Solid solubility Early Hume Rothery Rules (1934 to 1936)
From Prof. Bakers Lecture Notes….Alloying Factors (A.K. Early Hume Rothery Rules, 1934-1936 before he found about # e/atom...) Size Structure Electrochemical Valence Effect Size These factors matter, but the #e/atom rule is much more interesting and profound….
In case you forgot… A phase is a homogenous, physically distinct * and mechanically separable portion of the material with a given chemical composition and structure. * A bit in the eye of the beholder. E.g. The surface certainly has a different electronic (metal) and bonding (semiconductors) configuration. Is is a separate phase ? In the sense of Gibbs phase rules ?
1. Size Cu-Ni, Pt-Au HW 7.1 a) Look up the “size” of Cu, Ni, Pt, Au (Hint: Do you need to look up the ionic or the covalent radius ) b) Determine “size” difference in % c) Bonus for born metallurgists: The phase diagram for Si Ge is also a continuous solid solution. What is the % difference in atom “size” ? To expect a solid solution, do not exceed 15% That is, if : “ the other factors are favorable”
What happened to this author, an Asst.Prof of MS&E at Stanford ? The foundation of today’s high speed electronics is SiGe
Is this unusual ? Not as much as you might guess... Andy Grove, his famous predecessor started out with finding the Deal-Grove law for the oxidation of Silicon. The CEO and President of TSMC, worlds largest Si Foundry, got his Ph.D from us ! From the Intel Web Site: The chairman of the board
From Prof. Bakers notes on the size effect This means the following.
If you have a decomposition in the solid state the maximum temperature of the two phase field (its peak) is given by This is really not theory…. Rather a crude guess 1. Think of the shear modulus, not as force per areas but as energy per volume. Very useful in many metal problems; e.g. creep 2. the atomic volume. would be “Elastic energy per atom” 3. is the misfit. It goes in as a square like the extension of a spring.
4. If you divide energy by k you get a temperature. 5. Lucky guess. You should really use the B (bulk modulus) or E (Youngs modulus) as the stress field is not a shear stress field. Those are typically 3 times larger in metals 6. The factor 2 can be derived from the integration, but is lucky, cause we really need to worry about the degree of freedom once we use the kT concept as energy. 7. Applicability of continuum elastic theory at 14% strain is highly, highly doubtful
The expansion around a solute atom can be measured experimentally and is nowhere close to the elastic theory However, to our knowledge at least, EXAFS has allowed the first determination of the core effect (i.e. the displacement of the first shell surrounding each solute atom). Such an experiment has been performed on an Al-Cu solid solution (Fontaine et al 1979a) and has yielded the ratio db/b = -0.044 where b is the closest distance A1-AI, and db is the variation of b when one Cu atom substitutes an A1 one. The present work gives a similar measurement for the A1-0.83 at% Zn. As expected, the contraction of the first shell about each Zn atom in the solid solution is weak (db = -0.02 k 0.01 A, db/b = - 7.0 x This db/b ratio can be linked to the parameter’s shift (dalac = -0.120 for Al-Cu, -0.020 for Al-Zn solid solution) through a coarse elastic theory: strains about an impurity relax in the matrix which is treated as a continuous homogeneous isotropic elastic medium (Friedel 1955, Eshelby 1956, Blandin and Deplante 1963). This suggests that the change of x-ray lattice parameter in a crystal which is uniformly expanded by point defects is just what one would infer from the change of its macroscopic dimensions. The macroscopic volume change of the solid solution is the sum of the volume change of the point defect core (AVw = 47cb2db) and of the volume change due to the uniform dilatation (y - l)AVw caused by the so-called image stresses (y - 1 = 4 p x/3 - 0.5, p being the shear modulus and the compressibility coefficient). These are the salient elements of the calculation which lead to the relationship da/ac = (y/y) dbjb where y is the ratio of the atomic volume versus the spherical volume limited by the first shell. ‘y’ depends only on the crystalline structure and is equal to 0.169 for a FCC structure. Taking into account the elastic constants of the solute atom introduces very slight corrections : the value input in this relationship has to be ‘dber; which is very similar to the actual value db (dbeff = 1.0082 db for A1-Cu, dbeff = 0.955 db for Al-Zn). It is thus possible to compare the db experimental value with those computed from the slope of Vegard’s law. Table 1 shows the large discrepancy (about 300%) between the two sets of values. The assumption of a continuous elastic medium leads to a very underestimated value of db. These first available measurements of the core effect in solid solutions serve to emphasise the serious quantitative disagreement that can arise when a continuous model is used. The assumption of a continuous elastic medium leads to a very underestimated value of db
The Au-Ni phase diagram. The alloy decomposes in the solid state at 150 C (10hrs) via a spinodal decomposition. A spinodal starts as a small composition fluctuation over a large spatial region. Precipitation starts as a large composition fluctuation in a small space. You must have had this, but we will revisit it at a later date... The phase diagrams are more interesting, in particular 1 c !!!
