Sizing of colloidal particles withlight scattering: corrections for beginningmultiple scattering

Heimo Schnablegger and Otto Glatter

Static light scattering is widely used for sizing of particles with radii in the range of 50 nm up to severalmicrometers. These experiments usually require very low particle concentrations 1,10242 for preventionof multiple scattering. As a consequence, nonabsorbing samples that are suited for light-scatteringinvestigations must be transparent so that the transmittance of the incident light is typically above95%. Investigations of less translucent samples require corrective terms for the beginning of multiplescattering to retrieve the particle-size distribution successfully. We applied a computationally conve-nient first-order approximation for the multiple-scattering problem that has Hartel’s approach in its firststeps. When incorporated into our inversion technique, this approximation functions well for sampleswith transmittances above 30%. We present examples of applications to experimental data.Key words: Optical particle sizing, static light scattering, multiple scattering.

1. Introduction

The determination of particle sizes by the use of staticlight scattering usually requires samples of low con-centrations for fulfillment of the necessary conditionsfor single-scattering experiments. With these condi-tions fulfilled, the calculation of the size distributioncan be accomplished with the inversion of any single-scattering theory. In the case of homogeneous spheri-cal scattering objects the Lorenz–Mie theory is usu-ally applied.1–5 It assumes that the incident light inthe sample is scattered only once or, in practice, thatthe contributions of the multiply scattered light arenegligible.Previously we have demonstrated that the size

distribution6 and the refractive index7 of polydispersespherical particles can be calculated by the use of anumerical inversion of the single-scattering Lorenz–Mie theory. This functions well for samples with atransmittance of greater than 95%, which is equiva-lent to an optical thickness of smaller than 0.05.

When this work was performed the authors were with theInstitute of Physical Chemistry, University of Graz, Heinrich-strasse 28,A-8010 Graz, Austria. H. Schnablegger is now with theDepartment of Chemical Engineering, Massachusetts Institute ofTechnology, Room 66-469, Cambridge, Massachusetts 02139.Received 3 January 1994; revised manuscript received 3 January

1995.0003-6935@95@183489-13$06.00@0.

r 1995 Optical Society of America.

Thus the freedom to select an arbitrary concentrationrange is limited to highly diluted samples. Butoccasionally it is not possible or advisable to adjustthe concentration of the system under investigation,because the sample can undergo phase transitions.Also, in production-control applications the concentra-tions cannot be changed freely. Consequently, onealternative, which consists of extending the analysistechnique, is very desirable.In moderately or highly concentrated samples,

multiple scattering and interparticle interferencescome into play. We neglect the interference effects inthis paper because of the following consideration.When the concentration is increased, multiple-scattering contributions become effective before inter-particle interferences, because of the high scatteringefficiency of the particles that are typically investi-gated with light scattering. It is only in the case ofparticles with extremely weak optical contrast 1i.e.,relative refractive index very close to 1.02 and in thecase of highly charged systems that interparticleinterferences can become significant before multiple-scattering effects. This stands in severe contrast tothe situation in small-angle x-ray experiments, inwhich particle radii are usually much smaller than,say, 50 nm. Thus the scattering efficiency 1which iswell known to be approximately proportional to thesixth power of the particle radius2 is so small thatmultiple-scattering effects cannot be observed, evenwhen interparticle interferences are substantiallyaffecting the scattering pattern.

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3489

Basically, two different approaches to overcome theproblem of multiple scattering exist:

112 Amechanical possibility is the construction of avery thin sample cell to minimize the optical pathlength. When the absorption can be neglected, thetransmittance of the sample is a direct measure of thedegree of multiple scattering. The smaller the trans-mittance, the more multiple scattering is produced.Because the transmittance is exponentially depen-dent on the sample thickness, a reduction of thedimensions of the sample cell is certainly one efficientway to minimize the multiple-scattering contribu-tions.

122 Another possibility is a theoretical account formultiple scattering. One possible account is to com-pensate multiple scattering in the way that is pre-sented in this paper. The effect of multiple scatter-ing can be explained by a convolution process, whichis connected with a loss of structural information.This is the reason for the fact that a mathematicalcompensation can function only above a limitingtransmittance below which the structural informa-tion about the size distribution vanishes.To solve the inverse-scattering problem, the solu-

tion of the direct scattering problem has to be knownfirst. Once the direct problem has been solved, theinversion can be accomplished by means of the indi-rect transformation technique6,8 1see alsoAppendixA2.Because exact computations of the direct multiple-scattering problem are too time consuming, we haveapplied an approximation that has its foundations inHartel’s theory.9 Because this theory was designedfor a one-dimensional scattering atmosphere, we addedsome extensions to account for our particular experi-mental geometry. The resulting algorithm is a roughapproximation, but we think that we have found areasonable balance between range of applicabilityand computational expenses.The goal of this paper is to investigate improve-

ments that render possible the application of a simpli-fied multiple-scattering theory to the solution of theinverse-scattering problem. We have to emphasizethat we have been interested in finding a practicalrather than a perfect solution to the problem, andtherefore limitations for a successful inversion willexist. We want to illustrate these limitations interms ofmaximum resolution andminimum transmit-tance 1ormaximum optical thickness2with some exem-plary applications to experimental data. The com-puter program MMS 1FORTRAN source2 that resultedfrom these investigations is available from the au-thors.We confine our investigations to randomly polar-

ized incident light and to spherical and isotropicscattering obstacles. In principle the absorption1imaginary part of the relative refractive index2 of theparticles is taken into account, but the effect on thescattering function is almost negligible for manysamples. Because we calculate the transmittance asthe transmitted intensity of the solution divided by

3490 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

the transmitted intensity of the solvent, the absorp-tion of the solvent cancels and can be neglected.

