Biaxial Nematics Fact, Theory and Simulation
|
|
- Nelson Nelson
- 5 years ago
- Views:
Transcription
1 Soft Matter Mathematical Modelling Cortona 12 th -16 th September 2005 Biaxial Nematics Fact, Theory and Simulation Geoffrey Luckhurst School of Chemistry University of Southampton
2 Fact The elusive biaxial nematic Claims to discovery Recognition of the biaxial nematic
3 Biaxial Nematic N B molecular organisation Macroscopic definition Frank Symmetry of average tensorial property,t, such as the dielectric susceptibility or the quadrupolar splitting in NMR is biaxial. Principal components T XX T YY T ZZ for N B (n.b. T XX = T YY T ZZ for N U ) T The directors, n, l, m are identified with the principal axes of. η=3 T XX T YY / 2 T ZZ T XX T YY
4 Lyotropic Biaxial Nematic (potassium laurate (KL), 1-decanol, D 2 ) L J Yu, A Saupe, Phys.Rev.Lett., 1980, 45, Phase diagram
5 Identifying a Biaxial Nematic F P Nicoletta, G Chidichimo, A Golemme, P Picci, Liq.Cryst., 1991, 10, Discotic (N D ) Another lyotropic biaxial nematic (potassium laurate, decyl ammonium chloride, D 2 ) Cylindric (N C ) The primary 2 H NMR data Biaxial (N Bx ) η=0. 51 Biaxial (N Bx ) η=1 Biaxial (N Bx ) η=0.68
6 Thermotropic Biaxial Nematics? C 12 H 25 CH 2 C 6 H 13 C 6 H 13 C 12 H 25 CH 2 C 2 C 2 C 12 H 25 C 6 H 13 H H C 6 H 13 C 12 H 25 CH 2 C 6 H 13 C 6 H 13 C 2 H 5 H C 10 H 21 Cu C 10 H 21 H 25 C 12 H 25 C 12 C 12 H 25 H C 12 H 25 C 2 H 5 H 25 C 12 C 12 H 25
7 Thermotropic Biaxial Nematics 2004 V-shaped molecules L A Madsen, T J Dingemans, M Nakata, and E T Samulski, Phys. Rev. Lett., 2004, 92, η=0 η=0.11 Biaxiality in q η=0.11 n.b. for lyotropic biaxial nematics η η=0. 22
8 Thermotropic Biaxial Nematics 2005 Tetrapodes: J L Figueirinhas, C Cruz, D Filip, G Feio, A C Ribeiro, Y Frère and T Meyer, G H Mehl Phys. Rev. Lett., 2005, 94,
9 Tetrapodes: the NMR Evidence
10 Theory Molecular field approach Landau approach
11 General Notation Molecular Field Theory Single molecule potential U= u 2 mn 1 δ m2 1 δ n 2 1 δ p 2 R 2 pm R 2 p n Symmetry adapted functions Dependence on Euler angles 2 = 3 cos 2 β 1 /2 R 0 0 R = 3 8 sin 2 β cos 2 γ R = 3 8 sin 2 β cos 2 2 α R = cos 2 β γ cos 2 α cos 2 γ cos β sin 2 α sin 2 rder parameters are averages of R 2 mn Strength parameters u 200 u 220 u 202 u 222
12 rientational rder Parameters Biaxial molecule in biaxial phase Limit major ZZ 2 S zz = <(3l zz 1)/2> 1 molecular biaxial S xx ZZ - S yy ZZ = <3(l xz 2 l yz2 )/2> 0 phase biaxial S zz XX S zz YY = <3(l zx 2 l zy2 )/2> 0 phase biaxial (S xx XX S xx YY ) (S yy XX S yy YY ) = <3{(l xx 2 l xy2 ) - (l yx 2 l yy2 )/ 2}> 3 Axes: molecular xyz and space XYZ 8 8 R R R R 2 22
