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NASA NTRS2022

Diffusion in Melts and Magmas

Diffusion results from random motion of particles and entities. Diffusion in melts and magmas is due to thermally excited random motion of atoms, ions, and clusters, and plays a critical role in magmatic and volcanic processes. In melts and magmas, diffusion is one of the two mechanisms of mass transfer; the other being bulk flow (referred to as convection or advection). When both are present, diffusion refers to the dispersive motion relative to the mean bulk flow in a given reference frame (Richter et al. 1998). Diffusion plays critical roles in controlling magma mixing (Watson 1982; Koyaguchi 1985, 1989; Lesher 1994; Huber et al. 2009; Guo and Zhang 2020), mineral growth and dissolution rates in magmas (e.g., Watson 1982; Harrison and Watson 1983; Zhang et al. 1989; Newcombe et al. 2014; Macris et al. 2018), bubble growth and dissolution rate in magmas (Sparks 1978; Proussevitch and Sahagian 1998; Liu and Zhang 2000; Zhang 2013), and elemental and isotope fractionation during mineral 2 growth and dissolution (Jambon 1980; Richter et al. 1999, 2003; Watson and Muller 2009; Chopra et al. 2012; Watkins et al. 2014, 2017; Holycross et al. 2016, 2018). As a result, diffusion also plays an essential role in explosive volcanic eruptions and magma crystallization. Furthermore, diffusion has important applications in geospeedometry (Lasaga 1983, 1998; Zhang 1994, 2008; Trail et al. 2016; Zhang and Xu 2016). Experimental investigation of diffusion in geologically relevant silicate melts began to flourish in the 1970’s when micro-analytical measurements of diffusion profiles became available. In addition to the vast number of papers published since then, numerous books and reviews are available for diffusion in silicate melts. Hofmann et al. (1974) edited a book titled “Geochemical Transport and Kinetics” published by Carnegie Institution of Washington. This was the first landmark book summarizing the field. Lasaga and Kirkpatrick (1981) edited a book “Kinetics of Geochemical Processes” as volume 8 of the Reviews in Mineralogy (later becoming Reviews in Mineralogy and Geochemistry) series. Zhang and Cherniak (2010) edited “Diffusion in Minerals and Melts” as volume 72 of Reviews of Mineralogy and Geochemistry series, in which one chapter focused on diffusion theory, five chapters on diffusion in silicate melts (Behrens 2010; Lesher 2010; Liang 2010; Zhang and Ni 2010; Zhang et al. 2010), and other chapters were on experimental, analytical, and computational methods, and diffusion in minerals. Several textbooks covered the principles and applications of diffusion theories (Kirkaldy and Young 1987; Shewmon 1989; Cussler 1997; Lasaga 1998; Zhang 2008; Vrentas and Vrentas 2016), and two classic books covered the mathematics of diffusion (Carslaw and Jaeger 1959; Crank 1975). In preparing for this review chapter, we thought carefully about what to cover for this vast field, and decided to briefly go through the fundamentals of diffusion (more complete review can be found in Chakraborty 1995; Zhang 2008, 2010) and solutions to often-encountered 3 diffusion problems, and then focus on post-2010 diffusion studies on silicate melts and magmas. Here, melts refer to (mostly natural) silicate liquid, and magmas refer to crystal-bearing and/or bubble-bearing melts in which the continuous phase is the melt. There is a large body of work on diffusion in glasses, especially in the materials science literature, which is not covered in this review.

