Coherent Solvus Research Articles

The Cahn–Hilliard model of coherent lamellar microstructure: application to alkali feldspar

The temporal evolution of coherent lamellar microstructures is significantly influenced by their elastic properties, particularly when the exsolved phases are coherent. This research utilizes the Cahn–Hilliard model to examine the morphological and energetic evolution of binary alkali feldspar, integrating anisotropic elastic energy into the Gibbs energy equation. The Cahn–Hilliard model successfully simulated the orientation of lamellae observed in natural samples and the elastic strain was consistent with previous research. We also computed the coherent solvus from the annealing simulation of various precursor compositions and temperatures. The temperature difference (ΔT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T$$\\end{document}) between the strain-free solvus and the coherent solvus was ΔT=85∘C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T = 85\\,{}^\\circ \ ext {C}$$\\end{document}, which is slightly lower than previously reported values obtained from similar parameters. This discrepancy is likely due to the presence of non-planar lamellae at the onset of phase separation, which are more stable than planar ones. We also simulated the binodal curves of the coherent solvi for different precursor phase compositions. The computed solvi were not unique but varied depending on the precursor composition. Our model is flexible because it does not assume any specific shapes for the lamellar interfaces and is applicable to various coherent binary systems.

Coherent solvus of disordered alkali feldspar: experiment, atom probe tomography and thermodynamic model

At temperatures above about 600 °C, alkali feldspar forms a continuous solid solution between the Na and K end members. Towards lower temperatures a miscibility gap opens, and alkali feldspar of intermediate composition exsolves, forming an intergrowth of relatively more Na-rich and K-rich lamellae. During exsolution, the crystal structure usually remains coherent across the lamellar interfaces, a feature that may be preserved over geological times. Due to the compositional dependence of the lattice parameters, coherent intergrowth requires that the lamellae are elastically strained. The associated elastic strain energy counteracts exsolution, and the solvus delimiting the misciblity gap for coherent intergrowth lies below the solvus for strain free phase equilibria. To determine the coherent solvus, homogeneous gem quality alkali feldspar of intermediate composition was annealed at conditions falling into the two-phase region of the phase diagram. Thereby a coherent intergrowth of approximately 10–20 nanometers wide lamellae was produced. Lamellar compositions were determined with atom probe tomography defining points on the coherent solvus. In parallel, the coherent solvus was calculated using a thermodynamic mixing model calibrated on the same alkali feldspar as used for the exsolution experiments and accounting for the elastic strain energy associated with coherent lamellar intergrwoth. The experimentally determined and the calculated coherent solvus are in excellent agreement indicating that phase equilibria in coherent lamellar intergrowth of alkali feldspar are adequately described, providing a sound basis for the interpretation of phase relations in coherently exsolved alkali feldspar.

Coherent lamellar intergrowth in alkali feldspar

A thermodynamic analysis of coherent lamellar intergrowth resulting from the exsolution of initially homogeneous alkali feldspar is presented. In contrast to earlier treatments, where the simplifying assumption of zero strain in the lamellar interfaces was used, our treatment is more general. The elastic stresses and strains associated with coherent lamellar intergrowth of Na-rich and K-rich alkali feldspar are calculated by minimising the overall elastic energy of the lamellar microstructure. At given pressure and temperature, the elastic energy depends on the volume proportions of the two lamellar types, and thus on the composition of the homogeneous precursor feldspar. As a consequence, there is no single coherent solvus for alkali feldspar, but coherent solvi are different for different compositions of the homogeneous precursor phase. Experimentally observed lamellar orientations agree with those predicted by minimising the strain energy on a set of all possible lamellar orientations.

Spinodal decomposition in alkali feldspar studied by atom probe tomography

We used atom probe tomography to complement electron microscopy for the investigation of spinodal decomposition in alkali feldspar. To this end, gem-quality alkali feldspar of intermediate composition with a mole fraction of a_{text {K}}=0.43 of the K end-member was prepared from Madagascar orthoclase by ion-exchange with (NaK)Cl molten salt. During subsequent annealing at 550,^circ hbox {C} and close to ambient pressure the ion-exchanged orthoclase unmixed producing a coherent lamellar intergrowth of Na-rich and K-rich lamellae. The chemical separation was completed, and equilibrium Na–K partitioning between the different lamellae was attained within four days, which was followed by microstructural coarsening. After annealing for 4 days, the wavelength of the lamellar microstructure was approx 17,hbox {nm} and it increased to approx 30,hbox {nm} after annealing for 16 days. The observed equilibrium compositions of the Na-rich and K-rich lamellae are in reasonable agreement with an earlier experimental determination of the coherent solvus. The excess energy associated with compositional gradients at the lamellar interfaces was quantified from the initial wavelength of the lamellar microstructure and the lamellar compositions as obtained from atom probe tomography using the Cahn–Hilliard theory. The capability of atom probe tomography to deliver quantitative chemical compositions at nm resolution opens new perspectives for studying the early stages of exsolution. In particular, it helps to shed light on the phase relations in nm scaled coherent intergrowth.

