The dependence of optical properties on the structural composition of solar absorbers: Gold black

Solar Energy, Vol. 21. pp. 465--468 © Pergamon Press Ltd.. 1978. Printed in Great Brilain

O03~.X/78/12014W~651502.00[O

THE DEPENDENCE OF OPTICAL PROPERTIES ON THE STRUCTURAL COMPOSITION OF SOLAR ABSORBERS: GOLD BLACKt P. O'NEILL,~ A. IGNATiEVand C. DOLAND Department of Physics, Universityof Houston, Houston, TX 77004, U.S.A. (Received 20 February 1978;revision accepted 15 May 1978)

Abstract--Specimens of gold black were produced under controlled laboratory conditions so as to study the dependence of their optical properties on the particle-like nature of the material. A theoretical model incorporating the particulate nature of the gold black films has been applied to describe their optical properties. This theory is related to the experimentally measured optical properties and its general nature is contrasted to previous theories of the optical properties of gold black. l. INTROIN.JCTION

Recent interest in the utilization of solar energy has prompted this investigation into an understanding of the basic physical concepts underlying the absorption of solar radiation by a material. Materials known as "solar blacks" (gold black, carbon black, and many others) have been classically used as good solar absorbers[l]. These materials, generally stable only below ~30&C, are noted for their extremely low reflectivity throughout the solar spectrum, R < 1 per cent, and yet they are absorptive enough to provide complete extinction within a few microns of material. The basis for such ideal optical properties lies in the particulate, low density nature of the films[2]. However, the precise dependence of absorptance on microscopic structure is unknown. The analysis of the dependence of solar absorptance on absorber microscopic structure has been undertaken for a test solar black: gold black. The information obtained will be useful in the development of a hightemperature stable solar absorber through the concept of modifying the surface morphology of a material so as to obtain the optimal structural properties for maximum solar radiation absorption. 2. ~

A

L

Gold black films were deposited by inert-gas evaporation onto glass and metallic substrates and electron microscope grids as described previously[2,3]. The evaporations were done at Helium pressures of 1-20 torr and resulted in films consisting of loosely packed gold particles whose size increased from 40 to 95/~. The structural detail of the films was monitored by transmission electron microscopy (T.E.M,), and it was found that the films consist of crosslinked chains-of-spheres [3]. Median particle size was determined from the T.E.M. Micrographs[3] and from X-ray particle size broadening[4]. The film thickness t was determined using scanning electron microscopy (S.E.M.) as described previously[3]. tSupported in part by ERDA and The University of Houston Solar Energy Laboratory. :~NSFTrainee. SE vot. 21, No. 6 - 8

465

Spectrophotometric techniques[5] were used to measure the transmittance and reflectance of the gold blacks in the wavelength range of 0.35-2.4p, The measurements were made on a Beckman DK-2A spectrophotometer and were corrected for the substrate glass cover slips. These measurements have shown that the reflectance R of the gold blacks is less than i per cent at normal incidence over the noted wavelength range. Thus, the absorption coefficient a may be determined by: T(A) = e - ' ~ "

(1)

where T(~t) is the measured transmittance. 3. 'rH]~R£TICJd~ MODEL

Examination of a gold black T.E.M. micrograph readily establishes the fact that gold black consists of chains-of-spheres that interconnect so as to form completed (conducting) and interrupted (capacitive) strands. This structure led Zaeschmar and Nedoluha[6] to develop the capacitor theory of the optical properties of gold black. Using this theory, good agreement between theory and experiment is achieved for wavelengths between 3 and 100# by assuming that a single capacitor gap approximates the average gap between interrupted strands. However when the capacitor theory is applied in the solar spectrum (0,3-2.5#), the theory does not agree well with experiment [7]. l'he basic idea of a model based on gold black strands seems reasonable but requires modification. The model suggested in the following paragraphs achieves this modification by assuming that each gold black strand (chain-of-spheres) can be approximated by a spheroid with semimajor to semiminor axis ratio m, and that a distribution of various spheroid sizes (spheroids with Various values of m) exists for the complete gold black film. The theoretical model to be applied to the optical properties of gold black is a generalization of the theory presented by Genzel and Martin[8] and is referred to as continuum theory. When continuum theory is applied to. the case of spheroids (with various semimajor to semiminor axis ratios m) imbedded at random orientations in

466

P. O'NEILLet al.

air, it can be shown that: 1 - [ + d3 ~ f(m)[(l + (e - l)I~(m))-' + 2(1 + (~ - 1)L±(m))-'] (0 =

"

1 -,f + 1/3 ~ f(m)[(1 + (~ - l)14(m))-' + 2(1 + (,~ - 1)L±(m))-']"

(2)

m

where (¢) = dielectric function of the composite = (~t) + (~2); ¢ = dielectric function of the spherical particle material = ~, + i~2; f(m) is the packing density of spheroids with semimajor to semiminor axis ratio m; f = ~ [(m) = total packing density. = La(m) =

~'(m2- l)-~[m(m 2- l)-'nln[m +(m 2- l)~n]- I] for m > I 1/3 for m= 1 t

L±(m) = [1 -/qt(m)]/2.

