Generalized Dynamics of Soft-Matter Quasicrystals: Mathematical models and solutions

192 Pages·2017·5.45 MB·English

by  Tian-You Fan (auth.)

Springer Series in Materials Science 260
Tian-You Fan
Generalized
Dynamics of
Soft-Matter
Quasicrystals
Mathematical Models and Solutions
Springer Series in Materials Science
Volume 260
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Tian-You Fan
Generalized Dynamics
of Soft-Matter Quasicrystals
Mathematical Models and Solutions
123
Tian-You Fan
Beijing Institute of Technology
Beijing
China
ISSN 0933-033X ISSN 2196-2812 (electronic)
SpringerSeries inMaterials Science
ISBN978-981-10-4949-1 ISBN978-981-10-4950-7 (eBook)
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Preface
Aswell-known quasicrystalswith 12-foldsymmetryobservedsince2004inliquid
crystals, colloids, polymers and nanoparticles have been received a great deal of
attention.Inparticular,18-foldsymmetryquasicrystalsincolloidswerediscovered
in2011.Morerecentlythequasicrystalswith12-foldsymmetrywerealsofoundin
giant surfactants. The formation mechanisms of these kinds of quasicrystals are
connected closely with self-assembly of spherical building blocks by supramole-
cules, compounds, block copolymers and so on and are quite different from that
of the metallic alloy quasicrystals. They can be identified as soft-matter qua-
sicrystalsexhibitingnaturesofquasicrystalswithsoft-mattercharacters.Softmatter
liesinthebehaviourofintermediatephasebetweensolidandsimplefluid,whilethe
natureofquasicrystalsexhibitsimportanceofsymmetryastheyarehighlyordered
phase. These features are very complex yet extremely interesting and attractive.
Hence,theyhaveraisedagreatdealofattentionofresearchersinphysics,chemistry
and materials science.
All the observed soft-matter quasicrystals so far are two-dimensional qua-
sicrystals. It is well known that two-dimensional quasicrystals consist of only two
distincttypesfromtheangleofsymmetrytheory,onebeing5-,8-,10-and12-fold
symmetries, the other being 7-, 9-, 14- and 18-fold according to the symmetry
theory. Therefore, two terminological phrases can be defined such as the first and
second kinds of two-dimensional quasicrystals respectively. The two-dimensional
solid quasicrystals observed so far belong to the first kind ones only, while
soft-matter quasicrystals discovered up to now can be in both kinds. This may
implythatmanynewtypesofsoft-matterquasicrystalsinadditiontothosewith12-
and 18-fold symmetries may be observed in the near future. Hence, the interdis-
ciplinary studies on soft-matter quasicrystals present great potential and hopeful
research topics.
However, some difficulties exist in studying those new phases due to the
complexityoftheirstructuresandlack offundamental experimentaldata including
thematerialconstantstodate.Furthermore,thetheoreticalstudiesarealsodifficult.
Forexample,thesymmetrygroupsofsoft-matterquasicrystalsobservedorpossibly
to be observed have not yet been well investigated although there are some work
v
vi Preface
being done (the details are not be included in the book). In conjunction with this
issue, the study on constitutive laws for phasons and phonon–phason coupling are
still difficult.
