Class 5Class 5 –– Secondary BondsSecondary BondsClass 5Class 5 –– Secondary BondsSecondary BondsClass 5 Class 5 –– Secondary BondsSecondary BondsClass 5 Class 5 –– Secondary BondsSecondary Bonds

READING•Ibid.

Prof. M.L. Weaver

SECONDARY BONDINGSECONDARY BONDINGSECONDARY BONDINGSECONDARY BONDINGArises from interaction between dipoles•  Fluctuating dipoles

asymmetric electron ex: liquid H 2asymmetric electronclouds

+ ‐ + ‐ HH HH

H2 H2ex:  liquid H 2

• Permanent dipoles‐molecule induced

Adapted from Fig. 2.13, Callister & Rethwisch 8e.

secondary bonding

HH HHsecondary bonding

•  Permanent dipoles‐molecule induced

‐general case:Adapted from Fig. 2.15,Callister & Rethwisch 8e

secondary bonding

+ ‐ + ‐

‐ex: liquid HCl

Callister & Rethwisch 8e.

H Cl H Clsecondary bonding

Prof. M.L. Weaver

102

‐ex: polymer secondary bonding

Dipole ForcesDipole Forces Electrostatic interactions 

between adjacent molecules are called Dipole Forcesp

The most extreme form of di l  f   lt f  th  dipole forces result from the interaction of a hydrogen atom with a highly l i    electronegative atom resulting in hydrogen bonding.

Dispersion Forces (or Van der Waals forces)Dispersion Forces (or Van der Waals forces)

Interactions resulting from momentary concentration variations in the electron clouds of adjacent atoms

Th  i t ti    b i f  d  The interactions are brief and weak, but large molecules (such as polymers) have 

i i  f    opportunities for many simultaneous interactions

LL--J Potential Energy Model (continued)J Potential Energy Model (continued)

•To calculate the equilibrium separation (ro) between two atoms, or lattice constant, we differentiate equation (1) to find the minimum of Vo(r):

(2)7

613

12 16411240

rrdrdV

F o

•Solving for r allows ro to be determined: thus it’s the size parameter, , that determines ro.E i l t t i th f f ti F( ) i th ti f th d i ti t f E (1)

12.12 6/1 or

( )

(3)

rrdr

•Equivalent to saying the force function, F(r), is the negative of the derivative w.r.t. r of Eq. (1).

•Figure on left for inert gases shows this relationship.•The first derivative of the interatomic potential energy, with

di i h f di l Threspect to distance, gives the force-displacement curve. The positive forces are attractive and the negative forces are repulsive, and the zero force point is the equilibrium separation (ro).p ( o)•Good agreement between calculated and measured ro’s.•The L-J potential is a relatively good approximation and due to its simplicity often used to describe the properties of gases, and to model dispersion and overlap interactions in molecular

Prof. M.L. Weaver

and to model dispersion and overlap interactions in molecular models. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules, layered solids (graphite) and polymers.

LL--J Potential Energy Model (continued)J Potential Energy Model (continued)

•To calculate the cohesive energy per atom, U´, in eV/atom, we sum the potential energy (V) between each atom and all the other atoms in a crystal. The interaction energy, Uj, of the j-th atom with all the other N-1 atoms in the crystal is: where ri is distance to the i-th Natom with all the other N 1 atoms in the crystal is: where ri is distance to the i th atom.

•There is a similar term for every atom in the crystal, so the complete sum, U, is:F t f ½ i i l d d t id ti h i t ti t i

)(,1

N

jiiij rVU

N

jUU21

(4)

(5)Factor of ½ is included to avoid counting each interaction twice.•Assuming every atom is indistinguishable, we can rewrite the complete sum:

•Then cohesive energy per atom, U´ is total energy in eq. (6), divided by N:

j

j12

jNUU21

( )

(6)

1gy p , gy q ( ) y

•How do we actually compute the sum, Uj, in eq. (4) or eq. (7)?•We use simplest approximation first, i.e., since VDW forces are very weak, the nearest neighbor shells will significantly contribute to cohesive energy Thus we only need to compute the sum

jUU21' (7)

shells will significantly contribute to cohesive energy. Thus, we only need to compute the sum over the nearest neighbors. For our previous example, all inert gases have face-centered cubic (FCC) structure, so cohesive energy per atom is:

)(21'

12

1

i

irVU (8) or )(..21' rVNNU

Prof. M.L. Weaver

•For all 12 neighbors, V(ri)=V(ro)=-. Substitute into eq.(8), cohesive energy per atom:•We can also calculate this U’……•This approximation slightly underestimates the measured cohesive energy, since we have ignored all additional interactions (long range) with atoms outside of the nearest neighbor shell.

i

6'U(9)

FCC crystal structure FCC crystal structure representative of inert gasesrepresentative of inert gasesep ese ve o e g sesep ese ve o e g ses

# nearest neighbors or coordination number is 12:

FCC structure :FCC structure :

a2

The extendedcubic F lattice

Close-packed directions:

length = 4R = 2 a

a

1 2g

Pair energy (each atom’s contribution),Each atom is partitioned

1 2

Prof. M.L. Weaver

•This is a cubic face (F) center Bravais lattice. •Lattice + basis vectors = crystal structure.•12-fold coordination of each lattice point (same atom) is identical.

