Stochastic interfacial disturbances created by thermal noise and the importance of the interfacial viscosities

Stochastic Interfacial Disturbances Created by Thermal Noise and the Importance of the Interfacial Viscosities A Z A R D O K H T HAJILOO AND JOHN C. SLATTERY Department o f Chemical Engineering, Northwestern University, Evanston, Illinois Received April 23, 1984; accepted October 29, 1985 At the interface between two fluids at rest, small disturbances that are stochastic in space and time are created by thermal noise. These perturbations are represented as Fourier series in spatial position, the amplitudes of the various Fourier components being random functions of time. Thermal noise is taken to be a mutual force representable as the gradient of a potential. By analogy with the treatment of Brownian motion, this potential is modeled as a white noise. The spectral density of the amplitude of any given Fourier component is shown to be a strong function of the interfacial tension and of the viscosities of the adjacent phases. Under certain conditions, the spectral density is also a strong function of the Gibbs surface elasticity, of the interfacial viscosities, and of gravity. The results will be useful in refining a demonstrated technique for measuring ultralow interfacial tensions and interfacial tensions at elevated temperatures and pressures. © 1986AcademicPress,Inc.

INTRODUCTION

Smoluchowski (1) and Mandelstam (2) observed the scattering of the light reflected from an interface between two static fluids and identified this phenomena with the existence of small amplitude random waves at the interface. Mandelstam (2) related the mean value of the intensity of light scattered in a particular direction to the mean square of the amplitude of a Fourier component of these waves, which is inversely proportional to the interfacial tension but independent of the bulk viscosities and of the interfacial viscosities. (See Discussion for details.) Unfortunately, the time scale of the stochastic motion is too short to permit an accurate observation of the mean value of the intensity of the scattered light. The appearance of lasers and the development of spectroscopic techniques made it possible to measure statistical properties of the scattered light that could be simply related to the statistical properties of the capillary waves. Papoular (3), Meunier e t al. (4), and Bouchiat and Meunier (5) developed expressions for the

spectral density of the amplitude of any given Fourier component of these waves. Langevin and Bouchiat (6, 7) computed the spectral density assuming a form of surface viscoelasticity. Critiques of these developments are given under Discussion. After an initial series of experiments that demonstrated the viability of the technique (812), Pouchelon et al. (13) demonstrated its potential for measuring ultralow interfacial tensions. Kim et al. (14) not only measured ultralow interfacial tensions, but they also showed agreement between these measurements and those obtained using the sessile drop technique. There have been several studies of the parameters in a model for surface viscoelasticity (7, 15-17). In what follows, we present a new analysis of this technique in which it is explicitly recognized that these small amplitude capillary waves are the result of thermal noise, a stochastic mutual force. The effects of surface tension, of the Gibbs surface elasticity, of the surface viscosities, a n d . o f gravity upon the spectral density are examined. 325

Journal of Colloidand InterfaceScience, Vol. 112,NO.2, August1986

0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All rightsof reproductionin any form reserved.

326

HAJILOO AND SLATTERY STATEMENT OF PROBLEM

Consider two fluids at equilibrium in a closed, rigid, perfectly insulated container. Small amplitude capillary waves are created by thermal noise at the interface between these otherwise static fluids. The light from a laser beam reflected from this interface is dispersed by these capillary waves. We intend to relate some characteristics of this light dispersion to the interracial tensions and the two interfacial viscosities. We will require the following assumptions about the physical problem. (1) The mutual force per unit mass (18) bl{)) or thermal noise in phasej is representable as the gradient of a scalar potential ~(J)* that is a random function of time at any given position in space: b(J)* (t)

=

_~7~(j)* •

[1]

Here and in what follows the superscript...* is used to denote a dimensional quantity. (2) The fluids are incompressible and Newtonian: S (j)* ~ T (j)* "-[- p(J)* I

--- 2#(J)*D(y)*.

[2]

Here S (j)* is the viscous portion of the stress tensor in phase j; ~i)* is the stress tensor; p(J)* the pressure; I the identity tensor that transforms vectors into themselves; ~t(y)* the shear viscosity for phase j; D(j)* __- 1 [Vv(j)* + (Vv(J)*)T]

[31

is the rate of deformation tensor and v(y)* is the velocity vector. (3) The fluid-fluid interfaciai stress-deformation behavior can be represented by the linear Boussinesq surface fluid model (19, 20)

s.(o)" ~ T(~)* _ ~*p. = [(K* - e*)div(~)v(~)*].P + 2¢*D.(")*. [4] Here S (~)* is the viscous portion of the surface qv(~)* stress tensor; ~- the surface stress tensor; y* the interracial tension; P the projection tensor that transforms vectors defined on the .dividing Journal of Colloid and Interface Science, Vol. 112,No. 2, August 1986

surface into their tangential components; r* the interfacial dilatational viscosity; e* the interfacial shear viscosity; v(~)* the surface velocity vector;

p(~)* --- ½[.P. v(~)v (~)* + (v(~)v(~)*) T. .P]

[51

the surface rate of deformation tensor; V(,) the surface gradient operator and div(,) the corresponding surface divergence operation (21, 22). (4) The tangential components of velocity are continuous across the dividing surface. (5) The dividing surface is located in such a manner that [see footnote 1 of (23)] p(0~)*

p(2).L. < 1

[6]

where p(0~)* denotes a characteristic value of the total mass density of the dividing surface, o(2)* the density of the lighter adjacent bulk phase, and L~ a characteristic wavelength to be specified later. (6) Any mass transfer to or from the dividing surface is neglected. The adjacent phases are in equilibrium. The characteristic time for the interfacial disturbances is so small that we can neglect any transfer of surfactant between the interface and the adjacent phases either by diffusion or by adsorption. (7) The characteristic time for the interfacial disturbances is so small that we can neglect the effect of surface diffusion with respect to surface convection. (8) The two adjacent phases are unbounded in the z* direction but - L * ~< x* ~< L* - L * ~< y* ~< L*. The cross section of the container is square, measuring 2L* on a side outside the immediate neighborhood and influence of the walls. We do not require the velocity vector to go to zero at these walls nor do we impose a fixed contact angle at the three-phase line of contact. We will assume here that the positive direction along the coordinate z* points into the

327

STOCHASTIC INTERFACIAL DISTURBANCES

fighter phase, phase 2, in the opposite direction to gravity. GOVERNING EQUATIONS AND THEIR SOLUTIONS

Our first objective is to characterize the capillary waves generated by thermal noise. In terms of dimensionless variables, the equation of continuity for each phasej = 1, 2 requires div vO) = 0 [8]

By t*, we mean a characteristic time to be defined later. The jump mass balance or mass balance at the dividing surface is satisfied identically as the result of assumptions 5 and 6. The j u m p m o m e n t u m balance or force balance at the dividing surface becomes in terms of dimensionless variables (24, 25) V(~)7 + 2 H y f i + NdV~div(~)S (~)

+ NReNea(~D(1)-I ~p(2))~ + NuNcaS(2).~

and Cauchy's first law demands 0¥ (l) __

Ot

_~_ VV(I). ¥(1) ~

1

__V~/)(1)_~_ ~ R e

diV

- N~a~.

