Statistical Size Distribution Research Articles

Numerical simulation of droplet dispersion within meso-porous membranes

Analysis of membrane processes in fluid processing, and their main influencing operating conditions are relevant in a variety of industrial applications. Increasing regulatory scrutiny and environmental considerations are forcing industries across all sectors, from food and pharma to oil and gas, to further understand and optimise the handling and formulation of liquid systems for efficient process design. In a generic setup for emulsification and liquid formulation the flow and dispersion behaviour of a liquid oil droplet on its way through a porous water filled membrane is analysed. A set of high-resolution numerical simulations of a single oil droplet dispersed in water through a porous membrane structure with varying contact angles is performed. In this work cluster analysis of volume-of-fluid simulation results to obtain statistical droplet size distributions is conducted and further analysed to highlight the effect of the contact angle as well as pressure drop on the dynamics of the system. It is observed that based on the membrane surface activity the droplet behaviour changes from filtration with coalescence when the membrane is lipophilic to emulsification with droplet break-up when the membrane is lipophobic. Furthermore, the pressure drop is identified as a key factor for the dynamics of the droplet process and the frame in which it occurs. These results highlight that the membrane wettability is a determining factor for the emulsification or filtration effectiveness of a membrane for various applications.

Combustion of iron particles in solid propellants at elevated pressure

Metal fuels, such as aluminum (Al) and iron (Fe), can be added to composite solid propellants to improve their performance, such as specific impulse, density, and burning rate. In comparison to aluminum, iron can theoretically provide improved density specific impulse and higher flame temperatures; reduce condensed combustion product (CCP) concentration and the associated two-phase flow losses; and eliminate hydrochloric acid (HCl) in the exhaust products. A fundamental and quantitative understanding of metal particle aggregation and agglomeration processes in solid propellants is required to understand the underlying combustion mechanisms in these systems. In the current study, composite strand and laminate AP/HTPB/AP propellant samples loaded with Fe microparticles (∼45 μm in diameter) were burned at elevated pressures in an optically accessible strand bomb. Combustion processes were monitored with transient pressure diagnostics and a high-speed camera fitted with a high-magnification lens system (3.83 μm/pixel resolution) for the laminate propellant experiments. An automated image processing algorithm was developed to measure burning rates and ejected particle/agglomerate sizes and velocities. Time-resolved statistical distributions of both particle size and velocity are presented at elevated pressure for multiple laminate propellant experiments with a high degree of repeatability and low measurement error estimated as < ±5% and < ±1.5% for particle size and velocity, respectively. The incorporation of iron microparticles into the composite strand propellants yielded over a 20% increase in the global burning rate over the range of pressures evaluated (3.45–13.8 MPa). Similarly, the addition of iron to the fuel lamina in laminate propellant samples led to an approximately 30% increase in the global burning rate at the evaluated pressure (3.45 MPa). Additive particles were observed to eject near the oxidizer/fuel interface, or to melt, aggregate, coalesce, and agglomerate on the fuel lamina surface prior to ejection. Particle velocities are controlled by a balance of gravitational forces, drag forces imparted by expanding combustion product gases, and particle inertia. The observed combustion enhancements are attributed to the combined effects of catalytic mechanisms, increased radiation heat transfer, and local energy release from reacting iron particles. In addition, discussions on the image processing methods developed in the current study, corresponding potential sources of error, and prospective areas of improvement are provided. The experimental approach developed enables high-speed and high-magnification visualization of propellant combustion at high pressures and can be utilized to better understand the fundamental combustion behavior of energetic systems.

Fully spherical 3D datasets on sedimentary particles: Fast measurement and evaluation

AbstractRecently it became increasingly evident that the statistical distributions of size and shape descriptors of sedimentary particles reveal crucial information on their evolution and may even carry the fingerprints of their provenance as fragments. However, to unlock this trove of information, measurement of traditional geophysical shape descriptors (mostly detectable on 2D projections) is not sufficient; fully spherical 3D imaging and mathematical algorithms suitable to extract new types of inherently 3D shape descriptors are necessary. Available 3D imaging technologies force users to choose either speed or full sphericity. Only partial morphological information can be extracted in the absence of the latter (e.g., LIDAR imaging). In the case of fully spherical imaging, speed was proved to be prohibitive for obtaining meaningful statistical samples, and inherently 3D shape descriptors were not extracted. Here we present a new method by complementing a commercial, portable 3D scanner with simple hardware to quickly obtain fully spherical 3D datasets from large collections of sedimentary particles. We also present software for the automated extraction of 3D shapes and automated measurement of inherently 3D-shape properties. This technique allows for examining large samples without the need for transportation or storage of the samples, and it may also facilitate the collaboration of geographically distant research groups. We validated our software on a large sample of pebbles by comparing previously hand-measured parameters with the results of automated shape analysis. We also tested our hardware and software tools on a large pebble sample in Kawakawa Bay, New Zealand.

Effect of the Density Ratio on Emulsions and Their Segregation: A Direct Numerical Simulation Study

Using direct numerical simulation (DNS) in combination with the volume of fluid method (VoF), we investigate the influence of the density ratio between the carrier and dispersed phase on emulsions, where the baseline simulation approximately corresponds to the ratio of water-in-gasoline emulsions. For this purpose, homogeneous isotropic turbulence (HIT) is generated using a linear forcing method, enhanced by a proportional–integral–derivative (PID) controller, ensuring a constant turbulent kinetic energy (TKE) for two-phase flows, where the TKE balance equation contains an additional term due to surface tension. Then, the forcing is stopped, and gravitational acceleration is activated. The proposed computational setup represents a unique and well-controlled configuration to study emulsification and segregation. We consider four different density ratios, which are applied in industrial processes, to investigate the influence of the density ratio on the statistically steady state of the emulsions, and their segregation under decaying turbulence and constant gravitational acceleration. At the statistically steady state, we hold the turbulence constant and study the effects of the density ratio ρd/ρc, on the interface area, the Sauter mean diameter (SMD), and the statistical droplet size distribution. We find that all are affected by the density ratio, and we observe a relation between the SMD and ρd/ρc. Furthermore, we assume a dependence of the critical Weber number on the density ratio. In the second part of our work, we study the segregation process. To this end, we consider the change in the center of mass of the disperse phase and the energy release, to analyze the dependence of segregation on the density difference Δρ/ρd. We show that segregation scales with the density difference and the droplet size, and a segregation time scale has been suggested that collapses the height of the center of mass for different density ratios.

Assessment of the Critical Defect in Additive Manufacturing Components through Machine Learning Algorithms

The design against fatigue failures of Additively Manufactured (AM) components is a fundamental research topic for industries and universities. The fatigue response of AM parts is driven by manufacturing defects, which contribute to the experimental scatter and are strongly dependent on the process parameters, making the design process rather complex. The most effective design procedure would involve the assessment of the defect population and the defect size distribution directly from the process parameters. However, the number of process parameters is wide and the assessment of a direct relationship between them and the defect population would require an unfeasible number of expensive experimental tests. These multivariate problems can be effectively managed by Machine Learning (ML) algorithms. In this paper, two ML algorithms for assessing the most critical defect in parts produced by means of the Selective Laser Melting (SLM) process are developed. The probability of a defect with a specific size and the location and scale parameters of the statistical distribution of the defect size, assumed to follow a Largest Extreme Value Distribution, are estimated directly from the SLM process parameters. Both approaches have been validated using literature data obtained by testing the AlSi10Mg and the Ti6Al4V alloy, proving their effectiveness and predicting capability.

