Surface dilational viscosity and energy dissipation

Colloids and Surfaces A: Physicochemical and Engineering Aspects, 85 (1994) 21 l-219 0927-7757/94/$07.00 0 1994 -~ Elsevier Science B.V. All rights

Surface dilational viscosity and energy dissipation E.H. Lucassen-Reynders”, Unilever Research Laboratorium, (Received

4 October

J. Lucassenl Olivier van Noortlaan

1993; accepted

22 November

120, 3133 AT Vlaardingen,

The Netherlunds

1993)

Abstract Surface viscosity, like its three-dimensional counterpart, depends on the experimental conditions used for its measurement. In compression/expansion experiments, different definitions have been used for the “surface dilational viscosity”. Steady-state viscosity coefficients were first obtained for areas expanding at a constant relative rate, by van Voorst Vader et al. (Trans. Faraday Sot., 60 (1964) 1170). This coefficient is not generally equal to the viscous part of the dilational modulus measured for small periodic (wave) deformations at the same relative rate, even if the relaxation mechanism causing the viscous behaviour is the same in both cases (see Lucassen and van den Tempel, J. Colloid Interface Sci., 41 (1972) 491). A quantitative comparison is presented of the two cases if the relaxation mechanism is diffusional interchange of surfactant between the surface and the bulk solution. The relationship with the energy dissipation is evaluated for both cases. The surface viscosity defined for the small-deformation experiment is shown to be a pure viscous loss modulus, because it is directly proportional to the amount of mechanical energy dissipated as heat, The steady-state, large-expansion experiment yields a value which is not a true viscosity in this strict sense, because it also contains an elastic contribution which measures the amount of potential energy stored in the surface. Key words: Energy dissipation;

Surface dilational

viscosity

Introduction The term “surface dilational viscosity” has been surrounded by a certain amount of confusion even

experiments, transient effects are followed by a steady state in which the dynamic surface tension (cr) no longer changes with continuing expansion. A steady-state coefficient of dilational viscosity (K ) can be defined for this situation

[l-3]

among those who agree that it measures changes in the non-equilibrium surface tension during surface compression/dilation at a given rate. In prin-

KE

ciple, such viscosity can be produced by any relaxation mechanism driving the dynamic surface tension back to its equilibrium value. In practice, its measurement has been restricted to surfaces of surfactant solutions, where tension relaxation takes place by exchange of surfactant between surface and bulk solution. First experimental results were obtained for areas expanding at a constant relative rate by van Voorst Vader et al. [ 11. In such

where Au is the steady-state value of the surface tension deviation found when diffusion from the underlying solution just compensates for the depletion caused by the area A being expanded, at a constant relative rate (d In A/dt). However, surface dilational viscosity can also be defined from the imaginary part of the complex dilational modulus (E) measured for small periodic

A0

surface deformations *Corresponding author. ‘Present address: Gorlaeus Laboratories, Leiden P.O. Box 9502, 2300 RA Leiden, The Netherlands. SSnl

0927-7757(93102713-O

University,

(1)

d In Aldt

[4]

da E -= ~ = cd + iOjqd d In A

where

Ed is the dilational

elasticity

and ~7~is the

dilational viscosity for area variations of frequency to; here, the frequency plays the part of the relative rate of deformation (d In A/dt). The steady-state equal

coefficient

to the small-fluctuation

equilibrium

with deeper layers of the solution.

dynamic, small-deformation experiment then yields a dilational viscosity ~7~which can be expressed in terms

of surfactant

characteristic

K is not generally

the time scales of the diffusional

coefficient

experiment

qd mea-

The

parameters process

and

and the

[ 81

sured at the same relative rate, even if the relaxation mechanism causing the viscous behaviour is the same in both cases. Moreover, it has been argued that a “surface viscosity” caused by diffusional interchange with the bulk solution is not an intrinsic surface property [S], and that it is not a