S(l) ~

-

[9] 0v (2)

--

+ Vv(2)" v(2) = -V/)(2) + ~ N.

div S(2)" ~

H = L~H *

[101

[11]

with the understanding that the force per unit mass of gravity is representable as

v(J)* --

t~vq)*

~o(J) ~.

t~2(~o(j)* _ p~)

L~

p(J)* L~j 2

S(J) _ t~8 (j)* ~

/.t(j)*

x*

y*

z*

t*

z----L~

t~-t*

[13]

and as dimensionless parameters Np

/0(2)* p(1)*

N/,

/.t(2)* U(1),

¢*

(a=-

g*L~

[16]

e*

K*

g(1)*L~

N~ #OI.L~ N, vmL, N~a-3"*t~ (p(1)*(I)(ol)* _ p(2)*~])(2)*)t~:2

g*t~ 2 Ng~

L*

N¢=--

p(1),L~ 2

[171 Here H * is the mean curvature of the dividing surface, ~ is the unit normal to the dividing surface pointing into phase 2, ¢~0j)* a characteristic value of the thermal noise in phase j, p~' the pressure that would exist at the dividing surface in the absence of thermal noise, g* the magnitude of the acceleration of gravity, and 3"t the equilibrium interfacial tension or the interfacial tension at the concentration of surfactant corresponding to a planar interface. In view of assumptions 6 and 7, the j u m p mass balance for surfactant or the mass balance for surfactant at the dividing surface (24, 25) reduces to

O(S) + V(')0{; }" ('(')m ") + O{;}~V( .)V(') = 0

p(t)*L~2

N R e - U(0,tt

[15]

and as dimensionless parameters

[12]

In [8] through [10], we have introduced as dimensionless variables

7"

3"---3'0

p(l)*(I~(l)* _ p(2)*(I)(2)* (I) ~ p(l),t~(01)- -- p(2),tl~(02)*

Here

~(J)* =-p(J)* + p(J)*d)* + p(J)*d~(J)*

NRoNc.N~¢~ = O.

We define as dimensionless variables

Ot

b~g) = - V ¢ * .

" ~ - NReNcaNg(1 - N , ) ¢ ~

[14]

Ot

[18]

JournalofColloidandInterfaceScience,Vol. 112,No. 2, August1986

328

HAJILOO AND SLATTERY

in which we have introduced as dimensionless variables (~)* t~'u* (~) P(s) P(S)------ (~)*

[l-~--

P(s)o

v(~) -=

L~

/~V(~) *

[191

L8

[20]

in which 3q is the dimensionless Gibbs surface elasticity evaluated at the equilibrium surface concentration of surfactant. We seek a perturbation solution to this problem, taking as our perturbation parameter N+, a measure of the thermal noise generating the capillary waves. The zeroth pertubations of all variables, corresponding to the static problem in the absence of thermal noise, are identically zero. The first perturbations of [8] through [101 reduce to [21]

div ¥~J) = 0 0v~l)0t= -V~°~') + ~

= _V~o~2) +

div(Vv~')) N.

02U ~a)

+ N+) - OX 2

O~u~+)

+ N,-jy ou? ) + o w ? )

-

0~-

+

div(Vv]2)).

[22] [231

JournalofColloidandInterfaceScience,Vol. 112, No. 2, August 1986

o~v? )

+ lOu 2> . OwT)

~-x

~¢"L--~--z + --~-x )

[24]

3"1 op(s)l + (N. + N ) - Nca Oy Oy2

o2v~~)

O2u{~)

+ N'-~-xz + N" OxO--'---y -

OV]I' ..~ OW]I> [OI)~2> OW]2)~ Oz ~ - N"t--~-z + - ~ - y ]

Oahl + 02hi _ 2Nca(Ow~l) Ox2

fifTy2

[25]

c9w(12)~

t--~--z - N . -"~-z ]

+ NcaNm,(P(1'+- NpP~2)) - N~NReNp, X (1 - Np)hl - NcaNRe+ = O.

, (+)

3' = 1 - 3"1tO(s)- 1) + • • •

Ot

,~ (a) 3"10P(S)I _~_ (N.

N~. Ox

By PIll* we mean the surface mass density of (~)* surfactant S, p(s)0 the surface mass density of surfactant corresponding to a planar interface, and .* the velocity of a point on the interface whose surface coordinates are fixed. Equations [8] through [10], [15], and [18] are to be solved simultaneously consistent with the requirements that velocity be continuous at the dividing surface (assumptions 4 and 6), that very far away from the dividing surface both phases are static, that the bulk phases are Newtonian fluids (assumption 2), and that the interfacial stress-deformation behavior can be represented by the linear Boussinesq surface fluid model (assumption 3). In solving this set of equations, we will recognize that the interfacial tension y* is a function of the surface concentration of surfactant

vl C~(2)

Let u~, v~, and wx be the x, y, and z components of vt, the first perturbation of velocity; (~) let p(s)t be the mass density of surfactant. In view of [20], the x, y, and z components of the first perturbation of [ 15] demand at z = 0

[26]

Our understanding here is that the configuration of the interface takes the form z* = h*(x*, y*, t*) = h~(x*, y*, t*)N+ + • • •

[27]

with h i ~-

h~ LB"

[281

The first perturbation of [ 18] reduces to

o (o) Ou~) ov~~ P(s)----!+ + = 0. at ~

[29]

From [21] through [23], we can establish that

(+)

NRe ~ -- 92 (O2w~1)) = 0.

[30]

NRdVp fit - N~D2 (D2w~2)) = 0

[311

STOCHASTIC

INTERFAC1AL

329

DISTURBANCES

and

where 02

De - = -

(92

+--

OX 2

(92

+--

Oy 2

asz~-~:

[32]

OZ 2 "

wt l ) ~ 0 .

[37]

Seeking a solution of the form

In arriving at [30], for example, we have differentiated the x component of [22] with respect to x and the y component of [22] with respect to y, we have added the results, and we have eliminated u~ and v~ using [21]. We subsequently subtracted the derivative of this with respect to z from the equation obtained by operating

hi,

j),

°),

P(s)l

oo

=

~

[hmn(t),

(j) . . Wmn~t,

(a)

~AO(j)(

Z), Wmn(t), --rnnl, I, Z),

m,n = --oo

pl~Im.(t)]exp[iTr(mx + ny)LJ /L*] we conclude from [30] and [31] that (NR~ Ota _ D1)(Dlwm.) 2 2 (0 = 0

on the z component of [22]. In a similar fashion, we can take the derivative of [241 with respect to x, the derivative of [25] with respect to y, add the results, and take advantage of [21 ] to find that at z = 0

2

0.

02 - D2 = Oz 2 -

Oh, Ot

[331

_2xr

,D(I)

,l,R ....

aw] 1) --Oz

aw]2) Oz

[35]

Finally, very far away from the dividing surface, asz~oo: w]2 ) ~ 0 [36]

_

= ~

[42]

a 2 _

NR e

Ot]\ OZ ] [43]

a2Xr r,T ~(2) I vp2 * R e ~ m n

-

[34]

Because the tangential components of velocity are continuous across the dividing surface (assumption 4), we can use the equation of continuity [21 ] to say atz=0:

[41]

Here we have taken advantage of the orthogonality of the functions exp[iTr(mx + ny)L~/ L*]. Using the x and y components of [22] and [23], we can establish

=

Wl(,) . w.] 2 ) . w~) .

a 2

LU

In order to determine hi, w~j), w~~), and ~o~J)as functions of position and time, we must solve simultaneously [30], [31] as well as the x and y components of [22] and [23] consistent with [26], [29], [33] and the following boundary conditions. Since there is no mass transfer at the dividing surface (assumption 6), we have atz=0:

(2)

a 2 =- 7r2(m2 + 172)L , 2 .

0721

(W~') -- NuW]2)) =

2

[391

in which

~ 2 (~) 02 (~) ~ 0 3'1 [O O(s)l o-p(S)l~_ (ArK+ N,) - Nca k~-T-x 2 + OY2 ] Oz

-L-S+-0-U,+(0z 2

[38]

[441 In arriving at [43], for example, we have differentiated the x component of [22] with respect to x and the y component of [22] with respect to y, we have added the results, and we have eliminated u~ and v~ using the equation of continuity. With the assumption that we can represent cx3 =

E re,n= - o o

¢~mn(t)exp[iTr(mx + ny)LJ/L*] [45]

Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

330

HAJILOO AND SLATTERY

we find that [26] and [33] through [35] require atz=0

Bmn('r)

= limit -( -? 2

L T hmn(t)hmn(t + r)dt.