On the impact of differential diffusion between soot and gas phase species in turbulent flames

The molecular diffusivities of larger PAHs and soot particles approach zero leading to differential diffusion with gas-phase species. The present work systematically quantifies the impact on soot moments, soot related statistical correlations and Particle Size Distributions (PSDs) using a fully coupled transported joint probability density function (JPDF) method featuring a 78-dimensional joint-scalar space, including enthalpy, gas phase species with the PSD discretised using 62 size classes via a mass and number density preserving sectional method. Differential diffusion of soot (DDS) is treated via a gradual decline of diffusivity among soot sections maintaining realisability and the expected exponential decay of variance. The solution of the flow field features a time-dependent second moment closure and an elliptic solver. The turbulent non-premixed Sandia C2H4 flame from the International Sooting Flame (ISF) data base was selected as a target along with the KAUST (C2H4/N2) variant of the same flame. Results show that reduced soot diffusion leads to a significant increase in the soot volume fraction RMS and that the correlation coefficient between soot volume fraction and temperature is further reduced in a particle-size-dependent manner. Similar observations are made for correlations between the soot volume fraction and the mass fractions of gas-phase species such as CO, OH, H and C2H2. The results suggest that computational methods that presume explicit (e.g. flame structure related) correlations between such scalars and with soot face leading order modeling challenges. It is also shown that the correlation between CO and soot increases due to oxidation of soot and that DDS leads to a modest downstream shift of PSDs towards larger particles.

Breakdown of Reye’s theory in nanoscale wear

Building on an analogy to ductile fracture mechanics, we investigate the energetic cost of debris particle creation during adhesive wear . Macroscopically, Reye proposed in 1860 that there is a linear relation between frictional work and wear volume at the macroscopic scale. Earlier work suggested a linear relation between tangential work and wear debris volume also exists at the scale of a single asperity, assuming that the debris size is proportional to the micro contact size multiplied by the junction shear strength. However, the present study reveals deviations from linearity at the microscopic scale. These deviations can be rationalized with fracture mechanics and imply that less work is necessary to generate debris than what was assumed. Here, we postulate that the work needed to detach a wear particle is made of the surface energy expended to create new fracture surfaces, and also of plastic work within a fracture process zone of a given width around the cracks. Our theoretical model, validated by molecular dynamics simulations, reveals a super-linear scaling relation between debris volume (Vd) and tangential work (Wt): Vd∼Wt3/2 in 3D and Vd∼Wt2 in 2D. This study provides a theoretical foundation to estimate the statistical distribution of sizes of fine particles emitted due to adhesive wear processes.

Особливості розвитку ринку банківських послуг України: ефект розміру банку та закон Гібрата

Quantitative description of the development of the banking services market is important for making optimal decisions in the regulation of the banking sector. The importance of the question lies in the fact that such a description makes it possible to take into account the transformation of banks, sources of risks, to identify the main trends or regularities in the process of evolution. The aim of this article is to test Gibrat's law as a hypothesis for the development of the banking services market of Ukraine, by an empirical experiment where all existing banks are tracked over time on an annual basis and the distribution of banks by the size of assets or income is constructed annually. The work assumes the possibility of establishing a statistical size (assets or income) distribution of banks with certain average values and variances according to the lognormal distribution of banks. The research methodology is based on statistical analysis and verification of the correspondence of the empirical size distribution of banks to the logarithmically normal one provided by Gibrat's law. The study period covers the banking sector from 2014 to 2023 and includes all operating banks. The information base for the analysis was the annual official reporting data (report on the financial state) of banks, published by the National Bank of Ukraine and on the banks' official websites. The data set defines the assets and income of all banks and each bank separately, annually. It has been proven that Gibrat's law is not fulfilled for banks of Ukraine, the size distributions do not obey the Gaussian distribution law. The rate of growth of banks is not random (as it is assumed by Gibrat's hypothesis). The trend of gradual approximation of income distributions to the lognormal type and the potential possibility of implementation of Gibrat's law for the period of 2022–2023 has been revealed.

A reference methodology for microplastic particle size distribution analysis: Sampling, filtration, and detection by optical microscopy and image processing

AbstractMicroplastic (MP) contamination in natural water circulation is a concern for environmental issues and human health. Various types of polymer materials have been identified and were detected in MP analytic test procedures. Beyond MP polymer type, particle size and form play a major role in water analysis due to possible negative toxicologic effects on flora and fauna. However, the correct quantitative measurement of MP size distribution over several orders of magnitude is strongly influenced by sample preparation, filtration materials and processes, and microanalytical techniques, as well as data acquisition and analysis. In this paper, a reference methodology is presented aiming at an improved quantitative analysis of MP particles. An MP analysis workflow is demonstrated including all steps from reference materials to sample preparation, filtration handling, and MP particle size distribution analysis. Background‐corrected particle size distributions (1–1000 µm) have been determined for defined polyethylene (PE) and polyethylene terephthalate (PET) reference samples. Microscopically measured particle numbers and errors have been cross‐checked with the total initial mass. In particular, defined reference MP samples (PE, PET) are initially characterized and applied to filtration experiments. Optical microscopy imaging on full‐area Si filters with subsequent image analysis algorithms is used for statistical particle size distribution analysis. To quantify the effects of handling and filtration, several blind tests with distilled water are carried out to determine the particle background for data evaluation. Particle size distributions of PE and PET reference samples are qualitatively and quantitatively reproduced with respect to symmetry, and maximum and cut‐off diameter of the distribution. It is shown that especially MP particles with a radius of &gt;50 µm can be detected and retrieved with high reliability. For particle sizes &lt;50 µm, a significant interference with background contamination is observed. Data from blank samples allows a correction of background contaminations. Furthermore, for enhanced sampling statistics, the recovery of the initial amount of MP will be qualitatively shown. The results are intended as an initial benchmark for MP analytics quality. This quality is based on statistical MP particle distributions and covers the complete analytic workflow starting from sample preparation to filtration and detection. Microscopic particle analysis provides an important supplement for the evaluation of established spectroscopic methods such as Fourier‐transform infrared spectroscopy or Raman spectroscopy.

An Improved Method for the Evaluation and Local Multi-Scale Optimization of the Automatic Extraction of Slope Units in Complex Terrains

Slope units (SUs) are sub-watersheds bounded by ridge and valley lines. A slope unit reflects the physical relationship between landslides and geomorphological features and is especially useful for landslide sensitivity modeling. There have been significant algorithmic advances in the automatic delineation of SUs. But the intrinsic difficulties of determining input parameters and correcting for unreasonable SUs have hindered their wide application. An improved method of the evaluation and local multi-scale optimization for the automatic extraction of SUs is proposed. The Sus’ groups more consistent with the topographic features were achieved through a stepwise approach from a global optimum to a local refining. First, the preliminary subdivisions of multiple SUs were obtained based on the r.slopeunit software. The optimal subdivision scale was obtained by a collaborative evaluation approach capable of simultaneously measuring objective minimum discrepancies and seeking a global optimum. Second, under the selected optimal scale, unreasonable SUs such as over-subdivided slope units (OSSUs) and under-subdivided slope units (USSUs) were further distinguished. The local average similarity (LS) metric for each SU was designed based on calculating the SU’s area, common boundary and neighborhood variability. The inflection points of the cumulative frequency curve of LS were calculated as the distinguishing intervals for those unrealistic SUs by maximum interclass variance threshold. Third, a new effective optimization mechanism containing the re-subdivision of USSUs and merging of OSSUs was put into effect. We thus obtained SUs composed of terrain subdivisions with multiple scales, which is currently one of the few available methods for non-single scales. The statistical distributions of density, size and shapes demonstrate the excellent performance of the refined SUs in capturing the variability of complex terrains. Benefiting from the sufficient integrating approach of diverse features for each object, it is a significant advantage that the processing object can be transferred from general entirety to each precise individual.

