Semi-analytical Expressions Research Articles

Micro–macro approach of anisotropic damage: A semi-analytical constitutive model of porous cracked rock

A semi-analytical constitutive model of a porous cracked rock is proposed based on the micro–macro approach within a thermodynamic framework. The representative volume element (RVE) of a porous cracked rock is divided into two subproblems, P1 and P2. The porosity evolution equation of a spherical pore is proposed for P1, whereas P2 is an inelastic problem containing numerous anisotropic penny-shaped cracks. First, the relationship between microscopic and macroscopic strains is established through homogenization. This relationship is introduced into the macroscopic free energy equation. A crack propagation criterion based on the Griffith release rate is proposed, and the evolution equations of the shear and tensile crack densities are proposed based on the surface energy function. Subsequently, the Gaussian numerical integration method is used to obtain a semi-analytical explicit expression of the macroscopic stiffness tensor. Finally, based on the proposed crack propagation criterion, a strength-prediction method is developed. A series of uniaxial and triaxial experimental data for different rocks from other references is used to validate the proposed constitutive model and strength prediction method. The results obtained using the proposed model agree well with the experimental curves obtained from uniaxial tension and triaxial compression tests performed on different rocks. The macroscopic mechanical behavior and evolution of microscopic parameters for different rocks with an increase in confining pressure are effectively demonstrated by the proposed model. Strength prediction method can accurately describe the failure strength of different rocks. Furthermore, a sensibility analysis of model parameters revealed that initial porosity e0 and dilation coefficient ζ have a deteriorating effect on the failure strength, and material constant η has an enhancing effect on the failure strength.

Scalar induced gravitational waves from Chern-Simons gravity during inflation era

We investigate the scalar induced gravitational waves (SIGWs) in the Chern-Simons (CS) gravity with a dynamical scalar field during slow roll inflation. Due to the parity violation in the CS term, the SIGWs are generally polarized, which are effectively characterized by the degree of circular polarization. We derive the semianalytic expression to evaluate the power spectra and the degree of circular polarization of the SIGWs, which receive contributions from the general relativity and the parity-violating term, respectively. We find that the correction from the parity-violating CS term is negligible on large scales, which means that the degree of circular polarization of SIGWs is very small.

Analytic plenoptic camera diffraction model and radial distortion analysis due to vignetting.

Using a mathematical approach, this paper presents a generalization of semi-analytical expressions for the point spread function (PSF) of plenoptic cameras. The model is applicable in the standard regime of the scalar diffraction theory while the extension to arbitrary main lens transmission functions generalizes a priori formalism. The accuracy and applicability of the model is well verified against the exact Rayleigh-Sommerfeld diffraction integral and a rigorous proof of convergence for the PSF series expression is made. Since vignetting can never be fully eliminated, it is critical to inspect the image degradation it poses through distortions. For what we believe is the first time, diffractive distortions in the diffraction-limited plenoptic camera are closely examined and demonstrated to exceed those that would otherwise be estimated by a geometrical optics formalism, further justifying the necessity of an approach based on wave optics. Microlenses subject to the edge diffraction effects of the main lens vignetting are shown to translate into radial distortions of increasing severity and instability with defocus. The distortions due to vignetting are found to be typically bound by the radius of the geometrical defocus in the image plane, while objects confined to the depth of field give rise to merely subpixel distortions.

Jeffery’s paradox for the rotation of a single ‘stick–slip’ cylinder

We consider the problem of determining the two-dimensional fluid velocity due to the rotation of an infinitely-long ‘stick–slip’ cylinder in an otherwise quiescent Stokes flow. Stick–slip boundary conditions are introduced as a model of a rough superhydrophobic surface, via a distribution of alternating solid–liquid (stick) and gas–liquid (slip) interfaces. This leads to a mixed boundary-value problem for Stokes flow. Complex variable techniques are employed to transform the flow problem into a Hilbert problem, which involves finding a function analytic in a plane region assuming that on some portions of the boundary its real part is known, while on others its imaginary part is given. We solve the Hilbert problem to obtain semi-analytic expressions for all the pertinent fluid-dynamic quantities. We find that in the general aperiodic case there is no solution in which the velocity of the fluid vanishes at infinity. This is a form of Jeffery’s paradox, typically associated with viscous flow due to the counter-rotation of two equal rigid cylinders. Our work provides the first example of Jeffery’s paradox due to the rotation of a single cylinder.

