7 Equation of State and Stress Tensor In

Home , Stress (mechanics)

7 EQUATION OF STATE AND STRESS

TENSOR IN INHOMOGENEOUS

COMPRESSIBLE COPOLYMER MELTS

This chapter was previously published as:

'Equation of state and stress tensor in inhomogeneous compressible

copolymer melts. Dynamic mean- eld density functional approach'.

N.M. Maurits, A.V. Zvelindovsky, J.G.E.M. Fraaije,

Journal of Chemical Physics,

108 6, 1998, p. 2638-2650.

7.1 Summary

We have derived an expression for the global stress in inhomogeneous

complex cop olymer liquids. We apply the principle of virtual work to

the free energy as de ned in the dynamic mean- eld density functional

metho d. This metho d automatically provides the full stress tensor devi-

atoric and isotropic parts and hence an equation of state for inhomoge-

neous compressible cop olymer melts. The excluded volume interactions

and cohesive interactions b etween chains have b een explicitly taken into

account. Therefore the expressions for the stress and thermo dynamic

pressure have a wide range of validity. The connectivity of the chains

is automatically accounted for and the free energy adapts very well to

changes in the molecule prop erties. In the limiting case of homogeneous

systems it simpli es to known results. In order to study rheological prop-

erties of cop olymer melts and npT -ensemble simulations, the pressure and

stress comp onents have to be calculated at any given moment in time.

Weshowhow the pressure and stress can b e numerically evaluated during

simulations using a Green propagator algorithm, instead of having to cal-

culate the time-dep endent con guration distribution function explicitly

from a Smoluchowski equation. We provide illustrativenumerical results

that indicate how the pressure changes during microphase separation.

STRESS TENSOR 105

7.2 Intro duction

General

The dynamic mean- eld density functional metho d is a metho d for nu-

merically calculating the phase separation dynamics of cop olymer liquids

in 3D. It predicts morphologies and has recently been successfully used

to predict part of a Pluronic/water phase diagram. The metho d is

based up on a generalized time-dep endent Ginzburg-Landau theory for

19, 21, 22

conserved order parameter of the following general form:

@ r

D r; r r dr =

IJ 1 J 1 1

J =1

D r; r

IJ 1

dr + r;t ; 7.1

1 I

J 1

J =1

D r; r = r r; r r ; 7.2

IJ 1 r IJ 1 r

with particle concentration elds r I = 1;:::;Z, Onsager kinetic

co ecients , intrinsic chemical p otentials F = r F is the

IJ I I

free energy, = k T and noise elds r;t. The noise has a Gaus-

B I

sian distribution with moments dictated by a uctuation-dissipation the-

22{ 24

orem.

The goal of mesoscopic mo deling is to obtain a theory of ordering phe-

nomena in p olymer liquids, based on a 'molecular' description. We use

a free energy functional for a collection of Gaussian chains in a mean-

37, 132

eld environment. In this approach we try to retain as much as

p ossible of the underlying molecular detail, such as the architecture and

comp osition of the chain molecules. To this end, we do not use a phe-

nomenological expansion of the free energy in the order parameters, as

16{21

is commonly done in Ginzburg-Landau mo dels, but de ne an intrin-

sic free energy functional based on a density distribution function that

results from a minimization criterion. This intro duces external p oten-

tials as Lagrange multiplier elds. The density particle concentration

elds and external p otential elds are coupled one-to-one through the

density functional. In this approach the free energy immediately re ects

changes in the molecular prop erties which enables practical application

of the metho d. Some results of numerical calculations of phase separa-

tion in incompressible blo ck cop olymer melts and concentrated surfac-

tant solutions were already discussed in Refs. 28 and 82. The metho d

109

has been extended to include compressible melts and nonlo cal kinetic

133

co ecients.

In this pap er we have derived an expression for the global stress in

inhomogeneous complex cop olymer liquids which, in the limiting case of

homogeneous systems, is analogous to the expression for the stress tensor

that was derived by Doi and Edwards. This expression was obtained

from a con gurational average of microscopic stresses where the distri-

bution function results from a Smoluchowski equation in the presence of

homogeneous ow. To this end, we employ the principle of virtual work

106 STRESS TENSOR

on the free energy as de ned in the dynamic mean- eld density functional

metho d. This metho d provides an expression for the total stress tensor

and hence from its trace an expression for the thermo dynamic pressure,

i.e. an equation of state for inhomogeneous cop olymer melts. Before we

explain the details of the derivation, we rst discuss previous studies

on equations of state and stress tensors for inhomogeneous cop olymer

melts.

