Research article
Open Access
Analytical Solutions to the Problem of the Grain Groove
Profile
Tayssir Hamieh^{1,2*}, Zoubir Khatir^{1} and Ali Ibrahim^{1}
^{1}Systèmes et Applications des Technologies de l’information et de l’Energie (SATIE), Institut français des sciences et technologies des
transports, de l’aménagement et des réseaux (IFSTTAR), 25 Allée des Marronniers, 78000, Versailles, France
^{2}Laboratory of Materials, Catalysis, Environment and Analytical Methods (MCEMA) and LEADDER Laboratory, Faculty of Sciences and EDST, Lebanese University, Hariri Campus, Hadath, Beirut, Lebanon
^{2}Laboratory of Materials, Catalysis, Environment and Analytical Methods (MCEMA) and LEADDER Laboratory, Faculty of Sciences and EDST, Lebanese University, Hariri Campus, Hadath, Beirut, Lebanon
*Corresponding author: Tayssir Hamieh, SATIE, IFSTTAR, 25 Allée des Marroniers, 78000, Versailles, France and MCEMA, LEADDER, Faculty of Sciences and EDST, Lebanese University, Beirut, Lebanon, E-mail :
@ ;
@
Received: June 29, 2018; Accepted: July 17, 2018; Published: August 1, 2018
Citation: Hamieh T, Zoubir K, Ali I (2018) Analytical Solutions to the Problem of the Grain Groove Profile. Nanosci Technol 5(2): 1-10.
DOI: 10.15226/2374-8141/5/2/00157
AbstractTop
During the last sixty years, the problem of the formation of grain
boundary grooving in polycrystalline thin films, was largely studied,
analyzed and commented. The thermal effect on the properties
of the grain boundary grooving was first studied by Mullins in his
famous paper published in 1957 and then by other authors. This
paper constitutes a new contribution on the correction of Mullins
problem in the case of the evaporation-condensation and proposes a
more accurate solution of the partial differential equation governing
the geometric profile of the grain boundary grooving. The Mullins
hypothesis neglecting the first derivative (|y’|< < 1) in the main equation
was defeated by our new solution. In this paper, we proved that the
new proposed mathematical solution giving the solution y(x, t) is valid
for all x values without any approximation on the first derivative y’.
Introduction
The study of grain boundary grooves at the surface of
polycrystalline thin films is of vital importance, and especially
in nanomaterial sciences applied to electric and electronic
processes. Many studies on thin polycrystalline films showed that
when these films are submitted to a sufficient thermal treatment,
several holes can be formed conducting to the degradation
of materials [1-9]. The failure of the films resulted from the
formation of grain boundary grooves [9].
The thermal grooving along grain boundaries is certainly responsible for the failure initiation produced in the films at the intersection of grain boundaries. The grain boundary grooving is an important factor in two dimensional grain growth of thin films [10]. The deepening of grain boundary grooves at the surface tends to pin the boundaries and impede grain growth, ultimately causing stagnation of the evolution [10-13]. The morphological profile of the sloping sides is important because the local slope determines the driving force for grain growth required for the boundary to break free of the groove [10]. Mullins studied the thermal grooving, with an isotropic surface free energy, by either surface diffusion or evaporation–condensation processes [11].
It was observed that the first phase of aging is classically associated with the reconstruction of the metallization and the degradation of the bonding contact. The end of life is rather characterized by bond-wire heel-cracks and lift-off [14].
During the last sixty years, the problem of the thermal degradation and the formation of grain boundary grooving in polycrystalline thin films, was largely studied, analyzed and commented [15-24]. The thermal effect on the properties of the grain boundary grooving was first studied by Mullins in his famous paper published in 1957 and then by many other authors [15,18-24].
Chen proposed a stochastic theory of normal grain growth. His model was based on the concept that the migration of kinks and ledges should cause a Brownian motion of the grain boundary [25]. Chen proved that this motion results in a drift of the grain size distribution to larger sizes, and the kinetics of grain growth is related to the kinetics of kinks and ledges; specifically, via the rates of nucleation, recombination and sink annihilation [25].
Agrawal, R. Rajhave found that thin films of polycrystalline zirconia, deposited on a sapphire substrate become gradually discontinuous when annealed at high temperature [26]. They identified the nucleation and growth mechanism of cavities where zirconia grain boundaries meet the substrate, and proved a dependency of cavity nucleation and orientation on the interfacial energies.
The kinetics of cathode edge shrinkage and displacement coupled strongly with the grain boundary grooving was investigated by Ogurtani and Akyildiz using the novel mathematical model developed by Ogurtani in sandwich type thin film bamboo lines [26,27,28].
The study of grain boundary grooves is important in material processing and synthesis. When a vertical grain boundary ends at a horizontal free surface, a groove forms at the tip to reduce the combined grain-boundary and surface energies. Min and Wong studied the grain-boundary grooving by capillarity-driven surface diffusion with asymmetric and strongly anisotropic surface energies [29].
There is an important effect of the thermal treatment on the degradation of metallized films, and especially, on the metallization used in electronic components. Power electronic modules are key elements in the chain of power conversion. The application areas include aerospace, aviation, railway, electrical distribution, automotive, home automation, oil industry. These modules constitute an assembly of various materials (Figure 1). Generally, the power chips are carried on a ceramic substrate that must ensure good electrical insulation and good thermal conduction. This substrate is also welded on a sole to be cooled.
