Color effects of uniform colloidal particlesof different morphologies packed into films
Niels P. Ryde and Egon Matijevi6
The experimentally determined chromaticities and reflectance spectra of films consisting of uniformellipsoidal or spherical colloidal hematite particles are compared with calculated values and are found tobe in good agreement. The theoretical treatment of the light-scattering problem involves the Mie theoryfor the spheres and the T-matrix method for the ellipsoids. The reflectance spectra for the pigment filmsare calculated through the use of the Kubelka-Munk analysis.
Introduction
The ability to estimate theoretically the hues of finelydispersed pigments, either in suspensions or in films,is of both practical and fundamental importance.It has long been recognized that the color of a colloidparticle depends on its refractive index and size.'The angular scattering intensity can be used todetermine the size distribution of sufficiently uni-form dispersions of spherical particles, and con-versely the same quantity can yield the refractiveindex at a given wavelength and particle diameter.2
Because many pigments consist of colloidal miner-als, hematite (-Fe 2O3) dispersions consisting of uni-form particles represent convenient model systemsfor the evaluation of optical properties of such coloredmatter. Thus Kerker et al. 3 demonstrated the rela-tionships of the absorption index to the apparentcolor of single spherical hematite particles and ofsuch particles packed in films. Furthermore, goodagreement was found between measured and com-puted extinction spectra of hematite dispersions ofnarrow size distributions when the correct refractiveindex was used. 4
The objective of this study is to compare experimen-tal and calculated color properties of pigment filmsproduced by the deposition of colloidal hematite
N. P. Ryde is with Rohm & Haas, Spring House, Pennsylvania19477; E. Matijevid is with the Center for Advanced MaterialsProcessing, Clarkson University, Box 5814, Potsdam, New York13699-5814.
Received 3 January 1994; revised manuscript received 9 May1994.
particles of different size and morphology (spheresand prolate ellipsoids) on a white substrate. For thispurpose it is necessary for particles to be uniform inshape and of reasonably narrow size distribution.The procedures for the preparation of such hematitedispersions have already been described.5 6 The pro-grams given by Bohren and Huffman7 were used inthe calculations of scattering and absorption charac-teristics of spheres, and a procedure known as theT-matrix method was employed to obtain the sameinformation for ellipsoids as described by Barber andHill.8
Theory
The chromaticity of a dispersed pigment can readilybe obtained from transmission spectra, and that of apigment film from reflectance spectra. One maycalculate the latter for a given particle system byusing the scattering and absorption coefficients, Sand K, derived from the angular scattering functionsand scattering cross sections. The procedure used inthe theoretical analysis is as follows: (a) computethe angular scattering functions and the scatteringcross sections for a single particle; (b) calculate S andK, and thereafter the reflectance spectra; and (c)calculate the corresponding chromaticity coordinatesfrom the reflectance spectra. (The conversion tochromaticity is the same for both experimental andcalculated data.)
Light Scattering from a Single Particle
To evaluate the experimental data to be described, wefound it necessary to use the Mie theory because thea-Fe2O3 particles have a high complex refractiveindex. Furthermore, the method employed to com-pute the optical properties of ellipsoids can be viewed
as a generalization of the Mie theory. The equationsrequired for the computations of angular intensitiesand for the cross sections are numerous and cumber-some. The procedures are well documented in thebook by Bohren and Hufffman7 for spheres and in thebook by Barber and Hill8 for ellipsoids.
The extinction cross section of a single particlerepresents the total power removed from a unitintensity incident beam impinging on it, and it isrelated by
Cext = Csca + Cabs (1)
influence the reflectance, whereas light scattered inparallel directions is canceled out. The reflectancefrom a pigment film, being thick enough to ensurethat no light is scattered or absorbed by the back-ground, is given by
(3)
When the reflectance is influenced by the back-ground, the following expression is obtained for a filmof thickness d:
where the subscripts ext, sca, and abs representextinction, scattering, and absorption, respectively.The values of Csa can be calculated from
Csca = k2 f [il(O) + i2(0)]sin 0 dO, (2)
where i1 and i2 represent the vertically and horizon-tally polarized components of the scattered light.The scattered intensity by unpolarized light is thusgiven by
i. = (il + i2)/2.
The wave vector, k, is defined by k = 2rr/X, in which Xis the wavelength of the light in the surroundingmedium.
