A MECHANICAL AND THERMODYNAMICAL TMEORY OF NON-LINEAR RELAXATION BEHAVIOUR OF SOLIDS BY H. C. BRINKMAN AND F. SCHWARZL Centraal Laboratorium, T.N.O., Delft, Holland Received 15th January, 1957 A theory is worked out describing relaxation as a diffusion process of molecular groups over potential barriers. Mechanical and thermodynamical properties resulting from this model are calculated. On the basis of this theory deviations from linearity (i.e. proportionality between stress and strain) may be related to properties of the potential holes and barriers in which the diffusion process takes place. The reaction of a viscoelastic system on external forces shows retardation. It has often been suggested that such a behaviour may be interpreted on a molec- ular scale by the migration of inolecular groups from one equilibrium position to another.1 Such a picture is essentially a one-particle model analogous to the models introduced in the theory of liquids.The movement of certain kinds of molec- ular groups is considered. Their migration is assumed to be responsible for the mechanism of retarded deformation. The molecules surrounding these groups are not included explicitly in the calculation. Their influence is described by a potential field V(q) in which the movement of the groups takes place. The rate of migration of the molecular groups between two potential holes, separated by a barrier, is calculated. Up to now this has always been done by the theory of reaction rates under the very questionable assumption of thermal equilibrium between the particle density in the holes and on the barrier.In our theory the rate of migration is calculated on the basis of a diffusion theory. The consequences of this picture are discussed in detail. Even while starting from a very general model quite definite conclusions are reached due to the re- strictions imposed by thermodynamics. The model leads to a single retardation time for the resulting creep deformation; at larger stresses it shows non-linear viscoelastic behaviour, viz., a stress dependence of the retardation time and of the ultimate compliance. On the other hand, one must expect that our one particle model will describe the most simple retardation phenomena only. For the description of more com- plicated processes a distribution of retardation times is necessary. For linear behaviour, this is obtained by means of a distribution function of the parameters of the model.However, the combination of different retardation processes presents a difficult problem outside the region of linear behaviour which is as yet unsolved. On the other hand even our simple molecular picture shows traits also occurring in experiment, e.g. a decrease of the retardation time with increasing external force. We are confident therefore that the mechanical and thermodynamical properties resulting from our one particle model give some indication of the behaviour, even of complicated systems. THERMODYNAMICAL DISCUSSION OF THE RELAXATION PROCESS A viscoelastic bar under the action of a constant longitudinal force shows retarded deformation to a new equilibrium position.This creep may be inter- preted on a molecular scale as the migration of certain molecular groups to new 1112 NON-LINEAR RELAXATION BEHAVIOUR equilibrium positions. Before an external force is applied, the molecular groups are distributed according to a Boltzmann distribution in the potential field Y originating from the surrounding molecules. V(q) is a function of the position vector q. This potential field may show holes, separated by barriers. It is the migration of the molecular groups over these barriers which causes the phenomenon of retarded deformation. The external force tends to establish a new equilibrium distribution of the molecular groups, which are responsible for the creep of the material. These new equilibrium positions may be described as a new Boltzmann distribution of the molecular groups in a potential field V(q) - v(o, q).For a one-dimensional model v would be equal to uq ; in our case v is an as yet unknown function of the external force o and the position vector q. Some information on v may be ob- tained on the basis of statistical thermodynamics. Let the spatial distribution of the molecular groups considered be given by the density p(q, u, T, t ) . For the new equilibrium distribution p is given by where C is determined from the normalization condition b' pdq = 1 . The length of the bar I is related in some way to the distribution function p. It may be defined quite generally where g is an unknown function of CT and q. This means that all temperature influences apart from those acting via the Boltzmann factor in (1) have been neglected.For an experiment under a constant external force u the obvious thermodynamic functions to be defined are the entropy S and the enthalpy H : It should be noted that V, v and g are assumed to be independent of T. +a s =- FJ p logpdg, H = J (V-vlpdq. -a +a -cQ (4) When comparing two states of equilibrium for different o (and T ) the basic thermo- dynamic relation TdS = dH 4- Zdo (5) should be obeyed. By substitution