Elasto-Optic Coefficients of Borate, Phosphate, and Silicate Glasses

Article pubs.acs.org/JPCC

Elasto-Optic Coefficients of Borate, Phosphate, and Silicate Glasses: Determination by Brillouin Spectroscopy J. Galbraith,†,‡ L. Chapman,§ J. W. Zwanziger,*,†,‡,§ M. Aldridge,∥ and J. Kieffer∥ †

Department of Physics and Atmospheric Sciences, ‡Institute for Research in Materials, and §Department of Chemistry, Dalhousie University, Halifax, NS, B3H 4R2 Canada ∥ Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *

ABSTRACT: The elasto-optic tensor elements p12 and p44 of glasses in the series BaO−B2O3, PbO−B2O3, BaO−P2O5, PbO−P2O5, and PbO−SiO2 were measured using Brillouin spectroscopy. Densities, refractive indices, and elastic moduli were also determined. A model was also developed for the elasto-optic tensor elements based on a bond-polarizability approach, which was validated using first-principles calculations and then used to explain the compositional trends observed in the elasto-optic tensor elements. It was found that the shear elasto-optic element p44 could be explained in terms of the average bond polarizability anisotropy, while the dilation response p11 + 2p12 involved also the bulk cell polarizability response to volume changes.

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INTRODUCTION Photoelasticity is the general relation between stress or strain and changes in the optical response of a solid. The effect, for strain ϵkl, is typically expressed through (Δε−1)ij =

∑ pijkl ϵkl

photoelastic tensor. Brillouin spectroscopy is a particularly powerful tool that can be used to measure the entire elastooptic tensor of a glass. It is an inelastic light scattering technique that distinguishes the longitudinal and transverse acoustic modes of a transparent material, allowing individual elastic and elasto-optic tensor elements to be determined. There have been attempts to relate the Brillouin scattering of a glass to its composition or properties. Some of the investigations focused on commercially available glasses,7−10 and others considered binary, ternary, and doped glass systems, typically silicates.11−13 Many of the models attempted to correlate the elasto-optic coefficients with dielectric constant or refractive index, bond polarizabilities, and optical deformabilities among other material properties. However, no glasses with negative stress-optic response (i. e., positive p44) were measured, and no universal or predictive trends for pij were found. Thus, there is both fundamental interest and technological need for developing a predictive model for the full elasto-optic tensor of glass.3,4,14 To this end, the current work examines the influence of glass formers and additives with typically positive and negative stress-optic response (p44) on the entire elastooptic tensor. Brillouin scattering spectra have been collected for the binary systems BaO−B2O3, BaO−P2O5, PbO−B2O3, PbO− P2O5, and PbO−SiO2, and the effect of the glass former on the sign and magnitude of the elasto-optic tensor elements has been examined. The effect of the different additives, barium and lead oxide, on p12 and p44 are discussed, in the context of a bond polarizability model adapted from earlier work and studied with first-principles modeling.15

(1)

kl

where ε is the relative permittivity (dielectric) and pijkl is an element of the elasto-optic tensor. Because of symmetry, p can be expressed through Voigt notation with only 36 elements.1 Furthermore, glasses, like other isotropic materials, have only three symmetry-inequivalent nonzero elasto-optic tensor elements pij, only two of which are independent. The nonzero elements are p11, p12, and p44, and p44 = (p11 − p12)/2. The various elements of the elasto-optic tensor are of technological interest in different contexts. The shear element p44 is proportional to the stress-optic coefficient, which relates induced birefringence to applied stress load.2 Thus, for glass used in imaging applications with polarized light, it is typically of interest to find formulations where p44 = 0, that is, “zero stress-optic” glass. A predictive, empirical model has been developed by one of the present authors that relates p44 to structure,3 but it is of interest to expand this to a deeper, atomistic level of understanding. The p12 tensor element factors directly into the Brillouin gain coefficient, which relates to the magnitude of stimulated Brillouin scattering (SBS). SBS is a major limitation in the continued scaling to higher powers in fiber laser systems.4 While some recent work has focused on relating p12 to composition,5,6 much less is known than for p44 and again no atomistic model exists. Typical stress-optic measurements, where an external stress is applied and the phase shift of polarized light traveling through the glass is measured, can only resolve the shear element p44. Other techniques are necessary to measure the entire © XXXX American Chemical Society

Received: July 18, 2016 Revised: August 31, 2016

A

DOI: 10.1021/acs.jpcc.6b07202 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

