Inversion of light scattering data from fractals bythe Chahine iterative algorithm

Fabio Ferri, Marzio Giglio, and Umberto Perini

We used the nonlinear Chahine iterative inversion scheme to analyze size distributions of fractal objects andwe tested its usefulness by computer simulations. The data to be inverted are elastic light scatteringmeasurements at a number of angles. We chose the fractal dimension of the objects equal to 1.75 to duplicatecolloid aggregates growing in the diffusion limited aggregation mode. Even in the presence of a realistic levelof noise, the method offers good estimates of the average radius and of the spread of the distribution.

1. Introduction

The theory of fractals is a powerful way to describeand characterize disordered systems that is gainingpopularity since the original pioneering work of Man-delbrot.' Indeed, many systems have shown, oftenunexpectedly, to possess the dilation symmetry typicalof fractals, and in many cases such a property is ob-served over a substantial range. 2 3

Fractal objects are characterized by a nonintegerHausdorff dimension,4 which is sometimes connectedwith the physical mechanism controlling theirgrowth.5 In these cases a fruitful relationship betweengeometrical properties and physics is established. Ag-gregating colloids are examples of systems possessingthis property.6 Indeed, computer simulations haveindicated that the fractal dimension is connected tothe mode of aggregation, either a slow aggregationcalled reaction limited aggregation (RLA)7 or a fastone, diffusion limited aggregation (DLA).8 The firstgives rise to aggregates of dimension D = 1.75 while thesecond produces aggregates of higher dimension, D =2.1.

Experiments have confirmed this. The experimen-tal techniques employed are either TEM photogra-phy9 or light scattering, both static610 and dynam-ic.11'1 2 A fundamental class of problems still remains;namely, how to investigate the kinetics of the sizedistribution during the evolution of the aggregatingprocess.13 Among the most promising techniques, lowangle static light scattering seems a good candidate. 1 4

Umberto Perini is with CISE, P.O. Box 12081,20134 Milan, Italy;the other authors are with University of Milan, Italy.

Received 5 May 1988.0003-6935/89/153074-09$02.00/0.© 1989 Optical Society of America.

In the main, this paper analyzes the potential of lowangle scattering techniques in conjunction with an in-version algorithm that allows derivation of cluster sizedistribution from the scattered intensity distribution.The analysis is performed with the aid of computersimulations, and the inversion procedure is a modifiedversion of the Chahine nonlinear iterative algo-rithm. 15, 6

We start with a brief description of fractal objectsand stress the usefulness of the light scattering tech-nique for their analysis. Then we describe the appli-cation of the Chahine method to the specific problem,assuming realistic values for the number and geometryof the scattering channels. Simulations of the inver-sion procedure are then presented and we explicitlyassume that the input scattering data are affected bystatistical errors of plausible magnitude. The simula-tion is confined to fractals aggregating in the DLAmode since in this case the problem is somewhat sim-pler.

II. Fractal objects

Briefly recalling some of the basic concepts of fractalobjects, we introduce the definition of fractal dimen-sion adhering as close as possible specifically to aggre-gating clusters of colloidal monomers.

From a geometrical point of view the most peculiarfeature of a fractal is its random spatial arrangement,tenuously filling the space and characterized by irregu-lar ramifications that never quite make it fill the spacein a dense manner. An example of such aggregates isshown in Fig. 1, taken from Ref. (9). The picture is aTEM image of colloidal gold made up of 100-A mono-mers. From a physical point of view such an object ischaracterized by self similarity, or dilation symme-try' 7 1 8 ; it is therefore invariant with a change of scale,and its appearance does not change with magnifica-tion. Dealing with real objects, this property breaks

3074 APPLIED OPTICS / Vol. 28, No. 15 / 1 August 1989

Fig. 1. Transmission electron microscopy of a gold colloid9 of frac-tal dimension D = 1.75.

down when the scale is reduced to the size of themonomers or when it exceeds the overall size of thecluster itself. But this is of little relevance since theratio of the size of the cluster to the size of the mono-mer can be orders of magnitude.

From a mathematical point of view self similarityimplies that the second order correlation functiong(r,,r2) = (n(ri)n(r 2)) must satisfy the relation 7" 8

g(ari,ar 2 ) = f(a)g(r 1 ,r2 ), (2.1)

where n(r) is the monomer density at r and a is anarbitrary scaling factor. The function f(a) does notdepend on r. In our case the system is space invariantand is isotropic, so g depends only on the modulus r =r, - r21 and it can be shown that it must have the form

g(r) - r-0. (2.2)

So self similarity implies that g has power law behaviorand this law holds in the scale range discussed above.