T ~ oo

[54]

[a 2 + NeaNReNg(1 - No)]hmn "0 <1) N~ OW(2m)n~_ NeaNR e -F 2 N e a ( - - - ~ Oz]

× ,--runtY(I)-- N , , ~ ) n ) + Nc.NR~{mn = 0 [46] (1) ~/2"Y1 (a) OWran Nca P(S)mn~ a2(N~ + N,) a--~ - -

An expression for Bm,(r) that is more convenient for our present purposes than either [53] or [54] is (27) Bmn(r) = (N~) 2

. . . . . oo t

!

h m n f ( h 11. . . . . + ( ~02 +a]tWm~O)-'~ W(1) = W(2) mn

~V,Wm,) (2). = 0 , (~)

Ohmn

rnn = W m n

(1)

.

"~

Ot

[49]

Oz

(2) Wmn ~ 0 (1) W m n --~ O.

as z ~ - o o :

[50] [51]

Equation [29] together with the equation of continuity [21 ] and the fact that the tangential components of velocity are continuous across the interface require at z = 0 O P(S)mn (~')

Ot

., (l) O W mn - -

=

Oz

O.

[52]

Since ~ m n ( t ) is a random function of time, all dependent variables in this problem are also random functions of time. Our objective will be to determine certain statistical properties of these variables such as Bran(t, r), the correlation function in time of hmn(t), Bmn(t , 7) ~

(hmn(t)hmn(t

+

r))(N~) 2.

hm~(0)

• • .)dh'~l..

"dh'mn''"

[551

H e r e f ( h 1 1 , . . . , hmn, • " " ) is the joint probability density and (hmn(r)[h11(O) = h'11 . . . . . hmn(O) = h'mn, ' ' ' )

Similarly, [36] and [37] demand at z = 0 as z--~ oo:

(hmn(r)[h11(0) = h'~l . . . . . = h',,

[48]

(2)

Wren _ O W mn

Oz

[47]

t

hmn, " " - )

[53]

By (x), we mean the expected or mean value of x. The system that we are discussing here is stationary in the sense that Bm,(t, r) is independent of t and is only a function of r. For such a stationary process, a good estimate for the correlation function can be obtained by using the ergodic theorem [or a less general result, the law of large numbers (26)]: Journal of Colloid and Interface Science. Vol. 112,No. 2, August1986

is the conditional expectation or the expected (mean) value of the random variable hmn(r) assuming hi 1(0) = h'~l . . . . , hmn(0) = h ' , . . . . . Our understanding in writing [55] is that the series for h~ in [38] has been truncated with a finite number of terms. Before computing Bmn(-r), we will determine in the next two sections expressions for the conditional expectation and the joint probability density appearing in [55]. CONDITIONAL EXPECTATION In computing the conditional expectation of hmn('r), we will require two additional assumptions. (9) The conditional expectation of hmn(7") is independent of the velocity distribution that existed at t = 0. (10) As in the case of Brownian motion, the scalar potentials for the thermal noise can be represented as white noise, which implies among other things that (28) (t~mn(t)) = O.

[561

In view of assumption 9, we will seek solutions to [39], [40], [43], and [44] consistent with

STOCHASTIC INTERFACIAL DISTURBANCES boundary conditions [46] through [52] and initial conditions

att=0:

h.,,=h',

allm. n

331 -(i) ~ Wmn

as z ~ - ~ : Sp(s)mn

[57]

Oz

[68]

0

O.

[69]

Solutions to [59] and [60] consistent with [67] and [68] are

and att=O:

w• (J) ,~=O

all m, n and j = 1, 2. [58]

The Laplace transforms in time of [39], [40], [43], and [44] are

(SNRe + a2 -- O2 ][ae ~ -- O~)~(lm)~ =

fl mn u) = A exp(az) + C exp(bz)

[70]

if(2) mr/ = B exp(-az) + D exp(cz)

[71]

with

b = Va 2 + SNRe (root with positive real part)

[59]

[72]

c = -1/a 2 + SNRdVo/N

(SNRdV° + N f l 2

02][a2 az2l \

-

02 ]+,2, : -

,.°

(root with negative real part). [60]

Boundary conditions [65] and [66] demand A + C = B + D

u .*Re. mn

--

-- SNRe

[61 ]

= N.~-N.a

2-sN~Re

~

[62]

in which the Laplace transform of a variable is denoted by . The Laplace transforms of [46] through [52] are at z = 0 NeaNReNg(1 -

aA + bC = - a B + cD.

[75]

0 Cm'.

X t,--.~ ,~o) -

= Sh,~. -- h;~.

- 2a2N~B - N.(a 2 + c2)D = 0.

[76]

~(2) --ran

+ [a 3 + aN~aNRgVg(1 - No) + 2ab X (1 - Nu)Ncas]C + asN~uNRe~mn

Equations [74] through [77] may be solved for A, B, C, and D. Using these results as well as [65] and [70], we can calculate

[65] h m n = ! (~((lm)n ~-

[66]

Oz oo:

[641

r~(2)

,,ran = ~,'mn

as z ~

+b 2

+ [a 3 + aN~aNRdVg(1 - No)]h'mn = 0. [77]

+ a 2 (ff,~ - N. --m.,~(2)~= 0

Oz

+

+ N~aNReS2]A + NpNcaNReS2B

a23"1 ~(a) 01~(ml)n N~a /)(S)mn q- a2(N. + iV,) 0----~

0,~3(I)

+ a2b N . + N ~ + - ~ - ~

2a2~

[a 3 + aNeaNReNg(1 - No) + 2a2Nca(1 - N.)s

- N~NRe

Np~(2)mn) + Nc~NRe~m~ = 0 [63]

~(~ m~ = ~2~ rv mn = ~

3", _11 [a3(N~+N~+Ncas,+

Finally, [63] tells us

No)lhm.

+ 2Nc~(1 - N u) - ~ z

+

[74]

Equations [64] and [69] require

a2NoNRe~)~

[a 2 +

[73]

0

s

h'~.)

1

0

[67]

=-(A+C+h'~.) s Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

332

HAJILOO AND SLATTERY = h ' n {1 + [a 3 + aN~aNReNg(1 - N o ) l J } s + aNeaNReJg~mn

[78]

can be expressed as S(hm,(t)) = ~

where

aNcaNRe

f;+ f0 -iv

,o

eS(t-u-v)J(u)O,,n(v)dudvds

J = J ( s ) =- (a + c)(b - a)[aZ(N. + N,) + a + b - N . ( e - a)][det(Ko)] -l

[79]

and det(Kij) is the 3 × 3 determinant whose entries are K~ = - a -

where J(u) and Om, (v) are the variables whose Laplace transforms are j and ~mn- Noting that [Proof follows using the Fourier integral theorem (29).]

f~

°eiy(t-u-V)dy = 27rS(t- u - v)

c

[83]

[84]

oo

K12-- b-

c

in which 6(x) is the Dirac delta function, we can integrate [83] twice to find

K13 = - - a + c

(

S(hmn(t)) = aNcaNRe

K21~- a 3 N~ + N~ + Nc~ s !

J(t - V)Om~(v)dv.

[851

+ 2a 2 - N , ( a 2 + c 2)

(

11)

Since the future beyond time t will have no influence on S(hmn(t)), [85] reduces to

K22=-a2b N. + N. +-~ a

+ a 2 + b 2 _ N , ( a 2 + c a)

S(hmn(t)) = aNcaNRe

2

J(t - V)Cbmn(v)dv

1£23 ==-N~(c 2 - a 2)

[86]

/£31 --= a 3 + aN~aNR~Ng(1 - No)

or alternatively

+ 2aZN~.(l - N . ) s + N~aNReS 2

K32 -~ a 3 + aNcaNReNg(1 - No)

S(hmn(t)) = aNcaNRe n=O

17

cb

t7

+ 2ab(1 - N . ) N c . s K33 ~ N ~ c a N R e $2.

