Physica B 45 (21) 316 321 Contents lists available at ScienceDirect Physica B journal homeage: www.elsevier.com/locate/hysb On the retrieval of article size from the effective otical roerties of colloids A. García-Valenzuela a,,c.sánchez-pérez a, R.G. Barrera b, E. Gutiérrez-Reyes b a Centro de Ciencias Alicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Aartado Postal 7-186, 451 México Distrito Federal, Mexico b Instituto de Física, Universidad Nacional Autónoma de México, Aartado Postal 2-364, 1 México Distrito Federal, Mexico article info Keywords: Effective refractive index Colloids Particle sizing Particle refractive index Turbid media Coherent wave abstract We analyze the retrieval of the radius and comlex refractive index of colloidal articles from the measurement of the effective otical roerties of a dilute colloid. We oint out some necessary recautions for the measurement of the effective refractive index of colloids and discuss the main sources of uncertainty. & 21 Elsevier B.V. All rights reserved. 1. Introduction Since many years ago, it has been acknowledged that when an electromagnetic wave is incident on a turbid colloidal medium, an average or coherent wave travels through the colloid with an effective wave vector [1]. From this effective wave vector, an effective refractive index, n eff, of the colloid may be defined. Clearly, the effective refractive index will deend on the hysical roerties of the colloidal articles, and therefore, its correct measurement could be used to characterize the colloidal articles. This is one of the ractical interests on the understanding and use of the effective refractive index of turbid colloids. Particle sizing is an old roblem with a large variety of imortant alications and has been addressed by many authors over the years [2]. Several techniques have been develoed and many commercial aaratus are in the market today. However, all techniques used today are still limited in one way or another and several challenges remain [2]. In articular, turbidimetry techniques are attractive from a ractical oint of view because the instrumentation and exerimental methodology are less comlicated than other techniques and they can be used for less diluted colloids [3]. The main limitation of the inversion of turbidimetry techniques is that one needs to know with good recision on the refractive index of the articles in order to estimate correctly the article size, and is articularly critical when articles are small comared to the wavelength. It is not difficult to show that in turbidimetry techniques one measures the imaginary art of the effective refractive index but not its real Corresonding author. Tel.: +52 55 5622 862x1118; fax: +52 55 5622 8651. E-mail address: augusto.garcia@ccadet.unam.mx (A. García-Valenzuela). art. Measuring the real art of n eff in addition to its imaginary art comliments turbidimetric measurements and ermits retrieving the size of medium and small articles (with resect to the wavelength of radiation). However, the hysical nature of the effective refractive index of colloids remained rather obscure until recently. In Ref. [3] it is shown that the effective electromagnetic resonse of a colloidal medium is actually non-local and satially disersive. The range of non-locality in a colloid is given by the diameter of the articles. It was also shown in Ref. [3] that the non-local disersion equation for transverse modes in a dilute colloid has a fundamental solution given to a good aroximation by the so called van de Hulst formula. This formula was used in the ast by several authors, however its non-local origin was not understood at the time, causing in many cases errors or confusion when using it. For instance, when the article size in the colloid is not very small comared to the wavelength of radiation, the effective refractive index can not be used safely in Fresnel formulas to calculate the coherent reflection coefficients. Therefore, measurements of the effective refractive index of colloids using techniques relying on the measurements of the reflectance of light may incur in large errors. However, with a few recautions n eff may be measured safely by refraction of the coherent wave [4]. Also in Ref. [4], it was shown that it is ossible to retrieve the radius and refractive index of colloidal articles from the measurement of the real and imaginary arts of n eff er unit of the volume fraction at a single frequency when the articles are moderately sized with resect to the wavelength and the imaginary art of n is smaller than about 1 3 (weakly absorbing articles). In this aer, we extend the methodology roosed in Ref. [4] to highly absorbing articles. Additionally, we analyze the main sources of uncertainty. 921-4526/$ - see front matter & 21 Elsevier B.V. All rights reserved. doi:1.116/j.hysb.21.1.26
