A new calculation of the work of formation of bubbles and drops

doi: 10.1098/rspa.2005.1506, 3313-3334461 2005 Proc. R. Soc. A

Jeffery Lewins bubbles and dropsA new calculation of the work of formation of

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A new calculation of the work of formationof bubbles and drops

BY JEFFERY LEWINS

University of Cambridge, Cambridge CB3 0AG, UK([email protected])

The thermodynamic basis of surface tension for bubbles and drops is reformulated totake into account the variation in surface tension for very small, growing or incipientbubbles and drops. After assessing the magnitudes involved by means of conventionaltheory, a model is put forward for this variation that avoids the supposition of extremepressures at small sizes in the simple model. The parameter of this new model is then setby consideration of the properties of real gases, determined by analysis of the incipientbubble and then extended, it is argued, to the incipient drop. van der Waals’ model of areal gas is used to show how this combines with the Lewins model in a self-consistentapproach to the thermodynamics of surface tension for bubbles and drops. The changesfrom the simple theory will then allow a revision of the dynamic statistical theory ofauto-nucleation and a reassessment of the critical sizes for imposed or heterogeneousnucleation, based on a simple and self-consistent thermodynamic model.

Keywords: surface tension; nucleation; van der Waals

RecAcc

1. Introduction

The formation of bubbles and drops is of significance in a wide range oftechnology. A thermodynamic treatment acknowledges that only limits orbounds to the behaviour are described and that the real goal is to provide adynamic theory for the growth of drops and bubbles which must, however, beconsistent with these thermodynamic limits. Of particular interest is the criticalsizes 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 oftension’. We believe that this step is unnecessary (Lewins 2004a) and that a moredirect approach is valid. Even so, for the incipient or inchoate bubble or drop it isinsufficient to assume that the surface tension parameter s is a constant,independent of size. Surely at very small sizes the material on one side of abubble becomes influenced by the dense liquid on the other side as much as by itsown side. Similarly, the drop will be so small as to lose definition as a liquid withconsequent reduction of surface tension. Conventional analysis missing this pointwill use a radius of tension that is much larger than any realistic representation

Proc. R. Soc. A (2005) 461, 3313–3334

doi:10.1098/rspa.2005.1506

Published online 31 August 2005

eived 20 October 2004epted 25 April 2005 3313 q 2005 The Royal Society


Table 1. Nomenclature

romanA surface areaa, b, c van der Waals constantsf specific Helmholtz functionG Gibbs functiong specific Gibbs functionm massp drop or bubble pressureP imposed pressurer radius of drop or bubbleR ideal gas constantS entropys specific entropyT temperatureU internal energyu specific internal energyV volumev specific volumeW stored workx non-dimensional radius

greekb super or sub-pressurization fractionP pressurization ratior densitys surface tension parameterg spinodal excess ratio

superscript

* metastable or non-dimensional

subscriptg gascr thermodynamic critical point valuesfm formationi incipientl liquidnp non-physicalPy Poyntings surfacesat saturationspn spinodal0 linear to originN plane

J. Lewins3314


of the bubble or drop size. In consequence, the work of formation and the criticalsize will be wrongly calculated.

Furthermore, a simple model using an ideal gas for a bubble suffers the defectof predicting infinite pressures at zero radius. We offer, therefore, a developmentof 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

Proc. R. Soc. A (2005)


3315Nucleation theory


(Lewins model) that provides a self-consistent account of bubble and dropgrowth from a thermodynamic viewpoint, a macroscopic theory. Just as van derWaals is an empirical model requiring the setting of three parameters, the Lewinsmodel calls for the setting of one parameter. The self-consistent model, however,obtains this last parameter from the gas spinodal properties predicted by the vander Waals model or indeed any other real gas model. It is hoped that theresulting thermodynamic framework will serve to fine tune the more detailedmolecular arguments of statistical thermodynamics and fluctuation theory.

2. Fundamental theory

The thermodynamic analysis rests on examining an expression for the energy of asystem of mass m in the form either of liquid and bubble, or gas and drop, toinclude the work of formation in creating a bubble or drop from zero size until itexists at some finite radius (Reiss 1997). This expression is then to be tested forstability. The real system of mass m is to have a uniform temperature T and animposed 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? Thisdepends upon an assumption that we can manipulate the original state of thesystem, 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 andquasi-static fashion. If so, the increase in entropy of the system is measurable.There is more difficulty, however, in understanding how we might measure theaddition of internal energy.

We actually expect there to be some statistical fluctuation redistributingenergy that will bring about the process. As such, this is not a matter for classicalthermodynamics. For the analysis to proceed we have to offer at least aconceptual way in which the bubble, say, can be brought about. Thus, we putforward a highly impracticable ‘thought experiment’ in which a fine capillarytube is inserted in the system and supplied with the gaseous form at suchcontrolled pressure and uniform temperature that the bubble can be forced togrow in a reversible fashion. Any addition of gaseous mass is compensated for byremoving liquid to maintain the definition of the system. Work will need to bedone, up to a point at any rate, to grow the bubble and this work in principle canbe measured. With such a measurement we might claim that the gedankexperi-ment justifies the assumption that the system has measurable internal energyduring the process.

