Equilibrium Crystal Shapes by Virtual Work
MIKA REIVINEN
Tampere University Dissertations 513
MIKA REIVINEN
Equilibrium Crystal Shapes by Virtual Work
ACADEMIC DISSERTATION To be presented, with the permission of
the Faculty of Built Environment of Tampere University,
for public discussion in the auditorium RG202 of Rakennustalo, Korkeakoulunkatu 5, Tampere,
on 17 December, at 12 o’clock.
ACADEMIC DISSERTATION
Tampere University, Faculty of Built Environment Finland
Responsible supervisor and Custos
Professor Reijo Kouhia Tampere University Finland
Supervisor Professor emeritus Eero-Matti Salonen Aalto University Finland
Pre-examiners Professor Harm Askes University of Sheffield United Kingdom
University Lecturer Antti Kuronen University of Helsinki
Finland
Opponent Professor Anders Eriksson KTH Royal Institute of Technology Sweden
The originality of this thesis has been checked using the Turnitin OriginalityCheck service.
Copyright ©2021 author
Cover design: Roihu Inc.
ISBN 978-952-03-2194-9 (print) ISBN 978-952-03-2195-6 (pdf) ISSN 2489-9860 (print) ISSN 2490-0028 (pdf)
http://urn.fi/URN:ISBN:978-952-03-2195-6
PunaMusta Oy – Yliopistopaino Joensuu 2021
PREFACE
The research problem concerns the determination of equilibrium crystal shapes.
The outcome of the study is based on the research carried out first in the School of Engineering at Aalto University (former Helsinki University of Technology) and later at Tampere University.
I have been in a very privileged position since I have got extensive help and sup- port throughout the entire research process. Among many excellent teachers I have had during my studies in mechanics, I want to especially mention and give thanks to professor emeritus Eero-Matti Salonen. I am sincerely grateful for the supervision of this dissertation. His role has been significant in the present study as well as in many of my previous studies.
In addition, I wish to thank Dr.Tech. Vesa Vaskelainen and Dr.Tech. Heikki Junes for introducing the research problem, which reaches far out of the conven- tional research done by the community of classical mechanics. I also wish to thank Dr.Tech. Igor Todoshchenko for many helpful discussions considering faceting in connection with low temperature physics. Further, I thank M.Sc. Reeta Reivinen for her useful comments concerning the manuscript.
I also thank the pre-examiners, professor Harm Askes and docent Antti Kuro- nen, for getting acquainted with my research and for the comments I received. I am also grateful to Dr Minna Chudoba for her help to finish the work. Finally, I want to thank professor Reijo Kouhia for the possibility to finish the study at Tampere University.
Espoo, October 26, 2021 Mika Reivinen
ABSTRACT
This dissertation introduces a novel approach to the determination of equilibrium crystal shapes (ECS). The formulation is based on the principle of virtual work. The principle is generally used in mechanics, but the crystal surface physics seems to be a new application area. The emphasis is put on the proper detection of the facets appearing on the crystal’s surface. Both two-dimensional prismatic crystals and ar- bitrary shaped crystals in three dimensions are considered. The approach places re- strictions neither on the shape of the crystal nor on the shape of the substrate. The effect of gravity can be included as well. Thus, the virtual work approach seems to be the most general and straightforward basis for discrete methods in ECS problems.
The validity of the method has been justified by several numerical example cases.
TIIVISTELMÄ
Va¨ito¨skirjatyo¨ssa¨ on kehitetty uusi menetelma¨ tasapainossa olevien kiteiden muo- don ma¨a¨ritta¨miseksi. Tyo¨ssa¨ on sovellettu mekaniikassa yleisesti ka¨yto¨ssa¨ olevaa virtuaalisen tyo¨n periaatetta. Keskeisena¨ tavoitteena on ollut kiteen ja sita¨ ympa¨ro¨i- va¨n nesteen rajapintaan mahdollisesti muodostuvien tasomaisten alueiden havaitse- minen. Kehitettya¨ menetelma¨a¨ on sovellettu seka¨ kahdessa dimensiossa rajoittuen sa¨rmio¨ma¨isiin kiteisiin etta¨ yleisessa¨ kolmidimensioisessa tapauksessa. Menetelma¨ ei aseta rajoituksia tarkasteltavan kiteen tai alustan muodolle. Painovoiman vaikutus kiteen muotoon voidaan myo¨s ottaa huomioon. Virtuaalisen tyo¨n periaatteen sovel- taminen na¨ytta¨isi saavutettujen tulosten perusteella tarjoavan monipuolisen ja yleis- ka¨ytto¨isen ratkaisutavan pintaja¨nnitysongelmiin. Menetelma¨n toimivuus on var- mistettu numeeristen esimerkkitapausten avulla.
