We demonstrate a new approach for determining particle-size distribution in concentrated suspensions from spectral measurement of isotropic scattering coefficients by use of steady-state, continuous-wave photon-migration techniques. Successful recovery of particle-size distribution for TiO_2 suspensions in the form of log-normal functions is achieved through a regularized inverse algorithm, into which a synthesized scheme of Marquardt and Tikhonov regularizations has been incorporated. Our results for dense TiO_2 suspensions with three different particle concentrations are in excellent agreement with the size distribution as measured with x-ray sedimentation

Jiang, Huabei

Marquez, Guillermo

Wang, Lihong V.

Caltech Authors - Main

394 OPTICS LETTERS / Vol. 23, No. 5 / March 1, 1998Particle sizing in concentrated suspensions
by use of steady-state,
continuous-wave photon-migration techniques
Huabei Jiang
Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634–1911
Guillermo Marquez and Lihong V. Wang
Biomedical Engineering Program, Texas A&M University, College Station, Texas 77843–3120Received October 14, 1997
We demonstrate a new approach for determining particle-size distribution in concentrated suspensions from
spectral measurement of isotropic scattering coefficients by use of steady-state, continuous-wave photon-
migration techniques. Successful recovery of particle-size distribution for TiO2 suspensions in the form of
log-normal functions is achieved through a regularized inverse algorithm, into which a synthesized scheme of
Marquardt and Tikhonov regularizations has been incorporated. Our results for dense TiO2 suspensions with
three different particle concentrations are in excellent agreement with the size distribution as measured with
x-ray sedimentation.  1998 Optical Society of America
OCIS codes: 350.4990, 290.4210, 170.5280.Particle sizing in nondilute suspensions is important
in areas such as environmental remote sensing and on-
line control of chemical- and pharmaceutical-based in-
dustrial processes. Among the existing particle-sizing
techniques, optical methods such as dynamic light
scattering and optical-turbidity or extinction-spectrum
measurements have been particularly attractive be-
cause of their portability and potential for rapid mea-
surement.1,2 Although researchers have used these
optical techniques to obtain size information about
scattering particles noninvasively, the use of these
techniques is restricted to diluted suspensions and of-
ten requires extensive calibrations with the suspen-
sions that they are intended to monitor.
To overcome the problems associated with the
present optical techniques for particle sizing, Jiang
et al. recently proposed a new optical method based
on frequency-domain measurements of photon migra-
tion in scattering suspensions.3 Since this technique
depends on multiply scattered light, it is particularly
suitable for nondilute suspensions and has great
potential for on-line process monitoring. In addition,
the technique does not require calibrations with the
suspensions.
However, the measurements that we conducted were
based on a frequency-domain Ti:sapphire laser sys-
tem, which does not provide a light beam with a wide
spectrum range. Further, the procedures for optical
data collection were relatively slow, since multiwave-
length measurements were needed. In this Letter we
experimentally demonstrate particle sizing in concen-
trated TiO2 suspensions for what we believe to be
the f irst time, using cw-based photon-migration tech-
niques. This method allows faster and simpler opti-
cal measurements than frequency-domain techniques,
with the same accuracy of measurements of optical
properties in a scattering medium.4 – 6 In addition, the0146-9592/98/050394-03$10.00/0cost of a cw-based system is lower than that of its
frequency-domain counterpart. Using a regularized
inverse algorithm, we reconstructed particle-size dis-
tributions of TiO2 suspensions with three different con-
centrations. The results are in excellent agreement
with the size distribution as measured with x-ray sedi-
mentation.
Particle-size distributions are obtained with a
regularized inverse algorithm under Mie scattering
theory, as reported in Ref. 3. Our approach casts par-
ticle sizing as an optimization problem in which the op-
timization parameters are coefficients in a probability
function, such as an a priori log-normal or Gaussian
distribution function. The details of our inverse