LL--J Potential Energy Model (continued)J Potential Energy Model (continued)

•To correct errors from our neglect of long range interactions, we start by expressing all of the interatomic distances, ri, as multiples of the shortest one, ro.

•Thus eq. (8) becomes: therefore determining cohesive energy/atom:

)(21'

1

N

ioi rVU (10)

N

ir

N

ir ii

U1

6

1

122

(11) is a dimensionless parameter dependent on crystal structure

amounts to evaluating the sums:

•Minimizing eq. (11) to find equilibrium separation (ro) results:

12

112 /1

N

iiA

6

16 /1

N

iiA (12)

6/1

122

AAro (13)g q ( ) q p ( o)

•For FCC structure, A12=12.13 and A6=14.45, thus ro=1.09which differs from the nearest neighbor estimate we calculated in eq. (3), ro=1.12, by only 3%.

6

Ao

•When the corrected value for U′ is computed using new value for ro & eq.(11): 6.8U (14)

Prof. M.L. Weaver

LL--J Potential Energy Model (continued)J Potential Energy Model (continued)•The differences between the values computed using the N.N. and L.R. interaction models are summarized:

parameter nearest neighbor

long range differenceneighbor

ro 1.12 1.09 3%

U´ -6 -8.6 30%

•What this means for FCC structures in terms of U´ is 12 N.N. only contribute 70% of the total cohesive energy, and final 30% is supplied by the rest of crystal (next N.N. and so on…..).

•Relatively good agreement between measured and calculated parameters for inert gases.

•The L-J pair potential model has been successfully used to describe bonding in other systems.•For example the interaction between different segments on polymers: one LJ particle may•For example, the interaction between different segments on polymers: one LJ particle may represent a single atom on the chain (explicit atom model), a CH2 segment (united atom model), or segment consisting of several CH2 units (coarse-grained model). The united atom model has been shown to successfully reproduce explicit atom results for polymer melts.

Prof. M.L. Weaver

Significance of LSignificance of L--J Potential ModelJ Potential Model•While the L-J pair potential eq (1) is intended as a physical model for VDW bonds it is often•While the L-J pair potential, eq. (1), is intended as a physical model for VDW bonds, it is often extended empirically to model other types of crystals by adjusting the constants and .•Assuming that regardless of physical mechanism governing interactions among atoms, the total energy of the system can be treated as sum of attractive and repulsive pair-wise contributions.•Constants and are chosen such that pair-wise potentials reproduce known properties with accuracy. •Empirical models constructed in this way can then be used to compute physical properties that are difficult, tedious or impossible to measure by experiment. While quantitative accuracy canare difficult, tedious or impossible to measure by experiment. While quantitative accuracy can not be expected, the relative energies computed from such models are often qualitatively meaningful and instructive.•For example, a L-J model for Cu (FCC metal) was used to compute the surface energy/unit area ( ) f ti f f l Th f •Anisotropic is proportional to # bonds broken/area:() as a function of surface normal. The surface energy can be defined as the work required for the creation of a unit area of surface. Creating new surfaces requires bond breaking. For a surface with a unit cell that contains

Anisotropic is proportional to # bonds broken/area:

(111)<(001)<(110)

gN atoms in an area A, the is:where U´ is bulk potentialenergy/atom, Vo is the L-J potential defined in eq (1) and r is distance between ith

A

rVUNi j

ijo

1

)(

(15)

Prof. M.L. Weaver

potential defined in eq. (1) and rij is distance between ith

and jth atom. Creating a surface breaks symmetry of the lattice so positions do not have same environment, thus need to compute sum over atom pairs.

Return to Ionic Bonding ModelReturn to Ionic Bonding Model(general model)(general model)

rZZke

c rV 212

)( •The pair potential for an ionic bond is to add the attractive Coulomb potential energy:to L-J potential, eq.(1) from last class, to get:

61221

24)(

rrrZZkerV

(1)

(g )(g )

•The total lattice energy is written by modifying eq.(1) to include the Madelung constant (), n1and n2 stoichiometry, and the number of nearest neighbor (NN) contributions to L-J portion of the energy [long range attractive contributions can be added but in most cases are negligible for ionic bonding].

rrr (1)

•New expression for lattice energy as a function of separation for binary crystals is: •Crystal structure sensitive parameters are and # of NN.•Compound chemistry sensitive parametersare Z Z and

6122121

24

2)()(

rrNN

rZZnnkerV

(2)

are r, Z1, Z2, n1, n2, and .•Since varies over a small range, variations in the chemistry of the compound affect total lattice energy much more than variations in the structural configuration.

