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Copyright American Scientific Publishers

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All rights reserved

Printed in the United States of America

Advanced Science FocusVol 1 pp 367ndash368 2013(wwwaspbscomasfo)

Comments on ldquoThe Gibbs Equation versus theKelvin and the Gibbs-Thomson Equations toDescribe Nucleation and Equilibrium ofNano-MaterialsrdquoLasse Makkonen

VTT Technical Research Centre of Finland Box 1000 02044 VTT Espoo Finland

The arguments recently presented against the Kelvin equation and Gibbs-Thomson equation (J Nanosci Nano-technol 12 2625) are shown to be invalid and the related ldquomodified equationsrdquo incompatible with the theory ofthermodynamics

KEYWORDS Kelvin Gibbs Gibbs-Thomson Nucleation Nano-Thermodynamics

An intriguing article was recently published in which thehistorical foundations of surface thermodynamics of smallparticles were claimed to require revision1 The Kelvinequation and the Gibbs-Thomson equation were claimedto be wrong and the difference in the behavior of nano-systems compared to macro-systems was identified to bedue to their high specific surface area and not to the highcurvature of their interfaceUnorthodox theoretical papers which attempt to alter the

foundations of science are often difficult to get publishedin journals In this sense Kaptayrsquos paper1 is refreshingHowever its conclusions are flawed as will be discussedin the followingWhen one treats an interface by the theory of thermo-

dynamics surface excess terms need to be added into afundamental equation for the bulk material For the Gibbsfree energy the fundamental relation reads

dG=minusSdT +Vdp+dn (1)

where S is entropy T temperature V volume p pres-sure chemical potential and n the number of molesin the system The terms in a fundamental equationmust be consistent with the mathematical structure ofGibbsrsquo thermodynamics2 outlined eg by Hermann3 andBottomley et al4 Accordingly in surface thermodynamicssurface excess energy is taken into account by addingthe term dA into the right hand side of Eq (1) Here Ais surface area

Email lassemakkonenvttfiReceived 8 November 2013Accepted 11 November 2013

In Kaptayrsquos paper1 the molar Gibbs energy (Jmol)denoted here by GK is defined as

GK = G

n=G

(Vm

V

)(2)

where Vm is the molar volume The condition of thermody-namic equilibrium is then set as (Eqs (3) (11) and (19b)in Ref [1])

dGK = 0 (3)

and this condition is used to derive thermodynamicequationsHowever the condition of Eq (3) violates the first law

of thermodynamics The first law of thermodynamics onwhich Eq (1) is based states that energy is conserved notenergy divided by volume Accordingly in thermodynam-ics equilibrium is defined by

dG= 0 (4)

not by Eq (3)The difference between the conditions of equilibrium

given in Eqs (3) and (4) can be demonstrated for a spher-ical nanoparticle with radius r Equation (3) gives

GK

r= Gn

r= 0 (5)

Differentiating Eq (5) shows that for Eq (3) to hold

G

r= G

n

n

r(6)

According to the theory of thermodynamics at equilibriumEq (4) Gr = 0 Inserting this in Eq (6) shows that for

Adv Sci Focus Vol 1 No 4 2013 2330-076020131367002 doi101166asfo20131054 367

Delivered by Publishing Technology to Universidad de VigoIP 1931463273 On Wed 07 May 2014 110645

Copyright American Scientific Publishers

The Gibbs Equation versus the Kelvin and the Gibbs-Thomson Equations to Describe Nucleation and Equilibrium of Nano-Materials Makkonen

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ION

a non-zero G the condition of Eq (3) results in nr = 0at the true equilibrium This means that the volume doesnot change with radius which is untrue This shows thatEq (3) is an incorrect condition of equilibriumKaptay1 submits three arguments against the Kelvin

equation and the Gibbs-Thomson equation The first (i) isthat based on the Gibbs energy term due to Kelvin (Eq (7)in Ref [1]) ldquonanocrystals and thin films are expectedto have the same thermodynamic properties as the bulkphaserdquo The origins of surface energy are in the lowerpotential of the atoms in the bulk material compared tothose at the surface When a film gets very thin the bulkdoes not exist in its usual meaning Thus the classicalthermodynamics involve no such expectationArguments (ii) and (iii) in Ref [1] are based on show-

ing that the Kelvin equation and Gibbs-Thomson equa-tion can be derived by an incorrect derivation It is notmade quite clear why that derivation is incorrect In anycase such a proof is logically inconclusive because anincorrect derivation may lead to a correct result Cor-rect ways to derive these classical equations are discussed

eg in Ref [5] and can be found in the textbooks ofthermodynamicsTo summarize Kaptay1 defines the condition of ther-

modynamic equilibrium by the Gibbs energy per vol-ume instead of the Gibbs energy Differentials of thesetwo concepts with respect to the radius of curvatureare not the same Therefore the analysis in Ref [1]results in erroneous forms of the Kelvin equation andthe Gibbs-Thomson equation and consequent wrong con-clusions The Kelvin equation and the Gibbs-Thomsonequation are valid and set the metastable equilibrium ofnanoparticles

References and Notes1 D J Bottomley L Makkonen and K Kolari Surf Sci 603 97

(2009)2 J W Gibbs The Scientific Papers of J Willard Gibbs Thermody-

namics Ox Bow Woodbridge (1993) Vol 13 R Hermann Geometry Physics and Systems Marcel Dekker

New York (1973) Chap 64 G Kaptay J Nanosci Nanotechnol 12 2625 (2012)5 L Makkonen Langmuir 16 7669 (2000)

368 Adv Sci Focus 1 367ndash368 2013