Particle sizing by laser diffraction spectrometry in the anomalous regime Karl A. Kusters, Johan G. Wijers, and Dirk Thoenes The application of laser diffraction spectrometry to determine the size distributions of particles in the anomalous diffraction regime, i.e., particles with a refractive-index ratio close to one, has been examined. From a computer simulation, using the Mie theory and the geometrical optics approximation, it could be concluded that for suspensions with a refractive-index ratio near 1, the corresponding scattering matrix is required for calculation of the correct particle size distribution, even in the case of particles that are much larger than the wavelength of the incident light. In a system with the refractive-index ratio almost at unity, a suspension of ice crystals in a sucrose solution, the ice particles were sized by means of optical microscopyand laser diffraction spectrometry, and the results were compared. Key words: Laser diffraction particle sizing, anomalous diffraction, particle sizing errors associated with the Fraunhofer diffraction assumption, scattering matrix, geometrical optics approximation, optical microscopy, sizing of ice crystals in aqueous solutions. I. Introduction Forward lobe light scattering offers a powerful nonin- trusive technique for the size analysis of particulate suspensions.', Unlike other optical techniques, laser diffraction spectrometry does not require single parti- cles to be measured successively to obtain a size distribution. Instead the interaction between light and the ensemble of all illuminated particles is ana- lyzed. Analysis is rapid and makes on-line measure- ments possible. This technique is particularly useful for studying crystallization, aggregation, and disper- sion phenomena. The exact description of light scattering by spheri- cal isotropic particles is given by the Mie solution of the Maxwell equations for electromagnetic radia- tion. 3 ' 4 In the submicrometer range, i.e., for particles smaller than the wavelength X of light, it is claimed that the Mie theory is required for an adequate description of the forward-scattering patterns. For particles with a diameter x larger than the wave- length , Fraunhofer diffraction is often assumed. However, only scattering by opaque particles or parti- cles with a large real refractive-index ratio m, i.e., the The authors are with the Department of Chemical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Received 9 April 1990. 0003-6935/91/334839-09$05.00/0. e 1991 Optical Society of America. ratio of the refractive index of scattering particles to that of the fluid,results in diffraction patterns that are adequately described by Fraunhofer diffraction theory. For a system that gives rise to anomalous diffraction, i.e., a system that consists of particles that are large with respect to the wavelength of light and with m close to 1, the refraction of light through the particle contributes considerably to forward scat- tering. 5 This work was aimed at quantifying the errors that occur when this refraction is ignored, and the laser diffraction apparatus is applied to systems for which it is not calibrated. 6 To evaluate these errors we simulated the scattering of particles with different relative refractive indices m and several size distribu- tions. From the computed light energy distribution the particle size distribution was calculated using the standard scattering matrix with which the laser diffraction apparatus has been equipped. The com- puted particle size distribution is compared with the original one. We also performed some laser diffraction measure- ments in a suspension of ice crystals in a sucrose solution, a system with a refractive-index ratio near 1. The measured scattering patterns were processed with the standard matrix and with the matrix espe- cially computed for this particulate system with the corresponding refractive-index ratio. The ice particles were also sized using optical microscopy. The results are compared with those obtained by laser diffraction spectrometry. 20 November 1991 / Vol. 30, No. 33 / APPLIED OPTICS 4839
Transcript
Particle sizing by laser diffraction spectrometry inthe anomalous regime
Karl A. Kusters, Johan G. Wijers, and Dirk Thoenes
The application of laser diffraction spectrometry to determine the size distributions of particles in theanomalous diffraction regime, i.e., particles with a refractive-index ratio close to one, has been examined.From a computer simulation, using the Mie theory and the geometrical optics approximation, it could beconcluded that for suspensions with a refractive-index ratio near 1, the corresponding scattering matrix isrequired for calculation of the correct particle size distribution, even in the case of particles that are muchlarger than the wavelength of the incident light. In a system with the refractive-index ratio almost atunity, a suspension of ice crystals in a sucrose solution, the ice particles were sized by means of opticalmicroscopy and laser diffraction spectrometry, and the results were compared.
Key words: Laser diffraction particle sizing, anomalous diffraction, particle sizing errors associatedwith the Fraunhofer diffraction assumption, scattering matrix, geometrical optics approximation, opticalmicroscopy, sizing of ice crystals in aqueous solutions.