Of course it is not practicable to measure the internal energy or the work offormation bringing about the growth. This indeed is the point of having a surfacetension model that can be used to estimate it instead. The work of formation canbe equated to the expression for free work at an equilibrium. The free work willinclude the energy stored in the surface. We assume, therefore, that it is properto represent the real system as having a combination of properties such asUKTSCPV although this is not a conventional Gibbs function, since theinternal properties are not uniform.

The growth of the real system is to be modelled as a bubble, say, of uniformpressure p with its real three-dimensional capillary surface region replaced by atwo-dimensional surface of tension radius r. The pressure may be determined by



J. Lewins3316


a number of techniques and our model is to account for the pressure differencepKPZDP.

The real volume V is represented in the model by the volume of liquid V1 plusthe volume of gas Vg. The masses, internal energies and entropies in these modelvolumes are determined by the model pressures and temperature. Since, theirsum is unlikely to equate to the real mass, Gibbs (1948) proposes to associatewith the two-dimensional surface not only an internal energy and entropy butalso a mass msZmKmlKmg such that all the real properties, including thevolume, are maintained in the model.

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

The modelling process consists then of selecting a radius r that should berepresentative of the real size of a bubble or drop together with a surface tensions so that we have consistently DPZ2s=r, Kelvin’s formula1 derived frommechanical equilibrium of a spherical bubble or drop. In saying that the surfacetension s varies with radius we must acknowledge two quite different variations.The first is the modelling step: where do we chose to locate the two-dimensionalsurface to represent the actual system state? The actual system properties mustbe independent of this choice. The second concerns the actual change in thesystem as a bubble or drop grows. The radius of our two-dimensional surface willthen change accordingly.

The surface tension s might be measured in the plane case by extending thesurface and is a force per unit length. Equally a small extension requires energyso that s also represents the area specific free work at constant temperature orHelmholtz function (it is not the surface energy as such). In the plane case anarea of AZ4pr2 would then have stored energy sA (Lewins 2000). Actuallysurface tension is generally measured on curved surfaces such as the effect of afine capillary tube in elevating or depressing a liquid, the distortion of a sphericaldrop 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 amisunderstanding in the classical formulation in the representation of the surfaceenergy term. In certain limited circumstances this error may be ignored but itleads to the meta-physical and unnecessary introduction of the argument about aradius of tension.

Where others write sA we write instead for the energy stored in the surfacethe expression WsZ

ÐA0 sðA0ÞdA0Z8p

Ð r0 sðr 0Þr 0 dr 0. Only if s is independent of r

the expression will reduce to 4pr2s and s can be identified with the area specificHelmholtz function or area specific free-energy at constant temperature.Otherwise we have fsZ

ÐsðA0ÞdA0=AZ �s, a radially averaged value.

With this replacement for the bubble, the energy statement for free-work orwork of formation becomes

Wfm ZUKTSCPVlCpVg ZWsCmg for a bubble; ð2:1aÞ

Wfm ZUKTSCpVlCPVg ZWs Cmg for a drop: ð2:1bÞ

1 Also credited to Helmholtz and Laplace





In this expression for the available work or work of formation we have the systeminternal energy offset by the (reversible) heat required at constant temperaturebut incremented by the mechanical energy to establish the two model volumes attheir respective imposed pressures. On the right-hand side, we have the ‘freework’ available in the form of the Gibbs functions (which represent onlyvolumetric work) plus the work stored in the surface. The surface value of g isdefined by the equality of chemical potential in all three phases: gsZglZgg at theassumed equilibrium

These equations can be rearranged so that the left-hand side is independent ofthe choice of radius in the model:

UKTSCP½VlCVg�Kmg Z 8p

ðr0sðr 0Þr 0 dr 0K½pKP�Vg for a bubble; ð2:2aÞ

UKTSCP½VlCVg�Kmg Z 8p

ðr0sðr 0Þr 0 dr 0K½pKP�Vl for a drop: ð2:2bÞ

It is seen that we have now interpreted the element of area dA not as part of thesurface 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,dAZ d

dr ð4pr2ÞdrZ8pr dr. Note that, in this view, s 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, wherewe shall draw the two-dimensional surface for a given real bubble or drop.As long as the ratio 2s(r)/r retains its same value to represent DP we might drawthe surface where we like. If we want the surface in the model to properlyrepresent the bubble size, so that subsequent evaluations are useful, we haveto use the corresponding surface tension. It would seem that the two volumes,Vl,Vg, depend upon the bubble size. But we are analysing a real system in a fixedstate and choosing a bubble size in a two-dimensional model to represent it.In the real system in fixed state, the total volume VZVlCVg is fixed just asmZmlCmgCms the total mass is fixed. While the two volumes would changewith choice of radius, they do not change independently and the consequentchange of masses in these volumes, based on the assumed densities of gas andliquid at their common temperature and respective pressures, is taken up bychanges in the mass ascribed to the surface. This surface term is left out of theargument in some texts but is clearly crucial.

