1. Introduction

The Gaussian beams are widely used for laser-assisted manufacturing and materials processing [1]. This type of beams exhibits diffractive spreading, resulting in rapid decrease of the peak irradiance as a function of the longitudinal, that is the beam propagation, distance [2]. Also, the irradiance decreases monotonically in the transverse, that is the radial, direction and, therefore, heats the substrates with monotonically decreasing temperature distribution. Special beam shaping optics are needed to modify the Gaussian beam profile for creating localized peak temperatures. The Bessel beams, on the other hand, provide a natural means of heating the substrates with undulating temperature distribution. The zeroth order Bessel function of the first kind, ${J_0}$, represents the electric field amplitude of a nondiffracting zeroth order Bessel beam. Since the irradiance is proportional to the square of the electric field amplitude and the Jo function is not square integrable, an integration of the irradiance over the entire transverse plane would yield infinite laser power in the beam [3,4]. Physically an ideal zeroth order Bessel beam consists of an infinite number of circular rings around a central maximum irradiance and the energy in each ring is approximately equal to the energy contained in the central ring. The energy of such a beam would be infinite which is unrealistic and this ideal beam is, therefore, difficult to achieve in practice. Indebetouw [5] showed that approximately nondiffracting beam can be synthesized using an axicon lens. The irradiance of such approximate Bessel beams varies along the propagation direction [6] and the Rayleigh range is longer than that of a Gaussian beam [7]. Typical laser sources generally produce Gaussian beams which can be passed through axicon and biconvex lenses to form a hollow beam with half-Gaussian irradiance profile. When the hollow beam is focused with a lens, a new pattern is obtained at the focal place for the electric field amplitude which is square integrable and the corresponding beam is known as the Bessel-Gaussian (BG) beam. The formation of BG beams with different irradiance profiles and their heating effects on solid materials have been studied by Zeng et al. [8,9], showing an efficient utilization of laser heating with BG beams of certain irradiance profiles. Recently Castillo-Orozco et al. [10,11] used BG beams to evaporate microdroplets of nanoparticle suspensions and sinter the nanoparticles for depositing thin films on both rigid and flexible substrates.

Lasers offer precise control for spatial and temporal heating of materials with little damage to the substrates. In addition, nanoparticles have better mechanical, electrical, and optical properties as well as lower sintering temperature than to the corresponding bulk materials [12]. Depositing nanomaterials to produce thin lines or films is of interest for fabricating optoelectronic, energy and electronic components [13] such as light-emitting diodes [14], ultra-flexible electrodes [15] and solar cells [16]. Fabrication of nanostructures on flexible and stretchable substrates is also of interest in the development of wearable electronic and photonic devices [17]. Numerous deposition techniques have been studied by others without using lasers. An et al. [18] used layer-by-layer deposition to produce nanosheets of two-dimensional metal carbides in between polymer sheets. Thomas et al. [19] fabricated densely packed micro and nanoscale polymer features using a melt processing approach. Chemically synthesized semiconductor nanowires of ZnO or carbon nanotubes on flexible conductive substrates have been investigated for electronic and photonic device applications [20,21]. Conventional inkjet printing method has been employed to pattern microelectrodes of semiconductor materials on flexible substrates [22]. Shin et al. [23] developed an electrostatic force-assisted Ag paste dispensing technique to print Ag fingers on solar cells. An oven was used in their process to cure the paste and, consequently, the fabrication time was relatively long. Castillo-Orozco [10] employed a Nd:YAG laser BG beam and deposited thin films of metal (Ag) and semiconductor (Si) nanoparticles on flexible substrates. Metallization of the precursor Ag paste to form continuous Ag lines is a crucial step for fabricating efficient Si solar cells [24]. Good electrical contact between Si and Ag lines with low line resistance and high aspect ratio enhances the efficiency of solar cells by reducing resistive and optical losses [25]. The pastes of Ag nanoparticles are used for manufacturing the contact lines for its low sintering temperature compared to the bulk Ag material [26,27].

In the current study, an experimental setup is presented to show certain optical elements that produce a BG beam on the substrate surface. This setup was utilized to deposit Ag lines on indium tin oxide (ITO)-coated silicon substrates using an Ag nanoparticle paste as a precursor. The primary focus of this paper is to study the thermal effect of the BG beam on the temperature distribution in the substrate. For this purpose, an expression is obtained for the irradiance of the BG beam by applying the Fresnel diffraction model, and this irradiance is used as a heat source in a heat conduction model to analytically solve for the temperature distribution. Since the nanoparticles sinter above a certain temperature, this constraint is applied to the temperature distribution to predict the width of the deposited Ag line as an illustration for the practical application of the thermal model. Most of the results have previously been published in a dissertation [28]. Those results are presented in this paper more succinctly by highlighting the similarity between the beam pattern and temperature profiles, and the evolution of different heating regimes under various processing conditions.