The frequency o measures the time scale of the experiment, and i is the diffusion parameter which

true viscosity either [6]. Since the values obtained are the result of surface composition changes, some authors prefer to label the viscosity coefficients defined in Eqs. (1) and (2) as “apparent” or “compositional”. A well-known criterion applied in three-dimen-

relates the time scale of the diffusion to the time scale of the experiment

sional rheology is that viscosity is a measure for the irreversible dissipation of mechanical energy into heat, by any relaxational mechanisms operative in a system. The purpose of this paper is to clarify the status of the above viscosity coefficients by examining their relationship with the rate at which energy is dissipated in the dynamic and steady-state experiments.

where D is the diffusion coefficient of the surfactant. The limiting modulus Em measured in the absence of diffusion, finally, characterises the surfactant at a concentration c and equilibrium value r of the adsorption. Equation (3) is valid for any surface equation of state, i.e. for any analytical

Energy dissipation in dynamic and steady-state experiments

For simplicity, we shall limit calculations to situations where diffusional interchange of surfactant between surface and bulk solution is the only relaxational mechanism operative in the system. This means that we can neglect surface shear viscosity and any transverse surface viscosities that may be defined theoretically [7]. It also implies an absence of energy barriers slowing down the kinetics of adsorption: we consider the dynamic values of surface tension and adsorption to be in local equilibrium with the concentration immediately below the surface, although they are not in

process (r&r)

(4)

relationship between surface tension G and the adsorption I-. This relationship determines the numerical values of the limiting modulus eO and of the diffusional time scale in Eq. (4). but not the analytical form of the dilational viscosity frequency spectrum in Eq. (3). Such a general relationship, independent of the surface equation of state. cannot be derived for the spectrum of the steady-state coefficient K defined in Eq. (1). In such an experiment, the subsurface concentration and the adsorption may be so far from equilibrium that they can no longer be described by differentials except in the limit of very slow deformations (see Discussion section). Beyond this limiting range, a specific surface equation of state must be introduced in order to evaluate the rate dependence and to compare it to that of the small-deformation viscosity in Eq. (3). For demonstration purposes, the equation of state corresponding to SzyszkowskiCLangmuir adsorption suffices, implying that the limiting modulus and the diffu-

E.H. Lucassen-Reynders

sional

and J. Lucassen/Colloids

time scale can be calculated

no-~=

Swfaces

A: Physicochem

from

-RTPln(l-T/P)

Eng. Aspects 85 (1994) 211-219

213

“Viscosity” (mN s/m)

(5) 20

where CT’is the surface tension of the empty surface, r” and a are constants characteristic of the surfactant covering

it,

R is the gas constant

temperature. In terms of parameters, van Voorst Vader for convective diffusion in the at the following result for the KO=

and

T is the

these characteristic et al. [l] accounted steady state, arriving rate dependence

Aa,,=RTl=

x In

L4,t

1

ist - 1 + Jr,:

+ 2M 1 - c/u)/( 1 + c/u) + 1 1 (7)

where 0 is the relative rate of deformation (d In A/dt), the subscript st denotes the steady state, and ist is a steady-state diffusion parameter comparable to that defined for the small-deformation experiment in Eq. (4) a( 1 + c/a) ist =

ra

20

(8)

J-nH

In order to compare

the steady-state coefficient viscosity, we only need the Langmuir values for the limiting

K to the small-deformation to substitute modulus

E,, and

the diffusion

parameter

i into

Eq. (3)

Ed= RTT"c/a;

(=

a( 1 + c/a)’

rm

(9)

Numerical examples for the small-deformation and steady-state regimes described by Eqs. (3) and (7) are illustrated in Figs. l-3. For comparison, the absolute value (1~1) of the viscoelastic modulus for the small-deformation experiments is also represented

(10)

Reduced Concentration,

c/a

Fig. 1. Surface dilational coefficients for small and large area variations, as a function of surfactant concentration, at a given relative rate of area variation. Numerical example for surfactant with characteristic parameters a = 0.01 mol mm3 and r” = 4x10~6molm~2;D=5x10~‘om2s~‘;O=1s~’;w=1s~’. -- -, viscosity qd; . . . . . . I6l/w.

