## Abstract

The thermodynamic basis of surface tension for bubbles and drops is reformulated to take into account the variation in surface tension for very small, growing or incipient bubbles and drops. After assessing the magnitudes involved by means of conventional theory, a model is put forward for this variation that avoids the supposition of extreme pressures at small sizes in the simple model. The parameter of this new model is then set by consideration of the properties of real gases, determined by analysis of the incipient bubble and then extended, it is argued, to the incipient drop. van der Waals' model of a real gas is used to show how this combines with the Lewins model in a self-consistent approach to the thermodynamics of surface tension for bubbles and drops. The changes from the simple theory will then allow a revision of the dynamic statistical theory of auto-nucleation and a reassessment of the critical sizes for imposed or heterogeneous nucleation, based on a simple and self-consistent thermodynamic model.

## 1. Introduction

The formation of bubbles and drops is of significance in a wide range of technology. A thermodynamic treatment acknowledges that only limits or bounds to the behaviour are described and that the real goal is to provide a dynamic theory for the growth of drops and bubbles which must, however, be consistent with these thermodynamic limits. Of particular interest is the critical sizes of drops and bubbles, below which they would tend to shrink and disappear, above which they would tend to grow from a metastable mother-fluid.

The standard thermodynamic analysis (Guggenheim 1967; McGlashen 1979) has the complication of deriving—and then abandoning—a concept of ‘radius of tension’. We believe that this step is unnecessary (Lewins 2004*a*) and that a more direct approach is valid. Even so, for the incipient or inchoate bubble or drop it is insufficient to assume that the surface tension parameter *σ* is a constant, independent of size. Surely at very small sizes the material on one side of a bubble becomes influenced by the dense liquid on the other side as much as by its own side. Similarly, the drop will be so small as to lose definition as a liquid with consequent reduction of surface tension. Conventional analysis missing this point will use a radius of tension that is much larger than any realistic representation of the bubble or drop size. In consequence, the work of formation and the critical size will be wrongly calculated.

Furthermore, a simple model using an ideal gas for a bubble suffers the defect of predicting infinite pressures at zero radius. We offer, therefore, a development of the thermodynamic theory with an empirical model for real fluid behaviour (van der Waals) and a model of the variation of surface tension with radius (Lewins model) that provides a self-consistent account of bubble and drop growth from a thermodynamic viewpoint, a macroscopic theory. Just as van der Waals is an empirical model requiring the setting of three parameters, the Lewins model calls for the setting of one parameter. The self-consistent model, however, obtains this last parameter from the gas spinodal properties predicted by the van der Waals model or indeed any other real gas model. It is hoped that the resulting thermodynamic framework will serve to fine tune the more detailed molecular arguments of statistical thermodynamics and fluctuation theory.

## 2. Fundamental theory

The thermodynamic analysis rests on examining an expression for the energy of a system of mass *m* in the form either of liquid and bubble, or gas and drop, to include the work of formation in creating a bubble or drop from zero size until it exists at some finite radius (Reiss 1997). This expression is then to be tested for stability. The real system of mass *m* is to have a uniform temperature *T* and an imposed pressure *P* at its surface. The real system also has a volume *V*.

Does the real system have definable entropy *S* and internal energy *U*? This depends upon an assumption that we can manipulate the original state of the system, in metastable equilibrium with uniform pressure as well as temperature, in such a way as to bring about the growth of bubble or drop in a reversible and quasi-static fashion. If so, the increase in entropy of the system is measurable. There is more difficulty, however, in understanding how we might measure the addition of internal energy.

We actually expect there to be some statistical fluctuation redistributing energy that will bring about the process. As such, this is not a matter for classical thermodynamics. For the analysis to proceed we have to offer at least a conceptual way in which the bubble, say, can be brought about. Thus, we put forward a highly impracticable ‘thought experiment’ in which a fine capillary tube is inserted in the system and supplied with the gaseous form at such controlled pressure and uniform temperature that the bubble can be forced to grow in a reversible fashion. Any addition of gaseous mass is compensated for by removing liquid to maintain the definition of the system. Work will need to be done, up to a point at any rate, to grow the bubble and this work in principle can be measured. With such a measurement we might claim that the *gedankexperiment* justifies the assumption that the system has measurable internal energy during the process.