2. Experimental setup for BG beam formation and Ag line deposition

Figure 1 illustrates an experimental setup, showing a continuous wave CO₂ laser of wavelength $\lambda = 10.6\mu m$ and Gaussian irradiance distribution. This beam is passed through an axicon lens and a biconvex lens to form a hollow beam with outer-half-Gaussian irradiance distribution [8], and then focused with a hollow parabolic mirror on Ag lines, resulting in an irradiance profile that is referred to as the BG beam in this study. One of the advantages of BG beams is that its maximum intensity is much higher than that of Gaussian beams of the same power. Thus, much higher intensity can be applied to materials using BG beams. In addition, the BG beam is useful for deposition processes since it has longer Rayleigh range within which the irradiance distribution does not vary significantly. A stable thermal process, therefore, can be established even if the focal spot is not exactly on the substrate surface. This fault-tolerant feature is beneficial since the height of the Ag line may vary due to mechanical vibration of the linear stage or fluctuations in the flow rate of nanosuspension. The BG beams provide a large tolerance in the height direction to enable sintering the Ag lines uniformly. The mirror is hollow because it has a vertical hole starting from its top surface to the center of the reflective parabolic surface. This mirror with focusing capability, particularly the hole in the parabolic mirror, provides a convenient space to access the laser-material interaction region for measuring temperature using an infrared camera coaxially with the laser beam. Alternatively, the hollow space can be utilized to coaxially dispense nanosuspensions to the center of the beam. The laser spot diameter on the substrate surface was adjusted to be 300$\mu m$, which was the smallest spot size in the experiment setup of this study and this size was measured from the illuminated spot on a sensitive paper. From the BG beam profile, the laser irradiance at the beam radius of 150$\mu m$ is determined to be one-fifth of the maximum irradiance. The power, P, was varied from 2.3W to 11.8W to deposit Ag lines without causing any thermal damage to the silicon substrates used in this study. An Ag nanoparticle suspension, which was 75wt% LIFT (Laser-Induced Forward Transfer) printable silver ink from Sigma-Aldrich, was delivered to an ITO-coated Si substrate using a needle of inner and outer diameters 51$\mu m$ and 159$\mu m,$ respectively. The ink was delivered behind the BG beam and the substrate was moved at the speed V (Fig. 1). The flow rate of the ink, Q, was controlled with a syringe pump in the range of 100-200 nL/min and the substrate speed, V, was varied from 0.92 mm/s to 6 mm/s for depositing Ag lines under different conditions.

 

Fig. 1. Schematic of optical elements and laser track for Bessel-Gaussian beam formation during silver line deposition on ITO-coated substrates.

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The depositions were achieved in two steps, ink drying and nanoparticle sintering. For the case of ink drying, the silver ink was supplied to the substrate with a capillary tube of a syringe pump at the flow rate (Q) of 100nL/min while the substrate speed (V) was 6mm/s relative to the laser beam. The height (H_ct) of the tip of the capillary tube and the inclination angle $({{\theta_{in}}} )$ of the tube were 50µm and 40°, respectively, from the substrate surface. As a liquid ink line is drawn due to the motion of the substrate, the liquid was simultaneously evaporated to dry the line with the laser power (P_d) 2.3W. The diameter $({D_L})$ of the laser focal spot was 300µm on the substrate surface and the center of the laser beam was ahead of the tip of the capillary tube by a horizontal distance $({{d_{ct}}} )$ 1.5mm. After this drying step, the Ag nanoparticles were sintered with the laser power (P) 11.8W and V at 1mm/s. The width and height of the sintered Ag lines were measured with an optical microscope at 500 magnification and an optical profilometer respectively.

3. Methodology

3.1 Diffraction modeling of Bessel- Gaussian irradiance distribution

Figure 2 shows that a Gaussian beam is incident on an Axicon lens that transforms the incident beam into an annular beam, and a biconvex lens transforms the annular beam into a collimated outer half-Gaussian hollow beam [8] as shown at the plane AB. The outer radius and thickness of this hollow beam, ${R_{out}}$ and R₀ respectively, can be expressed as ${R_{out}} = L \cdot \tan ({\beta - \alpha } )$ and ${R_0} = {R_G}[{1 - \tan \alpha \cdot \tan ({\beta - \alpha } )} ]$., where L is the distance between the axicon and biconvex lenses, $\alpha $ and $\beta $ are the angles of incidence and refraction at conical surface of the axicon lens as depicted in Fig. 2, and $\sin \beta = n \cdot \sin \alpha $ th n being the refractive index of the lens material which is ZnSe in this study. The inner radius of the hollow beam, ${R_{in}}$, is now obtained as ${R_{in}} = {R_{out}} - {R_0}$.

 

Fig. 2. Laser propagation path for determining geometrical features of the beam.

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The irradiance of the outer half-Gaussian beam is given by the following expression:

The hollow parabolic mirror of focal length f reflects and focuses the incident outer-half-Gaussian hollow beam. The phasor of this reflected beam is given by the following Rayleigh-Sommerfeld diffraction formula [30]:

 

Fig. 3. Coordinate systems used in the Rayleigh-Sommerfeld diffraction formula for calculating the distance from the points on the parabolic mirror to the points on the plane of observation.