between v and g is of (1) in the definitions (3), (4) and (5) the following relation obtained g = avpo. (6) The internal energy U is +a U = H + oZ= [ -03 ( Y - v + o2)pdq. a. (7) We are interested in materials for which the instantaneous deformation due to the process considered is small as compared with the retarded deformation, i.e.a change of Z is mainly due to a change of p.H . C. BRINKMAN A N D F . SCHWARZL 13 If we restrict ourselves to this case g in (3) is a function of q only and the follow- ing expressions are obtained for g, v and U +a g = f ( q ) ; v = of(q) ;,U = 1 Vpdq. (8) -a THE DIFFUSION THEORY The migration of molecular groups in the field V - v is described as a diffusion process, governed by the Smoluchowski diffusion equation.:! In order to simplify the resulting formulae the calculations will be restricted to a one-dimensional model (co-ordinate 4). No essential features are lost by this restriction. The Smoluchowski equation reads In our special case the force Fis defined by whilefis a friction factor, defined as the ratio between Fand the systematic velocity of the molecular groups.This friction factor ensures the thermal coupling be- tween the molecular groups considered and the rest of the system which does not appear explicitly in the calculations. Iff is large enough, temperature equilibrium (i.e. a Boltzmann distribution of the momenta of the molecular groups) remains established in a first approximation, even while the diffusion process is set in motion.2 The diffusion process for our special model was treated by one of us 3 with the following results. Two states, 1 and 2, are defined, corresponding to two potential holes 1 and 2. The numbers of particles in these states are +a n1 = J@ pdq and 112 = / pdq, -a 4t where qr indicates the value of q at the top of the potential barrier, separating the two holes.This definition of two separate states has a physical meaning only, if the height of the barrier is large compared with kT. The following equations for n1 and 112 may be derived from (9) 3 (12) 0 . - ni = n2 = K1nl - K2n2, where the reaction rates Ki (i = 1, 2) are found as 3 By means of (12) and (13) the change of 121 and 122 may be calculated if V and v are known. The derivation of (13) is based on the assumption that the height H of the barrier is so large as compared with kT that the particles in each hole are approximately distributed according to Boltzniann’s formula, while the density p on the barrier is so low that it may be neglected. These assumptions amount to v - v exp (- I-) 11 in hole 1 , =p 0 on the barrier, (14) exp (- --) v - v in hole 2 , 1214 NON-LINEAR RELAXATION BEHAVIOUR where nl and 122 are solutions of (12) normalized by the relation y11 -f n2 = 1, while 11 and 12 are defined as 4 =I exp (- V - V -+ (i = 1,2).hole i Substitution of (14) in (3)-(5) yields the length and the thermodynamic pro- perties of the bar as functions of time. EXPLICIT CALCULATION OF THE DEFORMATION AND THE THERMODYNAMIC FUNCTIONS FOR A SPECIAL MODEL The general formulae of the previous sections are now applied to a simple, rectangular model consisting of two holes and a barrier (cf. fig. 1). For the deformation v of this field the following assumption is made C ~ O in hole 1, v = 2 Cp L on the barrier, C20 in hole 2, where C1, Cz and Ct may be functions of q, but not of D (cf.(8)). FIG. 1.-Potential energies of holes and barrier. The introduction of functions of q would be physically justified only for a model in which the detailed structure of the holes as a function of q is considered. Therefore the C are treated as constants in our rectangular model. The mechan- ical and thermodynamical functions may now be expressed in the constants of the model (cf. fig. 1) and n1 and 122. In the results the following abbreviations, related to the molecular entropies and free energies in the states 1 , 2 and t, are used : S1 = k log d l , 5’2 = k log d2, S* = k log b, F1 = - TS1, F2 = - TS2 + h, Ft = - TSt + H, + = (F2 - F1)/2, $ = Ft - (F1 + F2)/2. (17) For the sake of clearness the free enthalpy, Fi - Ciu, is given in fig.2 as a function of the co-ordinate q in the deformed state. For the undeformed state an analogous figure is obtained by putting (T = 0. It should be observed that the molecular free energies and enthalpies, introduced above, do not include the terms related to the particle density 121 or 112.H . C . BRINKMAN AND F . SCHWARZL 15 The following expressions are obtained for the total internal energy and the total entropy U = nzh, S = - k(n1 log nl + n2 log n2) + nlS1 + n2S2. 121 = nlo + (ny - n10) (1 - exp (- t/r}, (1 8) (1 9) where nlo, n y are the values of nl in the undeformed, deformed state of equilibrium. The differential eqn. (12) yields an expression for nl, t F-CC F,-C,a F,-c*c s-C,a 4 ? a d, q-- b FIG. 2.