Spinodal decomposition results in two continuos phases 1. Interesting mechanical properties ! 2. Intentionally dissolve one phase and you can make Corning lab glass ware consisting of nearly pure SiO 2. 3. Dissolve one phase, and make a sponge like nano metal that has unusual mechanical properties. Get much attention. 4. Discover that the old metal de-alloying is driven by spinodals (electrochemists do not read about phase transitions) and you will have a letter to Nature * * several might be required to get tenure at Harvard. Who would have thought they do metallurgy at Harvard ?
2. Electrochemical value The concept of electronegativity was introduced by Pauling in a book published by Cornell press. A measure on how easily an element will give up an electron. If the elements have similar electronegativity, they will make a solid solution, if they have a different electronegativity, a intermetallic compound
From Prof. Baker’s notes This book extract is a fancy way of saying that if the electro negativity difference increases, the alloy becomes more and more covalent.
This is just a graphic way of saying that when you get a more stable “coherent compound” (because you mix elements with larger differences in electronegativity) then the existence range of the solid solution goes down. I hope you recall enough of Mike Thompson’s Thermo to follow this :-)
http://home.c2i.net/astandne/help _htm/english/ A very handy website ! Includes software to calculate binary phase diagrams ! Ni = 1.91 Al = 1.61 Cu = 1.90 Au = 2.54
Extension by Darken and Gurry (of Thermo fame) 1963 Size and electro negativity of various Ag alloys
3. Higher valence metal more soluble in low valence than reverse. Well, yes, but...
It’s best to look at these “rules” as a primitive approach to QM. Which is much more powerful, even if used on a simple level (“hopping integrals, moment theorem, Hund’s rule etc..”) The “rules”, essentially are, that similar stuff likes to dissolve in similar stuff.
Extension: The “size” factor can be explained with elastic theory, but requires assumptions that do not always hold (e.g. Al has a very soft potential, and does not behave like a rigid sphere. Transition metals do). Look in Baker’s notes if interested in the formula. But “size” has one interesting consequence Under compression ! Ideal place for a small substitutional atom !!! Empty space under tension Ideal place for big atoms to segregate --- or whatever is not soluble
Coherent Guinier Preston zones in AA 7050 correctly heat treated Al AA7050 is an aluminum- zinc-copper- magnesium alloy used in many aerospace applications including various fuselage structures, wing coverings, landing gear supports, and rivets. Solid Solution® Guinier-Preston (G-P Zones)® ' (MgZn2)® (MgZn2)
HW-7.2 1. Look up the “size” of Mg and Zn atoms 2. Are they bigger or smaller than Al ? 3. If there is a size difference, what would be a natural place for the G-P zones to start to precipitate out ? An interesting tidbit : Age hardening Al alloys were discovered in 1906, by Alfred Wilm, a German metallurgist employed by the Prussian War Ministry to find a lightweight materiel to supplant brass in bullet casings. He termed his new metal Duraluminum. The discovery was way, way before dislocations, dislocation glide, obstacles to dislocation glide, difference between coherent and incoherent particles as glide obstacles and etc etc were known !.
From Professor Bakers lecture notes This is view of metallurgists that do not want to use QM. Nothing wrong with it, but since you took Quantum Chemistry you can use a better concept !
