ANOMALOUS FLOW I N GLASSES
235
AYOMALOUS FLOW I N GLASSES NELSON W . TAYLOR Departozenl of Ceramics, The Pennsylvania State CoZZege, State Culleye, Pennsylvania Received January 7 , IO@ DISTIKCTIVE FEATURES O F GLASS
Deformation and flow are properties which have been found very useful in the study of the constitution of glassy or vitreous matter. The so-called “glassy state” appears to be continuous with the liquid state, but differs from it in that the properties of glasses are dependent upon their history. Thus while femperczf.ure, pressure, and composition serve to describe the condition of an ordinary liquid, the additional factor of rate of cooling or “thermal history” is found to have a marked influence on the properties of glass. This is not surprising in view of the very high viscosity, in excess of 10” poises, in consequence of which chemical equilibrium is reached very slowly. At room temperatures most silicate glasses‘ are so rigid that they show almost no secular changes, although some very sensitive tests furnish evidence of slow adjustments of share and volume. Consequently it is possible to prepare samples of glass of a given composition with different properties, depending on whether the samples have been quenched (Le., quickly cooled) or annealed. DENSITY AND THERMAL HISTORY
Changes in density resulting from variations in heat treatment were first observed by A. Q. Tool (16, 17, 18) and coworkers a t the United States Bureau of Standards. They heated cubes weighing a few grams at 350’ to 550OC. for flint glass and 450’ to 750°C. for borosilicate glass for periods varying from 1 hr. to 264 hr., and cooled them in air. In general, the higher the temperature from which the glass was chilled the lower was the specific gravity, the deviation from normal attaining a few tenths of 1 per cent. The effect was later confirmed by Salmang and von Stoesser (9), who quenched small samples of glass very rapidly from a series of temperatures ranging from 100OC. to nearly 160O0C. The specific gravity, measured in every case at room temperature, ranged from 2.511 to 2.493, a change of 0.008. That this effect is not due to strain was shown by these authors with an experiment on the well-known Prince Rupert drop, which is made when molten glass is quenched in water, and which is under such strain that it frequently flies to pieces if scratched. The density of such a drop before and after powdering was 2.4439 and 2.4446, respectively. Thorough annealing and slow cooling, however, raised the density to 2.4620, a change of 0.0181 as compared with 0.0007. While these changes in density due to annealing are not large in magnitude, nevertheless they are definite, and in a direction to be ’The term “glass” is taken to cover only those materials which soften and harden reversibly on heating and cooling, respectively. It does not include those organic plastic bodies which decompose on heating. The most important glasses from a comniercial viewpoint are inorganic, e.g., silicates, borates, phosphates, or combinations of these,
236
XELSOX
V', TAYLOR
expect,ed if association or polymerization of the simplest ionic constituents of the glass were taking place a t or near the annealing temperatures. It is a wellknown fact that chilled pieces of glass, windshields, goggles, etc., are many times more resistant to mechanical shock than similar glass, slowly cooled, but this has more to do Xvith the distribution of strain in the surface layers than with physicochemical changes in the silicate network. Densit>y changes resulting from heat, treatment are by no means confined to silicate glasses. A sample of unannealed B203 glass, investigated by Spaght and Parks (lo), occupied a volume of 3.419 cc. at room temperature. The same glass sample, carefully annealed, occupied a volume of 3.373 cc. a t room temperat'ure. This volume decrease of 1.3 per cent is a consequence of molecular rearrangements. VISCOSITY AXD THERMAL HISTORY
Since viscosity is a rheological property, it is of int,erest to see how it is influenced by the thermal treatment of glasses. Viscosity is structure-sensitive to a high degree; consequently in that range of temperature where annealing takes place rapidly but not too rapidly for measurement, it is found that viscosity is a function of time. For every temperature there is an equilibrium viscosity, characteristic of the "stabilized glass," but hours, days, or even years may be necessary before equilibrium is reached. During this period the viscosity may increase or decrease gradually, depending on whether the glass was previously at a higher or a lower temperature. Figure 1, taken from a paper by Lillie ( G ) , illustrates this behavior. For ordinary soda-lime-silica glasses, 21 -9-70 per cent, the temperatures at which these effects are observable are near 500°C. For soda-silica glasses, 33 per cent sodium oxide, the corresponding temperatures are near 425°C. For a heavy barium crown optical glass they may be observed near 550°C. The range of temperature in which such phenomena may be readily observed depends on the temperature coefficient of viscosity, which at these temperatures doubles for every 6' to 8' drop in temperature. Thus a temperature rise of 100OC. will increase the rate of approach to equilibrium by a factor of about 10,000. The initial viscosities of the glass shown in figure 1 differ by a factor of 70, but even larger differences may be obtained where forced cooling or a lower temperature of stabilization is applied prior to the actual measurement. VISCOSITY h S D STRUCTURE O F SILICATE MELTS