-Free enthalpies of holes and barrier in the deformed state (undeformed state for CJ = 0).The retardation time 7(T-l = K1 + K2) and the final elongation A1 of the fb2 exp ($ - ua/kT) bar are found as (20) 7 -7 2 kT cash ($ - wa/kT) ’ where cc = c, - (Cl + C2)/2, w = (C2 - c,>/2. (2 2) Using (22), the following expressions for n10 and n? are found : (23) exp ( d / W . $2 = exp [($ - wa)/kT] 2 cosh ($/kT) ’ 2 cash [(# - i t ~ ) / k T ] ’ n10 = The retardation time r contains the friction factor f, originating from the dif- fusion theory. The expression # - ua in the exponent gives the difference between the free enthalpy of the transition state t and the mean free enthalpy of states 1 and 2. The expression C$ - wa in the denominator gives the difference between the free enthalpies of states 1 and 2. It is interesting to compare (20) with the expression for the reaction rate result- ing from the so-called transition state method 4 where K = 1[r = (kT[h) exp (- AF/kT), AF -- Ft -- F1.(24)16 NON-LINEAR RELAXATION BEHAVIOUR Eqn. (24) is derived by assuming thermodynamic equilibrium between the top of the barrier and a single hole. On the contrary, our equation is derived from a discussion of a diffusion process, while the questionable assumption of equilibrium between hole and barrier is avoided. Moreover, our treatment includes the forward as well as the backward diffusion process. DISCUSSION OF THE RETARDATION TIME AND OF THE FINAL ELONGATION A viscoelastic material shows linear behaviour if the retardation time r is independent of cr, and the elongation A1 is proportional to cr.A development of expressions (20) and (21) for r and gives r =TO 1 + k-T tanh - - - [ ""( kT w where + (gr{-& --; + (tanh&)l- U -tanh ['-" + ($)2 tanh kT + + W W (cosh +/kT)2 kT A1 = f@ exp ($lkT) T O = 2kT cosh (+/kT) A1 with respect to (J kT -A} + . . .]. (25) is the retardation time for a = 0." Now a material for which the retardation time has a term linear in cr (cf. (25)) does not even show linear behaviour for small values of a. Moreover, it behaves differently for extension and for compression. A linear region may be defined if the factors uw/kT in (25) and (crw/kT)2 in (26) vanish, i.e., if 4 = O and u = O . (28) As may be seen from fig. 2, the condition = 0 means that the free enthalpies of the two holes in the undeformed state (a = 0) are equal. This may be realized by various combinations of length and breadth of the potential holes of fig.1. The introduction of the free enthalpies in fig. 2 greatly simplifies the discussion as it combines depth and breadth of the holes in a single variable. The condition u = 0 means that the free enthalpy change caused by the deformation has the same absolute value but the opposite sign for holes 1 and 2. A model obeying conditions (28) will be called a symmetrical model. The development of r and A1 is much simplified for such a model A / = w(aw/kT+ . . . ), (30) is the retardation time under zero stress for a symmetrical model." The value of u for which crw/kT = 1 (32) may now be called the limit of linearity. a the material shows non-linear behaviour.Indeed for this order of magnitude of * This retardation time may be determined by means of recovery measurements after arbitrary stress histories.H. C. BRINKMAN AND F . SCHWARZL 17 In fig. 3 and 4 the retardation time 7 and the final deformation Al are given as functions of (T for zi = 0 (antisymmetrical free enthalpy change, cf. fig. 2) and for various values of q4 (different values of initial enthalpy difference of the holes, -4 -3 -2 -1 0 2 3 4 5 R’ COMPRESSION - -TENSION FIG. 3.-Retardation time as a function of external force for various free energy differences of the holes. 1 A 1 W - 0 -1 -2 ........ .... 2 1 ... ._.. 2 ._.-.-. -.- ,C.- .: 0 -1 - 2 / . / ./. /’ ./. - wu k T 7/. - . -.- _ _ __--/ I I I I I 1 I I - 4 - 3 - 2 - 1 0 1 2 3 4 5 =9 kT COMPRESSION - - TENSION FIG.4.-Final elongation as a function of external force for the cases of fig. 3. cf. fig. 2). From fig. 3 it may be observed that for positive $ the value of T in- creases with increasing u. This behaviour may be understood by observing that for positive 41 the retardation is mainly determined by the larger migration rate K2. This rate decreases with increasing CT and the retardation time increases. Fur negative $ the argument applies to KJ and the retardation time decreases.18 NON-LINEAR RELAXATION BEHAVIOUR The final deformation (fig. 4) shows a curious dissymmetry. The reason is that for positive q5 the particles are mainly in hole 1, even for the undeformed state. Compression does not change much in this situation. For negative 4 the particles are mainly in hole 2 and they remain so at extension. Obviously phenomena related to the asymmetry of the holes are observable for anisotropic materials (e.g. fibres) only. The authors are indebted to their colleagues Ch. A. Kruissink, A. J. Staverman and especially D. Polder for many discussions. 1 cf. Burte and Halsey, Textile Res. J., 1947, 17, 465. This paper contains a model similar to ours. However, the diffusion theory is not used for the calculation of the migration rates, while no thermodynamical discussion of the model is given. 2 Kramers, Physica, 1940, 7, 284. 3 Brinkman, Physica, 1956, 22, 29, 149. 4 Pelzer and Wigner, 2. physik. Chem., By 1932, 15, 445. Eyring, J. Chem. Physics, 1935, 3, 107. Evans and Polanyi, Trans. Faraday Soc., 1935, 31, 875.