CONTENTS
1 Introduction . . . 13
1.1 Background and research environment . . . 14
1.2 Objectives and scope . . . 19
1.3 Research approach . . . 19
1.4 Research process and dissertation structure . . . 20
2 Theoretical Foundation . . . 23
2.1 Equilibrium crystal shapes and virtual work . . . 23
2.2 Discrete formulation . . . 24
2.3 Enhanced scheme . . . 25
3 Research Contribution . . . 27
3.1 Two-dimensional ECS problem . . . 27
3.2 Jump conditions . . . 28
3.3 Surface stiffness and torque terms . . . 28
3.4 ECS problem in three dimensions . . . 29
3.5 Penalty based solution . . . 29
3.6 Summary . . . 30
4 Discussion . . . 31
References . . . 33
Appendix A Errata . . . 37
Publication I . . . 43
Publication II . . . 61
Publication III . . . 75 Publication IV . . . 89 Publication V . . . 113
ORIGINAL PUBLICATIONS
Publication I M. Reivinen, E.-M. Salonen, I. Todoshchenko and V. Vaske- lainen. Equilibrium Crystal Shapes by Virtual Work.Journal of Low Temperature Physics170 (2013). Copyright Holder Springer, 75–90. DOI:10.1007/s10909-012-0666-8.
Publication II M. Reivinen and E.-M. Salonen. Surface tension problems with distributed torque.Contact and Surface, Siena. Ed. by J. de Hos- son and C. Brebbia. Copyright Holder Transactions of the Wes- sex Institute eLibrary. 2013, 75–85. DOI:10.2495/SECM130071. Publication III E.-M. Salonen and M. Reivinen. Notes on Surface Stiffness.
Journal of Low Temperature Physics174 (2014). Copyright Holder Springer, 76–86. DOI:10.1007/s10909-013-0914-6.
Publication IV M. Reivinen, E.-M. Salonen, I. Todoshchenko and V. Vaske- lainen. Equilibrium Crystal Shapes by Virtual Work in 3D.
Journal of Low Temperature Physics180 (2015). Copyright Holder Springer, 394–415. DOI:10.1007/s10909-015-1313-y. Publication V M. Reivinen, E.-M. Salonen, I. Todoshchenko and V. Vaske-
lainen. An Enhanced Facet Determination Scheme in 3D.
Journal of Low Temperature Physics186 (2017). Copyright Holder Springer, 63–73. DOI:10.1007/s10909-016-1648-z.
Author’s contribution
Publication I The author contributed to the text, designed the algorithms and carried out the numerical implementation and evaluation of the method.
Publication II The author contributed to the text.
Publication III The author took part in writing.
Publication IV The author contributed to the text, designed the algorithms and carried out the numerical implementation and evaluation of the method.
Publication V The author contributed to the text and implemented and evalu- ated the numerical algorithms.
1 INTRODUCTION
The study concerns the problem of equilibrium crystal shapes (ECS). According to Herring the problem is " . . . determining what shape a small crystal must assume if its surface free energy is to be a minimum for a given volume. The surface free energy of any body is an integral of the form
γ(n)dS (1.1)
extended over the surfaceSof the body, where the specific surface free energyγ is, for anisotropic bodies, a function of the orientation of the unit outward normalnat each surface point"[1]. This principle of minimum surface free energy for crystals and droplets was proposed by J.W. Gibbs already in 1878.
A crystal is a solid material whose constituents, atoms or molecules, are arranged in a ordered microscopic structure, forming a crystal lattice that extends in all di- rections. Due to this microscopic structure, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces (facets) with specific, characteristic orientations, and sharp edges between the faces. Examples of crystals are snowflakes, table salt and helium in low temperatures.
The process of crystal growth is called crystallization or solidification. As a crys- tal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces. The flat surfaces tend to grow larger and smoother.
The macroscopic shape of a solid may resemble a crystal, but the microscopic atomic arrangement determines whether the solid is a crystal or not. Polycrystal is such a non-crystalline solid consisting of many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics and ice. Amorphous solids, for example glass, are another class of matter which do not have a periodic structure of atoms.
1.1 Background and research environment
The research problem originates from scholar contacts between the Department of Mechanics and the Low Temperature Laboratory at Helsinki University of Technol- ogy. The ECS problem has a close connection to faceting, which is of great interest in crystal surface physics. Established methods for ECS problems are the Wulff con- struction and Landau-Andreev method.
