Density dependence of refractive index of nanoparticle-derived titania films on glass

Thin Solid Films 558 (2014) 86–92

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Density dependence of refractive index of nanoparticle-derived titania films on glass Anja Matthias a, Nevena Raićevic b, Romeo Donfeu Tchana a, Detlef Kip b, Joachim Deubener a,⁎ a b

Institute of Non-Metallic Materials, Clausthal University of Technology, Zehntnerstraße 2a, 38678 Clausthal-Zellerfeld, Germany Faculty of Electrical Engineering, Helmut Schmidt University, Holstenhofweg 85, 21043 Hamburg, Germany

a r t i c l e

i n f o

Article history: Received 18 October 2013 Received in revised form 20 February 2014 Accepted 24 February 2014 Available online 28 February 2014 Keywords: Titania Nanoparticle sol Density to refractive index relationship Mixture models Porosity

a b s t r a c t In order to test the relationship of refractive index to mass density within a wide range, porous titania films up to 800 nm thickness were prepared on silica glass by repetitive dip-coating and thermal curing of anatase sols having primary particle sizes below 7 nm. Profilometry showed a decrease in film thickness and an increase in the index of refraction of up to 50%, if the curing temperature was increased from 100 °C to 1000 °C. The decrease in film thickness was related to an increase in mass density, which directly acts on the optical polarizability and thus the (effective) refractive index of the film. In particular, mass density–refractive index calculations were performed using linear (Arago–Biot, Gladstone–Dale) and nonlinear mixture models (Drude, Lorentz– Lorenz), assuming either air- or water-filled pores, while anatase and rutile fractions were determined by Xray diffraction. These investigations were verified using refractive index and mass density data from literature. For each model noticeable deviations from the expected trend were evident. We show that an empirical power law expression holds for the Lorentz–Lorenz theory and permits to calculate effective density and porosity of titania thin films from effective refractive index with high accuracy. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Titania (TiO2) is one of the most interesting and widely studied semiconductors with a large band gap. Deposited as thin films, TiO2 is of great interest for a large range of applications like gas sensors [1,2], solar cells [3,4], photocatalyst [5–7], anti-reflection [8,9] and self-cleaning coatings [10]. The refractive index n(λ) and optical dispersion of titania thin films were found to depend strongly on deposition methods and conditions. Values for n(λ = 550 nm) of 2.23– 2.35 [11], 2.0–2.3 [12], 2.3–2.52 [13] for electron beam evaporation, 2.42–2.56 [11], 2.45–2.73 [14] 2.31–2.37 [15], 1.53–1.54 [16] for magnetron sputtering, 2.56–2.72 [17] for chemical vapor deposition, 2.32–2.49 [18] for atomic layer deposition and 1.93–2.12 (at λ = 600 nm) [19], 2.18 [20], 2.26–2.31 [21], 2.20–2.43 [22], 2.02–2.11 [23], 2.34 [24] for sol–gel spin- and dip-coating were reported. In these films titania is present as amorphous material, possessing brookite, anatase, and rutile phases as well as mixtures of these modifications. In most works a deviation from the refractive index of the pure TiO2 polymorphs (mean refractive index n d at wavelength 550 nm is 2.566 for anatase [25], 2.654 for brookite [25] and 2.742 for rutile [26]) was related to a heterogeneous film structure. In particular, n(550 nm) b 2.45 (dense amorphous titania [11]) was assigned to porebearing films. The properties of these nanocomposites were mostly ⁎ Corresponding author. Tel.: +49 5323 72 2463. E-mail address: [email protected] (J. Deubener).

http://dx.doi.org/10.1016/j.tsf.2014.02.078 0040-6090/© 2014 Elsevier B.V. All rights reserved.

