method for the determination of the optical properties of...

2648 J. Opt. Soc. Am. A/Vol. 10, No. 12/December 1993

Method for the determination of the optical properties ofhighly conjugated pigments

J. L. Musfeldt*

Department of Chemistry, University of Florida, Gainesville, Florida 32611

D. B. Tanner

Department of Physics, University of Florida, Gainesville, Florida 32611

A. J. Paine

Xerox Research Centre of Canada, Mississauga, Ontario, L5K 2L1, Canada

Received December 30, 1992; revised manuscript received May 4, 1993; accepted June 2, 1993

Measurements of the near-normal solid-state-pellet reflectance of five commercially available color pigmentsare used to calculate the frequency-dependent complex index of refraction with the Kramers-Kronig relation-ship. Although it is conceptually simple, the Kramers-Kronig analysis is very sensitive to the choice of ex-trapolation constants that describe the reflectance behavior outside the accessible frequency range, especiallyin the ultraviolet. We develop a method to obtain accurate optical property data in the visible region by mini-mizing the difference between the absorbance calculated from the refractive index and the separately measuredabsorption spectrum through an appropriate choice of the high-frequency extrapolation exponent. Root-mean-square differences as low as 3% can be achieved.

PACS numbers: 78.20.-e, 78.20.Dj, 78.20.Ci, 7840.Ha.

1. INTRODUCTION

Highly conjugated molecules have received considerableattention in the past several years.' 6 Interest has fo-cused on the strong electronic features of (r,i.*) origin inthe visible region of the spectrum and the relationship ofchemical structure to perceived pigment color. Owing tothe intense optical absorption processes, these materialshave found applications in various optical devices, as wellas for coloring agents in textiles, paints, and plastics.",2' 5

The measurement and the manipulation of the uniqueabsorption characteristics of dyes and pigments is one ofthe major thrusts of color science. In a parallel manner,there has also been a strong interest in the prediction ofoptical properties of these materials, especially in thevarious solid-state media in which they are commonlyemployed.2'7 However, predictive methods for pigmentshave been hampered by a variety of complex problems.Whereas the optical properties of dyes are primarily dueto the absorption characteristics of the isolated molecules,the color properties of pigments are due to a combinationof absorption and scattering by pigment aggregates. Inpractice, the scattering effects are further complicated bythe aggregate size distribution.8 Thus, in order to build amore comprehensive model for pigment particle colorationproperties, predictive methods must be able to forecast thescattering properties as well.

A promising model for the simulation of pigment colorproperties uses Mie scattering theory to predict the ab-sorption and the angular dependence of the scattered ra-diation.' 2 Such a calculation requires both the real andimaginary parts of the refractive index of the pigment

particles as input parameters, in addition to an accurateknowledge of the pigment particle size distribution. Ascolor scientists well know, the optical data are not gener-ally available-even for common commercial pigments.Hence we sought a method of determining the frequency-dependent real and imaginary parts of the complex refrac-tive index with an accuracy adequate for input to a Miescattering calculation. It should be noted that althoughMie scattering is derived strictly for the scattering of aplane wave by an isolated, homogeneous sphere, these con-straints can be relaxed somewhat for practical use,'0 per-mitting the application of this theory to pigment materials.However, because the simulation of intrinsic pigmentcolor properties with Mie scattering is a part of our long-range modeling effort (and not the main subject of thispaper), these aspects will not be discussed in detail here.

In this paper we present reflectance measurementson pigment pellets, the Kramers-Kronig analysis to ob-tain the phase shift on reflection, and ultimately thefrequency-dependent optical constants. The merit of thismethod is that, unlike many other techniques, it providesa framework for the estimation of both the real and imagi-nary parts of the complex index of refraction. Althoughconceptually simple, the Kramers-Kronig analysis is verysensitive to the extrapolations describing the reflectancebehavior outside the commonly accessible frequencyrange. 131 5 This sensitivity emphasizes the importantcontribution of high-energy interband transitions to thetotal oscillator strength of conjugated dye and pigmentmaterials. Hence we developed a method to aid in theselection of a reasonable extrapolation behavior in theinterband frequency regime.

0740-3232/93/122648-10$06.00 © 1993 Optical Society of America

Musfeldt et al.

Vol. 10, No. 12/December 1993/J. Opt. Soc. Am. A 2649

0

II

0(a)

(b)

* 2Si02 xMoO3x=7-24

R=H

R = -SO2NH(CH2)17CH3

(c)

0OCH3 ii OCH3

fNH -S02 A =N-CH-C-NH CI

OCH3 OCH3

(d)

Fig. 1. Chemical structure of the pigments used in this study.The Chemical Abstracts Service registry numbers for eachpigment are (a) Hostaperm Pink, 00980-26-7; (b) Fanal Pink,75627-12-2; (c) PV Fast Blue, 00147-14-8; Neopen Blue,39702-40-4; (d) Novoperm Yellow, 12225-18-2.

This method can be briefly summarized as follows: AKramers-Kronig analysis of the power reflectance of thepellet samples is used to obtain a predicted bulk pigmentrefractive index and absorption coefficient, while trans-mittance measurements on the dilute pigment/toner filmsyield the effective absorption coefficient of these films.On account of the sensitivity of the Kramers-Kroniganalysis to the choice of interband extrapolation, we usethe film spectra to fix the exponent of a power-law ex-trapolation by closely matching the absorption coefficientsfrom these two sets of data. Using this procedure, wewere able to obtain a reliable (and Kramers-Kronig-consistent) determination of both the real and imaginaryindices of refraction in the visible-frequency region. Theusefulness of this method is demonstrated on a represen-tative set of commercially available pigments.