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Setting the origin of the Cartesian coordinates ($\epsilon $, η, z) at the center of the hole on the parabolic mirror, the coordinates of ${P_1}$ can be expressed in terms of $({\epsilon ,\eta ,\bar{ + }\mathrm{\xi }} )$ with $\mathrm{\xi } = \sqrt {{\epsilon ^2} + {\eta ^2}} \cdot \tan {\theta _p}$ and, therefore, ${U_{PM}}({{P_1}} )\equiv {U_{PM}}({\epsilon ,\eta ,\bar{ + }\xi } )$. Here the negative and positive signs are for the points ${P_1}$ located above and below the horizontal plane (Fig. 3(a)), respectively, since the z axis is negative above the horizontal plane. Therefore, the distance vector between the light source point P₁ and the observation point P₀ is $\vec{R} = \overrightarrow {{P_1}{P_0}} = ({x - \epsilon } ){\hat{e}_x} + ({y - \eta } ){\hat{e}_y} + ({z \pm \xi } ){\hat{e}_z}$, the unit vector along $\vec{R}$ is ${\hat{r}_R} = \frac{{({x - \epsilon } ){{\hat{e}}_x} + ({y - \eta } ){{\hat{e}}_y} + ({z \pm \xi } ){{\hat{e}}_z}}}{{\sqrt {{{({x - \epsilon } )}^2} + {{({y - \eta } )}^2} + {{({z \pm \xi } )}^2}} }}$, and the magnitude of $\vec{R}$ is $R = \sqrt {{{({x - \epsilon } )}^2} + {{({y - \eta } )}^2} + {{({z \pm \xi } )}^2}} $. So the inclination factor in Eq. (4) can be written as $\cos \left\langle {\hat{n},{{\hat{r}}_R}} \right\rangle = \hat{n} \cdot {\hat{r}_R} = \frac{{({y - \eta } )\sin a + ({z \pm \xi } )\cos {\theta _p}}}{{({z \pm \xi } )\left[ {1 + {{\left( {\frac{{x - \epsilon }}{{z \pm \xi }}} \right)}^2} + {{\left( {\frac{{y - \eta }}{{z \pm \xi }}} \right)}^2}} \right]}}$. The distance between the source plane ($\epsilon $η) and the observation plane (xy) is generally large compared to the distances in the x and y directions, especially, when the laser is focused onto the neighborhood of the origin of the x and y axes. Under this condition, the Fresnel approximation [30] can be applied and the inclination factor becomes $\cos \left\langle {\hat{n},{{\hat{r}}_R}} \right\rangle \approx \cos {\theta _p}$ in this study.

The phasor U_PM(P₁) in Eq. (4), i.e., ${U_{PM}}({\epsilon ,\eta ,\bar{ + }\xi } )$, can be determined by introducing a phase factor to Eq. (3) and applying coordinate transformation to ${\rho _v}({\epsilon^{\prime\prime},\eta^{\prime\prime}} )$. This additional phase corresponds to the optical path length traveled by the light from the plane AB to the surface of the parabolic mirror (Figs. 2 and 3(b)). The plane AB of Fig. 2 is included in Fig. 3(b) for clarity and two typical paths of the light are shown to highlight the phase delay distance, e.g., F₁F₂ = d₂, from the plane AB to the point of incidence on the mirror. Also shown in Fig. 3(b), the paths of the light reflected off the mirror toward a horizontal focal plane at the point F. The annular laser spot is oriented vertically at the plane AB with coordinates $({\epsilon^{\prime\prime},\eta^{\prime\prime}} )$ for various points on the spot. When the light from this spot is incident on the mirror at an angle of incidence θ_im, a slanted annular laser spot is formed with a certain curvature conforming to the shape of the mirror surface. The points on this slanted and curved annular spot can be described by the coordinates ($\epsilon ^{\prime},\eta ^{\prime},\zeta ^{\prime}$) with the origin at the center of the hole of the mirror. The plane $\epsilon ^{\prime}\eta ^{\prime}$ is tangential to the curved mirror surface at the center of the hole and $\zeta ^{\prime}$ represents the variation of the curved surface relative to this tangential plane. To simplify the integration in Eq. (3) as discussed earlier, the slanted and curved annular spot is projected further onto a horizontal $\epsilon \eta $ plane.

