Role of Interfacial Entropy in the Particle-Size Dependence of Thermophoretic Mobility A. Arango-Restrepo *,† and J. M. Rubi ‡ Departament de Física de la Mat´ eria Condensada, Facultat de Física, Universitat de Barcelona, Avinguda Diagonal 647, 08028 Barcelona, Spain (Received 7 May 2019; accepted 5 July 2020; published 24 July 2020) We show that changes in the surface tension of a particle due to the presence of nonionic surfactants and impurities, which alter the interfacial entropy, have an impact on the value of the thermophoretic mobility. We have found the existence of different behaviors of this quantity in terms of particle size which can be summarized through a power law. For particles that are small enough, the thermophoretic mobility is a constant, whereas for larger particles it is linear in the particle radius. These results show the important role of the interfacial entropic effects on the behavior of the thermophoretic mobility. DOI: 10.1103/PhysRevLett.125.045901 Particles movement induced by temperature gradients [1] known as the Soret effect has been the subject of many experimental and theoretical studies in recent years [2–5] due to its importance in areas as diverse as soft condensed matter, biophysics, microgravity, and nanoscience. Thermophoresis may, for example, be used for the control of colloids and macromolecules [6] and to implement effective particle separation methods [7] and focusing techniques [8]. Studies on thermophoresis have also shown their importance in the deposition of micro- and nanoparticles [9] in laminar [10] and turbulent [11] pipe flows, removing and collecting aerosol particles [12] and in biotechnological applications [13,14]. Crucial to the study of the motion of the particles is the knowledge of the thermophoretic mobility D T , the propor- tionality factor between the thermophoretic velocity ⃗ v T and the temperature gradient ∇T : ⃗ v T ¼ −D T ∇T , and whether it depends on the particle size or not. There is no general consensus on this question [15]. Experiments performed with polystyrene solid particles of sizes between 40 nm and 2 μm in Tris buffer solution, show a linear dependence of the thermophoretic mobility on the particle radius a [16] whereas others made with n-alkane liquid particles in water or surfactant ranging from 5 to 16 nm support the fact that the mobility does not depend on the particle radius [17,18]. An experiment carried out with latex particles in a solution containing tetrabutylammonium perchlorate with particle radius ranging from 100 to 400 nm [19], supported by simulation results of rigid particles with radius in the interval 36 to 154 nm [20] shows a decreasing behavior of the thermophoretic mobility as a function of the particle radius. These different behaviors of D T may be due to the fact that more than just a single driving force determines the thermophoretic force in experimental systems, each with a different size dependence. The thermophoretic force may result from the temperature response of the core material of the particles relative to that of the solvent, the possible presence of electrical double layers, and from the distri- bution of surfactant and fluid molecules at the interface which affects the interfacial entropy. To analyze the origin and the role of the interfacial entropic effects in thermo- phoresis is the main objective of this Letter. The thermophoretic velocity can be obtained from hydrodynamics by computing the force exerted by the solvent on a particle moving with a given velocity in the presence of a temperature gradient [21–24]. The force contains a thermophoretic contribution proportional to the temperature gradient [25]. A general expression valid when the particle is a drop, a bubble, or a solid particle with a monolayer of adsorbed solvent [26–28] was given in Ref. [23]: ⃗ F ¼ −μ −1 ⃗ u þ D T μ −1 ∇T , where μ is the mobil- ity. Under the hydrodynamic approach, D T is proportional to the derivative of the surface tension with respect to the temperature γ T ≡ dγ =dT and to the particle radius (a) and is a function of the viscosities and thermal conductivities of the inner and outer fluids. In our analysis, we show how the distribution of the nonionic solvent and surfactant molecules adsorbed at the interface may depend on the size of the particle. Since this distribution affects the interfacial entropy, it may bring about a dependence of the surface tension and, conse- quently, of the γ T factor on particle size. We thus find that for sufficiently large particles, the ratio between the numbers of fluid and surfactant molecules on the particle surface does not depend on the radius (a). Therefore, γ T does not depend on a and so the thermophoretic mobility is a linear function of a. On the contrary, for small particles that ratio may depend on the radius of the particle due to entropic effects which lead to a different behavior of γ T as a function of a. To show how the distribution of fluid and surfactant molecules at the interface affects the thermophoretic velocity, we will consider the stationary movement of a drop immersed in a fluid subjected to a temperature PHYSICAL REVIEW LETTERS 125, 045901 (2020) 0031-9007=20=125(4)=045901(6) 045901-1 © 2020 American Physical Society