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Transcript
3.012 Fundamentals of Materials Science
Fall 2003
Lecture 7: 09.24.03 Examples of work important in materials science and
engineering
Today:
LAST TIME ............................................................................................................................................................ 2
POLARIZATION OF MATERIALS1 ............................................................................................................................ 3
MAGNETIC WORK1 ................................................................................................................................................ 5
Types of magnetic materials .............................................................................................................................................................. 5
Work of magnetizing a paramagnetic material .................................................................................................................................. 5
Magnetic materials in materials science & engineering .................................................................................................................... 6
CHEMICAL WORK .................................................................................................................................................. 7
Examples of chemical work in single-phase systems ......................................................................................................................... 7
Chemical work and internal energy in multi-phase/multi-component systems .................................................................................. 8
REFERENCES ....................................................................................................................................................... 10
Reading:
Supplementary Reading:
Lecture 7 – forms of work
ALLEN AND THOMAS? ZEMANSKY?
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Last time

Thusfar, the only form of work we have discussed, for sake of simplicity, has been reversible work performed by
hydrostatic pressures. However, many other forms of work are important in the design and engineering of
materials- today we will discuss a few examples of these. Just as different materials require different terms to
account for all their forms of energy in the fundamental equation for internal energy, different forms of work have
different expressions for calculations with the first law.
Lecture 7 – forms of work
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Polarization of materials1

Dielectrics are materials where the molecules may orient or displace their center of positive and negative charge
in response to an electric field, but which cannot directly support an electric current because the electrons are not
free to leave the molecules. (Such materials are thus insulators). Materials in which positive and negative
charges can become slightly displaced from one another in the presence of an electric field are said to be
polarizable.
o The slight displacement of charge centers in dielectrics placed in an electric field gives rise to a net
charge on the external surfaces of the material.
(© W.C. Carter1)

Polarization of materials by an electric field is a form of work. The rate of work is given by:
o
Here, V is the volume of the system, E is the electric field, and D is the electric displacement. The
electric displacement measures the total polarization per unit volume. It is composed of 2 terms: the
contribution due to the polarization of the electromagnetic wave itself (o E ), and the contribution due to
displacement of charges within the material ( P ):

Lecture 7 – forms of work

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
Applying the equation for the electric displacement, we obtain for the differential of work:

This is a simple linear isotropic model for polarization in dielectric materials.
Examples of Dielectric materials

Provide some interesting examples Check Callister CD 18.19/18.20

Technological applications:
o Dielectrics are ubiquitous as capacitors in electronics.
o Other examples.
Lecture 7 – forms of work
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Magnetic work1

The work performed on magnetic materials by a magnetic field has similarities in form to the description of the
effect of an electric field on polarizable materials. One important difference is that electric dipoles always align
with the direction of the electric field, while magnetic dipoles (magnetic moments) in materials may align with an
external magnetic field or perpendicular to the magnetic field.
Types of magnetic materials
Ferromagnets
 These are the materials you commonly think of as magnets. They maintain a magnetization in the absence of an
externally applied magnetic field. Very few ferromagnetic materials exist; most ferromgnets contain iron, cobalt,
or nickel. Ferromagnets tend to align their magnetic moments with an externally applied magnetic field.
Non-ferromagnets
 Non-ferromagnets do not sustain their own magnetic field in the absence of an externally applied field. They are
sub-classed into paramagnetic and diagmagnetic materials:
o Paramagnetism: Magnetic dipoles induced in the material tend to align with the externally applied
magnetic field.
o Diamagnetism: Magnetic dipoles induced in the material tend to align anti-parallel to the externally
applied magnetic field.
Work of magnetizing a paramagnetic material