2. Theory

Before we outline the basic concepts of our inversiontechnique, let us explain the simplified multiple-scattering theory as it is used in this paper.Solving the direct multiple-scattering problem

means solving the equation of radiative transfer.10For a scattering atmosphere this equation is anintegrodifferential equation,

21

µ1s ·=2F1r, s2 5 F1r, s2 2

1

4p e p1s, s82F1r, s82dvs8,

112

where F1r, s2 is the intensity at a point r in thedirection s, µ is the extinction coefficient, and p1s, s82 isthe phase function or redistribution function thatdefines the scattering in the direction s that is causedby a light beam coming from the direction s8. Theintegral on the right-hand side of Eq. 112 is theso-called source function, and it sums up scatteredintensities that are incident from all solid angles vs8.In practice, Eq. 112 can be solved for all k-timesscattered intensitiesFk1k 5 1, 2, . . . , `2, which subse-quently can be summed up to obtain F.As described by van de Hulst,11 several different

ways to solve the equation of transfer exist. Unfortu-nately all these exact solution methods are computa-tionally very expensive. Hence we decided to use afirst-order approximation whose first steps are basedon the concepts of Hartel.9

A. Hartel’s Theory

Although the equation of radiative transfer 3Eq. 1124was unknown in 1940, Hartel’s theory can directly bederived from that equation. Let us assume a one-dimensional semi-infinite scattering atmosphere start-ing at the origin 1z 5 02 of a Cartesian coordinatesystem. Equation 112 can thus be written for thek-times scattered intensity component Fk,

2dFk1z, u, f2

µdz5 Fk1z, u, f2 2

1

4p

3 eu50

p ef50

2p

I11u,f, u8,f82Fk211z, u8,f82

3 sin u8du8df8, 122

where u and f denote the direction of the emittedscattered radiation in spherical coordinates relativeto the direction of the incident light beam 1z axis2.In Eq. 122 the redistribution function is set equal tothe single-scattering form factor I11u, f, u8, f82, whichis normalized to unit solid angle and multiplied withthe single-scattering albedo. In the case of a conser-vative 1i.e., lossless2 atmosphere the albedo is equal toone. Hartel’s main assumption is the factorization ofthe intensity Fk1z, u, f2 in normalized angular

distribution Ik1u, f2 and distance-of-propagation-depen-dent intensityQk1z2,

Fk1z, u, f2 5 Qk1z2Ik1u, f2. 132

This assumption is applicable only for scatteringatmospheres with negligible extinction. Because thisis usually not at all true, we have to make correctionsafterward. Nevertheless, when Eq. 132 is applied toEq. 122, it reduces to a system of k ordinary lineardifferential equations,

2dQk1z2

µdz5 Qk1z2 2 Qk211z2, 142

because Ik1u, f2 cancels on both sides. Note that wehave applied the relation

Ik1u, f2 51

4p eu50

p ef50

2p

I11u, f, u8, f82Ik211u8, f82

3 sin u8du8df8, 152

which corresponds to one of the two half-steps in themethod of successive orders described by van de Hulst1p. 48 of Ref. 112. This half-step is simply a convolu-tion of the form factor over all solid angles. With Eq.152 the form factors Ik1u, f2 must be successivelycalculated starting with the concept that the formfactor for a parallel incident light beam can bedescribed with a delta function, I01u, f2 5 d1u, f2. Inthe case of azimuth-independent scattering, 1i.e., un-polarized light2 the single-scattering form factor I11u2can be expanded in a series of Legendre polynomialsPn1cos u2,

I11u2 5 on50

`

AnPn1cos u2. 162

For spherical particles the expansion coefficients Ancan be calculated semianalytically.12,13 With thisexpansion it was shown by Hartel that Eq. 152 can besimplified so that successive computations becomeunnecessary. With the abbreviation Bn 5

An@3A012n 1 124 and after normalization to units ofsteradians, the result is

Ik1u2 51

4p on50

`

12n 1 12BnkPn1cos u2. 172

The second half-step in the method of successiveorders is the integration over the optical path length,which corresponds to integration of Eq. 142. Thesolution was found by Hartel,

Qk1z2 51µz2k

k!exp12µz2, 182

implying that the intensity of the incident light beamis Q01z2 5 exp12µz2 and that the boundary conditionsare Qk102 5 0 1k 5 1, 2, . . . , `2, which are typical for a

semi-infinite atmosphere. This result describes thepropagation of light in only one direction, whichmakes Hartel’s theory a so-called one-flux method.Examples of the two-flux method can be found inTheissing14 and Hecht.15Finally, the summation of all k-times scattered

intensities yields the total intensity F1z, u2,

F1z, u2 5 ok51

`

Qk1z2Ik1u2

51

4p on50

`

12n 1 125exp32µz11 2 Bn24

2 exp12µz26Pn1cos u2, 192

which was obtained by Hartel, who applied the rule

ok51

` 1µzBn2k

k!5 exp1µzBn2 2 1. 1102

Note that the summation starts with k 5 1 and notwith k 5 0, because the intensity of the primary beamis not included.In this presentation we use Eq. 192 as the fundamen-

tal equation describing the approximate intensity ofmultiply scattered light, which 1because of the as-sumed one-dimensional atmosphere2 is only definedon the line of the source beam at the distance z insidethe scattering atmosphere and radiating in the direc-tion u 1independent of the azimuth f2. Clearly, Eq. 192cannot be used to describe the experimentally acces-sible intensity, because the assumptions imposed onthe scattering geometry do not comply with ourexperimental setup. Especially the integration overthe optical path length 3Eq. 1424 does not account for thegeometry of the finite and cylindrical scattering atmo-sphere. Nevertheless, Eq. 192 provides a computation-ally fast way to estimate F1z, u2, which we regard as asuitable basis to model the expected experimentalintensity is1u2. But let us first extend it to polydis-perse scattering objects.

