Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 28

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The approximation of the logarithm is good for Nbr  1 , i.e. for astrong degree of folding of the molecular chain. We obtainS mol2È1È r Ø ØNk É ln 2 ÉÙ Ù,2 Ê Nb Ú ÚÊso that the entropy of the molecule is maximal when its end-to-end distanceis small.The understanding of the rubber molecule does a lot for grasping thenotion of entropy and its growth property, more – perhaps – than theunderstanding of gases.

Let us consider:The basic a priori axiom is: Equal probability of all realizations ormicrostates.Thus each and every microstate occurs just as frequently asany other one during the course of the thermal motion. This means inparticular that the fully stretched-out microstate shown in Fig. 4.10 occursjust as frequently as the partially folded microstate of the figure with theend-to-end distance r < Nb. This means also, however, that a foldeddistribution ^N, N ` with r < Nb occurs more frequently than the fullystretched distribution ^N, 0`, because the former can be realized by moremicrostates, while the latter has only one realization.r=Nb°°r°°Fig.

4.10. Fully stretched and folded realizations of the chain moleculeKinetic Theory of Rubber115Therefore, if the chain molecule starts out straight – with W = 1, i.e.Smol = 0 – the thermal motion will very quickly mess it up, and kick themolecule into a distribution with many microstates and eventually – withoverwhelming probability – into the distribution with most microstates,which we call equilibrium. In equilibrium we therefore have N+ = N-= 1/2 sothat r is zero. During that process the entropy Smol grows from zero tok ln N N!N! ! .

Thus the entropy growth is the result of a random walk of the22chain between its microstates.Of course, we can prevent this growth. If we wish to maintain the straightmicrostate, – or any r in the interval 0 < r < Nb – we need only give themolecule a tug at the ends each time when the thermal motion kicks it. And,if the thermal motion kicks the molecule 1012 times per second – a reasonable number – we may apply a constant force at the ends. That is the natureof entropic forces and of entropic elasticity.

And that is the nature of theforce needed to keep a rubber molecule extended. If r << Nb, the entropy islinear in r2, see above, and the force is proportional to r with the factor ofproportionality linear in T, the temperature: The more vigorous the thermalmotion is, the bigger is the entropic force. Mechanicians like to speak of theentropic spring; its hallmark is an elastic modulus proportional to T.It is often said that the value of the entropy of a distribution is a measurefor the disorder in the arrangement of its particles.

This interpretation ismost easily understood for the rubber molecule. Indeed, the stretched out,orderly distribution of Fig. 4.10 has zero entropy, because it can only berealized in one single manner. The disordered, folded distribution haspositive entropy.

And the most disorderly distribution with very manypossible realizations has the maximum value of entropy. Therefore thegrowth of entropy toward equilibrium involves a growth of disorder.Having said this I like to stress that order and disorder are not well-definedphysical concepts. To be sure, in the present context the notions jibe with ourintuition, but they do not always do that. Thus a cubic lattice in an alloy – judgedwell-ordered on intuitive grounds – has a higher entropy than the more disorderlymonoclinic lattice. For that reason the cubic phase is often the high-temperaturephase, because for higher temperature the entropy becomes more important in thefree energy, cf.

Chap. 5. This apparent violation of the equivalence of entropy anddisorder can be explained, but the explanation does not employ the notion ofcrystallographic order or disorder.A rubber bar consists of a network of rubber molecules all with differentlength vectors (ș1,ș2,ș3) and different lengths r T12 T 22 T 32 , as shown inFig. 4.7. Thus the entropies in the un-stretched and stretched states aregiven, respectively, by116 4 Entropie as S = k ln WS0³Sz ( T1 , T 2 , T3 )dT1dT2 dT3 and Smol 0³Smolz ( T1 , T 2 , T3 )dT1dT2 dT3 ,where z0(ș1,ș2,ș3)dș1dș2dș3 and z(ș1,ș2,ș3)dș1dș2dș3 are the numbers ofdistance vectors in the interval dș1dș2dș3 at ș1,ș2,ș3.The determination of the functions z0(ș1,ș2,ș3) and z(ș1,ș2,ș3) is again dueto Kuhn in 1936.73 For the argument he ingeniously employed the inversionof Smol = klnW : He assumed the number z0(ș1,ș2,ș3)dș1dș2dș3 to beproportional to the number W = exp{Smol/k} of realizations of chains withrT 2 T 2 T 2 and obtained123z0 (θ1 , θ 2 , θ3 )ÎÑ θ12 θ 2 2 θ32exp Ï 32 Nb 2ÑÐ2πNb 2nÞÑß,Ñàwhere n is the total number of chains.

