Molecular Dynamics Study

Memorial International Collaboration between IFS Tohoku University and ITAM SB RAS for 20 Years1 ‒Advanced coating process and powder synthesis using plasma flows and arcs‒ Hideya NISHIYAMA2 and Oleg P. SOLONENKO3 Summary The memorial international collaborative research activities between IFS and ITAM SB RAS as shown in Fig.1 are described in this report referring to our book and our review paper. Our collaboration has been continuing for 20 years since 1995 under the research exchange agreement. The main research topics are focused on control of coating process in plasma spraying and control of nano- and micro powder processes in thermal plasma flow systems. Our approach is based on plasma flow dynamics and multiphase flow including nano and micro powders. The original discipline of control characteristics of particle laden plasma jet by applying electromagnetic field is explained. Furthermore, control of coating thickness in plasma spraying is also explained based on splat formation model. Subsequently, spheroidization process of micro-particle and doping process for nano particle using DC-RF hybrid plasma flow system is explained. Finally, functional nano-and micro powder process using original transfer and non-transfer arcs is explained by showing typical results.

from the stand point of functional fluid and multiphase

1. Introduction

flow. However, Nishiyama did not have any experience Our collaboration research has started just after

and knowledge of plasma spraying. Then, this was his

Solonenko visited Nishiyama laboratory in May 1995.

first motivation to ask Solonenko, authority of plasma

Nishiyama had been involving in the numerical

spraying in Russia, to drop in Nishiyama laboratory on

modeling and simulation of particle laden plasma jet in

the occasion of his participation in the International

the applied magnetic field. Nishiyama’s research

Thermal Spray Conference 1995 in Kobe just after the

objective was to investigate the control characteristics of

great earthquake.

particle laden plasma jet by applying the magnetic field

(a)

for the plasma spraying process. Nishiyama’s approach was focused on the clarification of electromagnetic effect on the dynamics of the inflight particle and plasma jet

1. Manuscript received on June 1, 2015 and partly published in Nano-Mega Scale Flow Dynamics in Complex Systems, The 21st

(b)

Century COE Program, International COE of Flow Dynamics, Lecture Series Vol.12, (2007), Tohoku University Press, and Journal of Visualization, Vol.18 (2015), pp.1-15. 2. Professor, Institute of Fluid Science, Tohoku University, Japan. 3. Professor, Khristianovich Institute of Theoretical and Applied

Fig.1

Mechanics, SB RAS, Russia

1

(a) IFS and (b) ITAM SB RAS.

Rep. Inst. Fluid Science, Vol. 27 (2015)

Since then, we have been collaborating each other.

coating process in plasma spraying by applying

Nishiyama was in charge of numerical modeling and

electromagnetic

field.

The

second

part

is

the

numerical simulation on particle laden plasma jet in the

experimental study on the control of micro powder

applied magnetic field. Master student and Dr. Takehiko

spheroidization process and solution nano powder

Sato (presently Professor in IFS) joined to my research

process in a DC-RF hybrid plasma flow system.

since 1995 and 1998 respectively. On the other hand, Solonenko was in charge of modeling of splat formation on the substrate based on the fundamental experiment. Furthermore,

he

provided

the

database

of

thermossphysical properties for numerical simulation on particle laden plasma jet. Finally the integrated research by combining fluid engineering and material science was successfully achieved. In the second part, control of nano and micro powder process was conducted by using DC-RF hybrid plasma flow system. This system was originally established by supporting from Ministry of Education, Science and Culture Japan in 1998. Since 2009 Ph.D. student Mr. J. Jang

(formly,

JSPS

PD),

had

been

Fig.2

ITAM First Prime.

conducting

Fig.3

Fluids Science

Research Award.

experimental study of powder spheroidization process by controlling thermofluid field with water droplet injection,

2. Control of Coating Process in Plasma Spraying

mixing gases and giving sinusoidal flow. Since Solonenko had many experiences for different kinds of

Plasma spraying is to accelerate and melt the injected

powder processes using arc and plasma jet, I asked him

micro particles in the plasma jet, then to make the fully

to guide the method of powder process. Furthermore,

molten droplets impinge on the substrate for coating

Solonenko gave the good advice to Mr. J. Jang, when

formation [3, 4]. The coating thickness distribution in a

conducting solution photocatalytic nano powder process

few hundreds micron scale is determined by key physical

in a DC-RF hybrid plasma flow system. Owing to his

parameters such as impact particle velocity, impact

excellent research guidance, Mr. J. Jang has successfully

particle temperature and particle diameter in micron size

defended Ph.D. thesis in September 2013.

prior to impact onto the substrate [4]. Especially, particle

Our obtained research results were presented and

velocity and particle temperature are determined by

advertised in the international symposiums on AFI

momentum and energy transfer exchanges between

(Advanced Fluid Information), ICFD (International

macroscopic plasma jet and micron size particle during

Conference on Flow Dynamics, 21st Century COE

micro/milisecond

Program, Global COE Program), ISPC (International

application of electromagnetic field to the microsize

Symposium

JSME

particle laden plasma jet is expected to be effective

symposiums etc. by us. Luckly, we received the ITAM

method to control the in-flight and impact particle

First Prime in 2003 and Fluids Science Research Award

behavior on the substrate by Lorentz force and Joule

in 2012 for the much contribution to the international

heating during the small exchange times of momentum

collaboration and fluid science, as shown in Figs.2 and 3.

and energy [6, 7].

on

Plasma

Chemistry),

and

in-flight

[5].

Furthermore,

the

In this report, the obtained results in the international the stand point of functional plasma and multiphase flow

2.1 Particle plasma jet in the electromagnetic field Figure 4 shows the schematic illustration of a plasma

based on our previous publications [1, 2]. The first part is

spraying model to produce thermal barrier coating (TBC)

the modelling and numerical simulation on control of

and diamond coating, for example [1]. It is clearly shown

collaboration between IFS and ITAM are reviewed from

that there are many complex factors interacted with each 2


other in plasma spraying. The objective is to accelerate

(7) Vector potential and induction electric field have

in-flight particle and to enhance heat transfer from

only an azimuthal component.

plasma flow in temporal scale of momentum and heat

2.1.1 Plasma flow

exchanges

RF

plasma flow are described using Eulerian approach,

electromagnetic field. The optimum spraying condition

which includes Lorentian force, electrostatic force, Joule

is that the particle prior to impact on the substrate should

heating and radiation loss [6].     (  u)  0 , t  ( u)    ( uu) t   p      c ( E  u  B)  j  B,

respectively

by

applying

an

be fully molten state. Numerical simulation of a plasma spraying process from ceramic particle injection to a coating formation on the substrate is conducted by integrating particle-laden plasma flow model, metallic and ceramic splat deformation model and coating

Governing equations for

(1)

 e    (e  p)u t    (T )  j  ( E  u  B)  Qr  ΦD , p  RT

formation model to clarify and optimize a whole plasma spraying process [8, 9].