A. García-Valenzuela et al. / Physica B 45 (21) 316 321 317 2. Sizing dielectric articles from the effective refractive index of colloids The van de Hulst formula for the effective refractive index at a frequency o of a dilute colloid made of articles of radius a and refractive index n (o), may be written as, n eff ðoþn m ðoþ 1þi 2 r k 3 T S a ðþ ; ð1þ m where n m ðoþ is the refractive index of the matrix where the articles are susended which we assume to be a real quantity, k m =n m k where k is the wave number in vacuum, r T is the number density of articles, and S a ðþ is the forward scattering amlitude of articles or radius a at the frequency o. In the laboratory, one can measure and control the volume fraction occuied by the articles, f, and not the total number density of articles r T. These are related by f=r T V where V ¼ 4a 3 =3 is the volume of one article. The contribution of the colloidal articles to the refractive index may be written in terms of the volume fraction occuied by the articles as Dn eff 3 2 n mf S aðþ ð2þ x 3 m where x m =k m a is the so called size arameter of the articles. The effective refractive index increment in Eq. (2) is in general a comlex quantity, Dn eff ¼ Dn eff þidn eff. The imaginary art of it takes into account attenuation of the coherent wave due to scattering and absortion. The scattering losses convert the coherent radiation into diffuse radiation whereas absortion losses convert the coherent radiation into heat. Let us suose we have a dilute colloid made of articles of radius a and real refractive index n, embedded in a homogeneous substance (the matrix) with a real refractive index n m. Let us define the effective colloid arameters, Dn eff f and Dn eff f : ð3þ To see whether we may retrieve a and n from the measurement of the effective roerties vector, x=(, ), we can lot curves of versus for constant n and different values of a. If these curves are well searated in the (, )-sace, then the inversion of n and a is unique and well conditioned. We obtain n simly by recognizing to which curve the exerimental oint x belongs and a from its osition along the curve. An examle is illustrated here in Fig. 1. We lot two curves of vs. for n =1.7 and n =1.8 and varying a at stes of Da=1 nm from a= to 5 nm, assuming a wavelength of l=635 nm. For larger article radius, both curves tend to the origin and cross each other and the inversion of a and n is no longer ossible [4]. When the articles absorb light their refractive index is a comlex quantity, n ¼ n þin. At a secific wavelength we may view n and n as indeendent variables. In Fig. 1b we comare the curves of Fig. 1a with the same two curves but for articles with an imaginary art of their refractive index of n ¼ :1. Clearly one may not retrieve the value of a nor n if one does not know that n ¼ :1. In Ref. [4] it was shown that if n o1 3 we can ignore it and retrieve correctly a and n from the measurement of the effective roerties, x=(, ). When n is larger we cannot longer ignore it and we have three unknowns in the inverse roblem. Therefore, we need at least three effective otical roerties of the colloid to solve for the three article s arameters..6.4.2..6.5.3.2. n'' = n' = 1.7 3. Extended methodology for highly absorbing articles The imaginary art of the effective refractive index of a colloid, in the dilute regime, is roortional to the so called extinction cross section of the articles C ext. We have, Dn eff ¼ fc ext=2k V [5]. It is well known that the extinction cross section can be divided as the sum of a scattering and an absortion cross section, that is, C ext =C sca +C abs [5]. Then, we can slit the imaginary art of the effective refractive index in two comonents, one due to scattering and one due to absortion. That is Dn eff ¼ðDn eff Þ sca þðdn eff Þ abs a max = 5nm n' = 1.8 n' = 1.8 Δa = 1nm n'' =..1.2.3.4.5 n' = 1.7 