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:
c= 1 R = y"( x ) [ 1+y ( x ) 2 ] 3 2        (1) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGJbGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadkfa aaGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadMhacaGGIaWaaeWaa8 aabaWdbiaadIhaaiaawIcacaGLPaaaa8aabaWdbmaadmaapaqaa8qa caaIXaGaey4kaSIaamyEamaabmaapaqaa8qacaWG4baacaGLOaGaay zkaaWdamaaCaaaleqabaWdbiaaikdaaaaakiaawUfacaGLDbaapaWa aWbaaSqabeaapeWaaSaaa8aabaWdbiaaiodaa8aabaWdbiaaikdaaa aaaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGOaGaaeymaiaabMcaaaa@516E@
Where R is the curvature radius at point M y'( x )= y x MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaai4jamaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaGa eyypa0ZaaSaaa8aabaWdbiabgkGi2kaadMhaa8aabaWdbiabgkGi2k aadIhaaaaaaa@407F@ and y( x )= 2 y x 2 . MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaaiyhGmaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaGa eyypa0ZaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYa aaaOGaamyEaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qa caaIYaaaaaaakiaac6caaaa@436B@
The mathematical equation governing the evaporationcondensation problem can be written here as:
y t =C(T) y"( x ) ( 1+y ( x ) 2 )        (2) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamiD aaaacqGH9aqpcaWGdbGaaiikaiaadsfacaGGPaWaaSaaa8aabaWdbi aadMhacaGGIaWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaa8aa baWdbmaabmaapaqaa8qacaaIXaGaey4kaSIaamyEamaabmaapaqaa8 qacaWG4baacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaa kiaawIcacaGLPaaaaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGZaGaaeykaaaa@523A@
Where C(T) a constant of the problem depending on the temperature T, given by:
C( T )=μ P 0 ( T )γ( T ) ω 2 2πmkT       (3) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGdbWaaeWaa8aabaWdbiaadsfaaiaawIcacaGLPaaacqGH9aqp cqaH8oqBdaWcaaWdaeaapeGaamiua8aadaWgaaWcbaWdbiaaicdaa8 aabeaak8qadaqadaWdaeaapeGaamivaaGaayjkaiaawMcaaiabeo7a Nnaabmaapaqaa8qacaWGubaacaGLOaGaayzkaaGaeqyYdC3damaaCa aaleqabaWdbiaaikdaaaaak8aabaWdbmaakaaapaqaa8qacaaIYaGa eqiWdaNaamyBaiaadUgacaWGubaaleqaaaaakiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGaaeykaaaa@534B@
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:
y t =C(T)y"( x )     (4) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamiD aaaacqGH9aqpcaWGdbGaaiikaiaadsfacaGGPaGaamyEaiaackcada qadaWdaeaapeGaamiEaaGaayjkaiaawMcaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaabsdacaqGPaaaaa@48AD@
With the boundary conditions:
{ y(x,0)=0 y'( 0,t )=tantanθ=m      (5) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaqaabeqaaiaadMhacaGGOaGaamiEaiaacYcacaaIWaGa aiykaiabg2da9iaaicdaaeaapeGaamyEaiaacEcadaqadaWdaeaape GaaGimaiaacYcacaWG0baacaGLOaGaayzkaaGaeyypa0JaciiDaiaa cggacaGGUbGaciiDaiaacggacaGGUbGaeqiUdeNaeyypa0JaamyBaa aacaGL7baacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG 1aGaaeykaaaa@531C@
This problem is well-known in the conduction of heat in solids. It can be resolved by the following variable change:
u= x 2 Ct       (6) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1bGaeyypa0ZaaSaaa8aabaWdbiaadIhaa8aabaWdbiaaikda daGcaaWdaeaapeGaam4qaiaadshaaSqabaaaaOGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeikaiaabAdacaqGPaaaaa@4204@
and one obtains the derivatives of u as a function of x and t:
