Research article Open Access
Analytical Solutions to the Problem of the Grain Groove Profile
Tayssir Hamieh1,2*, Zoubir Khatir1 and Ali Ibrahim1
1Systèmes 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
2Laboratory 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.
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).
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
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:
$γ GB 2 γ SV =Sinθ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeq4SdC2damaaBaaaleaapeGaam4raiaadkea a8aabeaaaOqaa8qacaaIYaGaeq4SdC2damaaBaaaleaapeGaam4uai aadAfaa8aabeaaaaGcpeGaeyypa0Jaam4uaiaadMgacaWGUbGaeqiU dehaaa@43F2@$
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:
Where R is the curvature radius at point M $y\text{'}\left(x\right)=\frac{\partial y}{\partial x}$ and $y”\left(x\right)=\frac{{\partial }^{2}y}{\partial {x}^{2}}.$
The mathematical equation governing the evaporationcondensation problem can be written here as:
Where C(T) a constant of the problem depending on the temperature T, given by:
where γ is the isotropic surface energy, Po(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 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:
With the boundary conditions:
This problem is well-known in the conduction of heat in solids. It can be resolved by the following variable change:
and one obtains the derivatives of u as a function of x and t:
Now, using $\frac{\partial y}{\partial t}=\frac{\partial y}{\partial u}\frac{\partial u}{\partial t}$ and one obtains the following derivatives:
Then, equation (4) becomes as a function of u:
and
The solution of differential equation (12) is given by:
$y'( u )=A e − u 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbiaadMhacaGGNaWaaeWaa8aabaWdbiaadwhaaiaawIcacaGL PaaacqGH9aqpcaWGbbGaamyza8aadaahaaWcbeqaa8qacqGHsislca WG1bWdamaaCaaameqabaWdbiaaikdaaaaaaaaa@40BC@$
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:
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:
The constant Cst can be determined by the boundary condition y(∞,t)=0. This gives:
With $\underset{0}{\overset{\infty }{\int }}{e}^{-{u}^{2}}du=\frac{\sqrt{\pi }}{2}$ one obtains: $Cst=-2m\sqrt{Ct}\frac{\sqrt{\pi }}{2}$ and equation (15) can be written:
$y=2m Ct ∫ 0 x 2 Ct e − u 2 du−2m Ct π 2 =−m πCt [ 1− 2 π ∫ 0 x 2 Ct e − u 2 du ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaeyypa0JaaGOmaiaad2gadaGcaaWdaeaapeGaam4qaiaa dshaaSqabaGcdaGfWbqabSWdaeaapeGaaGimaaWdaeaapeWaaSaaa8 aabaWdbiaadIhaa8aabaWdbiaaikdadaGcaaWdaeaapeGaam4qaiaa dshaaWqabaaaaaqdpaqaa8qacqGHRiI8aaGccaWGLbWdamaaCaaale qabaWdbiabgkHiTiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaaaaGc caWGKbGaamyDaiabgkHiTiaaikdacaWGTbWaaOaaa8aabaWdbiaado eacaWG0baaleqaaOWaaSaaa8aabaWdbmaakaaapaqaa8qacqaHapaC aSqabaaak8aabaWdbiaaikdaaaGaeyypa0JaeyOeI0IaamyBamaaka aapaqaa8qacqaHapaCcaWGdbGaamiDaaWcbeaakmaadmaapaqaa8qa caaIXaGaeyOeI0YaaSaaa8aabaWdbiaaikdaa8aabaWdbmaakaaapa qaa8qacqaHapaCaSqabaaaaOWaaybCaeqal8aabaWdbiaaicdaa8aa baWdbmaalaaapaqaa8qacaWG4baapaqaa8qacaaIYaWaaOaaa8aaba WdbiaadoeacaWG0baameqaaaaaa0WdaeaapeGaey4kIipaaOGaamyz a8aadaahaaWcbeqaa8qacqGHsislcaWG1bWdamaaCaaameqabaWdbi aaikdaaaaaaOGaamizaiaadwhaaiaawUfacaGLDbaaaaa@6C33@$
In conclusion, the solution of approximated Mullins problem will be written as:
In conclusion for this part, the Mullins solution of approximated equation supposing $|y\text{'}|\ll 1$ , is given by:
Where erfc is the complementary error function.
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 εo 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 y1 (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
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.