WCu composites fabrication and experimental study of the shielding efficiency against ionizing radiation

A comprehensive study of the present work aimed to investigate the radiation shielding features for W–Cu composites materials with high density. Isostatic hot pressing technique was applied to obtain W85 wt%Cu15 wt% and W75 wt%Cu25 wt% composites with various thickness (0.6, 0.9, 1.2, 1.5 and 2.7 cm). Composites were the pollycristalline samples with well packed microstructure. The statistical grain size distribution performed that the size of an agrolemerated copper grains increases with the Cu content rising. X-ray diffraction (XRD) analysis showed that the main spectrum lines of the composites are corresponded to the bcc phase of tungsten and fcc phase of copper. The linear attenuation coefficient (LAC) was detected experimentally using a NaI detector at various incoming photon energy. The highest values of LAC are found 2.11 and 4.33 cm−1 for W75 wt%Cu25 wt% and W85 wt%Cu15 wt%, respectively, at low energy 0.266 MeV while the lowest values (0.44 and 0.48 cm−1 for W75 wt%Cu25 wt% and W85 wt%Cu15 wt%, respectively) are observed at high energy 1.25 MeV. From the obtained results, it can deduce the discussed W–Cu composites are suitable to be used in several applications of radiation protection.

The effective mechanical properties of solids with distributed rough cracks

A statistical multi-step self-consistent (SMSSC) micromechanical model for predicting elastic and plastic properties of a three-dimensional Representative Volume Element (RVE) is proposed in this research. A body with randomly distributed rough penny-shaped microcracks is considered for the first time in which the distribution of cracks is included through different well-known statistical distributions. Initially, the solution for a rough cohesive penny-shaped crack is developed, and the relationship between crack nominal length and roughness is derived. Consequently, Crack Opening Displacement (COD) and the corresponding volume crack opening are calculated as a function of surface roughness. Next, the SMSSC is used to study the effects of nominal crack length distribution on a fractured medium’s mechanical properties with a known statistical distribution of crack sizes. This is done by determining the energy release and evaluating its share in material degradation (elastic, plastic). The elastic properties calculated using SMSSC indicated that the fractality of crack trajectories leads to lower values of volume crack opening and, consequently, resulted in a lower level of stiffness degradation compared with previous smooth crack approximations. It is also reached that the statistical distribution of rough cracks influences the mechanical properties due to the interrelation between length and roughness. Regarding the plastic behavior, the proposed micromechanical methodology is implemented to construct a capped pressure-sensitive yielding potential function that explicitly highlights the influence of crack size statistical distribution on the yielding of cracked solids. It is evident that the statistical distribution of crack sizes which are interrelated with crack roughness, can significantly alter the corresponding yield surface of fractured media.

Statistical capillary tube model for porous filter media: An application in modeling of gasoline particulate filter

A new statistical capillary tube model is presented to predict both filtration efficiency and permeability of particulate filters. The model considers the porous structure of the filter media as an assembly of curved capillary tubes with a statistical size distribution. The key model descriptors of pore microstructure are porosity, pore size, pore size distribution, and tortuosity. The model was validated by permeability and size dependent filtration efficiency measurements of bare and washcoated gasoline particulate filters (GPFs), using porosimetry data of the filter media as model inputs. Comparing with a more widely used spherical model for particulate filters with typical porosity range of 30–60%, the capillary tube model has similar input parameters but provides additional permeability prediction as well as better filtration predictions over a wider porosity range including the lower porosity of washcoated filters. Furthermore, it offers a more visually straightforward and intuitive correlation between filtration paths and filter performance. This allows convenient adoption of the capillary tube model in numerous applications that are currently based on spherical model assumption and provides improved guidance for filter optimization. Beyond ceramic automotive exhaust filters, this geometric model offers a new framework for examining porous filter media in general.

Multiscale modeling of ferroelectrics with stochastic grain size distribution

Macroscopic properties of ferroelectrics are controlled by processes on the microscale, in particular the switching of crystal unit cells and the movement of domain walls, respectively. Besides these microscopic levels, the grains of a polycrystalline material constitute the mesoscopic scale. Interactions of grains with statistically distributed orientations, as a consequence of mechanical and electrostatic mismatch, give rise to for example, residual stress which in turn affects domain switching. A multiscale modeling thus has to incorporate at least three interacting scales. In this context, the condensed method has recently been elaborated as an efficient tool with low computational cost and effort of implementation. It is extended toward statistical distributions of grain sizes in a representative material volume element and amended with regard to the modeling of domain evolution. Each of the few parameters of the constitutive approach has a unique physical meaning and is adapted to available experimental values of macroscopic quantities of barium titanate taken from various sources.

Dynamical assessment of altitudinal trend of drop size distributions in a tropical region of Nigeria

A comparative assessment of different statistical drop size distributions in a tropical location in Nigeria is presented in this paper. The wet season months of the year 2014 rain data, measured using vertically-pointing Micro Rain Radar (MRR) installed at the Department of Physics, The Federal University of Technology Akure, Nigeria was considered. The data were archived at the Communication Research Group, Department of Physics, The Federal University of Technology Akure. The drop sizes were estimated from the fall velocity. The stratiform rainfall type was considered in this study due to its preponderance of occurrence. Three heights (160 m, 1440 m, and 2240 m) were reviewed for this study. These heights were given due consideration because of the density of the data. All the statistical distributions considered which include Lognormal, Gamma, and Weibull distributions performed better at the 160 m height compared to other heights. It was however observed that the Gamma and Weibull distribution performed better than the Lognormal distribution at this height. The results obtained in this study will be fundamental for the application of remote sensing and the propagation field in this area. The results will be useful to radio communication engineers to properly design a good fade margin in a radio communication system in this region.

Metastable high entropy alloys: An excellent defect tolerant material for additive manufacturing

Fatigue failure is ubiquitous in structural components. In additively manufactured (AM) components, the processing induced defects limit the fatigue performance. Further, the stochastic nature of defects in laser-powder bed fusion (L-PBF) make it difficult to predict the fatigue life in these components. In this work, we explored exceptional work hardening (WH) of a metastable Fe40Mn20Co20Cr15Si5 high entropy alloy (CS-HEA) to obtain high fatigue-resistance with L-PBF. Further, a fatigue life estimation tool based on the statistical size distribution of microstructural entities such as grains, pores and solid-state inclusions and, their mutual interaction was used to estimate the fatigue life of as-printed material. Upon deformation, CS-HEA exhibited γ (f.c.c.) → ε (h.c.p.) martensitic transformation and subsequent twinning in ε (h.c.p.) phase. Such deformation behavior resulted in sustained WH and is specifically beneficial in the vicinity of critical pores. A high normalized fatigue strength of 0.65 with respect to the yield strength was thus obtained in as-printed condition. Further, the model accurately predicted extended crack initiation life for CS-HEA. The current work therefore provides guidance towards developing defect-tolerant alloys for L-PBF and presents a tool for estimation of fatigue life of AM alloys with unconventional WH behavior.