Figure 1.1 A cross section
The basic setting consists of a crystal (C) surrounded by liquid (L) and resting in equilibrium on a solid surface wall (substrate) (W) (Fig. 1.1). The task is to determine the positionrof the interface surface between the crystal and the liquid phases. u1 andu2are surface coordinates. The shape of the wall surface is considered as given.
The interfacial energy density (surface tension) between crystal and liquid depends in general on the orientation of the interface surface. In the two-dimensional case, the orientation is given by an orientation angle, and in three dimensions by two spherical angles. The interface touches the wall along a boundary, and the position of some part of it may be fixed in advance depending on the problem statement.
Usually the position of the whole boundary is originally unknown. The contact angle on the boundary depends on the interfacial energy densities on the crystal- liquid, wall-crystal and wall-liquid interfaces. The volume of the crystal− or the area in two-dimensional case−is assumed to be given.
Wulff construction. Let us consider the two-dimensional Wulff construction fol- lowing the reference by Nozieres[2]. The solution was provided by Wulff in 1901 [3]. The local equilibrium condition for crystals with cylindrical geometry is
γ+γ
R =constant, (1.2)
whereγ˜=γ+γis the surface stiffness consisting of the surface energy densityγand of its second derivative with respect to the orientation angleθ(Fig. 1.2). The radius of the curvature is R = ds/dθ. In this two-dimensional case, with one curvature only, the equation can be written as
ds
dθ =γ˜(θ) (1.3)
The constant in (1.2) is absorbed in the scale of arch parameter s. The solution of (1.3) can be found by determining the envelope of the perpendicular to OM drawn through M. The mathematical details of the determination of the envelope are dis- cussed in[4].
Figure 1.2 The Wulff construction for cylindrical interfaces [2]
From Fig. 1.2 (OM=γ, OP=r)
r = γ(θ)
cos(φ−θ) (1.4)
Derivative of (1.4) is
r= dr
dθ = 1
cos(φ−θ)[γ−γtan(φ−θ)]. (1.5) Setting dr/dθ=0 gives the first equation of (1.64) in[2]
φ=θ+arctan(γ
γ) (1.6)
From equation (1.6) and Fig 1.2 (MP=γ) r =
γ2+γ2 (1.7)
Radius of curvature is now
R= ds dθ =
r2+r2φ2. (1.8)
From (1.8), (1.4), (1.5) and derivative of (1.6) it follows
R=γ+γ. (1.9)
The consideration resulted in the local equilibrium condition. A typical polar plot of surface free energy for a crystal and the Wulff construction based on it is shown in Fig. 1.3[1].
Landau-Andreev method. For three-dimensional geometries applies the Landau- Andreev formulation[5]. The crystal-liquid interface is described in Cartesian co- ordinates,z=z(x,y). The notations
p= ∂z
∂x, q=∂z
∂y (1.10)
are introduced for the derivatives determining the direction of the surface at each point. The surface tension can now be expressed as a function γ = γ(p,q). The equilibrium shape is determined by the condition
γ(p,q)
1+p2+q2dxdy=min, (1.11)
Figure 1.3 A typical polar plot and corresponding Wulff construction [1]
and the constancy of volume produces the auxiliary condition
zdxdy=const. (1.12)
This variational problem leads to the differential equation
∂
∂x
∂ f
∂ p+ ∂
∂y
∂ f
∂q =2λ, (1.13)
where
f(p,q) =γ(p,q)
1+ p2+q2 (1.14)
is the interface free energy per projected area. The solution of (1.13) is[5]
f =λ(p x+q y−z), (1.15)
which represents the equation of the envelope of the family of planes p x+q y−z= 1
λγ(p,q)
1+p2+q2, (1.16)
where pand q act as parameters. This result can also be interpreted as the Wulff construction in three dimensions[2],[5].
Further studies on ECS. It is interesting to note that also many further studies concern ECS and the underlying Wulff constructions. While the Wulff construction is a solution of a macroscopic optimization problem based on an a priori knowledge of the orientation dependent surface tension, the results about a rigorous micro- scopic justification of the Wulff construction for the two-dimensional Ising model at low temperatures and under periodic boundary conditions is discussed by Do- brushin et. al. in[6]. Ising model consists of discrete variables that represent mag- netic dipole moments of atomic spins that can be in one of two states, and the model can be used in studies of phase transitions.
Approximate solutions for ECS in the case of lattice gas with orientation depend- ing surface tension are presented by Miracle-Sole in[7]. The results are applied to study the shape of the equilibrium crystal and, in particular, the shape of the facets of the crystal. The lattice gas model is widely used in physical chemistry. The ba- sic idea of the model is that the volume available to the fluid is divided into cells of molecular size, which are usually arranged in a regular lattice.