We shall return again to a study of the ionic processes which occur when glass is being stabilized, as well as to a consideration of the elastic phenomena which occur when stabilized glass is subjected to stress. First, however, we shall try to determine why silicate glasses have viscosities so extremely high as compared with water and the simpler organic liquids, and why temperature has such a marked influence on the magnitude of the viscosity. Primarily, the difference in molecular structure between a silicate melt and an organic liquid like ethyl alcohol or carbon tetrachloride is that the latter are composed of discrete molecules between which the forces of attraction are weak (although they may be very strong within the molecule), while in silicate melts
,
ANOMALOUS FLOW I N GLASSES
237
a continuous structure exists in the sense that the ions build up a sort of lattice of alternate charges with no regions of distinct weakness. When viscous shear occurs, the process is not so localized as in the case of molecular liquids, and the critical "activation energy" is larger. This means a large numerical value for B in the well-known de Gunman (3) equation q = The interionic forces are very strong in silicates because of the high valences of the Si4+and 02ions, and are strong even where cations of lower valence are present. In a sodalime-silica glass the weakest bonds are those between Na+ and 0'-. In addition, the coordination number of silicon is 4, Le., silicon has four oxygen neighbors. This leads to strong bonds and to the formation of a continuous network
FIG.1. Viscosity-time curves for two samples at 486.7"C. The upper curve is for a sample previously treated at 477.8" for 64 hr.; the lower curve is for a newly drawn sample. Courtesy of H. R. Lillie.
in the more siliceous melts (2, 20). The character of the silicate ions in glasses can be inferred from the known structure of silicate crystals having the same si1icon:oxygen ratio. When this ratio is 1:4, as in orthosilicate crystals, such as Mg2SiO4, separate Si0:- ions exist, not connected directly with one another, but linked through cations such as Mg", Ca", or.Na+. When the silicon: oxygen ratio is 1:3, long chains or closed rings having the general formula Si,0s,2n- are found in crystals, and presumably they also occur in glass. These larger ions lead to higher viscosities. When the si1icon:oxygen ratio is 1:2.5, sheet structures are formed, as in micas or clay minerals. By analogy, similar though probably warped structures may exist in glasses having the same silicon: oxygen ratio. When the si1icon:oxygen ratio is 1:2, three-dimensional networks
238
X3LSOh* W. TAYLOR
such as that of quartz, tridymite, or cristobalite occur in crystals, and we may expect something similar in glasses. In fact, fused silica has t,he highest viscosity of any silicate melt, 11-hen such melts are compared at some constant temperature. Most commercial glasses, free of B203,show an atomic silicon: oxygen ratio of about 1:2.5. When ions such as B3-, or A13', substitute for Si4, in the four-fold coordination, the continuity of the ionic network remains unbroken, but the Tkcosity a t the higher temperatures is somewhat reduced as compared with the simple silicates. Cnder certain conditions, boron may have three oxygen neighbors, and aluminum six, with corresponding changes in the viscosity and its temperature coefficient. The viscosity of commercial glasses a t melting and fining temperatures is about lo2 poises; while being handled by automatic blowing machines, about 10' poises; a t the annealing temperature, about l O I 3 poises. These may vary about tenfold, but they indicate the order of magnitude of the viscosity a t which these various operations are carried on. Measurements have been made by the writer on glass fibers less than 100°C. below the annealing temperature, showing viscosities of about loL5poises. At room temperature the value must be extremely large, and because of the very long t'ime needed to reach equilibrium, as discussed previously, it is doubtful if true values of the viscosity will ever be obtained on cold glass. I n general, the replacement of an ion by one of higher valence, e.g., Ea' by Ca", leads to higher viscosities. The changes resulting from the replacement of an ion by a larger one of the same valence, e.g., Sa' by K', are often not quite so simple. We should expect that the larger ion would have weaker attract,ion for 0'-, t,hus leading to lower viscosities, or a t least to a lower temperature coefficient, and this is in fact the case (14). The partial substitution, hoivever, may cause abnormally low viscosities. Such effects may be due to volunie expansions on mixing, Le., to the volume not being a linear function of composition. It is well known that' viscosity is very sensitive to volume changes resulting from the application of pressure (4),and therefore probably also to 1-olume changes arising from chemical substitution. A systematic correlation of volume and viscosity in silicate melts would be of great interest. I n systems showing such a tremendous range of viscosity as 10' to loL5poises, were applicable it would be surprising indeed if the simple relation q = over the whole range. An equation of this form is satisfactory for limited temperature ranges, say 