IEEE Transactions on Terahertz Science and Technology
Curwen, Christopher A.; Kawamura, Jonathan H.; Hayton, Darren J.; Addamane, Sadhvikas J.; Reno, John L.; Williams, Benjamin S.; Karasik, Boris S.
We report high-resolution frequency study and phase locking have been performed on a terahertz (THz) quantum-cascade vertical-external-cavity surface-emitting laser (QC-VECSEL) operating around 2.5 THz. A subharmonic diode mixer is used to down convert the THz signal to a 100 MHz intermediate frequency that is phase locked to a stable 100 MHz microwave reference. Between 90% and 95% of the QC-VECSEL signal is locked within 2 Hz of the multiplied RF reference, and amplitude fluctuations on the order of 1%–10% are observed, depending on the bias point of the QC-VECSEL. The bandwidth of the locking loop is ~1 MHz. Many noise peaks in the IF signal are observed, likely corresponding to mechanical resonances in the 10 Hz–10 kHz. These peaks are generally -30 to -60 dB below the main tone and are below the phase noise level of the multiplied RF reference that ultimately limits the phase noise of the locked QC-VECSEL.
Zheng, Jeffrey X.; Fiagbenu, Merrilyn M.A.; Esteves, Giovanni E.; Musavigharavi, Pariasadat; Jariwala, Deep; Stach, Eric A.; Olsson, Roy H.
Ferroelectric Al1−xScxN has raised much interest in recent years due to its unique ferroelectric properties and complementary metal oxide semiconductor back-end-of-line compatible processing temperatures. Potential applications in embedded nonvolatile memory, however, require ferroelectric materials to switch at relatively low voltages. One approach to achieving a lower switching voltage is to significantly reduce the Al1−xScxN thickness. In this work, ferroelectric behavior in 5-27 nm films of sputter deposited Al0.72Sc0.28N has been studied. We find that the 10 kHz normalized coercive field increases from 4.4 to 7.3 MV/cm when reducing the film thickness from 27.1 to 5.4 nm, while over the same thickness range, the characteristic breakdown field of a 12.5 μm radius capacitor increases from 8.3 to 12.1 MV/cm. The 5.4 nm film demonstrates ferroelectric switching at 5.5 V when excited with a 500 ns pulse and a switching speed of 60 ns.
Cerjan, Alexander W.; Loring, Terry A.; Cheng, Wenting; Ying Chen, Ssu; Prodan, Camelia; Prodan, Emil
Topological metals are conducting materials with gapless band structures and nontrivial edge-localized resonances. Their discovery has proven elusive because traditional topological classification methods require band gaps to define topological robustness. Inspired by recent theoretical developments that leverage techniques from the field of C*-algebras to identify topological metals, here, we directly observe topological phenomena in gapless acoustic crystals and realize a general experimental technique to demonstrate their topology. Specifically, we not only observe robust boundary-localized states in a topological acoustic metal, but also re-interpret a composite operator—mathematically derived from the K-theory of the problem—as a new Hamiltonian whose physical implementation allows us to directly observe a topological spectral flow and measure the topological invariants. Our observations and experimental protocols may offer insights for discovering topological behaviour across a wide array of artificial and natural materials that lack bulk band gaps.
This report updates the high-level test plan for evaluating surface deposition on three commercial 32PTH2 spent nuclear fuel (SNF) canisters inside NUTECH Horizontal Modular Storage (NUHOMS) Advanced Horizontal Storage Modules (AHSMs) from Orano (formerly Transnuclear Inc.) and provides a summary of the surface sampling activities that have been conducted to date. The details contained in this report represent the best designs and approaches explored for testing as of this publication. Given the rapidly developing nature of this test program, some of these plans may change to accommodate new objectives or requirements. One goal of this testing is to collect defensible and detailed dust deposition measurements from the surface of dry storage canisters in a marine coastal environment to guide chloride-induced stress corrosion cracking (CISCC) research. Another goal is to provide data for the validation of computational fluid dynamics (CFD) based deposition modeling. To facilitate surface sampling, the otherwise highly prototypic dry storage systems will not contain SNF but rather will be electrically heated to mimic the decay heat and thermal hydraulic environment. Test and heater design is supported by detailed CFD modeling. Instrumentation throughout the canister, storage module, and environment will provide extensive information about the thermal-hydraulic behavior of horizontal dry cask storage systems. Manual sampling over a comprehensive portion of the canister surface at regular time intervals will offer detailed quantification and composition of the deposited particulates from a realistic storage environment. Discussions of a potential host site for the Canister Deposition Field Demonstration (CDFD) are ongoing. Until a host site is chosen, testing of key CDFD hardware components including the heater assemblies, power skid, and remote data acquisition system will continue. Functional testing of the finalized heater assemblies and test apparatus started this fiscal year. These initial heater tests have shown the assemblies are performing within design specifications. Staged surface sampling of a mockup of a canister outside the AHSM on a transfer skid was also performed. Refinements to the sampling procedures and techniques were captured from observation of these activities and lessons-learned debriefs. These updated sampling procedures and techniques are planned to be tested again in the field using the mockup in order to assure personnel are using the most accurate and repeatable methods possible prior to deployment for actual CDFD testing.