Coarsening kinetics of coexisting peristerite and film microperthite over 104 to 105 years

Alkali feldspar crystals from the Klokken syenite pluton (South Greenland) have a coarse patch perthite exsolution texture, on the scale of hundreds of micrometers, produced by dissolution-reprecipitation reactions at ~500 °C. We have discovered that Ab-rich patches (Ab92An7Or1) subsequently exsolved by spinodal decomposition to give peristerite, with a periodicity of 17 nm. This is the first time a peristerite has been characterized by TEM within a perthite, and the first peristerite to be found outside regional metamorphic rocks and pegmatites. The coexisting Or-rich patches, Ab16Or84(An0.3), unmixed over the same temperature interval to yield coherent film microperthite with periodicities ≤1 μm. Because the peristerite and film perthite from a single hand specimen from the Klokken intrusion have shared exactly the same cooling history, they provide a novel opportunity to test the utility of diffusion coefficients obtained in the laboratory for prediction of coarsening rates over 105 yr time scales, in the light of a calculated cooling trajectory for the pluton. We find good agreement between observed and predicted periodicities for the film perthite, which began to exsolve by coherent nucleation at the coherent solvus for ordered feldspars at ~420 °C, ~32 000 yr after crystal growth, and took a further 13 000-39 000 yr to coarsen to a periodicity of 1 μm. Peristerite exsolution began at the conditional spinodal at ~450 °C, but on the basis of the best existing experimental data would have taken ~520 000-4 600 000 yr to coarsen, incompatible with the cooling history. We speculate on changes to these “best” parameter values to describe NaSi-CaAl interdiffusion that would lead to peristerite coarsening rates compatible with the cooling history of the Klokken intrusion. We find that an interdiffusion expression:

First-principles theory of the coherency strain, defect energetics, and solvus boundaries in thePbTe-AgSbTe2system

Using first-principles data for the elastic properties of PbTe, ${\text{AgSbTe}}_{2}$, and related compounds, we extend our previous theoretical study of the thermodynamics of ${\text{PbTe-AgSbTe}}_{2}$ and present an in-depth analysis of the effects of elastic strain on the thermodynamics of ordering and coherent solvus boundaries. We find that the substitutional site preference for Pb in ordered ${\text{AgSbTe}}_{2}$ and the large asymmetry of the ${\text{PbTe-AgSbTe}}_{2}$ miscibility gap share a common physical origin in the peculiar defect energetics of ${\text{AgSbTe}}_{2}$. In particular, we find that Pb substitution on Ag sites has approximately the same energy cost as a complex defect consisting of Pb substitution on an Sb site combined with an ${\text{Sb}}_{\text{Ag}}$ antisite defect. Configurational entropy contributions strongly favor the latter, explaining why Pb substitutes almost exclusively on the Sb sites in ${\text{AgSbTe}}_{2}$. Coherency strain is shown to increase the solubility limits by a factor of $\ensuremath{\sim}2$ relative to the bulk values both for Ag,Sb in PbTe and for Pb in ${\text{AgSbTe}}_{2}$.

Nanotunnels and pull-aparts: Defects of exsolution lamellae in alkali feldspars

We have studied defects associated with flat, lens-shaped perthitic albite lamellae in alkali feldspars using SEM and TEM. In orthoclase phenocrysts from the Shap granite, Cumbria, NW England, bulk composition Or 70.20 Ab 29.05 An 0.85 , no dislocations were found even in optically “fresh” parts of grains. Instead, dissolution inferred to be localized on edge-dislocation loops has created tiny “nanotunnels” typically T effect of the ~1% An in the feldspars is taken into account for commencement of coherent exsolution. The orientation of the Pericline twins was fixed shortly after coherent exsolution began. Previous work has indicated that edge dislocations would start to form during cooling at ≤400 °C, so that nanotunnels form at still lower T . Dry heating experiments were carried out to establish the stability of the defects and the homogenization behavior of the exsolution lamellae. Na ↔ K exchange is rapid on heating above the coherent solvus and chemical homogenization of lamellae is complete after 24 h at 700 °C. In contrast, nanotunnels persist for >148 h at 1000 °C and >5748 h at 700 °C. Below the coherent solvus, exsolution lamellae thin on heating, leaving nanotunnels stranded in the orthoclase matrix. Microtextures related to Si-Al ordering patterns in the framework, such as Albite twins, are not eliminated, forming ghost-like lamellar strain patterns in chemically homogeneous feldspar. The presence of nanotunnels in optically “fresh” alkali feldspars shows that not only granites but also granulite-facies rocks have been pervasively affected by fluids at low T . Both nanotunnels and pull-aparts have important implications for feldspar reactivity in the upper crust, for 18 O exchange, and for transport of 40 Ar both in nature and in laboratory step-heating.

