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A system of three-dimensional complex variables

Some results of a new theory of multidimensional complex variables are reported, including analytic functions of a three-dimensional (3-D) complex variable. Three-dimensional complex numbers are defined, including vector properties and rules of multiplication. The necessary conditions for a function of a 3-D variable to be analytic are given and shown to be analogous to the 2-D Cauchy-Riemann equations. A simple example also demonstrates the analogy between the newly defined 3-D complex velocity and 3-D complex potential and the corresponding ordinary complex velocity and complex potential in two dimensions.

Martin, E. Dale↗

Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices

We consider conjugate gradient type methods for the solution of large sparse linear system Ax equals b with complex symmetric coefficient matrices A equals A(T). Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. We investigate conjugate gradient type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices. We propose a new approach with iterates defined by a quasi-minimal residual property. The resulting algorithm presents several advantages over the standard biconjugate gradient method. We also include some remarks on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

Freund, Roland↗

Lanthanide Complexes as a Test for Evidence of Life

The objective of this research is to advance the understanding of the interaction of lanthanide metals with biological organic molecules and to develop a technique to detect these compounds in the solid state and in situ in Mars and other planetary bodies. The detection of these complexes should provide evidence of life past or present. In addition, detection of the metals alone will provide important information about the geological history of a planetary body. Lanthanides were chosen as our focus of interest because they form very stable complexes with organic molecules in solution and they produce intense luminescence in the ultraviolet and visible spectra. The rare earth complexes available are mostly synthetic for diverse applications in medicine. There is not much work done on the complexes that form in nature. Lanthanides have many applications and they are mined aR over the world, however, since the interest has been only in the elements, the analytical techniques employed destroy any organic ligands that may be present. In order to determine if and which lanthanide complexes form in nature and their concentration, soil samples have been collected from areas rich in soluble lanthanide compounds like phosphates and also rich in vegetation. The soil samples will be analyzed and the lanthanide complexes if present will be isolated and characterized. A spectrometer to detect the lanthanide complexes in situ and in the solid state will be designed. In this workshop, the research approach and its implications will be discussed.

Benavides, Jeannette↗

ROSAT HRI and ASCA Observations of the Spiral Galaxy NGC 6946 and its Northeast Complex of Luminous Supernova Remnants

Analysis of 80 ks ASCA (Advanced Satellite for Cosmology and Astrophysics) and 60 ks ROSAT HRI (High Resolution Image) observations of the face-on spiral galaxy NGC 6946 are presented. The ASCA image is the first observation of this galaxy above approximately 2 keV. Diffuse emission may be present in the inner approximately 4' extending to energies above approximately 2-3 keV. In the HRI data, 14 pointlike sources are detected, the brightest two being a source very close to the nucleus and a source to the northeast that corresponds to a luminous complex of interacting supernova remnants (SNRs). We detect a point source that lies approximately 30" west of the SNR complex but with a luminosity -1115 of the SNR complex. None of the point sources show evidence of strong variability; weak variability would escape our detection. The ASCA spectrum of the SNR complex shows evidence for an emission line at approximately 0.9 keV that could be either Ne IX at approximately 0.915 keV or a blend of ion stages of Fe L-shell emission if the continuum is fitted with a power law. However, a two-component, Raymond-Smith thermal spectrum with no lines gives an equally valid continuum fit and may be more physically plausible given the observed spectrum below 3 keV. Adopting this latter model, we derive a density for the SNR complex of 10-35 cm(exp -3), consistent with estimates inferred from optical emission-line ratios. The complex's extraordinary X-ray luminosity may be related more to the high density of the surrounding medium than to a small but intense interaction region where two of the complex's SNRs are apparently colliding.