The object can be quantitatively characterized interms of its fractal dimension by assuming that wedraw around any monomer a sphere of radius R (ignor-ing the monomers too close to the periphery). As R ismade larger, we calculate the mass (number of mono-mers) contained in the sphere. We repeat the opera-tion for all the monomers and we take the average ofthe mass M(R). The following relation holds:

M(R) f| g(r)dr = RD,

dimension of the object depicted in Fig. 1 and to deter-mine the fractal dimension of strange attractors occur-ring in the dynamics of nonlinear chaotic systems.20-23

In the latter case the trajectories in a multidimensionalphase space are constructed from digitized signalsfrom the experiments, and thus, the actual coordinatesof the points belonging to the attractor are accuratelyknown. Although, the integral technique can be uti-lized, the algorithm is very time consuming. Indeed,the computer time to perform the calculation increasesunfavorably as the number of representative pointsincreases.

As we indicate in the next section, static light scat-tering provides an elegant and direct method of evalu-ating the fractal dimension, one ideally suited to theaggregating colloids we wish to investigate when theindividual points (the monomers) are much smallerthan the wavelength of light.

11. Light Scattering from Fractals

Using the correlation integral method is a cumber-some way to investigate the kinetics of the aggregationprocess. To follow the process, one must stop thereaction at chosen instants after the aggregation pro-cess has started. Then for each of the samples soobtained, TEM pictures of colloidal clusters are taken,from which the coordinates of each monomer belong-ing to the cluster must be derived. The TEM picture,however, being a 2-D image, does not allow analyzingfractals larger than two (2D). On the other hand, lightscattering operates in real time, is noninvasive, anddimensions up to three (3D) can be investigated.

Let us see how light scattering can be used for ourpurposes. As is well known, the scattered intensityI(q) as a function of the scattering wave vector q =2(2r/X) sin (0/2) (here X is the wavelength of light inthe medium and 0 is the scattering angle) is given by2 4

I(q) - f(q)S(q), (3.1)

where f(q) is the form factor and S(q) is the structurefactor. Because the monomers are much smaller indiameter than the wavelength of light, the form factoris virtually a constant, so the scattered intensity coin-cides with the structure factor which in turn is con-nected to the "center of mass" correlation functiong (r) 24(2.3).

where the exponent D is defined as the fractal dimen-sion of the aggregate and is connected to the previ-ously introduced exponent 3 by the relation

D = 3-F. (2.4)

The quantity D is called dimension since it obviouslycoincides with the usual integer dimension when D =1,2,3.

Equation (2.3) is the basis for an algorithm that hasbeen successfully employed to evaluate the fractal di-mension. It is called the correlation integral method19and it works best when one has a quantitative descrip-tion of the actual position of each individual monomer.For example, it has been used to compute the fractal

I(q) = f g(r) exp(iqr)dr. (3.2)

In Eq. (3.2) g(r) represents the actual correlationfunction of a fractal object. As we have already seen inEq. (2.2), g(r) has a power law behavior for small valuesof r, but it must decrease to zero when r becomescomparable with the physical size of the fractal objectitself. For a fractal object of radius R for g(r) we takethe following expression borrowed from the theory ofcritical phenomena:2 5

g(r) exp(-r/R)I (3.3)

where R actually coincides with the gyration radius ofthe object. According to Eq. (3.2), the scattered inten-

1 August 1989 / Vol. 28, No. 15 / APPLIED OPTICS 3075

sity is given by the so-called Fisher Burford function25

S(q) = 1[1 + (Rq)SJD/2

(3.4)

if one ignores a slowly varying function in the regionclose to qR = 1.26 It is interesting to note that thepower law behavior at large q is characterized by anexponent that coincides with the actual fractal dimen-sion of the object. This behavior does not depend onthe particular cutoff function chosen but is caused bythe characteristic power law behavior of the correla-tion function. In a sense, the complex algorithm of thecorrelation function integral is replaced by the experi-mentally simpler analysis of the asymptotic behaviorof the scattered intensity.