Youxue Zhang↗
NASA NTRS1999

Containerless, Low-Gravity Undercooling of Ti-Ce Alloys in the MSFC Drop Tube

Previous tests of the classical nucleation theory as applied to liquid-liquid gap miscibility systems found a discrepancy between experiment and theory in the ability to undercool one of the liquids before the L1-L2 separation occurs. To model the initial separation process in a two-phase liquid mixture, different theoretical approaches, such as free-energy gradient and density gradient theories, have been put forth. If there is a large enough interaction between the critical liquid and the crucible, both models predict a wetting temperature (T(sub w)) above which the minority liquid perfectly wets and layers the crucible interface, but only on one side of the immiscibility dome. Materials with compositions on the other side of the dome will have simple surface adsorption by the minority liquid before bulk separation occurs when the coexistence (i.e., binoidal) line in reached. If the interaction between the critical liquid and the crucible were to decrease, T(sub w) would increase, eventually approaching the critical consolute temperature (T(sub cc)). If this situation occurs, then there could be large regions of the miscibility gap where non-perfect wetting conditions prevail resulting in droplets of L1 liquid at the surface having a non-zero contact angle. The resulting bulk structure will then depend on what happens on the surface and the subsequent processing conditions. In the past several decades, many experiments in space have been performed on liquid metal binary immiscible systems for the purpose of determining the effects that different crucibles may have on the wetting and separation process of the liquids. Potard performed experiments that showed different crucible materials could cause the majority phase to preferentially wet the container and thus produce a dispersed microstructure of the minority phase. Several other studies have been performed on immiscibles in a semi-container environment using an emulsion technique. Only one previous study was performed using completely containerless processing of immiscible metals and the results of that investigation are similar to some of the emulsion studies. In all the studies, surface wetting was attributed as the cause for the similar microstructures or the asymmetry in the ability to undercool the liquid below the binoidal on one side of the immiscibility dome. By removing the container completely from the separation process, it was proposed that the loss of the crucible/liquid interaction would produce a large shift in T(sub w) and thus change the wetting characteristics at the surface. By investigating various compositions across the miscibility gap, a change in the type and amount of liquid wetting at the surface of a containerless droplet should change the surface nucleating behavior of the droplet - whether it be the liquid-liquid wetting or the liquid-to-solid transition. Undercooling of the liquid into the metastable region should produce significant differences in the separation process and the microstructure upon solidification. In this study, we attempt to measure these transitions by monitoring the temperature of the sample by optical pyrometry. Microstructural analysis will be made to correlate with the degree of undercooling and the separation mechanisms involved.

Robinson, M. B.↗
NASA NTRS2012

The Pursuit of K: Reflections on the Current State-of-the-Art in Stress Intensity Factor Solutions for Practical Aerospace Applications