New Method for the Correction of Mullins Solution
We recall below the general equation of the evaporationcondensation problem:
$∂y ∂t =C(T) y"( x ) ( 1+y' ( x ) 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamiD aaaacqGH9aqpcaWGdbGaaiikaiaadsfacaGGPaWaaSaaa8aabaWdbi aadMhacaGGIaWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaa8aa baWdbmaabmaapaqaa8qacaaIXaGaey4kaSIaamyEaiaacEcadaqada WdaeaapeGaamiEaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaaGccaGLOaGaayzkaaaaaaaa@4C63@$
Using the same notations given above, one writes:
The three combined equations (19) then give:
The second order differential equation is then given by the expression (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
The first integration of equation (22) as a function of the variable u gives:
$y ' ( u )= 2 3 m Ct ( 3 e − u 2 + m 2 e −3 u 2 )+Cst MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaWdbiaacEcaaaGcdaqadaWdaeaapeGa amyDaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacaaIYaaapa qaa8qacaaIZaaaaiaad2gadaGcaaWdaeaapeGaam4qaiaadshaaSqa baGcdaqadaWdaeaapeGaaG4maiaadwgapaWaaWbaaSqabeaapeGaey OeI0IaamyDa8aadaahaaadbeqaa8qacaaIYaaaaaaakiabgUcaRiaa d2gapaWaaWbaaSqabeaapeGaaGOmaaaakiaadwgapaWaaWbaaSqabe aapeGaeyOeI0IaaG4maiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaa aaaakiaawIcacaGLPaaacqGHRaWkcaWGdbGaam4Caiaadshaaaa@527E@$
Using the initial condition: $y\text{'}\left(0\right)=2m\sqrt{Ct}$ , one obtains the constant value Cst:
$Cst=− 2 3 m 3 Ct MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGdbGaam4CaiaadshacqGH9aqpcqGHsisldaWcaaWdaeaapeGa aGOmaaWdaeaapeGaaG4maaaacaWGTbWdamaaCaaaleqabaWdbiaaio daaaGcdaGcaaWdaeaapeGaam4qaiaadshaaSqabaaaaa@4089@$
and equation (22) becomes:
As a function of x and t, we will obtain:
$y ' ( x,t )= m 3 ( 3 e − x 2 4Ct + m 2 e − 3 x 2 4Ct )− m 3 3 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaWdbiaacEcaaaGcdaqadaWdaeaapeGa amiEaiaacYcacaWG0baacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aaba Wdbiaad2gaa8aabaWdbiaaiodaaaWaaeWaa8aabaWdbiaaiodacaWG LbWdamaaCaaaleqabaWdbiabgkHiTmaalaaapaqaa8qacaWG4bWdam aaCaaameqabaWdbiaaikdaaaaal8aabaWdbiaaisdacaWGdbGaamiD aaaaaaGccqGHRaWkcaWGTbWdamaaCaaaleqabaWdbiaaikdaaaGcca WGLbWdamaaCaaaleqabaWdbiabgkHiTmaalaaapaqaa8qacaaIZaGa amiEa8aadaahaaadbeqaa8qacaaIYaaaaaWcpaqaa8qacaaI0aGaam 4qaiaadshaaaaaaaGccaGLOaGaayzkaaGaeyOeI0YaaSaaa8aabaWd biaad2gapaWaaWbaaSqabeaapeGaaG4maaaaaOWdaeaapeGaaG4maa aaaaa@5781@$
The second integration of equation (23) leads to:
Equation (24) represents the new solution of the general problem given by equation (21)

Using the boundary condition y(ul)=0, for u=ul corresponding to xl=2√Ct ul, one obtains the following equation:
$0= 2 3 m Ct ( 3 ∫ 0 u l e − u 2 du+ m 2 ∫ 0 u l e −3 u 2 du )− 2 3 m 3 Ct u l + y ( 0 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIWaGaeyypa0ZaaSaaa8aabaWdbiaaikdaa8aabaWdbiaaioda aaGaamyBamaakaaapaqaa8qacaWGdbGaamiDaaWcbeaakmaabmaapa qaa8qacaaIZaWaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiaadwha paWaaSbaaWqaa8qacaWGSbaapaqabaaaneaapeGaey4kIipaaOGaam yza8aadaahaaWcbeqaa8qacqGHsislcaWG1bWdamaaCaaameqabaWd biaaikdaaaaaaOGaamizaiaadwhacqGHRaWkcaWGTbWdamaaCaaale qabaWdbiaaikdaaaGcdaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGa amyDa8aadaWgaaadbaWdbiaadYgaa8aabeaaa0qaa8qacqGHRiI8aa GccaWGLbWdamaaCaaaleqabaWdbiabgkHiTiaaiodacaWG1bWdamaa CaaameqabaWdbiaaikdaaaaaaOGaamizaiaadwhaaiaawIcacaGLPa aacqGHsisldaWcaaWdaeaapeGaaGOmaaWdaeaapeGaaG4maaaacaWG TbWdamaaCaaaleqabaWdbiaaiodaaaGcdaGcaaWdaeaapeGaam4qai aadshaaSqabaGccaWG1bWdamaaBaaaleaapeGaamiBaaWdaeqaaOWd biabgUcaRiaadMhapaWaaWbaaSqabeaaaaGcpeWaaeWaa8aabaWdbi aaicdaaiaawIcacaGLPaaaaaa@6809@$
Therefore, the equation (25) is obtained:
Where ho is the groove deep.

