Precipitation hardening is the foundation of modern high-performance alloys. The strength of nickel superalloys in jet engines comes from a dense dispersion of γ’ particles nanometers in diameter. The creep resistance of 9%Cr ferritic steels in power plant boilers depends on MX carbonitrides and M₂₃C₆ carbides that must be stable for 100,000 hours at service temperature. The age hardening of 2000, 6000, and 7000 series aluminum alloys — the structural material of every commercial aircraft — is entirely a precipitation phenomenon.
In all these systems, the engineering question is not just “will this phase precipitate?” — that is answered by equilibrium thermodynamics. The question is “when does it precipitate, how fast does it grow, what size will the particles reach after a given heat treatment, and how does the particle size distribution evolve as the microstructure coarsens over time?” These are kinetic questions, and they require a model that couples thermodynamics with transport kinetics in a physically rigorous way.
TC-PRISMA is Thermo-Calc’s answer. It is the Precipitation Module — an Add-on to Thermo-Calc that simulates the concurrent nucleation, growth, and coarsening of precipitates in multicomponent alloys under arbitrary heat treatment conditions. This guide explains what TC-PRISMA is, the physical theory it implements, what it produces, where it is applied, and how it fits into the broader Thermo-Calc ecosystem.
What Is TC-PRISMA?
TC-PRISMA (originally developed as a standalone program before integration into Thermo-Calc in 2016) is an Add-on Module to Thermo-Calc that simulates the kinetics of diffusion-controlled multi-particle precipitation in multicomponent, multiphase alloy systems. It requires:
- A Thermo-Calc license
- At least one thermodynamic database (e.g., TCFE for steels, TCNI for nickel alloys, TCAL for aluminum alloys)
- At least one mobility database (e.g., MOBFE, MOBNI, MOBAL) — providing the diffusion kinetics
Like DICTRA, TC-PRISMA operates at the intersection of CALPHAD thermodynamics and CALPHAD-assessed kinetics. The thermodynamic database provides the driving force for nucleation, the equilibrium composition of the precipitate, and the solubility of the precipitate in the matrix. The mobility database provides the diffusion coefficients that control how fast atoms move to feed precipitate growth.
The fundamental difference from DICTRA: DICTRA simulates long-range diffusion and moving phase boundaries — concentration profiles over distances of microns to millimeters. TC-PRISMA simulates the short-range kinetics of many precipitate particles simultaneously — how a population of particles nucleates, grows, and coarsens within a matrix phase. DICTRA does not consider diffusion inside precipitates; TC-PRISMA’s KWN model assumes that the diffusion that matters is in the matrix surrounding each particle, not inside it.
The Physical Framework: Langer-Schwartz and KWN
TC-PRISMA’s computational engine is the Kampmann-Wagner Numerical (KWN) method, an extension of the earlier Langer-Schwartz (LS) and Modified Langer-Schwartz (MLS) approaches.
Why KWN?
The earliest analytical models of precipitation kinetics (Lifshitz-Slyozov-Wagner theory for pure coarsening; simpler nucleation-growth models) assumed specific functional forms for the particle size distribution (PSD) — typically a self-similar distribution that evolves in a predictable way. This assumption fails during the nucleation and growth stages, which are transient and far from self-similar, and it misses the physics of concurrent nucleation, growth, and coarsening that occurs in real heat treatment conditions.
The Kampmann-Wagner Numerical approach resolves this by:
- Discretizing the particle size distribution directly — rather than assuming its shape, KWN tracks the actual number of particles in each size class at each timestep
- Solving the continuity equation of the PSD — the PSD evolves according to a continuity equation driven by nucleation rate (source of new particles at the critical radius), growth/dissolution rate (shifting particles between size classes), and coarsening (Ostwald ripening, which shifts the distribution to larger sizes at the expense of small ones)
- Treating coarsening as emergent — unlike models that treat coarsening as a separate stage requiring ad hoc assumptions, in KWN coarsening emerges naturally from the same equations. Small particles below the critical radius dissolve; large particles grow. The crossover between growth-dominated and coarsening-dominated behavior occurs automatically.
This mathematical framework directly tracks the time evolution of the full particle size distribution — number density, mean radius, volume fraction, and the complete PSD shape — throughout the entire precipitation process.
Nucleation: Classical Theory with Multicomponent Extensions
The Nucleation Barrier
Precipitate nucleation begins with the fluctuational formation of small clusters of atoms with the composition and structure of the new phase. These clusters are thermodynamically unfavorable until they exceed a critical size — below this size, the surface energy cost of creating the precipitate-matrix interface outweighs the volume free energy gain from forming the stable phase.