Schlegel, E.↗

Studying the Iron Line Complex in the Bright Seyfert Galaxy NGC 5506

This grant was to support the reduction and analysis of our approved XMM observation of the nearby Seyfert 2 galaxy NGC 5506. The observation has been carried out simultaneously with a BeppoSAX observation of the same source. The proposal was aimed to study in detail the Compton reflection component and the complex Iron K line of this source, combining the still unique capability of BeppoSAX in hard X-rays (to strongly constrain the reflection component, and then the intrinsic nuclear continuum), and the sensitivity of XMM at the energy of the Iron Line complex. NGC 5506 is one of the brightest AGN in hard X-rays and has been intensively studied in the past. GINGA detected the complex iron line as well as the reflection component. Both ASCA (spectroscopically) and Rossi-XTE (through variability analysis) suggested that the FeK line is complex, possibly made up of several distinct components. The centroid of the FeK complex in a subsequent BeppoSAX observation was bluer than the 6.4 keV energy of the relatively low-ionization iron Kalpha transition. NGC 5506 has been observed simultaneously by NewtonXMM and BeppoSAX on February 2-3 2001. we have reduced and analyzed both the NewtonXMM and the BeppoSAX data, and have written and published a paper on our results (appeared in Volume 377 (page 31) of A&A-Letters). Our main results can be summarized as follows: (a) we confirm that the FeK line is complex, and for the first time disentangle its components: we find that at least two components made up the FeK complex, one neutral and narrow, at 6.4 keV (rest energy), and another one either broader and highly ionized, at about 6.7 keV (rest frame), or, in turn, made up of two narrow and unresolved components from the He-like and the H-like ions of Fe; (b) the two possible solution for the high-ionization Fe-K component, are statistically indistinguishable. However, physically, a blend of two narrow lines from photoionized matter seems to be preferable to emission of a relativistically broadened line from an ionized accretion disk; (c) The bulk of the Compton reflection continuum originates in neutral matter, and is therefore associated with the narrow FeK line at 6.4 keV: they are most likely emitted by distant matter.

Nicastro, Fabrizio↗

Theoretical Investigation of OCN(-) Charge Transfer Complexes in Condensed Phase Media: Spectroscopic Properties in Amorphous Ice

Density functional theory (DFT) calculations of cyanate (OCN(-)) charge-transfer complexes were performed to model the "XCN" feature observed in interstellar icy grain mantles. OCN(-) charge-transfer complexes were formed from precursor combinations of HNCO or HOCN with either NH3 or H2O. Three different solvation strategies for realistically modeling the ice matrix environment were explored, including (1) continuum solvation, (2) pure DFT cluster calculations, and (3) an ONIOM DFT/PM3 cluster calculation. The model complexes were evaluated by their ability to reproduce seven spectroscopic measurements associated with XCN: the band origin of the OCN(-) asymmetric stretching mode, shifts in that frequency due to isotopic substitutions of C, N, O, and H, plus two weak features. The continuum solvent field method produced results consistent with some of the experimental data but failed to account for other behavior due to its limited capacity to describe molecular interactions with solvent. DFT cluster calculations successfully reproduced the available spectroscopic measurements very well. In particular, the deuterium shift showed excellent agreement in complexes where OCN(-) was fully solvated. Detailed studies of representative complexes including from two to twelve water molecules allowed the exploration of various possible solvation structures and provided insights into solvation trends. Moreover, complexes arising from cyanic or isocyanic acid in pure water suggested an alternative mechanism for the formation of OCN(-) charge-transfer complexes without the need for a strong base such as NH3 to be present. An extended ONIOM (B3LYP/PM3) cluster calculation was also performed to assess the impact of a more realistic environment on HNCO dissociation in pure water.