Equation (25) shows that the groove deep ho strongly depend on the time t, the grain length ul, 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:
$∫ 0 X e − u 2 du= π 2 [ 1−erfc( X ) ] ∫ 0 X e −3 u 2 du= π 2 3 [ 1−erfc( 3 X ) ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbmaawahabeWcpaqaa8qacaaIWaaapaqaa8qacaWGybaan8aa baWdbiabgUIiYdaakiaadwgapaWaaWbaaSqabeaapeGaeyOeI0Iaam yDa8aadaahaaadbeqaa8qacaaIYaaaaaaakiaadsgacaWG1bGaeyyp a0ZaaSaaa8aabaWdbmaakaaapaqaa8qacqaHapaCaSqabaaak8aaba WdbiaaikdaaaWaamWaa8aabaWdbiaaigdacqGHsislcaWGLbGaamOC aiaadAgacaWGJbWaaeWaa8aabaWdbiaadIfaaiaawIcacaGLPaaaai aawUfacaGLDbaaaeaadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGa amiwaaqdpaqaa8qacqGHRiI8aaGccaWGLbWdamaaCaaaleqabaWdbi abgkHiTiaaiodacaWG1bWdamaaCaaameqabaWdbiaaikdaaaaaaOGa amizaiaadwhacqGH9aqpdaWcaaWdaeaapeWaaOaaa8aabaWdbiabec 8aWbWcbeaaaOWdaeaapeGaaGOmamaakaaapaqaa8qacaaIZaaaleqa aaaakmaadmaapaqaa8qacaaIXaGaeyOeI0IaamyzaiaadkhacaWGMb Gaam4yamaabmaapaqaa8qadaGcaaWdaeaapeGaaG4maaWcbeaakiaa dIfaaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaaa@69E6@$
with $X=\frac{x}{2\sqrt{Ct}}$ , equations (24) and (25) can be written as:
$\left\{\begin{array}{c}{y}^{}\left(u\right)=\frac{2}{3}m\sqrt{Ct}\left[\frac{3\sqrt{\pi }}{2}\left[1-erfc\left(u\right)\right]+\frac{\sqrt{\pi }}{2\sqrt{3}}{m}^{2}\left[1-erfc\left(\sqrt{3}u\right)\right]\right]-\frac{2}{3}{m}^{3}\sqrt{Ct}u+{y}^{}\left(0\right)\\ {y}^{}\left(0\right)=-{h}_{0}=\frac{2}{3}{m}^{3}\sqrt{Ct}{u}_{l}-\frac{2}{3}m\sqrt{Ct}\left[\frac{3\sqrt{\pi }}{2}\left[1-erfc\left({u}_{l}\right)\right]+\frac{\sqrt{\pi }}{2\sqrt{3}}{m}^{2}\left[1-erfc\left(\sqrt{3}{u}_{l}\right)\right]\right]\end{array}$
As a function of (x,t), one obtains:
Taking into account the Mullins solution, one can write the new solution as:
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.
Comparison between Mullins Solution and the New Solution
The following expression gives the ratio of Mullins solution yMullins on the corrected solution ycorrected :
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 (yMullins /ycorrected) 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 (yMullins)/ycorrected) 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:
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 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 ho 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 xl.

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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