Here this paper introduces a publicly available PyTorch-ABAQUS deep-learning framework of a family of plasticity models where the yield surface is implicitly represented by a scalar-valued function. In particular, our focus is to introduce a practical framework that can be deployed for engineering analysis that employs a user-defined material subroutine (UMAT/VUMAT) for ABAQUS, which is written in FORTRAN. To accomplish this task while leveraging the back-propagation learning algorithm to speed up the neural-network training, we introduce an interface code where the weights and biases of the trained neural networks obtained via the PyTorch library can be automatically converted into a generic FORTRAN code that can be a part of the UMAT/VUMAT algorithm. To enable third-party validation, we purposely make all the data sets, source code used to train the neural-network-based constitutive models, and the trained models available in a public repository. Furthermore, the practicality of the workflow is then further tested on a dataset for anisotropic yield function to showcase the extensibility of the proposed framework. A number of representative numerical experiments are used to examine the accuracy, robustness and reproducibility of the results generated by the neural network models.
A significant amount of uncertainty exists regarding potential human exposure to laboratory biomaterials and organisms in Biosafety Level 2 (BSL-2) research laboratories. Computational fluid dynamics (CFD) modeling is proposed as a way to better understand potential impacts of different combinations of biomaterials, laboratory manipulations, and exposure routes on risks to laboratory workers. Here, in this study, we use CFD models to simulate airborne concentrations of contaminants in an actual BSL-2 laboratory under different configurations. Results show that ventilation configuration, sampling location, and contaminant source location can significantly impact airborne concentrations and exposures. Depending on the source location and airflow patterns, the transient and time-integrated concentrations varied by several orders of magnitude. Contaminant plumes from sources located near a return vent (or exhaust like a fume hood or ventilated biosafety cabinet) are likely to be more contained than sources that are further from the exhaust. Having a direct flow between the source and the exhaust (through-flow condition) may reduce potential exposures to individuals outside the air flow path. Designing a BSL-2 room with ventilation and airflow patterns that maximize through-flow conditions to the return/exhaust vents and minimize dispersion and mixing throughout the room is, therefore, recommended. CFD simulations can also be used to assist in characterizing the impacts of supply and return vent locations, room layout, and source locations on spatial and temporal contaminant concentrations. In addition, proper placement of particle sensors can also be informed by CFD simulations to provide additional characterization and monitoring of potential exposures in BSL-2 facilities.
Pb-Zr-Ti-O (PZT) perovskites span a large solid-solution range and have found widespread use due to their piezoelectric and ferroelectric properties that also span a large range. Crystal structure analysis via Rietveld refinement facilitates materials analysis via the extraction of the structural parameters. These parameters, often obtained as a function of an additional dimension (e.g., pressure), can help to diagnose materials response within a use environment. Often referred to as in-situ studies, these experiments provide an abundance of data. Viewing structural changes due to applied pressure conditions can give much-needed insight into materials performance. However, challenges exist for viewing/presenting results when the details are inherently three-dimensional (3D) in nature. For PZT perovskites, the use of polyhedra (e.g., Zr/Ti-O6 octahedra) to view bonding/connectivity is beneficial; however, the visualization of the octahedra behavior with pressure dependence is less easily demonstrated due to the complexity of the added pressure dimension. We present a more intuitive visualization by projecting structural data into virtual reality (VR). We employ previously published structural data for Pb0.99(Zr0.95Ti0.05)0.98Nb0.02O3 as an exemplar for VR visualization of the PZT R3c crystal structure between ambient and 0.62 GPa pressure. This is accomplished via our in-house CAD2VR™ software platform and the new CrystalVR plugin. The use of the VR environment enables a more intuitive viewing experience, while enabling on-the-fly evaluation of crystal data, to form a detailed and comprehensive understanding of in-situ datasets. Discussion of methodology and tools for viewing are given, along with how recording results in video form can enable the viewing experience.
