J. Chem. SOC., Faraday Trans. I, 1984,80, 3295-3305 Capillary Phenomena Part 25.-Thermodynamic Equilibrium and Stability of Capillary Systems in a Gravitational Field BY ERNEST A. BOUCHER School of Chemistry and Molecular Sciences, University of Sussex, Brighton BN1 9QJ Received 17th February, 1984 A unified analysis is given of the thermodynamics of systems in a gravitational field, appropriate for sessile and pendent drops and fluid bridges, as well as for fluids in porous media, solids detaching from surfaces and liquid lenses. Mechanical and physicochemical phenomena are described. Stability criteria as well as equilibrium changes are discussed. By permitting transport of substances between phases and compression effects a scheme is presented which can deal with a wide range of systems, including those in various gravitational fields.Examples, mainly using certain of the mechanical aspects, e.g. for solids at surfaces, for pendent drops and for fluid bridges, have been given separately. There are three theoretical approaches to the analysis of the equilibrium and stability of capillary systems such as pendent drops beneath a solid, fluid bridges between two solids or fluid/fluid interface displacement within a porous solid: (i) the phenomenological approach where, instead of experimental observations, one interprets the results of numerical computation starting with the equilibrium fluid/fluid interface shapes, (ii) the thermodynamic approach entailing more explicit quantities such as interfacial areas in free-energy expressions, with general criteria applied to detect the onset of instability, and (iii) the variational approach where selected (arbitrary, but convenient) perturbations from an equilibrium configuration are examined as a test for instabilities. These divisions are arbitrary, but they broadly reflect aspects of recent studies : they should lead to consistent conclusions and all three are thermodynamic in nature.Fluid/fluid displacement can also be dealt with by model computer simulation and by the application of network models and percolation the0ry.l. The phenomenological approach has been used extensively and successfully in recent years, especially for systems having axially symmetric fluid/fluid interface shapes in the presence of a gravitational The reasons for this success are (a) that second-order differential equations analogous to those solved numerically to give interface meridians by Powalky for the Bashforth and Adams monograph6 of a hundred years ago can now be solved rapidly and with ease and (b) that the data from this method are often amenable to self-evident interpretation. The thermodynamic approach is capable of showing explicitly how quantities such as the gravitational free energy and the surface free energy contribute to the overall beha~iour.~ The variational approach has proved itself of value in testing for stability limits (onset of irreversible changes of a large-scale nature such as pendent-drop detachment) or more precisely for confirming stability limits deduced otherwise,8 since alone the method is difficult to apply.g Even more difficult to use is the rigorous method of testing 32953296 CAPILLARY THERMODYNAMICS configurations neighbouring one known to be at equilibrium. lo The variational method has the advantage that the stability of a configuration can be tested with respect to symmetrical and unsymmetrical perturbations, but in practice this distinction of type cannot be made.The present study is based on the premise that a thermodynamic analysis should be capable of describing thermodynamic systems in a unified manner, leading to quantities relating to an equilibrium path and criteria for instabilities, either concerning fluid flow or displacement, or with respect to the physicochemical phenomena of condensation (evaporation) and the diffusion of chemical species across an interface.With all except simple systems some numerical computation will be needed to evaluate quantities, i.e. as an extension of the original phenomenological studies. Additional unification of approach has been provided by giving an analysis applicable to gravity-free systems as well as those subjected to the terrestrial gravitational field or to varied field strengths.ll In manipulating a capillary system it is important to acknowledge the nature of the mechanical device as well as the intrinsic properties of the capillary system in isolation. l2 The analysis and techniques developed in this paper can in principle be extended to deal with systems in a centrifugal field. Note that many other capillary systems can be treated in the manner described: only a few examples are