30" or 40"C., but over the whole scale a plot of log q us. 1/T is concave upward, Le., toward the log q axis. What this probably means in molecular or ionic terms is that the number and kind of bonds which must break t o pwmit viscous shear are themselves functions of temperature. The writer has Yhown (13) that a t .low temperatures only the sodium-oxygen hond is actiw in the shear process in soda-lime-silica glasses (11), for the removal of lime is without significant effect on the activation energy for viscous flow. At higher temperatures there is sufficient thermal energy to break stronger ionic bonds, and this will of course affect the slope of the log q VS. 1/T curve. -4 silicate melt a t high temperatures is a complex mixture of many molecular or ionic species. Statistically speaking, a t a given temperature certain species
AiXOhfALOUS FLOW I N GLASSES
239
will be favored, while a t lower temperatures certain larger or more complex species will predominate. The environment of any one atom is constantly changing, and the more rapidly the higher the temperature. As Morey (8) expresses it, “The equilibrium is kinetic and statistical, and will of necessity change with temperature.” ELASTIC PROPERTIES OF STABILIZED GLhSS
Studies of the elastic properties of stabilized glasses in the annealing range of temperature have helped to clarify our understanding of the glassy state. An elastic body may be defined as one which returns to its original shape after the removal of an externally applied distorting force; since this definition does not contain time as a variable, the elastic effect may be either sudden, or delayed, or any combination of the two.
z 0
H
TIME FIG.2. Typical flow curve a t constant temperature. Elongation uersus time a t constant temperature; a definite load added at B , and removed a t &.
A viscous body may be defined as one which is deformed continuously a t a constant rate under the influence of an external force. Upon removal of the force, a permanent deformation exists, the magnitude of which is proportional to the magnitude of $he applied external force and to the duration of its application. Elastzco-viscous properties are those properties which arise from a combination of elastic and viscous effects consistent with the above definitions. The terms elastic distortion, viscous flow, and elastico-viscosity, when used in this paper, will conform to these definitions. A stabilized glass may be defined as a glass in which the effects of its thermal history have been removed by thorough heat treatment a t the temperature of the test, until (insofar as present laboratory methods permit of disclosure) it has assumed the equilibrium property values characteristic of the glass a t that temperature. A typical curve depicting the progress of elongation of a stabilized glass fiber which hangs vertically in a constant-temperature furnace and to which a weight is attached is shown in figure 2. The ordinates are the over-all elongations and
240
NELSON TV. TAYLOR
the abscissas ale the corresponding times. The cycle may be traced as follows: The fiber is heat-treated a t the temperature of test until a constant rate of flow, A B , is attained, due to the weight of the fiber itself and the attached hanger. Upon application of a definite increased load to the stabilized fiber, an instantaneous elastic elongation, BC, occurs. This is followed by a combination of delayed elastic elongation and true viscous flow which results in an elongation cuiye, CD, whose rate decreases until it attains finally a constant value, KDE. I'pon removal of load, an immediate elastic contraction, E F , occurs. This is followed by a period, FG, during which the delayed elastic contraction opposes the tiue viscous flow owing to the load of the lower hanger. The resulting
6F 4
a
s
W
TIME FIG. 3. Effect of tempeiatuie on the over-all elongation-time ciiive for a stabilized glass sample.
curve, F G N , shows a rate which decreases through zero and iq followed by the attainment of the previous constant rate of elongation (LGH parallel to A B ) . It should be emphasized that the elastic effects are completely reveraihle and reproducible. The effect of temperatuie on the over-all elongation-time curve is qhown in figure 3. It is to be noted that the instantaneous elastic elongation increases as the temperature of test is increased, and the time necessary for the delnyed elastic distortion to be completed becomes shorter as the temperature is raised. The over-all elongation-time data may be resolved (figure 4) into three component parts: ( 1 ) an immediate elastic elongation, completed very rapidly; ( 2 ) a delayed elastic elongation, proceeding more slowly and approaching its limiting value asymptotically; and (3) a viscous flow, proceeding a t a constant
ANOMALOUS FLOW IN GLASSES
24 1
rate. The magnitudes will depend, of course, upon the composition of the glass, the temperature,. and the applied load. In terms of these components, the over-all elongation a t any time map be represented in the simplest case by their sum, as expressed in the following formula : E$ = Z &(I - e-*') k,t
+
+
where E l = over-all elongation a t time t , I = instantaneous elastic elongation, lo = total delayed elastic elongation, e = base of natural logarithms, k = rate constant for delayed elastic process, or fractional change in unstretched length per unit of time, t = time, and k , = constant rate of viscous flow.