Experimental study of two-phase flow structure in churn-turbulent to annular flows

The local phase distribution of adiabatic air-water two-phase flow in churn-turbulent to annular flows was experimentally investigated using the Two-Sensor Droplet-Capable Conductivity Probe (DCCP-2). The experiments were performed in a 2.54 cm inner diameter vertical round pipe at downward flows. Twelve flow conditions in churn-turbulent to annular flows were chosen for the current study. The inlet superficial gas velocity ranged from 5.68 m/s to 17.19 m/s, and the inlet superficial liquid velocity ranged from 1.21 m/s to 2.64 m/s. The DCCP-2 is able to distinguish droplets, liquid ligaments, bubbles, and continuous gas. The radial profiles of volume fraction, frequency, axial interfacial velocity, and chord length of droplets, ligaments and bubbles were measured. The experimental results showed that gas volume fraction had a center-peak distribution, and the maximum gas volume fraction in centerline could exceed 95%. The volume fraction of droplets was very low, only on the order of 0.1%. Droplet concentration also showed a center-peak distribution. Both ligament and bubble volume fraction showed a near-wall peak distribution. The ligament volume fraction in the near wall region could reach 80%, which indicated the existence of a film region near the wall and likely includes surface waves on the film. The volume fraction profile of gas, droplets, ligaments, and bubbles showed the tomography structure in churn-turbulent to annular flows. The effects of flow conditions on phase distribution were also studied. An interesting feature was that both the increase of gas flux and the increase of liquid flux tended to increase the bubble volume fraction. The statistical distribution of droplet size was found to be invariant of radial position. The droplet velocity at centerline was generally larger than the superficial gas velocity, which can be attributed to the gravitational effects at downward flows.

Nano/micro plastics – Challenges on quantification and remediation: A review

Nano and micro plastics (NP/MPs) represent one of the most challenging classes of micro-pollutants, with occurrence across all ecosystems and size distributions ranging from the nanometre to the millimetre scale. These broad composition and size distribution ranges limit the efficiency of detection methods, often inherently focused on a single and narrow class of NP/MPs sizes. In addition to their demonstrated native toxicity, NP/MPs may act as efficient carriers of pollutants and pathogens onto their surface, facilitating the transfer and penetration of other classes of hazardous materials. This comprehensive review presents the key challenges related to soil, air and water NP/MPs: (i) sampling and extraction, (ii) using defined synthetic NP/MPs for spiking environmental samples and for model studies, and (iii) characterisation. A major challenge discussed in this paper relates to the lack of relevant characterisation strategies for the NP/MPs materials, enabling simultaneous identification, quantification and generation of statistical size distributions. This critical discussion ends with a series of well-informed propositions to support the systematic assessment of the impact of NP/MPs materials, spanning from materials science and characterisation, as well as environmental and chemical engineering.

Exploring Contingency Skill Scores Based on Event Sizes

AbstractSpace weather forecasts are generally made for events with an arbitrary size threshold imposed on an event statistical size distribution which is likely described by a power law. This is the case for solar energetic (E &gt; 10 MeV) particle (SEP) events, which have a differential power law exponent of γ = 1.2. Event forecasts are usually evaluated by skill scores using a contingency table that matches the forecasted events against observed events independently of the event sizes. Each observed event is either a forecasted hit or a miss, and each forecasted event is either an observed hit or a false alarm. However, for SEP events and most other space weather parameters the event size is a critical factor for the user. It is more important that large events be well forecasted than threshold events. In addition, false alarms may be useful when they match observed events just below the forecast threshold. We explore a forecast evaluation scheme to incorporate the event size within the usual format of a binary contingency table to evaluate model performance. The scheme is applied to three different input options of a recently published evaluation of the Proton Prediction System (PPS) for SEP events to show differences between numbers‐based and intensity‐based skill scores of the PPS. We demonstrate how identical skill scores can result from models with extremely different performances of event intensity forecasts. The scheme requires model validation and would benefit from testing with other space weather applications.

