Surface Rheology I. The Planar Fluid Surface

Surface Rheology I. The Planar Fluid SurfaceD. A. Edwards and D. T. Wasan Citation: J. Rheol. 32, 429 (1988); doi: 10.1122/1.549974 View online: http://dx.doi.org/10.1122/1.549974 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v32/i5 Published by the The Society of Rheology Related ArticlesWall slip and spurt flow of polybutadiene J. Rheol. 52, 1201 (2008) Particle–particle and particle-matrix interactions in calcite filled high-density polyethylene—steady shear J. Rheol. 48, 1167 (2004) Cyclic generation of wall slip at the exit of plane Couette flow J. Rheol. 47, 737 (2003) Dynamics of end-tethered chains at high surface coverage J. Rheol. 46, 427 (2002) Slip at polymer–polymer interfaces: Rheological measurements on coextruded multilayers J. Rheol. 46, 145 (2002) Additional information on J. Rheol.Journal Homepage: http://www.journalofrheology.org/ Journal Information: http://www.journalofrheology.org/about Top downloads: http://www.journalofrheology.org/most_downloaded Information for Authors: http://www.journalofrheology.org/author_information

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Surface Rheology I. The Planar Fluid Surface

D.A. EDWARDS* and D.T. WASAN Department ofChemicalEngineering, Illinois Institute of Technology,

Chicago, Illinois 60616

Synopsis

The intrinsic rheology of a fluid-fluid interface has been found significant tomultiphase flows for which surfactant material is present and the surface-tovolume ratio is large, such as in emulsification and demulsification, foam andemulsion stability, foam rheology, the dynamics of small bubbles or droplets, andinterfacial mass transfer. An understanding of surface rheology therefore haspractical relevance to engineering technologies wherein such phenomena occur:examples include the enhanced recovery of oil and the processing of food materials. Herein, a rheological theory is developed for a planar fluid surface whichsupercedes previous dynamic theories by explicitly acknowledging the nonlocalnature of the three-dimensional interfacial transition zone. Beyond providinginsight into the nature of the surface excess pressure tensor without the complicating features accompanying surface curvature, the treatment will provide afoundation for subsequent articles which will examine the rheology of curvedsurfaces possessing curvature as large as occurs in microemulsion systems. Inthe equilibrium state, the surface excess pressure tensor is shown to result fromthe highly inhomogeneous nature of the phase interface, whereas in the dynamic state a linear constitutive assumption results in the classical surface properties of surface viscosity and surface tension.

INTRODUCTION

The rheology of the fluid-fluid surface has been investigatedextensively since the first investigations of Plateau.'

Prior to Erickson," Oldroyd," Scriven," and Slattery," the common approach to surface rheology was to neglect interactions between the surface and bulk phases, considering the surface as anisolated two-dimensional body, although a quantitative relation between interfacial and bulk deformational character had been provided by Boussinesq" many years before the work of Erickson."

*Current Address: Department of Mechanical Engineering, Technion-IsraelInstitute of Technology, Haifa, 32000 Israel.

© 1988 by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc.Journal of Rheology, 32(5), 429-445 (1988) CCC 0148-6055/88/050429·17$04.00


430 EDWARDS AND WASAN

The surface rheological theory of Boussinesq," which waspresented in a coordinate system coinciding with the principalrate-of-strain axes, was extended by Erickson" and Oldroyd" tosurfaces with arbitrary coordinates and then by Scriven" to timedependent deformable surfaces. Slattery" extended the theory ofScriven" to deformable surfaces undergoing mass exchange withthe bulk fluid phases.

The surface rheological theory which evolved from Boussinesq"to Slattery" was based upon a linear, viscous surface stress model.Extension to viscoelastic surface stress behavior has followed. 7-9

A shortcoming, however, in the early theory of surface rheologywas the treatment of the surface as a distinct "surface phase"which satisfied two-dimensional continuity, linear momentumconservation, and angular momentum conservation with dynamicexchange from the bulk phase. Conceptually, the theory was hindered with foreign concepts such as the "surface velocity" and the"surface mass density." Beyond conceptual difficulties, the theorycould not be generalized to surfaces of very large surface curvature which exist, for example, in microemulsion systems, nor didit permit a relation between the true three-dimensional dynamicsof the interfacial region and the "assigned" dynamics of the monolayer approach.

Eliassen" and later Murphy" presented mechanical surfacetheories based upon three-dimensional interfacial character utilizing principally the static works of Buff and Saltzburg.":" However, although extending the equilibrium theory of surfaces tolarge surface curvature (Murphy, Ref. 11), neither presented acomplete surface excess rheological theory.