Of course it is not *practicable* to measure the internal energy or the work of formation bringing about the growth. This indeed is the point of having a surface tension model that can be used to estimate it instead. The work of formation can be equated to the expression for free work at an equilibrium. The free work will include the energy stored in the surface. We assume, therefore, that it is proper to represent the real system as having a combination of properties such as *U−TS*+*PV* although this is not a conventional Gibbs function, since the internal properties are not uniform.

The growth of the real system is to be modelled as a bubble, say, of uniform pressure *p* with its real three-dimensional capillary surface region replaced by a two-dimensional surface of tension radius *r*. The pressure may be determined by a number of techniques and our model is to account for the pressure difference *p*−*P=*Δ*P*.

The real volume *V* is represented in the model by the volume of liquid *V*_{1} plus the volume of gas *V*_{g}. The masses, internal energies and entropies in these model volumes are determined by the model pressures and temperature. Since, their sum is unlikely to equate to the real mass, Gibbs (1948) proposes to associate with the two-dimensional surface not only an internal energy and entropy but also a mass such that all the real properties, including the volume, are maintained in the model.

If the bubble and indeed its surface in either two or three dimensions is to be in equilibrium with the mother fluid, then they must have the same temperature and the same chemical potential or, in this case of a pure substance, the same specific Gibbs function .

The modelling process consists then of selecting a radius *r* that should be representative of the real size of a bubble or drop together with a surface tension *σ* so that we have consistently , Kelvin's formula1 derived from mechanical equilibrium of a spherical bubble or drop. In saying that the surface tension *σ* varies with radius we must acknowledge two quite different variations. The first is the modelling step: where do we chose to locate the two-dimensional surface to represent the actual system state? The actual system properties must be independent of this choice. The second concerns the actual change in the system as a bubble or drop grows. The radius of our two-dimensional surface will then change accordingly.

The surface tension *σ* might be measured in the plane case by extending the surface and is a force per unit length. Equally a small extension requires energy so that *σ* also represents the area specific free work at constant temperature or Helmholtz function (it is *not* the surface energy as such). In the plane case an area of *A*=4*πr*^{2} would then have stored energy *σA* (Lewins 2000). Actually surface tension is generally measured on curved surfaces such as the effect of a fine capillary tube in elevating or depressing a liquid, the distortion of a spherical drop by gravity or the progress of Rayleigh waves disturbing a plane surface.

However, we are not dealing with a plane case. We believe that there is a misunderstanding in the classical formulation in the representation of the surface energy term. In certain limited circumstances this error may be ignored but it leads to the meta-physical and unnecessary introduction of the argument about a radius of tension.

Where others write *σA* we write instead for the energy stored in the surface the expression . Only if *σ* is independent of *r* the expression will reduce to 4*πr*^{2}*σ* and *σ* can be identified with the area specific Helmholtz function or area specific free-energy at constant temperature. Otherwise we have , a radially averaged value.

With this replacement for the bubble, the energy statement for free-work or work of formation becomes(2.1a)(2.1b)In this expression for the available work or work of formation we have the system internal energy offset by the (reversible) heat required at constant temperature but incremented by the mechanical energy to establish the two model volumes at their respective imposed pressures. On the right-hand side, we have the ‘free work’ available in the form of the Gibbs functions (which represent only volumetric work) plus the work stored in the surface. The surface value of *g* is *defined* by the equality of chemical potential in all three phases: *g*_{s}=*g*_{l}=*g*_{g} at the assumed equilibrium

These equations can be rearranged so that the left-hand side is independent of the choice of radius in the model:(2.2a)(2.2b)It is seen that we have now interpreted the element of area d*A* not as part of the surface of the (fixed) real bubble, as in the treatment of the ‘radius of tension’, but as the increment of area of the model of the bubble as it grows, . Note that, in this view, *σ* would only be the Helmholtz area-specific free-energy if it were constant, down to zero size.

Is the left-hand side really independent of bubble radius? Of course the volume grows as the bubble grows but at this point we are choosing the radius, where we shall draw the two-dimensional surface for a given real bubble or drop. As long as the ratio 2*σ*(*r*)/*r* retains its same value to represent Δ*P* we might draw the surface where we like. If we want the surface in the model to properly represent the bubble size, so that subsequent evaluations are useful, we have to use the corresponding surface tension. It would seem that the two volumes, *V*_{l},*V*_{g}, depend upon the bubble size. But we are analysing a real system in a fixed state and choosing a bubble size in a two-dimensional model to represent it. In the real system in fixed state, the *total* volume V=*V*_{l}+*V*_{g} is fixed just as *m*=*m*_{l}+*m*_{g}+*m*_{s} the total mass is fixed. While the two volumes would change with choice of radius, they do not change independently and the consequent change of masses in these volumes, based on the assumed densities of gas and liquid at their common temperature and respective pressures, is taken up by changes in the mass ascribed to the surface. This surface term is left out of the argument in some texts but is clearly crucial.