Park, Jin-Young↗

Cosmic Complexity

What explains the extraordinary complexity of the observed universe, on all scales from quarks to the accelerating universe? My favorite explanation (which I certainty did not invent) ls that the fundamental laws of physics produce natural instability, energy flows, and chaos. Some call the result the Life Force, some note that the Earth is a living system itself (Gaia, a "tough bitch" according to Margulis), and some conclude that the observed complexity requires a supernatural explanation (of which we have many). But my dad was a statistician (of dairy cows) and he told me about cells and genes and evolution and chance when I was very small. So a scientist must look for me explanation of how nature's laws and statistics brought us into conscious existence. And how is that seemll"!gly Improbable events are actually happening a!1 the time? Well, the physicists have countless examples of natural instability, in which energy is released to power change from simplicity to complexity. One of the most common to see is that cooling water vapor below the freezing point produces snowflakes, no two alike, and all complex and beautiful. We see it often so we are not amazed. But physlc!sts have observed so many kinds of these changes from one structure to another (we call them phase transitions) that the Nobel Prize in 1992 could be awarded for understanding the mathematics of their common features. Now for a few examples of how the laws of nature produce the instabilities that lead to our own existence. First, the Big Bang (what an insufficient name!) apparently came from an instability, in which the "false vacuum" eventually decayed into the ordinary vacuum we have today, plus the most fundamental particles we know, the quarks and leptons. So the universe as a whole started with an instability. Then, a great expansion and cooling happened, and the loose quarks, finding themselves unstable too, bound themselves together into today's less elementary particles like protons and neutrons, liberating a little energy and creating complexity. Then, the expanding universe cooled some more, and neutrons and protons, no longer kept apart by immense temperatures, found themselves unstable and formed helium nuclei. Then, a little more cooling, and atomic nuclei and electrons were no longer kept apart, and the universe became transparent. Then a little more cooling, and the next instability began: gravitation pulled matter together across cosmic distances to form stars and galaxies. This instability is described as a "negative heat capadty" in which extracting energy from a gravitating system makes it hotter -- clearly the 2nd law of thermodynamics does not apply here! (This is the physicist's part of the answer to e e cummings' question: what is the wonder that's keeping the stars apart?) Then, the next instability is that hydrogen and helium nuclei can fuse together to release energy and make stars burn for billions of years. And then at the end of the fuel source, stars become unstable and explode and liberate the chemical elements back into space. And because of that, on planets like Earth, sustained energy flows support the development of additional instabilities and all kinds of complex patterns. Gravitational instability pulls the densest materials into the core of the Earth, leaving a thin skin of water and air, and makes the interior churn incessantly as heat flows outwards. And the heat from the sun, received mostly near the equator and flowing towards the poles, supports the complex atmospheric and oceanic circulations. And because or that, the physical Earth is full of natural chemical laboratories, concentrating elements here, mixing them there, raising and lowering temperatures, ceaselessly experimenting with uncountable events where new instabilities can arise. At least one of them was the new experiment called life. Now that we know that there are at least as many planets as there are stars, it is hard to imagine that nature's ceasess experimentation would not be able to produce life elsewhere -- but we don't know for sure. And life went on to cause new Instabilities, constantly evolving, with living things in an extraordinary range of environments, changing the global environment, with boom-and-bust cycles. with predators for every kInd of prey, with criminals for every possible crime, with governments to prevent them, and instabilities of the governments themselves. One of the instabilities Is that humans demand new weapons and new products of all sort, leading to serious investments in science and technology. So the natural/human world of competition and combat is structured to lead to advanced weaponry and cell phones. So here we are In 2012, with people writing essays and wondering whether their descendents will be artificial life forms travelling back into space. And, pondering what are the origins of those forces of nature that give rise to everything. Verllnde has argued that gravitation, the one force that has so far resisted our efforts at a Quantum description, is not even a fundamental force, but is itself it a statistical force, like osmosis. What an amazing turn of events! But after all I've just said, I should not be surprised a bit.

Mather, John C.↗

Understanding Crew Decision-Making in the Presence of Complexity: A Flight Simulation Experiment

Crew decision making and response have long been leading causal and contributing factors associated with aircraft accidents. Further, it is anticipated that future aircraft and operational environments will increase exposure to risks related to these factors if proactive steps are not taken to account for ever-increasing complexity. A flight simulation study was designed to collect data to help in understanding how complexity can, or may, be manifest. More specifically, an experimental apparatus was constructed that allowed for manipulation of information complexity and uncertainty, while also manipulating operational complexity and uncertainty. Through these manipulations, and the aid of experienced airline pilots, several issues have been discovered, related most prominently to the influence of information content, quality, and management. Flight crews were immersed in an environment that included new operational complexities suggested for the future air transportation system as well as new technological complexities (e.g. electronic flight bags, expanded data link services, synthetic and enhanced vision systems, and interval management automation). In addition, a set of off-nominal situations were emulated. These included, for example, adverse weather conditions, traffic deviations, equipment failures, poor data quality, communication errors, and unexpected clearances, or changes to flight plans. Each situation was based on one or more reference events from past accidents or incidents, or on a similar case that had been used in previous developmental tests or studies. Over the course of the study, 10 twopilot airline crews participated, completing over 230 flights. Each flight consisted of an approach beginning at 10,000 ft. Based on the recorded data and pilot and research observations, preliminary results are presented regarding decision-making issues in the presence of the operational and technological complexities encountered during the flights.