B. Polydisperse Systems

Equation 192 can be directly applied to systems with aparticle-size distribution. The single-scattering func-tion I181u2 of an ensemble of spheres with relativerefractive index m and with radii distributed accord-ing to the distribution functionD1R2 is

I181u2 5 eR50

`

D1R2w1R2I11u, R, m2dR, 1112

where w1R2 is a weighting function that allows for thecalculation of number, volume, and intensity distribu-tions. w1R2 5 1 is the number distribution, w1R2 51@R3 is the volume distribution, and w1R2 5 1@A01R2is the intensity distribution. A01R2 is the first coeffi-cient in the series expansion of Eq. 162. SubstitutingI11u, R, m2 in Eq. 1112 with the right-hand side of Eq.162 and interchanging summation with integration

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readily reveals that the coefficients An in Eq. 162 can besubstituted for by new coefficients An8,

An8 5 eR50

`

D1R2w1R2An1R2dR, 1122

with which Eq. 192 can be applied to polydispersesystems.For a polydispersity analysis, however, it is conve-

nient to represent the size distribution D1R2 as aweighted series of q cubic B splines16–18 wi1R2 1see alsoAppendixA2,

D1R2 5 oi51

q

aiwi1R2, 1132

where the ai are the as-yet-unknown coefficients thatare determined in a nonlinear least-squares proce-dure in Subsection 2.E. With the use of the Bsplines, the definition of the coefficients An8 in Eq. 1122becomes

An8 5 oi51

q

aign,i, 1142

where the matrix gn,i contains the coefficients of theith transformed B-spline function,

gn,i 5 eR50

`

w1R2wi1R2An1R2dR. 1152

C. Geometry of the Light-Scattering Cell

Asmentioned above, Hartel’sF1z, u2 in Eq. 192 1approxi-mately2 describes the multiply scattered intensity atthe distance z inside a semi-infinite one-dimensionalscattering atmosphere. Strictly speaking, in a three-dimensional atmosphere it is valid on the path of theincident primary beam only. To estimate the experi-mentally expected intensity is1u2, we have to find anenergy-balanced concept for the description of inten-sity contributions coming from some distance off thebeam. This means that to account for a specialscattering geometry we have to improve the energybalance, which is not yet designed for our particularexperimental setup. We have to find a good estimatefor the fractions of radiant energy that arrive at thedetector. One fraction comes directly from the pri-mary beam, from the so-called single-scattering vol-ume, and the second fraction comes from scatteredlight, which is distributed all over the atmosphereand occasionally falls onto the line of sight of thedetector. Because with our goniometer we do notmeasure absolute intensities, we are considering thefractions of radiant energy relative to each other only.We find it very helpful to regard F1z, u2 in Eq. 192 as

the product of the normalized form factor FN1z, u2multiplied by the total intensity 31 2 exp12µz24, whichis proportional to the radiant energy. One can ob-tain this expression for the intensity by integrating

3492 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

F1z, u2 over all solid angles and, for demonstrationpurposes, assuming the single-scattering albedo to beequal to one. With this assumption of a losslessatmosphere, a light beam of unit incident intensity,which travels a distance z, produces multiply scat-tered light with the radiant energy proportional to31 2 exp12µz24, and the intensity of the primary beamis attenuated by the factor exp12µz2. This is thebasis of our so-called concept of effective path length,where we consider the gains and the losses of the realand the virtual source beams that are traveling insidethe atmosphere. With the effective path length wecan approximately account for any special scatteringgeometry with a minimum of additional expenditureof computation time. The form factor FN1z, u2, whichis always connected with a corresponding intensityterm of 31 2 exp12µz24, is considered to functionmerelyas a corrective term for intensities in other directionsthan the one along 1u 5 02. The form factor becomesgradually smoother with increasing z. This is due tothe increasing number of convolution processes thatoccur more frequently with increasing path length.A schematic view of the scattering geometry is

depicted in Fig. 1. It is basically the cross section of acylindrical sample cell. The circle with radius drepresents the sample boundaries as viewed fromabove. The two big arrows symbolize the primarybeam and the direction of observation, both definingthe scattering angle u. The primary beam withintensity i0 enters the sample at point A 1see Fig. 12and produces scattered light that travels all possibleways within the sample cell and occasionally leavesthe sample cell at point C in the direction of detectionand with the as-yet-unknown intensity is1u2.Although the primary beam and the line of sight ofthe detector are of finite widths 1see also the descrip-tion of the experimental setup2, they can be treated asinfinitely narrow lines because their dimensions can-

Fig. 1. Scattering geometry of a cylindrical sample cell withradius d. A narrow primary beam enters the sample at point Aand leaves at point D. At point B8 the effective intensity of themultiply scattered light is represented by an equivalent virtualsource beam characterized by the parameters z1, z2, and a. Theexperimental intensity is detected along the1x axis pointing to thedetector.

cel when relative intensities are considered. Also, itbecomes clear that in this presentation we consider atwo-dimensional atmosphere, which is the plane ofintersection of the cylindrical sample with the scatter-ing plane. The scattering plane is defined by thedirection of the incident beam and by the line of sightof the detector.We consider the intensity distribution at every

point within the sample to be in equilibrium with thesource, which is the whole line of the primary beamstarting at point A and ending in point D. Further-morewe consider that the equilibrium intensity distri-bution at every point can be represented by equiva-lent virtual source beams. Because we are usingHartel’s F1z, u2, we have to think in terms of inte-grated intensity contributions, which depend on theeffective path lengths of these virtual source beams.F1z, u2 is already integrated over one dimension 3seethe integration of Eq. 1424 and, to model a two-dimensional scattering atmosphere, only one addi-tional integration is necessary, the integration overthe line of sight of the detector.In Fig. 1 one virtual source beam is depicted in a

direction inclined at an angle 12a2. The sign ischosen in accordance with the sign of the x axis, whichis the line of sight of the detector, with its origin at thecenter of the sample cell. The distances z1 and z2represent the two parts of the effective path length,which contribute to the experimental intensity is1u2 atthe point B8. z1 characterizes the intensity status ofthe source beam at that particular point, and z2characterizes the producing capacity of this beam.The intensity status is merely the relative intensity ofa beam after it travels a certain equivalent 1or effec-tive2 distance. The producing capacity, on the otherhand, is the relative amount of scattered intensitythat can be produced by the same source beam bytraveling the rest of the equivalent 1or effective2distance inside the scattering atmosphere. In thisway we account for the finite dimensions of thescattering cell. 3In the integration of Eq. 142 noboundary conditions concerning the sample thicknessare considered.4 Because the virtual source beam ispart of the scattered light species, its intensity statusis proportional to 31 2 exp12µz124, whereas the inten-sity of the real source beam 1primary beam, a 5 02 isproportional to exp12µd2. Similarly, we define theproducing capacity of the virtual source beam 1atpoint B82 to be proportional to 31 2 exp12µz224 and thatof the primary beam 1at point B2 to be proportional to31 2 exp12µd24.When considering the producing capacity, we have

to be aware of the fact that the amount of scatteredlight that is actually produced depends on the pathlength along the 1x axis starting at point B8.Therefore we have to distinguish between two cases,where the path length 1d 2 x2 is longer or shorter thanthe distance z2. When the path length is longer, thecapacity of the source is used up and the beam isattenuated for the rest of the path, which is31d 2 x2 2 z24 long. On the other hand, when z2 is

shorter, then the whole producing capacity cannot beused, because the source beam is already outside theatmosphere. Because of these two different func-tional behaviors, extinction in the one region andnonproduction in the other one, the subsequent inte-gration over all contributions along the line of sight ofthe detector must be split into two regions.Let us briefly describe how these regions can be

found. It can be shown that z2 is always lower thanor equal to 1d 2 x2 for all angles 0 # u # p, when x # 0.The proof for this is easy, if one known the ancienttheorem that whenever two secants intersect a circlesuch that the point of intersection of the two secantslies within the circle, the products of the fractionalchord lengths are the same for all secants; i.e., z1z2 5

