3.012 Fundamentals of Materials Science
Fall 2003
Lecture 24: 12.01.03 Lattice Models of Materials
Today:
LAST TIME .............................................................................................................................................................................................. 2
LATTICE MODELS FOR TRANSLATIONAL DEGREES OF FREEDOM.............................................................................................................. 3
Lattice model of the ideal gas ............................................................................................................................................................ 3
LATTICE MODELS OF SOLID SOLUTIONS: THE REGULAR SOLUTION ....................................................................................................... 6
The free energy of mixing................................................................................................................................................................... 6
Entropy of mixing in the lattice model and the ideal solution ............................................................................................................ 6
The energy of mixing .......................................................................................................................................................................... 8
REFERENCES ........................................................................................................................................................................................... 9
Reading:
Supplementary Reading:
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Planning Notes:
HOMEWORK PROBLEMS:
Lecture 24 – Lattice Models of Materials
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3.012 Fundamentals of Materials Science
Fall 2003
Last time
Lecture 24 – Lattice Models of Materials
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3.012 Fundamentals of Materials Science
Fall 2003
Lattice models for translational degrees of freedom

We introduced statistical mechanics as a set of tools for calculating macroscopic thermodynamic quantities from
molecular models. These models can be derived either from quantum mechanics (e.g. the Einstein solid) or from
simpler non-quantum models. There are many cases when the quantum nature of available energies in the
system of interest does not dominate and we can use a non-quantum mechanical lattice model to capture the
important aspects of the material’s behavior.
Lattice model of the ideal gas

Consider the case of an ideal gas of N molecules confined in some volume V. A lattice model can be used to
estimate the available translational states for the gas- and thus to estimate the entropy and partition function. We
start by imagining a lattice composed of M sites whose size roughly matches the size of the individual gas
molecules, and whose volume is equivalent to the real volume V. We model the available translational states for
the gas molecules as they zoom through the space of the container as the complete set of ways that the N
molecules can be distributed on the lattice.
o
This is a counting problem: how many distinguishable ways can I array N molecules on M lattice sites?
 The total number of permutations for filling the lattice is determined in the following way:
 The first molecule placed on the lattice can be put in M possible sites
 The next molecules can be placed on M-1 different sites
 The next molecule has M-2 possible sites
 Etc.
 Thus the total number of ways to place all of the N molecules is:
W  M(M 1)(M  2)...(M  (N 1)) 
(Eqn 1)


However, this factorial expression overcounts the number of distinguishable
ways of arranging the molecules by a factor of N!. Correcting this expression,
the number of distinguishable translational states is:
W 
(Eqn 2)

M!
(M  N)!
M!
N!(M  N)!
From this expression we can calculate the translational entropy of the gas:

Lecture 24 – Lattice Models of Materials
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3.012 Fundamentals of Materials Science


M!
S  k ln W  k ln 

N!(M  N)!
(Eqn 3)
o
(Eqn 4)
The number of lattice sites is directly related to the volume V:
M V




(Eqn 7)


(Eqn 9)



MM
S  k ln  N
 M ln M  N ln N  (M  N)ln( M  N)
(M N ) 
N (M  N)

 N 
M  N 
S  kN ln   (M  N)ln 

 M 
 M 


S 
M  N 
 N 
   k1 ln M  ln( M  N)  
 k ln 1 
M U ,N
M  N 
 M 

Plugging into (Eqn 7):
(Eqn 11)

S 
S  dM  S   N 
           Av 
V U ,N M U ,N dV  M U ,N vMW 
Differentiating (Eqn 9):
(Eqn 10)

S 
P
  
V U ,N T
To evaluate S/M, we apply Stirling’s approximation to X: (Dill p. 57, and equation 2.3)
(Eqn 8)

Where NAv is Avogadro’s number, v is the specific volume of the gas (cm 3/g), and MW is the
molecular weight of the gas (g/mole).
This gives the entropy in terms of the volume and number of molecules:
From the entropy we can directly compute the pressure of the ideal gas:
(Eqn 6)


N Av
vMW


M!
S  k ln W  k ln 

N!(M  N)!
(Eqn 5)