The thermal grooving along grain boundaries is certainly responsible for the failure initiation produced in the films at the intersection of grain boundaries. The grain boundary grooving is an important factor in two dimensional grain growth of thin films [10]. The deepening of grain boundary grooves at the surface tends to pin the boundaries and impede grain growth, ultimately causing stagnation of the evolution [10-13]. The morphological profile of the sloping sides is important because the local slope determines the driving force for grain growth required for the boundary to break free of the groove [10]. Mullins studied the thermal grooving, with an isotropic surface free energy, by either surface diffusion or evaporation–condensation processes [11].
It was observed that the first phase of aging is classically associated with the reconstruction of the metallization and the degradation of the bonding contact. The end of life is rather characterized by bond-wire heel-cracks and lift-off [14].
During the last sixty years, the problem of the thermal degradation and the formation of grain boundary grooving in polycrystalline thin films, was largely studied, analyzed and commented [15-24]. The thermal effect on the properties of the grain boundary grooving was first studied by Mullins in his famous paper published in 1957 and then by many other authors [15,18-24].
Chen proposed a stochastic theory of normal grain growth. His model was based on the concept that the migration of kinks and ledges should cause a Brownian motion of the grain boundary [25]. Chen proved that this motion results in a drift of the grain size distribution to larger sizes, and the kinetics of grain growth is related to the kinetics of kinks and ledges; specifically, via the rates of nucleation, recombination and sink annihilation [25].
Agrawal, R. Rajhave found that thin films of polycrystalline zirconia, deposited on a sapphire substrate become gradually discontinuous when annealed at high temperature [26]. They identified the nucleation and growth mechanism of cavities where zirconia grain boundaries meet the substrate, and proved a dependency of cavity nucleation and orientation on the interfacial energies.
The kinetics of cathode edge shrinkage and displacement coupled strongly with the grain boundary grooving was investigated by Ogurtani and Akyildiz using the novel mathematical model developed by Ogurtani in sandwich type thin film bamboo lines [26,27,28].
The study of grain boundary grooves is important in material processing and synthesis. When a vertical grain boundary ends at a horizontal free surface, a groove forms at the tip to reduce the combined grain-boundary and surface energies. Min and Wong studied the grain-boundary grooving by capillarity-driven surface diffusion with asymmetric and strongly anisotropic surface energies [29].
There is an important effect of the thermal treatment on the degradation of metallized films, and especially, on the metallization used in electronic components. Power electronic modules are key elements in the chain of power conversion. The application areas include aerospace, aviation, railway, electrical distribution, automotive, home automation, oil industry. These modules constitute an assembly of various materials (Figure 1). Generally, the power chips are carried on a ceramic substrate that must ensure good electrical insulation and good thermal conduction. This substrate is also welded on a sole to be cooled.
Figure 1: High power IGBT module
The assembly technologies are various. This includes materials
and process for insulation or passivation, interconnections, and
die attach. The topside interconnections in power semiconductor
devices, consisting of the metallization and the wire bonds, are
subjected in operation to high functional stresses. This is the
result of an important difference between the coefficients of
thermal expansion (CTE) of the materials in contact: metallization
and wire bonds (aluminum) and dies (silicon). The metallization
layer (around 5 μm) deposited on the chips becomes a lot more
distorted than the silicon with temperature, leading to high tensile
and compressive stresses and thus to large inelastic strains [30].
It has been reported that two main types of degradation can take
place in the topside of power chips under the effect of thermo
mechanical cycles: metallization reconstruction and degradation
of bonding contacts (Figure 2 and 3) [30-32]. The last one may
itself be either heel-cracks or cracks propagation followed by liftoff
[33]. Various works have been conducted to propose scenarios
of degradation mechanisms using thermal and power cycling
tests [34-36]. Although it is quite clear that the wire-bond liftoff
contributes mainly to the module failure [37], this link is not
obvious with the metallization degradation [35].
Figure 2: Topside metallization: before and after aging
Figure 3: a) Heel cracks, b) Lift off
Mullins developed the two cases of the evaporationcondensation
and the surface diffusion by formulating in both
cases the mathematical problem of the partial differential
equation governing the geometric profile of the grain boundary
grooving [15-17].
The method consisted in using the general relation of the curvature R at any point (x, t) of the profile, where y = y(x, t) and R is a function of the two first derivatives y’ and y” as a function of x. All mathematical developments proposed by Mullins [15, 16], Mullins and al. and many other researchers were based on the approximation given by |y’|≪ 1 [11,20,22-24].
This paper is a contribution to a better understanding of the effects of stress parameters on the degradation mechanisms of the top side interconnections. In this paper, we reconsidered the Mullins problem by considering the problem of the evaporationcondensation proposing a new mathematical solution and obtaining an analytical solution giving the solution y (x, t) taking into account the boundary conditions independently of any conditions of y’(x).
The method consisted in using the general relation of the curvature R at any point (x, t) of the profile, where y = y(x, t) and R is a function of the two first derivatives y’ and y” as a function of x. All mathematical developments proposed by Mullins [15, 16], Mullins and al. and many other researchers were based on the approximation given by |y’|≪ 1 [11,20,22-24].
This paper is a contribution to a better understanding of the effects of stress parameters on the degradation mechanisms of the top side interconnections. In this paper, we reconsidered the Mullins problem by considering the problem of the evaporationcondensation proposing a new mathematical solution and obtaining an analytical solution giving the solution y (x, t) taking into account the boundary conditions independently of any conditions of y’(x).
