Langmuir 1986, 2, 79-82
79
former gas atmosphere (see eq 10).
knowledged.
Acknowledgment. We are grateful to the National Science Foundation for the financial support of this work. Valuable discussions with Glen Connell are also ac-
Registry No. HzO,7732-18-5; CO, 630-08-0; COz, 124-38-9; SiOz,7631-86-9;TiOz, 13463-67-7;A1,0,, 1344-28-1;ZnO, 131413-2; MgO, 1309-48-4.
Energetics of the Marangoni Instability Benjamin R. Irvin Program in Chemistry and Physics, St. Andrews Presbyterian College, Laurinburg, North Carolina 28352 Received July 16, 1985. In Final Form: September 4, 1985 A thermodynamic analysis of a simple model of the Marangoni instability is developed in order to account for the conversion of potential energy (either chemical or thermal) into kinetic energy of convection. Elucidation of the energy coupling mechanism yields some new insights on the role of dissipative processes in the interface.
Introduction Marangoni instability may be defined as the onset of convection a t a fluid interface driven by self-amplifying fluctuations in interfacial tension. Self-amplification may occur if the convective flow couples with existing gradients of temperature or concentration in the adjacent bulk fluids. The first theoretical model to establish the possibility of the instability was provided by Pears0n.l His analysis was limited to consideration of the dynamics of only one of the bulk phases. Sternling and Scriven2 performed a more thorough analysis in which the driving gradient was in the concentration of a surface-active solute, rather than in temperature as in the Pearson analysis. Both fluid phases were explicitly accounted for in the hydrodynamic and solute transport equations. These authors established the conditions required for instability based upon physical properties of the two fluids and parameters relating to the transport of solute and ita effect on surface tension. Numerous refinements and extensions of the SternlingScriven model have appeared in the 25 years since its publication. These studies, by focusing primarily on the determination of the loci of marginal stability, have paid relatively little attention to the mechanisms and energetics of the phenomenon. Investigation of these latter aspects, besides ita intrinsic interest, could yield an understanding of the processes involved that would guide further modeling efforts as well as help to place Marangoni convection within the domain of nonequilibrium thermodynamic theory. Previous work by this author has elucidated some of the mechanistic and energetic features of a simple model of Marangoni con~ection.~A particular problem that was addressed was the quantitatively accounting for the source of kinetic energy in convection. The solution of this problem and its implications for models of Marangoni instability are the subjects of this paper. It is shown that the energy transformation that drives convection cannot be derived from such models unless surface thermodynamic quantities are included in the energy and/or solute conservation equations and that, when this is done, relations that demonstrate the balance between chemical or thermal potential energy and hydrodynamic kinetic energy are obtainable. (1) Pearson, J. R. A. J. Fluid Mech. 1958, 4, 489. (2) Sternling, C. V.; Scriven, L. E. AZChE J. 1959,5,514. (3)Irvin, Benjamin R. Ph.D. Dissertation, The Florida State University, Tallahassee,FL,1983.
0743-7463/86/2402-0079$01.50/0
Thermodynamics of Marangoni Convection The following analysis applies strictly to convection driven by surface-tension gradients provided that (i) the interface is planar, (ii) convection is in the form of twodimensional roll cells, (iii) the perturbation in solute concentration (or temperature) has the same spatial periodicity as the convection pattern (the normal case of convective transport), and (iv) the bulk phases at the interface are in local thermodynamic equilibrium with each other and with the interface. Implications of these provisions are discussed below. Figure 1depicts the geometry of the system and conditions i-iii. I t will be helpful also to restrict consideration to the case of marginally stable convection (i.e., the disturbance neither grows or decays). Then attention can be focused on the mechanism of energy transformation in a steady-state system. Any model that describes Marangoni instability must include a relationship between surface-tension gradients and the variation of solute concentration, c, or temperature in the adjacent bulk fluids. In all previous models these relationships have been merely constitutive. Thus cc = dy/ac, (Y = 0) (subscript a refers to fluid a) and S;. = dy/dT (Y = 0) would be taken as definitions of the constant proportionality between variations in concentration and temperature and the resulting change in surface tension. These definitions are reasonable approximations in the analysis, but as essentially empirical relationships, they tend to obscure the thermodynamics of energy coupling. Since the discontinuity in shear stress a t the interface is directly related to the gradient of surface tension, and the gradient of bulk concentration or temperature is determined by the solution of the convection-diffusion problem, the constants Cc and tT fully characterize the link between the thermodynamic fields, c(X,Y,t) and T ( X , Y , t ) and , the hydrodynamic velocity field. It is not necessary to describe this connection any further in order to solve the stability problem. These empirical parameters, however, can be easily cast into more fundamental thermodynamic terms: -ay = - - =a7- ap0 ac, ap0 ac,
-rapo
ac,
(1)
and
where
r is the surface excess concentration of solute, po
0 1986 American Chemical Society
80 Lurigvjuir, Vol. 2, No. 1, 1986 high c ( o r T ) Y
J
7 low c ( o r T)
b
I uo Figure 1. (a) Coordinate system and labels for convecting system. Net transport of solute (or heat) is from phase a to phase b. The periodic variation of the concentration in phase a near the interface, c,(6Y), and of the surface velocity, uo, is shown in (b). Solute concentration (or temperature) will be elevated at point 2 if the bulk diffusion coefficient (or thermal conductivity) is smaller for phase a than for phase b.