1d 2 x21d 1 x2. With this theorem, one can see imme-diately that the condition z2 # 1d 2 x2 is alwaysfulfilled when z1 $ 1d 1 x2. With z1 5

1d2 1 x2 1 2dx cos u21@2 and by squaring, we obtain thecondition 0d2 1 x2 1 2dx cos u 0 $ d2 1 x2 1 2dx.Since d $ 0 and x2 $ 0, we can subtract 1d2 1 x22 fromboth sides and divide by 2d without changing thesign. The resulting condition is 0x cos u 0 $ x, which isfulfilled for all x # 0 when 21 # cos u # 1, QED.With the concept of effective path length, we under-

stand that the origin of every virtual and real sourcebeam must be at the point A, where the incidentprimary beam enters the sample. This is due to thefact that only in this case can the integration, which isautomatically included in the expression F1z, u2, coverthe whole path length 10 # z # 2d2 of the primarybeam inside the sample. In principle, point D couldalso be used as the origin for virtual source beams, butwe did not use it, because Hartel’s theory accounts forlight fluxes in the forward direction only.Let us proceed and model the experimental inten-

sity is1u2. The primary beam with intensity i0 travelsthrough the sample with radius d. In the center ofthe cell the primary beam intersects the line of sightof the detector, defining the scattering volume for thesingle-scattering limit. The intensity status of theprimary beam at this point is i0 exp12µd2, and theproducing capacity is 31 2 exp12µd24, with the corre-sponding angular correction FN1d, u2. The observedintensity, which is caused by the primary beam,is thus i0 exp12µd 231 2 exp12µd 24FN1d, u2 5 i0 exp12µd2F1d, u2. Because the residual path length BD ofthe primary beam is equal to the length BC of thesignal beam, no additional considerations have to bemade, except that the single-scattering volume isproportional to 1@sin u, which yields the result i0 exp12µd2F1d, u2@sin u.In the case of the beginning of multiple scattering

the detector also perceives the contributions of scat-tered light, which is distributed over the whole samplevolume. As described above, we represent the scat-tered light distribution some distance off the primarybeam with an equivalent virtual source beam. Atthe point of intersection, B8, this source beam has theintensity status i031 2 exp12µz124 with the correspond-

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3493

ing angular correction factor FN1z1, a2. Thus theintensity status is equal to i0F1z1, a2. The producingcapacity in the same point B8 is 31 2 exp12µz224when x# 0 and 51 2 exp32µ1d 2 x246 when x $ 0. The corre-sponding angular correction factors are FN1z2, u 2 a2and FN1d 2 x, u 2 a2, respectively. As explainedabove, z2 is smaller than 1d 2 x2 for all x , 0.Therefore the additional attenuation factor ofexp52µ31d 2 x2 2 z246 has to be applied in that region12d # x , 02.Finally, one can obtain the total intensity is1u2

perceived by the detector from the whole line of sight12d # x # d2 by summing up all contributions, yield-ing

is1u2 5 i0 exp12µd2F1d, u2@sin u 1 i0Ae2d

0

F1z1, a2

3 F1z2, u 2 a2exp52µ31d 2 x2 2 z246dx

1 e0

d

F1z1, a2F1d 2 x, u 2 a2dxB/2d, 1162

where

z1 5 1d2 1 x2 1 2dx cos u21@2, 1172

z2 5 1d2 2 x22@z1, 1182

cos1a2 5 1d 1 x cos u2@z1, 1192

cos1u 2 a2 5 1x1 d cos u2@z1. 1202

Equation 1162 consists of the contribution from theordinary single-scattering volume, i.e., directly fromthe primary beam 1first term2, and of the contributionof the scattered light distributed over the wholesample volume 1second term2.

D. Reflected Light from the Sample Cell

To apply the above-presented calculations to experi-mental data, a varying amount of reflected light fromthe glass wall opposite the direction of detection hasto be taken into consideration. As we have shown ina previous paper,7 these contributions become appar-ent when the optical thickness of the sample is smalland especially at high scattering angles 1backwarddirection2, where the intensity coming from the scatter-ing volume is of comparable magnitude to the re-flected intensity. Figure 2 schematically shows thescattering geometry that leads to the mathematicaldescription of the scattering problem applied in thispaper.The intensity is81u2 arriving at the detector consists

of the directly scattered intensity is1u2 and the re-flected intensity is1ur2 from the glass wall opposite tothe direction of detection,

is81u2 5 is1u2 1 crRpexp122µd2is1ur2, 1212

where the angle ur is equal to 1180° 2 u2. Becausethe path of the reflected light is 2d longer than the

3494 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

path of the directly scattered light, the correspondingattenuation exp122µd2 appears in Eq. 1212. Simi-larly, only Rp percent of the intensity falling onto thesurface of the sample cell is being reflected. Rp isFresnel’s reflectivity coefficient for normal incidence,

Rp 5 1ns 2 ngns 1 ng2

2, 1222

where ns and ng are the refractive indices of thesolvent and the glass wall, respectively. In the theo-retical case that the glass wall and the solvent havethe same refractive index 1i.e., ns 2 ng 5 02, Rp be-comes zero, which means that no light is beingreflected at all and is81u2 5 is1u2. From this we canconclude that the reflected-light contributions can beneglected when the refractive index of the solventmatches that of the sample cell and also when thesample is of high optical thickness.The coefficient cr in Eq. 1212 theoretically equals 1.0

for ideal experimental conditions. In our computa-tional procedure we let cr be optionally adjustable tocompensate all deviations from the ideal model.These deviations may come, for example, from aninexact refractive index of the glass ng, from slightmisalignments of the sample cell, and from the absorp-tion of the solvent, which cannot be experimentallycompensated for the reflected light. Similarly, crcompensates for the contribution of light that isreflected more than once.