In eqn (2), Lu(L±) are depolarization factors[9] for a spheroid in a static, uniform electric field parallel (perpendicular) to its semimajor axis. Equation (2) applies at optical frequencies to a random orientation of spheroids (gold black appears to be isotropic in S.E.M.) provided the wavelength is smaller than the semimajor axis a. This limitation is violated at the shortest wavelengths (0.35/z) for m ~ 20. However, if the strands are aligned not at random, but along orthogonal axes normal to the incident electric field as shown in Fig. 1, the strand size limitation does not apply (provided the semiminor axis b is small). Thus, the physical significance of the longer strands (m > 20) is that they are conducting paths lying orthogonal to the incidentelectric field. Applying eqn (1) to gold black by associating the gold black strands (or chains-of-spheres) with a distribution of spheroids and using the fact that f,~ 1, eqn (2) reduces to: (~,(A)) = 1

(3b)

m

gl(m, A) = [82(m) + 2&(m)Li(m)~t(A) + L~(m) x

I,(x~l=]-'; &(m)

=

1-

L , ( m ) ; i = II or .L; p(m)Am = f(m)l.f.

Z

1' = 1.71'Au(l + Ila)

(4)

where YAu= d.c. collision frequency for bulk gold = 4.16 x I0 ~3sec-~; 1 = mean free path in bulk gold = 340 ,~; a = particle size in ,~ (from X-ray broadening or T.E.M.). The factor 1.7 arises due to the fact that y is an optical rather than a direct current collision frequency[12]. Thus, the dielectric function consists of Drude and interband terms:

(3a)

(ez(A)) = ~2(A)//3 ~ [g,(m, A) + 2g±(m, A)] o(m)Am where

In order to apply eqn (3) to gold black, a microscopic model of the gold strands is required to provide ~(A). For this, the dielectric function of bulk gold[10] was used with the exception that 1' (electron collision frequency) was calculated such that the small size of the gold particles could be accounted for as suggested by Kriebeg and Fragstein [ 11]:

~,(to) = P - ~op2/(to 2 + 1'2)

(Sa)

• 2(to) = top2y/(to(to2+ 72)) + (2' (to).

(5b)

Where for bulk gold P =7; too =plasma frequency= l a x 10'rsec-'; to =frequency (sec-~)=2~rdA; c = speed of light in vacuum; A = wavelength; ¢21(to)=interband contribution to the dielectric function. Equation (2) can now be examined quantitatively. The absorption coefficient a is proportional to (~2(A))/A for (¢~(A)) = 1.0. Thus, for an array composed only of particles of given m, a(m, ,~) = [Ez(,~)l,~]igu(m, ,~)+ 2g~.(m, ~)1. f.

Spheroids porolle I to g'

(6)

This coefficient is plotted in Fig. 2 as a function of 1.0

,~X

perpendicular to

Fig. 1. Orientation of spheroids of various m that is equivalent to a system of randomly oriented spheroids.

,~

O0

3

m=l

(Spheres)

4

m=5

m:lO

.5 ,8 .7 .891.0

m=20

1.5

20

3.0

h(bLm) Fig. 2. Absorption coefficient a (normalized to unity) associated with an array of spheroids of given m.

The dependence of optical properties on the structural compositionof solar absorbers wavelength A for arrays composed of randomly oriented spheroids of various axis ratios m. Each curve has been normalized in Fig. 2 such that a = 1 at its peak and it should be noted that each spheroid (each value of m) produces a different absorption peak. Figure 2 also indicates that a distribution of spheroids is capable of producing absorption over the entire solar spectrum. The longer spheroids (larger m's) produce broader absorption, peak at a longer wavelength, and have higher absorption magnitudes.