Young, Steven D.↗

Using the System Complexity Metric (SCM) to Compare CO2 Removal Systems

A fundamental cause of difficulty in large engineering projects is their inherent complexity. An impression of complexity occurs if a system is simply difficult to understand, where there is no obvious mental model that correctly predicts its behavior. Higher system complexity is usually associated with higher cost and higher failure rate. Complexity is perceived if a system has many diverse components, multiple interactions and feedback loops, transients and dynamic behavior, and unanticipated failure modes. Identifying and removing these signs of complexity should improve performance and reduce the cost and failure rate. Complexity can be directly measured by the number of components and their interactions. The System Complexity Metric (SCM) is defined as the sum of the number of parts in a system, N, plus the number of the one-way interconnections between them, I. SCM = N + I. The SCM is easily determined by direct inspection of the system block diagram. SCM can be used to compare systems and to guide their redesign to reduce cost and failure rate. Carbon dioxide removal systems are analyzed using SCM, cost, and failure rate. As in previous work, cost is directly proportional to SCM and that failure rate increases as a power of SCM for large differences in SCM. The SCM ranking of carbon dioxide removal systems is the same as their ranking in detailed analysis and practice.

Harry W. Jones↗

Using the System Complexity Metric (SCM) to Compare CO2 Reduction Systems

A fundamental cause of difficulty in large engineering projects is their inherent complexity. An impression of complexity occurs if a system is simply difficult to understand, where there is no obvious mental model that correctly predicts its behavior. Higher system complexity is usually associated with higher cost and higher failure rate. Complexity is perceived if a system has many diverse components, multiple interactions and feedback loops, transients and dynamic behavior, and unanticipated failure modes. Identifying and removing these signs of complexity should improve performance and reduce the cost and failure rate. Complexity can be directly measured by the number of components and their interactions. The System Complexity Metric (SCM) is defined as the sum of the number of parts in a system, N, plus the number of the one-way interconnections between them, I. SCM = N + I. The SCM is easily determined by direct inspection of the system block diagram. SCM can be used to compare systems and to guide their redesign to reduce cost and failure rate. Carbon dioxide reduction systems are analyzed using SCM, cost, and failure rate. As in previous work, cost is directly proportional to SCM and that failure rate increases as a power of SCM for large differences in SCM. The SCM ranking of carbon dioxide reduction systems is the same as their ranking in detailed analysis and practice.

Harry W. Jones↗

Impact of Actinide Complexation on Ligand Radiolysis

Although the actinides boast many unique physical and chemical properties, their inherent susceptibility to radioactive decay are what make them truly interesting elements to study. The absorption of ionizing radiation from actinide decay leads to the formation of a variety of transient and steady-state radicals, ions, and molecular radiolysis products that can lead to significant changes in the surrounding environment, and ultimately dictate steady-state actinide redox distributions and the longevity of molecules designed for actinide complexation. Radiolysis of the latter leads to complexant destruction and the concomitant formation of degradation products that can complicate actinide studies and processes. However, the radiation chemistry of most actinide complexants have been studied in the absence of the actinides they were designed to complex, which can lead to inaccurate conclusions on longevity and degradation product distributions, as metal ion complexation has been historically shown to influence a ligand’s radiolytic behavior. Consequently, bridging this knowledge gap is important for actinide science. Here, I will discuss the impact of actinide complexation on the steady-state and time-resolved radiation-induced reactivity of a variety of complexants, including, tributyl phosphate (TBP) and N,N,N',N'-tetraoctyl diglycolamide (TODGA) .

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Large-Area Lasing in Nanoscale Complex Media: The Critical Role of Local Dielectric Environment

Controlling how electromagnetic waves interact with complex media is critical for applications in imaging and focusing. Such lightwave interactions with complex media can lead to dramatic optical effects like lasing. While much work in random lasing focus on understanding how gain and scattering co-operatively generate lasing, little work has focused on how to manipulate the lasing threshold without modifying the structural disorder. Here, a simple, mostly unexplored, strategy is demonstrated that employs atomic layer deposition (ALD) to tune the local near-field environment while preserving the underpinning disorder—controlling lasing in a nanoscale complex medium on a large scale (>cm 2 ). The nanoscale complex medium is a quasi-2D system of coupled zinc oxide nanospheres with overall thickness deep in the sub-wavelength regime (≈λ/4). Near-ultraviolet femtosecond spectroscopy probes the broadband response of the gain nanomaterial, details how ALD process fundamentally modifies the fast-picosecond and slow-nanosecond carrier dynamics, and informs on the relevant timescales critical for lasing. Full-field electromagnetic simulations provide critical insights about how near-field dielectric environment modifies the nanostructure's scattering cross-section, which ultimately results in enhanced lasing. Importantly, these results highlight a simple path to control how electromagnetic waves interact in a complex medium, a key step toward large-scale implementation of complex lasers.

