1 Introduction

Thermocapillary convection induced by a thermal action of the laser irradiation in a thin liquid layer on a solid substrate [1,2,3,4,5,6,7,8,9,10] is of great importance in a wide range of applications, including the laser diagnostics of liquids [7,8,9, 11,12,13,14,15,16,17], nondestructive evaluation control of solids [18,19,20], and the liquid-assisted laser cleaning of solid surfaces [21, 22]. An important feature of the laser-induced thermocapillary convection is that in the thin liquid layer in an area of stationary laser spot a concave deformation (called as a thermocapillary dimple) of the free liquid surface appears and the partially reflected from this deformation laser irradiation forms on a remote screen a circular interference pattern [7,8,9,10]. Da Costa and co-workers in their pioneering works [7,8,9] on investigation of the laser-induced thermocapillary deformation of the heavy oil layers noticed that thermal, mechanical and optical properties of liquids could be determined from the interference pattern [7,8,9]. Later, Bezuglyi and co-workers [6, 11,12,13,14,15,16,17,18,19,20] found out that the stationary diameter of the pattern (named subsequently as a photothermocapillary signal, PTC signal for short) depends on various parameters of an experimental arrangement (thickness of liquid layer and substrate, power and intensity distribution in a laser beam) and thermophysical properties of liquid and substrate. Basing on these findings a number of the non-contact photothermocapillary methods for measuring the thickness of transparent liquid layer on flat substrates [11, 12], viscosity of liquids [13], as well as a new method for measuring the contact angles of liquids on solid surfaces [14] have been developed. In [15, 16] a lag time between the switch-on moment of the laser beam and appearance of the PTC signal on a screen was detected. The deviation of the lag time from its quadratic change with the thickness of liquid layer was considered as a basis for development of a method for measuring the thermal diffusivity of liquids [15] and for monitoring of organic impurities on water surfaces [16]. Note that the liquids used in all mentioned experiments were optically transparent, and the thermal effect of the laser beam was achieved by absorption of the laser irradiation by solid substrates such as ebonite or carbolite.

Several subsequent papers [18, 19] reported a new photothermocapillary approach to the non-destructive evaluation control of solid samples, based on analysis of the stationary PTC signal coming from the thermocapillary dimple of a transparent liquid layer on a solid substrate with different hidden flaws. A possibility to detect in solid samples of caverns and highly heat-conductive foreign inclusions under varnish and paint coatings was shown [18, 19].

Despite of comprehensive and longstanding studies of the laser-induced thermocapillary convection the influence of the thermal conductivity of solid substrates has been overlooked until now. This influence, however, may prove to be a very crucial because the laser irradiation is absorbed by a substrate resulting in a two-dimensional heat source at the interface between liquid and solid. The solid substrate definitely serves as a heat sink dissipating the energy from the heat source in accordance with its thermal conductivity. As it was shown in [23] temperature distribution on the free surface of liquid layer rested on a solid substrate depends on the thermal conductivity of this substrate. In turn the changing the surface temperature in our case leads to the changing geometrical parameters of the thermocapillary dimple and, consequently, the PTC signal. Thus, the thermal conductivity of substrate can affect not only sensitivity but also applicability of the proposed methods of the laser diagnostics of materials [6, 11,12,13,14,15,16,17,18,19,20]. Note that the thermal conductivity of substrates plays a key role in wide range of physical problems, including the spreading of liquids on solid substrates [24] where thermal resistance between liquid and solid influences the spreading rate and the flow pattern.

In this paper, we investigate the sensitivity of the laser-induced thermocapillary effect to the thermal conductivity of solid substrates using the stationary PTC signal. Moreover, we anticipate that results obtained may be taken as a basis for a new laser method for the thermal conductivity measurements of solid materials.

