THE ENTROPY OF A FLEXIBLE MACROMOLECULE BY H. C. LONGUET-HIGGINS Dept. of Theoretical Chemistry, Cambridge University Received 31st January, 1958 An approximate calculation is made of the configurational entropy of a flexible macromolecule. The value of the entropy per link is found to be of the same order of magnitude as the rotational entropy of a small rigid molecule. This result is used to investigate the association between a flexible molecule and a rigid lattice. It is found that dissociation occurs sharply at a certain critical temperature, and that the kinetics of this process yield an apparent activation energy proportional to the number of external bonds which are broken in detaching the molecule from the lattice. 1. INTRODUCTION In discussing the behaviour of macromolecules, either singly or in aggregation, it is necessary to make a clear distinction between two sorts of question.On the one hand one may wish to know how the potential energy of a macromolecule, or a pair of macromolecules, depends upon the precise configuration; on the other hand one may be interested, for example, in the thermodynamic force be- tween two segments of a large molecule whose configuration is not specified in detail, or between two macromolecules about which only statistical information is available. These two types of question fall respectively within the spheres of quantum mechanics and statistical mechanics, and are to a considerable extent logically independent, although the thermodynamic properties of an assembly depend in the last resort upon the detailed manner in which the quantum mechanical potential energy varies with configuration.Broadly speaking, the physical chemistry of macromolecules seems to be in- telligible on the basis of two main generalizations. First, the intran~olecular and intermolecular potential energies of macromolecules are almost certainly additive with respect to the structural units which they contain, the individual terms being of the same origin and having the same range as for small molecules. (Specific long-range interactions of a purely mechanical nature would be im- possible to reconcile with the present theory.) Secondly, the thermodynamic behaviour of macromolecules is dominated by the extremely large number of geometrical configurations which are consistent with a given value of the energy.This multiplicity of accessible configurations is particularly important in flexible polymers, and is reflected in the very large entropy difference between the crystal- line and fluid (or amorphous) phases. In brief, then, it is entropy rather than energy differences which distinguish the properties of flexible polymers from those of small or fully rigid molecules. The application of these basic ideas to physical systems has shed light on phenomena as diverse as the elasticity of rubber-like materials, the stability of colloids, the thermodynamical properties of polymer solutions, and the residual entropy of glasses. To take a single example : there are more ways of arranging the links in a chain if the ends are close together than if the ends are far apart ; hence, a decrease in entropy accompanies the extension of such a chain, and this is the essential explanation of rubber elasticity.In this particular theory, how- ever, it is not necessary to inquire about the absolute entropy of the randomly oriented chain ; it is enough to estimate entropy differences. Nevertheless, there are circumstances in which it is useful to have an estimate of the absolute entropy associated with the configurational disorder of a long chain moleculc. 86H. C. LONGUET-HIGGINS 87 In this paper I shall, therefore, attempt an absolute calculation of the configura- tional entropy in a simple special case. It will transpire that this entropy is pro- portional to the number of links in the chain, a result which is not intuitively self-evident, but has important physical consequences.Furthermore, the value of the entropy per link can be estimated to order of magnitude, and is shown to determine the conditions under which a flexible macromolecule will undergo a particular type of order-disorder transition. 2 - - , el.e2, 0, . . . el . e2, 2 - a, e2. e3, . . . 0, e2.e3, 2--cc , . . . . . . . . . . . . en,l. en, 2 - 01 2. A SIMPLE PROBLEM Consider a large molecule (see fig. la) consisting of n + 1 atoms of mass rn, joined by n bonds of equilibrium length 1 and force constant K. The two bonds at an atom are freely jointed, that is to say, there are no angular terms in the potential function. The interaction between non-bonded atoms is neglected. This model, though a drastically simplified representation of a real macromolecule, can be investigated with reasonable rigour and I shall use it to obtain an estimate of the configurational entropy, which is of particular interest in the present con- nection.Strictly speaking, in order to calculate the entropy it would be neces- sary to calculate the energies of the vibrational-rotational eigenstates, or to carry out an equivalent calculation. This would be a quite impossible task for a