13 Molecular Field Theory N Boccara, R Medjani, L de Seze, J.Phys., 1977, 38, Phase diagram (molecular biaxiality ε = λ 6 ) Geometric mean approximation 1 u 220 = u u λ= u u Bounds on ε of 0 and 3 correspond to uniaxial molecules ε = 1is the maximum biaxiality giving the Landau point
14 Molecular Field Theory A M Sonnet, E G Virga and G E Durand, Phys. Rev., E, 2003, 67, Phase Diagram I 1/β N U N B u 220 =0 λ ~u 222 /u 200 λ
15 Landau Theory Uniaxial nematic of uniaxial molecules Key order parameter for nematic-isotropic transition: S A=A 0 BS 2 CS 3 DS 4 Free energy expansion of invariants Assumption: B=b T T Dependence on S and magnitude of other coefficients related to experimental quantities
16 Landau Theory Biaxial phase of biaxial molecules rder parameters Major S= 3 cos 2 β 1/2 = R Molecular biaxial T = sin 2 β cos 2 γ = 8 3 R Phase biaxial U= sin 2 β cos 2 α = 8 3 R 2 2 0
17 Landau Theory Phase / molecular biaxial V = cos 2 β cos 2 α cos 2 γ cos β sin 2 α sin 2 γ =2 R 2 2 Uniaxial nematic S 0 T 0 U=0 V =0 Biaxial nematic S 0 T 0 U 0 V 0
18 Landau Theory D W, Allender, M A Lee, N Hafiz, Mol. Cryst. Liq. Cryst., 1985, 124, Free energy A N A I = A S 2 /2 T 2 U 2 V 2 /18 B S 3 /4 3 SU 2 /2 3 ST 2 /2 SV 2 /12 TUV C 1 S 2 /2 T 2 U 2 V 2 /18 2 C 2 VS/2 2TU 2 D 1 S 2 /2 T 2 U 2 V 2 /18 S 3 /4 3 ST 2 /2 3 SU 2 /2 SV 2 /12 TUV D 2 SV /3 2 TU SV 2 /4 TV 2 /2 U 2 V /2 V 3 /108...
19 Landau Theory Free energy (continued)... E 1 S 2 /2 T 2 U 2 V 2 /18 3 E 2 S 2 /2 T 2 U 2 V 2 /18 SV /3 2TU 2 E 3 S 3 /4 ST 2 /2 3 SU 2 /2 TUV S V 2 /12 2 E 4 S 2 V /4 U 2 V /2 T 2 V /2 3 STU V 3 /108 2 E 5... Assumptions A is linear in temperature ~ ( T T * ) B, C 1, C 2, D 1, D 2, E 1, E 2, E 3, E 4, E 5 are temperature independent.
20 Landau Theory for Biaxial Nematics Phase diagram C 1 = 0, C 2 = 1, D 1 = 1, D 2 = 1, E 1 = 1, E 2 = 1, E 3 = 1, E 4 = 1, E 5 = -7
21 Landau Theory for Biaxial Nematics Problems: (b) Large number of unknown parameters (c) Single characteristic temperature, might have expected more (d) Four nematic phases are predicted Phase rder Parameter I 0 - N U1 1 S N U2 N B N B * 2 S, T 2 S, U 4 S, T, U, V
22 Strategy of Katriel et al. J Katriel, G F Kventsel, G R Luckhurst, T J Sluckin, Liq. Cryst., 1986, 1, (1) Free energy A= 1 2 u 200 P 2 2 k B T f β ln 2 f β sin βdβ =U TS (4) rder parameter P 2 = P 2 cos β f β sin βdβ (6) Entropy a functional of distribution function f(β) S=S [ f β ] But A is not yet a function of P!