{ u x = u x u t =2C u 3 x 2          (7) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaqaabeqaa8qadaWcaaWdaeaapeGaeyOaIyRaamyDaaWd aeaapeGaeyOaIyRaamiEaaaacqGH9aqpdaWcaaWdaeaapeGaamyDaa WdaeaapeGaamiEaaaaaeaadaWcaaWdaeaapeGaeyOaIyRaamyDaaWd aeaapeGaeyOaIyRaamiDaaaacqGH9aqpcqGHsislcaaIYaGaam4qam aalaaapaqaa8qacaWG1bWdamaaCaaaleqabaWdbiaaiodaaaaak8aa baWdbiaadIhapaWaaWbaaSqabeaapeGaaGOmaaaaaaaaaOGaay5Eaa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeikaiaabEdacaqGPaaaaa@546C@
Now, using y t = y u u t MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamiD aaaacqGH9aqpdaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaey OaIyRaamyDaaaadaWcaaWdaeaapeGaeyOaIyRaamyDaaWdaeaapeGa eyOaIyRaamiDaaaaaaa@464C@ and   y x = y u u x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcWaaSaaa8aabaWdbiabgkGi2kaadMhaa8aabaWdbiabgkGi 2kaadIhaaaGaeyypa0ZaaSaaa8aabaWdbiabgkGi2kaadMhaa8aaba WdbiabgkGi2kaadwhaaaWaaSaaa8aabaWdbiabgkGi2kaadwhaa8aa baWdbiabgkGi2kaadIhaaaaaaa@4777@ one obtains the following derivatives:
y t =2C u 3 x 2 y u      (8) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbmaalaaapaqaa8qacqGHciITcaWG5baapaqaa8qacqGHciIT caWG0baaaiabg2da9iabgkHiTiaaikdacaWGdbWaaSaaa8aabaWdbi aadwhapaWaaWbaaSqabeaapeGaaG4maaaaaOWdaeaapeGaamiEa8aa daahaaWcbeqaa8qacaaIYaaaaaaakmaalaaapaqaa8qacqGHciITca WG5baapaqaa8qacqGHciITcaWG1baaaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeikaiaabIdacaqGPaaaaa@4DC4@ y'( x )= y x = u x y u       (9) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbiaadMhacaGGNaWaaeWaa8aabaWdbiaadIhaaiaawIcacaGL PaaacqGH9aqpdaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaey OaIyRaamiEaaaacqGH9aqpdaWcaaWdaeaapeGaamyDaaWdaeaapeGa amiEaaaadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIy RaamyDaaaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG OaGaaeyoaiaabMcaaaa@4F2A@ y"( x )= 2 y x 2 = ( u x ) 2 2 y u 2       (10) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbiaadMhacaGGIaWaaeWaa8aabaWdbiaadIhaaiaawIcacaGL PaaacqGH9aqpdaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbi aaikdaaaGccaWG5baapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqa baWdbiaaikdaaaaaaOGaeyypa0ZaaeWaa8aabaWdbmaalaaapaqaa8 qacaWG1baapaqaa8qacaWG4baaaaGaayjkaiaawMcaa8aadaahaaWc beqaa8qacaaIYaaaaOWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbe qaa8qacaaIYaaaaOGaamyEaaWdaeaapeGaeyOaIyRaamyDa8aadaah aaWcbeqaa8qacaaIYaaaaaaakiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGXaGaaeimaiaabMcaaaa@56D2@
Then, equation (4) becomes as a function of u:
2 y u 2 =2u y u       (11) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOm aaaakiaadMhaa8aabaWdbiabgkGi2kaadwhapaWaaWbaaSqabeaape GaaGOmaaaaaaGccqGH9aqpcqGHsislcaaIYaGaamyDamaalaaapaqa a8qacqGHciITcaWG5baapaqaa8qacqGHciITcaWG1baaaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeymaiaa bMcaaaa@4D01@
and
y''( u ) y'( u ) =2u     (12) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaMbqaaaaa aaaaWdbmaalaaapaqaa8qacaWG5bGaai4jaiaacEcadaqadaWdaeaa peGaamyDaaGaayjkaiaawMcaaaWdaeaapeGaamyEaiaacEcadaqada WdaeaapeGaamyDaaGaayjkaiaawMcaaaaacqGH9aqpcqGHsislcaaI YaGaamyDaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabg dacaqGYaGaaeykaaaa@49A6@
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 y u = y x x u MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamyD aaaacqGH9aqpdaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaey OaIyRaamiEaaaadaWcaaWdaeaapeGaeyOaIyRaamiEaaWdaeaapeGa eyOaIyRaamyDaaaaaaa@4654@ , we obtain:
y u =2 Ct y x      (13) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamyD aaaacqGH9aqpcaaIYaWaaOaaa8aabaWdbiaadoeacaWG0baaleqaaO WaaSaaa8aabaWdbiabgkGi2kaadMhaa8aabaWdbiabgkGi2kaadIha aaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabo dacaqGPaaaaa@49F4@