I. Introduction
Forward lobe light scattering offers a powerful nonin-trusive technique for the size analysis of particulatesuspensions.', Unlike other optical techniques, laserdiffraction spectrometry does not require single parti-cles to be measured successively to obtain a sizedistribution. Instead the interaction between lightand the ensemble of all illuminated particles is ana-lyzed. Analysis is rapid and makes on-line measure-ments possible. This technique is particularly usefulfor studying crystallization, aggregation, and disper-sion phenomena.
The exact description of light scattering by spheri-cal isotropic particles is given by the Mie solution ofthe Maxwell equations for electromagnetic radia-tion.3'4 In the submicrometer range, i.e., for particlessmaller than the wavelength X of light, it is claimedthat the Mie theory is required for an adequatedescription of the forward-scattering patterns. Forparticles with a diameter x larger than the wave-length , Fraunhofer diffraction is often assumed.However, only scattering by opaque particles or parti-cles with a large real refractive-index ratio m, i.e., the
The authors are with the Department of Chemical Engineering,Eindhoven University of Technology, P.O. Box 513, 5600 MBEindhoven, The Netherlands.
Received 9 April 1990.0003-6935/91/334839-09$05.00/0.e 1991 Optical Society of America.
ratio of the refractive index of scattering particles tothat of the fluid, results in diffraction patterns thatare adequately described by Fraunhofer diffractiontheory. For a system that gives rise to anomalousdiffraction, i.e., a system that consists of particlesthat are large with respect to the wavelength of lightand with m close to 1, the refraction of light throughthe particle contributes considerably to forward scat-tering.5
This work was aimed at quantifying the errors thatoccur when this refraction is ignored, and the laserdiffraction apparatus is applied to systems for whichit is not calibrated.6 To evaluate these errors wesimulated the scattering of particles with differentrelative refractive indices m and several size distribu-tions. From the computed light energy distributionthe particle size distribution was calculated using thestandard scattering matrix with which the laserdiffraction apparatus has been equipped. The com-puted particle size distribution is compared with theoriginal one.
We also performed some laser diffraction measure-ments in a suspension of ice crystals in a sucrosesolution, a system with a refractive-index ratio near1. The measured scattering patterns were processedwith the standard matrix and with the matrix espe-cially computed for this particulate system with thecorresponding refractive-index ratio. The ice particleswere also sized using optical microscopy. The resultsare compared with those obtained by laser diffractionspectrometry.
The measurement of scattering patterns was carriedout by means of a Malvern 2600 particle sizer. Theoptical arrangement is shown schematically in Fig. 1.Particles are allowed to move across a parallel laserbeam. The scattered light is focused by a Fouriertransform lens onto 30 semicircular photosensitiverings placed at predetermined radii in the detectorplane. The annular rings are approximately logspaced,7 and the dimensions of the detector are listedby Dodge8 and Hirleman.9 The particle sizer isequipped with six lenses with corresponding sizeranges as shown in Table I. The size classes used inthe laser diffraction instrument are defined by thedetector dimensions and focal length through therelation':
XXWi)= rr* in~t[) FL1 *1.357, (1)
Table 1. Focal Length of Lens and Corresponding Upper Limits ofSize Classes in Micrometers
Focal Length of Lens (mm)Number ofSize Class 63 100 300 600 800 1000
where X is the wavelength of He-Ne light in vac-uum = 0.6328 pum,
r(i) is the distance of the detector element i fromthe center of the detector,
x(i) is the particle size representing the ith sizeclass, and
FL is the focal length of the Fourier-transformlens.
If successive detector elements are united into pairs,we arrive at 15 size classes for each lens. A 16th sizeclass comprises all particle sizes smaller than thelower limit of the 15th size class.