algorithms can be found in Ref. 3; we present a brief
overview here for context.
It is known that particle-size distribution in a
multiple-scattering medium is correlated with the
isotropic scattering coeff icient of the medium by the
following relationship7,8:
s1 2 gdmssld ­ ms0sld
­
Z ‘
0
3Qscatsx, n, ld f1 2 gsx, n, ldg
2x
3 ffsxddx , (1)
where s1 2 gdms is the isotropic scattering coeff icient,
g is the mean cosine of the scattering angle from a
single particle, Qscat is the scattering efficiency, n is the
refractive index of the medium, and l is the wavelength
of light; f sxd (x is the diameter of the particles) is the
particle-volume size distribution, and f is the total
volume fraction of the particles. Both g and Qscat can
be computed with Mie theory.8
For concentrated suspensions, we usually have
a priori knowledge about the number of modes and 1998 Optical Society of America
March 1, 1998 / Vol. 23, No. 5 / OPTICS LETTERS 395their size distributions. Generally, particulate sus-
pensions can be characterized as Gaussian or
log-normal distributions. We use log-normal dis-
tributions for TiO2 suspension in this study9: f sxd ­
expf2sln x 2 ad2y2b2gy
p
2p bx, where parameters
a and b denote the mean and the variance of the
distribution, respectively. Using our regularized op-
timization methods, we obtain the following system of
equations, which leads to an inverse problem solution3:
sT T T 1 aI dDz ­ T T fs ms0do 2 s ms 0dcg , (2)
where the Jacobian matrix T represents the sensitivity
of isotropic scattering coeff icients measured at multi-
ple wavelengths, the elements of which are numerically
computed with Eq. (1). I is the identity matrix, and
a can be a scalar or a diagonal matrix. The vectors
sms 0do and sms0dc contain the observed and the computed
values of isotropic scattering coefficients at the wave-
lengths of interest, respectively. Dz is the vector that
updates parameters a, b, and c: Dz ­ sda, db,dcdT
(we choose f as the third parameter). Thus the ba-
sic idea in determining a, b, and c is to measure the
isotropic scattering coeff icient of the suspensions at
multiple wavelengths. Then the particle-sizing task is
to perform the estimates [which are updated and im-
proved through solution of Eq. (2)] of parameters a, b,
and c that are required for the measured scattering
coeff icients to be sustained. Inasmuch as the matrix
T T T is known to be ill conditioned,2,3 a way to regu-
larize or stabilize the decomposition of T T T is needed.
We used a hybrid technique that synthesizes standard
Marquardt and Tikhonov regularization schemes.3
The experimental system, shown schematically in
Fig. 1, was based on oblique-incidence ref lectrome-
try.4 – 6 White light (Oriel 75-W xenon arc lamp) was
delivered, and the diffuse ref lectance was collected
with a fiber-optic probe made from black Delrin and
600-mm-diameter, low-loss optical f ibers. The source
fiber was oriented at a 45– angle of incidence, and
the nine collection f ibers, arranged in a linear ar-
ray, collected the diffuse ref lectance. To correct for
slight variations in collection eff iciency from one de-
tection fiber to the next, we calculated a correction
factor for each fiber based on a calibration procedure
using standard phantoms. The detection system was
composed of an imaging spectrograph (Oriel Multispec
257), a CCD camera (Princeton Instruments 1530P),
and a personal computer that automatically recorded
the spectrum of the collected light through the wave-
length range 400–800 nm. The spectral resolution of
the measurements was 0.5 nm.
The fiber probe was placed on the surface of each
concentrated suspension. We attached an exposed
x-ray f ilm to the top of each suspension to approxi-
mate a matched boundary condition for diffusion the-
ory. Four measurements were performed with the
probe oriented at 0–, 30–, 60–, and 90– with respect
to an arbitrary reference orientation. The isotropic
scattering spectra were evaluated for each diffuse re-
f lectance measurement, and the results were averaged.
The concentrated suspensions were mixtures of TiO2,
India ink, and polyacrylamide gel,10 which produced
TiO2 concentrations of 0.88, 1.17, and 1.46 wt %, re-spectively. Each mixture was poured into a plastic
container that was 8 cm in diameter by 6 cm in height
before gelling. In Fig. 2 we present the isotropic scat-
tering spectra measured for the TiO2 suspensions with
three different concentrations.