In spite of these problems, there are potential efforts to undertake the study on
these topics.For example,thesoft-matter quasicrystals asa new ordered phase are
connected with broken symmetry or symmetry breaking, like those discussed in
solid quasicrystals. Thus, the elementary excitations such as phonons and phasons
are important issues in the study of quasicrystals based on the Landau phe-
nomenological theory. For soft-matter quasicrystals, furthermore, another elemen-
tary excitation, i.e. the fluid phonon must be considered besides phonons and
phasons. According to the Landau school, liquid acoustic wave is fluid phonon
(refer to Lifshitz EM and Pitaevskii LP, Statistical Physics, Part 2, Oxford:
Butterworth-Heinemann, 1980). This is suitable for describing the liquid effect of
soft-matter quasicrystals, which can be seen as complex liquids or structured liq-
uids. The elementary excitations—phonons, phasons and fluid phonon—and their
interactionsconstitutethemainfeatureofthenewphase.Theywillbediscussedas
amajorissueinthebook.Theconceptofthefluidphononisintroducedinthestudy
ofquasicrystalsforthefirsttime.Relatedtothis,theequationofstateshouldalsobe
introduced. With these two key points and referencing the hydrodynamics of solid
quasicrystalsthedynamicsofSoftMatterquasicrystalscanbeestablished,butwith
an important distinction compared with that of solid quasicrystals. The present
hydrodynamicscannotbelinearizedduetothenonlinearityofequationofstate.To
overcome the difficulty arising from other aspects in theory, we can draw from
study of solid quasicrystals (For example, Lubensky TC, Symmetry, elasticity and
hydrodynamics in quasiperiodic structures, in Introduction to Quasicrystals, ed by
Jaric M V, Boston: Academic Press, 199–289, 1988; Hu CZ et al, Symmetry
groups, physical property tensors, elasticity and dislocations in quasicrystals,
Rep.Prog.Phys.,63(1),1-39,2000;FanTY,MathematicalTheoryofElasticityof
Quasicrystals and Its Applications, Beijing: Science Press/Heidelberg:
Springer-Verlag, 1st edition, 2010, 2nd edition, 2016). This shows that the theory
ofsolidquasicrystalsisabasisforthepresentdiscussion,whichprovidesaninitial
glimpse from the viewpoint of quantitative analysis to the rich phenomena of
soft-matter quasicrystals.
Some applications are given by describing the matter distribution, deformation
and motion of soft-matter quasicrystals. The mathematical principle and its appli-
cationsrequiretheassistanceofotherareasofknowledge,apartofwhichisbriefly
listed in the first six chapters of the book (more details can refer to Chaikin J and
Lubensky TC, Principles of Condensed Matter Physics, New York: Oxford
UniversityPress,1995),andtheothersareintroducedinduethecomputation.The
computational results are preliminary and very limited so far, but verified partially
themathematicalmodel,andexploredincertaindegree todistinguish thedynamic
behaviour between soft-matter and solid quasicrystals to some extent. In addition,
the specimens and flow modes adopted in the computation might be intuitive,
observable and verified easily. However, it does not mean that they belong to the
most important samples.
Preface vii
The author thanks the National Natural Science Foundation of China and
Alexander von Humboldt Foundation of Germany for their support over the years
and Profs. Messerschmidt U in Max-Planck Institut fuer Mikrostrukturphysik in
Halle, Trebin H.-R. in Stuttgart Universitaet in Germany, Lubensky T.C. in
University of Pennsylvania, Cheng, Stephen Z.D. in University of Akron in USA,
WensinkH.H.inUtrechtUniversityandinHolland,LiXian-FanginCentralSouth
University and Chen Wei-Qiu in Zhejiang University in China for their cordial
encouragement and helpful discussions.
Beijing, China Tian-You Fan
December 2016
Contents
1 Introduction to Soft Matter in Brief .... .... .... .... ..... .... 1
References. .... .... .... ..... .... .... .... .... .... ..... .... 4
2 Discovery of Soft-Matter Quasicrystals and Their Properties . .... 5
2.1 Soft-Matter Quasicrystals with 12- and 18-Fold
Symmetries... .... ..... .... .... .... .... .... ..... .... 5
2.2 Characters of Soft-Matter Quasicrystals .. .... .... ..... .... 8