n'' =.1 Δa = 1nm a max = 5nm..1.2.3.4.5 Fig. 1. Curves of versus for constant n and varying values of a from to 5 nm at stes of Da=1 nm: (a) for real values of n =1.7, 1.8 and (b) for comlex values of n =1.7+.1i, 1.8+.1i. where ðdn eff Þ sca ¼ fc sca =2k V and ðdn eff Þ abs ¼ fc abs =2k V. As already said, the measurement of the coherent transmittance of a collimated beam readily gives Dn eff, whether it has contributions from scattering or absortion or both. However, it is also ossible to measure the otical absortion by the articles indeendently of the scattering of the articles. Such measurements are in rincile ossible by either hotothermal techniques [6] or by integrating-cavity techniques [7]. In either case one could determine exerimentally what fraction of the coherent beam was absorbed and therefore can determine ðdn eff Þ abs. Then, let us define the third effective otical roerty ð4þ
318 A. García-Valenzuela et al. / Physica B 45 (21) 316 321 Δa = 1nm n = 1.8+i.1.12 1..8.6 n' =.2 n'' = 3.1 n'' = 3.15 n'' = 3.2 n' =.15 Δ a = 5 nm λ = 635 nm n m = 1.36 n = 1.7+i.1.8 x 3 x 3.4.6 n = 1.8 n = 1.7 a max = 5nm.3.4..2 a max = 1 nm x1 2 4 n' =.1 2 4 6.3 x2. x1. Fig. 2. Curves of x 3 versus (, ) for constant n =n +in and varying a from to 5 nm at stes of Da=1 nm. The bottom curves are for n =1.7 and 1.8 and the uer ones for n =1.7+.1i and 1.8+.1i. needed in the case of highly absorbing articles as x 3 x 3 ðdn eff Þ abs f ð5þ To see whether the three measurements x=(,, x 3 ) are sufficient to determine a, n and n in the case of highly absorbing articles, we may lot, using Eq. (2), curves of constant n and n for different values of a in the (,, x 3 ) sace. If these curves are well searated from each other, it means that the values of a, n and n may be inverted from exerimental measurements. In Fig. 2 we lot the same curves of Fig. 1b but in the three dimensional sace (,, x 3 ). We can areciate that the four curves are well searated from each other and thus the inversion of the articles arameters is unique and well conditioned. Another examle is considered in Fig. 3 for metallic articles. We lot curves of constant n and n varying a from a= to a=1 nm at stes of 5 nm at a wavelength of l=635 nm. The values of n and n for each curve are close to those for gold articles at this wavelength. We can areciate in Fig. 3 that, again, all the curves are searated from each other, and thus, the inversion of the article s arameters is well conditioned and unique in this examle. In the next section we oint out some asects on the measurement of Dn eff that need secial care. 4. Necessary recautions for the measurements of n eff The exerimental determination of the effective refractive index of a turbid colloid requires several recautions. First, one must determine n eff from measurements of the coherent intensity only. Therefore, one must subtract any contribution of diffuse light that may be suerimosed to the coherent ower collected by the detector. Second, one must bear in mind the non-local origin of the effective refractive index, and this rules out the unrestricted use of common techniques based on reflectance measurements [4]. The imaginary art may be obtained from the measurement of the attenuation of the coherent intensity as it travels through the bulk of the colloid and the real art may be obtained from the measurement of the refraction angle of the coherent comonent of light at a colloidal interface. Fig. 3. (a) Curves of (,, x 3 ) for constant n and n and varying a. Symbols on each curves are at stes of Da=5 nm from a= to 1 nm. Curves with oen circles, squares and triangles are for n ¼ :1; :15 and.2, resectively. For each value of n we show three curves for n ¼ 3:1; 3:15 and 3.2. (b) Is the same as (a) but from a different ersective. For the determination of the imaginary art of n eff, one must ensure that the transmission coefficients of the coherent light entering and leaving the colloid does not affect the measurement. This is so because we do not have a general and safe way to calculate these transmission coefficients. On the other hand, these transmission coefficients deend also on the real art of n eff which are to be determined as well. Exerimentally, this can be accomlished by measuring the transmittance factor of the coherent beam, as illustrated in Fig. 4a, for two cells of different