Marangoni Convection during Free Electron Lar Nitriding of Titanium
DANIEL HO CHE,SVEN MU LLER,GERD RAPIN,MICHELLE SHINN,
ELVIRA REMDT,MAIK GUBISCH,and PETER SCHAAF
Pure titanium was treated by free electron lar(FEL)radiation in a nitrogen atmosphere.As a
result,nitrogen diffusion occurs and a TiN coating was synthesized.Local gradients of inter-
facial tension due to the local heating lead to a Marangoni convection,which determines the
track properties.Becau of the experimental inaccessibility of time-dependent occurrences,
finite element calculations were performed,to determine the physical process such as heat
transfer,meltflow,and mass transport.In order to calculate the surface deformation of the gas-
liquid interface,the level t approach was ud.The equations were modified and coupled with
heat-transfer and diffusion equations.The process was characterized by dimensionless numbers
such as the Reynolds,Peclet,and capillary numbers,to obtain more information about the
acting forces and the coating development.Moreover,the nitrogen distribution was calculated
using the corresponding transport equation.The simulations were compared with cross-
ctional micrographs of the treated titanium sheets and checked for their validity.Finally,
the process prented is discusd and compared with similar lar treatments.
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DOI:10.1007/s11663-009-9243-1
ÓThe Author(s)2009.This article is published with open access
I.INTRODUCTION
T HE nitriding and carbonizing of surfaces are well-known methods for improving tribological properties of different metallic compounds,in particular,of titanium and its alloys.The established methods of metals surface treatments are plasma and gas nitriding.[1]The linked process,mainly diffusion,have been described by many authors.They are bad on the diffusion-like process of the n
itrogen in the matrix in millicond time regimes.Alternatively,it is possible to treat the surface with lar radiation,to synthesize hard coatings directly; in general,this has been done by Nd:YAG[2]and CO2[3] lars.In this work,coatings veral microns in thick-ness were synthesized.In other experiments,[4,5]TiN coatings were generated for thefirst time by means of a free electron lar(FEL).Due to its high power and the flexibility in its temporal shaping,this type of lar could be the right tool.The coatings show quite good tribological properties such as hardness.In order to understand the various physical process,it is necessary to make in-situ investigations.The nitrogen transport and corresponding coating properties are determined by the time of treatment,for which diffusion will be assisted by the Marangoni convection for extended time regimes. Thefilm thickness mainly depends on the melting depth and the nitrogen profile resulting from the ratio of the diffusive and convectional transport.
The modeling of similar lar treatments is thoroughly discusd in the literature;different approaches are available for describing process such as welding,deep penetration welding,drilling,cladding,and alloying.It is sometimes necessary to u moving interface ap-proaches to describe the physics of the conditioning. Different numerical models could be ud to describe the physics of gas-liquid interfaces,especially for lar melt pools;the volume-of-fluid,[6]Lagrangian–Eule
r,[7]and level t[8]methods are among the most popular.The latter method was ud to describe the keyhole devel-opment of iron treatment,[9]the cladding of stainless steel,[10]and solidification.[11]
II.THEORETICAL BACKGROUND
A.Beam Properties of FEL
Experiments were performed at the Jefferson Lab (Thomas Jefferson National Accelerator Facility,New-port News,VA).The FEL operates like a synchrotron and can be adjusted in different time regimes and wavelengths.Detailed information is available in Ben-son et al.[12]Figure1shows the temporal pul structure in a continuous wave mode.It is a quence of approximately200-fs puls with a frequency of approx-imately4.7MHz.Alternatively,it is possible to switch to the puld mode with macropuls of some hundreds of microconds full width at half maximum(FWHM) at frequencies of10to60Hz.That tup was ud in other investigations.[4]
DANIEL HO CHE and SVEN MU LLER are with the Universita t Go ttingen,II.Physikalisches Institut,37077Go ttingen,Germany. GERD RAPIN is with the Institut fu r Numerische und Angewandte Mathematik,37083Go ttingen,Germany.MICHELLE SHINN is with the Thomas Jefferson National Accelerator Facility,Free Electron Lar Group,Newport News,VA23606.ELVIRA
REMDT,MAIK GUBISCH,and PETER SCHAAF are with the TU Ilmenau,Institut fu r Werkstofftechnik,FG Werkstoffe der Elektrotechnik,98684 Ilmenau,Germany.Contact e-mail:dhoeche@uni-goettingen.dez Manuscript submitted June24,2008.
Article published online May2,2009.