treated by effective media approximations. Different mixing models were used to predict porosity, densification and pore-filling media (air and water) from effective optical properties. For sol–gel derived titania films Hu et al. [27], San Vincente et al. [19], Ahn et al. [22] and Mathews et al. [23] applied the Drude (also referred as Yoldas) model, while Hostetler et al. [20] and Vitala et al. [28] used a Lorentz–Lorenz relation, while Taylor et al. [29] calculated porosity from the Maxwell–Garnet equation. In most of these studies the microstructure of porous films was predicted without comparison between theory and experiment. Furthermore, changes in the deposition conditions used in these studies resulted in similar optical properties, i.e., the differences in the effective refractive index were relatively small (Δneff ≈ 0.1–0.3), which makes it difficult to test the above mixture models by analyzing trends in the density dependence of neff [30,12]. Thus, the present study aims in providing a relatively broad range of refractive indices by using a single preparation method. For this, low- and high-index TiO 2 films ranging from n eff = 1.75 to n eff = 2.62 were prepared using anatase sols with primary particle size below 7 nm and applying simple variations (temperature, time) in the curing conditions. In order to make theoretical predictions of porosity to allow for a comparison with experimental data, independent determinations of the film density are necessary, which were collected from gravimetric, Rutherford backscattering and X-ray absorption data reported in the literature. This compilation of densities covers a large range of the effective refractive index (Δ neff ≈ 1.5) for titania structures and thin films.

A. Matthias et al. / Thin Solid Films 558 (2014) 86–92

Assuming a polycrystalline random structure in the titania film, nD of Eqs. (1)–(5) can be derived from single crystal data by

2. Characterization of microstructure using effective optical properties 2.1. Mixture models

nD ðλÞ ¼

Effective medium approximations of nanocomposite and mesoporous thin films are derived from calculation and are based on relative volume fractions of their components. For ideal mixing of a dispersed pore volume (refractive index nP) in a dense titania film (refractive index nD) the effective refractive index neff of the porous film is given by
 f neff ¼ ð1−P Þf ðnD Þ þ P f ðnP Þ;

ð1Þ

where f (ni) is the specific refractive index of a certain model, P is the volume fraction of pores (porosity), and (1 − P) is the volume fraction of pore-free titania (i.e. packing density or filling factor). Numerous mixture models have been reported to treat a heterogeneous thin film as homogeneous with an effective refractive index. Among these Arago–Biot, Gladstone–Dale, Drude (or Yoldas), Lorentz–Lorenz, Heller, Wiener, Maxwell–Garnett, volume averaging theory and Bruggeman are most frequently employed [31,32]. The use of these principal mixture rules results in either a linear refractive index-porosity dependence or in a negative or positive deviation from a linear additivity of neff vs. P. Representatives of the former are Arago–Biot [f (ni) = ni] and Gladstone–Dale [f (ni) = (ni − 1)], while members of the latter are Drude [f (ni) = (ni2 − 1)] and Lorentz–Lorenz [f (ni) = (ni2 − 1)/(ni2 + 2)] models. Compilations of these models are given in [31–33]. Using Eq. (1) for volume conservation the Arago–Biot and Gladstone–Dale models lead to a porosity P¼

neff −nD : nP −nD

ð2Þ

Application of the Drude model results in

P¼

neff 2 −nD 2 nP 2 −nD 2

;

