Dendritic Phase Diagram Analysis

• Dendritic phase diagram is a mapping tool that defines branching growth regimes in alloys and excitable media using key control parameters.
• It delineates boundaries between dendritic, eutectic, and mixed morphologies by correlating variables like alloy composition, temperature gradient, and pulling velocity.
• Phase-field modeling coupled with kinetic analysis enables precise microstructural engineering and predictive control in both metallurgy and neuroscience.

A dendritic phase diagram delineates the regions of distinct growth morphologies arising in systems with branching, tree-like structures subject to phase transitions, most notably in solidifying alloys or in excitable media such as neuronal dendritic arbors. These diagrams map the stability and transitions between dendritic, eutectic, cellular, and planar phases as a function of key physical or control parameters, serving as a foundational tool in predicting and controlling microstructural outcomes in both materials science and neuroscience.

1. Fundamental Concepts and Definitions

The dendritic phase diagram specifies the boundaries separating different growth regimes in systems supporting ramified, dendritic structures. In alloy solidification, the main control variables typically include alloy composition, the temperature gradient ($G$), and the solidification (pulling) velocity ($v$). The diagram often maps these as axes to demarcate regions of pure dendritic growth, pure eutectic growth, and coupled or mixed morphologies (e.g., dendritic-eutectic coexistence). In active dendritic arbors within excitable media, control variables such as spike propagation probability and spike duration define transitions between quiescent and self-sustained active phases, with the diagram illustrating where persistent, avalanche-like activity becomes possible (Seiz et al., 2023, Gollo et al., 2013).

2. Alloy Solidification: Morphological Phase Boundaries

The phase behaviour of dendritic versus eutectic growth in binary alloys (e.g., Al–Cu, Ni–Nb) is governed by the interplay between chemical composition, $G/v$, and thermal/kinetic parameters. The dendritic phase diagram for Al–Cu systems utilizes the alloy mole fraction of Cu in the liquid ($c_0$) on the horizontal axis and the ratio $G/v$ (temperature gradient to pulling speed) on the vertical axis. For each value of $G$ (e.g., $6.18,\,24.7,\,99\,\text{K\,mm}^{-1}$), there exists a boundary curve $c_{0,\text{crit}}(G/v)$ that separates:

• Pure eutectic (E) region: Right of the boundary curve.
• Mixed dendritic + eutectic (D+E) region: Left of the boundary curve.

At large $G/v$ (low velocity), the critical composition approaches the thermodynamic solidus, whereas at low $G/v$ (high velocity), the window for pure eutectic narrows and can vanish entirely (Seiz et al., 2023). This boundary is determined by equating empirical laws for dendrite-tip undercooling (Burden–Kurz type) and eutectic-front undercooling (Jackson–Hunt scaling):

$v$0

with $v$1, $v$2, $v$3, and $v$4 as specific fit coefficients.

Key boundary features include:

• At $v$5, $v$6;
• At $v$7, $v$8.

Phase boundaries derived from two-dimensional phase-field simulations show pure-eutectic runs to the right of these curves and D+E coexistence to the left, in full agreement with empirical and simulation data (Seiz et al., 2023).

3. Orientation-Driven Morphological Transitions in Alloys

Morphological phase diagrams for dendritic growth in alloys under directional solidification, especially relevant to additive manufacturing (AM), examine the role of orientation selection and growth velocity. These diagrams are typically constructed in the $v$9 plane, where $G/v$0 is the misorientation angle between the thermal gradient and the crystalline axes, while $G/v$1 is the Péclet number (with $G/v$2 growth velocity, $G/v$3 primary arm spacing, $G/v$4 liquid diffusivity).

Distinct regions in the $G/v$5 diagram include:

• Aligned/axial dendrites ($G/v$6, $G/v$7): Growth along the temperature gradient, four-fold symmetric tips.
• Tilted dendrites ($G/v$8, $G/v$9): Single-needle tips; tilt angle determined by the Deschamps–Georgelin–Pocheau (DGP) law:

$c_0$0

with $c_0$1, $c_0$2.

• Columnar seaweed ($c_0$3, $c_0$4): Tip splitting, alternately left/right.
• Degenerate seaweed ($c_0$5, $c_0$6, low $c_0$7): Disordered, highly branched fronts.
• Planar front ($c_0$8): No cellularity; occurs beyond the Mullins–Sekerka stability limit (Tiwari et al., 12 Jan 2025).