Thus, we can argue that the left-hand side of the expression is indeedindependent of the radius chosen for the model at any particular state. Now, theargument that the right-hand side is then also independent of radius ondifferentiating yields the Kelvin formula:

8psrK4pr2DP Z 0 or DP Z 2sðrÞr

; ð2:3Þ

and we have consistency of mechanical and thermodynamic assessments whetheror not the surface tension varies with radius as the bubble grows. The result isvalid for an equilibrium situation, although widely used in non-equilibriumstudies that should really include inertial terms in any dynamic solution. It is



J. Lewins3318


important to note that equation (2.3) only determines a ratio s/r from themeasurable (in principle) pressure difference. If we want the radius to representthe size of the bubble adequately, it is up to us, the model-maker, to use a valueof s consistent with the circumstances bringing about the pressure difference DP.

3. Bubble and drop critical size-elementary model

It is useful to review the predictions of bubble and drop critical size in anelementary model. The theory just given can be used in the relations attributedto Poynting (Lewins in press) that exploit the common chemical potential of thephases, here the common specific Gibbs function in the liquid and gas phases. Weuse the simplest model in which the gas is an ideal gas satisfying PVZRT, theliquid incompressible with specific volume v l and the surface tension constant sN.Our example will show that, for a specific model and given conditions, thatinclude the specification of s, there is only one ‘equilibrium’ radius. The model issufficiently simple as to obtain the equilibrium condition, where the work offormation is a maximum, analytically.

These assumptions are used in analysing both the drop and the bubble at itsequilibrium size although the development has to be separate for these two. Bothturn on equating the change of specific Gibbs functions from the saturationpressure Psat, where the two bulk phases in equilibrium at the given temperaturehave equal values gl T ; psatðTÞð ÞZgg T ; psatðTÞð Þ. Since dgZKsdTCvdp thechange at constant temperature involves an integration over the specific volumeto the required pressure. Although many writers employ the supersaturationratio PhP=psat, we prefer an account in terms of the fractional sub- or super-pressurization b. For a bubble PZ1=ð1KbÞ and for a drop PZ1=ð1CbÞ.

(a ) The bubble at equilibrium

We have psatðTÞOPO0 and the specific Gibbs functions yieldðPpsat

v ldpZ

ðppsat

vgdp or Kv l½psatKP�ZRT

ðppsat

dp

pZRT ln

DPCP

psat

� �: ð3:1Þ

Define psatZð1CbÞP, where for small b this might be calculated via theClausius–Clapeyron equation for given superheated liquid but in any case isa measurable property given the system pressure and temperature. Then, putr�hv l=vgðPÞ!1 a non-dimensional density ratio, gas to liquid at the systempressure. Thus,

Kv lP

RTbZ ln

1CDP=P

1Cb

� �hKbr� and PPy Z psate

Kbr� zpsat½1Kbr��; ð3:2Þ

where PPy (Poynting) is the equilibrium or critical pressure, seen to be at orbelow the saturation pressure. If br� is indeed small enough to approximate theexponential we have

DP Z2s

rZ psate

Kbr�KPzPð½1Cb�½1Kbr��K1ÞzPbð1Kr�Þ: ð3:3Þ





Define a non-dimensional radius, based on the plane value for surface tensionwriting xZ rP

2sN. Then, the critical or Poynting radius is given by

1

xZ ð1CbÞeKbr�K1zbð1Kr�Þ or bxPy Z

1

1Kr�; ð3:4Þ

valid if either b or r� or br� 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 bP or psatKP.Notably, we have not assumed that b is necessarily small in the analysis of thebubble. We discuss whether this is stable or unstable equilibrium later.

(b ) The drop at equilibrium

0!psatðTÞhð1KbÞP!P. Nowðppsat

v ldpZ

ðPpsat

vgdp whence v l½pKpsat�ZRT

ðPpsat

dp

pZRT ln

P

psat

� �ð3:5Þ

and

Kr�DP

PCb

� �Z lnð1KbÞ or assuming b/1

DP

Pzb

1Kr�

r�:

If r�/1 then

bx Zr�

1Kr�zr� and PPyzpsatC

PKpsatr�

: ð3:6Þ

Thus, the excess pressure in the drop at critical size is of order bP=r�ZðPKpsatÞ=r�and, thus, very much larger than for the corresponding bubble. Indeed, if the gasdensity is very small, the predicted pressure in the drop at equilibrium, thePoynting value, may be unphysically large, casting doubt on this model using anideal gas, incompressible liquid and constant surface tension. Again, we haveestablished a drop size at equilibrium without yet considering the nature of thatequilibrium. Note that for the drop 0!b!1 given PO0 but if r�/1, close to thecritical temperature, and b/0 then x/N in this model. But if b/1; x/Kr�=lnð1KbÞ/0. A value of the relative density approaching one-half appears togive anomalous behaviour, but who would believe the model of incompressibilityso close to the critical point temperature?

Both these relations, for drop and for bubble, are essentially the same as thoseobtained by Poynting and Kelvin. Although the values of the super- or sub-pressurized fraction b and the density ratio r� are distinct for the two cases, whenthey are comparable it is seen that the critical size of the drop at comparableconditions is far smaller than the critical size of the bubble. This is even moreextreme for the masses contained in the drop or bubble which will go as r�2/1,an observation of significance for auto-nucleation theory assessing the number ofmolecules that must have sufficient fluctuation of their energy to establish thiscritical size, and for the volumes, going as r�3. It is also notable that the theorygives the critical pressure in a drop or bubble without reference to the surfacetension.