2.3 Some Concepts Concerning Possible Hydrodynamics on
Soft-Matter Quasicrystals . .... .... .... .... .... ..... .... 9
2.4 First and Second Kinds of Two-Dimensional Quasicrystals .... 9
2.5 Motivation of Our Discussion in the Book.... .... ..... .... 11
References. .... .... .... ..... .... .... .... .... .... ..... .... 11
3 Review in Brief on Elasticity and Hydrodynamics of Solid
Quasicrystals .. .... .... ..... .... .... .... .... .... ..... .... 13
3.1 Physical Basis of Elasticity of Quasicrystals, Phonons and
Phasons.. .... .... ..... .... .... .... .... .... ..... .... 13
3.2 Deformation Tensors .... .... .... .... .... .... ..... .... 16
3.3 Stress Tensors and Equations of Motion.. .... .... ..... .... 17
3.4 Free Energy Density and Elastic Constants.... .... ..... .... 19
3.5 Generalized Hooke’s Law. .... .... .... .... .... ..... .... 21
3.6 Boundary Conditions and Initial Conditions... .... ..... .... 22
3.7 Solutions of Elasticity.... .... .... .... .... .... ..... .... 23
3.8 Generalized Hydrodynamics of Solid Quasicrystals . ..... .... 23
3.8.1 Viscosity of Solid .... .... .... .... .... ..... .... 24
3.8.2 Generalized Hydrodynamics of Solid Quasicrystals.... 25
3.9 Solution of Generalized Hydrodynamics
of Solid Quasicrystals.... .... .... .... .... .... ..... .... 26
3.10 Conclusion and Discussion.... .... .... .... .... ..... .... 27
References. .... .... .... ..... .... .... .... .... .... ..... .... 27
ix
x Contents
4 Equation of State of Some Structured Fluids . .... .... ..... .... 31
4.1 Overview on Equation of State in Some Fluids .... ..... .... 31
4.2 Possible Equations of State.... .... .... .... .... ..... .... 33
4.3 Applications to Hydrodynamics of Soft-Matter
Quasicrystals.. .... ..... .... .... .... .... .... ..... .... 33
References. .... .... .... ..... .... .... .... .... .... ..... .... 34
5 Poisson Brackets and Derivation of Equations of Motion
of Soft-Matter Quasicrystals... .... .... .... .... .... ..... .... 35
5.1 Brown Motion and Langevin Equation... .... .... ..... .... 35
5.2 Extended Version of Langevin Equation . .... .... ..... .... 35
5.3 Multivariable Langevin Equation, Coarse Graining.. ..... .... 36
5.4 Poisson Bracket Method in Condensed Matter Physics.... .... 37
5.5 Application to Quasicrystals... .... .... .... .... ..... .... 39
5.6 Equations of Motion of Soft-Matter Quasicrystals .. ..... .... 39
5.6.1 Generalized Langevin Equation.. .... .... ..... .... 40
5.6.2 Derivation of Hydrodynamic Equations of
Soft-Matter Quasicrystals... .... .... .... ..... .... 40
5.7 Poisson Brackets Based on Lie Algebra.. .... .... ..... .... 44
References. .... .... .... ..... .... .... .... .... .... ..... .... 48
6 Oseen Flow and Generalized Oseen Flow .... .... .... ..... .... 51
6.1 Navier–Stokes Equations . .... .... .... .... .... ..... .... 51
6.2 Stokes Approximation ... .... .... .... .... .... ..... .... 52
6.3 Stokes Paradox.... ..... .... .... .... .... .... ..... .... 52
6.4 Oseen Modification. ..... .... .... .... .... .... ..... .... 52
6.5 Oseen Steady Solution of Flow of Incompressible Fluid Past
Cylinder . .... .... ..... .... .... .... .... .... ..... .... 53
6.6 GeneralizedOseenFlowofCompressibleViscousFluidPasta
Circular Cylinder .. ..... .... .... .... .... .... ..... .... 60
6.6.1 Introduction..... .... .... .... .... .... ..... .... 60
6.6.2 Basic Equations.. .... .... .... .... .... ..... .... 60
6.6.3 Flow Past a Circular Cylinder... .... .... ..... .... 61
6.6.4 Quasi-Steady Analysis—Numerical Solution..... .... 62
6.6.5 Conclusion and Discussion . .... .... .... ..... .... 66
References. .... .... .... ..... .... .... .... .... .... ..... .... 67
7 Dynamics of Soft-Matter Quasicrystals with 12-Fold Symmetry..... . 69
7.1 Two-Dimensional Governing Equations of Soft-Matter
Quasicrystals of 12-Fold Symmetry . .... .... .... ..... .... 69
7.2 Simplification of Governing Equations... .... .... ..... .... 73
7.2.1 Steady Dynamic Problem of Soft-Matter
Quasicrystals with 12-Fold Symmetry. .... ..... .... 73
7.2.2 Pure Fluid Dynamics.. .... .... .... .... ..... .... 74
7.3 Dislocation and Solution.. .... .... .... .... .... ..... .... 74