widths but otherwise identical. The ratio of these two measurements will deend only on the attenuation of the coherent beam on the difference in cell s widths, and will be indeendent of the transmission coefficients at the cell s interfaces. In an exeriment to determine the real art of n eff, one can use the refraction of a well defined otical beam at the interfaces of a colloid. For instance, one may use a rismatic cell filled with the colloid and measure the deflection angle of a beam transmitted trough it. The so called differential refractometer, illustrated here in Fig. 4b, may be articularly convenient for measuring Dn eff. The deflection angle will be obtained from the shift in osition of the intensity maximum or the intensity centroid of the light sot of the coherent beam at some detection lane. For a given otical beam and colloidal rism, one must carefully calculate the lateral dislacement and deflection angle of the coherent beam
A. García-Valenzuela et al. / Physica B 45 (21) 316 321 319 5. Uncertainty on article sizing due to errors in determining f χ P Scattering & heat n m d n' eff + in'' eff T w P ex( 2k n'' eff d) Fig. 4. (a) Coherent transmittance through a rectangular cell. The transmission factor T w must be cancelled with a second measurement at a different width. (b) Schematic illustration of the refraction of a light beam in a differential refractometer. transmitted through the colloidal rism, which in general deend on both, the real and imaginary arts of the effective refractive index. For very well collimated beams which start to diffract at distances large comared with the dimensions of the colloidal cell it can be shown that the angle of deflection deends only on the real art of n eff [8]. However, one must be careful on subtracting the lateral shift of the beam that occurs as it traverses the rismatic cell with the colloid [8]. If the beam is not so well collimated, as it could be with non-laser beams, the deflection angle may deend also on the imaginary art of n eff. The latter effect may be thought as being due to uneven attenuation of the coherent beam due to diffraction. We have already alied the roosed methodology in our laboratory to sizing small glass and olystyrene articles in water. The radii of the articles were in the range between 2 nm and 6 nm. To obtain f in the colloidal samles we first withdraw a know volume, v s, of the colloid with a recision syringe and weight it. Then we let the water evaorate and weight the remaining solids. Knowing the density of water we calculated the volume of evaorated water, v w, and obtained the volume fraction occuied by the articles in the colloid as f=(v s v w )/v s. We measured Dn eff and Dn eff using a rismatic cell and a rectangular one resectively. We erformed the measurements at l=475 nm using a solid state laser. First we made measurements with ure water and then added the colloidal articles and obtained Dn eff and Dn eff. We found good agreement between the retrieved article radii with the roosed methodology and those measured by a commercial aaratus based on dynamic light scattering and with TEM micro-hotograhs of the colloidal articles. Details of these and other measurements will be ublished elsewhere. In our measurements the largest exerimental relative error was on the determination of the volume fraction occuied by the articles. In the next section we analyze the uncertainty on the retrieval of a and n arising from the uncertainty on f. θ l Δx Let us suose the relative error on the measurements of the real and imaginary arts of the effective refractive index are small comared to the relative error on measuring f. If the three arameters, and x 3 were obtained for the same colloidal samle, the error in f is the same for the three measurements and their uncertainties are correlated. The uncertainty on the value of x j is given by dx j x j ðf =df Þ for j=1, 2 and 3. Then we can write, dx ¼ xðdf =f Þ, where x=(,, x 3 ). Now, when surfaces of constant a are searated from each other in the (,, x 3 )-sace, there is a well defined inverse function a(x). Then, if for a given colloid, we determine exerimentally x ¼ x 7dx, the uncertainty in retrieving a is given by, da ¼ ra dx. In the x-sace, ra is a vector erendicular to the surfaces of constant a. Its direction may be found from the cross roduct of two vectors