2 Formation of the thermocapillary dimple and the PTC signal

This section gives a brief description of the mechanism of the laser-induced thermocapillary convection and the PTC signal formation, see Fig. 1. When the heating laser beam, P h, is incident on an absorbing solid substrate covered with a thin transparent liquid layer a heat source at the solid–liquid interface arises. The heat from this source distributes both into the depth of substrate and into the bulk of liquid layer. As a result, after a lag time [9, 15] written as

$$\tau_{\text{d}} = h_{\text{l}}^{2} /4\kappa_{\text{l}}$$
(1)

the temperature of the free liquid surface increases that leads to a local decrease in the surface tension,

$$\gamma = \gamma_{0} + \gamma^{\prime}_{\text{T}} (T - T_{0} ),$$
(2)

where h l is the thickness of the liquid layer, κ l is the thermal diffusivity of the liquid; 0 is the surface tension of the liquid at a reference temperature T 0; \(\frac{\partial \gamma }{\partial T} \equiv \gamma^{\prime}_{\text{T}}\) is the thermocapillary coefficient, which is negative for most pure liquids. Consequently, a radially outward surface tension gradient along the free liquid surface arises:

$$\frac{\partial \gamma }{\partial r} = \frac{\partial \gamma }{\partial T}\frac{\partial T}{\partial r} > 0,$$
(3)

which is balanced by the velocity gradient in the bulk layer

$$\mu \frac{{\partial v_{\text{r}} }}{\partial z} = \frac{\partial \gamma }{\partial T}\frac{\partial T}{\partial r},$$
(4)

where \(\frac{\partial T}{\partial r}\) is the radial gradient of the surface temperature, v r is the radial velocity of the liquid, z is the vertical coordinate, and μ is the dynamic viscosity of the liquid. This process induces the radially outward thermocapillary flows, which lead to a lowering the liquid level and appearance of a thermocapillary concave deformation of the liquid layer in the irradiated spot. A negative capillary pressure below the curved surface causes the reverse bottom flow towards the heated spot, wherein it conjoins with the radially outward surface flows resulting in the thermocapillary toroidal convective vortex. Interestingly, that the deformed liquid interface represents a kind of optical element consisting of two mirrors separated by an inflection line: the one is a central concave mirror and the other is an annular convex mirror. A probe laser beam with the power P p ≪ P h reflected from this optical element forms a circular interference pattern on a remote screen, called as the PTC signal, Fig. 1. Note that in contrast to the classical Newton’s fringes the PTC signal has the bright and wide outer ring, see Fig. 1. As it was found in [15, 16], the PTC signal goes through several stages: a signal delay lasting τ d, then a fast increase of its diameter, which is related with the development of the thermocapillary convection and the dimple formation in the liquid layer and, finally, a slow increase of the PTC diameter up to reaching the stationary diameter, D st. The latter indicates onset of the stationary thermocapillary dimple and the steady-state thermocapillary flows in the layer. The diameter of the PTC signal is directly related to the curvature of the thermocapillary dimple [6].

Fig. 1
figure 1

Illustration of the thermocapillary deformation of a liquid layer caused by heating of a substrate with a laser beam and the formation of the photothermocapillary (PTC) signal by a probe laser beam. Temperature distribution (T-field) in both solid and liquid media as well as the velocity flows (v-field) in the liquid layer is depicted

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3 Experimental methods and materials

3.1 Substrates and fluidic cells preparation

The solid materials in a wide range of the thermal conductivity values from 0.13 to 117 W∕(m K) were used as substrates. The thermophysical properties of those materials are listed in Table 1. The substrates in the form of disks of 60 mm in diameter and 20 mm in thickness were prepared as follows: using ebonite, PMMA and Teflon samples the disks were machined on a lathe. Metallic substrates such as Wood’s metal, Rose’s metal, Pb, Sn, and Zn were prepared by melting of granules of those materials in cylindrical duralumin dishes. After solidification, the working surfaces of those substrates were smoothed by polishing with a soft abrasive paper. The disks (~50 mm in diameter and 6 mm thick) of KRS-5 (thallium bromoiodide crystal), BK7, quartz glass, KBr, NaCl, and CaF2, placed in duralumin dishes were used as glass substrates. A marble slab of surface area of 100 mm2 and 20 mm thickness was also used as a substrate. For convenience, in further experiments all substrates were conditionally divided into two types: the low heat-conductive (LHC) substrates with k s ≤ 9.7 W/(m K), and the highly heat-conductive (HHC) substrates with k s ≥ 9.7 W/(m K).