87

ð2no þ ne Þ ; 3

ð7Þ

where no and ne are the ordinary (2.5915 and 2.6433 at 550 nm) and the extraordinary (2.5138 and 2.9387 at 550 nm) refractive indices of anatase [25] and rutile [26], respectively. For mixed anatase–rutile films a mean value based on their relative volume fractions is widely accepted. In general these mixture models do not account for the actual polarization of the incident light, or for the local size, shape and spatial distribution of the pores in the thin film. Hence, refractive index predictions for given pore morphologies can differ considerably from numerical solutions [35,36]. Consequently, calculation of ρeff from effective refractive index using mixture models requires information on the true mass density or on the specific microstructure of the film. 2.2. True mass density of titania thin films and structures In the case of titania thin films true mass density ρ has been determined independent of effective optical theory by combining metric (thickness) with gravimetric experiments [30,13], using Rutherford backscattering intensities [11,17] and X-ray absorption [37]. Together with the corresponding effective refractive indices neff, which were determined in these studies using ellipsometry and optical transmittance spectra, the data permit to verify the underlying mixing models. These tests will provide the basis for estimates on the effective density of nanoparticle derived films within this study. Specific volumes of natural (anatase [38], brookite [39] and rutile [40]) and modeled (density functional simulations of columbite, rutile, brookite, anatase, ramsdellite, bronze and hollandite [41]) titania structures, as determined from their unit cell parameters, were included as well for the data compilation shown in Fig. 1. For these pure titania materials the refractive indices nD from Refs. [25,26] and [42] were applied. Fig. 1 reveals that the dependence of the refractive index [effective (neff) and bulk (nD)] on

ð3Þ

while the porosity obtained from Lorentz–Lorenz model is



 2 2 2 neff −nD nP þ 2   2 : P¼  2 neff þ 2 nP −nD 2

ð4Þ

The dimensionless Eq. (1) can be formulated in an alternative way by introducing the mass density of pure components, which is referred as specific refraction R (in units of volume per mass)
 f neff =ρeff ¼ ð1−P Þf ðnD Þ=ρD þ P f ðnP Þ=ρP ¼ ð1−P ÞRD þ PRP

ð5Þ

with ρeff, ρD and ρP being the mass density of the porous film, pore-free titania, and pores, respectively. For f (ni) = ni − 1 and water-filled pores, Eq. (5) is equal to the Gladstone–Dale equation in its widely used form: neff −1 ¼ Reff ¼ ð1−P ÞRTiO2 þ PRH2 O ; ρeff

ð6Þ

which is used to calculate mass density from refractive index of titaniabearing minerals. In optical mineralogy values of RTiO2 = 0.397 cm3 g−1 and RH2O = 0.34 cm3 g−1 at 598 nm are employed [34].

Fig. 1. Dependence of the index of refraction n (λ = 550 nm) on true mass density ρ of titania thin films and crystal structures in double logarithmic scales. Thin film data as prepared by: sputtering and ion plating (anatase) [11], filtered arc deposition (amorphous, anatase, rutile) [17] and electron beam evaporation and post heating (amorphous, anatase) [30,12,11,13]. Crystal data: anatase [38,25], rutile [41,26], brookite [39,25], modeled TiO2 structures: columbite, rutile, brookite, anatase, ramsdellite, bronze and hollandite (density functional simulation (DFT) for 2.1 eV) [41,42]. Photonic crystal data: Inverse OPAL structures (anatase) at 14,900 cm−1 [58,37]. Inset shows details for ρ N 2 g cm−3. The dashed line is intended as a visual guide.