Equations of state

Equations of state for chain molecules are usually limited to homogeneous

systems and often contain severe simpli cations regarding connectivity.

In Ref. 83, the connectivity is taken into accountbyallowing only a lim-

ited numb er of degrees of freedom less than 3 for chain elements, by use

of the so-called Prigogine parameter c. In this cell mo del, the Prigogine

parameter determines the expression for the free volume p er chain. Sev-

eral authors have tried to improve the cell mo del by including geometry

134

factors in the free volume expression, by using di erent expressions

135

for the mean p otential energy per mer or by including the p ossibility

of empty sites. The lattice- uid mo del develop ed in Ref. 90 that puts

chains on a lattice in order to b e able to calculate the partition function,

is improved in Ref. 92 by including molecular shap e di erences and in

Ref. 91 by including entropy contributions.

136, 137

The lattice theories of Flory and Huggins are generalized to

continuous space in Ref. 95. The complete disregard for chain structure

is rst improved in Ref. 138 by relating the compressibility factor of an

n-mer uid to compressibility factors of monomer and dimer uids at

the same volume fraction. In Ref. 139 the theory in Ref. 138 is further

improved by including structural information for a diatomic uid, us-

ing site-site correlation functions at contact, which results in a mo di ed

thermo dynamic p erturbation theory. These studies are all restricted to

athermal homogeneous melts.

Monte Carlo simulations of lattice mo dels for p olymer melts are em-

ployed in Refs. 140 and 141 to measure the chemical p otential by a

particle-insertion metho d. This enables calculation of the osmotic pres-

sure by thermo dynamic integration.

Stress tensor

Expressions for the stress in a visco elastic material usually serve the

132, 142

purp ose of analyzing theoretical and exp erimental results. Many

authors study stress-strain relationships in order to access the rheological

prop erties of the material. The metho ds that are used to derive expres-

sions for the stress given a Hamiltonian date back to the sixties and

seventies. A few di erent approaches can b e discerned.

STRESS TENSOR 107

In Ref. 21 sto chastic equations are given for low molecular weight

uids. These equations or other hydro dynamic equations valid for u-

143

ids in a two-phase region can b e combined with the uid momentum

balance equation. This allows for a relationship to be derived between

the Hamiltonian of the system and the divergence of the stress tensor.

If the Hamiltonian is a spatial average of a lo cal function, a compatible

143{ 150

lo cal stress tensor can be derived which is unique up to a diver-

gence free eld. The lo cal stress tensor can be spatially averaged to

145, 151, 152

obtain an expression for the average shear stress. Similar ex-

pressions are also used in inhomogeneous systems as a measure of the

147, 153, 154

spatial anisotropy of domains distorted by shear ow. The av-

erage shear stress may also b e expressed in terms of the nonequilibrium

147, 151, 155

structure factor. The total stress of the material is often com-

p osed of this 'anisotropy' factor, a separate viscous contribution and a

146, 147, 153, 154, 156, 157

hydro dynamic pressure contribution.

Another approach to visco elasticity is to consider the visco elastic

p olymeric material as a collection of particles susp ended in a viscous

solvent. The stress is then comp osed of the usual viscous solvent part

and a microscopic contribution due to the forces b etween interacting par-

37,126,158,159

ticles.

A more thermo dynamic approachistaken in Ref. 160 for solutions of

ro dlike p olymers, in Ref. 155 for homogeneous p olymer melts, in Ref. 161

for liquid crystals and in Ref. 162 for critical liquids. Here the principle of

virtual work is applied to nd the stress tensor by calculating the elastic

resp onse of a material to a small deformation.

In this pap er we generalize the expression for the stress tensor to

the case of compressible inhomogeneous complex cop olymer liquids. The

dynamical free energy A as considered in Ref. 37 accounts for entropy

e ects of ideal b eads and energy e ects due to the p otential eld H :

A dR k T ln + H : 7.3

Here, is the distribution function of the b eads. According to the prin-

ciple of virtual work, the stress due to the particles is related to a

change in the dynamical free energy. The formal application of the prin-

ciple of virtual work is not hard since the dynamical free energy is simple

ideal p olymers only and an expression for the variation of the distribu-

tion function can b e obtained from a simpli ed Smoluchowski equation

that is dominated by the velo city gradients due to the deformation. This

37, 159

leads to

= hF R i ; 7.4

m m

which is also found by considering the microscopic forces b etween b eads.