Table 1 Thermophysical properties of the substrate materials
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The fluidic cells were constructed by attaching to all these substrates the stopper rings of PVC tape of 50–60 mm in diameter using an epoxy resin. To avoid an outflow of the working liquid, which may be caused by a capillary action, the height of the rings was around 10 mm. To ensure the same absorption of the heating laser beam by substrates with different optical properties, on their working surfaces the writing-ink (“Edding T25”, Germany) covering the area of 20 mm in diameter was deposited. After the ink was evaporated the dry coating film of around 10 µm thickness, completely absorbed the laser radiation was formed. Below, this coating film is called as the heated area.

3.2 Experimental setup and method

The experimental setup is shown in Fig. 2. The beam of He–Ne laser (P h = 21 mW maximum value, wavelength λ = 633 nm, beam diameter d = 2.5 mm) serving as the heating beam for generation of thermocapillary convection is directed normally to the substrate and absorbed by the heated area. The power of the heating beam was varied depending on experimental requirements using a calibrated optical wedge. The probe laser beam (He–Ne laser, P p = 0.3 mW, λ = 633 nm, d > 5 mm) defocused with a spherical mirror is directed to the thermocapillary dimple. The probe beam being reflected from the free liquid surface forms the PTC signal on a screen. The experimental procedure was as follows. Silicon oil (μ = 4.6 mPa s, γ = 19.7 mN/m, \(\gamma^\prime_{\text{T}}\) ≅ 0.06 mN/m K, k l = 0.12 W/m K, C p = 1.5 kJ/(kg K), ρ = 920 kg/m3, β = 1.05 × 10−3 1/K [31]) was used as working liquid. A thin layer of silicon oil was deposited in a fluidic cell and its thickness was controlled using the method of the calibrated wires [12,13,14,15]. To exclude the influence thermo-gravitational convection in the process of the thermocapillary deformation, the thicknesses of layers were chosen to be below a critical thickness \(h_{\text{l}} < h_{\text{c}} = \left( {\frac{{\gamma^{\prime}_{\text{T}} }}{\rho g\beta }} \right)^{1/2}\) [32], which for this silicon oil is h c = 2.5 mm. For the LHC substrates, the liquid layer thickness was varied in the range of 550, 760 and 960 µm. In case of the HHC substrates due to conductive heat losses across the substrate, the liquid layer thickness was decreased down to 100 µm. After the layer was formed, the shutters of the heating and the probe lasers were opened. When the PTC signal reached a stationary state, its diameter, D st, was measured with a ruler. Then both shutters were closed and the system was left for up to 10 min to cool down to room temperature and damping the thermocapillary flows in the layer. To control reproducibility the experiment was repeated five times for each substrate. Error of measurements did not exceeds 5%.

Fig. 2
figure 2

Schematic view of the experimental setup. 1 The heating He–Ne laser (max power 21 mW), 2 the probe He–Ne laser (max power 0.3 mW), 3 shutters, 4 calibrated optical wedge, 5 mirrors, 6 spherical mirror

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4 Experimental results and discussion