[80]

In view of [56], we observe that

Taking the inverse Laplace transform, we find

1 f~+i~o eSt[tm~ds

hm,(t) = ~ - ~ o.-i~

for t > 0. [811

The real number u is arbitrary, so long as the line along which the integration is to be performed lies on the fight of all singularities of the argument in the complex plane. From [78] and [81], we see that the stochastic portion of h,~n(t) S ( h m . ( t ) ) =-- ~-~ aNc~NRe

e~t]~m.ds

[82] Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

( S ( h m n ( t ) ) ) = 0.

[881

Given [88], we conclude from [78] that for r>0 =

1 f~+i® eS, hmn

× {1 + [a 3 + aN~,NR~Ng(1 - Np)]J}ds. [891 Because B , ~ ( r ) , defined by [53], is independent of t for this stationary random process, it must be a symmetric function o f t . By [55], the conditional expectation must also be a symmetric function of r and for r < 0

333

STOCHASTIC INTERFACIAL DISTURBANCES l

~v+ioo

p

__ e_,~ hmn = 27ri ~.-io~ s

(hm.(r))

× {1 + [a s + aN~aNR~Ng(I - N p ) ] J } d s . [90] An expression valid for all r can be obtained by adding [89] and [90] (29): h ' . ~,+ioo _1 (e" + e -~) (hmn(r)) = - ~ ~p-ioo S x {1 + [a s + aN.NRW~(1 - NAlJ}ds. [91]

For all cases tested, the two singularities of the argument of this integral lie to the left of the imaginary axis, either on the negative real axis or symmetric with respect to the negative real axis. For this reason, we are free to take ~, = 0 and write [91 ] as

formed state at time t = 0 and that corresponding to the equilibrium plane interface; AP* the difference between the potential energies corresponding to this deformed state and to the equilibrium plane interface; T* the temperature; and k* the Boltzmann constant. The proportionality constant is determined by requiring

f?

. . . . . .

f(hl, ....

, hmn, • " ")

× dh'~l" " " dh'mn . . . .

1.

[96]

We can compute AA* = y*

;f{I L*

hm, fro I (ei~o~+ e_i~o~) (hrnn(r)) = ~ o0 l-~

L*

1 + (Oh*~ ~,Ox* ]

(°h*211'2-1}ex*d,*

+ \Oy*/J

× {1 + [a s + aNeaNReNg(1 - Np)]J}dw. [92]

Note that from [79]

-2

oo

[93]

J(iw) = J ( i w )

where the overbar denotes the complex conjugate. In view of [93], Eq. [92] yields the desired conditional expectation

= 2"r*L*2(N~)2

~

[97]

a2(hm.) 2

m,tl=-cx3

and

(hmn(z)lhH(O) = h'll . . . . .

hmn(O) = hmn, " " ")

kay] J

3-1=L L L~,Ox]

h*(p(l)*

AP* = L*

0(2)*)

L*

1 = - h'mn[a 3 + aN~,NReNg(1 - No)]

× g*z*dz*dx

*dy*

7r

= 1 (p(l)* _ p(2)*)g,L~4

×

Im(J(io~))exp(kor)do~.

hZdxdy L

[94]

oo

= 2(p0)* - p(2)*)g*L~ 2L*2(N¢)2 c~

JOINT PROBABILITY DENSITY

By Boltzmann's principle (30), the joint probability density for our isolated, isothermal system takes the form t

f(hll .....

L

t

hmn, " " ")

exp(

X

k*r ~

/

(hmn)2

rn, n= -0o

= 2 y ' L * 2(N¢)2NcaNR~g(1 - No) ×

AA* + AP*~

~

~

(hmn) 2.

[981

re,n= -oo

[95]

Here AA* is the difference between the Helmholtz free energy corresponding to this de-

Substituting [97] and [98] in [95], we see that the joint probability density can be rearranged in the form Journal of Colloid and Interface Science,

Vol. 112, No. 2, August 1986

334

HAJILOO AND SLATTERY 0.3

and in this way determine

1 ~2"r*L*2(N,) z

~t~0"2

Ni~'*e ~

× [a 2 + NCaNReNg(l - Np)]l 1/2. J SPECTRAL DENSITY OF

_>I03

i

92

i

96 t.,O

1O0

Cm,(~o) =-

104

Bm,(r)exp(-kor)dr.

FIG. 1. C** as a function of ¢ofor a liquid-gas system 10, T1 = 0 , and various values of N.+,. The solid lines represent [104] and the dashed lines [1231.

In view o f [55], [94], [991, [100], a n d [102], this takes the explicit form

Cmn(W )

~"

2(N¢)2[a 3 + aNcaNReNg(1 - Np)]

× 1 Im(](i~o)) h'.,

03

• • .)

2-y'L* 2(N~)2 k'T*

"~ exp

[103]

oo

w i t h Nea = 1 0 - 4 , N g ~

f(h'l, .....

hmn(t)

Rather than B,~.(r), the correlation function in time o f h,~n(t) defined by [53], we will find it m o r e convenient to speak in terms o f the spectral density o f h,~.(t), the Fourier transform o f Bmn('r):

o.,

0

[102]

(h~l . . . . .

N ~'

f°

. . . . . . oo

h',,

(h~,)2f

ao

• • . )dh'~ . • • d h ' , .

• •

re,n= - N

0.3

X [a 2 q- NcaNReNg(1 - No)](hrnn) 2} = f~,(h'~,). • "fmn(h'mn)" " "

[991

in which

f,~n(hmn) =- DmneXp

{

23'*L* 2(N,I,)2 t_)

k'T*

× [a 2 + Nc,NRdVg(1 - Np)](h'mn)2}.

[100]

R e m e m b e r that in arriving at [55], the series for hi in [38] was truncated with a finite n u m ber o f terms. Since the joint probability density has taken the form o f the p r o d u c t o f the probability densities o f independent r a n d o m variables in [ 1001, we can require

f

~

t

t

fmn(hmn)dhmn = 1

[101]

oo

Journal o f Colloid and Interface Science, Vol, 112, No. 2, August 1986

0.1

0

92

i

96

i

1O0

104

FIG. 2. C** as a function of ¢ofor a liquid-gas system with N= = 10-4, N~ ~ 10, N,+, = 0, and various values of 3'1. The solid lines represent [104] and the dashed lines [123].

335

STOCHASTIC INTERFACIAL DISTURBANCES 0.3

0.3

)i,=0

0,2

0.2

0

YB =t

0,1

0.1

i 96

92

0

s 100

FIG. 3. C** as a function of ~o for a liquid-gas system with N¢~ = 10-4, Ng ~< 10, N~+, = 100, and 3'1 ~< 1, computed from [104].

= 2(N.)2[a 3 + aNcaNRdVg(1 - Np)] × 1

Im(J(i09))

09

F

( h m, n ) "zf m . ( h m n, ) d h m . ,

o9

ak*T*

1

23,*L.2 09

Im(J(i09)).

[104]

+ ~)).

G*.(r)exp(-i09r)d~"

J

51

[106]

55

[123].

Here wo is the dimensionless frequency of the incident light and F* is assumed to depend only on the wavelength of the incident beam,

0.4

z

[105]

Arguing by analogy with the scattering of light by a diffraction grating, Langevin and Meunier (32) have suggested that the spectral density of the electric field P*.(w) ~

CO

FIG. 4. C*~* as a function of w for a liquid-gas system with N~ = 10-3, N~ ~ 10, N~+, = 1, and various values of "r~. The solid lines represent [104] and the dashed lines

Experimentally, we can measure the correlation function in time of the intensity of the scattered light (9-1 2, 14), which is related to the correlation function in time G*n(r*) of the electric field E * . ( t * ) of the scattered light (13,31) G*m.(~') -- (E*.(t)E*...(t

2=9

27

104

x 0.2

0.1

r 9

oJ

1=0

11

o~

and Cm.(~) are proportional to one another, P*mn(09 + 600) = F*Cmn(W).

