Non-equilibrium surface thermodynamics. Measurement of transient dynamic surface tension for fluid—fluid interfaces by the trapezoidal pulse technique

Colloids and Surfaces, 57 (1991) 335-342 Elsevier Science Publishers B.V., Amsterdam

335

Non-equilibrium surface thermodynamics. Measurement of transient dynamic surface tension for fluid-fluid interfaces by the trapezoidal pulse technique G. Loglio”, U. Tesei”, N. Degli InnocentP, “Dipartimento

di Chimica Organica, Universita’di

Italy bZentralinstitut

fiir Organische

Rudower Chaussee

(Received

Chemie, Akademie

R. Millerb and R. Cini” Firenze,

Via G. Capponi 9, 50121 Firenze,

der Wissenschaften

der DDR,

5, 1199 Berlin, Germany

18 June 1990; accepted

13 August 1990)

Abstract The trapezoidal pulse perturbation is shown to be an attractive procedure for the measurement of dynamic properties of fluid-fluid interfaces. The surface response characterizes the surface relaxation behavior well. Theoretically, this procedure has foundations within the framework of linear non-equilibrium thermodynamics. From the experimental point of view, the procedure offers the possibility of checking the experimental reliability and exhibits practical aspects due to the short duration of the surface response. Benefits are also evident in data processing.

INTRODUCTION

A unified modern formulation for equilibrium and non-equilibrium surface thermodynamics has been developed by Defay et al. [ 11. The most important observable physical quantity for the study of surface thermodynamics is the surface tension, y= ( dG/dA)T,p,,, that is, the partial derivative of the Gibbs energy of the whole system with respect to the area of a formed surface (at constant temperature T,pressure p and amount of each component n). For non-equilibrium systems, together with other bulk and surface properties, surface tension changes as a function of time, y= y( t) (dynamic surface tension). The time dependence of y( t ) has been described in various forms in the literature [ 21. Considering systems composed of two multicomponent fluid phases separated by an interface, we restricted our attention to cases in which y (t ) results from small amplitude changes of surface area disturbing a pre-existing equilibrium. Under this condition, we have pointed out the importance of treating interfacial dynamic properties in terms of the principles of linear non-equilib-

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Elsevier Science Publishers

B.V.

336

rium thermodynamics [ 31. Also, we emphasized that experimental techniques to measure y(t) and methods of data analysis have to be devised according to the above-mentioned principles. The square pulse is a typical functional form relevant to the input perturbation which has already been considered in the thermodynamics of irreversible processes [ 41. In this paper, we show the inherent benefits of adopting the square pulse (or, in practice, the trapezoidal pulse) for the institution of a measurement procedure in the case of transient dynamic surface tension at fluid-fluid interfaces. THEORETICAL

ASPECTS

In the linear case, non-equilibrium properties of adsorption layers at fluid interfaces are quantitatively described by the surface thermodynamic modulus

[VI e*(iW)=A(d2G/dA2)T,p,n =B{Ay(t)}/B{d

(1) In A(t)}

where i is the imaginary unit, COthe angular frequency and 9 the Fourier transformation operator. Upon rearranging Eqn ( 1) , we have dy(t)=~~‘{~*(io)~9{dInA(t)}}

(2)

or

b(t)

=s

9-‘{E*(iCL)),r}.AlnA(t-r)dr

(3)

0

The trapezoidal pulse is represented by [ 71 0,

t-co o
@c Ml, -@t+@(t,+2t,),

AlnA(t)= i

t, t2+2t1

0,

(4)

where t = 0, tl, tl + t2 and t2 + 2tI are the time instants at the trapezium vertices and 8=d In A/dt= (l/tl) ln(l-AA/A,)=constant (AA/A0 being the fractional area change (Fig. 1) ). The Fourier transformation of Eqn (4 ) is F{AlnA(t)}=

[S/(iw)2]

{l-exp(-iiwt,)-

exp[ -io(tI+t2)]

+exp[ -io(t2+2tl)]}

(5)

337

time

/

5

Fig. 1. Schematic diagram of the trapezoidal pulse to define the characteristic time instants tl and t-2.

