On Cyclical Phase Transformations in Driven Alloy Systems
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DOI: 10.1007/s11661-007-9379-z
- Cite this article as:
- Lee, J. Metall and Mat Trans A (2008) 39: 964. doi:10.1007/s11661-007-9379-z
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Abstract
Cyclical phase transformations occurring in driven materials syntheses such as ball milling are described in terms of a free energy minimization process of participant phases. The oscillatory flow behavior of metals with low stacking fault energies during hot working is taken as a prototype in which a ductile crystalline phase sustains undulation in its free energy, due to the alternate succession of work-hardening and work-softening mechanisms. A time-dependent, oscillatory free energy function is then obtained by solving a delay differential equation (DDE), which accounts for a time lag due to diffusion. To understand cyclical transitions on an atomistic scale, work is extended to molecular dynamics simulations. Under shear deformation, a two-dimensional nanocrystal shows cyclical transitions between an equilibrium rhombus and a nonequilibrium square phase. Three-dimensional simulations show crystalline-to-glass transitions at high strain rates, but very high shear strain rates are found to lead to a latticelike network structure in the plane perpendicular to the shear direction, with strings of atoms parallel to the shear direction.
1 Introduction
Material synthesis via external dynamic forcing is called a “driven process.” Perhaps, the best known driven process is mechanical alloying—a high-energy ball-milling process—but other examples include severe plastic deformation and irradiation. An interesting feature observed in several driven processes is temporal oscillations in phase fractions. For example, during mechanical alloying, the microstructure of a binary alloy of elemental Al_{50}Zr_{50} powders is shown to vary cyclically between a crystalline and an amorphous state,[1] while Co powders are shown to display changes between a fcc and a hcp structure.[2] In order to shed some light on the mechanism(s) of such cyclical phase transformations, we present a few findings obtained with both thermodynamic and atomic modeling approaches.
Mechanical alloying is a nonequilibrium thermodynamic process used to synthesize a variety of both equilibrium and nonequilibrium materials; examples include nanocrystalline solids, intermetallics, and glassy alloys. From a thermodynamic viewpoint, a system under external dynamic forcing experiences continuous energy input, and neither the formation of nonequilibrium structures nor the recovery of equilibrium structures through the decomposition of nonequilibrium structures is unexpected. Temporal oscillations in phase structure (or in any other property) under driven conditions are phenomena that occur far from equilibrium. Consequently, the term “equilibrium,” used frequently to describe the microstructure, is incorrect, as the system would continue to evolve even if the external driving force were removed. Whether a driven system undergoes sustained oscillations or reaches a steady-state microstructure, the total free energy, including that of the loading system, tends toward a minimum.
Systems with Cyclical Variations in Property under Driven Conditions
Number | Alloy System | Dynamic Force | Property | Observations |
---|---|---|---|---|
1 | Al_{50}Zr_{50} | mechanical alloying[1] | structure | C ↔ A cyclic phase transformations* |
2 | Co_{50}Ti_{50} | mechanical alloying[5] | structure | C ↔ A cyclic phase transformation |
3 | Co_{3}Ti | mechanical alloying[6] | structure | C ↔ A cyclic phase transformation |
4 | Cu_{33.3}Zr_{66.7} | mechanical alloying[7] | structure | C ↔ A cyclic phase transformation |
5 | Cu_{33.3}Zr_{66.7} | electron irradiation[8] | structure | C ↔ A ↔ C with increase in dose |
6 | Co | mechanical alloying[2] | structure | hcp + fcc ↔ hcp ↔ fcc + hcp ↔ fcc |
7 | Zn | mechanical alloying (cryogenic)[9] | hardness | three decaying cycles between 0.8 and 0.3 GPa |
8 | Cu | mechanical alloying[10] | enthalpy | two cycles between 0.4 and 1.2 kJ/mole |
9 | Cu | hot torsion[11] | torque | cyclic variations between 0.38 and 0.52 N·m |
10 | austenitic steel | hot torsion[12] | stress | cyclic variations between 14 and 35 MPa |
2 The DDE Approach
2.1 Oscillatory Flow Behavior during Hot Working
Hot working is an important thermomechanical process in which many metals and alloys are worked into a final or an intermediate product at high temperatures. Metals such as copper or austenitic steels with low stacking fault energies feature both diffusional flow and dislocation motion during hot working. As such, the true stress-true strain relationship depends on the strain rate (hereupon, the use of the term “true” for both stress and strain is dropped). At low strain rates, the stress-strain curve displays an oscillatory behavior with multiple peaks. As the strain rate increases, the number of peaks on the stress-strain curve decreases and, at high strain rates, the stress rises to a single peak before reaching a steady-state value. It is understood that dynamic recovery is responsible for the stress-strain behavior with zero or a single peak, whereas dynamic recrystallization causes the oscillatory nature. In the past, most predictive models are based on either modified Johnson–Mehl–Avrami kinetic equations or probabilistic approaches.[13, 14, 15, 16, 17, 18, 19, 20] In this work, a DDE is used for modeling this type of stress-strain behavior: a delay time due to diffusion is taken into consideration, and it is expressed in terms of a critical strain for nucleation for recrystallization.
