Statistical Theories as Applied to the Glassy State
Application of statistical mechanics td ncnequiiibriukri phdniirnena in inorganic glasses is reviewed, including ionic conductivity, diffusion, internal friction, viscosity, and fracture. Emphasis is placed on the connection between the physical assumptions of the various theories and the structure of glass. The theories discussed all involve several arbitrary parameters which are usually determined by requiring that the theoretical curves “fit” experimental points. Although i n some cases the different theories explaining the same phenomena appear very different, many theories can be correlated with one another and many points of similarity found. In some cases the different results obtained by individual investigators are due to differences in terminology rather than method. The correlation of various theories is suggested as a means of reducing the number of arbitrary parameters in the theories.
a
HE principal aim of this paper is to consider the application
of statistical mechanical methods to nonequilibrium phenomena in inorganic glasses. No attempt is made to review or analyze the foundations of statistical mechanics itself, or to discuss the many papers on the application of this powerful tool to condensed structures other than glasses, or the papers that deal with macroscopic statistical methods such as the distribution of flaws in an elastic continuum, General aspects and advantages of the statistical mechanical approach and aspects of the glassy state that are pertinent to subsequent discussions are discussed, as well as the application of statistical methods to diffusion, internal friction, ionic conductivity, and flow and fracture, and some of the correlations among these phenomena. The bibliography is intended to be representative of the papers on this subject only since 1941. As most of the papers reported herein cover the basic equations, no attempt is made to copy the formal algebraic steps leading from physical assumptions to the final equations. However, an effort has been made to include a few references to classical papers in the field of chemical kinetics. THE STATISTICAL APPROACH
All statistical approaches (13, 23) to nonequilibrium processes are based on the idea that the molecules comprising any material are in continuous restless motion such that the chemical bonds joining the molecules are being incessantly broken and reformed. The principal hypothesis made in order to relate this submicroscopic behavior to observable phenomena is that configurations and changes of configurations are determined by the rate a t which these bonds are broken less the rate at which these bonds are reformed. If these two rates are equal, there is no observable change in the body. If the breaking rate is ever so slightly greater than the reforming rate, the overwhelming molecular activity (about 10’8 vibrations per second) will evidence itself macroscopically in some phenomenon such as viscosity or internal friction. I n the absence of external loads, the chance that a bond is to be broken (or reformed) is the chance that a bond ha8 a pre154
scribed energy, and this chance is given by the Boltzmann exI
pression, e - E , where f represents the intrinsic change of energy (or activation energy) which is necessary to break the bond. This energy is independent of the load and is related to the chemical binding energy of the molecules themselves, In applying this general approach to a particular phenomenon, the basic problem is the correlation of the loads or external fields of the phenomenon with the intrinsic change of energy, f. When this is done, the net rate of breaking bonds is identified with load or field. The correlation of load or field with activation energy is ordinarily done by adding the work done by the applied load or field to the energy, f , in such a fashion that the rate of fracturing bonds is increased and the rate of reforming bonds is decreased. A number of unique features of the statistical approach give it certain advantages over other analytical methods, First, time is automatically introduced into the thermodynamic equations of state. Such time-dependent problems as nucleation, annealing, stress relaxation, creep, and devitrification are, therefore, particularly amenable to the sCatistical approach. For materials that are “history-sensitive,” all properties involve time and its neglect in thermodynamic equations may be a serioua omission. I n the statistical approach, the fundamental equation involves a rate, so that time can be separated from othervariables by one integration. Secondly, by the statistical approach, a theory may be evolved” without regard to the exact structure of the material. From this viewpoint, the rate of change of population from state one to. state two is dependent upon the distance between energy minima and the height of the potential energy barrier separating the two. states. The rate does not depend directly upon the souree of this barrier. For materials where the structure is not precisely known--e.g., liquids-the statistical approach is convenient, because it provides theories in spite of the uncertainty concerning relative positions of the atoms of the material. Thirdly, the fundamental rate equation of the statistical approach is easily adapted to distribution functions. Far materials where the positions of particles are to be described by distribution functions, this approach is useful, in fact, often mandatory,