In relatively recent years, surface excess rheological theorieshave been developed by Deemer and Slattery" and Goodrich": inboth surface excess theories there remain two significant limitations: (1) the theories assume apriori the validity of local continuum laws in the interfacial region; and (2) the theories do notapply to surfaces with large surface curvature for which the curvature radii of the surface approach the thickness of the surfaceitself.*

*Deemer and Slattery 14 retain higher order curvature terms in their theory, butas will be discussed in the following paper, Part II, they have not fully accountedfor the effects of large surface curvature.


SURFACE RHEOLOGY I 431

In this and the following articles of this series we will presenta dynamic surface excess theory, which again assumes threedimensional interfacial character, but also acknowledges nonlocalinteractions in the molecular scale interfacial region. The theorywill not attempt to bridge the entire void currently existing between interfacial statistical theory and macroscopic interfacialcontinuum theory, but rather, by considering the interfacial region as nonlocally continuous, that is, defined by spatially continuous field variables dependent both upon local and nonlocalposition, will offer a departure from the previously assumed localnature of the interface. Our primary goal in this first article isto reveal a relation between nonlocal intermolecular forces inthe 3D interfacial region and the surface excess stress tensor.Secondary to this, we wish to provide an internally consistenttheory, necessary particularly in the following reports dealingwith large surface curvature, for including nonlocal interactionsbetween interfacial zones.

SURFACE EXCESS LINEAR MOMENTUM

Consider then the very thin region of contact between the twofluid phases I and II (Fig. 1). The region III is distinguished fromthe contiguous bulk phases by nonuniformities in fluidic properties, such as mass density, as the "thinness" of the region isroughly equivalent to the length scale of short-range intermolecular interactions. A quantitative value of this "thinness" may be assmall as 10 A, but diverges to much larger values near the criticaltemperature.

PHASE I

----PHASE II

PHASE I

PHASE III

"SURFACE"A---

Fig. 1. Conceptual enlargement of the phase interface.



Local position in the interfacial region will be denoted by theEulerian position vector r

r = xi, + yi, + ziz ,

bounded within the domain

(1)

-l::sz::sl, (2)

where the volume element V = 2lL 2 (Fig. 2) is chosen of sufficient dimension such that the boundary aV lies far beyond thescale of nonlocal intermolecular interactions, and hence themathematical surfaces z = ±l lie completely outside the regionof inhomogeneity.

The nonlocal continuum statement ofthe conservation oflinearmomentum in V may then be expressed as

Iv d 3r { :tp(r)v(r) + V . p(r)v(r)v(r) - F(r) - V . p(r)} = 0,

(3)

where v(r) is the local fluid velocity, V is the local gradient operator, F(r) is the local external body force vector at r (originating

PHASE !II

PHASE II

Fig. 2. Microscopic "segment" of interfacial volume.



(5)

outside the volume element V) and p(r) and P(r) are the localmass density and pressure tensor, respectively, at r dependentthrough nonlocal intermolecular interactions, upon all other nonlocal positions in V.

Due to the nonlocal dependence of p and .P, linear momentumis conserved over V though not at every local point r in V, nevertheless a "local" form of (3) may be expressed in terms of thelocalization residual G(r) as'?

a .- p(r)v(r) + V' • p(r)v(r)v(r) - F(r) - V' • p(r) = G(r)iJt

(4)

where

Iv d 3r G(r ) = O.

The localization residual G(r), which may be interpreted as the"body-force" density vector at r arising from nonlocal intermolecular interactions in V, contains the complete microscopicinformation of nonlocal intermolecular interactions in the interfacial region, and by (5), vanishes to a spatial average over thedomain V. (Note that G is contrasted to F in that F originatesbeyond aV and G originates within the boundary aV).

For reasons that will become apparent in the following sectionlcp. Eq. (34)], the localization residual will be assumed to possessa direction colinear with the dominant inhomogeneity in the interface, vis.

G(r) = Giz • (6)

Examining explicitly the i, component of (4) (which is regardedas the "true" balance of linear momentum in the interface), integrated over the local normal coordinate, we have

JI dz {i pv + V' . pvv - F --I at y Y Y

~PXY - "»: - ~PZy - Gy } = O. (7)ax iJy dZ

Macroscopically, however it is conventional to neglect themicroscopic dynamics of the interfacial region and to consider



JI

----l------------- '01

GENERIC FIELD QUANTITYJ

___ A

_________~----- '0-1

Fig. 3. Difference between true and macro-fields.