2. EXPERIMENT

A. Pigment SamplesFive pigments were chosen as samples on which to test ourmethod: PV Fast Blue B2G-A, Hostaperm Pink E, andNovoperm Yellow from Hoechst, and Fanal Pink D4830and Neopen Blue NB802 from BASE The chemical struc-ture of each pigment is shown in Fig. 1. Note that PVFast Blue is the well-studied copper phthalocyanine.' 61 2

For each of these materials the pigment particle sizewas measured to be -100 nm. Note that the particle sizeis small compared with the wavelength of light in thesemeasurements. In addition, the loose pigment particlesize is appreciably less than the bulk optical absorptiondepth at the wavelengths of strong absorption, such as thecolor band. These considerations suggest that the extinc-tion is likely to be dominated by absorption. The extentto which the pigments are agglomerated by the pelletiza-tion procedure (described below) is discussed briefly inSection 5.

We made pellets from loose pigment powders in order tohave samples suitable for reflectance studies. Thepowders were compressed at 0.88 GPa in an evacuated die;the pressure was maintained for 2 min and then released.Pellets were typically 1-2 mm thick and 1.6 cm in diame-ter. A comparison of pellet densities to bulk densities(measured with an AccuPyc 1330 instrument) indicatesthat the pressed pigment pellets are, on average, 93% asdense as a bulk sample. X-ray diffraction spectra beforeand after pelletization confirmed that the pigment crys-talline polymorphs were preserved during the pressuretreatment. The infrared spectra of the loose powder andthe pressed pellets also displayed similar vibrational fre-quencies, implying that the pigment chemical and crystal-lographic properties were not changed by the pelletization.

B. Reflectance MeasurementsNear-normal (specular) reflectance measurements wereperformed on the pigment pellets from 500 to 45 000 cm-'at room temperature. The midinfrared spectra werecollected on a Bruker 113-V Fourier-transform infraredspectrometer. Measurements at higher energy were madeon a modified Perkin-Elmer grating spectrometer, whoseoperation has been described in detail elsewhere.22 Toimprove the accuracy of the reflectance in the near ultra-violet, we coated the samples afterward with 200 nm ofaluminum and measured the reflectance of the coatedsample. This auxiliary measurement permitted a first-order correction to the spectra for the scattered light fromthe uneven pellet surface. A correction was also made forthe known reflectance of aluminum. It is our experiencethat this procedure corrects for a majority of the surfaceirregularities in this frequency range.

C. Transmittance MeasurementsThe total transmittance spectra (specular and diffuse)were taken on xerographic solid-area images consisting ofwell-dispersed pigment (1-5% loading, uniform coverage)fused in a polymeric resin that was supported on a 100-gm-thick transparent plastic substrate. We call thesesamples pigment/toner films. These samples were mea-sured on a Milton Roy Diano MatchScan II integratingsphere instrument by E. N. Dalal and S. Blaszak at XeroxWebster Research Center. In this mode, scattering ef-fects were minimized by the large detector collectionangle (0-90°). The specular transmittance P is collected0-3.4° from the normal; the diffuse transmittance N iscollected from the remainder of the detection angle.There is little backscatter or reflectance, so the totaltransmittance (P + N) gives a good measure of the ab-sorptive component, independent of scattering in thesefilms. The measured optical density was -1.0 at strongly

Musfeldt et al.


absorbing wavelengths and nearly zero at nonabsorbingwavelengths. These data were recorded over the 380-700-nm range with a 10-nm bandpass. The effectiveabsorption coefficient is related to the measured trans-mittance by

T (1 - R)2 exp(-aefffh)1 - R 2 exp(-2aefffh)

and the extinction coefficient K is

K () = 2[R(w)] 2sin[O(w)]1 + R(W - 2[R()] 12cos[O(w)] (7)

The absorption coefficient a can be obtained as

(1)

where aff is the effective absorption coefficient, R is thereflectance of the pigment/toner composite system, h isthe thickness, and f is the volume fraction of pigment inthe sample. Neglecting the contribution of the denomina-tor because the reflectance of the film is low, we can writethe effective absorption coefficient of these films as aeff =-(1/hf)ln[T/(1 - R)2] = -(1/C)ln T, where the constantC contains the sample thickness, the pigment loading inthe binder, and any contribution from the binder itself tothe absorption.

3. METHOD OF ANALYSIS

A. Kramers-Kronig AnalysisFrom a Kramers-Kronig transform of the reflectancedata we calculate the index of refraction n(w) and the ex-tinction coefficient K(C) as functions of frequency. Theseproperties are fundamental material characteristics, asthey effectively define its optical response to light.

The Kramers-Kronig relations are based on the require-ment of causality and the fact that the real and imaginaryparts of a response function are always related through adispersion relation. 3 -5 For reflectance measurementsthe response function r() is the product of an amplitudeand a phase:

r(Q) = Ef/EindCd = p(Q)exp[i6(Q)], (2)

where E is the incident or reflected electric-field vector.The measured reflectance is

R(Q) = IErn/Eincid12 = p2(&), (3)

and the Kramers-Kronig integral relates the phase shiftto the reflectance:

( ) = ln[R(d')] - ln[R(w)] do. (4)

Although our data were collected over a wide frequencyrange (500-45 000 cm-'), extrapolation beyond the fre-quency interval of interest is necessary, as the integral inEq. (4) extends from zero to infinity. The details of theseextrapolations and the effect on the resulting n and K inthe visible-frequency regime are the subject of Subsec-tion 3.B.

Using

(n - 1) + K 5r(w) = (n + 1) + K

one can obtain the real and imaginary parts of the com-plex refractive index. The refractive index n is

n ( = + R~cs1 - R(w) (6)n~) 1 + R(w) - 2[R(w)]`1cos[6(w)]' (6

a(w) = 41TKc, (8)

where w is the frequency in wave numbers (cm-'). Thus,by a Kramers-Kronig analysis of reflectance data, we canobtain the optical constants of a material.