given.The aims of this paper are to give a thermodynamic account of equilibrium and stability in terms relevant to a variety of capillary systems. In particular, compression of phases, condensation and transfer of matter between phases are permitted with systems in a gravitational field. The ensuing free-energy expressions contain terms in surface areas, centres of mass and phase volumes which are discussed as they relate to mechanical and physical processes. Axial symmetry is not assumed, although this has made it possible to compute interface shapes and attendant quantities for the examples cited, e.g. pendent drops and fluid bridges. The dependence of stability limits on the mechanical properties of the device used to manipulate capillary systems is an important aspect to emerge from these studies.BASIC MECHANICAL MODELS Fig. 1 shows a schematic model enabling the fluid-bridge configuration to be altered by a controlled movement by means of mechanical links to external forces applied to the pistons or to the upper solid. Constraining solids could more generally be subjected to an applied torque13 as well as to a tensile force. Clearly similar arrangements could be used to produce pendent and sessile drops or other large-scale capillary systems. The fluid bodies have an axis of symmetry about which rotation of the meridian curve generates the fluid/fluid interface configuration. Many of the thermodynamic relationships already developed4* 7 7 l4 are applicable to fluids con- strained by irregular solids.We begin with the more abstract arrangement, without implied symmetry, shown in fig. 2. The boundary of the thermodynamic system can be assigned convenient properties. For the present it will be assumed to be rigid and impermeable but capable of permitting heat transfer to keep the temperature constant and of permitting the mechanical linkages to manipulate the phases. Fig. 3 shows three representative types of horizontal zone which could exist. A zone is a horizontal element of the system at gravitational potential v/ = gz, of height dz and volume dv. Zone 3(a) consists of two fluid phases a and /I, of volume dva and d d , which when the system is manipulated by incremental external displacements undergo changes 6ua and BUD. A sensibly chosen Gibbs dividing surface of area dAaP and tension yap separates a and /I within dv.E.A. BOUCHER 3297 mechanical l i n k c y l i n d e r and p i s t o n p cylinder and p i s t o n a Fig. 1. Idealized representation of a system of springs and pistons for manipulating fluid bridges and other capillary systems. I - - - I I 1 I I 1 I I I I I I I I I I I I I I I I I I I . . . . . . . . . . . . . . . . . . . . . bv . I f Ppring Z* . 4: m e c h a n i c a l l i n k . . . . . . . . . . . . . . . . . . . . . I I Fig. 2. Representation of a horizontal zone dz at position z relative to a fluid supply and mechanical link at position zt . The capillary system is bounded by the fence denoted ( . . . ) and the springs are within the boundary denoted (---).3298 CAPILLARY THERMODYNAMICS \ Fig.3. Types of horizontal zone in a capillary system: (a) consisting only of fluid phases a and p, (b) consisting of fluids a and p within a porous solid s with the alp interface making edge contact and contact with a smooth solid and (c) a representation of three fluids a, p and E meeting as in a liquid lens (the horizontal scale is distorted). Suppose fluids a and p in cylinders at a height where the gravitational potential is y/t undergo incremental volume changes Dva.t and D&t. It is assumed here that only one fluid phase exists in a cylinder and that new phases are not nucleated: in the most straightforward example, condensation involves a single component. A portion 6va7 t and Sd. 7, of each phase displacement, is associated with the change within du.A change can severally or collectively involve: (i) pure displacement (disp) of the alp interface, with or without area change 6A@, e.g. for incompressible liquids a and /I, (ii) displacement of the a/p interface because of condensation and/or evaporation (cond) of one or more components of a phase, (iii) displacement of the alp interface accompanied by compression or expansion (comp) or (iv) changes because of transfer by diffusion (dim of chemical components from one phase to another so as to attain and maintain physiochemical equilibrium. Since dv is constant, all changes are subject to 6va+6@ = 0 (1) and changes caused by condensation and evaporation, or compression and expansion, can be identified by the abbreviations above with the sign of the change indicating which of the pair is involved.Generally, for cases (ii)-(iv), 6va9 7 # Sva and Sd. 7 # dvp; only for change (i) will these volume changes at zt and z be equal. However, the corresponding mass changes are necessarily equal, e.g. Smayt = 6ma. In general 6v = Gu(disp) +dv(cond) + Sv(comp) + &(dim (2) where the right-hand terms apply to a zone. We shall for the moment neglect adsorption phenomena and will not specifically account for diffusion of components between phases. The abbreviation disp is used for the sum of all contributions to interface displacement. The external work Dw done on the system by moving pistons located at height zt is Dw(ext) = - F t D u a ~ t -Pp>tD&t. (3) A portion of this given by dwt = - p , t & a , t -pP,tg,P,t (4) is associated with changes within zone dz where the hydraulic pressures are pb[ andE.A. BOUCHER 3299 Pp and the gravitational potential is w. Within the zone, the work done on the system is dw = (p" - PB) dvyllsp) - P"[SV"~ t + Guy*)] - P ~ [ ~ V B ? t + dvP(d@)] + (w - v/t) (6ma. + + 6mp.t). ( 5 ) In deriving eqn (5) it has been assumed that dua(disp) = -Gva(disp), but dwayt and 6vB7t are not of equal magnitude, except for incompressible fluids in the absence of condensation and inter-phase diffusion. Little error will usually ensue from assuming densities pa and fl to be constant in the vertical direction, but an equation of state might be needed for compressible phases in some circumstances. When 6va9 7 = - t , and the only change is due to pure interface displacement, i.e.dva9t = -dva, 6w = (P- PB) dva+ (w - v/t) (dma +dmQ (6) where all quanitities except the arbitrarily chosen y/t pertain to a horizontal zone dz. An equivalent expression was derived previously15 for pendent-drop growth for incompressible phases. De Donder extended the thermodynamic methods of Gibbs by using the concept of entropy production and by introducing a state function called the affinity (see Prigogine and DefaylG and Sanfeld17). Clausius used the term uncompensated heat and Rayleigh recognized entropy production caused by viscosity effects. If 6,s is the entropy received from the surroundings and diS is the entropy produced within the (7) system, dS = d,S+diS; 6,s = dq/T,d,S 3 0. For a zone, the Helmholtz free energy d F undergoes a change 6F.From the first and second laws of thermodynamics d F = dw-TdiS-SdT (8) where d w is the work expended in forming the unperturbed state within the zone. Then the change 6F is given by The zones are thin enough that the Helmholtz free energy for the whole capillary system of volume u allows the summation over all zones to be replaced by integration, F = JvdF, and the change for the whole system is dF=dw-TdiS-SdT. (9) D F = [(dF+dq-dl;] = 6F. (10) zones sv We can write eqn (5) in the slightly simpler form dw = (P" - PB) Sva(t€i@) - ~ a d v ~ - Ppdvp + ( I+Y - v/t) (ha + dmB) d~ = (P" - PB) d~~(il@) - P"GV~ - P@UB + ( t , ~ - I,v~) (dma + 6,B) - ~6~ s (1 1) ( 1 2) so that, from eqn (9) and (1 1) at constant temperature, for a zone whence the complete differential of the Helmholtz free energy of the entire system is D F = (P" - PB) Gva(*) - sp"dua - sPp6vB+ [ ( v f - w ) (6ma + dms) - TD, S.(13) On integrating some of the terms in eqn (13) in the vertical direction to include the whole system we obtain D F = - P d v a - P@vB+ (P" - PB) dva(disp) + (yo- v/t) D(ma - mB) s -(rna+mB)D(v/*-t,vt)- 7'DiS. (14) sss3300 CAPILLARY THERMODYNAMICS The first two terms are only important when compression of one or other phase occurs. The potential rym is that at the centre-of-mass of the appropriate phase. The term representing integration over the pressure difference across the interface with respect to the volume displaced by movement of the alp interface will be simplified later. GIBBS MODEL We now suppose that there are present in the capillary system phases a, p, .. ., a, components 1,2, . . . , c and interfaces aB, . . ., au, between different adjacent phases. The Helmholtz free energy is then supposed to be completely described by (15) We adopt but extend the suggestion of Guggenheim18 for dividing a system into horizontal ‘phases’ of uniform gravitational potential ry (nothing to do with his sandwich model of adsorption), noting that several conventional phases can exist within a zone at potential ry, and in particular that F = F(T, Va, Va, . . . Vw, A@. . .AaW, n:. . .ny, . . ., nz.. .np). dF= d P + d p + d F etc. (16) where t~ refers to the interface(s). Consequently, from eqn (1 5), for a zone a a8 + c , z [P:+(ry-ryt)M,ldn: (17) phases components a C where Mc is the molar mass of component c and dn: is the incremental change in the number of moles of component c: as far as the chemical potential ,@ is concerned it does