(8)

bulk fields valid up to an imaginary dividing surface (see Fig. 3).Assuming a significant error in this macroscopic "dividingsurface" approach, we may utilize (4), the correct balance of linear momentum in the interfacial region (assuming the validity ofthe nonlocal continuum description), to assign the error (or excess) in the macroscopic approach to the artificially defined dividing surface.

The balance of y-component linear momentum using the di-viding surface approach may be written

JI dz {i pv + V . pvv - F --I at Y y y

a - a - a - lID II }-Px - -P - -P + fL 2 8(z) = 0ax Y ay YY az 2Y Y

where an overbar indicates the macroscopically perceived value ofthe barred quantity, e.g.



_ {pI(I)p = p"(-l)

O~z~l

-l ~ z ~ 0(9)

(11)

and where the term B(z)liP0'11, with B(z) the Dirac Delta Function and

IIP"yll = P~~IA - - P~yIA + (10)

the jump condition of the normal stresses, arises from the derivative of the "step" or jump in the normal stress across the mathematical dividing surface (Fig. 4) as shown in the Appendix.

Subtracting (8) from (7) yields the error or excess incurred inthe balance of y-component linear momentum in the interfacialregion via the dividing surface or macroscopic approach, and maybe written as

a a a_psvs + _psvsv s + _psvsv s _at y ax x y ay Y Y

F~ - ~P~y - ~P~y = -IIPzyllax ay

where we have used the following relations (valid assymptoticallyas lfl: ~ 0)

vxCz) = v~(l) = v~(-l) = v~

vy(z) = vW) = v~l(-l) = v;,

Fig, 4. Discontinuity in normal pressure tensor,

(12)



and where we have defined

p" = t (p - p)dz-I

as the surface excess mass density,

F~= t (Fy-Fy)dz-I

(13)

(14)

as the surface excess external force vector (y-component) and

P~{3 = L(Pa{3 - Pa(3) dz (15)

as the surface excess contact stress tensor where a: = x,y andf3 = x,y.

But p" is generally a very small quantity (Fig. 5), so that, particularly for the case of present interest, where viscous stress inthe interfacial region is large, surface inertial stresses are negli-

pI ~I

.!,Il

Fig. 5. Conjectured inertial and viscous stress profiles.



gible. Equation (11) may then be written, neglecting inertialterms as

F~(x,y) + ~P~y(x,y) + ~P~y(x,y) = IIP'y(x,Y)II. (16)ax ay

The i, and i, components may be derived in an identical mannerto (16) to yield the general surface vector equation

where

Vs • PS(x,y) + FS(x,Y) = i, 'IIPII, (17)

and

P' = P~ixix + P~yixiy + P~xiyix + P~yiyiy, (18)

F" = F~ix + F~iy + F~iz (19)

(20).., . a . aV s = l x - + l y - .ax ay

Equation (17) is the general local surface excess linear momentum equation for the "surface" A. In the absence of "excess"rheological behavior in the interfacial region (17) reduces to theclassical condition for the singular planar surface

i, '111'11 = o. (21)

Two characteristics of the surface stress condition (17) whichwill prove important in the subsequent reports may be noted.Firstly, the nonlocal interfacial force G does not explicitly appear in (17) owing to the placement of the interfacial boundariesbeyond the region of inhomogeneity. In the following paper whereinterfacial zones may be highly curved, the region of inhomogeneity will in general extend beyond the interfacial zone andhence G will play the explicit role of a normal surface excess force:this is the second novel characteristic of(17) [see (19)].The normalcomponent of the surface excess force is typically neglected inanalyses of interfacial stress, as inclusion of this component contradicts the Young-Laplace condition in hydrostatic equilibrium.Beyond the role which the normal surface excess force will playin the large surface curvature theory, normal surface excess forcebecomes equivalent to the disjoining pressure for two planar surfaces in close proximity."



The Interfacial Pressure Tensor

The application of (17) as a boundary condition to bulk flowrequires a constitutive relation for P' or more directly, through(15) for the interfacial pressure tensor p(r).

Consider then (4) with

G(r) = - '7 . g(r)

and

F(r) = -'7 . f(r)

so that, neglecting inertial terms

p(r) = f(r) + g(r) + A,

(22)

(23)

(24)

where A is an undetermined constant dyadic.It is apparent that the constitutive expression for P derives

fundamentally from the intermolecular interaction force densityG, or alternatively the potential g.