Figure 1 illustrates at a given relative

the concentration dependence rate of area expansion and/or

compression (0 = 1 and o = 1). All three coefficients (K, qd and IE~/co) pass through a maximum, because at low concentrations there is not enough material for the deformation to produce a measurable deviation in surface tension, while at high concentrations there is enough material but the deviation is levelled out by diffusion from the solution. Quantitatively, the steady-state parameter (KG) is closer to the complete viscoelastic modulus 1~1 than it is to the viscous part of the modulus (0~~). The differences show up more clearly in the ratedependence of the three coefficients: the smalldeformation viscosity is the odd one out in the high-frequency region (Fig. 2). This is even more

“Viscosity”

(mN s/m)

Deviation

K

1000

(Large

from

Equilibrium

(mN/m)

lOI

AA)

,.,_...’

,...

Modulus

. . . .

ICI/W (Small

--

q,

(Small

AA) (Small

81

AA)

/E i

,:.

A A) ;’

:

_’ Empty

Surface

100

10

,---5

‘.

1

@qd

.\ (Smell ‘\

0.00001 0.1 L 0.0001

0.001

0.01

0.1

1

d In A / dt, Frequency

Fig. 2. Rate dependence of dilational surfactant concentration. Numerical parameter values as in Fig. 1.

10

100

0.001

0.01

0.1

‘.

1

d In A / dt, Frequency

*. -

AA) -_

10--__

100 _.

(l/s)

(l/s)

coefficients, at a given example for c.!n = 1:

evident in Fig. 3, showing the rate dependence of surface tension deviation do, the dynamic expressed as KU, and that of wqd and of 1~1. At high rates of deformation, the steady-state value of the dynamic tension approaches the tension of the empty surface, go, i.e. da approaches the equilibrium surface pressure. The complete viscoelastic modulus (~1 also continues increasing until it reaches a plateau, given by the limiting modulus Ed. The viscous part of it, however, reduces to zero while the other two coefficients keep increasing. The results of Eqs. (3) (7) and (lo), illustrated in Figs. 1-3, can be summarised in two points. (i) In the slow-deformation, diffusion-controlled range the three parameters converge, apart from a constant factor. As noted before 191, this is explained by the large effect of diffusion here ([ >> l), resulting in very small deviations from equilibrium. (ii) At higher rates of deformation, however,

Fig. 3. Dynamic surface tensions spectra of Fig. 2.

corresponding

where the effect of diffusion the steady-state parameter and lus increasingly diverge from the modulus, which vanishes reach plateau values.

to the viscosity

becomes negligible. the complete moduthe viscous part of when the other two

We suggest that the fundamental difference between K and /ld is that the former. like the complete modulus 1~1. measures the sum of the viscous and the elastic response of the system, while the latter measures the purely viscous response to the surface deformation. This can be clarified by deriving the relationship of the two parameters with the rates of energy dissipation in the two experiments.

For sufficiently small surface deformations, the evaluation can be performed in complete analogy with the three-dimensional case [lo]. We calculate the rate at which work is being done by surface

E.H. Lucassen-Reynders

stress

tensor

bounding envisaged

and J. LucassenJColloids

components

tensor

because

line

elements

in the x direction.

components

the

component

shear

gXY and

modulus

gYX vanish

is supposed

to be

gXx

0 XX= Ceq + ay/ax + r/,&I,/& where t is the distance

(11)

by which a surface element

is displaced out of its position at rest, oeq is the equilibrium value of the surface tension and u, is the surface element’s velocity in the x direction. Equation ( 11) is equivalent to Eq. (2), and, alternatively, can be formulated in terms of changes in area (A) for any small compression or expansion do = EVA In A + qd d In A/dt the

215

of area (Udiss) takes place at the rate

sinusoidal

wave

(12)

motion

(17)

The surface

negligible, and the tensor component gJY cannot do work because there are no velocity components in the y direction for the plane wave considered. Thus, the only term to be accounted for is the

For

Eng. Aspects 85 (1994) 21 l-21 9

an enclosed surface area. The experiment involves a longitudinal surface wave

made to propagate stress

flij on

Surfaces A: Phgsicochem.

considered

we

have 5 = {() cos(kx + OX)