B. Method of ExtrapolationThe integral in Eq. (4) can be broken into three parts:

Cormin jamas

0(w) = J I(W)dw + J I(W)dw + J I()do, (9)° °J~~~'min °ma.

where I(w) is the integrand and (Omin and C)max are theminimum and maximum frequencies of available data, re-spectively. The middle term is evaluated numerically bytrapezoidal integration.2 3 The two other integrals arecalculated with extrapolation procedures.

High-Frequency ExtrapolationThe high-frequency extrapolation is meant to account fortwo distinctly different regimes: the interband region,where there is some (unknown) contribution to the opticalresponse from interband transitions, and the higher-energy free-electron region. 315 It is common to modelthe reflectance with power laws in because then thethird integral in 'Eq. (9) can be evaluated analytically byexpansion in a power series.'3 In the interband region,from 45 000 to 1.0 x 106 cm-', the reflectance is modeledas

R(w) = Rf(com m./co)E, (10)

where Rf and ma, are the reflectance and the frequencyof the last measured data point, respectively, and E is anadjustable exponent, typically taking on a value betweenzero and four. Above WA = 10 X 106 cm', free-electronbehavior

(11)R(W) = Rf'(wA/w) 4

was assumed.' 3 '15 Here Rf' is defined as

Rf' = Rf (Omax/oA)E, (12)

and CoA is the frequency at which free-electron behavior isobeyed. Hence the high-frequency extrapolation of theKramers-Kronig analysis has two adjustable parameters:E, the interband exponent, and CoA, the frequency of free-electron behavior. A reasonable choice of these two pa-rameters is critical to the success of this method.

Figure 2 displays the reflectance of a typical pigmentsample in which several high-frequency (interband) ex-trapolations in Eq. (10) are shown (E = 0,1,2,4). Whenthe exponent is large, the cutoff is very steep and signifi-cantly less area is included in the calculation of the phaseintegral. This choice results in the assignment of most ofthe oscillator strength to energies below W)A. Con-

Musfeldt et al.


FREQUENCY (cm- 1 )

104106

zI-U

uJ-JU-w

0.16

0.12

0.08

0.04

105

10

103 102

102 103 104

Fig. 2. Typical reflectance spectrum at room temperature. Thesent various extrapolations at high and low frequencies.

versely, if the exponent is artificially low, a significantportion of the total oscillator strength may be moved toenergies far greater than that of the color band.

It is expected that highly conjugated materials shouldhave intense absorptions in the ultraviolet and beyond,and thus a significant amount of spectral weight at highfrequency. The difficulty lies in determining how muchoscillator strength lies above CoA. Hence we have devel-oped two criteria for the optimal choice of E in these ma-terials. In the first the exponent E is chosen to minimizethe difference between the absorption coefficient as calcu-lated from the Kramers-Kronig analysis of reflectanceand the effective absorption coefficient as inferred fromthe specular and diffuse transmittance measurements(P + N) of the pigments in the binder. As an alternativecriterion one could choose to minimize the differencebetween the position of the minimum in the Kramers-Kronig-determined a(co) on the high-energy side of thecolor band and the corresponding minimum in the aeff(w)spectrum of the pigment in the toner with the necessaryrestriction that the calculated phase in Eq. (4) be positive.The motivation for considering this second criterion isthat this minimum is strongly affected by the extrapola-tion, whereas the location of the color-band maximum isweakly dependent on the extrapolation. However, thefirst criterion gave the more reliable color property resultsoverall, hence that method and the corresponding resultswill be presented here.

Low-Frequency ExtrapolationThe method of extrapolation at low energy in Eq. (4) mustalso be addressed. For nonconducting materials it iscommon to approximate R(w) as a constant to zero fre-quency'"'` (Fig. 2). To test the applicability of this ap-proximation for pigment and dye materials, we measuredthe infrared reflectance and included these data in thecalculation of the phase integral. The additional data hadlittle effect on the resulting optical constants in thevisible-frequency range because there are no major elec-tronic transitions at low energy. Thus we conclude that,for nonconducting samples such as conjugated dyes andpigments, the low-frequency reflectance (below the optical

ELENGTH (nm)solid curve corresponds to measured data, and the dashed curves repre-

absorption threshold) can be assumed constant, with neg-ligible error to the optical constants in the visible region.

C. Validity of the Dispersion RelationsIn this subsection it is our goal to motivate the use of theKramers-Kronig dispersion relations to analyze the powerreflectance of our bulk pigment samples and to obtain thefrequency-dependent response functions. It is clear thatwith a pellet composed of compressed pigment powder wecan expect to measure only average optical properties, notthe principal components of the dielectric tensor, as mightbe obtained from single-crystal samples. However, formany applications involving pigment or dye materials themolecules are in a random orientation, so that a poly-crystalline average is appropriate. (This is the case withdilute pigment/toner samples.) Consequently, data ob-tained from such a technique are likely to be extremelyuseful for understanding the color properties of pigmentsin a wide variety of applications.

Because the extinction is the sum of both absorptionand scattering effects,'0"2 the amount of internal (diffuse)scattering in these bulk pellet samples is an importantconcern. There is a large volume of literature on thissubject that suggests that an explicit inclusion of scat-tering effects might be important for our analysis."" 2

However, we believe that diffuse scattering is small inthese samples for several reasons. First, in the cases dis-cussed here the apparent density of the pigment pellet isvery close (93% on average) to the bulk (crystal) density.In the high-density limit diffuse scattering should be mini-mal because the number of scattering centers (the voids)is small.24 Second, the pigment particle size (and thus,one expects, the size of the voids) is small compared withthe wavelength. Finally, in the highly absorbing regions,such as in the highly absorbing or relaxational regimesnear the color band, the penetration depth is small be-cause of strong damping inside the sample. Consequently,internal inhomogeneity is less of a concern at theseenergies. In any event the success of the Kramers-Kronig analysis, as shown below, seems to justify ourneglect of bulk scattering in the samples used for the re-flectance spectra.