With the condition boundary y’(0,t) =m, one obtains A=2m Ct MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbGaeyypa0JaaGOmaiaad2gadaGcaaWdaeaapeGaam4qaiaa dshaaSqabaaaaa@3B8B@ . Using the other condition boundary y (x, 0) = 0, the solution of the differential equation (13) becomes:
y=2m Ct 0 u e u 2 du+Cst, u= x 2 Ct      (14) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaeyypa0JaaGOmaiaad2gadaGcaaWdaeaapeGaam4qaiaa dshaaSqabaGcdaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaamyDaa qdpaqaa8qacqGHRiI8aaGccaWGLbWdamaaCaaaleqabaWdbiabgkHi TiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaaaaGccaWGKbGaamyDai abgUcaRiaadoeacaWGZbGaamiDaiaacYcacaqGGaGaamyDaiabg2da 9maalaaapaqaa8qacaWG4baapaqaa8qacaaIYaWaaOaaa8aabaWdbi aadoeacaWG0baaleqaaaaakiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeikaiaabgdacaqG0aGaaeykaaaa@5757@
The constant Cst can be determined by the boundary condition y(∞,t)=0. This gives:
2m Ct 0 e u 2 du+Cst=0       (15) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaamyBamaakaaapaqaa8qacaWGdbGaamiDaaWcbeaakmaa wahabeWcpaqaa8qacaaIWaaapaqaa8qacqGHEisPa0WdaeaapeGaey 4kIipaaOGaamyza8aadaahaaWcbeqaa8qacqGHsislcaWG1bWdamaa CaaameqabaWdbiaaikdaaaaaaOGaamizaiaadwhacqGHRaWkcaWGdb Gaam4CaiaadshacqGH9aqpcaaIWaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeynaiaabMcaaaa@5172@
With 0 e u 2 du= π 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaeyOhIukan8aabaWd biabgUIiYdaakiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaamyDa8 aadaahaaadbeqaa8qacaaIYaaaaaaakiaadsgacaWG1bGaeyypa0Za aSaaa8aabaWdbmaakaaapaqaa8qacqaHapaCaSqabaaak8aabaWdbi aaikdaaaaaaa@453D@ one obtains: Cst=2m Ct π 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGdbGaam4CaiaadshacqGH9aqpcqGHsislcaaIYaGaamyBamaa kaaapaqaa8qacaWGdbGaamiDaaWcbeaakmaalaaapaqaa8qadaGcaa WdaeaapeGaeqiWdahaleqaaaGcpaqaa8qacaaIYaaaaaaa@4180@ and equation (15) can be written:
y=2m Ct 0 x 2 Ct e u 2 du2m 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:
y( x,t )=m πCt [ 1 2 π 0 x 2 Ct e u 2 du ]       (16) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjkaiaa wMcaaiabg2da9iabgkHiTiaad2gadaGcaaWdaeaapeGaeqiWdaNaam 4qaiaadshaaSqabaGcdaWadaWdaeaapeGaaGymaiabgkHiTmaalaaa paqaa8qacaaIYaaapaqaa8qadaGcaaWdaeaapeGaeqiWdahaleqaaa aakmaawahabeWcpaqaa8qacaaIWaaapaqaa8qadaWcaaWdaeaapeGa amiEaaWdaeaapeGaaGOmamaakaaapaqaa8qacaWGdbGaamiDaaadbe aaaaaan8aabaWdbiabgUIiYdaakiaadwgapaWaaWbaaSqabeaapeGa eyOeI0IaamyDa8aadaahaaadbeqaa8qacaaIYaaaaaaakiaadsgaca WG1baacaGLBbGaayzxaaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabIcacaqGXaGaaeOnaiaabMcaaaa@5DAE@
In conclusion for this part, the Mullins solution of approximated equation supposing | y' |1 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaabdaWdaeaapeGaamyEaiaacEcaaiaawEa7caGLiWoacqWIQjsp caaIXaaaaa@3D15@ , is given by:
{ y( x,t )=m πCt erfc( x 2 Ct ) y'( u )=2m Ct e u 2 ,withu= x 2 Ct        (17) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeGabaaabaWdbiaadMhadaqadaWdaeaapeGa amiEaiaacYcacaWG0baacaGLOaGaayzkaaGaeyypa0JaeyOeI0Iaam yBamaakaaapaqaa8qacqaHapaCcaWGdbGaamiDaaWcbeaakiaadwga caWGYbGaamOzaiaadogadaqadaWdaeaapeWaaSaaa8aabaWdbiaadI haa8aabaWdbiaaikdadaGcaaWdaeaapeGaam4qaiaadshaaSqabaaa aaGccaGLOaGaayzkaaaapaqaa8qacaWG5bGaai4jamaabmaapaqaa8 qacaWG1baacaGLOaGaayzkaaGaeyypa0JaaGOmaiaad2gadaGcaaWd aeaapeGaam4qaiaadshaaSqabaGccaWGLbWdamaaCaaaleqabaWdbi abgkHiTiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaaaaGccaGGSaGa ae4DaiaabMgacaqG0bGaaeiAaiaadwhacqGH9aqpdaWcaaWdaeaape GaamiEaaWdaeaapeGaaGOmamaakaaapaqaa8qacaWGdbGaamiDaaWc beaaaaaaaaGccaGL7baacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeikaiaabgdacaqG3aGaaeykaaaa@6BC8@