Wulff shape in statistical mechanics and the minimization of the total surface free energy associated to the crystal-medium interface is the object of the mathematically oriented article by Miracle-Sole in[8].
A prominent section of the ECS research concerns the faceting transition accom- panied by roughening transition of surfaces. The faceting transition is a shape tran- sition, which takes place at the roughening transition temperatureTR. For tempera- tures lower thanTR, the facet may exist. The transition is studied by developing the (temperature and orientation dependent) free energy expansion for a vicinal surface of a facet. The vicinity means here that the surface orientation is close to the facet.
The orientation is often given by the surface gradient[9],[10],[11],[12],[13],[14], [15]. In addition to the orientation dependent approximation for the surface energy density, the studies often have also an atom scale aspect to the crystal growth and shape.
The properties of nanostructured materials depend crucially on the behaviour of grain boundaries, and the transition between faceted and rough grain boundaries can drastically change the properties of a whole polycrystal. The construction of three- dimensional Wulff diagrams is applied to describe the grain boundaries faceting and roughening behaviour by Straumal et.al. in[16].
1.2 Objectives and scope
The aim of the study is to derive a general-purpose virtual work based formulation for the minimisation of the surface free energy integral (1.1). The study problem originated from the requirement to determine the equilibrium shape of helium crys- tals in low temperatures, but it is relevant always when the global presentation of the surface energy densityγas a function of the surface orientation, and the volume of the domain, are given. Attention has been paid especially to detect the faceting taking place on the crystal’s surface.
The objective has also been to overcome certain shortcomings associated with the established methods, which means for example the ability to take into the account the effect of gravity. Also, the treatment of crystals with double-valued surfaces in vertical coordinates, i.e. the crystal’s surface cannot be expressed as a single-valued functionz=z(x,y), had to be included into the formulation.
The effect of temperature is not considered in the present study, although the surface energy density, and therefore the shape of crystal, depends on the tempera- ture: As the temperature increases, there is a tendency that originally sharp crystal corners and edges between facets become smoother[9] [17] [18]. For temperatures higher than the roughening transition temperatureTR, the facets do not exist.
The equilibrium crystal shape is sought for crystals having fixed volume, and thus the process of crystal growth falls outside the scope of the study.
1.3 Research approach
According to[2], the interface problem can be considered as a "chemical" problem, but in the Publications I−V, however, the shape determination problem has been treated solely as a mechanical problem employing familiar concepts and methods of classical mechanics, especially the principle of virtual work. Instead of atomic scale,
the crystal interface is treated as a continuum, like the fluid interface.
The virtual work based results have been formally derived in a similar way as in various areas of structural mechanics, although some of the assumptions of solid mechanics are not valid with crystals. Contrary to conventional problems of me- chanics, the real displacement of a single material particle on the crystal-fluid inter- face cannot basically be followed. This fact does not restrain the application of the principle of virtual work since real displacements are not considered.
In mechanics, and especially in structural mechanics, the virtual displacement δr is interpreted as a variation of a displacement vectoruassociated to a certain material particle, i.e. δr= δu. Thus, for example, "the virtual strain" can be in- terpreted as a variation of the ordinary strainε=∂u/∂x, which can be written as δε=δ(∂u/∂x) =∂ δu/∂x according to the rules of calculus of variation. The difference between problems of ordinary mechanics and crystals comes from the fact that the displacements associated to the particles are not available, and the manipula- tions have to be done using only the "weight functions"δr. They can be interpreted as "small" displacements, and the corresponding virtual strains are evaluated using the theory of small displacements, but not strictly speaking by doing a variation of any displacement expression. To emphasize the difference between the present ap- proach and the ordinary virtual work applications, the termvi r t ual mov e me nt instead of the usual termvi r t ual d i s p l ac e me nt has frequently been used in the Publications.
The volume constraint of the crystal is taken into consideration using the La- grange multiplier method. The corresponding Lagrange multiplier is interpreted as the pressure in the crystal, and it is an unknown constant. The known pressure in the liquid is taken according to the hydrostatic pressure distribution depending on the assumed constant density of the liquid.
1.4 Research process and dissertation structure
The first steps towards the present formulation were taken already in[19], where the equilibrium shape of the liquid-vapour interface is considered in two dimensions, and special emphasis is put on the proper representation of the virtual work terms emerging from the free boundary. However, a constant surface tension is assumed.