Each detector ring effectively measures the energyof the scattered light at a particular angle to theincident beam. The particle size distribution is theninferred through a mathematical procedure that at-tempts to match the measured and calculated energydistributions over the detector rings. A brief descrip-tion of the fitting procedure used by the Malvern 2600particle sizer is given in Appendix A. The fittingprocedure is a model-independent iterative tech-nique; i.e., it does not constrain the volume distribu-tion to follow a common analytic expression, such asthe two-parameter Rosin-Rammler or the lognormaldistribution functions. The fitting error, which isbeing minimized, is the logarithm of the sum of thesquared differences between the measured and calcu-
lated light energy patterns. When the measured andcalculated light energy patterns are normalized sothat their peak values are equal to 2047, a good fit isobtained if the fitting error is < 4. The deconvolutionproblem can be written as a set of linear equationsexpressed in matrix notation as
L = A * Q (2)
where L is the light energy vector,Q is the particle size volume distribution vec-
tor, andA is the scattering matrix.
The total light energy measured on a detector ringis developed from the contributions of all the particlesizes present. Only the amount of this contributionvaries with size. This information is stored in thescattering matrix A. The scattering matrix has 15rows (15 detector element pairs) and 16 columns (16size classes). Matrix coefficient a(i,j) represents thelight energy on the adjacent detector elements 2i - 1and 2i, scattered by size class j and normalized on avolume basis:
a(ij) = (ij)Ix(j)3, (3)
parallelmonochromatic light
flow cell transflens
Fig. 1. Schematic of the 2600 Malvern parti
where l(i, j) is the light energy scattered by the sizefocal length classj on the detector elements 2i - 1 and 2i.
> The fitting procedure assumes uniform weightingFL on a volume basis within each size class.
The scattering matrix is dependent on two parame-ters:
(1) The choice of a focusing lens, which fixes theparticle size range and particle size classes.
ar (2) A relative refractive index of the scatteringform particles.
detector
Ole sizer. The Malvern 2600 particle sizer was equipped with
only one standard matrix for each lens, which corre-sponds with one particular refractive-index ratio.Later versions of the 2600 particle sizers have beenequipped with two scattering matrices per lens; onebased on Fraunhofer diffraction and the other onanomalous diffraction for the case of m _ 1.2. Toevaluate the refractive-index ratio for which thestandard matrices have been calculated, scatteringmatrices were computed for m ranging from 1.01 to1.5 and compared with the corresponding standardmatrices. The computation of the scattering matricesis discussed in the next section. The standard matrixproved to be most similar to the matrix computedwith a refractive-index ratio of 1.2.
111. Computation of the Scattering Matrices
For computation of the matrix coefficient a(i, j) of Eq.(3), the adjacent detector elements 2i - 1 and 2i wereconsidered to consist of ten subrings and every sizeclass of four subclasses. If a(i,j)gh represents thelight energy scattered on subring g of the detectorelements 2i - 1 and 2i by a unit volume fraction ofsubclass h of size class j, a(ij) is given by thefollowing summation:
4 10
a(ij) = a(iJ)gh, (4)h=1 g=1
wherea(ij)gh = l(1,J)ghIX ()h'-
The light energy l(i,j)g'h is defined as the lightintensity scattered by the subclass h of size X(i)h ontosubringg times the area of the subring. The scatteredlight intensity was computed with the Mie theory. Weapplied the Mie subroutine as presented by Bohrenand Huffman.10 Because the computing time for theMie theory increases considerably with the size param-eter a(= rr * x/X),"1 the Mie theory was approximatedwith geometrical optics (GO) to reduce the computingtime for large particles (x >> ). For large particlesthe incident beam of light may, in the GO approach,be considered as consisting of separate rays of light. Itis possible to distinguish between the rays hitting theparticle and the rays passing near the particle, giving
10°
K2J (Z 10-'
102
10-
10-
rise to two distinct phenomena: (1) reflection andrefraction, (2) Fraunhofer diffraction. We used theexpressions for the GO approximation as derived byGlantschnig and Chen.2 To demonstrate the mainfeatures of the GO approach, we use the followingapproximate expression for the so-called angularintensity function:
iGO(X, O, m) = 2 * a4 * [J,(z)Iz]2 + a2 * f (m, a), (5)
diffraction refraction
where J1(z) is the first-order Bessel function of thefirst kind and z = a * sin(e).
f ( a) 4 m \4 [m cos('/2) - 1i [m - cos(-a/2)3M - 1) cos('&/2) 2 + 1 - 2m cos(-/2)]V
[ + ] (6)
Figure 2 presents the Fraunhofer diffraction functionas a function of z = a * sin(ea). Figure 3 represents therefraction function f (m, 6) for several values of mand for the scattering angle range corresponding tothe 63-mm lens. Refraction becomes more importantwhen Im - 11 becomes smaller. The GO approachwas used in the computation of the scattering matri-ces only for a > 400, which resulted in matrixcoefficients matching those computed with the Mietheory within 2%.
