Supplementary Material
This document contains supplementary material related to the following manuscript:
Manuscript Title: "Critical Sizes for the Stabilization of Coherent Precipitates"
Authors: Arun Kumar, Monika Gautam and Anandh Subramaniam
Journal: Journal of Applied Physics
The effect of spherical domain on the energy landscape and critical size (r*) has been considered in this
supplementary material. In spherical domains the amount of solute is limited and hence it is better to visualize
the precipitate as an inclusion (of various sizes) with a certain misfit with the matrix (and not as a growing
precipitate).
A schematic of the numerical model used to simulate the stress state of the spherical Fe (with 2wt.% Cu)
inclusion in Cu-2wt.% Fe spherical matrix, is shown in Figure 1. Isotropic material properties used in the
simulations are same as mentioned in the original manuscript. The stress state of the coherent inclusion is
simulated by imposing eigen-strains in region P (marked in Figure 1) corresponding to the lattice misfit present
at the interface between the inclusion and the matrix. An interfacial misfit dislocation loop is simulated by
imposing eigen-strains in region D (marked in Figure 1) corresponding to the insertion of a disc of atoms in the
inclusion.
The domain is meshed with 4-noded bilinear quadrilateral elements of mesh size b×b. Displacement
boundary conditions are imposed as in Figure 1 and axisymmetry is along the y-axis. Simulations of the stress
state and the computation of the energies are performed by assuming linear elasticity theory and the finite
element model is implemented using ABAQUS/Standard FEM software (Version 6.81, 2008). Inset to Figure 1
shows the orientation of interfacial misfit dislocation loop with respect to the crystallographic directions/plane.
The critical size (r*), is determined by plotting the strain energy of the coherent (EC) and semicoherent (ESC)
inclusion as a function of its radius (rp). The value of r*, corresponds to the cross-over of the two curves. Special
attention is paid to the nature of these plots (and further to the intersections of these curves beyond r*), in
comparison to such plots for large scale domains. The abovementioned process is repeated for various values of
the domain radius (R). The critical domain size (R*) is determined from the plot of r* versus R; where R* is the
radius of the domain below which the EC and ESC curves do not intersect (i.e. there is no r*).
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FIG.1. Schematic of the finite element model, used for the simulation of the stress state of an inclusion and an
interfacial misfit edge dislocation loop, in a finite domain of radius 'R'. Boundary conditions are also marked in
the figure. Stress-free strains are imposed in regions P & D to simulate the stress state of a coherent inclusion
and an interfacial misfit dislocation loop respectively. The value of 'r p' is increased to simulate a growing
inclusion. Inset shows the crystallographic orientation of the inclusion and the dislocation loop.
Figure 2 shows the stress state of the system (plot of y contours) in the absence (Figure 2a) and presence
(Figure 2b) of an interfacial misfit edge dislocation loop (for R = 120b, rp = 80b). Domain deformations (with
deformation scale factor 10) due these stresses are to be noted.
Figure 3 highlights the fact that the strain energy of an inclusion of fixed radius (rp) in a finite domain is
lower than that of the same inclusion in an infinite domain. In Figure 3, the strain energy of the coherent system
with an inclusion of radius 30b is plotted with domain size (R). It is seen that the energy of the system
decreases steeply with a decrease in domain size (R) below 160b (~403Å); wherein nanoscale effects become
prominent (same as mentioned in the main manuscript).
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(a)
(b)
FIG. 2. Plot of the FEM simulated y stress contours in a symmetrical half of the domain: (a) corresponding to
the formation of a coherent inclusion of rp = 80b, (b) semi-coherent inclusion with an interfacial misfit
dislocation. Domain size (R) is 120b and the region shaded grey corresponds to the inclusion. The stress plots
are in the deformed configuration (deformation scale factor = 10). The undeformed domain is marked with a
dashed line.
FIG. 3. Reduction in the strain energy of a coherent inclusion (of fixed radius 30b) with decreasing domain size
'R'. Percentage reduction in energy for a domain of size 60b is also marked (point P). Nanoscale effects become
prominent in the region shaded grey.
Figure 4 shows the variation in the strain energy with the radius of the inclusion for system without (EC)
and with (ESC) a misfit dislocation loop (for a domain size of 160b ( 403Å)). Plot of theoretical equation (1) (in
main manuscript) is also shown for comparison. It is seen that for small inclusions the system behaves like an
infinite one and the FEM results match reasonably well with the theoretical equation (as seen in inset to
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Figure 4). For larger sizes of the inclusion, finiteness of the domain and domain deformations play an
increasingly important role. This results in two important effects: (i) change in the curvature of the E-rp plot
after a certain rp (labeled as rpinfl ), (ii) decrease in the energy of the system with increasing rp, beyond a certain rp
(labeled as rpmax ). The curves (a) and (c) intersect at two points (as in the case for cylindrical domain, shown in
the original manuscript), which gives the value of the usual r* and new critical size r2* . Beyond r2* is a second
region of the stability of the coherent state. The energy of the semi-coherent state is lower for inclusion sizes
between r* and r2* .
Figure 5 shows the variation of normalized critical sizes ((a) r */R and (b) r2* /R) with domain size. In the
region enclosed by curves (a) and (b) the semi-coherent state is stable. This is similar to a phase diagram where
the regions of stability of the coherent state and semi-coherent state have been delineated. As the domain size
(R) decreases, r*/R and r2* /R approach each other and at R = 100b the curves (a) and (b) in Figure 5 converge.
The E versus rp plot for R = 100b is shown in inset-1 (of Figure 5). It is seen that energy curve for the coherent
inclusion becomes tangential to the curve for the inclusion with a misfit dislocation loop (i.e. there is no
intersection). For R < 100b, the E-rp plot for the coherent inclusion is always lower than that for the semicoherent inclusion (i.e. the coherent state is stabilized for all rp). Example of such E-rp plot is shown in inset-2
for R = 90b. The value of R = 100b has been labeled as R*, which is the critical domain size below which a
coherent inclusion is stabilized for all rp. The r* and r2* values have been normalized with 'R', to accommodate
for the fact that there is an increase in r2* due to the increasing domain size (i.e. the coherent state become stable
when the inclusion size approaches that of the domain). The results can be summarized as follows: (i) R > 100b,
ESC < EC for rp  (r*, r2* ), (ii) R = 100b, ESC > EC for all rp except for one rp (where r* = r2* ), (iii) R < 100b, ESC >
EC for all rp.
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FIG. 4. Variation of strain energy with the radius of the inclusion (rp): (a) coherent inclusion (EC, FEM), (b)
coherent inclusion (theoretical Equation (1)), (c) inclusion with a misfit dislocation loop (ESC, FEM). The points
of intersection of the curves (a) & (c) correspond to r * and r2* (details explained in text). Domain size is 160b.
The region where the semicoherent state is stable has been shaded in grey. Inset shows the plot of E C (FEM) &
EC (Eq. (1) of main manuscript) for rp  [10b,70b].
FIG. 5. Plot of variation of normalized critical sizes ((a) r */R and (b)
r2* /R) with domain size ‘R’. The plot is
akin to a phase diagram where the semi-coherent state is stable between curves (a) & (b)– hashed region of plot.
E-rp curves (for coherent & semi-coherent inclusions) are included for R = 100b (in Inset-1) and R = 90b (in
Inset-2)
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