Using our reconstruction algorithm, we recovered
the particle-size distributions for all three TiO2 sus-
pensions, as shown in Fig. 3. The curves with the
symbols denote the values reconstructed from the
photon-migration measurements, and the short-
dashed curve represents the size distribution obtained
from the sedimentation measurement for comparison.
As can be seen, excellent agreement between our
results and those for the sedimentation was achieved.
Since we used the concentrated suspensions (whereas
a dilute solution was used in the sedimentation
measurement) and a log-normal distribution in our
reconstruction, errors in agreement between the
results from photon migration and sedimentation can
be expected. To reconstruct a typical particle-size
distribution, as shown in Fig. 3, we needed only a
few seconds on a SunSparc 5 workstation. We found
that the inversion results were not very sensitive to
the choice of the initial estimates of the reconstructed
parameters (a, b, and c), which were as much as
100% greater than the true parameters, depending of
course on the optical-property measurements. Since
measured data at 800 wavelengths between 400 and
800 nm were input into our reconstruction algorithm,
the inverse solution procedure was quite stable,
whereas the solution procedure was unstable when we
used frequency-domain photon-migration techniques
(measurements were made at only 15 wavelengths).3
Fig. 1. Schematic of the experimental apparatus. White
light was coupled to the oblique-incidence optical f iber
probe. A source fiber delivered light to the gel phantoms
at an angle of 45–, and the diffuse ref lectance was collected
by nine collection fibers. PC, personal computer.
396 OPTICS LETTERS / Vol. 23, No. 5 / March 1, 1998Fig. 2. Isotropic scattering spectra determined by the
method of oblique-incidence ref lectometry.
Fig. 3. Particle-size distribution as a function of diame-
ter for the TiO2 samples as inverted from the diffuse
ref lectance measurements and as measured from the
sedimentation.
It is also interesting to note that, although the shape of
particles in TiO2 suspensions is not perfectly spheri-
cal, the spherical particulate-based Mie theory still
provides accurate results from our inverse algorithm.This accuracy could be due to the use of the hybrid
regularization in our iterative optimization procedure,
which provides a compensation mechanism for the
geometric discrepancy that occurred in the theory.
In summary, we have used cw photon-migration
measurements of isotropic scattering for particle sizing
in concentrated suspensions. Our study shows that
fast, simple cw photon-migration measurements in a
wide wavelength range, coupled with an eff icient in-
verse algorithm, can provide the particle-size distri-
butions of concentrated suspensions in a matter of
seconds. This result indicates the potential of the
method described above for on-line process monitoring.
We acknowledge financial support from the
Greenville Hospital System–Clemson University
Biomedical Cooperative [for H. Jiang (HJ)], the Clem-
son University Research Grant Committee (HJ), the
Whitaker Foundation [L. V. Wang (LVW)], and the
National Institutes of Health (grants R29CA68562 and
R01CA71980 to LVW).
References
1. T. Allen, Particle Size Measurement (Chapman & Hall,
New York, 1990).
2. J. Wang and F. R. Hallet, Appl. Opt. 35, 193 (1996).
3. H. Jiang, J. Kao, and E. Sevick-Muraca, Appl. Opt. 36,
3310 (1997).
4. L.-H. Wang and S. L. Jacques, Appl. Opt. 34, 2362
(1995).
5. S.-P. Lin, L.-H. Wang, S. L. Jacques, and F. K. Tittel,
Appl. Opt. 36, 136 (1997).
6. G. Marquez, L.-H. Wang, S.-P. Lin, J. A. Schwartz, and
S. L. Thomsen, Appl. Opt. 37, 798 (1998).
7. H. Van de Hulst, Light Scattering by Small Particles
(Dover, New York, 1983).
8. G. Bohren and D. Hoffman, Absorption and Scattering
of Light by Small Particles (Wiley, New York, 1983).
9. E. L. Crow and K. Shimizu, Lognormal Distributions
(Marcell Dekker, New York, 1988).
10. M. Lee, ‘‘New gel phantoms simulating optical proper-
ties of biological tissue,’’ M. S. thesis (Texas A&N Uni-
versity, College Station, Tex., 1997) .


Particle sizing in concentrated suspensions by use of steady-state, continuous-wave photon-migration techniques

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Particle sizing in concentrated suspensions by use of steady-state, continuous-wave photon-migration techniques

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