Thus, we can argue that the left-hand side of the expression is indeed independent of the radius chosen for the model at any particular state. Now, the argument that the right-hand side is then also independent of radius on differentiating yields the Kelvin formula:(2.3)and we have consistency of mechanical and thermodynamic assessments whether or not the surface tension varies with radius as the bubble grows. The result is valid for an equilibrium situation, although widely used in non-equilibrium studies that should really include inertial terms in any dynamic solution. It is important to note that equation (2.3) only determines a ratio *σ*/*r* from the measurable (in principle) pressure difference. If we want the radius to represent the size of the bubble adequately, it is up to us, the model-maker, to use a value of *σ* consistent with the circumstances bringing about the pressure difference Δ*P*.

## 3. Bubble and drop critical size-elementary model

It is useful to review the predictions of bubble and drop critical size in an elementary model. The theory just given can be used in the relations attributed to Poynting (Lewins in press) that exploit the common chemical potential of the phases, here the common specific Gibbs function in the liquid and gas phases. We use the simplest model in which the gas is an ideal gas satisfying *P*_{V}*=RT*, the liquid incompressible with specific volume *v*_{l} and the surface tension constant *σ*_{∞}. Our example will show that, for a specific model and given conditions, that include the specification of *σ*, there is only one ‘equilibrium’ radius. The model is sufficiently simple as to obtain the equilibrium condition, where the work of formation is a maximum, analytically.

These assumptions are used in analysing both the drop and the bubble at its equilibrium size although the development has to be separate for these two. Both turn on equating the change of specific Gibbs functions from the saturation pressure *P*_{sat}, where the two bulk phases in equilibrium at the given temperature have equal values . Since the change at constant temperature involves an integration over the specific volume to the required pressure. Although many writers employ the supersaturation ratio , we prefer an account in terms of the fractional sub- or super-pressurization *β*. For a bubble and for a drop .

### (a) The bubble at equilibrium

We have and the specific Gibbs functions yield(3.1)Define , where for small *β* this might be calculated via the Clausius–Clapeyron equation for given superheated liquid but in any case is a measurable property given the system pressure and temperature. Then, put a non-dimensional density ratio, gas to liquid at the system pressure. Thus,(3.2)where *P*_{Py} (Poynting) is the equilibrium or critical pressure, seen to be at or below the saturation pressure. If *βρ*^{*} is indeed small enough to approximate the exponential we have(3.3)Define a non-dimensional radius, based on the plane value for surface tension writing . Then, the critical or Poynting radius is given by(3.4)valid if either *β* *or* *ρ*^{*} or *βρ*^{*} is ≪1.

This defines the radius of the bubble if it is in equilibrium with the liquid. The excess pressure in the bubble at critical size is then of order *βP* or *p*_{sat}−*P*. Notably, we have *not* assumed that *β* is necessarily small in the analysis of the bubble. We discuss whether this is stable or unstable equilibrium later.

### (b) The drop at equilibrium

. Now(3.5)andIf *ρ*^{*}≪1 then(3.6)Thus, the excess pressure in the drop at critical size is of order and, thus, very much larger than for the corresponding bubble. Indeed, if the gas density is very small, the predicted pressure in the drop *at* *equilibrium*, the Poynting value, may be unphysically large, casting doubt on this model using an ideal gas, incompressible liquid and constant surface tension. Again, we have established a drop size at equilibrium without yet considering the nature of that equilibrium. Note that for the drop 0<β<1 given *P*>0 but if *ρ*^{*}→1, close to the critical temperature, and *β*→0 then *x*→∞ in this model. But if . A value of the relative density approaching one-half appears to give anomalous behaviour, but who would believe the model of incompressibility so close to the critical point temperature?