As for the number z(ș1,ș2,ș3)dș1dș2dș3Kuhn assumed thatz (θ1 , θ 2 , θ3 )È1Øz0 É θ1, λθ 2 , λθ3 ÙÊλÚholds, so that the components of the length vectors are deformed exactly asthe edges of the (incompressible) rubber bar, whose deformation in thedirection of the force is given by L = ȜL0.Thus Kuhn obtainedS1 È2 3ØS0 nk É λ 2 Ù2 Êλ 2Úand PnkTL01ØÈÉÊ λ 2 ÚÙ .λThe latter formula represents the thermal equation of state of a rubber barwhich gives the load as a function of the temperature Tand the stretchȜ = L/Lo. The (P,Ȝ)-relation is obviously non-linear.This formula marks the beginning of polymer science74 and the field ofnon-linear elasticity.

Its derivation provides a deep insight into the73 W.Kuhn: “Beziehungen zwischen Molekülgröße, statistischer Molekülgestalt undelastischen Eigenschaften hochpolymerer Stoffe” [Relations between molecular size,statistical molecular shape and elastic properties of high polymers] Kolloidzeitschrift 76,p.

258 (1936).74 Modern representations of the field may be found in the monograph by P.J. Flory:“Principles of Polymer Chemistry.” Cornell University Press, Ithaca (1953). The book hasgone through many editions and re-printings in later years.Gibbs’s Statistical Mechanics117thermodynamic mechanism of polymer elasticity. That is its lastingsignificance, although as a quantitative description of rubber it is less thanperfect, particularly for bi-axial loading.75Gibbs’s Statistical MechanicsOur students are mildly bored when we use ideal gases to explain the powerof statistical arguments.

And indeed, the development of the atomic orstatistical theory of gases in equilibrium is inextricably entwined with old,old conjectures, observations, and measurements of ideal gas properties.Thus the thermal equation of state p = p(v,T) for ideal gases was knownbefore Daniel Bernoulli explained the phenomenon of pressure in terms ofmoving atoms, cf. Insert 4.1. And Bernoulli’s argument implied, that theinternal energy of an ideal gas depends on T only, which was – much later –conjectured by Clausius and experimentally confirmed by Joule and Kelvin.In a manner of speaking everything about gases was known to thestudents – one way or another – before its molecular interpretation wasdiscovered; hence the boredom.This is not so for rubber! In that case statistical arguments have given usa thermal equation of state P = P(L,T) that had not been known before.

So,in my experience this is the point where the students perk up – those ofthem who are capable of such a reaction – and they demand a da capo: Letus maximize entropy, they might say, given bySk ln W with WN!,· N xc !xcand calculate the thermal equations of state of a liquid (say), or of a metal!This is not a bad proposition but, alas, it is impractical and we cannotsatisfy this reasonable request. Let us consider:In a liquid the N atoms Į = 1,2,… N interact, – each one with all others.Therefore there is a kinetic and a potential energy. The latter is the sumover all pair-potentials ĭ(| xȕ – xĮ|). We haveUµ1Ç α 1 2 cα 2 2 Ç α,β 1ĭ | xα − xβ| ,NNor, in terms of Nxc75For a discussion of rubber elasticity in some detail and a discussion of the limitations ofthe Kinetic Theory of Rubber the reader is referred to the booklet “Rubber and RubberBalloons – Paradigms of Thermodynamics” by I.

Müller and P. Strehlow, SpringerLecture Notes in Physics, Springer, Heidelberg (2005).118 4 Entropie as S = k ln WU¦Nxcxcµ 2 1c ¦ N xc N x cccĭ (| x x c |) .22 xc , x cccThus, when we maximize S under the constraints of fixed N and U, theequilibrium distribution Nxcequ must be determined from the system76ȵØ1 k ln N xc equ α β É c 2 Ç N x „c „ equĭ (| x x „ |)Ù = 02 x „c „Ê2Úwith as many equations as there are cells in (x,c)-space. These equationscannot be solved analytically for Nxcequ. Therefore we cannot calculateequilibrium values for U and S; so we are stymied right at the beginning.Nor did Gibbs solve the problem with statistical mechanics. However, heshifted the difficulty from the beginning of the argument to the end byexpressing U and S in terms of a single function, the partition function.

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