Symmetry of the pressure tensor in macromolecular fluidsR. D. Olmsted and R. F. Snider Citation: The Journal of Chemical Physics 65, 3423 (1976); doi: 10.1063/1.433594 View online: http://dx.doi.org/10.1063/1.433594 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/65/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extended fluid models: Pressure tensor effects and equilibria Phys. Plasmas 20, 112112 (2013); 10.1063/1.4828981 Continuum percolation in macromolecular fluids J. Chem. Phys. 113, 9310 (2000); 10.1063/1.1319657 A generalization of the pressure equation for fluids to macromolecular systems J. Chem. Phys. 85, 2910 (1986); 10.1063/1.451051 The pressure tensor in nonuniform fluids J. Chem. Phys. 83, 3633 (1985); 10.1063/1.449170 Pressure tensor in lamellarly structured fluids J. Chem. Phys. 74, 6388 (1981); 10.1063/1.440977

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Symmetry of the pressure tensor in macromolecular fluids'" R. D. Olmsted and R. F. Snider

Department of Chemistry, University of British Columbia, Vancouver, Canada V6T 1 W5 (Received 16 June 1976)

For molecules possessing internal angular m<:>mentum, irreversible thermodynamics shows that the relaxation of the internal angular momentum occurs via the antisymmetric component of the pressure tensor. Yet it is common in fluid dynamics, even for fluids of orientable molecules, to assume that the pressure tensor is symmetric. An attempt to clarify this anomaly for solutions of macromolecules was made by Curtiss, Bird, and Hassager [Adv. Chern. Phys. 35, 31 (1976)]. They found in their kinetic theory treatment, that after making several approximations common in polymer theory, the pressure tensor is symmetric. The purpose of this paper is to show that within the usual question of uniqueness, their conclusion regarding the symmetry of the pressure tensor depends on only two things: (i) that the interatomic (between "beads") potential is spherical and (ii) that the fluid's densities (of mass and momentum) are localized at the bead points. If instead, the densities are localized at the polymer center of mass, there arises an inherently antisymmetric component of the pressure tensor. This localization is considered to be more in keeping with the molecular nature of such systems.

L INTRODUCTION

The fluid dynamical description of a system requires, among other things, the solution of the equation of mo­tion

a(Mu)/at = - V" (Muu) - V'. P (1)

for the stream velocity u and mass density M. This in turn requires an explicit expression for the momentum flux tensor P. This paper discusses formal expres­sions for the pressure tensor P based upon the exact von Neumann (quantum Liouville) equation governing the evolution of the system. These expressions are, by necessity nonunique, since it is V'. P that is uniquely defined, and not P itself. Thus P is determined only up to the curl of a tensor. Within this nonuniqueness, particular attention is paid to the symmetry properties of P, since this governs the number of independent phenomenological coefficients necessary for a descrip­tion of the fluid.

A recent comprehensive treatm·ent of the statistical mechanics of macromolecular solutions has been given by Curtiss, Bird, and Hassager1 (CBH). One of their fundamental goals was to determine the symmetry of the pressure tensor, albeit after the introduction of several approximations. They demonstrated that the form of P given by their Eq. (12.9)" is symmetric. This equation embodies the assumptions that (1) fric­tion coefficients are introduced, (2) all forces are of short range, (3) the solution is dilute in the concentra­tion of macromolecules, (4) there is homogeneous flow, (5) acceleration terms are omitted, and (6) there is equilibration in momentum space. Actually the sym­metry of P depends on none of these approximations but rests upon the nature of the potential between the atoms and upon the localization procedUre inherent in the def­initions of the local densities of observables. The CBH theory assumes that individual mass points (beads) in the molecules interact with one another through spher­ical, two body potentials. This pertains to intramolec­ular as well as intermolecular bead-bead interactions. The localization of observables (mass and momentum) is with respect to the centers of mass of the atoms as opposed to the centers of mass of the (macro)molecules

themselves. These conditions alone ensure that the resultant pressure tensor is symmetric.

An alternate view makes the same assumptions about the nature of the interatomic potentials but adopts a dif­ferent localization procedure. Conceptually, it is the velocity of the center of mass of a molecule that is an invariant of the free molecule motion, and thus (should?) be considered as an independent observable. Molecular observables should thus be localized with re­spect to the centers of mass of the molecules them­selves. Under these conditions, the naturally occur­ring pressure tensor has an antisymmetric component. 2

This component may vanish with the imposition of addi­tional assumptions or in particular fluid flows, but in general, it does not vanish. Of course, a transforma­tion can be made to symmetrize the pressure tensor, but this requires a different interpretation of the stream velocity, that is, the stream velocity is no longer the average velocity of the molecular centers of mass. 3

II. THE CBH FORMULATION

The basic terminology and nomenclature in this dis­cussion is that used by Curtiss, Bird, and Hassager1

(CBll), but the development is quantum mechanical rath­er than classical.