F is the -comp onent of the nonhydro dynamic forces acting on b ead

m, R is the -comp onent of the p osition of b ead m and the con gura-

tional average hi is taken with resp ect to . If the interaction p otential

between beads is harmonic Gaussian chain mo del, the particle contri-

108 STRESS TENSOR

bution to the stress is given by

N 1

= hR R R R i : 7.5

n+1 n n+1 n

n=1

Since only incompressible uids are considered, isotropic stress contri-

butions are dropp ed. Excluded volume potentials only a ect the stress

expression through the distribution function . A basic assumption in

the derivation is that even in the highly entangled state the 'short-time'

p olymer dynamics which completely determine the stress tensor is gov-

erned by the Rouse mo del.

Present pap er

We apply the principle of virtual work to the free energy functional as

7, 28

de ned in the dynamic mean- eld density functional theory to ob-

tain an expression for the stress tensor in compressible inhomogeneous

cop olymer melts. In the limit of homogeneous systems, we obtain an

expression analogous to Eq. 7.5. Moreover, employing the principle of

virtual work immediately leads to an expression for the diagonal comp o-

nents of the stress tensor as well and therefore to an expression for the

pressure equation of state.

Weshow that our mathematical pro cedures are consistentby deriving

the pressure not only by using the principle of virtual work, but also by

calculating the change in free energy that results from an in nitesimal

volume change. To our knowledge there have b een no attempts to de ne

an equation of state that includes molecular detail for general inhomo-

geneous compressible cop olymer melts. In our mo del the connectivity of

the chains is automatically accounted for and the free energy adapts very

well to changes in the molecule prop erties. The free energy functional

contains both excluded volume and cohesive interactions. The inhomo-

geneity of the system intro duces an external p otential as a Lagrange

multiplier eld from an optimization criterion for details see Ref. 28.

This external potential do es not explicitly contribute to the expression

for the stress tensor but changes the distribution function as will b e ex-

plained in Section 7.3.

Previous simulations of phase separation in blo ck cop olymer melts

using dynamic density functional theory see e.g. Ref. 28 were always

p erformed in an nV T -ensemble, since the volume of the simulation b ox

remained constant. Simulations in an npT -ensemble will increase the

relevance of numerical exp eriments compared to lab oratory exp eriments,

which are usually p erformed under constant pressure conditions. We in-

tend to use a pressure coupling algorithm during simulations to keep the

pressure constant. We show that the pressure for a homogeneous melt

may be calculated analytically and can hence be used as the reference

pressure in a pressure coupling algorithm in the near future. Weshortly

discuss an algorithm for numerically evaluating the pressure during a

STRESS TENSOR 109

simulation that uses Green propagators. As a pro of of concept, we show

some numerical results that indicate how the pressure changes during mi-

crophase separation. The Green propagator algorithm avoids the use of a

Smoluchowski equation to determine the con guration distribution func-

tion and can just as well be used to calculate the stress. The expression

for the stress will be used in a future publication to study stress-strain

relationships in a sheared melt.

7.3 Theory

Dynamic mean- eld density functional theory

We shortly rep eat the main part of the theory of the mesoscopic dynamics

algorithms in order to be able to explain the detailed derivation of the

stress tensor and the results of the numerical calculations. Weconsidera

melt of volume V ,containing n diblo ck cop olymers mo deled as Gaussian

chains, each of length N = N + N . There are two concentration

A B

elds r and r. Given these concentration elds a free energy

A B

functional F [] can be de ned as follows:

nid

F []= ln U r r dr + F []: 7.6

I I

nid

Here F [] is the contribution from the nonideal interactions and is

the partition functional for the ideal Gaussian chains in the external eld

U ,

] R [H + U

0 0

s s

s =1

dR R ; 7.7 e N

1 N

N 1

1 3

where N is a normalization constant equal to , R =

i 3 2

X ;Y ;Z is the position of the i-th b ead in the chain and H is the

i i i

Gaussian chain Hamiltonian cf. H in Eq. 7.3

2 G

R R ; 7.8 H =

s s1

2 a

s=2

with a the Gaussian b ond length parameter. The dynamical free energy

as used in Doi and Edwards Eq. 7.3 is equal to the rst term in

Eq. 7.6 except that the external eld now in uences the distribution

function. The free energy functional is derived from an optimization cri-

terion whichintro duces the external p otential as a Lagrange multiplier

eld. Notice that the inhomogeneity of the melt is re ected in the value

of the external p otentials. The external p otentials and the concentra-

tion elds are related via a bijective density functional for ideal Gaussian

chains:

[U ]r = r R

I s

s =1

[H + ] R U

0 0

s s

s =1

e dR R : 7.9

1 N

110 STRESS TENSOR

K 0

Here is a Kronecker delta with value 1 if b ead s is of typ e I and 0 oth-

0 0

erwise, exp [H + U R ]= is the single-chain con guration

s s

s =1

distribution function.

The nonideal free energy functional is formally split into two parts:

nid c e

F []=F []+F []; 7.10

e c

where F contains the excluded volume interactions, and F the cohesive

e c

interactions. The intrinsic chemical p otentials and are de ned by

I I

the functional derivatives of the free energy:

c e

F F

= U r+ +

r r

I I

c e

= U r+ r+ r : 7.11

I I

For the cohesiveinteractions we employatwo-b o dy mean- eld p otential:

Z Z

c 0 0 0

F []= jr r j r r drdr ; 7.12

IJ I J

V V

0 0

where jr r j= jr r j is a cohesiveinteraction b etween beads

IJ JI

of typ e I at r and J at r , de ned by the same Gaussian kernel as in the

ideal Gaussian chain Hamiltonian

0 0

e : 7.13 jr r j

In Ref. 109 wehave shown that a very simple cell mo del can b e used for

excluded volume e ects. In this case

X X

e 3

rln1 r F []= dr: 7.14

J K K

J K

c e

Here is the b ead volume. Notice that the mo dels for F and F are not

109

unique. However, the principle of virtual work can be applied to any

mo del for the free energy and the resulting expression for the stress tensor

will be mo del dep endent. For practical applications, a Helfand p enalty

107, 109

p otential can be used, which allows for small density uctuations

around the mean bulk density. In this case the nonideal free energy

functional is given by

Z Z

0 0 0 nid

jr r j r r drdr F [] =

IJ I J

V V

X X

dr; 7.15 r +

I I I

I I

where is a compressibility parameter, is the density of comp onent

+ is an I in the homogeneous melt and =

IJ JI II JJ IJ

exchange interaction parameter.

STRESS TENSOR 111

In equilibrium ris constant. This yields the familiar self consis-

nid

tent eld equations for Gaussian chains, given a prop er choice for F .

When the system is not in equilibrium the gradientoftheintrinsic chem-

ical p otential r acts as a thermo dynamic force which drives collec-

tive relaxation pro cesses. When the Onsager co ecients are constantthe

sto chastic di usion equations are of the following form

= r J ; 7.16

J = M r + J ; 7.17

I I I

where M is a mobility co ecient and J is a noise eld, distributed

according to a uctuation-dissipation theorem. Nonlo cal forms for the

Onsager co ecients have been studied in Ref. 133.

Principle of virtual work

A complex liquid or visco elastic material p ossesses b oth elastic and vis-

cous prop erties. If we apply an external force to a p olymeric material

the resp onse of the system is related to the internal stress which is, in

principle, a complex interplay between viscous and elastic stresses. In

principle the viscous and elastic stresses can not easily b e separated and

several constitutive equations have been prop osed to relate the stress

tensor to the history of the applied strains or strain rates via complex

163, 164

stress relaxation functions. Constitutive equations are needed to

calculate the ow of a uid when an external force is applied. If the sys-

tem is in the regime of linear visco elasticity, the stress is small and the

constitutive equation is a linear relationship between stress and strain.

However, the scop e of this pap er is not to show how to calculate the

dynamic b ehaviour of a cop olymer melt in an external ow eld. Wewill

p ostp one this discussion to a future publication.