on that surface, say, N ¼ @x=@n and M ¼ @x=@n. Then we may write,ra ¼jraj ^n a, where the unit vector ^n a is given by, ^n a ¼ðN MÞ=N M. The magnitude of the gradient of a at x may be calculated as jrajda=ds where ds is the distance in x-sace between the surface for articles of radius a+da and that for articles of radius a at oint x. Now, consider the differential vector in x-sace, dx ¼ð@x=@aÞda. If we roject this vector along the direction ^n a we get the corresonding value of ds for a given value of da, that is, ds ¼ dx ^n a. Thus, we get, da ¼½ð@x=@aÞ ^n a Š 1 ds, and therefore, jraj¼1=ð@x=@aþ ^n a. Using the formulas just given we may write, the relative uncertainty on the retrieved value of the article radius due to the uncertainty in the article s volume fraction as, da=a ¼ Fðdf =f Þ, where the factor F is given by x ^n a =½að@x=@aÞ ^n a Š. Similar formulas may be obtained for the relative uncertainty on retrieving n and n. For instance, dn =n ¼ Gðdf =f Þ where the G factor is given by, G ¼ x ^n n =½n ð@x=@n Þ ^n n Š and where ^n n is the corresonding unit vector of n n ¼ð@x=@aÞð@x=@n Þ. We illustrate in Fig. 5a the normal vector ^n a and the tangent vector N in the 2D-sace (, ) for the case of dielectric articles. Numerical evaluation of the factors F and G in the case of dielectric articles smaller than one wavelength and with low refractive index contrast with the matrix show that these are less than one. In other cases these factors may be smaller or larger than one. In Fig. 5b we lot some values of the F and G factors calculated for an examle of metallic articles. In this articular examle, the magnitude of the factor F is always less than.5, but that of the G factor is larger than one, meaning that the relative error in retrieving the article radius will be smaller than that on retrieving the real art of the refractive index of the articles. Another imortant source of uncertainty will come in the case of colloids that have a wide size distribution. In the next section we briefly analyze the effect of the size distribution when estimating the size of colloidal articles with the roosed methodology. 6. Polydiserse colloids The methodology described above assumes a monodiserse colloid, that is, it is assumed that the colloidal articles have a narrow size distribution. We may consider a colloid to be monodiserse when the most robable article radius, the average articles radius, and the radius of a article with the average volume, are close to each other in terms of the required recision for the size determination. If the size distribution is not narrow, we refer to the colloid as being olydiserse. In this case we will need to erform additional measurements to obtain information on the article s size distribution. When articles are not too small it may be enough to erform the same measurements but at other wavelengths.
32 A. García-Valenzuela et al. / Physica B 45 (21) 316 321 N dx a a + da curves of constant n S a ðþ¼½ ibk 3 m a3 þ 2 3 b2 k 6 m a6 þ:::š where b ¼ðm 2 1Þ=ðm 2 þ2þ and m is the relative refractive index n /n m. Corrections to the imaginary and real arts of the later exansion of S a ðþ are of order Oðk 5 m a5 Þ and Oðk 7 m a7 Þ, and higher, resectively. For instance, let us suose a log-normal article size distribution,! 1 nðaþ¼ a ffiffiffiffiffiffi ex ln2 ða=a Þ ð8þ 2 ln s 2ln 2 s ˆn a curves of constant a where s and a are the width and article radius arameters resectively. a is close but somewhat larger than the most robable radius. Using (8) to calculate /S a ðþs=/a 3 S yields, ReðDn eff Þ¼ 3 2 n mf b and ImðDn eff Þ¼n m f b 2 k 3 m a3 exð1 2 ½62 3 2 Šln 2 sþ ð9þ Numerical value.5 n =.17 + i3.1. -.5-1. -1.5-2. -2.5 Factor F Factor G n m =1.33-3. 