In-flight particle velocity and particle temperature are obtained by the equations of motion and heat balance of a particle using Lagragian approach [6]. Only Stokesian drag is acted on micro size particle in equation of motion. 2.1.2 Particle behavior du p mp dt







Fig.4 Schematic illustration of a plasma spraying model.

mpc p

To derive the governing equations, the following

  d p2

(1) Plasma flow is continuous ideal gas, laminar, optical

thin

in

dt h f T  T p   p  s T p4  Ta4

 







Here, drag coefficient CDf and heat transfer coefficient h f include non-continuum effect and temperature dependent transport properties in the particle boundary layer into. C Df

spraying process. and

(2)

dT p

T p  T pm and T p m  T p  T pb 

assumptions are introduced for simplicity in plasma

compressible



d p2  C Df u  u p u  u p  q p E ,

8

local

thermodynamic equilibrium (LTE) with temperature dependent thermodynamic and transport properties. (2) Flow, temperature and electromagnetic fields are



two-dimensionally axisymmetric. (3) All injected particles are sphere and with same mean



24 1  0.15Re 0p.687 Re p

hf 

diameter, and the particle internal temperature is uniform due to small Biot number. particle, furthermore, particle collision is neglected under the particle dilute concentration.

f Kn

(5) Thermodynamic and transport properties both in

3



s

s

   

 0.45

1 3

 0.45 f Kn ,



0.38

 C p  1   f Kn ,  C ps     2  a   s  4  1  Kn * ,   a  1   s  Prs

Re p 

(6) Nozzle and chamber are electrically insulated.

1 2

dp 

solid and in liquid phases of particle are as function of a particle temperature.





f    2  0.6 Re p Pr 

ρ μ      ρs μs

(4) There is one-way coupling between plasma flow and

   

   

0.6

 f d p u  up f

(3)


The criterions of complete melting state for optimum plasma spraying and complete evaporation state of particles are respectively as follows. t

0 Qndt  mp 

T pm

Tp0

c p dT  m p L pm  Q pm

t

Tpb

0 Qn dt  Q pm  m p T

Tp  Tpm ,

(4)

c p d T  m p L pb

pm

2.1.3 Induction electromagnetic field

The vector

potential equation derived from Maxwell’s equations and Ohm’s law [10].

 2 Ac  iμ0 σAc  0 , E c  iAc , Bc    Ac ,

  2 f RF ,

(5)

Ar , t   Ac r e it ,

Br , t   Bc r e it ,

E r , t   E c r e it , j   ( E  u  B)

2.1.4 Thermofluid field

Table 1 shows the

Fig.5

Axial plasma velocity isocontors: (a) p = 100 kPa; (b) p = 50 kPa.

numerical operating conditions for plasma spraying [1]. Figure 5 shows the axial plasma velocity isocontours of an impinging plasma jet [1, 8]. The plasma flow is accelerated, furthermore, the supersonic region is observed under the low-pressure condition due to the compressible effect [8]. Figure 6 shows the temperature isocontours of an impinging plasma jet [1, 8]. Although the velocity of the plasma flow is very high, the temperature is effectively Table 1

Plasma spraying conditions.

Working gas Gas flow rate (g/s)

Ar 0.1

DC torch input power (kW)

10

RF coil current (A) Frequency (MHz)

300 13.56

Operating pressure (Pa)

5  10 4

Nozzle angle (  ) Particle material Particle diameter (  m)

4.5 Al2O3 20, 30, 40

Particle injection velocity (m/s) RF induction coil

2.5 with/without

Spraying distance (mm) Substrate temperature (K)

70 350

Fig.6 Plasma temperature isocontours: (a) without RF, p = 50 kPa; (b) with RF, p = 50 kPa.

4


increased in the downstream region by applying the RF

important factors governing the splat formation process:

electromagnetic field due to the active Joule heating [8].

roughness and temperature of substrate, properties of splat-substrate interface, contact temperature, thermal

2.2 Fundamentals of metal and metal oxide splats formation In the advanced technologies of thermal

resistance between the flattening droplet and substrate,

(including plasma) spraying there is in most cases no

droplet material splashing, formation of oxide layer, etc.

surface chemistry and wettability, cooling rate of droplet,

possibility of in-process control of coating quality. This circumstance greatly impedes the coating design and optimization. To the knowledge of the present authors, until now there is no concept that is physically grounded and

approved

in

practice,

which

would

allow

approaching the design and optimization of specific technologies of thermal spraying (the plasma, flame, detonation and arc spraying, etc.) from the unified standpoint. The spraying process improvement is usually realized by conducting a limited set of technological experiments on a specific facility using the powder material of this or that fraction composition with

Fig.7

Schematic representation of the lamellar

subsequent studying the characteristics of sprayed

structure of plasma coating formed from the melted

coatings.

particles on a rough substrate. 1  the interface between the coating and the base, 2  the interface between the

Taking into account a fairly large number of factors

sprayed layers, 3  the interface between the splats.

determining the result of the spraying process, the “trial and error” method is completed in a number of cases by planning an experiment, diagnostics or modeling the

Consequently, the reliability of the results of

in-flight particles characteristics in high-temperature jet

modeling the structure of the sprayed coating is

and prior to impact onto the base surface. The attempts

determined, generally speaking, mainly by the accuracy

of performance of through complex experiments in

of the computation of the individual splats formation

thermal spraying for the purpose of studying the

under the given values of the key physical parameters

physical

(KPPs) prior to collision of particle with the base: the

processes,

“generation

of

occurring

high-temperature

in

the

chain

flow  powder

velocity,

material injection  formation of technological dusted

temperature,

and

size

of

particle,

and

temperature of the base.

flow  splats and coating formation”, are not numerous and mainly of a methodological interest because of their

modeling the coating formation, let's assume that it is

high labor efforts at development of specific technology.

necessary to conduct the modeling of the process of

For evaluating the computing costs required for

It is a well-known fact [11], that in a plasma spray

the coating fragment formation with transverse sizes L x and L y of thickness h , and the sprayed

process, even at the highest possible degree of jet loading with powder at which the particles still undergo complete melting, the coating is formed via successive splat laying

powder is assumed monodisperse with the particle size D p . Thus, if the volume of pores in the coating is

on impacts of individual powder particles onto the base

neglected, in a first approximation, then the estimate of

(substrate or previously deposited layer) (see Fig.7).