TIME FIQ.4. Analysis of the elongation-time curve a t constant temperature and constant stress. 1, instantaneous elastic elongation; 2, delayed clastic elongation; 3, viscous flow; 4, over-a11 elongation [4 = (1 2 3)l.
+ +
The rate of viscous flow, k,, may be determined by inspection or by application of the theory of least squares to the linear part of the over-all elongationtime curve, Le., after the delayed elastic effect is completed. Determination of k , is necessary for the calculation of viscosity. The curve representing the delayed elastic elongation can be readily obtained by subtracting the viscous flow and the instantaneous elastic elongation from the over-all elongation. Delayed elastic efect-rate
constants
The delayed elastic component of the over-all elongation-time curve may be readily obtained. In figure 5 (inset), lo is the total delayed elastic elongation, and 2 is the distance from equilibrium a t time t . If dl/dt is the rate of elongation
242
NELSON W. TAYLOR
a t time t , it can be shown that for a stabilized fiber - d l / d t = id. Integration of this espression gires log,, l / l o = -kt/2.303 aiid
I//, =
e-kt
The rate constants, IC, were determined from the delayed elastic elongation-time data for the various glasses at each temperature of test by application of the method of least squares. If these equations are satisfactory, a plot of
TIME (min) FIG.5 . Delayed elastic elongation w m u s time for certain soda-silicn g!ns3es. showing effect of temperatwe and obedience to the exponential decay I a n .
loglo l/Zo versus t should give a straight-line relation. Figure 5 shows graphically a test of these equations. The relaxation time is that time necessary for the delayed elasticity to decay t o l / e of its original value, i.e., to approximately 40 per cent. That is, I becomes &/e and the equation reduces to l/e = e-"
from which t = relasation time = l / k ~ This definition is essentially that given by Clerk 1\Iaxwl!. The calculated values of the relaxation times for certain soda-silica glasses are given in table 1 and shown graphically in figure 6.
243
ANOMALOUS FLOW IN GLBSSES
It is to be noted that for a given temperature the relaxation time increases considerably with increase of Si02 content, and, for any given relaxat,ion time, the necessary temperature is higher as the Si02 content is increased. The relaxation time may be compared qualitatively with the “half-life period” of radioactive decay, except that in this latter case the quantity is reduced to 1/2 rather than l/e of its original value. TABLE 1 Relaxatzon time of soda-szlzca alasses as a function of lemowature 113) GLASS
i
TEMPERATURE
RELAXATION TIME
“C.
minulss
lOa/T
427.8 446.6 456.0
12.7 1.6
0.7
1.427 1.389 1.372
435.0 442.9 452.5
7.1 3.3 1.2
1.412 1.397 1.378
427.7 442.2 446.8 452.5 455.9
10.2 4.8 2.6 1.5 1.2
1.427 1.398 1.389 1.378 1.372
Glass SS (23 5 per cent N h O )
442.2 446.8 452.5 456.0 459 * 3
10.0 5.0 2.9 1.9 1.3
1.398 1.389 1.378 1.372 1.365
Glass KO.1 (19.8 per cent N h O ) . . . . . . ,
442.4 447.0 452.5 456.0 459.3 466.0 471.2
100.0 59.0 28.6 17.2 0.1 3.7 1.9
1.398 1.389 1.378 1.372 1.365 1.353 1.344
Glass SSC (33.0 per cent Na,O) . . . . . . . , .
‘I
Glass X (ca. 30 per cent NaaO).. . . . , , . .
Glass No. 2 (27.8 per cent Na,O). . . . . . .
i
1
I t may readily be seen from an inspection of figure 6, which is a semilogarithmic plot, that a t room temperature relaxation times for these particular processes will be of the order of millions of years.
Young’s modulus ilnother striking feature, brought out in a previous investigation (15) and verified in later work (13), is the relation between Young’s modulus and the temperature. Table 2 contains the values of the modulus, E, (calculated from the instantaneous elastic elongation) and E2 (calculated from the total elastic elongation).
244
NELSON W. TAYLOR
X study of table 2 confirms the observation that Young’s modulus depends upon the time. If the delayed elasticity be taken into consideration, it appears that the value of Young’s modulus is independent of the temperature. It will be noted that the two calculated vahes of Young’s modulus become equal at the higher temperatures where there is apparently no delayed elasticity.
.
100.0
no
j.220
1.260
1,300IOYT-
1.360
1.400
1.460,
FIQ.6. Relaxation time versus temperature for the delayed elastic process in certain Boda-silica glasses.
