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65c2abace9ebbb4db9cf1a24 | 0 | First-principles predictions of linear optical spectra of molecules in the condensed phase remains a challenging problem in computational chemistry. The linear absorption and fluorescence lineshapes of chromophores often have vibronic contributions, making an explicit quantum mechanical treatment of the nuclei necessary. Additionally, the condensed phase environment can strongly influence optical spectra, both through environmental polarization effects and direct chromophoreenvironment interactions, such as hydrogen bonding in protic solvents. Slow, generally anharmonic, collective chromophore-environment motion also poses a challenge and is often of importance in complex biological systems. Robust computational approaches capable of capturing these effects from first principles are desirable, as experimental condensed phase optical spectra are often highly congested, making it challenging to establish structure-property relationships from experimental data alone. Additionally, accurately modeling linear spectroscopy often forms the first step in trying to interpret more complicated non-linear experiments used to probe excited state relaxation dynamics in complex systems. A commonly used framework to model optical spectra of molecules is the Franck-Condon (FC) approach. It relies on approximating the ground-and excited state potential energy surfaces (PESs) as harmonic around their respective minima. Nuclear wavefunctions are then a) Electronic mail: [email protected] harmonic oscillator wavefunctions of vibrational modes and the intensity of vibronic peaks is directly related to ground-and excited state wavefunction overlaps following Fermi's golden rule. The methodology accounts for changes in curvature between the ground-and excited state PES, as well as Duschinsky mode-mixing effects. The underlying harmonic nuclear Hamiltonian (referred to as a Generalized Brownian Oscillator Model or GBOM, see Appendix A 1) can be directly parameterized from ground-and excited state geometry optimizations and frequency calculations implemented in many electronic structure methods. Additionally, the exact optical spectrum for the GBOM Hamiltonian can be computed analytically. The method performs very well in predicting linear spectra for small, rigid molecules in non-polar solvents, where the harmonic approximation is expected to hold. However, the approach can struggle in semi-flexible molecules undergoing anharmonic motion, and cannot account for strong solute-solvent interactions, as the condensed phase environment is generally represented collectively through polarizable continuum models (PCMs). To compare directly to experiments, solvent broadening effects have to be accounted for through approximate broadening parameters, or by invoking timescale separation arguments. These shortcomings limit the applicability of the FC approach in large semi-flexible molecules and systems where the chromophore and the condensed phase environment undergo slow, coupled motion. |
65c2abace9ebbb4db9cf1a24 | 1 | An alternative approach to constructing linear spectra in the condensed phase is the cumulant method, where the linear response function is directly constructed from the fluctuations of the excitation energy sampled along ground-state molecular dynamics (MD) simulations in thermal equilibrium. Direct solute-solvent interactions, anharmonic effects, collective chromophore-environment motion and environmental polarization effects 10 are all accounted for in the MD sampling, making it a highly promising, albeit computationally expensive, approach in biological systems and pigment-protein complexes. Additionally, the computational cost of sampling energy gap fluctuations can be significantly reduced using machine learning (ML) techniques. However, in practical calculations, the cumulant expansion is generally truncated at second order, corresponding to mapping energy gap fluctuations onto a bath of linearly coupled harmonic oscillators (Brownian Oscillator Model or BOM). This truncation is only exact for systems where energy-gap fluctuations follow Gaussian statistics, and already leads to errors in the harmonic GBOM Hamiltonian including changes in PES curvature and Duschinsky mode mixing effects, that are captured exactly by the FC approach. The inclusion of higher order cumulants is vital in computing the correct lineshapes many systems, and we have recently demonstrated that a third order correction term can be constructed directly from MD, yielding improvements of spectral shapes in systems with small non-Gaussian contributions to the energy gap fluctuations. However, the approach has the significant shortcoming of being numerically unstable, yielding unphysical spectra with negative absorbances and divergent lineshapes for moderate to strong non-Gaussian fluctuations. These divergences can be traced to the discarding of higher order terms in the cumulant expansion and have been observed previously in 1D anharmonic model potentials. Since it is in general difficult to determine a priori whether the inclusion of a third order cumulant correction will improve the spectral lineshape or cause unphysical spectral contributions, the benefit of extending the cumulant method beyond second order in realistic condensed phase systems is questionable. |
65c2abace9ebbb4db9cf1a24 | 2 | In this work, we formulate and rigorously test an alternative approach to go beyond the second order cumulant expansion. The method relies on applying a dampening factor to the third order cumulant correction that approximately accounts for the effect of neglected higher order cumulants in cancelling divergences in the third order lineshape. We demonstrate that the dampening factor Φ is a function of the non-Gaussian nature of the energy gap fluctuations quantified via the skewness and kurtosis of the distribution. This function Φ can be directly parameterized using the exactly solvable model system of the GBOM Hamiltonian. We show that the approach rigorously removes unphysical features from linear absorption spectra in both harmonic and anharmonic model systems, as well as realistic condensed phase systems sampled from MD. Additionally, the approach yields an excellent agreement with the FC method for the GBOM Hamiltonian including Duschinsky mode mixing and changes in PES curvature, thus opening up the possibility of accounting for these effects from first principles in simulations of molecules embedded in condensed phase environments. |
65c2abace9ebbb4db9cf1a24 | 3 | Here, Ĥg and Ĥe denote nuclear Hamiltonians on the ground-and excited state potential energy surface (PES) respectively, |e⟩ and |g⟩ are pure electronic states, and ν e and ν g are nuclear wavefunctions. Under this form, the linear response function can be expressed as a trace over nuclear degrees of freedom in the condensed-phase system 52 |
65c2abace9ebbb4db9cf1a24 | 4 | where ρ g is the ground-state equilibrium density matrix of nuclear degrees of freedom and Veg = V † ge is the transition dipole operator between electronic states |e⟩ and |g⟩. If an exact expression for Eqn. 3 can be obtained for a system of interest, the linear absorption spectrum can be constructed through its Fourier transform: 52 |
65c2abace9ebbb4db9cf1a24 | 5 | Here, α is a constant factor necessary when comparing directly to experimental results, 36 but that will be set to 1 for the remainder of this work without loss of generality. Proposing approximate forms for Eqn. 3 that retain accuracy while remaining tractable in complex condensed phase systems is of key importance to the simulation of linear absorption spectra. |
65c2abace9ebbb4db9cf1a24 | 6 | Here, E e i and E g i denote the eigenvalues corresponding to the ith eigenstate of the nuclear Hamiltonians Ĥe and Ĥg respectively, P(i) is the Boltzmann population of the ith energy level on the ground state PES in thermal equilibrium and ω ji is the energy required to transition from state i on the ground state PES to state j on the excited state PES. For model systems where the eigenstates of the nuclear Hamiltonians can be computed exactly, the spectral decomposition yields the exact linear response function under the Condon approximation. This form provides a crucial benchmark within the scope of this work as a means to quantify the performance of approximate cumulant-based schemes (see Sec. II B). However, the sum-over-states formalism in Eqn. 5 has a number of drawbacks, especially when modeling realistic molecular systems in the condensed phase. |
65c2abace9ebbb4db9cf1a24 | 7 | First, Eqn. 5 requires the exact nuclear wavefunctions ν e j , {ν g i } to be known and the wavefunction overlap ν e j |ν g i to be easily evaluated for all i,j. In practice, this means that the full PESs of the molecule in the condensed phase have to be approximated through a model Hamiltonian. The most popular choice, invoked in the Franck-Condon (FC) method, is to approximate the PESs as harmonic around their respective minima. Second, evaluating Eqn. 5 directly, for example by parameterizing a GBOM (see Appendix A 1), means that the condensed phase environment can often only be accounted for approximately in the resulting spectrum, either through collective representations of the solvent environment through a PCM, or by invoking a timescale separation between solute and solvent degrees of freedom. These shortcomings render a direct evaluation of the spectral decomposition expression potentially impractical in condensed phase systems, especially for systems with strong solute-solvent interactions and chromophores with low frequency anharmonic modes ill described by a harmonic model PES. |
65c2abace9ebbb4db9cf1a24 | 8 | such that g i (t) denotes the ith order cumulant of the energy gap fluctuation operator and ∞ i=2 g i (t) ≡ g(t) is known as the lineshape function. Being of substantial importance of this work, the general form of the second and third order cumulant contributions to the lineshape function are presented here: |
65c2abace9ebbb4db9cf1a24 | 9 | A key distinction must be drawn between how the cumulants may be obtained in the context of realistic condensed-phase systems versus model Hamiltonians. For chromophores embedded in complex environments, the quantum correlation functions required to construct the cumulant expansion are generally inaccessible. Instead, cumulants are constructed through classical MD simulations in conjunction with calculations of the vertical excitation energy along the generated trajectory. From these vertical energies, classical correlation functions can be computed, and the approximate quantum correlation functions can be reconstructed with the help of quantum correction factors (QCFs). For the one-and two-time correlation functions necessary to construct the second and third order cumulant, the QCFs used in this work take the following form in the frequency domain: C qm (ω) |
65c2abace9ebbb4db9cf1a24 | 10 | with ω = ω + ω ′ . Substituting the Fourier representation of Eqns. 14 and 15 into Eqn. 12 and Eqn. 13, approximate expressions for g 2 (t) and g 3 (t) may be obtained that can be computed directly from energy gap fluctuations sampled along MD trajectories. To the best of our knowledge, there are no generally applicable correction factor analogs for higher order correlation functions. Thus the MD-type construction of the cumulant response is limited to the third order, and the implication of such truncation of the cumulant expansion will be explored in detail in this work. |
65c2abace9ebbb4db9cf1a24 | 11 | In contrast, for simplified model Hamiltonians, it is often possible to construct exact closed-form expressions of the second, third, and higher order cumulants. Additionally, higher order cumulants for model systems can be constructed numerically. These model systems can provide powerful insights into the types of errors made in evaluating low-order cumulant expansions using QCFs. In this work, we focus on the GBOM Hamiltonian 39 (see Sec. A 1), for which analytical expressions for g 2 (t) and g 3 (t) based on quantum and classical correlation functions and QCFs, as well as the exact result corresponding to the infinite order cumulant expansion can be computed analytically (see SI Sec. I and Ref. 39). |
65c2abace9ebbb4db9cf1a24 | 12 | For general molecular Hamiltonians, all orders of cumulants contribute to the spectral lineshape and a truncation at any finite order introduces errors. However, in the limit that the energy gap fluctuations obey Gaussian statistics, all cumulants beyond g 2 (t) vanish, such that exact response is recovered at the second order cumulant approximation. This behavior is observed for a system where all modes coupled to the electronic transition are harmonic and equal in ground and excited state curvature, resulting in an energy gap operator that is linear with respect to nuclear coordinates (See SI Sec I). Such a simplified model system is known as the Brownian oscillator model (BOM). |
