TABLE 10. Density of neodymium-doped laser glasses TABLE 11. Spectrochemical analyses of neodymiumdoped laser glasses, weight percent 12. Chemical Composition Chemical analyses were made on samples from each of the five laser glasses. A preliminary general spectrochemical analysis was conducted to determine the elements present. The results are given in table 11. From this table it is seen that all of the glasses are silicates and that Glass A, B, C, and D are mixed alklai glasses. Glass E is different in that it contains a single alkali, lithium. On the basis of the preliminary spectrochemical analysis, determinations of glass composition with respect to individual elements were made, classing those elements indicated to be at the 0.1 to 1 percent level or higher as major constituents. Table 12 summarizes the pertinent analytical data on the major constituents of the five laser glasses, along with the analytical methods employed in each case. The errors in the values of the major constituents of the glasses are estimated to be no greater than 1 in the last significant figure. The trace amounts of Fe2O3 found in the glasses are reported in this table because the absorption in the vicinity of the lasing wavelength is usually attributed to the Fe2O3 content. These values are believed to have a relative precision of ± 20 percent. Neutron activation analyses were also made on a sample of each of the glasses, primarily to determine trace impurities, but also as a check on other methods. The results obtained are given in table 13. Here, again, the errors in the values of the major constituents are TABLE 12. Major constituents of neodymium-doped laser glasses, weight percent 0.52 estimated to be no greater than ±1 in the last significant figure. For the amounts reported in parts per million, the standard deviation is estimated to be about ±5 percent in most cases. For the lower reported values of Sm there is severe interference from Nd and these values are probably no better than 20 percent. For the lower reported values for Eu and Ce the signalto-background ratio was poor and these values also are probably no better than ±20 percent. All results are reported as the oxide rather than the element to facilitate comparison with results from other methods. 13. Summary As an aid in laser design, measurements have been made on important physical and chemical properties of five commercially available, neodymium-doped laser glasses. The measurements include thermo-optic properties, photoelasticity, refractive index, optical homogeneity, transmittance, thermal conductivity, hardness, density, and chemical composition. Calculations have been made of the thermal change in refractive index at constant volume, because this parameter is important in the study of self-focusing of laser light. The combined facilities of the Inorganic Materials, Optical Physics, Polymers, Building Research, Mechanics and Chemistry Divisions within the National Bureau of Standards were utilized for making the measurements and compiling the data included in this report. The authors acknowledge the contribution of colleagues as follows: F. W. Rosberry for measurements of homogeneity, G. Dickson for determining the elastic constants, A. Feldman and D. Horowitz for determining the wavelength variation of the stressinduced birefringence, D. Flynn and W. L. Carroll for 14. References [1] Quelle, F. W., Jr., Appl. Optics 5, 633 (1966). [2] Blume, A. E., and Tittel, K. F., Appl. Optics 3, 527 (1964). [3] Sims, S. D., Stein, A., and Roth, C., Appl. Optics 5, 621 (1966). [4] Sims, S. D., Stein, A., and Roth, C., Appl. Optics 6, 579 (1967). [5] Epstein, S., J. Appl. Phys. 38, 2715 (1967). [6] Baldwin, G. D., and Riedel, E. P., J. Appl. Phys. 38, 2726 (1967). [7] Dishington, D. H., Hook, W. R., and Hilberg, R. P., Proc. IEEE 55, 2038 (1967). [8] Welling, H., and Bickart, C., J. Opt. Soc. Amer. 56, 611 (1966). [9] Guiliano, C. R., Appl. Phys. Letters 5, 137 (1964). [10] Chiao, R. Y., Townes, C. H., and Stoichéff, B. P., Phys. Rev. Letters 12, 592 (1964). [11] Olness, D., J. Appl. Phys. 39, 6 (1968). [13] Thornton, J. R., Fountain, W. D., Flint, G. W., Crow, T. G.. Appl. Optics 8, 1087 (1969). [14] Snitzer, E., Appl. Optics 5, 1487 (1966). [15] Military Specification, Glass, Optical, MIL-G-174A, 5 Nov. 1963. [16] Rosberry, F. W., Appl. Optics 5, 961 (1966). [17] Rogers, A. F., and Kerr, P. F., Optical Mineralogy (McGraw-Hill Book Co., Inc., New York and London, 1942). [18] Weir, C., Spinner, S., Malitson, I. H., Rodney, W., J. Res. Nat. Bur. Stand. (U.S.), 58, 189–194 (April 1957) RP2751. [19] Rodney, W. S., and Spindler, R. J., J. Res. Nat. Bur. Stand. (U.S.), 53, 185-189 (Sept. 1954) RP2531. [20] Tilton, L. W., J. Res. Nat. Bur. Stand. (U.S.), 2, 909–930 (1929) RP64. [21] Dodge, M. J., Malitson, I. H., and Mahan, A. I., Appl. Opt. 8. 1703 (1969). [22] Private communication. L. E. Sutton, Institute of Basic Standards, National Bureau of Standards, Washington, D.C. [23] Saunders, J. B., J. Res. Nat. Bur. Stand. (U.S.), 35, 157-218 (Sept. 1945) RP 1668. [24] Krishnan, R. S. Progress in Crystal Physics, Vol. I (Interscience Publishers, New York and London, 1958). [25] Ramachandran, G. N., Proc. Ind. Acad. Sci. [A] 25, 498 (1947). [26] Ramachandran, G. N., Proc. Ind. Acad. Sci. [A] 25, 286 (1947). [27] Waxler, R. M., and Weir, C. E., J. Res. Nat. Bur. Stand. (U.S.). 