the necessary amount of splats for complete filling of the coating volume V  L x L y h will be presented

The formation of metal and metal oxide splats from melted particles impacting onto substrate under thermal

as

spraying conditions was examined and analyzed in

N p  6V /(πDp3 )  6Lx Ly h /(πDp3 ) .

several recently published reviews [12‒21]. The authors have considered and discussed the effects due to the most

That is if the coating fragment has the base area 5


S  1 cm ( L x  L y  1 cm) and the thickness h  100  m, then at the size of sprayed particles Dp  10, 25, 50, and 100  m it is necessary to carry out the modeling of the process of splats formation at the 7 6 5 4 amount of N p  1.910 , 1.210 , 1.510 , and 210 , respectively. Detailed modeling of the process of a single splat formation generally requires the computation of a 3D or 2D unsteady boundary-value problems with a free boundary for the Navier — Stokes equations together with the equations of the conjugate convective-conductive heat transfer as well as phase transitions in the spreading 2

the delay of the beginning of solidification. The latter is explained by incorrect calculation of the contact temperature 'droplet-substrate' when crude mesh is used.

droplet, and in a number of cases also in the substrate [21 ‒31], etc. If it is assumed that one second of the CPU

Fig.8

time is required for such computation of single splat

in the stagnation point of the impinging melt versus the

formation,

underestimated

dimensionless time for the calculations carried out on

considerably by at least one order (even at the modeling

grids with the reference size of the triangle being: 0.01

of the process of splat formation on a smooth substrate),

(1); 0.03 (2) and 0.05 (3) of drops diameter.

which

is

a

quantity

Non-dimensional thickness of the solidified layer

then the computing time will amount, respectively, to

t calc  5280, 336, 48 and 5.5 hours. That is why today the latter approach presents an independent theoretical rather than practical interest as modeling of several dozen or hundred thousand droplets interacting with the substrate or with the previously deposited coating layer will entail huge computational costs. The numerical solution of the given problem in the equivalent cylindrical approximation of splats' shape for any combination of materials of the particle and the substrate and KPPs values, typical for thermal spraying, does not cause any principal difficulties. Nevertheless, such calculations and furthermore the calculations pretending to study the models in more complete formulation need to be carried out on rather detailed, preferably unstructured grids. So, the results of finite-element modeling of impact of the nickel drop of diameter Dp  50  m with the copper substrate (Fig.8), in approximation of the equivalent cylinder [32], are given in paper [33]. Droplet velocity and temperature were u p0  100 m/s and Tp0  2500 K, accordingly. The initial temperature of the substrate was Tb0  293 K. Apparently from the data presented in Fig.8, the delay of the beginning of solidification process for the 2nd and 3rd variant of calculations appeared to be comparable with the duration of the solidification process. Therefore, at modeling of splats formation process on rather crude mesh an incorrect physical conclusion can be drawn on

An alternative to direct computing of each splat formation is the use of experimentally validated theoretical solutions or semi-empirical dependencies, allowing to predict with an accuracy sufficient for practice the thickness and diameter of splats in a wide range of the KPPs, which accelerates substantially (by several

orders

of

magnitude)

the

computational

procedure of modeling the lamellar structure of coatings at given operation conditions [34‒45]. In this connection, it should be noted [46] here that, presently, a number of publications are available whose authors

attempt

theoretical

and

experimental

generalization of splat diameters in thermal spraying of powder materials [47‒63]. In those publications, authors use a dimensionless representation of splat diameter Ds  Ds / Dp , so called spreading factor, in the form Ds  aRe We (Table 2). Then, on the assumption of a cylindrical shape of the splat and with due account of the balance of particle mass prior to and after the impact, the non-dimensional splat thickness hs  hs / Dp can be

hs  represented with the dependence 2 2 2  (2 / 3a )  Re We . As it is seen from Table 2, in generalizing their own experimental data on D s the authors of [50‒55] adjust the value of the coefficient a in the solution by Madejski [49] while considering the viscous flow in the vicinity of the stagnation point.

6


Table 2

Values of coefficients in the generalizing

droplet; (3) spreading of the droplet over the solid base

dependences for splat diameter proposed by different

surface, and subsequent cooling and solidification of the

authors. # Author(s) 1 McPherson [47], Akao [48]

spread layer; (4) spreading of the droplet accompanied

2

Madejski [49]

3 4 5 6 7 8

Liu et al. [50] Trapaga, Szekely [51] Bertagnolli et al. [52] Yoshida [53] Watanabe et al. [54] Montavon, Coddet [55] Jones [56] Gasin, Uryukov [57] Chandra, Avedisian [58] Pasandideh–Fard et al. [59] Kurokawa, Toda [60] Cheng [61] Collings et al. [62] Li et al. [63]

9 10 11 12 13 14 15 16

a



β

0.613

0

0.39

1.29 0.577 1.04 1.0 0.925 0.83 0.82

0.2 0 0.2 0.2 0.2 0.2 0.2

0 0.5 0 0 0 0 0

0.1

0.2

0

1.16 0.922

0.125 0.25

0 0

0.687

0.25

0

0.5

0.25

0

0.96 0.816 0.408 1.21

0.095 0 0 0.125

0.084 0.25 0.5 0

with simultaneous local submelting of the base, followed by subsequent cooling and solidification of both.

(a)

Fig.9

(c) Qualitative

(b)

representation

(d) of four

basic

thermophysical scenarios of the splats formation. Here, z p is the current coordinate of the droplet vertex in the droplet deformation process,  and  are the coordinates of solidification front in the spreading droplet and front of melting in the substrate, accordingly.