65c2abace9ebbb4db9cf1a24 | 13 | A departure from Gaussian statistics is quickly met in the realistic case of mismatched ground-and excited state frequencies in harmonic vibrational modes (as present in the GBOM Hamiltonian), as well as in the presence of any anharmonicity. It can be shown analytically (see SI Secs. I and II) or numerically (SI Sec. III), that the introduction of higher order cumulants leads to an improved agreement with the exact response function in the short timescale limit, but that higher order terms become increasingly more volatile and divergent in the long timescale limit. In principle, a complete summation of the cumulant expansion to infinite order would lead to a cancellation of divergent terms and well-behaved response functions for all timescales. However, a truncation at finite order can yield unphysical lineshapes, as can be demonstrated even in the case of the harmonic GBOM Hamiltonian (SI Sec. II and Ref. 39), and only the second order cumulant approximation is guaranteed to yield linear response functions that do not diverge as t → ∞ for arbitrary Hamiltonians. This divergent behavior makes the benefit of evaluating higher order cumulants in realistic systems, such as by introducing a third order correction that can be computed in condensed phase systems from MD, questionable. In previous work, we have demonstrated numerically that monitoring the deviations of the energy gap fluctuations from Gaussian statistics, by calculating the skewness µ (3) of the distribution, can serve as an indicator of the importance of higher order cumulant contributions. It was found that for skewness values beyond 0.3, the third order cumulant correction becomes numerically unstable and unreliable due to the neglecting of higher order terms; In these cases the numerically stable, albeit inaccurate, second order cumulant approximation should be used instead. |
65c2abace9ebbb4db9cf1a24 | 14 | In this work, we instead introduce and justify the novel hypothesis that while the exact nature of neglected higher order cumulants in the expansion cannot be known in arbitrary systems, some information of their net effect on the lineshape can be inferred. By quantifying the degree of non-Gaussian behavior in the energy gap fluctuations, through metrics such as skewness µ (3) and excess kurtosis µ (4) , one may approximately reintegrate the effects of the neglected cumulants in cancelling divergences of the lineshape. Specifically, we propose that an improved, and most importantly more stable, approximation to the exact lineshape compared to a pure third order cumulant expansion can be obtained through |
65c2abace9ebbb4db9cf1a24 | 15 | To illustrate the points made in Sec. II B 2, we demonstrate the effect of truncating the cumulant expansion at low order on a simple 2-mode harmonic model system described by the GBOM Hamiltonian (see Appendix A 1 and Appendix A 5 a for model parameters). For this Hamiltonian, the second order cumulant expansion is no longer exact, as the ground-and excited state PESs can have different curvatures and individual modes can be coupled. The resulting spectra for a specific model parameterization can be found in Fig. . |
65c2abace9ebbb4db9cf1a24 | 16 | In the chosen parameterization, energy gap fluctuations are moderately non-Gaussian (with µ (3) = 0.33 and µ (4) = 0.22). The second order cumulant approximation fails to reproduce the correct vibronic progression, both underestimating the separation between vibronic peaks and predicting wrong intensities. When adding the third order correction to the lineshape, both peak intensities and positions are much improved, but at the cost of spurious unphysical spectral regions of negative absorbance. This spurious behavior of the truncated third order cumulant expansion is commonly observed for sufficiently non-Gaussian fluctuations. We note that, while deviating significantly from the exact lineshape, the second order cumulant approximation is guaranteed to yield a positive-definite and thus physically meaningful absorption spectrum for arbitrary non-Gaussian energy gap fluctuations (see SI Sec. II). |
65c2abace9ebbb4db9cf1a24 | 17 | The performance of the truncated cumulant expansion can be rationalized when considering the lineshape function g(t) determining the spectrum (see Fig. )). The third order cumulant approximation yields a short timescale correction towards the exact lineshape function, causing an improvement in the placement and intensity of vibronic progressions in third-order spectra of mildly non-Gaussian systems (See SI Sec. II A). However, this short time correction causes a divergence in the real part of the lineshape function proportional to t as t → ∞, with additional divergent terms present in the imaginary part (See SI Sec. II and Ref. 39). By inspection of Eqn. 5, it may be noted that an exact lineshape function should not exhibit this divergence. Characteristics of the divergent terms, such as the fact that the real part of the lineshape function can become negative in the short timescale (see Fig. )) can be seen as the origin of spurious negative spectral features in the resulting absorption spectrum. Thus, this spurious asymptotic behavior is a clear target for seeking methodological improvements to the computation of cumulant-based spectra. If the goal is to produce the most accurate, physically applicable spec-trum possible, the inclusion of a correction that causes unphysical negative absorbances is undesirable, even if it leads to observed improvements in position and intensity of vibronic features. Therefore, it becomes of high interest to develop a methodology that retains the corrections afforded by the third order cumulant approach while reliably safeguarding against any unphysical spectral features. Ultimately, any technique developed must also be freely applicable to MD-type simulations of molecules in the condensed phase in order to be useful in realistic systems. |
65c2abace9ebbb4db9cf1a24 | 18 | We propose that a low order cumulant expansion can be significantly improved by introducing a dampening factor Φ to the third order contribution g 3 (t), as outlined in Eqn. 16. To demonstrate this concept, we again turn to the harmonic model system in Fig. . We first define a goodness-of-fit metric for the cumulant spectrum with respect to the exact spectrum: |
65c2abace9ebbb4db9cf1a24 | 19 | The metric κ is simply taken as the unsigned difference between the cumulant and the exact spectrum, averaged over n numerical grid points along the frequency axis. We evaluate σ cumul using Eqn. 16, where the parameter Φ is systematically varied from 0 (corresponding to a pure second order cumulant spectrum) to 1 (corresponding to an undampened cumulant expansion truncated at third order). A plot of the resulting κ(Φ) can be found in Fig. ). We find that κ(Φ) is a convex function, and thus by selecting the prefactor value corresponding to the minimum of κ(Φ), one may optimize the performance of the third order cumulant approximation with respect to the exact spectrum. Selecting the optimal dampening factor Φ when constructing the third order cumulant lineshape following Eqn. 16 results in a spectral lineshape in very close agreement to the exact spectrum, both in the intensity and positioning of vibronic peaks (see Fig. B and SI Sec. V). Additionally, the unphysical negative absorbance is removed from the spectrum. While Fig. shows results for a specific parameterization of the GBOM Hamiltonian, we find that similar results can be obtained over a wide parameter range, suggesting that the functional form of a dampened cumulant response proposed in Eqn. 16 is widely applicable. |
65c2abace9ebbb4db9cf1a24 | 20 | The results in Fig. are obtained for a harmonic model system. In this work, we argue that 1) the general functional form of Eqn. 16 provides a pathway of improving cumulant lineshapes even in realistic systems with more complex (generally anharmonic) potential energy surfaces as encountered in condensed phase systems sampled with MD; and that 2) the optimal dampening factor Φ can be expressed as a function of the non-Gaussian features of the underlying energy gap fluctuations, such that Φ ≡ Φ µ (3) , µ (4) . |
65c2abace9ebbb4db9cf1a24 | 21 | The above points can be justified by the following observations. First, as demonstrated in Fig. , the third order cumulant correction does improve the lineshape function in the short timescale limit, but introduces oscillations with amplitudes growing linearly in time, leading to an overestimation of oscillatory terms at longer timescales (see SI Secs. II, III and V). Therefore, a net effect of higher order cumulant terms must be a dampening of the linear divergence observed in the third order cumulant correction. This finding can be confirmed by evaluating higher order cumulants for model systems numerically (see SI Sec. III). Second, the timescale of divergences in the third order cumulant contribution, and thus the importance of higher order cumulant terms, is directly related to the degree of non-Gaussian behavior in the underlying energy gap fluctuations. Thus, we expect Φ to be a function of µ (3) and µ (4) . |
65c2abace9ebbb4db9cf1a24 | 22 | To obtain a functional form of Φ(µ (3) , µ (4) ) we apply an iterative approach as outlined in the schematic of Fig. . A large set of GBOM Hamiltonians is constructed through the random sampling of parameters across a broad, yet realistic, domain relevant to molecular systems (see SI Sec. VI). For a given GBOM, the ideal value for the dampening factor Φ is determined by varying Φ be-FIG. . Schematic for generating predictive prefactor plot. A large number of GBOMs is sampled to produce a predictive grid of Φ as a function of µ (3) and µ (4) that may then be applied to molecular dynamics (MD) simulations in realistic condensed phase systems. |
65c2abace9ebbb4db9cf1a24 | 23 | tween 0 and 1, and finding the minimum of the resulting convex function κ(Φ), as outlined in Fig. . Additionally, the degree of non-Gaussian behavior in the energy gap fluctuations is evaluated by determining skew and excess kurtosis via Eqn. 22, thus obtaining a single point of the function Φ(µ (3) , µ (4) ) in statistical space. Repeating this process across a large number of unique GBOMs with randomly sampled parameters, we then construct a continuous function Φ(µ (3) , µ (4) ) by fitting an analytic function to the sampled data. This allows us to predict the ideal dampening factor Φ for arbitrary systems from the measures µ (3) and µ (4) of the underlying non-Gaussian energy gap fluctuations alone, with the aim of improving the lineshape in realistic systems sampled from MD. Thus regardless of the model system or realistic condensed phase molecule that generated an energy gap fluctuation distribution, the degree of dampening which would occur from the inclusion of higher order cumulants is now being inferred from quantifiable properties of the underlying distribution. In Sec. IV, we rigorously test the performance of the Φ(µ (3) , µ (4) ) derived from the GBOM sampling on a range of model systems and real molecules in the condensed phase. |
65c2abace9ebbb4db9cf1a24 | 24 | To construct a plot of Φ µ (3) , µ (4) as outlined in Sec. II D, ≈ 125, 000 individual GBOM parameterizations (see Appendix A 1) were sampled. The number of modes in the GBOMs was varied systematically from 2 to 50. The ground and excited state frequencies and shift vectors were sampled from uniform distributions over a physically realistic range of molecular vibrations. The n-mode Duschinsky rotation matrix relating ground-to excited state normal modes was constructed by filling the off-diagonal elements of each row with randomly generated values such that their sum equals a predetermined value; this value being sampled from a uniform distribution as well. The coupling matrix was subsequently made unitary through a Gram-Schmidt orthonormalization of column vectors. Full details of the distributions of model parameters and how they relate to realistic molecular systems can be found in SI Sec. VI A. |
65c2abace9ebbb4db9cf1a24 | 25 | For all systems sampled, a fixed solvent response was coupled to each GBOM to obtain realistically broadened spectral lineshapes (see Appendix A 3). It is emphasized that the solvent parameters need not be varied to explore the statistical nature of the prefactor, as the solvent modes are assumed to follow Gaussian statistics and are thus described exactly by the second order cumulant approximation. Since in the cumulant approach, no distinction has to be made between whether fluc-tuations arise from solvent or chromophore degrees of freedom, a stronger or weaker coupling to solvent environment can be interpreted as adding or removing noninteracting Gaussian modes from the underlying "chromophore" GBOM. Therefore, the effect of stronger solvent coupling through the addition of more Gaussian fluctuations is directly incorporated in the GBOM sampling scheme. |