69A, (Phys. and Chem.), No. 4, 325–333 (July-Aug. 1965). [28] Waxler, R. M., and Napolitano, A., J. Res. Nat. Bur. Stand. (U.S.), 59, 121–125 (Aug. 1957) RP 2779. [29] Feldman, A., Phys. Rev. 150, 748 (1966). [30] Vedam, K., Proc. Ind. Acad. Sci. [A] 31, 450 (1950). [31] Dickson, G., and Oglesby, P. L., J. Dent. Res. 46, [6] Part 2. 1475 (1967). [32] Quelle, F. W., Damage in Laser Glass, ASTM STP 469 (American Society for Testing and Materials, 1969) pp. 110-116. [33] Kerr, E. L., IEEE J. Quantum Electronics QE-6, 616 (1970). [34] Wachtman, J. B., Jr., Mechanical and Thermal Properties of Ceramics, Nat. Bur. Stand. (U.S.), Spec. Publ. 303, 272 pages (May 1969). [35] Brock, T. W., Draft of Tentative Method of Test for Knoop Indentation Hardness of Glass. Being considered for approval as an ASTM method by ASTM Committee C14.04. [36] Glaze, F. W., Young, J. C., and Finn, A. N., J. Res. Nat. Bur. Stand. (U.S.), 9, 799–805 (1932) RP 507. (Paper 75A3-660)) JOURNAL OF RESEARCH of the National Bureau of Standards - A. Physics and Chemistry Effect of Environment on Viscous Flow in Inorganic Oxide Glasses Joseph H. Simmons and Pedro B. Macedo* Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234 (February 9, 1971) Results from viscosity and shear structural relaxation measurements conducted above the liquidliquid phase transition of a series of immiscible inorganic oxide glasses are analyzed. A model is proposed which relates the temperature dependence of the complex modulus and viscosity to the behavior of microstructure in the glass resulting from supercritical fluctuations in composition. It is suggested that the critical microstructure induces differences in local environment in the glass which in turn cause the appearance of distributions of relaxation times. The model is formulated using elementary fluctuation theory and the resulting equations are successfully compared to the data. Key words: Critical point theories; immiscibility; phase transitions; structural relaxation; transport 1. Introduction The investigation of viscous flow in glass-forming liquids has been approached from a variety of direc tions. These can be classified into the empirical equation of Fulcher [1],' the thermodynamic approaches of Gibbs [2], Turnbull and Cohen [3], and Goldstein [4], and the kinetic approaches recently characterized by consideration of the structural relaxation mechanisms [5–7]. In the case of inorganic oxide glasses, the investi gations of structural relaxation have been undertaken by means of ultrasonic spectroscopy. Recently, ultrasonic [8] and viscosity [9] measurements on vitreous BO3 have yielded evidence conflicting with the predictions of the thermodynamic approaches, and thus have indicated the need for systematic studies of viscous flow by structural relaxation in such glass formers [9]. Before examining this evidence, let us first introduce the concept of a distribution of relaxation times. dependent modulus may in turn be expressed in terms where is the angular frequency. The frequencyof a sum of relaxation functions. This implies that the stresses are additive. In the case of liquids having simple distributions of relaxation times, such as those in this paper, the relaxation phenomena may be characterized in terms of two independent parameters which are the width and the most probable value of time of any existing distribution of relaxation times. While the choice of additive stresses appears arbitrary at first, we will show later that it is consistent with the physical interpretation of our model. The viscosity is now written as follows: n' (w) = G x x x T g(In 7)d In 7 (2) Viscous relaxation investigations by means of ultrasonic spectroscopy consist of studying the interaction of high frequency sound waves with matter as a function of the time of interaction. As the observation time, which is the inverse of the ultrasonic frequency, approaches the structural relaxation time, interaction between the flowing, or moving species and the sounds. is the zero-frequency limit of n'(w), waves leads to a frequency-dependent complex modulus. The frequency-dependent shear viscosity is given by the imaginary part of the shear modulus, 1+ (WT)2 where G is the instantaneous shear modulus, 7 is a relaxation time, and g(In 7) is known as the distribution of relaxation times. The "static" viscosity, *Also at the Vitreous State Laboratory, Catholic University of America, Washington, D.C 20017. Figures in brackets indicate the literature references at the end of this paper. x ns = x Gx. [* 7g (In 7)d In T. (3) The instantaneous shear modulus and the distribution of relaxation times can be investigated by shear ultra sonic spectroscopy provided the material is fluid enough to relax in a time scale comparable to the period of the ultrasonic waves, and one is equipped with instruments sensitive enough to detect the highly damped shear waves. The evidence conflicting with the thermodynamic approaches first came from ultrasonic measurements [8]. Log-Gaussian distributions of relaxation times were found to represent the data well for vitreous B2O3. At high temperatures, when the static viscosity followed an Arrhenian behavior, it