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

The calculation of accurate arbitrary-order sensitivities of eigenvalues and eigenvectors is crucial for structural analysis applications, including topology optimization, system identification, finite element model updating, damage detection, and fault diagnosis. Current approaches to obtaining sensitivities for eigenvalues and eigenvectors lack generality, are complicated to implement, prone to numerical errors, and are computationally expensive. In this work, a novel methodology is introduced that uses hypercomplex automatic differentiation (HYPAD) and semi-analytical expressions to obtain arbitrary-order sensitivities for eigenfrequency problems. The new methodology exhibits no sign of truncation nor subtractive cancellation errors regardless of the order of the sensitivity, it is general, and can obtain any high-order sensitivities with the simplicity of first-order computations. A numerical example is presented to verify the accuracy of the method, where the free vibration of a homogeneous cantilever beam is studied. For this problem, up to third-order sensitivities of the eigenvalues and eigenvectors with respect to the material and geometrical parameters were obtained, considering the cases of close and distinct eigenvalues. The results were verified using analytical equations, showing excellent agreement for the eigenvalues and the eigenvectors. The new method promises to facilitate the computation of sensitivities for eigenfrequency problems into routine practice and commercial software.

Semi-analytical Expression of G-mode Period Spacing: The Case of Brunt–Väisälä Frequency with Not a Jump but a Ramp

To decipher complex patterns of gravity-mode period spacings observed for intermediate-mass main-sequence stars is an important step toward a better understanding of the structure and dynamics in the deep radiative region of the stars. In this study, we apply the JWKB approximation to derive a semi-analytical expression of the g-mode period spacing pattern, for which the gradient in the Brunt–Väisälä frequency is taken into account. The formulation includes the term P −1 B ⋆, where P and B ⋆ represent the g-mode period and degree of the structural variation, the latter of which especially is related to the steepness of the gradient of the Brunt–Väisälä frequency. Tests with one-dimensional stellar models show that the semi-analytical expression derived in this study is useful for inferring the degree of the structural variation B ⋆ with an accuracy of ∼10 % in the case of relatively massive intermediate-mass models with the mass M larger than 3 M ⊙. The newly formulated expression will possibly allow us to put further constraints on, e.g., mixing processes inside intermediate-mass main-sequence g-mode pulsators such as β Cep, SPB, and γ Dor stars that have been principal targets in asteroseismology.

Numerical model supplemented by thin-layer analysis for diffusiophoresis of a particle incorporating finite ion size effects

The impact of finite-sized ions on the diffusiophoresis of a charged colloid subjected to a concentration gradient of electrolyte solution consisting monovalent or multivalent ionic species, is studied. In diffusiophoresis, the ion concentration is of O(1M). In this non-dilute electrolyte solutions, the ion–ion steric interaction is important. We have adopted the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model to account for the ion steric interactions and the Batchelor–Green expression for the relative viscosity of suspension. We have solved the standard model numerically considering ions as point charge (PNP-model), the modified Nernst–Planck equations incorporating the ion steric interaction with constant viscosity (MNP-model), and modification of the MNP-model by incorporating the viscosity variation with the ionic volume fraction (MNPV-model). Semi-analytical expressions for mobility based on a linear perturbation technique under a thinner Debye length is presented for PNP- and MNP-models. In the MNP-model, counterion saturation in the Debye layer due to the ion steric interaction enhances the surface potential by attenuating the shielding effect, diminishes the surface conduction, and magnifies the induced electric field. These in combination create a larger mobility at a thinner Debye length compared with the PNP-model. This increment in mobility attenuates when the MNPV-model is considered. The MNPV-model is more appropriate to analyze the finite ion size effects, and it is found to yield the mobility values more close to the experimental data compared with the MNP- and PNP-model. The semi-analytical expressions for mobility based on the PNP- and MNP-models agree with the corresponding exact numerical solutions when the surface potential is in the order of thermal potential. However, a large discrepancy between the simplified expression and the exact numerical results is found for a concentrated electrolyte in which the induced electric field is large.

Imaging faint sources with the extended solar gravitational lens

We consider resolved imaging of faint sources with the solar gravitational lens (SGL) while treating the Sun as an extended gravitating body. We use our new diffraction integral that describes how a spherical electromagnetic wave is modified by the static gravitational field of an extended body, represented by series of multipole moments characterizing its interior mass distribution. Dominated by the solar quadrupole moment, these deviations from spherical symmetry significantly perturb the image that is projected by the Sun into its focal region, especially at solar equatorial latitudes. To study the optical properties of the quadrupole SGL, we develop an approximate solution for the point spread function of such an extended lens. We also derive semianalytical expressions to estimate signal levels from extended targets. With these tools, we study the impact of solar oblateness on imaging with the SGL. Given the small value of the solar quadrupole moment, the majority of the signal photons arriving from an extended target still appear within the image area projected by the monopole lens. However, these photons are scrambled, thus reducing the achievable signal-to-noise ratio during the image recovery process (i.e., deconvolution). We also evaluate the spectral sensitivity for high-resolution remote sensing of exoplanets with the extended SGL. We assess the impact on image quality and demonstrate that despite the adverse effects of the quadrupole moment, the SGL remains uniquely capable of delivering high-resolution imaging and spectroscopy of faint, small, and distant targets, notably terrestrial exoplanets within $\ensuremath{\sim}30--100\text{ }\text{ }\mathrm{parsec}$ from us.

