Precipitation hardening
Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and some steels and stainless steels. In superalloys, it is known to cause yield strength anomaly providing excellent high-temperature strength. Precipitation hardening relies on changes in solid solubility with temperature to produce fine particles of an impurity phase, which impede the movement of dislocations, or defects in a crystal's lattice. Since dislocations are often the dominant carriers of plasticity, this serves to harden the  material. The impurities play the same role as the particle substances in particle-reinforced composite materials. Just as the formation of ice in air can produce clouds, snow, or hail, depending upon the thermal history of a given portion of the atmosphere, precipitation in solids can produce many different sizes of particles, which have radically different properties. Unlike ordinary tempering, alloys must be kept at elevated temperature for hours to allow precipitation to take place. This time delay is called "aging". Solution treatment and aging is sometimes abbreviated "STA" in specifications and certificates for metals. Two different heat treatments involving  precipitates can alter the strength of a material: solution heat treating and precipitation heat treating. Solid solution strengthening involves formation of a single-phase solid solution via quenching. Precipitation heat treating involves the addition of impurity particles to increase a material's strength. Kinetics versus thermodynamics. This technique exploits the phenomenon of supersaturation, and involves careful balancing of the driving force for precipitation and the thermal activation energy available for both desirable and undesirable processes. Nucleation occurs at a relatively high temperature (often just below the solubility limit) so that the kinetic barrier of surface energy can be more easily  overcome and the maximum number of precipitate particles can form. These particles are then allowed to grow at lower temperature in a process called "ageing". This is carried out under conditions of low solubility so that thermodynamics drive a greater total volume of precipitate formation. Diffusion's exponential dependence upon temperature makes precipitation strengthening, like all heat treatments, a fairly delicate process. Too little diffusion ("under ageing"), and the particles will be too small to impede dislocations effectively; too much ("over ageing"), and they will be too large and dispersed to interact with the majority of dislocations. Alloy design. Precipitation  strengthening is possible if the line of solid solubility slopes strongly toward the centre of a phase diagram. While a large volume of precipitate particles is desirable, a small enough amount of the alloying element should be added that it remains easily soluble at some reasonable annealing temperature. Elements used for precipitation strengthening in typical aluminium and titanium alloys make up about 10% of their composition. While binary alloys are more easily understood as an academic exercise, commercial alloys often use three components for precipitation strengthening, in compositions such as Al(Mg, Cu) and Ti(Al, V). A large number of other  constituents may be unintentional, but benign, or may be added for other purposes such as grain refinement or corrosion resistance. In some cases, such as many aluminium alloys, an increase in strength is achieved at the expense of corrosion resistance. The addition of large amounts of nickel and chromium needed for corrosion resistance in stainless steels means that traditional hardening and tempering methods are not effective. However, precipitates of chromium, copper, or other elements can strengthen the steel by similar amounts in comparison to hardening and tempering. The strength can be tailored by adjusting the annealing process, with lower initial  temperatures resulting in higher strengths. The lower initial temperatures increase the driving force of nucleation. More driving force means more nucleation sites, and more sites means more places for dislocations to be disrupted while the finished part is in use. Many alloy systems allow the ageing temperature to be adjusted. For instance, some aluminium alloys used to make rivets for aircraft construction are kept in dry ice from their initial heat treatment until they are installed in the structure. After this type of rivet is deformed into its final shape, ageing occurs at room temperature and increases its strength, locking  the structure together. Higher ageing temperatures would risk over-ageing other parts of the structure, and require expensive post-assembly heat treatment because a high ageing temperature promotes the precipitate to grow too readily. Types of hardening. There are several ways by which a matrix can be hardened by precipitates, which could also be different for deforming precipitates and non-deforming precipitates. Deforming particles: Coherency hardening occurs when the interface between the