The QM point of view (I will repeat this until you learn it !) 1. To first order, a metal is a box in which we have a free electron gas. What matters is how dense the gas is. That is electrons per atom. 2. If you add more electrons per atom, you “blow up the Fermi sphere like a balloon” 3. Sooner or later you will “hit a zone boundary” because the ever shorter wavelength will diffract from some set of lattice planes. So you get energy gaps. You can avoid “hitting a zone boundary” by switching to a different crystal structure (I.e. phase). If we need to deal with the lattice in more detail than its periodicty, we do it with hopping integrals and the momentum theorem 4. In dealing with the energy gaps, you need to consider if the electron gas can have oscillations such that it minimizes its need “ to jump the gap”. The natural oscillation of the electron gas is 2 k Fermi.
I discussed this with Richard Hennig and he came up with a possible reason for 2k Fermi The filled Fermi sphere The largest scattering vector you can have in the e-gas is 2k Fermi.. The e-gas interacts “with itself” via scattering. 2k Fermi is the shortest oscillation it can set up..
The Ti- alloys are interesting, because they are used a lot in aerospace, so let’s dig a little deeper here !
Pure Titanium (not alloyed) switches from hcp (c/a 1.587) to bcc at ~ 890C (the transition T is controversial) Therefore, Ti has very similar cohesive energy in both phases. Most commercial Ti alloys (high strength, good formability) are mixtures of So, as a practical metallurgist: “how would you generate this mixture? “ Part of the normal hexagonal cell. The half up atom is “one of the three” in the more conventional picture. Hcp metals have awful mechanical deformation properties.. Why ?
Ti has an atomic size and electronegativity similar to many of the common alloying elements Zr BTW: This makes it difficult to install precipitation hardening into Ti….. :-(
I hope you remember Lecture 6 and the famous Cu 3 Au order - disorder transition !
In Fe, where we can stabilize the high T, fcc, phase to RT and below by adding Ni (remember HW set 1!) In fact, we can mix the two phases...you remember those “dual phase stainless steel” I hope used in better scissors ! (if not Google it) In Ti, we can stabilize the high T bcc phase down to lower temperatures by adding by adding Mo, V, W Cu, Mn, Fe, Ni, Co, Ta and H!!!! Ti can dissolve 60 at% H !!!!. The alpha phase is stabilized by Al, O, N and Ga Sn, Zr, Si do nothing (much). The Zr simply shifts the natural phase transition a bit higher. Phase mixtures
H-W 7.2 Using Google, investigate the storage of hydrogen in Ti. a) What phase of Ti do you want to have before you add H ? b) Why ? C) How can you coax Ti in the desired phase ? Hydrogen in titanium, International Journal of Hydrogen Energy, 24 (1999) 565-576, O. N. Senkov and F. H. Froes.
The phase can transform into the phase but we do not want to go there…. Too advanced for this class Yes, there used to be metallurgy here !
Practical Metallurgy : Use of Ti in jet engines. One would think that Ti is ideal for jet engines, as it is strong and light…. Alas you must consider all failure modes…. Let see class !
During the test a piece of Ti stripe with a mass of approx. 12 grams (1/2 oz) was ignited. It was a very violent short conflagration accompanied by temperatures reaching up to 3300°C /7/. Such high temperatures destroy the sample. Consequences of a titanium fire in an aeroengine. Nickel alloy blades have been burnt away. Photograph courtesy of Dr M. Hicks, RR. Ti fire Ti is used only for the compressor housing, cause T, 400 C. Unfortunately, the clearance between the rotor vanes and the Ti stator has to be very small (to get good compression) and hence any eccentricity (from wear out to bird strikes) that make the tip rub against the Ti compressor housing will set off a lovely Titanium Fire...
Well…. You should all know this ! If not I will go over it !
A few words about modern theory It would kill us, if we would have to calculate the band structure of every possible phase (fcc, bcc, ) and minimize the energy to find the lattice constant. Consider that we have to enter all electron potentials into the potential of the Schroedinger equation and that the potential would change if one electrons would move... Modern theory does NOT need to calculate band structures. It turns out that ALL, that is ALL we need to know is how the charge is distributed in the metal. Repeat: Once we know how the density of the electron gas fluctuates with position we know everything about the ground state energy. * This is certainly easier to calculate that E(k) ! Unfortunately, excited states are an other matter...