Where erfc is the complementary error function.
However, knowing that ( 0,t )= ε 0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaaGimaiaacYcacaWG0baacaGLOaGaayzkaaGa eyypa0JaeyOeI0IaeqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaa aa@3ECF@ , one deduces the value of the groove depth:
ε 0 ( t )=m πCt =tanθ πCt      (18) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH1oqzpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8aa baWdbiaadshaaiaawIcacaGLPaaacqGH9aqpcaWGTbWaaOaaa8aaba Wdbiabec8aWjaadoeacaWG0baaleqaaOGaeyypa0JaamiDaiaadgga caWGUbGaeqiUde3aaOaaa8aabaWdbiabec8aWjaadoeacaWG0baale qaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaa bIdacaqGPaaaaa@508B@
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:
{ 1+y' ( x ) 2 =1+ ( 1 2 Ct y u ) 2 y t =2C u 3 x 2 y u y"( x )= 2 y x 2 = ( u x ) 2 2 y u 2        (19) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeWabaaabaWdbiaaigdacqGHRaWkcaWG5bGa ai4jamaabmaapaqaa8qacaWG4baacaGLOaGaayzkaaWdamaaCaaale qabaWdbiaaikdaaaGccqGH9aqpcaaIXaGaey4kaSYaaeWaa8aabaWd bmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaWaaOaaa8aabaWdbi aadoeacaWG0baaleqaaaaakmaalaaapaqaa8qacqGHciITcaWG5baa paqaa8qacqGHciITcaWG1baaaaGaayjkaiaawMcaa8aadaahaaWcbe qaa8qacaaIYaaaaaGcpaqaa8qadaWcaaWdaeaapeGaeyOaIyRaamyE aaWdaeaapeGaeyOaIyRaamiDaaaacqGH9aqpcqGHsislcaaIYaGaam 4qamaalaaapaqaa8qacaWG1bWdamaaCaaaleqabaWdbiaaiodaaaaa k8aabaWdbiaadIhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGcdaWcaa WdaeaapeGaeyOaIyRaamyEaaWdaeaapeGaeyOaIyRaamyDaaaaa8aa baWdbiaadMhacaGGIaWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPa aacqGH9aqpdaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaa ikdaaaGccaWG5baapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqaba WdbiaaikdaaaaaaOGaeyypa0ZaaeWaa8aabaWdbmaalaaapaqaa8qa caWG1baapaqaa8qacaWG4baaaaGaayjkaiaawMcaa8aadaahaaWcbe qaa8qacaaIYaaaaOWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqa a8qacaaIYaaaaOGaamyEaaWdaeaapeGaeyOaIyRaamyDa8aadaahaa Wcbeqaa8qacaaIYaaaaaaaaaaakiaawUhaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabMdacaqGPa aaaa@7FD5@
The three combined equations (19) then give:
2C u 3 x 2 y u =C ( u x ) 2 2 y u 2 1+ ( 1 2 Ct y u ) 2       (20) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHsislcaaIYaGaam4qamaalaaapaqaa8qacaWG1bWdamaaCaaa leqabaWdbiaaiodaaaaak8aabaWdbiaadIhapaWaaWbaaSqabeaape GaaGOmaaaaaaGcdaWcaaWdaeaapeGaeyOaIyRaamyEaaWdaeaapeGa eyOaIyRaamyDaaaacqGH9aqpcaWGdbWaaSaaa8aabaWdbmaabmaapa qaa8qadaWcaaWdaeaapeGaamyDaaWdaeaapeGaamiEaaaaaiaawIca caGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakmaalaaapaqaa8qacq GHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadMhaa8aabaWdbiab gkGi2kaadwhapaWaaWbaaSqabeaapeGaaGOmaaaaaaaak8aabaWdbi aaigdacqGHRaWkdaqadaWdaeaapeWaaSaaa8aabaWdbiaaigdaa8aa baWdbiaaikdadaGcaaWdaeaapeGaam4qaiaadshaaSqabaaaaOWaaS aaa8aabaWdbiabgkGi2kaadMhaa8aabaWdbiabgkGi2kaadwhaaaaa caGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqGWaGa aeykaaaa@6418@
The second order differential equation is then given by the expression (21):
y"( u )=2uy'( u )( 1+ 1 4Ct y' ( u ) 2 )       (21) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaaiOiamaabmaapaqaa8qacaWG1baacaGLOaGaayzkaaGa eyypa0JaeyOeI0IaaGOmaiaadwhacaWG5bGaai4jamaabmaapaqaa8 qacaWG1baacaGLOaGaayzkaaWaaeWaa8aabaWdbiaaigdacqGHRaWk daWcaaWdaeaapeGaaGymaaWdaeaapeGaaGinaiaadoeacaWG0baaai aadMhacaGGNaWaaeWaa8aabaWdbiaadwhaaiaawIcacaGLPaaapaWa aWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawMcaaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabgda caqGPaaaaa@55AF@