1 2 3 4 5 6 7 Particle radius, a Fig. 5. (a) Schematic illustration of constant a and constant n curves in the (, ) sace. The normal and tangent vectors ^n a and N are indicated. (b) Numerical evaluation of the factors F and G for metallic articles with n =.17+3.1i in water for various values of a. Therefore, in this case, the retrieved value of the articles refractive index is actually n but the retrieved value of the articles radius is a e ¼ a exð 32 2 ln2 sþ. We may comare the retrieved equivalent radius with the radius of the average volume: a v ¼ a exð 3 2 ln2 sþ. We can see that the equivalent radius ρ [μm -4 ] 5 4 3 2 1 σ = 1.1 a = 8 nm n = 1.6 average radius (8 nm) radius of average volume (81 nm) estimated radius (82 nm) If the colloid is olydiserse, the effective refractive index may be written as, n eff ðoþn m ðoþ 1þi 2 Z 1 rðaþs k 3 a ðþ da ; ð6þ m where r(a) da is the number density of articles with radius between a and a+da. We may write r(a)=r T n(a) where r T is the number density of all articles regardless of their radius and n(a) is the size distribution function. In this case the volume fraction occuied by the articles is given by f ¼ r T /V S where /V S ¼ 4/a 3 S=3 is the average volume of one article. The average in this case is taken over the size distribution function n(a), that is, / S ¼ R ðþnðaþ da. The refractive index increment in the case of a olydiserse colloid may be written in terms of the volume fraction occuied by the article, f, as, Dn eff i 3 2 n mf /S aðþs k 3 m /a3 m S Therefore, with olydiserse colloids we will actually obtain an equivalent article radius a e and refractive index n e that solves the real and imaginary arts of the equation S ae ðþ=a 3 e ¼ /S a ðþs=/a 3 S. In general, a e will coincide with neither the most robable radius, nor the average radius, nor the radius of the average volume. For instance, for very small articles, it is not difficult to show that we will retrieve an equivalent article radius that is larger than the average radius and the radius of the average volume. For very small articles we may aroximate [5], ð7þ ρ [μm -4 ] 1 8 6 4 2 σ = 1.4 a = 8 nm n = 1.6 4 8 12 16 Particle radius [nm] Particle radius [nm] average radius (85 nm) radius of average volume (95 nm) estimated radius (17 nm) 4 8 12 16 Fig. 6. Log-normal size distribution for f=.2, a =8 nm, and (a) s=1.1, (b) s=1.4. The average radius and the radius of the average volume are indicated. Also the equivalent radius retrieved for a colloid of articles with n =1.6 in water (n m =1.33) is indicated.
A. García-Valenzuela et al. / Physica B 45 (21) 316 321 321 is exð3ln 2 sþ times larger than a v. Thus, for s=1.1, 1.2, 1.3 and 1.4 the retrieved radius is 2.8%, 1%, 23% and 4% larger than the radius of the average article volume. Of course, for other size distribution functions the relation between a e and a v will be different, but in any case, the value of a e will be well within the interval of article sizes, but towards the region of larger articles. For larger articles we can exect a e to be closer to a v and eventually it could be smaller. As an examle, in Fig. 6a and b we show a log-normal size distribution for f=.2 and a =8 nm, with s=1.1 and 1.4, resectively. We indicate in the figures the average radius, the radius of the average volume, and the equivalent articles radius retrieved for articles of refractive index n =1.6 in water. 7. Summary and conclusions It is ossible to retrieve the radius and refractive index of small and moderately sized, weakly absorbing articles, from the measurement of the real and imaginary arts of the effective refractive-index increment Dn eff er unit volume fraction occuied by the articles. If one also measures the contribution from otical absortion to the imaginary art of Dn eff it is ossible to retrieve the size and comlex refractive index of highly absorbing articles. We analyzed the uncertainty on the retrieved arameters due to the uncertainty on determining the volume fraction of the articles and discussed the case when the colloidal articles have a wide size distribution. Acknowledgments We acknowledge financial suort from Dirección General de Asuntos del Personal Académico de la Universidad Nacional Autónoma de México through grant PAPIIT IN 1239. References [1] L.L. Foldy, Phys. Rev. 67 (1945) 17. [2] Theodore Provder and John Texter, editors; Particle Sizing and Characterization ACS Symosium Series 881, American Chemical Society and Oxford University Press, 24. [3] R.G. Barrera, A. Reyes-Coronado, A. García-Valenzuela, Phys. Rev. B 75 (18422) (27) 1. [4] A. García-Valenzuela, R.G. Barrera, E. Gutierrez-Reyes, Ot. Exress 16 (24) (28) 19741. [5] C.F. Bohren, Absortion and Scattering of Light by Small Particles, Wiley, New York, 1983. [6] D.P. Almond, P.M. Patel, Photothermal Science and Techniques, Chaman & Hall, London, UK, 1996. [7] D.M. Hobbs, N.J. McCormick, Al. Ot. 38 (1999) 456. [8] A. García-Valenzuela, Ot. Lett. 34 (14) (29) 2192.