23 Strategy of Katriel et al. A=k B T f β ln 2 f β sin βdβ 1 u 200 P 2 2 =U TS 2 Maximise entropy term subject to given P (1) f(β) a function of auxiliary parameter η f β = 1 Z η exp ηp cos β 2 (3) Partition function (5) P a function of η Z η = exp P 2 = ln Z η η ηp 2 cos β sin βdβ A is now a function of ηand P
24 Strategy of Katriel et al. A=k B T f β ln 2 f β sin βdβ 1 u P2 2=U TS (1) Invert equation: (3) A was a function of P and η P 2 = ln Z η η Expand P 2 in a power series in η (6) A is now a function of P Invert power series to required order Expand A in power series in P A=A P 2
25 Result A= 5 2 k B T T P k B T P k B T P T =u 200 /5 k B
26 Landau Theory for Biaxial Nematics A molecular field approach Energy U N = 1 2 u 2 mn 1 δ m2 1 δ n 2 1 δ p 2 R 2 pm R 2 p n U N = 1 2 u 200 S 2 2 U 2 4 u 220 ST 2UV 4 u 222 T 2 2 V 2 n.b. expansion coefficients are components of supertensor and not scalars
27 Simulation Rod-Disc Mixtures Rod-Disc Dimers Flexibility
28 The phase diagram Mixtures of rods and discs P Palffy-Muhoray, J R Bruyn, D A Dunmur, J.Chem.Phys., 1985, 82,
29 Rod Disc Dimers I D Fletcher, G R Luckhurst, Liq.Cryst., 1995, 18, J J Hunt, R W Date, B A Timimi, G R Luckhurst, D W Bruce, J.Am.Chem.Soc., 2001, 123, P H J Kouwer, G H Mehl., Mol.Cryst.Liq.Cryst., 2003, 397,
30 The Phase Behaviour of Rod-Disc Dimers M A Bates, G R Luckhurst, PCCP, 2005, 7, The Lebwohl-Lasher lattice Model Anisotropic interactions Torsional potential ε a > 0 symmetry axes orthogonal ε a < 0 symmetry axes parallel d U ij RR = ε RR P 2 r i r j U ij RD =ε RD P 2 r i d j U ij RD =ε RD P 2 d i r j U ij DD = ε DD P 2 d i d j U RD tors =ε a P 2 r d i i r
31 Parameterisation Scaling Parameter ε RR Relative anisotropy e.g. T =k B T /ε RR ε =ε DD /ε RR Controls the molecular biaxiality Geometric mean or Berthelot approximation ε RD = ε DD ε RR ½ Scaled torsional strength ε a =ε a /ε RR
32 Rigid Rod-Disc Dimer Phase diagram as function of relative anisotropy ε DD /ε RR T * I N u N u N B
33 Flexible Rod-Disc Dimers Phase diagram for different torsional strengths ε a ε a /ε RR ε a =0 ε a = 4 ε a = 6 ε a = 7
34 Flexible Rod-Disc Dimers rientational order parameters R Q AA ε a = 7 and D Q AA [ S α XX, S α YY ε =2.0 ε =1. 75, S α ZZ ] α= R or D
35 Acknowledgements Fact Gerd Kothe (Freiburg) Jacques Malthete (Paris) Klaus Praefcke (Berlin) Bakir Timimi (Southampton) Theory Tim Sluckin (Southampton) Ken Thomas (Southampton) Simulation Martin Bates (York) Silvano Romano (Pavia)
Sep 15th Il Palazzone SNS - Cortona (AR)
Repertoire o Sep 5th 25 Il Palazzone SNS - Cortona (AR) Repertoire of nematic biaxial phases. Fulvio Bisi SMMM - Soft Matter Mathematical Modeling Dip. di Matematica - Univ. di Pavia (Italy) fulvio.bisi@unipv.it
More informationBiaxial Phase Transitions in Nematic Liquid Crystals
- Biaxial Phase Transitions in Nematic Liquid Crystals Cortona, 6 September 5 Giovanni De Matteis Department of Mathematics University of Strathclyde ra.gdem@maths.strath.ac.uk Summary General quadrupolar
More informationChem8028(1314) - Spin Dynamics: Spin Interactions
Chem8028(1314) - Spin Dynamics: Spin Interactions Malcolm Levitt see also IK m106 1 Nuclear spin interactions (diamagnetic materials) 2 Chemical Shift 3 Direct dipole-dipole coupling 4 J-coupling 5 Nuclear
More informationHigh-resolution Calorimetric Study of a Liquid Crystalline Organo-Siloxane Tetrapode with a Biaxial Nematic Phase
High-resolution Calorimetric Study of a Liquid Crystalline Organo-Siloxane Tetrapode with a Biaxial Nematic Phase George Cordoyiannis, 1 Daniela Apreutesei, 2 Georg H. Mehl, 2 Christ Glorieux, 1 and Jan
More informationphases of liquid crystals and their transitions
phases of liquid crystals and their transitions Term paper for PHYS 569 Xiaoxiao Wang Abstract A brief introduction of liquid crystals and their phases is provided in this paper. Liquid crystal is a state
More informationI seminari del giovedì. Transizione di fase in cristalli liquidi II: Ricostruzione d ordine in celle nematiche frustrate
Università di Pavia Dipartimento di Matematica F. Casorati 1/23 http://www-dimat.unipv.it I seminari del giovedì Transizione di fase in cristalli liquidi II: Ricostruzione d ordine in celle nematiche frustrate
More informationPhase diagrams of mixtures of flexible polymers and nematic liquid crystals in a field