For spheres, quantities i, i2, Cabs, and Csca werecomputed with the program CALLBH,7 which is aFORTRAN routine based on the Mie theory. For ellip-soids, the FORTRAN programs T18 and T28 yielded il, i2 ,Cabs, and Csca; this approach is based on the T-matrixmethod. The latter begins with an expansion of theincident, scattered, and internal fields in terms of thespherical vector harmonic functions. In calculatingil and i2 for an ellipsoid, one must consider theorientation of the particle with respect to the incidentbeam. Because all computations are performed tosimulate pigment films, it is assumed that the semima-jor axis of the prolate ellipsoid lies parallel to thesubstrate (i.e., perpendicular to the incident beam).The i and i2 functions then represent an average overall such in-plane orientations.
(4)
where Rg is the reflectance of the background. Foran ideal white background Rg = 1, and for an idealblack background Rg = 0. Equations (3) and (4) arevalid for nonglossy films. Additional corrections arerequired to account for glossy interfaces.10 The ab-sorption coefficient, K, for spheres is given by3
3Cabs
27rr 3 (5a)
where r is the radius of the sphere. For ellipsoids,
Cabs
27wab2(5b)
where a and b are the semimajor and semiminor axesof the ellipsoid. The scattering coefficient, S forspheres and for ellipsoids reads3
9=sca(1 - COS 0)S = I ~~~~~~(6a)1.67rr-
9Csa(1 - cos 0)(6b)
16rrab2
respectively. One obtains the relations for ellipsoids[Eqs. (5b) and (6b)] by replacing the volume of asphere by the volume of the ellipsoid. The quantitycos 0 is called the asymmetry factor, which representsthe entire scattered light projected in the direction ofincidence, and it is defined by2
Light Scattering from a Pigment Film
The approach given by Kubelka and Munk9 is com-monly used in the estimation of the reflectance ofpigment films. The derivation of the appropriateexpression is summarized by Judd.10 Here it isassumed that only the component of the diffuse lightperpendicular to the pigment film will significantly
k 2 |;[i 1 (O) + i2(0)]cos 0 sin 0 do
(7)cos 0 =
k2 | [i1(O) + i2(0)]sin do
If this quantity is greater than zero, then more lightis scattered in the forward direction than in reverse.
The physical characteristics that are used to describecolors are the dominant wavelength, luminance, andpurity (see, e.g., Billmeyer and Saltzman"1 ), whichcorrespond in a general way to the psychologicalattributes of hue, brightness, and saturation. Thedominant wavelength and the purity are used to-gether to obtain the so-called chromaticity diagrams.In all calculations that follow, the Commission Inter-national de l'Eclairage system has been used for thecalculations of the tristimulus values X, Y and Z,which are defined by
x= R(X)E(X)Y(X)dX,
= R(X)E(X)y(X)dX,
= R(X)E(X)i(X)dX. (8)
The functions E, x-, y-, and z can be found in standardtables (e.g., those in Ref. 11). The energy distribu-tion function, E, is tabulated for different types oflight sources. The source corresponding to simu-lated overcast sky at daylight (D65) was used in thisstudy. The reflectance spectra R(A) are suppliedeither from experiments or calculations. The chro-maticity coordinates, x andy, are obtained from
Xx X+ Y+ Z
yY X + Y + Z' (9)
Experimental
The spherical hematite dispersions were preparedaccording to a previously described procedure. 5
Accordingly, solutions of FeCl3 in concentrations of1.2 x 10-2, 1.8 x 10-2, and 2.4 x 10-2 mol dm-3containing 1.0 x 10-3 mol dm-3 HCl were aged at100 C for 24 h. Four dispersions of ellipsoidalhematite particles were obtained by aging, at 100 'Cfor 3 days, these solutions, each containing 2.0 x 10-2
mol dm-3 FeCl3 solution and varying amounts of
NaH 2 PO 4 , i.e., 1.0 x 10-4, 2.0 x 10-4, 3.0 x 10-4, and4.2 x 10-4 mol dm- 3, respectively. All dispersionswere purified by the removal of the supernatantsolutions by means of centrifugation, followed byrinsing the precipitate with distilled water. The sizedistributions of ellipsoidal and spherical particleswere evaluated from scanning and transmission elec-tron micrographs, respectively.
For chromaticity and reflectance measurements, afraction of the suspension was filtered through a0.4-pum millipore membrane, and the collected solidswere then dried in a desiccator for 24 h. The chroma-ticities were evaluated by means of a Milton RoyColorscan II instrument, which is a double-beamscanning reflectance spectrophotometer that can yieldboth reflectance spectra and chromaticity diagrams.The instrument was calibrated with a white tileceramic standard having chromaticity coordinatesx = 0.3138 andy = 0.3310.