High-performance methylammonium-free ideal-band-gap perovskite solar cells

•A SnCl2•3FACl complex to assist MA-free Sn-Pb perovskite growth is reported•Film quality is enhanced, with improved microstructure and reduced residual stress•Near ideal-band-gap device efficiency is improved to ∼20% with enhanced stability To reach the ideal band gap for single-junction perovskite solar cells (PSCs), it is generally necessary to use about 25–30 mol% Sn in the Sn-Pb mixed hybrid perovskites. Synthesis of such Sn-Pb compositions has been difficult, with uncontrolled structural and defect characteristics. Consequently, it has been challenging to achieve simultaneously both high efficiency and good long-term operational stability for Sn-Pb-based ideal-band-gap PSCs. In this work, we show that the use of a SnCl2⋅3FACl complex additive can significantly enhance the methylammonium-free Sn-Pb perovskite film quality, with improved microstructure, reduced defect density, and suppressed development of residual stress. With this approach, we demonstrate near ideal-band-gap (∼1.34 eV) methylammonium-free Sn-Pb-based PSCs with high efficiency (∼20%) and promising operational stability of maintaining >80% of initial power-conversion efficiency over 750 h under maximum-power-point tracking. The development of mixed tin-lead (Sn-Pb)-based perovskite solar cells (PSCs) with low band gap (1.2–1.4 eV) has become critical not only for pushing single-junction devices toward the maximum efficiency given by the Shockley-Queisser limit, but also for enabling all-perovskite tandem devices beyond this limit. However, achieving high power-conversion efficiency (PCE) and long-term device operation stability simultaneously remains a significant challenge for Sn-Pb-based PSCs. Here, we demonstrate near ideal-band-gap (∼1.34 eV) methylammonium-free Sn-Pb-based PSCs with high efficiency (∼20%) and promising operational stability of maintaining >80% of initial PCE over 750 h under maximum-power-point tracking. The key to this success is the use of a SnCl2⋅3FACl complex additive that improves the microstructure and reduces the development of residual stress in the Sn-Pb perovskite thin films, which in turn enhances the efficiency and stability of the Sn-Pb-based ideal-band-gap PSCs. The development of mixed tin-lead (Sn-Pb)-based perovskite solar cells (PSCs) with low band gap (1.2–1.4 eV) has become critical not only for pushing single-junction devices toward the maximum efficiency given by the Shockley-Queisser limit, but also for enabling all-perovskite tandem devices beyond this limit. However, achieving high power-conversion efficiency (PCE) and long-term device operation stability simultaneously remains a significant challenge for Sn-Pb-based PSCs. Here, we demonstrate near ideal-band-gap (∼1.34 eV) methylammonium-free Sn-Pb-based PSCs with high efficiency (∼20%) and promising operational stability of maintaining >80% of initial PCE over 750 h under maximum-power-point tracking. The key to this success is the use of a SnCl2⋅3FACl complex additive that improves the microstructure and reduces the development of residual stress in the Sn-Pb perovskite thin films, which in turn enhances the efficiency and stability of the Sn-Pb-based ideal-band-gap PSCs. Perovskite solar cells (PSCs) have emerged as a disruptive photovoltaic (PV) technology that has been researched heavily since their invention in 2009.1Dunlap-Shohl W.A. Zhou Y. Padture N.P. Mitzi D.B. Synthetic approaches for halide perovskite thin films.Chem. Rev. 2019; 119: 3193-3295Crossref PubMed Scopus (292) Google Scholar, 2Snaith H.J. Present status and future prospects of perovskite photovoltaics.Nat. Mater. 2018; 17: 372-376Crossref PubMed Scopus (395) Google Scholar, 3NREL best research-cell efficiency Chart.https://www.nrel.gov/pv/cell-efficiency.htmlDate: 2019Google Scholar The most efficient PSCs reported thus far use Pb-based halide perovskites, generally with band gaps in the range of 1.5–1.7 eV.4Jiang Q. Zhao Y. Zhang X. Yang X. Chen Y. Chu Z. Ye Q. Li X. Yin Z. You J. Surface passivation of perovskite film for efficient solar cells.Nat. Photon. 2019; 13: 460-466Crossref Scopus (2206) Google Scholar,5Jeon N.J. Noh J.H. Yang W.S. Kim Y.C. Ryu S. Seo J. Seok S.I. Compositional engineering of perovskite materials for high-performance solar cells.Nature. 2015; 517: 476-480Crossref PubMed Scopus (4632) Google Scholar This band-gap range is substantially higher than that most suitable for single-junction solar cells for maximum allowed efficiency according to the Shockley-Queisser (S-Q) limit (∼1.34 eV),6Polman A. Knight M. Garnett E.C. Ehrler B. Sinke W.C. Photovoltaic materials: Present efficiencies and future challenges.Science. 2016; 352: aad4424Crossref PubMed Scopus (1106) Google Scholar as well as for the bottom cell in all-perovskite tandem devices to go beyond the S-Q limit.7Zhao D. Chen C. Wang C. Junda M.M. Song Z. Grice C.R. Yu Y. Li C. Subedi B. Podraza N.J. et al.Efficient two-terminal all-perovskite tandem solar cells enabled by high-quality low-bandgap absorber layers.Nat. Energy. 2018; 3: 1093-1100Crossref Scopus (272) Google Scholar To address this challenge, significant effort has been devoted to developing various halide-perovskite materials with lower band gaps.8Tong J. Song Z. Kim D.H. Chen X. Chen C. Palmstrom A.F. Ndione P.F. Reese M.O. Dunfield S.P. Reid O.G. Carrier lifetimes of >1 μs in Sn-Pb perovskites enable efficient all-perovskite tandem solar cells.Science. 2019; 364: 475-479Crossref PubMed Scopus (472) Google Scholar, 9Yang Z. Yu Z. Wei H. Xiao X. Ni Z. Chen B. Deng Y. Habisreutinger S.N. Chen X. Wang K. Enhancing electron diffusion length in narrow-bandgap perovskites for efficient monolithic perovskite tandem solar cells.Nat. Commun. 2019; 10: 1-9Crossref PubMed Scopus (67) Google Scholar, 10Wei M. Xiao K. Walters G. Lin R. Zhao Y. Saidaminov M.I. Todorovic P. Johnston A. Huang Z. Chen H. et al.Combining efficiency and stability in mixed tin-lead perovskite solar cells by capping grains with an ultrathin 2D layer.Adv. Mater. 2020; 32: 1907058Crossref Scopus (83) Google Scholar, 11Ogomi Y. Morita A. Tsukamoto S. Saitho T. Fujikawa N. Shen Q. Toyoda T. Yoshino K. Pandey S.S. Ma T. CH3NH3SnxPb(1–x)I3 perovskite solar cells covering up to 1060 nm.J. Phys. Chem. Lett. 2014; 5: 1004-1011Crossref PubMed Scopus (750) Google Scholar, 12Hao F. Stoumpos C.C. Chang R.P. Kanatzidis M.G. Anomalous band gap behavior in mixed Sn and Pb perovskites enables broadening of absorption spectrum in solar cells.J. Am. Chem. Soc. 2014; 136: 8094-8099Crossref PubMed Scopus (985) Google Scholar, 13Yang Z. Rajagopal A. Jen A.K.Y. Ideal bandgap organic–inorganic hybrid perovskite solar cells.Adv. Mater. 2017; 29: 1704418Crossref Scopus (104) Google Scholar, 14Zong Y. Wang N. Zhang L. Ju M.G. Zeng X.C. Sun X.W. Zhou Y. Padture N.P. Homogenous alloys of formamidinium lead triiodide and cesium tin triiodide for efficient ideal-bandgap perovskite solar cells.Angew. Chem. Int. Ed. 2017; 56: 12658-12662Crossref PubMed Scopus (54) Google Scholar In the past few years, significant breakthroughs have been achieved in Sn-Pb alloyed low-band-gap PSCs with power-conversion efficiencies (PCEs) above 20%, along with significantly longer carrier-diffusion lengths.8Tong J. Song Z. Kim D.H. Chen X. Chen C. Palmstrom A.F. Ndione P.F. Reese M.O. Dunfield S.P. Reid O.G. Carrier lifetimes of >1 μs in Sn-Pb perovskites enable efficient all-perovskite tandem solar cells.Science. 2019; 364: 475-479Crossref PubMed Scopus (472) Google Scholar, 9Yang Z. Yu Z. Wei H. Xiao X. Ni Z. Chen B. Deng Y. Habisreutinger S.N. Chen X. Wang K. Enhancing electron diffusion length in narrow-bandgap perovskites for efficient monolithic perovskite tandem solar cells.Nat. Commun. 2019; 10: 1-9Crossref PubMed Scopus (67) Google Scholar, 10Wei M. Xiao K. Walters G. Lin R. Zhao Y. Saidaminov M.I. Todorovic P. Johnston A. Huang Z. Chen H. et al.Combining efficiency and stability in mixed tin-lead perovskite solar cells by capping grains with an ultrathin 2D layer.Adv. Mater. 