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BACKGROUND Brillouin scattering occurs when light waves propagating through a material scatter from the small-scale density fluctuations within the material. The scattered photon intensity is measured as a function of the frequency shift of light relative to the incident wavelength. The photon can either create a phonon (Stokes scattering) or absorb a phonon (anti-Stokes scattering) with frequency corresponding to the longitudinal or transverse acoustic modes of the material. The Brillouin spectra of a glass therefore consists of two sets of doublets. The elastooptic tensor elements are determined from the intensities of the shifted Brillouin peaks, while the sound velocities and elastic constants of the material are found from the values of the frequency shifts. The application of Brillouin spectroscopy to glass has been recently summarized, including theory, instrumentation, and applications.16 For horizontally (H) or vertically (V) polarized incident and scattered light, the value of the frequency shift in the Brillouin spectra are related to the longitudinal and transverse sound velocities as follows: Δω VV = qvL =

4πn ⎛⎜ θ ⎞⎟ sin v ⎝2⎠ L λ

Δω VH = qvT =

4πn ⎛⎜ θ ⎟⎞ sin v ⎝2⎠ T λ

T=

⎛ ρ (IVV )ex ⎞1/2 ⎛ n ⎞5⎛ n + 1 ⎞2 (v ) g g L g ⎜ ⎟ ⎜⎜ r ⎟⎟ ⎜ g =⎜ ⎟ ex ⎟ (p12 )r ( I ) n n 1 ( v + ρ ⎠ L)r ⎝ r VV r ⎠ ⎝ g ⎠ ⎝ r

(p12 )g

p44 p12

⎞2 ε 4 ⎛⎜ 2 θ ⎟ 2 p cos p cos + θ 12 ⎠ 2 ρvL 2 ⎝ 44

IVH = IHV ∝

ε 4p44 2 ρvT 2

cos2

θ 2

1/2 vT ⎛ IVH ⎞ ⎜ ⎟ vL ⎝ IVV ⎠

(7)

2 θ ⎛ ⎞2 vL ⎜ 2p44 cos 2 + p12 cos θ ⎟ IHH I (θ ) = = ⎜ θ ⎟ IVH vT ⎝ p44 cos 2 ⎠

(2)

(8)

As θ is varied by small amounts around 90°, the change in the intensity ratio is d I (θ ) dθ

θ= 90 °

⎛ p ⎞ ∝ ⎜⎜ −1 − 12 ⎟⎟ p44 ⎠ ⎝

(9)

Then, since |p12/p44| ≫ 1, I(θ) is an increasing (decreasing) function of θ if the signs of p12 and p44 are different (the same). The absolute value of p44 can be determined from an independent measurement of the stress-optic coefficient.

For Brillouin spectra collected at a scattering angle θ, measured for horizontally or vertically polarized incident and scattered light, the scattered intensities can be related to the elasto-optic tensor elements pij by

IHH ∝

=

The relative signs of p44 and p12 are determined by measuring the change in HH and VH spectra as the scattering angle θ is varied.7 The ratio of HH and VH intensities is

(3)

ε4 p 2 ρvL 2 12

(6)

where the notations g and r represent properties for the glass and reference samples, respectively, and ex represents experimentally measured Brillouin intensities. Once p12 is determined, the shear elasto-optic coefficient p44 can be found by comparing the longitudinal and transverse peaks of a sample:

C11 = ρvL 2

IVV ∝

(5)

where n is the refractive index of the sample (ε = n2).7,8,19 The elasto-optic coefficient p12 is then determined from

where q is the wavevector of the incident photons and vL and vT are the longitudinal and transverse sound velocities of the scattering medium. VV and VH give the relative polarizations of incoming and scattered light. The elastic constants are then

C44 = ρvT 2

⎛ 2 ⎞2 ⎛ 2n ⎞2 ⎜ ⎟ ⎜ ⎟ ⎝ n + 1⎠ ⎝1 + n ⎠

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METHODS Glass Preparation. The glasses in this study were prepared by conventional melt-quenching techniques. Commercial-grade PbO (≥99.9%, Sigma-Aldrich), BaCO3 (≥99%, Alfa Aesar), B2O3 (99%, Sigma-Aldrich), NH4H2PO4 (≥98%, SigmaAldrich), and SiO2 (purum p.a., Sigma-Aldrich) were weighed and mixed in stoichiometric amounts. The lead glasses were made in platinum crucibles, while alumina crucibles were used for the barium glasses. Lead borates and silicates were melted directly at 1000−1100 °C for 8−12 h. Lead phosphates were calcined at 550 °C for 24 h before being melted at 1000 °C for 1−2 h. Barium borates were melted directly at 1500 °C for 1−2 h. Barium phosphates were calcined at 550 °C for 22 h and 850 °C for 24 h, and melted at 1100−1300 °C for 12−36 h. For all samples, the melts were quenched on a brass plate, either at room temperature or heated to 200−500 °C. After being cast, the lead borates, silicates and phosphates were annealed at 350−500, 450, and 150−350 °C respectively, while the barium borates and phosphates were annealed at 650 and 450−550 °C. For all glasses, nominal compositions were confirmed by mass-loss and density measurements. The compositions of barium phosphates, lead borates, and lead phosphates were also confirmed using electron microprobe analysis. The barium phosphates were found by this method to contain 1−7 mol % Al contamination from the alumina crucible