Both these relations, for drop and for bubble, are essentially the same as those obtained by Poynting and Kelvin. Although the values of the super- or sub-pressurized fraction *β* and the density ratio *ρ*^{*} are distinct for the two cases, when they are comparable it is seen that the critical size of the drop at comparable conditions is far smaller than the critical size of the bubble. This is even more extreme for the masses contained in the drop or bubble which will go as *ρ*^{*2}≪1, an observation of significance for auto-nucleation theory assessing the number of molecules that must have sufficient fluctuation of their energy to establish this critical size, and for the volumes, going as *ρ*^{*3}. It is also notable that the theory gives the critical pressure in a drop or bubble without reference to the surface tension.

Given the unique solutions for drop or bubble size, it is seen that the more general problem of the growth of a drop or bubble is not simply a thermodynamic static process, but must be a dynamic process to which thermodynamics can contribute limits or bounds only. Indeed it is apparent that the equilibrium point is only one of unstable equilibrium. Once beyond this critical size, the bubble will continue to expand because the superheated liquid is itself metastable and once beyond the barrier offered by surface tension will flash to gas. Similarly, the drop will tend to take over the system as the stable phase compared to the metastable gas. Both process release ‘free work’ which is then dissipated.

## 4. The work of formation

We seek to understand the nature of bubble and drop growth, not just at the point of unstable equilibrium. To this end, one studies the thermodynamic work of formation. Consider now a system starting with a mass *M* of liquid at a pressure *P* and superheated so that, as before, *P*<*P*_{sat}. Through some unspecified mechanism a bubble is formed and grows, of mass *m* at radius *r* or equivalently . Supposing this growth is reversible, the reversible expansion work done by the surface of the system (not in the bubble) is given by . The reversible heat absorbed is *Q*=*T*Δ*S* and the entropy change, including the change of entropy of the bubble surface from a zero radius is , noting that the *specific* properties of the remaining liquid mass are unchanged. The internal energy change is similarly . Thus, we can write the work of formation of the bubble as it grows as(4.1)We are not suggesting that the bubble actually grows by such a reversible process but this is the way thermodynamics puts a bound to the ‘free-work’ required.

The expression may be rewritten in terms of Helmholtz functions and of Gibbs functions as(4.2a)and(4.2b)where we have Δ*P*=(*p*−*P*) the excess pressure in the bubble that will go to zero as the bubble reaches infinite radius or the plane case. However, equilibrium arguments show that in that case, for equilibrium, the system pressure would have to be at the saturation pressure so that the specific Gibbs functions of the two phases were then equal. Thus, equation (4.2*a*) is useful for computational purposes but must be interpreted with care. The usual interpretation of a Gibbs function is the free work at constant temperature and pressure but here the pressures differ.

## 5. Ideal gas model with no radial dependence of surface tension

We first assume an ideal gas, an incompressible liquid and a constant surface tension equal to the plane value. Then, exploiting the equality of specific Gibbs functions at the saturation pressure(5.1)The mass in the bubble at the current pressure goes as . We then have(5.2)This may be put again in non-dimensional form writing *x*=*rP*/2*σ*_{∞} to give(5.3)The peak corresponds to the (unstable) equilibrium point found from the Poynting relation. For zero density ratio the peak work of formation goes as *β*^{−2}. Figure 1 shows the work of formation normalized by putting together with further normalizations of *x*^{*}=*βx*. As a consequence, the peak for any *β*-value would go through the point (1,1) for a density ratio *ρ*^{*} of zero. The figure shows the peak fitting to .

A similar analysis for the formation of a drop, using the same assumptions and *β*≪1 yields a work of formation with a peak at the expected and . The critical work to form a drop is thus much reduced compared to the critical work for a bubble at comparable saturation ratios and densities.

## 6. The physics of small sizes; the Lewins model

Smallness of size has a further physical effect, however. If the bubble is of the order of a few molecules, then it is of the order of the capillary region. That is, a molecule on the capillary surface is not only attracted back by the mass of liquid behind it but is attracted across the bubble by the mass of liquid the other side. The effect of this is to reduce the cause of surface tension. The diminution will only show up for drops and bubbles of very small size. This observation is consistent with work done solving the Schroedinger wave equation for a few molecules of water to show that not until about eight or nine are gathered together will the system display the properties of liquid water, density, pressure, etc., and therefore of surface tension.

