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Formation of Splats from Suspension Particles with Solid Inclusions Finely Dispersed in a Melted Metal Matrix

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Abstract

A theoretical model has been developed to describe the splats formation from composite particles of several tens of micrometers in size whose liquid metal binder contains a high volume concentration of ultra-fine refractory solid inclusions uniformly distributed in the binder. A theoretical solution was derived, enabling evaluation of splat thickness and diameter, and also the contact temperature at the particle-substrate interface, under complete control of key physical parameters (KPPs) of the spray process (impact velocity, temperature, and size of the particle, and substrate temperature) versus the concentration of solid inclusions suspended in the metal-binder melt. Using the solution obtained, the calculations performed demonstrate the possibility of formulating adequate requirements on the KPPs of particle-substrate interaction providing a deposition of ceramic-metal coatings with predictable splat thickness and degree of particle flattening on the substrate, and also with desired contact temperature during the formation of the first coating monolayer.

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Abbreviations

t :

Time

\( r,\;\,z \) :

Radial and longitudinal coordinates

\( u_{z} ,\;u_{r} \) :

Velocity components in the cylindrical coordinate system

T :

Temperature

ρ, c :

Specific density and heat capacity

μ, \( \upnu = {\upmu \mathord{\left/ {\vphantom {\upmu \uprho }} \right. \kern-0pt} \uprho } \) :

Dynamic and kinematic viscosity

λ, \( a = {\uplambda \mathord{\left/ {\vphantom {\uplambda {\uprho c}}} \right. \kern-0pt} {\uprho c}} \) :

Thermal conductivity and diffusivity

\( E = \sqrt {\uprho c\uplambda } \) :

Thermal effusivity

σ:

Surface tension

L m :

Latent heat of melting

ζ, ξ:

Coordinates of melting/solidification front in cermet particle and in substrate

\( c_{\upzeta } \) :

Parameter characterizing the rate of solidification of the ceramic-metal droplet

d :

Diameter of ultra-fine inclusions in cermet particle

\( D \) :

Diameter of cermet particle, splat, etc.

h :

Thickness of layer, splat, etc.

s :

Volume concentration of the ultra-fine inclusions in a melted binder

\( We = {{\uprho_{\text{pm}}^{{({\text{l}})}} D_{\text{p}} u_{{{\text{p}}0}}^{2} } \mathord{\left/ {\vphantom {{\uprho_{\text{pm}}^{{({\text{l}})}} D_{\text{p}} u_{{{\text{p}}0}}^{2} } {\upsigma_{\text{pm}}^{{({\text{l}})}} }}} \right. \kern-0pt} {\upsigma_{\text{pm}}^{{({\text{l}})}} }} \) :

Weber number

\( Re = {{\uprho_{\text{pm}}^{{({\text{l}})}} D_{\text{p}} u_{{{\text{p}}0}} } \mathord{\left/ {\vphantom {{\uprho_{\text{pm}}^{{({\text{l}})}} D_{\text{p}} u_{{{\text{p}}0}} } {\upmu_{\text{pm}}^{{({\text{l}})}} }}} \right. \kern-0pt} {\upmu_{\text{pm}}^{{({\text{l}})}} }} \) :

Reynolds number

\( Pr = {{\upnu_{\text{pm}}^{{({\text{l}})}} } \mathord{\left/ {\vphantom {{\upnu_{\text{pm}}^{{({\text{l}})}} } {a_{\text{pm}}^{{({\text{l}})}} }}} \right. \kern-0pt} {a_{\text{pm}}^{{({\text{l}})}} }} \) :

Prandtl number

\( Pe = {{D_{\text{p}} u_{{{\text{p}}0}} } \mathord{\left/ {\vphantom {{D_{\text{p}} u_{{{\text{p}}0}} } {a_{\text{pm}}^{{({\text{l}})}} }}} \right. \kern-0pt} {a_{\text{pm}}^{{({\text{l}})}} }} \) :

Peclet number (Pe = Re·Pr)

\( Fo = {{a_{\text{pm}}^{{({\text{l}})}} t} \mathord{\left/ {\vphantom {{a_{\text{pm}}^{{({\text{l}})}} t} {D_{\text{p}}^{2} }}} \right. \kern-0pt} {D_{\text{p}}^{2} }} \) :

Fourier number

\( Ste_{\text{p}}^{{({\text{l}})}} = \upchi c_{\text{pm}}^{{({\text{l}})}} T_{{ 1 {\text{m}}}} /L_{\text{pm}} (s) \) :

Stefan number characterizing the phase transitions in composite material, \( \upchi = \uprho_{\rm pm}^{{({\text{l}})}} /\uprho_{1{\rm m}}^{{({\text{l}})}} ,\;L_{\rm pm} (s) = (1 - s)L_{1{\rm m}} \)

\( K_{\upvarepsilon }^{{ ( {\text{b,p)}}}} = E_{bm}^{(s)} /E_{\rm pm}^{{({\text{l}})}} = \sqrt {(\uprho c\uplambda )_{bm}^{(s)} /(\uprho c\uplambda )_{\rm pm}^{{({\text{l}})}} } \) :

Relative thermal effusivity of substrate material with respect to particle material, \( K_{\upvarepsilon }^{{ ( {\text{p,b)}}}} = 1/K_{\upvarepsilon }^{({\text{b,p}})} \)

\( f_{i,j}^{(\upalpha ,\upbeta )} = f_{i}^{(\upalpha )} /f_{j}^{(\upbeta )} \), i,j = p,b; α,β = s,l:

f is an arbitrary scalar function

\( f_{{{\text{p}},{\text{p}}}}^{{({\text{s}},{\text{l}})}} = f_{\text{pm}}^{{({\text{s}})}} /f_{\text{pm}}^{{({\text{l}})}} \) :

Ratio of the cermet particle properties in solid and liquid states at the melting point of metal binder

Fo :

Time

\( \bar{r},\,{\kern 1pt} \bar{z},\,{\kern 1pt} \bar{h},\,\bar{D} \), etc.:

Spatial variables

\( \upvartheta = T/T_{\text{pm}} \) :

Temperature (for cermet particle \( T_{\text{pm}} = T_{{1{\text{m}}}} \))

s and l :

Solid and liquid state

p and b:

Particle and base (substrate)

j = 1 and 2:

Properties corresponding to metal binder and solid inclusions

s:

Corresponds to splat parameter

0:

Initial value of parameter

m:

Parameters at the melting point of metal binder or substrate

c:

Corresponds to parameter at contact between particle and substrate

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Acknowledgments

The results reported in the present publication were obtained within Research Program No. 23 of the Presidium of the Russian Academy of Sciences for the years 2009-2011 (Project No. 7) and Interdisciplinary Integration Project No. 2 of the Siberian Branch of the Russian Academy of Sciences for the year 2012-2014.

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Correspondence to O. P. Solonenko.

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This article is an invited paper based on an oral presentation at the 5th International Workshop on Suspension and Solution Thermal Spraying (S2TS) 2011. This workshop was held in Tours, France, October 3-4, 2011 and was organized by CEA Le Ripault.

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Solonenko, O.P. Formation of Splats from Suspension Particles with Solid Inclusions Finely Dispersed in a Melted Metal Matrix. J Therm Spray Tech 21, 1135–1147 (2012). https://doi.org/10.1007/s11666-012-9821-7

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