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“Thermal Mirror” Method for Measuring Physical Properties of Multilayered Coatings

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In this study theoretical principles underlying the photothermal displacement (“thermal mirror”) method for measuring physical properties of opaque multilayered and functionally graded coatings with low thermal conductivity are analyzed. In this method, the specimen is locally heated by a power laser beam, and a two-dimensional transient temperature field is formed in a specimen. The physical basis for the photothermal displacement method is the non-stationary buckling and displacement of an irradiated surface due to a non-uniform thermal expansion. The surface is monitored by a low-power probe beam of a second laser, which is reflected from the specimen, i.e., the system operates as a convex “thermal mirror.” The photoinduced displacement varies with time, and the probe beam is reflected at a different angle depending on the slope of the displacement. The deflection angle is measured as a function of time by a position sensor, and the results of these measurements are compared with the theoretical dependence of the deflection angle on time and physical properties of a coating. This dependence was determined analytically from the solution of the two-dimensional thermal elasticity problem. It is shown that for the specimen composed of a substrate and a coating it is feasible to determine the properties of the coating, e.g., the thermal diffusivity and coefficient of linear thermal expansion provided that the analogous properties of the substrate are previously measured or otherwise known.

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Abbreviations

c :

specific heat

d :

substrate thickness

E 0 :

total laser power absorbed by a specimen

F 0 :

Fourier number

G :

shear module

I 0 :

laser beam intensity absorbed by a coating

J i (x ):

ith order Bessel function of the first kind

k :

thermal diffusivity

r :

radial coordinate

r 0 :

radius of a laser beam

p, s :

parameters of Hankel (p) and Laplace (s) transforms with respect to r and t, respectively

t :

time

T :

temperature

\({\overline{\theta}(s,p,z)}\) :

Laplace–Hankel transform of temperature

u, w :

radial and axial components of displacement

V m ,V c :

volume content (m: metal, c: ceramics)

z :

axial coordinate

α:

coefficient of linear thermal expansion

\({\Delta}\) :

coating thickness

\({\varepsilon}\) :

laser beam deflection angle

λ:

thermal conductivity

\({\theta}\) :

Hankel transform of temperature

\({\nu}\) :

Poisson’s ratio

σ ij :

thermal stress

\({\sigma_{rr}, \sigma_{\Psi\Psi}}\) :

radial and tangential stresses, respectively

\({\tau_{0}}\) :

duration of a laser pulse

Subscripts:

 

0:

substrate

i =  1,...,n :

number of a layer in a coating

Superscript:

 

 + , −:

upper and lower surfaces of a layer in a coating, respectively

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Authors and Affiliations

  1. Department of Mechanical Engineering, Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, P. O. Box 653, Beer-Sheva, 84105, Israel

    T. Elperin & G. Rudin

Authors
  1. T. Elperin
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  2. G. Rudin
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Correspondence to T. Elperin.

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Elperin, T., Rudin, G. “Thermal Mirror” Method for Measuring Physical Properties of Multilayered Coatings. Int J Thermophys 28, 60–82 (2007). https://doi.org/10.1007/s10765-007-0157-3

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  • DOI: https://doi.org/10.1007/s10765-007-0157-3

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