The wavelength k during the material processing was 1.64microns.Equation [1]describes the Gaussian-like shape of a pul in a Cartesian system (x ,y ,I ).The raw beam approximately 6cm in diameter was focud to a spot size of approximately 600l m:
I mic t ;x ;y ðÞ¼I 0mic exp
Àx 2þy 2ÀÁ2r 2d
!Áexp Àt Àt 0ðÞ22r 2t "#½1
spatial:r d ¼d b
2:35¼255l m
temporal:r t ¼s
pul
2:35¼85fs
Due to the millicond time regime of the titanium treatment,the lar power at the subpicocond puls was averaged over the processing time.As a result,an averaged intensity I 0was calculated and ud for the modeling.The averaged power was measured by a commercial calorimeter and compared,respectively,synchronized to the modeling data.In order to get the time dependence at a fixed obrvation point (y =0),the moving lar beam intensity I mov was defined as
I mov ¼I 0exp Àx 22r d
sin p t
v s
½2
Equation 2contains the spatial distribution exp Àx 2
2r d
and the perpendicular movement of the lar beam with the scan velocity ud described with a sine function
sin p t
s :Transient welding simulations normally have to be done in three dimensions,in order to describe the problem correctly.In the prent ca,the scan velocity is much lower than the melt flow velocity;this allows the u of two-dimensional modeling.Table I shows the ud beam parameters.B.Experimental Setup
Titanium sheets (blank,1-mm thick,>99.98pct purity)were cut into pieces 15915mm 2in size.For
the lar treatments,the samples were placed in a chamber first evacuated and then filled with nitrogen (purity 99.999pct)to a pressure of 1.15atm.The focud beam reached the sample surface through a fud silica window.In order to treat the whole surface of the samples,the chamber was mounted onto a computer-controlled x -y table.A relative velocity v =24mm/s in the x direction was ud.Figure 2shows the processing scheme.
The experiments have been realized at the FEL ur facility.By means of specific optics,the raw bea
m approximately 6cm in diameter was redirected to the lens and focud to the ud spot size.In addition,a camera was installed for monitoring the treatments.In Figure 3,a snapshot taken during the irradiation is shown.After the treatments,scanning electron micros-copy (SEM)was performed with a LEO Supra 35Gemini (Carl Zeiss SMT AG,Oberkochen,Germany);all images were recorded with a quadral backscattering detector.
C.Free Surface Modeling
Becau of the forceful influence of the advection,convection,and conduction effects during the lar treatment,the surface deformation is not negligible.In order to model the surface tracking,the level t method was applied.A function /(x ,y ,t )was defined over the
1—Gaussian-like time shape of a micropul at the time ps (FWHM 0.2ps).
Table I.
FEL Beam and Scan Parameters
Parameter Value I 03Æ109W/m 2P 650W
I 0mic 2.2Æ1015W/m 2E mic 125l J f 4.68MHz d b 600l m
s mic 200to 400fs v s
2.4
cm/s
Fig.2—Scanning scheme of the nitriding treatment and the moving melt pool at the symmetry axis.
entire domain,to describe the interface according to the following criteria:
/x ;y ;t ðÞ¼>0:5nitrogen ¼0:5interface <0:5titanium 2
6
4½3
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The interface movement can be described by a simple partial differential equation,which can be solved numerically at the same time as the Navier–Stokes,heat-transfer,and diffusion equations.Olsson et al .[13,14]ud the following the expression:
@/@t þu Ár /¼c L r Áe r /Án ðÞn ðÞÀ/1À/ðÞr /r /j j
½4 Here,the vector u is the speed function of the surface tracking,which is a result of the Navier–Stokes equa-tion and the normal vectors of the interface n .The term on the right side that describes the interface thickness e was t to e >2h (3to 4mesh elements for improving convergence),in line with the sugges-tions of some authors.[8,15,16]The term h is the mesh size and c L is a stabilization
parameter that determines the repetition of the reinitialization for each time step.The c L was t equal to 1,in order to avoid mass loss during the calculation and keep the interface thickness constant.The interface normal vector is defined by
n ¼
r /r /j j /¼0:5;½5 which is important in computing the surface curvature j S ,given as
j S ¼Àr n j /¼0:5
½6
and in implementing the surface tension force.A smoothed,continuously differentiable delta function was defined to accommodate the boundary conditions at the interface:
d ¼6r /j j /1À/ðÞj j
½7
As a result,it is possible to transform boundary forces to volumetric ones.Due to the smeared-out int
erface,the material parameters along the interface were de-scribed by Eq.[8];f corresponds to the different physi-cal properties (density (q ),viscosity (g ),etc .)of titanium and nitrogen.