The elastic stress intensity factor (SIF, commonly denoted as K) is the foundation of practical fracture mechanics (FM) analysis for aircraft structures. This single parameter describes the first-order effects of stress magnitude and distribution as well as the geometry of both structure/component and crack. Hence, the calculation of K is often the most significant step in fatigue analysis based on FM. This presentation will provide several reflections on the current state-of-the-art in SIF solution methods used for practical aerospace applications, including a brief historical perspective, descriptions of some recent and ongoing advances, and comments on some remaining challenges. Newman and Raju made significant early contributions to practical structural analysis by developing closed-form SIF equations for surface and corner cracks in simplified geometries, often based on empirical fits of finite element (FE) solutions. Those solutions (and others like them) were sometimes revised as new analyses were conducted or limitations discovered. The foundational solutions have exhibited striking longevity, despite the relatively "coarse" FE models employed many decades ago. However, in recent years, the accumulation of different generations of solutions for the same nominal geometry has led to some confusion (which solution is correct?), and steady increases in computational capabilities have facilitated the discovery of inaccuracies in some (not all!) of the legacy solutions. Some examples of problems and solutions are presented and discussed, including the challenge of maintaining consistency with legacy design applications. As computational power has increased, the prospect of calculating large numbers of SIF solutions for specific complex geometries with advanced numerical methods has grown more attractive. Fawaz and Andersson, for example, have been generating literally millions of new SIF solutions for different combinations of multiple cracks under simplified loading schemes using p-version FE methods. These data are invaluable, but questions remain about their practical use, because the tabular databases of key results needed to support practical life analysis can occupy gigabytes of storage for only a few classes of geometries. The prospect of using such advanced numerical methods to calculate in real time only those K solutions actually needed to support a specific crack growth analysis is also tempting, but the stark reality is that the computational cost is still so high that the approach is not practical except for specific, critical application problems. Some thoughts are offered about alternative paradigms. Compounding approaches are some of the earliest building blocks of SIF development for more complex geometries. These approaches are especially attractive because of their very low computational cost and their conceptual robustness; they are, in some ways, an intriguing contrast and complement to the brute-force numerical methods. In recent years, researchers at NRC-Canada have published remarkable results showing how compounding approaches can be used to generate accurate solutions for very difficult problems. Examples are provided of some successes--and some limitations--using this approach. These closed-form, tabulated numerical, and compounding approaches have typically been used for simple remote loading with simple load paths to the crack. However, many significant cracks occur in complex stress gradient fields. This is a job for weight function (WF) methods, where the arbitrary stress distribution on the crack plane in the corresponding uncracked body (typically determined using FE methods) is used to determine K. Several significant recent advances in WF methods and solutions are highlighted here. Fueled by advanced 3D numerical methods, many new solutions have been generated for classic geometries such as surface and corner cracks with wide ranges of geometrical validity. A new WF formulation has also be developed for part-through cracks considering the arbitrary stress gradients in all directions in the crack plane (so-called bivariant solutions). Basic WF methods have recently been combined with analytical expressions for crack plane stresses to develop a large family of accurate SIF solutions for corner, surface, and through cracks at internal or external notches with very wide ranges of shapes, sizes, acuities, and offsets. Finally, WF solutions are much faster than FE or boundary element solutions, but can still be much slower than simple closed-form solutions, especially for bivariant solutions that can require 2D numerical integration. Novel pre-integration and dynamic tabular methods have been developed that substantially increase the speed of these advanced WF solutions. The practical utility of advanced SIF methods, including both WF and direct numerical methods, is greatly enhanced if the FM life analysis can be directly and efficiently linked with digital models of the actual structure or component (e.g., FE models for stress analysis). Two recent advances of this type will be described. One approach directly interfaces the FM life analysis with the FE model of the uncracked component (including stress results). Through a powerful graphical user interface, simplified FM life models can be constructed (and visualized) directly on the component model, with the computer collecting the geometry and stress gradient information needed for the life calculation. An even more powerful paradigm uses expert logic to automatically build an optimum simple fracture model at any and every desired location in the component model, perform the life calculation, and even generate fatigue crack growth life contour maps, all with minimal user intervention. This paradigm has also been extended to the automatic calculation of fracture risk, considering uncertainty or variability in key input parameters such as initial crack size or location. Another new integrated approach links the engineering life analysis, the component model, and a 3D numerical fracture analysis built with the same component model to generate a table of SIF values at a specific location that can then be employed efficiently to perform the life calculation. Some attention must be given to verification and validation (V&V) issues and challenges: how good are these SIF solutions, how good is good enough, and does anyone believe the life answer? It is important to think critically about the different sources of error or uncertainty and to perform V&V in a hierarchal, building-block manner. Some accuracy issues for SIF solutions, for example, may actually involve independent material behavior issues, such as constraint loss effects for crack fronts near component surfaces, and can be a source of confusion. Recommendations are proposed for improved V&V approaches. This presentation will briefly but critically survey the range of issues and advances mentioned above, with a particular view towards assembling an integrated approach that combines different methods to create practical tools for real-world design and analysis problems. Examples will be selectively drawn from the recent literature, from recent enhancements in the NASGRO and DARWIN computer codes, and from previously unpublished research