Diffusion properties of bulk fluids have been predicted using empirical expressions and machine learning (ML) models, suggesting that predictions of diffusion also should be possible for fluids in confined environments. The ability to quickly and accurately predict diffusion in porous materials would enable new discoveries and spur development in relevant technologies such as separations, catalysis, batteries, and subsurface applications. Here in this work, we apply artificial neural network (ANN) models to predict the simulated self-diffusion coefficients of real liquids in both bulk and pore environments. The training data sets were generated from molecular dynamics (MD) simulations of Lennard-Jones particles representing a diverse set of 14 molecules ranging from ammonia to dodecane over a range of liquid pressures and temperatures. Planar, cylindrical, and hexagonal pore models consisted of walls composed of carbon atoms. Our simple model for these liquids was primarily used to generate ANN training data, but the simulated self-diffusion coefficients of bulk liquids show excellent agreement with experimental diffusion coefficients. ANN models based on simple descriptors accurately reproduced the MD diffusion data for both bulk and confined liquids, including the trend of increased mobility in large pores relative to the corresponding bulk liquid.
In this study, we explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation, we use the Hamilton–Jacobi–Bellman equation. It is well known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high-dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.
Chong, Lina; Gao, Guoping; Wen, Jianguo; Li, Haixia; Xu, Haiping; Green, Zach; Sugar, Joshua D.; Kropf, A.J.; Xu, Wenqian; Lin, Xiao M.; Xu, Hui; Wang, Lin W.; Di Liu, Jia
Discovery of earth-abundant electrocatalysts to replace iridium for the oxygen evolution reaction (OER) in a proton exchange membrane water electrolyzer (PEMWE) represents a critical step in reducing the cost for green hydrogen production. We report a nanofibrous cobalt spinel catalyst codoped with lanthanum (La) and manganese (Mn) prepared from a zeolitic imidazolate framework embedded in electrospun polymer fiber. The catalyst demonstrated a low overpotential of 353 millivolts at 10 milliamperes per square centimeter and a low degradation for OER over 360 hours in acidic electrolyte. A PEMWE containing this catalyst at the anode demonstrated a current density of 2000 milliamperes per square centimeter at 2.47 volts (Nafion 115 membrane) or 4000 milliamperes per square centimeter at 3.00 volts (Nafion 212 membrane) and low degradation in an accelerated stress test.
Oriented attachment (OA) of nanoparticles is an important crystal growth pathway in the synthesis of hierarchical structures. Although a significant understanding of OA has been made, the effect of atomistic misalignment and the roles of solvent/particle and particle/particle interactions on the structure-energy relationship during an OA remain elusive. In this study, we perform molecular dynamics simulations to calculate the potential of mean force (PMF) profile for gibbsite particle translation on a gibbsite slab with 1 or 2 intervening water layers (1W or 2W). The structures of the gibbsite surfaces and the confined water are analyzed to determine how the number and type of hydrogen bonds (H-bonds) influence the free energy profile during the translation. The PMF profile exhibits a periodicity of length 5.078 Å, consistent with the gibbsite unit cell size along the translation direction. The changes in the surface-water and water-water hydrogen bond network and water and surface OH groups’ orientations during the translation are strongly coupled with the changes in the PMF profile in the 1W case. However, when increasing the number of intervening water layers from 1W to 2W, the particle/slab misalignment becomes a dominant factor controlling the behavior of the PMF profile. We also establish a method to quantify misalignment between the particle and the slab, which exhibits a strong correlation with the free energy for the 2W case. These results shed more light into the roles of particle/slab misalignment and hydrogen bond network in the OA of mineral particles in aqueous solution.
