Download Bonding in solids 1

Bonding in solids
Many different types of interaction are important
– electrostatic (ionic)
– covalent
– Van de Waals
Ionic bonding favors high symmetry structures
with high coordination numbers
Covalent bonding favors low coordination
numbers
Bonding solids overview
Ionic model
–
–
–
–
More realistic approaches
–
–
–
–
Ionic radii
Electrostatic bond strength
Radius ratio rules
Lattice energies
» Born-Harber cycles
Mooser-Pearson plots
Bond valence bond strength correlations
Crystal Field Stabilization Energy (CFSE)
Inert pair effect
Band theory
Ionic model
In
its simplest form it treats ions as hard spheres of
well defined size
– More sophisticated treatments allow that the ions are not
hard spheres and that they do not have a precisely defined
size
In
reality, ions such as Al3+ and O2- do not have +3
and –2 charges in the solid state
While the ionic model is not a very realistic picture it
is simple and it can provide a useful guide to
structural and thermodynamic trends
Ionic radii
The distances between nearest neighbor anions and
cations can be considered to be the sum of an anion
and cation radius
How do we determine the radii?
– many different methods used
– preferred method uses a crystallographic determination of
electron density
Different methods give different answers
– never mix values of radii from different sources
The experimental electron density
distribution in LiF
Electron density variation
between Li+ and F-. Note the
variation has a very flat bottom.
M, G and P indicate the true
minimum and the Goldshmidt and
Pauling ionic radii.
The variation of ionic size with
coordination number
From Shannon and Prewitt,
Acta Cryt. B25, 725 (1969) and
B26, 1046 (1970).
Data based on rF- = 1.19Å and
rO2- = 1.26Å
Trends in ionic radii
•
Ionic radii increase going down a group
Ionic radii decrease with increasing charge for any isoelectronic
series of ions
• Na+, Mg2+, Al3+, Si4+
Ionic radii increase with increasing coordination number
Ionic radii decrease with increasing oxidation state
For lanthanide 3+ ions with a given coordination number there is
a steady decrease in size on going across the series left to right
• Lanthanide contraction
A similar decrease in size is seen on going across transition
metals series but it is not always a smooth decrease
General principles of ionic bonding
Ions are charged elastic spheres
Held together by electrostatic forces so cations are surrounded
by anions and vice versa
In order to maximize the attractions, cations are surrounded by
as many anions as possible provided that the cation maintains
contact with all the anions
Next nearest neighbor interactions are repulsive. So ionic
structures tend to have a high symmetry and the maximum
volume possible
– This minimizes the repulsions
Structures are locally electrically neutral
– The valence of an ions is equal to the sum of the electrostatic bond
strengths between it and all of its opposite charge neighbors
Electrostatic bond strength
For a cation Mm+ surrounded by n anions, Xx-, the
e.b.s. is given by
– e.b.s. = m / n
For the anion, the cation e.b.s. must balance the
charge on the anion
– Σ (m / n) = x (sum over nearest neighbors)
This rule precludes certain structures
– you can never have three SiO4 units sharing a common
corner
Use of ebs
ebs
can be used to rationalize why some types of
polyhedral linkage do not occur
What determines a crystal structure?
We are not currently able to predict the structure
of new complex materials with certainty
There are some tools available to help us make an
educated guess for simple materials
There is a large amount of structural data that can
be used as a guide
Structure prediction
The so called “Radius ratio rules” are often used to
make predictions of preferred structure type or to
rationalize an observed change in structure type
Just about the simplest approach possible
Radius ratio rules make use of idea that cation will
have as many anions around it as possible as long as
the cation can still touch all of the anions
Radius ratio rules
It is possible to predict the type of ion
coordination that you will get if you know the
ratio of the cation to anion size
r+/r- values
> 0.732
Preferred coordination
number
8 – cubic coordination
0.414 – 0.732
6 – octahedral coordination
0.225 – 0.414
4 – tetrahedral coordination
How the limiting values are calculated
Use of radius ratio rules
Can be used to explain trends
Failure of radius ratio rules
Not reliable for
absolute prediction
- Ions are not hard spheres,
ion size varies with
coordination number,
radius ratio varies
depending whose ionic
radii you use
Distorted structures
The radius ratio rules were based on the notion
that structures were unstable if the cations could
rattle around inside their coordination polyhedra
– this is not universally valid
BaTiO3, PbTiO3, LiNbO3, KTiOPO4 (KTP) etc.
