Study of the time dependence of physical properties in the transformation range of glass is complicated by the "memory effect" and the inherent nonlinearity which are characteristic of structural relaxation. A multiparameter model of structural relaxation is presented that differs from earlier models in that it takes account of both these effects. This model fits available experimental data well; these data were obtained for the most part by observing the evolution of properties (such as density or refractive index) following a step change in temperature. The present model also permits prediction of the physical properties of glass subjected to arbitrary and more complex temperature-time histories. It should, therefore, also be useful in the rational design of heat treating processes such as annealing. InkoductionHE approach of glass structure toward its equilibrium T state has been studied extensively both for practical reasons and for a better understanding of the glass transition itself. Some practical applications are the annealing of flat glass, in which control of residual stresses is sought, and the annealing of optical glass to obtain a homogeneous structure and a high degree of uniformity of refractive index. Studies of relaxation processes are of fundamental importance in describing the structural state of a nonequilihrium glass. Experiments on the response of the structure of glass to heat treatment are intrinsically difficult, since the structure is both temperature-and time-dependent. In such experiments, structural evolution is studied by observing the temperature-and time-dependent changes in a physical property such as density. Although data on the relaxation of physical properties of glass are available, their interpretation is made difficult by the nonlinear character of the relaxation process.A plausible explanation of the nonlinearity of structural relaxation is that the changing structure alters viscosity and hence the rate of relaxation. Tool' proposed a model that sought to explain this aspect of structural relaxation; he introduced the concept of fictive temperature, T,, as a measure of the structural state of glass and expressed viscosity as a function of the actual and fictive temperatures. Tool's equation is Thus, the rate of change of the fictive temperature is proportional to its deviation from the actual temperature, and the proportionality factor depends on both T and TI. The bracketed factor is inversely proportional to viscosity and expresses the exponential dependence of viscosity* on the actual and fictive temperatures. Thus, viscosity and relaxation rate are as sensitive to changes in structure as they are to changes in temperature. This relation explains the marked nonlinearity served as the influence of past heat treatments on the future behavior of glass. Goldstein: who discussed this point in detail, has shown that "nonexponential relaxation," "spectrum effect," and "memory effect" refer to the same type of relaxation behavior; these terms may be used interchangeably.Ritlan...

Finally, it is interesting to consider to what extent these experimental observations support and, in turn, are explained by Narayanaswamy's quantitative treatment of volume relaxation effects in tempering.8 His calculations follow the thermal history of the glass as it is being cooled and use data on volume relaxation rates to determine the fictive temperatures at which different layers of glass set. His Fig. 8 illustrates how a distribution of fictive temperatures develops in tempered glass. From such differences between the fictive temperatures of different layers and the structural expansivity (Plzq-Pscass) of the gtass, density distributions were calculated. These distributions are shown in the right half of Fig. 4 by the individual point symbols superimposed on the (continuous) plots of experimental data.* Agreement between the two is evidently very good. V. ConclusionA schlieren technique was used to measure refractive index and density distributions in specimens of tempered glass and its fragments. It was found that temper stresses are not wholly released even in small fragments, but density distributions in truly stress-free tempered glass could be obtained readily from the measured density and stress distributions.The results present a clear picture of density distributions in tempered glass, both before and after fracture; and the difference between them is accounted for by the intervening release of stress. Thus surface layers of tempered glass are more dense (and have a higher refractive index) than the interior. This is due to compression of the surface layers and dilatation of the interior by temper stresses.*The calculations of Ref. 8, like the schlieren experiments, yield only distributions of density within a given specimen, not absolute densities, which had to be measured separately.However, density differences are smaller than might be inferred from the magnitude of temper stresses. The reason for this is that intrinsically, i.e. in its stress-free state, tempered glass has a nonuniform density distribution and its surface layers are less dense than its interior.Experimental results on terminal density and stress distributions do not, by themselves, shed light on the tempering process. However, questions regarding the interaction of density changes and the simultaneous genesis of temper stresses have meanwhile been answered theoreticallys; the present data on stress-free density distributions support and, in turn, are explained by this quantitative treatment of volume relaxation effects in tempering.Acknowledgments: Acknowledgment is made to Paul Acloque, former Director of Fundamental Research of Saint-Gobain, whose writings first aroused the writer's interest in the question of possible structural effects in tempering. This is the question addressed experimentally in the pment paper and answered theoretically in the following paper by 0. S. NarayanasWamy, whose long-time collaboration on this topic is also acknowledged. The writer is indebted to Albert Naplitano, formerly of the National Bur...

Laboratory simulation of the industrial process of annealing sheet glass has yielded data on the genesis of stresses in initially stress‐free glass. The experimental results differed from expectations based on classical annealing theory in that stresses began to develop in the annealing range even when the glass was being cooled at a constant rate, i.e. even in the absence of any changes of temperature gradients within the glass. Typically, these stresses account for 40% of the total residual stress in glass annealed according to a linear schedule. The remaining 60% are the well‐known thermoelastic stresses that arise later in the annealing process from the decay of temperature gradients in the glass. The stresses observed to arise in glass as it is being cooled at a constant rate are attributed to volume relaxation effects which, in parts of the annealing range, generate stresses rapidly enough that they are not dissipated by stress relaxation. A mathematical model of annealing is proposed that takes account of both stress and structural relaxation. The model fits the experimentally observed evolution of stresses during linear cooling. It also suggests that (with the activation energies of stress and structural relaxation about the same) the actual rate, at any given temperature, of structural relaxation is about 4 times lower than that of stress relaxation.

Thermorheological simplicity (TRS) in the glass transition is derived from the principle of temperature (actual and fictive)‐time equivalence. Methods of formulating structural relaxation models consistent with TRS are given. Nonlinear dependence of properties on glass structure (Tf) and temperature (T) is taken into account by modifying the traditional definition of fictive temperature (Tf).

Lee, Rogers, and Woo presented a theory for calculating the stresses produced in glass by tempering. Unlike earlier treatments, theirs takes account of the viscoelastic properties of glass. The purpose of this paper is to evaluate this theory by comparison with experimental data. Discrepancies were found to exist between calculated and observed stress distributions. For cases in which glass is quenched from initial temperatures well above the strain point, these discrepancies result from the numerical formulation of the theory, not from any shortcomings of the theory itself. A modification of this formulation brought calculated and observed stresses into good agreement. For quenching from lower initial temperatures, the present theory could not be brought into agreement with experimental data. The reasons for this are briefly considered in the context of the structural heterogeneity of tempered glass and its possible effects on residual stresses.