The efficient utilization of lignin, the direct source of renewable aromatics, into value-added renewable chemicals is an important step towards sustainable biorefinery practices. Nevertheless, owing to the random heterogeneous structure and limited solubility, lignin utilization has been primarily limited to burning for energy. The catalytic depolymerization of lignin has been proposed and demonstrated as a viable route to sustainable biorefinery, however, low yields and poor selectivity of products, high char formation, and limited to no recycling of transition-metal-based catalyst involved in lignin depolymerization demands attention to enable practical-scale lignocellulosic biorefineries. In this study, we demonstrate the catalytic depolymerization of ionic liquid-based biorefinery poplar lignin into guaiacols over a reusable zirconium phosphate supported palladium catalyst. The essence of the study lies in the high conversion (>80 %), minimum char formation (7–16 %), high yields of guaiacols (up to 200 mg / g of lignin), and catalyst reusability. Both solid residue, liquid stream, and gaseous products were thoroughly characterized using ICP-OES, PXRD, CHN analysis, GC-MS, GPC, and 2D NMR to understand the hydrogenolysis pathway.
Using first-principles density functional theory (DFT) methods and size-converged supercell models, we analyze the electronic and atomic structure of magnetic $3d$ transition metal dopants in cubic gallium nitride (c-GaN). All stable defect charge states for Fermi levels across the full experimental gap are computed using a method that correctly resolves the boundary condition problem (without a jellium approximation) and eliminates finite-size errors. The resulting computed defect levels are not impacted by the DFT band-gap problem, they span a width consistent with the experimental gap rather than being limited to the single-particle DFT gap. All defects with electronically degenerate (half-metal) $T$d ground states are found to have significant distortions, relaxing to $D$2d structures driven by the Jahn-Teller instability. This leads to insulating ground states for all substitutional $3d$ dopants, refuting claims in the literature that +$U$ or hybrid functional methods are required to avoid artificial half-metal results. Interpreting the $d$n atomic occupations within a crystal-field model and exchange splittings, we identify a systematic trend across the $3d$ transition metal series. Approaches to estimate excited-state energies as observed in photoluminescence from defect centers are assessed, ranging from a Koopmans-type single-particle energy interpretation to relaxed total energy differences in fully self-consistent DFT. The single-particle interpretations are found to be qualitatively predictive and the calculations are consistent with the limited available experimental data across the $3$d dopant series. These results provide a baseline understanding to guide future studies and a conceptual framework within which to interpret new results.
Balogun, Shuaib A.; Lively, Ryan P.; Losego, Mark D.; Ren, Yi; Steiner, Adam M.
Vapor phase infiltration (VPI) is a post-polymerization modification technique that infuses inorganics into polymers to create organic–inorganic hybrid materials with new properties. Much is yet to be understood about the chemical kinetics underlying the VPI process. The aim of this study is to create a greater understanding of the process kinetics that govern the infiltration of trimethyl aluminum (TMA) and TiCl4 into PMMA to form inorganic-PMMA hybrid materials. To gain insight, this paper initially examines the predicted results for the spatiotemporal concentrations of inorganics computed from a recently posited reaction–diffusion model for VPI. This model provides insight on how the Damköhler number (reaction versus diffusion rates) and non-Fickian diffusional processes (hindering) that result from the material transforming from a polymer to a hybrid can affect the evolution of inorganic concentration depth profiles with time. Subsequently, experimental XPS depth profiles are collected for TMA and TiCl4 infiltrated PMMA films at 90 °C and 135 °C. The functional behavior of these depth profiles at varying infiltration times are qualitatively compared to various computed predictions and conclusions are drawn about the mechanisms of each of these processes. TMA infiltration into PMMA appears to transition from a diffusion-limited process at low temperatures (90 °C) to a reaction-limited process at high temperatures (135 °C) for the film thicknesses investigated here (200 nm). While TMA appears to fully infiltrate these 200 nm PMMA films within a few hours, TiCl4 infiltration into PMMA is considerably slower, with full saturation not occurring even after 2 days of precursor exposure. Infiltration at 90 °C is so slow that no clear conclusions about mechanism can be drawn; however, at 135 °C, the TiCl4 infiltration into PMMA is clearly a reaction-limited process, with TiCl4 permeating the entire thickness (at low concentrations) within only a few minutes, but inorganic loading continuously increasing in a uniform manner over a course of 2 days. Near-surface deviations from the uniform-loading expected for a reaction-limited process also suggest that diffusional hindering is high for TiCl4 infiltration into PMMA. In conclusion, these results demonstrate a new, ex situ analysis approach for investigating the rate-limiting process mechanisms for vapor phase infiltration.
