Quantum size effects on gas transport

Abstract

Fabrication of micro and nano electromechanical systems have led to the possibility of small scales devices such as, sensors, actuators, biomedical devices, pumps, propulsion system and micro engines for power generation based on new physical effects. Therefore, it is important to understand the influence of new effects on transport properties of gases confined in small scale structures. In nano scale, quantum mechanical effects become important and the transport models based on the concepts of the classical mechanics must be modified by considering the quantum mechanical concepts. In nano-scale, even at thermodynamic equilibrium, density distribution is not uniform and there is a boundary layer near to the boundaries where the density goes to zero. This layer is called quantum boundary layer since the thickness of the layer is proportional to the Planck’s constant. Due to quantum boundary layer, gas particles fill an effective volume which is less than the geometric one. Due to wave characteristics of particles, there is a nonlocal repulsive interaction between boundaries and particles. Therefore particles tend to accumulate in the inner part of the domain and it causes a higher local density than the classical one for the interior regions. Thickness of this boundary layer is in the order of thermal de Broglie wavelength of the particles and it defines a characteristic length scale. Therefore, quantum size effects appear when the size of the domain becomes comparable with the thickness of the boundary layer. Due to non-local interaction between boundaries and particles, particles feel the boundaries when the distance is in the order of thermal de Broglie wave length. Thus, the potential acting on the wave-like particles is different than the potential acting on the particle-like ones. The true potential can be represented by adding an effective quantum potential to the classical one. In an effective potential approach, one replaces the quantum distribution function by a classical distribution function with a modified potential. Thus, all the quantum effects on local density are modeled through the effective quantum potential. Transport behavior of gases confined in nano scale is considerably different than that in macro scale, due to different kinds of size effects. Classic scaling approaches generally assumes that physical constants and material properties remain independent of the scale. However, this assumption breaks down when the wave character of gas particles is considered or the length scale of the system approaches to the characteristic length scale of the mechanism that controls the property of interest. When the dimension of the system becomes very small in one direction, then the component of the wave vector of the particle in this direction becomes strongly quantized. Consequently, in nano scale,it is important to examine the quantum size effects on transport coefficients In this study, conductivity, diffusion and thermal conductivity coefficients of a Maxwellian gas are derived by considering the quantum size effects and the Boltzmann transport equation under the relaxation time approximation for 3D rectangular geometry. The particle-particle and particle-boundary collisions based transport processes are examined individually. In the case of particle-boundary collisions dominated transport regime, it is shown that quantum size effects are stronger in comparison with those in particle-particle collisions dominated one. It is seen that quantum potential gradient can be a driving force for the transport processes. In a domain filled by a Maxwellian gas under a temperature gradient, the temperature dependence of quantum potential causes quantum potential gradient as a secondary driving force and these driving forces may cause a new type of convection which can be called quantum convection. Consequently, it is shown that conductivity; diffusion and thermal conductivity coefficients depend on size and geometry of the transport domain due to quantum size effects. Therefore, shape and size of the domain become additional control parameters on the transport processes. Because of these control parameters, new transport processes can be defined and new micro and nano engines can be designed by using the quantum size effects. Furthermore, it is shown that the universal Wiedemann-Franz law, which is the ratio of the thermal conductivity to the electrical conductivity, loses its universality in nano scale due to quantum size effects.

Özet

Yarıiletken teknolojilerinde kullanılan mikro fabrikasyon tekniklerinin gelişimi, yeni fiziksel etkilerin gözlendiği sensörler, tıbbi cihazlar, güç üreten mikro makineler, ısı değiştiricileri, pompalar, valfler, mikro-nano pipetler ve mikro iticilerin üretilmesini de günümüzde mümkün kılmıştır. Nano sistemlerde termodinamik denge durumunda dahi yoğunluk dağılımı homojen olmamakta ve sınırlarda yoğunluğun azalarak sıfır değerine ulaştığı bir sınır tabaka oluşmaktadır. Sınır tabaka kalınlığı Planck sabiti ile orantılı olduğundan kuantum sınır tabakası olarak adlandırılır. Kuantum sınır tabakası nedeniyle gaz parçacıkları geometrik hacimden daha küçük olan etkin bir hacmi doldururlar. Kuantum sınır tabakasının kalınlığı ısıl de Broglie dalga boyu mertebesindedir ve sınırlar ile parçacıklar arasındaki uzaktan etkileşme sebebiyle parçacıklar sınırları ısıl de Broglie dalga boyu mesafesinde hissederler. Bundan dolayı, parçacıkların dalga karakterleri nedeniyle etkilendikleri potansiyel, tanecik karakterine sahip parçacıkların etkilendiği klasik potansiyelden farklıdır. Bu nedenle potansiyel, klasik potansiyele etkin kuantum potansiyel olarak adlandırılan bir başka potansiyelin eklenmesiyle temsil edilir. Bu çalışmada; dikdörtgen geometriler için relaksasyon zamanı yaklaşımı altında Maxwellian gazların transport katsayıları kuantum ölçek etkisi altında incelenmiştir. Parçacık-parçacık ve parçacık-duvar çarpışmaları sonucu meydana gelen transport süreçler ayrı ayrı incelenerek iletim, difüzyon ve termal iletkenlik katsayıları 3 boyutlu dikdörtgen geometriler için çıkarılmıştır. Transport katsayıların, kuantum ölçek etkileri nedeniyle, transport ortamının ölçek ve geometrisine bağlı olduğu görülmüştür. Ayrıca, evrensel bir yasa olan Wiedemann-Franz yasasının nano ölçekte kuantum ölçek etkileri nedeniyle evrenselliğini kaybettiği bulunmuştur.