LECTURE #19 : 3.11 MECANICS OF MATERIALS F3 INSTRUCTOR : Professor Christine Ortiz OFFICE : 13-422 PONE : 452-384 WWW : http://web.mit.edu/cortiz/www REVIEW : INTRODUCTION TO TE MOLECULAR ORIGINS OF MECANICAL PROPERTIES QUANTITATIVE TREATMENT OF INTERATOMIC BONDING : TE LENNARD-JONES POTENTIAL
SUMMARY : LAST LECTURE I. Basic Definitions : elasticity & elastic moduli Young s modulus shear modulus bulk modulus L o length scales : macroscopic, microscopic, molecular II. Questions : 1. What is the molecular origin of the elastic moduli? i.e. What provides the internal resistance to external mechanical forces and deformations and allows materials to hold their shape? 2. Why do different materials have vastly different elastic moduli? 3. Why do some materials have one elastic moduli (E=constant, isotropic) while other s have different ones in different directions (anisotropic, E=E(θ)? III. Answers : 1. The TYPE of internal cohesive forces/ bonding/ molecular structure holding the material together. 2. ow these forces are ARRANGED within the material (e.g the placement, packing, and location). 3. The NUMBER of bonds/unit volume or unit area IV. Thermodynamic Contributions to Elastic Moduli : Molecular (1) Energetic (2) Entropic Origin or Enthalpic macroscopic result linear elasticity oo ke's Law σ=eε nonlinear elasticity Ru bber Elasticity strain range small strains large strains material metals, ceramics, crystalline materials polymers, rubber networks V. Enthalpic Origin of Elastic Moduli: distortion of chemical and physical bonds : types lattice strain in crystalline materials P I. II. apply load remove load P L L o
Summary of Types of Bonding Type of Interaction Classification Characteristics Schematic Covalent : Metallic : Ionic : ion-ion Polar Interactions : charge-dipole dipole-dipole hydrogen bonding Polarization Interactions : charge-nonpolar (induced or instantaneous dipole) dipole-nonpolar (induced dipole) Dispersion or London Interactions : (*also called charge-fluctuation, electrodynamic, induced-dipoleinduced dipole forces) nonpolar-nonpolar ydrophobic : Entropic Elasticity : Primary or Chemical Bonds : usually characterized as individually strong outer orbital e- cooperatively shared between two or more atoms so that discrete nature of atoms is lost quantum mechanical in origin Secondary or Physical Interactions : usually characterized as individually weak no e- sharing, more subtle attraction between (+) and (-) charges, discrete nature of atoms preserved typically exhibits : lack of specificity lack of directionality lack of stoichiometry Special Interactions : not really true bonds non-directional e- are localized directional (i.e. oriented at welldefined angles to each other) only metals atoms are involved e- are completely delocalized and mobile throughout entire material non-directional coulombic in origin, occurs between oppositely charged species electron transfer from one atom to another force between an ion and a dipole or two dipoles where the (+) charge attracts the (-) charge (purely electrostatic) -bonding : a special type of dipole-dipole interaction that results from the bonding between a atom which is partially (+) charged and a highly electronegative atom such as O, F, N, Cl, (directional) an ion or dipole in the vicinity of a nonpolar atom or molecule causes instantaneous polarization and electrostatic attraction the (+) nucleus of a nonpolar atom attracts the (-) charged electrons of another nonpolar atom resulting in instantaneous, induced, dipoles and fluctuating electron clouds quantum mechanical in origin attraction between nonpolar molecules in aqueous solution caused by their inability to form - bonds with O so as to minimize the disruption of - bonds in O entropy-driven attractive, recoiling force produced via extensional deformation macromolecules e-ee- e- e-+ e-+ e- e- + e- e-+ e- + e- + e- cation anion + - + d - d + d - d - d - d + O d + d - d + + d - d + d + d + O O d - O d + d - d - d + d + O O O C O C O C O O O
Atomistic Basis for Elasticity: One Example : Crystalline Materials
Atomistic Basis for Elasticity: One Example : Crystalline Materials r e r e r f
Atomistic Basis for Elasticity: covalent bond : outer orbitals cooperatively shared + e- e- e- + r e + e- e- e- + r e r f lattice strain disturbs electronic configuration
Consider an Individual Bond interatomic distance, r (nm) + + interatomic force, F (nn) interatomic (bond) energy, W (kj/mol)
Interaction Parameters interatomic distance, r (nm) + + interatomic force, F (nn) interatomic (bond) energy, W (kj/mol)
Molecular Origin of Repulsive Component 2 w(r)(kbt) 1.5 1.5 ard-sphere Repulsion n= σ.1.2.3.4.5.6.7 r(nm)
Molecular Origin of Repulsive Component 2 w(r)(kbt) 1.5 1.5 Soft Repulsion B=1-134 Jm 12 n=12 σ Soft Repulsion.1.2.3.4.5.6.7 r(nm)
Molecular Origin of Attractive Component w (r)(kbt) -.1 -.2 -.3 -.4 -.5 -.6 -.7 -.8 -.9-1.2.4.6.8 1 London dispersion interaction r(nm) A=1-77 Jm 6 m=6
Complete Interaction Potential : The Lennard-Jones Potential 1.8.6 A=1-77 Jm 6 B=1-134 Jm 12 n=12, m=6 w (r)(kbt).4.2 -.2 -.4 -.6 -.8-1.2.4.6.8 1
Complete Interaction Potential : The Lennard-Jones Potential 1.8.6 A=1-77 Jm 6 B=1-134 Jm 12 n=12, m=6 w (r)(kbt).4.2 -.2 -.4 -.6 -.8-1.2.4.6.8 1
Interaction Force :.1.5 F(r)(nN) -.5 -.1 -.15 -.2.2.4.6.8 1 r(nm)
Interaction Force :.4.2 w(r) (kbt) -.2 -.4 -.6 -.8.1.2.4.6.8 1 F(r)(nN).5 -.5 -.1 -.15 -.2.2.4.6.8 1 r(nm)
Interaction Force :.1.5 F(r)(nN) -.5 -.1 -.15 -.2.2.4.6.8 1 r(nm)
Interaction Force :.1.5 F(r)(nN) -.5 -.1 -.15 -.2.2.4.6.8 1 r(nm)