The work performed on a magnetic material by an externally applied magnetic field is given by:
dw  VH  dB
(Eqn 3)
o
Where V is the volume of the system, H is the applied magnetic field, and B is the magnetic induction.
EXPLAIN UNITS ON QUANTITIES!
o Analogous to the case of the electric displacement in polarizable materials, the magnetic induction B can
 be broken down as:
B  oH  M
(Eqn 4)
o
Where µo is the permeability of vacuum and relates the magnetization of empty space to the applied field,
and I is the induced magnetic field density in the system. Similar to the case of polarizing a dielectric
material, the first term is due to the magnetization of the applied field itself, and the second term
 measures the field induced in the material.
o The induced magnetic field density can be modeled as a linear response to the applied magnetic field:
M  oH
(Eqn 3)
o
The magnetic susceptibility  measures the tendency of the material to respond to the applied field with
formation of magnetic dipoles:

o
Expanding our definition for the magnetic induction above:
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B  o H  M
 o H (1  )
(Eqn 4)
 H
o
Where the µ is the permeability of the material, analogous to the permittivity in polarization. Finally, the
expression for the rate of work is:

(Eqn 5)
o
dw  VH  dH  VHdH
…where the second equality again arises for an isotropic material where the response is always exactly
aligned with the direction of the externally applied magnetic field.

Magnetic materials in materials science & engineering

Examples and their uses
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Chemical work

Chemical work in materials can occur when the internal energy of a system changes in response to changes in
the composition of an open system (remember: open system = system which can transfer matter or heat across
its boundaries). We have already mentioned the chemical potential, which is the driving force for chemical work.

What is chemical work and the chemical potential?
o
The chemical potential is a force resisting the addition or removal of molecules to a system; chemical
work is performed to move molecules against this force.
 In terms of ‘F dx’, a driving force multiplied by a displacement:
 µA = thermodynamic driving force on molecules of type A in the system
o the chemical potential drives a change in NA – physically, introduction or removal
of A molecules in the system
 dNA = ‘displacement’, a change in the number of molecules of type A in response to the
chemical potential

o
This term is analogous to the chemical component of the internal energy.
DISCUSS NATURE OF CHEMICAL POTENTIAL? SOURCE OF DRIVING FORCE?
Examples of chemical work in single-phase systems
Absorption of water by a sponge
o
Imagine a dry sponge placed into the middle of humid room.1 What will happen to the sponge? Is there a
specific force driving water vapor into the sponge from the air? What determines when the sponge stops
absorbing water?
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
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When the dry sponge is first placed in a humid room, a chemical potential exists driving water to
enter the sponge – dN has a positive value as water molecules are absorbed into the sponge
from the air. When an equilibrium amount of water is absorbed, uptake will stop.
Chemical work and internal energy in multi-phase/multi-component systems

Suppose now that we consider a closed multi-phase system, that cannot exchange molecules with its
surroundings. Our glass of water with ice in it will work as an example, if we seal the top of the glass. Even
though the system (ice + water) cannot exchange molecules with its surroundings, chemical work can be
performed by exchanging molecules between the phases present within the system, or creating new phases
within the system.
o
o

E.g., some of the ice could melt and become liquid water (change in the relative amounts of the two
phases)
Or, some of the liquid water could boil and become vapor (introduction of a new phase)
When a multiphase system is also comprised of multiple components, which is an important case in materials
science & engineering, then we can also have phases change their compositions by exchanging molecules:
Keeping track of the chemical potential in multi-component, multi-phase systems
n

A

When a material is comprised of the most general case, containing C components and P phases, then the
chemical work term contains contributions for each component in each phase. The general form for writing the
reversible chemical work is:

…where we have separate sums over every component in every phase in the material. The notation µkj refers to
the chemical potential of component j within phase k. The chemical work, as we will soon see, has the same form
as the chemical contribution to the internal energy. The sums account for the chemical energy parameters in
each phase of a material. For example, if we consider the two-component, two-phase system shown
schematically above, we have the following parameters:
 phase:
moles of A atoms in  phase
Lecture 7 – forms of work
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

A
 phase:
moles of A atoms in  phase
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n B
Fall 2003
n B
moles of B atoms in  phase
 A
 B
chemical potential of A atoms in  phase
chemical potential of A atoms in  phase

A
B
moles of B atoms in  phase
chemical potential of A atoms in  phase
chemical potential of A atoms in  phase


In addition to moving molecules around, the chemical
potential accounts for chemical reactions: changes of one
species into another or the appearance of a new
 species due to chemical reaction. We will discuss this last
important form of chemical work later in the term.
Lecture 7 – forms of work
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References
1.
Carter, W. C. (2002).
Lecture 7 – forms of work
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