was seen that attributing a single relaxation time to the flow units was sufficient to account for the frequency dependence of the modulus. As the temperature was reduced, however, the shear viscosity departed from the Arrhenius curve, and a broadening distribution of relaxation times was required to fit the data. It was seen that the broadening occurred concurrently with the departure of the viscosity from Arrhenian behavior. Use of the Eyring Rate-Equation led to an analysis of the distribution of activation energies. The latter was found to be Gaussian and to broaden with decreasing temperature as did the distribution of relaxation times. The mean activation energy, however, did not change with temperature over the range of measurements. In conclusion, contrary to the assumption implicit in most phenomenological equations of the thermodynamic approaches, the non-Arrhenian behavior of the viscosity was not associated with an increase in the average activation energy with decreasing temperature, but rather with the appearance of a distribution of energies; some indeed higher, but others lower than the average. These lower activation energies could not be explained within the framework of existing viscosity theories. Further support for these conclusions came from measurements of the viscosity of B2O3 in the range 1010 to 1014 P [9], in which the viscosity appeared to revert to Arrhenian behavior at low temperatures. The authors could find no reasonable fits of this extended data by any of the viscosity theories, as the parameter [In ŋs/¿ (1/T)] = Eapp representing an apparent activation energy, approached a temperatureindependent behavior near T, while all existing theories predicted a continued increase with decreasing temperature. In this case, a temperatureindependent apparent activation energy implies Arrhenian viscosity behavior. They thus concluded that the temperature dependence of the viscosity was not controlled by structural effects, such as free volume, and configurational entropy, but rather by some activation energy effects, represented by [ở În ŋs/ở (1/T)]. A successful theory would have to lead to a constant parameter [ În îs/ở (1/T)], or activation energy spectrum near T. It became apparent that a microscopic model was needed which could explain the appearance of a symmetric distribution of activation energies, and give physical significance to both the distribution of relaxation times and the temperature dependence of the viscosity. In analyzing the origin of a spectrum of relaxation times, one cannot differentiate between the occurrence of the same nonexponential relaxation for all flow units, and the weighted sum of varying but exponential relaxation effects caused by a varying environment [11]. This problem was considered in the analysis of annealing experiments on several inorganic glasses [12, 13]. In the case of a borosilicate crown glass [12] both volume relaxation and ionic conduction were measured. Of the various models present in the literature, only a modified version of Fröhlich's model [14] attributing the existence of a distribution of activation energies to a distribution of environments. could fit both sets of data consistently. In these measurements, it was found that if the distribution were represented by two relaxation times, the fast volume relaxation could be related to the fast electrical conductivity relaxation. The results implied that there are definite regions in these materials which are associated with the different relaxations. The connection between a distribution of activation energies and a distribution of environments was postulated long ago by Fröhlich, but until recently, due to a lack of sufficiently detailed electron micrographs and high temperature x-ray instrumentation, no structure was observed in molten oxide glasses with known distributions of relaxation times. The model presented in the annealing investigations ruled out the possibility of nonexponential decays, but only indicated the possibility of applying some environmental model to the analysis of structural relaxation in inorganic oxide glasses. The problem in interpreting ultrasonic and other relaxation experiments arises from the fact that a distribution of relaxation times does not necessarily indicate any particular molecular relaxation process. As Goldstein [11] clearly suggested, calculation of a distribution of relaxation times in the analysis of a response function is only a mathematical transform and cannot carry physical significance by itself. One must first begin with a model for the molecular mechanisms for structural relaxation and then derive a resulting set or distribution of relaxation times. For this reason, we have chosen to investigate the viscous relaxation process in a series of inorganic oxide glasses with predictable distributions of environments and to attempt to analyze the results in terms of the related microstructure. Microstructure in molten oxides can best be controlled by selective doping, or by approaching the critical point of an immiscible system. Since x-ray data or electron micrographs describing the structure of normal glasses are lacking, immiscible systems offer the best solution due to the possibility of specifying the temperature dependence of the