The fact that e (r), once known, tells us all about the ground state, was discovered in 1964 by Hohenberg and Kohn. In DFT theory, each electron moves independently of each other electron. The interaction with all the other electrons is taken care of by an effective potential. What that is, later. The ground State Energy, E ground is E ground = F ( (r)) Unfortunately, although the function F is unique, we don’t know what function it is, nor do we know the function (r). Let’s see how a modern computer would calculate the problem
1. Start with a guessed electronic charge distribution (r) 2. Take a potential that takes care of all other electrons (here it’s Hartree, V H ) 3. Add that potential to the V(r) from the nuclei. Call the result V eff (r) 4. Solve Schroedinger find (r) NOT energy - that would require much more work ! 5. Calculate (r), loop back To calculate (r), with electron-electron interaction
The Hartree Potential is lousy.. it ignores the fact that electrons likes “to clear its neighborhood of same energy electrons with same spin “ (the “exchange hole”). So how do we get a better effective potential for the electron - electron interaction ? We need to add a correction for the “exchange hole” How do we get this correction. We cheat creatively ! We minimize E ground = F ( (r)) for an electron gas in a jellium (homogeneously smeared out positive charge). That can be solved as (r) is a constant in r.
The local density approximation In reality (r) is not uniform. But we simply assume that in each sufficient small volume we can get away with it ! That is we assume, that whatever (r) is, the exchange correction is that of the jellium model. There were lots of people who thought this would never work in the core, where the electron density varies wildly But it did…. The results agreed with the experiment !
DFT, Density Functional Theory The calculation, runs as before, but now we have a better potential to the electron electron interactions into account. The correction is the V xc (r) added to the Hartree term. V n is the potential of the nucleus. To speed things up a pseudo potential is used.
The pseudo potential is a simplified potential that “simulates the inner core electrons circling in closed shells” These electrons don’t do much, except making life difficult for (r). There are two ways of looking why this is so 1.The first (more formal) way is that the free electron state must be orthogonal to the core atomic wave functions. 2. The second (more deep) concept is that the exchange principle, [that two electrons with the same quantum numbers can be in the same place at the same time] keeps electron from the core.
Either view shows that the charge carried by (r) is reduced in the core. 1.In the first picture, the wave function wiggles like crazy to meet the orthogonal condition. 2. In the second picture, the exchange energy between the core electrons and the free electron wave “weakens the latter. I am putting more details into the appendix. In either case, it is – once you analyze it – the hostility between electrons in the same quantum state that reduces the electron density in the core.
In the free electron gas (valence electrons) we took care of the “local reduction in density” between “hostile” electrons of the electron gas by adding the term V ex. Note: this is the repulsion between the electrons “of the valence electron gas”. Mutually, between themselves. Now, the valence electron gas is encountering a “new, other kind of gas – the “electron gas” made up of the core electrons. To model the reduced density, we invent a fake potential that repulses electrons. This is much easier that dealing with the exhange energy in a mathematical formulation.
Indeed, coulomb attraction and exchange energy almost completely cancel. The potential is frequently simplified to The “empty core” potential contains only one adjustable parameter w Coulomb tails, set by charge of metal atom
How Kohn got the pseudo potential for Si (simplified) 1.Select a value for w 2.Calculate elastic constants for Si (known) 3.If wrong, change w 4.Keep doing this until elastic constants are correct. Creative new step 1.Calculate the pressure at which Si will transform from dc to fcc phase - not known, never measured ! 2.BINGO ! Accurate Pseudopotentials for K-S DFT are available
Professor Ruoff, to this day, marvels at the fact that Kohn (Berkeley) using DFT predicted at what pressure Si and Ge should transform before the experiment was done The proof of a good theory that it predicts something unknown correctly (like Einstein’s prediction that gravity would deflect photons). Modern DFT theory now deals with gradients.. too far out for this course!
The End Hint: Spinodal Decomposition Who is this man ???