The classical nucleation rate equation implemented in TC-PRISMA is:
J = Z × β × N × exp(−ΔG / kT) × exp(−τ / t)*
where:
- Z is the Zeldovich factor (accounts for the probability that a critical nucleus grows rather than dissolves)
- β is the atomic attachment rate (how fast atoms add to a critical nucleus — depends on diffusivity, which TC-PRISMA takes from the mobility database)
- N is the number density of available nucleation sites
- ΔG* is the critical free energy barrier for nucleation (from thermodynamics — depends on the chemical driving force and the interfacial energy)
- τ is the incubation time (time to establish the quasi-steady-state nucleation rate)
The multicomponent nature of real alloys enters through β: the atomic attachment rate is controlled by the slowest-diffusing element required to build the precipitate composition. In a nickel superalloy with Al and Ta in γ’, the attachment rate is limited by the slower-diffusing substitutional element, not by the faster interstitial-like diffusers. TC-PRISMA’s β accounts for these multicomponent diffusivity differences explicitly, using mobility data from the paired mobility database.
Nucleation Sites
Where precipitates nucleate is as important as when. TC-PRISMA supports five categories of nucleation sites:
Bulk (homogeneous): Nucleation occurs anywhere within the matrix volume. Requires the highest driving force (highest undercooling or supersaturation) because there is no reduction in the nucleation barrier from defects.
Grain boundaries: Nucleation on the interface between two grains. The grain boundary energy partially offsets the interfacial energy cost of forming the precipitate, reducing ΔG* and enabling nucleation at lower driving forces. The number of available grain boundary sites depends on grain size and grain aspect ratio.
Grain edges (triple junctions): Nucleation at the line where three grains meet. Three grain boundaries contribute to offsetting the nucleation barrier — lower ΔG* than grain boundaries alone. Fewer available sites than grain boundaries, but nucleation is more favorable at each one.
Grain corners (quadruple points): Nucleation at the point where four grains meet. Lowest nucleation barrier (most grain boundary energy available to offset it), but fewest available sites. At the lowest driving forces where no other site type can nucleate, grain corners activate first.
Dislocations: Nucleation on dislocation lines within grains. Dislocation cores act as pipes with locally elevated diffusivity and as strain fields that interact with the misfit strain of the coherent precipitate. The number of available dislocation nucleation sites is specified as a dislocation density (m⁻² — a user-specified microstructural parameter).
The physical progression: As supersaturation increases (deeper undercooling from equilibrium), nucleation sites activate in order from lowest to highest barrier: grain corners → grain edges → grain boundaries → dislocations → bulk. TC-PRISMA can simulate nucleation on multiple simultaneous site types — for example, when both grain boundary carbides and bulk carbides nucleate competitively in a steel.
Non-spherical nuclei and wetting angle: For heterogeneous nucleation on planar substrates (grain boundaries), the nucleus is not spherical — it forms lens-shaped caps on either side of the grain boundary, characterized by a wetting angle. TC-PRISMA allows specification of the wetting angle, which controls the effective volume of the nucleus and the fraction of the spherical surface area actually in contact with the parent phase.
Growth and Coarsening
Growth Kinetics
Once a stable nucleus exists (radius > critical radius), it grows by diffusion of solute from the surrounding matrix to the precipitate-matrix interface. TC-PRISMA models the growth rate as controlled by the diffusion of solute in the matrix phase — not within the precipitate — using a mean-field approximation where each particle grows in an infinite matrix with the mean composition.
The growth rate depends on:
- Supersaturation: The difference between the actual matrix composition and the equilibrium solubility at the interface (given by the Gibbs-Thomson equation for a curved particle surface)
- Diffusivity in the matrix: From the mobility database
- Thermodynamic factor: From the thermodynamic database — converts composition gradients into chemical potential gradients that actually drive diffusion in multicomponent systems
The Gibbs-Thomson Effect
A critically important subtlety: the equilibrium solubility of the precipitate-forming species at the surface of a curved particle is higher than the bulk equilibrium solubility. A small particle “dissolves” more readily than a large one — this is the Gibbs-Thomson effect (also called the capillarity effect). TC-PRISMA applies the Gibbs-Thomson correction using the exact exponential form of the modified Langer-Schwartz theory:
x_eq(r) = x_eq(∞) × exp(2γVm / RTr)
where γ is the interfacial energy, Vm is the molar volume of the precipitate, and r is the particle radius. This correction is essential for correctly predicting coarsening behavior, where large particles grow at the expense of small ones because the small particles have higher effective solubility at their curved surface.