f ¼f Ti þf N Àf Ti ðÞ/
½8
The initial level t function /0was t to y À0.001=0,according to the geometry.D.Heating and Evaporation
The classical heating of metals during lar irradiation can be described with the normal heat transport equation:q T ðÞc p T ðÞ
@T
@t
¼r Áj T ðÞr T ðÞþQ int d À~u q T ðÞc p T ðÞr T ½9
which contains the source term
Q int ¼a ext 1ÀR s ;l T ðÞÀÁI mov ~r ;t ðÞ
ÂÃ
Àh t T ÀT 0ðÞÀr B e B T 4ÀT 4
0ÀÁÀL ev J ev |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Q loss
½10
In this equation,the melting behavior,surface temper-ature,and heat loss Q loss are calculated.The specific heat and the thermal conductivity shown in Figures 4and 5were calculated by the data from the National Institute of Standards and Technology.[17]The c p was modified to accommodate the pha transition.There-fore,the latent heats were added as a strong increa in specific heat at the pha-change temperature,shown by the rectangles in Figure 4.
scissor sistersThe mathematical description of pha changes is a difficulty formally known as the Stefan problem.I
n our ca,there are two free boundaries (the top and bottom of the melt pool)for a parabolic equation.To solve the problem,the smoothed Heaviside function H (T ,B )was ud;B is the width of the temperature region in
which
Fig.3—Experimental tup consisting of X -Y table,lens,N 2cham-ber,and N 2
supply.
H (T ,B )changes from 0to 1and was assumed to be 50K:
f T ðÞ¼f sol Àf sol þf liq ÀÁ
ÁH T ;B ðÞ½11 Concerning our problem,function 11was ud for
the reflectivity R s,l and the mass density q (T ).For the reflectivity of the solid and liquid pha of titanium,the values were interpolated from the experimental data of Xie.[18]The radiation and convective heat loss to the surroundings were taken into account by the well-known formulations for convective heat transfer and the Stefan–Boltzmann law.The values of the heat-transfer coefficient h t and the emissivity e are shown in Table II .Additionally,the external absorption parameter a ext was defined to account for heat loss due to the optics,chamber,and soiling and was prescribed equal to 0.32.The high temperature leads to the evaporation and ablation of titanium in the ambient atmosphere.At the surface,a saturation pressure P S (T S )originates;this was calculated by the integrated Clausius–Clapyron equation:
P S T S ðÞ¼P 0exp L ev T S ÀT b ðÞ
R g T S T b ;½12
where L ev is the latent heat of evaporation,P 0the reference pressure (1atm)at the normal boiling point T b ,T S the surface temperature,and R g the universal gas constant.In the literature,varying values of parameters are available due to the different compositions of the alloys and other modifications.Conquently,the data of pure titanium (Table II )were taken as far as possible.The evaporation flux was calculated according to the Langmuir equation:
J ev ¼nP S T ðÞ
英文版简历ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p M Ti R gTi T
p ;
½13
and is usually an order of magnitude higher than the real physical rate at 1atm pressure.As a result,n was t to be 0.1.[26,27]Furthermore,it is possible to calculate the heat loss at the interface due to evaporation (the mathematical description is shown in Eq.[9]as the last term of Q loss ).
E.Hydrodynamics
For the asmbly of the transport equation,the maximum flow velocity in the melt pool has to be calculated empirically [28]as
V max %@c @T dT dy W 0:5
0:664q 0:5l 0:5eff 2=3½14 This expression allows for an estimation of the flow
motion and the expected surface deformation.In Table III ,all ud values are shown.As a result,V max was calculated to be approximately 2m/s,which is on the order of magnitude of such materials-processing techniques.