65c2abace9ebbb4db9cf1a24 | 26 | After sampling 125,000 GBOMs, evaluating µ (3) and µ (4) and computing the ideal prefactor Φ for each GBOM using the metric κ(Φ), resulting data of Φ(µ (3) , µ (4) ) was then averaged across cells of ∆µ (3) × ∆µ (4) , with ∆µ (3) = 0.026 and ∆µ (4) = 0.030. A cubic bivariate spline was fit through the resulting data to create a continuous function for Φ (see SI Sec. VI B), that can then be used to predict the appropriate correction factor for arbitrary systems. Only cells of ∆µ (3) × ∆µ (4) sampled with at least three distinct GBOMs were considered in the spline fit. We emphasize that predicted prefactor values outside of the contour of Fig. are extrapolated from the collected model system data using the spline fitting. While prefactors obtained in this fashion are likely reliable in regions close to the sampled bounds, the third order cumulant correction should be discarded entirely for systems exhibiting extreme values of |µ (3) | or |µ (4) |, as even the corrected third order cumulant approach likely becomes unreliable due to the missing higher order cumulant contributions. We will explore how well the parameter space explicitly sampled by the 125,000 GBOMs corresponds to skewness and kurtosis values in realistic condensed phase systems in Sec. IV C. |
65c2abace9ebbb4db9cf1a24 | 27 | For all GBOMs sampled, the second and third order cumulant lineshapes were evaluated using the analytical expressions for the exact quantum correlation functions, that are generally inaccessible in realistic condensed phase systems. This was done so that the prefactor Φ(µ (3) , µ (4) ) was independent of errors introduced in the lineshape through constructing cumulants from classical correlation functions using QCFs. We have performed tests (see SI Sec. V A) to confirm that the errors introduced through QCFs are generally small in comparison to the errors introduced by a low-order truncation of the cumulant expansion. Thus the Φ(µ (3) , µ (4) ) derived for exact quantum correlation functions in the GBOM Hamiltonian is expected to perform well for condensed phase systems sampled in MD, where lineshape functions have to be evaluated from classical correlation functions using QCFs. |
65c2abace9ebbb4db9cf1a24 | 28 | To sample the fluctuations of the coumarin dye, mixed quantum mechanical/molecular mechanical (QM/MM) dynamics of the molecule in a 30 Å solvent sphere in open boundary conditions were performed, with the QM region confined to the chromophore. QM/MM dynamics were run using the interface between Amber 81 and TeraChem, and the force field parameters for the Toluene solvent was generated using AmberTools. The QM region was treated using density-funtional theory (DFT) at the CAM-B3LYP 83 /6-31+G* 84 level of theory. |
65c2abace9ebbb4db9cf1a24 | 29 | A timestep of 0.5 fs was used throughout and the system was kept at 300 K using a Langevin thermostat with a collision frequency of 1 ps -1 . A 50 ps pure MM equilibration was carried out, before switching to QM/MM dynamics for 22 ps. The first 2 ps of the QM/MM trajectory were discarded to allow for additional equilibration upon switching the chromophore Hamiltonian from an MM to a QM representation. |
65c2abace9ebbb4db9cf1a24 | 30 | Along the QM/MM trajectory, vertical excitation energies were computed in 2 fs intervals using time-dependent density-functional theory (TDDFT) as implemented in the TeraChem code, with the same basis set and functional as for the ground state dynamics. A total of 10,000 individual vertical excitation energies were then used to compute classical correlation functions C |
65c2abace9ebbb4db9cf1a24 | 31 | The spline fit of Φ µ (3) , µ (4) resulting from sampling ≈125,000 GBOMs can be found in Fig. , whereas the raw data prior to fitting the spline can be found in SI Sec. VI. Close to the Gaussian limit (µ (3) = 0, µ (4) = 0), the third order contribution should only provide a small correction to the second order cumulant response function, as all higher order cumulants strictly vanish for Gaussian fluctuations. As such, we expect that the largest prefactor values of Φ µ (3) , µ (4) ≈ 1 should reside close to the Gaussian limit, as is indeed observed. In this region, the undampened third order cumulant approximation provides the best approximation to the exact spectrum. |
65c2abace9ebbb4db9cf1a24 | 32 | Additionally, we observe that neither the statistical region sampled by the GBOMs, nor the spline fit to Φ(µ (3) , µ (4) ) are symmetric around the origin of µ (3) = 0, µ (4) = 0. Thus, the volatility of the third order cumulant approximation does not depend solely on the magnitude of non-Gaussian behavior (|µ (3) |, |µ (4) |), and specifically the sign of skewness of the energy gap fluctuations imparts a substantial effect on the value of the dampening factor Φ. This finding can be rationalized by examining the relative performance of the undampened (Φ=0) third order cumulant approximation against the second order cumulant approximation. We define the relative metric κ rel = (κ 3 -κ 2 )/κ 2 , where κ 2 and κ 3 correspond to the metric of Eqn. 23 evaluated for the second order and undampened third order cumulant approximation respectively. A plot of κ rel across the range of sampled GBOMs can be found in Fig. . As can be seen, systems with energy gap fluctuation statistics presenting with a negative skew always yield a third order cumulant spectrum that outperforms the second order cumulant approximation. It may be demonstrated analytically (see SI Sec. II. A) for a system with uncoupled GBOM modes (corresponding to the Duschinsky rotation matrix being equal to the identity matrix, [J] = [I]), that a negative skew is indicative of a relaxation in vibrational frequency upon excitation (ω e < ω g ). This causes a divergence proportional to t 2 in the real part of the third order cumulant lineshape function, leading to a dampening of the overall response function and a physical linear absorption spectrum. Conversely, as ω e > ω g a positive skew occurs in the simple model system. As ω e /ω g becomes increasingly large, the lineshape function diverges proportionally to -t 2 , causing an unphysical divergent response function. Additionally, oscillatory divergences with linearly growing amplitudes appear more dominantly in the short timescale relevant to linear optical response and these terms can cause unphysical features in the resulting spectral lineshape. Interestingly, our calculations indicate that an analogous statement holds for GBOMs with couplings between individual modes as described by the Duschinsky rotation: A negative skew is indicative of a stable third order cumulant approximation that improves over the second order cumulant lineshape. For positive skewness values, truncating a cumulant expansion at third order almost universally leads to a deterioration of the spectrum in comparison to the second order approximation. |
65c2abace9ebbb4db9cf1a24 | 33 | Evaluation of the performance of the prefactoroptimized third order cumulant approach versus the second order cumulant approach with an analogous relative metric κ rel (see Fig. ) reveals a substantial expansion in statistical space where the third order cumulant approximation may be applied to good effect. Specifically, introducing the dampening factor Φ significantly improves the performance for systems with a positive skewness value in the energy gap fluctuations. As is demonstrated in Sec. IV C, this expansion of the third order cumulant method to positive skew, low kurtosis energy gap fluctuations is key for obtaining reliable corrections to the lineshape in real molecular systems in the condensed phase sampled with MD. |
65c2abace9ebbb4db9cf1a24 | 34 | We also note that there is a low skewness, high kurtosis region where the optimized third order cumulant method under-preforms the second order cumulant approximation. This behavior may be rationalized in the following way: Within this region, one observes both numerically unstable GBOM parameterizations and GBOMs for which the unaltered third order cumulant and second order cumulant are in close agreement. With respect to the first effect, we observe nonphysical artifacts in this region not only for the third order cumulant approximation, but also with FC and ensemble computational methods which are generally assumed to be stable. This suggests that this region likely corresponds to model parameterizations that lie outside the domain of realistic molecular systems. This observation is reinforced by the fact that none of the real molecular systems sampled with MD investigated in this study (see SI Sec. VII and Sec. IV C) reside within this domain of statistical space. In fact, with all molecules studied for this work we find that their energy gap fluctuations reside in regions where the optimized third order cumulant approximation is predicted to out- preform the second order cumulant approximation. |
65c2abace9ebbb4db9cf1a24 | 35 | In Fig. B), we focus on the relative performance between the dampened and the undampened third order cumulant approximation. We note that in the negative skew region, the dampened third order cumulant approximation is indistinguishable from the pure third order cumulant approximation in average performance, again indicating that the pure third order cumulant approximation systematically improves the lineshape in this region. In the positive yet small skew limit, it is generally observed that physically reasonable GBOMs only have a very small third order correction, resulting in very little effect in the overall response function by conditioning the cumulant expansion through the ideal dampening factor Φ. In the larger valued positive skew region, we observe a strong and consistent improvement of the dampened third order cumulant approximation over the pure third order cumulant approximation, with Φ becoming small for large skewness values. We find these results consistent with the principles that led us to propose the functional form of Eqn. 16: As we reach regions with a more strong departure from Gaussian behavior, the size of the third order cumulant contribution must increase. In parallel, the (unaccounted for) higher order cumulant contributions must increase in size as well, requiring a larger degree of dampening. Once the function Φ(µ (3) , µ (4) ) has been parameterized by fitting a bivariate spline to the sampled data points of ≈125,000 GBOMs, the prefactor conditioning of the third order cumulant approximation can now be applied in a predictive manner. To do so, we apply the prefactor method to a set of GBOMs where, rather than selecting the ideal dampening factor from minimizing the metric κ(Φ), we evaluate Φ directly from the statistical properties of the energy gap fluctuations, namely the skew µ (3) and excess kurtosis µ (4) , without the need to refer to the exact analytical solution of the spectral lineshape. Three example GBOMs can be found in Fig. (information on model parameters is provided in Appendix A 5 b). As can be seen, the fitted spline function for Φ(µ (3) , µ (4) ) yields an appropriate dampening factor for all GBOM parameterizations in different areas of statistical space. In all cases, the dampened third order cumulant lineshape is much improved over the pure third order cumulant lineshape, removing strong divergences and unphysical negative absorbances in the case of Fig. ) and C). The dampened lineshape also provides a significant improvement over the second order cumulant approximation, both in terms of the position and intensity of vibronic peaks. While Fig. shows three representative GBOM parameterizations, we observe a similar performance over a wide range of model parameterizations studied. The fact that the correction factor can be applied with good accuracy to the same class of models it was parameterized for is not entirely surprising, but serves at a good empirical demonstration that Φ can be truly represented as a function of the non-Gaussian features of the energy gap fluctuations, rather than explicit parameters of the underlying Hamiltonian. In the context of GBOMs, this implies that the optimal prefactor within a region of non-Gaussian fluctuations obtained from a set of randomly sampled GBOMs can still be effectively applied to a GBOM that may strongly vary in underlying parameters, as long as it shares similar energy gap fluctuation statistics. What remains to be demonstrated is that a dampening factor Φ(µ (3) , µ (4) ) derived for a simple set of harmonic model can be applied to more general (anharmonic) systems. However, the good performance of the dampening factor shown in Fig. is highly promising, as it suggests that failures of the cumulant approach in correctly capturing Duschinsky mode mixing effects and changes in PES curvature upon excitation can be effectively cured even in complex condensed phase systems, where the FC method cannot be applied. |