PHYSICAL REVIEW E VOLUME 58, NUMBER 5 NOVEMBER 998 Phase diagrams of mixtures of flexible polymers and nematic liquid crystals in a field Zhiqun Lin, Hongdong Zhang, and Yuliang Yang,, * Laboratory of
More informationAn introduction to Solid State NMR and its Interactions
An introduction to Solid State NMR and its Interactions From tensor to NMR spectra CECAM Tutorial September 9 Calculation of Solid-State NMR Parameters Using the GIPAW Method Thibault Charpentier - CEA
More informationPHASE BIAXIALITY IN NEMATIC LIQUID CRYSTALS Experimental evidence for phase biaxiality in various thermotropic liquid crystalline systems
Book_Ramamoorthy_74_Proof2_November, CHAPTER 5 PHASE BIAXIALITY IN NEMATIC LIQUID CRYSTALS Experimental evidence for phase biaxiality in various thermotropic liquid crystalline systems KIRSTEN SEVERING
More informationProblem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1
Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More informationSpin Interactions. Giuseppe Pileio 24/10/2006
Spin Interactions Giuseppe Pileio 24/10/2006 Magnetic moment µ = " I ˆ µ = " h I(I +1) " = g# h Spin interactions overview Zeeman Interaction Zeeman interaction Interaction with the static magnetic field
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationStability and Symmetry Properties for the General Quadrupolar Interaction between Biaxial Molecules
Stability and Symmetry Properties for the General Quadrupolar Interaction between Biaxial Molecules Epifanio G. Virga SMMM Soft Matter Mathematical Modelling Department of Mathematics University of Pavia,
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationHomework 1/Solutions. Graded Exercises
MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both
More informationLiquid Crystals. Martin Oettel. Some aspects. May 2014
Liquid Crystals Some aspects Martin Oettel May 14 1 The phases in Physics I gas / vapor change in density (scalar order parameter) no change in continuous translation liquid change in density change from
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationarxiv: v1 [cond-mat.soft] 13 Sep 2015
Abstract arxiv:1509.03834v1 [cond-mat.soft] 13 Sep 2015 Investigations of the phase diagram of biaxial liquid crystal systems through analyses of general Hamiltonian models within the simplifications of
More information16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity
6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =
More information3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship
3.2 Hooke s law anisotropic elasticity Robert Hooke (1635-1703) Most general relationship σ = C ε + C ε + C ε + C γ + C γ + C γ 11 12 yy 13 zz 14 xy 15 xz 16 yz σ = C ε + C ε + C ε + C γ + C γ + C γ yy
More informationAlgebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
More information3D and Planar Constitutive Relations
3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace
More informationProblem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:
Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein
More informationMaxwell s Equations:
Course Instructor Dr. Raymond C. Rumpf Office: A-337 Phone: (915) 747-6958 E-Mail: rcrumpf@utep.edu Maxwell s Equations: Physical Interpretation EE-3321 Electromagnetic Field Theory Outline Maxwell s Equations
More informationLOWELL WEEKLY JOURNAL
Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q
More informationMolecular simulation of hierarchical structures in bent-core nematics
Molecular simulation of hierarchical structures in bent-core nematics Stavros D. Peroukidis, Alexandros G. Vanakaras and Demetri J. Photinos Department of Materials Science, University of Patras, Patras
More informationOrientation of Piezoelectric Crystals and Acoustic Wave Propagation
Orientation of Piezoelectric Crystals and Acoustic Wave Propagation Guigen Zhang Department of Bioengineering Department of Electrical and Computer Engineering Institute for Biological Interfaces of Engineering
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More informationPhase transitions in liquid crystals
Phase transitions in liquid crystals Minsu Kim Liquid crystals can be classified into 5 phases; Liquid, Nematic, Smetic, Columnar and Crystalline according to their position order and orientational order.