(4)

where ε is the relative permittivity (dielectric constant) of the glass, ρ is its density, and vL and vT are the longitudinal and transverse sound velocities, respectively.17,18 Typically, spectra are collected at a right angle scattering geometry (θ = 90°). To determine the absolute values of the elasto-optic coefficients from the Brillouin spectra, a few special considerations are necessary. The Brillouin intensities of the glass samples need to be compared to those of a reference material with known density, refractive index, and elastic and elasto-optic properties.7−10,13 Toluene or fused quartz are often used as references. The transmission of light at the air-glass interface also needs to be accounted for. For normal incidence, the scattered light intensities in eq 4 must be weighted by the transmissivity, B


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fused quartz used to compute the ratios in eq 6 include index of refraction 1.48, density 2.202 g/cm3, and p12 = 0.286.9 νL for the reference sample was computed from each day’s reference runs using the reference index of refraction and eq 2. Longitudinal and transverse peaks for reference and sample Brillouin spectra were fit using the Fityk software package.20 Background noise was subtracted from the spectra where necessary. Spectral lines were fit with Voigt functions for glass, because the structural inhomogeneity of glass routinely imparts a significant Gaussian character to the line shape. Lorentzian functions were used for the fits to toluene spectra. The area under the curve, normalized by the number of scans, gives the Brillouin intensity, while the position of the center of the peak relative to the Rayleigh peak gives the frequency shift, measured in GHz. Each Brillouin spectrum shows Stokes and anti-Stokes scattering; both absolute frequencies and intensities are used to determine properties of interest. Uncertainties in the measured quantities were estimated from multiple repeated measurements of the spectra. First-Principles Calculations. Elasto-optic tensor elements were computed for model compounds using first-principles (density functional theory, or DFT) methods. The DFT calculations were carried out using the ABINIT software package, a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors (URL http://www.abinit.org). This code package provides an implementation of DFT for periodic systems using a planewave basis and pseudopotentials.21,22 Norm-conserving pseudopotentials of optimized Vanderbilt type were used,23 together with local-density approximation (LDA) exchange and correlation functionals. A planewave energy cutoff of 45 Ha together with a k-point spacing at least as tight as 0.027 Å−1 were used. These values were sufficient to converge the cell stress to better than 0.01%. The choice of LDA was made specifically because in this study, crystalline PbO was one of the most interesting model compounds to examine, and PbO is a layered material. It has been found that when dispersion interactions are important in a layered solid but dispersion corrections are not made explicitly, then LDA outperforms GGA functionals in obtaining reasonable geometries.24 The elasto-optic tensor elements were computed using the following procedure. First, for a given experimental crystal structure, all forces and stresses were relaxed to obtain residual stress of less than 10−3 GPa, which as usual with LDA led to a contraction of lattice constants by about 1−3%. The clampedion dielectric tensor was then computed for this model, which corresponds to ε∞. Such a dielectric tensor models the material response at frequencies far above phonon frequencies but below electronic absorption, which is the regime where optical glasses are generally of interest. The dielectric calculation was made within ABINIT using the density-functional perturbation theory formalism.25,26 Then, finite strains were applied to the unit cell, the ion positions relaxed, and the dielectric recomputed. In this way the elasto-optic elements could be computed by a finite-difference approach. Four strains ϵkl of values −0.01, −0.005, 0.005, and 0.01 were used for each symmetry-distinct direction, and the elasto-optic element computed from a linear fit of ΔBij to ϵkl, where