65c2abace9ebbb4db9cf1a24 | 36 | To test the validity of the prefactor mapping on molecular systems which contain strongly anharmonic modes, we construct model systems under the scheme outlined in Appendix A 2, where the chromophore is approximated through a set of harmonic modes described as a GBOM and a single anharmonic mode described as a Morse oscillator. The results of pure and dampened cumulant approximations are once again assessed against the exact spectrum. The results for a specific model parameterization can be found in Fig. , with the exact model parameters specified in Appendix A 5 c. |
65c2abace9ebbb4db9cf1a24 | 37 | As found in our previous work, both second and third order cumulant expansions struggle to replicate higher order vibronic progressions in the Morse spectrum. However, we find that the predicted prefactor Φ removes nonphysical characteristics of the third order spectrum entirely. While the degree of improvement for this anharmonic model is not appreciable in comparison to the harmonic systems studied in Fig. , conditioning through the dampening factor Φ remains an effective way to safeguard against unphysical spectral lineshapes. Furthermore, the ability to apply a prefactor parameterized for harmonic systems to a system which now contains anharmonic PESs supports the hypothesis that the dampening factor is indeed independent of the underlying Hamiltonian of the physical system, and instead depends on the non-Gaussian features of the energy gap fluctu-ations alone. This suggests that the fitted dampening factor Φ(µ (3) , µ (4) ) can be applied to complex condensed phase systems with energy gap fluctuations sampled directly from MD simulations. |
65c2abace9ebbb4db9cf1a24 | 38 | The ultimate objective of the optimization approach developed within this work is to apply the inference gained in model systems to the simulated absorption spectra of realistic molecular systems in the condensed phase, where the exact spectrum can no longer be computed. For these complex systems, the aim of evaluating the parameterized dampening factor Φ(µ (3) , µ (4) ) is to both improve the computed lineshapes and reduce computational cost. |
65c2abace9ebbb4db9cf1a24 | 39 | MD-type calculations required to sample energy gap fluctuations in the condensed phase to construct cumulant spectra are computationally expensive compared to calculations on simple model systems (generally requiring the computation of tens of thousands vertical excitation energies along the trajectories). Additionally, computing reliable third order cumulant corrections requires significantly more data than second order cumulant spectra, as two-time correlation functions of the energy gap have to be converged. Thus, there is merit in being able to predict the overall stability of the third order cumulant approximation based on statistics of energy gap fluctuations alone, which can be computed cheaply from preliminary data sets. If the skew and excess kurtosis of energy gap fluctuations of a molecule fall into a region of low confidence in Fig. , it can be concluded that expending the extra computational cost to construct third order cumulant corrections is unlikely to yield improved spectra. In other systems, specifically those with moderate positive skew in the energy gap fluctuations, we expect a prefactor-conditioned MD-based third order cumulant approach to yield improved lineshapes, both by removing unphysical negative spectral features and by improving the underlying vibronic fine-structure of the transition. |
65c2abace9ebbb4db9cf1a24 | 40 | To examine the usefulness of the dampening factor Φ(µ (3) , µ (4) ) in realistic systems, we first demonstrate that typical condensed phase systems sampled from MD fall into the statistical range of non-Gaussian energy gap fluctuations sampled by the randomly parameterized GBOMs used in this work. Such an analysis is provided for nine selected molecules in SI Sec. VII. Both isolated molecules in vacuum and chromophores in condensed phase environments are included in the data set. For all nine molecules, it is found that the non-Gaussian features of the energy fluctuations, µ (3) and µ (4) , fall into the region sampled by the GBOMs. Additionally, they all fall into a region where the optimized third order cumulant approximation is predicted to outperform the second order cumulant approximation, thus indicating that the dampening factor Φ can likely be used to improve spectral lineshapes in these realistic systems. Here, we focus on a single selected system in more detail, namely coumarin-135 in toluene, due to the fact that vibronic peaks are well resolved in the lineshape, and an experimental spectrum is readily available. We emphasize that to produce what one may define as a sufficiently accurate spectrum for a molecule in the condensed phase using the MD-based cumulant method involves additional challenges when compared to the study of simple model systems where the exact spectrum can be readily computed. Vibronic lineshapes are often found to differ considerably depending on the level of theory used for modeling the ground-and excited state potential energy surfaces with TDDFT. Thus, any discrepancies with respect to the experimental spectrum cannot be easily ascribed to errors introduced in the low order cumulant expansion alone. For this reason, we more broadly focus on observed changes in the vibronic fine structure under the different cumulant approximations, rather than direct a quantitative comparison to the experimental lineshape. |
65c2abace9ebbb4db9cf1a24 | 41 | Experimental and simulated MD-based cumulant spectra for coumarin-153 can be found in Fig. . We note that available experimental spectrum in Ref. 86 uses hexane as the non-polar solvent, rather than toluene as used in our calculations. Due to the fact that both are non-polar solvents with similar dielectric constants, we expect the spectra in the two solvents to match each other closely. Additionally, we note that the weakly interacting nature of the solvent means that this system is likely well-described by more commonly used and computationally affordable Franck-Condon type approaches. While strongly interacting solvents provide a more suitable application for the cumulant method, as direct solute-solvent interactions and slow collective chromophore-environment motions cannot be straightforwardly included in the Franck-Condon method, the well-defined vibronic finestructure of coumarin-153 in weakly interacting solvents allows for a more detailed analysis of the performance of the corrected cumulant approach and is therefore presented here as a test case. |
65c2abace9ebbb4db9cf1a24 | 42 | The results in Fig. indicate that the pure third order cumulant spectrum for this mildly non-Gaussian system already exhibits negative contributions to the spectral lineshape. For the underlying non-Gaussian statistics of the energy gap, a prefactor of Φ = 0.43 is predicted for this system. When applying the dampening factor to the spectral lineshape we indeed successfully remove the unphysical negative absorbance from the onset of the spectrum at 2.8 eV. |
65c2abace9ebbb4db9cf1a24 | 43 | Focusing on the spectral shape, we note that the pure third order cumulant approach predicts a much more pronounced vibronic fine structure than the second order cumulant approach. Additionally, while in the second order cumulant approximation and in the experimental spectrum, the second vibronic peak has the highest intensity, the third order cumulant approach erroneously predicts the first vibronic peak to be more bright. The dampened third order cumulant method produces a spectrum with a more resolved vibronic fine-structure and improved peak separation compared to the second order cumulant approach, but still predicts a high intensity for the second vibronic peak. In general, the dampened spectrum is in good agreement with the experimental spectrum, with only a minor overestimation of the first vibronic peak. The origin of remaining discrepancy with the experimental spectrum is unclear, but can potentially be ascribed to inaccuracies in the (TD)DFT description of the system Hamiltonian, rather than the truncation of the cumulant expansion. |
65c2abace9ebbb4db9cf1a24 | 44 | We take the fact that the prefactor Φ(µ (3) , µ (4) ) successfully removes non-physical features from the third order cumulant spectrum in a realistic condensed phase system as an additional proof of the wide applicability of the dampened cumulant response developed in this work (see Eqn. 16). No details of the underlying system Hamiltonian are needed to obtain the ideal dampening factor Φ, and only non-Gaussian features of the energy gap fluctuations determine this quantity. We have demonstrated that constructing Φ(µ (3) , µ (4) ) from a stochastic sampling of simple, exactly solvable model systems provides a recipe for improving cumulant spectra truncated at low order in complex condensed phase systems. |
65c2abace9ebbb4db9cf1a24 | 45 | In this work, we have outlined an approach to account for third order corrections to the linear absorption spectrum computed in the widely-used cumulant framework. The approach takes into account the effect of moderately non-Gaussian energy gap fluctuations without exhibiting unphysical divergences and regions of negative absorbance in the resulting spectra. The method promises to yield more accurate and robust linear spectra in model systems and chromophores embedded in complex condensed phase environments alike. |
65c2abace9ebbb4db9cf1a24 | 46 | The key insight in this work consists of introducing a dampening factor Φ(µ (3) , µ (4) ) that is applied to the bare third order cumulant correction, where Φ is taken to be an explicit function of the skew and excess kurtosis, measures of the non-Gaussian nature of the energy gap fluctuations. We have rationalized the functional form of this correction factor by noting that the third order cumulant term improves the spectral lineshape in the short timescale, but exhibits divergences at longer timescales. These divergences would be cancelled by higher order cumulant contributions that have to be neglected in any practical application to condensed phase systems. Thus the factor Φ approximately accounts for the collective dampening contribution to the lineshape of higher order cumulants. |
65c2abace9ebbb4db9cf1a24 | 47 | We have shown that the functional form of Φ(µ (3) , µ (4) ) can be parameterized by stochastically sampling model parameters of the GBOM, a harmonic model Hamiltonian that is widely applied to the prediction optical spectra of semi-rigid molecules. By constructing Φ for a model system where the exact spectrum cam be computed analytically, we were able to parameterize an ideal dampening factor as a function of the non-Gaussian fluctuations only, rather than the parameters of the underlying system Hamiltonian. With this predictive plot in place, we were able to find optimal dampening factors Φ in MD-type simulations of molecular systems in the condensed phase. |
65c2abace9ebbb4db9cf1a24 | 48 | We have demonstrated that the parameterized dampening factor rigorously removes unphysical and divergent lineshapes in the third order cumulant approximation, both for model systems and for condensed phase systems sampled with MD. In the GBOM Hamiltonian, we have also shown that the approach improves the agreement with the exact spectrum, both compared to the second order and the pure third order cumulant approach, for a wide range of model system parameterizations. For the realistic condensed phase system of coumarin-135 in toluene, our method yields a final spectrum in excellent agreement with experiment. The results presented indicate that the approach outlined in this work provides an efficient, computationally affordable pathway for correcting the main shortcomings of the second order cumulant approach in condensed phase systems, namely the inability to account for non-Gaussian fluctuations introduced by anharmonic effects, Duschinsky mode mixing effects and changes in the PES curvature upon excitation. Since our parameterization of Φ only relies on the skewness and kurtosis of the underlying energy gap fluctuations, we expect the method to be widely applicable to condensed phase systems sampled with MD. The dampened third order cumulant approach developed in this work has been implemented in the open-source software package MolSpeckPy developed within our group. Appendix A: Model systems |
65c2abace9ebbb4db9cf1a24 | 49 | In this work, we turn to two exactly solvable model systems. The Generalized Brownian Oscillator Model (GBOM) is a harmonic system that will be used to infer the functional form of the dampening factor Φ µ (3) , µ (4) , whereas the 1D Morse oscillator is invoked to explore the effectiveness of the dampening factor in systems that go beyond the harmonic approximation. |
