SUPPLEMENTARY INFORMATION
“Surface Morphological Stabilization of Stressed Crystalline Solids by
Simultaneous Action of Applied Electric and Thermal Fields”
Dwaipayan Dasgupta, Georgios I. Sfyris, M. Rauf Gungor, and Dimitrios Maroudas
Department of Chemical Engineering, University of Massachusetts Amherst,
Amherst, MA 01003 – 3110
In this supplementary document, we present in considerable detail some aspects of the
analysis of surface morphological stabilization that were outlined in the main article.
Specifically, we provide a detailed description and discussion of (i) the solution of the thermal
boundary-value problem (BVP); (ii) all the possible orientations of the applied fields, namely,
the applied electric field and thermal gradient for a crystalline solid under uniaxial tension; and
(iii) the linear stability analysis taking into account the temperature dependence of the material’s
properties. This description and discussion does not include any findings of the study that were
not reported in the main article; it merely clarifies further certain points in the analysis of the
main article.
1
Solution of the Thermal Boundary-value Problem
As stated in the main article, we study the surface morphological stability of a singlecrystalline elastic conductor in uniaxial tension and under the simultaneous action of an electric
field and a temperature gradient. We take into account the resulting surface electromigration
(EM), surface thermomigration (TM), and stress-driven surface diffusion in addition to the
curvature-driven surface diffusion. The elastic solid extends infinitely in the y-direction of a
Cartesian frame of reference, its surface is traction-free, and both its surface and its base are
electrically and thermally insulated.
The problem is solved in a two-dimensional (2D)
representation assuming no morphological variation along z. In the main article, the linearized
height evolution equation for this problem, Eq. (3), was derived; this was based on the governing
equations, Eqs. (1) and (2) of the main article, for the evolution of the surface morphology,
h(x,t), and the solution of the three BVPs for the three contributions to the surface flux, namely,
an elastostatic, an electrostatic, and a thermal BVP. The solutions to the elastostatic and the
electrostatic BVPs have already been presented in the literature, in Refs. S1 and S2, respectively.
Here, we formulate and solve the relevant thermal BVP.
The temperature field is determined by solving Laplace’s equation,  2T  0 , subject to
the imposed thermal gradient in x and to the insulating boundary conditions (BCs)
nˆ  T  0 at y  hx,t and
(S1)
yˆ  T  0 at y   .
(S2)
In this boundary-value problem, the governing equation assumes that (1) the thermal
conductivity of the material is uniform and (2) there is no heat generation due to the Joule effect.
2
In the materials and systems under consideration in this study, these are reasonable assumptions
for the problem. Assumption (1) is based on the fact that the temperature dependence of the
thermal conductivity of the material is weak over the temperature range of interest; this
temperature range is discussed further in the last section of this supplementary document.
Assumption (2) relies on the fact that Joule heating is negligible in the systems of interest under
the conditions considered; this has been demonstrated satisfactorily in our previous studies of
void dynamics in metallic thin films under electromigration conditions (see Ref. 14 of the main
article and references cited therein). BCs (S1) and (S2) take into account the morphology of the
surface that has been perturbed from its planar state. Specifically, the initially planar surface of
the body is disturbed according to a low-amplitude plane-wave perturbation as described in the
main article. The position vector of an arbitrary point on the solid surface can then be written as
r  xxˆ  h( x)yˆ ,
(S3)
where the surface morphology y  hx,t is considered at any certain time instant. We define the
unit vectors normal and tangential to the surface as
nˆ  hx xˆ  yˆ and
sˆ  xˆ  hx yˆ .
(S4)
In order to solve the above thermal BVP, Laplace’s equation with Eqs. (S1)-(S4), for the
case of a perturbed surface morphology from the planar state, we introduce a corresponding
small perturbation Tˆ in the temperature field T x, y; t  . It should be mentioned that the BVP is
solved in the quasi-steady state approximation due to the separation of time scales (the time scale
for thermal conduction is much shorter than that for surface morphological evolution); therefore,
3
the time dependence of T is merely parametric. With the perturbation Tˆ , the temperature field T
can be expressed as
