Thermodynamic and mechanical properties of curved interfaces : a discussion of models

M. Oversteegen

Research output: Thesisinternal PhD, WU


<p>Although relatively much is known about the physics of curved interfaces, several models for these kind of systems seem conflicting or internally inconsistent. It is the aim of this thesis to derive a rigorous framework of thermodynamic and mechanical expressions and study their relation to previous models.</p><p>In chapter 2 interfaces are described mathematically. It turns out that their curvatures can generally be determined by two independent coefficients, <em>viz.</em> the total and the Gaussian curvature. These degrees of freedom of a system must be accounted for in the thermodynamic expression for the internal energy and are conjugated to the bending stress and torsion stress, respectively. The curvatures can then be taken as intensive variables, as has been done by Gibbs, or as extensive variables, as proposed by Boruvka and Neumann. The two ways of accounting for curvature leads to different definitions of the interfacial tension. In the former way the curvatures can be fixed when changing the interfacial area, whereas in the latter the area times the curvature must be constant upon variation of the interfacial area. Consequently, the interfacial tension according to Boruvka and Neumann incorporates bending as well as stretching work. Hence, for homogeneously curved interfaces, the difference between the two interfacial tensions is the bending work.</p><p>It follows from a quasi-thermodynamic description that the interfacial work according to Gibbs can be described mechanically as the volume integral of the excess pressure profile. Writing the volume element in terms of the curvatures, the interfacial tension according to Gibbs can be expressed in terms of the zeroth, first, and second bending moments, respectively. Using their thermodynamic definitions, the bending and torsion stress can also be given mechanically, i.e., in terms of the excess pressure profile. Subsequently, using the relation between the interfacial tension of Gibbs and that of Boruvka and Neumann is expressed in terms of the bending moments. The newly derived equations differ significantly from those known in the literature. However, it is shown that the Laplace equations of capillarity derived from either the thermodynamic or the mechanical route are consistent.</p><p>The mechanical and thermodynamic notion of `pressure' are scrutinized in chapter 3. The mechanical or virial route to the pressure is reviewed as a result of the forces exerted by the momenta and interactions of the particles per unit area. The mechanical pressure turns out to be a tensor quantity and is used to recover results known in the literature. Since the interactions cannot be assigned unambiguously to one position in space, the local pressure is found to be equivocal.</p><p>A lattice model allowing spatial gradients is elaborated. The grand potential density of a system, which is the work of changing the volume of the system reversibly, is identified as the scalar thermodynamic pressure. For a bulk system, the grand potential density recovers the Kamerlingh-Onnes virial expansion of the pressure and has the same features as the reduced van der Waals pressure. Moreover, it has been shown that in the continuous limit the Helmholtz energy of the lattice gas can be written as the Landau expression for the free energy. For an inhomogeneous system of monomers, pressure profiles are found from the grand potential density that have similar features as those found from the virial route. That is, in the vicinity of an interface both tensile and compressive regions are observed. In the model by Szleifer <em>et al.</em> the tensile, i.e., negative region of the locale pressure is omitted. Since that region may be necessary to obtain low interfacial tensions for some systems, an important feature of their `pressure' has been ignored. Since the reference state of the energy of the lattice model can be chosen freely, it is concluded that the thermodynamic pressure can neither be given unambiguously.</p><p>The bending and torsion stress of a monomer liquid-vapour interface are determined from their mechanical expressions using two definitions of the local pressure. The expressions as derived in chapter 2 turn out to give unique consistent results, whereas the expressions known in the literature give ambiguous outcomes for the thermodynamically well-defined parameters. The latter is physically unacceptable. Since, unlike the virial route to the pressure, the thermodynamic pressure of the lattice model yields by definition a unique expression for the grand potential, it is concluded that this lattice model is a useful tool to model curved interfaces.</p><p>A phenomenological description of the curvature dependence of the interfacial tension is given in chapter 4. Up to first order in the curvature, the change of the interfacial tension is determined by the Tolman length. A second order description is given by the Helfrich equation, which, in turn, is determined by the bending modulus and the saddle-splay modulus. These Helfrich constants turn out to be the (derivatives of the) bending and torsion stresses of the planar interface, respectively. As a consequence of the different mechanical expressions for the bending and torsion stress, the Helfrich constants cannot be obtained from the properties of the planar interface only but also require the curvature dependence of the bending moments. This difference with the equations known in the literature can be traced back to the difference of the definition of the pressure from either a virial or thermodynamic route. It is shown that for a simple liquid-vapour interface the extra terms are needed when the pressure is found from the grand potential density. Only then are the Tolman length and the mechanically obtained Helfrich constants consistent with a parabolic fit to the interfacial tension.</p><p>The Helfrich constants of the simple liquid-vapour interface can be determined as a function of the intermolecular interactions. It is shown that a van der Waals density functional theory and its asymptotic expressions reproduce the Helfrich constants found from the lattice model in the vicinity of the critical point. Away from the critical point the square gradient of the van der Waals theory is not sufficient to account for the changes in the density profile across the interfacial region.</p><p>The phase behaviour of a bilayer membrane is considered in chapter 5. In order to model surfactants, the lattice gas model is extended to chain molecules. It is thought that each segment of the chain emerges from its predecessor such that the end of the chain can be considered as a diffusing particle obeying the Fokker-Planck equation. The grand potential density is again identified as an (ambiguous) local pressure. By choosing proper interactions, the formation of surfactant vesicles can be modelled.</p><p>For this study, the non-ionic surfactant C <sub>12</sub> E <sub>5</sub> is modelled. The interfacial tension of the vesicle is determined as a function of its radius. The resulting Helfrich constants determined both mechanically and from a parabolic fit to the interfacial tension are consistent. Keeping the hydrophobicity of the tail group constant, the Helfrich constants of the vesicle are obtained as a function of the hydrophilicity of the head group. It is found that for very hydrophilic head groups the bending modulus has an almost constant positive value, whereas the saddle-splay modulus is negative. This is thus interpreted that the membranes are relatively rigid. When the hydrophilicity decreases, the bending modulus becomes less positive and the saddle-splay modulus less negative. This renders less rigid bilayers, allowing large collective fluctuations, i.e. undulations, of the membranes. Hence, owing to steric hindrance, the spacing between a set of bilayers increases with decreasing hydrophilicity. For moderate hydrophilicity, the bending modulus is decreasingly positive. However, the saddle-splay modulus becomes positive which favours the formation of handles between the undulating bilayers. When the hydrophilicity is relatively low, the Helfrich constants seem to diverge because the head groups do not longer hydrated and the system phase-separates into surfactant and solvent rich phases. Since all these phases have been observed experimentally, it is concluded that the lattice model is a potentially valuable tool to study surfactant systems.</p><p>More information can be found on<A HREF="">the author's homepage</A>.</p>
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Lyklema, J., Promotor, External person
  • Leermakers, Frans, Promotor
  • Barneveld, P.A., Promotor, External person
Award date18 Jan 2000
Place of PublicationS.l.
Print ISBNs9789058081780
Publication statusPublished - 2000


  • thermodynamics
  • interface
  • capillary rise

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