When surface relaxation is dominated by the translational-diffusion mechanism, the modulus E*(iw) is expressed by a two-parameter model [ 81 (solution-gas interface, one-component system) e*(iO)=

(COJiw)/(Jiw+J2wo)

(6)

in which E,,= - dy/d In r and o,, = ( dc/dl”)2 O/2 (T and c being surface and bulk concentration respectively, and D the diffusion coefficient). By inserting Eqns (5) and (6) into Eqn (2), we obtain the surface response to trapezoidal pulse excitation, for the diffusion mechanism 4 4(t)=

(t) = ( 8cO/2wO) exp (20,t) erfc (20,t) l/2 + (2tO~~)/(27CW0)1’2-~EO/2W0 , o
dY2(t)=dY1(t)-dY1(t-tl)

,

i

4J,(t)=&(t)+dyl(t-

(7)

t,
dy,(t)=dy,(t)-dyl(t-(t,+t,))

(t2+2t1))

tl+&t,+2t,

EXPERIMENTAL

We used a time-resolved surface viscoelastometer [ 91. Essentially, the measurement apparatus is a computer-driven Wilhelmyplate Langmuir trough. Barrier motion (in the form of an elastic ring deformation ) is governed by a compiled-BASIC program, via (1) an Analog Devices RTI-817 (real time interface) board, (2) a motor controller and (3) a stepping motor. Experiment parameters are set by a screen menu (Fig. 2 ).

338

MEASUREMENT OF INTERFACIAL DILATIONAL PROPERTIES Department of organic Chemistry, University of Florence Displacement > TRAP

<

Function STEP

Initial Final

4095

SENSOR

]

OUTPUT

Shortening Shortening

Initial Area Final Area Average Area Fractional-Area Change Duration Time 1 Duration Time 2 steps :1493 cycles t ACTUAL VALUES Modify

TRI

EXP

Dim. Diam.

Pts in 1 in SfZl. by + -

/

(Select by Left/Right

Data

SIN

:

0.1812 : 0.9277

cm cm

:20.5647 :19.5361 :20.0504

cm' cm' cm'

: : : :

s s

0.0500 7.000 100.0

1

Arrow)

PRT SCR QUIT

se1.

STATUS by U/D Arrc

PARALLEL

PORT

- to confirm

Fig. 2. Laser printer copy of computer screen: software menu and values of experiment parameters.

A fractional interfacial area change (i.e., an external stimulus) is applied to the system as a function of time, say dA (t)/A, and the subsequent variation dy( t) is measured in synchronism with the stimulus. Excitation and response curves are read into the computer, stored in a disk file and displayed on a strip-chart recorder. The surfactants used were (1) n-dodecyldimethylphosphine oxide (DC&PO) obtained from Procter and Gamble, (2) sodium bis (2-ethylhexyl) -sulfosuccinate (DESS ) from Fluka and (3 ) n-octadecyltrimethylammonium bromide (STAB) from Fluka [lo]. Water was doubly distilled from alkaline permanganate, in the usual way, and checked with respect to surface purity. RESULTS

AND DISCUSSION

Figures 3-5 represent (manually digitized) curves of dy response to trapezoidal AA/A pulse excitation, for an aqueous solution of (1) a non-ionic surfactant (DC,,PO), (2) an anionic surfactant (DESS) and (3) a cationic surfactant (STAB). Rectangular symbols are the experimental points, the solid line is the least-squares fit curve, from Eqn. (7)) and the dots represent the baseline. The concentrations for the three surfactants are one order of magnitude below the respective critical micellar concentrations (c.m.c. ). Above the c.m.c., the surface relaxation mechanism is no longer purely diffusive because of the influence of the monomer-micelle equilibrium. The model parameters E,,and w, are surface thermodynamic properties. Their values, obtained from the fits for the different surfactants, depend (i) on the

339

time

/

s

Fig. 3. Time evolution of dy response to a fractional area trapezoidal pulse perturbation (AA/ A,,=0.05, tl =6.5 s). Comparison between observed values and calculated values using the diffusion model. Data for an aqueous solution of a non-ionic surfactant (DC,,PO). Room temperature, T=2O”C, relative humidity, h=90%. Fit parameters: eo=27.3 mN m-l, ~,=2.2*10-~ rad s-l.

Fig. 4. Same as Fig. 3. Data for an aqueous solution of an anionic surfactant (DESS). Room temperature, T=22”C, relative humidity, h=98%. Fit parameters: t,=33.9 mN m-‘, o,= 2.0. IO-’ rad s-r.

concentration c and (ii) on the surface activity dy/dc (this last quantity ultimately depends on the amphiphilic structural characteristics of each surfactant). Taking into consideration the theory and results reported above, we can summarize the experimental benefits of the trapezoidal pulse procedure. Comparison with the frequency-domain procedure In general, for all transient procedures, information about surface relaxation properties is obtained by looking only at the time-evolution of the dy response.

-I

!.50 time / s

Fig. 5. Same as Fig. 3. Data for an aqueous solution of a cationic surfactant (STAB). Room temperature, T=21”C, relative humidity, h=90%. Fitparameters: ~=34.8 mN m-l, w,,= 1.8*10W4 rad SC’.