We propose an empirical model for the reduction in the capillary effect for very small bubbles and drops. This will contain one parameter to be determined by further argument. It calls for a surface tension that falls rapidly as bubble size approaches zero but approaches the plane value at larger radii. A reasonable representation of the behaviour of surface tension for modelling purposes would then be(6.1)An expression in the area or even the volume is possible but we chose the simplest case, the linear radial dependence. The effect of the variation of surface tension with bubble size and area means that the surface term in the work of formation now has to be integrated2 such that(6.2)so that as and . It is seen that the surface work term has been reduced in this empirical model.

## 7. Ideal gas model with Lewins model: radial dependence of surface tension

We now introduce the radially-dependent surface tension to the model, retaining the non-dimensional definitions based on the *plane* case. The effect of the variation means that the work of formation becomes(7.1)with *x*_{0}=*r*_{0}*P*/2*σ*_{∞}.

Figure 2 shows the surface tension for this model, and figure 3 shows the resulting pressure. Note that the limiting excess pressure at zero radius is , since(7.2)The asymptotes at the origin and at infinity, useful for inverting to find the radius, are, respectively,(7.3)If the parameter *x*_{0} goes to zero we recover the constant surface tension model (with ideal gas). Figure 3 shows that the Lewins model for surface tension puts a finite size to the excess pressure at zero radius, in itself a distinct advantage.

Figure 4 shows the effect, therefore, on the work of formation of a bubble at a specified superheat value in the ideal gas model for various choices of the parameter *r*_{0} or the equivalent *x*_{0} of radially dependent surface tension. Again, when the Lewins parameter is zero, the departure of the peak from (1, 1) is due to the finite density ratio *ρ*^{*}. The value 0.17 chosen here matches the van der Waals case used later. It is seen that as the parameter of the Lewins model increases, both the critical work of formation and its location decrease. The critical work is reduced by up to 50% by the drop in surface tension, and critical size by up to 10% for the values used. However, the model still uses an ideal gas.

To set the value of the Lewins parameter we turn to a real gas model, rather than a ideal gas model, in which the maximum pressure in a gas is limited by attractive forces between molecules. This maximum pressure can be identified in any particular model and can be used to give an upper bound, at least, to the Lewins parameter. In this paper we shall use the simplest real fluid model, the van der Waals model with its predictions of spinodes.

## 8. The van der Waals model

Thermodynamics can provide a framework for this extended treatment of surface tension at small radius and hence the formation of bubbles and drops, by consideration of what is called the spinodal or turning-point. To this end, we outline the classic theory of the van der Waals gas as developed by Maxwell, amongst others.

The van der Waals equation is perhaps the simplest model of departure from the ideal gas that represents the attractive forces between molecules (van der Waals forces) in an analytical form. It predicts the nature of a critical point, the equilibrium between gas and liquid phases (with help from Maxwell) and the metastable properties of gas and liquid. To the extent that real *P–V–T* data match this equation of state, it may be used as a realistic model. The equation may be written in the first instance as(8.1)On the right, the first term represents the kinetic pressure due to the thermal energy and momentum. In this term, *b* represents the minimum volume that unit mass of the molecules can be compressed into; this is greater than the size of the molecules in that it includes essential ‘dead’ space between them. The parameter *c* is analogous to the gas constant *R*. The second term is the correction due to the attractive forces. The constant *a* is to represent the force between two molecules at some effective range. One factor of 1/*v* is proportional to the density and thus to the number of other molecules close to one molecule that is attracted by them. The further factor of 1/*v* gives the number of molecules per unit volume each feeling this attraction.

The equation is seen to reduce to the ideal gas equation for large specific volumes and high temperatures. But isotherms of constant temperature have a different shape below the ideal gas region. In particular, there is a point of inflexion where the derivatives . The local behaviour is similar to the isotherms measured in carbon dioxide by Andrews and thus represent the critical point,3 that highest pressure and temperature enabling gas and liquid phases to be identified coexisting in equilibrium.

### (a) Normalized van der Waals equation

It is convenient to reformulate the equation in terms of the three critical point values, writing; so that the equation in non-dimensional form is(8.2)Below this critical point the isotherms pass through two extrema called spinodes. The spinodes are given by the cubic equation with the third root in the unphysical region 0<*v*<1/3.(8.3)In a *P–V–T* space these spinodes would form spinodal lines. They are the theoretical equivalent of Wilson points and Wilson lines which are the *observed* limits of metastable behaviour affected by other considerations such as nuclei or even statistical fluctuations.