Formulation of the Problem
It was observed that a two dimensional metallic film remains
flat for any temperatures and for a very long time. When the
temperature increases, the metal atoms move causing grooves
at the grain boundary surface. The metallic atoms can diffuse
at the surface or in the volume. Atoms can also evaporate into
the vapor phase or condensate. Mullins developed the grooving
process produced in solid surfaces [11]. Grain-boundary
migration controls the growth and shrinkage of crystalline grains
and is important in materials synthesis and processing. A grain
boundary ending at a free surface forms a groove at the tip, which
affects its migration [18]. In polycrystalline thin films, grain
boundary grooving through the thickness of the film is a common
failure mode that strongly affects their properties. The grooving
forms and develops at the point of intersection when the grain
boundary ends at a free boundary in order to reduce the total
free energy [10]. As example, we give on Figure 4 the symmetric
profile of grain boundary grooving of a thin film.
It was experimentally shown that when thin polycrystalline films are annealed, they tend to break up by the formation
It was experimentally shown that when thin polycrystalline films are annealed, they tend to break up by the formation
Figure 4: Profile y(x,t) of the grain boundary groove.
and growth of grooves [7, 8, 38-45]. Smaller and larger holes
are usually located at the intersection points of three-grain
boundaries [9]. The break up in thin films of some metals as
copper on sapphire was initiated at processing defects in the film
,in contrast of cavities found at grain boundaries on zirconia on
alumina polycrystalline films [7,26].
Galina and Fradkov studied the problem of the grain boundary hole development by assuming that there is evaporation/ condensation as the transport mechanism rather than surface diffusion as generally supposed [46,47]. Mullins studied the thermal grooving mechanisms relative to the evaporation/ condensation and surface diffusion phenomena [15]. Mullins supposed for a polycrystalline solid at equilibrium, a symmetric grain boundary groove profile and then the ratio of grain boundary energy per unit area, γGB, to surface energy per unit area, γSV, is related to the groove angle, θ by this relationship:
$$\frac{{\gamma}_{GB}}{2{\gamma}_{SV}}=Sin\theta $$
Galina and Fradkov studied the problem of the grain boundary hole development by assuming that there is evaporation/ condensation as the transport mechanism rather than surface diffusion as generally supposed [46,47]. Mullins studied the thermal grooving mechanisms relative to the evaporation/ condensation and surface diffusion phenomena [15]. Mullins supposed for a polycrystalline solid at equilibrium, a symmetric grain boundary groove profile and then the ratio of grain boundary energy per unit area, γGB, to surface energy per unit area, γSV, is related to the groove angle, θ by this relationship:
Note that tanθ is equal to the slope of the profile y(x,t) at x = 0.
When studying the evolution of grain boundary groove profiles in the cases of the evaporation/condensation and surface diffusion, Mullins assumed that: (1) the surface diffusivity and the surface energy, γ_SV, were independent of the crystallographic orientation of the adjacent grains and (2) the tangent of the groove root angle, θ, is small compared to unity [15]. Mullins also supposed an isotropic material. The assumption (tanθ << 1) was used in all papers’ Mullins to simplify the study of the mathematical partial differential equation. The polycrystalline metal was supposed (3) in quasi-equilibrium with its vapor. The interface properties doesn’t depend on the orientation relative to the adjacent crystals. The grooving process was described by Mullins using the macroscopic concepts (4) of surface curvature and surface free energy. The matter flow (5) is neglected out of the grain surface boundary.
We propose in this paper to study the grain boundary groove profiles in polycrystalline metal and to give an analytical solution relative to the only case of evaporation/ condensation, more precise than of the solution found by Mullins that supposed very small slops for all x values [15].
On Figure 4, we give the profile y(x,t) of the grain boundary groove in metal polycrystal.
By using the notion of curvature c at any point M(x; y(x, t)) given by the following relation:
$$c=\frac{1}{R}=-\frac{y"\left(x\right)}{{\left[1+y{\left(x\right)}^{2}\right]}^{\frac{3}{2}}}\text{(1)}$$
When studying the evolution of grain boundary groove profiles in the cases of the evaporation/condensation and surface diffusion, Mullins assumed that: (1) the surface diffusivity and the surface energy, γ_SV, were independent of the crystallographic orientation of the adjacent grains and (2) the tangent of the groove root angle, θ, is small compared to unity [15]. Mullins also supposed an isotropic material. The assumption (tanθ << 1) was used in all papers’ Mullins to simplify the study of the mathematical partial differential equation. The polycrystalline metal was supposed (3) in quasi-equilibrium with its vapor. The interface properties doesn’t depend on the orientation relative to the adjacent crystals. The grooving process was described by Mullins using the macroscopic concepts (4) of surface curvature and surface free energy. The matter flow (5) is neglected out of the grain surface boundary.
We propose in this paper to study the grain boundary groove profiles in polycrystalline metal and to give an analytical solution relative to the only case of evaporation/ condensation, more precise than of the solution found by Mullins that supposed very small slops for all x values [15].
On Figure 4, we give the profile y(x,t) of the grain boundary groove in metal polycrystal.
By using the notion of curvature c at any point M(x; y(x, t)) given by the following relation:
Where R is the curvature radius at point M
$y\text{'}\left(x\right)=\frac{\partial y}{\partial x}$ and
$y\u201d\left(x\right)=\frac{{\partial}^{2}y}{\partial {x}^{2}}.$
The mathematical equation governing the evaporationcondensation
problem can be written here as:
$$\frac{\partial y}{\partial t}=C(T)\frac{y"\left(x\right)}{\left(1+y{\left(x\right)}^{2}\right)}\text{(2)}$$
Where C(T) a constant of the problem depending on the
temperature T, given by:
$$C\left(T\right)=\mu \frac{{P}_{0}\left(T\right)\gamma \left(T\right){\omega}^{2}}{\sqrt{2\pi mkT}}\text{(3)}$$
where γ is the isotropic surface energy, P_{o}(T) the vapor
pressure at temperature T in equilibrium with the plane surface
of the metal characterized by a curvature c = 0, ω is the atomic
volume, m is molecular mass, μ the coefficient of evaporation and
k is the Boltzmann constant.