Coarsening (Ostwald Ripening)
As precipitation progresses and supersaturation is consumed, the system enters a coarsening-dominated regime. Individual particles neither predominantly grow nor dissolve — instead, the distribution shifts to larger mean radii and lower number density at constant volume fraction, driven by the Gibbs-Thomson effect.
In the KWN framework, coarsening is not separately modeled. It emerges automatically: particles smaller than the instantaneous critical radius (determined by the current mean matrix composition and interfacial energy) dissolve; particles larger than the critical radius grow. The critical radius itself increases as the matrix composition approaches equilibrium, so progressively larger particles become unstable and dissolve.
The classical LSW (Lifshitz-Slyozov-Wagner) theory predicts that in the pure coarsening limit, mean radius grows as r̄ ∝ t^(1/3) — the cube-root law. KWN recovers this behavior asymptotically while correctly handling the transient nucleation and growth stages that LSW cannot describe.
Inputs and Outputs
What TC-PRISMA Needs
System definition:
- Elements and thermodynamic database (identical to standard Thermo-Calc calculation setup)
- Paired mobility database
Matrix phase: The parent phase from which precipitates form (e.g., FCC_A1 austenite for carbide precipitation in steel, or γ-FCC for γ’ precipitation in nickel superalloys)
Precipitate phase(s): One or more precipitate phases to simulate. TC-PRISMA can simultaneously track multiple precipitate types — for example, γ’ and γ” in nickel superalloys, or M₂₃C₆, M₇C₃, and MX in Cr-Mo steels. Each precipitate requires:
- Phase name (from the thermodynamic database)
- Interfacial energy (γ, in J/m²)
- Nucleation site type and density
- Initial particle size distribution (typically none — starting from a supersaturated single-phase matrix)
Temperature-time profile: The heat treatment cycle. Can be:
- Single isothermal hold
- Linear ramp + hold (solution treat → quench → age)
- Complex multi-step cycle (industrial heat treatment protocol)
- Any arbitrary temperature as a function of time
Key user-controlled parameters:
- Interfacial energy (γ): The most critical and least well-constrained input. TC-PRISMA can estimate γ from thermodynamic data using the Becker model, but this estimate is approximate. For quantitative predictions, γ is typically calibrated against one experimental dataset, then the model is used predictively for other conditions.
- Grain size / dislocation density: For heterogeneous nucleation site density calculations
- Initial matrix composition: Usually the alloy’s nominal composition (assuming a fully homogeneous solution annealed starting point)
What TC-PRISMA Produces
Particle Size Distribution (PSD) evolution: The complete PSD at any time during the simulation — number of particles per unit volume in each size class. This is the most fundamental output, from which all other outputs are derived.
Mean particle radius (r̄) vs. time: The average size of the precipitate population. Directly measurable by TEM or SAXS and the most commonly compared experimental output.
Number density (N_V) vs. time: Total number of particles per unit volume. Peaks early in the precipitation sequence (when nucleation rate is high) and decreases during coarsening.
Volume fraction (f_V) vs. time: Fraction of the microstructure occupied by the precipitate. Approaches the equilibrium volume fraction (from Thermo-Calc) as precipitation completes.
Composition of matrix and precipitate vs. time: As precipitation proceeds, the matrix is depleted in precipitate-forming elements. TC-PRISMA tracks this depletion self-consistently.
Nucleation rate vs. time: How many new particles are nucleating per unit volume per unit time. Peaks early, then drops to zero as supersaturation is consumed.
TTT diagrams (Time-Temperature-Transformation): Computed by running multiple isothermal simulations at different temperatures and finding the time to reach a specified fraction of equilibrium volume fraction (e.g., 1%, 5%, or 50% of equilibrium). The “C-curve” shape of the TTT diagram emerges naturally from the competition between thermodynamic driving force (increases at lower T) and diffusivity (decreases at lower T).
CCT diagrams (Continuous Cooling Transformation): TC-PRISMA can compute precipitation kinetics during continuous cooling, producing CCT diagrams that show what precipitates form and when for a given cooling rate.
Representative Applications
γ’ Precipitation in Nickel Superalloys
The γ’ phase (Ni₃Al, L1₂ crystal structure) is the strengthening precipitate in virtually all nickel-based superalloys used in gas turbine engines. Its volume fraction, mean radius, and size distribution directly determine creep strength, tensile yield strength, and fatigue resistance.