The position and momentum operators of the center of mass of the ith molecule of species 01 (referred to as "mOlecule OIi") are denoted by r Ol ; and POI;' Although the system may consist of several species 01, each molecule OIi is composed of atoms (mass pOints or beads) jJ., 1/, • ••• The symbol OIiv is used for the vth atom of molecule OIi. There are N atoms in the system and the N-atom density operator P<N) is normalized to 1,

(2)

The average (<I» of an N-atom observable <I> is given by

(<I» = Tr1 ... N <I>p(N) • (3)

The time evolution of the state of the system is gov­erned by the von Neumann (quantum Liouville) equation

(4)

The Journal of Chemical PhYSics, Vol. 65, No.9, 1 November 1976 Copyright © 1976 American Institute of Physics 3423

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3424 R. D. Olmsted and R. F. Snider: Pressure tensor in macromolecular fluids

where,c is the commutator superoperator of the total Hamiltonian H(N),

(5)

Assuming that the interaction between atoms is pair­wise additive, the total Hamiltonian is written as

(6)

The potential V", iv, 6i I' acts between atoms Cii v and (3 j Il,

and is assumed to depend only on the magnitude of the separation of the atoms;

(7)

Moreover, the atoms are assumed to be structureless so Ha;v is just the kinetic energy operator for atom Ciiv.

Two additional superoperators, ,c"'iv and 'COIiv,MIJ. are defined as

and

As a result, the total Liouville operator can be ex­pressed as

'\\' 1 '1\' .,c = L.., ,caiv + -2 L.., 'Caiv,Sit< •

aill o:iv 6iIJ.

(8)

(9)

(10)

Attention now is directed to the definitions of the local mass and momentum densities, and the calculation of the fluid equation of motion. According to the CBH the­ory, the mass density Ma of species Ci at point r at time t is defined by

(11)

where mav is the mass of atom Ciiv and r",;v is the posi­tion operator for the same atom. Note the localization that M '" measures the probability of finding an atom at point r. We emphasize that M", is the average density of those atoms which comprise molecules of the Cith species. Similarly, the stream velocity Uc> of species Ci is defined as

M",u", = L([ p",;vo(r",iv - r)].) iv

(12)

where P"'lv is the momentum operator for the vth atom of molecule Cii. Again this is described as the average local velocity of those atoms comprising molecules of species Ci. The total stream velocity u is then given by

MU=LM",u", = L ([p",ivo(r",iv - r)]s) • (13) 0: ad"

The subscript "s" in Eqs. (12) and (13) indicates the ap­propriately symmetrized operator,4 namely,

[p",ivo(r",iv - r)]s = t[p",ivo(r",iV - r) + o(r",;v - r)p",iv 1 . (14)

From the definition of the stream velocity, Eq. (13) and the von Neumann equation, Eq. (5), the evolution of the stream velocity is given by

ia(Mu)/at = Tr1 ... N L (P",ivo(raiv - rlls aiv

(15)

Evaluation of the commutators then gives

a(Mu)/at = - V' .(L;: (m",vr1[p"'ivPo<ivo(r"'iV - r)L) ""V

(16)

The total force F~':.i,6j), defined by CBH in Eq. (4.9) as

(17)

has been split into intramolecular contributions F~~i,Cd) and intermolecular contributions F~':.i,M); CiU, {3j. The gradient V' is with respect to the gross position vector r

(18)

On comparison with Eq. (1), the right-hand side of Eq. (16) is used to define the pressure tensor. But this is not unique! The curl of any tensor can be arbitrarily added. Thus if P is a pressure tensor such that Eqso (1) and (16) are identical, then for arbitrary tensor T,

p'=P+V'xT (19)

can also be used as the pressure tensor, again making Eqs. (1) and (16) identical. The difference between P and P' is considered to be physically irrelevant. Thus we show below, that one identification of the pressure tensor is symmetric, and so conclude that any pressure tensor consistent with Eqs. (1) and (16) is mathematical­ly and physically equivalent to a symmetriC pressure tensor. In particular, the pressure tensor obtained by CBH is consistent with Eqs. (1) and (16). It is thus equivalent to a symmetric pressure tensor although it does not appear to be symmetric. There is then no surprise if, after reasonable physical approximations, their pressure tensor is found to be symmetric.