36 MATERIALS SCIENCE↗

Characterization of Quantum States Based on Creation Complexity

Abstract The creation complexity of a quantum state is the minimum number of elementary gates required to create it from a basic initial state. The creation complexity of quantum states is closely related to the complexity of quantum circuits, which is crucial in developing efficient quantum algorithms that can outperform classical algorithms. A major question unanswered so far is what quantum states can be created with a number of elementary gates that scales polynomially with the number of qubits. In this work, it is first shown that for an entirely general quantum state it is exponentially hard (requires a number of steps that scales exponentially with the number of qubits) to determine if the creation complexity is polynomial. Then, it is shown that it is possible for a large class of quantum states with polynomial creation complexity to have common coefficient features such that, given any candidate quantum state, an efficient coefficient sampling procedure can be designed to determine if the state belongs to the class or not with arbitrarily high success probability. Consequently, partial knowledge of a quantum state's creation complexity is obtained, which can be useful for designing quantum circuits and algorithms involving such a state.

74 ATOMIC AND MOLECULAR PHYSICS↗

Synergistic foam stabilization and transport improvement in simulated fractures with polyelectrolyte complex nanoparticles: Microscale observation using laser etched glass micromodels

Inaccessibility to direct pore scale observation in hydrocarbon recovery of tight shale formations poses a great challenge to water-energy nexus initiatives and necessitates the use of high throughput technologies to emulate environmentally friendly processes. Herein, we employ a precise glass micromodel fabrication and visualization method to isolate the supercritical CO 2 bubbles surrounded by CO 2 -water lamella prepared in saline produced water stabilized with molecular complexation of zwitterionic surfactants (ZS) and polyelectrolyte complex nanoparticles (PECNP). The Selective Laser Enhanced Etching (SLE) technique was selected for micromodel simulation of high-pressure flow. Two representative designs, (1) fracture/micro-crack network 28 and (2) fracture/matrix were etched on fused silica glass with a laser printing machine and scCO 2 foam was injected to study the foamability, propagation, stability, and fluid loss properties. The highly monodispersed and uniformly distributed array of scCO 2 bubbles were detected in flow of scCO 2 foam in highly saline brine containing ionic complexes of positively charged PECNPs and ZS, whereas foam flow with the lamella containing ZS in fractures offered a noticeably large and polydisperse array of scCO 2 bubbles. scCO 2 bubble motion and deformation were traced, and local description of foam flow was visually examined. The confined array of scCO 2 bubbles stabilized by ZS in microcracks was affected by bubble growth and coalescence, whereas the super-populated array of monodispersed scC O2 bubbles with lamella containing complexes of PECNP and ZS were able to fill the channels with stable configurations within the timeframe of comparative stability measurements. The ability of complex fluid to prevent the formation damage was evaluated through fluid loss visualization in micromodels. Probing scCO 2 foam transport in homogenous porous media revealed smaller volume leak-off for scCO 2 foam containing PECNP-ZS ionic complexes.

04 OIL SHALES AND TAR SANDS↗

Evaluating three evapotranspiration estimates from model of different complexity over China using the ILAMB benchmarking system

The land surface models range in complexity of terrestrial evapotranspiration, yet it is unknown how model complexity translates to accuracy of modeled evapotranspiration estimates. Here, we use the International Land Model Benchmarking system to assess ET estimates from three models of varying complexity driven by the same forcing datasets: an earth system model, a terrestrial biosphere model, and a stand-alone ET model. The performance assessment includes both temporal and spatial evaluation, and different plant functional types across China. Our results indicate that the most complex model, an earth system model, performed best against the benchmarking datasets and metrics. Terrestrial biosphere model performed best in simulating inter-annual variability of ET, while earth system model performed best in simulating the seasonal cycle. The more complex models (earth system model and terrestrial biosphere model) perform better in forest, shrub and crop ecosystems, while the simpler model (stand-alone ET model) perform better in grass ecosystems. Our study demonstrates the impact of model complexity on ET estimates and highlights directions for future ET model improvements.

