Quantum size effects on thermodynamics of gases and the lateral forces

Abstract

Today, especially parallel to the progress in semiconductor technology, the developing nanotechnology makes the mechanical systems and structures in micro/nano scale possible. Mechanical structures in micro/nano scale such as gas turbines, pumps, mixers, heat exchangers, valves etc., bring the following topics into consideration: how vary thermodynamic properties of gases, how it can be modeled, how one can make use of these differences, how new devices and technologies can be developed in this scale. In this perspective, the subject of quantum size effects (QSE) on thermodynamics of gases at micro/nano scale is a new research area and it has many potential applications from the technologies of genetic, space and energy to the military ones. In nano scale, thermodynamic properties (TP) of gases differ from those in macro scales. One of the reasons of this difference is the QSE, which become important when the thermal de Broglie wavelength of particles ($lambda_T$) is not negligible in comparison with the characteristic length of the system (L=V/A, V volume, A surface area). In such a case, the continuum approximation for the energy eigenvalues of particles becomes invalid and the discrete nature of energy eigenvalues causes QSE, which are noticeable in nano scale. QSE make the thermodynamic state functions depend on geometry (shape) and size of the system, and QSE causes some new and interesting behaviors, which are not observed in macro scale. Some of them are anisotropic gas pressure, gas diffusion due to size and geometry differences, thermosize effects like thermoelectric effects and the disappearance of the additivity property of extensive quantities. In literature, the dependence of TP of gases on the shape and size of the confinement domain have recently been examined for the global properties only. These studies consider only rectangular, cylindrical and spherical geometries with a single domain, for which the analytical solution of the Schrödinger equation (SE) is possible, and they are limited with the classical ideal gases obeying Maxwell- Boltzmann statistics. All TP of a system can be calculated when the partition function (PF) is known. PF consists of a summation over the energy eigenvalues of the stationary SE. For macro systems, it is a very good approximation to replace the summation by integration to calculate the PF. For micro/nano systems, however, it is necessary to use a more precise formula (such as Poisson, Euler-Maclaurin or Abel-Plana formula). The precise calculation of the PF is the key work in determining the QSE on TP of small systems. A complete analytical solution of the stationary SE is possible only for a rectangular domain. Even for spherical and cylindrical domains, some approximations should be made to obtain an analytical expression for the energy eigenvalues. For a domain of an arbitrary shape, it is impossible to solve the SE and find the eigenvalues. Fortunately, Weyl’s conjecture gives a precise formula for an asymptotic behavior of the eigenvalue spectrum of the stationary SE. Therefore, it is possible to determine the density of states of eigenvalue spectrum in an asymptotic form for a domain of an arbitrary shape. Consequently, the influence of the confinement geometry on the global TP of an ideal gas can be generalized by using the Weyl’s conjecture. In this study, global TP of an ideal Maxwellian gas confined in a domain of an arbitrary shape are derived by using Weyl’s conjecture for the density of states. It is seen that the results obtained by this generalization agree with the results for the domains in which the SE can be solved analytically. It is obtained that as a result of QSE; the global TP of gases confined in nano scale are considerably different than those in nano scale. As an example, an infinitely thin and movable wall which separates a rectangular box filled by an ideal gas into two parts is subjected to a repulsive lateral force due to quantum size effects. When the local gas density distribution is examined, it is seen that there is a layer near to the boundaries of the domain and the gas density goes to zero within this layer due to non-local interactions of the particles with the boundaries. Lateral forces can be explained by considering this boundary layer.

Özet

Nano ölçekte elektronik yapıların yanı sıra mekanik yapıların da üretilebilir hale gelmesiyle birlikte bu ölçekte gazların termodinamik özelliklerinin ne şekilde değiştiği, nasıl modellenebileceği, bu değişimlerden nasıl yararlanılabileceği, olası yeni davranışlara dayalı yeni cihazların/teknolojilerin nasıl geliştirilebileceği gibi sorular da gündeme gelmeye başlamıştır. Nano ölçekte gazların termodinamik özellikleri makro ölçektekinden farklılık gösterir. Bu farklılığın nedenlerden biri olan Kuantum Ölçek Etkileri (KÖE); sistemin karakteristik boyutu (L=V/A, V: hacim, A: yüzey alanı) yanında, parçacıkların ısıl de Broglie dalga boyunun ($lambda_T$) ihmal edilemediği durumlarda önem kazanır. Böyle bir durumda, parçacıkların enerji değerlerine ilişkin süreklilik yaklaşımı da geçerliliğini yitirir ve kesiklilik, sistemin davranışlarında özellikle nano ölçekte belirgin hale gelen KÖE’nin ortaya çıkmasına yol açar. KÖE, termodinamik hal fonksiyonlarını sistemin geometri (şekil) ve ölçeğine bağımlı hale getirerek makro ölçekte karşılaşılmayan yeni ve ilginç davranışlara yol açar. Bunlar arasında anizotropik gaz basıncı, ölçek ve geometri farkından kaynaklanan gaz difüzyonu, termoelektrik etkilere benzer termoölçek etkilerinin ortaya çıkması ve kütleye bağımlı büyüklüklerin (extensive quantities) toplanabilirlik (additivity) özelliğinin ortadan kalkması sayılabilir. Bu çalışmada, Weyl varsayımından hareketle keyfi bir ortamda tutuklanmış Maxwell gazların bütünsel termodinamik özellikleri KÖE göz önüne alınarak türetilmiştir. Buradan hareketle makro ölçekte gözlenmeyen bir davranış olarak ideal bir gazda dahi yanal kuvvetler oluşabileceği tespit edilmiştir. Maxwellian bir gazın yerel yoğunluk dağılımı incelendiğinde, ortam sınırları civarında gözlenen bir sınır tabaka nedeniyle yoğunluğun termodinamik denge durumunda dahi homojen olmadığı ve bu sınır tabaka yardımıyla yanal kuvvetlerin de açıklanabileceği anlaşılmıştır.