DOC PREVIEW
MIT 3 11 - Molecular Origins of Elastic Moduli

This preview shows page 1-2-3-4 out of 12 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

LECTURE #23 :3.11 MECHANICS OF MATERIALS F03INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : 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-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1000.20.40.60.81r(nm)w(r)(kBT)A=10-77Jm6m=6Londondispersion interaction-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1000.20.40.60.81r(nm)w(r)(kBT)A=10-77Jm6m=6Londondispersion interactionSUMMARY : Molecular Origins of Elastic ModuliI.Atomistic Basis for Elastic Moduli• “lattice strain”- uniform distortion of interatomic bonds, e.g.covalent bonds; disturbing outerorbital electron cloud• represent an individual bond by a linearelastic Hookean spring which connectstwo atoms represented by hard spheresFatomic=kbondδδδδbondFatomic=interatomic forcekbond=bond stiffnessδbond=(rf-re)= bond displacementre=equilibrium bond lengthrf=strained bond length II. Interatomic Parameters : rfrerfreree-e-e-rf++ree-e-e-rf++00.511.520 0.1 0.2 0.3 0.4 0.5 0.6 0.7r(nm)w(r)(kBT)Hard-Sphere Repulsionn=∞σ00.511.520 0.1 0.2 0.3 0.4 0.5 0.6 0.7r(nm)w(r)(kBT)Hard-Sphere Repulsionn=∞σHard-Sphere Repulsionn=∞σSoft Repulsion00.511.520 0.1 0.2 0.3 0.4 0.5 0.6 0.7r(nm)w(r)(kBT)Soft RepulsionB=10-134Jm12n=12σSoft Repulsion00.511.520 0.1 0.2 0.3 0.4 0.5 0.6 0.7r(nm)w(r)(kBT)Soft RepulsionB=10-134Jm12n=12Soft RepulsionB=10-134Jm12n=12σ-23mn = Boltzmann's constant = 1.38 × 10 J/K, T = absolute temperature (K)-A BU(r) or (r) = U (r) + U (r) = = - F(r)drrrBBInteratomic Potential or Bond Energy (J or J/mol or k T) :kInterato+∫attractive repulsiveW22= interatomic separation distance= constants determined by the type of intera-dU(r)F(r) = k(r)drdr-d U(r) dF(r)k(r) =drdrr (nm)A,B,m,nmic (Bond) Force (nN) : Interatomic (Bond) Stiffness (nN/nm) : ==∫ctionSUMMARY : LJ PotentialI. Lennard-Jones Potential BLJLJ7Bss12661213m = 6, n = 12m = 6, n = 12rr-A BU( ) = =4Err-6A 12BF( ) = rrE = "binding energy," "bond dissociation energy," or depth of potential wellr = distance at which U(r ) exhibits σσ −  ++sRUPTUREe eeoooand inflection point, F(r ) min imum Fr = equilibrium bond length, distance at which U(r ) = minimum, F(r ) = 0r = = distance at which U(r ) 0, F(r )σ===→∞Interatomic Force versus Separation Distance Curver(nm)-0.8-0.6-0.4-0.200.20.400.20.40.60.81w(r) (kBT)-0.02-0.015-0.01-0.00500.0050.010 0.2 0.4 0.6 0.8 1F(r)(nN)r(nm)-0.8-0.6-0.4-0.200.20.400.20.40.60.81w(r) (kBT)-0.02-0.015-0.01-0.00500.0050.010 0.2 0.4 0.6 0.8 1F(r)(nN)ro, re, rsEBFRUPTUREIIIIIVVIIanharmonic(asymmetric)SUMMARY : Rubber ElasticityI. 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, νx=#strands/m3≈1021strands/cm3• entropic origin of random coil conformation :Ω=number of available chain conformations<r2>1/2=root-mean-square end-to-end distance=n1/2a<>= statistical mechanical time averager=instantaneous chain end-to-end separation distanceII. 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 segmentsLcontour= Lc=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 next2. segments are connected by revolving pivots,free rotation at the bond junctions3. no self-interactions, overlap of different parts of chain allowed4. no enthalpic deformations, bond length stays constanta1rana2a3an-1an-2≈≈≈≈a1rana2a3an-1an-2≈≈≈≈single random coilpolymer chainpolymer network<r2>1/2At T>0°K the chain is continually in motiondue to rotation about backbone bondsFFx1x2x3FFFFx1x2x3FFx1x2x3FFmacroscopicmolecularSUMMARY : LJ PotentialII. 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↑disorder and entropy↓# of available configurations, Ω↓elastic restoring force, Felastic=-externally applied force, F ↑2. General Statistical Mechanical Formulas :3. Gaussian Formulas For Stretching a Single Polymer Chain :FrFelasticFelastic≈≈≈≈r1F1xyzB= number of chain conformationsP(r) = probability of finding a free chain end a radial distance, r, away from fixed chain end (origin) ~ ΩS(r) = configurational entropy = k lnP(r)A(r) = Helmholtz free ener ΩB22gy = U(r) - TS(r) = - Tk lnP(r)-dA(r)f(r) = entropic elastic force = drdF(r) -d A(r)k(r) = (global) entropic chain stiffness = = drdr0322223222B22BB2cBB2cBB2c4b r 3P(r) exp[ b r ] where b =2na4b rS(r) = k ln exp[ b r ]3k T 3k TA(r)=r=r2L a2na3k T 3k Tf(r) = - r = rLana3k T 3k Tk(r) = = cons tan tLana ππ=−−=Linear ElasticityAssemble Strands into a Networkλλλλ2λλλλ1λλλλ3macroscopicallydeform rubber cubex1x2x3rr’F1>0x1x2x3molecularlevel deformationF=0(r1’ r2’ r3’)(r1r2r3)Assemble Strands into a Networkλλλλ2λλλλ1λλλλ3macroscopicallydeform rubber cube<r2>1/2r’F1>0x1x2x3molecularlevel deformationF=0x1x2x3(r1’ r2’ r3’)(r1 r2 r3)Assemble Strands into a Networkλλλλ2λλλλ1λλλλ3macroscopicallydeform rubber cube<r2>1/2r’F1>0x1x2x3molecularlevel deformationF=0x1x2x3(r1’ r2’ r3’)(r1 r2 r3)CONSTANT VOLUME CONSTRAINTλλλλ1111λλλλ2222λλλλ3333x3333x2222x1111[]222Bx123123kTA321 GAUSSIAN CONSTANT VOLUME DEFORMATIONνλλλλλλ∆= + + −=Stress Equations[]222Bx123123kTA321 GAUSSIAN CONSTANT VOLUME DEFORMATIONνλλλλλλ∆= + + −=Stress versus Strain Equations for Uniaxial Deformationλλλλ1111λλλλ2222λλλλ3333x3333x2222x1111[]222Bx123123kTA321 GAUSSIAN CONSTANT VOLUME DEFORMATIONνλλλλλλ∆= + + −=Uniaxial Deformation: Comparison ofTheory with


View Full Document

MIT 3 11 - Molecular Origins of Elastic Moduli

Documents in this Course
Load more
Download Molecular Origins of Elastic Moduli
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Molecular Origins of Elastic Moduli and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Molecular Origins of Elastic Moduli 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?