[107]

FIG. 5. C*~*nas a function of w for a liquid-gas system with N= = 10-2, Ng ~< 10, iV,+, = 1, and various values of 3q. The solid lines represent [104] and the dashed lines [123]. Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

336

HAJILOO AND SLATTERY 0.7

NK, ~ >_10 z

N K.~ >102

0.6

"o 0.5

X

4

CA

0.4

~,/ t

0.3

~

/ a 102~

0.2

i

2 oJ

2

3

FIG. 6. C** as a function of ~o for a liquid-gas system with N~ = 0.1, Ng ~ 10, 3'~ < 0.1, and various values of N,+,. The solid lines represent [104] and the dashed lines [123].

the angle of incidence, and the scattering angle. An explicit expression for F* is unnecessary, if [107] is used to write instead

i

~o

3

FIG. 8. C** as a function of o~ for a liquid-gas system with N~ = 1, Ng = 10, 7~ ~< 1, and various values of N,+,. The solid lines represent [ 104] and the dashed lines [ 123].

/'*.(o~ + o~o)

C,..(~)

P*mn(6Oa q'- 600)

Cmn(~R)

[108]

where ~R is an arbitrary dimensionless reference frequency. In the same manner, Huang

2.1

3 1.4 o

t.~ 2

0.7

1

0 0.1

I

,

1.1

2.1

Ca)

0 3.1

FIG. 7. C'm*. as a f u n c t i o n o f 00 f o r a l i q u i d - g a s system

N~ = 1, Ng = 0, and all values of N,+, and 71. The solid line represents [104] and the dashed line [123].

with

Journal of Colloidand InterfaceScience,Vol. 112,No. 2, August1986

i

0.12

0.32 ~a

0.52

FIG. 9. C** as a function of w for a liquid-gas system with N,~ = 10, Ng = 0, and all values of N,+, and ~q. The solid line represents [104] and the dashed line [118].

STOCHASTIC INTERFACIAL DISTURBANCES 0.2

337

0.2

¢D 0.1

t.) 0.1

\

0 0.1

1.1

2.1

Co

3,1

i

50

--

i

60

0J

70

80

FIG. 10. C**~ as a function o f ~ for a liquid-gas system with N~a = 10, Ng = 1, and all values of N,+, and 3"1. The solid line represents [104] and the dashed line [123].

FIG. 12. C** as a function of~0 for a liquid-liquid system with N o = 10 -4, N, = 1, N o ~ 0.9, N ~ < 10, and all values of 3"~ and N~÷~. The solid line represems [104] and the dashed line [ 123].

(33) has suggested relationships similar to [107] and [108] between [B,.n(r)] 2 and the correlation function in time of the intensity of the scattered light.

The interfacial tension and in some cases the sum of the interfacial viscosities can be determined through measurements of P*n(OO + wo), plotted as

N K ÷ ~ ~-iO 3

N K * ~ -> I 0 ~

[

% 5

x

X t._) 4

3

,S/

1 2

i 0,1

1.1

2.1

i

50

i

3.1

FIG. 1 l. C** as a function o f ~ for a liquid-gas system with N,, = 10, Ng = 10, 3"1 ~< 1, and various values ofN,+,. The solid lines represent [104] and the dashed line [123].

6d

i

(t./

70

80

FIG. 13. C*,*, as a function of~o for a fiquid-hquid system with Nc~ = 10-4, Ng = 10, No = 0.9, Ng ~ 10,3'i = 0, and various values of N,+,. The solid lines represent [104] and the dashed line [ 123]. Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

338

HAJILOO AND SLATTERY or L~ - - -L* - (m 2 + n2)_1/2.

y,=i

[111]

7r

T h e characteristic time t~ is fixed b y d e m a n d ing NRe = 1 [l 12]

5 0

or

×

p(l)*L~ 2

t~ 3

/~tl)*

[113]

As a result o f these definitions, we will find it convenient to observe /.t(l) . 2

Nea = p(l). y . L ~ i

50

60

i

CO

70

r* + E* N,+, -- N, + iV, = tt(l).L~

BO

FIG. 14. C** as a function oft0 for a liquid-liquid system with Nc~ = 10-4, N, = 10, No = 0.9, Ng ~ 10, N,+, = 0, and various values of 3'~.

g*p(0*2Z~3 Ng = it0). 2

[114]

DISCUSSION

P*.(o~ + o~o)

Callen and Greene (36, 37) have proposed an alternative derivation o f the conditional

P*n(WR + WO) and interpreted t h r o u g h [108] as

Cm.(O~) Cm.(~R)

0.3

"

The interfacial tension and the s u m o f the interfacial viscosities m a y be determined by fitting this plot o f experimental data with [ 104], our derived expression for Cmn(O~). T h e parameters m and n are defined to the extent that (2, 34, 35) (m 2 + n2)1/2 = 2L*ni X* (sin 08 - sin 0i).

[109]

Here X* is the wavelength o f the incident light, ni the index o f refraction in the incident medium, 0s the angle o f scattering measured with respect to the n o r m a l to a plane interface, and 0i the angle of incidence measured with respect to this same normal. For convenience, we define L~ by requiring a = 1

[1101

Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

0.2

~9

0.1

11

FIG. 15. C** as a function ofw for a liquid-liquid system with N~ = 10-2, iV, = 10, Np = 0.9, Ng~<10, and all values of 71 and N,+,. The solid line represents [104] and the dashed line [ 123].

STOCHASTIC INTERFACIAL DISTURBANCES 10

339

0.2

i

% X (jE

0.1

*E t)

0

, 4

= 6 O3

i 8

0 1

10

FIG. 16. C*~* as a function of w for a liquid-liquidsystem with N~ = 1, N~ = 10, No = 0.9, Ng ~< 10, and all values of "I'1and N~+,. The solid line represents [104] and the dashed line [123].

e x p e c t a t i o n r e q u i r e d here. C o n s i d e r a n artificial system similar to the o n e d e s c r i b e d above. But r a t h e r t h a n a stochastic, d i m e n -

I 31

i

(.~

61

91

FIG. 18. C** as a function ofw for a liquid-liquid system with N~ = 10-3, N~ = 100, Np = 0.9, N, ~ 10, and all values of-yt and N,+,. The solid line represents [ 104] and the dashed line [123],

sionless difference • in force p o t e n t i a l s p e r u n i t v o l u m e generating capillary waves at the interface, a constant ~ ' m a i n t a i n s the interface in a s t a t i o n a r y configuration whose first pert u r b a t i o n in N , is

h'l =

~

h'mnexp[iTr(mx + ny)L~ /L*].

m,n= - o o

Ill5]

2

% ×

:0 (.9

i 31

OJ

61

91

FIG. 17. C** as a function of w for a liquid-liquidsystem with N~ = 10-4, N, = 100, No = 0.9, Ng ~ 10, and all values of 3'1 and N,+,. The solid line represents [104] and the dashed line [ 123].

A t t i m e t = 0, this force p o t e n t i a l is r e m o v e d a n d the system decays to e q u i l i b r i u m . Callen a n d G r e e n e (36) argue w i t h o u t p r o o f t h a t the value o f the f u n c t i o n o b s e r v e d in this system is equal to the c o n d i t i o n a l e x p e c t a t i o n for the real p r o b l e m with which we are c o n c e r n e d for positive values o f r. This can be p r o v e d b y observing that their initial value p r o b l e m is n e a r l y identical with that described b y [39], [40], [43], [44], [46] t h r o u g h [52], [57], a n d [58]. T h e o n l y difference is t h a t ~mn does n o t a p p e a r in [46] for their initial value p r o b l e m . E q u a t i o n [94] for the c o n d i t i o n a l e x p e c t a t i o n follows i m m e d i a t e l y . Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

340

HAJILOO A N D SLATTERY TABLE I Mean and Standard Deviation for N~+, as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 1 with N~ = 10-4, Ng ~< 10, and 3q = 0 o"= 0.5%

a = 1%

N,+.

(N,+.)

:~.

(N,+.)

.....