65c2abace9ebbb4db9cf1a24 | 50 | The GBOM is a convenient, numerically robust model in which the second order cumulant approximation is no longer exact, but in which the exact second and third order cumulants can be evaluated analytically (see SI Sec. II and Refs. 39 and 60). Furthermore, the exact response function (Eqn. 5) can be computed analytically, enabling us to examine in detail the errors introduced by truncating the cumulant expansion at some finite order. |
65c2abace9ebbb4db9cf1a24 | 51 | where {ω g,i } and {ω e,i } are the set of ground-and excited state vibrational frequencies, ∆ is the adiabatic energy gap between the two electronic surfaces and {q i } and {r i } are ground and excited state normal mode coordinates respectively. The coordinates are related to each other by a linear transformation in terms of a shift-vector (k) between the ground and excited state minima, and the mode-mixing Duschinsky rotation matrix (J): |
65c2abace9ebbb4db9cf1a24 | 52 | A main shortcoming of the GBOM as applied to realistic systems is that it does not exhibit any anharmonicity in the potential energy surface. To probe the influence of anharmonic effects on computed lineshapes, we turn to the 1D Morse oscillator with the system Hamiltonian taking the following form: Ĥg (q, p) = p2 2 + D g [1e -αg q] 2 (A4) Ĥe (q, p) = p2 2 + D e [1e -αe(q-k) ] 2 + ∆. (A5) |
65c2abace9ebbb4db9cf1a24 | 53 | Here, D is the well depth of the potential, k is the displacement between the ground and excited potential minima, and α g and α e are the anharmonicity parameters for the ground-and excited state. Analytical expressions for the Morse oscillator wavefunctions exist, such that the exact response function for this Hamiltonian can be constructed directly through Eqn. 5. Additionally, cumulants based on the exact quantum correlation function can also be constructed numerically. The exact analytical expressions for the Morse oscillator wavefunctions are numerically unstable, meaning that the parameters this model have to be chosen somewhat carefully to guarantee well-defined results. As such, it less apt for the high-throughput screening of parameter space desired in this study. Instead, we construct a few selected Morse oscillator parameterizations inspired by realistic molecular systems to test whether the conclusions obtained for the GBOM Hamiltonian can be carried over to the anharmonic case. In these model calculations, we combine the Morse oscillator with a GBOM to model a chromophore with a few harmonic and a single anharmonic mode. The anharmonic mode is taken to be decoupled from the harmonic ones, such that the total response function can be written as: |
65c2abace9ebbb4db9cf1a24 | 54 | where λ env is the solvent reorganization energy quantifying the strength of the solvent coupling and ω c is a where well depths (D g,e ) are in units of Hartree energy, anharmonic factors (α g,e ) are in units of a -1 0 and the displacement (k) is in units of a 0 , where a 0 is the Bohr radius. |
65c2abace9ebbb4db9cf1a24 | 55 | See the supplementary material for the analytic expressions of the second and third order cumulants of a GBOM Hamiltonian, an analysis of their asymptotic behavior and an inspection of higher order cumulants for a number of model systems. Stochastic sampling techniques used to construct the prefactor plot and an analysis of the statistical properties of simulated molecular systems are provided as well. |
65f8aff366c138172942ac5a | 0 | Lithium-ion batteries have become a norm for energy storage these days. Many advances to increase their efficiency have been proposed. In the realm of material science, electrode, electrolyte, and separator are interesting design elements. Specifically focusing on anodes, several materials are being explored such as graphitic materials, metals, metal oxides, and metal phosphide. A key material explored is silicon. The obvious reason for silicon being a promising candidate is its high theoretical specific capacity (4200 mA-h/g for Li 22 Si 5 |
65f8aff366c138172942ac5a | 1 | 2 ) compared to the current graphite anode (372 mA-h/g 3 ). Furthermore, silicon is the secondmost abundant element in the earth's crust, making it a readily available material. However, its practical implementation is limited by a main bottleneck, the huge volumetric expansion after lithiation (310% for Li 22 Si 5 ) which causes a large build-up of stress, resulting in the pulverization of the material and immediate capacity loss during cycling. An example is the loss of 70% of the capacity of silicon anode made from 10 µm silicon powder within the first five cycles. There has been a surge in research output to tackle this bottleneck over the past decade. Specifically, one focus of research is on using micro-nano sized particle silicon anodes as they have been shown to resist cracking and accommodate the volumetric change. However, the improvement is still not sufficient for practical application. Other cutting edge research concentrates on modifying the anode by the formation of alloys, composites, core-shell structures, 20,21 films and porous systems. On the fundamental aspect, unavoidable volume expansion in each crystalline grain of silicon upon lithiation creates important mechanical stress in the material, responsible for rupture. A major mechanism to relieve this stress would be to allow facile sliding at the grain boundaries. Previous experimental observations of GB sliding have been predominantly limited to metallic materials such as Al, Cu, Sn, Zn, and Mg. To our knowledge, there has been no work in the literature for GB sliding in silicon. This comes from the fact that silicon has strong covalent bonds, which may render GB sliding highly activated and inoperative at near ambient temperature. In this manuscript, we study GB sliding in silicon using first-principle atomistic simulations. We show that GB sliding in silicon is activated and that doping silicon with aluminum markedly facilitates GB sliding, and is, therefore, a potential solution to improve the mechanical properties and durability of silicon anodes upon cycling. We show that the small amount of aluminum segregates in the grain boundaries (GBs) of silicon and greatly facilitates GB sliding. The prevalence of the Σ3 {111} GB in polycrystalline materials has been well-documented in numerous previous studies and is revealed in our GB characteristics quantification by electron backscattered diffraction (Supporting Information Figure ). Recognizing its significance, we have consistently employed the Σ3 {111} GB in all our simulations. We have devised a unique model to perform grain boundary sliding simulations. Our investigative approach encompasses the utilization of basin hopping, a global optimization technique to understand the segregation of aluminum in the GB. Through this computational framework, we seek to gain insights into the influence of aluminum on GB sliding behavior, hence enabling the reorganization of the polycrystalline silicon anode without mechanical failure during lithiation. To validate the simulation results, micro-sized polycrystalline silicon is experimentally doped with 5 wt.% (4.9 mol.%) aluminum and mixed by high-energy ball milling. Using our charge-discharge cycling test of half cells, we show that aluminum-doped silicon anode exhibits improved capacity retention than the undoped counterpart. We believe that this is an innovative and cost-effective way of improving the cyclic stability of silicon anodes. |
65f8aff366c138172942ac5a | 2 | All electronic energy calculations are performed with Density Functional Theory (DFT), implemented using the Vienna ab initio simulation package (VASP). The atomic simulation environment (ASE) is used in conjunction with VASP to develop our custom automation scripts in Python which are available at GitHub. The electron-ion interactions are treated using the projected augmented wave (PAW) method. The exchange-correlation effects are incorporated using the Perdew-Burke-Ernzerhof (PBE) functional. The Brillouin-zone integration is performed using Monkhorst pack 40 k-point grids of 15x9x1 for all the calculations. |
65f8aff366c138172942ac5a | 3 | To improve the convergence of the calculation with respect to the k-points, tetrahedron smearing with Blöchl corrections is used. The valence electrons are considered as a set of plane waves according to the Bloch theorem with a cutoff energy of 300 eV. All the structures are geometrically optimized using the conjugate gradient algorithm until the force on each atom is less than 0.01 eV/ Å. |
65f8aff366c138172942ac5a | 4 | The Σ3 (GB) can be described as a twist boundary characterized by a layered structure. In this GB, each grain's top and bottom layers are derived from bulk silicon with a relative twist angle of 60°between them. We have implemented the bicrystal model of GB. Subsequently, layers from each grain are systematically stacked atop one another, a process facilitated by our custom Python script. One notable advantage of this approach is the precise control it provides over the number of layers within each grain, allowing for tailored investigations. |
65f8aff366c138172942ac5a | 5 | where E I GB is the interface energy of the GB, E GB is the energy of the GB supercell, n Si is the number of silicon atoms in the GB, E BulkSi is the per atom energy of bulk diamond cubic silicon, n Al is the number of aluminum atoms in the GB, E BulkAl is the per atom energy of bulk fcc aluminum and A is the interface area of the GB. The factor 2 is incorporated to indicate the presence of two GBs in the unit cell. |
65f8aff366c138172942ac5a | 6 | In order to obtain the most stable configuration for the aluminum segregation in the GB, we use the Basin Hopping algorithm which is based on the canonical Monte Carlo technique, where the algorithm alters the coordinates of the current structure to a new structure according to a predefined constraint and then geometrically optimizes it. The optimized new structure can be accepted or rejected based on the Metropolis criterion. The results are interpreted using aluminum insertion energy as defined in Equation cluster on an alumina substrate under varying hydrogen pressures. 43 |
65f8aff366c138172942ac5a | 7 | The dominant phenomenon observed during mechanical operations at grain boundaries is known as GB sliding. To gain insights into how the segregation of aluminum at GB influences this phenomenon, a comprehensive study of GB sliding becomes imperative. As a result, we have developed a model through our custom Python script. In our approach, we have fixed one layer within each of the grains, preventing any atomic position relaxation. Employing a systematic, serial methodology, we introduce relative displacements between the layers of one grain to the other. Subsequently, the resulting atomic structure is subjected to geometric optimization. These fixed layers effectively serve as constraints, maintaining the deformation within the GB region. |
65f8aff366c138172942ac5a | 8 | The morphology and elemental distribution are observed using a scanning electron microscope (Thermo Scientific Apreo SEM) with energy-dispersive X-ray spectroscopy (Oxford Instruments X-Max 80 EDS detector). The powder samples are mounted in the epoxy and polished to reveal the cross sections for electron back-scattered diffraction (EBSD). Then, EBSD is acquired with Oxford Instruments Symmetry EBSD detector on SEM at 10-15 kV and 13-26 nA. The grain size and GB characteristics are analyzed through HKL Channel 5 |
65f8aff366c138172942ac5a | 9 | The Our investigation first aimed at understanding the underlying principles governing aluminum segregation within this system. An important observation is that aluminum exhibits a significant preference for the substitutional site (0.99 eV) over the interstitial site (2.77 eV) in bulk silicon. Furthermore, our findings indicate that it is energetically more favorable for aluminum to occupy the grain boundary substitutional site (0.9 eV) compared to the bulk site (0.99 eV). This intriguing behavior implies that aluminum's segregation tendency is profoundly influenced by the grain boundary environment. |
65f8aff366c138172942ac5a | 10 | It is noteworthy that despite the positive insertion energy of aluminum within silicon, a prediction aligned with the phase diagram, 46 the unique conditions imposed by the ball milling process, employed in the sample preparation, facilitate the exploration of higher energy states and the creation of metastable structures with well-dispersed aluminum. Consequently, this plays a pivotal role in driving the observed aluminum segregation phenomenon. |