The isotherms are continuous and do not represent the jump in specific volumes between liquid and gas that we would expect. This horizontal jump is added by means of a construction credited to Maxwell (Lewins 2003). This construction is to locate the level of the two-phase pressure that will join the small specific volume of liquid, left, to the large specific volume of gas, right, on an isotherm necessarily below the critical point. We know that the specific Gibbs functions (or chemical potential) are equal for liquid and gas at this pressure. Thus, an integral from left to right along the horizontal isotherm of will have . But this will also be the case if the contour follows the van der Waals isotherm. Thus, the area of the loop below the saturation line is equal to the area above, figure 5. The numerical implementation of this construction is given in Lewins (2003).

### (b) Behaviour at the spinode

The section of the isotherm between the two spinodes, although predicted as an extension of the equation of state, is an absolutely unstable region which cannot exist in nature in stable form. Konorski (1990) uses this argument cogently in an excellent paper on nucleation. He adduces similar work by Rusanov (1960) and Skripov (1972). But the segment leading up to the spinode from the saturated value is potentially achievable and represents the metastable states of liquid and gas we have been using, now in a more realistic model for the equation of state.

It follows that, at a given temperature, the gas spinode represents the maximum overpressure available in a bubble. Above this pressure, the system can only exist as a liquid. We have to say, therefore, that the very large excess pressures predicted by the Kelvin formula and an ideal gas model, with constant surface tension at small bubble size, are unfounded. What we should expect then is that the surface tension to be used for the incipient bubble is essentially zero, as the bubble is first formed but only at this limited overpressure. There is some uncertainty in this interpretation; clearly a bubble does not form at *zero* radius, since it would not contain one molecule until of finite radius. If we could predict the radius *r*_{i} of the smallest possible or *incipient* bubble, that might contain say 10 molecules, then we can use the spinodal excess pressure in Kelvin's quasi-equilibrium formula to say what the now radial-dependent surface tension will be for a true bubble radius:(8.4)However, thermodynamics does not of itself predict the incipient radius *r*_{i} and this would be a matter for physical modelling of the behaviour of real molecular systems.4 Our empirical model might reasonable join these two points, zero and incipient, by a straight line which would extrapolate to represent the surface tension at small radii until it curved to take up its constant asymptotic value. We can now model the surface tension and excess pressure such that this latter never exceeds the spinodal value. And in figure 3 we saw how the excess pressure now depends upon radius.

We can get some support for the assumption of a linear dependence of surface tension on radius by examining the behaviour at a spinode. At the spinode a change of radius leading to a change of specific volume does not, however, lead to a change of pressure and, therefore, of excess pressure. Thus,(8.5)and we have the postulated linear behaviour.

With the Lewins empirical expression for the radial behaviour of surface tension, with a radius that more realistically represents the size of the bubble, we can express the excess pressure in the bubble as(8.6)where the free parameter *r*_{0} is chosen such that the excess pressure relates to the maximum or spinodal pressure at the origin (differentiation shows that the excess pressure decreases with increasing radius):(8.7)If we put *P*_{spn.g}=γ*P* then the corresponding non-dimensional radius is given by (*γ*−1)*x*_{0}=1. Our *ansatz* then is to say that the maximum pressure inside the bubble occurs at its incipient point, effectively zero, and is the gas spinodal pressure. This much of the Lewins model, the linear behaviour at very small sizes of incipient bubbles is then not empirical but based on the thermodynamic behaviour at the spinode. At the very least, this provides a bound to the pressure; our model says this indeed is the pressure in the initial bubble. The slope can be predicted by any real gas model predicting gas spinodal values, not just by van der Waals.

## 9. Work of formation: Lewins–van der Waals model

### (a) Work of formation of a bubble

The work of formation (equation 7.1) is expressed in terms of differences of specific Gibbs at constant temperature:(9.1)With this empirical model for surface tension and with the van der Waals model for the fluid gas and liquid which sets the spinodal parameter in the Lewins model, the work of formation becomes(9.2)Here(9.3)We are thus able to plot the non-dimensional work of formation. Figure 6 illustrates this for a particular assumption of super-heated liquid and the formation of a bubble. The figure shows that at the equilibrium or Poynting size, the work of formation is reduced from the original model (of interest in auto-nucleation theory) but that the introduction of the van der Waals model actually increases the Poynting radius (for imposed nucleation).