Short-Range Order in Ni-Rich Ni–Ti Studied by Diffuse Neutron Scattering

Diffuse wide-angle neutron scattering has been measured for 58 Ni-5.8 at% Ti at room temperature after quenching from 777 K, and for 5 8N i-9.6 at% Ti at 1103 K. The fitted sets of short-range order and static displacement parameters are similar to those found in a previous investigation of short-range ordered Ni-8.9 at% Al. Applying a Bragg-Williams model, the coherent solvus of the γ phase is well reproduced. Neither the displacement parameters nor the phonon dispersior branches measured along (ξξξ) for the sample containing 9.6 at% Ti show any signs of instability of the short-range ordered face centered cubic structure.

Coherent phase equilibria: Effect of composition-dependent elastic strain

Recent investigations on coherent equilibria have shed much light on the nature of phase stability. For example, equilibrium phase concentrations are shown to depend on the alloy composition, and two-phase coexistence can terminate at a critical coherency strain, termed a Williams point. This work, however, presents that when lattice parameter, hence coherency strain, is taken to be a function of phase composition, Williams point can disappear, i.e. the two-phase termination occurs only at an infinite elastic strain. In particular, if lattice parameter depends linearly on phase composition following Vegard's law, the equilibrium phase compositions are found to be independent of alloy composition, similar to the incoherent equilibrium case, but the chemical potentials vary with alloy composition, underlying breakdown of the usual Gibbs phase rule. When the analysis is extended to a binary regular solution obeying Vegard's law, complete nature of both coherent solvus and coherent spinodal is revealed in accordance with Cahn's prediction made a generation ago.

Low-temperature coherent exsolution in alkali feldspars from high-grade metamorphic rocks of Sri Lanka

In the alkali feldspars of the amphibolite- and granulite-facies rocks of Sri Lanka, a late-stage, final exsolution event is observed which produced film lamellae and fine-scale spindles. These were investigated by optical, microprobe, single-crystal, transmission electron microscopy and atomic resolution microscopy techniques. The lamellae and spindles exsolved below the coherent solvus at temperatures as low as 300 to 350° C. Precession photographs and ARM micrographs show that the intergrowth is perfectly coherent. In sections ‖ (010) the rhombic section of the Pericline twins corresponds to analbite or high albite. The albite lamellae and spindles nucleated and grew at low temperatures in a metastable disordered structural state within a tweed-orthoclase matrix and became periodically twinned analbite or high albite, which subsequently developed only a slight increase in Al, Si order. The relationship between twin periodicity and lamellar width, predicted for coherent intergrowths by Willaime and Gandais (1972), is obeyed. In Or-rich grains, in which coherent exsolution is the only exsolution event, the film lamellae tend to be restricted to the rim, the fine-scale spindles to the centre of the grains. The films nucleated heterogeneously at grain boundaries and grew towards the grain centres. Fine-scale spindles probably nucleated homogeneously in the interior part of grains. Heterogeneous nucleation and coherent growth are not mutually exclusive.

Determination of the δ coherent solvus in the Al-Li system by small-angle X-ray scattering

Abstract The δ' coherent phase solvus is determined in the range 20° to 200°C by X-ray small-angle scattering. The data represent an upper limit to the coherent boundary and agree well with the higher temperature data on δ' and δ solvus previously reported in the literature. The radius of G.P. zones, calculated at several aging temperatures by SAXS, is in reasonable agreement with the values observed by other authors using transmission electron microscopy.

The effect of oxygen on the structure and mechanical behavior of Aged Ti-8 Wt pct Al

It is shown that for a Ti-8 wt pct Al alloy aged at a temperature high in the two-phase region (695°C) to precipitate the ordered α2 phase, an increase in oxygen content from 600 ppm to 1200 ppm decreases the fracture strain from 20 to 1 pct elongation at room temperature and slightly increases the yield strength. The fracture mode is changed from dimpled rupture to predominantly cleavage. Further increase in oxygen content to 3000 ppm does not produce significant additional changes in ductility or yield strength. It is demonstrated that oxygen additions alter the position of the α/α + α2 coherent solvus, resulting in formation of coherent α2 in specimens containing ⪞ 1000 ppm oxygen aged at 968 K (695°C). For a given aging time the volume fraction of α2 increases with increasing oxygen up to 1300 wt ppm and then levels off. The changes in mechanical behavior are attributed to the presence of α2. The experimental evidence suggests that oxygen partitions preferentially into α2.