Musfeldt et al.


D. Least-Squares OptimizationWe used a nonlinear least-squares fitting procedure to op-timize the choice of high-frequency extrapolation parame-ters. The objective of this procedure was to minimize thedifference between the absorption spectra calculated fromthe Kramers-Kronig analysis of pigment pellet reflec-tance and that measured in a dilute suspension of pigmentin resin by appropriate adjustment of the Kramers-Kronighigh-frequency extrapolation parameters.2 5 In principle,one can manipulate both the exponent E and the free-electron value (OA simultaneously to produce the best fit.However, because CoA is far from the region of interest, weselected E as the optimization variable of choice and fixedCoA at 1 X 106 cm' (124 eV).

The optimum value of E is the one that minimizes theleast-squares difference (rms error) between the observedspectra and the computed absorption coefficient a(Co)according to the following procedure: For each value ofE we

(1) Perform the Kramers-Kronig analysis, calculatingthe phase from Eq. (4) and obtaining the complex refrac-tive index n(C) + iK(Co).23

(2) Convolve n(ou) and K(Co) with a 10-nm bandpass toobtain n(A) and K(A) corresponding to the visible-rangespectra (380-700 nm at 10-nm increments).

(3) Compute the absorption coefficient a according toEq. (8).

(4) Find the scaling constant C, which relates a to theobserved thin-film transmission Tb,(A) = /1O (correctedfor the 6% reflectance of the xerographic solid-area imagesamples), such that -ln[Thb,(A)] = Ca(A), where C is an ad-justable coefficient containing the thickness h and theloading of the pigment in the binder, f for the thin-filmsamples. This adjustment is readily accomplished byleast-squares methods, which gives an error estimate forC, r2, and a frequency-averaged rms fitting error,2 6 theequation for which is

{2 [Ca(A, E)] + ln[Tob,(A)] 2}

rms = 100% x -ln[TObs(min)]

Kramers-Kronig-consistent) determination of both thereal and the imaginary index of refraction in the visible-frequency region.

4. RESULTS

A. Reflectance MeasurementsThe solid-state reflectance spectra of the five pigmentsused in this study are shown in Fig. 3. The spectra aredominated by several strong electronic absorption bandsin the visible and near-ultraviolet frequency region origi-nating from (, ir*) transitions, features that are similarin character to the spectra of other highly conjugatedmaterials. 45"5" 7

The visible spectra of most highly conjugated pigmentsand dyes consist of a single main absorption of (r, 7r*) ori-gin; this strong excitation is often referred to as the colorband. The wavelength, the width, the shape, and the in-tensity of the color band are the primary factors in deter-mining the optical properties of the pigment.2 8 In manychemical systems a shoulder is observed on the high-energy side of the color band; it is attributed to a vibra-tional feature of the first excited electronic state.4 6 Themagenta and blue pigments in this study clearly displaythis feature. For porphyrin-based materials the colorband and its higher-energy shoulder are referred to as Qaand Qs bands, respectively. 29

Several additional electronic features of (r, ir*) originare observed in the near ultraviolet. Numerous spectralstudies have shown that the exact position, shape, andintensity of these bands are extremely sensitive to the

0.5

(13)

The fitting error was scaled to the extreme value of theabsorption coefficient so that the results are presented asa percentage of the maximum in a(A).

(5) Choose the exponent E and the scaling factor Csuch that the rms error between the two spectra is a mini-mum. Our method uses a well-known least-squares mini-mization procedure.2 6

To summarize, the Kramers-Kronig analysis of thepower reflectance of the pellet samples leads to a pre-dicted bulk pigment refractive index and absorption coef-ficient, whereas the transmittance measurements on thedilute pigment/toner films yield an effective absorptioncoefficient. Owing to the sensitivity of the Kramers-Kronig analysis to the choice of interband exponent E, thefilm spectra are used to determine the exponent in theiterative manner described above. This adjustment ismade in order to account properly for the missing oscilla-tor strength above the highest measurement frequency.Using this procedure, we are able to obtain a reliable (and

0.4

wC.z

-JU-w

0.3

0.2

0.1

FREQUENCY (104 cm-1)

1 2 3 4 5

0 ' I I L I I .

500020001000 600500 400WAVELENGTH

Fig. 3. Room-temperature reflectancefrom 5000 to 200 nm.

300 200(nm)of pure pigment pellets

Musfeldt et al.


400 450 500 550 600 650 7CWavelength (nm)

(a)

400 450 500 550Wavelength (nm)

(b)

2

0

.04C

1.51

0.5

'0

600 650 700

PV Fast Blue

, ----- .n....................

\ ~~~observe/11calculated

i.' _ _-

400 450 500 550 600 650 70

Wavelength (nm)

(d)

2.0. .

1.5C0

'a

0.0

U

Novoperm Yellow.

...n..

0 /observed

calculated \s

400 450 500 550 600Wavelength (nm)

(e)

C0

0.0A0

Neopen Blue

2 n

1.5 -

calculated I0.5 ~~~~observed0.5

O _ _ _ _ _ _ _ _ _ _ _ _/

400 450 500 550 600

Wavelength (nm)

650 700

(C)

chemical structure of the pigment molecule.4 6 "6 27 Ourmeasurements also support this conclusion, as evidencedby the different electronic spectra of the various pigmentspresented in Fig. 3. The spectral excitations in thesolid state are also much broader than those reported forpigment materials in solution or in the gas phase.6 " Inour materials the broadening is most evident above25 000 cm-' and is probably due to interactions among ad-jacent molecules and the broadening effects inherent inthe solid state, where electronic overlap between moleculesmay lead to band formation.'