### (b) Work of formation of a drop

Matters are rather different for the formation of a drop. We have seen that in the simple model, the excess pressure in the drop is substantially higher than the saturation pressure or indeed the gas spinodal pressure. It does not *fall* to the *liquid* spinodal pressure; the behaviour there has little to do with the drop nucleation process. The liquid spinodal behaviour, where the limiting ratio of surface tension to radius is constant, is immaterial and we turn to a more involved argument to justify using the slope appropriate to the *gas* spinode for the drop case.

We ask the question whether the surface tension, varying with radius even at constant temperature, should be different for liquid and bubble system and for drop and gas system. Detailed molecular analysis might suggest reasons why at the same curvature (but of opposite sign) the surface behaviour might differ but a simple picture, consistent with the two-dimensional surface model being used, leads us to suppose that it will be adequate to use the same surface tension *σ*(*r*) for drop as for bubble.

Consider a drop in a gas so tenuous as to neglect the attractive forces. A surface tension exists because the attractive forces in the drop draw it together with an inward radial force. But if the surrounding gas were replaced with the same density liquid, surface tension would disappear and matter in the original surface layer would be in equilibrium under isotropic forces. Thus, the surrounding liquid exerts an equal and opposite attractive force radially outwards. Take away the original drop and one is left with a bubble displaying the same surface tension.

We feel justified, therefore, in using the same surface tension for drop as bubble at a particular radius (and temperature). Now the parameter *r*_{0} of the bubble model is not a size as such but an asymptotic interception of the small radius behaviour with the large radius behaviour. The incipient bubble size, that smallest size at which it can be thought to have the properties of a bubble and corresponding surface tension, will lie on the small radius asymptote and its radius will be smaller than *r*_{0} itself. It is in the linear range.

Considering the incipient size of a drop, it is shown in the simple analysis that the size of a drop, at the Poynting equilibrium, is much smaller than the corresponding size of a bubble. We can reasonably expect, therefore, that the incipient size of a drop is smaller than the incipient size of a bubble-say the space occupied by nine water molecules in a nonamer is smaller than the largely empty space occupied by a bubble with at least one molecule in it to represent the properties of the fluid. Then the drop radius, being smaller, will also lie on the asymptote. We can thus employ the parameter fitted from the bubble gas spinodal analysis to set the Lewins model in the drop analysis. The non-dimensional work of formation is now(9.4)Figure 7 illustrates the resulting shift of critical work of formation and critical size in a specified example using the Lewins–van der Waals model.

These results can be compared to the predictions of the ideal gas model with constant surface tension. At comparable values, the simple model predicts a non-dimensional critical radius of 0.2, whereas here we have 0.14. Since, , the Lewins–van der Waals model at 0.015 predicts a maximum work that is only half the value of the elementary model with an ideal gas and constant surface tension.

We note that the maximum pressure predicted in the incipient drop is the gas spinodal pressure. We might rationalize this mildly surprising result by saying that at the point of birth, the inchoate fluid cannot be identified clearly as either gas or liquid; thus in both cases there is the same pressure.

## 10. Conclusion

We have offered an empirical model for the behaviour of the surface tension of drops and bubbles extending to their incipient size. A reformulation of the thermodynamic foundations clarifies the nature of the variable surface tension parameter and we argue that this must be allowed to go to zero at the very small sizes at which bubbles or drops can be said to grow or be incipient. The model used thus falls off exponentially to the origin leaving a linear behaviour, supported by the known behaviour at spinodes.

The parameter of the Lewins model is set by the maximum gas pressure at the gas spinode. Certainly this is a necessary and close upper bound and our model takes this as the actual value, thus avoiding the elementary predictions of an infinite pressure in the inchoate bubble. We argue that the gas spinode is a reasonable thermodynamic representation of what, in a more realistic dynamical picture, is the nature of the fluid at the point where it changes phase from liquid to start bubble growth. Arguing that the same surface tension at a given radius applies to drop and bubble, we use the same fitting for a model of the incipient drop. This parameter can be predicted by any real fluid model that predicts gas spinodal behaviour. Figures 8 and 9 illustrate the (irreversible) path proposed for bubble and drop growth in the van der Waals model.