2020; 32: 1907058Crossref Scopus (83) Google Scholar, 11Ogomi Y. Morita A. Tsukamoto S. Saitho T. Fujikawa N. Shen Q. Toyoda T. Yoshino K. Pandey S.S. Ma T. CH3NH3SnxPb(1–x)I3 perovskite solar cells covering up to 1060 nm.J. Phys. Chem. Lett. 2014; 5: 1004-1011Crossref PubMed Scopus (750) Google Scholar, 12Hao F. Stoumpos C.C. Chang R.P. Kanatzidis M.G. Anomalous band gap behavior in mixed Sn and Pb perovskites enables broadening of absorption spectrum in solar cells.J. Am. Chem. Soc. 2014; 136: 8094-8099Crossref PubMed Scopus (985) Google Scholar For such high-PCE Sn-Pb-based PSCs, the band gap of Sn-Pb perovskites is typically ∼1.25 eV, which is frequently studied for use as the bottom cell in a tandem device.8Tong J. Song Z. Kim D.H. Chen X. Chen C. Palmstrom A.F. Ndione P.F. Reese M.O. Dunfield S.P. Reid O.G. Carrier lifetimes of >1 μs in Sn-Pb perovskites enable efficient all-perovskite tandem solar cells.Science. 2019; 364: 475-479Crossref PubMed Scopus (472) Google Scholar, 9Yang Z. Yu Z. Wei H. Xiao X. Ni Z. Chen B. Deng Y. Habisreutinger S.N. Chen X. Wang K. Enhancing electron diffusion length in narrow-bandgap perovskites for efficient monolithic perovskite tandem solar cells.Nat. Commun. 2019; 10: 1-9Crossref PubMed Scopus (67) Google Scholar, 10Wei M. Xiao K. Walters G. Lin R. Zhao Y. Saidaminov M.I. Todorovic P. Johnston A. Huang Z. Chen H. et al.Combining efficiency and stability in mixed tin-lead perovskite solar cells by capping grains with an ultrathin 2D layer.Adv. Mater. 2020; 32: 1907058Crossref Scopus (83) Google Scholar The perovskite composition for these high-efficiency Sn-Pb PSCs often has a Sn:Pb ratio of about 0.5:0.5 to 0.6:0.4. To reach the ideal band gap for single-junction devices, it is necessary to reduce the amount of Sn to about 25–30 mol% in hybrid perovskites.13Yang Z. Rajagopal A. Jen A.K.Y. Ideal bandgap organic–inorganic hybrid perovskite solar cells.Adv. Mater. 2017; 29: 1704418Crossref Scopus (104) Google Scholar, 14Zong Y. Wang N. Zhang L. Ju M.G. Zeng X.C. Sun X.W. Zhou Y. Padture N.P. Homogenous alloys of formamidinium lead triiodide and cesium tin triiodide for efficient ideal-bandgap perovskite solar cells.Angew. Chem. Int. Ed. 2017; 56: 12658-12662Crossref PubMed Scopus (54) Google Scholar, 15Prasanna R. Gold-Parker A. Leijtens T. Conings B. Babayigit A. Boyen H.-G. Toney M.F. McGehee M.D. Band gap tuning via lattice contraction and octahedral tilting in perovskite materials for photovoltaics.J. Am. Chem. Soc. 2017; 139: 11117-11124Crossref PubMed Scopus (332) Google Scholar Synthesis of these related compositions has been challenging, with uncontrolled structural and defect characteristics. Consequently, it has been very challenging to achieve simultaneously both high efficiency and good long-term operational stability for Sn-Pb-based ideal-band-gap PSCs.13Yang Z. Rajagopal A. Jen A.K.Y. Ideal bandgap organic–inorganic hybrid perovskite solar cells.Adv. Mater. 2017; 29: 1704418Crossref Scopus (104) Google Scholar,14Zong Y. Wang N. Zhang L. Ju M.G. Zeng X.C. Sun X.W. Zhou Y. Padture N.P. Homogenous alloys of formamidinium lead triiodide and cesium tin triiodide for efficient ideal-bandgap perovskite solar cells.Angew. Chem. Int. Ed. 2017; 56: 12658-12662Crossref PubMed Scopus (54) Google Scholar,16Li N. Zhu Z. Li J. Jen A.K.Y. Wang L. Inorganic CsPb1−xSnxIBr2 for efficient wide-bandgap perovskite solar cells.Adv. Energy Mater. 2018; 8: 1800525Crossref Scopus (139) Google Scholar,17Hu M. Chen M. Guo P. Zhou H. Deng J. Yao Y. Jiang Y. Gong J. Dai Z. Zhou Y. Sub-1.4 eV bandgap inorganic perovskite solar cells with long-term stability.Nat. Commun. 2020; 11: 1-10PubMed Google Scholar In this study, we show that, in addition to the quality of the Sn-Pb perovskite thin films, the residual stresses in the thin films appear to play an important role in influencing both device stability and efficiency. By utilizing a novel Sn-halide complex (SHC) additive, SnCl2⋅xFACl (x is optimized to 3; 5 mol% addition; FA, formamidinium), the residual stress is effectively reduced in the methylammonium (MA)-free, Cs-FA-based Sn-Pb halide perovskite thin film of composition (FAPbI3)0.7(CsSnI3)0.3 (or Cs0.3FA0.7Sn0.3Pb0.7I3) with a near-ideal band gap of ∼1.34 eV. It is found that SnCl2⋅3FACl enables the formation of a high-quality perovskite-substrate interface during the room-temperature solution precipitation stage. The use of this additive also reduces the defect density by 2 orders of magnitude, and it further improves the microstructures and optoelectronic properties of the thin films. Using this approach, we are able to push the PCE of the resulting PSCs to near 20%, which is the highest for MA-free Sn-Pb-based PSCs with a near-ideal band gap. More importantly, we demonstrate a promising operational stability of maintaining >80% of initial PCE over 750 h under continuous operation at about 45°C with maximum power point (MPP) tracking under one-sun-intensity illumination. Our study attests to the potential for the realization of stable and efficient Sn-Pb-based ideal-band-gap PSCs. For fabricating the PSC devices, we have chosen the MA-free perovskite composition of Cs0.3FA0.7Sn0.3Pb0.7I3, not only for its near ideal band gap (Figure S1) but also to avoid the deleterious effect of MA+ volatility that is widely hypothesized in limiting the long-term operational stability of PSCs.18Turren-Cruz S.-H. Hagfeldt A. Saliba M. Methylammonium-free, high-performance, and stable perovskite solar cells on a planar architecture.Science. 2018; 362: 449-453Crossref PubMed Scopus (541) Google Scholar Here we have used a new thin-film tailoring method for fabricating perovskite thin films in the presence of SnCl2⋅3FACl. Briefly, a stoichiometric perovskite precursor solution with 5 mol% excess SnCl2⋅3FACl additive was first spin-coated, followed by thermal annealing (see Supplemental information for details). PSCs were prepared with an “inverted” device structure comprising glass/indium tin oxide (ITO)/poly(3,4-ethylenedioxythiophene):polystyrene sulfonate (PEDOT:PSS)/perovskite/C60/bathocuproine (BCP)/Ag, which is typically used for high-efficiency Sn-Pb low-band-gap PSCs. The cross-sectional scanning electron microscopy (SEM) image of a standard device stack is shown in Figure S2. The typical current density-voltage (J-V) curves, and the corresponding external quantum efficiency (EQE) spectra, for PSCs made with and without the SnCl2⋅3FACl additive are shown in Figures 1A and 1B , respectively. The PSC made with SnCl2⋅3FACl additive shows a reverse-scan PCE of 18.3% with a short-circuit current density (JSC) of 29.1 mA/cm2, an open-circuit voltage (VOC) of 0.787 V, and a fill factor (FF) of 79.9%. The forward scan of the same device shows a PCE of 17.6%, and the stabilized power output (SPO; Figure 1A, inset) efficiency is 18.3%. For comparison, the PSC without the SnCl2⋅3FACl additive shows a PCE of 14.7% for reverse scan, 13.1% for forward scan, and SPO efficiency of 14.2%. The detailed device parameters are given in Table S1. The significant improvement in device performance is attributed largely to the higher VOC and FF. All these observations are statistically reproducible based on PV parameters from 20 devices for each device condition (Figure 1C). There is a negligible impact of the additive on JSC, which is consistent with the device EQE spectra (Figure 1B) and the optical absorption results (Figure S1). The slight increase in the band gap with additive is consistent with the blue shift of the onset of the EQE spectrum on the long wavelength side. However, there is also a slight change of the EQE spectra in the shorter wavelength range. These changes are relatively small with opposite effects, leading to comparable JSC values in the two devices. Note that the PSCs prepared with SnCl2⋅xFACl, with x ranging from 1.5 to 4.5, all showed enhanced device performance compared with the PSCs without the additive (Figure S3), but x = 3 gave the highest performance improvement. In addition, 5 mol% SnCl2⋅3FACl addition was found to be optimum (Figure S4). We observe that the large performance improvements are not achieved simply by using individual additive components, i.e., SnCl2 