Towards photophoresis with the generalized Lorenz-Mie theory

Highlights

• We introduce optical forces from thermal origins with the aid of the generalized Lorenz-Mie theory (GLMT)

• It is, to the best of the authors’ knowledge, the first formal analysis to describe photophoresis with the GLMT stricto sensu

• Analytical formulas are presented for the cartesian components of the asymmetry vector, which is proportional to photophoretic forces

• Results are valid both in the slip-flow and in the continuous regime and for any arbitrary shaped beam

Abstract

Based on the adjoint boundary value problem proposed decades ago by Zulehner and Rohatschek [1], analytic and closed-form expressions for the photophoretic forces exerted by arbitrary-shaped beams on homogeneous and low-loss spherical particles is derived in both the free molecular and slip flow regimes. To do so, the asymmetry vector for arbitrary refractive index particles is explicitly calculated by expanding the internal electromagnetic fields with the aid of the generalized Lorenz-Mie theory (GLMT). The approach here proposed is, to the best of the authors’ knowledge, the first systematic attempt to incorporate the GLMT stricto sensu into the field of photophoresis and might as well be extended, e.g. to spheroids and find important applications, among others, in optical trapping and manipulation of microparticles, in geoengineering, particle levitation, optical trap displays and so on.

Introduction

The determination of radiometric or photophoretic forces (${\mathbf{F}}_{\text{ph}}$) is not always an easy task. Because of that, the scientific community in the area of photophoresis suffers from the lack of an analytical theory capable of predicting such forces for light beams with arbitrary electromagnetic field profiles. In fact, the ’standard’ solution procedure involves dealing simultaneously with the heat conduction equation and the Navier-Stokes equation with appropriate boundary conditions on the particle surface. Such conditions are based on physical grounds and dependent upon the Knudsen number $Kn=\ell /a$, where $\ell$ is the mean free path of molecules in the host fluid and $a$ the radius of the illuminated particle.

After solving this set of equations for quantities such as temperature distribution in the particle and in the fluid, gas pressure and velocity fields, and photophoretic velocities, photophoretic forces are then evaluated from the stress tensor.

For plane waves and spherical particles with low losses, analytical solutions exist for what is known as the asymmetry factor $J_1$ (and, consequently, for ${\mathbf{F}}_{\text{ph}}$, since they are proportional to each other), in the slip-flow and free molecular regimes. The formalism involves expansions of the electromagnetic fields internal to the particle using the Mie theory [2]. Qualitatively, however, it is known in advance that the resulting photophoretic forces will point either parallel (positive photophoresis) or anti-parallel (negative photophoresis) to the Poynting vector. For light beams with arbitrary spatial field profiles, we find most of the times attempts to approximate or use numerical methods [3], [4], [5], [6].

Boundary conditions depend upon the Knudsen number $Kn$. For $Kn>>1$ (free molecular regime), the particle is much smaller than the mean free path $\ell$ of the gas and kinetic theory of gases applies. In 1967, using this theory, Hidy and Brock found an expression for the photophoretic force in this regime by assuming a solid, non-volatile and non-radiative homogeneous sphere [7]. Such an analysis was further improved by Tong in 1973, who introduced the additional effect of radiation from the surface of a black body caused by heating [8], and by subsequent works [2], [9], [10], [11], [12]. For $Kn<1$ or $Kn<<1$ (slip-flow or continuous regime, respectively), the particle is larger or much larger than $\ell$ and the mechanical transport of the particle is given in terms of a continuous medium approach with appropriate slip-flow boundary conditions, the photophoretic force being then a direct consequence of thermal creep [13], [14]. In 1928, Hettner presented the first expressions for ${\mathbf{F}}_{\text{ph}}$ in the continuous regime, assuming solid and non-volatile homogeneous spheres [15]. Also, a few decades after Rosen and Orr proposed an order of magnitude estimation for ${\mathbf{F}}_{\text{ph}}$ [16] based on specific expressions for the temperature gradient at the surface of the particle previously deduced by Rubinowicz [17] and relying upon spheres illuminated only over a single hemisphere ($z<0$). In a notorious work, Yalamov, Kutukov and Shchukin carried out a systematic study of the theory of photophoretic movement for volatile aerosols, considering the pressure on the particle surface caused by the asymmetric evaporation of the substance from the sphere [18]. Another interesting work was also published by Reed in almost the same period [19], who theoretically investigated photophoretic forces in the low Knudsen number regime for opaque particles, comparing his theoretical predictions with the most recent experimental results so far available [8], [20]. The dependence of the photophoretic force as a function of the size parameter was analyzed numerically by Arnold and Lewittes [21] and analytically by Mackowski [2] with the aid of the Mie theory for expressing the internal electric and magnetic fields in terms of partial waves. Studies involving photophoretic forces in the intermediate region $Kn<1$ and how two extreme cases $Kn>>1$ and $Kn<<1$ link to each other were initially carried out by Reed [19] and Mackowski [2]. In all previous works, as well as in the majority of publications to date, theoretical analysis has been restricted to uniform plane wave illumination (see, for instance, Refs. [14], [22], [23] for the period before 2013, to be complemented by Refs. [24], [25], [26], [27], [28] and references therein.). Very recently, photophoretic longitudinal and transverse asymmetry factors for dielectric and magnetodieletric cylinders and aggregates, including reflection from planar boudaries and corner spaces, have been investigated by Mitri, including incidence by waves and light-sheets with arbitrary polarization and incidence angle [29], [30], [31], [32].