and it is likely that the other Ba-containing glasses also contain small Al contamination. The glass samples were cut with a low-speed diamond saw to have a square base of 10 × 10 mm with a maximum variation of 1 mm on either length. The height of each sample was cut to be >4 mm to ensure that no scattered light was blocked by the edges of the sample holder. Three faces of the glass perpendicular to the square base were polished to 1 μm optical transparency using silicon carbide paper and diamond paste of decreasing particle size. Density, Elastic Moduli, Refractive Index, and StressOptic Coefficient Measurements. Densities were measured using Archimedes method with >99% ethanol as the immersion fluid. Sound velocities vT and vL were measured using a Panametric ultrasonic thickness gauge, as well as from the frequency shift of the transverse and longitudinal Brillouin peaks. Wavelength-dependent refractive indices were measured using an M-2000 Woollam ellipsometer; here, the amplitude and phase of the reflected polarized light were fit to a Cauchy model. The interpolated value of n at 532 nm was used for the analysis of the Brillouin spectra. The stress-optic coefficients of the glasses were measured by applying a stress to the glass and measuring the phase shift of light according to the Sénarmont compensator method.3 The values of C were determined at 532 nm; the shear elasto-optic tensor element can then be found from C=−

n3 p 2G 44

(10)

where G is the shear elastic modulus. Collection and Analysis of Brillouin Spectra. The Brillouin spectra of the glasses were collected at the University of Michigan using a Sandercock six-pass Tandem Fabry−Perot interferometer (TFPI) with 512-channel binning. Polarized laser light with wavelength 532 nm and tunable power set to 145 mW was passed through a beam splitter to separate the light into two beams. The first reference beam was focused on the TFPI, while the second sample beam followed a path through the glass sample. Light scattered from the sample at 90° relative to the incident light was focused on the interferometer. For each glass sample, the intensity IVV was measured at 5− 10 spots along the height of the glass cube, with only 25−100 scans necessary to resolve the Brillouin peaks. The intensity IVH was collected at 1−5 spots along the height of the glass. Since p44 is typically much smaller than p12, many more scans were necessary to resolve the Brillouin peaks in the VH geometry, ranging from 500 to 5000 scans. The HH and VH spectra for glasses with nonzero p44 were measured at θ = 85°, 90°, and 95°, with 100−1000 scans necessary to resolve the peaks. The sign of the slope of IHH/IVH gives information on the relative sign, and then the absolute sign of p44 is determined from independently measured stress-optic coefficients at 532 nm. Prior to measurements for each glass sample, VV measurements of fused quartz and toluene were taken, typically with 25−50 scans per measurement, using reference samples of identical size. Both gave equivalent results within uncertainty for the each glass sample, however, only results based on the fused quartz reference are presented here. The reference intensity and frequency shift used in the calculation of the elasto-optic tensor element p12 was taken from the average and standard deviations of that day’s measurements to minimize variations due to sample placement. The reference data from

ΔBij = [ε−1(ϵkl)]ij − [ε−1(0)]ij C

(11) DOI: 10.1021/acs.jpcc.6b07202 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C is the difference between the inverse dielectric tensors of the strained and unstrained models.

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RESULTS Refractive Index, Density, Elastic Moduli, and StressOptic Coefficient. Refractive indices measured at 532 nm, densities, and sound velocities determined from the ultrasonic method are found in Table 1. As the amount of additive Table 1. Refractive Indices, Densities, and Sound Velocities of Lead and Barium Borates, Phosphates, and Silicates family

x

n

ρ (g/cm3)

vL (km/ s)

vT (km/s)

xBaO −(1 − x)B2O3

0.20 0.25 0.30 0.35 0.35 0.40 0.45 0.50 0.55 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.40 0.45 0.50 0.55 0.60 0.50 0.55 0.60

1.59(1) 1.60(1) 1.61(1) 1.62(1) 1.55(1) 1.56(1) 1.58(1) 1.59(1) 1.60(1) 1.78(1) 1.81(2) 1.86(1) 1.93(1) 1.99(3) 2.03(3) 2.04(2) 1.67(5) 1.69(1) 1.71(1) 1.76(1) 1.80(2) 1.85(1) 1.92(1) 1.95(1)

2.89(1) 3.091(5) 3.375(5) 3.60(1) 3.26(1) 3.351(5) 3.482(5) 3.652(5) 3.84(1) 4.598(5) 5.037(5) 5.324(5) 5.67(1) 5.992(5) 6.278(5) 6.514(5) 4.036(5) 4.338(5) 4.65(1) 5.023(5) 5.422(5) 5.649(5) 5.991(5) 6.392(5)

5.15(1) 5.24(1) 5.30(1) 5.27(1) 4.17(1) 4.49(1) 4.37(1) 4.28(1) 4.30(1) 4.33(1) 4.12(1) 3.99(1) 3.73(2) 3.57(3) 3.33(3) 3.16(3) 3.48(1) 3.44(1) 3.35(1) 3.29(1) 3.26(1) 3.38(2) 3.26(1) 3.09(1)

