LECTURE 23 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www REVIEW INTERATOMIC BONDING THE LENNARD JONES POTENTIAL RUBBER ELASTICITY I DERIVATION OF STRESS VERSUS STRAIN LAWS FOR RUBBER ELASTICITY SUMMARY Molecular Origins of Elastic Moduli I Atomistic Basis for Elastic Moduli lattice strain uniform distortion of interatomic bonds e g covalent bonds disturbing outer orbital electron cloud represent an individual bond by a linear elastic Hookean spring which connects two atoms represented by hard spheres Fatomic kbond bond Fatomic interatomic force kbond bond stiffness bond rf re bond displacement re equilibrium bond length rf strained bond length re rf e II Interatomic Parameters e e re rf Interatomic Potential or Bond Energy J or J mol or kBT kB Boltzmann s constant 1 38 10 23 J K T absolute temperature K A B U r or W r Uattractive r Urepulsive r m n F r dr r r dU r Interatomic Bond Force nN F r k r dr dr d 2 U r dF r Interatomic Bond Stiffness nN nm k r dr dr 2 r nm interatomic separation distance A B m n constants determined by the type of interaction 0 0 2 0 4 0 6 0 8 0 0 1 2 0 2 1 0 5 Soft Repulsion 1 5 Soft Repulsion B 10 134Jm12 n 12 1 0 5 0 0 1 0 2 0 3 0 4 r n m 0 5 0 6 0 7 0 4 London dispersion interaction 0 5 0 6 0 7 A 10 77Jm6 m 6 0 8 0 9 0 0 0 3 w r k BT Hard Sphere Repulsion n 1 5 w r k B T w r k B T 2 0 0 1 0 2 0 3 0 4 r n m 0 5 0 6 0 7 1 r nm 1 SUMMARY LJ Potential I Lennard Jones Potential 6 12 A B U LJ m 6 n 12 6 12 4EB r r r r 6A 12B 13 r7 r EB binding energy bond dissociation energy or depth of potential well rs distance at which U rs exhibits and inflection point F rs min imum FRUPTURE re equilibrium bond length distance at which U re minimum F re 0 ro distance at which U ro 0 F ro FLJ m 6 n 12 ro re rs 0 4 w r k B T 0 2 0 0 2 0 0 2 0 4 0 6 0 8 1 anharmonic asymmetric EB 0 4 0 6 0 8 0 0 1 V F r nN 0 0 0 5 0 0 0 0 5 0 0 0 1 0 0 1 5 0 0 2 I IV 0 2 FRUPTURE 0 4 0 6 II III r nm 0 8 1 Interatomic Force versus Separation Distance Curve SUMMARY Rubber Elasticity I Structure of Polymer Networks single random coil structure and properties of polymers and elastomers polymer network polymer chain N degree of polymerization large crosslink network strand conformation spatial arrangement of atoms crosslink density x strands m3 1021 strands cm3 entropic origin of random coil conformation number of available chain conformations r2 1 2 r2 1 2 root mean square end to end distance n1 2a At T 0 K the chain is continually in motion statistical mechanical time average due to rotation about backbone bonds r instantaneous chain end to end separation distance F x1 molecular F macroscopic x2 F x3 F II Freely Jointed Chain FJC Model Two molecular level parameters a statistical segment length or local chain stiffness determined by chemical structure n number of statistical segments Lcontour Lc na contour length or length of fully extended chain Assumptions 1 random walk of rigid segments all angles a3 between statistical segments are equally probable a2 and each segment is uncorrelated to the next a1 2 segments are connected by revolving pivots free rotation at the bond junctions r 3 no self interactions overlap of different parts of chain allowed 4 no enthalpic deformations bond length stays constant an 2 an 1 an SUMMARY LJ Potential II Freely Jointed Chain FJC Model Cont d 1 Qualitative Description of Single Chain Stretching links rotate so as to uncoil and extend polymer chain along stretching axis F disorder and entropy of available configurations elastic restoring force Felastic externally applied force F Felastic r Felastic 2 General Statistical Mechanical Formulas number of chain conformations P r probability of finding a free chain end a radial distance r away from fixed chain end origin S r configurational entropy k B lnP r z A r Helmholtz free energy U r TS r Tk B lnP r dA r 0 f r entropic elastic force dr dF r d 2 A r k r global entropic chain stiffness dr dr 2 r1 y 3 Gaussian Formulas For Stretching a Single Polymer Chain P r 4b3 r 2 exp b 2 r 2 where b x 3 2na 2 4b3 r 2 S r k B ln exp b 2 r 2 3k T 3k T A r B 2 r 2 B r 2 2na 2Lc a 3k T 3k T f r B2 r B r na Lc a 3k T 3k T k r B2 B cons tan t na Lc a F1 Linear Elasticity Assemble Strands into a Network x3 3 macroscopically deform rubber cube 2 1 x1 F1 0 x2 r1 r2 r3 r x3 r1 r2 r3 F 0 r x2 x1 molecular level deformation Assemble Strands into a Network F1 0 r1 r2 r3 x3 x3 r r1 r2 r3 F 0 3 r2 1 2 2 x2 x1 molecular level deformation x1 1 macroscopically deform rubber cube x2 Assemble Strands into a Network F1 0 r1 r2 r3 x3 x3 r r1 r2 r3 F 0 3 r2 1 2 2 x2 x1 molecular level deformation x1 1 macroscopically deform rubber cube x2 CONSTANT VOLUME CONSTRAINT k B T x 12 2 2 32 3 GAUSSIAN 2 1 2 3 1 CONSTANT VOLUME DEFORMATION A x1 1 2 x 2 x3 3 Stress Equations k B T x 12 2 2 32 3 GAUSSIAN 2 1 2 3 1 CONSTANT VOLUME DEFORMATION A Stress versus Strain Equations for Uniaxial Deformation k B T x 12 2 2 32 3 GAUSSIAN 2 1 2 3 1 CONSTANT VOLUME DEFORMATION A x1 1 2 x 2 x3 3 Uniaxial Deformation Comparison of Theory with Experiment x1 7 1 1 MPa 6 1 11 2 x2 5 x3 1 11 2 4 3 2 crosslinks strand F F 1 0 1 2 3 4 5 1 6 7 8 9