1 A perspective on the microscopic pressure stress tensor history current understanding and future challenges

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A perspective on the microscopic pressure (stress) tensor: history,

current understanding, and future challenges

Kaihang Shi,1, ‡, † Edward Smith,2, † Erik E. Santiso,1 and Keith E. Gubbins1, †

1Department of Chemical and Biomolecular Engineering, North Carolina State University,

Raleigh, North Carolina, USA

2Department of Mechanical and Aerospace Engineering, Brunel University London, Uxbridge,

London, UK

‡Current address: Department of Chemical and Biological Engineering, Northwestern

University, Evanston, Illinois, USA

†Authors to whom the correspondence should be addressed: kaihangshi0@gmail.com;

edward.smith@brunel.ac.uk; keg@ncsu.edu.

Abstract

The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic

levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid

mechanics, biophysics, and thermodynamics. In this perspective paper, we review methods to

calculate the microscopic pressure tensor. Connections between different pressure forms for

equilibrium and non-equilibrium systems are established. We also point out several challenges in

the field, including the historical controversies over the definition of the microscopic pressure

tensor; the difficulties with many-body and long-range potentials; the insufficiency of software

and computational tools; and the lack of experimental routes to probe the pressure tensor at the

nanoscale. Possible future directions are suggested.

3 
 
Contents 
1. Introduction ................................................................................................................................... 4 
2. Fundamental equations for the pointwise pressure tensor .................................................................. 8 
3. Microscopic pressure tensor in equilibrium systems ....................................................................... 11 
3.1 Local pressure tensor in different geometries ............................................................................ 11 
3.2 Thermodynamic route to the pressure tensor and its equivalence to the mechanical route in 
thermodynamic equilibrium .......................................................................................................... 18 
4. Microscopic pressure tensor in non-equilibrium systems ................................................................ 22 
4.1 Ensemble average and the time evolving phase space ............................................................... 22 
4.2 A single trajectory in time ....................................................................................................... 23 
4.3 Streaming velocity and the kinetic term ................................................................................... 24 
4.4 Localization of the pressure tensor in space .............................................................................. 27 
4.5 A moving reference frame ....................................................................................................... 33 
4.6 Coordinate transforms ............................................................................................................ 35 
4.7 Statistical uncertainty of different pressure methods.................................................................. 37 
5. Challenges and future directions ................................................................................................... 38 
5.1 Controversies over the microscopic pressure tensor .................................................................. 38 
5.2 Complex systems interacting with many-body and long-range potentials .................................... 42 
5.3 Software and computational tools ............................................................................................ 45 
5.4 Experimental measurements of microscopic pressure tensor ...................................................... 47 
6. Concluding Remarks .................................................................................................................... 49 
Appendix ........................................................................................................................................ 52 
A1. Linking the IK-contour pressure to the MoP form .................................................................... 52 
A2. The Noll form of Pressure ...................................................................................................... 55 
Acknowledgment ............................................................................................................................ 56 
Conflict of Interest ........................................................................................................................... 56 
Data Availability ............................................................................................................................. 56 
Reference ........................................................................................................................................ 56 
 
  

1. Introduction

The pressure, , defined as the average force  per unit area acting on a surface of area ,

is one of the primary state variables, together with the temperature, , and composition, that

determine the thermodynamic properties of a homogeneous system of molecules at equilibrium.

In such a system the average force and the pressure are the same in all directions,  is a scalar, and

is well-defined even at the micro-scale. Statistical mechanics informs us that , where

superscript  indicates the contribution due to the kinetic energy of the molecules, and  indicates

the configurational contribution, i.e. that from the intermolecular forces and any external field. For

condensed phases the configurational contribution to  is usually dominant, especially at low

temperatures. For a perfect gas at equilibrium in the absence of an external field, or a real

equilibrium gas at low enough density that the influence of intermolecular forces can be neglected,

the configurational contribution is negligible, and , where  is the number

density at position  and  is the Boltzmann constant. For such a gas this relation holds even

when the gas is non-uniform in density or composition, as long as the concept of a local average

density, , is valid.

For more general situations, for example an inhomogeneous dense fluid, a nanoscale fluid

or solid, or a non-equilibrium system, the pressure  is a second-order tensor, depending on the

direction of both the force and of the surface it acts on. In general,  has 9 components , where

 is the force per unit area in the -direction acting on a surface element normal to the -

direction. These components depend on position, , and for non-equilibrium systems they will also

depend on time, . Off-diagonal components are the shear pressures (shear stress) and the diagonal

components are the direct pressures. In some specific types of systems, the number of non-

vanishing components of  may be less than 9. For an inhomogeneous fluid that is at equilibrium

and not under strain, for example, the off-diagonal components vanish and there are only 3 non-

vanishing components. Also, the condition of mechanical (hydrostatic) equilibrium (the average

rate of change of linear momentum vanishes) often provides relations between the remaining non-

vanishing components.