Further, the case where the surface tension is no more constant but depends
in a prescribed way on the orientation of the interface is discussed in [20]. The non-constancy of the surface tension means that for equilibrium certain distributed torque must act on the interface. Corresponding relation between torque and surface tension is derived.
The variation of plane areas is considered in[21]. The need for this appears in the determination of positions of interfaces between different material phases when the principle of virtual work is applied, and the cross-sectional area appears as a kine- matic constraint.
Crystals on non-horizontal and arbitrary shaped wall are considered in[22].
The publication[23]continues the study presented in[19]and[20]concerning the case of discontinuous distributed torque.
Finally, an enhanced facet determination scheme in two-dimensional case is in- troduced in[24].
The dissertation consists of five additional research papers concerning the ECS problem in two- and three-dimensional cases. Publication I examines the non-constan- cy of the surface tension in crystals and presents a formulation for the determination of equilibrium crystal shape in two-dimensional case, where the surface energy den- sity depends on only one orientation angle. Consequently, the formulation applies only for prismatic crystals. Corresponding discrete solution method is described.
Publication II extends the non-constancy of the surface tension to three dimen- sions. The non-constancy involves again the torque terms in the formulation. Dis- continuities in torque may result in edges on the interface. Corresponding jump conditions at the edges are derived in three dimensions, and the conditions in two dimensions follow as a special case.
Publication III considers certain terms associated with the concept of surface stiff- ness based on the reference[2], and introduces a straightforward approach for the determination of actual crystal shapes in three dimensions.
Publication IV extends the formulation presented in Publication I to three di- mensions and describes the corresponding discrete solution method.
Finally, Publication V presents an enhanced facet determination scheme to over- come difficulties encountered in numerical examples treated in Publication IV.
Equilibrium crystal shapes and virtual work, the discrete method and enhanced facet determination scheme are discussed in Chapter 2 of this overview. Theoreti- cal and computational results of the study are reviewed in Chapter 3. Concluding
remarks are presented in Chapter 4.
2 THEORETICAL FOUNDATION
The treatment of the ECS problem in this study is based on the use of the principle of virtual work [25]. In ECS problem the task is to find the position of the inter- face (excess layer) between crystal and liquid phases. It is assumed that a continuum surface with infinitesimal thickness forms the interface, and the solid (crystal) and the liquid phases both consist of the same material, e.g. helium. For real crystals, the interface between different phases changes its position due to melting or solid- ification. Thus, the interface in a new position consists generally no more of the material particles of the old position, and hence it cannot be considered as a closed material system. In the present approach, however, any real movements of the phase interface can be ignored. From the mathematical point of view, the virtual displace- ments are just weight functions of a weak form, and they do not represent any real displacements. Thus, the material system of the interface, bounded by a solid wall, is considered at its each assumed configuration as a system, to whom the principle of virtual work is applied at each step of the iteration process.
In this chapter, the principle of virtual work and the corresponding discrete for- mulation are shortly reviewed. Finally, an enhanced facet determination scheme is presented.
2.1 Equilibrium crystal shapes and virtual work
For the ECS problem the virtual work equation can be written as
δW ≡δWint+δWext+δWbound=0. (2.1) The equation is valid for any virtual displacement of the interface. The first term (int) refers to the virtual work of the internal forces due to surface tension. The second term (ext) refers to the external forces, i.e. the virtual work from pressure
difference between crystal and liquid, and the work done by the distributed torque acting on the crystal’s surface. The last term (bound) stands for the virtual work associated with the free boundary under the influence of interfacial energy densities on the wall-crystal and wall-liquid interfaces.
In two-dimensional case, the torque m required for the equilibrium is propor- tional to the derivative of surface tension with respect to direction angleψ
m=−γ(ψ). (2.2)
The contributions to the virtual work from internal and external forces and from boundary are discussed in detail in the two-dimensional case in Publication I, and in three dimensions in Publications III and IV.
2.2 Discrete formulation
The discretization is performed following the ordinary finite element technique, e.g.
[26]and[27]. In two dimensions, the discretization makes use of two-noded line ele- ments, while three-noded triangular elements are employed in the three-dimensional case. For a triangular element, the generalized coordinates used are (normally) the nodal Cartesian coordinatesxi,yi and zi (i =1,2,3). For the free boundary being located, for example, on the planez = c ons t ant, coordinate zi is excluded. The virtual work contribution obtains the form
δW =Xiδxi+Yiδyi+Ziδzi, (2.3) where Xi, Yi and Zi are generalized forces, andδxi, δyi and δzi corresponding virtual displacements. Summation convention is applied.
The generalized forces have also contributions from internal and external forces and of the terms emerging from the free boundary. Detailed expressions for the generalized forces in the two-dimensional case are given in Publication I, and the derivation of the generalized forces in the three-dimensional case is discussed in some extent in Publication IV.