Upon comparison between Eqs. (1) and (16), it is rea­sonable to identify the "kinetic" part5 of the pressure tensor as

PK =/2;. (m",)-1 [(P"'iV -mc>vu) (P"'iv -m",vu)o(r",iv - rll,) . \O:IV (20)

The remaining parts of the pressure tensor can only be identified after writing the force terms as gradients. This is now done. By use of the "modified Taylor the­orem", compare CBH Eq. (7.10),

J j(x)o(x-r-R)dx= J j(x)o(x-r)+R' :r

x r j(x)o(x- r - ~R) dxd~ , o

(21) valid for sufficiently well behaved j (x), the intramolec­ular force term can be written as

J. Chern. Phys., Vol. 65, No.9, 1 November 1976

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R. D. Olmsted and R. F. Snider: Pressure tensor in macromolecular fluids 3425

x o [t(raiIL +raiv) - r+t ~(raiv - r aiIL )]) (22)

Thus the intramolecular contribution to the pressure tensor is here identified as

xoWraiv+raiIL)-r+H(raiV-raiIL)]). (23)

Upon following an identical procedure, the intermolecu­lar contribution P:I is written as

xo[t(raiV+raJ,.)-r+t ~(raiV-raJIL)]). (24)

Since the interatomic potentials are spherical,

(25)

for all ai and f3j. This means that P~ and ph are sym­metric. In addition, PK is symmetric, making the total pressure tensor p' = PK + p~+ ph symmetric. As dem­onstrated, this depends upon the nature of the inter­atomic forces and the localization procedure, and not upon the additional approximations made in the CBH discussion.

The above method of re-expressing the force terms as a gradient has made maximum use of the symmetry under interchange of the /l and II atomic labels. Use of the modified Taylor theorem, Eq. (21), is a formally exact expansion of the delta function o(raiv - r) about the geometric center of atoms /l and II. This differs from the CBH procedure which expands o(raiv-r) about rai> the center of mass of molecule ai. For example, the CBH procedure uses the modified Taylor theorem to express the intermolecular force term as

- V" Ii d t ('"""' (r - r )F(al,ai) " W alv ai VIL o aivlJ.

(26)

The first term can be shown to vanish by interchanging the /l and II indices, giving an identification for PI-This expression for PI is different from that obtained from Eq. (23). However, Eqs. (22) and (26) are equally valid, both being based upon Eq. (21). Therefore, the pressure tensors obtained from Eq. (22) and Eq. (26) are either identical or differ at most by a divergence free tensor, see Eq. (19). Strictly analogous comments hold for the intermolecular force term.

III. THE CENTER OF MASS FORMULATION

To demonstrate the role that localization plays in the specification of the pressure tensor, the above deriva­tion is repeated with the same interatomic forces but a different localization procedure. It was remarked that the local mass and momentum densities as defined by CBH are average local properties of the atoms com­priSing the molecules. A different physical interpreta­tion is obtained by associating the mass and momentum densities with the molecular properties of the system rather than with the atomic properties. Thus the local mass denSity of species a, Ma , is defined by

(27)

the stream velocity Va of species a by

MaVa=(~ [PaivO(ral- r)]s)=(~ [Paii5(rai - r)]s) ,

and the total stream velocity V by (28)

(29)

Here, r al and p",; are the pOSition and momentum oper­ators of the center of mass of molecule ai, while m", is the mass of a molecule of species a. The interpreta­tion of Ma is quite different than that of M"" Ma is as­sociated with the probability of finding an atom at point r, whereas M", is associated with the probability of finding a molecule at that point. Similarly, u", is the average velocity of the atoms comprising molecules of species a while va is the average velocity of the mole­cules themselves. The stream velocity v'" is that cor­responding to the (basically independent) translational motion of the macromolecules; u'" contains both the translational motion of the macromolecules and their internal motions. Details of the differences have been discussed elsewhere2 for the case of diatomic mole­cules.

With the definition of V given by Eq. (29) and the von Neumann equation, Eq. (5), the equation of motion for V is written as

i8(MV)/8t = Tr1 ••• N ~ [p",;o(r al -r)]s(L: £aiIL "" aJIL

(30)

Note that the operator for the momentum denSity is a molecular operator but that the full N-atom Liouville operator and denSity operator are used, and that the trace is over all N atoms. This is a mixture of an atomic and molecular representation. 2 The equation of motion is calculated in this way to emphasize the effects of the localization. The commutators in Eq. (30) can be evaluated with the result that

+ / L: o(r ai - r)F~~I,aJ») \Q:ill,8JIJ.