is the solute chemical potential at the interface, and s, is the interfacial entropy density. The first equality follows from the Gibbs adsorption equation and the second results from the identity between surface tension and surface free energy density in a one-component systeme4 This significance of the j+ parameters has been largely neglected, although Smensen showed how relations similar to eq 1 could be used to estimate Cc from equilibrium measurements with an ideal solution appr~ximation.~ Energy and Mass Conservation at the Interface. The original analysis of Marangoni convection induced by solute transfer employed boundary conditions in which solute adsorption was neglected.2 The interface was treated as a mathematical plane whose only property was surface tension (surface viscosity was introduced, but was deleted from the solution of the problem because of the analytical difficulties it caused). In studies of thermally induced convection, a separate balance equation for surface entropy has apparently never been considered. In the following discussion, attention will be restricted to the isothermal, solute-transfer case. It is then shown how the thermal instability is fully analogous. More recent treatments of the solute-transfer instability have included complete conservation equations for adsorbed solute (e.g., ref 5). The equation suitable in the present case (two-dimensional flow, plane interface) is
Iruin
neutrally stable disturbance and will be dropped with that additional provision. With the transfer of matter there is an associated transfer of chemical energy. Equation 3 may be converted to a conservation equation for chemical energy through multiplication by M ~ ( Xthe ) , solute chemical potential function at the interface. The observation may be immediately made that if adsorption is neglected (r = 0), no chemical energy is lost from the bulk phases as solute is transported across the interface. Since it seems intuitively clear that the energy source for convection must be derived from the bulk gradients of solute concentration, this apparently benign neglect will eliminate the possibility of accounting for that source. The expression for the kinetic energy flux density into the bulk phases is based on the fluid mechanical formula that relates the discontinuity in the tangential component of stress across the interface to the force exerted by the surface tension gradient. Multiplying by the velocity at the interface yields
where qa and q b are the dynamic viscosities of the bulk fluids, ui is the X component of the velocity in phase i, and 8, is a surface viscosity coefficient whose exact nature is not important here. Without the surface velocity factor, eq 4 is the central coupling condition between the hydrodynamic field, whose momentum source appears on the left as a discontinuity in the stress field, and the concentration field that determines the spatial variation in surface tension on the right. The surface viscosity term, as well as the surface diffusion term in eq 3, serves to introduce dissipation into the interfacial dynamics. In order to see the connection between eq 3 and 4,it is necessary to integrate the flux densities over at least one half-cell (from 1 to 2 in Figure 1). Given the symmetry of the pattern, every half-cell will give an identical result. The net diversion of chemical energy from the bulk phase is
Integrating each term by parts, using the facts that uo vanishes and bO(x)passes through extrema at the half-cell boundaries, one obtains Echem
=
From eq 4, the net flux of kinetic energy from the interface to the bulk phases is
By use of eq 1and integration of the second term by parts, eq 7 is transformed to where D iis the solute diffusion coefficient in phase i and uois the X-component of the fluid velocity at the interface (i.e., the surface velocity). Equation 3 represents the local rate of diversion of solute flux from bulk diffusion to the surface per unit area. The three terms on the right represent accumulation, convective transport, and surface diffusion, respectively. The first term vanishes for the (4) Adamson, A. W. 'Physical Chemistry of Surfaces", 4th ed.; Wiley: New York, 1982; Chapter 3.