The

conclusion

is that

the

surface

area

U pot =

+

;

dk5o)2

which energy can be used to propagate mechanical disturbances over the surface. Thus, the parameters yld and Ed, defined in Eq. (2) and measured in small-amplitude wave experiments, are a true viscosity and elasticity, respectively. The argument presented is valid for any relaxation mechanism that drives the surface tension to its equilibrium value, and this includes diffusional relaxation by interchange with an adjoining bulk solution. Energy dissipution in steudy-state

v, = a
(14)

The amount of energy dissipated in one wave period and one wavelength, per unit of length in is given by

(15) After substituting we find

Eq. (14) and integrating

Udiss= -2qdWkX2(g This means

that the dissipation

(18)

(13)

where &, is the amplitude of the horizontal displacement, k is the wave number and, in the first approximation

the y direction

dilatational

viscosity defined in Eq. (2) indeed measures the rate at which mechanical energy put into the system is irreversibly lost as heat. A similar argument serves to show that the elastic part of the dilational modulus in Eq. (2) measures the amount of potential energy (I+,,,~)stored in a unit of surface

Eq. (1.5)

(16) of energy per unit

For

any

large-deformation

experiment experiment,

the

above calculation on the basis of stress tensor and surface velocity is much more complicated than for small sinusoidal variations. There is an alternative, simpler way of finding the energy dissipated in the steady state, where the surface tension and the adsorption and the subsurface concentration are all constant. We calculate the energy dissipation by subtracting the potential energy left after the experiment is halted from the total amount of energy put into the system. The latter energy is equal to the work (W) done on the system by the barrier expanding the surface at constant relative rate. Per unit of time we have dW __ = Ao,~ % dt

= KH2A,,

(19)

The potential free energy

energy

(dF)

the amount

is equal to the increase

of the system,

and is related

in

(23)

to

of work (Wre,) that can be recovered

when the steady-state expansion is stopped. When the moving barrier is released, it will move backwards until an equilibrium area (A,,) is reached where the surface tension is back at its equilibrium

In this expression, the values of f,, and r,, are constant for a given surfactant solution subjected to a given rate of steady-state expansion ((I). Thus, the expression in brackets is constant, but A,, is

value (c,~). The work recovered

variable stopped

wards barrier

motion

during

this back-

is given by

different times. Differentiation of AF with respect to time, therefore. is straightforward

A St wP_,

(a

=

because the steady-state expansion can be at different areas A,, corresponding to

-

geq)

dA

(20)

i Am

The value of W,,, depends on how 0 is allowed to vary after the expansion is stopped. For the present purpose we are interested only in the masir?zunz recoverable work, because this represents the increase in free energy, AF. The maximum value for the recoverable work is obtained from Eq. (20) if no further energy is dissipated after the expansion ceases, i.e. if there is no decrease in u by diffusion of surfactant towards the surface. This means that the barrier is considered to move backwards in a way which prevents diffusion from taking place. During this hypothetical process, the total amount of adsorbed surfactant (TA) remains at the value it has at the time of cessation of the steady-state expansion dA=

-(A/T)

dI-=

(AT),, dT - 1

(24) This is the rate at which the free energy increases during the steady-state experiment. This rate is to be subtracted from the rate at which mechanical energy is put into the system. The latter is given by Eq. (19) or, in terms of the adsorptions

(25) The rate of energy dissipation now obtained by combining and dividing by the area A,,

(21)

This enables us to formulate AF, as given by Eq. (20), in terms of the adsorption values in the steady state and at equilibrium, using Eq. (5) for the surface tensions

(26) In the

we find for AF

final

adsorption means of 1 -&Jr”

After integration

per unit of area is Eqs. (24) and (25)

step

we correlate

r,, to the viscosity

= (1 - c,/ra)

the

steady-state K by

coefficient

exp(KtQRTT”)

(27)

This gives us the required relationship between the rate of energy dissipation and the steady-state coefficient K, to be compared to Eq. (17) for the small-deformation wave experiment. It is clear that the relationship in the large-deformation case is far more complex than that in Eq. (17): unlike tld, the coefficient K is not generally proportional to

E.H. Lucassen-Reynders and J. LucassenlColloids Surfaces A: Physicochem. Eng. Aspects 85 (1994) 211-219

the rate of energy dissipation. tionality

is found

the deviations

Such simple propor-

only in the limiting

from equilibrium

case where

are very small, i.e.

when the effect of diffusion is very large (i >> 1). In this limit, Eqs. (26) and (27) are reduced to -

dudis

K

p

dt

Beyond this slow-deformation limit, part of the energy is stored in the surface, as given by Eq. (24), and K measures the sum of dissipated and stored energy.

