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1 CE 530 Molecular Simulation Lecture 16 Dielectrics and Reaction Field Method David A Kofke Department of Chemical Engineering SUNY Buffalo kofke eng buffalo edu 2 Review Molecular models intramolecular terms stretch bend torsion intermolecular terms van der Waals electrostatics polarization Electrostatic contributions may be very long ranged monopole monopole decays as r 1 dipole dipole as r 3 in 3D Need to consider more than nearest image interactions direct lattice sum not feasible Ewald sum approximate field due to charges by smearing them smeared charge field amenable to Fourier treatment correction to smearing needs only nearest neighbor sum 3 Capacitors Infinite electrically conducting parallel plates separated by vacuum d A area of each plate 4 Capacitors Infinite electrically conducting parallel plates separated by vacuum Apply a potential difference V 1 2 across them 1 d 2 A area of each plate 5 Capacitors Infinite electrically conducting parallel plates separated by vacuum Apply a potential difference V 1 2 across them Charges move into each plate setting up a simple uniform electric field 1 E0 Vd e z d 2 A area of each plate 6 Capacitors The total charge is proportional to the potential difference Q A f2 d 1 Q CV eA Capacitance C 0 d Infinite electrically conducting parallel plates separated by vacuum Apply a potential difference V 1 2 across them Charges move into each plate setting up a simple constant electric field 1 E0 Vd e z d 2 A area of each plate Dielectrics A dielectric is another name for an insulator non conductor Dielectrics can polarize in the presence of an electric field distribute charges nonuniformly Capacitor with a dielectric inside 7 Dielectrics A dielectric is another name for an insulator non conductor Dielectrics can polarize in the presence of an electric field distribute charges nonuniformly Capacitor with a dielectric inside dielectric sets up an offsetting distribution of charges when field is turned on 1 E 2 8 Dielectrics 9 A dielectric is another name for an insulator non conductor Dielectrics can polarize in the presence of an electric field distribute charges nonuniformly Capacitor with a dielectric inside dielectric sets up an offsetting distribution of charges when field is turned on 1 Capacitance is increased E V d e e z C Q V C0 same charge on each plate but potential difference is smaller is the dielectric constant 1 2 10 Polarization The dielectric in the electric field exhibits a net dipole moment M Every microscopic point in the have a dielectric also will dipole moment The polarization vector P is the dipole moment per unit volume in general P P r for sufficiently small fields P is proportional to the local E this is a key element of linear response theory P c E c electric susceptibility 11 Origins of Polarization Polarization originates in the electrostatic response of the constituent molecules In ionic systems charges migrate N M qiri i 1 In dipolar systems molecular dipoles rotate N r M mi i 1 In apolar systems dipoles are induced E 12 Polarization Charges Does polarization imply a nonzero charge density in the dielectric No For uniform P the charge inhomogeneities cancel Zero net charge But for nonuniform P there will be a net charge due to polarization Non zero net In general r pol P P1 P2 charge 13 Surface Charge At the dielectric surface the polarization is discontinuous i e it is non uniform non zero net charge at surfaces charge per unit area is the component of P normal to surface Net negative charge Net positive charge Relation between physical constants describes dielectric constant s pol P n increased capacitance due to surface charges electric susceptibility describes polarization due to electric field and which leads to the surface charges Electrostatic Equations for Dielectrics 1 14 Basic electrostatic formula E r Separate free charges from polarization charges E r free r pol r P free Apply linear response formula E r free c E free relates to the electric with no dielectric vacuum 1 c E field r free The relation follows E0 r free 1 c E E0 e 1 c eE E0 Note on notation 0 permittivity dim less dielectric constant or relative permittivity 15 Electrostatic Equations for Dielectrics Equations are same as in vacuum but with scaled field eE r free eE constant 0 All interactions between free charges can be scaled accordingly Coulomb s law in a linear dielectric q1q2 r 2 er boundaries F Dielectric q1 and q2 are free charges free independent of dielectric implies continuity of E I II but only normal component I II eI E eII E EPI EPII n 16 Thermodynamics and Dielectric Phenomena Fundamental equation in presence of an electric field d b A Ud b b PdV bmdN b E dM r m M total dipole moment i Analogous to mechanical work P V creation of dipole moment M in constant field E requires work E M E M More convenient to set formalism at constant field E Legendre transform d b A d b A E M Ud b b PdV bmdN M d bE P M V Ud b b PdV bmdN VP d bE Electric susceptibility is a 2nd derivative property Pz 1 2 A c 2 Ez V Ez T V N take z as direction of field 17 Simple Averages 4 Heat Capacity from Lecture 6 Example of a 2nd derivative property E 2 b A 2 Cv k b 2 T b V N V N k b 2 1 N N bE dr dp Ee b Q b Expressible in terms of fluctuations of the energy Cv k b 2 2 E E 2 Note difference between two O N2 quantities to give a quantity of O N Other 2nd derivative or fluctuation properties isothermal compressibility kT 1 V V P T N 18 Statistical Mechanics and Dielectric Phenomena Formula for electric susceptibility Pz 1 2 A c 2 Ez V Ez T V N 1 1 N N N b E b Ez M z dr d w dp M e e z V Ez Q E z b 2 M z2 M z V b 2 2 M M DV Susceptibility is related to fluctuations in the total dipole moment this is sensible since describes how loose the charges are how easily they can appear orient in response to an external field Connection to Ewald Sum Surface charges on dielectric produce a nonnegligible electric field throughout Local regions inside dielectric feel this field even though they have no polarization charge Surface charges arise when the system has uniform polarization 19 dipole moment per unit volume In simulation of polar systems at any instant the system will exhibit an instantaneous dipole This dipole is replicated to infinity But in principle it stops at some surface Should the resulting field influence the simulated system The boundary conditions at infinity are relevant 2 4pV 2 U q 12 2 e k 4a r k k 0 k k 0 term corresponds to this effect


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