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CE 530 Molecular Simulation
Lecture 16
Dielectrics and Reaction Field Method
David A. Kofke
Department of Chemical Engineering
SUNY Buffalo
[email protected]
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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
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Capacitors
(Infinite) electrically conducting
parallel plates separated by vacuum
d
A = area of each plate
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Capacitors
(Infinite) electrically conducting
parallel plates separated by vacuum
Apply a potential difference V = f1 - f2
across them
f1
d
f2
A = area of each plate
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Capacitors
(Infinite) electrically conducting
parallel plates separated by vacuum
Apply a potential difference V = f1 - f2
across them
Charges move into each plate, setting
up a simple uniform electric field
+ + + + + + + + + + + + + + f1
E0  Vd eˆ z
d
- - - - - - - - - - - - - -
f2
A = area of each plate
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Capacitors
The total charge is proportional
to the potential difference
Q/ A
 f2  f1
d
Q  CV
Capacitance C 
0 A
(Infinite) electrically conducting
parallel plates separated by vacuum
Apply a potential difference V = f1 - f2
across them
Charges move into each plate, setting
up a simple constant electric field
d
+ + + + + + + + + + + + + + f1
E0  Vd eˆ z
d
- - - - - - - - - - - - - -
f2
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
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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
+ - + + - + + - + + - + + - + + - + + - + f1
E
- - + - - + - - + - - + - - + - - + - - + f2
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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
+ - + + - + + - + + - + + - + + - + + - + f1
E  Vd/  eˆ z
- - + - - + - - + - - + - - + - - + - - + f2
 Capacitance is increased
• C = Q/(V/) =  C0
• same charge on each plate, but potential difference is smaller
•  is the dielectric constant ( > 1 )
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Polarization
 The dielectric in the electric field exhibits a net dipole
moment M
+
+
+
+
+
+
+
 Every microscopic point in the
dielectric also will have a
- +- +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
P  E
  electric susceptibility
• this is a key element of linear response theory
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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
M   i
i 1
 In apolar systems, dipoles are induced
E
+
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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
• In general
-  pol    P
P1
P2
+- ++-+
++
Non-zero net
charge
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Surface Charge
 At the dielectric surface, the polarization is discontinuous
• i.e., it is non uniform
• non-zero net charge at surfaces
+++
+++
+++
+++
+++
+++
+++
+++
+++
+++
+++
Net negative charge
Net positive charge
• charge per unit area is the component of P normal to surface
 pol  P  nˆ
 Relation between physical constants
• dielectric constant  describes increased capacitance due to
surface charges
• electric susceptibility  describes polarization due to electric
field, and which leads to the surface charges
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Electrostatic Equations for Dielectrics 1.
 Basic electrostatic formula
E  
 Separate “free” charges from polarization charges
  E  ( free  pol )
 ( free    P)
 Apply linear response formula
  E   free    (  E) 
   (1   )E   free
 free relates to the electric field with no dielectric (vacuum)
  E0  free
 The relation follows
(1   )E  E0 
  1 
 E  E0 
Note on notation
 [=] 0  “permittivity”
 dim’less  “dielectric constant” or
“relative permittivity”
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Electrostatic Equations for Dielectrics
 Equations are same as in vacuum, but with scaled field
  ( E)  free   ( E)  0
constant 
 All interactions between “free” charges can be scaled
accordingly
• Coulomb’s law in a (linear) dielectric
F
q1q2
rˆ
2
r
q1 and q2 are free charges
 Dielectric boundaries
• free independent of dielectric
• implies continuity of E
I
II
I I
II II
but only normal component  E   E
E I  E II
n̂
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Thermodynamics and Dielectric Phenomena
 Fundamental equation in presence of an electric field
d (  A)  Ud    PdV   dN   E  dM
M = total dipole moment
 i
• Analogous to mechanical work PDV, creation of dipole moment M
in constant field E requires work E . DM = E . M
 More convenient to set formalism at constant field E
d  A  d  ( A  E  M )
Legendre transform
 Ud    PdV   dN  M  d  E
 Ud    PdV   dN  VP  d  E
P = M/V
 Electric susceptibility is a 2nd-derivative property
 Pz 
1   2 A 
 
  2

 Ez T ,V , N V  Ez 
take z as direction of field
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Simple Averages 4. Heat Capacity
(from Lecture 6)
 Example of a “2nd derivative” property
2
 E 
2   (  A) 
Cv   
 k  
2 
 T V , N



V , N
 k  2
 1
N
N
 E
dr
dp
Ee
 Q (  ) 
 Expressible in terms of fluctuations of the energy
2
Cv  k  2  E 2  E 


Note: difference between two O(N2)
quantities to give a quantity of O(N)
 Other 2nd-derivative or “fluctuation” properties
• isothermal compressibility
1 V
 T    
V  P T , N
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Statistical Mechanics
and Dielectric Phenomena
 Formula for electric susceptibility
 Pz 
1   2 A 
 