not matter where this substance came from, but in the gravitational term an arbitrary source position of potential ryt has been chosen.From eqn (17), for two phases a and p consisting of pure components of molar mass Ma and MB, d F = -SdT-Pdva-PBdvb+yafidAaB + + (ry - ryt ) Ma] dna + ha+ (ry - yt ) Mq dnb. (1 8) Some of the consequences of taking the gravitational field into account can be deduced. The gravitational field does not affect the partial differential quantities for component i, aPi/ap = av/ani; api/dT = -as/an, = V i - - -si. (19) One component, i, can achieve physiochemical equilibrium at constant temperature, volume and area within this gravitational zone, i.e.where d(dF),, v a , ~ , A a ~ = 0, as follows : (i) within two conventional phases at the same gravitational potential P9 = P& v(a) = wco> (20) (ii) within the same conventional phase at two different gravitational potentials y’ and pf( ry’) + Mi y’ = pf( ly”) + Mi ry” ryff or the equivalent dpf+Midry = 0E. A. BOUCHER 3301 and (iii) within two different phases a and /3 located at two different chemical potentials ry' and ry" Because we are dealing with capillary systems, the applied pressures in two conventional phases at the same gravitational potential will not usually be equal. Finally, from this line of reasoning a form of Pascal's laws of hydrostatics can readily be deduced, since at constant temperature eqn (19) and (22) give1* ,u: + Mi v'( a) = pf + Mi ~"(p).(23) a P p y = - p i ; pi = Mi/Vi. (24) Returning to the free-energy expression, it would for some purposes be sufficient to use only eqn (17), but we now investigate the nature of the incremental variation 6(dF) in dF, using the two-phase system of pure components for which at constant temperature 6(dF) = - P d v a - Ppdvp+ y@dA@+(ry- ryt) drna+(y- v/t) drnp+paSna+p~6np (25) where ama = Madnu etc. Eqn (25) still applies only to a horizontal zone. It is seen that eqn (12) was obtained by doing mechanical work on the system. If it is supposed that eqn (25) should be its equivalent then this expression alone accounts for physicochemical effects through the presence of the chemical potentials of the components. The dva and dub terms in eqn (25) are associated with the work done on compression of the phases.Equating eqn (1 2) and (25) T6, S = (P - P9 6 v a ( G ) - ya@dAap - padnu - ppdnp. (26) If in the zone dz the pressure difference across the a/P interface is given by the Laplace equation in terms of the mean surface curvature J where the potential is ry, we can write eqn (26) as Td, S = yapJSva(*) - yap6Aap - pa6na - pp6np. (27) When the change within zone dz involves condensation of a to p, i.e. both phases are of the same component, dna = -6np and in the absence of compression duo! = -6vp, giving Everett and Hayneslg derived an exactly analogous expression for vapour adsorption in a porous solid, regarded as overall gravity free.The same equation was later usedz0 to deduce the criteria for the mechanical and physicochemical equilibrium and stability of fluid/fluid displacement in porous solids. A thin horizontal zone behaves as though it were gravity free. Tdi S = yJdva - y6Aap+ - p 9 ana. (28) EQUILIBRIUM AND STABILITY CONDITIONS We continue to examine equilibrium and stability conditions using the system of two fluid phases, rather than the more general system to which eqn (1 7) applies. At constant temperature the complete differential of the Helmholtz free energy for the whole system is given by integrating eqn (25): DF = - [ P d v a - [P&bp + yamAaD J J3302 CAPILLARY THERMODYNAMICS from which DF = - P d v a - PbduB + yaSDAaB i s + (yo - v/t) D(ma + mB) - (ma + mB) D( yo - v/t) + [paGna + [p@nb.(30) J J On the supposition that eqn (14) and (30) should be compatible for capillary system, the entropy production by irreversible processes can be as TD, S = s(P - PB) dva(d@) - yaBDAaB- Jpadna - spbd.8. the entire expressed (31) Assuming that within each zone the local pressure difference (P - PB) across the a//? interface satisfies the Young-Laplace principle, eqn (3 1) can be written with integration over the local mean surface curvature J for each zone dz: Entropy production in the system can occur by the flow of phases a and /? to achieve overall equilibrium, not just local equilibrium, by area changes (generalized below) and by the transport of substances between phases by condensation (or evapora- tion) and diffusion so that the chemical potentials pa and become equal, eqn (20), when overall physicochemical equilibrium is attained.To see more clearly the implications for achieving local and overall equilibrium we write eqn (31) in a less concise form without