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and Glass-
-Ceramics Finally, when the statistical approach is used, a material may be treated as though it were monatomic for, from a statistical point of view, t8heprecise nature of the fundamental unit of the structure is unimportant. The unit may be considered to be an atom, a group of atoms (such as an SiOatetrahedron), a molecule, an inleratomic bond, or even a grain or crystallit,e. For materials that have a complicated structure, or materials that are composed of a number of different elements in some complicated fashion, the statistical approach is convenient. THE GLASSY STATE
It is generally held among writers on glass technology that the glassy state is most simply characterized as an extended random network, lacking both symmetry and periodicity, as originally postulated by Zachariasen (40). From x-ray analyses, it is known that the relative positions of the atoms in a glass are simi1a.r to the arrangement found in liquids-that is, they show short-range order but no long-range order. Horn-ever, the mechanical properties of cold glass show that t.he atoms have very definite average positions about which they oscillate and that the forces between atoms must be of the same character as those found in the crystalline state and in contrast to those found in the ordinary liquid state. Nevertheless, the lack of symmetry and periodicity in the arrangement of atoms in glasses is in contrast to the regular arrangement of the crystalline state, and leads to several consequences which are characteristic of glasses alone. Two necessary characterizations of the glassy state are given by Zachariasen. 1. THEPARTICLES THATCONSTITUTE THE VITREOUSSTRUCTURE ARE RANDONLY LOCATED.This randomness precludes an exact definition of the structure, and the positions of the particles must, be described by distribution funct.ions. As the particles are randomly dispersed, a large number of them are not in positions corresponding to lowest energy stat.es. The tot.al energy of the vitreous structure is, however, approaching a minimum a t a very slow, but finite, rate which means that many propert,ies of glass are history-sensitive. 2. THEENERGY COSTEKTOF THE AMORPHOUS NETWORKIs ()SLY SLIGHTLY GREATER THAN T H A T O F THE CORRESPOXDING CRYSTAL NETFORK. The result of this is that only relatively few oxides are capable of forming stable glasses. Those oxides that can are called network formers (N.W.F.)-e.g., SiOzwhile those that cannot are called network modifiers (X‘.K.M.)e.g., NazO. Although the distinction between these two classes of oxides is more of degree than kind, most constituents found in the silica glasses fall clearly into one class or the other (the exceptions are AI203and Fez03). Warren (SQ),Stevels (52)) and others have advanced the socalled network theory to account for glasses that are constituted of a mixture of these two types of oxides. It is held by the network theory that the network-forming oxides of such a multicomponent glass form the glassy network proper, while the network-modifying cations attach themselves individually to the network in a random fashion. It is sufficient to note here that, according t,o the network theory of glass, the two classes of ions have dissimilar functions in regard to the physical properties of glass, even to the extent of often being independent of each other. Undoubtedly, the semidependence of the network-modifying class of ions to the glassy network proper is one of the essential features of the glassy state. This is substant,ially true whether or not the glass has network-modifying ions, for constituents that are not a part of a glass will be independent of the glassy network. Some of the properties of glass particularly affected by motion of the network-modifying class of ions are: electrical conductivity, diffusion, color, dielectric properties, internal friction, and molar refraction. Other properties of glass, which are particularly affected by the motion of the network-forming class of ions, are:
January 1954
viscosity, fracture, thermal conductivity, and coefficient of expansion. NOR- both classes of ions affect anv given property, more or less, but in many cases, the effect of one type of ion predominates to such an extent that anv effects of the other type can often be ignored. The above characterizations are essential features of the glassy structure and while from the theoretical point of view it is desirable t o make the most elementary assumptions in developing a theory to explain a property of glass, any simplifications so made should not disregard either of these features. Thus any model of the glassy structure should include: (1) an allowance for the randomness of the glassy structure, and (2) separate mechanisms for the network-modifying ions and the network-forming ions. These two structural features of glass are serious obstacles to the formulation of any theory relating structure to properties, and, a t present, can be treat,ed only by statistical methods. Indeed, every aspect of the glassy structure indicates that it is more amenable to statistical techniques than other condensed structures. STATISTICAL THEORIES OF THE TRANSPORT OF VETWORKMODIFYING IONS