In the absence of surface dissipation, the second terms of both eq 6 and 8 vanish, and it is seen that Echem = Ekin. This result establishes the quantitative transformation of chemical potential energy into kinetic energy in neutrally stable Marangoni convection driven by mass transfer, subject only to the general provisions outlined above. It is thus independent of any particular form of the flow and
Langmuir, Vol. 2, No. I, 1986 81
Energetics of the Marangoni Instability concentration fields and of whatever relation between chemical potential and concentration is applied. The first, nondissipative term in eq 6 and eq 8 is positive, since it represents the positive kinetic energy flux into the fluids. In eq 6 the diffusion term is positive since d r / d p > 0 for an adsorbed solute. This part of the chemical energy extracted from the bulk diffusion flux is not available for conversion to kinetic energy, as it represents irreversible degradation of chemical potential energy due to diffusion within the interface. The dissipative term in eq 7 is negative, indicating that surface viscosity diminishes the rate a t which kinetic energy is supplied to the fluids. Physical Interpretation of the Energy Transformation. The integrand po(X)[d(I’uo)/dX]in eq 5 accounts for the diversion of chemical energy from the bulk diffusive transport that is made available to drive convection. In a small disturbance model, the surface excess concentration may be represented as I? = ro+ 6I’ where 6r is the perturbation of the steady-state (without convection) value of r. The integrand then becomes pO(X)[ro(auo/dx)+ m d u o / a x ) + uo(d6r/ax)i (9) Since uo and 6r are both proportional to the magnitude of the disturbance, the second and third bracketed terms are of the second order of smallness and may be neglected in a linear approximation. The expression ro(du0/dX) represents the change in I’ due to convergence or divergence of the surface flow. In the steady state, this change is exactly compensated by adsorption or desorption, with bulk diffusion maintaining the concentration pattern according to eq 3. In the typical case of J? > 0, the surface flow diverges in the region of elevated solute concentration and thus of elevated chemical potential, while it converges in the region of lower concentration and chemical potential. (This phase relationship between the velocity field and the concentration perturbation is necessary for instability and arises naturally in the solution of the convection-diffusion equations. See ref 3 for more detailed discussion.) Net adsorption accompanies divergence, while desorption accompanies convergence. Overall, “high energy” solute is removed from the diffusion flux by adsorption and the adsorbed solute is convected toward the converging flow region releasing some of its chemical energy as kinetic energy and is finally returned to the diffusion flux by desorption as “low energy” solute. The mechanism for the conversion of chemical to mechanical energy apparently involves the expansion and contraction of surface elements, forced by adsorption and desorption as the convecting elements attempt to maintain equilibrium with the local bulk-phase concentration. Although eq 6 and 8 describe the mechanochemical coupling in a global way (since they result from integrating over a convection cell), the local analysis of energy balance appears to require a formulation of the energetics of a nonequilibrium surface engaged in mass transfer and nonuniform flow. Thermally Induced Convection. The case of convection induced by temperature gradients and thermal energy diffusion is analogous to the mass-transfer case. Equation 2 indicates that the interfacial entropy density, s,, plays a role similar to that of the surface excess concentration. As the conjugate variable, temperature replaces chemical potential. Thermal energy conservation at the interface is given by (cf. eq 3)
where the K~ are the thermal conductivities of the two fluids. One difference between eq 10 and 3 is the absence of a dissipative term in eq 10, corresponding to the surface diffusion term in eq 3. Although surface heat conduction may be a real process, it is not well-established and consideration of it is omitted here. The analogue of eq 5-6, here describing the diversion of thermal energy from bulk-phase conduction, is
while eq 8 becomes
with the use of eq 2. Again it is clear that the nondissipative terms demonstrate the equivalence of extracted thermal energy and applied kinetic energy at the interface. The Surface Engine. The energy transformation described above parallels that encountered in an ideal heat engine following the Carnot cycle. In both cases, entropy is extracted from a reservoir at a high temperature and rejected to a reservoir at a low temperature. The difference in the amounts of heat transferred in each step accounts for the work done by the engine. In the mass transfer case, the most familiar analogue is the electrochemical concentration cell. Ions are “extracted” by redudion in the solution with higher concentration and “rejected” by oxidation into the solution with the lower concentration. Again, the difference in the energies involved in each half-reaction appears in the performance of electrical work. The operative principle is the same throughout: matter, charge, or entropy is transferred from a region of high potential (p or T ) to one of low potential by a nondissipative route. The Carnot engine involves a working fluid subject to a cyclic process, while the surface engine and the concentration cell operate with a continuous, unidirectional flow. Limitations of the Model. The conditions of validity of the foregoing analysis were listed at its outset and should be examined to see how seriously they restrict the generality of the coupling principle that has been established. First, the assumption of a planar interface simplifies the mass and energy conservation equations as well as the integration steps in the derivation. The integrations could be taken along a deformed surface, however, with no essential change in the results, with all quantities evaluated here at the Y = 0 plane taken instead at the location of the deformed interface. While the possibility of deformation introduces new terms into the conservation equations, these could still be neglected as long as the steadystate case is considered. It should be noted that those stability analyses that have included surface deformation (e.g., ref 5) found it to have little impact on the results except in the limiting cases of vanishing surface tension or very long wavelength. Second, the restriction to two-dimensional flow allows the analysis of surface quantities to be limited to one dimension of spatial variation. There is no reason to believe that the same principles applied here could not be extended to include two dimensions of spatial periodicity at the cost of a great deal of formal simplicity. The third condition, requiring congruency of the periodicities of the (5) Ssrenson, T. S. In “Dynamicsand Instabilities of Fluid Interfaces”; Sorenson, T. S.,Ed.; Springer-Verlag: New York, 1979. pp 1-74.