IV. Simulation of Forward Scattering
The scattering of particles was simulated for differentrelative refractive indices (ranging from 1.01 to 2.0)and several size distributions (Rosin-Rammler, bimo-dal, etc.). The simulated scattering patterns weresubsequently deconvoluted using the standard ma-trix, and the computed particle size distributionswere compared with the original ones.
m1.05
500
f(m, 0)
10
0 2 4 6 8 10 12 14
z
Fig. 2. Fraunhofer intensity distribution.
----- 1.1
----- 1.15
†1.2
-- 1.3
---- 1.5
- 1.90 1 2 3 4 5 6 7 8 9 10 11 12 13
Scattering angle (degrees)
Fig. 3. Refraction function f (m, 0) versus scattering angle.
Note: In the case of a Rosin-Rammler distribution the model parameters X and N are given. VMD and SMD are the volume and Sautermean diameters, respectively, as defined by Eqs. (7) and (8). The last column lists the values of the focal length FL of the lenses for whichscattering patterns have been computed using the corresponding size distribution.
The various types of size distribution used in thesimulation are listed in Table II. The Rosin-Rammlersize distribution function is given by Y(x) = exp[-(x/X)N], where Y(x) is the volume fraction of particleswith a diameter larger than x, X is a representativemean diameter so that 36.8% of the total particlevolume is greater than X, and N determines thewidth of the distribution with N = o for a monodis-perse size distribution. The values of X and N of theRosin-Rammler distributions are given in Table II.The broad and bimodal distributions, which are alsoindicated in Table II, were arbitrarily chosen and areshown in Figs. 4 and 5.
By proper choice of the lens the resolution inparticle size can be optimized for the particularsystem of interest. Larger particles diffract lightthrough smaller angles, and to increase the resolu-tion in particle size a lens with a larger focal length isrequired to unravel the scattering at these smallerangles. As can be seen from Table II, the simulation isprimarily confined to use of the 63- and 300-mm lens.For the other lenses only the errors in the deconvolu-tion of the scattering patterns computed from distri-butions 6 and 7 were examined.
Figures 6 and 7 illustrate the variation in scattered
25.0
20.0
0)EJ
15.0
10.0
5.0
0.0
light with a change in the refractive-index ratio forthe 63- and 300-mm lenses, respectively. In Fig. 6 thescattering patterns for the 63-mm lens, computedwith several matrices for a Rosin-Rammler distribu-tion with X = 10 m and N = 4.0, are shown. Thelight energy patterns differ strongly after the maxi-mum at the eighth detector element. The scatteringof particles with m = 1.20 is most similar to the lightenergy pattern computed with the standard matrix. Arefractive-index ratio smaller than 1.20 gives rise toan increase in the amount of light striking the outerdetector elements, whereas there is a rapid decreasein light energy on the outer segments predicted bycalculations with Fraunhofer diffraction. The differ-ences in the light energy patterns on the outersegments can be explained with GO. The scatteringafter the maximum in the light energy distribution ismainly caused by the refraction of light by the largeparticles. This refraction is larger as the refractive-index ratio moves closer to 1. At smaller values ofIm - 11 a more intense scattering is therefore ob-served at the outer detector elements.
In Fig. 7 the scattering patterns for the 300-mmlens and a Rosin-Rammler size distribution of X =200 pim and N = 2.2 are shown. The scattering
25
20
a)EZj
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
size class no.
distribution distributionno. 7 no. 8
Fig. 4. Distributions 7 and 8 from Table II.
15
10
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
size class no.