The first statistical theory directly applied to the problem of transport of network-modifying ions was proposed by Maurer (28) in 1941, who used the well-known methods of ionic crystal theory to explain the phenomenon of ionic conduction in glasses. He essentially assumed glass to be a onodimensional crystal full of “Schottky holes,” in which a certain fraction of the total number of network-modifying ions is located. He formulated the net rate a t which the ions jumped from hole to hole, to obtain an expression relating current and voltage which shows that glass is an ohmic conductor only at low fields. His expression,
where i is current, B is voltage, a and A are constants, and ‘p is the activation energy, predicts that a t high potentials the current increases exponentially with voltage, as was experimentally reported much earlier by Poole (29). Maurer experimentally verified the hyperbolic sine law over wide ranges of voltage and showed that the jump distance of the network-modifying ion between equilibrium positions is a function of temperature. Stevels (%), using a more realistic picture of the glassy structure. but the same statistical method as Maurer, derived equations relating the dependence of resistivity on temperature. His assumptions were: ( a ) The jump distance is independent of temperature and equal to the average distance between interstices (the cristobalite lattice constant in the case of silica glasses); ( b ) the activation energy is independent of temperature for most glasses, but varies with the temperature for a few glasses where the network is very rigid (such as found with the borates); and (c) the “holes” in the network through which the ion jumps are limited in number but randomly oriented in space. His final equations for resistivity as a function of temperature are Inp=A+-ll_
RT
for the cases where
‘p,
the activation energy, is constant, and
In p = A
+ BT + CT2
(3)
for cases where ‘p is a function of temperature These two equations are verified by experiment. Cox (6) has incorporated a derivation of ionic conductivity in his theory of viscosity, discussed in the following section. His theory yields an equation of the form,
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where A , p, D’,and C are constants. This equation, which is reducible to the form of Equation 2, is the only one relating resistivity and temperature which can be used both above and below the transformation region. All other theories (and equations) are limited to one range or another. Stuart and Anderson (34.4)’using the Eyring reaction rate theory, later derived Equations 2 and 3. Their derivation differs from Stevels’ only by the assumption that the jump distance varies with temperature, as originally reported by Maurer, instead of the activation energy varying with temperature. These theories are in substantial agreement. They all regard the jump distance as the distance between positions of equilibrium of the network-modifying ions, or the lattice constant of the network. I n addition, Anderson and Stuart ( 1 ) have advanced a semiempirical theory (involving no arbitrary constants) which relates the activation energy of ionic conductivity to the physical constants of glass. Thus, the statistical approach has been successful in explaining ionic conductivity, as the two statistical parameters have been identified with the structural and physical constants of glass. It has been known for some time that the migration of networkmodifying ions under a rapidly varying stress is a source of internal friction in glasses a t high temperatures and low frequencies. Korn (21) assumed a general statistical model incorporating two stable equilibrium positions, and used Eyring’s reaction rate theory t o explain migration-induced sound absorption in solids. He derived two equations defining internal friction for the isothermal and adiabatic cases. In the isothermal case, his equation for internal friction is the usual form,
(4) where A is a constant, w is the frequency, and the relaxation time r is defined in terms of the activation energy, q, by (5)
r = rOedRT
From his application of this analysis to internal friction in silica glass, Korn concludes that the mechanism of the absorption is the stress-induced diffusion of network-modifying ions from one cavity to the next with the breaking of a silica-oxygen bond as proposed by von Riitger (38). Johnson, Bristow, and Blau (19) have recently conducted an extensive experimental investigation of the diffusion of networkmodifying ions in silica glasses using radioactive tracer techniques. They applied Eyring’s reaction rate diffusion equation ( 1 4 )to analyze their results and found that the activation energy of this process is substantially the same as found for the case of ionic conductivity and internal friction in equivalent glass systems. Fitzgerald (11) has recently discussed the relation between the activation energies of diffusion, internal friction, and ionic conductivity, and concluded that all three phenomena involve the same mechanism of network-modifying transport (see Table I).