The stress tensor , of which the ij -comp onent describ es the i-

comp onent of the force per unit area p erp endicular to the x -axis, can

be de ned by considering the work done by internal stresses if a body

is undergoing a smal l deformation the principle of virtual work, see

e.g. Ref. 37. Supp ose the deformation displaces the p oint r on the

material to r + r . Here, = @ u =@ r is the -comp onent of

165

the unsymmetrized strain tensor and u is the displacement vector.

Then it can be shown that the related change in free energy F is given

by:

F = V: 7.18

Hence, by applying a small deformation to a body and calculating the

change in free energy in the limit of this deformation going to zero, we

can calculate the stress tensor given the free energy F . Notice that

the deformation must be applied slowly enough to ensure that b efore

and after the deformation the system is in thermo dynamic equilibrium,

i.e. the distribution function changes but remains optimal.

112 STRESS TENSOR

The stress tensor as de ned in 7.18 can b e rewritten in terms of the

165

symmetrized strain tensor u + :

F 1

: 7.19

V u

Notice that we have explicitly indicated the volume here.

The stress tensor and pressure in inhomogeneous compressible cop oly-

mer melts

Calculation of the stress tensor from rst principles

The ij -comp onent of the stress tensor can be derived by employ-

ing the principle of virtual work 7.18. We consider a cubic volume

V with a density pattern f g. If we consider the sp ecial deformation I

1/3 1/3 ~ V ρ V α ρ x,y,z x',y',z' y V1/3 V1/3 1/3 1/3 VVx (a) z

V1/3 ρ ρ~ V1/3 x,y,z x',y',z' y V1/3 V1/3 1/3 1/3 (b) VVx (1+α)

Figure 7.1 a Virtual deformation applied to a cubic volume V in order to

calculate . The volume V is preserved. b Virtual deformation applied to a

cubic volume V in order to calculate . The volume V is not preserved.

of the cubic volume as sketched in Fig. 7.1a, is the only element of

the unsymmetrized strain tensor which is not zero. Since is small

we have = tan = . This deformation allows us to calculate .

xy xy

Since the system is in thermo dynamic equilibrium b efore and after the

STRESS TENSOR 113

deformation but not necessarily in a free energy minimum, the distri-

bution function changes to remain optimal. Hence the densities and

the external p otentials are di erent b efore and after the deformation.

In App endix 7.6 we show that the principle of virtual work applied

to the nonideal free energy 7.10 due to the cohesiveinteractions using

the cell mo del leads to

nid nid nid

F = F [~] F []

Z Z

= jr r j

IJ 1 2

V V

x x y y r r dr dr : 7.20

1 2 1 2 I 1 J 2 1 2

The excluded volume interactions, which are lo cal, only in uence the

isotropic stress see also Ref. 37 b ecause the deviatoric stress is com-

pletely determined by intra- and intermolecular nonlocal forces.

The principle of virtual work applied to the ideal free energy leads to

see App endix 7.6

P P

N N

U R R R

0 0

s s

s =1 s=2

F = e

X X Y Y dR dR

s s1 s s1 1 N

s=2

* +

= X X Y Y : 7.21

s s1 s s1

s=2

Here, the ensemble average hi is taken with resp ect to the Boltzmann

P P

N N

0 0

weightexp[ R R U R ] which includes the

2 s s1 s s

s=2 s =1

external p otential so that the inhomogeneity of the system is automat-

ically taken into account. Since, by de nition, we have for the sp ecial

deformation under consideration

F = V = V ; 7.22

xy xy xy

we nd the nal result

* +

= X X Y Y

xy s s1 s s1

a V

s=2

Z Z

jr r j

IJ 1 2

2a V

V V

x x y y r r dr dr : 7.23

1 2 1 2 I 1 J 2 1 2

Notice that explicitly dep ends on the entire density pro le f g

xy I

but only implicitly on the external p otential fU g via the Boltzmann

weight. The other nondiagonal comp onents of the stress tensor are very

similar; for the term X X Y Y must be replaced by

xz s s1 s s1

X X Z Z and for by Y Y Z Z etcetera.

s s1 s s1 yz s s1 s s1

The second part of results from the nonlo cal mean- eld interactions,

114 STRESS TENSOR

the rst part of results from the connectivity of the chains ori-

entation of the b ond vectors and is analogous to the result obtained

for homogeneous concentrated p olymer solutions. Notice however that

the con gurational average in Eq. 7.23 is taken with resp ect to the

Boltzmann factor that includes the external p otential which accounts

for the inhomogeneity in the system. The distribution function in

Eq. 7.3 is obtained from a Smoluchowski equation for a macroscopic

velo city eld using a Rouse dynamics mo del. In Section 7.4 we show

how Eq. 7.23 can be calculated in our framework of a generalized

time-dep endent Ginzburg-Landau mo del without explicitly considering

a Smoluchowski equation for . If the Helfand p enaltypotential instead

of the cell mo del is considered, the interaction parameters havetobe

replaced by exchange interaction parameters .