Steady state concentration in an amperometric biosensor using mathematical modeling

This paper presents an analytical solution to the model. By solving the system of nonlinear reaction-diffusion equations using Akbari–Ganji’s method (AGM), we can derive the semi-analytical expressions for substrate and oxidized and reduced mediator concentrations. These models provide information about enzyme electrode mechanisms and their kinetics. These modeling results can be helpful for sensor design, optimization, and determining how the electrode will react. Further, we studied the effects of many physical and biological parameters on model prediction. The mathematical equations, which are nonlinear partial differential equations, are solved using AGM. Also, we derived the semi-analytical results for concentrations for saturated and unsaturated kinetics. It is verified that the proposed solution is validated by comparing it with numerical solutions.

The divergence of mean phenotypes under persistent directional selection.

Numerous organismal traits, particularly at the cellular level, are likely to be under persistent directional selection across phylogenetic lineages. Unless all mutations affecting such traits have large enough effects to be efficiently selected in all species, gradients in mean phenotypes are expected to arise as a consequence of differences in the power of random genetic drift, which varies by approximately five orders of magnitude across the Tree of Life. Prior theoretical work examining the conditions under which such gradients can arise focused on the simple situation in which all genomic sites affecting the trait have identical and constant mutational effects. Here, we extend this theory to incorporate the more biologically realistic situation in which mutational effects on a trait differ among nucleotide sites. Pursuit of such modifications leads to the development of semi-analytic expressions for the ways in which selective interference arises via linkage effects in single-effects models, which then extend to more complex scenarios. The theory developed clarifies the conditions under which mutations of different selective effects mutually interfere with each others' fixation and shows how variance in effects among sites can substantially modify and extend the expected scaling relationships between mean phenotypes and effective population sizes.

The semi-analytic solution to the magnetic field of frequency-band-selecting eddy current testing for local corrosion flaw in large diameter pipeline.

Local corrosion flaws (LCFs) widely exist in large diameter pipelines (LDPs) (e.g. steam, oil/gas pipeline) due to flow-accelerated corrosion, environmental corrosion, and material aging. Frequency-band-selecting eddy current testing (FBS-ECT), as an improved eddy current technique (ECT), has rich harmonic components in exciting signals as a pulsed eddy current technique (PECT), as well as more effective utilization of electromagnetic energy by selecting a frequency band. It is expected that FBS-ECT has promising applications in detecting and evaluating the LCFs in LDP. Therefore, a solution to the magnetic field of FBS-ECT for LCF detection in LDP is of great concern to reveal the features of the FBS-ECT signal and the parameters of the LCF. Extending our previous work, the semi-analytic expression of the magnetic flux density for FBS-ECT in regions 1 and 2 (where the magnetic sensor is located) is deduced according to the superposition principle and TREE, the computation accuracy is validated by numerical simulation; next, a semi-analytic solution to the magnetic field of FBS-ECT for LCF in LDP is proposed; then, the sampling distance in FBS-ECT is optimized by considering the magnitude of △Bz and the sensitivity of △Bz with the LCF depth; finally, a case study is carried out to verify the optimal strategy and find the mapping relationship between the radius/depth and the FBS-ECT signal. The proposed semi-analytic solution for the magnetic field of FBS-ECT is not only helpful to understand the theoretical relationship of the magnetic field and LCFs with higher computation efficiency but also lays a theoretical foundation to promote FBS-ECT to apply in the characterization of other flaws in metal components.

Imperfect narrow escape problem.

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Metasurface supporting quasi-BIC for optical trapping and Raman-spectroscopy of biological nanoparticles.