The bubble forms (figure 8) from liquid at the metastable state (‘m’) with pressure *P* that is less than the corresponding saturation pressure *p*_{sat}. The bubble itself makes its first appearance (‘o’) in a limiting sense at the gas spinode (spn), where it has the extreme metastable properties of the spinode. This limit is in the sense of the thermodynamic limit of vanishing radius that may be called the inchoate radius. Accepting a molecular view, beyond thermodynamics, that there will be a small but finite size when the fluid can really be said to be in a gaseous state, we identify the incipient radius of this smallest bubble as a nearby point on the metastable line (‘i’). If the bubble grows, it may reach its equilibrium point, also on the metastable line, a point of unstable equilibrium where the bubble (and its surface) can be said to be in equilibrium with the mother liquid-the Poynting point (‘Py’). Unstable growth beyond this leads to transfer further along the metastable line (if dynamic effects can be ignored) through the saturation pressure and on to the system pressure *P* when all liquid has been transformed (or the bubble has burst) to an end point (‘e’).

Similarly for the growth of a drop, figure 9, the mother fluid (‘m’) is a metastable gas at a pressure *P* that is now *higher* than its corresponding saturation pressure *p*_{sat}. The drop appears in the sense of a thermodynamic limit at its zero or inchoate size (‘o’) at a liquid pressure equal to the gaseous spinodal pressure (spn). Close by is the incipient size (‘i’) and then the unstable equilibrium or Poynting size (‘Py’). Beyond this, again, the drop is unstable and will grow until the system is completely transformed and will end as liquid (‘e’) at the system pressure *P*.

That there is a small but finite size to the incipient bubble or drop is not modelled here as such but the error in starting the work integral at zero instead is very small; what is significant to the prediction of Poynting sizes is the slope parameter of the Lewins model. We note that in this model we find that the incipient bubble and the drop are both at their most dense, the density falling as they grow. This latter is strictly only a thermodynamic bound for the real dynamic and irreversible process. Analysed in the Gibbs grand canonical ensemble of statistical mechanics, where temperature and chemical potential are retained but molecule density and energy fluctuate for example, a statistical fluctuation bringing about auto-nucleation might see an energy change without temperature change. But we might hope that we have a better representation of both the Poynting size and the maximum work involved in forming drops and bubbles.

Can thermodynamics be taken to apply in the process represented by the growth of the bubble when there is only the one point of equilibrium and that unstable? We have argued (just) that the formation of the bubble up to its Poynting point could be undertaken slowly and reversibly by means of a miniscule pipette that was somehow supplied with metastable steam, initially at the gas spinodal value, with compression work to form the bubble. This would be a hypothetical instrument rather like Van't Hoff's Reaction Box. Beyond the Poynting point, then certainly the bubble runs away and it would be difficult to conceive a pseudo-stationary process.

When analysed in conjunction with van der Waals equation of state, again an empirical model, a self-consistent theory can be advanced for a thermodynamic account of the work of formation. This shows some modest changes in the predicted size of the equilibrium system and its corresponding maximum work of formation that may be of interest to those studying auto-nucleation and indeed imposed or inhomogeneous nucleation. No doubt the parameters can be improved but already the model provides a self-consistent account that avoids suggesting infinite pressures and reduces the critical energy to be supplied by fluctuation, compared to simpler models. Even though the changes it predicts from the simple model are modest, they will of course be significant in nucleation theory where rate changes are exponentially sensitive to the necessary quantity of critical work. It is hoped that this framework will not only help understanding but allow finer tuning of the nucleation models proposed by Tolman (1948) and those following his statistical theory, or the quantum mechanic approach of say Clary and his school Liu *et al*. (1996)

## Acknowledgments

I am grateful to Professor John Young and Sir Brian Pippard for stimulation.

## Footnotes

↵Also credited to Helmholtz and Laplace

↵Many authorities have not made this correction but just write

*σ*(*A*)*A*, if indeed allowing*σ*to vary. We would perhaps agree on the differential*σ*(*A*)d*A*, but how is this to be integrated? We see no operational justification for integrating over the surface of the established bubble but can see an operational meaning for the integration from zero size, analogous to the expression for the reversible work done by a volumetric change↵With this in mind we shall now refer to the maximum in the work of formation as the Poynting point and not a critical value.

↵Nine molecules forming a drop offers a symmetric pattern. Then, the hexamer with eight molecules may well be the smallest aggregation of water molecules offering an attraction to make up nine and the characteristics of a drop.

- Received October 20, 2004.
- Accepted April 25, 2005.

- © 2005 The Royal Society