simplifying terms in dna and dnb: n + [(v/ - v/t) D(ma +mB) - (ma +mB) D(y/ - v/f)] r r We have assumed sufficient symmetry that separate potentials $(a) and yo@ are not required. The most general equilibrium condition is that DiS = 0. However, it will usually be found convenient to examine a system separately for equilibrium and for the stability of that equilibrium with respect to mechanical and physicochemical pheno- mena. The stability of a system can be decided by the requirement that DiS > 0 as equilibrium is attained.Conversely, one can formally identify the condition DiS < 0 as signifying that the system being perturbed from equilibrium is stable. PHYSICOCHEMICAL AND MECHANICAL FEATURES Often mechanical and physicochemical effects are not only separated by their involving different thermodynamic quantities, but by the timescale of events. Fluid flow can be very rapid, whereas diffusion of components of the same system can occur relatively very slowly. One must of course realize that these phenomena affect one another, and that in a practical system judgements must be made about the magnitude of effects and the timescales involved. Included in the category of physicochemical phenomena are the adsorption of chemical components at interfaces, the influence of adsorption on interfacial tensions and therefore on spreading pressures being wellE. A.BOUCHER 3303 recognized. Furthermore it is tacitly assumed that no electrostatic effects are involved which might modify interface shapes. One of the most commonly discussed physicochemical effects in capillarity is the vapour pressure of component one, say, of a phase separated from a liquid phase by a curved surface. The chemical potential of component one in the liquid phase pi( T, v / ) will equal that, &'(T,p*), in the vapour phase where p* is the fugacity. The vapour pressure p relative to that p o above a plane surface at the same temperature is given for an incompressible liquid, with vapour imperfection represented by the virial coefficient B, by4 RTln @ / P O ) = - (pl -pv)gzv: + (p -po) (0: - B) (34) where liquid and vapour densities are p1 and pv and v: is the partial molar volume of one in the liquid phase.The vapour pressure p(z) over a curved surface in a gravitational field, e.g. up the side of a pendent drop, varies with z in exactly the same way as that above a plane liquid surface varies with height in the absence of a capillary system. Eqn (17) contains the sum of terms of the type yaPdAaP, but for simplicity other interfacial areas did not appear in subsequent equations. Many systems involve two fluid phases and one or two solids. Interfacial area changes can be combined into a quantity called the effective area Aeff,14 provided Young's equation of wetting is (35) accepted : ysB = ysa + yap cos 8 where 8 is the contact angle between the a/P interface and the solid taken through the a fluid phase, the denser.The general acceptance of eqn (35) is not being advocated since in some practical circumstances it might be advisable to avoid, if possible, problems of contact-angle hysteresis and controversy over microscopic as opposed to macroscopic contact angles. The surface free-energy change for a system consisting of solids s, and s,, and fluid phases a and p, and with contact angles 8, and 8,, can be written as (36) D P = y@DAaP+ yaslDAasl + y&DA& + yaS2DAaS2 + akDAPS2. Defining the effective area as Aeff = I"/y@, noting that one is usually justified in writing DAaP1 = -DAaS1 with a similar expression for s,, the application of eqn (34) for each solid gives ~ ~ e f f = D A ~ P - DAQS~ cos 8, - DA~S, cos 8,.(37) It will usually be convenient to use reduced or dimensionless rather than actual quantities, e.g. by dividing actual quantities of dimension I" by av where a2 = 2yaB/Apg: Ap = p a - / . To evaluate the ensuing reduced quantities involves the separate task of numerical computation which has recently included examples of pendent drops subjected to variation in the gravitational field,119 21 the detachment of rods from liquid surfaces and fluid drainage22 and the properties of fluid bridges between Detailed examples will not be discussed here: the present treatment unifies the accounts just cited and provides a framework for extensions while showing the basic assumptions of the model. However, attention is drawn to additional features revealed by consideration of examples.A pendent drop of liquid a at rest will according to eqn (33) satisfy ya'BDAaP = - maD( v/ - v/t) (38) since the equilibrium condition is that Di S = 0 and the phases are at constant volume3304 CAPILLARY THERMODYNAMICS Fig. 4. (i) Indirect area change accompanying