Mullins [10] supposed that the coefficient of evaporation μ is equal to the unit.
Mullins [10] supposed that the coefficient of evaporation μ is equal to the unit.
Mullins Approximation
To resolve this differential equation, Mullins was constraint to
suppose that |y’|≪ 1 that means that the slope y’(x,t) at any point
of the curve y (x, t) is very small behind 1 and can be neglected.
Equation (4) can be written as:
$$\frac{\partial y}{\partial t}=C(T)y"\left(x\right)\text{(4)}$$
With the boundary conditions:
$$\{\begin{array}{l}y(x,0)=0\\ y\text{'}\left(0,t\right)=\mathrm{tan}\mathrm{tan}\theta =m\end{array}\text{(5)}$$
This problem is well-known in the conduction of heat in
solids. It can be resolved by the following variable change:
$$u=\frac{x}{2\sqrt{Ct}}\text{(6)}$$
and one obtains the derivatives of u as a function of x and t:
$$\{\begin{array}{l}\frac{\partial u}{\partial x}=\frac{u}{x}\\ \frac{\partial u}{\partial t}=-2C\frac{{u}^{3}}{{x}^{2}}\end{array}\text{(7)}$$
Now, using $\frac{\partial y}{\partial t}=\frac{\partial y}{\partial u}\frac{\partial u}{\partial t}$
and $\frac{\partial y}{\partial x}=\frac{\partial y}{\partial u}\frac{\partial u}{\partial x}$
one obtains the following
derivatives:
$$\frac{\partial y}{\partial t}=-2C\frac{{u}^{3}}{{x}^{2}}\frac{\partial y}{\partial u}\text{(8)}$$
$$y\text{'}\left(x\right)=\frac{\partial y}{\partial x}=\frac{u}{x}\frac{\partial y}{\partial u}\text{(9)}$$
$$y"\left(x\right)=\frac{{\partial}^{2}y}{\partial {x}^{2}}={\left(\frac{u}{x}\right)}^{2}\frac{{\partial}^{2}y}{\partial {u}^{2}}\text{(10)}$$
Then, equation (4) becomes as a function of u:
$$\frac{{\partial}^{2}y}{\partial {u}^{2}}=-2u\frac{\partial y}{\partial u}\text{(11)}$$
and
$$\frac{y\text{'}\text{'}\left(u\right)}{y\text{'}\left(u\right)}=-2u\text{(12)}$$
The solution of differential equation (12) is given by:
$$y\text{'}\left(u\right)=A{e}^{-{u}^{2}}$$
With A a constant of the problem.
Knowing that $\frac{\partial y}{\partial u}=\frac{\partial y}{\partial x}\frac{\partial x}{\partial u}$ , we obtain:
$$\frac{\partial y}{\partial u}=2\sqrt{Ct}\frac{\partial y}{\partial x}\text{(13)}$$
Knowing that $\frac{\partial y}{\partial u}=\frac{\partial y}{\partial x}\frac{\partial x}{\partial u}$ , we obtain:
With the condition boundary y’(0,t) =m, one obtains
$A=2m\sqrt{Ct}$. Using the other condition boundary y (x, 0) = 0, the
solution of the differential equation (13) becomes:
$$y=2m\sqrt{Ct}\underset{0}{\overset{u}{{\displaystyle \int}}}{e}^{-{u}^{2}}du+Cst,\text{}u=\frac{x}{2\sqrt{Ct}}\text{(14)}$$
The constant Cst can be determined by the boundary
condition y(∞,t)=0. This gives:
$$2m\sqrt{Ct}\underset{0}{\overset{\infty}{{\displaystyle \int}}}{e}^{-{u}^{2}}du+Cst=0\text{(15)}$$
With $\underset{0}{\overset{\infty}{{\displaystyle \int}}}{e}^{-{u}^{2}}du=\frac{\sqrt{\pi}}{2}$
one obtains: $Cst=-2m\sqrt{Ct}\frac{\sqrt{\pi}}{2}$
and
equation (15) can be written:
$$y=2m\sqrt{Ct}\underset{0}{\overset{\frac{x}{2\sqrt{Ct}}}{{\displaystyle \int}}}{e}^{-{u}^{2}}du-2m\sqrt{Ct}\frac{\sqrt{\pi}}{2}=-m\sqrt{\pi Ct}\left[1-\frac{2}{\sqrt{\pi}}\underset{0}{\overset{\frac{x}{2\sqrt{Ct}}}{{\displaystyle \int}}}{e}^{-{u}^{2}}du\right]$$
In conclusion, the solution of approximated Mullins problem
will be written as:
$$y\left(x,t\right)=-m\sqrt{\pi Ct}\left[1-\frac{2}{\sqrt{\pi}}\underset{0}{\overset{\frac{x}{2\sqrt{Ct}}}{{\displaystyle \int}}}{e}^{-{u}^{2}}du\right]\text{(16)}$$
In conclusion for this part, the Mullins solution of
approximated equation supposing $\left|y\text{'}\right|\ll 1$
, is given by:
$$\{\begin{array}{c}y\left(x,t\right)=-m\sqrt{\pi Ct}erfc\left(\frac{x}{2\sqrt{Ct}}\right)\\ y\text{'}\left(u\right)=2m\sqrt{Ct}{e}^{-{u}^{2}},\text{with}u=\frac{x}{2\sqrt{Ct}}\end{array}\text{(17)}$$
Where erfc is the complementary error function.