Let us see how polydispersity affects the distribu-tion of the scattered intensity as a function of thetransferred momentum q. The scattered intensity perunit solid angle by a single cluster of radius of gyrationR is given by 24

I = AM 2(R)S(qR), (3.5)

where A is a constant that does not depend on R or q,M(R) is the molecular weight of the cluster, and S(qR)is the structure factor that we take as a homogeneousfunction of the product qR [see Eq. (3.3)]. In Eq. (3.5)S(qR) is normalized so that S(O) = 1. For a polydis-perse sample described by a number distributionN(R), the total intensity IT(q) is

IT(q) = A f N(R)M(R)S(qR)dR. (3.6)

Equation (3.6) is a first kind Fredholm integral equa-tion where IT(q) is a known quantity provided by theexperiment, M2(R)S(qR) is the kernel, and N(R) is theunknown function. Equation (3.6) is valid when wehave dilute solutions in which all the clusters can beconsidered as noninteracting particles. Actually,IT(q) is measured at a finite number of wave vectors, qi(i = 1,2,. .. ,m), within a finite range (qmin,qmax). Be-cause of the limited extension of the range (qminqmax),the range of the clusters that can be simultaneouslyanalyzed is finite and, as a consequence, the distribu-tion N(R) can be studied only over a finite range(RminRmax). Furthermore, since IT(q) is known onlyfor m values of q, Eq. (3.6) is transformed into a set of mrelations connecting the integral in Eq. (3.5) to themeasured values I(qj). Therefore, it is possible toestimate the N(R) at m different points only.27

Let us divide the integration range (Rmin,Rmax) intom subranges or classes of width 5j and associate each ofthem with a radius Rj. To cover as large a radius rangeas possible and preserve the same 6j/Rj percentageaccuracy for each class, it is convenient to scale bj andRj according to a geometrical progression. In this casethe values bj and Rj are

bj = 6,a'-1 Rj = Rla'', (3.7)

where a is the geometric ratio that depends on therange (Rmin,Rmax) and on the number of the classes.Assuming that inside each subrange 6 the distribution

PhotodetectorLaser Collimator Lens

Sample

Photodetector

I I ,q =,q - 1q.1

Fig. 2. Low angle light scattering setup. The lower part of thefigure shows the annular sensors of the detector array and the

collection geometry.

N(R) is flat, it is possible27 to approximate the integralEq. (3.6) with the following set of equations:

(qi= Z N(R) M2 (R)S(qiR)dR (i = 1,2,. . m), (3.8)

where the integration is over the j subrange and Rj is avalue inside 5j. Equation (3.8) is a set of m linearalgebraic equations over the unknowns N(Rj) whosesolution gives the histogram that approximates thewanted distribution N(R).

If the thickness of the subranges 5j is thin enough it ispossible (suitably choosing the values for R) to ap-proximate Eq. (3.8) with

MIT(qi) = NJ(R)S(qiR),

j=i(3.9)

where N = N(Rj)bj. Here N represents the totalnumber of clusters inside the class j and, from now on,we shall refer to the particle distributions. How thetwo ranges [qmin,qmax] and [Rmin,Rmax] are related toeach other is discussed in the next section.

IV. Chahine Technique for the Inversion of ScatteringData

The inversion scheme of the scattering data is basedon an iterative nonlinear algorithm originally pro-posed by Chahine in 1968 for the analysis of data inatmospheric physics.15"6 The Chahine technique,though lacking the mathematical rigor and the ele-gance of other methods, offers the advantages of beingeasily understood from a physical point of view, ofconverging quickly, and of remaining reasonably sta-ble in the presence of experimental noise. This tech-nique has been successfully utilized in solid particleanalysis by means of elastic light scattering.2 8

The Chahine technique capitalizes on the possibilityof establishing even a loose correspondence between agiven value of transferred momentum q and a well-defined class of clusters Rj. Let us consider a cluster ofgyration radius R that scatters light following Eq. (3.1)and assume that the scattered radiation is collectedaccording to the optical scheme shown in Fig. 2. Thephotodetector is an array of thin annular sensors which

3076 APPLIED OPTICS / Vol. 28, No. 15 / 1 August 1989

LensAB

k,

Part c e I

II I

In Fig. 3 we show the structure factor S(x) and thecorresponding function H(x) for fractal dimension D =1.75.

The existence of a maximum allows us to establish aloose correspondence between the radius of cluster Rand a radial position on the photodetector. For anygiven cluster of radius Rj there will be an annularsensor collecting the largest fraction of the overallintensity scattered by that cluster. The desired corre-spondence is then

RS 3

210

Fig. 3. Log-log plot of the Fisher-Burford structure factor S(x) andthe signal sequence H(x) as a function of the adimensional variable x

= qR. The curves refer to fractal dimension D = 1.75.

are placed in the focal plane of the lens at differentdistances from the optical axis. Each annular sensorcorresponds to a wave vector given by

qj= 2(27r/X) sin(ri/2F), (4.1)

where X is the wavelength in the medium, r is theaverage radius of the ith sensor, and F is the lens focallength. The current signal i(q,R) out from the sensorassociated with the scattering wave vector q and col-lecting light in the range q - Aq,q + Aq is