TABLEI. ACTIVATIONENERGYFOR VARIOUSN.W.M. TRANSPORT PHENOMENA^ Activation energy, kcal./mole a
Ionic Conduction
Internal Friction
Diffusion
21
21
20
All three measurements were t a k e n for the same glass by Fitzgerald (11).
between the particles. Nevertheless, any discussion of present statistical theories of viscosity of glasses can scarcely omit the theories of the liquid state put forward by Andrade (I),Frenkel ( l a ) , and Eyring (10). A4ndradevisualizes a liquid to be composed of close-packed particles as in a solid, except that the particles have a very large amplitude of vibration. Assuming that the momentum is transmitted across a layer one atomic distance thick, he finds an expression for viscosity as a function of the frequency of vibration of the atom, the atomic mass, and the interatomic separation. Alternative forms of Andrade’s equation are obtained using various temperature-dependent expressions for the frequency. Frenkel obtains an expression for viscosity by equating the frictional coefficient of the Einstein diffusion equation to the frictional coefficient in the expression of Stokes’ law which defines the velocity of a sphere through a viscous medium. Eyring visualizes a liquid as a binary mixture of holes and identical molecules in which the viscosity process consists of a molecule adjacent to a hole moving through the hole to a neighboring position. The motion of the molecule is visualized by Eyring to be controlled as a rate process involving an energy barrier. A11 three of these theories lead to equations of the form, 7 = Aeb/kT
where the constant A contains the size of the flow unit and b is some form of activation energy. Attempts to apply this simple equation directly to glasses have not met with as much success as applications to normal liquids. Seddon (30)attempted to fit this equation to several glasses, but he found A and b to be rapidly varying functions of temperature-Le., he found that two arbitrary constants are insufficient for an equation of viscosity of the glassy state. Jones (20) offered the interesting hypothesis that the flow properties of glass can be explained by the above expression if one assumes a continuous distribution of activation energies. By Fourier methods any given distribution of activation energies can be transformed into a viscosity-temperature function and vice versa. This approach is used by many investigators in the field of mechanical properties of high polymers. However, although such an approach is undoubtedly consistent with the structure of glass, it has not as yet yielded any clear picture of the general relations between structure and flow properties, so it is not discussed here. Douglas (8) approached the problem of formulating a statistical model of viscosity more directly in terms of the known structure of glass rather than simply adopting an established theory of viscosity of liquids. Douglas developed his theory on the basis of the random features of the glassy network. From the results of x-ray diffraction experiments, it is known that the interatomic distances (or angles) of the particles of glass have a large variability. Because of this variability, there are many atoms which are separated from their nearest neighbors by distances larger than the equilibrium distances of adjacent particles in a perfectly ordered state. Using the usual expression for potential energy as a function of interatomic spacing, Douglas shows that for those particles with large interatomic spacings, there are two alternative positions of equilibrium. The random nature of the glassy state, therefore, predicts that there is a fractional number of network-forming ions, w,which will have alternative positions He showed that if there are only two possible spacings the temperature dependence of w is given (9) by
STATISTICAL THEORIES OF VISCOSITY
All existing statistical theories of the glassy state ow?! a great deal to the various theories of viscosity of liquids. AS pointed out by Douglas (7), these theories can apply only to normal liquids which are monatomic and have nondirected forces acting 156
where C and D are arbitrary parameters. Douglas visualizes the flow process as the jumping of networkforming ions having dual equilibrium positions from one alter-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 1
and Glass-
-Ceramics nate position to the other. This jump provides a large spacing, and consequently dual equilibrium positions, for the next ion which can then jump, and so on. The energy barrier between alternate positions is the activation energy of the viscosity process. Because of the variability of interatomic distances, there will be a variation in the magnitude of the activation energy for those particles having the choice of alternate position. However, in order t o simplify his mathematics, Douglas assumes that the activation energy between all alternate positions has a single value. Thus, his model is approximately the Eyring model of a liquid. Douglas’ equation for viscosity becomes 7 =
AT - eb/kT W
and substituting the expression for w, the expression becomes q = A T e b / k T (1
+ CeD/T)
(6)
