3.012 Fundamentals of Materials ScienceFall 2003

Lecture 7: 09.24.03 Examples of work important in materials science and engineering

Today:

Last time......

Polarization of materials1

Magnetic work1

Types of magnetic materials......

Work of magnetizing a paramagnetic material......

Magnetic materials in materials science & engineering......

Chemical work......

Examples of chemical work in single-phase systems......

Chemical work and internal energy in multi-phase/multi-component systems......

References......

Reading: ALLEN AND THOMAS? ZEMANSKY?

Supplementary Reading:-

PLANNING:

Connections to structure- check Allen and Thomas, Zemansky

Halliday and Resnick Ch. 37

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.

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.
  • 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.
  • SKETCH ALIGNMENT OF POLAR MOLECULES TO CONTRAST WITH DISPLACEMENT SHOWN IN FIGURE BELOW

(© W.C. Carter1)

  • Polarization of materials by an electric field is a form of work. The rate of work is given by:

(Eqn 1)

  • Here, V is the volume of the system, is the electric field, and 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), and the contribution due to displacement of charges within the material ():

(Eqn 2)

  • o is the permittivity of free space and measures the polarization of an electromagnetic wave traveling though vacuum.  is the permittivity of the material (= o(1+)). The dielectric susceptibility, , measures the polarizability of the material.
  • Note that , , and are vectors, since polarization and electric fields have an associated direction.
  • Applying the equation for the electric displacement, we obtain for the differential of work:

(Eqn 1)

  • For an isotropic material, the electric displacement is always in the same direction as the electric field applied- thus the dot product .If the polarization work is reversible (no conversion of the energy to heat, and system stays in equilibrium), then the differential dw can be integrated to determine the total work in the process of polarization:

(Eqn 2)

  • 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:
  • Dielectrics are ubiquitous as capacitors in electronics.
  • Other examples.

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:
  • Paramagnetism: Magnetic dipoles induced in the material tend to align with the externally applied magnetic field.
  • 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:

(Eqn 3)

  • Where V is the volume of the system, H is the applied magnetic field, and B is the magnetic induction. EXPLAIN UNITS ON QUANTITIES!
  • Analogous to the case of the electric displacement in polarizable materials, the magnetic induction B can be broken down as:

(Eqn 4)

  • 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.
  • The induced magnetic field density can be modeled as a linear response to the applied magnetic field:

(Eqn 3)

  • The magnetic susceptibility  measures the tendency of the material to respond to the applied field with formation of magnetic dipoles:
  •  > 0dipole aligns with magnetic field(paramagnetic)
  •  < 0dipole aligns anti-parallel with magnetic field(diamagnetic)
  • Expanding our definition for the magnetic induction above:

(Eqn 4)

  • 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)

  • …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

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?
  • 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
  • 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
  • (Eqn 6)
  • 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
  • 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?
  • 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.
  • 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
  • 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:

(Eqn 5)

  • …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: phase:

moles of A atoms in  phasemoles of A atoms in  phase

moles of B atoms in  phasemoles of B atoms in  phase

chemical potential of A atoms in  phasechemical potential of A atoms in  phase

chemical potential of A atoms in  phasechemical 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.

References

1.Carter, W. C. (2002).

Lecture 7 – forms of work1 of 11 8/27/03