Fall 2003

 N   N 
S 
M   N 
   k Av ln 1  k ln 1 
V U ,N
V   M 
vMW   M 
Finally we get the result for the pressure of the ideal gas:
(Eqn 12)

(Eqn 13)

S 
M   N 
P  T   kT ln 1 
V U ,N
V   M 
 MkT  N  1 N   NkT
P  
 1   ...
 V  M  2 M   V
Lecture 24 – Lattice Models of Materials
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Thus we find that this simple lattice model arrives at the equation of state for the ideal gas.
CHEMICAL POTENTIAL OF THE IDEAL GAS FROM THE LATTIC MODEL?
Lecture 24 – Lattice Models of Materials
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Lattice Models of Solid Solutions: The Regular Solution

In our discussion of the thermodynamics of solutions, we introduced a model for the chemical potential of ideal
solutions:
i  i,0 (T,P)  RT ln X i
(Eqn 14)
o
We saw how the free energy behavior of solutions with such a chemical potential gives rise to phase
diagrams that mimic some real solutions reasonably well. However, a question unanswered is- where did
this model come from? And what should the chemical potential look like in a solution where a miscibility
 in the phase diagram? What causes a miscibility gap, in molecular terms? Answers to these
gap exists
questions can be found in simple lattice models.
The free energy of mixing

Generally, when considering a solid or liquid solution, one is interested in the change in free energy that occurs
when the two pure components become intimately mixed by forming a solution. To start formulating the model,
we imaging a cubic lattice filled by the two solution components, which are modeled as spheres with equivalent
molecular volumes. The free energy of mixing is the change in free energy when a given number of molecules
of the two pure components are mixed in the lattice:

We will use an ensemble for this system with fixed (T,V,N) rather than (T,P,N) to simplify the calculations. In
experimental systems of fixed (T,V,N), we minimize the Helmholtz free energy in order to satisfy the second law.
The Helmholtz free energy of mixing is:
Fmix  Fsolution  Fpure_ components
(Eqn 15)
Entropy of mixing in the lattice model and the ideal solution


Generalizations of this theory have been developed which utilize different lattices, or which allow the two molecules to
have significantly different sizes.
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In this simple lattice model, we again assume that only translational degrees of freedom matter in the
determination of the possible states- the states of the system are simply defined by the number of ways the
molecules can be arranged on the lattice. We take N for the total number of molecules (N = NA + NB). To
determine the entropy of mixing, we need the number of distinguishable states W for the mixed and unmixed
components. For the unmixed pure components, there is only one distinguishable state (all lattice sites occupied
by either A or B), so the entropy is 0 (S = k ln 1 = 0).
For the solution, the number of ways to arrange NA A atoms and NB B atoms is:
W solution 
(Eqn 16)
N!
N A !N B !
o

Where N! is the total number of microstates (arrangements of the atoms on the lattice) and the factorials
NA! And NB! In the denominator correct for over-counting of indistinguishable arrangements.
The entropy of mixing is then given by:

Smix  Ssolution  S pure_ components  k ln W solution  k ln
(Eqn 17)
N!
 k(N ln N  N A ln N A  N B ln N B )
N A !N B !
 k(NA ln N  NB ln N  NA ln NA  NB ln NB )  k(NA ln XA  NB ln XB )

The mixing entropy can also be expressed in terms of XB alone, for example:


(Eqn 18)
Smix  Ssolution  S pure_ components  k ln W solution  k ln
N!
 k(N ln N  N A ln N A  N B ln N B )
N A !N B !
Which has the following form when Smix is plotted vs. XB:
o

Free energy and chemical potential of the ideal solution

Ideal solutions are those solutions in which there is no interaction between the components- thus their total free
energy of mixing is composed only by contributions due to the entropy of mixing:
ideal
Fmix
 TSmix 
(Eqn 19)

Differentiating the free energy of mixing, we can determine the chemical potential in the ideal solution:
o FILL IT IN

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The energy of mixing

Most real solutions are not ideal. Molecules form bonds with one another that change the internal energy of the
system. However, the lattice model can account for these interactions.

Lecture 24 – Lattice Models of Materials
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References
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