Elastic after-eflects in potash-silica glasses JJ‘hen the elongation curves for potash-silica glasses are examined, a new effect is observed. The data for the early part of the delayed elastic elongation fall below the best straight line which represents the plot of log l/lo vs. t. This negative departure of the log Illo ~ a l u e sis clearly depicted in figure 7. One
TABLE 2 Young's modulus based on instantaneous
( E l ) and
total (E*)elastic extension EI
E1
Glass SSC ..... . .
. . . . . . .. . . .. . .
Glass X . .. , . . . . . . . . . . . , . . . . . . .
Glass No. 2 . . , , . .. . . . . . . . , . . . . ,
Glass SS... . , , , . . . . . . . , .
Glass No. 1 . ., ..... . . . . . . , . . , . .
'C.
kg.
per nm.'
kR. per nm.l
103
2.3 2.1 2.2
x x x
103 103 103
103 103 103
2.4 2.4 2.3
x
103 103
x x x x
103 103
2.8 x 103 2.6 X 1Oa 2.6 x 103 2.3 x 103
x x x x x
103
3.6 3.2 2.9 2.5 4.0 3.7 3.1 2.8
x x x x
427.8 446.6 459.0
3.8 2.7 2.2
x x x
435.0 442.9 452,5
3.7 3.3 2.3
x
427.8 446.8 459.0 466.0
4.4 3.3 2.8 2.3
442.2 446.8 456.0 459.0 466.0
4.0
442.4 452.5 459.0 471.0
x
x
103 103
103 103
103
103 103 103 103 103 103
103
2.8 2.7 2.6 2.8 2.5
x
x
x
x x x x
103
103 103 103 103 103
2.3 x 103 2.4 X 103 2.3 X lo3 2.3 X 10a
FIG.7. Delayed elastic elongatiun versus time for certain pot.ash-silica glaases, showing apparent lack of obedience t o the exponential decay law for the carly part of the relaxation. 245
24G
NELSOK W. TAYLOR
rate constant, k , probably does not describe the whole course of the delayed elastic process, but two or more “unimolecular” processes apparently are operating with different characteristic rates, and the observed phenomena represent the summation of these processes operating simultaneously. If the first process is distinctly more rapid than the second, the characteristic rate constants may be evaluated by examining the early and late data independently. This was
FIG.8. Evaluation of the constants for the fast and the slow relasations in the delayed elastic process for a potash-silica glass. Each process obeys the exponential law, but the two are operating simultaneously.
done in the following manner: From the original curves of elongation vs. time, a list of values of the unstretched length, I , for various times, t , was prepared; a plot was made of log I us. t , and an extrapolation was made back to the zero time (when the additional load was added t o the fiber). The early points mere neglected in making this extrapolation t o get log lo, and weight was given t o those later points which were self-consistent in falling on a straight line (figure 8). Values of log 1 corresponding to the early part or possibly the first quarter or first third of the duration of the delayed elastic process may be read from the plot.
AKOiIIALOUS FLOW IN GLASSES
247
These cciculated I values are less than those actually observed for the early time period. The differences then are plotted in the form, log I’ us. t , and they fall satisfactorily on a straight line the slope of which is considerably greater than that of the first plot, thus indicating a molecular process which goes much more rapidly than the other. This procedure is exactly like that used by Taylor in analyzing the data on the birefringence of glass (11). The entire delayed elastic elongation a t any time, t , would thus consist of two terms lo(l - e--Et) and I:(1 - .e-“’), instead of only-the first term. More such terms could be added if needed. This means, in a qualitative sense, that the glass probably contains molecular groups of various sizes. The application of an additional load to a fiber is considered to bring about an elastic elongation by turning or orienting these molecular groups so that the longest axis of each tends to become parallel to the fiber direction. The smallest groups will respond quickly to the added load and will give a large rate constant in the equation, -dl/dt = k’l, or in the corresponding expression, 2: (1 - e-” ’) ; larger groups will respond more slowly and will give different values for k an$ la. Although this complexity first became evident in potash-silica glasses, a reexamination of the data on soda-silica glasses shows that it is also present but to such a small degree that it had been overlooked. The soda-silica glasses apparently are more homogeneous in their molecule types, a t least in the annealing range of temperature. Cnless all molecules in a silicate melt or glass have exactly the same size, which would be contrary to statistical and kinetic theory, the several sizes (and masses) will lead to several constants for the rate of orientation when the piece of glass is subjected to stress. The delayed elastic phenomena therefore point to a sort of molecular picture for the glass. A close analogy is found in the behavior of a polar liquid in an electric field. Each dipole responds in its own way t o the orienting force, and the factors, such as dipole moment and temperature, affect the various relaxation times. With elastic phenomena, the “shape” of the molecule probably plays a big part in controlling the amount of orientation and in the consequent fiber elongation, and the sizes and ionic charges control the relaxation times, which of course would also be reduced by increased temperature. Under isothermal conditions, the equation for the entire elongation a t any instant is expressed by the following equation:
E, =
z
+ Z;(I - e-”‘) + lO(1 - e-’’) + k,t
or more exactly by