The projection angle θ_p depends on the position of point P₁ due to the curvature of the parabolic mirror. The effect of curvature is neglected in this study for large focal lengths of the parabolic mirror, resulting in a slanted planar annular spot on the mirror surface, and therefore the projection angle θ_p can be considered constant. Under this approximation, the projection of the slanted annular beam onto the horizontal ($\epsilon \eta $) plane will have the same shape as the vertically oriented annular spot with the same inner and outer radii R_in and R_out, respectively, when θ_im = π/4. For an arbitrary value of θ_im, the coordinate transformation from the vertical plane AB to the slanted planar plane and then to the horizontal $\epsilon \eta $ plane can be represented by $({\epsilon^{\prime\prime},\eta^{\prime\prime}} )\to ({\epsilon^{\prime},^{\prime}} )\to ({\epsilon ,\eta } )$, which yields ${\rho _v}({\epsilon^{\prime\prime},\eta^{\prime\prime}} )= \sqrt {\epsilon ^{\prime}{^{\prime}2} + \eta ^{\prime}{^{\prime}2}} = \sqrt {\epsilon {^{\prime}2} + \eta {^{\prime}2}{{\cos }^2}{\theta _{im}}} = \sqrt {{\epsilon ^2} + {\eta ^2}{{\cot }^2}{\theta _{im}}} = {\rho _h}({\epsilon ,\eta } )$ with ρ_h transforming the radial distance ρ_v from the vertical plane to the horizontal plane. Now the phasor U_PM can be written as follows by modifying Eq. (3) with a phase factor due to the phase delay distance d₂:

Now the phasor at the point ${P_0}({x,y,z} )$ can be calculated by applying Eq. (5) to the Rayleigh-Sommerfeld formula as given in Eq. (4):

The integral with respect to ϕ can be evaluated using the integral representation of the zeroth-order Bessel function of the first kind, i.e., ${J_0}(x )= \frac{1}{\pi }\mathop \smallint \nolimits_0^\pi {e^{ - ix\cos ({\phi - \theta } )}}d\phi $, and applying the Fresnel aroximation to $\xi $, i.e., $\xi = \rho \tan {\theta _p} \ll z$, to simplify Eq. (8) as:

Equation (13) influences the heat flux on the substrate surface and thus affect the temperature distribution in the material.

3.2 Bessel-Gaussian laser heating of moving substrates

A portion of the laser beam is reflected, and the rest of the energy is absorbed at the substrate surface when the transmittance of the substrate is zero. The absorbed energy manifests as heat flux on the surface of the substrate and provides the thermal energy for evaporating the liquid of the nanoparticle suspension and sintering the nanoparticles to deposit a thin film on the substrate surface. The irradiance of the BG beam is used to determine the heat flux as a boundary condition in the following heat conduction model. This model is applied to the case of depositing Ag lines on a moving ITO-coated Si substrate using an Ag ink. The temperature distribution of the model is used for selecting the laser processing parameters to heat without any thermal damage to the Si substrate and also predict the width of the deposited Ag lines.

Shown in Fig. 4 is the schematic of the conduction model in the Cartesian coordinates with the heat flux due to laser irradiation at the top surface of the substrate. The Ag ink is delivered to the right side of the laser spot and this unsintered ink passes through the spot as the substrate moves in the negative x direction with velocity V, resulting in the sintered Ag line on the left side of the spot. The effects of heating the Ag ink and the substrate motion are considered in the boundary condition for the heat flux at the top (z = H) surface of the substrate as discussed later in this section. The effect of ITO, however, is neglected since it is very thin (1–3 µm). Therefore, the governing equation for the temperature distribution, T(x,y,z,t), in the Si substrate can be written as follows by considering temperature-dependent thermophysical properties:

 

Fig. 4. Heat flux due to laser irradiation and the Cartesian coordinate system used in heat conduction model.

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The energy of a quantum of the CO₂ laser is much lower than the bandgap energy of the Si substrate and, therefore, the substrate does not absorb the laser energy. The laser beam is absorbed by the Ag ink and the Ag nanoparticles during the sintering process and this mechanism is incorporated in the model by considering the reflectance of silver in the boundary condition (14g). In this study, however, the width of the Ag line is much smaller than the diameter of the laser spot as illustrated in Fig. 4 and, therefore, only a small area of the incident laser spot is utilized for heating the Ag line and substrate. Denoting the area of the Ag line by A_Ag which is illuminated by the laser beam and the area of the entire laser spot by A_LS, the power contained in A_Ag would be P·A_Ag/A_LS. Since P is included in the term I_h(x,y,t), the line illumination factor, A_Ag/A_LS, is introduced in the boundary condition (14g). The area of the Ag line is estimated by considering the volumetric flow rate at which the Ag ink is delivered to the substrate and the speed of the substrate, resulting in A_Ag ≈ 0.0153mm², where the spreading of the ink on the substrate surface due to surface tension and the viscous resistance to the spreading are neglected. The heat conducts from the Ag line into the Si substrate, resulting in a hot Si surface adjacent to the line and, therefore, an area average emissivity, $\bar{\epsilon } = 0.216{\tilde{\epsilon }_{Ag}} + 0.784{\tilde{\epsilon }_{Si}}$, is considered in this study, where ${\tilde{\epsilon }_{Ag}}$ and ${\tilde{\epsilon }_{Si}}$ are the emissivities of Ag and Si respectively.