T ( x, y; t )  T0  Tx x  Tˆ , Tˆ  T  y; t  exp ikx ,
(S5)
where Tˆ mimics the plane-wave perturbation of the film surface morphology that was
introduced in the main article. In Eq. (S5), Tx is the imposed (constant) temperature gradient,
set by the two fixed temperature levels at the domain boundaries in x, T0 and T with T0  T .
For a planar surface morphology, the solution to the BVP is T ( x, y; t )  T0  Tx x .
Substituting into Laplace’s equation the expression for the temperature field, T, from Eq.
(S5) yields for the plane-wave amplitude, T ( y; t ) , the following second-order ordinary
differential equation
d 2T
 k 2T  0 .
2
dy
(S6)
Applying BC (S2) gives
T  C exp ky .
(S7)
To within leading order in the amplitude  of the surface shape perturbation (as defined in the
main article), the integration constant C in Eq. (S7) is determined by applying BC (S1) and
taking into account that for a low-amplitude perturbation exp  khx, t   1 . This gives
C  i Tx t  .
(S8)
4
Combining Eqs. (S5), (S7), and (S8) yields the final expression for the temperature field along
the perturbed surface as
T ( x, h( x, t ); t )  T0  Tx x  it  exp ikx .
(S9)
Equation (S9) has been used in the main article as the leading-order solution T ( x; t ) in the
surface morphological stability analysis.
Possible Orientations of Applied Fields
In order to examine the possibility for synergy or competition between the applied
electric and thermal fields, we distinguish among the various possible orientations of these fields.
There are four such cases, which are depicted in the schematic of Fig. S1: (1) E  E xˆ and
Tx  Tx x̂ , (2) E  E xˆ and Tx   Tx x̂ , (3) E  E xˆ and Tx  Tx x̂ , and (4)
E  E xˆ and Tx   Tx x̂ , where x̂ is the unit vector along x. In Fig. 1(b) of the main
article, it was demonstrated that case (1) leads to destabilization of the planar surface
morphology, cases (2) and (3) require a stronger electric-field strength to stabilize the planar
surface than in the absence of an imposed thermal gradient, while case (4) is the most effective
one for surface stabilization leading in synergy (toward planar surface stabilization) of the two
externally applied fields.
5
FIG. S1. Schematic representation of the four possible orientations of the applied electric and
thermal fields. A surface morphology perturbed from the planar one by a plane wave is depicted.
The perturbation amplitude has been amplified for clarity.
Linear Stability Analysis with Temperature-dependent Material Properties
The temperature difference across the material (T∞ - T0) is taken to be on the order of 10
K. The dynamic length scale l used in the linear stability theory of the main article is defined as
l  M  2 ; a typical value of l for aluminum is l = 1m (for γ ~ 1 J/m2, σ∞ = 140 MPa, and M =
70 GPa). For such values of l, an elastic solid with dimensions on the order of 1 mm can be
assumed to be semi-infinite for all practical purposes. It should be mentioned that, due to the
6
small physical dimensions in many technological applications, a temperature difference of a few
K can generate a temperature gradient T x as high as 500 K/cm. The temperature dependence
of the thermal conductivity, the electrical conductivity, and the mechanical properties is of major
importance in the accurate solutions of the 3 BVPs in our study; for the relatively narrow
temperature range that is of relevance here, these weakly temperature dependent properties can
be taken to be approximately constant [S3] and have been treated as such in our analysis. As a
result, the solutions of the 3 BVPs that are used to determine the surface flux of Eq. (1) of the
main article remain the same (i.e., as given in the main article, neglecting the temperature
dependence of the material properties).
Nevertheless, the Arrhenius-type temperature
dependence of the surface diffusivity is strong in spite of the narrow temperature range under
consideration and needs to be taken into account. In such a case, the two key questions to be
addressed are how such a temperature dependence affects (i) the criticality condition derived
from Eqs. (4) in the main article and (ii) the dynamical response of the perturbed surface
morphology. These questions have been answered in the main article.
Here, we examine in detail how the temperature dependence of the surface diffusivity
affects the dispersion relation of Eq. (4a) of the main article and leads to Eq. (6) of the main
article.
Following the main article, we write Ds ( , T )  D s ,min T  f ( ) and analyze D s ,min T  / T
and 1 / T T / x , which are terms that appear explicitly in the surface flux expression of Eq. (1)
of the main article. Taylor expansion about T  T0 and truncation yields
D s ,min T 
T

