Table Of ContentI. Mu¨ller P. Strehlow
Rubber
and Rubber Balloons
Paradigms of Thermodynamics
1 3
Authors
IngoMu¨ller PeterStrehlow
TechnischeUniversita¨tBerlin Physik.-Techn.Bundesanstalt
Thermodynamik Abbestr.2-12
Straßedes17.Juni 10587Berlin,Germany
10623Berlin,Germany [email protected]
[email protected]
I.Mu¨ller,P.Strehlow,RubberandRubberBalloons,ParadigmsofThermodynamics,Lect.
NotesPhys.637(Springer-VerlagBerlinHeidelberg2004),DOI10.1007/b93853
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Balloons,ParadigmsofThermodynamics
Contents
1 Stability of Two Rubber Balloons ........................ 1
2 Kinetic Theory of Rubber ................................ 7
2.1 Rubber Elasticity is “Entropy Induced” ................... 7
2.2 Entropy of a Rubber Molecule ........................... 10
2.3 The Rubber Molecule with Arbitrarily Oriented Links....... 12
2.4 Entropy and Disorder, Entropic Elasticity ................. 14
2.5 Entropy of a Rubber Bar ................................ 14
2.6 Uniaxial Force-Stretch Curve ............................ 16
2.7 Biaxial Loading ........................................ 17
2.8 Application to a Balloon ................................ 19
2.9 Criticism of the Kinetic Theory of Rubber................. 20
3 Non-linear Elasticity...................................... 21
3.1 Deformation Gradient and Stress-Stretch Relation .......... 21
3.2 Material Symmetry and Isotropy ......................... 22
3.3 Material Objectivity .................................... 24
3.4 Representation of an Isotropic Function ................... 24
3.5 Incompressibility. Mooney-Rivlin Material ................. 26
3.6 Biaxial Stretching of a Mooney-Rivlin Membrane........... 27
3.7 Free Energy ........................................... 27
3.8 Rubber Balloon with the Mooney-Rivlin Equation.......... 28
3.9 Surface Tension ........................................ 29
3.10 Cylindrical Balloon ..................................... 30
3.11 Treloar’s Biaxial Experiment............................. 32
4 Biaxial Stretching of a Rubber Membrane – A Paradigm
of Stability, Symmetry-Breaking, and Hysteresis.......... 35
4.1 Load-Deformation Relations
of a Biaxially Loaded Mooney-Rivlin Membrane............ 35
4.2 Deformation of a Square Membrane
Under Symmetric Loading............................... 35
4.3 Stability Criteria ....................................... 37
4.4 Minimal Free Energy ................................... 38
4.5 Free Enthalpy, Gibbs Free Energy ........................ 39
VI Contents
4.6 Symmetry Breaking..................................... 41
4.7 Hysteresis ............................................. 43
4.8 Non-monotone Force-Stretch Relation..................... 44
4.9 Hysteresis and Breakthrough. Dissipation.................. 46
4.10 A Three-Dimensional View .............................. 47
4.11 Comparison with Treloar’s Dead Loading Experiment ....... 48
4.12 Experimental Evidence for Bifurcation .................... 50
5 Stability of a Single Balloon .............................. 51
5.1 Pressure-Radius Characteristic
and Filling-Radius Characteristic......................... 51
5.2 Pressure-Filling Curve .................................. 52
5.3 Free Energy-Radius Relation of the Balloon................ 52
5.4 Stability and Non-monotonicity .......................... 53
5.5 Suggestive Stability Criterion ............................ 57
5.6 A Hydrostatic Device for Controlling the Inflation
of a Balloon ........................................... 61
6 Stepwise Inflation of a Balloon............................ 63
6.1 Stages of One Stroke of Inflation ......................... 63
6.2 Partial Equilibria....................................... 64
6.3 Stepwise Inflation for a Stiff Spring ....................... 66
6.4 Stepwise Inflation for a Soft Spring ....................... 66
6.5 Dissipation ............................................ 67
6.6 Minima of Available Free Energy ......................... 67
7 Two Balloons – Revisited................................. 69
7.1 Equilibria in the (r ,r )-Diagram ......................... 69
1 2
7.2 Investigation of Stability ................................ 71
7.3 Energy Landscape ...................................... 73
7.4 Simultaneous Inflation-Deflation Experiment. Hysteresis..... 76
8 Many Balloons – The Emergence
of a Pseudoelastic Hysteresis ............................. 79
8.1 Simultaneous Inflation of Four Balloons ................... 79