A fundamental drawback of the dependences given in

Simultaneously, for each of the above-mentioned

Table 2 is that all of them ignore the hydrodynamic and

basic thermophysical scenarios of the splat formation

thermophysical features of the splat formation process.

process, two characteristic regimes of heat exchange

Testing reported dependences in a broad range of

between the melt and the solid wall (solidification front

than 600) obtained under full control of the KPPs of the

 (r, t ) in the particle or substrate surface, z  0 ) are possible depending on the ratio  ν /  a  Pr between the thicknesses of hydrodynamic  ν and thermal  a

process (velocity, temperature, and size of droplet;

boundary layers in the melt spreading on the base [19]. If

temperature of polished substrate) [19, 46] showed their

the melt has a low viscosity and a high thermal

limited appropriateness in predicting splat diameter values at Prandtl number Pr   p / ap  1 (  p and

droplets, then the thickness of the viscous sublayer in the

droplet-substrate interaction parameters performed using a representative set of experimental metal splats (more

conductivity ( P r  1 ), which case is typical of metal liquid particle constitutes only a small fraction of the

ap are the kinematic viscosity and thermal diffusivity of the melt, accordingly). 2.2.1 Hydrodynamical and thermophysical features of splats formation As it was noted in [46], in the plasma spray process, depending on the values of the KPPs and thermophysical properties of the particlesubstrate material pair, the splat formation process at the stage of enforced spreading of the melt droplet should follow one of the four basic thermophysical scenarios (Fig.9): (1) spreading and simultaneous solidification of the droplet on the solid base; (2) spreading and simultaneous solidification of the droplet, and local submelting of the base at the contact spot with the

thermal boundary layer (for all metal-melt droplets, we have P r ~ 102 ). According to [19], the relative thickness of the viscous layer in the melt flow in the vicinity of stagnation point can be represented as  ν / Dp ~ 2 / Re . Hence, at Reynolds numbers R e  104 , which values are typical of thermal spraying of metal powders, we have  ν / Dp  0.02 . Thus, the viscous melt flow is realized in a layer with a temperature close to the wall temperature

Tw since the temperature drop across the viscous layer is negligible (Fig.10). It follows from the aforesaid that the hydrodynamic features of the flow in the viscous layer exert no 7


thickness of the solidified splat in the vicinity of stagnation point ( 2r  Dp ,  ~ 1 ) is defined by the (l ) non-dimensional time Fo  Fo ( Fo  apm t / Dp2 is the (l ) Fourier number, apm is thermal diffusivity at melting

point) at which the vertex of the ideally spreading melt droplet meets the solidification front (Fig.11, a). An Relative thicknesses of the thermal (  a ) and

approximate theoretical solution to this problem,

hydrodynamic (  v ) boundary layers for the cases in

obtained with taking into account solidification of

which splats are formed from metal ( P r  1 ) or metal

droplet and possible submelting of substrate and defining

oxide droplets ( P r  1 ).

the final thickness and diameter of formed splat, and also

Fig.10

the contact temperature, was derived, for instance, in [46, 64]. If, on the contrary, the melt has a high viscosity and a low thermal conductivity ( P r  1 ), which case is typical, for instance, of metal oxide droplets, then the thermal boundary layer, across which the temperature along the outer normal to the wall drops from Tw to Tp0 , lies within the near-wall viscous melt flow, while in the region over the latter flow, at sufficiently high Reynolds and Weber numbers we have a flow resembling the (a)

potential spreading of a liquid with initial temperature

Tp0 (on neglect of the radiative heat loss from the free surface of the liquid) (see Fig.10). Hence, at P r  1 the regularities of the heat exchange between the melt and the substrate are fully defined by the hydrodynamic features of the flow in the viscous sublayer; the latter entails the necessity to consider a model heat-transfer problem for a viscous liquid normally impinging onto an isothermal wall. Here, the splat formation process is two(b) Fig.11

stage (Fig.11 (b), involving: 1) ideal enforced spreading

Qualitative representation of hydrodynamic

of the melt over the viscous layer forced away from the

features of the splat formation process.

substrate surface by the counter-moving solidification front; during this stage, the droplet vertex moves at a constant velocity u p0 until it finally, at some moment

substantial influence on a heat exchange between the wall and melt, whose temperature drops from Tp0 to

Fo  Fo1 , meets the outer boundary of the viscous layer; 2) viscous enforced spreading, starting immediately on accomplishment of the first stage and ending at some moment Fo  Fo , at which the solidification front reaches the free boundary of the viscous layer. Approximate theoretical solutions of this problem can be found, for instance, in [65, 66]. Below there are presented the theoretical fundamentals and their experimental validation of metal/metal oxide splats formation used in [67‒72], etc. for prediction of the structure and properties of the

Tw  Tpm . In turn, for sufficiently high Reynolds and Weber numbers in the region over the viscous layer we have a flow resembling the potential spreading of the liquid. It is across this flow that the predominant drop of temperature, defining the heat flux into the wall, occurs. Hence, here, except for splats forming on the surface of thin foils, we have to consider a model heat transfer problem for an ideal liquid normally impinging onto a semi-infinite obstacle. In the latter case, the splat formation process accomplishes in one stage, and the 8


sprayed coatings in the technological chain “plasma

characterizes the parameter at melting point of the

torch  injection of powder particles-dusted plasma

corresponding material. Under the function of f i,(j ,  ) we understand the following ratio f i,(j ,  )  f i ( ) / f j(  ) ,

jet  coating”. The main attention was paid to the modeling of the lamellar structure (Fig.7) formed from

where i, j = p, b;  = s, l; f is an arbitrary scalar

fully melted metal/metal oxide particles taking into account all KPPs prior to droplet-base interaction. The

function. Reducing Eqs.(6) and (7) to dimensionless form we

obtained results have enabled to develop the concept of

shall have

  (s)   p(l)   d 1 l)  p    ,  (l) (s,  p,p  z   z   d Fo Ku p    0    0  

intelligent integrated plasma laboratory for studying the basic phenomena in plasma spraying [73, 74]. 2.2.2 One-stage metal splats formation

The

possibility of simultaneous processes of solidification,

d Sp,b   Fo Ku (l) p

melt spreading over the solidified layer of the particle and substrate submelting is the characteristic peculiarity

the vertex z p (t ) of spreading droplet. Apparently, moment

of

complete

solidification

0   (t )  zp (t )  Dp . Here we shall restrict ourselves to the consideration of parameters z p (t ) and  (t ) , as well as the coordinate  (t ) of the front of possible submelting the substrate, only for the symmetry axis r  0 of the particle-substrate system ( z -axis is directed towards the spreading droplet) and we consider that the particle has solidified completely, if the relation  (t * )  zp (t * ) takes place at the time t*. Hence, according to our consideration, the finall thickness of the flattened and solidified particle (splat) will be equal to hs   (t * ) . Further the main attention will be paid to the prediction of possible particle and substrate phase transitions, the study of which in the equilibrium conditions is connected with solution of the Stefan's equations for  ,  ,  (s) Tp(s)   (l) Tp(l)     ,  p  p      z  z   0    0

 p(l) Lp

d dt

b(s) Lb

d  (s) Tb(s)   b dt  z 

 T (l)   (bl ) b  ,  z   0   0



of

melting,

(s) p



/ z







 (1  c ) /  ,

(10)

 (bm  c ) /  ,

(11)

0

(l ) b / z   0

where c is the contact temperature between particle and substrate calculated on the basis of heat fluxes equality at interface z  0 , i.e. assuming the ideal contact,

(6)

(1  c ) /   (b,l ,ps ) (bm  c ) /  .