J. Lewins3320


Given the unique solutions for drop or bubble size, it is seen that the moregeneral problem of the growth of a drop or bubble is not simply a thermodynamicstatic process, but must be a dynamic process to which thermodynamics cancontribute limits or bounds only. Indeed it is apparent that the equilibrium pointis only one of unstable equilibrium. Once beyond this critical size, the bubble willcontinue to expand because the superheated liquid is itself metastable and oncebeyond the barrier offered by surface tension will flash to gas. Similarly, the dropwill tend to take over the system as the stable phase compared to the metastablegas. 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 thepoint of unstable equilibrium. To this end, one studies the thermodynamic workof formation. Consider now a system starting with a mass M of liquid at apressure P and superheated so that, as before, P!Psat. Through someunspecified mechanism a bubble is formed and grows, of mass m at radius r orequivalently xZrP=2sN. Supposing this growth is reversible, the reversibleexpansion work done by the surface of the system (not in the bubble) is given byPDVZPm½vgKv l�. The reversible heat absorbed is QZTDS and the entropychange, including the change of entropy of the bubble surface from a zeroradius is DSZssðrÞ4pr2Cm½sgKsl�, noting that the specific properties of theremaining liquid mass are unchanged. The internal energy change is similarlyDUZusðrÞ4pr2Cm½ugKul�. Thus, we can write the work of formation of thebubble as it grows as

Wfm ZmfP½vgðpÞKv lðPÞ�C ½ugðpÞKTsgðpÞ�K½ulðPÞKTslðPÞ�gCA usKTss½ �:ð4:1Þ

We are not suggesting that the bubble actually grows by such a reversibleprocess 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 Gibbsfunctions as

Wfm ZmfP½vgðpÞKv lðPÞ�C ½fgðpÞKflðPÞ�gCAfs ð4:2aÞ

and

Wfm Zmf½ggðpÞKglðPÞ�KvgðpÞDPgCAfs; ð4:2bÞ

where we have DPZ(pKP) the excess pressure in the bubble that will go to zeroas the bubble reaches infinite radius or the plane case. However, equilibriumarguments show that in that case, for equilibrium, the system pressure wouldhave to be at the saturation pressure so that the specific Gibbs functions of thetwo phases were then equal. Thus, equation (4.2a) is useful for computationalpurposes but must be interpreted with care. The usual interpretation of a Gibbsfunction is the free work at constant temperature and pressure but here thepressures differ.





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

We first assume an ideal gas, an incompressible liquid and a constant surfacetension equal to the plane value. Then, exploiting the equality of specific Gibbsfunctions at the saturation pressure

ggðpÞKggðpsatÞZðppsat

vgdpZRT lnp

psat

� �and glðPÞKglðpsatÞZKv l½psatKP�:

ð5:1ÞThemass in thebubble at the currentpressure goes asmZ 4

3 pr3=vgðpÞZ 4

3 pr3p=RT.

We then have

Wfm ZmRT ln1CDP=P

1Cb

� �K

DP

pCv l½psatKP�

� �CAfs

Z4

3pr3P 1C

DP

P

� �ln

1CDP=P

1Cb

� �K

DP

PCr�b

� �C4pr2sN: ð5:2Þ

This may be put again in non-dimensional form writing xZrP/2sN to give

WfmðxÞZ32

3ps2

P3x2 ð1CxÞ½lnð1C1=xÞKlnð1CbÞ�Cr�xbC

1

2

� �: ð5:3Þ

The peak corresponds to the (unstable) equilibrium point found from thePoynting relation. For zero density ratio the peak work of formation goes as bK2.Figure 1 shows the work of formation normalized by putting W �Z 3

16P3

ps2Nb2W

together with further normalizations of x�Zbx. As a consequence, the peak forany b-value would go through the point (1,1) for a density ratio r� of zero. Thefigure shows the peak fitting to W �

crzð1Kr�ÞK2.A similar analysis for the formation of a drop, using the same assumptions and

b/1 yields a work of formation W �fmZðbxÞ2 3K2bx r�

1Kr�

h iwith a peak at the

expected x�ZbxZr�=ð1Kr�Þ and W �crZ

r�

1Kr�

� �2zr�2. The critical work to form

a drop is thus much reduced compared to the critical work for a bubble atcomparable 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 theorder of a few molecules, then it is of the order of the capillary region. That is, amolecule on the capillary surface is not only attracted back by the mass of liquidbehind 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 willonly show up for drops and bubbles of very small size. This observation isconsistent with work done solving the Schroedinger wave equation for a fewmolecules of water to show that not until about eight or nine are gatheredtogether will the system display the properties of liquid water, density, pressure,etc., and therefore of surface tension.



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60

0.5

1.0

1.5

2.0

2.5

non-dimensional radius, x*

non-

dim

enso

nal w

ork

of f

orm

atio

n, W

*

Figure 1. Work of formation in ideal gas model. Increasing density ratio: 0, 0.1, 0.2, 0.3.