65f8aff366c138172942ac5a | 11 | Using the algorithm detailed in Section II, we conducted simulations to investigate GB sliding initially using the model of Figure and 2 where the GBs are separated by 11 silicon layers. The simulation spanned 20 steps, with each step corresponding to a relative displacement of 0.773 Å (equivalent to one-fifth of the unit cell length in the x-direction, measuring 3.866 Å). Figure illustrates the initial steps characterized by elastic deformation, where no significant bond restructuring occurs, signifying the absence of sliding. This elastic deformation accumulates stress within the structure, eventually leading to stress release through a sliding event between the relative displacement of 8.506 Å and 9.279 Å (steps 11 and 12 respectively). Notably, this sliding occurs between layers 5 and 6 as can be seen in Figure , as well as 17 and 18, which are the grain boundary layers, as depicted in Figure . |
65f8aff366c138172942ac5a | 12 | Here, E[min] corresponds to the interface energy of the GB without deformation, while E[max] corresponds to the maximum interface energy of the GB along the deformation. To obtain an accurate value of ∆E, we performed ten GB sliding simulations (with a smaller step corresponding to a displacement of 0.077 Å) between the relative displacement of 8.506 Å and 9.279 Å as shown in Figure . A lower barrier indicates greater ease of sliding and more effective stress release. However, in our current case, the sliding barrier measures 0.33 eV/ Ų, which is 220 times that of the interface energy of the undeformed GB. This implies a high degree of rigidity in sliding and a consequently elevated risk of structural cracking. |
65f8aff366c138172942ac5a | 13 | Apart from the energetics, since the atoms in layers 0, 11, and 21 are fixed as shown in Figure , they experience a non-zero force when the GB is deformed. The average magnitude of force on the fixed atoms is represented in Figure . This force drops at a relative displacement of 9.279 Å which coincides with the GB sliding. However, the structure and its energy does not return to that of the undeformed state as can be seen in Figure and. Sliding stops at a partially deformed structure, indicating that the stress release is not effective during sliding. |
65f8aff366c138172942ac5a | 14 | In order to check our hypothesis, firstly, we used basin hopping as described in Section II to understand the segregation of aluminum in the grain boundary, for up to 4 aluminum per GB. Aluminum is allowed to substitute in place of GB silicon atoms. The number of aluminum substituted and their positions are chosen randomly in each step. Each basin hopping is run for 100 steps. The results of basin hopping for the 4 aluminum per GB case are represented in Figure . From Figure , we can observe that higher energy states are attained throughout the run, switching from one to the other, enabling a wide exploration window. Moreover, a broad range of energies ( 0.7 eV -1.1 eV) are covered in the run as can be seen in Figure . The resulting global minimum has a characteristic feature, i.e., all the aluminum are in a single layer. This is a consequence of the phase diagram of siliconaluminum, where aluminum has very low solubility in silicon, and therefore aluminum atoms prefer to accumulate together. After identifying the optimal aluminum segregation configurations for different numbers of aluminum, our subsequent step involved conducting GB sliding simulations using these configurations. Each simulation consisted of 20 steps, with each step corresponding to a relative displacement of 0.773 Å, similar to the one used for pure Si GB. The results from GB sliding simulations for the scenario with four aluminum atoms per GB are visually presented in Figure . Several significant differences between GB sliding with and without aluminum are discernible: (a) The sliding barrier is remarkably lowered to 0.0385 eV/ Ų, representing about 12% of the case without aluminum as can be seen in Figure . Moreover, this value is merely 40% of the interface energy of the undeformed GB with aluminum. (b) |
65f8aff366c138172942ac5a | 15 | The peak force of the system with aluminum has dropped by 66 % in comparison to the system without aluminum as shown in Figure . This reduction suggests that the presence of aluminum mitigates the rigidity of the GB, enabling sliding, contributing to effective stress relief, and, ultimately, the prevention of mechanical failure; (c) In contrast to sliding without aluminum, the frequency of sliding is notably higher. Sliding occurred at a relative displacement of 3.634 Å, compared to a relative displacement of 9.279 Å in the absence of aluminum, as depicted in Figure . This increased sliding frequency plays a crucial role in averting the accumulation of excessive stress in the material and (d) Following each sliding event, the GB rapidly returned to its completely undeformed state. This observation indicates substantial stress alleviation due to the presence of aluminum. These findings collectively validate our hypothesis that aluminum facilitates GB sliding, thereby reducing the likelihood of mechanical failure within the material. The observed outcomes are contingent on the aluminum content per grain boundary (GB). However, the influence of the number of aluminum atoms at the GB on the sliding behavior exhibits a nuanced pattern, graphically represented in Figure . The introduction of one aluminum per GB yields a substantial 49% reduction in the sliding barrier. The introduction of second and third aluminum per GB further decreases the barrier by 18% and 21% respectively. However, when the aluminum content increases from 3 to 4 atoms per GB, the sliding barrier is more or less stable, and diminishes with a mere decrease of 0.3%. This suggests that beyond a certain threshold, additional aluminum content has only a minor impact on GB sliding. Another crucial factor to consider is the concurrent increase in interface energy with rising aluminum content. This counters the reduction in the sliding barrier attributed to aluminum's presence by making the initial interface formation more challenging. Therefore, selecting the optimal aluminum content in the silicon anode entails a delicate balance between reducing the sliding barrier and managing the associated increase in interface energy. |
65f8aff366c138172942ac5a | 16 | The results are graphically represented in Figure . Our analysis revealed that the maximum force required to initiate GB sliding remained consistent across all the structures with the same number of aluminum per GB. This observation implies that the number of layers within the GB has no discernible impact on the sliding behavior. However, there are interesting differences between the structures with and without aluminum as represented in Figure . First, the maximum displacement of grains before sliding increases with the number of layers and is hence not an intrinsic parameter describing the sliding. This is because a specific displacement per layer is required to induce the shear (or force) for GB sliding. |
65f8aff366c138172942ac5a | 17 | Furthermore, from Figure , the reduction in slope indicates that the maximum relative displacement of grains before the sliding event occurred became a weaker function of the number of layers when aluminum is added to the system. A notable observation for systems with aluminum is, that even though there is an increasing amount of rigid Si-Si bonds as the number of layers per GB increases, this has no significant impact on the sliding indicating that the aluminum at the GB plays a major role in facilitating sliding. In the presence of Al at the GB, a displacement between grains results in an equivalent shear at the GB interface because this is the weakest part. In the case of pure silicon, the deformation is distributed within the grain and at the GB interface. Understanding Bonding in the GB using COHP |
65f8aff366c138172942ac5a | 18 | To delve into the underlying chemical mechanisms governing the impact of aluminum on grain boundary (GB) sliding, we utilized Crystal Orbital Hamilton Population (COHP) analysis, facilitated by the Local-Orbital Basis Suite Towards Electronic Structure Reconstruction (LOBSTER) code. This analytical approach provides insights into the atomic bonding interactions within a given structure, with particular emphasis on the quantification of bond strength through Integrated COHP (ICOHP) values. A higher ICOHP value signifies a stronger bond. In the absence of aluminum, the sole bonds present in the GB are Si-Si bonds, which exhibit an ICOHP value of 4.380. However, when aluminum is introduced at the GB, the landscape shifts. Some Si-Si bonds are supplanted by Al-Si bonds, characterized by an ICOHP value of 3.774, indicative of weaker binding. Furthermore, the bonding states for Si-Si are comparatively higher than the bonding states of Al-Si (refer the green area in Figure ). This contrast highlights the weaker nature of Al-Si bonds in comparison to their Si-Si counterparts. Consequently, this disparity in bond strength facilitates more facile bond reconstructions during GB sliding, a phenomenon akin to a lubricating effect induced from the presence of aluminum. |
62b9e8400bba5d053b774c26 | 0 | Li-ion batteries (LIBs) constitute a technological breakthrough with a large-scale impact on our societies. The current state of the art LIB is highly optimized and its density is close to the theoretical one, however, a next generation battery with higher capabilities can constitute a step further towards decreasing the reliance on fossil fuels. All-Solid-State Battery (ASSB) resembles a potential candidate for the upcoming energy storage devices. ASSBs may not just be superior with larger energy density and power performance because it also hinders (theoretically) dendrite formation giving the possibility of using Li metal anodes. They also provide higher safety aspects since the inflammable carbonatebased liquid electrolyte is not used. |
62b9e8400bba5d053b774c26 | 1 | There are several types of solid electrolytes (SE) that can work as a medium for ionic (lithium ion) diffusion and as separators in SSBs. These types can be categorized into polymers, oxides, sulfides, hydrides, halides, borates, and thin films. SSBs are not sufficiently mature and still under development. Composite electrodes made of blends of active material (AM) and SE particles give the promise of SSBs with high energy density. Still, there are many issues to overcome. One issue is the limited contact surface area between AM and SE particles, which blocks the lithium ions from reaching the AM through SE. Another one is the short ionic and electronic percolation pathways within the domain of the composite electrode, which causes low conductivity and high resistivity throughout the system. Sulfide SEs are considered lithium superionic conductors because of their high conductivity (~10 -2 S cm - 1 ). Compared to oxide SEs, they offer better performance and more deformability, yet higher sensitivity to air and less (electro)chemical stability. They also undergo degradation in the presence of water to produce H2S gas. As a result, they can only be handled under an argon atmosphere. Chen et al. showed the stability of Li6PS5Cl in the dry room under oxygen, with minor conductivity losses due to carbonate formation. Even with the good deformability of the sulfide SE, there are still many complications in the manufacturing process. Difficulties in mixing, high air sensitivity and high temperature synthesis of the different components of the composite electrodes are the main issues facing the commercialization of SSBs. Thus, wet processes for the preparation of composite electrode films are preferred as a more feasible and scalable process for the manufacturing of the ASSB positive electrodes. However, inadequate attention has been paid in the past to the liquid chemistry of thiophosphates (main Li ion conducting components in sulfide SE). Currently, more research is being conducted on the liquid chemistry of sulfide SE. The liquid-phase processing of sulfide SE has more advantages in terms of time, synthesis temperature and scalability. More investigation and research are needed in order to facilitate scaling up the wet manufacturing process. |