E. Inverse Problem

The inversion of the multiple-scattering problem; i.e.,the calculation of the size distribution D1R2 from thescattering curve is81u2 is a nonlinear least-squaresproblem. It consists of the calculation of the qsize-distribution parameters ai1i 5 1, 2, . . . , q2 of Eq.1132, the scaling parameter i0 of Eq. 1162, and, option-ally, the reflectivity parameter cr of Eq. 1212.

Fig. 2. Scattering geometry for the description of the reflected-light contributions. In the center of the cylindrical glass cell thelaser light is scattered in all directions. The intensity arriving atthe detector consists of the directly scattered intensity withscattering angle u and the reflected intensity from the glass wallopposite to the direction of detection with scattering angle ur 5

180° 2 u.

The relative refractive index of the particles andthe optical density 1or the transmittance2 of the sampleare assumed to be known, and therefore they areinput parameters. The relative refractive index canbe determined, e.g., from single-scattering experi-ments,7 and the optical density can be estimated fromtransmittance experiments. i0 scales the scatteringfunction to experimental intensity units. It has to bekept variable because we do not measure absoluteintensities. As we explained above, the cr value isnot known exactly, and thus we keep it variable.We solve the nonlinear least-squares problem with

themethod of linearization.19 For this purpose we fitthe experimental function y1u2 with our previouslyobtained theoretical multiple-scattering functionis81u2 5 f 1u, c2,

y1u2 5 f 1u, c2 1 e1u2, 1232

where e1u2 is the statistical noise of the experimen-tal data in y1u2 and the parameter vector c containsthe q size-distribution parameters ai, the scalingparameter i0, and the reflectivity parameter cr. Thuswe have N 5 q 1 2 unknown parameters c 5

1a1, a2, . . . , aq, i0, cr2. The fitting function f 1u, c2 inEq. 1232 can be expanded to the first order in a Taylor’sexpansion at the starting values c 5 c0,

y1u2 5 f 1u, c02 1 oi51

N

≠f 1u, c2@≠ci 0c5c01ci 2 c0i2 1 e1u2,

1242

which can be rewritten in a condensed notation,

yD 5 A8cD 1 e, 1252

where we have used the abbreviations

yD 5 y1u2 2 f 1u, c02, 1262

cD 5 c 2 c0 1272

and the matrix A8 contains the partial derivatives ofthe fitting function with respect to the N parametersci at the starting values c0i. The elements Aji8 of thematrix A8 are

Aji8 5 ≠fj1c2@≠ci 0c5c0, 1282

where fj1c2 is the fitting function f 1u, c2 sampled at thejth angle.Thus the method of linearization yields a linear

approximation of the nonlinear least-squares problem.The coefficients cD can be calculated as outlinedbelow. The intermediate solution c1 of the nonlinearleast-squares problem is c0 1 cD. This intermedi-ate solution vector cannot be regarded as the finalsolution, but it can serve as a new starting-valuesvector for a new iteration, thus yielding c2. Thesecalculations have to be repeated until the solutionvector remains constant, which indicates that theprocedure has converged and that the best-fitting

solution of the nonlinear least-squares problem hasbeen reached. No guarantee or proof for conver-gence can be given, but up to now convergence alwayscould be achieved.The calculation of the difference vector cD in Eq. 1252

is a linear least-squares problem that one can solve byminimizing the norm of the statistical noise e,

6e6 5 6A8cD 2 yD6 5 min, 1292

where 6e6 is the abbreviation for 1oi ei221@2. The

experimental standard deviations si can be includedinto the calculations by multiplying Eq. 1252 with adiagonal matrix containing 1@si as diagonal elements.This matrix is commonly known as the square root ofthe inverse covariance matrix.Because the least-squares problem is ill posed,

constraints have to be imposed on the solution vectorcD. If no constraints are applied, artificial oscilla-tions will spoil the solution function. These oscilla-tions are caused by an increased propagation ofstatistical noise. To remove these oscillations, weapply a regularization technique that has been ap-plied successfully to similar problems.6,7,20 We areusing the constraint of a smooth solution functionD1R2 and the positivity constraint; i.e., the solutionfunctionD1R2must have only positive values.The constraint of a smooth solution function can be

included when the constraint matrix B and the con-straint vector h are mixed with the least-squarescondition Eq. 1292, thus yielding

001A8

lB2cD 2 1 yD

lh200 5 min 1302

in an augmented matrix notation. B is the matrixrepresentation of the first derivative of the solutioncoefficients,

B 5 321 1 0 0 . . .

0 21 1 0 . . .

0 0 21 1 . . .···

······

···· · ·4 , h 5 1

0

0

0···

2 , 1312

and h is the zero vector, indicating that the firstderivative should be a minimum and close to zero.The mixing parameter l is the so-called Lagrangemultiplier, which levels the weight of the constraint,and its estimation is the most important task of thisregularization technique.6,8The positivity constraint is incorporated byminimi-

zation of Eq. 1302 subject to

G1cD 1 c 02 $k, 1322

where G contains sampled cubic B splines and k is azero vector. The sampling points are positioned atthose radii at which the solution function obtainedwith 1cD 1 c02 has to be greater than or equal to zero.If the spline functions would not overlap, the matrixG would be the identity matrix. Because k is the

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3495

zero vector, Eq. 1322 can be rewritten as

GcD $ 2Gc0, 1332

which means that the ith intermediate solution GcDiof each iteration can be negative, but the final solu-tionGcDn after n iterations is always positive.For the solution of the least-squares problem 3Eq.