8.2 Ten Balloons........................................... 80
8.3 Many Balloons ......................................... 83
8.4 Phase Transition and Pseudoelasticity..................... 84
9 Stability of the Spherical Shape........................... 87
9.1 Rotational But Not Spherical Symmetry?.................. 87
9.2 A Modicum of Surface Geometry ......................... 87
9.3 Balance of Forces....................................... 89
9.4 Motion, Stretches, Surface Stress, and Pressure............. 90
9.5 Equilibrium Conditions ................................. 91
Contents VII
9.6 Testing Stability of the Spherical Solution ................. 92
9.7 Discussion ............................................. 95
10 Stress-Induced Crystallization of Rubber ................. 97
10.1 Phenomenon........................................... 97
10.2 The “Mechanism” of Crystallization ...................... 98
10.3 Free Energy of Crystallizing Rubber ...................... 100
10.4 Hysteresis and Residual Stretch .......................... 102
11 History of Rubber and Its Use............................ 105
References.................................................... 111
Chapter 1 (Young-Laplace Equation) .......................... 111
Chapter 2 (Kinetic Theory of Rubber)......................... 112
Chapter 3 (Non-linear Elasticity) ............................. 114
Chapter 4 (Plane Square Membrane) .......................... 115
Chapter 5 Through 8 (One, Two, and Many Balloons)........... 116
Chapter 9 (Stability of the Spherical Shape).................... 118
Chapter 10 (Crystallization and Limited Extensibility)........... 119
Chapter 11 (History) ........................................ 120
Index......................................................... 121
1 Stability of Two Rubber Balloons
Rubber balloons, that usually adorn children’s parties, can also enchant the
scientist. Indeed, there is much more than fun and games to be had with
balloons;theyformasuitablesubjectformathematicalstudiesandaninter-
esting paradigm of methods of modeling in physics and chemistry.
First of all, balloons consist of rubber, a remarkably unique elastic ma-
terial with an extreme extensibility − superior to that of all other solids −
andperfectlyresilient.Inasensetobeexplainedlatertheclosestrelativesof
rubber amongst natural materials are the ideal gases, like air. Both, rubber
and gas, are the only examples of entropic materials.
Inordertocatchthereader’sattentionandraisehiscuriosityfortherich
properties of rubber and rubber balloons, we perform the experiment shown
in Fig.1.1. Two balloons inflated to the same size are fixed to the ends of a
connectingpipe,cf.Fig.1.1 .However,thatsituationprovestobeunstable
top
when the tap is opened; air flows from one balloon to the other one until
the stable equilibrium is reached, which is demonstrated in the bottom part
of Fig.1.1, where one balloon is small and the other one is large. The initial
symmetry of the arrangement has been broken spontaneously.
From this we must not, however, conclude that a small balloon always
inflates the large one with which it can exchange air. The stability depends
on the amount of filling and the process shown in Fig.1.1 will only occur
for intermediate fillings. Indeed, with either more air or less air in the two
balloons,thelargeroneinflatesthesmalleroneuntil asymmetricequilibrium
situation is attained and both are equal in size, see Figs.1.2 and 1.3. We
shall study that phenomenon in due time and understand it completely as a
question of stability.
The early research relevant to balloons does not concern balloons at all
butdroplets,bubblesand,inparticular,soapbubbles.Rubberballoonswere
unknown before the 1820’s, when Michael Faraday glued one together − or
so they say. To be sure balloons are different from bubbles but there is some
similarity; certainly both enclose air in a spherical membrane. Therefore it
is somewhat pertinent to balloon-research that, 200 years ago, Young and
Laplace related the pressure difference [p] = p−p between the inside and
0
the outside of a soap bubble to the radius r. Their result is now known as
the Young-Laplace formula and it reads
I. Mu¨ller and P. Strehlow: Rubber and Rubber Balloons, Paradigms of Thermodynamics,
Lect.NotesPhys.637,(2004),pp.1–5
http://www.springerlink.com/ (cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Stability of Two Rubber Balloons
Fig. 1.1. Symmetrybreaking.Theinitialequilibriumisunstable.Top:Tapclosed.