Equilibrium Interfacial Pressure Tensor

In equilibrium, and in the absence of an external potential, wehave from (6) and (24)

p(z) = -pI + {g(z) - gl},

where p is the scalar hydrostatic pressure in the bulk fluid

pel) = P(-I) = -pI

and

gl = g(l) = g(-l).

We have also used

(25)

(26)

(27)

(28)

Equilibrium Intermolecular Force Density

Edelen" has developed an explicit form of the equilibrium intermolecular force density G for the case of isotropic intermolecular interactions, which we will recapitulate here, as theresult is both illuminating to the nature of the nonlocal residual



quantity and also allows a more explicit definition of the surfaceexcess pressure tensor in equilibrium.

We let r: represent a second position vector in V and k(r; r) theforce exerted upon a molecule at r by a molecule at r'. For simplicity we consider the single component case where intermolecularinteractions are completely isotropic. The extension to multicomponent systems and anisotropic interactions would appearstraightforward.

Then rewrite k(r; r') in terms of a correlation function 1] as

k(r;r') = 1](ir' - rl) ( - rlr - r

or with N denoting the number of molecules per mass

f 1](lr' - ri)G(r) = N d 3r 'p(r') (r' - r).

v [r' - r]

Define

R = r: - r

and let

JA J1T J21Tv, R 2 sin 8d8def>dRo 0 0

(29)

(30)

(31)

(32)

represent the subspace in V containing the complete nonlocalinteractions between the local point r and the nonlocal environment (Fig. 6). Here also R = IRI.

Expanding the density in a Taylor's series expansion about localposition as

1p(r') = p(r) + R . Vp(r) + 2(R' V)2p(r) + ... , (33)

it follows directly that

G(r) = i, 4;N fR 21] (R){R :: + ~ R 2 :~ + .. .} dR

(34)

assuming the dominant inhomogeneity is in the normal i,coordinate.



Fig. 6. Exaggeration of nonlocal subspace Vo in V.

The source of the nonlocal intermolecular force vector in theinterfacial region is then seen in (34) to be directly related to theinhomogeneity of the region.

Equilibrium Surface Excess Pressure Tensor

We note from (30), with also

'V(r' - r) = - r-rjr - r

that

G(r) = -NV Iv d 3r 'p(r ' )h(lr ' - rl)

where

h(y) = rT/(w)dw,

so that by (22)

g(z) = INIv p(r')h(jr' - rlld 3r',

(35)

(36)

(37)

(38)



is the equilibrium intermolecular potential in the interface arising from isotropic intermolecular interactions.

Isotropic interactions then naturally lead to an isotropic pressure in the interfacial region from (25)

where

p(z) = -(p + {g(z) - gl})I,

g(z) = g(z)I.

(39)

(40)

[In (39) we see clearly that for large intermolecular potential g(z)(large density gradients in the interfacial region) the interfacialpressure becomes large and negative].

The surface excess pressure tensor follows from (39) and (15) as

P' = (Is. (g(z) - gl) dz (41)

or

r- = Isr{g(z) - g/}dz.-I

We may define equilibrium surface tension as

P' = aJs ,

so that

(J = r{g(z) - gl}dz.-I

(42)

(43)

(44)

Equations (42) and (44) represent definitions of the surface excess pressure tensor and surface tension for the single-componentisotropic-interaction case, and are primarily intended to illustratethe method by which a constitutive form for P' is obtained usingthe nonlocal surface excess approach implicit in (17).

Nonequilibrium Surface Excess Pressure Tensor

Development of a nonequilibrium constitutive relation for P'i"

using (24) is a far more difficult matter. Such a development willnot be attempted here. The primary intention of the present section is but to indicate the constitutive assumption implicit (to the



present theory) in the Boussineeq-Scrivenv" dynamic stressboundary conditions at a planar fluid interface.

Let

p(r) = -Pt(r)Is - pi.i, + (K - ~ f.L)V' • v(r)I

+ f.L(V'v(r) + Vvt(r)), (45)

where the surface idemfactor is defined as

(46)

Pt i13 the tangential component of the interfacial pressure (weassume implicitly here anisotropic interactions in the interfacialregion), K and f.L are viscosity coefficients, and t denotes the transpose operator.