0 1 10 *1000

5.3 N 10-3 0.992 9,98 1005

1.6 × 10-2 0,034 0.773 *8 × 106

1.8 X 10-2 0,985 9.96 1010

3.8 X 10-2 0.068 1.50 "1.6 × 107

Note. See text for explanation of asterisk.

It appears that there is an alternative solution to this same initial value problem

h,..(t), .:,..t,, ~j> ""

k'T* Cmn(W) --

23,,L, 2 [1 + NeaNg(1 - No)] -1

X S1S2(S1 q- $2)(S 2 -~- 092)-1($2 ~- 602)-1

z), w~).(t), ~>~>.(t)

where Sl and s2 are solutions of

= (h,, v~(lJ)(z), ¢v((1, ~5~J))exp(s,t)

det(K0) = 0.

+ (hi, v~(2J)(z), v~(2~), ~b(zJ))exp(szt) [116] but it leads to a trivial solution when the initial condition [58] is imposed. In order to avoid this trivial solution, Papoular (3) and Meunier et aL (4) replace [58] by

[I 19]

In the range of the parametric study in the next two sections, sj and s2 are either complex conjugates Sl = --Sr + isi

[120]

s2 = --Sr -- isi

at t = 0:

Ohmn

--

Ot

= 0

[1 181

[117]

in discussing the case where the effects of the Gibbs elasticity, of the interfacial tension gradient, and of all surface stress-deformation behavior can be neglected. The resulting expression for the spectral density of hmn(l) is

or they are both real and negative. When [120] applies, [118] reduces to k'T* Cmn(W ) = T , L , 2 N [1 + NcaNg(1 - No)]-'Sr(SZr + s~)

× [~04+ 2(s] - s Z ) J + (s~ + sZ)Z]-1. [121]

TABLE II Mean and Standard Deviation for 3q as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 2 with Nca = 10-4, Ng ~< 10, and N,+, = 0 = 0,5%

0 O. 1 *0.5 *1

6.7 × 10-5 0.099 0,499

0.995

~ = I%

0.019 O. 134 *40.6 "0.71 × 103

Note. See text for explanation of asterisk. JournalofColloidandInterfaceScience,Vol. 112,No. 2, August1986

1.3 × 10-4 0.099 0,498

0.989

0,038 *0.269 *80.6 "1.4 × 103

STOCHASTIC INTERFACIAL DISTURBANCES TABLE III Mean and Standard Deviation for ']/1 as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 3 with N~a= 10-4, Ng ~ 10, and N,+, = 100 a = 0.5%

0 *1

0.007 1.004

a = 1%

0.85 × 103 *3.2 × l03

0.005 1.006

1.7 × l03 *0.64 × l04

Note. See text for explanation of asterisk. This is similar to the spectral density of the Brownian motion of a harmonic oscillator (26). If instead of [116], we look for a solution of the form

hmn(t), "Wmntt ,(j),"z), W(m)n(t), ~D~)n(t ) = (hl, ff~J)(z), v ~ ), ~b~J))exp[(-Sr + isi)t] [122] we can satisfy neither [58] nor [117]. The corresponding Lorentzian (40) spectral density of

hmn(t) k'T* Cmn(w)

4"y'L* 2

x [1 + N.Ng(1 - N,)l-lsr ×

[ Sr2 + (W1 -- Si)2 + S~ + (~1 + Si)21

[1231

is c o m m o n l y employed in discussing the case where the effects of the Gibbs elasticity, of the interfacial tension gradient, and of all surface

341

stress-deformation behavior can be neglected (15, 40). The quality of this approximation is examined in the next two sections. Bouchiat and Meunier (5) have presented three different solutions for the case where the effects of the Gibbs elasticity, of the interracial tension gradient, and of all surface stress-deformation behavior can be neglected. In the first, they follow Callen and Green (36) in computing the conditional expectation, but they fail to satisfy all of the initial conditions (see [117]). In the second, they again follow Callen and Green (36) in computing the conditional expectation, but they do not recognize that the solution of their initial value problem is equal to the conditional expectation only for positive values of t. (Their Eq. [42] is correct, but it does not follow from their Eq. [41 ]. A factor of 2 is missing.) In their third solution, they used an expression proposed by Callen and Green (36, 37) that incorporates a correct result for the conditional expectation. This approach is identical with that used in the second solution, if their second solution had been carried out correctly. Langevin and Bouchiat (6, 7) follow the second solution of Bouchiat and Meunier (5) in computing the spectral density for a form of surface viscoelasticity (38) (including the same error described above). They however do not include the effects of the Gibbs elasticity or of the interfacial tension gradient (Ref. (7), their first equation on p. 415). In addition, although their description o f surface stress-deformation behavior does belong to the general class of surface material behavior (simple sur-

TABLE IV Mean and Standard Deviation for 3'1 as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 4 with N,~ = 10-3, Ng ~< 10, and N,+, = 1 o- = 0 . 5 %

0 0.1 *0.5 *1

9.9 × 10-6 0.099 0.49 0.99

,r = 1%

0.025 0.029 *1.34 "18

1.9 × 10-5 0,099 0.49 0.99

0.054 0.058 *2.67 *37

Note. See text for explanation of asterisk.

Journal of Colloid and InterfaceScience. V o l .

1 1 2 , N o . 2, A u g u s t

1986

342

HAJILOO AND SLATTERY TABLE V

([hm(t)] 2) = Bmn(O)

Mean and Standard Deviation for ~/1 as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 5 with N= = 10-2, NG ~< 10, and N,+, = 1 ~r = 0.5%

(N )2

f °

-- oD

oo

(h',)Z f(h'~, . . . . .

tr=l%

h',,

• • .)

× d h ' ~ . • • dh',,,,. • .

0 0.1 0.5 1

1.9 × 10-4 0.100 0.50 0.99

0.026 0.028 0.11 0.89

3.9 × 10-4 0.1 0.50 0.99

0.053 0.057 0.22 1.77

f°

= (N~) 2

' : , , (hmn)'f(hmn)dhmn

oo

k'T*

- 4~,,L, 2 [1 + N~Ng(1 - Np)]-'. Note. See text for explanation of asterisk.

[124] face materials) discussed by Slattery and Ram a m o h a n (39), it is not a special case o f surface fluid behavior (simple surface fluid) and it does not include as a special ease the Boussinesq surface fluid, Eq. [4]. I n this sense, their work does not consider the effects o f the surface viscosities. However, the linearized j u m p m o m e n t u m balance of Langevin and Bouchiat (Ref. (6), their Eq. [2]) reduces to equations similar to [24] and [25] (for two-dimensional liquid-gas systems, not including the effects o f thermal noise, o f the Gibbs elasticity, or the interfacial tension gradient), if their surface elasticity and transverse viscosity are set to zero. In contrast, Mandelstam (2) was concerned with the m e a n square o f the amplitude o f a Fourier c o m p o n e n t . By [53], [55], [99] through [102], [110], and [112], this m a y be expressed as

Notice that by [ l l 0 ] t h r o u g h [114] this expression depends u p o n neither the bulk viscosities nor the interfacial viscosities. Equation [ 124] can be e m p l o y e d as the basis for an experiment to measure the interfacial tension (41). RESULTS L i q u i d - G a s Interfaces

We consider here only the limiting case o f liquid-gas systems: N , ~ 0, Np ~ 0. For simplicity, we plot 2-y'L* 2 C**(~o)-

k'T*

C,,n(o~)

[125]

as functions ofNca, N,+,, 3'1, a n d Ng, A typical order o f magnitude for the dimensionless Gibbs surface elasticity 3q is 10 -1 (15-17). We

TABLE VI Mean and Standard Deviation for N,+, as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 6 with N= = 0.1, N~ ~< 10, and ~/1 ~<0.1 a = 0.5%

0 1 *10 *100

3.2 × 10-a 1.009 10.20 117

a = 1%

4.6 X 10-2 0.150 22.3 "1.6 × 105

Note. See text for explanation of asterisk. Journal o f Colloid a n d Interface Science, Vol. 112, N o . 2, A u g u s t 1986

6.48 X 10-3 1.018 10.41 120

9.3 × 10-2 0.305 *47.8 *3 × 106

STOCHASTIC INTERFACIALDISTURBANCES

343

TABLE VII Mean and Standard Deviation for N,+, as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 8 with N~ = 1, N~ = 10, and ~,~ < 1 = 0.5%

a = 1%

N,+.