Optical trapping combined with Raman spectroscopy have opened new possibilities for analyzing biological nanoparticles. Conventional optical tweezers have proven successful for trapping of a single or a few particles. However, the method is slow and cannot be used for the smallest particles. Thus, it is not adapted to analyze a large number of nanoparticles, which is necessary to get statistically valid data. Here, we propose quasi-bound states in the continuum (quasi-BICs) in a silicon nitride (Si3N4) metasurface to trap smaller particles and many simultaneously. The quasi-BIC metasurface contains multiple zones with high field-enhancement ('hotspots') at a wavelength of 785 nm, where a single nanoparticle can be trapped at each hotspot. We numerically investigate the optical trapping of a type of biological nanoparticles, namely extracellular vesicles (EVs), and study how their presence influences the resonance behavior of the quasi-BIC. It is found that perturbation theory and a semi-analytical expression give good estimates for the resonance wavelength and minimum of the potential well, as a function of the particle radius. This wavelength is slightly shifted relative to the resonance of the metasurface without trapped particles. The simulations show that the Q-factor can be increased by using a thin metasurface. The thickness of the layer and the asymmetry of the unit cell can thus be used to get a high Q-factor. Our findings show the tight fabrication tolerances necessary to make the metasurface. If these can be overcome, the proposed metasurface can be used for a lab-on-a-chip for mass-analysis of biological nanoparticles.

Modeling of spontaneous emission in presence of cylindrical nanoobjects: the scattering matrix approach

We propose a method of analysis of spontaneous emission of a quantum emitter (an atom, a luminescence center, a quantum dot) inside or in vicinity of a cylinder. At the focus of our method are analytical expressions for the scattering matrix of the cylindrical nanoobject. We propose the approach to electromagnetic field quantization based of eigenvalues and eigenvectors of the scattering matrix. The method is applicable for calculation and analysis of spontaneous emission rates and angular dependences of radiation for a set of different systems: semiconductor nanowires with quantum dots, plasmonic nanowires, cylindrical hollows in dielectrics and metals. Relative simplicity of the method allows obtaining analytical and semi-analytical expressions for both cases of radiation into external medium and into guided modes.

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.

Optical conductivity of an anharmonic large polaron gas at weak coupling

In a polar solid, electrons or other charge carriers can interact with the phonons of the ionic lattice, leading to the formation of polaron quasiparticles. The optical conductivity and optical absorption spectrum of a material are affected by this electron-phonon coupling, most notably leading to an absorption peak in the midinfrared region. Recently, a model Hamiltonian for anharmonic electron-phonon coupling was derived [M. Houtput and J. Tempere, Phys. Rev. B 103, 184306 (2021)] that includes both the conventional Fr\"ohlich interaction as well as an interaction in which an electron interacts with two phonons simultaneously. In this article, we calculate and investigate the optical conductivity of the anharmonic large polaron gas, and we show that an additional characteristic absorption peak appears due to this one-electron--two-phonon interaction. We calculate a semianalytical expression for the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ at finite temperatures and weak coupling using the Kubo formula. The electronic and phononic contributions can be split and treated separately, such that the many-body effects of the electron gas may be taken into account through the well-known dynamical structure factor $S(\mathbf{k},\ensuremath{\omega})$. From the resulting optical conductivity, we calculate the polaron effective mass, an estimate for the electron-phonon scattering times, and the optical absorption spectrum of the anharmonic polaron gas. It is shown that the effects are negligible for four common III-V semiconductors (BN, AlN, BP, AlP) in the zinc-blende structure, which justifies the commonly used harmonic approximation in these materials. We show that alongside the well-known polaron absorption peak at the phonon energy $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{LO}}$, the one-electron--two-phonon interaction leads to an additional absorption peak at $2\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{LO}}$. We propose this absorption peak as an experimentally measurable indicator for non-negligible one-electron--two-phonon interaction in a material, since the height of this peak is proportional to the strength of this anharmonic interaction.

Vibrations of a strip with delamination in the framework of the one-parameter Aifantis model of gradient elasticity theory

The problems on in-plane and anti-planar steady-state vibrations of an isotropic elastic strip with delamination at the lower boundary has been investigated. The goal of the study is to analyze the stress-strain state in the crack tips areas and to construct a crack opening function being the main mechanical characteristics in the crack theory problems. The problems under study have been solved in the framework of the nonclassical gradient elasticity theory (GET) on the basis of the one-parameter model proposed by Aifantis. The boundary integral equations (BIE) are obtained with respect to crack opening functions or their derivatives. The analysis of BIEs is carried out, regular and irregular parts are distinguished, the obtained BIEs with singular (e.g., with hypersingular, with cubic singularity) integrals are solved via collocation methods, approximating Chebyshev polynomials, quadrature formulas for singular integrals. For the in-plane problem solution, the simplified Ru-Aifantis method has been applied. The Ru-Aifantis method allows to divide the initial boundary value problem into two sub-problems - the classical linear elasticity theory (LTE) problem and the simplified boundary value problem for finding the gradient solution which includes the solution found via the classical theory. For each of the problems, semi-analytical expressions for the functions of crack opening have been constructed, and the analysis of the stress-strain state in the area of crack tips has been carried out. The problems have also been solved in the case of a crack with small relative length, the analysis of BIE depending on small parameters ratio has been carried out, and explicit expressions for the crack opening functions have been obtained. Numerical calculations have been performed; the applicability conditions for the asymptotic method are determined, and a comparative analysis of the results obtained on the basis of GET and LTE models, depending on the values of the gradient parameter and the delamination length, is realized.