A vector in space Components of the vector in some coordinate system Components of the same vector in an other coordinate system
The value of the components of a vector depend on the coordinate system. The vector itself, does not change ! In particular, its length, L, is always the same ! No matter what coordinate system use. That means there must be a function of the coordinates the value of which does not change, regardless what coordinate system we pick Can you guess the function ?
The function a x 2 + a y 2 Its value does not depend on the representation which you choose. In the case of a vector, this is self-evident. In higher dimensions, or for more complicated objects is less clear. Functions the value of which does not depend on the coordinate system we choose are called invariants. For example, the stress tensor has 3 invariants, known as I 1, I 2, I 3. The familiar van Mieses failure criterion can be written concisely as F(I 1,I 2 )=0. Which is a nice way to show that you do not need to use a special coordinate system. You need to find special coordinate systems if you use the Tresca yield criterion
Extensions: A vector V in a higher dimensional space (say ten dimensions) has the invariant I 1 I 1 = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 + a 7 2 + a 8 2 + a 9 2 + a 10 2 Which you can look as some generalized length. In a more concise form you can write this as I 1 = V * V Which you know as the scaler product, also known as inner product If you think of the wave function as a vector (and will see you can) than I = (r) * (r) also as an invariant ! But the meaning is not length, it’s charge density
How to think of (r) as a vector No matter what form (r) has you can expand it into a Fourier series (r) = a 0 + a 1 cos(x) + a 2 cos(2x) + a 3 …. + b 1 sin(x) + b 2 sin(2x)… Now you must make a leap and think of cos(x) and so on.. as an axis in a coordinate system. There are many of such axis so this is a multidimensional space. The a 0, a 1, …b 1, b 2.. (in this view) are the co-ordinates of the vector. The cos(nx) and sin are base vectors (or if you wish, unit vectors along the axis, or if you wish a base set). This view makes it immediately clear, that the value of
… that the value of (r) * (r) Can not depend on the coordinate system we use. The coordinate system are here sin and cos, or more generally e inx (the use of 2 is in the eye of the beholder), But we could “use any other base set” set is we can expand in series that use reciprocal lattice vectors (very useful when interactions between and the crystal lattice need to be investigated) or we could expand into a series that uses LCAO (Linear combination of Atomic Orbials) * The latter are interesting beasts, but first lets consider why we always use rectangular coordinates (or if we think about the basis vectors not as unit vectors along x, y, z, but functions like sin and cos, we they have to be orthogonal) * Much of modern metal QM is concerned about the energy of defects. Assume you want to calculate the formation energy of a vacancy. Such a thing is easier with LCAO then with Bloch waves. You would have to generate a “local” variation in electron density by adding lots of Bloch waves of different wavelength. It is much easier to start with the AO, hopping integrals, and that a neighbor is missing
A vector in an non-rectangular coordinate system There are two ways to determine the “components” of this vector Drop line at right angles the two axis from the tip of the vector (a) Draw lines parallel to the axis to the tip of the vector (b). This is simplified (I am dropping a scaling factor), but clearly there are two ways to define components While both are valid and have their uses, we certainly do not get involved with this problem. Given base vectors a 1, a 2,…..a n the coordinate system is rectangular if the scalar (inner) of the a base vector with all others is zero. a b
The same is true for vectors 1 and 2. If the inner product of 1 2 is zero, the functions are orthogonal. This is true regardless of what “coordinate system” we use, that is if we use (r) * (r) or (k) * (k) or (LCAO) * (LCAO) In modern, metal QM one starts with atomic orbitals AO.. The AO form an extremely useful base set to calculate the interaction of a free electron (aka conduction electron, aka a plane wave) with an atom. To calculate the interaction, we just look at the “coordinates” of in the AO base set. That is, we decompose the plane, which when viewed as a vector is sum of LCAO. That view shows immediately that each AO component of the plane wave must obey the exclusion principle that two particles with the same quantum numbers (including spin) can not be in the same place at the same time
The resulting avoidance (exchange energy) can be modeled by either 1.Requiring that is orthogonal to the atomic functions or 2.Inventing a repulsive potential that pretty much cancels the attractive Coulomb potential The two conditions are equivalent. The second one is easier to implement, and known as pseudopotential
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