In two dimensions, the additional system equation concerning the area constraint A=AC with the unknown pressure inside the crystal appears. Correspondingly, the three-dimensional case retains a volume constraint,V =VC.
When the total discrete system is considered, the generalized coordinates (or move- ments) are denoted byqi. They are defined in principle anew for each current sys- tem configuration. Normally, three movementsΔxk,Δyk andΔzkmeasured from a current generic nodekinside the mesh are employed. However, at a free boundary node, only two or just one movements are applied.
The virtual work for the interface model with respect to a current configuration obtains the form
δW =
Ndof
i=1
Qiδqi, (2.4)
whereQiis thei:th generalized force corresponding to generalized coordinate, and Ndofis the total number of coordinates. The generalized forces must vanish, and the volume constraint must be satisfied. Thus, the system equations are
Qi=0,i=1, ...,Ndof,
V =VC. (2.5)
The solution of (2.5) is strongly non-linear, and the system equations are solved iter- atively by a Newton-Raphson solution method version as discussed in Publication I. For a proper convergence, a sufficient small reduction coefficient for the displace- mentsqiat the early stage of the iteration process was sometimes applied.
2.3 Enhanced scheme
In three dimensions, the surface tensionγgenerally depends on two spherical orien- tation angles,γ =γ(ψ,ϕ). A facet may appear on the crystal surface when there is singularity in the second derivative of the surface tension with respect to the angles ψorϕ, which means that there is a jump in torque acting on crystal’s surface. It is found that numerical difficulties result especially from singularities appearing simul- taneously with respect to both orientation angles. To improve the ability to model planar facets properly, penalty terms in the neighbourhood of the singularities giv- ing contributions to the generalized forcesXi,Yi,Ziin virtual work expression (2.3) are applied. Let the normal direction to a facet be given by the values of orientation anglesψ=ψi andϕ =ϕi. When, during the iterations, the triangular element of
the discrete model comes almost parallel to the actual facet, the following quadratic penalty term is applied
pe= 1
2α(ψ−ψi)2+1
2α(ϕ−ϕi)2. (2.6)
A mechanical interpretation to this penalty term is the potential energy of two ro- tational springs having spring constantsα. Publication V deals with penalty terms in the three-dimensional case applying the theory presented in[28]. Penalty terms in two dimensions are discussed in more detail in[24].
3 RESEARCH CONTRIBUTION
Publication I concerns the ECS problem in two dimensions, for which a virtual work based formulation and corresponding discrete method is derived. Publications II - V extend the treatment to three dimensions.
3.1 Two-dimensional ECS problem
Publication I considers prismatic crystals, for which the virtual work terms are de- rived in detail. Special attention is put on the terms emerging from the free bound- ary. The corresponding strong forms, i.e. the field equations and boundary condi- tions, are deduced from the virtual work terms. The discrete solution method, based on line elements, is presented.
The numerical example cases are mostly borrowed from[29], and they are in- tended to give an idea about the working of the discrete method. The numerical examples cover both crystals with constant interfacial energy density and crystals with surface orientation dependent interfacial energy density. The constant interfa- cial energy density means that no torque appears on the crystal’s surface, and con- sequently, the smooth solution resembles in shape a soap bubble. Thus, the well- known Laplace equation is valid at the liquid-vapour interface, and the contact angle obeys the Young’s equation[30]. Further, the crystals having orientation dependent interfacial energy density demand a certain torque, which may also have jumps in its value. These singular points−the cusps onγ-plot−correspond facets on actual crystals. It was found, that due to those discontinuities, a proper convergence can- not be achieved resulting in poor crystal shapes. However, sufficient convergence can be obtained, if the jump in torque is replaced by a linear approximation in a small neighbourhood (one or two percent of the element size) of the singular point.
Consequently, the facet appears as a shallow arch instead of a straight line in the numerical solution.
Some of the two-dimensional numerical example cases considered in Publication I have also analytical solutions, otherwise the validation has been performed using for example the Wulff’s construction. The first example cases in Publication I having constant interfacial energy densities showed clearly the effect of the gravity on the equilibrium shape. The effect of gravity was studied also in the case of non-constant surface tension.
3.2 Jump conditions
Publication II continues the study on the distributed torque and the jump condi- tions relating to non-constant surface tension. The distributed torque is required for the equilibrium, and if it has discontinuities in its value, the solution is neces- sarily no more smooth, but edges can emerge on the interface. The treatment in Publication II is based on the weak form of the virtual work equation arriving fi- nally at the strong form, which connects the edge geometry with the surface tension and the torque. The corresponding jump conditions at the edges are derived in the three-dimensional case, and the jump conditions in two dimensions are obtained as a special case. Finally, the demonstration using the discrete formulation and numerical example presented in Publication I showed that the jump conditions are satisfied.