Polymorphism and phase transitions in sodium diuranate, Na2U2O7, are investigated with density functional perturbation theory (DFPT). Thermal properties of crystalline α-, β- and γ-Na2U2O7 polymorphs are predicted from DFPT phonon calculations, i.e., the first time for the high-temperature γ-Na2U2O7 phase (R3̄m symmetry). The standard molar isochoric heat capacities predicted within the quasi-harmonic approximation are for P21/a α-Na2U2O7 and C2/m β-Na2U2O7, respectively. Gibbs free energy calculations reveal that α-Na2U2O7 (P21/a) and β-Na2U2O7 (C2/m) are almost energetically degenerate at low temperature, with β-Na2U2O7 becoming slightly more stable than α-Na2U2O7 as temperature increases. These findings are consistent with XRD data showing a mixture of α and β phases after cooling of γ-Na2U2O7 to room temperature and the observation of a sluggish α → β phase transition above ca. 600 K. A recently observed α-Na2U2O7 structure with P21 symmetry is also shown to be metastable at low temperature. Based on Gibbs free energy, no direct β → γ solid-solid phase transition is predicted at high temperature, although some experiments reported the existence of such phase transition around 1348 K. This, along with recent experiments, suggests the occurrence of a multi-step process consisting of initial β-phase decomposition, followed by recrystallization into γ-phase as temperature increases.
Wexler, Robert B.; Sai Gautam, Gopalakrishnan; Bell, Robert T.; Shulda, Sarah; Strange, Nicholas A.; Trindell, Jamie T.; Sugar, Joshua D.; Nygren, Eli; Sainio, Sami; McDaniel, Anthony H.; Ginley, David; Carter, Emily A.; Stechel, Ellen B.
Modeling-driven design of redox-active off-stoichiometric oxides for solar thermochemical H2 production (STCH) seldom has resulted in empirical demonstration of competitive materials. We report the theoretical prediction and experimental evidence that the perovskite Ca2/3Ce1/3Ti1/3Mn2/3O3 is synthesizable with high phase purity, stable, and has desirable redox thermodynamics for STCH, with a predicted average neutral oxygen vacancy (VO) formation energy, Ev = 3.30 eV. Flow reactor experiments suggest potentially comparable or greater H2 production capacity than recent promising Sr-La-Mn-Al and Ba-Ce-Mn metal oxide perovskites. Utilizing quantum-based modeling of a solid solution on both A and B sub-lattices, we predict the impact of nearest-neighbor composition on Ev and determine that A-site Ce4+ reduction dominates the redox-activity of Ca2/3Ce1/3Ti1/3Mn2/3O3. X-ray absorption spectroscopy measurements provide evidence that supports these predictions and reversible Ce4+-to-Ce3+ reduction. Our models predict that Ce4+ reduces even when it is not nearest-neighbor to the VO, suggesting that refinement of Ce stoichiometry has the possibility of further enhancing performance.
The propagation of self-sustained formation reactions in sputter-deposited Co/Al multilayers is known to exhibit a design-dependent instability. Multilayers having thin bilayers (<55 nm period) exhibit stable propagating waves, whereas those with a larger period react unstably. The specific two-dimensional (2D) instability observed involves the transverse propagation of a band in front of a stalled front commonly referred to as a “spin band.” Previous finite-element studies have shown that these instabilities are thermodynamically driven by the forward conduction of heat away from the flame front. However, the magnitude of that loss is inherently tied to the bilayer design in traditional bimetallic multilayers, which couples any proposed stability criteria to a varying critical diffusion distance. This work utilizes a recently developed class of materials known as “inert-mediated reactive multilayers” to decouple the thermodynamic and kinetic contributions to propagating wave stability by reducing the stored chemical energy density in normally stable bilayer designs. By depositing an inert product phase (B2-CoAl) within the mid-plane of Co and Al reactant layers, spin instabilities arise as a function of both diluted volume and critical diffusion distance. From there, a stability criterion is determined for Co/Al multilayers based on enthalpy loss from the reaction zone, and its physical significance is explored.
