This document summarizes research on modeling instabilities in immiscible multilayer systems using phase field simulations. It describes how a Cahn-Hilliard equation can track the evolution of instabilities over time as an order parameter evolves. Simulation results show two stages of destabilization: initial instability propagation followed by coarsening. Increasing the solubility of individual phases reduces defect instabilities. The timescale of instability is influenced by factors like mobility, gradient energy, and interfacial thickness. Initial instabilities may propagate vertically, potentially stabilizing vertical microstructures. More research is needed to model additional factors like stress and 3D effects.

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Tracking Microstructure Evolution
Stages
Modeling instabilities in immiscible multilayers
Nathan Z. Zhao, Dina V. Yuryev, Michael J. Demkowicz
Instabilities exist and degrade usability in multilayer systems
Three Dimensions
Aspects of 2D simulations preserved, but many new morphological
complexities arise
FEM Simulation Parameters
𝜎 ∝ 𝛼∆𝑓
λ ∝
𝛼
∆𝑓
𝜎 ∝ λ∆𝑓
λ: interfacial thickness
𝜎: interfacial energy
∆𝑓: height of the double
well
𝛼: gradient energy
coefficient
𝝏∅
𝝏𝒕
= 𝜵 ∙ (𝑴(𝒄𝒊 𝒓, 𝒕 )𝜵
𝜹𝑭[𝒄]
𝜹𝒄𝒊(𝒓, 𝒕)
)
𝜇 =
𝛿𝐹
𝛿𝑐𝑖(𝒓, 𝑡)
= −𝛼𝛻2
∅ +
𝜕𝑓 ∅, 𝑇
𝜕∅
Non-constant mobility
Constant mobility
Tracking the Evolution of the Instability
∅: Order Parameter
𝜇: Chemical Potential
𝛼: Gradient Energy coefficient
F: Total Free Energy functional
f: Free Energy Densities
𝝏∅
𝝏𝒕
= 𝑴𝜵 𝟐
𝜶𝜵 𝟐
∅ +
𝝏𝒇 ∅, 𝑻
𝝏∅
Cahn-Hilliard Equation
System with Bulk Transport of Phases: Constant Mobility
Coarsening Layer Evaporates onto
Adjacent Layers (Ostwald Ripening)
System Without Bulk Transport of Phases: Gaussian Mobility
No Ripening, highly unstable: Two Stages to Destabilization of Microstructure
Potential Applications in Materials Processing?
But there are no ways to produce vertically deposited layers on a substrate.
Most layer processes deposit layers horizontally over a large area (sputtering or ARB)
High fan angle, high interface thickness
Characteristics of S1 Instability Propagation
Bulge Geometry independent of all simulation parameters, but does evolve in time
S1 Instability Propagation
Citations
Copper-Niobium Multilayers
• Promising material for energy and defense applications
• Very low mutual miscibility
𝐹 𝑐 = (𝑓ℎ𝑜𝑚 + 𝑓𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒 + 𝑓𝑜𝑡ℎ𝑒𝑟)𝑑𝑉
𝜏 ∝ 𝑘𝑀−1
𝜎−1
λ−𝑛
: 𝑛 > 1, 𝑛 ∈ [3,4], k is a scaling constant that may be a
function of other parameters
• Models evolution of a system with a tendency to phase separate and minimize
interfacial energy
• Order parameter is conserved  mass is conserved for ∅ = c(x)
𝑀 𝑐 = 𝐾𝑒−𝑎𝑐2
+ 𝐵
Optimized Mobility Function
K: control rate of coarsening
a: control rate of ripening
B: modulate bulk transport in pure
phases (best if B = 0)
Concentration
S1: Initial instability propagation S2: Coarsening of S1 features
Low fan angle: small interface thickness
tan(𝜃 𝑓𝑎𝑛) ∝
𝑟(𝜆)ℎ𝑜𝑟𝑖𝑧
𝑟(𝜆) 𝑣𝑒𝑟𝑡
Ratio of vertical/horizontal
instability propagation
𝑓ℎ𝑜𝑚 𝑐 = 0.25 (1 − 𝑐2
)
2
Free Energy :
[1] Zheng A., & Carpenter J.S. (2014). An interface driven Rayleigh instability in high-aspect-ratio bimetallic nanolayered composites. Applied
Physics Letters, 105, 111901
[2] Misra A. & Hoagland R.G., Effects of elevated temperature annealing on the structure and hardness of copper/niobium nanolayered films.