54 ENVIRONMENTAL SCIENCES↗

Compositionally-driven surface nanostructuring on refractory compositionally complex alloys under low energy helium bombardment

Additive manufacturing enables user-defined control of the compositional complexity, opening new design spaces for complex concentrated alloys (CCAs). Refractory CCAs may offer enhanced performance in the divertor region of a fusion reactor environment where plasma-facing materials will be subject to high temperatures, low energy He and D particles, and 14 MeV neutrons. In this work, specimens with the nominal composition of NbTaMoTi, NbTaMo, NbTaTi, and NbTa were fabricated via directed energy deposition (DED) then bombarded with 40 eV He ions to a fluence of 2x10 26 m −2 at ∼ 1000 K. Post irradiation, the surface morphology and composition were examined with electron microscopy and x-ray photoelectron spectroscopy to offer information on the spatial distribution of surface nano-structuring. Sub-surface He bubbles driving the nano-structuring were examined with electron microscopy techniques. Analysis indicates the local composition directly influences the He bubble and tendril size, while the region with the highest complexity showed the shortest nano-structuring. Molecular Dynamics simulations complement the experimental results, showing comparable helium bubble growth and migration as a function of compositional complexity. This work demonstrates the compositional dependence of surface nanostructure formation of compositionally complex alloys, important for future design of complex plasma facing materials in fusion reactors.

Complex concentrated alloys↗

The Complexation Properties of Self-Defensive Microgel-Modified Antimicrobial Surfaces

The complexation of cationic antimicrobials with polyanionic microgels on a biomaterial surface can render that surface self-defensive against bacteria by killing those bacteria which physically contact the antimicrobial-loaded microgels. This killing has been attributed to the contact-driven transfer of antimicrobial from a microgel to a challenging bacterium, though much remains unknown about this process. Here, in this study, we use a combination of experiments and computational modeling to identify key aspects of the complexation phenomena which influence the self-defensive properties. We synthesize poly(acrylic acid) (PAA) microgels (∼2–5 μm diameter) via membrane emulsification and electrostatically deposit them onto polycaprolactone (PCL) coupons or onto glass to form a discontinuous submonolayer. Subsequent microgel loading with colistin or with Sub5 antimicrobial peptide (AMP) causes microgel deswelling. Under physiological conditions Sub5 remains stably sequestered whereas colistin is quickly released. Coarse-grained molecular dynamics (CGMD) simulations confirm stronger Sub5/PAA complexation. CGMD calculations also indicate that Sub5 forms dimers and higher-order structures, a prediction confirmed experimentally by Small-Angle X-ray Scattering (SAXS). Supramolecular structure entropically enhances the complexation strength because of enhanced counterion release per complexation event, and this finding can help identify other antimicrobials well suited for such a nonelutive yet self-defensive strategy. CGMD simulations also show that Sub5 has a higher complexation strength with the Staphylococcus aureus membrane than it does with PAA, confirming that there is a thermodynamic driving force for antimicrobial transfer. Such self-defensive surfaces significantly reduce S. aureus colonization (over 90% reduction relative to unmodified controls) in an in vitro hematogenous contamination model and remain cyto-compatible as evidenced by mesenchymal stem cell spreading and proliferation.

36 MATERIALS SCIENCE↗

EF-Hand Battle Royale: Hetero-ion Complexation in Lanmodulin

The lanmodulin (LanM) protein has emerged as an effective means for rare earth element (REE) extraction and separation from complex feedstocks without the use of organic solvents. Whereas the binding of LanM to individual REEs has been well characterized, little is known about the thermodynamics of mixed metal binding complexes (i.e., heterogeneous ion complexes), which limits the ability to accurately predict separation performance for a given metal ion mixture. In this paper, we employ the law of mass action to establish a theory of perfect cooperativity for LanM-REE complexation at the two highest-affinity binding sites. The theory is then used to derive an equation that explains the nonintuitive REE binding behavior of LanM, where separation factors for binary pairs of ions vary widely based on the ratio of ions in the aqueous phase, a phenomenon that is distinct from single-ion-binding chemical chelators. We then experimentally validate this theory and perform the first quantitative characterization of LanM complexation with heterogeneous ion pairs using resin-immobilized LanM. Importantly, the resulting homogeneous and heterogeneous constants enable accurate prediction of the equilibrium state of LanM in the presence of mixtures of up to 10 REEs, confirming that the perfect cooperativity model is an accurate mechanistic description of REE complexation by LanM. We further employ the model to simulate separation performance over a range of homogeneous and heterogeneous binding constants, revealing important insights into how mixed binding differentially impacts REE separations based on the relative positioning of the ion pairs within the lanthanide series. In addition to informing REE separation process optimization, these results provide mathematical and experimental insight into competition dynamics in other ubiquitous and medically relevant, cooperative binding proteins, such as calmodulin.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