(N,+,)

....

(N,+.)

ale.÷,

0 *1 "10 *100

3.1X 10-2 1.06 10.6 t62

0.57 "1.97 *335 *7 X 106

6.4 X 10-2 1.11 11.4 410

1.19 *4.2 *830 *6 X 108

Note. See text for explanation of asterisk. will therefore restrict our illustrations to 3'1 ~<1. Figures 1, 2, 4, and 5 indicate that the spectral density is a significant function of the interfacial viscosities and of 3'1 for Nca < 10 -2. Figure 3 shows that, with increasing Nca, the effect of'y~ on the spectral density disappears. With systems for which Nca ~< 10-z, it may be possible to use this technique to determine the interfacial tension, ARK+,,as well as 3'1. As Nca increases beyond 10 -1, first the effect of 3"1 and then that of N~+, disappear. There is no effect ofNg ~< I0 for No, < 10-1. Figures 7 through 11 indicate that there is a significant effect of gravity for N~ >~ 1. As Ng increases, the effects of the interfacial viscosities begin to reappear. It is obvious from Figs. 1 through 11 that the Lorentzian form [123] is generally a poor approximation to the actual spectrum unless Nc~ < 10-4.

Similar to the liquid-gas case, the Lorentzian spectrum [123] is generally a poor approximation.

Propagation of Errors Errors in the experimental spectral density data propagate in computing NK+, and 3'1. In order to examine the effect of such experimental errors upon the calculated values of these parameters, we have carried out the following computer experiments. For each spectral density curve represented in Figs. 1 through 6, 8, and 11, we chose 100 equally spaced points. (The effects of N~+, and ~t essentially disappear in the other figures.) We then generated a random error bar for each of these points, the mean value of which was zero and the standard deviation of which was either 0.5 or 1% of the m a x i m u m spectral density on the curve. New sets of noisy data TABLE VII1

Liquid-Liquid Systems Figures 12 through 18 illustrate the effects of N= and N, for only one density ratio: N o = 0.9. The effects of the dimensionless Gibbs surface elasticity "n and of the dimensionless sum of the interfacial viscosities NK+~virtually disappear for these systems. They reappear only when Nca and N, are sufficiently small. Both Ncaand iV. are important parameters; Ng is of little significance.

Mean and Standard Deviation for N~+, as the Fitted Parameter in the Computer Experiment for the Results Shown in Fig. 11 with N~ = 10, Ng = 10, and "rl < 1 rr=0.5%

a=

l%

N,+,

(N,+.)

....

(N,+.)

....

*0 *1 *I0 *100

1.3 X 10-2 1.03 10.46 230

*6.4 "24.1 *3650 "108

2.7 × 10-2 1.06 10.96 236

"13.1 *50. *8450 *8 X l0s

Note. See text for explanation of asterisk. Journal of Colloid and Interface Science, Vol.

112, N o . 2, A u g u s t 1986

344

HAJILOO AND SLATTERY

were generated by adding the old values to these generated error bars. New values of either N,+, or 3'1 and the corresponding standard deviafions (42) then were computed by fitting [ 104] to these noisy data. Since this computer experiment was for illustrative purposes only, we determined only one parameter, either N,+, or 3"1,in using [ 104] to fit these data, assuming the other was known. In reality, the experimentalist may wish to determine both of these parameters as well as Nc~, in which case the standard deviation for each parameter must be estimated. Tables I through VIII represent the results of this computer experiment. The first column of each table represents the parameter of the original curve that was distorted to represent the noisy data. Columns 2 and 3 show the mean value and the standard deviation of the fitted parameter corresponding to the specified standard deviation on the error bar of the spectral density data. The rows in Tables I through VIII that are marked by asterisks correspond to a large standard deviation, indicating that under these conditions meaningful values of the parameter cannot be estimated in this manner. The results have the same character that we observed above in examining the figures. Tables I, II, IV, and V and Figs. 1, 2, 4, and 5, respectively, indicate that the spectral density is a significant function of N,+, and of 3"1 for Nc~ ~< 10-2. Table III and Fig. 3 show that, with increasing N~a, the effect of 3"1 on the spectral density disappears. With systems for which Nca ~< 10-2, it may be possible to use this technique to determine N~a as well as N~+, and 3"1, depending upon the quality of the spectral density data. As N~a increases beyond 10-1, first the effect of 3q and then that of N~+, disappear.

b bE) k(J)* u(t) B Bran(r)

c C Cm~(w) C** (o)) D D2 DE

Dmn .D(j).

a

A AA*

NOMENCLATURE

defined by [42]; see also [110] coefficient in [70] difference in Helmholtz free energy (see [971)

Journal of Colloidand InterfaceScience, Vol. 112,No. 2, August 1986

[102]

rate of deformation tensor in phase j, defined by [3] D.(,). surface rate of deformation tensor, defined by [5] E*,(t*) electric field of the scattered light f (h'l . . . . , joint probability density (see [55], h'., [95], [96], and [99]) *oo)

finn(h%) F* g.

G%(3') h* hi

hm,(t) H*

I. J

APPENDIX:

defined by [72] force per unit mass of gravity mutual force per unit mass in phase j coefficient in [71] correlation function in time of hmn(t), defined by [53], [54], and [55] defined by [73] coefficient in [70] spectral density of hmn(t), defined by [103] defined by [125] coefficient in [71 ] operator defined by [32] operator defined by [41 ] coefficient in [100], defined by

97(S)

k* KO L*

probability density function (see [100]) coefficient in [ 107] magnitude of the acceleration of gravity correlation function of E ' n , defined by [105] configuration of the interface defined by [28] Fourier coefficient of hi, defined by [38] mean curvature of the dividing surface (see also [ 16]) identity tensor Laplace inverse of J(s) defined by [79] Boltzmann constant matrix defined by [80] width of interface (see assumption 8)

STOCHASTIC

L~ m n ni

Nca N~

characteristic wavelength, defined by [111] integer representing Fourier coefficients, limited by [ 109] integer representing Fourier coefficients limited by [109] index of refraction in the incident medium capillary number, defined by [ 17] dimensionless group, defined by [ 17]

NRo N. N. NK+,

N~ N, N, P~ p(J)*

g

AP* P*,(o~) /)(J)*

/)(~)* t, z) S(J)*

S(~) *

t*

T* T(J)*

INTERFACIAL

DISTURBANCES

T (*)* u*

u~j) u~~) v(j)* v (~)*

345

surface stress tensor velocity of a point on the interface whose surface coordinates are fixed (see also [19]) x component of the first perturbation of velocity in phase j x component of the first perturbation of the surface velocity velocity vector in phasej (see also [13]) surface velocity vector y component of the first perturbation of velocity in phase j y component of the first perturbation of the surface velocity z component of the first perturbation of velocity in phase j z component of the first perturbation of the surface velocity Fourier coefficients of w~j), defined by [38] Fourier coefficients of w]~), defined by [38] rectangular coordinate (see also [131) rectangular coordinate (see also [13D rectangular coordinate (see also [13])

