REVIEW : INTERATOMIC BONDING : THE LENNARD-JONES POTENTIAL & RUBBER ELASTICITY I DERIVATION OF STRESS VERSUS STRAIN LAWS FOR RUBBER ELASTICITY

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1 LECTURE #3 : 3.11 MECHANICS O MATERIALS 03 INSTRUCTOR : Professor Christine Ortiz OICE : PHONE : WWW : REVIEW : INTERATOMIC BONDING : THE LENNARD-JONES POTENTIAL & RUBBER ELASTICITY I DERIVATION O STRESS VERSUS STRAIN LAWS OR RUBBER ELASTICITY

2 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 atomic =k bond δ bond atomic =interatomic force k bond =bond stiffness δ bond =(r f -r e )= bond displacement r e =equilibrium bond length r f =strained bond length II. Interatomic Parameters : Interatomic Potential or Bond Energy (J or J/mol or k T) : k -A B U(r) or W (r) = U attractive(r) + U repulsive(r) = + = - (r)dr m n r r -du(r) Interatomic (Bond) orce (nn): (r) = = k(r)dr dr -d U(r) d(r) Interatomic (Bond) Stiffness (nn/nm): k(r) = = dr dr r (nm) = interatomic separation distance A,B,m,n = constants determined by the type of interaction B -3 B = Boltzmann's constant = J/K, T = absolute temperature (K) r e r rff + e- e- e- + r e r f w(r)(kbt) Hard-Sphere Repulsion n= σ r(nm) w(r)(kbt) Soft Repulsion Soft Repulsion B= Jm 1 n=1 σ r(nm) w (r)(kbt) London dispersion interaction A=10-77 Jm 6 m=6 r(nm)

3 SUMMARY : LJ Potential I. Lennard-Jones Potential U ( m = 6, n = 1) = = 4E LJ ( m = 6, n = 1) = LJ 7 -A B r r -6A 1B + 13 r r B σ σ r r 1 6 E B = "binding energy," "bond dissociation energy," or depth of potential well r s = distance at which U(r s) exhibits and inflection point, (r s) = minimum = r e = equilibrium bond length, distance at which U(r e) = minimum, (r e) = 0 r = σ = distance at which U(r ) = 0, (r ) o o o 0.4 r o, r e, r s RUPTURE 0. w(r) (k B T) E B anharmonic (asymmetric) -0.6 (r)(nn) RUPTURE V IV I II Interatomic orce versus Separation Distance Curve III r(nm)

4 SUMMARY : Rubber Elasticity I. Structure of Polymer Networks structure and properties of polymers and elastomers : N=degree of polymerization (large) crosslink network strand conformation (spatial arrangement of atoms) crosslink density, ν x =#strands/m strands/cm 3 entropic origin of random coil conformation : Ω=number of available chain conformations <r > 1/ =root-mean-square end-to-end distance=n 1/ a <>= statistical mechanical time average r=instantaneous chain end-to-end separation distance polymer network single random coil polymer chain <r > 1/ At T>0 K the chain is continually in motion due to rotation about backbone bonds macroscopic molecular II. reely-jointed Chain (JC) Model Two molecular level parameters : a= statistical segment length or local chain stiffness (*determined by chemical structure) n= number of statistical segments L contour = L c =na= contour length or length of fully extended chain Assumptions : 1. random walk of rigid segments, all angles between statistical segments are equally probable and each segment is uncorrelated to the next. segments are connected by revolving pivots, free rotation at the bond junctions a 3 a 3. no self-interactions, overlap of different parts of chain allowed 4. no enthalpic deformations, bond length stays constant a 1 r a n- a n-1 a n

5 SUMMARY : LJ Potential II. reely-jointed Chain (JC) Model (Cont d)_ 1. Qualitative Description of Single Chain Stretching : links rotate so as to uncoil and extend polymer chain along stretching axis disorder and entropy # of available configurations, Ω elastic restoring force, elastic =-externally applied force, elastic r elastic. General Statistical Mechanical ormulas : Ω = number of chain conformations P(r) = probability of finding a free chain end a radial distance, r, away from fixed chain end (origin) ~ Ω S(r) = configurational entropy = kblnp(r) A(r) = Helmholtz free energy = U(r) - TS(r) = - Tk z BlnP(r) -da(r) 0 f(r) = entropic elastic force = 1 dr d(r) -d A(r) k(r) = (global) entropic chain stiffness = = dr dr r 1 3. Gaussian ormulas or Stretching a Single Polymer Chain : 3 4b r 3 P(r) = exp[ b r ] where b = π na 4b r π 3k T A(r) = r 3k T = r 3 S(r) = kbln exp[ b r ] B B na Lca 3k T 3k T 3k T 3k T B B f(r) = - r = r na La c B B k(r) = = cons tan t na = La c x Linear Elasticity y

6 Assemble Strands into a Network λ 3 macroscopically deform rubber cube λ 1 >0 (r 1 r r 3 ) (r 1 r r 3 ) r =0 r molecular level deformation

7 Assemble Strands into a Network 1 >0 (r 1 r r 3 ) (r 1 r r 3 ) =0 λ 3 r <r > 1/ λ molecular level deformation macroscopically deform rubber cube

8 Assemble Strands into a Network 1 >0 (r 1 r r 3 ) (r 1 r r 3 ) =0 λ 3 r <r > 1/ λ molecular level deformation macroscopically deform rubber cube

9 [ λλλ = 1] 1 3 CONSTANT VOLUME CONSTRAINT kbtν A = λ1 λ λ3 3 GAUSSIAN + + CONSTANT VOLUME DEORMATION λ λ 3

10 [ λλλ = 1] 1 3 Stress Equations kbtν A = λ1 λ λ3 3 GAUSSIAN + + CONSTANT VOLUME DEORMATION

11 Stress versus Strain Equations for Uniaxial Deformation kbtν A = λ1 λ λ3 3 GAUSSIAN + + CONSTANT VOLUME DEORMATION [ λλλ = 1] 1 3 λ λ 3

12 Uniaxial Deformation: Comparison of Theory with Experiment 7 6 σ 1 (MPa) // 1 1/λ crosslinks 1// 1 strand