13024 with inequality constraints 3Eq. 13224 we usestandard routines of Lawson and Hanson,21 which arealso applied in the most common program,CONTIN.22 The routines of Lawson and Hanson arelisted as FORTRAN programs, and they are discussed indetail in their book.21

3. Results and Discussion

In this section we try to find the answers to two basicquestions:

112 The information content of a scattering func-tion vanishes with increasing degree of multiplescattering. This is due to the fact that the multiple-scattering process is, in principle, a convolution pro-cess 3see Eqs. 152 and 11624. When we argue in terms ofa series expansion of Legendre polynomials, eachconvolution process reduces the number of thosecoefficients that are greater than a predefined thresh-old level determined by the amount of statisticalnoise. Coefficients smaller than that level cannot berecovered, because they are indistinguishable fromnoise. From this point of view we want to find outthe minimum transmittance at which, in an idealexperiment, i.e., assuming our simplified multiple-scattering theory is correct, a size distribution can besuccessfully retrieved. Because our simulation andinversion procedures use the same mathematicalexpressions, we cannot identify errors that arise fromour simplified multiple-scattering model. However,we find it very important to answer the questionconcerning themaximum feasible resolution by invert-ing this nonlinear ill-posed problem.

122 Once we have found a theoretical limit, wewant to verify this limit with experimental data andthus investigate the quality of our approximation.If, in practice, the theoretical limit cannot be re-gained, we have clear evidence that the approxima-tions are too rough, and improved theories have to beapplied.

A. Application to Simulated Data

To check the performance of the computational proce-dure, a size distribution of Lorenz–Mie spheres wassimulated. The distribution was a bimodal Gauss-ian volume distribution of radii with centers at 500and 600 nm, with an amplitude ratio of 2:1, and withstandard deviations of 37.5 nm each. The particleswere assumed to have a relative refractive index of1.2. We simulated three volume concentrations12.140 3 1022, 2.600 3 1022, and 3.381 3 1022

mL@mL2 so that the transmittances were 15.029%,10.000%, and 5.008% of the incident radiation, respec-tively. The simulations were performed for a cylin--

3496 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

drical sample cell with a radius of 11 mm. Thescattering curves were sampled at 71 points between10° and 150° of scattering angle, and 1% statisticalnoise was added. Furthermore, reflected-light contri-butions with cr 5 1.00 3ns 5 1.332, ng 5 1.530, see Eq.12224were included. In all simulations, we assumed a632.8-nm free-space wavelength and a refractive in-dex of the solvent of 1.332.The reconstruction of the corresponding volume

distributions Dv1R2 and of the additional parameters1i0 and cr2 was carried out under the assumption thatthe relative refractive indices of the scattering objectsand the transmittances of the samples were knownquantities. Basically, all size distributions were suc-cessfully retrieved, but they revealed a graduallyincreasing loss of resolution. Within the range be-tween 15% and 5% transmittance 11.9 and 3.0 opticalthickness2, the information about the shape of the sizedistribution was fading away. Additional simula-tions showed that the width and the position of thedistribution could be recovered even for lower trans-mittances, but the two separate peaks could not beresolved any more.The best-fitting values of cr were 0.737 115%2, 0.541

110%2, and 0.004 15%2 instead of the simulated value ofcr 5 1, which indicates that the contributions of thereflected light are losing significance because they aresmall relative to the statistical noise; i.e., the contribu-tions of reflected light are approaching zero, andtherefore the absolute value of cr becomesmeaningless.In this case the best-fitting cr value cannot and neednot be determined with high precision, because theactual cr value does not significantly influence theretrieved size distributions.The other question we examined was the problem of

finding an appropriate value for the transmittance.The transmittance can be experimentally deter-mined, but the obtained values are sometimes errone-ous, because the detected zero-angle intensities alsocontain contributions of scattered light. Thereforewe investigated if it is possible to extract the actualvalue of the transmittance from the scattering curves.A simultaneous determination of the transmittance

by extension of the nonlinear least-squares algorithmfailed because of the introduction of quasi-singulari-ties. The reason for this behavior is the fact that thetransmittanceT affects the scattering curves in nearlythe same way as the overall scaling parameter i0 does.A decrease of T increases the smearing effect but alsoincreases the scattered intensity, which could be doneequally well by merely increasing i0. However, thesmearing effect is not sufficiently pronounced, so thatthe nonlinear least-squares algorithm cannot distin-guish between intensity scaling by i0 and intensityscaling plus smearing by T. Therefore a collision ofthese parameters is inevitable, and the result is adiverging least-squares procedure.Hence we solved the least-squares problem in a

different way. We solved it a couple of times, buteach time with a different value of T as a constant

input parameter. With that the least-squares proce-dure converges and an optimum mean deviation isreached for every T value. If these mean deviationvalues would not show aminimumwithin the range of10 # T # 12, we could be assured that a simultaneousdetermination of T and the size distribution D1R2 isdefinitely impossible. But this is not the case, as wecan demonstrate in Fig. 3.In Fig. 3 we show the result for the same size

distribution as above, with a simulated transmittanceof 0.0702. Figure 3 depicts the mean deviation sur-face as a function of the transmittance T and thestability parameter l 3see Eq. 13024. Each curve repre-sents the mean deviation for a set of T values at aconstant l value. The top and the bottom curvescorrespond to the highest and the lowest l values,respectively. These relatively flat curves indicate aweak dependence of the fit on the applied T value, andthis illustrates the difficulty of finding exact optimumT values by searching for the minimum in the meandeviation surface. We found the best-fitting transmit-tance of T 5 0.072 6 0.002 with the sensitivitymethod.7 The simulated and the extracted T valuesare in agreement, and therefore we find that, inprinciple, it is possible to determine a size distribu-tion from multiple light-scattering data, even whenthe transmittance of the sample is unknown or mea-sured with considerable uncertainty.