Figure 3 shows D st of the PTC signal versus k s in the logarithmic scale obtained for the LHC substrates for three thicknesses of the silicon oil layer at the power of the laser beam equals to 21 mW. For the HHC substrates, D st versus k s for the layer of 100 µm thickness at the powers of the laser beam 21, 18 and 17 mW are shown in the logarithmic scale in Fig. 4. According to both Figs. 3 and 4, the dependency of D st on k s follows a power law \(D_{\text{st}} \sim k_{\text{s}}^{ - n}\), where the power exponent n increases from 0.672 ± 0.002 to 0.836 ± 0.003 when decreasing the layer thickness at constant P h (Fig. 3) and lays in between 0.794 ± 0.001 and 0.899 ± 0.002 when changing the power of the heating laser beam at constant h l (Fig. 4). In the range of the high values of k s the diameter of PTC signal for both types of the substrates reaches only a few millimeters. The PTC interference pattern has several bright and blurry rings that makes it difficult to be measured. This change in the pattern is related with substantial decreasing the depth of the thermocapillary dimple. When further increasing k s the circular interference structure of the PTC signal transforms to a light spot with the gauss distribution of the intensity, called as a zero-signal. This limit case corresponds to the reflection of the probe laser beam from a flat liquid layer that is the layer with zero curvature of the free surface. Thus, it is possible to conclude that there is a critical value of \(k_{\text{s}}^{*}\) at which the thermocapillary deformation of the free surface of the liquid layer becomes fully damped. The critical \(k_{\text{s}}^{*}\) in turn is related to the thickness of the liquid layer and the power of the heating laser beam. For instance, at P h = 21 mW for the layers of 960 and 550 µm the zero-signal appears at \(k_{\text{s}}^{*} = 6.5\;{\text{W/(m}}\;{\text{K}})\) (the case is shown in Fig. 3 no experimental points obtained at \(k_{\text{s}} \ge k_{\text{s}}^{*}\)) and \(k_{\text{s}}^{*} = 16\;{\text{W/(m}}\;{\text{K}})\) (Rose’s metal is not shown in Fig. 3), respectively. For the layer of h l = 550 µm at lowest power of the heating beam P h ≅ 1 mW the thermocapillary dimple decays starting from \(k_{\text{s}}^{*} = 1.4\;{\text{W/(m}}\;{\text{K}})\) (see Fig. 5). In the case of h l =  100 µm the zero-signal is detected on duralumin substrate with \(k_{\text{s}}^{*} = 160\;{\text{W/(m}}\;{\text{K}})\) at P h =  21 mW.

Fig. 3
figure 3

The diameter of the PTC signal as a function of the thermal conductivity of substrates for different thicknesses of the liquid layer: 550, 760 and 960 µm at the power of the heating beam 21 mW. The power exponents of fitting lines are shown in the legend

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Fig. 4
figure 4

The diameter of the PTC signal as a function of the thermal conductivity of the metallic substrates for thicknesses of the liquid layer 100 µm at the powers of the heating beam 17, 18 and 21 mW. The power exponents of fitting lines are shown in the legend

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Fig. 5
figure 5

The dependency of the diameter of the PTC signal on the power of the heating laser beam for substrates with different thermal conductivities: ebonite—0.13 W/(m K), Teflon—0.25 W/(m K), KRS-5—0.32 W/(m K) and quartz glass—1.4 W/(m K)

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It should be noted that there is another limit case, when the diameter of the PTC signal increases infinitely until disappearing, so no signal is observed on the screen in the end. This corresponds to the thermocapillary rupture of the liquid layer, when the solid substrate occurs to be free from a liquid in the irradiated area and, therefore, diffusely reflects the laser beam. In the current experiments, this situation was observed for h l =  100 µm at P h = 17, …, 21 mW on substrates with k s =  0.13, …, 6.5 W/(m K).

Obviously, the reason for both the damping of the thermocapillary convection in the liquid layer and the thermocapillary rupture of the layer is the domination of one of the heat transfer mechanisms: the conductive heat transfer into the solid substrate or the convective heat transfer into the liquid. An estimation of the Peclet number Pe = vdρC p/k l for silicon oil with a typical velocity of the thermocapillary flow v ~ 1 cm/s and the width of the heated area equal to the beam diameter d = 2.5 mm, gives the value Pe ≈ 300. The latter means that the heat flow into the liquid layer is caused by the convective heat exchange, whereas the heat flow into the solid substrates occurs due to the conductive heat transfer. In the range of substantial changing of D st for both types of substrates, the convective heat transfer dominates over the conductive one and the thermocapillary convection in the layer is pronounced. When the thermal conductivity of substrate becomes high enough the conductive heat transfer starts to dominate decreasing the temperature of the heated area. As a result, temperature of the free surface of the liquid layer decreases; hence, the radial gradient of temperature along the free surface (see Eq. 4) becomes insufficient to generate the thermocapillary stresses required for the layer deformation.