(12)

The heat flux to solid substrate is evaluated using the (7)

solution of boundary value problem for heat conduction equation in a half-space at fixed surface temperature

 b(s)      z    0

T is the temperature. From now on superscripts “s”, “l” of the variables correspond to the solid and liquid state of the material, and subscripts “p”, “b” correspond to particle and substrate (base). An additional subscript “m” heat







where  ,  , L are density, thermal conductivity and latent

(9)

(l) (s) (l ) Lp /  bm Lb are the Ku (pi )  Lp /[cpm Tpm ] and Sp,b   pm StefanKutateladze's criterion and the parameter characterizing the relative difficulty of particle and substrate materials melting/solidification; ratio of thermal conductivities are calculated at melting temperature of the corresponding materials. Equations (8) and (9) can be effectively resolved, if appropriate approximations for all derivatives in their right sides will be derived. For p( s ) / z  0 and b(l ) / z  0 we shall use the condition of relative infinitesimality of thickness of the layers  ,  . In this case one can use the following approximations:

front of equilibrium solidification  (t ) moves towards the

  (s)   b(l)   l)  b  (s,   , (l,l) b,p  b, p  z    z      0   0 

where z  z / Dp ,    /D p ,    / Dp ,   T /Tpm ,

of molten particle-surface interaction. In this case the

before

(8)

respectively,

(l , s ) ap, b



bm  b 0  /Fo ,

(13)

(l , s ) where ap, is the ratio of the thermal diffusivities of b

particle and substrate materials at their melting points;

b 0 is the initial temperature of the substrate. 9


To determine the heat flux from the spreading melt to

multiplier q c (Fo, Pe) characterizes the contribution of

z   (Fo) , we consider the unsteady-state axisymmetric nonisothermal potential flow of incompressible ideal fluid in the vicinity of stagnation point. The liquid having the temperature  p(l0) and the velocity up 0 at infinity, and thermophysical properties independent of the temperature impinges on a normally placed obstacle, the temperature of which is equal to pm  1 , and then spreads along its. Let's introduce the cylindrical coordinate system with origin in the stagnation point and z-axis inward-directed

melt convection in the vicinity of the stagnation point.

the moving surface

Taking into account that characteristic dimensionless time Fo d of the particle spreading is determined from the condition Pe Fod  1 , one can find the effective mean value q c (Pe) of the convection contribution ( Pe  0 ) in the heat flux in the vicinity of stagnation point 1 /Pe

0



quasi-stationary velocity field [75]

c  0.26 .

neglecting the thermal conductivity in liquid in

Thus, the required approximation for the derivative

r -direction we shall have the following boundary value problem for symmetry axis (r  0) :

 Fo

 Pe z

p(l ) z



 2 p(l ) z 2

has the appearance



(l ) p

,

c  0 , c  0 are the constants. This, conversely, provides the constancy with time of the contact temperature c . Using Eqs. (8), (9) and (12) it is possible to derive the expressions for parameters c , c and c . Parameter c characterizing the dynamic of the melt solidification is the positive root of the equation

d .

(17)

0

From solution Eq.(17) it is easy to derive the

c2  Pc Q  0 ,

derivative determining the heat flux from spreading liquid to the wall (front of solidification)



(l ) p

/z



 0 

          z  z  0     0 ,

P

 p(l0) 1

Q

  p(l ) 

πFo

q c (Fo, Pe)

where qc (Fo, Pe)  2 Pe Fo/(1  e2 PeFo ) . One can see that the first multiplier on the right side of

the

expression

determines

(19)

Where

form



(18)

 (Fo)  c Fo ,  (Fo)  c Fo ,

the solution of the problem can be represented in the

 2 / 4

(l ) 1   c p 0  1 .  π Fo

from the conditions Eqs.(10) and (11) it follows

  z / (1  e 2 PeFo ) /(2 Pe) ,

e

 0



for Eqs.(10)‒(13) and (18) is closed. One can show that

(16)

where Pe  Dpup 0 / is the Peclet number. Introducing the self-simulated variable

 1



Hence the system of Eqs.(8) and (9) with due account

(l ) apm

p(l0)  1 

/ z

(15)

p(l ) ( F o0,)  1 , p(l ) (Fo, z )  p 0 , z   ,

p(l ) ( )

1 x /(1  e x ) dx  1.26 2 0

It is obvious, in the absence of convection ( Pe  0 ) the factor q c (0)  1 , i.e. at Pe  0 qc  1  c , where

(14)

Using the aforesaid dimensionless variables and

p(l )

2 Pe Fo d Fo 1  e 2 PeF o

2

to the spreading metal melt ( P r  1 ) characterized by

u z  2z , u r  r ,   up 0 /(2Dp ) .



qc (Pe)  Pe

the

heat

(20)

2[d 2 (2 Sp,b  (bl,,ps ) )  d1 Sp,b ] Ku (pl ) [Sp,b  (b,l ,ps ) ]

,

(21)

2[2d 2 (d 2  d1 ) Sp,b  (b,l ,pl ) Ku (pl ) (1  bm )] [Ku (pl ) ]2[Sp,b  (b,l ,ps ) ]

,

(22)

p) d1  K(b, (bm  b 0 ) / π , d 2  (1   c )(p(l0)  1) / π , ε p) (l , s ) K (b,  (b,sp,l ) ap, is the ratio of the effusivities of ε b

transfer

substrate and particles materials.

conditioned by heat conduction, and the second

Parameter 10

c

characterizing

the

dynamic

of


substrate submelting is defined as

Q

c  Sp,b[c  (d2  d1) /Ku (pl ) ] . Finally, the value of the contact temperature c is

2(p,sp,l ) (1  b 0 )  (1   c )(p 0  1)   1  . p)  Ku (pl ) (1  b 0 ) K (b,   ε

(29)

Analysing the coefficient Q in expression Eq.(29)

calculated from the expression

we shall obtain the condition of the particle solidification

c  [c  c (b,l ,ps )bm ] /[c  c (b,l ,ps ) ] .

absence

(23)

p) (p 0  1) /(1  b 0 )  K (b, /(1   c ) . ε

The scenario of the melt droplet-substrate interaction

(30)

considered (Fig.9 (b)) is important from the theoretical

The model developed allows to predict the dynamics

point of view but it occurs relatively rarely at thermal

of equilibrium solidification of the metallic droplet at its

spraying in comparison with scenario 1 (Fig.9 (a)) when

interaction with the base. In particular, the obtained

formation of splats is performed under the conditions of

solutions make it possible to estimate one of the most

melt droplet spreading and simultaneous solidification on

important in thermal spray technology parameter, i.e. the

a solid substrate.