Impedance Response of Ionic Liquids in Long Slit Pores

We study the dynamics of ionic liquids in a thin slit pore geometry. Beginning with the classical and dynamic density functional theories for systems of charged hard spheres, an asymptotic procedure leads to a simplified model which incorporates both the accurate resolution of the ion layering (perpendicular to the slit pore wall) and the ion transport in the pore length. This reduced-order model enables qualitative comparisons between different ionic liquids and electrode pore sizes at low numerical expense. We derive semi-analytical expressions for the impedance response of the reduced-order model involving numerically computable sensitivities, and obtain effective finite-space Warburg elements valid in the high and low frequency limits. Additionally, we perform time-dependent numerical simulations to recover the impedance response as a validation step. We investigate the dependence of the impedance response on system parameters and the choice of density functional theory used. The inclusion of electrostatic effects beyond mean-field qualitatively changes the dependence of the characteristic response time on the pore width. We observe peaks in the response time as a function of pore width, with height and location depending on the potential difference imposed. We discuss how the calculated dynamic properties can be used together with equilibrium results to optimise ionic liquid supercapacitors for a given application.

Continuous absorption and emission of light by the heteronuclear rare gas (ArXe)+ and (KrXe)+ ions

Studies of radiative transitions between ground and charge transfer excited terms of heteronuclear rare-gas (ArXe)+ and (KrXe)+ ions with small and moderate dissociation energies (D0 ArXe+ =179 and D0 KrXe+ =389 meV) are reported. Potential energy curves and dipole transition matrix elements are evaluated using ab initio calculations based on the CASSCF method followed by the NEVPT2 treatment and perturbative description of relativistic effects as implemented in the ORCA suite. We develop an efficient approach for the calculations of Boltzmann-averaged cross sections, rate constants, and absorption and emission coefficients associated with the integral contributions of bound-bound and bound-free transitions from a manifold of rovibrational states of the (RgXe)+ ion. A similar approach is used for the description of the free-free and free-bound transitions between different electronic terms of a quasimolecular (RgXe)+ ion temporarily formed during a collision of Ar or Kr atoms with Xe+ ions. Our method is based on the quantal version of the theory of non-adiabatic transitions in molecular and quasimolecular systems combined with the approximation of a quasicontinuum for rovibrational energy levels. This allowed us to obtain a semianalytic expression for the integral absorption properties particularly relevant at elevated temperatures. We perform a comprehensive analysis of the identified radiative processes in local thermodynamic equilibrium plasmas of rare gas Rg/Xe mixtures containing both atomic, Xe+, and molecular, RgXe+, ions and determine their role in the formation of the absorption and emission spectra in a wide range of wavelengths and gas temperatures. The results obtained are in good agreement with the available experimental data.

Graphene multilayers for coherent perfect absorption: effects of interlayer separation.

We present a model study to estimate the sensitivity of the optical absorption of multilayered graphene structure to the subnanometer interlayer separation. Starting from a transfer-matrix formalism we derive semi-analytical expressions for the far-field observables. Neglecting the interlayer separation, results in upper bounds to the absorption of 50% for real-valued sheet conductivities, exactly the value needed for coherent perfect absorption (CPA), while for complex-valued conductivities we identify upper bounds that are always lower. For pristine graphene the number of layers required to attain this maximum is found to be fixed by the fine structure constant. For finite interlayer separations we find that this upper bound of absorption only exists until a particular value of interlayer separation (Dlim) which is less than the realistic interlayer separation in graphene multilayers. Beyond this value, we find a strong dependence of absorption with the interlayer separation. For an infinite number of graphene layers a closed-form analytical expression for the absorption is derived, based on a continued-fraction analysis that also leads to a simple expression for Dlim. Our comparison with experiments illustrates that multilayer Van der Waals crystals suitable for CPA can be more accurately modelled as electronically independent layers and more reliable predictions of their optical properties can be obtained if their subnanometer interlayer separations are carefully accounted for.