B. Experimental Setup

Our experiments have been carried out with a light-scattering instrument with cylindrical cells only.An instrument with a planar cell is in developmentbut is not yet ready for experimental verification ofour technique.The light-scattering experiments have been done

with a laboratory-built laser goniometer consisting ofa 50-mW linearly polarized He–Ne laser with a 1:1

Fig. 3. Mean deviation surface as a function of the transmittanceT and the stability parameter l 3see Eq. 13024. Each curve repre-sents the achieved mean deviation with a set of T values as inputparameters and with l kept constant. The top curve correspondsto the highest l value.

beam expander to produce a well-defined beam of1.8-mm diameter, an adjustable cylindrical index-matching bath with temperature control and a remov-able beam stop 1Rayleigh horn2 inside, an adjustablesample holder 1glass cell2, and detection optics with aspatial filter and a photomultiplier tube on a stepping-motor-driven swivel arm. At the entrance of thebeam expander a quarter-wave plate is mounted toswitch between linear and circular polarization with-out changing the alignment. The technical specifica-tions are

Wavelength 632.8 nmAcceptance angle of detector 60.3°Scattering volume at 90° .4.7 1in cubic

millimeters2Range of available angles 7.5°–150°Angular accuracy 0.01°n632.820 of index-matching bath 1.4559 6 0.0001n632.820 of index-matching fluid

1Decalin21.4732 6 0.0001

n632.820 of glass cell 1.4616 6 0.001Inner radius of index-matchingbath

40.0 1mm2

Outer radius of index-matchingbath

42.5 1mm2

Inner radius of glass cell 11.3 1mm2Outer radius of glass cell 12.6 1mm2

As a check of the alignment quality of the scatteringdevice, we present the experimental scattering func-tion 1Fig. 42 of pure toluene between 7.5° and 150°.The recorded intensity values 1arbitrary units2 aremultiplied with sin u to compensate for the varyingscattering volume at different observation angles.Thus the resulting scattering function should be aconstant or at least a set of uncorrelated data points ifthe index-matching bath, the glass cell, and theincident laser beam are perfectly centered. The

Fig. 4. Experimental scattering function of toluene between 7.5°and 150° as a check of the alignment quality of the scatteringdevice. The scattering function should be a constant or at least aset of uncorrelated data points if the light-scattering device isperfectly aligned.

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3497

increase of intensity at the small-angle side is due tothe residual primary beam, which could not be en-tirely extinguished by the beam stop. The contribu-tions of the residual primary beam do not spoildifferential measurements 1sample intensity minussolvent intensity2 as long as they do not dominate thesignal.The actual measurements are done with circularly

polarized light to simulate random polarization.The scattering volume of static light-scattering experi-ments is large enough so that no differences can bedetected between experiments with circularly andrandomly polarized light.

C. Application to Latex Particles

We used sulfate-polystyrene latex suspensions fromInterfacial Dynamics Corporation to test the computa-tional procedure. We prepared suspensions of mono-disperse latex particles with different radii. Thenominal radii were 200 nm 163.8%2 and 317 nm163.1%2.The first series of experiments was performed on

317-nm latex suspensions of varying concentrations.The suspensions were diluted with doubly distilledand filtrated water. The final nominal concentra-tions at measuring time were 2.0 3 1024, 5.0 3 1023,and 2.0 3 1022 mg sulfate-polystyrene particles permilliliter, with a nominal density of the polymer of1.055 mg@mL. The corresponding experimentallydetermined transmittances of the samples 1with athickness of 2d 5 22.45 mm2 were 0.9870 6 1.4 31024, 0.7306 6 1.3 3 1024, and 0.2870 6 9.9 3 1025,where the standard deviations represent the reproduc-ibility of the transmittance measurement.The second series of experiments was done with a

mixture of both lattices characterized above. Thesuspensions were mixed with a volume ratio of 2:1;i.e., the volume content of the 200-nm particles was 2times the volume content of the 317-nm particles.The final nominal concentrations of sulfate-polysty-rene particles at measuring time were 2.0 3 1024,5.03 1023, 2.03 1022, and 5.03 1022 mg@mL, and thecorresponding transmittances were 0.9809 61.5 3 1024, 0.7914 6 1.2 3 1024, 0.4082 6 6.7 3 1025

and 0.1040 6 2.0 3 1025, respectively.The light-scattering experiments 1blank solvent and

samples2 covered the angular range between 7.5° and150.0°. After subtraction of the blank curve, weobtained the differential scattering curves that aredepicted in Fig. 5 1symbols2; Fig. 51a2 contains thescattering curves of the monodisperse samples, andFig. 51b2 contains the scattering curves of themixtures.With increasing concentration, the curves are markedwith crosses, pluses, triangles, and diamonds. Theintensities are displayed as obtained from the mea-surement. Only the scattering function of the fourthconcentration in Fig. 51b2 1diamonds2 has been multi-plied by a factor of 3.0 for presentation purposes.The curves represent the fit to the experimental data.The scattering curves of the lowest-concentrated

samples 1crosses2 can be regarded as single-scattering

3498 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

curves and can be inverted with the single-scatteringLorenz–Mie theory. Figures 61a2 and 61b2 1crosses2show the resulting size distributions, which we ob-tained with the best-fitting refractive index of 1.6000 60.0033 of the scattering objects. The determinationof the refractive index is published elsewhere.7 Themaxima of the volume size distributions are at 208and 333 nm, which are 8 and 16 nm greater than thenominal radii, respectively. The differences can beexplained by small inevitable polydispersities of thesamples that lead to the well-known phenomenonthat the weight or volume average of the molecularweight of a polymer is always greater than or equal toits number average.The effects of multiple scattering become rather

obvious in the size distributions 3Figs. 61a2 and 61b24when the calculations are done with the higher-concentrated samples 1crosses and triangles2 and with

Fig. 5. Differential scattering curves of increasingly concentratedlatex suspensions 1symbols2 and the fitting functions 1curves2.The concentrations of both 1a2 the monodisperse lattices and 1b2 the2:1 mixture of two monodisperse lattices were 2.0 3 1024 1crosses2,5.0 3 1023 1pluses2, 2.0 3 1022 1triangles2, and 5.0 3 1022

1diamonds2mg@mL sulfate-polystyrene particles.

the assumption of single scattering. In this caseartificial peaks appear at small radii. It is worthnoting, though, that the main peaks at 208 and 333nm remain unchanged at the same positions.When we apply the multiple-scattering correction,

the results improve substantially, as can be seen inFigs. 71a2 and 71b2, in which the same symbols havebeen used as in Figs. 61a2 and 61b2. The calculationshave been done with the experimentally determinedtransmittances of the samples. We note that theratio of the areas under the two peaks is 2.0 6 0.1,which is in excellent agreement with the volume ratioof the latex suspension as it was prepared. Thereconstructed size distributions from the second1pluses2 and the third 1triangles2 concentrations of themixture 3Fig. 71b24 fit the size distribution 1crosses2obtained from the single-scattering experiment al-most perfectly. The fourth concentration in Fig.71b2 1diamonds2 has been included for demonstration