condensation of vapour phase into a droplet of liquid phase and (ii) the extension of area A" by a directly linked forcef. and composition. When subjected to a change dc in the gravitational field denoted by [g, eqn (38) becomes in reduced terms (39) aAa@ - -2vyC7(z'--zt) 1 - ac ac V , T since ry' = 2% and vapa = ma. By the above stability criteria, the change signified by eqn (39) will be stable if The advantage of introducing the variable gravitational field, permitting a unified treatment of systems subjected to terrestrial- and micro-gravity, has been dis- cussed.ll9 21 It is also now clear from eqn (14) and (30) that a distinction has been made between mass transfer in a gravitational field and volume changes.Previ0usly,~3 systems were in effect at a constant total volume. An explicit distinction can also be made between two types of area change. Fig. 4(i) and (ii) show, respectively, the condensation of liquid into a droplet with concomitant area change and an area change for a planar interface. In the case of condensation the equilibrium change (P" - P) do" is exactly balanced by the work done in areal extension ydA. There is no analogous volume change when the plane interface is extended and the work done requires direct coupling to an externally applied force.FREE-ENERGY CHANGES The work done on a capillary system or the finite Helmholtz free-energy difference AFaccompanying a change from state I to state I1 can be broken down in the following manner : A F = J;*DF' (41) is given by integrating eqn (30) along an equilibrium path. When there is no transfer of substance between phases and no overall volume changes A F = ya@AAa@ + A[(ty ' - tyt) (ma + mq]. (42) An interesting relationship is obtained by expressing the (often ficticious) mean surface curvature J t at zt to that, J ' , at the centre of mass, z', of say phase a: ya@Jf = yaBJ'+@a-#)g(z'-~t) (43) and using the simplifying conditions D(ma-mmP) = (pC'--pp) Doa, Dua,t = -Do@,+ and Dua.t = Doa (44)E.A. BOUCHER 3305 the differential equivalent of eqn (42) leads directly to a condition for an equilibrium displacement4 (45) CONCLUDING REMARKS The overall effect of doing work on a capillary system will be that the existing a phase changes position dz' with an energy change pavagdz', and that the new material has been moved vertically a distance (z'-zt) with an energy change paduag(z'-zt), accompanied by changes to p in the opposite sense, in addition to which there has been an area change dAeff with energy change yaBdAeff, and energy changes may have occurred because of volume changes (e.g. by compression) to the a and /3 phases. The necessary distinction between methods used to manipulate systems, leading to ideal extremes of stress and strain control, with intermediate cases depending on the effective mechanical stiffness, has led to a demonstration of the value of considering the entire system.22 For example, the raising of a rod with its lower edge contacting a surface by pulling on a spring attached to the upper end of the rod, fig.1 and 2, would introduce the energy stored in the spring for a displacement into the free-energy expressions. Instability, i.e. rod detachment, depends on the spring constant and is signified in a cusp in a plot of AFtot against displacement. It is clear that one could relate the isolated cases so far examined to catastrophe theory. It is also evident that we have found mechanical instability criteria which are exactly analogous to those described by Thompson24 for mechanical structures.The majority of this analysis was carried out while the author was Visiting Professor at Carnegie-Mellon University, Pittsburgh. Support of the Universities and the encouragement of colleagues during that period, 198 1-82, is gratefully acknowledged. J. Koplik, J . Fluid Mech., 1982. 119, 219. R. Chandler, J. Koplik, K. Lerman and J. F. Willemsen, J. Fluid Mech., 1982, 119, 249. J. F. Padday, Pure Appl. Chem., 1976,48,485. E. A. Boucher, Rep. Prog. Phys., 1980. 43, 497. S. Hartland and R. Hartley, Axisymmetric FluidlLiquid Interfaces (Elsevier, Amsterdam, 1976). F. Bashforth and J. C. Adams, An Attempt to Test the Theories of Capillary Action (Cambridge University Press, Cambridge, 1983). D. H. Michael, Annu. Rev. Fluid Mech., 1981, 13, 189. E. Pitts, J. Fluid Mech., 1976, 76, 461. ' E. A. Boucher, Proc. R. Soc. London, Ser. 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Boucher and T. G . Jones, J. Chem. SOC., Faraday Trans. I , 1983, 79, 2529. 23 E. A. Boucher and M. J. B. Evans, J. Chem. Soc., Faraday Trans. 1 , in press. 24 J. M. T. Thompson, Philos. Trans. R . Soc. London, Ser. A, 1979, 292, 1. (PAPER 4/276)