For alloys with low stacking fault energies, both cross slip and climb are difficult, due to large stacking faults, and this, in turn, reduces the rate of dynamic recovery through dislocation motion.[17] Consequently, the dislocation densities can reach the critical level for the onset of recrystallization. At low strain rates, there is sufficient time for the recrystallizing grains to grow before they become saturated with high dislocation densities. The flow curve then becomes oscillatory, due to recurrent cycles of recrystallization. With an increase in the strain rate, however, the difference in stress between recrystallizing and old grains diminishes, resulting in a reduced driving force for grain growth and a rendering of smaller grains in the alloy. Therefore, there emerge fewer peaks with concurrent deformation, due to high strain rates. Eventually, there will be a single peak at a sufficiently high stain rate. In a temporal sense, microstructural evolution at low strain rates is inhomogeneous, but becomes progressively homogeneous as the stain rate increases.
In their classic work,[13] Luton and Sellars correctly proposed that multiple stress peaks should appear when the ratio of the characteristic strain for nucleation of recrystallization, ε_{n}, to the characteristic strain for completion of recrystallization, ε_{x}, is large. If ω/θ is taken to be ε_{x} on the basis that θ is a high-temperature modulus and ω is a steady-state stress, the ratio, ε_{n/}ε_{x}, is then equivalent to λ = θε_{n}/ω. This ratio is the reason the stress starts to display an oscillatory behavior when λ is greater than a critical value, 1/e = 0.37. Low strain rates provide adequate time for the recrystallizing grains to grow before they become saturated with high dislocation densities. The final (average) grain size at the steady state increases with a decrease in the strain rate, resulting in a lower σ_{ss}. Thus, ω should decrease with a decrease in the strain rate, \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \). Taking temperature effects into consideration, let us utilize a creep power-law type and write ω = \( A\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{q} \exp (qQ/{\text{R}}T){\text{ }} = \gamma \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{q} \), where Q is the activation energy for self diffusion and q ∼ 0.2.[23] As an example, with γ = 4 and q = 0.2, ω is converted to yield \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \), which is shown in the last column of Figure 1.
Input and Output Parameters for the DDE Curves in Figure 2
ω (ksi) | σ_{0} (ksi) | θ (ksi) | ε_{n} | \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \)(s^{−1}) | \( \gamma = \omega /\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{{0.2}} \) | λ (=θε_{n}/ω) | \( \tau ( = \varepsilon _{n} /\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }) \) |
---|---|---|---|---|---|---|---|
10.8 | 5.5 | 75.0 | 0.1 | 1.1 | 10.6 | 0.69 | 0.09 |
8.7 | 5.1 | 75.5 | 0.1 | 0.4 | 10.4 | 0.87 | 0.25 |
7.3 | 4.6 | 73.5 | 0.08 | 0.14 | 10.8 | 0.81 | 0.57 |
5.5 | 4.0 | 73.0 | 0.07 | 0.035 | 10.8 | 0.93 | 2.00 |
4.3 | 3.0 | 70.0 | 0.055 | 0.0037 | 13.2 | 0.90 | 14.9 |
3.2 | 2.3 | 65.0 | 0.05 | 0.0011 | 12.5 | 1.02 | 45.5 |
— | (average) | (72.0) | (0.076) | — | (11.4) | — | (s) |
2.2 Cyclical Phase Transformations between Crystalline and Glass Phase
During hot working, metals experience a rolling power (energy/volume/time) that is proportional to the strain rate. A fraction of the rolling power is converted to stored mechanical energy, and much is dissipated as heat. The efficiency depends on the material and the rolling device, and defects such as dislocations or grain boundaries represent the stored mechanical energy. In addition, there occurs recovery, such as dislocation annihilation or recrystallization of new grains. Thus, the thermodynamic state of the deforming metal should be expressed in terms of free energy, which requires the detailed nature of the crystalline defects and the entropic state of the metal, both configurational and vibrational. To a first-order approximation, however, the true stress, σ, should be a good representation for the free energy per unit volume, G. In developing Eq. [11], σ = aμρ^{0.5}as used, but if other defects such as grain boundaries are taken into account, the approximation seems reasonable, provided that there arise no significant temperature changes that might cause transitions in the crystal structure.