particles and the matrix is coherent, which depends on parameters like particle size and the way that particles are introduced. Small particles precipitated from supersaturated solid solution usually  have coherent interfaces with the matrix. Coherency hardening originates from the atomic volume difference between precipitate and the matrix, which results in a coherency strain. The associated stress field interacts with dislocations leading to an increase in yield strength, similar to the size effect in solid solution strengthening. Modulus hardening results from the different shear modulus of the precipitate and the matrix, which leads to an energy change of dislocation line tension when the dislocation line cuts the precipitate. Also, the dislocation line could bend when entering the precipitate, increasing the affected length of the dislocation line. Chemical strengthening is  associated with the surface energy of the newly introduced precipitate-matrix interface when the particle is sheared by dislocations. LIke modulus hardening, the analysis of interfacial area can be complicated by dislocation line distortion. Order strengthening occurs when the precipitate is an ordered structure such that bond energy before and after shearing is different. For example, in an ordered cubic crystal with composition AB, the bond energy of A-A and B-B after shearing is higher than that of the A-B bond before. The associated energy increase per unit area is anti-phase boundary energy and accumulates gradually  as the dislocation passes through the particle. However, a second dislocation could remove the anti-phase domain left by the first dislocation when traverses the particle. The attraction of the particle and the repulsion of the first dislocation maintains a balanced distance between two dislocations, which makes order strengthening more complicated. Non-deforming particles: In non-deforming particles, where the spacing is small enough or the precipitate-matrix interface is disordered, dislocation bows instead of shears. The strengthening is related to the effective spacing between particles considering finite particle size, but not particle strength, because once the particle is strong  enough for the dislocations to bow rather than cut, further increase of the dislocation penetration resistance won't affect strengthening. Theory. The primary species of precipitation strengthening are second phase particles. These particles impede the movement of dislocations throughout the lattice. You can determine whether or not second phase particles will precipitate into solution from the solidus line on the phase diagram for the particles. Physically, this strengthening effect can be attributed both to size and modulus effects, and to interfacial or surface energy. The presence of second phase particles often causes lattice distortions. These lattice distortions result when the  precipitate particles differ in size and crystallographic structure from the host atoms. Smaller precipitate particles in a host lattice leads to a tensile stress, whereas larger precipitate particles leads to a compressive stress. Dislocation defects also create a stress field. Above the dislocation there is a compressive stress and below there is a tensile stress. Consequently, there is a negative interaction energy between a dislocation and a precipitate that each respectively cause a compressive and a tensile stress or vice versa. In other words, the dislocation will be attracted to the precipitate. In addition, there is a positive interaction energy  between a dislocation and a precipitate that have the same type of stress field. This means that the dislocation will be repulsed by the precipitate. Precipitate particles also serve by locally changing the stiffness of a material. Dislocations are repulsed by regions of higher stiffness. Conversely, if the precipitate causes the material to be locally more compliant, then the dislocation will be attracted to that region. In addition, there are three types of interphase boundaries (IPBs). The first type is a coherent or ordered IPB, the atoms match up one by one along the boundary. Due to the difference in  lattice parameters of the two phases, a coherency strain energy is associated with this type of boundary. The second type is a fully disordered IPB and there are no coherency strains, but the particle tends to be non-deforming to dislocations. The last one is a partially ordered IPB, so coherency strains are partially relieved by the periodic introduction of dislocations along the boundary. In coherent precipitates in a matrix, if the precipitate has a lattice parameter less than that of the matrix, then the atomic match across the IPB leads to an internal stress field that interacts with moving  dislocations. There are two deformation paths, one is the coherency hardening, the lattice mismatch is Where formula_3 is the shear modulus, formula_4 is the coherent lattice mismatch, formula_5 is the particle radius, formula_6 is the particle volume