More informationTensor Visualization. CSC 7443: Scientific Information Visualization
Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its
More informationLecture 7. Properties of Materials
MIT 3.00 Fall 2002 c W.C Carter 55 Lecture 7 Properties of Materials Last Time Types of Systems and Types of Processes Division of Total Energy into Kinetic, Potential, and Internal Types of Work: Polarization
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationAMC Lecture Notes. Justin Stevens. Cybermath Academy
AMC Lecture Notes Justin Stevens Cybermath Academy Contents 1 Systems of Equations 1 1 Systems of Equations 1.1 Substitution Example 1.1. Solve the system below for x and y: x y = 4, 2x + y = 29. Solution.
More informationStrain Transformation equations
Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
More informationCrystal Relaxation, Elasticity, and Lattice Dynamics
http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität
More informationGG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS
GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why
More informationLinearized Theory: Sound Waves
Linearized Theory: Sound Waves In the linearized limit, Λ iα becomes δ iα, and the distinction between the reference and target spaces effectively vanishes. K ij (q): Rigidity matrix Note c L = c T in
More informationNIELINIOWA OPTYKA MOLEKULARNA
NIELINIOWA OPTYKA MOLEKULARNA chapter 1 by Stanisław Kielich translated by:tadeusz Bancewicz http://zon8.physd.amu.edu.pl/~tbancewi Poznan,luty 2008 ELEMENTS OF THE VECTOR AND TENSOR ANALYSIS Reference
More informationModule #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.
HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp. 7-36 Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass The Stress Tensor z z x O
More informationChapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)
Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5
More informationIOAN ŞERDEAN, DANIEL SITARU
Romanian Mathematical Magazine Web: http://www.ssmrmh.ro The Author: This article is published with open access. TRIGONOMETRIC SUBSTITUTIONS IN PROBLEM SOLVING PART IOAN ŞERDEAN, DANIEL SITARU Abstract.
More informationBrazilian Journal of Physics ISSN: Sociedade Brasileira de Física Brasil
Brazilian Journal of Physics ISSN: 003-9733 luizno.bjp@gmail.com Sociedade Brasileira de Física Brasil Simonario, P. S.; de Andrade, T. M.; M. Freire, F. C. Elastic Constants of a Disc-Like Nematic Liquid
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationElectromagnetic Properties of Materials Part 2
ECE 5322 21 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.edu Lecture #3 Electromagnetic Properties of Materials Part 2 Nonlinear
More informationElectromagnetism II Lecture 7
Electromagnetism II Lecture 7 Instructor: Andrei Sirenko sirenko@njit.edu Spring 13 Thursdays 1 pm 4 pm Spring 13, NJIT 1 Previous Lecture: Conservation Laws Previous Lecture: EM waves Normal incidence
More informationCHM 6365 Chimie supramoléculaire Partie 8
CHM 6365 Chimie supramoléculaire Partie 8 Liquid crystals: Fourth state of matter Discovered in 1888 by Reinitzer, who observed two melting points for a series of cholesterol derivatives Subsequent studies
More informationDynamics of Glaciers
Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers
More informationApplications of Eigenvalues & Eigenvectors
Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal
More informationNeatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.
» ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z
More informationLandau description of ferrofluid to ferronematic phase transition H. Pleiner 1, E. Jarkova 1, H.-W. Müller 1, H.R. Brand 2
MAGNETOHYDRODYNAMICS Vol. 37 (2001), No. 3, pp. 254 260 Landau description of ferrofluid to ferronematic phase transition H. Pleiner 1, E. Jarkova 1, H.-W. Müller 1, H.R. Brand 2 1 Max-Planck-Institute
More information1 Arithmetic calculations (calculator is not allowed)
1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More information1 Stress and Strain. Introduction
1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may
More informationRECENT TOPICS IN THE THEORY OF SUPERFLUID 3 He
RECENT TOPICS IN THE THEORY OF SUPERFLUID 3 He Lecture at ISSP Tokyo 13.5.2003 ErkkiThuneberg Department of physical sciences, University of Oulu Janne Viljas and Risto Hänninen Low temperature laboratory,
More informationQuasi-Harmonic Theory of Thermal Expansion
Chapter 5 Quasi-Harmonic Theory of Thermal Expansion 5.1 Introduction The quasi-harmonic approximation is a computationally efficient method for evaluating thermal properties of materials. Planes and Manosa
More information3 Constitutive Relations: Macroscopic Properties of Matter
EECS 53 Lecture 3 c Kamal Sarabandi Fall 21 All rights reserved 3 Constitutive Relations: Macroscopic Properties of Matter As shown previously, out of the four Maxwell s equations only the Faraday s and
More informationEnergetic and Entropic Contributions to the Landau de Gennes Potential for Gay Berne Models of Liquid Crystals
Polymers 213, 5, 328-343; doi:1.339/polym52328 Article OPEN ACCESS polymers ISSN 273-436 www.mdpi.com/journal/polymers Energetic and Entropic Contributions to the Landau de Gennes Potential for Gay Berne
More informationAttraction and Repulsion. in Biaxial Molecular Interactions. Interaction Potential
Attraction and Repulsion Interaction Potential Slide in Biaxial Molecular Interactions Epifanio G. Virga SMMM Soft Matter Mathematical Modelling Department of Mathematics University of Pavia, Italy Slide
More informationModel for the director and electric field in liquid crystal cells having twist walls or disclination lines
JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 12 15 JUNE 2002 Model for the director and electric field in liquid crystal cells having twist walls or disclination lines G. Panasyuk a) and D. W. Allender
More informationComputer Simulation of Chiral Liquid Crystal Phases
John von Neumann Institute for Computing Computer Simulation of Chiral Liquid Crystal Phases Reiner Memmer published in NIC Symposium 2001, Proceedings, Horst Rollnik, Dietrich Wolf (Editor), John von
More informationLecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI
Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still
More informationMATH 19520/51 Class 5
MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential
More informationMolecular simulation and theory of a liquid crystalline disclination core
PHYSICAL REVIEW E VOLUME 61, NUMBER 1 JANUARY 2000 Molecular simulation and theory of a liquid crystalline disclination core Denis Andrienko and Michael P. Allen H. H. Wills Physics Laboratory, University
More informationSOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM
Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,
More informationNumerical Simulation of Nonlinear Electromagnetic Wave Propagation in Nematic Liquid Crystal Cells
Numerical Simulation of Nonlinear Electromagnetic Wave Propagation in Nematic Liquid Crystal Cells N.C. Papanicolaou 1 M.A. Christou 1 A.C. Polycarpou 2 1 Department of Mathematics, University of Nicosia
More informationAM 106/206: Applied Algebra Madhu Sudan 1. Lecture Notes 12
AM 106/206: Applied Algebra Madhu Sudan 1 Lecture Notes 12 October 19 2016 Reading: Gallian Chs. 27 & 28 Most of the applications of group theory to the physical sciences are through the study of the symmetry
More informationLOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort
- 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [
More informationModel For the Director and Electric Field in Liquid Crystal Cells Having Twist Walls Or Disclination Lines
Kent State University Digital Commons @ Kent State University Libraries Physics Publications Department of Physics 6-15-2002 Model For the Director and Electric Field in Liquid Crystal Cells Having Twist
More informationRotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.
Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect
More informationCSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer
CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at
More informationPolynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:
Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a
More informationLittle Orthogonality Theorem (LOT)
Little Orthogonality Theorem (LOT) Take diagonal elements of D matrices in RG * D R D R i j G ij mi N * D R D R N i j G G ij ij RG mi mi ( ) By definition, D j j j R TrD R ( R). Sum GOT over β: * * ( )
More informationLecture #16. Spatial Transforms. Lecture 16 1
ECE 53 1 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.edu Lecture #16 Spatial ransforms Lecture 16 1 Lecture Outline ransformation
More informationTwo-dimensional flow in a porous medium with general anisotropy
Two-dimensional flow in a porous medium with general anisotropy P.A. Tyvand & A.R.F. Storhaug Norwegian University of Life Sciences 143 Ås Norway peder.tyvand@umb.no 1 Darcy s law for flow in an isotropic
More informationElasticité de surface. P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004).
Elasticité de surface P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004). The concept I Physical origin Definition Applications Surface stress and crystallographic parameter of small crystals Surface
More informationUnit IV State of stress in Three Dimensions
Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength
More informationMath 110: Worksheet 1 Solutions
Math 110: Worksheet 1 Solutions August 30 Thursday Aug. 4 1. Determine whether or not the following sets form vector spaces over the given fields. (a) The set V of all matrices of the form where a, b R,
More informationRock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth
Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of
More information6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant
DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional
More informationHyperfine interactions Mössbauer, PAC and NMR Spectroscopy: Quadrupole splittings, Isomer shifts, Hyperfine fields (NMR shifts)
Hyperfine interactions Mössbauer, PAC and NMR Spectroscopy: Quadrupole splittings, Isomer shifts, Hyperfine fields (NMR shifts) Peter Blaha Institute of Materials Chemistry TU Wien Definition of Hyperfine
More informationIntroduction, Basic Mechanics 2
Computational Biomechanics 18 Lecture : Introduction, Basic Mechanics Ulli Simon, Lucas Engelhardt, Martin Pietsch Scientific Computing Centre Ulm, UZWR Ulm University Contents Mechanical Basics Moment
More information' Liberty and Umou Ono and Inseparablo "
3 5? #< q 8 2 / / ) 9 ) 2 ) > < _ / ] > ) 2 ) ) 5 > x > [ < > < ) > _ ] ]? <
More informationUnusual Entropy of Adsorbed Methane on Zeolite Templated Carbon. Supporting Information. Part 2: Statistical Mechanical Model
Unusual Entropy of Adsorbed Methane on Zeolite Templated Carbon Supporting Information Part 2: Statistical Mechanical Model Nicholas P. Stadie*, Maxwell Murialdo, Channing C. Ahn, and Brent Fultz W. M.
More informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More informationLayer-Inversion Zones in Angular Distributions of Luminescence and Absorption Properties in Biaxial Crystals
Materials 2010, 3, 2474-2482; doi:10.3390/ma3042474 Article OPEN ACCESS materials ISSN 1996-1944 www.mdpi.com/jonal/materials Layer-Inversion Zones in Angular Distributions of Luminescence and Absorption
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationELASTICITY (MDM 10203)
LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering
More informationMechanics of materials Lecture 4 Strain and deformation
Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationMANY BILLS OF CONCERN TO PUBLIC
- 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -
More informationClosed-Form Solution Of Absolute Orientation Using Unit Quaternions
Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding
More informationOWELL WEEKLY JOURNAL
Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --
More informationAn Overview of the Oseen-Frank Elastic Model plus Some Symmetry Aspects of the Straley Mean-Field Model for Biaxial Nematic Liquid Crystals
An Overview of the Oseen-Frank Elastic Model plus Some Symmetry Aspects of the Straley Mean-Field Model for Biaxial Nematic Liquid Crystals Chuck Gartland Department of Mathematical Sciences Kent State
More informationVariational principles in mechanics
CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of
More information