is not difficult to understand. When the beam-to-particle diameter ratio tends to zero, the beam diameter is very small compared with the particle size so that the effect of beam profile should cancel out and hence χ ( Λ ) tends to 1:
⎧ lim χ ( G) ( Λ ) = 1 ⎪Λ →0 lim χ ( Λ ) = ⎨ . Λ →0 χ ( U) ( Λ ) = 1 ⎪⎩ Λlim →0
suspension of particles passes through the cuvette with a pathlength of ΔZ = 16.5 mm . Transmitted light is collected on a photodiode by a convex lens. The signals may be sent to a personal computer and then processed. Alternatively, they can be treated by a true RMS chip and then sent to the computer. With this setup, the transmission fluctuation spectrum is acquired for a monodisperse system ( x = 2.2 mm ) at a concentration of 1.4% in volume. The measured results are shown in Fig. 5 in closed dots. Simulated results (in open dots) and the theoretical curve (in line) are also given in the figure so as to make a comparison with the measurements.
Fig. 4
Scheme of the optical setup.
0.1
-0.05
ln(ETS)
-0.25
(31)
0.08
( ) ( )]
χχ(G( ) (ΛΛ ) − χχ( (U) () ΛΛ) 1 1 ⎡ χ((GG)) ( Λ ) + χ (U(U) ()Λ )⎤ Λ + χ Λ⎦ 2 ⎣χ 2
[
U
Model
-0.2
-0.3
0.12
( ) ( )
Measurement
-0.15
-0.35
G
10
Simulation
-0.1
⎧ lim χ ( G) ( Λ ) = 0 ⎪ Λ →∞ lim χ ( Λ ) = ⎨ . Λ →∞ χ ( U) ( Λ ) = 0 ⎪⎩ Λlim →∞
0.10
1
0
(30)
On the other hand, as the beam diameter is much larger than the particle diameter, the transmission fluctuations can be completely smoothened so that χ ( Λ ) tends to 0:
245
Fig. 5
0.06
Λ
Comparison between experimental results, simulations and theoretical curve. Experiments and simulations are performed with parameters as follows: the pathlength is 16.5 mm; the particle concentration is 1.4% in volume and the particle diameter is 2.2 mm; the diameter of the uniform beam varies in a range of 0.22 mm to 8 mm. Theoretical curve is calculated with Eq. (26).
0.04
5. Conclusions 0.02 0.00 0.01
Fig.3
0.1
1
Λ Λ
10
100
The relative deviation of the transition functions for a Gaussian beam and a circular uniform beam.
4. Experimental Evidence Measurement is performed with a simple optical setup portrayed in Fig. 4 (Kräuter, 1995). The He−Ne laser is expanded to a circular beam with variable diameters. The
By changing the beam diameter, the spatial resolution of the transmission signals varies accordingly, whereby the transmission fluctuation spectrum is obtained. For a monodisperse suspension, the reversal point of the spectrum is in the vicinity of Λ = D x = 1 , which can be used to evaluate the particle size. From Eqs. (26) and (29), and Fig. 5, we can find that the extreme values of ln(ETS) at Λ → 0 and Λ → ∞ are the logarithm of the average transmission and its double respectively, from which the particle concentration may be calculated. As to a polydisperse suspension, the transmission fluc-
CHINA PARTICUOLOGY Vol. 1, No. 6, 2003
246
tuation spectrum is given in Eq. (26), whereby the effects from different particle size fractions superimpose linearly to form the ln(ETS), making it possible to reduce the PSDcalculation to a linear inversion problem solved by known methods. It is proven that, compared to the TFS with variable temporal averaging, the transmission fluctuation spectrometry with variable spatial averaging has a high resolution for particle size analysis and is independent of flow velocity. Due to the simplicity of the optical arrangement, this method is predestined for low-cost in-situ particle size measurements.
Acknowledgement The work presented here is performed under DFG Grant Ri 533/7-1 and Ri 533/7-2, which is gratefully acknowledged.
ts
averaging time interval in statistics
u v x Zml
integration variable particle flow velocity particle diameter thickness of the monolayer
β
monolayer density
δ(
χ(
ω
area of monolayer beam intensity distribution particle volume concentration
A B CV
D e(
beam diameter expectancy
)
e (Tml2 )
ETS F{ }
expectancy of transmission square through a monolayer expectancy of transmission square of the total suspension expectancy of transmission square Fourier transform
H(
transition function of the transmission spectrum
( )
e Ttot2
)
Heav (
)
Heaviside function
i j
layer number number of particle fraction in PSD
J1
1 order Bessel function
ml Nml
monolayer number of monolayers
P PSD K r K r0 K rk
structure parameter of monolayer particle size distribution coordinates coordinate of beam center
st
)
Dirac delta function
)
pathlength dimensionless averaging time beam-to-particle diameter ratio averaging time transition function
ΔZ Θ Λ τ K
Nomenclature
time transmission transmission fluctuation spectrometry transmission of monolayer
t T TFS Tml
space reciprocal
References Breitenstein, M. (2000). Grundlagenuntersuchung zur statistischen Partikelgerößenspektrometrie mittels Auswertung der Transmissionsfluktuation von Licht in diepersen Systemen (pp.33-92). Doctoral thesis, Brandenberg Technical University of Cottbus, Germany. Breitenstein, M., Kräuter, U. & Riebel, U. (1999). The fundamentals of particle size analysis by transmission fluctuation spectrometry, part 1: a theory of temporal transmission fluctuations in dilute suspensions. Part. Part. Syst. Char., 16, 249-256. Gergory, J. (1985). Turbidity fluctuations in flowing suspensions. J. Colloid Interface Sci., 105(2), 357-371. Kräuter, U. (1995). Grundlagen zur in-situ Partikelgrößenanalyse mit Licht und Ultraschall in konzentrierten Partikelsystemen (pp.155-162). Doctoral thesis, Karlsruhe. Matsui, Y., Tambo, N., Ohto, T. & Zaitsu, Y. (1993). Dual wavelength photometric dispersion analysis of coagulation and flocculation. Water Sci. Technol., 27, 153-165. Ohto, T., Zaitsu, Y. & Tambo, N. (1993). Advances in photometric dispersion analyzer with dual wavelength light. Water Sci. Technol., 27, 257-260. Wessely, B., Stinz, M. & Ripperger, S. (1995). In-line-fähiges Meßverfahren zur Überwachen von Flockungsprozessen in einem weiten Konzentrations-bereich. Chem. Ing. Tech., 67, 754-757.
th
coordinate of the k particle center Manuscript received August 19, 2003 and accepted October 27, 2003.