Chapter 2 ATOMIC STRUCTURE AND INTERATOMIC BONDING INTERATOMIC BONDS Comparison of Different Atomic Bonds INTERATOMIC BONDS (1) INTERATOMIC BONDS (2)
FA – attractive force is defined by the nature of the bond (e.g. Coulomb force for the ionic bonding)
FR – atomic repulsive force, when electron shells start to overlap
Thus the net force FN (r) = FA + FR
In equilibrium: FN (r0) = FA + FR =0 Let us consider the same conditions B but in the term of potential energy, E. E R r n By definition: E Fdr r r r E F dr F dr F dr E E N N A R A R dE if F 0 N 0 E has extremum N dr N
more specifical ly at r r0, FN 0 , system in equilibrium
and EN possesses minimum ELASTIC DEFORMATION
1. Initial 2. Small load 3. Unload
bonds stretch
return to initial F F Linear- elastic
Elastic means reversible! Non-Linear- elastic Strain Stress Versus Strain: Elastic Deformation
Typical Stress-Strain Diagram for one-dimensional tensile test Elastic Region
Hooke's Law: s = E e E [N/m2; GPa] is Young’s modulus or modulus of elasticity PROPERTIES FROM BONDING: E (1)
• Bond length, r • Elastic modulus, E
F F r
• Bond energy, Uo
Energy (r) = U(r) Elastic modulus
unstretched length F L r o = E r Ao Lo U o = “bond energy” PROPERTIES FROM BONDING: E (2)
• Elastic modulus, E Elastic modulus
F L = E Ao Lo
Energy
unstretched length
ro E is larger if Uo is larger. r smaller Elastic Modulus
larger Elastic Modulus Atomic Mechanism of Elastic Deformation
ro- equilibrium
E~(dF/dr)ro
Weaker bonds – the atoms easily move out from equilibrium position PROPERTIES FROM BONDING: E (3) • Elastic modulus, E Elastic modulus
F L = E Ao Lo
The “stiffness” (S) of the bond is given by: S=dF/dr=d2U/dr2 S (d2U/dr2 ) Energy o ro U
unstretched length • E ~ S=curvature of U at ro ro r smaller Elastic Modulus Show that E=(So/r0) larger Elastic Modulus Modulus of Elasticity for Different Metals Young’s modulus Young’s modulus is a numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction.
Higher E – higher “stiffness” YOUNG’S MODULI: COMPARISON
Graphite Metals Composites Ceramics Polymers Alloys /fibers Semicond 1200 1000 800 Diamond 600 Si carbide 400 Tungsten Al oxide Carbon fibers only Molybdenum Si nitride Steel, Ni CFRE(|| fibers)* E(GPa) 200 Tantalum <111> Platinum Si crystal Cu alloys <100> Aramid fibers only 100 Zinc, Ti 80 Silver, Gold Glass-soda AFRE(|| fibers)* 60 Aluminum Glass fibers only Magnesium, 40 Tin GFRE(|| fibers)* Concrete Composite data based on 9 20 GFRE* reinforced epoxy with 60 vol% 10 Pa CFRE* GFRE( fibers)* of aligned 10 Graphite 8 CFRE( fibers)* carbon (CFRE), 6 AFRE( fibers)* Polyester aramid (AFRE), or 4 PET PS glass (GFRE) 2 PC Epoxy only fibers. PP 1 HDPE 0.8 0.6 Wood( grain) PTFE 0.4
0.2 LDPE HOT TOPIC PROPERTIES FROM BONDING: a . • Coefficient of thermal expansion, a coeff. thermal expansion
L = a(T2-T1) Lo material CTE (ppm/°C) a is larger if Uo is smaller. silicon 3.2 alumina 6–7 copper 16.7 tin-lead solder 27 E-glass 54 S-glass 16 epoxy resins 15–100 silicone resins 30–300 • a ~ symmetry at ro Relationships between properties
. Expansion coefficient and melting point Engineering materials – the same dependence PROPERTIES FROM BONDING: TM
• Melting Temperature, Tm The melting point of a solid is the temperature at which it changes state from solid to liquid at atmospheric pressure. Energy (r) At the melting point the solid and liquid phase exist in equilibrium.
The Lindemann criterion states that melting is expected when the root mean square vibration amplitude exceeds a threshold value. Assuming that all atoms in a crystal vibrate with the same frequency ν, r the average thermal energy can be estimated using the equipartition theorem: o r where m is the atomic mass, ν is the frequency, u is the average vibration
amplitude, kB is the Boltzmann constant, and T is the absolute temperature smaller Tm If the threshold value of u2 is c2a2 where c is the Lindemann constant and a is the atomic spacing, then the melting point is estimated as larger T m
Tm is larger if Uo is larger *DNA melting temperature
The Tm is defined as the temperature in degrees Celsius, at which 50% of all molecules of a given DNA sequence are hybridized into a double strand and 50% are present as single strands. Note that ‘melting’ in this sense is not a change of aggregate state, but simply the dissociation of the two molecules of the DNA double helix. Relationships between properties
. Modulus and melting point Engineering materials – the same dependence SUMMARY: PRIMARY BONDS
Ceramics Large bond energy large Tm (Ionic & covalent bonding): large E small a
Metals Variable bond energy moderate Tm (Metallic bonding): moderate E moderate a
Polymers Directional Properties Secondary bonding dominates (Covalent & Secondary): small T small E large a SUMMARY: BONDING
Type Bond Energy Comments Ionic Large! Nondirectional (ceramics)
Variable Directional Covalent large-Diamond (semiconductors, ceramics small-Bismuth polymer chains)
Variable Metallic large-Tungsten Nondirectional (metals) small-Mercury Directional Secondary smallest inter-chain (polymer) inter-molecular