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true mass density is in general nonlinear. The slope δ[log(neff, nD)] / δ[log(ρ)] is ≈ 0.24 for ρ = 0.3 g cm− 3 and increases with increasing density towards ≈0.64 for ρ = 4 g cm−3. This trend can be explained from the fact that in dilute structures (ρ b 1) like in inverse titania opals dipoles are too far from one another (e.g. from wall to wall) to exert noticeable effects on the local electric field, while in concentrated structures (ρ N 1) the dipole–dipole interaction is stronger and the local electric field experienced by an individual molecule is increasingly deviating from the applied field. 3. Experimental 3.1. Preparation of TiO2 thin films Hombikat XXS 100 sol (Sachtleben Chemie, Duisburg, Germany) of primary anatase particle size b7 nm (Scherrer) and 6 wt.% TiO2 was diluted in pure ethanol (≥ 99.9%, Sigma Aldrich) in a ratio of 1:6. The suspension was stirred at 20 °C and 55% RH for 1 h. Silica glass plates (type II, GVB, Herzogenrath, Germany) of size 60 × 30 × 1 mm3 were dipped into the sol and pulled upward using a withdrawal speed of 2 mm s−1. Before dipping the silica glass plates were cleaned using deionised water and acetone and subsequently masked, to ensure single sided coating. For the first series each plate was coated 10 times and annealed for 15 min between successive coatings at temperatures Ta = 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 °C (preheated box furnace lined with silica glass) to reach a total thickness of N390 nm. Coated substrates were labeled accordingly as H100, H200, H300, H400, H500, H600, H700, H800, H900 and H1000. The second series of glasses were coated 10 times and annealed at 500 °C for 15 min between successive coatings. After the last coating annealing at 500 °C was performed using dwell times ta = 1, 2, 5, 20, 50, 100, 200 and 500 h. These samples were labeled accordingly as H500/1, H500/2, H500/5, H500/20, H500/50, H500/100, H500/200 and H500/500. Preparation of the titania sol and the coating of silica glass plates were performed under clean room conditions (class 2; ISO 14644-1).

thickness was obtained from height differences of scans across the film-substrate edge at five different positions using a profilometer (Tencor P1 Texas Instruments, Dallas, USA). 3.4. Optical spectroscopy Transmittance T(λ) of titania-coated glasses in the wavelength range from 200 nm to 2500 nm was measured using a UV–Vis-NIR spectrometer (Lambda 950, Perkin Elmer, Massachusetts, USA) equipped with an integrating sphere (150 mm). Both thickness d and effective refractive index neff (λ) were obtained from fitting the optical interference pattern using software Spektrum (LZH Hannover, Germany) based on the envelop method [45–47]. In particular from the transmission spectrum envelopes around the transmission maxima TM and transmission minima Tm are constructed and considered as continuous spectra of TM(λ) and Tm(λ), respectively. The expression for the refractive index of the film is given by [46] 2

neff ¼ N þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 −ns 2

ð9Þ

where N ¼ 2ns

T M −T m ns 2 þ 1 þ 2 TMTm

ð10Þ

and ns is the refractive index of the substrate. The thickness of the film is calculated from the refractive index n1(λ1) and n2(λ2) of two adjacent maxima (or minima) according to [46] d¼

λ1 λ2 : 2ðλ1 n2 −λ2 n1 Þ

ð11Þ

The Sellmeier equation [48] was used to determine the optical dispersion: 2

neff ðλÞ ¼ 1 þ

3.2. X-ray diffraction

B1 λ2 λ2 −C 1

ð12Þ

where B1 and C1 are Sellmeier coefficients. Low angle (incidence angle = 0.5°) X-ray diffraction patterns were collected using a XRD system (PANalytical X'Pert Pro MPD Diffractometer, Almelo, Netherlands) in ω/2θ-geometry using Cu-Kα radiation (45 kV, 40 mA source) equipped with a PIXel RTMS detector. The XRD patterns were recorded in a 2θ range from 10° to 70° with a step size of 0.03° and a counting time of 5 s. The lattice parameters and the crystal size were determined by Rietveld refinement and Scherrer broadening using X'Pert High Score Plus software. The crystal volume fraction of anatase VA was derived from the integrated intensities of the (101)-peak of anatase and (110)-peak of rutile, i.e. IA(101) and IR(110) [43,44]:   ð1−W A ÞρA −1 V A ð% Þ ¼ 100 1 þ W A ρR

ð8Þ

with WA = [1 + 0.884(IA / IR)]−1 and ρA and ρR for densities of bulk anatase (3.89 g cm−3 [38]) and rutile (4.25 g cm−3 [40]), respectively. 3.3. Electron microscopy and profilometry A field emission scanning electron microscope (FESEM, Helios NanoLab 600, FEI, Eindhoven, Netherlands) was used to image cross sections and surfaces (top view) of titania coated glasses in secondary electron mode with voltage/current settings of 5 kV/21 pA and 2 kV/11 pA, respectively. Before imaging samples were sputtered with a thin film of carbon to reduce discharges. The mean grain size was determined using ImageJ software (http://rsb.info.nih.gov/ij/). Film