thickness of the flattened and solidified particle (splat) in

In this case the corresponding Stefan's problem

the vicinity of stagnation point ( r  Dp / 2 ). Taking into

consists of the Eq.(8) regarding unknown coordinate of

consideration the motion dynamics of the droplet vertex

front solidification  (Fo) in the spreading melt. Using the approximations Eqs.(10) and (18) and approximation

(top) z p (Fo) , in each particular case, the thickness of

for the derivative

the equation zp (Fo)   (Fo) . In a first approximation, one can assume the

 b(s)      z    z 0

(l , s ) ap, b0

π

c  b 0 / F o,

the flattened and solidified particle can be found from

(24)

following dynamics of the droplet vertex motion in time

z p (t )  Dp  u p 0 t ,

it is easily shown that

 p(s)     dz   

which can be represented to following dimentionless p)  (p,l ,ps ) K (b, (c  b 0 ) / π F o. ε

(25)

form

 0

z p (Fo)  1  Pe Fo .

The latter approximation was derived taking into account the conjugate heat condition at z  0 (interface

Hence, we can find the time Fo* corresponding to the instant of time when the front of droplet

between solidifying layer and substrate).

solidification will meet the free surface, by solving the

From the expressions Eqs.(10) and (25) one can find

c

c , if representation of  in the from Eq.(19) is used, the

representation

of

through

equation

the

1  Pe Fo  c

p) p) c  [1  c (p,l ,ps ) K(b, b 0 / π ] /[1  c (p,l ,ps ) K(b, / π]. ε ε

dependence

Substituting Eq.(26) into Eqs.(10) and (25), and then the

Fo*  [c ( 1  4 Pe/ c2  1) / 2 Pe] 2 .

obtained expressions in right side of Eq.(8), we shall

Taking into account that, by convention, the

have the quadratic equation related to c . Solving the

parameter

obtained equation we shall have

c  P[ 1  4Q / P2  1] / 2, p) π (p,sp,l ) Ku (pl )  2(1   c ) K (b, (p 0  1) ε p) π K (b, Ku (pl ) ε

Fo .

As a consequence we shall have the required

(26)

P

(31)

hs  1  Pe Fo*

(27)

corresponds to

,

(28)

(32)

dimensionless thickness ( hs  hs / Dp )

of the splat in the vicinity of the stagnation point, it is easy to derive the expression for dimensionless diameter of the splat in the case when its shape not far from the 11


disk. From the balance of particle volume prior to and

The principal diagram of the experimental facility

after impact with substrate we have

used by us is shown in Fig.12. Quartz glass tube (1), with (33)

external and inner diameters d1  6 mm and d 2  4 mm, respectively, one end of which was profiled with the

In order to validate of the theoretical solutions the

help of heating and drawing out in order to provide an

comprehensive model experiments on metallic splats

inner diameter of 0.05‒0.15 mm, was placed inside the

formation have been carried out [76‒78]. The primary

ohmic heater (2). Model materials (In, Sn, Pb, Zn and Ag,

purpose

detailed

purity of 99.9999) were fed into a glass tube placed in a

comparison of the thickness and diameter of deposited

heater. First gas line (Ar) containing manometer (3),

splats under full control of the KPPs (velocity u p0 ,

rubber bulb (4) and tap (5) was attached to upper part of

temperature Tp0 and size D p of metallic droplets

tube, emerged through heaters window. Input of argon

prior to their impact with a smooth substrate, and

through second gas line was performed tangentially in

temperature Tb0 of substrate. The additional model experiments have been also

order to remove remaining air, situated in annular gap

performed to illustrate the possibility of continuous

hollow copper tube with external and inner diameters

changing the diameter of splats and scenarios of their

d 3  12 mm and d 4  8 mm, respectively, and two heating sections rounded by nichrome wire of 0.5 mm diameter. Between them was placed positioner (6), to center the tube with material in heater and two chromel-kopel thermocouples (7) and (8) with lead-out on controllers (9) and (10). In order to prevent heat loss, front area of heater was heat-insulated with asbestoscement disk (11) with inner diameter of 4.5 mm and creates a step. In this area behind the step formation of droplet took place. While conducting the experiment the temperature of heaters' wall, measured by upper and lower thermocouples, were identical. The usage of distributed feeding the inert gas was performed as follows. First gas line was used for forced blowing of material with inert gas during its heating. After heating and melting of material sample placed inside the glass tube, melt overlapped the outlet crosssection of capillary. It was followed by the rise of pressure, registered with manometer (3). With the help of tap (5) gas line was shut off and after some time lag at given temperature, additional rising the pressure, was provided for droplets generation with the help of bulb (4). At the same time, second argon line was providing the blowing of external side of quartz tube with inert gas with flow rate GAr  0.005 g/sec to prevent the drop

Ds  2 /(3hs ) .

of

these

experiments

was

the

between glass tube and heater. As heater was used a

forming at increasing the contact temperature.

Fig.12 Schematic diagram of modeling experimental installation. 1 - quartz tube; 2 - ohmic heater; 3 - first gas line (Ar) containing manometer (3), rubber bulb (4) and crane (5); 6 - adjusted device for tube alignment with material in heater; 7, 8 - chromel-kopel thermocouples with output to PID controllers (9), (10) and (23); 11 heat-insulating asbestos-cement washer; 12 - the replaceable vertically located tube; 13 - airtight camera; 14 - thermo-controlled table with substrate fixed on it (15); 16 - manometer for pressure measurement in camera, regulated by means of hydraulic lock (17); 18 diode laser; 19 - shaping lens; 20 - photodiode set in focus of collecting lens (21) and a double-slotted diaphragm (22).