2.834(5) 2.872(5) 2.908(5) 2.871(5) 2.287(5) 2.491(5) 2.387(5) 2.309(5) 2.287(5) 2.394(5) 2.281(5) 2.198(5) 2.046(5) 1.918(5) 1.779(5) 1.688(5) 1.895(5) 1.846(5) 1.774(5) 1.735(5) 1.680(5) 1.901(5) 1.814(5) 1.704(5)

xBaO −(1 − x)P2O5

xPbO −(1 − x)B2O3

xPbO −(1 − x)P2O5

xPbO −(1 − x)SiO2

Figure 1. Typical Brillouin spectra, here for 20BaO-80B2O3 glass, showing scattered light intensity as a function of frequency shift. Inelastic scattering from longitudinal waves are measured in VV configuration (blue), while transverse waves are measured in VH configuration (red). The unshifted, and much more intense, elastically scattered Rayleigh line is at 0 GHz. The discrete data points are connected by lines as a guide to the eye.

separation between the values of p44 for for barium borates and phosphates with equal mol % additive. As x increases, leadcontaining glasses transition from negative to positive p44, while the elasto-optic coefficient of the barium-containing glasses remains negative regardless of additive content. This is consistent with the model of photoelasticity presented by Guignard et al.3 Furthermore, the values of p44 measured using Brillouin scattering are in agreement with those determined from the stress-optic coefficients of the glass. Inconsistencies are likely due to small inhomogeneities in the stress applied to the glass during the stress-optic measurements. The elasto-optic tensor elements p12 are plotted versus mol % of additive (x) in Figure 3. The values of p12 decrease as x increases. This trend is opposite that of p44, and is consistent with previously published results.8,11−13 Values of p12 show a dependence on the glass former present in the system. At the additive content of x = 0.3, barium phosphates have a larger value of p12 than barium borates. There is a similar separation between lead phosphates and lead borates. The coefficients also depend on the modifier. Barium phosphates have larger values of p12 than lead phosphates with equal additive content. The same is true of barium borate glasses. For all glass systems considered, the elasto-optic coefficients p12 are one to 2 orders of magnitude larger than values of p44. For all glasses measured, p12 is a positive value.

content is increased in the glass, the refractive indices and densities also increase. This increase is greater in lead glasses than barium glasses, as expected. The longitudinal and transverse sound velocities are larger in the barium glasses than in lead glasses with the same glass former. For the barium borates and phosphates, vL and vT do not vary greatly with composition. The sound velocities both decrease as x increases for the lead-based glasses. For all glasses, the ratio vT/vL ≈ 0.55 is independent of composition. Brillouin Results. A typical Brillouin spectrum is shown in Figure 1. The small difference in intensity (area) of the VV Stokes and anti-Stokes peaks and of the VH Stokes and antiStokes peaks is taken to be due to the thermal difference between excitation from the ground state (Stokes) and excited states (anti-Stokes). The sound velocities and elasto-optic tensor elements determined from Brillouin scattering spectra are compiled in Table 2. The values of vL and vT are in agreement with those measured using the ultrasonic method. The relationship between the shear elasto-optic coefficients p44 and mol % additive (x) is shown in Figure 2. A linear trend is apparent for each binary glass family, the slope of which is consistent between different glass formers and modifiers. The lead borates, phosphates, and silicates have overlapping elastooptic coefficients, notably at x = 0.5, while there is some

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DISCUSSION The Brillouin measurements shown above (Figures 2 and 3) yield respectively the two elasto-optic tensor elements p44 and p12. In an isotropic material there are only two independent elasto-optic elements, and only three nonzero elements: the third, p11, is related to the others by p11 = 2p44 + p12. We show p11 for the glasses studied here in Figure 4. The various glass modifiers studied here behave broadly similarly. First, both barium and lead addition correlates with an increase in p44, as has been seen before both for these and many other modifiers.27−29 Note that due to the minus sign in eq 10, the stress-optic coefficient, as is usually measured in glass, changes oppositely to p44. D


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Table 2. Transverse and longitudinal Sound Velocities and Elasto-Optic Tensor Elements Measured by Brillouin Scattering for Lead and Barium Borates, Phosphates, and Silicates family

x

vL (km/s)

vT (km/s)

p12

p44

xBaO −(1 − x)B2O3

0.20 0.25 0.30 0.35 0.35 0.40 0.45 0.50 0.55 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.45 0.50 0.55 0.60 0.50 0.55 0.60