4. Results 4.1. Microstructure and chemical composition The onset of the anatase-to-rutile transformation was specified by the appearance of diffracted intensities of both TiO2 polymorphs in the H700 film, while exclusiveness of rutile intensity peaks indicated completion of this reaction in titania films annealed at Ta ≥ 900 °C (Fig. 2). In contrast, only anatase was detected in titania films treated at temperatures ≤700 °C (series 1) and for times b 20 h (series 2). The lattice constants of anatase were found to depend only slightly on the annealing temperature, i.e. aA was negatively temperature correlated and cA was positively temperature correlated. In contrast, the unit cell parameters of rutile were almost constant (Table 1). Changes in the volume fraction and in the mean crystal size of anatase and rutile were opposed in Fig. 3. Annealing at 100 °C resulted in 5.5 ± 0.2 nm-sized crystals, which grew to sizes of 22.8 ± 0.9 nm at 800 °C. In the mixed anatase–rutile films mean sizes of rutile are larger than those of anatase, which points out a faster growth of rutile at T ≥ 700 °C. At the final dwell temperature of series 1 (Ta = 1000 °C) rutile crystals reached mean sizes of about 30 nm. Isothermal dwelling (series 2) at 500 °C for up to 500 h resulted in mixed anatase–rutile films, which consisted of larger rutile [(24.7 ± 0.9) nm] than anatase [(13.2 ± 0.6) nm] crystals (Table 1). Secondary electron microscopy images revealed a particulate microstructure of H500 films (Fig. 4). The mean grain size (≈12 nm) was in agreement with the size of the crystal domains as determined by XRD.

A. Matthias et al. / Thin Solid Films 558 (2014) 86–92

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Table 1 Mean crystal size, lattice parameters and volume fractions of anatase and rutile of titania films of series 1 (H100–H1000) and series 2 (H500/2–H500/500). Numbers in parentheses give uncertainty of the last digit. Titania film

H100 H200 H300 H400 H500 H600 H700 H800 H900 H1000 H500/2 H500/5 H500/20 H500/50 H500/100 H500/200 H500/500

Anatase

Rutile

DA (nm)

aA (pm)

cA (pm)

5.5(2) 6.0(3) 7.1(4) 8.0(4) 8.9(4) 9.9(5) 18.4(9) 22.8(9)

388(4) 370(4) 373(3) 374(1) 373(1) 374(1) 376(2) 378(1)

936(2) 936(2) 934(2) 936(1) 936(1) 936(1) 948(3) 953(3)

9.2(4) 9.6(1) 10.7(5) 11.2(4) 11.8(5) 12.9(6) 13.2(6)

378(1) 378(1) 378(1) 378(1) 378(1) 379(1) 379(1)

945(1) 921(1) 944(1) 943(1) 943(1) 946(1) 946(1)

DR (nm)

aR (pm)

cR (pm)

24.2(9) 26.1(9) 29.1(9) 29.9(9)

456(2) 459(1) 458(1) 459(1)

290(1) 296(1) 296(4) 296(1)

13.3(2) 16.1(8) 20.9(9) 22.3(9) 24.7(9)

464(1) 461(1) 461(1) 461(1) 460(1)

295(1) 297(1) 296(1) 296(1) 296(1)

VR 0 0 0 0 0 0 0.074(8) 0.283(6) 1 1 0 0 0.024(5) 0.030(5) 0.031(4) 0.044(3) 0.065(2)

5. Discussion

Fig. 2. X-ray diffraction pattern of titania films H100–H1000 (series 1) on silica glass and intensities of reference ICDD cards 21–1276 (rutile) and 21–1272 (anatase). Color key of figure (A): pure anatase (green), mixed anatase–rutile (blue) and pure rutile (red). Panel (B) shows broadening of the main diffraction peaks of anatase (101) and rutile (110).