from oxidation while its formation on capillary end. During implementation of free fall regime, drop, reaching some equilibrium size, defined by density and surface tension of melt, as well as by diameter of tube-capillary and thickness of its walls, detached from 12


ms is the mass of deposited splat ,  p(l ) is the density of particle material in a liquid state. Mass of splats was determined by direct weighting with the help of analytical balance Ohaus Pioneer PA214C with accuracy of 0.1 mg. To obtain the given size and velocity of the generated droplet the gas input system was arranged in the following way. With the help of the first gas line argon was blown onto the heated material. When a metal melt appeared inside the quartz pipe, the excessive pressure for formation of a single droplet at its lower end was created with the help of a vessel connected with the first gas line. When the droplet grew up to the necessary diameter it detached from the melt and accelerated in the gravity field and by the coaxial high-temperature jet generated with the help of the second gas line. During this process it was possible to control the particle velocity prior to its interaction with the substrate by changing the gas flow rate of the coaxial gas flow. In order to check the validity of developed theoretical basics corresponding to 2nd scenario of metallic splats formation, the model experiments were carried out at impact of droplets of lead with smoothed substrates of indium [64]. Results of comparison of experimentally measured and theoretically predicted diameters of splats are given in Fig.13 (a). X-axis in Fig.13 answers to

capillary by gravity. To

control

the

temperature

of

substrate

a

corresponding temperature-controlled table was made. It consisted of junction for substrates fixation, ohmic heater and

chromel-alumel

thermocouple,

flattened

and

polished thermal junction of which pressed against upper surface of substrate with spring loaded ceramic pressing. To maintain the given temperatures of melt and substrate and to increase the accuracy of their determining, the experimental setup was equipped with PID controllers ТРМ151-01 type (9), (10) and (23). The thermocouples' readings were transmitted to regulators inputs

and

were

compared

with

pre-installed

temperatures. Control over heaters was realized in order to reduce the deviation of current temperature values from the given. Furnace for melting of material sample was hermetically connected with the help of replaceable vertically oriented tubes (12) of different length with hermetical

chamber

(13),

in

which

temperature-

controlled table (14) with installed substrate (15) was placed. At that, the length of tube (12) determines the velocity of droplets during their free fall and impact with the substrate surface. To control the velocity of droplet prior to its impact with substrate and for direct observation of process, chamber measured with manometer (16) was regulated

nondimensional complex Pe Fo* , and ordinate axis - to nondimensional values of splats diameter (Fig.13 (a)),

with the help of hydraulic valve (17).

its thickness (Fig.13 (b)) and thickness of submelted

chamber was equipped with glass windows. Pressure in

To measure the droplet velocity prior to its impact

layer of base (Fig.13 (c)). The corresponding values of

with substrate a time-of-flight method was implemented.

contact temperature are given in Fig.13 (d). In Fig.13 (a)

With the help of diode laser (18) and collecting lens (19)

are also given the characteristic photos of splats, brought

a parallel light beam was generated, coming through the

to a common scale. It is possible to observe a quite

side windows of chamber in close proximity to the

satisfactory agreement of experimental and theoretical

substrates surface. Receiving part consists of photodiode

diameters of splats. The divergence (~ 10%) is caused by

(20), placed in a focus of collecting lens (21) and

assumption of theory about cylindrical shape of splats

double-slotted diaphragm (22) cutting two narrow

whereas the experimental splats on their periphery have

horizontal slots from beam with known distance L

either roller form, or irregular border related to "finger-

between them. During particle crossing the parallel beam,

shaping".

photodiode registered sequentially two impulses from

Check of applicability of the dependences [49, 56,

transiting particle, which are transmitted to oscilloscope memory. Droplet velocity was defined as up 0  L /  ,

61] widely used in the practice of thermal spraying was

where  is the time interval between impulses.

calculated by means of the dependences suggested in [56,

also carried out. In Fig13 (a) diameters of splats,

Diameter of droplet prior to impact with substrate was calculated by formula Dp 

3

6ms / p(l )

61] are given. It is possible to state their considerable

, where

divergence with experiment. Significantly bigger 13


dependence suggested in [49]. From data presented in Fig.13 (c) it is possible to draw a conclusion on inequivalent influence of overheating the droplets above

Tpm and heating of substrate on depth of its submelting that is caused by behavior of the contact temperature (Fig.13 (d)). Thus, the physical and mathematical model, taking into account all KPPs of "droplet-surface" interaction and allowing to derive of theoretical dependences for estimation of metal splats parameters, depth of substrate submelting in its contact spot with particle and temperature in contact "particle-surface", was suggested for the first time and experimentally validated. These dependences can be used at criteria generalization of experimental data. As it was mentioned above, as a rule, the process of metallic splats formation at thermal spraying occurs under the conditions of droplet flattening and simultaneous solidification on a solid substrate (scenario 1, bm  c  1 ). To the present day this case is one of the most extensively exemined both theoretically and experimentally as applied to the thermal spray (l ) technology when Reynolds number Re   pm Dp u p 0 / (l ) (l ) (l ) pm  1000 and Weber number We   pm Dp u p2 0 /  pm (l ) (l ) and  pm are the dynamic viscosity  100 . Here  pm and surface tension of the melt. The initial temperature of the droplet can be both higher or equal to the melting point of its material. The solution corresponding to the 1st scenario of metal splat formation presented above have allowed us to fulfil the theoretical generalization of the available experimental data characterising the thickness and diameter of splats without introducing any empirical constant. Figure 14 (a) illustrates the results of such criterion generalization of the experimentally measured diameter of more than 600 metallic splats. Besides, in Fig.14 (b) there are shown the results of comparison of the theoretically predicted thickness and calculated from the equation of mass balance of the experimental particles before and after their interaction with substrate based on the measured diameter of splats. One can see that the theoretical solution developed (solid lines) generalises quite satisfactorily the experimental data without introducing any empirical constant. At the same time, the

(a)

(b)

(c)

(d) Fig.13

Comparison of dimensionless experimental and

theoretically predicted diameters of splats as well as calculated according to dependences suggested in [56, 61] - (a); corresponding theoretical estimates of relative thickness of lead splats - (b), and submelted layer of indium substrates - (c), at various temperatures of base ( Tp0 = const) and temperatures of lead drops ( Tb0 = const);

the

corresponding

theoretical

values

of

temperatures in contact "particle - substrate" - (d). divergence with experiment is observed when using the 14


6

4

_ Ds _ Jones ( Ds=1.16Re 0.125 )

2

0 0.80

Sn Cu SS

In Sn Pb Zn - - -

PeFo*

0.84

0.88

0.92

0.96

1.00

(a) _ hs

(a)

0.20

0.15

0.10

0.05 _ Jone s (hs=0.495Re - 0.25)

0.00 0.80

0.85

0.90

0.95

Pe Fo * 1.00

(b) Fig.14

(b)

Criterion generalization of the experimental

Fig.15

dimensionless diameter (a) and thickness (b) of splats.