•Appropriate pair potentials for ionic materials must obviously contain an electrostatic component a short range repulsion and an attractive VDW component

component, a short-range repulsion and an attractive VDW component.•Ex.: Compute the equilibrium spacing (ro) of the Na and Cl atoms in NaCl and the lattice energy, V(r), of the crystal. Compare the computed values with the known values (next page).

Prof. M.L. Weaver

Return to Ionic Bonding ModelReturn to Ionic Bonding Model(practical model: Born(practical model: Born--MayerMayer--Huggins form of potential energy)Huggins form of potential energy)

•In the physical description of the ionic bond we used a term from the L-J model to account for short-range repulsions. This was chosen mainly for consistency. However, in practice, it is common to use a repulsive parameter of the form: similar in form to ijij /r

ijij eA)r(V (3)our previously defined repulsive force:

Aij and ij are empirically derived parameters for specific atom pairs (Table 7.17 below).

•The parameters are chosen so that a model using these values reproduces known properties, with th ti th t th d l ill th b bl f ti k ti

ijijrep eA)r(V ( )

/rR eF

CRC Handbook of Chemistry and Physics

the assumption that the same model will then be capable of computing unknown properties. •When combined with an electrostatic attractiveterm, this is the Born-Mayer-Huggins (BMH) form of the potential energy.

=ro (Å) =ro=V =V( ) f f p gy•Simple calculations can be carried out by hand,while calculations involving more complex structures or defects are usually carried out by computers using the BMH potentialcomputers using the BMH potential.

[39]=G.V. Lewis, Physica 131B (1985) 114.[40]=J.R. Walker and C.R.A. Catlow, J.Phys.C, 15 (1982) 6151.

Prof. M.L. Weaver

“attractive VDW component”

Return to Ionic Bonding ModelReturn to Ionic Bonding Model(BMH form continued)(BMH form continued)( )( )

•If we combine the repulsive parameter, eq.(3), with the Madelung attractive energy (ignoring VDW attractive part for the moment), we get the BMH form of the potential energy:

ijrijeNNA

rZZnnkerV /2121

2

2)()(

(4)

•We can compute the equilibrium lattice constant (separation), (ro) and lattice energy V(ro).•For example, from Table 7.17, a) calculate ro and V(ro) for MgO using only nearest neighbor repulsions how do the values compare with Table 7 1 in Rohrer

r2

repulsions, how do the values compare with Table 7.1 in Rohrer.

O-O repulsions and the attractive VDW component (-Cij/r6) from L-J model [secondary effects] have little influence on the equilibrium lattice constant (separation) (r ) and lattice energy V(r )have little influence on the equilibrium lattice constant (separation), (ro) and lattice energy V(ro).

•Ionic pair potential calculations can be used to determine defects (point and area) in ionic compounds, e.g., NiO grain/tilt boundaries have different surface energies than inside

i l i f i di d i l f FCC C

Prof. M.L. Weaver

grains; analogous to computing surface energies we discussed previously for FCC Cu.

Cohesive Energy (U′) and Cohesive Energy (U′) and InteratomicInteratomicSeparation Distance (dSeparation Distance (doo) in Metallic Bonding) in Metallic BondingSeparation Distance (dSeparation Distance (doo) in Metallic Bonding) in Metallic Bonding

=ro =ro

Prof. M.L. Weaver

Summary: BondingSummary: Bonding

TypeIonic

Bond EnergyLarge!

CommentsNondirectional (ceramics)Ionic

Covalent

Large!

Variable

Nondirectional (ceramics)

Directionallarge-Diamondsmall-Bismuth

(semiconductors, ceramicspolymer chains)

Metallic Variablelarge-Tungstensmall Merc r

Nondirectional (metals)

Secondary

small-Mercury

smallest Directionalinter chain (polymer)

Chapter 2 - 115

inter-chain (polymer)inter-molecular

Summary: Primary BondsSummary: Primary BondsCeramics

(Ionic & covalent bonding):

Large bond energylarge Tmlarge E

llsmall

Metals Variable bond energymoderate T(Metallic bonding): moderate Tmmoderate Emoderate

Polymers

(Covalent & Secondary):

Directional PropertiesSecondary bonding dominates

small T( y)

small Tmsmall Elarge

Chapter 2 - 116