ResultsThe size characteristics of four ellipsoidal particles(A-D) and of three dispersions of spherical hematiteparticles (E-G) are listed in Table 1. The scanningelectron micrographs in Fig. 1 illustrate the unifor-mity in shape and size of two of these samples. Therefractive index of hematite in air,3 4 required in thecalculations, is shown as a function of the wavelengthin Fig. 2.
Figure 3 gives the calculated scattering cross sec-tions for single particles of types A, B, and C. Only apart of the curves for C could be obtained, because inthis case it became increasingly more difficult to reachconvergence in the routines that compute the Tmatrix, which is also the reason for omitting calcula-tions of the D system. It is noteworthy that thedifficulties are more sensitive to the ratio of the majorto the minor axes than to the particle size. Theresults show an interesting feature in the range550-600 nm, where a sharp drop appears in theabsorbance cross sections, accompanied by a steepincrease in the scattering cross sections, resulting in adecrease of the calculated K/S values with a simulta-neous increase of R. The same types of plots areshown in Fig. 4 for single spheres (E-G), which arecharacteristic of rather small particles.
The experimentally obtained reflectance spectra forfilms of hematite particles coded A-G, together with
Table 1. Description of Preparation Conditions and Size Characteristics of Hematite Particles
FeCl3 NaH 2 PO4 HCl Aging d ± r da ( db a AspectSample (mol dm- 3 ) (mol dm- 3 ) (mol dm- 3 ) Days (nm) (nm) (nm) Ratio
A 2.0 x 10-2 1.0 x 10-4 3 335 33 220 18 1.5B 2.0 x 10-2 2.0 x 10-4 3 467 ± 37 216 13 2.2C 2.0 x 10-2 3.0 x 10-4 3 556 ± 38 180 16 3.2D 2.0 x 10-2 4.2 x 10-4 3 682 ± 67 166 12 4.1E 1.8 x 10-2 1 x 10- 3 1 61 9F 2.4 x 10-2 1 x 10-3 1 74 13G 1.2 x 10- 2 1 x 10- 3 1 53 8
Fig. 1. Scanning electron micrographs of hematite samples (a) Band (b) D, prepared under conditions given in Table 1.
3.0,
2.0
1 .0
0.040( 500 600 700
X/nmFig. 2. Complex refractive index for hematite in air.3 '4
Fig. 3. Calculated scattering and absorbance cross sections forsingle particles having refractive index of hematite of types A and B(Table 1). Cross sections for X > 570 nm are also shown forsample C.
the calculated ones for A, B, and E-G, are displayed inFig. 5.
The color quality of a film must also depend on thethickness of the pigment layer. For this reasonexperiments were carried out with varying amountsof hematite particles deposited on the membrane.Figure 6 shows the measured reflectance spectra ofellipsoidal particles A (Table 1) for four differentamounts of deposited solids, along with the calculateddata that use Eq. (4) for several layer thicknesses.It is noteworthy that reflectance does not changemuch at d > 0.5 [Lm, which also agrees very well withthe experimental trends. In these calculations it isassumed that the background substrate (polymerfilter membrane) behaves as an ideal white surface,i.e., Rg= 1.
3.5E-003
3.OE-003
E 2.OE-003
u .5E-003
1.OE-003
5.0E-004
0.E-000400 500 600 700
X/nFig. 4. Calculated scattering and absorbance cross sections forsingle spherical particles E-G having refractive index of hematite(Table 1).
Fig. 5. Experimental and calculated reflectance spectra for pig-ment films of hematite particles, described in Table 1. Theamounts of solids in milligrams per inverse square centimeters inrespective films: A, 2.0; B, 3.3; C, 2.6; D, 0.90; E, 0.90; F, 0.47; G,0.67.
Analogous plots for films of spherical particles aredisplayed in Fig. 7. There is little change in thereflectances of two samples containing differentamounts of the pigment. It appears that the agree-ment between the experimental and theoretical trendsimproves as the layer becomes thinner.
A comparison of Figs. 6 and 7 also points to someinteresting differences between films of ellipsoidaland spherical particles. In the latter case, the reflec-tance is much less dependent on the thickness of thefilm than for ellipsoidal particles used in these compu-tations. The smaller spherical particles appear moretransparent. The transparency of a pigment film is
100
80
60
40
20
0 '400 5oo 600
X/nm700
Fig. 6. Reflectance spectra of films of ellipsoidal hematite particleA (Table 1), calculated for different layer thicknesses d. Experi-mentally measured values are for films made up with 2.0 ( 0), 1.0(o), 0.67 (A), and 0.16 mg cm- 2 (0) of solids.