Comparison of theoretical (solid line) and

experimental (figures) diameters of splats. (a) - absolute values of splat diameter vs velocity of impacted droplet,

best dependence of H. Jones [56] (among the known

(b) - criterion generalization the same experimental data

dependences presented in the Table 2) deviates

[23].

essentially from the experimental data. Using the experimental data [14], we have checked in

splat characteristics that can be used for different

addition the validity of the theory developed for the droplets with diameter of Dp ~ 50 m character for

particular conditions characterized by the KPPs of the process (velocity u p0 , diameter D p , surface

typical conditions at plasma spraying.

temperature Tp0

and aggregate state of droplets;

Figure 15 illustrates a rather good agreement between

temperature Tb0 and surface state of the substrate). As

theoretically predicted and experimentally measured

it was shown above, for metal splats deposited onto

diameters of splats. 2.2.3 Two-stage metal oxide splats formation During the last decades, there has been a considerable

polished substrates ( Pr  1 ), this problem was

interest in the metal oxide coatings (Al2O3, ZrO2, TiO2,

characterizing

Cr2O3 , etc.) thermally sprayed onto solid surfaces. It was

plasma-sprayed metal oxide splats ( Pr  1 ) is presented

recognized that further progress in improving the quality

and experimentally validated. This solution allows one to

and structure of sprayed materials largely depends on the

predict the splat characteristics without introducing any

understanding of processes that occur during the

empirical constants. The theoretical approach developed

interaction of melted particles, carried by a high-

is based on a two-stage description of the splat formation

temperature jet, with substrates.

process (initially ideal and consequently viscous

successfully solved. Below, the theoretical solution [19, the

thickness

and

65,

66]

diameter

of

spreading of the melt over the simultaneously solidifying

To further improvement the plasma spray technology,

layer of the material).

it is required to derive a set of analytical solutions for 15


To devise such description, it is required to consider

which allow us to disregard the surface tension force

an axisymmetric and non-isothermal spreading of a

at the droplet flattening stage. The force due to

melted droplet over a substrate in a vicinity of the

surface tension plays a considerable role only at the

stagnation point. Here, only the transitional period

final stage of the splat formation process, this force

should be considered during which the melt flattens to

being primarily responsible for the final morphology

reach its final thickness. As it was mentioned above, if

of the peripheral regions.

viscosity of the melt is high and its thermal conductivity

7.

At determining the splat thickness in the vicinity of

is low ( Pr  1 ), the thermal boundary layer, in which the

the stagnation point ( 2r  Dp ), we assume all non-

temperature varies along the direction normal to the wall

stationary conjugated heat transfer processes in the

(solidification front) from Tpm to Tp0 , turns out to be

particle-substrate system, including phase transitions,

submerged into a viscous near-wall flow. Over the

to be one-dimensional ones. In this formulation, the

viscous near-wall flow, at sufficiently high Reynolds and

splat thickness hs is more rigidly related to the

Weber numbers we have a flow much the same as ideal

KPPs of the process and to the thermophysical

spreading of a liquid having an initial temperature equal

properties of the two materials, and for the

to Tp0 (here, the emission from the free surface is

above-indicated range of Reynolds and Weber

ignored). Hence, in this case the heat transfer regularities

numbers this thickness turns out to be almost

are fully determined by the hydrodynamic features of the

independent of the peripheral processes in the

flow in the viscous sub-layer, and we have to consider a

spreading particle ( 2r  Ds , Ds

model heat transfer problem describing a viscous liquid

diameter of splat). Having determined the splat

normally impinging on a non-isothermal wall. The

thickness, we may determine the splat diameter

physical model for the process of interest for the first

using the balance relation for the particle mass prior

time developed in [65] is based on the following

to and after the impact and assuming the final shape

assumptions:

of the splat to be almost cylindrical.

1.

2.

No melt overcooling occurs below the particle melting point Tpm , i.e., the droplet solidifies under

8.

4.

5.

equilibrium conditions. All thermophysical properties of the particle and

Fig.16), one being occupied by an external potential

substrate materials are temperature-independent.

near-surface viscous quasi-stationary layer having

They are taken at melting point (m) for solid (s) and

an effective thickness  v, eff .

flow and another one representing an internal

9.

We assume that the temperature distribution across

At the particle – base interface the ideal contact

the solidifying layer is always quasi-stationary, i.e.,

takes place. The substrate is a semi-infinite body. It can be

the temperature across the layer varies linearly from Tc to Tpm .

shown that, during the whole splat formation process,

10. Ceramic splats are formed in the following two

the depth to which thermal disturbance penetrates

stages: (1) the stage of ideal pressure droplet

the substrate is smaller than droplet diameter.

spreading, during which the front of solidification

During droplet flattening and solidification, the

the substrate remains unchanged.

 (t ) starts on the substrate surface while the top of the droplet z p (t ) keeps moving with a constant velocity u p0 until it reaches the external boundary  v of the viscous layer; and (2) the stage of viscous

effective contact temperature at the interface between the layer that undergoes solidification and 6.

In the course of its flattening and solidification, the droplet may be subdivided into two zones (see

liquid (l) state of the materials. 3.

is the final

Under thermal spraying, typical Reynolds and

pressure spreading that begins just after the first

Weber numbers satisfy the inequalities

stage; this stage is completed at the moment at

(l ) Re  Dp u p 0 /  pm  10 3 ,

which the solidification front meets the free boundary of the viscous layer.

(l ) (l ) We   pm Dp u p2 0 /  pm  10 2 ,

16


t1* , at which the top z p of the spreading droplet reaches the external boundary of the viscous layer (Fig.16 (b)). * In view of the aforesaid, we have zp (t1 )   ν,ef f  hs1 , (l ) /  , hs1   (t1* ) is the where  ν, eff    pm (l) t is its thickness of the solidified layer,  (t )  c apm instant position. As a first approximation, we can adopt an assumption that during this stage the droplet top moves with a constant velocity u p0 , and its current coordinate may be represented as

(a)

(b)

z p (t )  Dp  u p 0 t . Introducing the variables  

3

non-dimensional

(l ) 2 a pm  t

,  / 3

time (l ) apm z

and

space

, we obtain a

bi-quadratic equation for the duration of the first stage; (c) Fig.16

this equation, with due regard for [65], may be written as (1   ) Dp c      0,   (34)

Successive splat formation stages during impact

interaction of a metal-oxide droplet with a substrate.

(l ) 2   up 0 / 3  (apm ) ,

Since the solidification front velocity u s  hs / t s in

(l ) where   (2 /  )  /  pm ,  is the parameter that

the droplet under typical thermal spray conditions is far

characterizes the ideal flow of the melted material in the

lower than the characteristic droplet flattening velocity

vicinity of the stagnation point. The solidification rate c

u p0 , in deriving an approximate solution for the heat flux from the normally impinging melt to the equilibrium solidification front the motion of the interface between the phases may be ignored. Indeed, since hs