fraction, formula_7 is the burgers vector, formula_8 equals the concentration. The other one is modulus hardening. The energy of the dislocation energy is formula_9, when it cuts through the precipitate, its energy is formula_10, the change in line segment energy is The maximum dislocation length affected is the particle diameter, the line tension change takes place gradually over a distance equal to formula_5 . The interaction force between the dislocation and the precipitate is Furthermore, a dislocation may cut through a precipitate particle, and introduce more precipitate-matrix interface, which is chemical strengthening. When the dislocation is entering the particle and is within the particle, the upper part of the particle shears b with respect to the lower part accompanies the dislocation entry. A similar process occurs when the dislocation exits the particle. The complete transit is accompanied by creation of matrix-precipitate surface area of approximate magnitude formula_15, where r is the radius of the particle and b is the magnitude of the  burgers vector. The resulting increase in surface energy is formula_16, where formula_17 is the surface energy. The maximum force between the dislocation and particle is formula_18, the corresponding flow stress should be formula_19. When a particle is sheared by a dislocation, a threshold shear stress is needed to deform the particle. The expression for the required shear stress is as follows: When the precipitate size is small, the required shear stress formula_21 is proportional to the precipitate size formula_22, However, for a fixed particle volume fraction, this stress may decrease at larger values of r owing to an increase in  particle spacing. The overall level of the curve is raised by increases in either inherent particle strength or particle volume fraction. The dislocation can also bow around a precipitate particle through so-called Orowan mechanism. Since the particle is non-deforming, the dislocation bows around the particles (formula_23), the stress required to effect the bypassing is inversely proportional to the interparticle spacing formula_24, that is, formula_25, where formula_26 is the particle radius. Dislocation loops encircle the particles after the bypass operation, a subsequent dislocation would have to be extruded between the loops. Thus, the effective particle spacing for the second  dislocation is reduced to formula_27 with formula_28, and the bypassing stress for this dislocation should be formula_29, which is greater than for the first one. However, as the radius of particle increases, formula_30will increase so as to maintain the same volume fraction of precipitates, formula_24 will increase and formula_32 will decrease. As a result, the material will become weaker as the precipitate size increases. For a fixed particle volume fraction, formula_32 decreases with increasing r as this is accompanied by an increase in particle spacing. On the other hand, increasing formula_6 increases the level of the stress as a result  of a finer particle spacing. The level of formula_32 is unaffected by particle strength. That is, once a particle is strong enough to resist cutting, any further increase in its resistance to dislocation penetration has no effect on formula_32, which depends only on matrix properties and effective particle spacing. If particles of A of volume fraction formula_37are dispersed in a matrix, particles are sheared for formula_38 and are bypassed for formula_39, maximum strength is obtained at formula_40, where the cutting and bowing stresses are equal. If inherently harder particles of B of the same volume fraction are present, the level  of the formula_41 curve is increased but that of the formula_32 one is not. Maximum hardening, greater than that for A particles, is found at formula_43. Increasing the volume fraction of A raises the level of both formula_32 and formula_41 and increases the maximum strength obtained. The latter is found at formula_46, which may be either less than or greater than formula_47 depending on the shape of the formula_48 curve. Governing equations. There are two main types of equations to describe the two mechanisms for precipitation hardening: Dislocation cutting through particles: For most strengthening at the early stage, it increases  with formula_49, where formula_50 is a dimensionless mismatch parameter (for example, in coherency hardening, formula_50 is the fractional change of precipitate and matrix lattice parameter), formula_52 is the volume fraction of precipitate, formula_53 is the precipitate radius, and formula_54 is the magnitude of the Burgers vector. According to this relationship, materials strength increases with increasing mismatch, volume fraction, and particle size, so that dislocation is easier to cut through particles with smaller radius. For different types of hardening through cutting, governing equations are as following. For coherency hardening, formula_55, formula_56, where formula_57 is increased shear stress, formula_58 is