A difficulty in many applications is that the local (microscopic) pressure tensor at some

point  is not uniquely defined in non-equilibrium systems or in equilibrium ones that are

inhomogeneous. Although the kinetic contribution, , is well-defined, as noted above, the

configurational part,  is not, because the intermolecular forces themselves are non-local. Thus,

while the force between molecules  and , located at positions and , is well defined in general,

there is no well-defined way to assign a contribution to the force acting on a surface element at

some position . This arbitrariness in the force acting across the surface element  seems to have

first been stated explicitly by Irving and Kirkwood in 1950.1 There they stated the matter

succinctly:

“…all definitions (of the configurational pressure tensor) must have this in common – that the

stress between a pair of molecules be concentrated near the line of centers. When averaging over

a domain large compared with the range of intermolecular force, these differences are washed

out, and the ambiguity remaining in the macroscopic stress tensor is of negligible order.”

This point was discussed in a footnote to an appendix to Irving & Kirkwood’s paper, and so was

not noticed widely at the time. It was of little consequence to these authors, who were primarily

interested in transport processes at the macroscale. However, it is important for nanoscale systems,

such as small nanoparticles, drops and fluids confined within nanoporous materials or living cells,

and we expand on this later in this perspective. The non-uniqueness of the local pressure tensor,

and its consequences for inhomogeneous fluids and the properties of gas-liquid interfaces, was

investigated in the 1982 paper of Schofield and Henderson.2

While in this paper we shall mainly focus our discussion on the local pressure tensor, in

fields where the primary interest is in non-equilibrium phenomena and solid mechanics, the stress

tensor  is usually used in place of the pressure tensor.1 These two tensors differ only in

sign:1,3



(1)

The negative sign ensures that the stress tensor definition is consistent with Newton’s law of

viscous flow, and that the viscosity is positive. The terms “pressure tensor” and “stress tensor” are

used interchangeably in this paper and in much of the literature, the sign change being understood.

The molecular level local pressure/stress tensor has been the key to the depiction of the

mechanical and thermodynamic picture of many important phenomena. Its important role is

evidenced by a rapid growth of publications mentioning it (Figure 1). In biophysics, the local

stress tensor has been applied to understand the mechanical properties of lipid bilayer membranes

(see Figure 2a).4–7 The structure and mechanics of the lipid membrane play a critical role in the

function of proteins involved in processes of transport, signaling and mechano-transduction. The

local stress tensor also enables the quantification of the mechanical state of proteins in glassy

matrices.8 Such information is pivotal to a sophisticated design and control of the lyophilization

(freeze-drying) process for long-term storage and stabilization of labile biomolecules in the food

and pharmaceutical industries. In material science, the stress tensor has been related to the

structural deformation of the materials upon adsorption, and such a connection is useful for

materials characterization.9,10 For gas-liquid11–13 or liquid-solid14,15 nucleation, the pressure tensor

profile provides a mechanical picture of the nucleus interfacing with the surrounding environment;

such a profile is useful for calculating the Tolman length for interfacial free energy14 and for

understanding the distinct structure of the nucleus15 (see Figure 2b). The pressure tensor profile

across the interfacial region is also essential in a virial (or mechanical) route to the surface tension

(see examples for planar,3,16–19 spherical3,11,12,20 and cylindrical21 interfaces). For confined systems,

the knowledge of the microscopic pressure tensor paves the way for understanding phase

transitions in nanopores,22–24 and for developing sophisticated equations of state for confined

fluids.25,26 Recently, the microscopic pressure tensor has provided a mechanistic understanding of

high-pressure phenomena in confinement or near strongly wetting surfaces for advanced materials

synthesis and enhanced chemical processing.27 The high-pressure phenomena include enhanced

chemical reactions in pores that normally require a high pressure in the bulk,28,29 and the formation

and stabilization of high-pressure phases in nanopores.30–32 For simple non-reacting adsorption

systems, high (tangential) pressures that are about three to four orders of magnitude larger than the

bulk pressure were found in the adsorbed layers on carbon surfaces (see Figure 2c).33–35 This

compression effect is caused by strong attractive forces exerted on the adsorbate molecules by the

surface, which leads to a tightly packed adsorbed layer near the surface, and strong repulsions

between adsorbate molecules.36–39 For more complex systems, the mechanism behind the induced

high pressure in confinement is under active investigation.

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1Aperspectiveonthemicroscopicpressure(stress)tensor:history,currentunderstanding,andfuturechallengesKaihangShi,1,‡,†EdwardSmith,2,†ErikE.Santiso,1andKeithE.Gubbins1,†1DepartmentofChemicalandBiomolecularEngineering,NorthCarolinaStateUniversity,Raleigh,NorthCarolina,USA2DepartmentofMechanicalandAerosp...

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