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Adsorption and Desorption at Dynamic Nonequilibrium Interfaces: Interfacial Stagnation Flow Williarn B. Krantz,' Phillip C. Martin, and Lee F. Brown Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309
A dynamic Langmuir trough is described for studying the growth of nonequilibrium surfactant films formed in interfacial stagnation flow. This new instrument provides for the continuous measurement of the instantaneous drag force on these surfactant films. These measurements can be related to the thermodynamic state and adsorption/desorption characteristics of these molecular films. Several new models are developed to describe the growth of nonideally compressible surfactants, and those exhibiting adsorption or desorption. These new models include a rigorous analytical solution to the convective diffusion equation which allows for a surfaceenergy barrier resistance to desorption and for convective diffusion in the bulk phase. These models are applied to film-growth data for aqueous solutions of stearic acid, phenol, and octadecyltrimethylammoniumchloride.
Introduction This paper considers a phenomenon associated with the interface between a liquid and a gas or another immiscible liquid in relative moition. Under certain hydrodynamic configurations a stagnation point or line exists a t which the interfacial velocity is zero; these flows will be referred to as interfacial stagnation flows. The stagnation points or lines can be natural consequences of the hydrodynamic configuration, such as the rearward stagnation point on a moving drop or bulbble, or can be imposed, as when a physical barrier is placed in the interface. Numerous observations have been made indicating that a stagnant film of molecules will develop in the interface upstream from a hydrodynamic stagnation point or line if the liquid contains a solution of surface-active molecules. A stagnant film of surface-active molecules may be viewed as developing ,at a stagnation line in the gas-liquid interface shown in Figure 1 by the following mechanism. Surface-active molecules which flow toward the stagnation line are prevented from moving past the stagnation line and accumulate as a {stagnantfilm. The film is stagnant only in the sense that the interfacial velocity within the film is zero; it is dynamic in that this film can grow in time. The rate of growth of the stagnant film will depend upon the rate a t which surface-active molecules are added to the film, the rate at which these molecules leave the film, and the thermodynamic state of these molecules in the surface film. The stagnant film will continue to grow until the rate a t which molecules leave the film is equal to the rate a t which molecules are convected into it. Molecules can leave the film by desorption, since the film has a surface concentration exceeding its equilibrium concentration. Film growth also can be affected by many other factors; examples are compression of the film caused by viscous drag exerted by the substrate fluid, two-dimensional phase transitions within the film, formation of multimolecular films, collapse of the monolayer into bulk-phase fragments or lenses, and evaporaltion of the surface-active molecules. Engineers have been interested in stagnant-film formation in interfacial stagnation flow because of its occurrence in a variety of contacting equipment. Even in the purest systems simall amounts of surface-active impurities are always piresent and stagnant-film formation can adversely affect such operations as mass transfer, settling or rise times in gravity separators, and holdup in bubble columns. Indeed, stagnant-film formation has been observed on wetted-wall columns, drops and bubbles, 0019-7874/78/1017-0341$01 .OO/O
liquid jets, open channel flows, rotating-drum contactors, horizontal radial flows, and film flow over spheres. The authors' interest in stagnant-film formation in interfacial stagnation flow was generated for two reasons. This phenomenon is sensitive to the presence of surfactants a t ppm levels, and so may provide a new method for detecting trace amounts of surfactants. The potential application of stagnant films to measuring surfactant concentration has been presented (Krantz, et al., 1974). In addition to the potentialities for surfactant detection, stagnant-film formation on open-channel flow is an ideal configuration for studying adsorption and desorption from dynamic nonequilibrium interfaces under conditions of known bulk-phase hydrodynamics. This paper reexamines stagnant-film formation in interfacial stagnation flow both theoretically and experimentally; in particular, it considers the effects of adsorption and desorption on this phenomenon. Related Studies This literature review will be restricted to stagnant-film formation a t the gas-liquid interface in open-channel flow. For a variety of reasons, there has been little success in studying stagnant-film growth using film geometries other than horizontal open-channel flow; uncontrollable or unknown hydrodynamics, high shear rates with correspondingly very short films, and significant end or edge effects are all serious difficulties in other systems. The most recent comprehensive reviews of this subject are those of Merson (1964) and Mass (1967). It is convenient to divide the present review into modeling studies of stagnant-film formation and experimental studies. Modeling Studies. Theoretical studies of this phenomenon seek to predict both the rate a t which these stagnant films will grow and the ultimate steady-state length of the film. The various models proposed for stagnant-film growth differ primarily in their assumptions concerning three aspects of the film. First of these is the surface equation of state assumed for the surfactant in the compressed stagnant film; Le., the surfactant in the film may obey any of a variety of equations of state extending from a two-dimensional ideal-gas law for low surface pressures to completely incompressible solid-like behavior a t higher pressures. Another factor in which models differ is in the significance and form of any desorption flux from the stagnant film. Since the film usually has a surface concentration greater than its equilibrium value, desorption from the film is possible via several mechanisms.