For the delayed elastic process in these s,tabilized glasses, the rate of stress release is proportional to the stress, -dE/dt = kl, as suggested by Clerk Maxwell. The rate constants have been evaluated as well as their reciprocals, that is, the “relaxation times”. The temperature coefficient of the rate constants leads to activation energies in the potash-silica series ranging from 57,000 to 96,000 calories, values which are considerably lower than for corresponding soda-silica
248
XELSOK W. TSYLOR
glasses. The temperature coefficients of these rate constants lead to activation energies of 105,000 to 150,000, the higher energies being for glasses higher in silica. THE PRIKCIPLE
OF SUPERPOSITIOK-%EMORY”
EFFECTS
(12)
When an electric charge is placed upon a piece of glass between condcnser plates, the charging current, large a t first, decays with time according to the empirical expression i = At-“, where n is not an integer and also is not constant for small values of t . This current-time relationship, according to von Schweidler (19), may be better expressed as the summation of several terms each of the
+I
FIG.9 FIG.10 FIG.9. Current-time relation in the discharge of a glass condenser FIG.10. Reversal of the discharge current (curve l), shown as the result of two simple discharge curves (2 and 3) with different directions and different relaxation times. form i = A,e-k”‘, where A , is a constant having the dimensions of current, t is time, k , is a rate constant equal to the reciprocal of a relaxation time T,, and e is the base of natural logarithms. This equation is exactly like that for the elongation of a glass under constant stress if the viscous part, k,t, is omitted. In other words, dielectric absorption and elastic after-effects in glass appear to be closely related. “Hopkinson (see reference 7 ) found that, if a conderiser which absorbs a residual charge is charged for some time and then the sign of the e.m.f. is reversed for a shorter time, the first part of the discharge current corresponds in direction to the last charge imposed on the condenser but the direction of the discharge becomes reversed a t a later stage of the discharge and corresponds in direction to the first charge.” This phenomenon is illustrated in figure 9, taken from Guyer. Lord Kelvin is said to have remarked, “The charges come out of the glass in
ANOMALOUS FLOW IN GLASSES
249
the inverse order in which they go in,” while Hopkinson has stated ,“It seems safe to infer that the effects on a dielectric of past and present electric forces are superposable.” The writer presents curve 1 of figure 10 to show the result obtained by algebraic addition (Le., subtraction) of two simple exponential curves 2 and 3, where the former has a relaxation time roughly one-tenth of the latter. The reversal in direction of current flow is clearly shown by curve 1. In dealing with these dielectric effects in glass it appears justifiable to use the concept of weakly or strongly bonded complex ions or “molecules” in the random network which constitutes the glass. In such a random structure there are small and large “molecules,” these being complex silicon-oxygen groups which are separated from other such groups by weak bonds such as i\ja+-O--. Under the influence of an external orienting force the smaller ions or “molecules” would respond more quickly than the larger, presumably because they would have fewer bonds to break as they turn under the orienting influence of the applied field. Thermometers frequently show gradual drifts in their readings of the “icepoint” which are believed to be due to slow changes in the dimensions of the glass bulb. If this bulb slowly shrinks in size, the ice-point readings will gradually rise. On the other hand, if the bulb would first expand and then shrink, the ice-point readings would first fall and later rise. Figure 11’ shows such a result on a thermometer which was held for 2650 hr. a t 212”F., and then for an additional period of 1350 hr. a t 306’F. The ordinates show the change in the ice-point a t various periods of time. While the glass was a t 212’F. a slow shrinkage was taking place, owing to the combination or association of ions or molecules which characterized the glass a t higher temperatures. Before this association was complete, the thermometer was heated to 306’F. This rise in temperature had the effect of dissociating some ionic complexes which had formed a t 212”F., and this brought about a relatively rapid expansion of the glass with a corresponding drop in the ice-point. This was followed by a gradual shrinkage of the glass due to combination of other ions which characterize much higher temperatures and which had not time to combine completely a t 212°F. The association process a t 306°F. goes on much faster than the corresponding process a t 212”F., a s can be seen from the slopes of the two curves. It is evident from such results as that of figure 11, that the molecular or ionic structure of glass is quite complex, and that the various ions appear to respond independently to stresses imposed upon the glass. An exactly analogous elastic reversal has been observed in rubber (5) in an experiment performed by H. Kohlrausch in 1876. Quoting from Houwink, “A rubber thread (see figure 12) is twisted 2 X 360” to the right and held in that position for 18 hours (a). It is then released and twisted for 45’ beyond the position of equilibrium to the left and held so for 30 seconds (b). An elastic after-effect is observed first from (b) past the equifibrium to (c), for example, 1 Private communication from Dr. Bradford Noyes, Taylor Instrument Co., Rochester, New York.