which has the four arbitrary parameters A : b, C, and D. In the Douglas model, parameter w may be time-dependent, since the number of alternate positions may vary with time. In this model, the network-modifying ions affect the primary network by sealing off oxygen bonds and by loosening the networkLe., by increasing w. Douglas made a considerable step forward when he recognized that the activation energy of the viscous process is substantially due to the alternate positions of atoms generated by the random network of glass. Cox ( 5 ) developed a theory, analogous to that of Douglas, by assuming a continuous distribution of interatomic separations of the network, Cox replaced this continuous distribution by a two-cell histogram equivalent to Douglas’ large and small interatomic spacings. The bonds in cell A are distinguished as having greater strength but small interatomic distance, while those in cell B have less strength but larger interatomic distance. Cox derived an expression giving the relative number of bonds, n/Ar, in the highest energy state as a function of temperature (which is equivalent to the number of atoms with alternative positions in the Douglas theory),
n/N =
[
eD’/T
7 = Aeb/T
+ CeD/T
(8)
where again there are four arbitrary parameters. Although they made no attempt to correlate this expression with the structure of glass, they did show that such an equation is closely related to experimental values of viscosity. In fact, by this equation they were able to compute the viscosity in terms of composition of 200 glasses with close agreement to the experimental values. Their important contribution was to show that a four-parameter statistical equation of viscosity yields quantitative results. The most striking feature of all the previous theories of conductivity and viscosity is their close similarity, rather than their diversity. They differ only in a few details. One of the most evident differences is the presence or absence of the temperature as a multiplying factor in front of exponential. Some investigators, especially the British authors, derive equations of the form A’TebIT
(9)
while others, especially those following Eyring’s statistics, derive equations of the type Aeb/ T
(10)
The absence or presence of this T factor has been the source of some discussion by Cole (3) and others. Cole points out that the experimental data are best fitted by those equations not including the disputed T . Actually, the T factor is unimportant for two reasons: The effect of temperature in the exponential masks any variation caused by T as a multiplying factor, and the T , to a first approximation, does not properly belong in either equation. According to Maxwell-Boltzmann statistics, the fractional number of particles that exceed any arbitrary energy (say the energy required to fracture their bonds) per second is ve- f / k T
where P is some thermal vibration frequency near 1Wa. According to Eyring statistics, the rate of chemical reaction is
7
+T
Cox uses his expression for n/,V in a similar manner to that in which Douglas uses l/w,but he reasons that the probability that an ion can detach itself from the network and therefore flow involvm the simultaneous presence of more than one (say 2 ) of the large interatomic distance type. The probability that there exists a region where the bonds are all sufficiently weak, so that an applied shear stress can cause flow, is then (nlN)’, and Cox’s equation for viscosity is
(7) which contains five arbitrary parameters. It is important to note that the Cox and Douglas theories differ in one respect. The fundamental unit in the Douglas theory is a particle as contrasted to the Si-0 “linkage” (or bond) in the Cox theory. For this reason, Cox is unable to state explicitly the manner in which the flow of atoms occurs. Cox’s most important contribution is his recognition that the three-dimensional glassy network requires the cooperation of a number of Si-0 linkages if flow is to occur. Poncelet (68) postulated a three-dimensional structure containing randomly placed voids, which iricorporates the general features of the Douglae one-dimensional model, in order to account for the tensorial characteristics of stress and to relate such a stress to flow. He fuither gave a specific example of such a January 1954
structure and, using the Joffe law of interatomic attraction, he indicated the method of computation of activation energiee. Hodgdon and Stuart (IS)proposed an empirical mathematical model of glass to explain Viscosity which reduced to the viscosity equation
p e - f/kT h
where Eyring defines the k T / h factor as a universal frequency. There is no functional difference between the Eyring equation and the one involving Y , since to a first approximation the vibrational frequency of an atomic bond is proportional to temperature, and the order of magnitude of k T / h is that of Y. In the non-Eyring approach, the constant A ’ involves l / r , and Equation 9 more properly has the form
A’ (T) eb/kT MT)I which is reducible to the approximate form Aeb/kT
over any limited temperature range. A second difference among the various theories is the type of resulting equations. As an example, three equations of viscosity are listed. The Douglas equation is
while the Cox equation is
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
157
and the Hodgdon and Stuart equation is
n
= Aeb/T
+ Ceb'/T
which may be considered an approximate form of the Douglas equation. Although these equations appear different, in each case the authors reduce them to a set of two equations in order to compare them with experimental results: q M A'eb'IT
for high temperatures
q
for low temperatures.