Fig. 6. Volume distribution functionsDv1R2 of both 1a2 themonodis-perse lattices and 1b2 the 2:1 mixture of two monodisperse latticesdetermined from the experimental data 1Fig. 52 with the assump-tion of single scattering. With increasing concentration 35.0 3 1023

1pluses2 and 2.0 3 1022 1triangles2 mg@mL4, artificial peaks appearbecause of an increasing degree of multiple scattering.

purposes, to show the limit at which our method fails.The peaks shift, the widths of the peaks are broader,and artificial oscillations appear on both sides.Similarly, the third concentration 1triangles2 of themonodisperse sample 3Fig. 71a24 starts to show anartificial oscillation on the left-hand side. Its trans-mittance of 28.7% indicates the limit for our method.The calculated coefficients ai 3see Eq. 11324 can also

be used to reconstruct the corresponding single-scattering curves. We call these reconstructed curvesdesmeared curves because the calculation procedureis analogous to the removal of instrumental broaden-ing effects usually applied to small-angle x-ray scatter-ing data.8 In Figs. 81a2 and 81b2 we present as abyproduct the corresponding reconstructed single-scattering functions of the second 1pluses2 and thethird 1triangles2 concentrations, together with the fitto the corresponding experimental single-scatteringfunctions 1curves2. The lower curves correspond tothe second concentrations and fit very well, whereasthe curve of the third concentration 1triangles2 of themonodisperse sample 3Fig. 81b24 starts to deviate.

Fig. 7. Same as in Fig. 6 but calculated with the multiple-scattering correction presented in this paper. In comparison withthe results in Fig. 6, the effects of multiple scattering can besuppressed down to a limiting transmittance value of approxi-mately 30% 1triangles2.

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3499

From our simulations we expected that for idealconditions the reconstruction of a size distributionshould be possible down to a transmittance of approxi-mately 10% of the incident intensity. Yet the experi-ments taught us that, in practice, a transmittance ofapproximately 30% is the limiting value.

4. Conclusion

Let us summarize the results of our investigationswith the following four items:

112 We have found that the information content ofa simulated light-scattering curve drops significantlywhen the degree of multiple scattering reaches alimiting value. Therefore the application ofmultiple-scattering theories to the inverse problem can success-fully retrieve size-distribution features above this

Fig. 8. Reconstructed single-scattering curves 1symbols2 of both 1a2the monodisperse lattices and 1b2 the 2:1 mixture of two monodis-perse lattices obtained from the experimental scattering curves inFig. 5 in comparison with the scattering curve 1curves2 ofthe lowest-concentrated sample 1the reflected-light contributionsare already subtracted2. The concentrations were 5.0 3 1023

mg@mL 1pluses2 and 2.0 3 1022 mg@mL 1triangles2.

3500 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

limit only. For typical experimental conditions, andassuming that the applied theory is correct, one canspecify this limit by a transmittance of approximately10%, which corresponds to an optical thickness of 2.3.

122 With experimental light-scattering curves, how-ever, we have learned that, because of the underlyingtheoretical simplifications, the inversion can be suc-cessfully accomplished down to a limiting transmit-tance of only 30%. This is equivalent to an opticalthickness of 1.2.

132 Applications to experimental data showed thatit is possible to extend the maximum accessibleconcentration for light-scattering experiments from2 3 1024 to 2 3 1022 mg@mL. The calculations forthis paper have been done on a 66-MHz 486 IBM-compatible PC, and they took approximately 3 hrs ofcomputation time to complete. More accurate mul-tiple-scattering theories undoubtedly can improve thequality of the results, but these are inevitably con-nected with a higher computational expense.

142 Another very promising method to reduce mul-tiple scattering is in preparation. In particular aplane-parallel light-scattering cell that allows for thereduction of the sample thickness will clearly improvethe possibility of investigating colloidal samples ofeven higher concentrations.

Appendix A.

Let us explain the indirect transformation techniqueby means of the inversion of the single-scatteringproblem, which can be expressed with the followingFredholm integral equation of the first kind:

y1u2 5 e0

`

D1R2w1R2I11u, R, m2dR 1 e1u2, 1A12

where y1u2 is the experimental scattering function, e1u2is the statistical noise of the experimental data, D1R2is the size distribution that has to be calculated,I11u, R, m2 is the Lorenz–Mie single-scattering formfactor of a sphere with relative refractive index m,and w1R2 is a weighting function with which thevarying contribution of each particle size to the signalcan be compensated to calculate volume or numberdistributions.The indirect transformation method assumes that

the D1R2 can be described as a series of q cubic Bsplines16–18 wi1R2,

D1R2 5 oi51

q

aiwi1R2, 1A22

where the linear coefficients ai are as yet unknownand have to be determined. The B-spline functionswi1R2 are bell-shaped functions, and thus they providefor a smooth distribution function D1R2 that, inprinciple, can be of any shape. This makes thetechnique more general because it does not assumeany particular analytical form of the distribution.But it must be stated that narrow peaks in D1R2 can

be resolved only when the widths of the B splines arenarrow enough.Each wi1R2 can be transformed to the corresponding

theoretical scattering function ci1u, m2 1often calledreciprocal space2 because the direct scattering prob-lem is linear and can be solved exactly,

ci1u, m2 5 e0

`

wi1R2w1R2I11u, R, m2dR. 1A32

Now the transformed B splines ci1u, m2 can be used toapproximate the experimental scattering function y1u2by a series expansion,

y1u2 5 oi51

q

aici1u, m2 1 e1u2. 1A42

The coefficients ai can be calculated by the use of alinear least-squares algorithm, and with these coeffi-cients the size distribution D1R2 can be calculated bythe use of Eq. 1A22. By substituting ci1u, m2 in Eq. 1A42with the right-hand side of Eq. 1A32 and by interchang-ing the integral with the sum, one can easily checkthat the coefficients ai of Eq. 1A42 are identical to thoseof Eq. 1A22.

This work was supported by the OsterreichischerFonds zur Forderung der wissenschaftlichen Forsch-ung under grant P 7399-CHE. We express specialthanks to G. Scherf for carrying out the experiments.

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