~q+AqI2i(q,R) = A'M 2 (R) q S(qR) 27rq dq, (4.2)

fqAq/2

where A' = Au (here o- is the photodetector sensitivity)and Aq is the wave vector interval that depends on thethickness of the sensor. Equation (4.2) can be writtenby means of a function H(x) that depends on the adi-mensional variable x = qR:

i(q,R) = A' (2) X H(qR), (4.3)R2

wherex IAx12

H(x) = 27r J S(x) x dx. (4.4)fx-Ax/2

As it can be seen from Eq. (4.3), the dependence ofthe signal i(q,R) on the variable q is included in thefunction H(x) only. Therefore, the function H(x) de-scribes the behavior of the signal sequence out fromthe annular sensors for any radius of gyration R. Letus assume that the thickness of the different rings isarranged so that each sensor collects the scattered lightwith the same transferred momentum range Aq. Inthis case the integration range in Eq. (4.4) does notdepend on Ax. Assuming that D > 1, H(x) exhibits amaximum. In fact at small angles the scattered inten-sity is constant causing the signals to increase propor-tionally to the ring areas. At large angles the intensitydecays like a power law of exponent D > 1 and, there-fore, the signals decrease. The maximum falls in themiddle where a balance exists between the increase ofthe areas and the decrease of the scattered intensity.

where x* is the value of x for which H(x) has its maxi-mum.

We indicate with L(rj) the total signal coming fromthe jth ring due to all clusters belonging to the sample.Adding all the contributions we have

(4.6)L(rj) = E Nki'(rj,Rk),k=1

where Nk is the cluster number of class k and i'(rj,Rk) =i[q(rj),Rk]. Equation (4.6) is a set of m algebraic linearequations in the unknown Nk.

For a hint of the underlying mechanism of the Cha-hine inversion, assume that H(x) possesses an ex-tremely sharp peak. Then one could easily find outthe desired distribution since it would imply that eachsensor would receive light scattered by a cluster of agiven radius only. In other words, the coefficient ma-trix i'(rj,Rk) would be diagonal. Unfortunately H(x)has a rather broad peak (see Fig. 3). In the followingwe show how the existence of a peak for H(x) and itslocation can be used to establish an ordered iterationscheme.

Let us assume that we have measured the signalsequence out from the photodetector. These signalsare our input data for the inversion procedure and weindicate them by [L(rj)]in. Our task is to find a distri-bution Nj so that the signal sequence [L(rj)]ca1, calcu-lated according to Nj, matches the input data as closelyas possible.

The iteration proceeds as follows: Start by assum-ing a zeroth-order approximation N° which can be ofany shape; for example, a flat distribution over thewhole range. Then calculate the signal sequence ac-cording to Eq. (4.6) and compared it with the inputdata sequence. The two distributions will be differentand it is necessary to find a better approximation to thenext iteration. Generally, after p iterations the calcu-lated signal sequence is

m

[L(rj)]Pai = I Ni'(rj,Rk). (4.7)k=1

The iteration of order (p + 1) is then found correctingthe population Nj according to

N =P+ NP [L(rj)] (4.8)

This last equation is pivotal in the inversion proce-dure. The (p + 1) iteration is derived from the previ-

1 August 1989 / Vol. 28, No. 15 / APPLIED OPTICS 3077

103

10 _

10-

-110X (adimen.)

R 2(27r/X) sin(ri/2F)(4.5)

ous one by retouching the population in each Rj classaccording to the comparison between the input dataand those recalculated according to the p-iteration.The retouching is done by considering only the signalcoming from the annular sensor corresponding to theRj class.

To give a quantitative evaluation of the goodness ofthe fit, we introduce the quantity ring mean error(r.m.e.) defined as follows:

r.m.e. = / 1 [L(rj)] - [L(rj)]i 2N m j~~l l [L(rj)] in

15

0.0Eal

(4.9)

The r.m.e. describes the mean rms fractional devi-ation of the recalculated signal sequence from the in-put one. Note that the r.m.e. parameter allows us tohave an idea of the fit reliability. In the presence ofnoise, however, the reduced X squared (from now onindicated with X2) is a more utilized parameter toevaluate the fit. From the definition [Eq. (4.9)3 onecan note that, when the percentage of noise is the samefor each annular sensor, the r.m.e. and the X2 arerelated to each other by

X2= (r.m.e.)2 /T2,

0 oiL10 11

' OL10 1

size (microns)Fig. 4. Output distributions for various iteration numbers. Theinput data have been generated for a monodisperse sample with R =5 pm and fractal dimension D = 1.75. No noise was added to the

input data.