_ no. 9 no. 10 =* no. 1
Fig. 5. Distributions 9, 10, and 11 from Table II.
m= 1.05 m= 1.1 standard m= 1.2 Fraunhofermatrix diffraction
Fig. 6. Light energy patterns of the Rosin-Rammler size distribu-tion with X = 10 pim and N = 4, computed for several values of therefractive-index ratio m, the standard matrix, and the Fraunhoferdiffraction. Focal length FL = 63 mm.
patterns, computed with the matrices for the 300-mmlens for several m 1.1, are similar. The scatteringpattern computed with m = 1.04 deviates strongly,again at the large scattering angles. The particle sizeof the corresponding classes increases in proportion[Eq. (1)] to the focal length of the receiving lens.Because diffraction is proportional to X 4 and refrac-tion to x2 [Eq. (4)], the contribution of the refractionterm to forward scattering becomes less importantfor the larger lenses. The matrices for different valuesof m look more similar for larger focal lengthsbecause their major part can be adequately describedwith Fraunhofer diffraction, which is independent ofm. For I m - 11 < 0.05 the refraction is still apprecia-ble, so that the light energy patterns computed for
o 240001
. _Z
1 20000
W 1600a)
*0c 12000
01 800c° 400
01-
0.01 0.1 1
scattering angle (degrees)
15
10
1 10 100
particle size (um)
Fraunhofer Idistribution MM m = 1.05
diffraction no. 6Fig. 8 Volume distributions deconvoluted with the standardmatrix from scattering patterns computed using, respectively,Fraunhofer diffraction and the Mie theory/GO approximation form = 1.05 versus the original size distribution 6. Focal length FL =
100 mm.
these values of m give rise to more scattering on theouter detector rings.
The computed light energy distributions were de-convoluted using the standard matrix. Deviations inthe computed particle size distribution relative to theoriginal one proved to be directly related to differ-ences in the scattering patterns at large angles. Forthe 63- and 100-mm lenses the extra light energy onthe outer rings for 1 < m < 1.2 can only be accountedfor by larger volume fractions in the correspondingsize classes, i.e., the small particles [Eq. (1)]. For m >1.2 the opposite is true. The less scattering at largeangles results in smaller volume fractions for thesmaller particles, as shown in Fig. 8. For the 300-mmlens only the deconvolution of scattering patternscomputed with m < 1.05 results in biased distribu-tions. Figure 9 shows the overestimation of thevolume fraction of the fine particles when the scatter-ing pattern computed with m = 1.04 is deconvolutedwith the standard matrix.
Starting from a Rosin-Rammler distribution the
20
0 10
10
0
m= 1.05 m= 1.1 standard m= 1.2 Fraunhofermatrix diffraction
Fig. 7. Light energy patterns of the Rosin-Rammler size distribu-tion with X = 200 pum and N = 2.2, computed for several values ofthe refractive-index ratio m, the standard matrix, and the Fraunho-fer diffraction. Focal length FL = 300 mm.
5.0 10.0 50.0 100.0 500.0particle size (Oum)
= distribution M m = 1.04no. 10
Fig. 9 Volume size distribution deconvoluted with the standardmatrix from the scattering pattern computed with the Mie the-ory/GO approximation for m = 1.05 versus the original sizedistribution 10. Focal length FL = 300 mm.
fitting procedure does not return estimates for theRosin-Rammler parameters X and N, because it doesnot a priori assume a functional form to its solution.Instead the volume mean diameter (VMD) and theSauter mean diameter (SMD) and a relative spread,called SPAN, are presented:
n(x) * x=VMD = D43 - =nx *x - q(x) * x
0o
(7)
40
20
0
-20
-40
ln(x) *x 3 1 (8)SMD = D32 = :n(x) * x2 q(x)lx
SPAN = Dvso% - Dvlo% (9)Dv~o%
where n(x) is the number fraction of particles of sizeX,
q(x) is the volume fraction of particles of sizex, and
Dvf% is the particle size for which the f volumepercentage of the particles is smaller.