80
60
40
20
0400 500 600
X/nm700
Fig. 7. Same plots as in Fig. 6 for films of spherical hematiteparticles E. Experimentally measured values are for films madeup with 0.90 (0) and 0.71 mg cm-2 () of solids.
greatly affected by the scattering cross section of asingle hematite sphere. Figure 8 shows calculatedscattering efficiency, Qext = Cext/r2 rr, as a function ofthe particle diameter for different wavelengths. Theinvestigated dispersions correspond to the smallestsize shown in this diagram. The optical propertiesof large spheres are similar to those of ellipsoids (Fig.3), which is due to the larger dominant size of thelatter. The results shown in Figs. 6 and 7 indicatethat Eq. (3) may not be applicable to films of transpar-ent small particles (such as E-G), because the back-ground contributes significantly to the reflectancespectra.
The experimentally determined chromaticities arecompared with the calculated ones (for which reflec-tance spectra could be obtained) in Figs. 9 and 10.The intercept with the boundary of the line connect-
Fig. 8. Extinction efficiency Qext = Cet/r 2 rT as a function ofdiameter and wavelength, calculated according to the Mie theoryfor spheres having refractive index of hematite.
The color purity of the pigment depends on the- - l . distance of the experimental point from the bound-
ary.In all stated systems, there is a reasonable agree-
- -\ - ment between the calculated and experimental data..... .... This is especially true with respect to the dominant
wavelength, as summarized in Table 2, which also0.0 0.2 0.4 0.6 0.8 lists the corresponding purities. It is interesting to
X note that sample B is characterized by the longestChromaticity diagrams for films of ellipsoidal particles dominant wavelength. The same sample also dis-ircles represent experimental points. The dominant wave- played the longest wavelength at which the value ofgiven by the intersection with the boundary of the line the absorption cross section decreased sharply.
ng the neutral and experimental points. The calculated Figure 11 shows the effect of the film thickness on[cities are for system A (0) and for system B (), respec- the chromaticity of ellipsoidal and spherical particles,
the reflectances of which are given in Figs. 6 and 7.
ing the neutral point (obtained from the standard tilesupplied by the instrument company and located atx = 0.3138 and y = 0.3310, using light source D65)with the sample point yields the dominant wavelength.
1.0
0.8
1.0
0.8
0.6
y0.=
0.2
0.0
Fig. 10.
0.6
y0.4
0.2
0.0-If -1 1 ~~0.0 0.2 0.4 0.6 0.8
- - -. - -. - . ____ !Fig. 11. Chromaticity diagram for hematite particles in films of: j _ t different thicknesses as given in Figs. 6 and 7. Circles and squares
refer to measured quantities for ellipsoidal and spherical particles,respectively. The solid curve is calculated for ellipsoids of the
0.0 0.2 04 6 * same size but increasing film thickness, whereas the dashed curveX is for spheres of the samiie diamweter as E. Te eiid of both curves
Same plot as in Fig. 9, except for spherical particles E-G. nearest to the boundary refers to an infinitely thick layer.
Circles refer to experimental values of the ellipsoids,clearly indicating that the dominant wavelength isindependent of the thickness of the pigment layer butthat the color purity improves with the added amountof hematite. The solid curve was calculated for thesame system through the use of Eq. (4). The end ofthis curve nearest to the boundary represents aninfinitely thick film. In contrast, the dashed curverefers to a film of spherical particles corresponding tosample E, whereas the experimental data are shownwith squares. Again, the end closer to the boundaryof the calculated curve represents the infinite layerthickness. In this case both the color purity anddominant wavelength are strongly dependent on thethickness of the pigment layer.
Conclusions
The proposed procedure for the theoretical chromatic-ity evaluations of uniform, monodispersed colloidalparticles is in agreement with the experimental data.The method should make it possible to customizecolloidal systems to produce pigments of desiredcolors. The color matching between calculated andexperimental samples could also be used to assess thecomplex refractive index of specific colloidal samples.
Ellipsoidal particles exemplified by samples A andB also indicated that larger particles tend to give adeeper red color. However, this trend is not general,as seen from samples C and D. Obviously, the hue ofpigments of nonspherical particles depends on theirspecific dimensional characteristics. The relation-
ship of the morphology to optical properties of suchparticles is not trivial, and it has to be evaluated foreach case.
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