vI
s
31 v e - Q / R T
(11)
where v is the propagation velocity of a transverse pulse, and Q is the activation energy of the fracture process. I n the presence of a certain stress, V , Poncelet assume8 that the Joffe function (18),s, is replaced by (s - u),and the energy to overcome the net attractive forces of an ion pair-Le., the activation energy of the fracture process-is the area under the force curve between the two zeros of the curve,
and M
A"eb"lT
I t is apparent that there is no real functional difference between these types of equations. One sees why all of these theories fit the experimental data equally well. This situation is not surprising, when it is recalled that the statistical approach begins w t h the energy barrier, and cannot distinguish among various possible hypotheses concerning the source of that barrier The requirement of such a theory of viscosity, and also the common denominator of all of these theories, is the presence of four arbitrary parameters in the resulting equations. I n spite of the fact that these theories of viscosity are in good qualitative agreement with experiment, it must be candidly admitted that many types of equations with so many arbitrary parameters can be fitted to a set of experimental points with a fair degree of accuracy. I t would seem that no further progress can be made with statistical theories of viscosity until the arbitrariness of such theories is reduced. One way to reduce the number of arbitrary parameters is to correlate, in so far as possible, the parameters with the physical constants and properties of glass. Both Cos and Douglas have based their theories on the assumption that the randomness of the network-forming ions, leading to alternate positions separated by an energy barrier for these ions, accounts for one of the activation energy constants. Following their reasoning and recalling that two of these arbitrary constants represent activation energies, one is led to ask in what manner the activation energy terms are related to the presence of the network-modifying ions. As a matter of fact, while all these theories of viscosity have incorporated Zachariasen's random position criterion of glass, none has incorporated the network theory hypothesis-namely, that the network-modifying ions are practically independent of the primary structure. Thus, the above question cannot, at present, be answered. A possible explanation of the large number of constants (and possible interpretation of these constants) is that, for the case of viscosity, it is necessary to assume, a priori, a separate statistical mechanism for each class of ions. This would make physical sense, provided that both classes of ions made substantial, and largely independent, contributions to the observed deformations. STATISTICAL THEORIES OF FRACTURE AND FATIGUE
Poncelet (24, 26) in 1943 first originated a statistical "flawgenesis" theory in which he showed that flaws can be generated in a random structure under favorable circumstances. His approach was to postulate that the "Joffe curve" (or any other such curve giving the net attractive atomic forces as a function of interatomic distance) is displaced by the presence of certain stresses, so that a significant number of ion pairs will have a large probability of attaining the requisite potential energy to fly apart. Thus, after sufficient time, a flaw can be created by a certain load. Poncelet then shows that the opening of such a flaw causes a compression and transverse pulse to pass through the material, the latter causing the flaw to propagate in its fracture plane. Poncelet derives an expression for the rate of crack propagation,
158
Poncelet's expression for crack propagation is generally in accord with experimental results. His important contribution, however, was to show that statistical methods are sufficient to explain the low breaking strength of glass (as contrasted to the theoretical strength). In a later paper, Poncelet ($6,27) extended his theory of fracture propagation to the problem of static fatigue. Expressing his crack velocity, v,, as the rate of increase of crack depth, daldt, he derived an expression giving the time required for the crack to enlarge to a certain critical depth,
where a1 is the initial crack depth and a2 is some critical depth. Poncelet's final expression for time and stress is of the form log t = A - B log
c~
(14)