The VMD and SMD are related to the Rosin-Rammler parameters through the following equa-tions13:
VMD = X* F(1 + 1/N), (10)
SMD = X/(1 - 1/N), (11)
where F is the gamma function.The deviations in the deconvoluted size distribu-
tion are also expressed by the values of the VMD andSMD. If the same matrix is used in the deconvolutionas has been used to compute the scattering, the fittingprocedure returns the values of the VMD and SMDwithin 5%. When the scatter patterns are deconvo-
-60 0.01 0.1
m - 1Fig. 11. Relative errors in VMD and SMD after deconvolution ofthe scattering patterns computed for size distribution 6 with FL =100 mm and for various values of the refractive-index ratio m.Deconvolution is done with the scattering matrix based on Fraun-hofer diffraction.
luted with the standard matrix, the errors can beconsiderably larger. For example, Fig. 10 shows rela-tive errors in VMD and SMD of 15 and 20%, respec-tively, for small values of m - 1 for the case of FL =100 mm and distribution 6. The SMD is most affectedby the inappropriate scattering matrix choice, be-cause the value of the SMD is greatly determined bythe tail of the size distribution in the small particles.In Fig. 11 the errors involved with deconvolutionwith the matrix based on Fraunhofer diffraction arepresented, and they are even more severe than whendeconvoluting with the standard matrix.
Figures 10 and 11 concern distribution 6. Figure 12shows the relative errors in the VMD and SMD forvarious distributions along with the fitting error forthe case of m = 1.04 and FL = 300 mm. A smallerfitting error does not imply that the resemblancebetween the calculated and original size distributionis better. Reduction of the fitting error depresses therandom errors but not the systematic errors that areintroduced into the calculated size distribution due to
40
30
00)
20
10
0
-10 A Oi
. 0 AAA
-200.01 0.1 1
m - 1
Fig. 10. Relative errors in VMD and SMD after deconvofution ofthe scattering patterns computed for size distribution 6 with FL =100 mm and for various values of the refractive-index ratio m.Deconvolution is done with the standard matrix.
C
01)
10
0
-10
-20
-30
-40
-50
-60
00 0
0A A
*~~~ vA
AA:As AA A A A
1 2 3 4 5 6 7 8 9 10 11
distribution no.
fittingerror
16
14
1200)8 0
6 i-
4
2
0
0 VMD A SMD
Fig. 12. Relative errors in VMD and SMD and the fitting errorafter the deconvolution of scattering patterns computed for m =1.04 and for the various distributions with FL = 300 mm.Deconvolution is done with the standard matrix.
the inappropriate matrix choice in the deconvolution.Figure 12 also shows that the errors seem to beslightly larger when starting from the broad andbimodal distributions.
In Table III the values of the refractive-index ratiosbelow which the error in the SMD becomes largerthan 15% are presented as a function of the focallength of the lenses. The lower limit of m where thestandard matrix is still applicable in the deconvolu-tion with errors in the SMD smaller than 15%decreases with the focal length of the lens down tom = 1.04 for the 800- and 1000-mm lenses.
It can be concluded that for a focal length of 63 and100 mm, application of the standard matrix to sys-tems with a refractive-index ratio ranging from 1.1 to- 1.3 does not lead to significant errors in the shape
of the deconvoluted volume distribution. The relativeerror in the VMD and SMD to be expected lies within10 and 15%, respectively. For the larger size ranges(larger focal lengths) the application can be extendedto systems with m larger than 1.3. The lower limit ofapplication is dependent on the focal length of theFourier-transform lens and is shown in Table III.Computation of the particle size distribution for msmaller than this lower limit is to be done preferablywith the scatter matrix, which is especially calculatedfor the specific system.
For m < 1, the lower limit of application of thestandard matrix is given by the reciprocal of thevalues listed in Table III.
In the case of strong absorption (absorption in-dex 2 0.01) the refraction is zero. The forward-scattered intensity equals Fraunhofer diffraction. Forthe analysis of light scattering by strongly absorbingparticles, Fraunhofer diffraction will yield the bestsolution to the inversion problem. Thus, for thelarger lenses (> 100 mm) the standard matrix can beused. In the case of weak absorption (absorptionindex << 0.01), the fraction of refracted light that isabsorbed is a function of a., i.e., the size parameter.This means that for the 63- and 100-mm lenses thecorresponding matrices have to be computed for allcomplex refractive indices. For the larger lenses only,in the case of a refractive-index ratio smaller than thevalues indicated in Table III, the corresponding ma-trix has to be calculated.
V. Experiments
For the determination of the size of ice crystalsproduced in a crystallization unit for the freezeconcentration of sucrose solutions, two methods wereused: optical microscopy and laser diffraction spec-trometry.