5.19(12) 5.30(11) 5.36(10) 5.32(9) 4.22(13) 4.50(6) 4.45(12) 4.32(11) 4.36(11) 4.37(9) 4.20(8) 4.00(8) 3.80(8) 3.54(8) 3.32(4) 3.13(9) 3.43(12) 3.38(6) 3.34(11) 3.27(5) 3.29(1) 3.19(5) 3.06(9)

2.84(9) 2.91(9) 2.93(8) 2.90(8) 2.31(11) 2.54(9) 2.44(10) 2.34(8) 2.34(8) 2.51(16) 2.29(3) 2.21(5) 1.94(25) 1.92(2) 1.82(5) 1.71(7) 1.94(11) 1.84(10) 1.77(10) 1.75(9) 1.90(17) 1.79(6) 1.72(7)

0.296(15) 0.279(15) 0.251(7) 0.251(4) 0.311(7) 0.307(4) 0.296(8) 0.285(8) 0.267(5) 0.211(13) 0.213(19) 0.196(19) 0.192(16) 0.200(29) 0.185(26) 0.191(32) 0.252(25) 0.245(19) 0.215(18) 0.220(21) 0.205(11) 0.204(6) 0.198(7)

−0.0459(80) −0.0453(63) −0.0376(63) −0.0360(52) −0.0134(9) −0.0189(9) −0.0141(11) −0.0104(5) −0.0080(3) −0.0139(9) −0.0109(11) −0.0059(7) 0.0026(2) 0.0039(6) 0.0072(11) 0.0080(14) −0.0053(7) 0.0017(5) 0.0057(5) 0.0078(10) 0.0014(2) 0.0098(35) 0.0130(19)

xBaO −(1 − x)P2O5

xPbO −(1 − x)B2O3

xPbO −(1 − x)P2O5

xPbO −(1 − x)SiO2

Figure 2. Elasto-optic coefficient p44 shown as a function of additive content. Values from Brillouin experiments are given by filled symbols, while those derived from stress-optic coefficient measurements are given by unfilled hollow symbols.

Figure 3. Elasto-optic coefficient p12 shown as a function of additive content.

In order to begin to explain these trends we adapt the bond polarization model of Cardona et al. to the present case.15 In this approach, two polarizabilities are associated with each chemical bond. These polarizabilities, α∥ and α⊥, measure the bond polarizability in the bond direction and perpendicular to it, respectively. This ansatz requires some comment. First, we note that the notion of a “bond” in a solid is not itself an observable, as it is somewhat arbitrary as to which approach distances are short enough to be considered bonding. Second, the assignment of only two polarizabilities to each bond implies that they have cylindrical symmetry. Nonetheless, despite these caveats, we consider this approach potentially fruitful in that it will suggest which additives should be used to confer various elasto-optic properties, and so should help both to explain the current data as well as suggest routes to new useful glass compositions.

Second, the coefficient p12, which is much less commonly measured in glass, is seen to decrease with added modifier. Finally, p11, derived as noted above from p44 and p12, is evidently only weakly dependent on composition. As there are only two independent elasto-optic tensor elements, it is equivalent to consider other combinations besides p12 and p44, and for our purposes we will consider p44 and p11 + 2p12, which separate the strains into a pure shear response and a pure dilation response, respectively. The composition dependence of p11 + 2p12 is shown in Figure 5. This combination also decreases with added modifier, more strongly than does p12, due to the presence of two factors of p12. These factors are more significant than the dependence on the increasing p44, which has smaller magnitude. E


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where the final factor of 3 arises from the relation of volume under a small dilation V → V(1 + 3ϵ). Upon differentiating eq 13 by volume, we obtain finally p11 + 2p12 =

1 (3⟨αV ⟩ − ⟨α1⟩) ε(0)2

(16)

with ⟨αV ⟩ =

P =1−ε V

⟨α1⟩ = 3

∂P ∂V

p11 − p12 = −

Having partitioned all the polarizability to the bonds, the total polarizability is written15 Pij =

∑

+ (δij −

b

(13)

For small strains, the relation defining the elasto-optic tensor, eq 1, can be rearranged to 1 − Δεi = ∑ pij ϵj ε(0)2 j (14) where ε(0) is the unstrained permittivity, and Voigt notation is used for the tensor elements. For a pure dilation, the same strain ϵ is imposed in the three Cartesian directions. Thus, each component εi is enhanced by pii + 2pji, and one obtains from eq 14 p11 + 2p12 = −

1 Δε 1 ⎛⎜ ∂ε ⎞⎟ 3 = 2 ε(0) ϵ ε(0)2 ⎝ ∂V ⎠

(19)

⟨αV ⟩ = 4(⟨α ⟩ + 2⟨α⊥⟩)/3V

(20)