However, knowing that $\left(0,t\right)=-{\epsilon}_{0}$ , one deduces the value of the groove depth:
$${\epsilon}_{0}\left(t\right)=m\sqrt{\pi Ct}=tan\theta \sqrt{\pi Ct}\text{(18)}$$
However, knowing that $\left(0,t\right)=-{\epsilon}_{0}$ , one deduces the value of the groove depth:
The obtained calculations are presented on Figure 5
showing the variation of the profile y(x, t) of the grain groove
of Mullins approximation as a function of the distance x
Figure 5: Evolution of the profile y(x, t) of the grain groove of Mullins
approximation as a function of the distance x from the grain separation
surface for various groove angles θ from 1º to 45º.
from the grain separation surface for various angles θ. It is
obvious shown that when the groove angle θ increases, the
slope m at x = 0 increases and this increases the groove depth
ε_{o} as proved by Figure 5. However, all representative curves
obtained in this case are located under the x-axis meaning that
y(x,t) is negative for all x values without any inflexion point.
There is no maximum for these curves. This means that the
Mullins solution cannot describe correctly the experimental
observations shown in Figure 4.
On Figure 6, we plotted the variations of the derivative y’(x) of the grain groove profile for Mullins approximation as a function of the distance x from the grain separation surface for various angles θ. This derivative strongly depends on the groove angle. The smaller the groove angle is, the smaller the derivative y’ is. For θ. = 1°, all derivatives y^{1} (x) can be neglected relatively to 1 for all x values. In this nonrealistic case and only in this case, the Mullins approximation can be applied and the solution can describe the profile of the grain groove in the case of evaporation-condensation. However, Figure 6 shows that the Mullins derivatives for all groove angles are positive implying the non-existence of a maximum of groove profile that can be observed in the experiments.
It is clearly shown on Figure 6 that for a groove angle and greater than 5°, all derivative values y’ (x) are greater than 0.25 whatever the x value and become greater than 1 when θ ˃ 25°. In all cases, the Mullins approximation cannot be applied when the groove angle θ ˃ 5° and the Mullins condition |y’|≪1 becomes invalid in such cases.
On Figure 6, we plotted the variations of the derivative y’(x) of the grain groove profile for Mullins approximation as a function of the distance x from the grain separation surface for various angles θ. This derivative strongly depends on the groove angle. The smaller the groove angle is, the smaller the derivative y’ is. For θ. = 1°, all derivatives y^{1} (x) can be neglected relatively to 1 for all x values. In this nonrealistic case and only in this case, the Mullins approximation can be applied and the solution can describe the profile of the grain groove in the case of evaporation-condensation. However, Figure 6 shows that the Mullins derivatives for all groove angles are positive implying the non-existence of a maximum of groove profile that can be observed in the experiments.
It is clearly shown on Figure 6 that for a groove angle and greater than 5°, all derivative values y’ (x) are greater than 0.25 whatever the x value and become greater than 1 when θ ˃ 25°. In all cases, the Mullins approximation cannot be applied when the groove angle θ ˃ 5° and the Mullins condition |y’|≪1 becomes invalid in such cases.
Figure 6: Evolution of the derivative y’ (x) of the grain groove profile
of Mullins approximation as a function of the distance x from the grain
separation surface for various groove angles θ.
It is clearly shown on Figure 6 that for a groove angle and
greater than 5o, all derivative values y’ (x) are greater than
0.25 whatever the x value and become greater than 1 when θ
˃25°. In all cases, the Mullins approximation cannot be applied
when the groove angle θ ˃ 5° and the Mullins condition |y’|≪1
becomes invalid in such cases.
In order to more clarify the non-validity of the Mullins hypothesis in general, we give below on Table 1 the error
In order to more clarify the non-validity of the Mullins hypothesis in general, we give below on Table 1 the error
Table 1: Evaluation of the errors (in %) made when supposing |y′| << 1, for different groove angle values
x |
θ = 1º |
θ = 5º |
θ= 10º |
θ = 20º |
θ = 30º |
θ = 40º |
θ = 45º |
0 |
1.7 |
8.7 |
17.6 |
36.4 |
57.7 |
83.9 |
100 |
0.1 |
1.7 |
8.7 |
17.6 |
36.3 |
57.6 |
83.7 |
99.8 |
0.2 |
1.7 |
8.7 |
17.5 |
36 |
57.2 |
83.1 |
99 |
0.3 |
1.7 |
8.6 |
17.2 |
35.6 |
56.5 |
82 |
97.8 |
0.4 |
1.7 |
8.4 |
16.9 |
35 |
55.5 |
80.6 |
96.1 |
0.5 |
1.6 |
8.2 |
16.6 |
34.2 |
54.2 |
78.8 |
93.9 |
0.6 |
1.6 |
8 |
16.1 |
33.3 |
52.8 |
76.7 |
91.4 |
0.7 |
1.5 |
7.7 |
15.6 |
32.2 |
51.1 |
74.2 |
88.5 |
0.8 |
1.5 |
7.5 |
15 |
31 |
49.2 |
71.5 |
85.2 |
0.9 |
1.4 |
7.1 |
14.4 |
29.7 |
47.2 |
68.5 |
81.7 |
1 |
1.4 |
6.8 |
13.7 |
28.3 |
45 |
65.3 |
77.9 |
1.1 |
1.3 |
6.5 |
13 |
26.9 |
42.7 |
62 |
73.9 |
1.2 |
1.2 |
6.1 |
12.3 |
25.4 |
40.3 |
58.5 |
69.8 |
1.3 |
1.1 |
5.7 |
11.6 |
23.9 |
37.8 |
55 |
65.5 |
1.4 |
1.1 |
5.4 |
10.8 |
22.3 |
35.4 |
51.4 |
61.3 |
1.5 |
1 |
5 |
10 |
20.7 |
32.9 |
47.8 |
57 |
1.6 |
0.9 |
4.6 |
9.3 |
19.2 |
30.4 |
44.2 |
52.7 |
1.7 |
0.8 |
4.2 |
8.6 |
17.7 |
28 |
40.7 |
48.6 |
1.8 |
0.8 |
3.9 |
7.8 |
16.2 |
25.7 |
37.3 |
44.5 |
1.9 |
0.7 |
3.5 |
7.2 |
14.8 |
23.4 |
34 |
40.6 |
2 |
0.6 |
3.2 |
6.5 |
13.4 |
21.2 |
30.9 |
36.8 |
percentages made when supposing |Y´|<< 1, for different
groove angle values. Table 1 clearly shows that the error
percentage is higher than 10% from a groove angle θ
exceeding 6° the error dramatically increases to 17% for
θ= 10° and exceeds 100% since θ = 45°. This proves that the
Mullins approximation cannot be justified after θ = 6° and all
results of the literature based on the condition |Y´|<< 1 are
experimentally false.