The inclusion of arbitrary-shaped beams in photophoresis problems with homogeneous spheres will certainly lead us to work within the formalism of the generalized Lorenz-Mie theory (GLMT) [33]. In the GLMT strictu senso, the incident, scattered and internal fields are expanded over a set of orthogonal spherical wave functions, the coefficients of such expansions - the beam shape coefficients (BSCs) - carrying all the information regarding the spatial field distribution of the incident wave. Because any solution to Maxwell’s equations can be described within this context, we expect that any general theory on photophoresis for light-scattering by arbitrary-shaped beams and homogeneous spheres must inevitably incorporate GLMT into its mathematical foundations. In this path, Ambrosio has recently been able to extend the analysis beyond plane waves and dielectric particles, first by introducing arbitrary refractive index spheres in the case of plane wave illumination [34] and then by considering photophoretic forces exerted by on-axis axisymmetric beams [35], subsequently extended to higher-order Bessel beams by Wang et al. [36].

As stated by Fuchs [37] (also quoted in Ref. [14]), “The main difficulty in calculating the radiometric force on a particle is the determination of the temperature gradient in the particle itself.”. Lamb’s general solution, usually applied for plane wave illumination, might not be of much help beyond it [38], [39].

This paper deals with analytic solutions to the photophoretic forces in both slip-flow and free molecular regimes with the aid of the GLMT. It incorporates into the theory of photophoresis, for the first time in the literature to the best of the author’s knowledge, shaped beams beyond plane waves and arbitrarily located with respect to an opaque, non-radiative, non-volatile spherical scatterer. To do so, the method of the Adjoint Boundary Value Problem (ABVP) to the heat conduction equation proposed a few decades ago by Zulehner and Rohatschek [1] is here invoked in order to resolve for a vector generalization of $J_1$ called the asymmetry vector ${\mathbf{r}}_{\text{as}}$, thus allowing us to solve for the photophoretic forces without the need for explicitly finding the temperature distribution within the particle itself. Expressions for both longitudinal and transverse components of ${\mathbf{r}}_{\text{as}}$ exerted on arbitrary refractive index micro-spheres are then derived in terms of the BSCs, a feature which makes the present theory valid for any incident wave field in any optical regime (Rayleigh, Mie or geometric).

Section 2 presents a brief review on the method of calculation of ${\mathbf{F}}_{\text{ph}}$ for spherical particles and plane waves, including the main aspects of the ABVP to be adopted in the subsequent sections. Section 3 concerns the derivation of ${\mathbf{r}}_{\text{as}}$ for arbitrary beams with the aid of the GLMT, using the approach proposed by Zulehner and Rohatschek, for which ${\mathbf{F}}_{\text{ph}}\propto {\mathbf{r}}_{\text{as}}$. Here, both heat transfer from the particle and absorption of radiation within the fluid are neglected, and particles are restricted to non-volatile (solid) homogeneous spheres. Finally, conclusions are presented in Sec. 4.