Journal of Materials Research, 20(8),2047-2053
[3] A. Misra & Hoagland R.G. Thermal stability of self-supported nanolayered Cu/Nb films. Philosophical Magazine, 84(10), 1021-1028.
[4] B.G. Chirranjeevi, Abinandanan. A phase field study of morphological instabilities in multilayer thin films. Acta Materiala
[5] Ursell, T.S. Cahn-Hilliard Kinetics and Spinodal Decomposition in a Diffuse System. Caltech., Pasadena, CA pp 1-6
[6] Provatas,Nikolas & Elder, Ken, Phase-Field Methods in Materials Science and Engineering, Weinheim, Germany: Wiley-VCH, 2010
[7] Ballufi et. al. Kinetics of Materials, Hobkoen, NJ: Wiley&Sons Inc., 2005
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0.25
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0.35
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0.45
-1.5 -1 -0.5 0 0.5 1 1.5
FreeEnergy
"Normalized" Concentration
∆𝑓
𝑀 𝑐 𝑥, 𝑦, 𝑧 𝑑𝑉
𝑓𝑑𝑉
“Hole” type defect
Spheroidization of microstructure = significant loss in
strength, conductivity, etc.
Pinch off
Necking
Spheroidization
Analyze two global factors in instability
propagation (via FEM):
Transport characteristics, heterophase energy
Factors left out: strain/stress, anisotropic surface
energies, grain boundaries
AnWave Perturbations
References
Evolution of the initial instability (S1) shows promise for potentially
‘capturing’ and stabilizing a vertical microstructure
This work was supported by the CMSE Research Experience for Undergraduates Program, as part of the MRSEC
Program of the National Science Foundation under grant number DMR-14-19807, and by the MIT Materials Processing
Center
Ostwald Ripening
Vertical protolayer formation as initial instability propagates
“line” type defect
Protolayers are highly perforated for both annulus and hole defects
Motivation: Instabilities in Nanoscale Multilayer Systems
Seed defects, anchoring
defects
Stability governed by
contact with boundary
𝑓𝑖𝑛𝑡𝑒𝑟𝑓𝑎𝑐𝑒 =
𝛼
2
𝛻2
∅
Conclusion: increasing solubility of the multilayer’s individual phases
increases resistance against defect instabilities
Time Scale ≡ time it takes for layer to interfere with adjacent layer of same c(x)
y = 537.88x-0.994
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TimetoFirstPinch-off
Peak Mobility
Timescale vs Peak
Mobility
y = 2.3195x-0.998
R² = 0.99961
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TimetoFirstPinch-off
Gradient Energy…
Timescale vs Gradient
Energy Coefficient
y = 11.258x-3.453
R² = 0.99946
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Interfacial Thickness
Time Scale vs Interface Thickness
Coarsening Geometry is sensitive to interfacial thickness
y = 3586.2x-0.238
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InterfaceEnergy
Time
Evolution of Interface Energy at Long
Time Scales
Instability of ‘welded’ multilayers: higher
misalignment = more random propagation of
instability
Do instabilities ‘imprint’ themselves over
a large section of microstructure?
𝑡1 𝑡2 > 𝑡1
𝜆 = 0.9 𝜆 = 0.5 𝜆 = 0.3
Pinching cascade = vertical growth of
new microstructure
Initial Instability propagates
horizontally
Pinch off creates new
microstructure
45° misalignment
7° misalignment
[2] Misra, &Hoagland,
Wave Perturbations
More research needed
2D: ultra thin interface
3D: thick interface
[1] Zheng, Caprenter, et. al
𝜆 = 0.9 𝜆 = 0.5 𝜆 = 0.3