The obtained XRD, FESEM and UV–Vis-NIR results demonstrate a strong correlation between microstructure and optical properties of TiO2 films prepared from anatase sols by isochronal and isothermal annealing. The microstructural evolution is accompanied by an anatase-to-rutile transformation with an onset T a/ta condition of 700 °C/0.25 h for series 1 and 500 °C/20 h for series 2. Coexisting TiO2 phases and the continuous crystal transformation in the temperature range from 700 °C to 800 °C agree well with previous results on sol–gel derived thin films [50,51]. Associated with the anatase-to-rutile transformation, an increase in the slope of the temperature-size curve of anatase is evident (Fig. 3), which was previously reported for spray-coated titania films of large crystal sizes [51]. Size dependence of the transformation temperature was recently addressed by Satoh et al. [52] for nanocrystals b 2 nm. They found that initiating temperature Tonset and crystal size are inversely correlated by the equation T onset = 990 °C − 552 °C nm/D A . Considering the crystal size in the film H600 (DA = 9.9 nm) for rutile formation the latter equation overestimates Tonset by more than 300 K. The same discrepancy results if one compares the annealing temperature of the film

In contrast, particle growth and coarsening at temperatures above 500 °C resulted in grain sizes 20–50 nm in H700 and 60–300 nm in H1000, which are much larger than their constitutive crystal domains.

4.2. Optical properties Using the envelop method both film thickness and optical dispersion of titania films (series 1 and 2) were determined and compiled in Table. 2. From inspection of this data it can be seen that the thickness of the films decreases with increasing annealing temperature and dwell time while the refractive index increases with Ta and ta. However, the refractive index of film H1000 (2.62 at 550 nm) is still smaller than nD of the polycrystalline random structure of rutile (2.74 [26]), which indicates the presence of residual inter- and intra-granular porosity even after firing at 1000 °C. Thus, refractive index and density of the film have to be treated as effective optical properties. Further Table 2 shows that the latter property is proportional to the reciprocal value of the film thickness since the values of d determined from profilometry and by fitting the optical interference pattern were in close agreement. Herein it is anticipated that a porous film on glass densifies via shrinkage of its thickness d while lateral dimensions are constant [49].

Fig. 3. Mean crystal size of anatase DA and rutile DR (left ordinate) and anatase volume fraction VA (VA = 1 − VR, right ordinate) as a function of annealing temperature Ta of titania films on silica glass of series 1 (H100–H1000). Lines are intended as visual guides.

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Fig. 4. Secondary electron microscope images of titania films on silica glass (series 1) annealed at 500 °C [top view (A) and cross section (B)], 700 °C [top view (C) and cross section (D)] and 1000 °C [top view (E) and cross section (F)].

H600 with T onset (900 °C) of pyrogenic particles of equivalent size 7.7–10.5 nm [53]. We are fully aware that densification of the titania film by thermal annealing is accompanied by the removal of the residuals of the sol–gel synthesis (liquids, organic groups, etc.). It has been shown that the elimination of these residuals by evaporation and burning can improve densification of porous thin films [54]. Further, if filling of pores by residual materials will not be taken into consideration the use of mixture models underestimates porosity. Thus, atmospheric ellipsometric porosimetry of porous silica [55] and titania [56] films gave clear evidence that the adsorption of water to the films led to an increase in refractive index in an ambient (humid) atmosphere. Therefore, to take this effect into full account, we assume in our calculations besides air water to fill any pores. In order to calculate the volume fraction of pores P from neff the three mixture models in Section 2.1 were used. As mentioned before, the application of the Gladstone–Dale model results in a linear P-neff