Machine learning is on a bit of a tear right now, with advances that are infiltrating nearly every aspect of our lives. In the domain of materials science, this wave seems to be growing into a tsunami. Yet, there are still real hurdles that we face to maximize its benefit. This Matter of Opinion, crafted as a result of a workshop hosted by researchers at Sandia National Laboratories and attended by a cadre of luminaries, briefly summarizes our perspective on these barriers. By recognizing these problems in a community forum, we can share the burden of their resolution together with a common purpose and coordinated effort.
Measured salt compositions in dust collected over roughly the last decade from surfaces of in-service stainless-steel alloys at four locations around the United States are presented, along with the predicted brine compositions that would result from deliquescence of these salts. The salt compositions vary greatly from ASTM seawater and from laboratory salts (i.e., NaCl or MgCl2) commonly used on corrosion testing. The salts contained relatively high amounts of sulfates and nitrates, evolved to basic pH values, and exhibited deliquescence relative humidity values (RH) higher than seawater. Additionally, inert dust in components were quantified and considerations for laboratory testing are presented. The observed dust compositions are discussed in terms of the potential corrosion behavior and are compared to commonly used accelerated testing protocols. Finally, ambient weather conditions and their influence on diurnal fluctuations in temperature (T) and RH on heated metal surfaces are evaluated and a relevant diurnal cycle for laboratory testing a heated surface has been developed. Suggestions for future accelerated tests are proposed that include exploration of the effects of inert dust particles on atmospheric corrosion, chemistry considerations, and realistic diurnal fluctuations in T and RH. Understanding mechanisms in both realistic and accelerated environments will allow development of a corrosion factor (i.e., scaling factor) for the extrapolation of laboratory-scale test results to real world applications.
The burgeoning field of materials informatics necessitates a focus on educating the next generation of materials scientists in the concepts of data science, artificial intelligence (AI), and machine learning (ML). In addition to incorporating these topics in undergraduate and graduate curricula, regular hands-on workshops present the most effective medium to initiate researchers to informatics and have them start applying the best AI/ML tools to their own research. With the help of the Materials Research Society (MRS), members of the MRS AI Staging Committee, and a dedicated team of instructors, we successfully conducted workshops covering the essential concepts of AI/ML as applied to materials data, at both the Spring and Fall Meetings in 2022, with plans to make this a regular feature in future meetings. Here, in this article, we discuss the importance of materials informatics education via the lens of these workshops, including details such as learning and implementing specific algorithms, the crucial nuts and bolts of ML, and using competitions to increase interest and participation.
Protecting against multi-step attacks of uncertain start times and duration forces the defenders into indefinite, always ongoing, resource-intensive response. To allocate resources effectively, the defender must analyze and respond to an uncertain stream of potentially undetected multiple multi-step attacks and take measures of attack and response intensity over time into account. Such response requires estimation of overall attack success metrics and evaluating effect of defender strategies and actions associated with specific attack steps on overall attack metrics. We present a novel game-theoretic approach GPLADD to attack metrics estimation and demonstrate it on attack data derived from MITRE's ATT&CK Framework and other sources. In GPLADD, the time to complete attack steps is explicit; the attack dynamics emerges from attack graph and attacker-defender capabilities and strategies and therefore reflects 'physics' of attacks. The time the attacker takes to complete an attack step is drawn from a probability distribution determined by attacker and defender strategies and capabilities. This makes time a physical constraint on attack success parameters and enables comparing different defender resource allocation strategies across different attacks. We solve for attack success metrics by approximating attacker-defender games as discrete-time Markov chains and show evaluation of return on detection investments associated with different attack steps. We apply GPLADD to MITRE's APT3 data from ATT&CK Framework and show that there are substantial and un-intuitive differences in estimated real-world vendor performance against a simplified APT3 attack. We focus on metrics that reflect attack difficulty versus attacker ability to remain hidden in the system after gaining control. This enables practical defender optimization and resource allocation against multi-step attacks.