Discussion The simple relationship between the steady-state coefficient K and the energy dissipation in Eq. (28) is found only in the limit of very slow deformation, where all mechanical energy put into the system during the steady-state expansion is dissipated as heat by the diffusion process. The functional dependence of the dissipation rate on K and 0 in this limit is indeed similar to that on qd and o in the small-deformation experiment in Eq. (17). This is not surprising, since in this range K expresses small deviations from equilibrium, even though the area expansion may be very large. In the near-equilibrium range, no analytical surface equation of state is needed and the steady state is expressed by [ 1 l]

Ao,,=KO=-

RTr2 c

dl J 20

and the small-deformation

for 85,i,<< 1 parameters

demonstrate

very low deformation sions

for all three

that, in the limit of

rates, the analytical dilational

coefficients

expresare the

same, apart from a constant factor. Beyond this limiting range, the rate dependence of the smalldeformation modulus 1~1according to Eq. ( 10) can be represented by a single master curve in terms of concentration and of E0 and Tdif, independent

for Ordirf<< 1

-=

These expressions

217

equation-of-state parameters. The steady-state parameters do and K, however, cannot be so represented: their evaluation requires the concentration and the equation of state to be specified. This is illustrated for the surface tension deviation in the steady state by Fig. 4, showing that different concentrations follow different curves beyond the slow-deformation range, even if the equation of state is the same. The difference is probably caused by the different effects of convection generated by

1

AU/E,

I

_---__---

c/a = 0.1, ,” ,

I I

(29)

by

In this near-equilibrium range, the adsorption and the concentration are simply related to c0 and rdiR through Gibbs’ adsorption law, which results in

(31)

1o.5

10‘4

10-3 0.01

0.1

1

10

100

1000

CdIn A I W * Tdiff Fig, 4. tion in single

demonstrating that surface tension steady-state experiment be described a spectrum. Langmuir with a

the area deformation. (Convective effects in this type of experiment were discussed recently by Joos

spectrum At such

and Van Uffelen [ 121.) For any given surfactant

behaves as an insoluble monolayer because the time scale of the experiment is much smaller than

dilational parameters diverge with increasing

solution,

moreover,

the

K, 1~1 and vd increasingly rate of compression/expan-

only 6% is dissipated and 94% is stored. high rates of deformation the surface

that of the diffusion process. It may be concluded that the steady-state

dila-

sion because more and more of the energy added is stored as potential (elastic) energy. The parame-

tional coefficient K measures the sum of storage and loss of energy. as does the small-deformation

ter K measures the total energy added, as given by Eq. (19) while yld measures the fraction that is irreversibly lost (Eq. (17)). A numerical example of the increasing effect of storage is given in Fig. 5, as calculated from Eqs. (26) and (27) for the same parameter values as used in Figs. 1 and 2. The limiting range of very low rates of deformation (H, o<< l/Zdi,) where Eqs. (28)-(30) are valid appears to be rather small. At a rate equal to the reciprocal value of the diffusional time scale T'diff, the amounts of energy dissipated and stored are roughly equal; three decades further on in the

modulus

Energy,

in

m.Jl(m*s)

Stored / s /

m..TotalExpended/ s

100

10 s

1

0.1

0.01

0.001

L/--,1_.__

0.001

0.01

0.1 d InA

1 /

10

100

1000

dt (l/s)

Fig. 5. Rate of energy dissipation and energy storage in steadystate experiment, calculated from Eqs. (26) and (27). Numerical example for c/u = 1; parameter values as in Fig. I.