"'1"'IlJ

(31)

J. Chem. Phys., Vol. 65, No.9, 1 November 1976

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3426 R. D. Olmsted and R. F. Snider: Pressure tensor in macromolecular fluids

Once again the force term must be written ~s a gradient in order to identify the pressure tensor. The modified Taylor theorem is used in the form

( ~. o(rai-r)F~~i'Bi») cOII,81 J£

aiflli

= - tV'· f I d~ ( L (r ai - rBJ)F~~i,Bj) -I a iv,BJ I>

aiflli

xo[t(rai+rBJ)-r+t~(rai-rBj)l) ,

to write the pressure tensor as

(32)

P=PK+P y , (33)

with

PK"'(~ m~1 [(Pai - maV)(Pai - m",V)6(rai - r)l) (34)

and

p = 1:. II d~ / L (r - r )F(ai,BJl y- 4 -I \alv,BJI> al BJ VI>

aii/3j

(35)

The kinetic contribution PK is symmetric but Py has an antisymmetric component; the total force LVl>F~~i,8j) is not directed along the line of centers of the molecules, r ai - r BJ , producing a torque on the molecules due to their interactions. This torque gives rise to the anti­symmetric component p~ of Py (( is the Levi-Civita antisymmetric tensor of third rank)

P~=+t(Ofl d~/L (rai-rBj)XLF~~I,8j) -I \ aiBJ VI>

QitBj

xo[t(rai+rBj)-r+tHrai-rBj)J) . (36)

Another important difference is that the symmetric components of P and P are not the same. For diatom­ics this difference is related to accelerations of the orientational and vibrational degrees of freedom of the molecules. 2 A more obvious distinction is that P con­tains only intermolecular contributions while P contains both inter and intramolecular contributions.

IV. DISCUSSION

It has been stressed that the CBH pressure tensor is equivalent to a symmetric pressure tensor, the latter

symmetry being due to: i) the atomic (bead) localiza­tion of mass and momentum densities and ii) the spher­ical nature of the atom-atom forces. In contrast, a localization based on the macromolecular centers of mass implies the presence of an inherently antisym­metric component of the pressure tensor. The fact that both treatments are mathematically correct, but lead to different results, does not seem to have been pre­viously recognized.

There remains the obvious question as to which local­ization provides the better description of the fluid. As we see it, this is impossible to decide from the present viewpoint, but we speculate on what will be involved in seeking an answer to this problem. For small mole­cules, specifically diatomics, the nonsphericity of the intermolecular potential is usually small. Hence it seems reasonable that the orientational motions of the molecules only slightly perturb their translational mo­tions. Surely then, the lIIolecular localization is the best "first picture" of a gas of diatomics. For macro­molecules, the picture is not clear. A macromolecule in solution is never "free", so its internal (configura­tional) motions are always strongly coupled to the sol­vent. Thus a molecular localization does not appear to be a reasonable first approximation to the macromo­lecular motion, unless a solvent sheath might be en­visaged as accompanying the whole macromolecule, For diatomics, the atomic localization severely mixes the internai and translational motions, 2 so we expect that this is not a good picture for macromolecules ei­ther. We are left with the conclusion that the whole system, polymer plus solvent, must be treated together, and a localization of mass and momentum densities on a scale that is small compared to the "ize of the macro­molecule, may not be a useful concept,

*Work supported in part hy the National Research Council of Canada.

IC. F. CurtiSS, R. B. Bird, and O. Hassager, Adv. Chem. Phys. 35, 31 (1976).

2R . D. Olmsted and R. F. Snider, J. Chem. Phys. 65, 3407 (1976), preceding paper.

3The procedure for symmetrizing the pressure tensor is dis­cussed by J. A. McLennan, Physica (Utrecht) 32, 689 (1966). The effects of this procedure on the interpretation of the stream velocity is discussed in Ref. 2.

4H. Mori, Phys. Rev. 112, 1829 (1958); J. S. Dahler, ibid. 129, 1464 (1963).

5The split-up of the pressure tensor into kinetic and potential parts cannot rigorously be performed quantum mechanically owing to the noncommutivity of the potential and kinetic en­ergy operators. Here the words kinetic and potential are used only for classification purposes.

J. Chern. Phys., Vol. 65, No.9, 1 November 1976

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