Reynolds number, defined by [ 14] v]j~ dimensionless interfacial shear viscosity, defined by [17] v~~) dimensionless interfacial dilatational viscosity, defined by [17] w~j) defined by [ 114] perturbation parameter, defined w~*~ by [17] viscosity ratio, defined by [14] , "(J)t. Z) Wmn~t density ratio, defined by [ 14] pressure that would exist at the Wren(t) (') dividing surface in the absence of thermal noise x* pressure in phase j projection tensor that transforms y* vectors defined on the dividing surface into their tangential z* components difference in potential energy (see [98]) Greek Symbols spectral density of E*.(t*), defined by [106] 3'* interfacial tension modified pressure in phase j, de- 3"~. equilibrium interfacial tension or the fined by [ 11] interfacial tension at the concenFourier coefficient of ~o]J),defined tration of surfactant corresponding by [38] to a planar interface viscous portion of stress tensor in 3'1 dimensionless Gibbs surface elasticity phase j, defined by [2] (see also evaluated at the equilibrium sur[131) face concentration of surfactant viscous portion of surface stress and equal to -03"/0o{~)) (see also tensor, defined by [4] [2o]) time (see also [13]) 6(x) Dirac delta function characteristic time defined by ~* interfacial shear viscosity angle of incidence measured with re[113] 0i spect to the normal to the plane temperature interface stress tensor in phase j Journal of Colloidand InterfaceScience, Vol. 112, No. 2, August 1986

346 Os

K*

X* g(J)*

p(J)* p(o*)* (or)*

P(s)

P(s)0 O"

O'Nx+~ O'~/1

~(J)*

HAJILOO AND SLATTERY angle o f scattering measured with respect to the n o r m a l to the plane interface interfacial dilatational viscosity wavelength of the incident light shear viscosity for phase j unit n o r m a l to the dividing surface pointing into phase 2 density of phase j characteristic value o f the total mass density of the dividing surface surface concentration o f surfactant (see also [19]) surface concentration o f surfactant corresponding to a planar interface standard deviation o f the error bar of the spectral density as a percent o f the m a x i m u m for a given curve standard deviation o f N~+, standard deviation of ~/1 scalar gravitational potential, defined by [12] defined by [ 16] scalar potential for the thermal noise in phase j, defined by [ 1] characteristic value of the scalar potential for t h e r m a l noise in phase

J d~mn(t)

Fourier coefficients of ~, defined by

[451 O00

09R

dimensionless frequency o f the incident light arbitrary dimensionless reference frequency

Other Symbols

1

T

subscript denoting the first perturbation o f a variable superscript denoting a dimensional quantity superscript denoting the transpose of a tensor overbar denotes the c o m p l e x conjugate overtilde denotes Laplace transformation of variable

Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986

div(~) div V

surface divergence operation (21, 22) divergence operation gradient operation surface gradient operator (21, 22) expected (mean) value o f x ACKNOWLEDGMENTS

The authors are grateful for financial support received from Amoco Production Company, Exxon Education Foundation, Gulf Research and Development Company, and the Standard Oil Company (Ohio). Note added in proof Recent experimental evidence by R. B. Dorshow (Sohio Petroleum Company) confirms equation [ 104] for a liquid-liquid system. Interfacial light scattering on this system shows very good agreement using the theory presented and poor agreement assuming a Lorentzian fit. A manuscript is in preparation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14.

15. 16. 17. 18.

Smoluchowski, M. V., Ann. Phys. 25, 225 (1908). Mandelstam, L., Ann. Phys, 41, 609 (1913). Papoular, M., J. Phys. 29, 81 (1968). Meunier, J., Cruchon, D., and Bouchiat, M., C. R. Hebd. Seances Acad. Sci. Set. B 268, 92 (1969). Bouchiat, M. A., and Meunier, J., Z Phys. 32, 561 (1971). Langevin, D., and Bouchiat, M. A., C. R. Hebd. Seances Acad, Sci. Set. B 272, 1422 (1971). l_angevin,D., J. Colloid Interface Sci. 80, 412 (1981). Katyl, R. H., and Ingard, U., Phys. Rev. Lett. 20, 248 (1968). Bouchiat, M. A., Meunier, J., and Brossel, J., C. R. Hebd. Seances Acad. Sci. Set. B 266, 255 (1968). Bouchiat, M. A., and Meunier, J., C. R. Hebd. Seances Acad. ScL Ser. B 266, 301 (1968). Huang, J. S., and Webb, W. W., Phys. Rev. Lett. 23, 160 (1969). Bouchiat, M. A., and Meunier, J., Phys. Rev. Lett. 23, 752 (1969). Pouchelon, A., Meunier, J., Langevin, D., and Cazabat, A. M., J. Phys. Lett. 41,239 (1980). Kim, M. W., Huang, J. S., and Bock, J., "Interfacial Light Scattering Study in Microemulsions," SPE/ DOE 10788, 3rd Joint Symposium on Enhanced Oil Recovery of the Society of Petroleum Engineers, Tulsa, Okla., April 4-7, 1982. Hard, S., and Lfifgren, H., J. Colloid Interface Sci. 60, 529 (1977). Byrne, D., and Earnshaw, J. C., J. Phys. D 12, 1145 (1979). Hard, S., and Neuman, R. D., J. Colloid Interface Sci. 83, 315 (1981). Slattery, J. C., "Momentum, Energy, and Mass

STOCHASTIC INTERFACIAL DISTURBANCES

19. 20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

Transfer in Continua," 2nd ed. Krieger, Malabar, Fla., 1981. Boussinesq, J., C. R. Hebd. Seances Acad. ScL 156, 983 (1913). Scriven, L. E., Chem. Eng. Sci. 12, 98 (1960). Wei, L. Y., Schmidt, W., and Slattery, J. C., J. Colloid Interface Sci. 48, 1 (1974). Briley, P. B., Deemer, A. R., and Slattery, J. C., J. ColloM Interface Sci. 56, 1 (1976). Slattery, J. C., Chen, J. D., Thomas, C. P., and Fleming, P. D., J. Colloid Interface Sci. 73, 483 (1980). Slattery, J. C., Chem. Eng. Commun. 4, 149 (1980). Slattery, J. C., and Flumerfelt, R. W., in "Handbook of Multiphase Systems" (G. Hetsroni, Ed.), pp. 1224. Hemisphere, New York, 1982. Yaglom, A. M., "Stationary Random Functions." Dover, New York, 1973. Lindgren, B. W., "Statistical Theory," 3rd ed. Macmillan, New York, 1976. Schuss, Z., "Theory and Application of Stochastic Differential Equations." Wiley, New York, 1980. Spiegel, M. R., "Laplace Transforms," pp. 176, 201. McGraw-Hill, New York, 1965. Kestin, J., and Dorfman, J. R., "A Course in Statistical Thermodynamics," p. 184. Academic Press, New York, 1971.

347

31. Pouchelon, A., Chatenay, D., Meunier, J., and Langevin, D., J. Colloid Interface Sci. 82, 418 (1981). 32. Langevin, D., and Meunier, J., in "Photon Correlation Spectroscopy and Velocimetry" (H. Z. Cummins and E. R. Pike, Eds.). Plenum, New York, 1976. 33. Huang, J. S., Ph.D. dissertation. Cornell University, Ithaca, New York, 1969. 34. Chu, B., "Laser Light Scattering." Academic Press, New York, 1974. 35. Pike, E. R., in "Photon Correlation Spectroscopy and Velocimetry" (H. Z. Cummins and E. R. Pike, Eds.). Plenum, New York, 1976. 36. Callen, H. B., and Greene, R. F., Phys. Rev. 86, 702 (1952). 37. Greene, R. F., and Callen, H. B., Phys. Rev. 88, 1387 (1952). 38. Goodrich, F. C., Proc. R. Soc. London Ser. A 260, 490 (1961). 39. Slattery, J, C., and Ramamohan, T. R., Chem. Eng. Commun. 26, 193 (1984). 40. Edwards, R. V., Sirohi, R. S., Mann, J. A., Shih, L. B., and Lading, L.,Appl, Optics 21, 3555 (1982). 41. Lachaise, J, Graciaa, A., Martinez, A., and Roussei, A., Thin Solid Films 82, 55 (1981). 42. Himmelbau, D. M., "Process Analysis by Statistical Methods." Sterling SwiR, Austin, Texas, 1970.

Journal of Colloid and Interface Science, Vol. 112, No. 2, August 1986