TC-PRISMA models the entire γ’ precipitation sequence during cooling from solution temperature and subsequent aging:
- Nucleation rate as a function of cooling rate and composition
- Mean radius evolution during aging at 750-900°C
- Number density evolution — coarsening kinetics at service temperature (>900°C)
- Competition between γ’ and secondary phases (δ, η in IN718, TCP phases in Re-bearing alloys)
The interfacial energy of γ/γ’ (typically 10-30 mJ/m² — lower than most metallic systems because γ’ is coherent with the γ matrix) is the most important calibration parameter. Once calibrated, TC-PRISMA can predict how changes in alloy composition (Al content, Ti:Al ratio, Ta additions) affect the γ’ kinetics and final microstructure.
Carbide Precipitation in Creep-Resistant Steels
9-12%Cr ferritic-martensitic steels for power plant applications develop complex precipitation sequences during tempering and long-term service:
- M₂₃C₆ carbides (Cr-rich) — primary strengthening precipitate during tempering
- MX carbonitrides (V, Nb rich) — fine, stable precipitates critical for long-term creep strength
- Z-phase (Cr(V,Nb)N) — forms slowly at service temperature, consuming MX and reducing long-term creep resistance
TC-PRISMA simulates all three simultaneously. The competition between M₂₃C₆ and MX during tempering, the long-term coarsening of M₂₃C₆ (which degrades creep strength), and the nucleation of Z-phase within MX particles (particle transformation) can all be tracked. This allows TC-PRISMA to predict the microstructural evolution during 100,000-hour service exposure — a crucial design input for power plant components.
Age Hardening of Aluminum Alloys
6xxx (Al-Mg-Si) and 2xxx (Al-Cu-Mg) aluminum alloys are strengthened by precipitation during aging. TC-PRISMA simulates:
- β” and β’ precipitation in 6xxx alloys during T6 aging
- S phase (Al₂CuMg) and θ’ (Al₂Cu) precipitation in 2xxx alloys
- Natural aging vs. artificial aging kinetics
- Overaging — when the peak hardness is passed and particles coarsen
The mean radius and number density outputs from TC-PRISMA can be directly connected to strengthening models (dispersed barrier hardening theory) to predict hardness and yield strength vs. aging time — enabling optimization of aging protocols.
AlN Precipitation in Deep-Drawing Steels
A classical application with one of TC-PRISMA’s first published validations: precipitation of aluminum nitride (AlN) in low-carbon steels during hot rolling and coiling. AlN precipitation affects texture development and therefore the deep drawability of the final sheet. TC-PRISMA predicts when and where AlN precipitates during the thermomechanical process, enabling coiling temperature optimization.
Laves Phase in Additive Manufacturing of Stainless Steels
For laser powder bed fusion (LPBF) processed ferritic stainless steels (e.g., AISI 441 with Nb additions), TC-PRISMA has been used to model Laves phase (Fe₂Nb) precipitation during post-heat treatment. The model, calibrated with TEM and SEM measurements, can optimize post-AM heat treatment to achieve target Laves phase size and distribution for desired mechanical properties — an example of TC-PRISMA’s application in the growing additive manufacturing materials design space.
Interfacial Energy: The Critical Parameter
The interfacial energy γ between the precipitate and matrix is the single most important and least precisely known input parameter in TC-PRISMA. It enters the nucleation barrier as γ³ (in the critical nucleus energy ΔG* ∝ γ³/ΔG_v²), so errors in γ have a strong nonlinear effect on predicted nucleation rates.
Sources of γ estimates:
- Becker model / thermodynamic estimation: TC-PRISMA can estimate γ from thermodynamic data — the fraction of solute atoms at the interface and their binding energies. Provides a starting-point estimate, typically accurate within a factor of 2.
- Database-provided values: Some databases include assessed interfacial energy values for specific precipitate/matrix pairs
- Experimental calibration: The most reliable approach — fit γ to match mean radius or volume fraction at a specific temperature and time, then use the calibrated γ for prediction across other conditions
- Function of temperature or radius: For some systems, γ depends on temperature or particle size (coherent→semi-coherent transition). TC-PRISMA supports user-defined interfacial energy functions.
The coherency of the precipitate strongly affects γ. Fully coherent precipitates (crystal structure continuous with matrix, no interface dislocations, small lattice misfit) have very low γ — typically 10-50 mJ/m². Semi-coherent precipitates (coherency maintained by interface dislocations) have intermediate γ — 50-200 mJ/m². Incoherent precipitates (no crystallographic relationship with matrix) have high γ — 200-1000 mJ/m². This is physically reasonable: the interface energy reflects the crystallographic disruption at the boundary.