467

SAMPLE 14A .25

2O p {m) .15 llO

.05 aZStnLTSOF ~i'UCA~ON TO GOLDBLACK The distribution of spheroids (gold black strand lengths) in a gold black sample can be found by deter0 -"-~ I', ff 0 5 I0 15 20 25 (~ mining the p(m) of the model such that the experimental m absorption coefficient is fit by that predicted by the model equation (3b). A least squares fitting technique Fig. 4. Distribution of spheroids for a typical gold black sample. using a preselected set of discrete values of m chosen The histogramrepresents p(m) as determined using least squares fit to transmittance data of Fig. 3. The smooth curve shows a such that each associated absorption coefficient a(m, ,~) log-normal distribution function fit to the histogram. is linearly independent was utilized. For a typical gold black sample, the least square fitting inert-gas evaporated gold spheres forming the chains-ofof the transmittance (Fig. 3) resulted in the determination spheres strands which compose the gold black. of the distribution shown in Fig. 4. It should be noted that p(m) for 24< m 24). The area theory[6]. As a result, the capacitor model was also under the segment m = l to m = 24 was found to be applied to gold black in the solar spectrum[7] and =0.90. Therefore, the total area under the distribution showed significantly poorer comparison especially in p(m) was found to be ~-1.20 and is consistent with the light of having to envoke an unrealistically short electron ideal case in which this area is 1.0. collision time in the gold strands. The principal cause for The analytic form of the distribution of gold black the poorer agreement was the incompleteness of the gold strand lengths in our sample closely follows that of a black microscopic structure description utilized in the log-normal distribution (with independent variable n - l) capacitor model. Only completed (conducting) or interas shown in Fig. 4. For this distribution, the median rupted (capacitive) gold strands were used in that model. particle size is tfi = 4 and the geometric standard deviaIn comparison the spheroid model presented above more tion is tr = 3.5. The existance of a log-normal distribution fully describes the structure of the gold black in that it is consistent with the theory of coalescence[13] of small assumes a distribution of gold strand lengths. It does, however, reduce to the capacitor model upon the assumption of only 2 types of strands: (i) infinitely long 50 ones (m = oo) and (ii) those of one finite length. Under these conditions eqns (3a) and (3b) reduce to: 40

tA2 t.)

t= .84/~.L ~= 9OA 6.05 x 1014sec-I

30

~ 2o IE

0

I

.3

14

I

I

.5 .6

I

I

.8

I

1.0

I

1.5

I

I,

I

20 25 3.0

Fig. 3. Experimental (.) and theoretical (solid line) transmittance of a typical gold black sample vs wavelength, t is film thickness, ti is mean spherical particle size (gold black composedof chainsof-spheres) and y is colision frequency.

(~1) -~ 1.0

(7a)

(~2)--"[~2"f/3][pc + p~(1+ 2de, + d21el2y']

(7b)

where d = / 4 is related to a small capacitor gap between strands and from the value given in Ref. [6] corresponds to strands of approximately 35 spheres long (m = 35). pc and p~ are the fraction of completed and interrupted strands respectively. The applicability of the capacitor model to a description of optical properties in the IR as shown in Ref. [6] can be seen from Fig. 2. Contributions to the absorption coefficient from gold strands with m < 30 is negligible for A > 3.0#, i.e. measurements in the IR are not exception-. ally sensitive to the microscopic structure of the black, whereas, measurements in the solar spectrum are.

P. O'NEILL et al.

468

6. CONCLUSION The spheroid model of solar blacks presented here appears to provide more detailedknowledge of the effect of microscopic structure on gold black optical properties than was previously possible. Specifically, it indicates that absorption throughout much of the solar spectrum is due to the presence of a distribution of small 50-100 ,g, diameter gold strands or chains-of-spheres that are - 2 0 or less spheres long.

Acknowledgements--The authors wish to thank Dr. A. F. Hildebrandt for his interest and contribution to this work. They also thank Dr. W. W. Wendlandt for the use of the spectrophotometric instruments in his laboratory. RF.,FERI~CES

1. A. H. Pfund, J. Opt. Soc. Am. 23, 275 (1933). 2. L. Harris, The Optical Propenies o[ Metal Blacks and

Carbon Blacks (The Epply Foundation for Research) Newport, R. 1. Monograph Series No. 1 (1%7). 3. C. Doland, P. O'Neill and A. Ignatiev, J. Vac. Sci. Technol. 14, 259 (1977). 4. B. E. Warren, X-Ray Di~:raction. Addison-Wesley, Reading, Massachusetts (1967). 5. W. W. Wendlandt and A. G. Hecht, Reflectance Spectroscopy. Wiley, New York (1966). 6. G. Zaeschmar and A. Nedoluha, J. Opt. SOc. Am. 62, 348 (1972). 7. P. O'Neill, C. Doland and A. Ignatiev, Appl. Opt. 16, 2822 (1977). 8. L. Genzel and T. P. Martin, Sur[ace Sci. 34, 33 (1973). 9. J. A. Stratton, Electromagnetic Theory. McGraw-Hill, New York (1941). 10. M. L. Th/~ye, Phys. Rev. B2, 3060 (1970). 11. V. Kreibeg and C. V. Fragstein, Z. Phys. 224, 307 (1%9). 12. F. Abeles, In Optical Properties of Solids (Edited by F. Abeles), p. 93. North-Holland, Amsterdam (1972). 13. C. G. Granqvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1975).