0 1978 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 LIOUID-GAS INTERFACE
/ SUR FACE-ACTIVE MOLECULES
STAGNANl FILM STAG N A T l ON
FLUID FLOW
increasing function of surface concentration, which in turn increases with the surface pressure, which in its turn increases with the film length. As a result, models which allow for desorption can predict an ultimate steady-state length for the film. This model predicts a steady-state length given by
Lo = Figure 1. Mechanism for film growth in interfacial stagnation flow.
Finally, the freely flowing interface upstream from the stagnant film may not have sufficient time to equilibrate, and so some assumption must be made for the adsorption flux; i.e., the third aspect in which models differ is in the significance and form of any adsorption flux to the freely flowing interface upstream from the leading edge of the stagnant film. Merson (1964) and Merson and Quinn (1964) developed several models for film growth, some of which are applicable to the data presented here. Detailed derivations of the equations can be found in these references. The models will be labeled “Case I”, “Case 11”,etc., for later reference. As a matter of convenience for the interested reader who desires more details on any of the models presented here, the case numbers used here correspond to those catalogued in the thesis on which this paper is based (Martin, 1976). Since not all the situations modeled in the thesis are relevant to the present study, the cases presented here will not necessarily be numbered sequentially. Case I. Incompressible surfactant; negligible desorption from film; rapid attainment of equilibrium upstream from film. For this case the length of the stagnant film L is given as a function of time t by
L=
Ulrnrm
t
(r, rm)
(1)
-
where Ulm is the upstream surface velocity, rmis the surface concentration at the leading edge of the film, and I‘8is the constant surface concentration in the stagnant film. That is, this model predicts that an incompressible film will grow linearly in time. Case 11. Ideally compressible surfactant; negligible desorption from film; rapid attainment of equilibrium upstream from film. In this model, the surfactant film is assumed to obey a surface equation of state which is analogous to a two-dimensional ideal-gas law. Since the surfactant is now considered to be compressible,the growth rate will depend on the shear stress r, exerted by the substrate flow on the interface
in which k is Boltzmann’s constant. For an ideally compressible surfactant the film growth will be proportional to the square root of time. Case IV. Ideally compressible surfactant; surfaceenergy-barrier-controlled desorption from film; rapid attainment of equilibrium upstream from film. This case assumes a desorption flux of the form JD = kl(r, - r0), in which k l is the rate constant for desorption, and yields the following equation for film growth
The desorption rate from the film is a monotonically
(
)
2 Ul,r,k T ‘1’ klr,
(4)
Many additional models have been developed, but none directly pertinent to the present work. Details are in the theses of Merson (1964), Mass (1967), and Martin (1976). Of note is one in Merson which allows for slow upstream adsorption following a penetration-theory type of model. No models appear to have been developed, however, which account for nonideal surface behavior other than the limiting case of completely incompressible surfactants. Furthermore, no models have been developed which account for a bulk-diffusion resistance to desorption from the stagnant film. Models which incorporate a combined surface-energy barrier and bulk-diffusion resistance to desorption would also be of value. Finally, no models have accounted for simultaneous slow adsorption of the surfactant upstream from the stagnant film and desorption from the film. This paper presents new models incorporating these effects. Experimental Studies. Experimental studies seek to measure the rate of growth of the stagnant films, the final steady-state length of the film, the interfacial and bulkphase hydrodynamics both upstream from and beneath the stagnant film, and finally, the force exerted by the surface-film pressure on the stagnation barrier. Unfortunately, none of the prior experimental studies has measured all of these quantities for surfactants of known thermodynamic properties. Again, this review will be confined to experimental studies of stagnant-film formation on open-channel flows. Merson (1964) and Merson and Quinn (1964) studied the growth of stagnant films by observing the stopping point of talcum powder sprinkled onto the upstream surface. Surfactants such as stearic acid and calcium oleate were used to verify the incompressible growth model given by eq 1. The ideally compressible-growth model given by eq 2 was substantiated using solutions of dodecyltrimethylammonium chloride. The desorption of an unknown contaminant in reagent grade benzene appeared to follow the surface-energy-barrier-controlled-desorption model given by eq 3. Merson encountered considerable difficulty in measuring the growth rates from solutions of octanol using the talcum powder method due to the highly compressible nature of this surfactant. The talcum powder, rather than stopping abruptly a t the leading edge of the stagnant film, decelerated over a distance of several centimeters; this made precise determination of the stagnant-film length impossible. Merson made no attempt to determine whether the bulk-phase hydrodynamics agreed with the predictions for subcritical open-channel flow. Kenning and Cooper (1966) measured the velocity profiles upstream from an interfacial stagnation barrier in a relatively narrow channel (2.44 cm) with a deep film (1.35 cm). They observed circulatory motions in the interface such that the interfacial velocity was not zero as assumed in the film-growth models discussed above. These interfacial circulation currents arise because the substrate drag on the liquid-gas interface near the side walls of the channel is less than that in the center. Thus, the interfacial currents are downstream in the center of the channel and
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edge and far enough downstream from the upstream sweep, the free surface velocity, U,,, will be fully developed a t the boundary of the system. The material balance on this system then becomes
Figure 2. System defined for stagnant-film material balance.
upstream near the side wall. Kenning and Cooper developed a simple model to describe this circulatory flow which ignores the effect of interfacial shear viscosity. This model predicts that the interfacial circulation velocity decreases with increasing channel width and decreasing film thickness. A moire sophisticated model developed by Mass (1967), which accounts for interfacial shear viscosity, predicts that this property inhibits the interfacial circulation. Thus, by properly selecting the channel width and thickness, it is possible to eliminate the effect of interfacial circulation and to ensure that the zero interfacial velocity condition prevails. Mass (1967) extended Merson’s studies to include other surfactants. He developed a radioactive tracer technique to measure the film lengths. However, this method met with limited success due to interference effects from the background radiation of the bulk fluid. Mockros and Krone (1968) attempted to measure the force on the stagnation barrier by attaching it to a pendulum whose deflection was measured. This method was more suitable for measuring the force on films which had reached their steady-state length rather than for continuous force measurement of growing films. Mockros and Krone studied only the natural unknown contaminants in tap water and did not analyze the hydrodynamic conditions in their flow channel. The current status of experimental studies of this phenomenon is best summarized in a recent article of Cook and Clark (1973), who concluded that empirical correlations were of more value than the available theoretical models for predicting stagnant film lengths. In the present article new models are developed to remedy this, and careful experimental studies are presented which support some of these. It is our contention that prior experimental studies have met with1 limited success because insufficient information was avail able concerning the hydrodynamics and relevant physical properties of the surfactants studied. Past experimental studies have not always ascertained whether the channel hydrodynamics agreed with those assumed in the film-growth models. The studies which did verify the hydrodynamics carefully worked with surfactants of unknown physical properties. Further, no experimental techniques used thus far have been able to measure film growth for surfactants exhibiting a “soft” leading edge. Neither has a technique been developed for continuously measuring the film growth. An apparatus for accomplishing thlese goals will be described in the experimental design section.