D0 exp -Ea /k BT  D s ,min T0 
~

1  T  O  2
T
T0

7
 
(S10)
and
~
1 T
T

O 2 ,
T x
x
 
(S11)
where

 E

T  T0
~ T  T0
 O(1) , and    a  1  O1 .
 1 , T 
T  T0
T0
 k BT0 
(S12)
with the orders of magnitude in Eq. (S12) determined by the relatively narrow temperature range
under consideration. A representative value for the activation energy barrier (for copper) is
Ea  0.5 eV [S3]; with k B  8.6  10 5 eV/K , a T0  600 K confirms that ξ is O(1).
Using Eq. (S10) and Eqs. (1) and (2) of the main article yields the height evolution
equation


~
h
 
 Ds ,min T0  1  T f   s
  
t
x 
k BT0 


 Q * T 

  Es qs* 
 .

s T s 


(S13)
Combining Eq. (S13) with Eqs. (S11), (S12), and the definitions of the time scale and
dimensionless variables from the main article yields the dimensionless height evolution equation
~
~
2~
  
h
l 2  df
1  ~
x  Q * T  f   0   2  ~
x
~
*
*  T 







 Q  ~ 2 
  1  T   Es qs 

2
~
~

 ~
~
t
  d   0  ~x 
l

x
l

x
l
x
x 



~ 
 
1  ~
x  Q * T 
~

*

 f   0 ~  1  T   Es qs 

(S14)
 .
~
~
x 
l x
l x 







Linearization of Eq. (S14), i.e., retaining up to O() and O() terms only, yields
8
~
~
~
*
2
  2h
h
~  E qs*l 2 Q Tx l  df
~  4h

 2  f   0 1  T ~ 4
 1  T 

 d   0  ~
~
t




T
x
0

 x


 Q * Tx l 2  ~ ~ ~ 2
~
~   2 l ~ ~3
it k exp ik~~
 2 f   0 1  T   ~
t k exp ik ~
x  f   0 
x .



T
0
 M 







 

 
(S15)
~
~
Substituting into Eq. (S15) ~
t    0 exp ~
t  and using the definition of the length scale l yields
the dispersion relation
~  
~   df


~ 
 eff k 2  2 f   0k 3  f   0k 4  ,
 k  1  T  



0
d

 


~
~
(S16)
where  eff is defined in Eq. (4b) of the main article. It is evident that Eq. (S16) is identical with
Eq. (6) of the main article.
References
S1. D. J. Srolovitz, Acta Metall. 37, 621 (1989).
S2. G. I. Sfyris, M. R. Gungor, and D. Maroudas, Appl. Phys. Lett. 96, 231911(2010); G. I.
Sfyris, M. R. Gungor, and D. Maroudas, J. Appl. Phys. 108, 093517 (2010); G. I. Sfyris,
M. R. Gungor, and D. Maroudas, J. Appl. Phys., 111, 024905 (2012).
S3. C.-L. Liu, J. M. Cohen, J. B. Adams, and A. F. Voter, Surf. Sci. 253, 334 (1991); S. Aksöz,
Y. Ocak, N. Maraşh, E. Çadirli, H. Kaya, and U. Böyük, Exp. Therm Fluid Sci. 34, 1507
(2010).
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