J. Lewins3322


We propose an empirical model for the reduction in the capillary effect for verysmall bubbles and drops. This will contain one parameter to be determined byfurther argument. It calls for a surface tension that falls rapidly as bubble sizeapproaches zero but approaches the plane value at larger radii. A reasonablerepresentation of the behaviour of surface tension for modelling purposes wouldthen be

sðr;TÞZsNðTÞ½1KexpðKr=r0Þ�: ð6:1Þ

An expression in the area or even the volume is possible but we chose thesimplest case, the linear radial dependence. The effect of the variation of surfacetension with bubble size and area means that the surface term in the work offormation now has to be integrated2 such that

Afs Z 4psN

ðr02r½1KeKr=r0 �dr Z 4psN r2K2r20 1Kð1C r

r0ÞeKr=r0

� �h i; ð6:2Þ

so that as rr0/0;Afs/0 and r

r0/N;Afs/4psN r2K2r20

. It is seen that the

surface work term has been reduced in this empirical model.

2Many authorities have not made this correction but just write s(A)A, if indeed allowing s to vary.We would perhaps agree on the differential s(A)dA, but how is this to be integrated? We see nooperational justification for integrating over the surface of the established bubble but can see anoperational meaning for the integration from zero size, analogous to the expression for thereversible work done by a volumetric change WZ

Ðp dV :



1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1.0

1.2


radi

ally

dep

ende

nt s

urfa

ce te

nsio

n

Figure 2. An empirical model for radially dependent surface tension.



7. Ideal gas model with Lewins model: radial dependence ofsurface tension

We now introduce the radially-dependent surface tension to the model, retainingthe non-dimensional definitions based on the plane case. The effect of thevariation means that the work of formation becomes

W �fmðxÞZ x2 xC1KeKx=x0

� �ln

1C 1x 1KeKx=x0 1Cb

!CrdxbC

1

2CeKx=x0

" #

K3x20 1K1 ð1C x

x0

� �eKx=x0

� �ð7:1Þ

with x0Zr0P/2sN.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 isDPð0ÞZ2sN=r0, since

DP Z2s

rZ

2sNr

½1KeKr=r0 �z2sN1K1Cr=r0

r: ð7:2Þ

The asymptotes at the origin and at infinity, useful for inverting to find theradius, are, respectively,

2sNr0

1Kr

2r0

� �and 2sN

r0r: ð7:3Þ



2 4 6 8 10 120

0.2

0.4

0.6

0.8

1.0


non-

dim

ensi

onal

exc

ess

pres

sure

Figure 3. The variation of excess pressure in the empirical model. (The asymptotes 1K12 x and 1=x

lie either side of the true value, an aid to inverting the functional behaviour to obtain rZr(p).)

J. Lewins3324


If the parameter x0 goes to zero we recover the constant surface tension model(with ideal gas). Figure 3 shows that the Lewins model for surface tension puts afinite 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 aspecified superheat value in the ideal gas model for various choices of theparameter r0 or the equivalent x0 of radially dependent surface tension. Again,when the Lewins parameter is zero, the departure of the peak from (1, 1) is due tothe finite density ratio r�. The value 0.17 chosen here matches the van der Waalscase 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 isreduced by up to 50% by the drop in surface tension, and critical size by up to10% 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, ratherthan a ideal gas model, in which the maximum pressure in a gas is limited byattractive forces between molecules. This maximum pressure can be identified inany particular model and can be used to give an upper bound, at least, to theLewins parameter. In this paper we shall use the simplest real fluid model, thevan der Waals model with its predictions of spinodes.

8. The van der Waals model

Thermodynamics can provide a framework for this extended treatment of surfacetension at small radius and hence the formation of bubbles and drops, byconsideration of what is called the spinodal or turning-point. To this end, we



x0=0.6

0

0.5

1.0

1.5

non-

dim

enso

nal w

ork

of f

orm

atio

n, W

*

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6non-dimensional radius, x*

*

x0=0*

Figure 4. The work of formation for given superheat with an ideal gas model and the Lewins modelfor surface tension at varying parameters of x �

0: 0, 0.2, 0.4 and 0.6. The x�-axis is bx, and the y�-axisW �. The density ratio r� is 0.17. bZ0.01.



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 fromthe ideal gas that represents the attractive forces between molecules (van derWaals forces) in an analytical form. It predicts the nature of a critical point, theequilibrium between gas and liquid phases (with help from Maxwell) and themetastable properties of gas and liquid. To the extent that real P–V–T datamatch this equation of state, it may be used as a realistic model. The equationmay be written in the first instance as

P ZcT

vKbK

a

v2: ð8:1Þ

Onthe right, the first term represents the kinetic pressure due to the thermal energyand momentum. In this term, b represents the minimum volume that unit mass ofthemolecules canbe compressed into; this is greater than the size of themolecules inthat it includes essential ‘dead’ space between them. The parameter c is analogousto the gas constantR. The second term is the correction due to the attractive forces.The constant a is to represent the force between two molecules at some effectiverange. One factor of 1/v is proportional to the density and thus to the number ofother molecules close to one molecule that is attracted by them. The further factorof 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 specificvolumes and high temperatures. But isotherms of constant temperature have adifferent shape below the ideal gas region. In particular, there is a point of