101

(4.10)

where is the percentage error (taken to be the samefor all channels).

As we shall see in the next section, in the presence ofnoise a fairly good approximation (corresponding to avalue of X2 not too far from unity) can be achievedafter a finite number of iterations. After that the X2

value starts growing slowly or remains substantiallyconstant. The best choice of the reconstructed distri-bution Nj occurs when the above-mentioned condi-tions are attained.

V. Results of Computer Simulations

We now describe the results of the simulation tests.The routine is the same for all the tests: A hypotheti-cal number distribution Nj and the fractal dimensionof the aggregates is selected. We are then able tocalculate the sequence of the signals out from thesensor according to Eq. (4.6). In some cases we addnoise to the input signals. The fictitious data (theinput data) are then processed via the Chahine itera-tive procedure and the calculated distribution is thencompared with the input one.

We assume that the sensor can collect light at anglesspanning a range of - two decades and, as a conse-quence via Eq. (4.5), the range of particle size (Rmin,Rmax) covers about a factor of a hundred. In all thetests the size (radius of gyration) range is 0.5-50 Azm,the choice being arbitrary but not physically unrealis-tic.

We decided to simulate data typical of the DLAmode of aggregation, since in this case it is expectedthat the distribution Nj is narrow and possibly bell-shaped.29 It would have been more critical to considerthe RLA mode where one expects the distribution to bebroad and peaked at the small size end of the range.30

In this case a significant fraction of the signal couldcome from smaller particles falling outside the range

0.2

E

-c1

100 103

Fig. 5. Behavior of the number average radius (R), and of thestandard deviation a, as a function of the number of iterations for

the case described in Fig. 4.

we are actually considering, and, for the moment, weare not able to take this into account.

The first test was devised to check how rapidly andhow accurately the iterative procedure converges inthe simplest possible case. The input distribution wastaken as perfectly monodisperse, with the size of thecluster falling at midrange, and no noise was added tothe input data. The initial trial distribution was takenas flat over the entire range. We show in Fig. 4 asequence of distributions obtained as the number ofiterations is increased. The distribution narrows rap-idly at first, but then the narrowing-down processslows with the iteration. It should also be noted thatthe narrowing process is somewhat asymmetric, thetails toward the smaller size end being slower to taperoff. This behavior is seen in Fig. 5 where we show thenumber average radius of gyration (R), and the stan-dard deviation as a function of the iteration number.Again, note that the average radius approaches thetrue input value from below, as a consequence of thetails at the smaller size end of the range. In Fig. 6 weshow the r.m.e. vs iteration number, and one can ap-preciate in a quantitative manner how slowly, after the

3078 APPLIED OPTICS / Vol. 28, No. 15 / 1 August 1989

210

a;E .9 .

10°10°100

X2

10°

103

Fig. 6. Decrease of the r.m.e. as a function of the number of theiterations for the case described in Fig. 4.

100 103

Fig. 7. Behavior of the X2 for six sets of input data identical to thatof Fig. 4 but with a 3% rms noise added.

initial fast decrease, the residual error dies away withthe iteration number. Incidentally, Fig. 6 indicatesthe kind of an error we can expect when noise is addedto the input data. Indeed, it is reasonable to expectthat the quality of the results will not improve once ther.m.e. becomes comparable with the rms noise added tothe input data.

We now show the results obtained for the case de-scribed above, but with noise added to the input data.Six typical runs are presented, the rms noise levelbeing always 3%. In discussing these results it is in-structive to analyze first the behavior of the X2 (seeFig. 7) for the different runs as a function of the intera-tion number. One should note that the curves tend toexhibit a minimum at a finite number of iterations.Also, one can see that the value of X2 is closer to one ifthe iteration number at which the minimum is reachedis larger.

The results are consistent with Fig. 6 where one cannotice that approximately 100 iterations are necessaryto achieve a r.m.e. of the order of 3% and consequentlyan X2 of the order of one.

In Fig. 8 and Fig. 9 we show the behavior of (R)n anda, as a function of the iteration number. The curveschange somewhat from run to run. Curves bc,d,f at-tain an X2 value <6 at -100 iterations. The corre-

10I

C

ACV

100

100 103

Fig. 8. Behavior of (R)'n as a function of the number of iterationsfor the six cases of Fig. 7. Comparing with Fig. 7, one can notice thatthe (R), values are fairly stable in correspondence of the number of

iterations at which the X2 attain their minimum values.

