Materials and Biomaterials INTERPARTICLE AND INTERSURFACE FORCES II

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Materials and Biomaterials INTERPARTICLE AND INTERSURFACE FORCES II 3.052 Nanomechanics of Materials and Biomaterials LECTURE # 11 : INTERPARTICLE AND INTERSURFACE FORCES II Prof. Christine Ortiz DMSE, RM 13-4022 Phone : (617) 452-3084 Email : cortiz@mit.edu WWW : http://web.mit.edu/cortiz/www

Interparticle and Intersurface Forces : Review : Lecture #10 Interparticle and Intersurface Forces : I. Net Interaction Free Energy or Potential : W(D) (J or kBT) assumptions : (1) w(r)=-A/rn (2) “pairwise additivity” W(D)MOL-SFC =-2pAr/(n-2)(n-3)Dn-3 W(D)SPHERE-SFC =-4p2Ar2R/(n-2)(n-3)(n-4)(n-5)Dn-5 (D<<R) =-8p2Ar2R3/3(n-2)(n-3)Dn-3 (D>>R) W(D)SPHERE-SPHERE =-2p2Ar2R/(n-2)(n-3)(n-4)(n-5)Dn-5 (D<<R) (*equal radii) ~1/Dn (D>>R) W(D)planes/unit area =-2pAr2/(n-2)(n-3)(n-4)Dn-4 net interaction force (N): F(r)=-dW(D)/dD = k(r)dr net interaction stiffness(N/m) : k(r)=-d2 W(D)/dD2 = dF(r)/dr II. Contrasts Between Interatomic / Intermolecular Interactions and Interactions Between Bodies : Interactions between bodies : 1. are much longer range (i.e. decay much more slowly with D) 2. are size-dependent 3. are geometry-dependent III. Comparison with Experimental Data : retardation effects IV. Derjaguin Approximations : F(D)sphere-surface=2pRW(D)planes (D>>R) F(D)sphere-sphere=2pR1R2/(R1+R2)W(D)planes (D>>R) F(D)crossed-cylinders=2pR1R2/(sinq)W(D)planes (D>>R)

Contrasts Between Interatomic / Intermolecular Interactions and Interactions Between Bodies

Interaction Potential Between a Sphere and a Flat Surface z z=2R WSPHERE-SFC(D)=WMOL-SFC(D+z)(rp(2R-z)zdz) where : WMOL-SFC(D+z)=(-A2pr)/[(n-2)(n-3)(D+z)n-3] WSPHERE-SFC(D)= -A2pr/(n-2)(n-3)(D+z)n-3 (rp(2R-z)zdz) -A2p2r2/(n-2)(n-3) (2R-z)z/(D+z)n-3dz for D,z<<R WSPHERE-SFC(D)=-A4p2r2R/(n-2)(n-3)(n-4)(n-5)Dn-5 Dispersion VDW (n=6): WSPHERE-SFC(D)=-Ap2r2R/6D z=2R 2R-z z=0 sphere z=R R x x z=2R dz planar surface z D+z z=0 z=0 ' D z=2R z=0 CHORD THEORUM : x2=(2R-z)z (*p. 143 Israelachvili) area=px2 volume=px2dz # of atoms=rpx2dz= rp(2R-z)zdz

Interaction Potential Between a Sphere and a Flat Surface z z=2R WSPHERE-SFC(D)= -A4p2r2R/(n-2)(n-3)(n-4)(n-5)Dn-5 Dispersion VDW (n=6): WSPHERE-SFC(D)=-Ap2r2R/6D 2R-z sphere z=R R x x dz planar surface z D+z z=0 ' D CHORD THEORUM : x2=(2R-z)z (*p. 143 Israelachvili) area=px2 volume=px2dz # of atoms=rpx2dz= rp(2R-z)zdz

The Hamaker Constant

Experimental Measurement of Van der Waals Interactions (*Israelachivili, page 197)

Retardation Effects in Dispersion Interactions

Derjaguin Approximations

Derjaguin Approximations