These results lead us to reconsider the evaporationcondensation problem by proposing a new method taking into account the general equation without neglecting the first derivative y’(x). The new method consisting in the correction of Mullins solution is presented in the following section.
These results lead us to reconsider the evaporationcondensation problem by proposing a new method taking into account the general equation without neglecting the first derivative y’(x). The new method consisting in the correction of Mullins solution is presented in the following section.
New Method for the Correction of Mullins Solution
We recall below the general equation of the evaporationcondensation
problem:
$$\frac{\partial y}{\partial t}=C(T)\frac{y"\left(x\right)}{\left(1+y\text{\'}{\left(x\right)}^{2}\right)}$$
Using the same notations given above, one writes:
$$\{\begin{array}{c}1+y\text{\'}{\left(x\right)}^{2}=1+{\left(\frac{1}{2\sqrt{Ct}}\frac{\partial y}{\partial u}\right)}^{2}\\ \frac{\partial y}{\partial t}=-2C\frac{{u}^{3}}{{x}^{2}}\frac{\partial y}{\partial u}\\ y"\left(x\right)=\frac{{\partial}^{2}y}{\partial {x}^{2}}={\left(\frac{u}{x}\right)}^{2}\frac{{\partial}^{2}y}{\partial {u}^{2}}\end{array}\text{(19)}$$
The three combined equations (19) then give:
$$-2C\frac{{u}^{3}}{{x}^{2}}\frac{\partial y}{\partial u}=C\frac{{\left(\frac{u}{x}\right)}^{2}\frac{{\partial}^{2}y}{\partial {u}^{2}}}{1+{\left(\frac{1}{2\sqrt{Ct}}\frac{\partial y}{\partial u}\right)}^{2}}\text{(20)}$$
The second order differential equation is then given by the
expression (21):
$$y"\left(u\right)=-2uy\text{\'}\left(u\right)\left(1+\frac{1}{4Ct}y\text{\'}{\left(u\right)}^{2}\right)\text{(21)}$$
In this paper, we propose a new method of resolution
of equation (21) using the approximated Mullins solution:
$y\text{'}\left(u\right)=2m\sqrt{Ct}{e}^{-{u}^{2}}$
and replacing it in equation (21), one obtains
$$y"\left(u\right)=-4m\sqrt{Ct}u{e}^{-{u}^{2}}\left(1+{m}^{2}{e}^{-2{u}^{2}}\right)\text{(22)}$$
The first integration of equation (22) as a function of the
variable u gives:
$${y}^{\text{'}}\left(u\right)=\frac{2}{3}m\sqrt{Ct}\left(3{e}^{-{u}^{2}}+{m}^{2}{e}^{-3{u}^{2}}\right)+Cst$$
Using the initial condition: $y\text{'}\left(0\right)=2m\sqrt{Ct}$
, one obtains the
constant value Cst:
$$Cst=-\frac{2}{3}{m}^{3}\sqrt{Ct}$$
and equation (22) becomes:
$${y}^{\text{'}}\left(u\right)=\frac{2}{3}m\sqrt{Ct}\left(3{e}^{-{u}^{2}}+{m}^{2}{e}^{-3{u}^{2}}\right)-\frac{2}{3}{m}^{3}\sqrt{Ct}\text{(23)}$$
As a function of x and t, we will obtain:
$${y}^{\text{'}}\left(x,t\right)=\frac{m}{3}\left(3{e}^{-\frac{{x}^{2}}{4Ct}}+{m}^{2}{e}^{-\frac{3{x}^{2}}{4Ct}}\right)-\frac{{m}^{3}}{3}$$
The second integration of equation (23) leads to:
$${y}^{}\left(u\right)=\frac{2}{3}m\sqrt{Ct}\left(3\underset{0}{\overset{u}{{\displaystyle \int}}}{e}^{-{u}^{2}}du+{m}^{2}\underset{0}{\overset{u}{{\displaystyle \int}}}{e}^{-3{u}^{2}}du\right)-\frac{2}{3}{m}^{3}\sqrt{Ct}u+{y}^{}\left(0\right)\text{(24)}$$
Equation (24) represents the new solution of the general
problem given by equation (21)
Using the boundary condition y(u_{l})=0, for u=u_{l} corresponding to x_{l}=2√Ct u_{l}, one obtains the following equation:
$$0=\frac{2}{3}m\sqrt{Ct}\left(3\underset{0}{\overset{{u}_{l}}{{\displaystyle \int}}}{e}^{-{u}^{2}}du+{m}^{2}\underset{0}{\overset{{u}_{l}}{{\displaystyle \int}}}{e}^{-3{u}^{2}}du\right)-\frac{2}{3}{m}^{3}\sqrt{Ct}{u}_{l}+{y}^{}\left(0\right)$$
Using the boundary condition y(u_{l})=0, for u=u_{l} corresponding to x_{l}=2√Ct u_{l}, one obtains the following equation:
Therefore, the equation (25) is obtained:
$${y}^{}\left(0\right)=-{h}_{0}=\frac{2}{3}{m}^{3}\sqrt{Ct}{u}_{l}-\frac{2}{3}m\sqrt{Ct}\left(3\underset{0}{\overset{{u}_{l}}{{\displaystyle \int}}}{e}^{-{u}^{2}}du+{m}^{2}\underset{0}{\overset{{u}_{l}}{{\displaystyle \int}}}{e}^{-3{u}^{2}}du\right)\text{(25)}$$
Where h_{o} is the groove deep.