CraigMcClung, R.↗
NASA NTRS2012

No Need to Wait for the Clean Air Dividend

Controlling smog and soot is the classic win-win situation, so it's great that the world is finally waking up to the idea. WHAT if there was a way to simultaneously slow down climate change, save millions of lives, improve crop yields and contribute to sustainable development and energy security? It sounds too good to be true, but it is possible. It won't be free or easy, but with some effort and moderate investment, it can be done. The way to do it is to reduce emissions leading to two types of pollution: black carbon and ozone. These are the only pollutants that we know contribute to both global warming and poor air quality. Black carbon is essentially soot, emitted from incomplete combustion of fossil fuels and biomass. It warms the climate in two ways: by absorbing heat in the atmosphere - similar to the greenhouse effect - and by reducing Earth's albedo, or ability to reflect sunlight. Inhaled into the lungs, it leads to cancer and cardiovascular disease. Ozone in the atmosphere also acts as a greenhouse gas, while ground-level ozone is toxic to humans and plants, so leads to both premature death and reduced crop yields. Ozone is not emitted directly but is produced by the action of sunlight on other pollutants, which are known as ozone precursors. Since black carbon and ozone are important components of soot and smog, a great deal of effort has already been put into developing methods to reduce emissions. So effective technology is available, but needs wider implementation. The recommended control measures for black carbon include widespread and tight emission standards on diesel cars and trucks; improved solid fuel cooking stoves, brick kilns and coke ovens in the developing world; and a ban on the open burning of agricultural waste. Implementation of these measures would have a rapid impact on the climate and human health, and also have the added benefit of greatly reducing emissions of carbon monoxide, an important ozone precursor. A second key ozone precursor is methane, which is also a powerful greenhouse gas in its own right. Control measures include reducing leaks from natural gas pipelines and storage tanks, and capturing it from coal, gas and oil extraction, landfills and wastewater treatment plants. Aeration of rice paddies and manure management can also reduce methane releases. Captured methane can often be sold or turned into power. In Monterrey, Mexico, for example, electricity generated from methane collected from the city landfill powers the public transportation system. So such measures can be beneficial even when ignoring the health and climate effects, as they can contribute to energy security and often pay for themselves. According to calculations by me and my colleagues, phasing in all these measures over the next 20 years would reduce global warming by about 0.5 degC in 2050, half of the projected increase between now and then (Science, vol 335, p 183). Regional benefits would be even greater, as black carbon disrupts rainfall patterns and magnifies warming and melting of snow and ice in parts of the world including the Arctic and the Himalayas. On top of the climate benefits, cutting black carbon and ozone would prevent over 3 million premature deaths from air pollution, and increase yields of staple crops by roughly 50 million tonnes a year. Improved cooking stoves would also decrease the demand for firewood in the developing world, reducing deforestation and freeing up time for those who collect wood - primarily women and children - to pursue other activities such as education. Similarly, improved brick kilns now being used in parts of Latin America and Asia require half as much fuel as traditional ones and are less time-intensive for the operators. This means that in addition to their environmental benefits, these measures can contribute to sustainable and human development. Tackling black carbon and methane is clearly a great idea, so why hasn't it been done already? There are many barriers. The upfront costs of some measures can be prohibitive even when they eventually pay for themselves. But this can be overcome by mechanisms such as international financing of capital costs. For other measures, the costs are typically borne by a few while the benefits accrue to everybody. In such cases civil society and governments must get involved. Governments are starting to act. In February, the US, Canada, Sweden, Bangladesh, Ghana and Mexico launched the Climate and Clean Air Coalition to support implementation of measures like these. This coalition will hopefully expand and achieve rapid, widespread adoption of measures to cut black carbon and ozone. While the climate benefits will be substantial, it is important to note that these measures cannot substitute for cuts in carbon dioxide. Black carbon, ozone, carbon monoxide and methane stay in the atmosphere for a fairly short time - a few days for black carbon and about a decade for methane. They thus respond quickly to emissions changes and give us substantial leverage over near-term climate change. In contrast, carbon dioxide is very long-lived and so responds slowly to emissions changes. This means that cuts have little immediate impact, but it also means they must be made now to avoid disastrous changes later on. Controlling short-lived climate pollutants is thus an issue of fairness. Much as failure to reduce carbon dioxide emissions soon would condemn future generations to disastrous change, failure to reduce near-term climate change condemns those alive today to suffer worsening effects of the sort already seen. Some wonder if we really can do both. We can, and we must.

Shindell, Drew↗