The thermal conductivity data [31] of Si are expressed as a function of temperature, $k(T )= p\cdot {T^q}$ with p = 10^5.75 and q = -1.41 in the temperature range 300K to 1000K, and an average value of the thermal diffusivity is taken as α = 0.8cm²/s. The emissivities of Ag and Si are [32,33] ${\tilde{\epsilon }_{Ag}}$ = 0.793 and ${\tilde{\epsilon }_{Si}} = 0.67$, and the reflectance of Ag is R_Ag = 0.98. In our experimental set up, the convection heat transfer to the ambient air is due to natural convection. The heat transfer coefficient varies from 2 to 25W/(m²·K) for natural convection [34]. An intermediate value of 10W/(m²·K) has been chosen for h_i, i= 1,2,3,4, in this study. h_Ag is estimated by considering the effect of the coated silver ink as an additional heat convection coefficient, i.e., ${h_{Ag}} = \frac{{{\rho _{Ag}}{C_{p,Ag}}\; {H_{Ag}}V}}{L}$, where ${\rho _{Ag}}$, ${C_{p,Ag}}$ and ${H_{Ag}}$ are the density, specific heat capacity and height of the Ag film deposited on the substrate surface. The derivation of h_Ag is presented in Appendix A.

Equation (14a) is a nonlinear heat conduction problem which Kar et al. [35] solved by linearizing the problem using the Kirchhoff transform and then applying the finite-medium Fourier transform in the x, y and z directions. They determined the temperature distribution in stainless steel substrates for the case of laser heating with a Gaussian beam. In their study, the irradiance profile was a Gaussian function that exhibits a monotonically decreasing profile with a maximum irradiance at the center of the beam and very low irradiance at the edge of the beam. In the present study, however, the irradiance profile involves the Gaussian and Bessel functions, exhibiting an oscillatory profile with a principal maximum irradiance at the center of the beam and rings of subsidiary maximum irradiances surrounding the principal maximum. The effect of this Bessel-Gaussian beam on the temperature distribution is analyzed in this paper by following Kar et al.’s method [35]. The solution of the above thermal model, Eq. (14a) and boundary and initial conditions (14b-h), is summarized by the following temperature distribution:

4. Results and discussion

4.1 Model validation

Isotherms are plotted in Fig. 5 to show the temperature contours both on the top surface and along the thickness of the substrate for the case of incident laser power 11.8W, substrate speed 1mm/s, and Ag ink feed rate of 100nL/min with a capillary tube of inner diameter (ID) 51µm. The isotherms on the top surface are utilized to predict the width of the sintered Ag line by considering that the sintering occurs only above a certain threshold temperature. The threshold is set to be $180\circ{C}$ in this study since this temperature is within the range of the silver ink curing temperatures [26,27]. Applying this criterion to Fig. 5, the predicted Ag line width is found to be 64$\mu m$ for nanoparticle dispensing capillary tube of ID 51µm. The thermal model was applied to another case with the same sintering condition as above except for a different capillary tube of ID 108µm, and the predicted Ag line width was found to be 122$\mu m$.

 

Fig. 5. Temperature distribution in silicon substrate during Ag line deposition. Zoomed in view of the isotherms, showing the sintering threshold isotherm of $180\circ{C}$ for predicting the Ag line width. Process parameters for temperature calculation: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, $V = 1mm/s$, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$.

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To validate the model predictions, experiments were conducted using the apparatus shown in Fig. 1 for sintering Ag nanoparticles to deposit Ag films as a narrow line on a Si substrate. For the above-mentioned two cases of tube ID 51 µm and 108 µm, the experimental line widths (Fig. 6) range from 59 to 62$\mu m$ and 117 to 122$\mu m$, respectively, which agree with the predicted line widths fairly well. The widths were measured at different locations along the length of the line due to the nonuniformity in the width. The average line width and thickness are 60µm and 1.11µm, respectively, for the case of 51$\mu $m ID tube and 120µm and 1µm, respectively, for the larger tube of 108$\mu $m ID under the same drying and sintering conditions as for the smaller tube. Figure 6 shows optical microscopic and Secondary Electron Microscopic (SEM) images, and Electron Dispersive X-ray (EDX) data for the composition of various elements in the sintered Ag film. The optical micrographs of the Ag lines show that the widths vary slightly along the length of the lines and the SEM images exhibit nanosize clusters formed by the sintered Ag nanoparticles. The EDX spectrographs, on the other hand, demonstrate the purity of the process for depositing contaminant-free Ag lines, even though the process was carried out in an open space with ambient air. Dominant Ag peaks can be seen in both Figs. 6(a) and 6(b) with extremely small peaks of C atoms, indicating that the organic liquid of the Ag nanosuspension is almost completely evaporated. While no peaks of O₂ and Si atoms exist in Fig. 6(a), there presence in Fig. 6(b) may be due to the native oxide of Si in the original Si substrate. The presence of S peak is ascribed to contamination during cleaning and handling of the samples after the deposition process.

