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Analytical model for ultrashort pulse laser heating in a titanium nanofilm by implementing dual-phase-lag theory in mathematical analysis

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Abstract

Surface modification processes of any metallic structure with the help of laser irradiation are a universal practice. The profoundly focused laser beam is to irradiate on the substrate surface to modify the surface condition for the improvement of the tribological properties in heavy and rouged engineering applications. There is a wide application of pure or raw Ti in the field of the biomedical sector specifically in implants and artificial joint prosthesis. In the viewpoint of the above, the present work determines the thermal characteristic aspect in a pure Ti physical domain by developing a two-dimensional heat transfer approach with the dual-phase-lag (DPL) model under the influence of the ultrashort pulsed laser heating. The physical domain has a Ti nanofilm of the 4 nm length, 2 nm width, and 0.02 nm thickness. A DPL model is developed for analyzing the ultrafast heating, as it is the most potential heat transfer model. The present study has modelled a hybrid analytical analysis comprising of the Duhamel’s theorem and the finite integral transform method. This work highlights the essentiality of applications of the DPL heat conduction model over the conventional Fourier’s model based on the qualitative assessment. The selection of thermal relaxation time lags was carefully chosen from the existing experimental evidence due to the requirement to reach the melting point temperature of pure Ti. The peak temperature of laser irradiation declines with the increase in the optical penetration depth and laser pulse time. Finally, the competence of the present analysis is validated with the published numerical and experimental works from the engineering accuracy standpoint.

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Abbreviations

c:

Specific heat/J kg−1 °C−1

C:

Heat capacity/J m−3 °C−1

C1−C2 :

Constants for solution of differential equations, see Eq. (18)

F:

Non-dimensional time, refer Eq. (5)

Fp :

Non-dimensional laser pulse duration, see Eq. (5)

i:

Nonnegative integers (0, 1, 2, 3….)

I0 :

Laser energy density/intensity/J m−2

\(I_{0}^{\prime \prime }\) :

Non-dimensional energy density/intensity, see Eq. (5)

\(I_{0}^{\prime \prime \prime }\) :

Non-dimensional energy density/intensity in relation with initial condition, see Eq. (6c)

j :

Nonnegative integers (0, 1, 2, 3….)

k :

Thermal conductivity/W m−1ºC−1

L:

Thickness of the nanofilm considered along x-direction/m, see Fig. 1b

M1 :

M2, non-dimensional constants, defined in Eq. (17)

q:

Rate of heat conduction/W m−2

Q:

Volumetric external heat source/W m−3

ra and rf :

Absorptivity and reflectivity, respectively

rD :

Laser beam radius focused on substrate surface/m

RD :

Non-dimensional laser beam radius focused on substrate surface, see Eq. (5)

R:

Geometrical coordinates/m

t:

Time for laser exposure/s

tp :

Laser pulse duration/s

T:

Local temperature of the nanofilm surface/°C

Ti :

Initial temperature of the nanofilm/°C

Tpeak :

Peak temperature, see Fig. 3/°C

Veq and VeT :

Non-dimensional thermal relaxation time lag for heat flux and temperature gradient respectively, see Eq. (5)

x :

X-spatial direction/m

X:

Non-dimensional spatial direction, \({x \mathord{\left/ {\vphantom {x L}} \right. \kern-\nulldelimiterspace} L}\)

y :

Y-spatial direction/m

Y:

Non-dimensional spatial direction, \({y \mathord{\left/ {\vphantom {y L}} \right. \kern-\nulldelimiterspace} L}\)

α :

Thermal diffusivity/m2s−1

θ :

Non-dimensional local temperature of plate, see Eq. (5)

δ:

Absorption depth/m

Θ:

Anonymous transformed function of temperature based on Duhamel’s theorem, refer Eq. (9)

\(\overline{\Theta },\overline{\overline{\Theta }}\) :

Transformed function of temperature based on finite integral transform, see Eqs. (13) and (15), respectively

ρ :

Density/kg m−3

Ω:

Non-dimensional constant, see Eq. (5)

τ q and τ T :

Thermal relaxation time lag for heat flux and temperature gradient respectively

\(\xi\) :

Transformed time function based on Duhamel’s theorem, see Eq. (9)

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Authors and Affiliations

  1. Department of Mechanical Engineering, Jadavpur University, Raja S. C. Mallick Road, Kolkata, West Bengal, 700032, India

    Jaideep Dutta & Balaram Kundu

  2. Department of Mechanical Engineering, MCKV Institute of Engineering, 243, G. T. Road (N), Liluah, Howrah, West Bengal, 711204, India

    Jaideep Dutta

  3. Kazi Nazrul University, Asansol, Paschim Bardhaman, West Bengal, 713340, India

    Ranjib Biswas

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  1. Jaideep Dutta
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Correspondence to Balaram Kundu.

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Dutta, J., Kundu, B. & Biswas, R. Analytical model for ultrashort pulse laser heating in a titanium nanofilm by implementing dual-phase-lag theory in mathematical analysis. J Therm Anal Calorim 147, 7337–7352 (2022). https://doi.org/10.1007/s10973-021-11044-2

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