Self-thermophoresis of laser-heated spherical Janus particles

Abstract

An analytic framework is presented for calculating the self-induced thermophoretic velocity of a laser-heated Janus metamaterial micro-particle, consisting of two conducting hemispheres of different thermal and electric conductivities. The spherical Janus is embedded in a quiescent fluid of infinite expanse and is exposed to a continuous light irradiation by a defocused laser beam. The analysis is carried under the electrostatic (Rayleigh) approximation (radius small compared to wavelength). The linear scheme for evaluating the temperature field in the three phases is based on employing a Fourier-Legendre approach, which renders rather simple semi-analytic expressions in terms of the relevant physical parameters of the titled symmetry-breaking problem. In addition to an explicit solution for the self-thermophoretic mobility of the heated Janus, we also provide analytic expressions for the slip-induced Joule heating streamlines and vorticity field in the surrounding fluid, for a non-uniform (surface dependent) Soret coefficient. For a 'symmetric' (homogeneous) spherical particle, the surface temperature gradient vanishes and thus there is no self-induced thermophoretic velocity field. The 'inner' temperature field in this case reduces to the well-known solution for a laser-heated spherical conducting colloid. In the case of a constant Soret phoretic mobility, the analysis is compared against numerical simulations, based on a tailored collocation method for some selected values of the physical parameters. Also presented are some typical temperature field contours and heat flux vectors prevailing in the two-phase Janus as well as light-induced velocity and vorticity fields in the ambient solute and a new practical estimate for the self-propelling velocity.

Fig. 1

Schematic description of the Janus particle (JP) problem, where r is the spherical radius, the JP radius is a and $\theta$ is the spherical angle

Fig. 2

The variation of the temperature normalized by $T_{\mathit{ref}}=q_2{a}^2{k}_2/\left(3,k_o\right)$ along the particle’s (a) axis of symmetry and (b) the boundary axis between the two Janus particle’s halves, where $q_1=0$, $k_{\mathrm{O}}=0.6$ W/(m K) as of water and $k_2=$ 8 W/(m K) as of Beryllium copper

Fig. 3

Contours of the heat flux modulus normalized by $q_2{\mathit{ak}}_2/3$ and which are plotted for the Janus particles of Fig. 2

Fig. 4

The normalized heat flux that is plotted along the particle’s axis of symmetry x. The rest of the conditions are as in Fig. 3

Fig. 5

Heat flux vectors that are plotted for the cases of Fig. 3, where the vector modulus of each field was scaled for optimal presentation and each vector is located at its tail

Fig. 6

Contours of the stream function normalized by $D_{\mathrm{T}}{T}_{\mathrm{ref}}/a^3$, and which are plotted for the Janus particles of Fig. 5. $T_{\mathrm{ref}}$ is also defined in Fig. 2

Fig. 7

Velocity vectors that are plotted for the cases of Fig. 6, where the vector modulus of each flow field was scaled for optimal presentation and each vector is located at its tail. The velocity field is described in moving coordinates attached to the particle