 

Fig. 6. Deposited silver lines on ITO-coated substrate. Optical microscopic and SEM images, and EDX data for a silver line deposited with a capillary tube of inner diameter (a) 51$\mu $m and (b) 108$\mu $m. Process parameters for deposition: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, $V = 1mm/s$, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$.

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The model was also used to examine the effect of laser power on the sintered Ag line width as shown in Fig. 7. It includes the experimental line widths as well for comparison between the experimental data and theoretical predictions. When the experiments were run at the laser power 8.9 W, no deposition of Ag film occurred which is also predicted by the model. Although the Ag lines were deposited at 11.8 W in this study, Fig. 7 shows that lines of smaller width can be deposited below 11.8W but above certain threshold powers. Also Fig. 7 indicates that extremely narrow, e.g., a few micrometers or nanometers wide, Ag lines can be deposited with laser powers slightly above the threshold value. The threshold powers for sintering are 10.45 W and 11.3 W for capillary tubes of ID 108 µm and 51 µm, respectively. This change in the threshold power with tube ID is attributed to the variation in the width and thickness of the initial liquid (Ag ink) line produced on the substrate surface by different capillary tubes. A larger ID tube would produce wider and thinner liquid lines than a smaller ID tube under the same mass flow rate of the nanosuspension and the same substrate speed. and, correspondingly, fewer Ag nanoparticles would be present per unit volume of the liquid line in the case of nanosuspension dispensation with a larger tube. The laser is, therefore, required to heat less materials in thinner films to achieve the threshold sintering temperature which is taken as 180°C for Ag nanoparticles in this study. These two phenomena, the nanosuspension dispensation and the laser heating of fewer nanoparticles, influence the width and thickness of the final product which is formed after the initial liquid Ag line is sintered. The combined effect of these two phenomena may be the reason for lower threshold power in the case of larger ID tube. For the same laser power, e.g., 11.8 W, the larger ID tube yields wider and thinner sintered Ag line than in the case of the smaller ID tube as evident from the experimental data discussed earlier in this section.

 

Fig. 7. Validation of thermal model for silver line deposition and prediction of threshold laser power for sintering. Process parameters for deposition: Q = 100nL/min, Capillary tube inner diameter = 51µm and 108µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, $V = 1mm/s$, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$.

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For the same width of sintered Ag line, Fig. 7 shows that the larger ID tube requires lower laser power that the smaller tube. To produce the same width by both the tubes, the laser must heat the same width with a boundary of 180°C isotherm. Since a larger ID tube yields fewer nanoparticles per unit volume of the liquid line, lower power would be needed to create the required isotherm than in the case of a smaller ID tube. Figure 7 also shows that the larger ID tube produces wider Ag lines than the smaller ID tube for the same laser power. This result is due to the above-mentioned two phenomena as well. A larger ID tube produces wider Ag liquid lines than a smaller ID tube and, consequently, the concentration of nanoparticles is lower in the former case. A given laser power, therefore, will be able to heat the nanoparticles above the threshold sintering temperature 180°C in a wider region in the case of a larger ID tube.

4.2 Irradiance profile from the diffraction model

The laser beam patterns change significantly as the beam passes through different lenses and the parabolic mirror. Shown in Fig. 8 are the laser irradiance distributions as a function of the beam radius at the exit of the CO₂ laser device, after the biconvex lens and at the focal plane of the parabolic mirror. The laser comes out of the laser device as a Gaussian beam of diameter 4.06mm. The Gaussian beam is transformed to a hollow beam after passing through the axicon and biconvex lenses and the irradiance profile in the ring of this beam is outer half-Gaussian as discussed in section 3.1. The hollow beam is then reflected and focused by the parabolic mirror, resulting in a Bessel-Gaussian irradiance pattern due to interference of the light arriving from various points of the ring to the substrate surface. This BG beam consists of a principal maximum irradiance at the center of the beam and subsidiary maxima at different points in the beam. The effect of the subsidiary irradiances on the temperature distribution is discussed in the next section.

 

Fig. 8. Laser beam patterns at different locations of the laser path. (a) optical setup, (b) Incident Gaussian irradiance profile, (c) Outer half-Gaussian beam of inner and outer radii ${R_{in}} = 6.8mm$ and ${R_{out}} = 8.8mm$, respectively, and (d) Bessel-Gaussian beam with a principal maximum of radius ${R_B} = 0.15mm$.

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4.3 Parametric studies on laser heating and temperature distribution

The temperature distribution in the silicon substrate is examined for different process parameters by considering the length (L), width (W) and height (H) of the substrate as $60mm,30mm\; and\; 0.2mm,$ respectively. Two cases of laser heating are analyzed when the laser beam scans the top surface of the substrate, i.e., the laser beam moves relative to the substrate, in two different trajectories. In one case, the laser follows a symmetric track along the center line on the top surface and the trajectory is the locus of the coordinates (x, D, 0) for D = 15 mm and $0 \le x \le L$. In the second case, the laser follows an asymmetric track along an off-center line on the top surface and the trajectory is the locus of the coordinates (x, D, 0) for D = 10 mm and $0 \le x \le L$. The dimensionless distances of the track from the front (y = 0) surface of the substrate are ${D^\ast } = 0.5$ and ${D^\ast } = 0.33,$ where D* = D/W, for the symmetric and asymmetric tracks respectively. The laser power is set at 11.8W and the scanning speed at V = 1 mm/s for both cases.

