Casimir Forces Between Nanoparticles and Substrates
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CASIMIR FORCES BETWEEN NANOPARTICLES AND SUBSTRATES C. E. Rom´an-Vel´azquez1 , Cecilia Noguez2 , C. Villarreal2 and R. Esquivel-Sirvent2 1 Centro de Investigaci´on en Ciencia Aplicada y Tecnolog´ıa Avanzada, Instituto Polit´ecnico Nacional, Av. Legaria 694, Col. Irrigaci´ on, D.F. 11500, M´exico 2 Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´exico, Apartado Postal 20-364, Distrito Federal 01000, M´exico ABSTRACT We study the Casimir force between a nanoparticle and a substrate using a spectral representation formalism. We consider the interaction of metal nanoparticles with different substrates within the dipolar approximation. The force is calculated as a function of the distance between the particles and the substrate. The particles are made of gold or potassium spheres, and the substrate is titanium dioxide, sapphire or a perfect conductor. INTRODUCTION Recent advances in micro and nano devices have opened the possibility of studying quantum phenomena that occurs at these length scales. Such is the case of Casimir forces [1] predicted by the theory of quantum electrodynamics. The textbook example [2] consist of two parallel neutral conducting plates separated by a fixed distance. The plates will attract each other with a force per unit area of roughly one atmosphere when the plates are 35 nm apart. This force has been measured accurately in different ways. However, a truly parallel plate configuration has been measured only by Bressi et al. [3]. The difficulty of keeping the two plates parallel at separations of few nanometers makes it easier to measure the Casimir force between a large conducting sphere and a plane using microtorsional balances [4] or atomic force microscopes [5,6]. In this cases, comparison with the theoretical results obtained for the parallel plates is done using the proximity theorem [7]. The proximity theorem is a geometrical approximation that states that if the free energy per unit area E at a given distance between two parallel plates is known, the force between a sphere and a plane is 2πER, where R is the radius of the sphere. This approximation is valid provided the minimum separation between the sphere and the plane is much smaller than R. The theorem does not quantify or gives bounds for the ratio between R and the separation with the plane. Thus, the question improving the theoretical description of the proximity theorem is important from the theoretical and experimental point of view. In this work we present a calculation of the Casimir force between a sphere and a conducting plane using a spectral representation approach [8]. The Casimir force is calculated as a function of z, the minimum separation between the sphere and the plane. The force is studied as a function of the sphere’s radius, and the dielectric functions of the sphere and the substrate. THEORY Consider a homogeneous sphere of radius R, electrically neutral and with a local dielectric function s (ω). The sphere is suspended over a substrate which is also electrically neutral
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