82 Langmuir, Vol. 2, No. I , 1986
hydrodynamic and thermodynamic disturbances, is clearly critical for the results of eq 5-8 and 11-12. To question this assumption is probably pointless, however, since the intimate coupling of these two fields is at the very heart of the instability. It would be artificial to imagine instances in which their spatial variations were independent. Finally, the assumption of local equilibrium a t the interface is equivalent to the condition that there is no kinetic barrier to sorption or phase transfer of solute on the time scale of the convective motion. (It is no doubt a very safe assumption that thermal equilibrium is maintained at the interface.) While this assumption is almost universally made in models of solute-induced Marangoni convection, its violation would have a significant effect on the conditions for instability and on the logic of the derivation given here. The coupling mechanism described here assumes that there is no irreversibility (and thus no dissipation) involved in the transfer of solute to and from the surface elements as they convect and deform. If such irreversibility did exist, then less chemical energy would be available to drive convection and the instability would be damped. Furthermore, the use of Gibbs’ adsorption equation would no longer be strictly legitimate, thus breaking the firm thermodynamic link between the parameter and the surface excess concentration upon which the coupling mechanism relies. In spite of these difficulties, it seems reasonable that the principle of chemical to kinetic energy conversion would still apply and could be salvaged, again, at the price of much simplicity. In summary, the conditions imposed for the analysis were chosen so that the principal points could be made rigorously with a minimum of clutter in the resulting expressions. It appears that the physical coupling phenomenon modeled here should remain active even when these conditions are relaxed, though the formalism describing it would need to be extended and modified somewhat.
Conclusions The results of eq 5-8 show the role that surface viscosity and surface diffusion play in directly dissipating energy that would otherwise be avilable to drive the instability. It has been noted that surface viscosity has a pronounced effect on the onset of instability in the Marangoni problem, whereas surface diffusion has a minimal impact? Of course such claims depend entirely on the relative magnitudes of the transport coefficients for matter (the diffusion coefficient) and for momentum (the kinematic viscosity) in the interface. In general, momentum transport is much more efficient than diffusive matter transport in condensed matter, so one can explain the relative importance of these two factors on that basis.
Irvin The results of eq 5-8 demonstrate the energy coupling in a global but not in a local way, as noted above. The problem remains to show the detailed local balance of matter and energy as a function of position within a convection cell. The difficulty arises because the conversion of chemical to kinetic energy involves the expansion and contraction of convecting “surface elements” which are simultaneously exchanging matter with the adjacent bulk phases. These surface elements (however they might be defined) are created and destroyed in the process, yet they serve as temporary reservoirs of the energy that is being converted. Attempts to develop the equations that would describe the local dyanmics of the energy conversion have been unsuccessful. Resolution of this problem might enhance the power of the theoretical tools currently available to students of interfacial dynamics. It has been shown how the source of kinetic energy in Marangoni convection can be accounted for quantitatively through the use of the Gibbs adsorption equation. A paradoxical feature of the earliest stability analyses of the Marangoni effect was that they explicitly neglected the kinds of thermodynamic considerations that would allow their models to explain the conservation and transformation of energy in the physicochemical system. The surface tension forces were related to the concentration or temperature gradients “empirically”, without the benefit of grounding in thermodynamic reasoning. They thus appear almost as uexternaln forces, since their necessary connections with the thermodynamic functions of the system were not considered. The predictions in terms of instability conditions were not much damaged by this inconsistency, however, since the magnitude of the mass and energy diversions that drive the convection are generally quite small relative to the net f l ~ x e s .It~ may be that the energetic accounting is neglected because the technique of linear stability analysis is often first encountered in the realm of mechanics, not thermodynamics, and the emphasis is more on the forces than on the energies. The energy accounting approach taken here coould shed light on other variations of the Marangoni instability problem, such as interfacial chemical reactions, electrical effects, etc.6 Indeed, any spontaneously occurring hydrodynamic pattern will necessarily involve a source (or sources) for the kinetic energy that it embodies. The question of how that source couples to the flow field is always a legitimate one to ask,and the answer may inform a deeper understanding of the phenomenon. ~~
~
(6)Sanfeld, A.; Steinchen, A,, et al. In “Dynamicsand Instabilities of Fluid Interfaces”;Sorenson,T. S.,Ed.; Springer-Verlag:New York, 1979; pp 168-204.