Theoretical Development Consider the surface material balance taken on the system shown in Figure 2. Let L, be the length of surface over which the surface material balance is to be written, L,, the length of the flow channel, L, the instantaneous length of the stagnant film; 1, a coordinate measuring the distance downstream from the upstream sweep which creates a fresh interface, and x, a coordinate measuring the distance downstream from the leading edge of the stagnant film. By choosing L, far enough upstream from the leading
This balance states that the rate of accumulation of surface-active material in the system is equal to the rate a t which surface-active material is convected into the system (first term on the right-hand side) minus the rate a t which this material desorbs from the compressed stagnant film (second term) plus the rate a t which the freely flowing surface upstream from the leading edge of the stagnant film gains surface-active material by adsorption from the bulk (third term). Prior modeling studies appear to have considered a material balance on a system consisting of just the stagnant film. The latter system presents a conceptual problem in that it does not allow for convection of material into the film, since by definition the interfacial velocity must be zero a t the leading edge of the film. Nevertheless, the results of models based on the latter conception appear to be in error only for those situations in which the upstream adsorption is slow. To increase the range of the theoretical attack on film growth, nine new models have been developed based on eq 5. The ones of particular interest are presented below; details of the remaining models may be found in the thesis of Martin (1976). These models differ in the assumptions they use for the surface equation of state and in the significance and form of the desorption and adsorption fluxes in eq 5 . One assumption which is common to all the models treating compressible films (Cases VII, IX-XI, XIII, and XV) is that the substrate flow is fully developed throughout the length of the film. This means that the substrate shear stress 7, is not a function of position beneath the film. Appropriate treatment of the data, as illustrated later, can make this assumption applicable even to films with a significant region a t the leading edge where the substrate flow develops. Case VII. Surface film is assumed to obey the Volmer-Mahnert surface equation of state; negligible desorption from the film; rapid attainment of equilibrium upstream from film. The Volmer-Mahnert equation of state allows for compressible surface behavior which is nonideal since it accounts for the finite size of the adsorbed molecules. The equation is s(a - ao) = kT (6) where H is the surface pressure which is a function of the viscous shear stress exerted on the surface film by the substrate fluid and uo is the area occupied by one molecule of surfactant at infinite surface pressure. The assumptions for Case VI1 imply that J D = 0 and J A = 0, and that r(l=L,-Ld = rm.[Note that rmneed not be equal to ro,the Gibbs equilibrium surface concentration; rather, it is merely some constant concentration which is established a t the leading edge of the film. This comment applies to all the models developed here which specify the interfacial concentration to be rmat the leading edge of the stagnant film and has significant implications in interpretation of film-growth data.] The surface concentration r(x=L) is determined from the surface equation of state and from the surface pressure; the latter is a linear function of x for fully developed laminar flow between two flat plates. Equation 5 then can be integrated to yield the following
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implicit equation for L as a function of time
Equation 7 reduces to eq 2 for the limiting case of small surface pressures or short films. It reduces to eq 1 in the limit of very large surface pressures or very long films. Thus, Case VI1 includes the ideally compressible and incompressible models described by eq 1 and 2, respectively, as special limiting cases. Case IX. Ideally compressible surfactant; surfaceenergy-barrier-controlleddesorption and adsorption. This model implies that JD= k l ( r , - r0)and JA= k l ( r o- rl). The slow adsorption implies that the surface concentration upstream from the leading edge of the stagnant film will be a function of 1. A material balance in this upstream region yields
rl = ro+ (ri- ro)e-kll/"lm
(8)
in which r i is the surface concentration a t the upstream sweep where fresh surface is created. Equation 8 permits determination of rrnand r(l=L,-Ls) in eq 5. This model also assumes that U1, remains constant up to the leading edge of the film. This should involve little error as the boundary-layer transition region at the leading edge of the film is short in comparison with the total length of the upstream region. The ideally compressible behavior implies that within the stagnant surfactant film rx= I'+Lc-L) + r , x / k T . The resulting form of the surface material balance cannot be integrated analytically. However, an analytical solution can be obtained for conditions such that kl(Lc - Lo)/Ulrn