and FACl (Figure S5). To understand the origin of the significantly improved device performance associated with using the SnCl2⋅3FACl additive (5 mol%), a set of physical and optoelectronic properties of the thin films was characterized. The top-view SEM images in Figures 2A and 2B show that the apparent grain size is increased from a few hundred nanometers to about a micrometer with the additive. The statistical distribution of the apparent grain sizes of the pristine sample and that made with the SnCl2⋅3FACl additive are further compared in Figure S6. This can be attributed to the FACl component in the additive complex facilitating the grain-boundary migration during the thin film growth.1Dunlap-Shohl W.A. Zhou Y. Padture N.P. Mitzi D.B. Synthetic approaches for halide perovskite thin films.Chem. Rev. 2019; 119: 3193-3295Crossref PubMed Scopus (292) Google Scholar The indexed X-ray diffraction (XRD) patterns in Figure 2C confirm the pure perovskite phase in both thin films, with and without SnCl2⋅3FACl additive, but the thin film with the additive shows about 10-fold increase in the intensity on the same scale, and a 20% reduction in the FWHM (full width at half-maximum) of the characteristic 100 peak, which indicates improved crystallinity. Furthermore, the SnCl2⋅3FACl additive is also expected to passivate the grain boundaries by possibly decorating them with SnCl2 via a thermal decomposition process, similar to the effect of the SnF2⋅xFACl additive in a previous study.16Li N. Zhu Z. Li J. Jen A.K.Y. Wang L. Inorganic CsPb1−xSnxIBr2 for efficient wide-bandgap perovskite solar cells.Adv. Energy Mater. 2018; 8: 1800525Crossref Scopus (139) Google Scholar The grain-boundary passivated microstructure is confirmed using conductive atomic force microscopy (AFM) in Figure S7, where the grain boundaries are coated continuously with less conductive SnCl2 phases. The electrical measurement based on a field-effect transistor (FET) configuration shows that the “dark” carrier density of the thin films is reduced by 2 orders of magnitude with the use of the SnCl2⋅3FACl additive (Figures 2D–2F and S8), which is indicative of very low defect density in that thin film. These results correlate well with X-ray photoelectron spectroscopy (XPS) analysis of the corresponding perovskite thin films (Figure S9 and Table S2). The improved microstructure and crystallinity of perovskite thin films are often key factors that contribute to enhancing PSC performance. In this context, we also prepared another set of Cs0.3FA0.7Sn0.3Pb0.7I3 perovskite thin films and devices using a different additive, SnF2⋅3FACl (5 mol%). In comparison with the pristine thin films without any additive, the use of SnF2⋅xFACl additive also enhanced the crystallinity (Figure S10) and apparent grain size significantly, to a degree similar to what is observed in the SnCl2⋅3FACl additive case. The optical band gap and absorption are also similar for thin films prepared using SnCl2⋅3FACl and SnF2⋅3FACl additives (Figure S1). However, the SnF2⋅3FACl-additive-based thin film exhibited about 3-fold higher dark carrier density (Figure S8), compared with the SnCl2⋅3FACl-additive-based thin film. Furthermore, the SnF2⋅3FACl-additive-based device reached an SPO efficiency of only 16.6% (Figure S11 and Table S1), which is higher than the pristine device (14.2%, Figure 1A), but significantly lower than the SnCl2⋅3FACl-augmented device (18.3%, Figure 1A). These results indicate that improved morphology and crystallinity are not sufficient to account for the significant improvements in the PSC performance observed using the SnCl2⋅3FACl additive. Recently, stress (or strain) has been shown to affect the charge transport and chemical stability of perovskite thin films.19Xue D.-J. Hou Y. Liu S.-C. Wei M. Chen B. Huang Z. Li Z. Sun B. Proppe A.H. Dong Y. Regulating strain in perovskite thin films through charge-transport layers.Nat. Commun. 2020; 11: 1-8Crossref PubMed Scopus (125) Google Scholar, 20Zhu C. Niu X. Fu Y. Li N. Hu C. Chen Y. He X. Na G. Liu P. Zai H. Strain engineering in perovskite solar cells and its impacts on carrier dynamics.Nat. Commun. 2019; 10: 1-11PubMed Google Scholar, 21Zhao J. Deng Y. Wei H. Zheng X. Yu Z. Shao Y. Shield J.E. Huang J. Strained hybrid perovskite thin films and their impact on the intrinsic stability of perovskite solar cells.Sci. Adv. 2017; 3: eaao5616Crossref PubMed Scopus (367) Google Scholar, 22Jones T.W. Osherov A. Alsari M. Sponseller M. Duck B.C. Jung Y.-K. Settens C. Niroui F. Brenes R. Stan C.V. Lattice strain causes non-radiative losses in halide perovskites.Energy Environ. Sci. 2019; 12: 596-606Crossref Google Scholar, 23Chen Y. Lei Y. Li Y. Yu Y. Cai J. Chiu M.-H. Rao R. Gu Y. Wang C. Choi W. Strain engineering and epitaxial stabilization of halide perovskites.Nature. 2020; 577: 209-215Crossref PubMed Scopus (199) Google Scholar, 24Jariwala S. Sun H. Adhyaksa G.W.P. Lof A. Muscarella L.A. Ehrler B. Garnett E.C. Ginger D.S. Local crystal misorientation influences non-radiative recombination in halide perovskites.Joule. 2019; 3: 3048-3060Abstract Full Text Full Text PDF Scopus (89) Google Scholar Usually there is a biaxial tensile residual stress in solution-processed perovskite thin films, associated with the film formation process and the coefficient of thermal expansion (CTE) mismatch between the thin film and the substrate. In general, such strain is detrimental to PSC operation as it can increase charge recombination, decrease perovskite stability, and reduce mechanical reliability.21Zhao J. Deng Y. Wei H. Zheng X. Yu Z. Shao Y. Shield J.E. Huang J. Strained hybrid perovskite thin films and their impact on the intrinsic stability of perovskite solar cells.Sci. Adv. 2017; 3: eaao5616Crossref PubMed Scopus (367) Google Scholar,22Jones T.W. Osherov A. Alsari M. Sponseller M. Duck B.C. Jung Y.-K. Settens C. Niroui F. Brenes R. Stan C.V. Lattice strain causes non-radiative losses in halide perovskites.Energy Environ. Sci. 2019; 12: 596-606Crossref Google Scholar,25Ramirez C. Yadavalli S.K. Garces H.F. Zhou Y. Padture N.P. Thermo-mechanical behavior of organic-inorganic halide perovskites for solar cells.Script. Mater. 2018; 150: 36-41Crossref Scopus (50) Google Scholar To evaluate the possible effect of residual stress, Cs0.3FA0.7Sn0.3Pb0.7I3 perovskite thin films without additive (pristine) and with the SnCl2⋅3FACl additive (5 mol%) were characterized using the well-established XRD sin2ψ method, as illustrated in Figure S12.25Ramirez C. Yadavalli S.K. Garces H.F. Zhou Y. Padture N.P. Thermo-mechanical behavior of organic-inorganic halide perovskites for solar cells.Script. Mater. 2018; 150: 36-41Crossref Scopus (50) Google Scholar,26Yadavalli S.K. Dai Z. Zhou H. Zhou Y. Padture N.P. Facile healing of cracks in organic–inorganic halide perovskite thin films.Acta Mater. 2020; 187: 112-121Crossref Scopus (21) Google Scholar In Figure 3A and 3B, (220) interplanar spacing (d220) is plotted as a function of sin2ψ for Cs0.3FA0.7Sn0.3Pb0.7I3 perovskite thin films without and with the SnCl2⋅3FACl additive, respectively. A positive slope of the linear fit to the d220-sin2ψ data for the pristine thin film indicates the presence of biaxial tensile residual stress (or strain). In contrast, the thin film with the SnCl2⋅3FACl additive shows a much lower positive slope. The biaxial residual stress (σR) can be estimated from the sin2ψ data in Figures 3A and 3B using the following relation:27M. Birkholz. Thin Film Analysis by X-Ray Scattering. Berlin: Wiley-VCH.Google ScholarσR=(E<220>1+ν)(mdn),(Equation 1) where m is the slope of the linear fit to the data, dn is the d220 spacing at sin2ψ = 0 (y intercept), E<220> is Young's modulus in the <220> direction, and ν is the Poisson ratio. E<220> is estimated as 18.5 GPa, as shown in the Supplemental information. The typical ν value of 0.33 is assumed.28Feng J. Mechanical properties of hybrid organic-inorganic CH3NH3BX3 (B= Sn, Pb; X= Br, I) perovskites for solar cell absorbers.APL Mater. 