  2

 Ez T ,V , N V  E z 

1   1
N
N
N
  E  Ez M z 
dr
d

dp
M
e
e
z

V E z  Q ( E z ) 


 2
2
Mz  Mz 

V

  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
non-negligible electric field throughout
-
-
+
-
+
+
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-
+
+
+
• Local regions inside dielectric feel this field,
even though they have no polarization charge
• Surface charges arise when the system has
-+ -+ -+ -+ -+
uniform polarization
-+- -+- -+- -+- -+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!
+
+- +- +- +- +- + -+ - + -+ - + -+ - + -+ - + -+
+- +- +- +- +- + -+ - + -+ - + -+ - + -+ - + -+
+- +- +- +- +- + -+ - + -+ - + -+ - + -+ - + -+
+- +- +- +- +- + -+ - + -+ - + -+ - + -+ - + -+
+ + + + +
4 V  k 2 / 4
2
e

(
k
)
2
k 0 k
U q  12 
k = 0 term corresponds to this effect
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Boundary Conditions for Ewald Sum

- - - - -+-+ -+-+ -+-+ -+-+ -+-+
- +- +- +- +- +- -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ +-+-++ -++ -++ -++ -++ -++ -+ +- ++-+- -+- -+- -+- -+- -+- -+- -+- -+- +- +- + - + - + - + - + - -+ -+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ +-+- +-+- +-+- + - + - + - +- +- +- +- +- - + - + - + -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ +-+- +-+- +-+- +-+- + - + - +- +- + - + - + - + - + - - + - + - + - + -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +- +- +- +- +- +- +- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +- +- +- +- +- +- +- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +- +- +- +- +- +- +- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+ -+-+
+ + + + + +- +- +- +- +- + + + + +
-+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+
+- +- +- +- +-+-+ -+-+ -+-+ -+-+ -+-+
+ + + + +
-+-+
- +-+-+ -+-+
- +- +-+-+ -+-+ -+-+
- +- +- +-+-+ -+-+ -+-+ -+-+
+ + + +
+--++
+--+- +--++ +
+--+- +--+- +--++ + +
+--+- +--+- +--+- +--++ + + +
 Consider the simulated system
embedded in a continuum
dielectric
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Boundary Conditions for Ewald Sum
 Consider the simulated system
embedded in a continuum
dielectric
 An instantaneous dipole
fluctuation occurs often during
the simulation

M
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Boundary Conditions for Ewald Sum

- - - -
-
E
+
+
+
+ ++ + +
+
 Consider the simulated system
embedded in a continuum
dielectric
 An instantaneous dipole
fluctuation occurs often during
the simulation
 The polarization creates a
surface charge, and field E
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Boundary Conditions for Ewald Sum

-
-
-
+
+ ++ + +
+
-
- - - -
+
-
+
+
+
+
-
+
+ + ++ +
- - - -
 Consider the simulated system
embedded in a continuum
dielectric
 An instantaneous dipole
fluctuation occurs often during
the simulation
 The polarization makes a
surface charge, and field E
 The dielectric responds with a
counter charge distribution
-
-
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Boundary Conditions for Ewald Sum

-
-
-
+
+ ++ + +
+
-
- - - -
+
-
+
+
+
+
-
+
+ + ++ +
- - - -
 Consider the simulated system
embedded in a continuum
dielectric
 An instantaneous dipole
fluctuation occurs often during
the simulation
 The polarization makes a
surface charge , and field E
 The dielectric responds with a
surface charge distribution
 This produces a depolarizing
field that attenuates the surfacecharge field
-
-
Boundary Contribution to Hamiltonian
 The field due to the surface charge is
Ep 
2
P
3
complicated by spherical geometry
 The depolarizing field from the dielectric is
Ed  
4    1
P
3 2   1
 The contribution to the system energy (Hamiltonian) from its
interaction with these fields is
U pol  (E p  Ed )  P

2
3
 2(   1)  2
1 
P

2


1


 Common practice is to use conducting boundary, ' = 
U pol  0
Both effects cancel, and we can use the Ewald
formula as derived
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Reaction Field Method
 Alternative to Ewald sum for treatment of longrange electrostatic interactions
 Define interaction sphere  for each particle

 Direct interactions with other particles in 
 Treat material outside sphere as continuum
+

+
dielectric
• Dielectric responds to instantaneous dipole
+
inside sphere
• Contributes to energy as described before
U rf
 2(   1) 1
1 N
  i  
3
2 i 1
 2   1 rc

 j

ji
• Aim to make ' =  (results not sensitive to choice)
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Summary
 Dielectric constant describes increased capacitance associated
with insulator placed between parallel plates
 Insulator responds to an external field by polarizing
 Polarization leads to surface charge that offsets imposed field
 Charges at infinitely distant boundary have local effect due to
this offsetting field
 If the system is embedded in a conducting medium, it will
respond in a way the eliminates the surface-charge effect
 Susceptibility describes linear relation between polarization and
external field
• 2nd-derivative property related to fluctuations in polarization
 Reaction field method is a simpler and more efficient alternative
to the Ewald sum