62b9e8400bba5d053b774c26 | 2 | To speed up the process of commercialization of the SSBs, modeling can be employed as a fast and more accessible testing strategy, allowing for a wide spectrum of different assumptions to be explored in a relatively short time in contrast to slow and resource intensive experiments. It can provide insights to improve every step in the process. Moreover, these models can be used in the advanced analysis and process control of the system by giving real-time feedback and tuning on the fly manufacturing parameters, with the use of the appropriate sensors in the manufacturing line. On the other hand, the manufacturing process of battery electrodes is highly complex and convoluted since it involves a series of sequential steps impacted by a wide range of parameters. Deconvoluting the influence of each of the manufacturing parameters on the output electrode microstructure and the resulting electrochemistry is extremely complicated. Three-dimensional (3D) models, developed by the ERC-funded ARTISTIC project, are used as a powerful tool to study the influence of each parameter in isolation and their combination on the product electrode microstructure and its electrochemical performance. These tools are freely available online with user-friendly interface to give the ability to run simulations for users of all backgrounds. 3D physical models of LIB electrode manufacturing have proven to show good compromise between throughput and accuracy. Surprisingly, the available modeling literature on SSBs is less extensive, with a very limited number of studies focusing on manufacturing parameters. LIB physicsbased 3D models provide precise control for each of the manufacturing parameters unlocking a high capability to understand and examine the effect of the manufacturing on the microstructure of the electrode. The output microstructure can be embedded into heterogeneous 4D (time dependent 3D) electrochemical that links all of the manufacturing, microstructure and performance by ultimately predicting the experimental discharge curves from the initial set of manufacturing parameters. Within the frame of the ARTISTIC project, our group has set up a series of LIB models attributed to different steps of the manufacturing process starting from the slurry phase, its drying, and the dry electrode calendering and electrolyte filling. All these models can be injected directly into 4D electrochemical models to simulate the energy discharge (current potential curves) and electrochemical impedance spectroscopy for the resistance and conductivity of the electrode. |
62b9e8400bba5d053b774c26 | 3 | Coarse Grained Molecular Dynamics (CGMD) is a modeling technique which relies on Force Fields (FFs) to simulate the behavior of a system based on interactions among the individual particles. This simulation is performed by solving Newton's equations as function of these forces. In this work we consider the particles to be the NMC622 AM, the Li6PS5Cl SE and the carbon-binder domain (CBD) involved in the electrode wet manufacturing process. The CBD is a geometrical domain within the model used to represent a phase consisting of the carbon additives, binder and in some cases the solvent. We simulate the formation of the electrode slurry, its drying and the calendering of the resulting structure to form the electrode (Figure ). |
62b9e8400bba5d053b774c26 | 4 | Each one of the models requires two sorts of parameters. They are referred to here as the manufacturing parameters and the FF parameters. The manufacturing parameters are the chosen parameters used to determine the influence of each manufacturing process, whereas the FF parameters are variables in the Newtonian equation governing the interaction among the particles in the system. The simulation output is a 3D microstructure of the electrode considering Periodic Boundary (PB) conditions for the corresponding manufacturing step where the output of the previous phase is fed consecutively into the next one-the resulting microstructure of the slurry simulations into the drying model, then finally, the resulting microstructure from the drying model is injected in the calendering model. The solid to liquid ratio is defined as the mass of the dry material before the solvent addition over the mass of the slurry after the solvent addition. The active material (AM), SE, carbon additives and binder wt.%, The SE particle size distribution (PSD) are the manufacturing parameters measured in-house for the workflow (more information is provided in the Experimental Section). These values were decided beforehand and lead to acceptable results in terms of electrochemical results and coating homogeneity. Throughout the simulation process, all manufacturing parameters were as experimentally defined and kept without further changes. To fit the FF parameters, some of the simulation outputs (such as density, viscosity and porosity) are compared with the experiments after each manufacturing step., Within the CGMD simulations, the AM and SE are considered to be two distinct groups of spheres that represent particles in the 3D geometrical domain with a particle size distribution and a certain density. |
62b9e8400bba5d053b774c26 | 5 | Within the 3D domain, these are interacting pairwise among themselves and with the CBD according to the FFs. The CBD is modeled to be a group of spheres with a certain mass of carbon, binder (and solvent in the case of the slurry phase) with an effective density and nanoporosity of 50% which was found experimentally. Two FFs are used to represent the physiochemical properties of the slurry. These FFs are: (i) Lennard Jones (LJ) ; (ii) Granular Hertzian (GH) to represent the mechanical interactions of the granular media within the slurry phase. Both FFs are available in LAMMPS simulation software and reported in Eq. 1-2: |
62b9e8400bba5d053b774c26 | 6 | ELJ is LJ potential energy, where r is the distance between the center of two interacting particles, 𝑟 𝑐 is the cut-off distance at which there is no more interaction between the particles, ε is the depth of the potential well, more commonly known as dispersion energy, 𝜎 is the bond length. The GH FF is given by |
62b9e8400bba5d053b774c26 | 7 | 𝐹GH is GH forces, where 𝑅 𝑖 and 𝑅 𝑗 are the radius of the particles, θ is the overlap distance, 𝑘 n , 𝑘 t are the elastic constants, 𝛾 n , 𝛾 t viscoelastic damping constants and v n , v t are the relative velocity components between the two particles for the normal and tangential contact respectively. Where 𝑚 𝑒𝑓𝑓 is the effective mass of both particles, ∆𝑠 t is the tangential displacement vector between the two interacting particles, 𝑛 𝑖𝑗 is the unit vector that connects the centers of the two particles. Because the overlap is taken into account in the GH FFs, the experimental diameter of the AM and SE is increased slightly to count for the presence of the surrounding solvent . Lastly, an additional parameter is still needed for the GH simulations which is Xu, which is the friction coefficient that is defined as the highest ratio between the normal and tangential forces. |
62b9e8400bba5d053b774c26 | 8 | The computational workflow starts with the simulation of the slurry phase. To this end, it is known that the rheological properties of the slurry have a great impact on the manufacturing process. The quality and speed of the coating, viscosity and the resulting cathode microstructure all depend on the slurry microstructure and its physiochemical properties. Many studies demonstrate the interdependency between electrode characteristics and the properties of the slurry such as the porosity, conductivity and geometrical tortuosity. The slurry microstructure model is generated based on the experimental weight percentages of 69.0%, 27.6% and 3.4% for AM, SE and CBD, respectively. The diameter of the AM and SE are dependent on the starting materials as well, where the diameter of the AM is 10 µm and the particle size distribution of the SE constitutes of seven different diameters (1.5 µm, 2.5 µm, 3.5 µm, 4.5 µm, 5.5 µm, 6.5 µm and 7.5 µm), the methodology used to obtain these values is explained in the Experimental Section. To avoid overlapping of particles at the initial state, they are randomly placed in a very large simulation box (716 µm x 716 µm x 1000 µm). The initial velocities of the particles were set randomly according to a distribution that corresponds to a temperature of 300 K. For the initial generation of the particles and the slurry equilibration, the number of particles, pressure and temperature are set to be 300 K and 1 atm, respectively (which is known as NPT environment), throughout the CGMD simulation. After that, nonequilibrium CGMD (nE-CGMD) simulations are run to evaluate the shear viscosity (η) of the system against the applied shear rates (γ). |
62b9e8400bba5d053b774c26 | 9 | Besides all the above-mentioned FF parameters, the CBD diameter (dCBD) and its density (ρCBD) account for effective parameters that are used for further optimization of the models and the physicochemical evaluation of the slurry simulation. To optimize the FFs, the Bayesian optimization was conducted according to the reference until achieving results sufficiently close to the experiments, then it was improved by systemic manual fitting for each of the FFs. To address the parameterization correctly, the viscosity of the slurry is measured experimentally as a function of γ̇ as demonstrated in Figure . |
62b9e8400bba5d053b774c26 | 10 | The shear viscosity can give relative insights into the undergoing structural changes taking place during slurry coating due to the application stress deformation. This curve can be used to fit the FF parameters, which are unknown beforehand. In our case, the shear rate-viscosity curve and slurry density for the slurry, and the porosity of the dried and calendered electrode. The simulated density of the slurry is fitted to match the slurry density that was measured experimentally which was 1.90 g•cm -3 . From Figure , it can be remarked that the slurry model and the results of its nE-CGMD viscosity simulations and the experimental observables demonstrate acceptable agreement results making the models considered as a digital representative for further assumptions. After the fitting of the slurry density and viscosity, the slurry 3D microstructure model was utilized to account for the particles' initial position for the drying and calendering simulations. Where for the dried electrode simulations, the NPT environment is set to be 300 K and 1 atm throughout the simulation. The drying starts with the shrinkage of the CBD particles to 3 μm to mimic the evaporation of the solvent with the whole system being under 1 atm pressure as the slurry thickens to the dried electrode state under ambient room conditions. There was a difference in the porosity between simulations and experiments where the simulated porosity was 35.0% and the experimental porosity is 50.7%. This high experimental porosity causes low ionic conductivity and short Li ion pathways within the electrode. Unlike the traditional LIB where the liquid electrolyte fills the pores, in SSB the optimal porosity is zero. |
62b9e8400bba5d053b774c26 | 11 | The degree of electrode compression is kept as an extra freedom degree during calendering for further validation of the computational workflow. To verify whether the simulated microstructures are sufficiently representative to undergo qualitative and semi-quantitative tests to find the impact of the calendering on the electrode by having different compression degrees (0 %-30%) shown in Figure . The electrode's effective electrical conductivity (δ) and the geometric tortuosity (τ geo ), shown in Figure and, are two parameters chosen to have a qualitative indicator of the effect of the calendering degree on the electrode electrochemical performance. They are simple observables defined to account for both ionic and electronic properties of the simulated dried (0% compression degree) and calendered (5%-30%) electrodes. The simulations showed no effective electronic pathways along the z axis for 5% compressed electrode microstructure, so we were not able to perform the same calculations. |
62b9e8400bba5d053b774c26 | 12 | The electrodes δ was calculated by solving the Poisson equation in the simulation domain, applying a 1 V potential difference between opposite sides along the z direction (perpendicular to the calendering plane). Then, Ohm's law is used to obtain the δ. The electronic conductivities of the AM and the CBD phases were set to 0.005 S m -1 and 15.93 S m -1 , respectively. PB conditions were considered for the outer xz and yz planes. The τ geo values were calculated according to τ = √ |
62b9e8400bba5d053b774c26 | 13 | , where 𝜂 is the volume fraction occupied by the solid electrolyte and 𝐷 𝑒𝑓𝑓 is the effective diffusion coefficient. 𝐷 𝑒𝑓𝑓 is in turn calculated by solving Fick's first law in the solid electrolyte domain, with a concentration difference 𝛥𝑐 between the outer xy planes. 𝐷 𝑒𝑓𝑓 is obtained from the overall diffusive flux 𝑗 as: 𝐷 𝑒𝑓𝑓 = -𝑗 × length /𝛥𝑐. Since τ geo is a purely geometrical magnitude, it is independent of the values chosen for 𝛥𝑐 and the diffusion coefficient. Periodic boundary conditions were considered for the outer xz and yz planes. |