have ions that can rattle
Lattice energies and the
prediction of structural stability
It
is possible to calculate the thermodynamic
stability (∆Hf not ∆Gf) of an ionic solid using
relatively simple thermodynamic arguments
– As part of this calculation we have to now the
materials lattice energy
» Related to forces holding solid together
Lattice energy
For an ionic compound the lattice energy is
defined as the energy needed to break up
the solid into its constituent ions in the gas
phase
– MX(s) -----> M+(g) + X-(g)
Determined by a combination of long range
electrostatic interactions and short range
repulsions
Attractive and repulsive interactions
There are electrostatic interactions between
every pair of ions in the solid
– electrostatic energy = -Z1Z2e2/4πε0d
» overall electrostatic interaction energy for an ionic solid
is always favorable
The repulsive interactions are short range in
nature
– repulsive energy = b/dn
» n is usually quite large ~10
The balance between repulsive and
attractive forces
Madelung constant
The exact value of the electrostatic
component of the energy depends upon the
crystal structure
– For NaCl structure energy of one ion
» PE = -6e2/4πε0d + 12e2/1.41x4πε0d 8e2/1.71x4πε0d + 6e2/2x4πε0d - ...
» PE = -Ae2/4πε0d
A is the Madelung constant and depends upon the crystal
structure
The NaCl structure
Madelung constants
Total lattice energy
PE = -Ae2Z1Z2/4πε0d + B/dn
For a crystal at equilibrium the distance
between neighboring ions, d0, will be the
one that gives the lowest PE
U0 = NAZ1Z2e2(1 -1/n)/4πε0d0
– n is readily estimated so lattice energies can be
easily calculated using simple arguments
The Born - Lande equation
Total interaction energy between ions
– U = - e2 Z+ Z- N A / r + BN / rn
– to get equilibrium value differentiate with respect to r
and set dU/dr = 0
– other functional forms of repulsive part are sometimes
used
Lattice energy is
– U = - [e2 Z+ Z- NA / re ] (1 - 1/n)
– n can be experimentally determined
The Born -Meyer equation
Born Meyer equation is obtained when the
repulsive potential takes the form,
– V = B exp (-r/ ρ)
Born - Meyer equation
– U = - [e2 Z+ Z- NA / re ] (1 - ρ / re)
The Kapustinskii equation
Kapustinskii noticed that A / ν, is almost constant
for all structures
– ν is the number of ions in the formula unit
Variation in A / ν with structure is partially
canceled by change in ionic radii with
coordination number
U = [1200 ν Z+ Z- / (r+ + r-)][1 - 0.345/(r+ + r-)]
Extended calculations
Include zero point vibrational motion
Heat capacity of the solid
Van der Waals forces
Total correction ~ 10 kJ mol-1
Thermodynamic data for the
alkali metal halides
∆Uc = coulomb term, ∆uB Born repulsive term, ∆ULdd = London dipole-dipole term,
∆ULdq = London dipole-quadrupole term ∆UZ = zero point term
Trends in lattice energies
Lattice
energies go up as the charge on the ions go up
Lattice energies go up as the size of ions decreases
Effect of covalency
The
experimental lattice energies for
compounds that have a significant covalent
contribution to their bonding are often in poor
agreement with those calculated using the ionic
model
The use of lattice energies
Can be used to estimate heats of formation for
compounds
– is an unknown compound likely to be stable?
– will a compound disproportionate?
Can be used to estimate electron affinities
Can be used to estimate thermochemical radii
All these applications make use of a Born-Harber
cycle
The formation of ionic compounds
Energies of formation can be calculated by
considering the process of formation to occur in a
distinct series of steps
Consider forming NaCl(s)
–
–
–
–
–
atomize the metal
dissociate chlorine molecules
ionize the sodium
form ions from the chlorine atoms
bring the ions together to form solid NaCl
Born-Harber cycles
This step wise approach is often shown diagramatically
The stability of compounds
Born-Harber cycles along with lattice
energy calculation and experimentally
measured quantities such as ionization
energies allow the calculation of enthalpies
of formation for compounds that have never
been made
This allows you to rationalize why some
compounds form and others do not
Magnesium fluorides
Why is MgF2 the only stable magnesium fluoride ?