1101

C0~~~~~~~~~~~~~~~

.2

E f

-1

N. iterations100 103

Fig. 9. Behavior of the (a~,2 as a function of the number of itera-tions for the six cases of Fig. 7. As in Fig. 8 the a,, values are fairly

stable when the X2 reach their minimum (see Fig. 7.).

sponding values of (Ran and o-n are fairly stable andthe agreement with the input data is fair. Largervalues of X2 are attained in the other runs and accord-ingly the results are in poorer agreement with the inputdata. The results are also summarized in Table I. For

Table I. Summary of Results Obtained for a Monodisperse Sample with R= 5 m and Fractal Dimension D = 1.75

No. ofxin iterations (R)a

a 13.3 23 3.17 8.52b 3.47 385 4.37 1.15c 4.11 1000 4.85 0.21d 5.13 1000 4.89 0.56e 9.06 38 3.77 1.61f 5.66 1000 4.58 0.76

input 5.0 0.0average (b,c,d,f) 4.67 0.24 0.67 0.39

Note: A noise level of 3% rms was added to the input data. Foreach run we show the minimum value of X2 , the number of iterationsat which this value is attained, and the parameters (R),2 and a,calculated when the inversion procedure is stopped at that iterationnumber. In the first row we have indicated the theoretical valuesand a comparison can be made with averages values shown in the lastrow (the averages refer to the run where the minimum X2 hasattained a value lower than 6).

1 August 1989 / Vol. 28, No. 15 APPLIED OPTICS 3079

curves bc,d,f, we also report in Fig. 10 the distributionscalculated at the iteration numbers that gave the low-est value for the X2. Despite the irregular and indent-ed shape of the distributions, the calculated values arein fair agreement with the data. The indented appear-ance of the calculated distributions is a fairly commonfeature of the results obtained via the Chahine proce-dure and it is intimately connected with the structureof the procedure itself. Averaging the indented ap-pearance of the calculated distributions, however, pro-duces quite reasonable distributions.

We now analyze the results obtained for polydis-perse samples, and again we utilize the criterion ofstopping the iteration procedure when the X2 attainsits minimum value.

With the following tests we investigated the methodof discriminating between monodisperse samples andsamples with a moderate bell-shaped distributionwidth. The differences in the actual intensity distri-bution due to monodisperse and broadband distribu-tions can be minimal. In Fig. 11 we show the intensitydistribution due to a Gaussian distribution character-ized by an average radius R = 5 jAm and width 4.2 Am(continuous line). We superimpose on this curve thedistribution due to a monodisperse sample (dashedline). The value of the radius and total number ofparticles have been chosen so that the two curves sharethe same asymptotes. The difference between the twocurves is marginal and amounts to a few percent in thearea around the bend in the log-log plot. It is interest-ing to ask whether such a small difference can beappreciated by the inversion scheme.

The input data have been generated according to aGaussian distribution

(5.1)

60

ae

8

E

0

b100

- 385 It. 1

I . I .10 0110 1

c d100

WOO it. 1000 it.

0 . . -10 1 10

size (microns)10

Fig. 10. Output distributions at the iteration number that mini-mizes the X2 value. The correspondence between this set of curves

and that shown in Fig. 7 is indicated by the inset letters.

-510

Cs

._

-810

210

transf. momentum (cm' )105

Fig. 11. Plots of the theoretical intensity distribution for: (a) aGaussian distribution described by Eq. (5.1) (b) a monodispersesample where the radius and the concentration have been chosen so

that the curves share the same asymptotes.

10

where R = 5 m, a = 4.2 jAm and 6j are given by Eq.(3.7). It should be pointed out that the average radius(R) 0 and the standard deviation an calculated on thebasis of Eq. (5.1), and taking into account that we areconsidering a particle range of 0.5-50 jAm, are some-what different from R 0 and a. We find (R)n = 6.13 jamand a = 3.41 jam; these are the theoretical valuesagainst which the numbers generated by the inversionshould be compared.

In the first test no noise was added to the input data.In Fig. 12 we show the input distribution given by Eq.(5.1) and the output distribution obtained after 1000iterations. One can see that the reconstructed distri-bution matches the input distribution fairly well. Themost interesting case is when noise is added to theinput data. We investigated how the method workswith a noise level of 3% rms. As in the monodispersecase, the minimum values of X2 are attained after alimited number of iterations (see Fig. 13) but note thatthese values are smaller and closer to unity than in theprevious case.

We show in Figs. 14 and 15 the behavior of (R), anda,, vs iteration number. For each test both parametersbecome fairly stable as soon as the X2 reaches its

0

,0

10

0

size (micron)102

10 10

Fig. 12. (a) Distribution of the radii of gyrations Nj used to calcu-late the input data in the polydisperse case of Eq. (5.1); (b) distribu-tion of the radii of gyrations calculated with the inversion scheme

after 1,000 iterations (no noise added to the input data).