In this paper, we propose a new method of resolution of equation (21) using the approximated Mullins solution: y'( u )=2m Ct e u 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaai4jamaabmaapaqaa8qacaWG1baacaGLOaGaayzkaaGa eyypa0JaaGOmaiaad2gadaGcaaWdaeaapeGaam4qaiaadshaaSqaba GccaWGLbWdamaaCaaaleqabaWdbiabgkHiTiaadwhapaWaaWbaaWqa beaapeGaaGOmaaaaaaaaaa@4340@ and replacing it in equation (21), one obtains
y"( u )=4m Ct u e u 2 ( 1+ m 2 e 2 u 2 )       (22) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaaiOiamaabmaapaqaa8qacaWG1baacaGLOaGaayzkaaGa eyypa0JaeyOeI0IaaGinaiaad2gadaGcaaWdaeaapeGaam4qaiaads haaSqabaGccaWG1bGaamyza8aadaahaaWcbeqaa8qacqGHsislcaWG 1bWdamaaCaaameqabaWdbiaaikdaaaaaaOWaaeWaa8aabaWdbiaaig dacqGHRaWkcaWGTbWdamaaCaaaleqabaWdbiaaikdaaaGccaWGLbWd amaaCaaaleqabaWdbiabgkHiTiaaikdacaWG1bWdamaaCaaameqaba WdbiaaikdaaaaaaaGccaGLOaGaayzkaaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeOmaiaabMcaaa a@5699@
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'( 0 )=2m Ct MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bGaai4jamaabmaapaqaa8qacaaIWaaacaGLOaGaayzkaaGa eyypa0JaaGOmaiaad2gadaGcaaWdaeaapeGaam4qaiaadshaaSqaba aaaa@3ED0@ , 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:
y ' ( u )= 2 3 m Ct ( 3 e u 2 + m 2 e 3 u 2 ) 2 3 m 3 Ct      (23) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaWdbiaacEcaaaGcdaqadaWdaeaapeGa amyDaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacaaIYaaapa qaa8qacaaIZaaaaiaad2gadaGcaaWdaeaapeGaam4qaiaadshaaSqa baGcdaqadaWdaeaapeGaaG4maiaadwgapaWaaWbaaSqabeaapeGaey OeI0IaamyDa8aadaahaaadbeqaa8qacaaIYaaaaaaakiabgUcaRiaa d2gapaWaaWbaaSqabeaapeGaaGOmaaaakiaadwgapaWaaWbaaSqabe aapeGaeyOeI0IaaG4maiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaa aaaakiaawIcacaGLPaaacqGHsisldaWcaaWdaeaapeGaaGOmaaWdae aapeGaaG4maaaacaWGTbWdamaaCaaaleqabaWdbiaaiodaaaGcdaGc aaWdaeaapeGaam4qaiaadshaaSqabaGccaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGYaGaae4maiaabMcaaaa@5B92@
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:
y ( u )= 2 3 m Ct ( 3 0 u e u 2 du+ m 2 0 u e 3 u 2 du ) 2 3 m 3 Ct u+ y ( 0 )     (24) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaaaaOWdbmaabmaapaqaa8qacaWG1baa caGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbiaaikdaa8aabaWdbi aaiodaaaGaamyBamaakaaapaqaa8qacaWGdbGaamiDaaWcbeaakmaa bmaapaqaa8qacaaIZaWaaybCaeqal8aabaWdbiaaicdaa8aabaWdbi aadwhaa0WdaeaapeGaey4kIipaaOGaamyza8aadaahaaWcbeqaa8qa cqGHsislcaWG1bWdamaaCaaameqabaWdbiaaikdaaaaaaOGaamizai aadwhacqGHRaWkcaWGTbWdamaaCaaaleqabaWdbiaaikdaaaGcdaGf WbqabSWdaeaapeGaaGimaaWdaeaapeGaamyDaaqdpaqaa8qacqGHRi I8aaGccaWGLbWdamaaCaaaleqabaWdbiabgkHiTiaaiodacaWG1bWd amaaCaaameqabaWdbiaaikdaaaaaaOGaamizaiaadwhaaiaawIcaca GLPaaacqGHsisldaWcaaWdaeaapeGaaGOmaaWdaeaapeGaaG4maaaa caWGTbWdamaaCaaaleqabaWdbiaaiodaaaGcdaGcaaWdaeaapeGaam 4qaiaadshaaSqabaGccaWG1bGaey4kaSIaamyEa8aadaahaaWcbeqa aaaak8qadaqadaWdaeaapeGaaGimaaGaayjkaiaawMcaaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG0aGaaeykaaaa @6D58@
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:
y ( 0 )= h 0 = 2 3 m 3 Ct u l 2 3 m Ct ( 3 0 u l e u 2 du+ m 2 0 u l e 3 u 2 du )     (25) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaaaaOWdbmaabmaapaqaa8qacaaIWaaa caGLOaGaayzkaaGaeyypa0JaeyOeI0IaamiAa8aadaWgaaWcbaWdbi aaicdaa8aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaaGOmaaWdaeaa peGaaG4maaaacaWGTbWdamaaCaaaleqabaWdbiaaiodaaaGcdaGcaa WdaeaapeGaam4qaiaadshaaSqabaGccaWG1bWdamaaBaaaleaapeGa amiBaaWdaeqaaOWdbiabgkHiTmaalaaapaqaa8qacaaIYaaapaqaa8 qacaaIZaaaaiaad2gadaGcaaWdaeaapeGaam4qaiaadshaaSqabaGc daqadaWdaeaapeGaaG4mamaawahabeWcpaqaa8qacaaIWaaapaqaa8 qacaWG1bWdamaaBaaameaapeGaamiBaaWdaeqaaaqdbaWdbiabgUIi YdaakiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaamyDa8aadaahaa adbeqaa8qacaaIYaaaaaaakiaadsgacaWG1bGaey4kaSIaamyBa8aa daahaaWcbeqaa8qacaaIYaaaaOWaaybCaeqal8aabaWdbiaaicdaa8 aabaWdbiaadwhapaWaaSbaaWqaa8qacaWGSbaapaqabaaaneaapeGa ey4kIipaaOGaamyza8aadaahaaWcbeqaa8qacqGHsislcaaIZaGaam yDa8aadaahaaadbeqaa8qacaaIYaaaaaaakiaadsgacaWG1baacaGL OaGaayzkaaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae OmaiaabwdacaqGPaaaaa@706E@
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 [ 1erfc( X ) ] 0 X e 3 u 2 du= π 2 3 [ 1erfc( 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= x 2 Ct MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGybGaeyypa0ZaaSaaa8aabaWdbiaadIhaa8aabaWdbiaaikda daGcaaWdaeaapeGaam4qaiaadshaaSqabaaaaaaa@3BFA@ , equations (24) and (25) can be written as:
{ y ( u )= 2 3 m Ct [ 3 π 2 [ 1erfc( u ) ]+ π 2 3 m 2 [ 1erfc( 3 u ) ] ] 2 3 m 3 Ct u+ y ( 0 ) y ( 0 )= h 0 = 2 3 m 3 Ct u l 2 3 m Ct [ 3 π 2 [ 1erfc( u l ) ]+ π 2 3 m 2 [ 1erfc( 3 u l ) ] ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeGabaaabaWdbiaadMhapaWaaWbaaSqabeaa aaGcpeWaaeWaa8aabaWdbiaadwhaaiaawIcacaGLPaaacqGH9aqpda WcaaWdaeaapeGaaGOmaaWdaeaapeGaaG4maaaacaWGTbWaaOaaa8aa baWdbiaadoeacaWG0baaleqaaOWaamWaa8aabaWdbmaalaaapaqaa8 qacaaIZaWaaOaaa8aabaWdbiabec8aWbWcbeaaaOWdaeaapeGaaGOm aaaadaWadaWdaeaapeGaaGymaiabgkHiTiaadwgacaWGYbGaamOzai aadogadaqadaWdaeaapeGaamyDaaGaayjkaiaawMcaaaGaay5waiaa w2faaiabgUcaRmaalaaapaqaa8qadaGcaaWdaeaapeGaeqiWdahale qaaaGcpaqaa8qacaaIYaWaaOaaa8aabaWdbiaaiodaaSqabaaaaOGa amyBa8aadaahaaWcbeqaa8qacaaIYaaaaOWaamWaa8aabaWdbiaaig dacqGHsislcaWGLbGaamOCaiaadAgacaWGJbWaaeWaa8aabaWdbmaa kaaapaqaa8qacaaIZaaaleqaaOGaamyDaaGaayjkaiaawMcaaaGaay 5waiaaw2faaaGaay5waiaaw2faaiabgkHiTmaalaaapaqaa8qacaaI Yaaapaqaa8qacaaIZaaaaiaad2gapaWaaWbaaSqabeaapeGaaG4maa aakmaakaaapaqaa8qacaWGdbGaamiDaaWcbeaakiaadwhacqGHRaWk caWG5bWdamaaCaaaleqabaaaaOWdbmaabmaapaqaa8qacaaIWaaaca GLOaGaayzkaaaapaqaa8qacaWG5bWdamaaCaaaleqabaaaaOWdbmaa bmaapaqaa8qacaaIWaaacaGLOaGaayzkaaGaeyypa0JaeyOeI0Iaam iAa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqpdaWcaaWd aeaapeGaaGOmaaWdaeaapeGaaG4maaaacaWGTbWdamaaCaaaleqaba WdbiaaiodaaaGcdaGcaaWdaeaapeGaam4qaiaadshaaSqabaGccaWG 1bWdamaaBaaaleaapeGaamiBaaWdaeqaaOWdbiabgkHiTmaalaaapa qaa8qacaaIYaaapaqaa8qacaaIZaaaaiaad2gadaGcaaWdaeaapeGa am4qaiaadshaaSqabaGcdaWadaWdaeaapeWaaSaaa8aabaWdbiaaio dadaGcaaWdaeaapeGaeqiWdahaleqaaaGcpaqaa8qacaaIYaaaamaa dmaapaqaa8qacaaIXaGaeyOeI0IaamyzaiaadkhacaWGMbGaam4yam aabmaapaqaa8qacaWG1bWdamaaBaaaleaapeGaamiBaaWdaeqaaaGc peGaayjkaiaawMcaaaGaay5waiaaw2faaiabgUcaRmaalaaapaqaa8 qadaGcaaWdaeaapeGaeqiWdahaleqaaaGcpaqaa8qacaaIYaWaaOaa a8aabaWdbiaaiodaaSqabaaaaOGaamyBa8aadaahaaWcbeqaa8qaca aIYaaaaOWaamWaa8aabaWdbiaaigdacqGHsislcaWGLbGaamOCaiaa dAgacaWGJbWaaeWaa8aabaWdbmaakaaapaqaa8qacaaIZaaaleqaaO GaamyDa8aadaWgaaWcbaWdbiaadYgaa8aabeaaaOWdbiaawIcacaGL PaaaaiaawUfacaGLDbaaaiaawUfacaGLDbaaaaaacaGL7baaaaa@AAC8@
As a function of (x,t), one obtains:
{ y ( u )=m πCt erfc( u )+ m 3 3 πCt 3 [ 1erfc( 3 u ) ] 2 3 m 3 Ct u+m πCt + y ( 0 ) y ( 0 )+m πCt = 2 3 m 3 Ct u l +m πCt erfc( u l ) m 3 3 πCt 3 [ 1erfc( 3 u l ) ]  (26) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeGabaaabaWdbiaadMhapaWaaWbaaSqabeaa aaGcpeWaaeWaa8aabaWdbiaadwhaaiaawIcacaGLPaaacqGH9aqpcq GHsislcaWGTbWaaOaaa8aabaWdbiabec8aWjaadoeacaWG0baaleqa aOGaamyzaiaadkhacaWGMbGaam4yamaabmaapaqaa8qacaWG1baaca GLOaGaayzkaaGaey4kaSYaaSaaa8aabaWdbiaad2gapaWaaWbaaSqa beaapeGaaG4maaaaaOWdaeaapeGaaG4maaaadaGcaaWdaeaapeWaaS aaa8aabaWdbiabec8aWjaadoeacaWG0baapaqaa8qacaaIZaaaaaWc beaakmaadmaapaqaa8qacaaIXaGaeyOeI0IaamyzaiaadkhacaWGMb Gaam4yamaabmaapaqaa8qadaGcaaWdaeaapeGaaG4maaWcbeaakiaa dwhaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGHsisldaWcaaWdae aapeGaaGOmaaWdaeaapeGaaG4maaaacaWGTbWdamaaCaaaleqabaWd biaaiodaaaGcdaGcaaWdaeaapeGaam4qaiaadshaaSqabaGccaWG1b Gaey4kaSIaamyBamaakaaapaqaa8qacqaHapaCcaWGdbGaamiDaaWc beaakiabgUcaRiaadMhapaWaaWbaaSqabeaaaaGcpeWaaeWaa8aaba WdbiaaicdaaiaawIcacaGLPaaaa8aabaWdbiaadMhapaWaaWbaaSqa beaaaaGcpeWaaeWaa8aabaWdbiaaicdaaiaawIcacaGLPaaacqGHRa WkcaWGTbWaaOaaa8aabaWdbiabec8aWjaadoeacaWG0baaleqaaOGa eyypa0ZaaSaaa8aabaWdbiaaikdaa8aabaWdbiaaiodaaaGaamyBa8 aadaahaaWcbeqaa8qacaaIZaaaaOWaaOaaa8aabaWdbiaadoeacaWG 0baaleqaaOGaamyDa8aadaWgaaWcbaWdbiaadYgaa8aabeaak8qacq GHRaWkcaWGTbWaaOaaa8aabaWdbiabec8aWjaadoeacaWG0baaleqa aOGaamyzaiaadkhacaWGMbGaam4yamaabmaapaqaa8qacaWG1bWdam aaBaaaleaapeGaamiBaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgkHi Tmaalaaapaqaa8qacaWGTbWdamaaCaaaleqabaWdbiaaiodaaaaak8 aabaWdbiaaiodaaaWaaOaaa8aabaWdbmaalaaapaqaa8qacqaHapaC caWGdbGaamiDaaWdaeaapeGaaG4maaaaaSqabaGcdaWadaWdaeaape GaaGymaiabgkHiTiaadwgacaWGYbGaamOzaiaadogadaqadaWdaeaa peWaaOaaa8aabaWdbiaaiodaaSqabaGccaWG1bWdamaaBaaaleaape GaamiBaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaay5waiaaw2faaaaa aiaawUhaaiaabccacaqGOaGaaeOmaiaabwdacaqGPaaaaa@A69C@
Taking into account the Mullins solution, one can write the new solution as:
y ( x,t )= y Mullins ( x,t )+ m 3 3 πCt 3 [ 1erfc( 3 Ct x 2 ) ] m 3 3 x+m πCt + y ( 0 )   (27) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaaaaOWdbmaabmaapaqaa8qacaWG4bGa aiilaiaadshaaiaawIcacaGLPaaacqGH9aqpcaWG5bWdamaaBaaale aapeGaamytaiaadwhacaWGSbGaamiBaiaadMgacaWGUbGaam4CaaWd aeqaaOWaaWbaaSqabeaaaaGcpeWaaeWaa8aabaWdbiaadIhacaGGSa GaamiDaaGaayjkaiaawMcaaiabgUcaRmaalaaapaqaa8qacaWGTbWd amaaCaaaleqabaWdbiaaiodaaaaak8aabaWdbiaaiodaaaWaaOaaa8 aabaWdbmaalaaapaqaa8qacqaHapaCcaWGdbGaamiDaaWdaeaapeGa aG4maaaaaSqabaGcdaWadaWdaeaapeGaaGymaiabgkHiTiaadwgaca WGYbGaamOzaiaadogadaqadaWdaeaapeWaaOaaa8aabaWdbmaalaaa paqaa8qacaaIZaaapaqaa8qacaWGdbGaamiDaaaaaSqabaGcdaWcaa WdaeaapeGaamiEaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaaaiaa wUfacaGLDbaacqGHsisldaWcaaWdaeaapeGaamyBa8aadaahaaWcbe qaa8qacaaIZaaaaaGcpaqaa8qacaaIZaaaaiaadIhacqGHRaWkcaWG TbWaaOaaa8aabaWdbiabec8aWjaadoeacaWG0baaleqaaOGaey4kaS IaamyEa8aadaahaaWcbeqaaaaak8qadaqadaWdaeaapeGaaGimaaGa ayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaae4nai aabMcaaaa@73FE@
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 :
y Mullins y corrected = y Mullins ( x,t ) y Mullins ( x,t )+ m 3 3 πCt 3 [ 1erfc( 3 Ct x 2 ) ] m 3 3 x+m πCt + y ( 0 )    (28) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamyEa8aadaWgaaWcbaWdbiaad2eacaWG1bGa amiBaiaadYgacaWGPbGaamOBaiaadohaa8aabeaaaOqaa8qacaWG5b WdamaaBaaaleaapeGaam4yaiaad+gacaWGYbGaamOCaiaadwgacaWG JbGaamiDaiaadwgacaWGKbaapaqabaaaaOWdbiabg2da9maalaaapa qaa8qacaWG5bWdamaaBaaaleaapeGaamytaiaadwhacaWGSbGaamiB aiaadMgacaWGUbGaam4CaaWdaeqaaOWaaWbaaSqabeaaaaGcpeWaae Waa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjkaiaawMcaaaWdaeaa peGaamyEa8aadaWgaaWcbaWdbiaad2eacaWG1bGaamiBaiaadYgaca WGPbGaamOBaiaadohaa8aabeaakmaaCaaaleqabaaaaOWdbmaabmaa paqaa8qacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaacqGHRaWkda WcaaWdaeaapeGaamyBa8aadaahaaWcbeqaa8qacaaIZaaaaaGcpaqa