J. Lewins3326


inflexion where the derivatives vPvv

� �TZ v2P

vv2

� �TZ0. The local behaviour is

similar to the isotherms measured in carbon dioxide by Andrews and thusrepresent the critical point,3 that highest pressure and temperature enabling gasand 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 pointvalues, writing; T=Tcr/T ; v=vcr/v;P=Pcr/P so that the equation in non-dimensional form is

P Z8T

3vK1K

3

v2: ð8:2Þ

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 theunphysical region 0!v!1/3.

vP

vv

� �T

Z 0Z6

v3K

8T

ð3vK1Þ2: ð8:3Þ

In a P–V–T space these spinodes would form spinodal lines. They are thetheoretical equivalent of Wilson points and Wilson lines which are the observedlimits of metastable behaviour affected by other considerations such as nuclei oreven statistical fluctuations.

The isotherms are continuous and do not represent the jump in specificvolumes between liquid and gas that we would expect. This horizontal jump isadded by means of a construction credited to Maxwell (Lewins 2003). Thisconstruction is to locate the level of the two-phase pressure that will join thesmall specific volume of liquid, left, to the large specific volume of gas, right, onan isotherm necessarily below the critical point. We know that the specific Gibbsfunctions (or chemical potential) are equal for liquid and gas at this pressure.Thus, an integral from left to right along the horizontal isotherm of dgZKsdTCv dP will have

Ð gl dgZ0

. 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 isequal to the area above, figure 5. The numerical implementation of thisconstruction is given in Lewins (2003).

(b ) Behaviour at the spinode

The section of the isotherm between the two spinodes, although predicted asan extension of the equation of state, is an absolutely unstable region whichcannot exist in nature in stable form. Konorski (1990) uses this argumentcogently in an excellent paper on nucleation. He adduces similar work byRusanov (1960) and Skripov (1972). But the segment leading up to the spinodefrom the saturated value is potentially achievable and represents the metastablestates of liquid and gas we have been using, now in a more realistic model for theequation of state.

3With this in mind we shall now refer to the maximum in the work of formation as the Poyntingpoint and not a critical value.



0.5 1.0 1.5 2.0 2.5 3.0 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

v*

p*

gas spinode

–

+

van der Waals for zero pressure liquid spinode: T=27/32

Figure 5. The van derWaalsmodelwith spinode points. Horizontal line: saturation pressureZ0.4880.



It follows that, at a given temperature, the gas spinode represents themaximum overpressure available in a bubble. Above this pressure, the systemcan only exist as a liquid. We have to say, therefore, that the very large excesspressures predicted by the Kelvin formula and an ideal gas model, with constantsurface tension at small bubble size, are unfounded. What we should expect thenis 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 someuncertainty 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 predictthe radius ri of the smallest possible or incipient bubble, that might contain say10 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 befor a true bubble radius:

sðri;TÞZ 1

2riDPspn: ð8:4Þ

However, thermodynamics does not of itself predict the incipient radius ri andthis would be a matter for physical modelling of the behaviour of real molecularsystems.4 Our empirical model might reasonable join these two points, zero andincipient, by a straight line which would extrapolate to represent the surfacetension 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

4Nine molecules forming a drop offers a symmetric pattern. Then, the hexamer with eightmolecules may well be the smallest aggregation of water molecules offering an attraction to makeup nine and the characteristics of a drop.



J. Lewins3328


never exceeds the spinodal value. And in figure 3 we saw how the excess pressurenow depends upon radius.

We can get some support for the assumption of a linear dependence of surfacetension on radius by examining the behaviour at a spinode. At the spinode achange of radius leading to a change of specific volume does not, however, lead toa change of pressure and, therefore, of excess pressure. Thus,

sðrÞr

� �spn

Z const:; ð8:5Þ

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, wecan express the excess pressure in the bubble as

DP Z2sNr

1KeKr=r0h i

; ð8:6Þ

where the free parameter r0 is chosen such that the excess pressure relates to themaximum or spinodal pressure at the origin (differentiation shows that the excesspressure decreases with increasing radius):

Pspn:gðTÞKP Z2sNr0

: ð8:7Þ

If we put Pspn.gZgP then the corresponding non-dimensional radius is given by(gK1)x0Z1. Our ansatz then is to say that the maximum pressure inside thebubble occurs at its incipient point, effectively zero, and is the gas spinodalpressure. This much of the Lewins model, the linear behaviour at very small sizesof incipient bubbles is then not empirical but based on the thermodynamicbehaviour at the spinode. At the very least, this provides a bound to thepressure; our model says this indeed is the pressure in the initial bubble. Theslope can be predicted by any real gas model predicting gas spinodal values, notjust 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 ofspecific Gibbs at constant temperature:

gðpÞKgðPÞZðpPvdpZ

ðpPv K

24T

ð3vK1Þ2C

6

v3

� �hDgðp;PÞ

Z8T

3

1

3vK1Kln vK

1

3

� �� K

6

v2

� �pP

: ð9:1Þ

With this empirical model for surface tension and with the van der Waals modelfor the fluid gas and liquid which sets the spinodal parameter in the Lewins