To determine the diagonal comp onents of the stress tensor we consider

deformations as indicated in Fig. 7.1b. This particular deformation yields

since = and all other comp onents of the unsymmetrized strain

xx xx

tensor are zero. The details of the derivation of the diagonal comp onents

of the stress tensor are outlined in App endix 7.6.

In case of the cell mo del we get b oth an excluded volume and a

cohesive interaction contribution from the nonideal free energy to .

The ideal free energy now gives a contribution from the connectivity

of the chains and an ideal gas contribution We nd for the diagonal

comp onent of the stress tensor see App endix 7.6

* +

nN 3n

= + X X

xx s s1

V a V

s=2

Z Z

jr r j

IJ 1 2

2a V

V V

x x r r dr dr

1 2 I 1 J 2 1 2

1 r

I I

r dr: 7.24

1 r

I I

The stress tensor comp onents and are very similar in that

yy zz

2 2 2

X X must b e replaced everywhere byY Y and Z Z

s s1 s s1 s s1

resp ectively, etcetera. In case the Helfand penalty mo del is considered

the contribution from the nonideal free energy to the diagonal comp onent

of the stress tensor is given by:

Z Z

nid

= jr r j

IJ 1 2

2a V

V V

x x r r dr dr

1 2 I 1 J 2 1 2

X X

dr: 7.25 r

I I I

I I

In Ref. 81 we have shown that most phenomenological free energy

mo dels e.g. Oono-Puri or Cahn-Hilliard can be obtained from the free

energy 7.6 by a functional Taylor expansion that is truncated at dif-

ferent places. The free energy that is used in e.g. Refs. 144 and 148 to

STRESS TENSOR 115

derive an expression for the stress is also contained in Eq. 7.6. Since the

162

principle of virtual work can b e applied to any free energy functional

the expression for the stress tensor as found in Refs. 144 and 148 is au-

tomatically contained in our expressions 7.23 and 7.24. In principle,

a direct Taylor expansion of 7.23 or 7.24 combined with a tting

pro cedure should also yield phenomenological expressions for the stress.

This pro cedure is rather cumb ersome however and will not b e dealt with

in the present article.

Comparison trace of stress tensor and pressure

165, 166

The op erational de nition of pressure is given by

Tr = 3p: 7.26

For compressible materials p is also the thermo dynamic pressure. This

do es not hold for incompressible materials where the pressure is merely

a Lagrange multiplier see also the discussion in Ref. 109. In an in-

compressible uid, the isotropic part of the stress tensor is determined

by external conditions only and therefore irrelevantinthe discussion on

visco elastic prop erties of materials. The thermo dynamic pressure in

nV T -ensembles is de ned as follows

p : 7.27

N;T

We can check the consistency of our mathematical pro cedures for calcu-

lating the stress tensor by comparing the result for the pressure resulting

from the thermo dynamic de nition 7.27 to the result for the pressure

as obtained from 7.26. In App endix 7.6 we show for the cell mo del

that indeed + + = p as might b e exp ected.

xx yy zz

Homogeneous limiting case

Expression 7.62 allows us to analytically calculate the pressure of a

homogeneous cop olymer melt U = 0, r = . For the cohesive

I I

3 3

28 0 0 0

exp interactions we employ jr r j = r r

2 2

2a 2a

and hence:

P P

nN nN

J I

V V

n 1 1 n N N

I J

J I

p = : 7.28 + +

V 2 V

Here, N is the numb er of b eads of typ e I in one cop olymer. In a homo-

geneous cop olymer melt the nondiagonal comp onents of the stress 7.23

are all zero b ecause the distribution function is always assumed to be

optimal. In an npT -simulation, expression 7.28 provides us with a ref-

erence pressure which may be used in a pressure coupling algorithm. A

similar expression is easily obtained for the Helfand penalty p otential