Figures 9(a,b) and 9(c,d) show the isotherms on the top surface of the substrate for the asymmetric and symmetric cases, respectively. Two different times, t = 3 s and 36 s, are considered for each case, and these two times yield the locations of the laser beam center on the substrate surface at two different points (Vt, D) where the temperature is maximum. The temperature distribution is asymmetric for the case of ${\textrm{D}^{\ast }} = 0.33$, showing higher temperatures in the region between the laser track and the front (y = 0) surface of the substrate. This asymmetry is due to the slow convection and radiation heat losses, resulting in accumulated internal energy that raises the temperature. For the case of ${\textrm{D}^{\ast }} = 0.5$, on the other hand, the laser heating is a perfectly symmetric problem and, therefore, the temperature distribution is symmetric as shown in Figs. 9(c,d).

 

Fig. 9. Isotherms on the top surface of the substrate at selected positions of the moving laser beam center. (a, b) Asymmetric temperature contours at the points (3mm, 10mm) and (36mm, 10mm), and (c, d) Symmetric temperature contours at the points (3mm, 15mm) and (36mm, 10mm). Process parameters for temperature calculation: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, $V = 1mm/s$, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$.

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However, the temperature distribution in a small region of radius approximately 1.5 mm around the laser beam center is found to be symmetric regardless of the distance, D, of the scanning track from the y = 0 surface, since the effect of laser heating dominates over the convection and radiation heat losses in the neighborhood of the beam center. Figure 10(a) shows the temperature distribution along the laser scanning line on the top surface of the substrate when the beam center is at x = 30 mm and D = 10 mm, i.e., D* = 0.33, and Fig. 10(b) is the corresponding isotherm which exhibits symmetric temperature contour in the neighborhood of the beam center. Away from this neighborhood, however, the isotherms have convex or concave tips corresponding to the peaks or valleys of the temperature profile in Fig. 10(a). The BG irradiance profile of Fig. 8(d) is sketched on top of the temperature profile in Fig. 10(a) to illustrate that the peaks and valleys in the temperature profile correspond to the peaks and valleys in the BG beam profile. The wavy temperature profile is thus ascribed to the oscillatory irradiance profile.

 

Fig. 10. Temperature distribution at the top surface of the substrate in the neighborhood of the laser beam center for a low scanning speed V = 1 mm/s: (a) Wavy temperature profile along the laser scanning line, and (b) Temperature contour with deformed leading and trailing edges. Process parameters for temperature calculation: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$, D = 10mm (D* = 0.33) and t = 30s (t* = 0.5).

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It is also noticeable in Fig. 10 that the temperature contour is symmetric about the x = 30 mm and y = 10 mm lines over a small distance in these two directions. This symmetric feature generally occurs at low scanning speeds, such as 1 mm/s, since the conduction heat transfer is much faster than the advection mechanism at low speeds. The latter heat transfer mechanism arises from the relative motion between the laser and the substrate. When the scanning speed is increased to V = 40mm/s, the effect of advection increases, resulting in lower temperatures in the leading edge than in the trailing edge of the beam as shown in Fig. 11(a). Similarly, the isotherms become densely spaced in the leading side than in the trailing side of the beam as in Fig. 11(b). Thus, the influence of advection compared to the conduction heat transfer induces an asymmetric temperature field along the laser scanning line. Considering the isotherm of 86°C in Fig. 11(b), the distance (L_l) of the leading edge of this isotherm is much shorter than the trailing edge (L_t) from the laser beam center. It can also be seen from the isotherms that the asymmetry occurs about the x = 30 mm line. It can also be seen that the temperature contours are still symmetric about the y = 10 mm line whereas they are asymmetric about the x = 30 mm line. This direction-dependent temperature field is caused by the direction of the laser beam scanning relative to the substrate. The laser scanning occurs only in the x direction relative to the substrate since the substrate is moved in the -x direction (Fig. 4) in this study, resulting in advection only in the -x direction. Therefore, both conduction and advection cause heat transfer in the x direction, whereas only conduction occurs in the y direction, and this difference in the heat transfer mechanism produces the asymmetric and symmetric temperature fields about the above-mentioned x and y lines, respectively.

 

Fig. 11. Temperature distribution at the top surface of the substrate in the neighborhood of the laser beam center for a high scanning speed V = 40 mm/s: (a) Wavy temperature profile along the laser scanning line, and (b) Temperature contour with deformed leading and trailing edges. Process parameters for temperature calculation: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, P = 11.8W, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$, D = 10mm (D* = 0.33) and t = 0.75s (t* = 0.5).