dependence while Lorentz–Lorenz leads to a positive and Drude to a negative deviation from linear additivity. Fig. 5 shows that P varies considerably from one model to another. As a consequence of the anatase-to-rutile transformation P shifts at the onset condition from the air(water)-anatase curve to the air(water)-rutile curve. Particularly for neff = 1.7–1.9 one can see that porosity varies from 28% to 73%. We note that neff of the majority of the prepared films of series 1 and 2 lies within this interval. Consequently, tests of the mixture models have to be carried out to validate the results. To test the validity of the described models the effective density of the porous titania films of series 1 and 2 using P, Eq. (5), and the mass density values of pure anatase (3.89 g cm−3 [38]), rutile (4.25 g cm−3 [40]), water and air were calculated. Effective density was then plotted together with true mass density data from Fig. 1 vs. (n eff − 1) for

Table 2 Film thickness d measured by profilometry as well as Sellmeier coefficients B1 and C1 and film thickness d obtained from fitting of the interference patterns of the optical transmittance. Effective refractive index neff for the wavelength 550 nm is calculated using the Sellmeier equation. Numbers in parentheses give uncertainty of the last digit. Titania film

Profilometry d(nm)

Transmittance d(nm)

B1

C1 (μm2)

neff (550 nm)

H100 H200 H300 H400 H500 H600 H700 H800 H900 H1000 H500/1 H500/2 H500/5 H500/20 H500/50 H500/100 H500/200 H500/500

797(11) 758(8) 730(9) 749(7) 715(6) 630(5) – 420(5) 393(5) 407(4) 709(3) 700(4) 698(3) 687(4) 680(3) 669(3) 660(4) 651(4)

815 789 770 749 712 635 452 432 400 393 702 696 693 685 674 657 651 645

1.7837 1.8089 1.8470 1.8554 2.0562 2.2367 3.5194 4.2463 4.2959 4.6303 2.0719 2.0869 2.1329 2.1544 2.1658 2.1881 2.1932 2.2209

0.0415 0.0445 0.0413 0.0454 0.0448 0.0484 0.0643 0.0376 0.0558 0.0633 0.0447 0.0445 0.0440 0.0438 0.0437 0.0435 0.0434 0.0432

1.7514 1.7667 1.7717 1.7841 1.8476 1.9138 2.3388 2.4185 2.5036 2.6184 1.8560 1.8564 1.8695 1.8756 1.8788 1.8851 1.8865 1.8943

Fig. 5. Calculated pore volume fraction P vs. effective refractive index neff (λ = 550 nm) of titania films of series 1 and 2 according to Eqs. (2) to (4) for air and water filled pores, respectively. Lines show P-neff dependence of binary anatase-air, rutile-air, anatase-water, and rutile-water films using Gladstone–Dale (blue), Drude (red) and Lorentz–Lorenz (green) model. (Assuming bulk refractive index of pure materials: anatase nD = 2.5656 , rutile nD = 2.7418 [26], water nP = 1.33, air nP = 1)[25].

A. Matthias et al. / Thin Solid Films 558 (2014) 86–92

Gladstone–Dale (Fig. 6A), (neff2 − 1) for Drude (Fig. 6B) and (neff2 − 1) / (neff2 + 2) for Lorentz–Lorenz (Fig. 6C) model. Fig. 6A–C reveals that for each case noticeably deviations from the expected linear trend are

