Phase transformations are the processes by which a material changes from one phase (or mixture of phases) to another. These transformations are fundamental to materials science because they allow us to control microstructure and, consequently, the properties of materials. Nearly every heat treatment, solidification process, and aging procedure relies on our understanding of how and why phases transform.
Consider steel: the same composition of iron and carbon can yield dramatically different properties depending on how it is cooled from high temperature. Slow cooling produces soft, ductile pearlite. Rapid quenching creates hard, brittle martensite. The difference lies entirely in the phase transformation pathway. Understanding these transformations gives engineers the power to tailor material properties for specific applications.
Phase transformations can be broadly classified into several categories:
Every phase transformation involves two fundamental questions: (1) Will it happen? and (2) How fast will it happen? Thermodynamics answers the first question by determining which phases are stable under given conditions. Kinetics answers the second by describing the rates of nucleation and growth. A transformation may be thermodynamically favorable but kinetically sluggish, or vice versa. Mastering both aspects is essential for controlling microstructure.
The driving force for any phase transformation is a reduction in the Gibbs free energy of the system. At equilibrium, a system resides in its lowest free energy state. When conditions change (temperature, pressure, or composition), a different phase may become more stable, providing the thermodynamic impetus for transformation.
The Gibbs free energy is defined as:
Where: $G$ = Gibbs free energy, $H$ = enthalpy, $T$ = absolute temperature, $S$ = entropy
For a transformation to occur spontaneously, the change in Gibbs free energy must be negative:
At high temperatures, the $T\Delta S$ term dominates, favoring phases with high entropy (typically disordered phases like liquids or high-temperature solid solutions). At low temperatures, enthalpy dominates, favoring phases with strong bonding and low energy.
The equilibrium transformation temperature $T_e$ is where the free energies of two phases are equal. However, transformations rarely occur exactly at $T_e$. For a transformation to proceed at a measurable rate, we need a driving force, which requires departing from equilibrium.
Undercooling ($\Delta T$) is the difference between the equilibrium temperature and the actual temperature during cooling:
Where: $\Delta T$ = undercooling, $T_e$ = equilibrium transformation temperature, $T$ = actual temperature
Greater undercooling provides a larger driving force ($\Delta G_v$, the volume free energy change), but it also affects atomic mobility, which decreases at lower temperatures.
For solidification or solid-state transformations, the volume free energy change per unit volume can be approximated as:
Where: $\Delta G_v$ = volume free energy change, $\Delta H_f$ = latent heat of transformation, $\Delta T$ = undercooling
This expression shows that the driving force for transformation increases linearly with undercooling, a key concept for understanding nucleation behavior.
Nucleation is the first step in any phase transformation. It involves the formation of small clusters (nuclei) of the new phase within the parent phase. These nuclei must reach a critical size before they can grow spontaneously. Understanding nucleation is crucial because it often controls when and where transformations begin.
Homogeneous nucleation occurs uniformly throughout the parent phase, without any preferential sites. Consider the formation of a spherical nucleus of radius $r$. The total free energy change has two competing contributions:
The total free energy change for forming a spherical nucleus is:
Where: $\Delta G_v$ = volume free energy change (positive value, energy released per unit volume), $\gamma$ = interfacial energy per unit area
For small nuclei, the surface term dominates and $\Delta G > 0$. As the nucleus grows, the volume term eventually wins. The critical radius $r^*$ occurs at the maximum of the $\Delta G$ curve:
The activation energy barrier for nucleation is:
Nuclei smaller than $r^*$ tend to dissolve (shrinking reduces their free energy), while nuclei larger than $r^*$ grow spontaneously.
In practice, homogeneous nucleation is rare. Most nucleation occurs heterogeneously at preferential sites: grain boundaries, dislocations, inclusions, or container walls. These sites reduce the effective surface energy, lowering the nucleation barrier.
For nucleation on a flat surface, the barrier is reduced by a shape factor $S(\theta)$:
Where: $S(\theta) = \frac{(2 + \cos\theta)(1 - \cos\theta)^2}{4}$ and $\theta$ = contact angle
When $\theta = 180°$ (no wetting), $S(\theta) = 1$ and we recover homogeneous nucleation. When $\theta \to 0°$ (perfect wetting), $S(\theta) \to 0$ and nucleation occurs with essentially no barrier.
The nucleation rate (number of nuclei formed per unit volume per unit time) follows an Arrhenius-type expression:
Where: $N$ = nucleation rate, $K_1$ = pre-exponential factor, $\Delta G^*$ = activation energy for nucleation, $k$ = Boltzmann constant
This rate increases with undercooling (which decreases $\Delta G^*$) but eventually decreases at very low temperatures due to reduced atomic mobility.