The optimal set of optical constants for the pigments isdisplayed in the various plots of Fig. 4. In each plot n(w)displays the usual dispersive behavior expected at each ab-sorption band. Extrapolation of n(w) to zero frequency

Fig. 4. Solid curves: effective absorption spectrum fromspecular and diffuse transmittances of dilute pigment/tonersamples; dotted curves: real part of the complex index of refrac-tion n(w) as calculated from the Kramers-Kronig analysis;dashed curves: absorption spectrum Ca(c) as obtained fromthe Kramers-Kronig analysis. Plots: (a) Hostaperm Pink,(b) Fanal Pink, (c) Neopen Blue, (d) PV Fast Blue, (e) NovopermYellow.

also provides an estimate of the refractive index of eachpigment (Table 1). The frequency-dependent absorptioncoefficient in Fig. 4 has been calculated directly from K(W)

through Eq. (8), permitting easy comparison with the ob-served pigment absorption in the binder. As we discussed

Table 1. Zero-Frequency Refractive IndicesPigment n

Hostaperm Pink 1.83Fanal Pink 1.77Neopen Blue 1.70PV Fast Blue 1.91Novoperm Yellow 1.74

Musfeldt et al.

2

C0

0.

.0

4o

1.5

1

Hostaperm Pink, K .. ................ob vobserved -

caculated Ai

I~~ ~~ .. I_

0.5

2.5

2

1.5

00.

.0

Fanal Pink

...........~ ~ ~ ~ ~~~~~~~~~~............... ~ ~ ~ ~ ~ ..

//bserved \calculated \==S,/w/y , , _> __ __ _ __ __/

0.5

0 650 700

_ . . * . * * z. .. l l l l l l

O3

_ _ _ _

. . . . . .

1

1.C

0.!1


Table 2. Color-Band Center Frequencies and Absorption Maxima in K(co) from ReflectanceMeasurements on Pigment Pellets

w Color Band w Sideband a Color Band a SidebandPigment (cm-) (cm-') (105 cm',) (105 cm',)

Hostaperm Pink 17696 18640 0.90 0.88Fanal Pink 18383 19480 2.38 2.06Neopen Blue 14955 16040 0.67 0.71PV Fast Blue 14120 16040 1.69 1.88Novoperm Yellow 25285 1.30

in Subsection 3.C, this calculation assumes minimal scat-tering in the bulk samples.

The center frequencies and the absorption coefficientsof the major excitations in the visible-frequency range arepresented in Table 2. The absorption coefficients of thecolor-band maxima are of the order of 105 cm-', which iscommon among pigments and dyes. It should be notedthat, since the molar volume of the Neopen Blue is ap-proximately 4.6 times that of PV Fast Blue, the NeopenBlue is actually a stronger absorber than the PV Fast Blueon a per phthalocyanine-center basis. Thus the fullabsorptive character of the Neopen Blue pigment is notreadily apparent from the value in Table 2. The electron-withdrawing nature of the sulfonamide side chain is pre-sumably responsible for the slight blue shift of the NeopenBlue color band compared with that of PV Fast Blue.

B. Effect of Interband ExtrapolationFigure 5 displays the effect of the interband extrapolationon the frequency-averaged rms error of the predicted a(w)compared with the actual spectra of the pigment in thebinder. The minima are clear and well defined for all thesamples except the yellow pigment. The best E values forthe five pigments fall in the range 0.5-1.5 with no appar-ent pattern. We conclude, as might be expected, thatthese highly conjugated pigments have a significant frac-tion of their oscillator strength in the ultraviolet and be-yond. Figure 5 clearly shows that, unless the interbandoscillator strength is properly accounted for, the high-frequency extrapolation represents a large source of errorin the application of the Kramers-Kronig method to thesematerials.

In three cases (Novoperm Yellow, Hostaperm Pink, andPV Fast Blue) the minimum fitting error was obtainedwith all the phase angles positive (in the solid curve por-tion of Fig. 5). However, in the cases of Fanal Pink andNeopen Blue the optimum value of E from the least-squares fitting process produces some regions of negativephase in the visible region. Because negative phase isphysically unreasonable, our selected value of E includesthe constraint of positive phase over the 380-700-nm in-terval. This constraint shifted the selected E upwardslightly for Fanal Pink and Neopen Blue, giving the valuesindicated by the solid dots in Fig. 5. These values are rea-sonably close to the global minimum and can be taken asan indication of the size of the error bars in this analysis.

The results of our optimizations are summarized inTable 3. The predicted absorption spectra (as obtainedfrom a Kramers-Kronig analysis of the reflectance) agreewith the experimental spectra (obtained from transmit-tance measurements of thin films), with rms differences

of 3-15%. The magenta pigments were the best, withonly a 3% difference, whereas the yellow pigment was byfar the worst. Another parameter of interest is CIELABAE, which is defined as the square root of the sum of thesquare of the differences between the calculated and theobserved a*, b*, and L*, where a* and b define the hueangle and L* defines the chroma20 When AE is less than-3, color differences are imperceptible. Values forCIELAB AE, describing the difference between colorproperties predicted by the observed pigment/toner dataand those calculated with the n(w) and the K(@) obtainedby a Kramers-Kronig analysis of the power reflectance,are shown in the fourth column of Table 3. These datashow excellent agreement in two cases, with less good butacceptable agreement in two other cases. The final col-umn in Table 3 gives the product of the volume fractionpigment in the toner layer, f, and the path length h. Thisproduct would correspond to the value of the scaling con-stant C if the pigments were molecularly dispersed andthere were no effects that were due to particularity.Three of the pigments-Hostaperm Pink, Neopen Blue,and PV Fast Blue-seem to be in this category. The twoother pigments have computed C values that are approxi-

