3.012 Fundamentals of Materials Science
Fall 2003
Lecture 25: 12.03.03 Thermodynamics of Macromolecules and
Biomacromolecules
Today:
LAST TIME .............................................................................................................................................................................................. 2
THE WORK OF STRETCHING MOLECULES1................................................................................................................................................ 3
The fundamental equation written for rubber bands ......................................................................................................................... 3
WHAT RUBBER BANDS AND DNA HAVE IN COMMON .............................................................................................................................. 6
REFERENCES ........................................................................................................................................................................................... 8
Reading:
Supplementary Reading:
-
Planning Notes:
Rubber elasticity theory Dill p. 157
Hydrophobic entropy – Dill p. 277
HOMEWORK PROBLEMS:
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
1 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
Last time
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
2 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
The work of stretching molecules1
The fundamental equation written for rubber bands

We’ve now discussed a number of different molecular degrees of freedom that give rise to internal energy in
materials, and some of the models for these degrees of freedom. Today, we will finish up by analyzing a degree
of freedom unique to polymeric materials – both synthetic polymers and natural polymers like proteins and DNA.
This degree of freedom has to do with molecular elasticity and rubber bands.

Rubber bands are composed of molecular chains that have been linked together by covalent bonds at random
locations (called cross-links). Such tied-up molecular spaghetti has the extremely useful property of exhibiting
rubber elasticity, a property unique to polymeric materials. Rubber elasticity is the ability to undergo extremely
large deformations (e.g. stretching of a rubber band) with complete elastic recovery (snapping back to its original
shape).
Cross-linking
Covalent linkages
o

You know that as you stretch a rubber band more and more, a increasing force is resisting the
deformation you induce. We can ask some very interesting questions about the thermodynamics of a
rubber band: What is the source of rubber elasticity- what makes the rubber band snap back? Is it
entropy of energy (enthalpy)?
Recall that when we introduced the concept of thermodynamic driving forces, we pointed out that materials may
have unique driving forces- and to account for these we simply introduce new terms to the internal energy to
account for these new internal degrees of freedom. To answer these questions, we need to introduce a new term
for the elastic energy into the fundamental equation.
o The fundamental equation for the internal energy when there is no exchange of molecules with the
surroundings is given by:
dU  TdS  PdV
(Eqn 1)
o
(Eqn 2)
To this simple equation we introduce a new term that accounts for elastic forces on the material:
dU  TdS  PdV  f el dL



This new term is composed of the thermodynamic driving force fel, the elastic retracting force,
multiplied by the change in the length of the rubber band, dL. We’d typically carry out
experiments at constant temperature and pressure rather than under conditions of constant
entropy and volume, so let’s calculate the Gibbs free energy for the rubber band:
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
3 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
dG  d(U  PV  TS)  Sdt  VdP  f el dL
(Eqn 3)

G 
f el   
L T ,P

(Eqn 4)


Now, G = H – TS, so we can break down the elastic retracting force into an enthalpic (energy)
component and an entropic component:
G 
H 
S 
f el        T 
L T ,P L T ,P
L T ,P
(Eqn 5)
o
Because G is a state function, we know the elastic force is equal to:
(at constant temperature and pressure)
(Eqn 5) explains how the change in enthalpy per stretched length and entropy will add into the total
elastic retracting force. But how do we measure the two terms on the right? When we are looking for
measurable quantities, think Maxwell’s equations! Using (Eqn 3) we can write the following Maxwell
 relations:
 2G
 2G

 T L  L T
f el
S

T
L
(Eqn 6)
o (Eqn 6) suggests a very simple experiment. Hold a rubber band at a fixed stretched length L, and
measure how the retractive force fel depends on the temperature. The slope of a plot of fel vs. T is –
dS/dL. Shown below is some experimental data for rubbery polyethylene:


o
Note that the measured retraction force increases with increasing temperature. This means the
entropy is decreasing as the polymer is stretched. This is in contrast to the behavior observed for
metals- where stretching loosens the bonds between atoms and increases the entropy of the
system.
How can we measure the enthalpic component of the retractive force- and compare its magnitude to the
contribution from the entropy?
 Substituting (Eqn 6) into (Eqn 5), we get:
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
4 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
H 
f 
   f el  T el 
L T ,P
T P,L
(Eqn 7)



…which we can measure from our same stretching experiment.
GET MEASURED VALUES FOR ENTROPY VS. ENTHALPIC COMPONENT TO
SHOW DOMINANCE OF ENTROPY.

Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
5 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
What rubber bands and DNA have in common

Polymers are not only useful in countless technological applications, they are a fundamental component of all
living things, making up the proteins, polysaccharides, and polynucleic acids that make up biology. Thus, many
properties of synthetic polymers are shared by biomacromolecules. We can use models of the structure of
materials to make quantitative thermodynamic predictions of the behavior of long chain molecules, rubber bands,
and DNA molecules.
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
6 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
(Dill p. 621)

Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
7 of 8
2/12/16
3.012 Fundamentals of Materials Science
Fall 2003
References
1.
Dill, K. & Bromberg, S. Molecular Driving Forces (New York, 2002).
Lecture 25 – Thermodynamics of Macromolecules and Biomacromolecules
8 of 8
2/12/16