Problem: Calculate the critical radius for the solidification of pure copper at an undercooling of 200 K. Given: $\gamma = 0.177$ J/m², $\Delta H_f = 1.88 \times 10^9$ J/m³, $T_m = 1358$ K.
Solution:
First, calculate the volume free energy change:
$$\Delta G_v = \frac{\Delta H_f \cdot \Delta T}{T_m} = \frac{(1.88 \times 10^9)(200)}{1358} = 2.77 \times 10^8 \text{ J/m}^3$$
Now calculate the critical radius:
$$r^* = \frac{2\gamma}{\Delta G_v} = \frac{2(0.177)}{2.77 \times 10^8} = 1.28 \times 10^{-9} \text{ m} = 1.28 \text{ nm}$$
This critical nucleus contains only about 100 atoms, illustrating just how small critical nuclei are.
Once stable nuclei form, they grow by the addition of atoms from the parent phase. The rate and manner of growth depend on whether the transformation is controlled by processes at the interface or by long-range diffusion.
In interface-controlled growth, atoms can readily diffuse to the interface, but attachment to the growing phase is the rate-limiting step. The growth rate depends on the atomic mobility at the interface and follows:
Where: $v$ = growth velocity, $Q$ = activation energy for atomic motion across interface, $\Delta G$ = driving force
For small driving forces, this simplifies to a linear dependence on undercooling. Interface-controlled growth is common in pure metals and in transformations involving no composition change.
When the new phase has a different composition than the parent phase, atoms must diffuse over significant distances. The growth rate is then limited by how fast solute can be transported to or away from the interface.
For a planar interface, the growth rate can be approximated as:
Where: $D$ = diffusion coefficient, $x$ = characteristic diffusion distance, $C_0$ = bulk composition, $C_\alpha$ and $C_\beta$ = equilibrium compositions
As the transformation proceeds, the diffusion distance increases, causing the growth rate to decrease with time. This leads to parabolic growth kinetics where the interface position varies as $\sqrt{Dt}$.
Growth rate generally increases with temperature because diffusion is thermally activated. However, the driving force (undercooling) decreases as temperature approaches the equilibrium temperature. This creates a maximum in the growth rate at intermediate temperatures, similar to what we see in overall transformation kinetics.
The overall rate of a phase transformation depends on both nucleation and growth occurring simultaneously. The classic treatment of this problem was developed independently by Johnson, Mehl, Avrami, and Kolmogorov, leading to what is commonly called the JMAK equation (or simply the Avrami equation).
The fraction transformed $y$ as a function of time follows:
Where: $y$ = volume fraction transformed, $k$ = rate constant (temperature-dependent), $t$ = time, $n$ = Avrami exponent
This equation produces the characteristic S-shaped (sigmoidal) transformation curve when $y$ is plotted against time. Initially, transformation is slow (nucleation-dominated), then accelerates (growth of many nuclei), and finally slows as the untransformed regions become exhausted.
The exponent $n$ contains information about the nucleation and growth mechanisms:
| $n$ Value | Interpretation |
|---|---|
| 4 | Constant nucleation rate, 3D growth |
| 3 | Site saturation (all nuclei form at $t=0$), 3D growth; or constant nucleation, 2D growth |
| 2 | Site saturation, 2D growth; or constant nucleation, 1D growth |
| 1 | Site saturation, 1D growth |
Values between these integers indicate mixed modes or changing mechanisms during transformation.
Taking the double logarithm of the JMAK equation yields a linear form:
Plotting $\ln[-\ln(1-y)]$ versus $\ln t$ gives a straight line with slope $n$ and intercept $\ln k$. This is the standard method for extracting JMAK parameters from experimental data.
Problem: The isothermal transformation of austenite to pearlite at 600°C has $n = 3$ and $k = 5 \times 10^{-5}$ s$^{-3}$. Calculate (a) the time for 50% transformation and (b) the fraction transformed after 60 seconds.
Solution:
(a) For 50% transformation, $y = 0.5$:
$$0.5 = 1 - \exp(-kt^n)$$
$$\exp(-kt^n) = 0.5$$
$$kt^n = \ln 2 = 0.693$$
$$t = \left(\frac{0.693}{k}\right)^{1/n} = \left(\frac{0.693}{5 \times 10^{-5}}\right)^{1/3} = 24.0 \text{ s}$$
(b) For $t = 60$ s:
$$y = 1 - \exp[-(5 \times 10^{-5})(60)^3]$$
$$y = 1 - \exp(-10.8) = 1 - 2.0 \times 10^{-5} \approx 0.99998$$
The transformation is essentially complete after 60 seconds.