Equation (25) shows that the groove deep h_{o} strongly depend on the time t, the grain length u_{l}, the constant C and the slope m at x = 0 and then on the groove angle θ.
If one uses the complementary error function erfc:
$$\begin{array}{l}\underset{0}{\overset{X}{{\displaystyle \int}}}{e}^{-{u}^{2}}du=\frac{\sqrt{\pi}}{2}\left[1-erfc\left(X\right)\right]\\ \underset{0}{\overset{X}{{\displaystyle \int}}}{e}^{-3{u}^{2}}du=\frac{\sqrt{\pi}}{2\sqrt{3}}\left[1-erfc\left(\sqrt{3}X\right)\right]\end{array}$$
Equation (25) shows that the groove deep h_{o} strongly depend on the time t, the grain length u_{l}, the constant C and the slope m at x = 0 and then on the groove angle θ.
If one uses the complementary error function erfc:
with
$X=\frac{x}{2\sqrt{Ct}}$
, equations (24) and (25) can be written as:
$\{\begin{array}{c}{y}^{}\left(u\right)=\frac{2}{3}m\sqrt{Ct}\left[\frac{3\sqrt{\pi}}{2}\left[1-erfc\left(u\right)\right]+\frac{\sqrt{\pi}}{2\sqrt{3}}{m}^{2}\left[1-erfc\left(\sqrt{3}u\right)\right]\right]-\frac{2}{3}{m}^{3}\sqrt{Ct}u+{y}^{}\left(0\right)\\ {y}^{}\left(0\right)=-{h}_{0}=\frac{2}{3}{m}^{3}\sqrt{Ct}{u}_{l}-\frac{2}{3}m\sqrt{Ct}\left[\frac{3\sqrt{\pi}}{2}\left[1-erfc\left({u}_{l}\right)\right]+\frac{\sqrt{\pi}}{2\sqrt{3}}{m}^{2}\left[1-erfc\left(\sqrt{3}{u}_{l}\right)\right]\right]\end{array}$
As a function of (x,t), one obtains:
$\{\begin{array}{c}{y}^{}\left(u\right)=-m\sqrt{\pi Ct}erfc\left(u\right)+\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{3}u\right)\right]-\frac{2}{3}{m}^{3}\sqrt{Ct}u+m\sqrt{\pi Ct}+{y}^{}\left(0\right)\\ {y}^{}\left(0\right)+m\sqrt{\pi Ct}=\frac{2}{3}{m}^{3}\sqrt{Ct}{u}_{l}+m\sqrt{\pi Ct}erfc\left({u}_{l}\right)-\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{3}{u}_{l}\right)\right]\end{array}\text{(26)}$
Taking into account the Mullins solution, one can write the
new solution as:
${y}^{}\left(x,t\right)={y}_{Mullins}{}^{}\left(x,t\right)+\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{\frac{3}{Ct}}\frac{x}{2}\right)\right]-\frac{{m}^{3}}{3}x+m\sqrt{\pi Ct}+{y}^{}\left(0\right)\text{(27)}$
Equation (27) obviously shows the large difference
between the Mullins solution and the proposed solution.
On Figure 7, we draw the variations of the profile y(x, t) of the grain groove of the new solution as a function of the distance x from the grain separation surface for various angles θ. Two cases can be distinguished here:
The first one is relative to small angles θ from 1° to 15° (Figure 7 a), where we can see an identical curve with that of Mullins solution with a difference for the groove deep value. There is an over estimation in the values obtained by Mullins. The difference between our solution and Mullins solution can reach 20% for the groove deep in this case.
The second case concerns larger angles θ from 20° to 45° (Figure 7 b). Here, it can be observed an important difference between the classical and new solutions. The new solution can easily explain the physical presence of a maximum of y for a certain value of x. whereas, the curves obtained by Mullins are monotonous and do not present any change in the derivative . that is constantly positive whatever the x value.
On Figure 8, we give the evolution of the derivative y’ (x) of the grain groove profile obtained by the new solution for various groove angles θ from 1o to 45°. Here, there is no restriction concerning the value of the first derivative of the grain groove profile. The new results given by our solution confirm the experimental results given in literature showing a maximum of groove profile and then a decrease after this maximum [7, 9, 10, 14, 22-25]. The new derivative clearly shows a change in the sign of the solution y and its cancellation and then the presence of a maximum.
On Figure 7, we draw the variations of the profile y(x, t) of the grain groove of the new solution as a function of the distance x from the grain separation surface for various angles θ. Two cases can be distinguished here:
The first one is relative to small angles θ from 1° to 15° (Figure 7 a), where we can see an identical curve with that of Mullins solution with a difference for the groove deep value. There is an over estimation in the values obtained by Mullins. The difference between our solution and Mullins solution can reach 20% for the groove deep in this case.