Note that the power of the heating laser beam influences the sensitivity of the laser-induced thermocapillary effect to the thermal conductivity of substrates. In Fig. 5 the D st as a function of P h for the LHC substrates (such as ebonite, Teflon, KRS-5 and quartz glass) is presented for h l ≅  550 µm. As is seen in Fig. 5 the D st linearly increases with the power of laser beam, D st ∼ P h for all substrates used. It is interesting to note that according to results [10, 33] the depth of the thermocapillary dimple also linearly increases with the power of the laser beam (a water layer) [10] and with temperature of an electrical heater embedded in a substrate (silicon oil on ebonite) [33]. As it follows from Fig. 5 the slope of the dependency D st on P h decreases with the increase in the thermal conductivity of substrate. This suggests that the dissipation of power into a substrate increases that leads to lowering the temperature gradient at the free liquid surface required for development of the thermocapillary dimple.

As mentioned above, the PTC signal is influenced by hydrodynamical factor as well. In Fig. 3 it is clearly seen that D st for a given substrate decreases with the increasing of h l. Those results are quite expectative and are in accordance with the previous results [6]. Obviously when increasing the thickness of the layer, a resistance to the reverse bottom flow, which can be considered as flow with a parabolic velocity profile in a slit-like channel of 2h l/3 thickness, decreases. Therefore, the depth of the thermocapillary dimple becomes smaller and, consequently, the smaller D st of the PTC signal. The sensitivity of the thermocapillary effect to h l (determined as ratio \(\Delta D_{\text{st}} /\Delta h_{\text{l}}\), where ∆h l = 410 µm) as a function of k s for the LHC substrates at the constant P h is presented in Fig. 6. The values of \(\Delta D_{\text{st}} /\Delta h_{\text{l}}\) are negative because an increase in h l causes a decrease in D st. The absolute value of \(\Delta D_{\text{st}} /\Delta h_{\text{l}}\) decreases non-monotonically with the increasing k s. In the range of \(k_{\text{s}}^{*} > 3\;{\text{W/(m}}\;{\text{K}})\) the increment of the thickness has a negligible effect on D st. That is the influence of the thermal conductivity of substrates on the thermocapillary effect becomes prevalent in comparison with that of the layer thickness.

Fig. 6
figure 6

The sensitivity of the thermocapillary effect to the variation of the thickness of the layer as a function of the thermal conductivity of substrates at constant P h = 21 mW. The increment of the thickness of the liquid layer equals to ∆h l = 410 µm

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Interestingly that revealed in our experiments the reverse dependency \(D_{\text{st}} \sim 1/k_{\text{s}}^{n}\) correlates with those determined in a number of photothermal methods [34,35,36]. We suggest that the sensitivity of the PTC signal to the thermal conductivity of substrates may be used as a basis for a new photothermocapillary method for the thermal conductivity measurements of solid samples.

5 Conclusions

The influence of the thermal conductivity of solid substrates on the thermocapillary effect induced by the laser beam in a thin liquid layer deposited on those substrates was studied experimentally. For quantitative analysis, the diameter PTC signal–the circular interference pattern formed on a screen by the probe laser beam reflected from the free surface of the liquid layer deformed by the thermocapillary flows was plotted as a function of the thermal conductivity. It was revealed that D st of the PTC signal varies with k s as 1∕k ns in a wide range of k s values, where n is the rate of the changing D st, which depends on the thickness of liquid layer and the power of the heating beam. Increasing the k s of substrates leads the decreasing thermocapillary deformation of the layer, and consequently the decreasing D st, due rising the conductive losses of heat in substrate material. At some critical value of the thermal conductivity of substrate, the latter works as a heat sink lowering the thermal disturbance on the free liquid surface; as a result the thermocapillary deformation of the liquid layer becomes fully damped. A variation of D st for a given substrate with the power of the laser beam is similar to the variation of the depth of the thermocapillary dimple with temperature of the heater embedded flush with the substrate [33]. The sensitivity to the thickness of the liquid layer decreases non-monotonically with the k s increasing at the constant power of the heating beam. The findings obtained above clearly demonstrate that thermal conductivity of substrate is very crucial parameter affecting the laser-induced thermocapillary effect and has to be taken into account when planning the experiments or developing methods for laser diagnostics of material properties.