250
A-ELSOS W , TAYLOR
then the direction of the after-effect is changed and the thread twists to the left until a final position a t (d) is reached. The behavior of the thread does
HUNDREDS OF HOURS FIG.11. Change in the ice-point of a glass thermometer during aging. The reversals point to n complex elastic process. Cowtesy of Bradford Noyes.
FIG.
12
FIG.13
FIG.12. Elastic reversal bcd in a rubber rod; axis of rod normal to page FIG.13. Hydrostatic model showing a reversal effect due t o simullaueons fast and s!ow processes.
indeed give the impression that it possesses a ‘memory’. ’ I This behavior again illustrates the intimate relation which exists between the elastic and dielectric effects.
ANOMALOUS FLOW I N GLASSES
25 1
A simple hydrostatic model illustrating a reversal may be described. In figure 13, the central tube is connected to tube A by a capillary S through which a liquid can flow only slowly, and is connected to tube B by a large tube R through which flow is rapid. If a suitable volume of liquid is admitted slowly into the system through the opening 0, the level indicated in the central tube will rise from 1 to 2. If, now, the same quantity of liquid is withdrawn rapidly from 0, the liquid level in the central tube will fall first to 3 and then will reverse its direction and rise again to the original level. This analogy shows how a slowly and a rapidly operating mechanism can combine to produce reversals in physical behavior which, on first examination, would appear to be anomalous. The elastic and the dielectric evidence lend strong support to the concept that there are several sizes of complex silicate ions in glass, and that these various ionic units respond independently and with different rates to applied stress. This idea is in harmony with other physicochemical evidence and with the x-ray evidence of a random ionic network. THE ANNEALING OF GLASS
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The molecular phenomena which occur during the annealing of glass are of great interest. Let us assume that the effect of stress is to produce atomic distortions and also orientation of atomic groups (molecules or ions) in such a way as to tend to relieve the stress, just as dipoles may be induced and others oriented by an electric field. The shapes of the atomic groups are important, the greatest orientation being produced in groups of low symmetry (deviating most from the sphere). Evidence has been presented in the first part of this paper that quick cooling of a glass tends to preserve its high-temperature dissociated structure, resulting in an assortment of irregularly shaped ions which become squeezed or oriented in certain preferred directions depending on the cooling stresses. The strain-bearing elements are therefore mainly the dissociated units, but as these units combine to form larger and more symmetrical aggregates their “micro-strains” disappear, just as the esternal field of two bar magnets is weakened when they pair with one another. Such a process is essentially a bimolecular reaction and should obey the law -dx/dt = kx2,where z is the concentration of dissociated units. An empirical expression of this form -dS/dt = kz6* was found by Adams and Williamson (1) to give satisfactory representation of the variation of strain with time, as measured by the birefringence of optical glasses. The strain-bearing units which have been oriented by the cooling stresses may also relax independently and assume random position, so that the net effect is zero strain. This is an elastic adjustment. If this process occurs, the rate law should be like that for radioactive decay, namely, -dS/dt = k16, and this is in fact the form of the Maxwell equation, which states that the rate of release of strain is proportional to the amount of strain. Obviously each molecular or ionic species should have its characteristic rate constant and relaxation time. The writer has shown that these two equations together account quantitatively for the whole data on strain release in Pyrex chemically resistant glass, measured over a 2-yr. period. The annealing process therefore involves two mechanisms,-independent relaxation and interaction.