a8qacaaIZaaaamaakaaapaqaa8qadaWcaaWdaeaapeGaeqiWdaNaam 4qaiaadshaa8aabaWdbiaaiodaaaaaleqaaOWaamWaa8aabaWdbiaa igdacqGHsislcaWGLbGaamOCaiaadAgacaWGJbWaaeWaa8aabaWdbm aakaaapaqaa8qadaWcaaWdaeaapeGaaG4maaWdaeaapeGaam4qaiaa dshaaaaaleqaaOWaaSaaa8aabaWdbiaadIhaa8aabaWdbiaaikdaaa aacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyOeI0YaaSaaa8aabaWd biaad2gapaWaaWbaaSqabeaapeGaaG4maaaaaOWdaeaapeGaaG4maa aacaWG4bGaey4kaSIaamyBamaakaaapaqaa8qacqaHapaCcaWGdbGa amiDaaWcbeaakiabgUcaRiaadMhapaWaaWbaaSqabeaaaaGcpeWaae Waa8aabaWdbiaaicdaaiaawIcacaGLPaaaaaGaaeiiaiaabccacaqG GaGaaeikaiaabkdacaqG4aGaaeykaaaa@8D2B@
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:
y ' Mullins y ' corrected =1+ m 2 3 ( e 2 u 2 e + u 2 )     (29) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamyEaiaacEcapaWaaSbaaSqaa8qacaWGnbGa amyDaiaadYgacaWGSbGaamyAaiaad6gacaWGZbaapaqabaaakeaape GaamyEaiaacEcapaWaaSbaaSqaa8qacaWGJbGaam4BaiaadkhacaWG YbGaamyzaiaadogacaWG0bGaamyzaiaadsgaa8aabeaaaaGcpeGaey ypa0JaaGymaiabgUcaRmaalaaapaqaa8qacaWGTbWdamaaCaaaleqa baWdbiaaikdaaaaak8aabaWdbiaaiodaaaWaaeWaa8aabaWdbiaadw gapaWaaWbaaSqabeaapeGaeyOeI0IaaGOmaiaadwhapaWaaWbaaWqa beaapeGaaGOmaaaaaaGccqGHsislcaWGLbWdamaaCaaaleqabaWdbi abgUcaRiaadwhapaWaaWbaaWqabeaapeGaaGOmaaaaaaaakiaawIca caGLPaaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGYa GaaeyoaiaabMcaaaa@60BD@
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 ( x,t )= y Mullins ( x,t )+ m 3 3 πCt 3 [ 1erfc( 3 Ct x 2 ) ] m 3 3 x+m πCt h 0 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG5bWdamaaCaaaleqabaaaaOWdbmaabmaapaqaa8qacaWG4bGa aiilaiaadshaaiaawIcacaGLPaaacqGH9aqpcaWG5bWdamaaBaaale aapeGaamytaiaadwhacaWGSbGaamiBaiaadMgacaWGUbGaam4CaaWd aeqaaOWaaWbaaSqabeaaaaGcpeWaaeWaa8aabaWdbiaadIhacaGGSa GaamiDaaGaayjkaiaawMcaaiabgUcaRmaalaaapaqaa8qacaWGTbWd amaaCaaaleqabaWdbiaaiodaaaaak8aabaWdbiaaiodaaaWaaOaaa8 aabaWdbmaalaaapaqaa8qacqaHapaCcaWGdbGaamiDaaWdaeaapeGa aG4maaaaaSqabaGcdaWadaWdaeaapeGaaGymaiabgkHiTiaadwgaca WGYbGaamOzaiaadogadaqadaWdaeaapeWaaOaaa8aabaWdbmaalaaa paqaa8qacaaIZaaapaqaa8qacaWGdbGaamiDaaaaaSqabaGcdaWcaa WdaeaapeGaamiEaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaaaiaa wUfacaGLDbaacqGHsisldaWcaaWdaeaapeGaamyBa8aadaahaaWcbe qaa8qacaaIZaaaaaGcpaqaa8qacaaIZaaaaiaadIhacqGHRaWkcaWG TbWaaOaaa8aabaWdbiabec8aWjaadoeacaWG0baaleqaaOGaeyOeI0 IaamiAa8aadaWgaaWcbaWdbiaaicdaa8aabeaaaaa@6DA5@
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 πCt m 3 3 x l m πCt erfc( x l 2 Ct )+ m 3 3 πCt 3 [ 1erfc( 3 Ct x l 2 ) ] MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGObWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9iab gUcaRiaad2gadaGcaaWdaeaapeGaeqiWdaNaam4qaiaadshaaSqaba GccqGHsisldaWcaaWdaeaapeGaamyBa8aadaahaaWcbeqaa8qacaaI ZaaaaaGcpaqaa8qacaaIZaaaaiaadIhapaWaaSbaaSqaa8qacaWGSb aapaqabaGcpeGaeyOeI0IaamyBamaakaaapaqaa8qacqaHapaCcaWG dbGaamiDaaWcbeaakiaadwgacaWGYbGaamOzaiaadogadaqadaWdae aapeWaaSaaa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaWGSbaapaqa baaakeaapeGaaGOmamaakaaapaqaa8qacaWGdbGaamiDaaWcbeaaaa aakiaawIcacaGLPaaacqGHRaWkdaWcaaWdaeaapeGaamyBa8aadaah aaWcbeqaa8qacaaIZaaaaaGcpaqaa8qacaaIZaaaamaakaaapaqaa8 qadaWcaaWdaeaapeGaeqiWdaNaam4qaiaadshaa8aabaWdbiaaioda aaaaleqaaOWaamWaa8aabaWdbiaaigdacqGHsislcaWGLbGaamOCai aadAgacaWGJbWaaeWaa8aabaWdbmaakaaapaqaa8qadaWcaaWdaeaa peGaaG4maaWdaeaapeGaam4qaiaadshaaaaaleqaaOWaaSaaa8aaba WdbiaadIhapaWaaSbaaSqaa8qacaWGSbaapaqabaaakeaapeGaaGOm aaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@6DAC@
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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