structure from analysis of the supercritical composition fluctuations. The immiscibility phase transition is quite widespread in oxide glasses. Liquid-liquid phase separations occur in such systems by changes in composition associated with phases of widely different viscosities. Phase separation at the concentration which has the highest transition temperature, Te (top of the immiscibility dome), occurs by means of a pseudo secondorder phase transition which is characterized by the continuity of the free energy and its first derivatives 150 Å across the transition boundary. As a consequence, flow process and the supercritical fluctuations in the thermodynamically unstable fluctuations which composition. lead to demixing below Te, do not end abruptly when one raises the temperature to the critical point. These fluctuations in fact extend far into the supercritical T>T) region, where they are thermodynamically metastable, with associated well defined wavelengths and lifetimes. The resulting transient domains have some structural relationship to the subcritical immiscible phases, and thus are characterized by large viscosity differences. drastically change the environment of the flow species as the temperature is varied. Consequently, the presence of these fluctuations in composition is expected to have large effects on the viscous flow processes of the materials. Recently, light scattering and x-ray diffraction experiments have observed such fluctuations above the critical point [15, 16]. The effect is usually referred to as critical opalescence. Theoretical analyses first derived to describe gasliquid critical point phenomena, were applied to liquid-liquid phase transitions by replacing density fluctuations with fluctuations in composition [15]. Equations are thus available to describe the wavelengths, A, and lifetimes, 7, of the existent fluctuations as a function of temperature for T> Tc. The wavelength, or range of fluctuations varies from very small values far from the critical point, to macroscopic extent near Te. The range of fluctuations, A(T/Te), is expressed as: The classical exponent of 2 is shown here, but there has been strong evidence that a larger value such as 0.6 may be more appropriate for some liq. uids. Such a small difference has little effect on our model, so we will follow the classical equation. The parameter is a constant of the material, and has been shown to vary little among liquid-liquid nonpolymeric systems (see table 4 of ref. [17]). Taking the average value of 11 Å, which also coincides with results from low angle x-ray scattering measurements on PbO-B2O3-Al2O3, we may plot the most probable fluctuation wavelength as a function of reduced temperature in figure 1, and thus predict the size of the microstructure present in such critical systems above the solution temperature. It is now possible to investigate the effect of environmental microstructure on viscous flow by conducting structural relaxation measurements above the critical point of some immiscible oxide mixtures, and analyzing the resulting parameters in terms of reduced temperatures. We have recently reported results from an experimental investigation of the behavior of the viscosity [18] and the frequency-dependent modulus [19] above the critical temperature of a series of immiscible oxide glasses. We will review, here, the salient features of these results and then proceed to analyze the observed supercritical effects in terms of a proposed mechanism for the interaction between the viscous RANGE OF FLUCTUATIONS, A 100 Å The samples chosen for this study are four sodiumborosilicate oxides with similar high-temperature structures but widely varying critical temperatures. These characteristics were chosen to allow separation of effects due to the presence of supercritical composition fluctuations from the noncritical material behavior. Sample 1, with a composition of 70.5 percent SiO2, 22.7 percent B2O3, and 6.8 percent Na2O by mole is the critical composition (top of the dome) of its system [20], and has a critical temperature of 752 °C. Samples 2, 3, and 4 were made by adding 2.1 mol percent CaO; 1.8 mol percent Al2O3; and 1.05 mol percent CaO+ 0.9 mol percent Al2O3 respectively to the concentrations of sample 1. Samples 2, 3, and 4 have resulting transition temperatures of 830, 643, and 741 °C respectively. These samples clearly provide many advantages. First, the viscosities of the separated phases differ by several orders of magnitude, so that the composi tion fluctuations can be expected to induce effects of large magnitude on the shear viscous relaxation parameters. Then, the critical temperatures of the samples occur at such high viscosities as to allow investigation of the distribution of shear relaxation times by ultrasonic relaxation spectroscopy. The shear viscosity was measured by a rotation somewhat, between the samples, at high temperatures, viscometer [21]. The results [18] were found to vary reflecting the effect of doping (log viscosities of 2.41, 2.32, 2.71, and 2.46 at 1300 °C for samples 1, 2, 3, and 4 respectively). The viscosity curves were therefore normalized by the values at 1300 °C, in order to study only the critical point effects. This normalized viscosity: n* (T) = ns (T)/ns (1300 °C) (5) |