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8


non-

dim

enso

nal w

ork

of f

orm

atio

n, W

*

Figure 6. The work of formation of a bubble Lewins–van der Waals model. The density ratiois 0.17. bZ0.01.



model, the work of formation becomes

W �bubble Z

x3

PvgðpÞDggðp;PsatÞKDglðP;PsatÞ� �

Cx21

2CeKx=x0

� �

C3x20 1Cx

x0

� �eKx=x0K1

� �; ð9:2Þ

Here

r0 Z2sN

PspnKP; pZ

2sNr

1KeKr=r0h i

and pZ8vcrpcrT=Tcr

3vgKvcrK

3pcrv2cr

v2g: ð9:3Þ

We are thus able to plot the non-dimensional work of formation. Figure 6illustrates this for a particular assumption of super-heated liquid and theformation 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 actuallyincreases 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 inthe simple model, the excess pressure in the drop is substantially higher than thesaturation pressure or indeed the gas spinodal pressure. It does not fall to the liquid



J. Lewins3330


spinodal pressure; the behaviour there has little to do with the drop nucleationprocess. The liquid spinodal behaviour, where the limiting ratio of surface tensionto radius is constant, is immaterial and we turn to a more involved argument tojustify using the slope appropriate to the gas spinode for the drop case.

We ask the question whether the surface tension, varying with radius even atconstant temperature, should be different for liquid and bubble system and fordrop and gas system. Detailed molecular analysis might suggest reasons why atthe same curvature (but of opposite sign) the surface behaviour might differ but asimple 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 s(r)for drop as for bubble.

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

We feel justified, therefore, in using the same surface tension for drop asbubble at a particular radius (and temperature). Now the parameter r0 of thebubble model is not a size as such but an asymptotic interception of the smallradius behaviour with the large radius behaviour. The incipient bubble size, thatsmallest size at which it can be thought to have the properties of a bubble andcorresponding surface tension, will lie on the small radius asymptote and itsradius will be smaller than r0 itself. It is in the linear range.

Considering the incipient size of a drop, it is shown in the simple analysis thatthe size of a drop, at the Poynting equilibrium, is much smaller than thecorresponding size of a bubble. We can reasonably expect, therefore, that theincipient size of a drop is smaller than the incipient size of a bubble-say the spaceoccupied by nine water molecules in a nonamer is smaller than the largely emptyspace occupied by a bubble with at least one molecule in it to represent theproperties of the fluid. Then the drop radius, being smaller, will also lie on theasymptote. We can thus employ the parameter fitted from the bubble gasspinodal analysis to set the Lewins model in the drop analysis. The non-dimensional work of formation is now

W �drop Z

2x3

Pv lðpÞ½Dglðp; psatÞKDggðP; psatÞ�Cx2 1C2eKx=x0

h i

C6x20 1Cx

x0

� �eKx=x0K1

� �: ð9:4Þ

Figure 7 illustrates the resulting shift of critical work of formation and criticalsize in a specified example using the Lewins–van der Waals model.

These results can be compared to the predictions of the ideal gas model withconstant surface tension. At comparable values, the simple model predicts anon-dimensional critical radius of 0.2, whereas here we have 0.14. Since,

r�

1Kr�

� �2z0:04, the Lewins–van der Waals model at 0.015 predicts a maximum



0 0.05 0.10 0.15 0.20 0.25–6

–4

–2

0

2

4

6

8

10

12

14


non-

dim

enso

nal w

ork

of f

orm

atio

n, W

*(×1

0–3

)

Figure 7. The work of formation of a drop Lewins–van der Waals model.



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 sayingthat at the point of birth, the inchoate fluid cannot be identified clearly as eithergas 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 ofdrops and bubbles extending to their incipient size. A reformulation of thethermodynamic foundations clarifies the nature of the variable surface tensionparameter and we argue that this must be allowed to go to zero at the very smallsizes at which bubbles or drops can be said to grow or be incipient. The modelused 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 thegas spinode. Certainly this is a necessary and close upper bound and our modeltakes this as the actual value, thus avoiding the elementary predictions of aninfinite pressure in the inchoate bubble. We argue that the gas spinode is areasonable thermodynamic representation of what, in a more realistic dynamicalpicture, is the nature of the fluid at the point where it changes phase from liquidto start bubble growth. Arguing that the same surface tension at a given radiusapplies to drop and bubble, we use the same fitting for a model of the incipient



0.5 1.0 1.5 2.0 2.5 3.0 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

v

p

o+ i

+

m+ e+

Py+

gas spinode

>>

bubble formation process: T=27/32

Figure 8. The path for growth of a bubble. Dashed line: saturation pressureZ0.4880.

J. Lewins3332


drop. This parameter can be predicted by any real fluid model that predicts gasspinodal behaviour. Figures 8 and 9 illustrate the (irreversible) path proposed forbubble and drop growth in the van der Waals model.