On the supposition that a melt is an incompressible fluid,the Navier–Stokes equation can be ud to describe the fluid motion in the melt pool:q @u @t
þu r u ¼r ÁÀP hyd I þg r u þr u ðÞT
þF sur þF g
½15
Table II.Physical Properties of Titanium and Nitrogen Ud
in Simulations [19–25]
Parameter
Value Reference
Titanium T L 1941K 19T b 3560K 19L m 295.6kJ/kg 19L ev 8.8MJ/kg 20h t
1590W/m 2K 21R s at 1.64l m @0.5022R l at 1.64l m @0.4522e B 0.297
23a 52Æ106m À119M Ti 47.89g 19q Ti s 4520kg/m 319q Ti l
b l4110kg/m 323g Ti at T =2000K ~3.2mPas 23r
1.65N/m
23[(¶r )/(¶T )]À2.4Æ10À4N/mK 23b T 1.169Æ10À41/K 23E a
243kJ/mol 24Nitrogen q N2
~
1kg/m 319g N2at T =2000K ~60l Pas 25j N20.026W/mK 19c p N21040J/kgK 19P 0
105Pa
—
Table III.Physical Properties of Titanium Ud for Approximation [23,29]
Parameter Valuehappiness什么意思
Reference
q Ti 4.11g/cm 3
23g eff0.61gcm À1s À129W
0.25mm
—[(¶r )/(¶T)]À0.24dyn cm À1K À123[dT /dy]
0.8Æ105K/cm
—
containing the equation of continuity:
rÁu¼0½16 It contains the mass density q,the dynamic viscosity g, and the hydrodynamic pressure P hyd.Moreover the velocityfield u is divergence free.The last term on the right side describes the buoyancy force F g(Eq.[17]), which was applied in the Boussinesq approximation to take into account the volumetric change due to ther-mal expansion:
F g¼q g b T TÀT L
ðÞ½17 The force F sur at the surface is a superposition of the surface tension force,the Marangoni force and the re-coil pressure induced force:
F sur¼2rj S nÀ@r
@T
IÀnn
ðÞÁr TþnÁP recoil
!
d;½18
whereas the acting forces were multiplied by d to in-sure that they appear only at the interface.The r is the surface tension of liquid titanium,j S is the inter-face curvature,n is the normal vector of the interface, [(¶r)/(¶T)]describes the surface tension coefficient,I is the identity matrix,and P recoil corresponds to the recoil pressure,which can be calculated by[30]
P recoil¼0:55P S T S
ðÞ½19 The nitrogen mass transfer convectional diffusion equation together with the other transport equations describes the nitrogen concentration c,according to the expression
bec中级报名费@c x;y;t ðÞ
@t þrÁD TðÞr c x;y;t
ðÞ
ðÞ¼J N dÀuÁr c x;y;t
ðÞ
½20
japaneschoolchild21The nitrogen absorption massflux D m per surface ele-ment S m was calculated by means of comparisons with the experimental results.Therefore,Eq.[21]from Ponticaud et al.,[24]who performed similar investiga-tions,was modified.Due to the smeared-out interface, the boundaryflux has to be recalculated as a volumet-ric reaction rate,which was done by means of Eq.[7] in the manner of the level t approach:
J N¼d D m
S m
¼j0eÀE a RT
ðÞ½21
For simplification,an averageflux j0per surface ele-ment was calculated.The temperature dependence was taken into account via the activation energy E a of nitrogen in titanium.Wood and P
aasche[31]found a formulation for the temperature-dependent diffusion coefficient during their studies of the lar irradiation of titanium:
D TðÞ%0:12Áexp À45200
1:98ÁT
!
;½22
which reprents a typical Arrhenius behavior(values are in cm2/s).Dimensionless numbers are ud influid dynamics to describe comparable physical problems and to reprent specific information such asflow behavior and acting forces.Table IV summarizes all dimension-less numbers ud in our simulations,including their physical background.As a result,it is necessary to define the characteristic length L,which is t generally to be the half width of the melt pool.[32]
F.Geometry and Boundaries
Modeling of such process generally requires simpli-fications and assumptions.From the outt,one
must utilize the radial symmetry of the melt pool and compute only one side.An advantage of the level t method is the formulation of volumetric boundary conditions.As a result,only boundaries outside the interface have to be declared mathematically.The symmetry axes were t to slip conditions according Figure6.At the solid bound-aries of the model,the velocity was t to be zero(no slip).No-slip conditions at the solid-liquid interface were obtained by means of a viscosity jump at the melting temperature T L.On the gaous side,neutral flow conditions were employed.Figure6shows the ud triangular mesh of Lagrangian elements.It contains 6153elements with3135nodes.This leads to degrees of freedom of46,671.
Table IV.Dimensionless Numbers and Their Physical
Meaning of Ratios
Re¼q u max L
g
inertial/viscosity force
Ma¼@r
@T
q c P TÀT L有道翻在线翻译
有声童话ðÞL
gj
intensity of Marangoni force
Pe¼u j j q c P L
j
convective/conductive heat transport
Ca¼@r
@T
TÀT L
ðÞ
r
Marangoni/surface tension force
Pr¼g c P kinematic viscosity/conductibility
of temperature
We¼q L u j j2
r
inertial/surface tension force
Gr¼g b T TÀT L
ðÞq2L
8
ðÞ3
g2
buoyancy/viscosity force
Sc¼g
D q
convective/diffusive mass
transport
Fig.6—Geometry,boundary conditions,and mesh of the simula-tion.