Table ill. Values of the Refractive-Index Ratio of the Particulate SystemBelow which Deconvolution of the Scattering Patterns with the Standard
Matrix may Result in Relative Errors in SMD Estimates that areLarger than 15%
Fig. 13. Microscopic photograph of ice crystals.
A. Optical Microscopy
Pictures of ice crystals were taken using the equip-ment designed by Swenne."4 A small slurry streamwas circulated through a microscope cell. In the cell aportion of the circulated slurry can be immobilizedand photographed. Figure 13 shows that some agglom-eration occurred during the crystallization process.From the microscopic photographs size distributionsof the agglomerates were constructed.
B. Laser Diffraction Spectrometry
The volume percentage of ice in the suspension was5%. To avoid multiple-scattering effects the ice
suspension had to be diluted with mother liquor. Tolower the required dilution to a factor of 2, a continu-ous flow cell with an optical path length of only 3 mmwas constructed. In this way the melting of icebecause of dilution could be limited.
The two windows in the continuous-flow cell arepositioned as shown in Fig. 14 with a small angle p of0.4°. In this way the third reflection is focused justbelow the unreflected light beam because of inversionof the lens and does not fall onto the semicirculardetector.
Warren'5 presents a compilation of the complexrefractive indices of hexagonal ice from the UV to the
2nd reflection
3rd reflection
1st reflection
unreflected laser beam
Fig. 14. Relative position of windows in the continuous-flow cell.
microwave, which is valid for temperatures varyingfrom -600 to 0C. For X = 0.6328 pum the real part ofthe refractive index is 1.3084, and the imaginary partis 1.1 x 10-5 0. The refractive index of the sugarsolution was 1.37, giving a refractive-index ratio of0.955.
Because of the large size of the particles (100-500pum) measurements with the laser diffraction appara-tus were performed with an 800-mm lens. As dis-cussed in Section IV, serious errors in the analysisusing the standard matrix instead of the appropriateone are not expected for I m - 11 2 0.04 when usingthis large lens. Nonetheless the light scattering wasprocessed with both matrices to determine whetherthis statement is correct.
C. Results
Figure 15 shows an example of the volume distribu-tions computed with the two different matrices andthe volume distribution of the agglomerates, as deter-mined by optical microscopy. Note that the verticaland horizontal axes are presented in logarithmicform. Application of the standard matrix instead ofthe appropriate scattering matrix produces indeedonly slightly biased results, as was expected consider-ing the large lens that had to be used. To a firstapproximation the size distributions obtained withlaser diffraction and microscopy are in agreement.The differences in the volume fractions of the smallersizes originate from the following:
(1) The uncertainties in the agglomerate size distri-bution measured by microscopy. The microscopiccounting of agglomerates of the various sizes isinherently difficult and imprecise. Because m wasnear 1, the contrast between ice crystals and themounting medium is small, and particles smallerthan 80 m are hard to distinguish. Furthermorelarger particles are always in the region of focus, andthe small particles are not, resulting in an underesti-mation of the number of small particles present.
(2) The nonspherical shape of the agglomerates.The agglomerates show the most resemblance to
100
100)EJ
0.110 100 1000
particle size (wn)
_ OpticalMicroscopy
LDS E LDSm = 0955 standard
Fig. 15. Volume distribution of ice crystals and correspondingVMD's and SMD's determined with laser diffraction spectrometryand optical microscopy.
oblate spheroids. The aspect ratio of the agglomeratesvaries from 1.5 to 2.0. Because the ice crystals rise tothe top of the microscopic cell and tend to direct theirlargest cross section toward the microscopic eye, thesize distribution measured with optical microscopycorresponds merely to the largest cross sections of theagglomerates. The forward scattering of oblate sphe-roids"5 can be approximated by the average scatteringof an ensemble of spheres (i.e., circles), one for eachprojected cross section corresponding to each orienta-tion of the spheroid. Thus the size distribution mea-sured with laser diffraction corresponds to the pro-jected cross-sectional area averaged over allorientations of the agglomerates. For a fair compari-son with the laser diffraction result, the size distribu-tion obtained with optical microscopy should be spreadout to smaller sizes.