The bubble forms (figure 8) from liquid at the metastable state (‘m’) withpressure P that is less than the corresponding saturation pressure psat. Thebubble 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 isin the sense of the thermodynamic limit of vanishing radius that may be calledthe inchoate radius. Accepting a molecular view, beyond thermodynamics, thatthere will be a small but finite size when the fluid can really be said to be in agaseous state, we identify the incipient radius of this smallest bubble as a nearbypoint on the metastable line (‘i’). If the bubble grows, it may reach itsequilibrium point, also on the metastable line, a point of unstable equilibriumwhere the bubble (and its surface) can be said to be in equilibrium with themother liquid-the Poynting point (‘Py’). Unstable growth beyond this leads totransfer further along the metastable line (if dynamic effects can be ignored)through the saturation pressure and on to the system pressure P when all liquidhas 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 ametastable gas at a pressure P that is now higher than its correspondingsaturation pressure psat. The drop appears in the sense of a thermodynamic limitat its zero or inchoate size (‘o’) at a liquid pressure equal to the gaseous spinodalpressure (spn). Close by is the incipient size (‘i’) and then the unstableequilibrium or Poynting size (‘Py’). Beyond this, again, the drop is unstable and



oi

++

m+

e

+ Py+

gas spinode

<<

drop formation process: T=27/32

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

p

0.5 1.0 1.5 2.0 2.5 3.0 3.5v

Figure 9. The path of growth for a drop. Dashed line: saturation pressureZ0.4880.



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 notmodelled here as such but the error in starting the work integral at zero instead isvery small; what is significant to the prediction of Poynting sizes is the slopeparameter of the Lewins model. We note that in this model we find that theincipient bubble and the drop are both at their most dense, the density falling asthey grow. This latter is strictly only a thermodynamic bound for the realdynamic and irreversible process. Analysed in the Gibbs grand canonicalensemble of statistical mechanics, where temperature and chemical potential areretained but molecule density and energy fluctuate for example, a statisticalfluctuation bringing about auto-nucleation might see an energy change withouttemperature change. But we might hope that we have a better representation ofboth the Poynting size and the maximum work involved in forming drops andbubbles.

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



J. Lewins3334


When analysed in conjunction with van der Waals equation of state, again anempirical model, a self-consistent theory can be advanced for a thermodynamicaccount of the work of formation. This shows some modest changes in thepredicted size of the equilibrium system and its corresponding maximum work offormation that may be of interest to those studying auto-nucleation and indeedimposed or inhomogeneous nucleation. No doubt the parameters can beimproved but already the model provides a self-consistent account that avoidssuggesting infinite pressures and reduces the critical energy to be supplied byfluctuation, compared to simpler models. Even though the changes it predictsfrom the simple model are modest, they will of course be significant in nucleationtheory where rate changes are exponentially sensitive to the necessary quantityof critical work. It is hoped that this framework will not only help understandingbut allow finer tuning of the nucleation models proposed by Tolman (1948) andthose following his statistical theory, or the quantum mechanic approach of sayClary and his school Liu et al. (1996)

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

References

Gibbs, J. W. 1948 Collected Works. Yale University Press, reprinted 1948 esp., vol. 1. pp. 219–331and 331–337.

Guggenheim, E. A. 1967 Thermodynamics. An advanced treatment for chemists and physicists, 5thedn. Amsterdam: North Holland.

Konorski, A. 1990 Thermodynamic equilibrium limits of extremely small droplets in saturatedvapour. Forschung im Ingenieurwesen 56, 119–129.

Lewins, J. D. 2000 Plane surface tension. Int. J. Mech. Eng. Edu. 27, 217–229.Lewins, J. D. 2003 Enthalpy phase change predictions from van der Waals equation model,

accepted Int. J. Mech. Eng. Edu.Lewins, J. D. 2004a A reformulation of the fundamental thermodynamics of surface tension for

spherical bubbles and drops. J. Chem. Thermodyn. 36, 977–982. (doi:10.1016/j.jct.2004.07.022.)Lewins, J. D. In press. Thermodynamic nucleation theory. Int. J. Mech. Eng. Edu.Liu, K., Brown, M. G., Carter, C., Saykally, R. J., Gregory, J. K. & Clary, D. C. 1996

Characterization of a cage form of the water hexamer. Nature 381, 501–502. (doi:10.1038/381501a0.)

McGlashen, M. L. 1979 Chemical thermodynamics. New York: Academic.Reiss, H. 1965 Methods of thermodynamics, (reprinted Dover 1997).Rusanov, A. 1960 Thermodynamics of surface phenomena. Leningrad: Izdat. University. (in

Russian).Skripov, W. 1972 Metastable liquids. Moskva: Izd. ‘Nauka’, Red. Mat-Fiz. (in Russian).Tolman, R. 1948 Consideration of the Gibbs theory of surface tension. J. Chem. Phys. 16, 758.

(doi:10.1063/1.1746994.)


http://dx.doi.org/doi:10.1016/j.jct.2004.07.022

http://dx.doi.org/doi:10.1038/381501a0

http://dx.doi.org/doi:10.1038/381501a0

http://dx.doi.org/doi:10.1063/1.1746994


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A new calculation of the work of formation of bubbles and drops