The second case concerns larger angles θ from 20° to 45° (Figure 7 b). Here, it can be observed an important difference between the classical and new solutions. The new solution can easily explain the physical presence of a maximum of y for a certain value of x. whereas, the curves obtained by Mullins are monotonous and do not present any change in the derivative . that is constantly positive whatever the x value.
On Figure 8, we give the evolution of the derivative y’ (x) of the grain groove profile obtained by the new solution for various groove angles θ from 1o to 45°. Here, there is no restriction concerning the value of the first derivative of the grain groove profile. The new results given by our solution confirm the experimental results given in literature showing a maximum of groove profile and then a decrease after this maximum [7, 9, 10, 14, 22-25]. The new derivative clearly shows a change in the sign of the solution y and its cancellation and then the presence of a maximum.
Comparison between Mullins Solution and the New
Solution
The following expression gives the ratio of Mullins solution
y_{Mullins} on the corrected solution y_{corrected} :
$$\frac{{y}_{Mullins}}{{y}_{corrected}}=\frac{{y}_{Mullins}{}^{}\left(x,t\right)}{{y}_{Mullins}{}^{}\left(x,t\right)+\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{\frac{3}{Ct}}\frac{x}{2}\right)\right]-\frac{{m}^{3}}{3}x+m\sqrt{\pi Ct}+{y}^{}\left(0\right)}\text{(28)}$$
Figure 7: Evolution of the profile y(x, t) of the grain groove of the new
solution as a function of the distance x from the grain separation surface
for various groove angles θ. Case of small angles θ from 1o to 15o (a) and
larger angles θ from 20° to 45° (b).
Figure 8: Evolution of the derivative y’(x) of the grain groove profile
of the corrected solution as a function of the distance x from the grain
separation surface for various groove angles θ.
On Figure 9, we represent the variations of the ratio
(y_{Mullins} /y_{corrected}) as a function of the distance x from the grain
separation surface for various groove angles θ. The only case
that the two solutions are the same is that when the groove
angle θ = 1°. For all other values of θ = 5°, the two solutions
are different. The curves of Figure 9 prove that the Mullins
solution is so far from the reality. There is an over estimation
of y values given by Mullins solution. As example, we draw on
Figure 10, the evolution of the profile y (x) of the grain groove
for Mullins approximation and the new solution as a function
of the distance x from the grain separation surface for θ =
35°. The two obtained curves show an important difference
between the two cases.
Figure 9: Evolution of the ratio (y_{Mullins})/y_{corrected}) of Mullins solution on
the corrected solution for the grain groove profile as a function of the
distance x from the grain separation surface for various angles θ.
Figure 10: Evolution of the profile y (x) of the grain groove for Mullins
approximation and the new solution as a function of the distance x from
the grain separation surface for θ = 35°.
The ratio of the Mullins derivative y’_{(Mullins)} and the corrected
derivative y’_{corrected} is given by the following expression:
$$\frac{y{\text{'}}_{Mullins}}{y{\text{'}}_{corrected}}=1+\frac{{m}^{2}}{3}\left({e}^{-2{u}^{2}}-{e}^{+{u}^{2}}\right)\text{(29)}$$
To compare between the two derivatives of Mullins and
new solution, we draw on the Figure 11, the evolution of
the derivative y’(x) of the grain groove profile for Mullins
approximation and the new proposed solution as a function
of the distance x from the grain separation surface for θ = 45°.
The two obtained curves show an extreme deviation when the
distance x increases.
Figure 11: Evolution of the derivative y’(x) of the grain groove profile
for Mullins approximation and the new proposed solution as a function
of the distance x from the grain separation surface for θ = 45°
Conclusion
This original study proposed a mathematical correction to
the Mullins problem of the grain boundary groove profile relative
to the case of evaporation/condensation. We proved that the
solution found by Mullins in his famous paper is false [10]. This
validity of Mullins solution is only assured for non-realistic case
of a groove angle q ≈ 1° to 2°. The Mullins’s hypothesis expressed
by |y′| << 1 is is not satisfied. The obtained results proved that
the error percentage of Mullins solution reaches 100% for values
of the groove angle less than 45º. The Mullins solution over
estimates the real values of the solution y of the groove profile.
The new found solution is given by the following equation:
${y}^{}\left(x,t\right)={y}_{Mullins}{}^{}\left(x,t\right)+\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{\frac{3}{Ct}}\frac{x}{2}\right)\right]-\frac{{m}^{3}}{3}x+m\sqrt{\pi Ct}-{h}_{0}$
The new found solution is given by the following equation:
The result of the new solution is the confirmation of the
experimental profile tendency justifying the presence of a
maximum of distance x from the surface boundary. On the
other hand, our solution was able to predict the groove deep
h_{o} by the following equation:
${h}_{0}=+m\sqrt{\pi Ct}-\frac{{m}^{3}}{3}{x}_{l}-m\sqrt{\pi Ct}erfc\left(\frac{{x}_{l}}{2\sqrt{Ct}}\right)+\frac{{m}^{3}}{3}\sqrt{\frac{\pi Ct}{3}}\left[1-erfc\left(\sqrt{\frac{3}{Ct}}\frac{{x}_{l}}{2}\right)\right]$
It was proved that the groove deep strongly depend on the
problem constant C, the time t, the slope of the profile at the
origin or the groove angle θ and the grain length x_{l}.
As perspective, another study is preparing and concerns the general solution of the partial differential equation of the groove boundary profile, when combining the two cases relative to evaporation/condensation and surface diffusion.
As perspective, another study is preparing and concerns the general solution of the partial differential equation of the groove boundary profile, when combining the two cases relative to evaporation/condensation and surface diffusion.
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