(3) The optical inhomogeneity of the agglomer-ates. The agglomerates are not optically homoge-neous, as assumed in the scattering theories withwhich the scattering matrices are computed. To takeinto account the inhomogeneity of the agglomeratesLatimer 7 suggested use of the average value of therefractive index of the component particles and thespaces as an estimate of the refractive index of theagglomerates. With the number of particles in theagglomerates varying from 2 to 10, the porosity of theagglomerates is 0.3, resulting in a refractive-indexratio for the agglomerates of 0.97. This implies thatagglomerates give rise to more scattering on the outerdetector rings than isotropic ice particles of the sameoverall size. Again this results in higher fractions ofsmaller particles in the analysis by laser diffractionthan would be obtained in the microscopic analysis.
Considering the biasing errors mentioned in thissection and the small amount of particles counted(- 200), which makes the conversion from number tovolume distribution inaccurate (a relative error involume fractions of 7%), the volume distributionobtained with optical microscopy agrees reasonablywell with the one obtained with laser diffractionspectrometry. The determined VMD values weresimilar, and the SMD values differed because of thetail in the volume distribution at the small particles,which were measured by laser diffraction spectrome-try but not observed by optical microscopy.
VI. Conclusions
Laser diffraction particle size analyzers that assume afixed value of the refractive-index ratio are limited intheir application to systems with other refractive-index ratios. The Malvern 2600 laser diffractioninstrument referred to in Section II uses a scatteringmatrix corresponding to a refractive-index ratio of- 1.2. Serious errors can occur when the apparatus is
used to size systems with refractive-index ratios near1. Comparison with optical microscopy showed thatlaser diffraction can be a useful tool for fast on-linemeasurements in ice crystallization processes. Be-cause the refractive-index ratio of these systems isnear 1, it is advisable to process the measured scatter-
ing patterns with the corresponding scattering ma-trix instead of the standard matrix.
Appendix A: Fitting Procedure as Used in the Malvern2600 Particle Sizer
The fitting procedure is a least-squares analysis thatcalculates the size distribution that gives the closestfitting scattering pattern. The fitting error is thelogarithm of the least-squares error between themeasured and calculated scattered-light energy data.In the fitting procedure the measured and the calcu-lated light energy patterns are continually normal-ized with their peak values set to 2047. The volumesize distribution is not normalized with its cumula-tive sum equal to 1 until it is presented as the finalresult of the optimization routine.
The fitting procedure makes a first guess of the sizedistributions using the following equation:
q(i) = (i)/a(i, i), (Al)
where q (i) is the volume fraction of size class i,1 (i) is the light energy on detector rings 2i and
2i - 1, anda(i, i) is the diagonal scattering matrix coeffi-
cient.
The volume fraction of the 16th size class is extrapo-lated from the volume fractions of the 14th and 15thsize class:
q(16) = 1.5 * [2 * q(15) - q(14)]. (A2)
The purpose of the initial estimate is to give theoptimization routine a starting point close to theerror minimum. Equation (Al) implies that the lightenergy I (i) scattered on the detector elements 2i and2i - 1 is solely due to scattering by size class i. This isincorrect, and before entering the final fitting routinethe estimate is optimized by reducing the distribu-tion's width with the following iteration:
q (i)kI= q(i)k * q(i), (A3)
where k is the iteration number.This iteration is continued until the fitting error
obtains a minimum. Next the volume distributionenters the final iteration:
q(i) = q(i)' * 2 )m + 1)m] 2 (A4)
where l()m is the measured light energy on detectorrings 2i and 2i - 1 and
l(i), is the calculated light energy on detectorrings 2i and 2i - 1.
The volume fraction of the 16th size class ismultiplied with the following correction factor:
q(16)k+l = q(16) * 2 * 1(16) (A5)
where l(16)m = 2 * l(15)m - l(14)m and l(16),= 2 * l(15) - (14),.
This iteration procedure requires that the lightenergy maxima in the matrix fall close to the diago-nal. This is ensured by definition of the size classes byEq. (1). The iteration is repeated 8 times after whichthe fitting error is calculated and compared with theprevious one. If the fitting error is smaller than theprevious one, these iterations start again. Otherwisethe volume distribution that entered the last eightiteration steps will be presented as the solution to thelight-scattering problem. To minimize the computingtime, the iteration is also truncated if the number ofiteration steps exceeds 64 and the last calculatedvolume distribution is presented as the solution.
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