3080 APPLIED OPTICS / Vol. 28, No. 15 / 1 August 1989

I I-// II / I

I ..' I-, .,.1 . . ;1111 , , , - 1 Nj = 6j exp - 21 ,

2I- 1 [

101

x2

10°

100N. iterations

b c351

e

.4

160 It

20

i 135

size (microns).4

160 It.

20

Fig. 13. Behavior of the X2 for six sets of input data identical tothat of Fig. 12 but with a 3% rms noise added.

101

C

ACVa:

100

100 103

Fig. 14. Behavior of (R)0 as a function of the number of iterationsfor the six cases of Fig. 12. Comparing with Fig. 13, one can note thatthe (R ,n values are fairly stable in correspondence of the number of

iterations at which the X2 attain their minimum values.

101

C0

100N. iterations

100 1o3

Fig. 15. Behavior of the (a),, as a function of the number of itera-tions for the six cases of Fig. 12. As in Fig. 14 the are fairly stable

when the X2 reach their minimum (see Fig. 13).

minimum. The distributions obtained in correspon-dence with the lowest values of X2 are shown in Fig. 16and, in spite of their irregular and indented appear-ance, the values of R)O and a are in fair agreementwith the input ones. In Table II we have summarizedthe results of the different runs. As can be seen,

.4 20 .4 20

size (microns)

Fig. 16. Output distributions at the iteration number that mini-mizes the X2 value for the case of Fig. 13. The correspondencebetween this set of curves and that shown in Fig. 13 is indicated by

the inset letters.

averaging the runs where the minimum X2 is <6, givesdifferences between the calculated and input valuesthat are less than 1% for (R), and less than 2% for a.For the distributions shown in Fig. 16, the number ofiterations at which X2 has attained its minimum valueis shown in the insets.

VI. Conclusions

We have investigated the possibility of adapting thenonlinear Chahine iterative inversion scheme to theanalysis of elastic light scattering data from fractalobjects. In particular we have tested the capability ofthe method in deriving the number distribution ofgyration radii of assemblies of fractal clusters of givendimension. Monodisperse, peaked and bell-shapeddistributions have been considered, and the input data(scattering intensity vs angle) to be handled with theinversion scheme have been taken both with no noiseand with a realistic noise spread (3% rms). The simu-

Table II. Summary of Results Obtained for a Polydisperse SampleCharacterized by a Gaussian Distribution with a Number Average Radius

(R), = 6.13 ,um and a Width a,, = 3.41 Am

No. ofXmjn iterations (R) n an

a 2.68 160 6.20 3.31b 1.64 205 6.70 3.48c 3.87 160 6.30 3.49d 6.94 32 4.62 3.12e 2.07 86 5.86 3.48f 2.41 116 5.69 3.66

input 6.13 3.41average (a,bcd,e,f) 6.15 + .39 3.46 + .13

Note: A noise level of 3% rms was added to the input data. As inTable I the values for X2, (Rant a, refer to the iteration number atwhich the X2 attains its minimum. The results match fairly wellwith the input values.

1 August 1989 / Vol. 28, No. 15 / APPLIED OPTICS 3081

lations show that the technique is fairly effective andprovides, even in presence of noise, fair estimates ofthe average and the width of the distributions. Crite-ria for choice of the optimal number of iterations havebeen found. The reconstructed distributions have of-ten an indented appearance, but coarse grain averag-ing reproduces the basic features of the input distribu-tions fairly well. We also noticed that the convergenceof the method is more rapid with the iteration whenone analyzes intensity distributions due to broad dis-tributions of cluster sizes.

This work was performed at CISE.We acknowledge fruitful discussions with D. S. Can-

nell.

References

1. B. B. Mandelbrot, Fractals, Forms, Chance, and Dimension(Freeman, San Francisco, 1977).

2. F. Family and D. P. Landau, Kinetics of Aggregation and Gela-tion (North Holland, Amsterdam 1984).

3. H. E. Stanley and N. Ostrowsky, in On Growth and Form,NATO ASI Series (Martinus Nijhoff, Dordrecht, Holland,1986).

4. H. G. Schuster, "Deterministic Chaos: an introduction," Phy-sik (Verlag GmbH, Veinheim, FRG, 1984).

5. R. C. Ball, "DLA in real world, in On Growth and Form, NATOASI Series (Martinus Nijhoff, Dordrecht, Holland, 1986).

6. D. W. Schaefer, J. E. Martin, P. Wiltzius, and D. S. Cannell,"Fractal Geometry of Colloidal Aggregates," Phys. Rev. Lett.,52, 2371-2374 (1984).