Precipitation in CuTi and CuTiAl alloys; discontinuous and localised precipitation

Discontinuous precipitation in a Cu-4% Ti binary alloy is studied through transmission electron microscopy. Different types of contrast conditions are used and moire fringes and interfacial dislocations are revealed. The grain refinement that occurs during discontinuous precipitate growth is due to translational and rotational subgrain boundaries formed on account of the different habit planes of precipitates from the different nucleation sites. It is also shown that discontinuous precipitation is absent in a Cu-2.1% Ti-2.4% Al and a Cu-2.1% Ti-5% Al alloys over a wide range of aging temperatures. However, a pronounced tendency for localised precipitation is observed in these alloys. It is found that the ternary CuTiAl alloys have lower coherent solvus temperatures. Solute supersaturation is therefore relieved through precipitation of the equilibrium phase by heterogenous nucleation at grain boundaries, twin boundaries, dislocations and other structural imperfections where a lower activation energy barrier is present.

Continuous precipitation in Cu-rich Cu-Ti binary and Cu-Ti-Al ternary alloys

An investigation has been carried out to study precipitation in a Cu-4% Ti binary and a Cu-2.1% Ti-2.4% Al ternary alloys which develop modulated microstructures upon ageing. Precipitate-free zones near the grain boundaries were observed on ageing immediately below a critical temperature representing the transition from the coherent to the incoherent region of the coherent phase diagram. In addition, a pronounced tendency for elastic interaction between the precipitate particles was detected, giving rise to aligned groups of precipitates containing a maximum of eight or nine particles in each group. At least near this critical temperature, decomposition occurs in the metastable region of the coherent phase diagram rather than in the spinodal region. The formation of the modulated structures in these alloys is, therefore, considered to arise not necessarily through spinodal decomposition. Continuous precipitation in a Cu-2.1% Ti-5% Al alloy is shown to result in spherical particles with no alignment. The effect of Al addition on the coherent solvus temperature is discussed in terms of a vacancy solute interaction mechanism.

Coarsening kinetics of the exsolution microstructure in alkali feldspar

A maximum microline-low albite of 33 mole percent Or was prepared by alkali ion exchange at 1000° C and subsequently annealed in air at lower temperatures. The exsolution microstructure produced in these samples was studied by transmission electron microscopy. At or above 600° C the compositional difference of the coherent lamellae increases rapidly and reaches a maximum, corresponding to the coherent solvus, within a few days. The lamellar spacing (λ) increases more slowly and the coarsening rate is given by λ=λ0+kt 1/3; where t is annealing time, k is a constant for a given temperature, and λ0 is the lamellar spacing obtained by extrapolation to zero annealing time. λ0 is 65 A at 600° C and ca. 200 A at 700° C. The latter is near the maximum temperature at which exsolution is observed. The nonzero value of λ0 and its increase with increasing temperature is evidence that the exsolution occurs by a spinodal mechanism. Data for the coarsening kinetics at other temperatures should be useful for estimating the thermal history of cryptoperthites with coherent lamellae.

Influence of continuous precipitation upon the growth kinetics of the cellular reaction in an Al-Ag alloy

The influence of the prior formation of a continuous precipitate upon the growth kinetics of the cellular reaction has been evaluated in an Al-17.9 wt. % Ag alloy. The continuous precipitate, in the form of intragranular plates of the γ′ transition phase, was shown to have reduced the upper bound of the driving force for the cellular reaction from the silver content of the untransformed alloy to that corresponding to the coherent solvus. When this reduction (≥ 98 %) is taken into account, the growth of cells is found to be controlled by cell boundary rather than by volume diffusion on the basis of both the Turnbull and the Cahn theories of the cellular reaction. Changing the mode of heat treatment from the usual quenching-and-aging to that of isothermal transformation reduces both the rate of growth of cells and the proportion of cellular structure formed by about an order of magnitude and increases the interlamellar spacing by 50–100%. These effects appear to result from a further decrease in the driving force. This decrease is attributed to a higher rate of introduction of misfit dislocations into the broad faces of the γ′ plates constituting the continuous precipitate, and thus to smaller values of the coherent solvus.