cc0

Ir

Neopen Blue

PV Fast Blue

Hostaperm Pink EFanal PinkPermanent Yellow

0 0.5 1.0 1.5 2.0 3 4

KK E PARAMETERFig. 5. Dependence of the frequency-averaged fitting error onthe Kramers-Kronig (KK) extrapolation exponent for the fivepigments investigated. Note the scale change for E = 2 to 4.The solid curves correspond to positive phase angle 0 over the380-700-nm range, while the dashed curves correspond to atleast one negative 0 value. The solid dots indicate the selectedvalue of the interband extrapolation parameter that minimizesthe least-squares difference between the computed absorptioncoefficient a and the observed toner film transmission whileretaining positive 0 in the visible range.

Musfeldt et al.


Table 3. Summary of Results from Fitting ProcedureOptimal rms CIELAB C Actual fh

Pigment Exponent (to) AE r2 (nm) (nm)

Hostaperm Pink 0.79 2.6 2.7 0.9969 330 ± 11 307Fanal Pink 1.28 3.3 1.2 0.9949 197 ± 6 294

Neopen Blue 1.52 11.0 19.8 0.9614 447 ± 251 430PV Fast Blue 1.29 10.1 18.4 0.9672 157 ± 26 148

Novoperm Yellow 0.53 14.1 35.3 0.9176 242 ± 164 351

FREQUENCY (cm-1/1000)

22 20 18

-~~~~~~~~~~~0.E=1.0 / - 0.30 A

- --0 - - ------- 0__

I-~~~~~~~~~~~~~~~010.16

LU

O 0.08U.a 0.04

400 450 500 550 600 650 700WAVELENGTH (nm)

Fig. 6. Phase angle of Hostaperm Pink, O(o), as calculatedfrom the Kramers-Kronig analysis with different interbandextrapolations.

mately a third less than the actual value of fh, indicatingthat perhaps some of the pigment is aggregated and thusnot so highly absorbing as it would be if the pigment werecompletely dispersed.

Figure 6 illustrates how variations in the interband ex-trapolation exponent are manifested in the optical spectraof Hostaperm Pink. An underestimation of the oscillatorstrength at high energy results in a reduced contrastbetween the color-band maxima and the baseline (thechroma). Overestimation of E also distorts the predictedcolor properties as well because the position of the colorband is shifted slightly to the red as the oscillator strengthof the color band is pushed to lower energy. Conversely, atoo large value of the oscillator strength at high energy(a too small value of E) results in a negative phase through-out most of the frequency range of interest.

5. DISCUSSION

In the following subsections we briefly discuss the partic-ulars of each pigment in reference to their spectra appear-ing in the various plots of Fig. 4, the data in Table 3, andthe fitting process in general. For these samples the realpart of the refractive index, n, is greater than the imagi-nary part K at all the wavelengths considered here, andboth optical constants display strong dispersive behaviorat energies near the color band. This strong dispersion islikely to prevent applications of these materials based onindex, matching because of the inherent difficulty of ob-taining such a match. The good agreement between thetwo sets of data also indicates that the extinction of the

pigment samples is dominated by absorption, althoughrms deviations are between 3% and 15%. Several ex-amples are presented below to demonstrate the use of thistechnique on a wide range of materials. Our goal is notonly to highlight the obvious successes of this method butto illustrate the difficult cases as well.

A. Hostaperm PinkThe observed and calculated spectra of Hostaperm Pinkare displayed in Fig. 4(a). This pigment gave the best fitof the effective absorption spectrum (as obtained from thepigment/toner films) to the Kramers-Kronig-determinedabsorption coefficient a, with rms error less than 3% anda very low CIELAB AE. The Kramers-Kronig calcula-tions seem to generate a slight mismatch in the relativeintensity near 550 nm and a hint of an additional absorp-tion at 660 nm. On the whole, the agreement is quitegood, demonstrating the validity of our interband exponentoptimization procedure.

B. Fanal PinkA Kramers-Kronig analysis of the reflectance spectrumof this material also gave an excellent fit of a(w) to theeffective absorption spectrum of the pigment suspendedin toner resin [Fig. 4(b)] and the lowest CIELAB AE value(Table 3). Some deviations between the two spectra occurin the 380-500-nm range, where scattering effects thatare due to the particulate nature of the pigment wouldbe expected to become more important. That C is sig-nificantly less than fh is another indication of the onsetof particulate effects. The shape of the frequency-dependent absorption coefficient in the 380-500-nm rangemay also affect the overall quality of the fit. The colorband in Fanal Pink is followed by a sharp, deep minimumcentered at 450 nm, whereas the color band in HostapermPink is broader and shallower. Since the absorption datafor the pigment in the binder were obtained with a 10-nmbandpass, we are likely to obtain a better fit for broadenedfeatures, such as those in the Hostaperm Pink spectrum.On the whole, however, the results for the Fanal Pink pig-ment are very satisfactory.