Enthalpy contributions MgF
(kJmol-1)
MgF2
MgF3
Mg atomization
+150 +150
+150
F-F bond energy
+80
+240
Mg ionization
+740 +219
0
+9930
F electron affinity
-330
-660
-990
Lattice energy
-900
-2880 -5900
∆Hf
-260
-1040 +3430
+160
So 2MgF(s) MgF2(s) + Mg(s) ∆H = -520 kJmol-1
Enthalpy of formation for MgF(s)
Mg(g)
∆H atomization
0.5F2(g) F(g)
0.5 bond enthalpy
Mg(g) Mg(g)+ + e- 1st ionization enthalpy
F(g) + e- F(g)electron affinity
Mg(g)+ + F(g)- MgF(s) minus lattice energy
Add these up
Mg(s)
– Mg(s) + 0.5F2(g) MgF(s) Enthalpy of formation
Making compounds containing ions with
unusually low or high oxidation states
Compounds
containing cations in usually low
oxidation states are often unstable with respect to
disproporationation
– This tendency can be minimized by reducing the lattice
energy of the compound with the cation in the higher
oxidation state
» Use large anion
Compounds
containing cations in usually high
oxidation states are often unstable with respect to
decomposition giving a compound in a lower
oxidation state
– Maximize lattice energy of high oxidation state compound
» Use small high charge anion
Thermochemical radii
How do we obtain an ionic radius for an ion such as
CO32- ?
Measure heat of formation of carbonate compound
Estimate lattice energy using Born-Harber cycle
Calculate ionic radius using Kapustinskii equation
Electron affinities
We can not measure some of the required electron
affinity data directly
2e-(g) + S(g) --------> S2-(g)
2e-(g) + O(g) --------> O2-(g)
However, we can use a Born-Harber cycle to
estimate the electron affinity if we know all of the
other terms
Prediction of thermal stability
MCO3(s) -------> MO(s) + CO2(g)
The decomposition temperature depends
upon T= ∆H0 / ∆S0
∆H0 can be calculated with the help of some
lattice energy data
Solid State Metathesis Reactions can
be Very Exothermic
MoCl5(s) + 5/2Na2S(s) ---> MoS2(s) + 5NaCl(s) + 1/2S(s)
Reaction reaches 1050 ºC and is over in 300 ms
Empirical structure prediction
Radius ratio rules do not work very well
Find some other simple way of predicting
structure
– search database of known compounds looking for
features that allow us to predict structure
– can create stability field diagrams based on:
» ion size
» electronegativity and average principle quantum number
Stability field diagram for MX structures
Stability field diagrams for MX2 structures
Stability field diagram for A2BO4 structures
Stability field diagram for AIIIBIIIO3
structures (size only)
Stability fields for AIIIBIIIO3 structures
(size and ionicity)
Bond valence sum rules
Pauling’s e.b.s concept was a first step
towards associating a bond (cation anion
distance) with a valence
Other workers (particularly Brown)
expanded the concept
– calculate a valence associated with every cation
anion distance in a structure.
– For a particular anion or cation these should
sum to give the formal valence of the species
Bond valence and bond length
It seems reasonable that bond strength
should correlate with bond length
The form of the bond strength bond
length relationship
Two commonly used forms
– s = (r / ro)-N
– s = exp [(ro - r)/ B]
The second functional form is superior as B is
roughly the same for all cation anion pairs
– only one parameter, ro, for a given anion cation pair
see Brown and Altermat, Acta Cryst. B41, 244 (1985)
Applications of bond-valence bondlength relationships
Can be used to check new structures
Can be used to locate missing atoms
– good for things like hydrogen that are not easily
located using X-ray techniques
Can be used to examine site occupancies
– aluminosilicates
Non-bonding electron effects
The
structures and stability of many transition metal
containing solids are effected by the d-electron
configuration of the metal ion
– Preference for a particular site geometry due to Crystal
Field Stabilization Energy (CFSE) effects
– Distortions due to Jahn-Teller effect
The
structures and properties of many compounds
containing heavy post transition metal ions are effects
by the presence of a stereochemically active lone pair
Crystal Field Theory
Consider the ligands are point negative
charges or as dipoles. How do these charges
interact with the electrons in the d-orbitals?