1~1. The steady-state

parameter

K, there-

fore, can be termed a viscosity only in a loose sense, by analogy with common practice in the rheological description of three-dimensional viscoelastic liquids. The parameter kid. in contrast, is a true surface viscosity in the strict sense that it measures the dissipation into heat of mechanical work done against surface tension forces, through any relaxation mechanism that affects the surface tension. Diffusion of surface-active material to and from the solution, as evaluated here, may be considered to be a rather peculiar relaxation mechanism, because it dissipates energy not in the surface proper but in the adjoining layers of bulk phase(s). This detracts nothing from the formal correctness of applying the term surface dilational viscosity to tld evaluated for this mechanism in Eq. (3). It is also emphasised that, for any relaxation mechanism, surface viscosity refers only to the irreversible loss of surface energy; under experimental conditions the loss of energy through hnlli viscosity. in the inevitable flow of the adjoining bulk phases, may be larger than the amount of energy dissipated through sur;$~e viscosity. The advantage of the small-amplitude wave experiment for measuring the surface dilational modulus 1~1 is that it permits us to determine the elastic and viscous contributions separately and easily, without any knowledge of the analytical surface equation of state. The steady-state experiment, however, has the great advantage of giving information on surface behaviour far from equilibrium. as it directly measures the dynamic surface tension after large surface expansions. This is particularly relevant for technological applications such as, for example, emulsification, where break-up of large drops into a fine dispersion may

E.H. Lucmsen-Reynders

und J. Lucassen/Colloitis

Swfuces

A: Physicochem.

take place only after the large drop has first been extended

into

a long

cylindrical

thread

[13,14].

Such thread break-up after continuous extension is a more efficient mechanism of emulsification than is repeated

and

break-up

of a drop into two halves

5 6

conditions. In an impor[lS], thread formation

7

break-up

under quasi-equilibrium tant recent development

2 3 4

was shown

to be not

only

more

8

efficient but also more likely to be the predominant mechanism in many practical mixing devices. The

9

final size of the emulsion drops was found to depend on the bulk liquid viscosities in a markedly different manner for the two dispersion mechanisms, especially if the dispersed liquid was the more viscous one. The effect of the su$uce rheological behaviour induced by emulsifiers, however, has only partly been elucidated so far [ 141. In any future efforts to tackle this problem, the surface dilational viscosity K is likely to be a relevant

10

parameter.

15

References 1

F. van Voorst Vader, T.F. Erkens and M. van den Tempel, Trans. Faraday Sot., 60 (1964) 1170.

11

12 13 14

Eng. Aspects 85 (1994) 211-219

219

J. Boussinesq, Ann. Chim. Phys., 29 (1913) 349. B. Stuke, Chem. Ing. Tech., 33 (1961) 173. J. Lucassen and M. van den Tempel, J. Colloid Interface Sci., 41 (1972) 491. J.A. Mann and G. Du, J. Colloid Interface Sci., 37 (1971) 2. D.A. Edwards, H. Brenner and D.T. Wasan, Interfacial Transport Processes and Rheology, ButterworthHeinemann, Stoneham, MA, 1991, p. 232. F.C. Goodrich, J. Phys. Chem., 66 (1962) 1858. E.H. Lucassen-Reynders and J. Lucassen, Adv. Colloid Interface Sci., 2 (1969) 347. J. Lucassen and D. Giles, J. Chem. Sot., Faraday Trans., 71 (1975) 217. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon Press, New York, NY, 1959, p. 54. S.P.S. Andrew, in P.A. Rottenburg (Ed.), Proc. Int. Symp. Distillation, Institution of Chemical Engineering, London, 1960, p. 73. P. Joos and M. Van Uffelen, J. Colloid Interface Sci., 155 (1993) 271. G.I. Taylor, Proc. R. Sot. London, Ser. A, 146 (1934) 501. B.J. Carroll and J. Lucassen, in A.L. Smith (Ed.), Theory and Practice of Emulsion Technology, Academic Press, London, 1977, p. 29. J.M.H. Janssen

and H.E.H. Meijer, J. Rheol., 37 (1993) 597.