Physical mechanisms underlying transitions involve the competition between process anisotropy (thermal gradient direction) and crystalline anisotropy, as well as instabilities (Mullins–Sekerka) and kinetic solute-trapping.

4. Phase-Field Modeling and Order Parameters

The construction of dendritic phase diagrams in alloys utilizes multiphase, multicomponent phase-field models. In the case of Al–Cu systems, four order parameters describe microstructural variants and two chemical potentials govern solute transport. The coupled Allen–Cahn and Cahn–Hilliard equations capture interface motion and solute redistribution. Anisotropy is included explicitly via gradient energy terms and interfacial energy coefficients.

Boundary conditions:

• Sides: periodic.
• Bottom: no-flux (for both order parameter and chemical potential).
• Top: Dirichlet (imposed composition and temperature).

Nondimensionalization choices (from (Seiz et al., 2023)):

• Length: $c_0$9 m,
• Time: $G/v$0 s,
• Reference diffusivity: $G/v$1 m$G/v$2/s.

Key simulation variables are systematically scanned to map out phase boundaries and verify the resulting phase diagram’s robustness, including three-dimensional confirmations which reveal consistency of microstructural selection and the influence of interfacial anisotropy (Seiz et al., 2023, Tiwari et al., 12 Jan 2025).

5. Hysteresis and Kinetic Effects in Dendritic Morphologies

Dendritic phase diagrams are not purely equilibrium constructs: kinetic effects, such as velocity jumps during solidification, can induce morphological hysteresis. For instance, upon transient increase and subsequent reduction of pulling speed at fixed $G/v$3 and $G/v$4, a system can irreversibly transit from D+E coexistence to pure eutectic, and the reverse transition may not spontaneously occur unless a dendrite seed survives. This illustrates that the morphological selection exhibits history dependence and is not fully determined by instantaneous control parameters. Eutectic lamellar spacing, dominated by Jackson–Hunt scaling, is less affected by hysteresis—responding rapidly to velocity changes (Seiz et al., 2023).

6. Dendritic Phase Diagrams in Excitable Media

Phase diagrams can be extended to active dendritic arbors modeled as excitable media. In such contexts, the phase space is spanned by spike propagation probability ($G/v$5) and stochastic spike termination probability ($G/v$6). The phase boundary $G/v$7 separates:

• Quiescent phase ($G/v$8 at vanishing input): No self-sustained activity.
• Active phase ($G/v$9 at vanishing input): Persistent, spontaneous and avalanche-like firing.

A key requirement for the active phase is non-deterministic spike durations ($G$0), as deterministic spikes ($G$1) prevent returning waves necessary for sustained activity on loopless arbors (Gollo et al., 2013).

The order parameter $G$2 is the stationary firing rate at the root site. Approaching the critical line $G$3 from below, $G$4 with a mean-field exponent $G$5. The system's dynamic range and gain for weak input signals are maximized precisely at criticality.

7. Practical Implications and Microstructure Control

Dendritic phase diagrams have direct utility for microstructure engineering in metallurgy and additive manufacturing. They provide a predictive roadmap to tailor microstructural features by adjusting alloy chemistry, process velocities, thermal gradients, and deliberate crystal misorientation, enabling control over dendrite-arm spacing, secondary-phase precipitation, and resistance to defects such as hot tearing (Tiwari et al., 12 Jan 2025).

In excitable systems such as neuronal dendritic arbors, the phase diagram implies that computation at criticality ensures maximal sensitivity while avoiding saturation, supporting the hypothesis that analog computation in extended media is optimized at the edge of a phase transition (Gollo et al., 2013).

System	Key Axes	Order Parameter / Boundary
Alloy (Al–Cu)	$G$6 vs $G$7	$G$8 from $G$9
Dendrite AM	$6.18,\,24.7,\,99\,\text{K\,mm}^{-1}$0 vs $6.18,\,24.7,\,99\,\text{K\,mm}^{-1}$1	Morphology by DGP law, primary-arm scaling
Neuronal arbor	$6.18,\,24.7,\,99\,\text{K\,mm}^{-1}$2 vs $6.18,\,24.7,\,99\,\text{K\,mm}^{-1}$3	Active vs quiescent regimes

The unity of the dendritic phase diagram concept across physical and biological systems underscores its foundational role in understanding pattern formation, phase transitions, and the relationship between stochastic dynamics and structure selection.