TTT diagrams (also called isothermal transformation diagrams or IT diagrams) are powerful tools for understanding and predicting phase transformations. They show the time required for a transformation to begin and end at different constant temperatures.
A TTT diagram plots temperature on the vertical axis and time (usually logarithmic) on the horizontal axis. The diagram contains curves showing when transformation starts (typically 1% transformed) and when it finishes (typically 99% transformed). The region to the left of the start curve represents untransformed parent phase; the region to the right of the finish curve represents the fully transformed product.
TTT diagrams typically show a characteristic C-shape (or nose shape). This arises from the competing effects of driving force and diffusion:
To use a TTT diagram for isothermal heat treatment:
Many TTT diagrams show regions for different transformation products. For steel:
While TTT diagrams describe isothermal transformations, most practical heat treatments involve continuous cooling. CCT diagrams address this by showing transformation behavior during constant-rate cooling from the austenitizing temperature.
CCT diagrams look similar to TTT diagrams but have important differences:
To predict microstructure from a CCT diagram:
The critical cooling rate is the minimum cooling rate that avoids all diffusion-controlled transformations, producing a fully martensitic structure. On a CCT diagram, this is the cooling curve that just misses the nose of the pearlite/bainite curves.
Practical note: Alloying elements in steel shift the CCT curves to the right (longer times), reducing the critical cooling rate. This is why alloy steels are more "hardenable" than plain carbon steels.
Moderate cooling rates often produce mixed microstructures. For example, a cooling curve might pass through the pearlite region, then the bainite region, and finally cross the $M_s$ line. The resulting microstructure would contain pearlite, bainite, and martensite. The fractions of each depend on how much time the cooling curve spends in each transformation region.
The martensitic transformation is fundamentally different from diffusion-controlled transformations. It is a diffusionless, displacive transformation where atoms move cooperatively by less than one atomic spacing to create a new crystal structure. This transformation is responsible for the remarkable hardness achievable in quenched steels.
The martensite start temperature ($M_s$) is where martensite first begins to form upon cooling. The martensite finish temperature ($M_f$) is where transformation is complete. These temperatures depend strongly on composition:
Where element symbols represent weight percentages. This is an empirical formula for steels.
Carbon has the strongest effect: increasing carbon content dramatically lowers $M_s$. High-carbon steels may have $M_f$ below room temperature, leaving some retained austenite after quenching.
In steel, martensite has a body-centered tetragonal (BCT) structure. The tetragonality arises because carbon atoms, which fit into octahedral interstitial sites in FCC austenite, are trapped in the BCT structure and preferentially occupy one type of interstitial site, causing the unit cell to elongate in one direction.
The tetragonality ratio $c/a$ increases with carbon content:
This tetragonal distortion is responsible for the extreme hardness of martensite: it creates a highly strained lattice that strongly resists dislocation motion.
Martensite in steels appears in two main morphologies:
Heat treatment processes manipulate phase transformations to achieve desired properties. Understanding TTT and CCT diagrams allows us to design heat treatment schedules that produce specific microstructures.
Annealing involves heating steel into the austenite region and cooling slowly (typically furnace cooling). The slow cooling allows complete transformation to the equilibrium phases (ferrite + pearlite for hypoeutectoid steels, or pearlite + cementite for hypereutectoid steels). Annealing produces:
Normalizing involves austenitizing followed by air cooling, which is faster than furnace cooling. This produces finer pearlite than annealing, resulting in:
Quenching involves rapid cooling (in water, oil, or polymer solutions) to avoid diffusion-controlled transformations and produce martensite. The cooling rate must exceed the critical cooling rate for the specific steel composition. Quenching produces:
As-quenched martensite is too brittle for most applications. Tempering involves reheating quenched steel to temperatures below the eutectoid temperature (typically 150-650°C) to improve toughness while retaining some hardness.
During tempering, several processes occur:
Problem: A 1045 steel component requires a surface hardness of 50 HRC with reasonable toughness. Design an appropriate heat treatment.
Solution:
1. Austenitize: Heat to 850°C and hold for 1 hour per inch of thickness to ensure complete austenitization and carbon dissolution.
2. Quench: Use oil quenching for 1045 steel to achieve full martensitic transformation while minimizing distortion and cracking risk. (Water quench is too severe for this section size.)
3. Temper: From tempering charts for 1045 steel, tempering at approximately 350°C for 1-2 hours will yield approximately 50 HRC while improving toughness significantly over as-quenched martensite.