C. Neopen BlueThe spectra of the Neopen Blue pigment are shown inFig. 4(c). For this sample the fit error is approximatelythree times larger than it was for the two magenta pig-ments. Examination of Fig. 4(c) shows that the majordifference again appears in the 380-500-nm region of thespectrum, where the pellet reflectance datum predictsgreater absorption than is observed in the thin pigment/toner films. This deviation could result from scatteringlosses resulting from pellet features smaller than the 200-nm aluminum overcoat. Such a deviation may also result

Musfeldt et al.


from small inadequacies in the Kramers-Kronig extrapo-lation assumption, as the highly absorbing and relaxationregimes in this material are closer to the missing vacuum-ultraviolet data and so are more strongly affected by theextrapolation. Despite these shortcomings the Kramers-Kronig analysis reproduces the shape and the intensity ofthe color band (600-700-nm feature) very well.

D. PV Fast BlueThe spectra of PV Fast Blue (copper phthalocyanine) aredisplayed in Fig. 4(d). The optical properties of thismaterial have enjoyed special attention from both theexperimental 164s and theoretical 9 -2 perspectives. Thecolor-band features, as well as the higher-energy spectralcharacteristics [not shown in Fig. 4(d)], are in overall rea-sonable agreement with previous measurements, althoughthe features in the optical spectra of copper phthalo-cyanine are slightly red shifted and broadened comparedwith results reported by Edwards and Gouterman'6 andothers. 7 "8 This band is also red shifted compared withthe characteristic color-band position (670 nm) in otherphthalocyanine-based materials with D4h symmetry. 6

The inclusion of filler or extender in these commercialpigment samples is a possible explanation for thisdifference.

Although the magnitude of the fit error for this pigmentis comparable with that of the Neopen Blue sample, thetype of discrepancy is different and potentially moreserious for color calculations of a predictive nature. AsFig. 4(d) shows, the computed absorption spectrum is redshifted from the observed spectrum of the pigment in thebinder by 10-20 nm. Perhaps the denser packing in thepigment pellet introduces slight modifications to the elec-tronic spectrum that red shift the band position slightlyfrom its position in the binder. However, it is more likelythat this difference reflects the combination of measure-ment difficulties (both in the pellet form and in the binder)and the limitations of the Kramers-Kronig method forthis material.

E. Novoperm YellowThe spectrum of Novoperm Yellow [Fig. 4(e)] was the mostdifficult to analyze and is significantly different from thefour other conjugated pigments in this study. Rather thana strong color band with good contrast from a near-zerobaseline absorption, as found in the dilute transmittancestudies, the Kramers-Kronig analysis yields several broadand overlapped features in the visible energy region. Al-though the gross features of the observed absorption spec-trum are captured in the spectrum computed by theKramers-Kronig procedure, the Kramers-Kronig analysissuggests a secondary absorption near 530 nm in the bulkpigment that is absent in the effective absorption spec-trum of the dilute pigment/toner film. In addition, themain absorption peak in the optical regime is blue shiftedby 20 nm compared with what is observed in the dilutepigment/toner samples.

At the present time we have only tentative explanationsfor the discrepancy between optical constants as obtainedby a Kramers-Kronig analysis of bulk pigment reflectionand the direct absorption measurements on dilutepigment/toner films for Novoperm Yellow. One possibilityis that there may be a preferential orientation in the

pressed pellets in this anisotropic material.5 In this caseour reflectance measurements may have oversampled oneaxis, perhaps one with charge-transfer character, therebyproducing the deviation from the (randomly oriented)pigment/toner absorption that we observe in Fig. 4(e). Itis also possible that the feature at 530 nm may be a solid-state (rather than isolated molecule) effect, with theflexible nature of the pigment molecule providing an op-portunity for adjacent molecules to be in close proximityand thus for charge transfer between intermolecularquinoid rings.

The deviation between the position color bands of thecomputed and observed absorption spectra also points outthe inherent difficulties of the Kramers-Kronig methodfor materials that absorb in the violet: features nearer tothe high-frequency limit of the measurement are not soreliable as those away from it. This is the most likelyexplanation for the observed discrepancy in the color-bandfeature. The spectra shown in Fig. 4(e) illustrate whatis, in our experience, a worst-case scenario. This resultshould serve as a warning about the limitations of theKramers-Kronig technique-the results need to be inter-preted with care.

6. CONCLUSION

We have presented a method based o the Kramers-Kronig relationship to automate the determination of reli-able optical constants in the neighborhood of the colorband in highly conjugated pigments and dyes. The choiceof the high-frequency exponent is the key to a meaningfulanalysis of the optical properties of these pigments be-cause of the large spectral weight of the interband transi-tions in the ultraviolet-frequency range. We demonstratethe effect of various exponents on the character of thecolor-band absorption and develop criteria for the optimalchoice of the high-frequency interband exponent in thesehighly conjugated materials.

The major advantage of a dispersion-based method, suchas that outlined here, is that it provides a framework inwhich the real and imaginary parts of the complex indexof refraction are obtained in a self-consistent manner. Itis also conceptually simple and readily applicable tocommon use. Consequently, this technique is likely to seewide application in the field of color science.

The capabilities of our optimization method in calculat-ing n(w) and K(w) were demonstrated on a representativeset of five commercially available pigments. Our futureresearch centers on the use of real and imaginary refrac-tive indices obtained in this manner as input to Mie scat-tering calculations, with the ultimate goal of modeling thecolor properties of pigment/toner systems.

ACKNOWLEDGMENTS

The authors thank E. N. Dalal, S. Jeyadev, and I. Morrisonof Xerox Webster Research Center (WRC) for interestingthem in this project. We also thank T. L. Bluhm of XeroxResearch Centre of Canada (XRCC) for the x-ray diffrac-tion measurements, C. Kuo of XRCC for the bulk densitymeasurements, and E. N. Dalal and S. Blaszak of WRC forthe spectra of the toner film samples.

Musfeldt et al.


*Present address (corresponding author), Centre deRecherches en Physique du Solide, D6partement de Phy-sique, Universit6 de Sherbrooke, Sherbrooke, Qu6bec JiK2R1, Canada.