Let us apply Eq. [12] to cyclical phase transformations observed during ball milling. As Courtney and Lee[3] pointed out, we may classify the crystalline-glass transition into three groups: (1) stable crystalline phase state, (2) cyclical transition between crystalline and glass phase, and (3) stable glass phase state upon transition of an initially crystalline phase. Let us now release the implicit assumption used for the derivation of Eq. [12], that regardless of the state of G(t), the activation free energy required for the α to β (or any other different phase) transition is prohibitively large. Furthermore, we imagine that the transition is instantaneous whenever the free energy of the α phase is greater than that of the β phase (that is, some degree of supersaturation required for transition is neglected). The instantaneity suggests no kinetic barriers. Likewise, the same assumption applies for the reverse transition, from β to α.
Obviously, if G_{0β} is far greater than G_{0α}, as marked with a in Figure 3, the two free energy curves never cross each other, and the system should remain in a crystalline α phase. Milling elemental metal powders such as Cu or Zn should belong to this class. Conversely, if G_{0β} is close to (but still greater than) G_{0α}, for example, to the level marked with c in Figure 3, we should expect to observe a stable β phase, except for the onset of milling with the initial “α” powders. Some alloys, such as Al_{50}Zr_{50} and Cu_{33.3}Zr_{66.7} (Nos. 1 and 4 in Table I), show glass phases at low milling intensity but cyclical phase transformations between crystalline and glass phase at high milling intensity. Thus, Eq. [12] can describe why such interesting changes would occur with different milling intensities.
In describing the results of Figures 3 and 4, we have assumed no kinetic barriers for phase transitions. With more realistic kinetic barriers, oscillations are expect to decay faster and the system should early on reach a dynamic equilibrium at which the phase fractions attain their steady-state values.[3] The DDE model of Eq. [12] is a linear approximation with one single delay. Clearly, one may attempt to include multiple delays or nonlinear terms, or a mix of both, and could expect more interesting results, although the mathematical complexity increases. Finally, it should be noted that, in a driven process, the ground-state free energy density, G_{0}, of a phase is not necessary equal to the value at bulk. For example, some amorphous nanoparticles are shown to have higher stability, because melting points often decrease with particle size.[25]
3 Nonequilibrium Molecular Dynamics Studies
3.1 Two-Dimensional Case
Potential Parameters for Atomic Interactions for Two-Dimensional Study
Parameter | A-A | A-B | B-B |
---|---|---|---|
ε_{ij} | 1.140 | 0.210 | 1.000 |
σ_{ij} | 0.900 | 0.600 | 0.865 |
Mass | 3.5 | — | 1.0 |
At high temperatures, however, due to vibrational entropies, the S phase becomes stable, with four nearest-neighbor-like atoms. At the absolute zero, the atomic volume of the S phase is greater than that of the R phase by 7 pct. In order to enhance the stability of both phases and to prevent phase decomposition into pure A and pure B during a thermomechanical processing, the 16 A atoms are fenced around Kastner’s original 41-atom cluster. The free energy difference between the two structures is found by calculating the isothermal strain energy associated with the elastic deformation between the two structures. From the free energy information, the transition temperature, T_{c}, between the R and the S phases is found to be ∼791 K.[28]
3.2 Three-Dimensional Case
Potential Parameters for Atomic Interactions for Three-Dimensional Study
Parameter | Ni-Ni | Ni-Zr | Zr-Zr |
---|---|---|---|
ξ_{ij} | 0.6857 | 1.3707 | 1.4604 |
q_{ij} | 1.1890 | 2.2300 | 2.2490 |
d_{ij} | 2.4900 | 2.7610 | 3.1790 |
A_{ij} | 0.0241 | 0.1388 | 0.1239 |
p_{ij} | 16.999 | 8.3600 | 9.3000 |
ε_{ij} | 0.6857 | 1.3708 | 1.4604 |
σ_{ij} | 2.4900 | 2.7610 | 3.1790 |
mass | 1.6801 | — | 2.6103 |
In shearing deformation, however, the influence of the loading machine surface is not as simple as in compressive deformation. Fixed boundary conditions are often used at both the bottom and the top surfaces along the z direction, but they can cause undue influence on the momentum transfer for the atoms located in the proximity of the load surfaces, unless the sample dimension is very large.[31,32] As sketched in Figure 9(b), the sandwich-type periodic boundary condition with a pivoted center line permits sheared atomic motions with correct momentum transfer, and thus appears to alleviate the modeling difficulty, as will be demonstrated in the results. It is important that a microsample in a simulation should mirror a macroscopic system as correctly as possible.