62b9e8400bba5d053b774c26 | 14 | We present for the first time a physics-based computational workflow able to simulate in 3D the wet manufacturing process of SSB cathode electrodes using NMC622 as active material. The workflow has as inputs the materials and manufacturing process parameters used in our in-house experiments. It describes three major steps in the electrode manufacturing process: the slurry preparation, the drying and the calendering. This computational workflow, that was initially developed to simulate the manufacturing process of LIB electrodes, proved that is highly flexible to be extended and applied to diverse scenarios for a spectrum of energy storage chemistries: LIB, ASSB, Si/Gr, organic SIB batteries taking into account the wide range of manufacturing parameters associated each battery chemistry. |
62b9e8400bba5d053b774c26 | 15 | The simulations were fitted by the comparison of the simulated to the experimental results of slurry density, slurry viscosity-shear rate curve, dried and calendered electrode porosities. The well-fitted simulation results demonstrate that this workflow consisting of several models is a good representation of the real, manufacturing process to produce electrodes with similar features. Furthermore, the ratio between the electronic conductivity of the electrode to the tortuosity factor is utilized as an additional qualitative observable to study the electrode microstructure under different degrees of calendering. The analysis of this observable indicated an increase in the electrochemical performance with calendering. However, the simulations became unstable when compressing the electrodes to 30% of their initial thickness. This suggests difficulty of calendering to high degrees without cracking or other undesired effects, which are unaccounted for in the model. |
62b9e8400bba5d053b774c26 | 16 | This work further demonstrates the chemical neutrality nature of the ARTISTIC project electrode manufacturing models, which are not limited to one type of energy storage system, yet it can be extended to encompass the implementation of next generation battery manufacturing. These experimentally validated computational procedures give the opportunity of inspecting a large number of combinations of the manufacturing parameters to rapidly examine the accuracy of assumptions before starting experiments. |
62b9e8400bba5d053b774c26 | 17 | LAMMPS software was utilized to execute all the manufacturing simulations for this work-the slurry, drying and calendering. Approximately 40 AM, 420 SE and 4630 CBD particles were simulated accounting for more 5000 particles. The CBD particle size was 7.5 μm in the slurry phase and 3 μm for the dried and calendered electrodes. During the slurry simulations, the CBD particles were enlarged to encompass the carbon, binder and solvent, then the CBD particles were shrunk down to exclude the solvent after its evaporation during drying. |
62b9e8400bba5d053b774c26 | 18 | The Li6PS5Cl (NEI) SE powder was pretreated by a wet ball milling process to reduce the size of the agglomerates and this treatment was decided based on an experimental study assisted by Machine learning (ML) to decide the best milling conditions to produce Li6PS5Cl SE thin films . Wet ball milling experiments were executed using a ball milling machine (Pulverisette 7, Fritsch) with 10 mm-diameter zirconia milling media and a zirconia jar under the atmosphere of a dry room. P-xylene (≥ 99%, Sigma-Aldrich) was mixed with Li6PS5Cl powder and then milled at 300 rpm for 2 hours. After milling, the mixture was left to dry under a vacuum at 80°C for an overnight to obtain the wet milled SE (WM-SE) powder. Then, the PSD of the WM-SE was collected manually from several Scanning Electron Microscopy (Quanta 200F, FEI) images where the number of samples recorded was over 2000. |
62b9e8400bba5d053b774c26 | 19 | The dry composite of slurry consisted of NMC-622 (HEX-10, Umicore) AM, WMSE and Carbon additives (Carbon Super C45) mixed well with a mortar and pestle. the binder of styrene-butadiene-styrene copolymer (SBS) (Sigma -Aldrich) was dissolved in the solvent p-xylene (≥ 99%, Sigma-Aldrich) for 30 minutes at 70°C with continuous mixing. After that, the solvent was left to cool to ambient temperature then the dry composite was added to it and mixed for one hour to formulate the slurry. |
62b9e8400bba5d053b774c26 | 20 | The shear viscosity curve of the slurry was obtained using a rheometer (Kinexus lab+, Malvern Instruments) by applying immediately a shear rate between (0.1 -700 s -1 ), while its density was measured manually by dividing the weight of the slurry by the volume of its container with a known volume. |
626bdff9ebac3ac754e735e4 | 0 | Shaped by millions of years of evolution, enzymes have mastered an impressive arsenal of chemical transformations and, in doing so, conquered even the far reaches of chemical space. Enzymes have learned how to effect transformations, which go against the innate reactive preferences of substrates, transformations, which rapidly construct molecular complexity from simple precursors and transformations, which require highly precise control over transition states to obtain the desired product. This catalytic prowess has enabled biosynthetic access to natural products with complex and intriguing architectures as well as valuable biological activities. Understanding and exploiting these compounds and their biological activities has been an active area of research for several decades, with implications in medicinal chemistry and beyond. This, however, requires a viable route for access to said natural products, which continues to serve as a driving factor motivating the chemical total synthesis of complex natural products. However, without enzymes, accessing many natural products is less than straightforward with conventional chemical methods: Cross-coupling reactions typically require prefunctionalization, oxidative manipulations generally demand protecting groups, and asymmetric transformations often necessitate extensive catalyst screening and reaction optimization. Nonsurprisingly, the strategic use of enzymes in natural product synthesis has gained increased popularity in recent years. |
626bdff9ebac3ac754e735e4 | 1 | By enabling transformations in a fashion orthogonal to established chemical approaches, enzymes streamline the preparation of complex natural products and introduce new retrosynthetic disconnections and synthetic strategies. While dearomatization with Rieske dioxygenases 3 and lipase-catalyzed deracemization of early alcohol intermediates have found a solid foothold in contemporary natural product synthesis, several other classes of enzymes have recently made an appearance in routes toward natural products and their analogues. Owing to their versatility, enzymes have found applications in nearly all phases of (total) synthesis and continue to reshape the way organic chemists approach complex target compounds. As a primer to the state-of-theart concerning strategic uses of enzymes in natural product synthesis and the underlying concepts, this review highlights selected recent literature examples (covering 2020 to 2022) employing enzymes for the provision of chiral starting materials, endgame transformations, convergent synthesis and cascades facilitating complex reaction sequences (Figure ). |
626bdff9ebac3ac754e735e4 | 2 | First, the enzymatic generation of (typically chiral) starting materials or intermediates for total synthesis campaigns is an elegant method for the precise oxidative manipulation or deracemization of bulk material. Generally, this strategy aims to generate synthons which are either not easily available by purely chemical methods or are difficult to obtain in the desired optical purity. Here, enzymes facilitate synthesis by complementing the tools available through conventional organic chemistry. Although, from a conceptional standpoint, this approach follows in the footsteps of the early examples of enzymes in synthesis (which were largely limited to lipases), the scope of available enzymes has been broadened immensely, and so has the variety of available functionalizations and accessible synthons. At the forefront of these advancements, the Renata group has been leading the use of iron-dependent enzymes for the oxyfunctionalization of early intermediates. For instance, in their concise five-step synthesis of mitrekaurenone, 10 the use of a dioxygenase-reductase fusion enzyme enabled the precise introduction of a hydroxyl group on an otherwise scarcely functionalized terpene scaffold. In a similar fashion, the use of an engineered P450BM3 variant previously developed for steroid oxidation enabled the early hydroxylation of a sclareolide epimer en route to the meroterpenoid polysin, among other terpenoids. In both of these studies, the authors used readily available starting materials from the chiral pool and employed oxygenases for their further functionalization. In addition to these terpenes, the Renata group has pursued a modular chemoenzymatic synthesis of the tetrapeptide GE81112 B1, a bacterial translation inhibitor produced by Streptomyces sp. which is composed of four unnatural amino acids. In this synthesis, non-heme -ketoglutaratedependent hydroxylases gave access to hydroxylated amino acids, such as 3-hydroxy-L-pipecolic acid and 4-hydroxyl-L-citrulline which could be combined by peptide coupling after suitable protection. This strategy could also be leveraged to explore structure-activity relationships as the hydroxylated analogues of L-proline and L-norleucine could be obtained in an analogous fashion. By employing a dynamic kinetic resolution strategy with thermostable amino acid transferases, the same group recently also demonstrated the preparation of enantioenriched -methylated amino acids. This provided access to building blocks bearing vicinal stereocenters, which could, for instance, be elaborated by ester and amide formation to give jomthonic acid, a soil-derived natural product with antidiabetic and antiatherogenic activities. Starting from perhaps more exotic synthons, the groups of Huang and Chen recently used a Baeyer-Villiger monooxygenase to desymmetrize a dichlorinated ketone to give the corresponding chiral ester, setting the stage for a subsequent Prins reaction and olefination en route to a variety of prostaglandins, among them PGF2. Notably, in addition to the early use of a monooxygenase, the authors also employed a ketoreductase for the stereoselective hydrogenation of a late branching point intermediate. Finally, in their synthesis of viridicatin, the groups of Chang and colleagues used the ketoglutarate-dependent oxygenase AsqJ to install an epoxide on a diketodiazepine scaffold where ring-contraction could be triggered by Lewis acid catalysis. Further elaboration of the obtained brominated quinolinone under Heck conditions afforded a diversely substituted set of viridicatin analogues. |
626bdff9ebac3ac754e735e4 | 3 | Secondly, the elaboration of late-stage intermediates to the final product (commonly also termed an endgame) with an enzyme is a relatively new area of research, which profits from the ability of enzymes to discriminate between multiple reactive moieties and facilitate precise transformations on highly functionalized compounds. In contrast to the examples above, a surprisingly diverse array of enzyme functions has recently found applications in endgame transformations. While the Sherman group used a promiscuous P450 monooxygenase to diversify the scaffold of tirandamycin via repeated hydroxylation, several groups employed P450 enzymes for biaryl coupling reactions. For instance, the Gulder group furnished the bityrosine motif of arylomycin by enzymatic coupling following construction of the peptide backbone by solid phase synthesis. The groups of Seyedsayamdost and Tailhades and Cryle independently used OxyB, a P450 from vancomycin biosynthesis, to introduce a variety of different aryl connections into linear peptides with (sometimes only distant) structural similarity to the native substrate. Complementing the previously mentioned work with the oxygenase AsqJ, the Gulder group employed this enzyme for ring contractions yielding quinazolinones through a radical mechanism relying on the absence of stabilizing effects in non-native substrates. Aside from irondependent enzymes, flavoenzymes, prenyltransferases and SAM-dependent enzymes have recently enjoyed increased attention. For instance, the Renata group used a flavindependent halogenase for the regioselective installation of a chloride substituent in their total synthesis of fasamycin. The groups of Rudolf and Elshahawi independently employed promiscuous prenyltransferases for the late-stage diversification of arenes and arene-bearing peptides. By employing the methyltransferase PsmD for concurrent C-Hmethylation and stereoselective cyclization, the Pietruszka group gained access to physostigmine and analogues thereof featuring different acylation profiles. Finally, the groups of Tang, Houk and Garg used the strictosidine synthase from Catharanthus roseus for the final stereoselective Pictet-Spengler reaction to furnish strictosidine. |
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