3.3 Surface stiffness and torque terms
Publication III deals with the local equilibrium condition in terms of the surface stiffness, which according to Nozieres[2]is the most essential concept in studies of equilibrium crystal shapes. In the two-dimensional case the concept is quite clear, but in three dimensions there is a need to provide some illumination on the role of the so-called reference angles. In Publication III , the study of surface stiffness relating to the reference angles is widely based on the use of differential geometry according to Kreyszig[31]. It was found much more straightforward to base the crystal shape determination on a virtual work formulation where the torque expres- sions are kept as such, and not eliminated to produce the surface stiffness. Detailed terms for the torque components in the three-dimensional case were also derived.
For comparison, an alternative approach[10], based on the Landau-Andreev for- malism with the crystal surface given asz= z(x, y), is considered.
3.4 ECS problem in three dimensions
Publication IV extends the treatment to three dimensions by making use of the re- sults obtained in Publication III . The formulation is again based on the principle of virtual work, and the corresponding discrete method is presented. The numeri- cal examples contain crystal with constant surface tension, an axisymmetric crystal where the surface tension depends only on one direction angle, and finally a case where the surface tension depends on two direction angles. Correct crystal shapes with possible facets were obtained with reasonable accuracy during the iteration pro- cess. However, difficulties were encountered to detect the facets properly when the discontinuities in torque occurred simultaneously with respect to both orientation angles.
3.5 Penalty based solution
Publication V introduces an enhanced scheme making use of penalty terms to im- prove the ability to model facets more accurately at those particular points where the singularities in torque appear. The derivation of the corresponding penalty terms is discussed in detail. The enhanced formulation still employs the principle of virtual work developing the three-dimensional formulation further. The corresponding dis- crete method is also presented.
It should be emphasized that the penalty method is purely a numerical device to increase the accuracy of the facet determination, and the penalty terms are applied in calculations only near to the singular points of the surface tension, i.e. the ori- entation of the element of the discrete model is close to the one of the actual facet.
The only additional input data consist of direction angles corresponding to possible cusps in the γ-plot, i.e. the polar plot of the surface energy density. The working of the enhanced method is demonstrated by a specific example case, in which the singularities occur simultaneously with respect to both orientation angles[32].
3.6 Summary
A novel virtual work based approach for the solution of ECS problems is discussed and the corresponding discrete formulation evaluated in Publications I - V. In the community of crystal surface physics, the principle of virtual work seems to have been an unfamiliar concept so far.
A possible alternative approach could be the configurational mechanics, which considers the energy variations that go along with changes of the material configu- ration. Configurational forces are then energetically dual to these configurational changes[33]. A comprehensive presentation of the concept can be found in[34]. Configurational forces in the context of the evolution of two-phase materials, frac- ture mechanics and finite element method are discussed for example in[35], [36], [37],[38],[39],[40],[41]. The present study, however, begins with the two-dimen- sional problem, which can be solved applying quite elementary mechanics and stan- dard Newtonian forces. The two-dimensional formulation has then been straight- forwardly extended to three dimensions, but any reason to make use of the configu- rational scheme in this context has not been detected.
In all numerical example cases the initial configurations were far from the final one with respect to geometry and crystal volume, and still a satisfactory convergence was achieved. The discrete formulation was found to be able to detect automatically facets and the edges between facets, and the final crystal shapes were in good agree- ment with the results obtained by comparative methods. As the virtual work ap- proach does not need, say, potential energy concepts etc. [42], it seems to be the most general and straightforward basis for discrete methods in ECS problems.
The effect of gravity is also included to the novel formulation. The gravity creates a vertical pressure gradient in the liquid surrounding the crystal and may have a considerably effect on the crystal’s shape on a scale larger than the capillary length.
In the case of smaller crystals, the interfacial free energy dominates the shape, and the effect of gravity can be neglected. In practice, however, to resolve the fine structure of true ECS, also studies on macroscopic crystals are required, and therefore the possibility to take the effect of gravity into account supports the experimentation with crystals[43].
4 DISCUSSION
The outcome of this study is a novel formulation for the solution of ECS problems.
The formulation is based on conventional mechanics and the principle of virtual work. However, it is by no means obvious that the use of principle of virtual work is justified in connection with the ECS problems as discussed in Section 1.3.