REFERENCES AND NOTES

1. P. A. Lewis and T. C. Patton, eds., The Pigment Handbook(Wiley, New York, 1973), Vols. 1-3.

2. F. W Billmeyer, Jr., and M. Saltzman, Principles of ColorTechnology, 2nd ed. (Wiley, New York, 1981).

3. J. Fabian and H. Hartman, "Light absorption of organic col-orants: theoretical treatment and empirical rules," in Re-activity and Structure; Concepts in Organic Chemistry,K. Hafner, J. M. Lehn, C. W Rees, P. von Ragu6-Schleyer,B. M. Trost, and R. Zahnradnfk, eds. (Springer-Verlag, Berlin,1980), Vol. 12, pp. 1-19.

4. M. Gouterman, "Optical spectra and atomic structure," inThe Porphyrins, Part 3 of Physical Chemistry A, D. Dolphin,ed. (Academic, New York, 1978), pp. 1-159.

5. F. H. Moser and A. L. Thomas, Phthalocyanine Compounds(Reinhold, New York, 1963), pp. 292-307.

6. M. J. Stillman and T. Nyokong, 'Absorption and magneticcircular dichroism spectral properties of phthalocyanines,Part 1: Complexes of the dianion Pc(-2)," in Phthalo-cyanine: Properties and Applications, C. C. Leznoff andA. B. P. Lever, eds. (VCH Publishers, New York, 1989),pp. 133-290.

7. R. M. Johnson, "Color theory," in The Pigment Handbook,T. C. Patton, ed. (Wiley, New York, 1973), Vol. 3, pp. 229-288.

8. W Carr, "Pigment powders and dispersions: measurementand interprethtion of their physical properties," in The Pig-ment Handbook, T. C. Patton, ed. (Wiley, New York, 1973),Vol. 3, pp. 11-27.

9. G. Mie, "Beitraige zur Optik truber Medien, speziellkolloidalen Metallosungen," Ann. Phys. (Leipzig) 25, 377-445 (1908).

10. C. F. Bohren and D. R. Huffman, Absorption and Scatteringof Light by Small Particles (Wiley, New York, 1983).

11. W G. Egan and T. W Hilgeman, Optical Properties of Inho-mogeneous Materials (Academic, New York, 1979).

12. H. C. van de Hulst, Light Scattering by Small Particles(Dover, New York, 1981).

13. F. Wooten, Optical Properties of Solids (Academic, New York,1972), pp. 173-185.

14. E. L. Pankove, Optical Processes in Semiconductors (Dover,New York, 1971), pp. 87-91.

15. E. E. Bell, "Optical constants and their measurement,"in Handbuch der Physik, S. FlUgge, ed. (Springer-Verlag,Heidelberg, 1967), Vol. 25, pp. 1-58.

16. L. Edwards and M. Gouterman, "Vapor absorption spectraand stability: phthalocyanines," J. Mol. Spectrosc. 33, 292-310 (1970).

17. B. R. Hollebone and M. J. Stillman, 'Assignment of absorp-tion and magnetic circular dichroism spectra of solid, a phasemetallophthalocyanines," J. Chem. Soc. Faraday Trans. 2 74,2107-2127 (1978).

18. P. Day and J. P. Williams, "Spectra and photoconduction ofphthalocyanine complexes (I)," J. Chem. Phys. 37, 567-570(1962).

19. A. J. McHugh, M. Gouterman, and C. Weiss, Jr., "PorphyrinsXXIV Energy, oscillator strength and Zeeman splittingcalculations (SCMO-CI) for phthalocyanine, porphyrins andrelated ring systems," Theor. Chim. Acta 24, 346-370 (1972).

20. A. M. Schaffer and M. Gouterman, "Porphyrins XXV.Extended Hfickel calculations on location and spectral effectsof free base protons," Theor. Chim. Acta 25, 62-82 (1972).

21. A. M. Schaffer and M. Gouterman, "Porphyrins XXVIII.Extended Hfickel calculations on metal phthalocyanines andtetraporphins," Theor. Chim. Acta 30, 9-30 (1973).

22. K. D. Cummings, D. B. Tanner, and J. S. Miller, "Opti-'cal properties of cesium tetracyanoquinodimethanide,Cs2 (TCNQ)3," Phys. Rev. B 24, 4142-4154 (1981).

23. Note that the contribution of the numerical integral over thedata to the phase spectrum (the time-consuming step) needbe made only once.

24. G. L. Carr, S. Perkowitz, and D. B. Tanner, "Far-infraredproperties of inhomogeneous materials," in Infrared andMillimeter Waves, K. J. Button, ed. (Academic, New York,1985), Vol. 13, pp. 171-263.

25. A copy of this program may be obtained from the authors.26. P. R. Bevington, Data Reduction and Error Analysis for the

Physical Sciences (McGraw-Hill, New York, 1969).27. L. Bajema, M. Gouterman, and B. Meyer, 'Absorption and

fluorescence spectra of matrix isolated phthalocyanines,"J. Mol. Spectrosc. 27, 225-235 (1968).

28. J. Griffiths, "Developments in chemistry and technology oforganic dyes," in Critical Reports on Applied Chemistry,J. Griffiths, ed. (Blackwell, Oxford, 1984), Vol. 7, p. 2.

29. J. R. Platt, "Classification and assignments of ultravioletspectra of conjugated organic molecules," J. Opt. Soc. Am.43, 252-257 (1953).

30. See, for example, Ref. 2, p. 102, and Ref. 7, p. 247.31. Z. Zhang, M. M. Lerner, V J. Marty, and P. R. Watson, "Scan-

ning tunneling microscopy on pressed powders," Langmuir 8,369-371 (1992).

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