4. Verify: After treatment, measure hardness and perform impact testing if critical to ensure specifications are met.
Precipitation hardening (also called age hardening) is a heat treatment technique that produces a fine dispersion of precipitates within a metal matrix. These precipitates act as obstacles to dislocation motion, dramatically increasing strength. This mechanism is essential for high-strength aluminum alloys, nickel superalloys, and many other engineering materials.
Not all alloy systems can be precipitation hardened. The requirements are:
Step 1: Solution Treatment (Solutionizing)
The alloy is heated to a temperature where all solute dissolves into a single-phase solid solution, then held to ensure homogeneity.
Step 2: Quenching
Rapid cooling traps the solute in solution, creating a supersaturated solid solution (SSSS). The material is relatively soft at this stage.
Step 3: Aging
The supersaturated alloy is held at an intermediate temperature (or even room temperature for "natural aging"). Solute atoms diffuse and cluster, eventually forming precipitates. This is where hardening occurs.
Precipitation typically occurs through a sequence of metastable phases before reaching equilibrium. For the classic Al-Cu system:
$$\text{SSSS} \to \text{GP zones} \to \theta'' \to \theta' \to \theta \text{ (equilibrium)}$$
The degree of hardening depends on whether precipitates are coherent (atomic planes continuous across the interface), semi-coherent, or incoherent:
Maximum hardness occurs at an intermediate stage where the combined resistance from coherency strain and particle bypass is greatest.
Prolonged aging or aging at too high a temperature causes precipitates to coarsen (Ostwald ripening). The spacing between precipitates increases, making it easier for dislocations to bypass them. This "overaging" reduces hardness below the peak value.
Problem: A 2024-T4 aluminum component needs to be aged to maximum hardness (T6 condition). Design the heat treatment.
Solution:
1. Solution treatment: If not already in T4 condition, heat to 495°C and hold for sufficient time to dissolve all Cu and Mg into solution.
2. Quench: Rapidly quench in water to create supersaturated solid solution. (T4 condition means material was naturally aged at room temperature after quench.)
3. Artificial aging: Heat to 190°C and hold for approximately 10-12 hours. This temperature-time combination produces peak hardness in 2024 alloy by forming fine $\theta'$ and S' precipitates.
4. The resulting T6 temper provides yield strength of approximately 345 MPa, compared to about 290 MPa for the T4 condition.
When metals are deformed plastically, their microstructure becomes distorted and their stored energy increases. Subsequent heating can activate several processes that restore a more equilibrium microstructure: recovery, recrystallization, and grain growth. These processes are fundamental to metal processing and property control.
Recovery occurs at relatively low temperatures (below the recrystallization temperature). It involves:
During recovery, some stored energy is released, internal stresses decrease, and electrical conductivity improves. However, the original grain structure is retained, and only modest softening occurs.
Recrystallization is the formation of new, strain-free grains that consume the deformed microstructure. It is driven by the stored energy of cold work and proceeds by nucleation and growth.
Key characteristics:
The recrystallization temperature is defined as the temperature at which a cold-worked metal completely recrystallizes in one hour. It is not a fixed property but depends on several factors:
Where $T_m$ = absolute melting temperature. This is a rough approximation; actual values depend on purity and prior deformation.
Factors that lower the recrystallization temperature:
After recrystallization is complete, continued heating causes grain growth. Grain boundaries migrate to reduce the total grain boundary area (and thus the total interfacial energy).
Grain growth kinetics typically follow:
Where: $d$ = grain diameter, $d_0$ = initial grain diameter, $n$ = grain growth exponent (often $\approx 2$), $K$ = rate constant (temperature-dependent), $t$ = time
The rate constant follows an Arrhenius relationship:
Fine grain size is generally desirable because it increases strength (Hall-Petch relationship) and often improves toughness. Grain growth can be inhibited by:
Problem: A 70-30 brass sheet was cold rolled to 50% reduction and has a hardness of 160 HV. The material needs to be softened for further forming. The recrystallization temperature for this brass at 50% cold work is approximately 300°C. Design an annealing treatment that produces a fine, recrystallized grain structure.
Solution:
1. Temperature selection: Choose a temperature just above the recrystallization temperature to allow recrystallization but minimize grain growth. An annealing temperature of 350°C provides adequate driving force.
2. Time selection: At 350°C, recrystallization should be complete in 30-60 minutes for typical sheet thicknesses. Use 45 minutes as a starting point.
3. Cooling: Air cool after annealing. (Brass does not undergo phase transformations that would require controlled cooling.)
4. Expected results: Hardness should drop to approximately 70-80 HV, with a fine recrystallized grain structure suitable for further forming operations.
5. Caution: Higher temperatures or longer times would produce coarser grains, reducing formability and strength.
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