In contrast, in the case of the sinusoidal procedure, relaxation properties are derived from observation of the time-lag (phase angle) between AA/A excitation and dy response. However, except for uniform surface extension/contraction, the apparent phase angle also contains the Marangoni wave propagation time [ 111. This is difficult to evaluate and thus is a source of significant experimental error. In the case of pulsating bubbles and elastic ring deformations this effect can be neglected. Comparison with other time-domain procedures ( 1) As seen in Figs 3-5, the dy response curve to trapezoidal pulse excitation consists of four sections. Two of these exhibit a rapidly changing slope which characterizes the surface relaxation behavior (the fitting parameters, obtained from processing the two sections separately, must have the same values assuming that surface relaxation after extension and contraction is due to a single dominating mechanism). In contrast, the dy response ensuing after a stepexcitation shows a featureless long “tail” (thus, in the limit of long times, a different mechanism may overtake the initial one). (2) Since the final surface area is the same as the initial one, dy( t) returns back to zero value in a much shorter time than that relevant to step excitation. This fact allows several successive measurements to be repeated on the same sample. Thus, measurement reproducibility can be checked. Also, a possible baseline shift, if any, soon becomes apparent. It is therefore easy to estimate the measurement reliability and to discover possible sources of experimental error. (3) According to theory, the two dy steps in the opposite directions (per-

341

taining to extension and contraction) must show the same magnitude, whatever the relaxation mechanism. When the observed behavior does not comply with this symmetry rule, within experimental error, the measured curve is discarded as it is certainly affected by a concomitant spurious process, is an artifact or is due to a change of the relaxation mechanism. (4) When the surface relaxation mechanism is translational diffusion, observed values can be compared with the calculated ones by using the diffusion model which is mathematically expressed with a known functional form for the response to trapezoidal pulse, Eqn (7). Referring to Figs 3-5 we obtain a good agreement between experimental data and the theoretical curves. This proves that for the systems studied diffusion is the controlling relaxation mechanism. (5) As a remark, the rapid decay of the response curve allows data to be subjected without further treatment (apodization) to numerical Fourier transformation. It is understood that there is complementarity (not competition) in using the present trapezoidal pulse procedure and other time- or frequency-domain procedures, for the measurement of surface non-equilibrium properties. We note that such a kind of measurement is among the most demanding in physics due to the difficulty of reproducing the surface equilibrium state and to the multitude of processes that may affect the measurement signal. CONCLUSIONS

The comparative examination reported above suggests that the trapezoidal pulse procedure may be advantageously adopted for the measurement of transient dynamic surface tension, i.e. for the determination of non-equilibrium properties of adsorption films. The response curve conveys all information about surface relaxation properties, i.e. there is no need to refer to the input perturbation. The time evolution of dy (t ) can be attributed to a single particular mechanism dominating the surface behavior rather than to different subsequent mechanisms. Measurement reliability can be estimated. Curve fitting to data (or numerical Fourier transformation) can be easily performed. ACKNOWLEDGEMENTS

The authors thank the Akademie der Wissenschaften der DDR and the Consiglio Nazionale delle Ricerche for their financial support.

342 REFERENCES R. Defay, I. Prigogine and A. Sanfeld, Surface thermodynamics, in M. Kerker, A.C. Zettlemoyer and R.L. Rowe11 (Eds.), Colloid and Interface Science, Vol. 1, Academic Press, New York, 1977, p. 527. 2 R. Miller, Colloid Polym. Sci., 259 (1981) 1124. 3 G. Loglio, U. Tesei and R. Cini, Colloid Polym. Sci., 264 (1986) 712,1098. 4 H. Baur, Einfiihrung in die Thermodynamik der Irreversiblen Prozesse, Wissenschaftliche Buchgesellschaft, Darmstadt, 1984 (in German). 5 G. Loglio, U. Tesei and R. Cini, Ber. Bunsenges. Phys. Chem., 81 (1977) 1154. 6 I. Balbaert and P. Joos, Colloids Surfaces, 23 (1987) 259. 7 N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior. An Introduction, Springer-Verlag, Berlin, 1989. 8 J. Lucassen and M. van den Tempel, Chem. Eng. Sci., 27 (1972) 1283. 9 G. Loglio, U. Tesei and R. Cini, Rev. Sci. Instrum., 59 (1988) 2045. 10 U. Tesei, G. Loglio, N. Degli Innocenti and R. Cini, unpublished work. 11 I. Panaiotov, D.S. Dimitrov and M.G. Ivanova, J. Colloid Interface Sci., 69 (1979) 318.