TC-PRISMA and DICTRA Together
TC-PRISMA and DICTRA (the Diffusion Module) operate at complementary length scales and together cover the full kinetic landscape of solid-state transformations:
| DICTRA | TC-PRISMA | |
|---|---|---|
| Scale | Microns to millimeters | Nanometers to micrometers |
| Geometry | 1D spatial profiles | Mean-field (no spatial coordinate) |
| What it tracks | Composition profiles, moving boundaries | Particle size distribution, nucleation rate |
| Typical applications | Carburization, homogenization, interdiffusion | Age hardening, precipitate stability |
Sequential coupling: DICTRA provides the composition profile after a diffusion process (e.g., homogenization or carburization), and TC-PRISMA takes the resulting matrix composition at each depth as its starting condition, calculating how precipitates form and grow at each position. This enables full microstructure prediction through a complete thermomechanical process.
Both modules can be accessed through TC-Python for automated, high-throughput calculations.
TC-PRISMA vs. Other Precipitation Simulation Tools
MatCalc (TU Wien): The most capable alternative to TC-PRISMA for precipitation kinetics. Also uses a KWN-based framework, also coupled to CALPHAD thermodynamics (using its own databases or imported .tdb files). MatCalc has some capabilities TC-PRISMA lacks — notably, inner-particle nucleation (phase transformation within an existing precipitate) and subgrain boundary nucleation. TC-PRISMA has tighter integration with Thermo-Calc’s proprietary databases (TCFE, TCNI, TCAL) which are more widely cited in the literature.
PanEvolution (CompuTherm Pandat): Pandat’s precipitation module, conceptually similar to TC-PRISMA and using Pandat’s databases. A closer functional equivalent.
MICRESS: A phase-field code that can simulate precipitation with full spatial resolution. Computationally much more expensive than KWN-based tools, but captures particle shape evolution, spatial correlations between particles, and 3D microstructure. For most industrial applications, TC-PRISMA’s mean-field KWN is sufficient and far faster.
The key reason to choose TC-PRISMA for publication-grade research: access to TCFE, TCNI, and TCAL databases, which are the most extensively validated and most cited thermodynamic databases in the materials science literature. For work where reviewers scrutinize the thermodynamic database, Thermo-Calc’s databases carry the most credibility.
Limitations
Mean-field assumption: TC-PRISMA assumes each particle grows in a uniform matrix with the mean composition. It does not account for spatial correlations between particles, depletion zones around individual particles, or preferential nucleation at specific microstructural features in 3D. For very dense particle distributions, this assumption becomes less accurate.
Spherical particles: TC-PRISMA assumes spherical precipitates for the KWN calculation. Real precipitates are often plate-shaped (θ’ in aluminum alloys), needle-shaped (β” in 6xxx alloys), or cuboidal (γ’ in nickel superalloys). Non-spherical morphologies affect growth kinetics (different diffusion geometry) and coarsening rates. TC-PRISMA supports some non-spherical corrections through the wetting angle parameter for grain boundary nucleation, but full 3D shape evolution is not captured.
No internal precipitate diffusion: TC-PRISMA assumes the precipitate composition is determined by local equilibrium at its surface — no diffusion gradients within the particle. This is a good approximation for small particles but may fail for large particles at high temperatures where internal diffusion redistributes the composition.
Single matrix phase: TC-PRISMA simulates precipitation within one matrix phase at a time. For two-phase matrix systems (e.g., dual-phase steel where precipitates form in both ferrite and martensite simultaneously), the problem must be decomposed.
Summary
TC-PRISMA is the computational tool for quantitative prediction of precipitation kinetics in multicomponent alloys — from the first nanoseconds of nucleation through the extended coarsening of an aged microstructure. Its KWN framework correctly handles concurrent nucleation, growth, and coarsening without the simplifying assumptions that earlier analytical models required, and its coupling to CALPHAD thermodynamics and kinetics databases enables no-adjustable-parameter prediction of precipitation sequences in real multicomponent alloys.
For any materials engineering question about how heat treatment time and temperature control precipitate size, number density, volume fraction, or TTT/CCT behavior — TC-PRISMA is the tool of record. Its integration with DICTRA for long-range diffusion effects and with TC-Python for automated screening of alloy compositions and heat treatment schedules makes it the kinetic simulation workhorse of the Thermo-Calc ecosystem.
For TC-PRISMA and Thermo-Calc licensing assistance, contact via Telegram: t.me/DoCrackMe
Also see: Thermo-Calc DICTRA — Complete Guide to the Diffusion Module | TC-Python Guide — Automating Thermo-Calc with Python | Thermo-Calc 2026a — What’s New: Complete Release Overview | Thermo-Calc vs FactSage vs Pandat — CALPHAD Software Compared