very large molecule, but may fortunately be circumvented by virtue of the fact that almost all the vibrational modes will be of such high frequency that each atom completes many vibrational cycles in the time taken for the molecule to bend appreciably. It therefore becomes possible to use the " adiabatic " approxima- tion in which the vibrational energy is regarded as an effective potential governing the motion in the other degrees of freedom.A further simplification arises from the fact that in an arbitrary angular configuration the vibration frequencies are high compared with kT/h ; hence if for a given set of valency angles the vibrational frequencies are v1, v2, . . . Vn, then the effective potential for the bending modes is the zero-point energy 3 On the basis of these approximations the partition function may be evaluated in a straight-forward manner.* The vibrational degrees of freedom are first separated from the " free " modes, and the partition function for the latter is evaluated semi-classically. If ei denotes the unit vector along the ith bond, the final expression for the partition function is n hi.i- 1 = 0, (2.4) where88 ENTROPY OF A FLEXIBLE MACROMOLECULE and (2.5) An immediate question is: how does the partition function, as defined by (2.1) to (2.5), depend upon n, the number of bonds in the molecule? In order to see this it is necessary to consider the frequency distribution in an arbitrary configuration. Now the off-diagonal elements in (2.4) are the direction cosines of the valency angles, of which in an arbitrary configuration a large number will be close to n/2. Therefore the determinant will roughly factorize into smaller determinants associated with sections of the molecule at the ends of which the valency angles are close to right-angles. Hence to a good approximation the integrand in (2.1) may be expressed in the form n ZtnY (41 n Cvr(4/vol = [Z’V, (4P, (2.6) where 2” depends on the distribution of the valency angles but is independent of the length of the chain.Consequently the free energy, given by the equation (2.7) i= I Fn(T, Y ) = - k T In Zn(T, V ) includes a term proportional to n, the number of bonds. Writing the partition function in the form we see that there is a tendency for the molecule to adopt configurations in which the distribution of valency angles is such as to maximize Z “ { , (e)). It may easily be shown, for example, that at very low temperatures the vibrational zero- point energy tends to straighten out the molecule since in the straight configuration the integrand in (2.1) is at a maximum. This effect is, however, small and is ignored in, for example, the statistical theory of rubber elasticity, which is a fully classical theory.The effect will, in any case, be of much less importance than the “ excluded volume effect ” in favouring one group of configurations as against another. I shall therefore replace every frequency v i in (2.1) by a mean frequency v, regarding V as independent of the valency angles. This leads at once to a greatly simplified expression for the partition function, namely (2.9) The free energy may thus be regarded as comprising three independent terms ; the first is associated with the free motion of one end of the molecule ; the second is asscciated with the 2n free motions in which the angular co-ordinates change; the third arises from the vibrational zero-point energy.The same statement may be made about the entropy. k[ln Y + 1 + 8 In (2.rrmkT/h2)] arises from the translational freedom of the first atom in the chain. Secondly, the entropy associated with the bending modes is proportional to the number of links and may be written as nAS1, where First, an amount of entropy (2.10) and thirdly, the vibrational entropy vanishes in the approximation made here. Particular interest attaches to the magnitude of A,!?,. Since v is of the same order of magnitude as VO, AS1 will be close to the rotational entropy of a diatomic molecule of reduced mass rn and bond length 1. An order of magnitude for AS1 may beH. C . LONGUET-HIGGINS 89 obtained from the rotational entropy of N2 at 25°C ; this is about 10 cal/mole deg.AS1 will of course be reduced if the potential function contains terms restricting the bending motions; on the other hand an increase in mass of the repeating units will lead to an increase in AS1 if these units are freely hinged together. 3. A CO-OPERATIVE PHENOMENON Rather than discussing the general implications of these results I shall now draw attention to a physical situation in which the configuration entropy gives rise to a somewhat startling effect. Let us imagine that our model polymer can conveniently be accommodated on a one-dimensional lattice of n + 1 sites, the distance between successive sites being I (see fig. lb). That is to say, each atom experiences a short-range attraction to one of these sites, the binding energy having FIG. 1. a fixed value - E .The rth atom can only be pulled away from the rth site if the (r + 1)th atom has already been disconnected from the (Y + 1)th site. (The nearest familiar analogy is the zip fastener.) In order to determine the thermo- dynamic behaviour of this system one must evaluate the change in free energy AF1 which accompanies the separation of the last bonded atom from the lattice. When the last bonded atom is pulled away, two vibrations are replaced by two bending motions, with each of which is associated an energy +kT. The change in encrgy is therefore (3.1) Assuming that the atom in the bound state has no vibrational entropy, the entropy change on separation is A E ~ = kT -I- E .90 ENTROPY OF A FLEXIBLE MACROMOLECULE this being a direct consequence of eqn.