252
NELSON W. TAYLOli
The two mechanisms operate a t the same time but with different rates and with different temperature coefficients. Both of these processes result in dimensional changes in the glass, so that measurements of either optical birefringence or length can be used to determine when the glasv is completely annealed and stabilized. At any time t the strain birefringence, 6, per unit thickness of glass may be expressed as follows: 6 = 6,
+ 6,,,
where ~ 3 , ~=
where ,a,
and
6,
+ 6,,,.At)
= 6awo/(l
(-dj/dt = k,f) and 6,,,, = total strain released according to the Adams-Killiamson law (-dj/dt = k2j2). k , and k2 are characteristic stress-rate constants for the two processes. This study makes it clear that the past history of a glass sample has a profound effect upon its properties. On the one hand a glass may be “stabilized” and yet show a complex elastic character; on the other hand, if not properly annealed, it may be undergoing dimensional changes due to molecular rearrangements. These properties affect its utility and its behavior in service. = total strain released according to hlaxwell’s law
SUbIhlARY
Glass is a material having properties very dependent on thermal history. Yiscosity and elasticity are properties very sensitive to structure, consequently their study provides an insight into the nature of glasses. The dissociated character of glasses a t high temperature tends to be preserved on quick cooling, owing to the high viscosities, but as association proceeds the glass finally becomes stabilized. Such a glass shows a viscosity which at constant temperature is constant, Le., independent of time. The rate of association appears to follow a bimolecular reaction law, e.g., the rate of stress release is ,proportional to the square of the stress. When stabilized glass is subjected to a constant stress, the elongation is a summation of elastic adjustment and viscous flow. The former is itself complex, being a combination of several relaxation processes of different characteristic rates, each obeying the Maxwell law that the rate of stress release is proportional to the stress. Numerical data are given for soda-silica and potash-silica glasses. Superposition of independent relaxation processes of different characteristic rates and of opposite sign gives rise to anomalous reversals. Examples of such phenomena in the discharge current of condensers and in the aging of thermometers are described. .4“molecular” picture of a giass based upon the above-mentioned properties is presented.
STRUCTURAL ISOMERS I N SIMPLE RING COMPOUXDS
253
REFERENCES
(1) AD.4MS, I,. H., AND WILLIAMSON, E . D . : J. Franklin Inst. 190,597-631 (1920). (2) BRAGO, W. L.: Atomic S t r i ~ L ? ~ofr eMinerals. Cornel1 University Press, Ithaca, Kew York (1037). (3) D E GUZMAN, J . : Anal. sac. espafi. fis. qulm. 11, 353 (1913). (4) Dow, R.B.:Physics 6, 270-72 (1935). It.: Elasticity, Plasticity, and Structure OJ' Matter, pp. 165,186. Cambridge (5) HOUWINK, University Press, London (1937). (6) LILLIE,H. W.:J. Am. Ceram. SOC.16, 619 (1933). (7)LITTLETON, J. T . , A N D MOREY, G. W.: Electrical Properties of Glass, p. 100, Fig. 32. John Wiley and Sons, Inc., Kew York (1933). (8) ~ I O R EG. Y ,W.: Ind. Eng. Chem. 32, 1423-7 (1940). G , . ~ N DYON STOESSER, K.: Glastech. Ber. 8,463 (1930). (9) S ~ L M A XH., M .E.,AND PARKS, G. S.: J. Phys. Chem. 38, 103-10 (1934). (10) SPAGHT, N.W.:J. Am. Ceram. Soc.21,85-9(1938). (11) TAYLOR, N.W.:J. Applied Phys. 12, 753 (1941). (12) TAYLOR, (13) TAYLOR, N. W., AND DEAR,P. s.: J. Am. Ceram. S O C . 20, 296-301 (1937). (14) TAYLOR, N.W.,AND DORAN, R. F.: J. Am. Ceram. SOC.24,103-9 (1941). K. W., MCNAMARA, E. P., AND SHERMAN, JACK: J. Sac. Glass Tech. 21 (83), (15) TAYLOR, 61-81 (1937). (16) TOOL, A.Q., A N D EICHLIN,C. G.: J. Am. Ceram. Sac. 14,276 (1931). (17) TOOL,A.Q.,A N D HILL,E . E.: J. Sac. Glass Tech. 9, 185 (1925). (16) TOOL,A. Q., A N D VALASEK, J.: Bull. Am. Inst. Mining Eng. 1019.1945. (19) V O N SCHWEIDLER, E . : Ann. Physik 24, 711 (1907). W. H . : J. Am. Chem. Soc. 64,3841-51 (1932). (20) ZACHARIABEN,
OK T H E NUMBER OF STRUCTURAL ISOMERS IN SIMPLE RING COMPOUNDS.
I
TERRELL L. HILL MorZey Chemical Laboratory, Western Reserve University, Cleveland, Ohio
Received December 4 , 1949
The number of structural isomers possible in the case of substituted simple symmetrical rings can of course be obtained by enumeration for any particular case. Thus a table might be constructed giving the number of isomers as a function of n,the number of vertices in the ring, and of the type of substitution (e.g., X, Xz,Xt, , XY, XYZ, . . . , etc.). It is, however, of some theoretical interest to correlate or unify these numbers by means of as general functions as possible. Polya (1) has treated the general mathematical problem of symmetry in organic compounds and has applied his method, which is useful in finding the number of isomers for a given ring where the degree of substitution is generalized (e.g., successive hydrogens may be replaced by other groups), to derivatives of benzene, naphthalene, anthracene, phenanthrene, thiophene, paraffin hydrocarbons, etc. The alternative method of generalization (applicable to simple rings such as
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