⟨αQ ⟩ = 4(⟨α ⟩ − ⟨α⊥⟩)/3V

(21)

From eqs 17−19 and the measured data for pij, we may compute the polarizability parameters ⟨αV⟩, ⟨α1⟩, and ⟨αQ⟩ for each sample. These are plotted as a function of additive in Figures 6 and 7. First, ⟨αV⟩, Figure 6, is essentially the permittivity and shows therefore the same behavior as n and ε. It increases with the additives studied here, BaO and PbO, because of their high electron density, and the effect is strongest with PbO. Next, ⟨αQ⟩, Figure 7, which is related to p11 − p12 and therefore also to p44, shows the same behavior as p44 itself, scaled by the permittivity. Using eq 21, we can thus interpret the change in p44 and hence stress-optic response as arising from the increased bond anisotropy (⟨α∥⟩ − ⟨α⊥⟩) as Si−O, B−O, and P−O bonds are replaced by Ba−O and Pb−O bonds. Finally, ⟨α1⟩, Figure 8, involves the change in polarization with volume and hence conflates electronic properties with structure in a nontrivial way. In Cardona’s original formulation, this term was due only to the change in polarizability with bond length, but in our approach, starting from the (observable) elasto-optic elements, this term involves the entire cell. Its dependence on additives is fairly weak, but does show some trends, namely, a mild increase with additive in the BaO cases and a decrease in the PbO cases.

(12)

where the sum is over all bonds in the unit cell. Because the polarizabilities α∥, α⊥ depend on bond length, the overall polarization depends on the strain state of the cell. In fact the dielectric permittivity depends on the polarizability through εij = δij + Pij/V

1 ⟨αQ ⟩ ε(0)2

defining ⟨αQ⟩. In Cardona’s work, very similar relations (and others) were derived starting from the bond model. We take the opposite approach, defining ⟨αV⟩, ⟨α1⟩, and ⟨αQ⟩ in terms of elasto-optic tensor elements. The reason for doing this is that ε(0) and the elasto-optic elements are well-defined observables, which may be both measured and computed by first-principles, with no arbitrariness in the identification of bonding pairs. On the other hand, because the materials under study include a variety of different bonds, the ⟨α⟩ parameters derived as above should be understood as average bond polarizabilities, where the average is over the different bonds present in the sample. We may then, following Cardona et al.,15 associate average bond elements ⟨α∥⟩, ⟨α⊥⟩ with the above-defined polarizabilities through

Figure 5. Pure dilation elasto-optic coefficient p11 + 2p12 shown as a function of additive content.

b b R̂ i R̂ j )α⊥b]

(18)

where we have dropped the component subscripts for simplicity (so P here is polarizability). These results are similar to Cardona’s, but expressed directly in terms of the total polarizability and volume. The anisotropy is related similarly to the elasto-optic anisotropy through

Figure 4. Elasto-optic coefficient p11 shown as a function of additive content.

b b [R̂ i R̂ j α b

(17)

(15) F


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Figure 6. Effective polarizability parameter ⟨αV⟩ (see eq 17) as a function of glass additive.

Figure 9. Effective polarizability parameters for B2O3, SiO2 (α-quartz), MgO, BaO, and PbO (litharge). The abscissa is labeled by the bond type in each case.

for B2O3, SiO2 (α-quartz), MgO (included as a transition from light atoms to heavy atoms), BaO, and PbO (litharge). To obtain the average values shown here, the full computed elastooptic tensors were Voigt-averaged to obtain effective isotropic values.30 From the values in Figure 9, the individual bond values are extracted using eqs 20 and 21 and reported in Figure 10.

Figure 7. Effective polarizability parameter ⟨αQ⟩ (see eq 19) as a function of glass additive.

Figure 10. Effective bond polarizability parameters for B2O3, SiO2 (αquartz), MgO, BaO, and PbO (litharge). The abscissa is labeled by the bond type in each case.

The bond polarizabilities (Figure 10) indicate the strong differences between the bond types in these glasses and provide a qualitative explanation for the p44 and stress-optic behavior of these glasses. In particular, the B−O and Si−O bonds show polarizability only along the bond, and virtually none in the perpendicular direction. This result aligns with the fact that these are strong, covalent bonds, with large HOMO−LUMO gaps and low metallicity.31 The Ba−O and Pb−O bonds, on the other hand, show increasing perpendicular polarizability, to the extent where in PbO we have ⟨α⊥⟩ > ⟨α∥⟩. This behavior is why in oxide glasses with high lead content, one finds p44 > 0 and the stress-optic coefficient less than zero. At intermediate content, the response of the directed covalent bonds is canceled by the heavy metal oxide bonds, and zero stress optic glass is

Figure 8. Effective polarizability parameter ⟨α1⟩ (see eq 18) as a function of glass additive.