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The effect of speed is analyzed further in Fig. 12 to determine the dominant role of conduction or advection heat transfer mechanism in different ranges of the scanning speed, V. The maximum temperature in terms of V, ${T_{max}}(V )= T\left( {\frac{L}{2},D,H,\frac{L}{{2V}}} \right)$, which occurs at the laser beam center, is plotted as a function of the scanning speed for different laser powers. T_max(V) decreases as the scanning speed increases. Three ranges of the scanning speed are identified corresponding to three dominant modes of heat transfer mechanism, such as the conduction, advection and weakly heating regimes, using a temperature ratio to delineate these regimes. The temperature ratio is defined as ${T_r}(V )= \frac{{{T_{max}}(V )- {T_a}}}{{{T_{max}}({{V_{min}}} )- {T_a}}}$, where ${V_{min}} = 0.1mm/s$ is the lowest velocity used in the experiments of this study. The dominant conduction, dominant advection and weakly heating regimes are classified by considering ${T_r} \ge 0.99$, $0.1 \le {T_r} < 0.99$ and ${T_r} < 0.1$.

T_max(V) is nearly constant with ${T_r} \ge 0.99$ for all powers and $0.1mm/s\; \le V \le 6mm/s$. This regime is characterized as the dominant conduction regime since the slow motion of the substrate allows sufficient time to reach a balance between the laser heating and the heat conduction. Thus a steady state is achieved, which also explains the reason for the maximum temperature to be nearly constant. T_max(V), however, decreases significantly in the range $6mm/s\; < V \le 50m/s$ . This regime is characterized as the dominant advection regime since the fast motion of the substrate decreases the laser heating time in the laser-irradiated region. Consequently, the spreading of heat by conduction is less than the heat carried away from the irradiated region by the moving substrate, and a steady state is reached in this case due to a balance between the laser heating and the heat advection. Finally, T_max­(V) tends to the room temperature asymptotically when $V > 50m/s$ since high scanning speeds decrease the laser-substrate interaction time and thus reduce the laser energy input to the substrate as the substrate passes through the laser-irradiated region. This high-speed scanning range is classified as the weakly heating regime. The substrate attains different temperatures in these three regimes of heating and the temperature, in turn, affects the sintering of the nanoparticles and the width of the deposited thin film.

 

Fig. 12. Maximum temperature of the substrate in the dominant conduction, dominant advection and weakly heating regimes for different scanning speeds and laser powers. Process parameters for temperature calculation: Q = 100nL/min, ID = 51µm, ${H_{ct}} = 50\mu m$, ${\theta _{in}} = 40^\circ $, ${d_{ct}} = 1.5mm$ and ${D_L} = 300\mu m$.

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5. Conclusion

A Gaussian CO₂ laser is transformed to a Bessel-Gaussian beam using an axicon lens, a biconvex lens and a hollow parabolic mirror. A hollow collimated beam is incident on the mirror which reflects the beam as a hollow cone and the Bessel-Gaussian beam is formed at the focal plane of the mirror. The diffraction of light is analyzed using the Fresnel approximation to obtain an analytical expression for the irradiance distribution of the Bessel-Gaussian beam. This beam consists of a principal maximum irradiance at the laser beam center and subsidiary maxima that surround the principal maximum as rings. The irradiance distribution is applied as a heat flux boundary condition to a conduction model to analytically solve for the temperature distribution in a moving silicon substrate on which Ag nanoparticles are sintered for depositing Ag lines. The temperature profile exhibits a wavy pattern like the wavy pattern of the irradiance distribution, resulting in noncongruent isotherms consisting of convex and concave tips. The Ag line width obtained from the thermal model is found to compare well with experimental data.

Symmetric and asymmetric isotherms are formed on the substrate surface depending on the distance of the laser beam from one of the edges of the substrate. The isotherms are perfectly symmetric when the laser scanning path coincides with the line of symmetry on the substrate surface, and asymmetric isotherms occur when the laser scanning path deviates from the line of symmetry. Another aspect of the thermal model is that the moving substrate introduces advection which influences the substrate temperature in addition to the conduction heat transfer mechanism. When the laser scanning speed relative to the substrate is low, the conduction effect is stronger than advection and the isotherms are symmetric in both the scanning (x) and transverse (y) directions running through the laser beam center. At higher scanning speeds, however, the advection effect is stronger than conduction and asymmetric and symmetric isotherms appear in the scanning (x) and transverse (y) directions, respectively. This direction-dependent characteristics appear out of the advection versus conduction effects. Since the substrate moves in the -x direction, the advection occurs only in this direction causing the asymmetric isotherms. Due to the absence of advection in the transverse direction, only conduction occurs in this direction resulting in the symmetric isotherms. When the scanning speed is increased further, the laser-matter interaction time decreases which lowers the energy input to the substrate and, consequently, the substrate temperature does not increase significantly. The heating regimes in these three ranges of the scanning speed can be classified as the dominant conduction, dominant advection, and weakly heating effects.