2014; 2: 081801Crossref Scopus (230) Google Scholar The calculated residual stresses for Cs0.3FA0.7Sn0.3Pb0.7I3 perovskite thin films with and without SnCl2⋅3FACl additive are presented in Figure 3C. Overall, the tensile σR in the pristine thin film is estimated at 42–51 MPa, which is reduced to only 19–24 MPa for the SnCl2⋅3FACl case. In addition, it was found that replacing the SnCl2⋅3FACl additive with SnF2⋅3FACl can also reduce the residual stress to some extent, but not as much as SnCl2⋅3FACl (Figure S13). Based on these results, the reduced residual stresses in the thin film associated with the SnCl2⋅3FACl additive are likely to be an important factor contributing to the exceptional device performance, in addition to the microstructural considerations. To elucidate the possible mechanisms responsible for the low residual stresses due to the SnCl2⋅3FACl additive, mechanical delamination tests were performed, as described elsewhere.29Dai Z. Yadavalli S.K. Hu M. Chen M. Zhou Y. Padture N.P. Effect of grain size on the fracture behavior of organic-inorganic halide perovskite thin films for solar cells.Script. Mater. 2020; 185: 47-50Crossref Scopus (16) Google Scholar,30Rolston N. Printz A.D. Tracy J.M. Weerasinghe H.C. Vak D. Haur L.J. Priyadarshi A. Mathews N. Slotcavage D.J. McGehee M.D. et al.Effect of cation composition on the mechanical stability of perovskite solar cells.Adv. Energy Mater. 2018; 8: 1702116Crossref Scopus (81) Google Scholar Residual stresses in perovskite thin films typically develop when the perovskite phase crystallizes from the as-spun thin film during the thermal annealing process, where the perovskite thin film attaches to the substrate at high temperatures and is subsequently cooled down. The significantly higher CTE of perovskite compared with that of glass results in the tensile nature of the residual stresses in the thin film after cooling (Figure 3D).25Ramirez C. Yadavalli S.K. Garces H.F. Zhou Y. Padture N.P. Thermo-mechanical behavior of organic-inorganic halide perovskites for solar cells.Script. Mater. 2018; 150: 36-41Crossref Scopus (50) Google Scholar,31Rolston N. Bush K.A. Printz A.D. Gold-Parker A. Ding Y. Toney M.F. McGehee M.D. Dauskardt R.H. Engineering stress in perovskite solar cells to improve stability.Adv. Energy Mater. 2018; 8: 1802139Crossref Scopus (133) Google Scholar We find that the pure perovskite phase has formed immediately after the spin-coating process (without heat treatment) for both cases, without and with the SnCl2⋅3FACl additive (Figure S14). However, the structural integrity of the perovskite-substrate interface during this stage is strikingly different. In the additive-free case, it is found that the delamination occurs primarily at the perovskite-substrate interface, whereas the thin film fractures within the bulk thin film in the SnCl2⋅3FACl additive case (see Figure S15). This indicates that the SnCl2⋅3FACl additive promotes the bonding between the perovskite thin film and the substrate before heating, reducing the development of residual stresses upon the subsequent heating-cooling cycle. This proposed mechanism is illustrated schematically in Figure 3D. Figure 4A shows the J-V curves of the champion device based on Cs0.3FA0.7Sn0.3Pb0.7I3 using the additive SnCl2⋅xFACl (x = 3). It shows a PCE of 19.3% in reverse scan and 18.5% in forward scan, and SPO efficiency of 19.1% (Figure 4A, inset). The detailed PV parameters are shown in Table S3. The EQE spectra along with the integrated JSC are shown in Figure 4B, where the latter is 28.3 mA/cm2, which is within 3%–4% of the J-V measurement. To test the impact of the SnCl2⋅3FACl additive on the device stability, we evaluated the continuous operation (Figure 4C) with MPP tracking of unencapsulated Cs0.3FA0.7Sn0.3Pb0.7I3 devices under one-sun-intensity illumination, in nitrogen atmosphere, with ambient temperature of about 45°C (ISOS-L-1 stability32Khenkin M.V. Katz E.A. Abate A. Bardizza G. Berry J.J. Brabec C. Brunetti F. Bulović V. Burlingame Q. Di Carlo A. et al.Consensus statement for stability assessment and reporting for perovskite photovoltaics based on ISOS procedures.Nat. Energy. 2020; 5: 35-49Crossref Scopus (344) Google Scholar). In comparison to the pristine and SnF2⋅3FACl-additive-based devices (Figure S16), the SnCl2⋅3FACl-additive-based device showed much improved operational stability with a T80 ∼750 h, where T80 refers to the duration at 80% retention of the initial PCE. This continuous one sun operation stability is approaching that for PSCs based on Pb only, and it represents the best reported operational stability for high-efficiency (19%–20%) Sn-Pb ideal-band-gap PSCs. It is worth pointing out that the T90 is ∼690 h in Figure 4C. The observed increasing degradation rate starting at ∼650 h is likely associated with the use of the Ag contact, which can migrate to induce degradation.33Prasanna R. Leijtens T. Dunfield S.P. Raiford J.A. Wolf E.J. Swifter S.A. Werner J. Eperon G.E. de Paula C. Palmstrom A.F. et al.Design of low bandgap tin–lead halide perovskite solar cells to achieve thermal, atmospheric and operational stability.Nat. Energy. 2019; 4: 939-947Crossref Scopus (141) Google Scholar PEDOT:PSS is also known to cause device degradation in Sn-Pb PSCs.33Prasanna R. Leijtens T. Dunfield S.P. Raiford J.A. Wolf E.J. Swifter S.A. Werner J. Eperon G.E. de Paula C. Palmstrom A.F. et al.Design of low bandgap tin–lead halide perovskite solar cells to achieve thermal, atmospheric and operational stability.Nat. Energy. 2019; 4: 939-947Crossref Scopus (141) Google Scholar These contact layer optimization strategies are expected to further increase the device stability in the future. Finally, we further tested the effect of tuning the Cs content on device performance. By reducing the Cs content to 10%, the resulting perovskite Cs0.1FA0.9Sn0.3Pb0.7I3 has a band gap of ∼1.32 eV (Figure S17), which is also near the ideal band gap (∼1.34 eV) for single-junction devices. Figure S18 shows the device characteristics of the champion Cs0.1FA0.9Sn0.3Pb0.7I3 device by using the SnCl2⋅3FACl additive (5 mol%). The device efficiency is slightly higher (SPO = 19.5%; Table S3), but the stability is slightly reduced to a T80 of ∼430 h, compared with Cs0.3FA0.7Sn0.3Pb0.7I3 devices. Future work is needed to further examine the impact of varying Cs amount on the efficiency and stability of Sn-Pb-based ideal-band-gap PSCs. Our results suggest that adjusting/improving the Sn-Pb perovskite absorber layer is critical for enhancing Sn-Pb perovskite device efficiency and stability. Our results also indicate the importance of not only improving the thin-film microstructure but also reducing residual stresses in Sn-Pb thin films when developing Sn-Pb-based ideal-band-gap PSCs having simultaneous high PCE and long operational stability. In summary, we have demonstrated high-performance ideal-band-gap MA-free PSCs with high PCEs near 20% and long operational stability (T80 ∼750 h). The key to this development is the use of a complex additive of SnCl2⋅3FACl in the Sn-Pb perovskite thin film processing, which leads to not only better film crystallinity and favorable grain-boundary structures, but also significantly lower undesirable residual stresses in the resulting thin films. Additives with various concentrations and halide types (SnX2⋅FACl, X = Cl, F) are shown to profoundly influence residual stresses and the corresponding solar cell performance, which could be due to the physical effects (film microstructure, crystallinity, etc.) and the hydrogen bond strength brought forth by the halide contents distributed within perovskite thin films or at the interfaces with charge transport layers. Therefore, careful selection and tailoring of the SHC additive is the key to successful optimization of perovskite film growth that is a feature of additive-induced interfacial bonding to the substrate layer, during the early stage of film crystallization. Apart from the scope of solar cell application, this SHC additive, which is potentially versatile in composition, can also be tailored to meet specific requirements for resulting thin film properties and used for improving the performance of other electronic devices, such as light-emitting devices and photodetectors.