In order to develop some intuitive understanding of these parameters we have computed them from first-principles for a variety of model compounds of the compositions used in the glasses studied here. Detailed numerical results are provided in the Supporting Information, and here the trends are reported in bar graph form. Figure 9 shows ⟨αV⟩, ⟨α1⟩, and ⟨αQ⟩ computed G


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of the terms in the model and will require future work to express in a simple pictorial way. Nevertheless, the present work represents an important step in developing an atomistic understanding of photoelasticity and in so doing as a way to suggest new glass formulations with desired photoelastic properties.

obtainable. Our previous empirical model, in which bond length and coordination number were used to estimate metallic behavior and deformability,3 can now be understood as a simple proxy for the more detailed bond behavior uncovered here. In order to explain p11 and p12, the additional parameter ⟨α1⟩ (eq 16) must be considered. Here we note that in this model, the value of p11 + 2p12 is a competition between ⟨αV⟩ and ⟨α1⟩ (eq 18). The first term, ⟨αV⟩, is always positive, as it essentially reflects the permittivity and thus increases in the present cases with addition of Ba or Pb (see Figure 6). On the other hand, from Figure 9, we see that while ⟨α1⟩ is small and positive for the glass formers, it is strongly positive for BaO, and it is smaller and negative for PbO. This behavior is reflected in Figure 8 for the glasses, where ⟨α1⟩ is seen to increase upon BaO substitution and is roughly constant or decreases for PbO substitution. Thus, the elasto-optic dilation, p11 + 2p12, might be expected to decrease fairly strongly, as the weaker effect of Ba addition on increasing ⟨αV⟩ is more than offset by the strong positive ⟨α1⟩. Lead addition, however, features a stronger increase in ⟨αV⟩, due to lead’s higher electron density, and a small additional enhancement due to the smaller but negative ⟨α1⟩ term for PbO. In fact this only very roughly captures the behavior of the dilation term, which is closer to constant rather than increasing. Still, this model certainly seems to be of at least qualitative use in explaining features of the elasto-optic response of glass in terms of individual bonding structures. To conclude this discussion, we note that the above explanations depend on the transferability of the model parameters derived from simple compounds to glasses. For the Si−O bonds in α-quartz, for example, we would expect the transferibility to be very good, because the bond lengths and angles, and coordination numbers, are known to be quite similar between the crystal and glass. BaO and PbO might be less transferable, because, while the bond lengths and cation coordination numbers are similar between the crystal forms and Ba and Pb oxide glasses, the associated oxygen coordinations are definitely different. In particular, in BaO (FCC) the oxygen coordination is 6, while in oxide glasses it is nearly always 2 or sometimes 3, and in PbO (litharge) the oxygen coordination number is four. This discrepancy is anticipated to have the biggest effect on our estimations of the ⟨α1⟩ parameter, because as noted this parameter includes effects both of polarization and structure.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b07202. Additional experimental results on the glass samples obtained with static methods, Cauchy fit parameters for the index of refraction, and detailed results of the modeling studies (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Natural Sciences and Engineering Research Council of Canada (Canada Grant Number RGPIN 261987) is gratefully acknowledged. Computational facilities are provided by ACENET, the regional high performance computing consortium for universities in Atlantic Canada. ACENET is funded by the Canada Foundation for Innovation (CFI), the Atlantic Canada Opportunities Agency (ACOA), and the provinces of Newfoundland and Labrador, Nova Scotia, and New Brunswick.

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REFERENCES

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CONCLUSIONS In this contribution, the elasto-optic elements p12 and p44 are reported for a variety of oxide glasses modified with BaO or PbO. Trends in p44 were largely as expected based on the considerable body of knowledge of stress-optic response of glass, but the trends in p12 are largely new. Most importantly, a bond polarizability model was adapted to the present case, and explored from the macroscopic level by using the experimental data to derive effective bond values, and from the microscopic level by computing the values from first principles. In this way, a qualitative explanation could be given for the compositional trends in the elasto-optic elements. In particular, it was found that p44 can be well explained by the difference in parallel and perpendicular polarization components of the bonds. The element p12, or more precisely the dilation term p11+2p12, was described as an interplay between the bulk permittivity term ⟨αV⟩ and the polarization dependence on structure, as expressed through ⟨α1⟩. This latter term is the least transferable H


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Elasto-Optic Coefficients of Borate, Phosphate, and Silicate Glasses

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