7. W. D. Brown and R. C. Ball, "Computer Simulation of Chemical-ly Limited Aggregation," J. Phys. A: Math. Nucl. Gen., 18,L517-L521 (1985).

8. P. Meaking, "Formation of Fractal Clusters and Networks byIrreversible Diffusion-Limited Aggregation," Phys. Rev. Lett.51, 1119-1122 (1983).

9. D. A. Weitz and M. Oliveria, "Fractal Structures Formed byKinetic Aggregation of Aqueous Gold Colloids," Phys. Rev.Lett. 52, 1433-1436 (1984).

10. D. S. Cannell and C. Aubert, "Aggregation of Colloidal Silica,"On Growth and Form, NATO ASI Series (Martinus Nijhoff,Dordrecht, Holland, 1986).

11. D. A. Weitz, J. S. Huang, M. Y. Lin, and J. Sung, "Dynamics ofDiffusion-Limited Aggregation," Phys. Rev. Lett. 53,1657-1660(1984).

12. G. Bolle, C. Cametti, P. Codastefano, and P. Tartaglia, "Kinetics

of Salt-Induced Aggregation in Polystyrene Lattices Studied byQuasielastic Light Scattering," Phys. Rev. A; 35,837-841 (1987).

13. D. A. Weitz and M. Y. Lin, "Dynamic Scaling of Cluster-MassDistributions in Kinetic Colloid Aggregation," Phys. Rev. Lett.57, 2037-2040 (1986Y.

14. F. Ferri, "Diffusione di Luce da Aggregati Frattali," Ph.D. The-sis. U. Milan, Department of Physics, Italy, (1987).

15. M. T. Chahine, "Determination of the Temperature Profile inan Atmosphere from its Outgoing Radiance," J. Opt. Soc. Am.58, 1634-1637 (1968).

16. M. T. Chahine, "Inverse Problems in Radiative Transfer: De-termination of Atmospheric Parameters," J. Atmos. Sci. 27,960-967 (1970).

17. T. A. Witten Jr. and L. M. Sander, "Diffusion-Limited Aggrega-tion, a Kinetic Critical Phenomenon," Phys. Rev. Lett. 47,1400-1403 (1981).

18. T. A. Witten and L. M. Sander, 'Diffusion-Limited Aggrega-tion," Phys. Rev. B 27, 5686-5697 (1983).

19. P. Grassberger and I. Procaccia, "Characterization of StrangeAttractors," Phys. Rev. Lett. 50, 346-349 (1983).

20. J. Guckenheimer and G. Buzyna, "Dimension Measurements forGeostrophic Turbulence," Phys. Rev. Lett. 51, 1438-1441(1983).

21. A. Brandstater, J. Swift, H. L. Swinney, A. Wolf, J. D. Farmer, E.Jen, and P. J. Crutchfield, "Low Dimensional Chaos in a Hydro-dynamic System," Phys. Rev. Lett. 51, 1442-1445 (1983).

22. B. Malraison, P. Atten, P. Berge, and M. Dubois, "Dimension ofStrange Attractors: an Experimental Determination for theChaotic Regime of Two Convective Systems," J. Phys. Lett. 44,L-897-L-902 (1983).

23. M. Giglio, S. Musazzi, and U. Perini, "Low-Dimensionality Tur-bulent Convection," Phys. Rev. Lett. 53, 2402-2404 (1984).

24. C. Tanford, Physical Chemistry of Macromolecules, Wiley,New York (1961).

25. M. E. Fisher and R. J. Burford, "Theory of Critical Point Scat-tering and Correlations. The Ising Model," Phys. Rev. 156,583-622 (1967).

26. S. K. Sinha, T. Freltoft, and J. Kjems, "Observation of Power-Low Correlations in Silica-Particle by Small Angle NeutronScattering," in Kinetics of Aggregation and Gelation (NewHolland, Amsterdam, 1984).

27. C. T. H. Baker, The Numerical Treatment of Integral Equa-tions, Clarendon, Oxford (1977).

28. R. Santer and M. Herman, "Particle Size Distributions fromForward Scattered Light Using the Chahine Inversion Scheme,"Appl. Opt. 22, 2294-2302 (1983).

29. P. G. J. van Dongen and M. H. Ernst, "Dynamic Scaling in theKinetics of Clustering," Phys. Rev. Lett. 54, 1396-1399 (1985).

30. R. C. Ball, D. A. Weitz, T. A. Witten, and F. Leyvraz, "UniversalKinetics in Reaction Limited Aggregation," Phys. Rev. Lett. 58,274-277 (1987).

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