4 Discussion
The results of the molecular dynamics study presented here are only preliminary, and further work is needed in order to mimic realistic cyclical phase transformations occurring in a driven system. For example, with a large crystallite, one could expect crystalline defects, such as a two-dimensional square-rhombus interface or dislocations that may play crucial roles during cyclical transformations. The square-rhombus phase transition is an ideal case representing no kinetic constraint in the phenomenological description of Courtney and Lee.[3] For the three-dimensional compressive deformation mode, there have been numerous simulation results displaying transitions between the crystalline and glass phases, and the current work confirms that a glass phase can be nucleated from a free surface or a faulted region of a twining or a kinking.
For shear deformation, a new simulation scheme is introduced, to allow a correct atomic momentum transfer without undue influence from the loading machine surface. In many of the earlier investigations, fixed boundary conditions are used for the interface between the computation cell and the loading machine: it is simple, but the atomic momentum calculation in the neighborhood of the loading machine surface is apt to transfer incorrectly.[32] This can be viewed with Figure 9(b): imagine that the top and the bottom horizontal lines represent the interface planes between the loading machine and the computation cell. The atomic momenta along the z direction should diminish to zero, as atoms approach the machine wall. As a consequence, these “frozen” atoms at the interface region influence the dynamics of the interior atoms, unless the cell size along the z direction is very large. By no means is the new scheme perfect. As portrayed in Figure 14(a), it creates an extra plane with zero shear velocity at the center and, thus, a large cell size along the z direction is preferred. Nevertheless, it mimics momentum transfer compatible to the applied shear strain rate, by alleviating the unwarranted influence of the loading machine surface.
In the field of colloids with monodisperse particles 10 nm to 1 μm in diameter, there has been great progress on phase transition behavior in particle arrangement.[33,34] For example, using synchrotron X-ray scattering techniques, Sirota et al.[34] found that as the volume fraction of polystyrene spheres in the liquid mixture of methanol and water increases, the system undergoes structural changes from liquidlike to a colloidal crystal of, first, bcc and, then, fcc. Furthermore, the colloidal viscosity is also studied and is found to be strongly dependent on the applied shear strain rate, exhibiting Newtonian as well as non-Newtonian behavior. At high strain rates, shear thinning, i.e., a drastic reduction in viscosity, is observed, which indicates dynamic structures with stream lines of colloidal particles along the shear direction. It is interesting to note that a high shear strain rate can create a solid-state string phase even on an atomic scale, as demonstrated in Figure 16.
It should be noted that improved simulations are necessary, in that a constant atomic volume is used for the shear deformation. In reality, deformation involves both compression as well as shearing, and thus, the molar volume is not conserved. A better scheme must reflect both modes of deformation, perhaps even a hydrostatic pressure to mimic a ball-milling process. In our simulation, no nucleation of the three-dimensional crystalline phase out of the glass phase is observed, although one might consider that the string phase shown in Figure 16 could be a precursor of a three-dimensional crystalline phase. A glass phase represents a system of nonergodicity, which means that a time average does not necessarily represent an ensemble average. Therefore, it is not a surprise that simulations with only a few thousand atoms fail to show crystallization in a time of less than a nanosecond. In other words, sampling of a very minute sequence of a tiny amorphous cluster has too low a probability to exhibit such a transition.