Bottom: Tap open
Fig.1.2.Largefilling:Aninitiallyasymmetricsituation(top)becomesasymmetric
equilibrium upon opening the tap
4σ
[p]= . (1.1)
r
The coefficient σ is called surface energy or also surface tension.1
In later times, when the close relation between work and energy became
commonknowledge,theequation(1.1)wasderivedasfollows:Foranincrease
dr of the bubble radius, the surfaces − inner and outer surface − increase
1 For literature and literary references we refer the reader to Chap.12.
1 Stability of Two Rubber Balloons 3
Fig. 1.3. Small filling: Same process as in Fig. 1.2: A symmetric equilibrium sit-
uation will evolve from an asymmetric situation when the balloons can exchange
air
by dA = 2·8πrdr. That expansion of surface requires a certain amount of
work which we assume to be proportional to dA. Let us say that the work is
equal to σdA, where σ is a factor of proportionality, possibly dependent on
the extant r. The work is done by the pressure difference [p] which inflates
the volume V = 4πr3 by dV = 4πr2dr so that the work equals [p]dV. Thus
3
we have
4σ
[p]dV =σdA, and hence [p]= . (1.2)
r
Sowhataboutstability?LetusconcentrateonFig.1.2 andtestthe
bottom
stability of that situation in which p is equal in both ballons. In a thought
experiment we squeeze the right balloon (say), thereby decreasing its radius
and making air move to the left balloon, which will therefore grow. If in that
processthepressureinthesqueezedballoondecreasesandthepressureinthe
inflated one increases, the air will move back as soon as the squeezing stops;
theinitialsituationwillbereestablishedandthereforeitisastablesituation.
Thusstabilityrequiresthat d[p] >0holds.If d[p] weresmallerthanzero,the
dr dr
imposed shift would not heal itself, but become worse and thus expose the
initial state as unstable.
Actually,thethoughtexperimentdoesnotrequireanactiveexperimenter
to do the squeezing. Indeed, even if the balloons are initially exactly alike,
the inevitable thermal fluctuations of pressure will be enough to test their
stability.
The unstable situation can remind us of the predicament of Buridan’s
ass who should starve − according to the scholastic philosopher − in the
4 1 Stability of Two Rubber Balloons
middle of two equal bales of hay. Of course it does not starve, because the
tiniestfluctuationofitsheadwillmakeoneortheotherbalemoreattractive
spontaneously.
By (1.1) the stability condition d[p] >0 may be written in the form
dr
4σ 4dσ dσ σ
− + >0, hence > . (1.3)
r2 r dr dr r
A stability argument like this was first given by Gibbs and it implies that σ
mustdependonr,sincetheexperimentsofFigs.1.2and1.3haveshownthat
there are stable equilibria between balloons of equal size. To be sure we do
not expect the inequality (1.3) to hold for all values of r, since we have seen
in Fig.1.1 that two balloons with equal pressures and equal radii may also
be unstable.
Thequestionishow thesurfacetensionσdependsonr,andthatquestion
willbeansweredinthenexttwochapterswhichdealwiththeKineticTheory
ofRubber(Chap.2)andwithNon-linearElasticity(Chap.3).Thosetwothe-
oriesaremasterpiecesofmaterials’scienceintheirownright,faroutreaching
theirapplicabilitytorubberballoons.Therefore,inpresentingthesetheories,
we do a little more than we must for the purpose of explaining balloons; we
thus hope to be able to share some exciting results of thermodynamics, sta-
tistical mechanics and continuum mechanics with the reader.
Of course, we need not rely upon theory; there is always experiment and,
indeed, it is quite easy to measure the pressure difference [p] across the bal-
loon membrane as a function of its radius r. Figure1.4 shows a suitable
left
manometer:Weattachafullyinflatedballoontoitsinletandlettheairdrain
out slowly through a small opening in the connecting pipe. We monitor the
40 [p] (mbar)
35
30
25
20
15
10
drain
5
0 r/r
0
0 1 2 3 4 5 6 7
Fig. 1.4. Left:Amanometertomeasurethegraph[p]vs. r .Right:Thepressure-
r0
radius characteristic of the balloon
