Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor


The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

$$ P_{ZZ} = kT\rho \left( {Z_{1} } \right) + \pi kT\rho \left( {Z_{1} } \right)\int\limits_{ - d}^{0} {\rho \left( {Z_{2} } \right)} Z_{2}^{2} g_{Z,H} (d){\text{d}}Z_{2} - \frac{1}{2}\iint {\int\limits_{0}^{2\pi } {\phi^{\prime } \left( {\vec{r}_{2} } \right)\rho \left( {Z_{1} } \right)\rho \left( {Z_{2} } \right)g_{Z,H} (r_{2} )} }{\frac{{Z_{2}^{2} }}{{(R_{2}^{2} + Z_{2}^{2} )^{{\frac{1}{2}}} }}}R_{2} {\text{d}}R_{2} {\text{d}}Z_{2} {\text{d}}\Uptheta ;\quad \left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{2} } \right| > d $$

In which RΘZ are the cylindrical coordinate’s notations and \( \left| {\vec{r}_{2} } \right| = \sqrt {R_{2}^{2} + Z_{2}^{2} }\). Assuming the truncated-Lennard-Jones potential, the third term represents the attractive intermolecular potential energy contribution to the normal pressure tensor. We report solution of this equation for the truncated-Lennard-Jones confined fluid in nanoslit pores, and we demonstrate the role of attractive potential energy by comparing the results of Lennard-Jones and hard-sphere fluids. Our numerical calculations show that the normal pressure tensor has an oscillatory form versus distance from the walls for all confined fluids. The oscillations increase with reduced bulk density and decrease with fluid–fluid attraction. It also becomes broad and smooth with pore width at constant temperature and density. In comparison with hard-sphere confined fluids, the values of the normal pressure for LJ fluids at all distances from the walls are less than the hard-sphere fluids. This analytic pressure tensor equation is a useful tool to understand the role of attractive and repulsive forces in the normal pressure tensor and to predict phase behavior of nanoconfined fluids.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Barker JA, Henderson D (1976) What is “liquid”? Understanding the states of matter. Rev Mod Phys 48(4):587–671

    MathSciNet  Article  Google Scholar 

  2. Dong W, Chen XS, Zheng WM (2005) Thermodynamic pressure of a fluid confined in a random porous medium. Phys Rev C 72(1):012201

    Google Scholar 

  3. Eijkel JC, Berg AV (2006) Nanofluidics: what is it and what can we expect from it? Microfluid Nanofluidics 2(1):12–20

    Article  Google Scholar 

  4. Heidari F, Keshavarzi T and Mansoori GA (2010) DFT simulation of the behavior of Lennard-Jones fluid in nanoslit pore (to be published)

  5. Kalikmanov VI (2001) Statistical physics of fluid: basic concepts and applications. Springer, Berlin

    Google Scholar 

  6. Kamalvand M, Keshavarzi E, Mansoori GA (2008) Behavior of the confined hard-sphere fluid within nanoslits: a fundamental-measure density functional theory study. J Nanosci 7(4&5):245–253

    Google Scholar 

  7. Kang HS, Lee CS, Ree T, Ree FH (1985) A perturbation theory of classical equilibrium fluids. J Chem Phys 82:414. doi:10.1063/1.448762

    Google Scholar 

  8. Keshavarzi T, Sohrabi R, Mansoori GA (2006) An analytic model for nano confined fluids phase-transition: applications for confined fluids in nanotube and nanoslit. J Comput Theor Nanosci 3(1):134–141

    Google Scholar 

  9. Keshavarzi T, Sedaghat F, Mansoori GA (2010) Behavior of confined fluids in nanoslit pores: the normal pressure tensor. Microfluid Nanofluidics 8:97–104

    Article  Google Scholar 

  10. Lan SS, Mansoori GA (1975) Perturbation equation of state of pure fluids. Int J Eng Sci 14:307–317

    Article  Google Scholar 

  11. Mansoori GA (2005) Principles of nanotechnology: molecular-based study of condensed matter in small systems. World Science, Publication Co., Hackensack

    Google Scholar 

  12. Mansoori GA, Provine JA, Canfield FB (1969) Note on the perturbation equation of state of Barker and Henderson. J Chem Phys 41(12):5295–5299

    Article  Google Scholar 

  13. Yu W, Wu J (2002) Structures of hard-sphere fluids from a modified fundamental-measure theory. J Chem Phys 117(22):10156–10164

    Article  Google Scholar 

  14. Zhang GP et al (2007) First-principles simulation of the interaction between adamantane and an atomic-force-microscope tip. Phys Rev B 75:035413

    Article  Google Scholar 

  15. Zhou S (2000) Inhomogeneous mixture system: a density functional formalism based on the universality of the free energy density functional. J Chem Phys 113(19):8718–8723

    Google Scholar 

  16. Zhou S (2006) Thermodynamic perturbation theory in fluid statistical mechanics. Phys Rev E 74(3):031119

    Article  Google Scholar 

  17. Ziarani AS, Mohamad AA (2006) A molecular dynamics study of perturbed Poiseuille flow in a nanochannel. Microfluid Nanofluidics 2(1):12–20

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to G. A. Mansoori.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Heidari, F., Keshavarzi, T. & Mansoori, G.A. Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor. Microfluid Nanofluid 10, 899–906 (2011).

Download citation


  • Attractive intermolecular interaction
  • Density functional theory
  • Hard-Sphere fluid
  • Inhomogeneous fluid
  • Lennard-Jones fluid
  • Nanoslit pore
  • Nanoconfined fluid
  • Normal pressure tensor
  • Perturbation theory
  • Repulsive intermolecular interaction