The DDE analyses on the flow curve, for metals with low stacking fault energies, demonstrate that a cyclical or oscillatory behavior of certain materials properties—stress or free energy—in a driven system can be described with a delay or drag nature, due to diffusion. In Table II, the incubation time for the nucleation for recrystallization is calculated with a linear relationship, τ (\( = \varepsilon _{n} /\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \)), and the results display a significant increase from 0.09 to 45.5 seconds, as shown in the rightmost column. The incubation time, τ, is equal to ∼ l^{2}/D, where l is the mean diffusion distance and D is the self-diffusion coefficient.[35] If we take l as d/2, d being the mean subgrain size, and D ∼ 7.7 × 10^{−16} m^{2}/s (for γ-Fe at 1100 °C[36]), the incubation times from 0.09 to 45.5 seconds suggest mean subgrain sizes from about 17 to 374 nm. The increase in the subgrain size with a decrease in the strain rate, \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }, \) simply manifests the decrease in the steady-state stress, ω. For the alloys with high stacking fault energies, as noted earlier, easy cross slip or climb is sufficient to reduce dislocation densities below the level for the onset of dynamic recrystallization, and thus there arise no multiple peaks in the flow curve. For such a case, the incubation time, τ, becomes on the order of ∼0.1 seconds, if l is taken as the mean dislocation spacing, with ρ ∼ 10^{16}/m^{2}. Therefore, ε_{n} would be ∼10^{−4} at a typical slow strain rate \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } = 10^{{ - 3}} /{\text{s}} \), which is too small to cause oscillations on the flow curve for those alloys or metals. A similar conjecture is applied to the glass phase, which competes against the crystalline phase during a ball-milling process (Figure 3).
Cyclical phase transformations can be understood in terms of reaction-like competition between participant phases. For example, in a recent work of Johnson et al.,[37] thermodynamic balance equations are expressed to describe the time evolution of two competing phases undergoing ball milling. The state of the microstructure is then determined by the phase fraction, the deformation state of each phase, and the homogeneous temperature. One of the steady-state solutions shows spiral points suggesting cyclic amorphization behavior, in agreement with the DDE results of Eq. [12]. Beyond phenomenological treatments such as DDE, however, it is desirable to see how atomistic mechanics is working in cyclical transitions. Accordingly, some molecular dynamics simulations are initiated in this work, but much work is clearly needed to answer many of the puzzling questions associated with cyclical phase transformations in driven systems.
5 Summary
Cyclical phase transformations occurring in driven systems, such as hot working or ball milling, are described in terms of a minimization process of the free energies of participant phases. Based on the oscillatory flow behavior of metals with low stacking fault energies during hot working, it is argued that a ductile crystalline phase sustains undulation in its free energy, due to the alternate succession of work-hardening and work-softening mechanisms. A time-dependent free energy function is then obtained by solving a DDE, which accounts for a time lag due to diffusion. The oscillatory behavior of the function is shown to depend both on the recovery capacity and the delay time of a given phase, and thus allows us to understand the systems that would experience cyclical phase transformations. In an attempt to understand cyclical transitions on an atomistic scale, molecular dynamics simulations are also studied, for both two-dimensional and three-dimensional crystals undergoing shear deformation. The two-dimensional results of a nanocrystal shows cyclical transitions between an equilibrium rhombus and a nonequilibrium square phase. The results of the three-dimensional simulations show crystalline-to-glass transitions at high strain rates, and the nature of the transition is either homogeneous or heterogeneous, depending on the availability of defects, such as free surface or stacking faults. However, very high shear strain rates produce a string phase made of linear lines of atoms, when viewed along the shear direction, but of a latticelike network, when viewed perpendicular to the shear direction.
Acknowledgments
The work is prepared in memory of two great materials scientists, Drs. Hubert I. Aaronson and Thomas H. Courtney: for three decades, the author’s life was profoundly enriched by these two gentlemen. The author is also grateful to Professor William C. Johnson for the delightful discussions during the course of the work.