The focus is put on the ability to detect automatically possible facets appearing on the crystal’s surface. The global presentation for the surface energy density on the crystal-liquid interface is assumed to be known. It is generally given by two spherical orientation angles.
The shape determination problem is strongly non-linear and, for simplicity, the ideas were first evaluated in two dimensions. The two-dimensional approach applies only to prismatic crystals while the three-dimensional formulation covers arbitrary shaped crystals. The crystal shape solutions have brought out the possible angular behaviour on the crystal surface. However, attention had to pay to the singularities in the derivatives of the energy density. In the discrete method, the discontinuities in torque have been replaced by a linear approximation in a small neighbourhood of the singular point. In addition, difficulties were encountered when the discontinu- ities appeared simultaneously with respect to both orientation angles. To improve the treatment in this particular case, an enhanced penalty based facet determination scheme has been introduced.
The results suggest that the novel approach has a broad application area. Con- trary to the Landau-Andreev formulation, the present formulation is not restricted only to cases where the solution, i.e. the shape of the crystal surface, is single val- ued in vertical coordinate. Further, the substrate under the crystal has not to be horizontal, but also arbitrary shaped substrates are allowed. The effect of gravity is included to the present formulation, which supports experimentation with crystals.
In conclusion, the virtual work based approach seems to be the most general basis for discrete methods in surface tension problems. Further, potential applications of
the approach may also be found outside the ECS scene. Such may be nano-clusters since the shape of nano-clusters may affect their physical properties, and therefore some ideas presented in the present study may be usable in that connection.
The problem size, i.e. the number of degrees of freedom in all example cases, even in three dimensions, has been kept quite low. Yet, the iterative solutions have been time consuming. In this study, however, no special attention has been paid to the computational efficiency. The computer program used to solve the numerical example cases has been of a preliminary nature mainly to confirm the correctness of the theory. For future research carried out on larger problems, a more powerful program version is therefore required. This may be the case at certain temperatures when there are, for example, partly smooth surfaces and partly sharp edges between the facets. In smooth parts there is not only the crystal surface continuous, but so is its slope too. A large discrete model is needed to describe this kind of setting properly.
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A ERRATA
Publication I
Page 78, the formula (6):
h=
dx du
2
+
dz du
2
.
should be
h=
dxˆ du
2
+
dzˆ du
2
.
Page 79, the formula (14):
S
−γ R+dm
ds +pC− pL
n·δrds+
S
dγ ds + m
R
t·δrds + −γ
−t+ nˆ sinθ
+
γLW−γCW n
sinθ+mn ·δr
s=sa
+ −γ t+ nˆ
sinθ +
γLW−γCW n
sinθ−mn ·δr
s=sb
.
should be
S
− γ R+dm
ds +pC−pL
n·δrds+
S
dγ ds +m
R
t·δrds + −γ
−t+ nˆ sinθ
+
γLW−γCW n
sinθ+mn ·δr
s=sa
+ −γ t+ nˆ
sinθ +
γLW−γCW n
sinθ−mn ·δr
s=sb
=0.
Page 87:
. . . contact anglesψ=π/2
should be
. . . contact anglesθ=π/2
Publication II
Page 78, the formula (3):
δWext=
s
(pC−pL)n·δrds−
s
mn·dδr ds ds.
should be
δWext=
s
(pC−pL)n·δrds−
s
mn·dδr ds ds.
Page 78:
The scale factorh=
(dx/du)2+ (dz/du)2
should be
The scale factorh=
(dxˆ/du)2+ (dˆz/du)2
Page 78, the formula (6):
δWbound=−γnˆ·δr sinθ
s=sa
+ (γLW−γCW)n·δr sinθ
s=sa
−γnˆ·δr sinθ
s=sb
+ (γLW−γCW)n·δr sinθ
s=sb
,
should be
δWbound=−γnˆ·δr sinθ
s=sa
+ (γLW−γCW)n·δr sinθ
s=sa
−γnˆ·δr sinθ
s=sb
+ (γLW−γCW)n·δr sinθ
s=sb
,
Publication III
Page 78, after the formula (3):
Vectornis the unit vector . . .
should be
Vectornis the unit normal vector . . .
Publication IV
No errors detected.
Publication V
No errors detected.
PUBLICATIONS
PUBLICATION I
Equilibrium Crystal Shapes by Virtual Work
M. Reivinen, E.-M. Salonen, I. Todoshchenko and V. Vaskelainen
Journal of Low Temperature Physics170.(2013), 75–90. Copyright Holder Springer DOI:10.1007/s10909-012-0666-8
Publication reprinted with the permission of the copyright holders
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