Substituting (45) into (15) using also (12) yields

P' = aI, + (KS - jJ.!)V's • v'I, + f.LS(V'sv s + V'svst)

(47)

where we have defined

u= rCi5 - pt)dz-I

(48)

as the surface tension,

f.Ls = JI (f.L - Ti)dz-I

as the surface excess shear viscosity and

JI 1

KS = (K - K) dz + -f.L'~I 3

(49)

(50)

(51)

as the surface excess dilatational viscosity.Substituting (47) into (17) we have in component form, for con

stant KSand f.Ls

au a (au S au S

)IIP,yll = F~ + - + (KS+ f.LS)- _x + -yay ay ax ay

a (au S au S

)_ f.Ls- _x __Y

ax ay ax


and

SURFACE RHEOLOGY I

IIP... II = F~ + au+ (KS + P!).i-(au~ + au~)aX ax ax ay

a (au. au.)+ p,'_ _x _ _ Y

ay ay ax

IIPzzl1 = F~,

443

(52)

(53)

which are the classical Newtonian boundary conditions at a fluidinterface with an intrinsic viscous behavior [with the notableexception of the normal surface excess force in Eq. (53)].

CONCLUSIONS

We have presented a formulation of the dynamics of the planarfluid surface, elaborating specifically upon surfaces possessing aNewtonian stress behavior. It has been shown within the contextof continuum theory, that a fundamental relation exists betweenthe surface excess stress tensor and nonlocal intermolecular interactions in the interface. Further elucidation is clearly calledfor in the context of a statistical interfacial theory.

Special thanks are extended to Professor Howard Brenner for many helpfuldiscussions. This work was supported by the National Science Foundation. One ofus (D. E.) was also supported by an Amoco Foundation Doctoral Fellowship.

APPENDIX

In integral form, the linear momentum equation for the interfacial volumeelement V may be expressed as

f [!!.... pv - F] dV - ( m : PdA = 0, (Al)v Dt Jav

whereD[Dt is the convective derivative, aV is the bounding surface to the volumeelement V, and m is the unit outward normal to the bounding surface. To themacroscopic observer, the interfacial volume element V is a continuum element offluid bisected by a surface of discontinuity, hence with Green's transformation fora body possessing a surface of discontinuity" it follows that

(A2)



where we maintain the definitions used in Eqs. (8)-(10), omitting the overbarnotation. The interfacial volume element V may be defined by

J dV = JdxJyJI dz ,v x d -l

and the dividing surface A by

(A3)

(A4)

which requires that the dividing surface be the planar surface at z = o. With thepreceding equations then

or

fl dz[D pv - F - V·p - n'IIPI1iJ(Z)] = O.-I Dt

Equation (A5), expressed in component form, yields Eq. (8).

References

(A5)

(A5)

1. J. A. F. Plateau, Bull. Acad. Belg., Ser 2, 34, 404 (1872).2. J.L. Erickson, J. Rat. Mech. Anal., 1,521 (1952).3. S. G. Oldroyd, Proc. Cambridge Phil. Soc., 53, 514 (1957).4. L. E. Scriven, Chern. Eng. Sci., 12, 98 (1960).5. J.e. Slattery, Chem. Eng. Sci., 19,379 (1964),6. M. J. Boussinesq, Ann. Chim. Phys., 29, 349 (1913a).7. R.J. Mannheimer and R. S. Schechter, J. Colloid Interface Sci., 32, 225

(1970).8. J. W. Gardner and R. S. Schechter, in Colloid and Interface Science, Vol. IV,

ed. M. Kerker, Academic Press, New York, 1976, p. 421.9. J. W. Gardner, J. V. Addison, and R.S. Schechter, AIChE J., 24,400 (1978).

10. J. D. Eliassen, Ph.D. thesis, University of Minnesota, Minneapolis (1963).11. e. L. Murphy, Ph.D. thesis, University of Minnesota, Minneapolis (1966).12. F.P. Buff, J. Chern. Phys., 25,146 (1957).13. F.P. Buff and H. Saltzburg, J. Chem. Phys., 26,23 (1957).14. A. R. Deemer and J. e. Slattery, Int. J. Multiphase Flow, 4, 171 (1978).15. F.e. Goodrich, Proc, Royal Soc. London, A374, 341 (1981).16. A. Scheludko, D. Platikanov, and E. Manev, Disc. Faraday Soc., 40, 253

(1965).



17. D.G.B. Edelen, in Continuum Physics, Vol. IV, A.C. Eringen, ed., Academic Press, New York, 1976, p. 75.

18. J. C. Slattery, Momentum, Energy and Mass Transfer in Continua, McGrawHill, New York, 1972, p. 18.

Received February 2, 1986Accepted December 7,1987


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Surface Rheology I. The Planar Fluid Surface