Octahedral complexes
Two of the d-orbitals point towards the ligands
– Repulsion between the ligand electrons and electrons in
these two d-orbitals destabilizes them
Crystal field splitting
The crystal field splitting depends upon the
oxidation state of the metal, which row the
metal is from, and the ligand type
High oxidation state favors large ∆
Trend in ∆ is usually 5d > 4d > 3d
Effect of ligand is given by the spectrochemical
series
» I-< Br-< Cl-< F-< OH-< OH2< NH3< en< CN-< CO
High spin and low spin complexes
HS versus LS is determined by the relative size of
the ligand field splitting and the pairing energy
If ∆ is bigger than the pairing energy the complex
will be low spin
Tetrahedral complexes
Three of the d-orbitals point almost towards the
ligands. The other two point between the ligands
– Repulsion between the ligand electrons and electrons in
the three d-orbitals that almost point at the ligands
destabilizes them
Square planar complexes
Magnetic properties
The loss of degeneracy of the d-orbitals due
to crystal field splitting explains why some
complexes are diamagnetic and others are
paramagnetic
– e.g. Ni(CN)42- (square planar) is diamagnetic
– but NiCl42- (tetrahedral) is paramagnetic
Hydration energies
The double humped trend that is seen in the hydration
enthalpies of TM ions can be explained using the
Crystal Field Stabilization Energy
CFSE for high spin d4 is
= (+3/5 – 2/3 – 2/3 – 2/3)∆
Ionic radii for 3d metals
For
high spin ions
there is a “double
humped” trend in
ionic radii
– Due to crystal field
stabilization effects
Lattice energies of 3d oxides MO
Double
humped
trend due to CSFE
and high spin ions
CFSE and coordination preferences
The
CFSE for octahedral and tetrahedral sites is
different and the magnitude of the difference varies
with d-electron configuration
– Some metal ions show a strong preference for octahedral
coordination due to CFSE effects
Degree of inversion in Spinels
AB2O4 materials with the Spinel structure have one tetrahedral
and two octahedral sites per formula unit
The fraction of the A cations that are found in the octahedral sites
is referred to as the degree of inversion γ
– If all A cations are octahedral the material is an inverse Spinel, and if all A
cations are tetrahedral the Spinel is said to be normal
– Degree of inversion can often be rationalized using CFSE arguments
Inert pair effect
Many
heavy main group cations
that are in an oxidation state two
less than that normally displayed by
other group members show highly
distorted coordination environments
– Tl+, Pb2+, Sn2+, Sb3+, Bi3+
– Ions have a lone pair that can be
stereochemically active
» Lone pair occupies space around ion just
as lone pair on ammonia does
Coordination environment
of Pb2+ in PbO
Jahn-Teller effect
Jahn-Teller
theorem states that any species with an
electronically degenerate ground state will distort to
remove the degeneracy
– Compounds containing approximately octahedral Cu2+ (d9t2g6eg3), Mn3+ (d4 - t2g3eg1) and L.S. Ni3+ (d7 - t2g6eg1) often
display distorted coordination environments as the
distortion breaks the degeneracy of the octahedral ground
state
Jahn-Teller effect 2
Typical
distortion of an octahedron leads to 4 + 2
coordination with either 2 short or 2 long bonds
JT effect important in copper oxide superconductors
and manganese CMR materials
JT effect can also occur for tetrahedrally coordinated
species
JT effect not very strong for “octahedral” compounds
with degenerate ground state involving incomplete
occupancy of t2g orbitals
Examples of the Jahn-Teller effect
CuF2
has a distorted rutile structure
CuO shows almost square planar
coordination of Cu2+
Cs2CuCl4 shows a flattened tetrahedral
coordination of Cu2+ due to JT effect