(2.10). The free energy change is therefore independent of the serial number of the atom being detached and is given by AF1= AE, - TA&= E - kTln (3.3) At sufficiently low temperatures it is clear that AF1 will be positive, so that each atom will tend to associate itself with the lattice as soon as the previous atom is in position. The thermodynamically stable situation will therefore be that in which most of the molecule is bound to the lattice. As the temperature rises, how- ever, there will come a point at which the second term in (3.3) outweighs the first. Above a certain critical temperature therefore, given by AF, will become negative and there will be a tendency for the rth atom to detach itself as soon as the (r + 1)th atom has come adrift. In mathematical terms, if the number of molecules in which r atoms are bound to the lattice is denoted by Nr then it is easily seen that (3.5) the sequence iV1, N2, .. ., forming a geometrical progression in which the common ratio is greater or less than 1 according as AF1 is positive or negative. Conse- quently as the temperature rises through T, all the chains tend suddenly to detach themselves from the lattice, this transition being more abrupt the larger the value of n. An obvious question is : for what value of E will AF1 become zero at 300"K? Assuming AS, = 10 cal/mole deg. we obtain (3.6) A larger value of AS1 would imply a larger value of E; hence for T' to be in the neighbourhood of room temperature the binding energy of each atom to the lattice should lie in the range 1 to 10 kcal/mole, a range in which the energies of hydrogen bonds are found to lie.Indeed, it seems more than likely that the co-operative phenomenon here described is responsible for the extreme sensitivity of certain macromolecular assemblies to temperature and other intensive variables. Nr+1 = NI ~ X P (rAFiIkT), E = AE, - kT = T(AS1 - k ) = 2.4 kcaljmole. 4. A KINETIC APPLICATION The co-operative system discussed in the foregoing section invites a cursory kinetic discussion. Let us set up the kinetic equations the last step, namely the removal of the last atom from its lattice site, being irreversible. We look for a solution of the form (4.2) Nr = A, exp (- P t ) ; Y = 1,2, . . ., n + 1, where /3 is independent of r. Defining a unimolecular rate constant ko by (4.3) we see that it is related to #I by the equation,H .C. LONGUET-HIGGINS 91 The application of straightforward algebra shows that /3 and the A, are related by the eigenvalue equation k2 - - k2, being the lowest eigenvalue. Investigation of eqn. (4.5) shows that if k2 > kl the rate constant ko is given approximately by this equation being more accurate the smaller the value of k3 and the larger the value of iz. The apparent activation energy for disengagement of the chains is t herefme d In k3 d In (1 - kl/k2) d kT2d In ko .=kT2[,+ dT + n -1n - , (4.7) dT (::)I dT in which, for large n, the most important term is the last and has the value The apparent activation energy is therefore n times the energy required to detach a single atom and can be very large indeed if the chain is long.(The case kl > k2 need scarcely be considered, since the problem then reduces to that of detaching a single atom.) The physical reason for this large activation energy is to be found in the equilibrium distribution of the chains according to the number of atoms which are bound to the lattice. Under the conditions just considered (k2 > kl) there will be a predominance of chains with most of their atoms attached to the lattice. Detachment of almost every chain therefore requires the successive removal of nearly all the atoms from their attractive sites, and this is the origin of the large apparent activation energy. The kinetic behaviour of this system is not, incidentally, consistent with the generalization that a very large activation energy implies the simultaneous breaking of a large number of bonds; it is enough that the bonds be broken in a definite succession. It is interesting to note that the rate of denaturation of some proteins shows just the sort of temperature dependence encountered here. I am indebted to Dr. R. A. Sack for stimulating criticism. Note added in proof (17th March, 1958): An excellent general discussion of the residual entropy of linear polymers has been given by Temperley.1 Temperley has used a classical model with discrete configurations, but has not considered the quantum mechanical partition function explicitly. 1 Temperley, J. Res. Nat. Bur. Stand., 1956, 56, 55.