the shear  modulus of the matrix, formula_59 and formula_60 are the lattice parameter of the precipitate or the matrix. For modulus hardening, formula_61, formula_62, where formula_63 and formula_64 are the shear modulus of the precipitate or the matrix. For chemical strengthening, formula_65, formula_66, where formula_67 is the particle-matrix interphase surface energy. For order strengthening, formula_68 (low formula_69, early stage precipitation), where the dislocations are widely separated; formula_70 (high formula_69, early stage precipitation), where the dislocations are not widely separated; formula_72, where formula_73 is anti-phase boundary energy. Dislocations bowing around particles: When the precipitate is strong enough to resist dislocation penetration , dislocation bows and the maximum stress is given by Orowan equation. Dislocation bowing, also called Orowan strengthening, is more likely to occur when the particle density in the material is lower. where formula_57 is the material strength, formula_58 is the shear modulus, formula_54 is the magnitude of the Burgers vector, formula_78 is the distance between pinning points, and formula_53 is the second phase particle radius. This governing equation shows that for dislocation bowing the strength is inversely proportional to the second phase particle radius formula_53, because when the volume fraction of the precipitate is fixed, the spacing formula_78 between particles  increases concurrently with the particle radius formula_53, therefore formula_83 increases with formula_53. These governing equations show that the precipitation hardening mechanism depends on the size of the precipitate particles. At small formula_53, cutting will dominate, while at large formula_53, bowing will dominate. Looking at the plot of both equations, it is clear that there is a critical radius at which max strengthening occurs. This critical radius is typically 5-30 nm. The Orowan strengthening model above neglects changes to the dislocations due to the bending. If bowing is accounted for, and the instability condition in the Frank-Read mechanism is  assumed, the critical stress for dislocations bowing between pinning segments can be described as: formula_87 where formula_88 is a function of formula_89, formula_89 is the angle between the dislocation line and the Burgers vector, formula_91is the effective particle separation, formula_7 is the Burgers vector, and formula_5 is the particle radius. Computational discovery of new alloys. While significant effort has been made to develop new alloys, the experimental results take time and money to be implemented. One possible alternative is doing simulations with Density functional theory, that can take advantage of, in the context of precipitation hardening, the crystalline structure of  precipitates and of the matrix and allow the exploration of a lot more alternatives than with experiments in the traditional form. One strategy for doing these simulations is focusing on the ordered structures that can be found in many metal alloys, like the long-period stacking ordered (LPSO) structures that have been observed in numerous systems. The LPSO structure is long packed layered configuration along one axis with some layers enriched with precipitated elements. This allows to exploit the symmetry of the supercells and it suits well with the currently available DFT methods. In this way, some researchers have developed  strategies to screen the possible strengthening precipitates that allow decreasing the weight of some metal alloys. For example, Mg-alloys have received progressive interest to replace Aluminum and Steel in the vehicle industry because it is one of the lighter structural metals. However, Mg-alloys show issues with low strength and ductility which have limited their use. To overcome this, the Precipitation hardening technique, through the addition of rare earth elements, has been used to improve the alloy strength and ductility. Specifically, the LPSO structures were found that are responsible for these increments, generating an Mg-alloy that exhibited high -yield strength: 610 MPa at 5% of elongation at room temperature. In this way, some researchers have developed strategies to Looking for cheaper alternatives than Rare Elements (RE) it was simulated a ternary system with Mg-Xl-Xs, where Xl and Xs correspond to atoms larger than and shorter than Mg, respectively. Under this study, it was confirmed more than 85 Mg-Re-Xs LPSO structures, showing the DFT ability to predict known LPSO ternary structures. Then they explore the 11 non-RE Xl elements and was found that 4 of them are thermodynamically stable. One of them is the  Mg-Ca-Zn system that is predicted to form an LPSO structure. Following the previous DFT predictions, other investigators made experiments with the Mg-Zn-Y-Mn-Ca system and found that at 0.34%at Ca addition the mechanical properties of the system were enhanced due to the formation of LPSO-structures, achieving “a good balance of the strength and ductibility”.