91

evident. In detail, ρeff calculated using the Gladstone–Dale model (Fig. 6A) slightly underestimates the observed trend for titania thin films, while when using the Drude model (Fig. 6B) ρeff is found to be in close agreement with the experimental values within the considered density range from 1.4 g cm−3 to 2.2 g cm−3, in particular for the assumption of only partially water-filled pores (water should be assessable exclusively to open porosity). However, as one can see also for the latter model, a linear relationship over the entire density range cannot be established. Finally, a test of the Lorentz–Lorenz model (Fig. 6C) gives ρeff values, which are considerably smaller than the observed densities. Vice versa a strong nonlinearity is evident if the Lorentz–Lorenz model is used to correlate true mass density with refractive index. Against the backdrop of the nonlinear nature of neff -ρ function we finally test the dependence of the effective specific refraction Reff on true mass density. For this Reff was calculated as (n eff − 1) / ρ for Gladstone–Dale, (neff2 − 1) / ρ for Drude and [(neff2 − 1) / (neff2 + 2)] / ρ for Lorentz–Lorenz models. Fig. 7 shows that these ratios as a function of true mass density are not constant, here plotted in double-logarithmic scales. For dilute titania films (ρ b 1) they decrease with increasing true mass density, while at ρ N 1 the effective specific refraction increases again (Drude), becomes finally a constant (i.e. Gladstone–Dale constant: RTiO2 ≈ 0.4) or decreases further (Lorentz–Lorenz). If one anticipates from a physical standpoint that the local polarization in titania films and heterogeneous materials is given by a Lorentz– Lorenz type of interaction, the decrease of Reff with mass density can be described by a simple power-law expression [57],

Reff ρ ¼

neff 2 −1

!

neff 2 þ 2

¼ Aρ

B

ð13Þ

with constants A = 0.29 and B = 0.61 (ρ in g cm−3) obtained from the slope and intercept of the straight line fitted through the data as shown in Fig. 7. This result is in agreement with the analysis of Liu and Daum [57], who showed for ambient aerosols that Reff decreases with increasing mass density, leading to a sub-linear dependence (power exponent b1) of the specific refractive index on mass density. Eq. (13) thus opens up the possibility to calculate effective density and porosity of titania thin films from the specific index of refraction of the Lorentz–Lorenz model with high accuracy. While every effort has been made to ensure considering any accessible data we note that titania films prepared

Fig. 6. Effective refractive index (λ = 550 nm) scaled by neff − 1 (Gladstone–Dale model) (A), neff2 − 1 (Drude model) (B) and (neff2 − 1) / (neff2 + 2) (Lorentz–Lorenz model) (C) vs. calculated effective density of thin films of series 1 and 2 for air-filled and water-filled pores together with data of Fig. 1 (gray spheres). For clarity only the neff–ρeff dependence of binary anatase-air and anatase-water films are shown (dashed and dashed-dotted lines, respectively).

Fig. 7. Effective specific refraction Reff = (neff − 1) / ρ (Gladstone–Dale model), Reff = (neff2 − 1) / ρ (Drude model) and Reff = [(n2 − 1)/(n2 + 2)] / ρ (Lorentz–Lorenz model) as a function of true mass density (λ = 550 nm). The solid line is a best linear fit according to log Reff = (− 0.537 ± 0.003) − (0.389 ± 0.006) logρ, while the dashed lines are intended as visual guides.

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under other processing conditions than those used in Figs. 6–7 may lead to deviations from the calculated trend. 6. Conclusions It is found that the use of mixture models to treat a heterogeneous titania thin film as an effective homogeneous material yields to a wide range of different porosities. Therefore questions arise concerning the validity of these expressions. It is shown that the preparation of titania thin films by an anatase sol of primary particle size below 7 nm and curing temperatures up to 1000 °C can increase the effective refractive index from 1.75 to 2.62, which provides a broad basis for testing predictions of the effective media theory against the background of 67 pairs of published data on refractive index and true mass density. These tests revealed noticeable deviations from the expected trend, which increase in the order using Drude, Gladstone–Dale and Lorentz–Lorenz mixture models. Finally, in order to take into account the nonlinear dependence of the specific refractive index on mass density, an empirical power-law expression consistent with the Lorentz–Lorenz relation is proposed. Acknowledgments The support of the Deutsche Forschungsgemeinschaft (grants DE 598/17-1 and KI 482/13-1) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6]

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