where A and B are arbitrary constants. Poncelet's theory accounts for the excessive stress concentration introduced by sharp cracks and flaws; one of the constants in the above expression contains the Inglis stress concentration factor. Taylor ( 8 7 )was the first to apply chemical rate theory methods to static fatigue of glass. He pointed out that by this approach the rate of fracturing chemical bonds is
koe - Q / R T where ka is a constant having dimensions of frequency, and therefore the time required for fracture is simply the reciprocal of this expression
where Q is the activation energy of the fracture process. Taylor argued that the activation energy of the fracture process is related to the activation energy of breaking the chemical bond, W , by the expression,
Y
Q-W-
where Y is Young's modulus and u is the applied tensile stress. Taylor's expression for time and stress is
where A and B are constants. Taylor's theory was a step forward, because it predicted that the conditions of the ambient atmosphere would affect the activation energy of the fracture process. I t showed that statistical methods are capable of explaining a well-known experimental fact-the atmosphere significantly changes the time required for a specimen to fracture under a given stress. Taylor further suggested that the surface energy released by the fracture is related to the distance between equilibrium positions of the potential barrier. Unfortunately, he did not provide a mechanism relating surface energy to activation energy.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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*
-
and Glass-
-Ceramics The Poncelet and Taylor theories of fatigue have been criticized ( a l ) ,because they predict that a specimen of glass will ultimately fail for any load, however small. There is ample experimental evidence to show the existence of a fatigue stress limit below which a load will never produce fracture ( 1 7 ) . Gurney (16) has deduced from thermodynamics the theoretical existence of such a limit. Cox ( 4 ) later developed a theory, very much along the lines of the earlier Poncelet flaw-genesis theory, based upon the assumption that under given favorable conditions there is a chance that a crack will be generated in a brittle substance. His expression for the time required for the formation of a flaw a t a given stress is of the form
where a, A , Q,and 2 are arbitrary constants. By means of his theory, Cox was able to give a qualitative relation between fatigue and such phenomena as the distribution of breaking frequencies with stress, the effect of size of the specimen on the rupture modulus, the variation of the rupture modulus with temperature, and the velocity of crack propagation. Cox's theory suffers, however, because of the large number of arbitrary constants, none of which has been related to the physical constants of glass. It is superior to the previous theories in one important respect. It predicts the existence o f a static fatigue limit, Stuart and Anderson (35)have recently advanced a statistical theory which combines many of the features of the previous ones. This approach uses Griffith's concept of the pre-existence of flaws, the stress concentration effect as used by Poncelet, and Taylor's idea concerning the effect of ambient atmosphere on the activation energy. In addition, allowance is made for the difference in energy between the initial and fractured state by means of an unsymmetrical energy barrier. They found an expression for the rate a t which the bond8 changed from a stable state into a fractured state _ Q_ = NAe R T f ( a u , c , E )for au 2 E/2c dt and G' = 0 for (YU 5 E / 2 c
-
dt
where Q is the activation energy of the fracture process, E is the surface energy (with appropriate dimensions), and f( ( ~ u , c , E ) is a function involving energy of strain, (YU,the stress concentration factor, c, and the surface energy, E. The time of fracture is found by integrating the number of bonds, N , in the above expression from No to 0. The final expression for time and stress reduces to E In t = K1 - &u for (YU 2c
>>-
for By this theory,
Q
=
W
(YU -.f
- E/2
E -
2c (20)
where Q is the rate-determining energy of fracture, W is the energy required to break a primary network bond, and E is the surface energy required to form a new fracture surface. As E can be determined by the static fatigue limit, or by separate surface tension tests, the activation energy of the breaking bonds can be determined by fatigue experiments. This theory, like the one of Cox, predicts the existence of a static fatigue limit. The important contribution of this theory was the identification of the static fatigue limit with the surface energy of glass It is evident that the equations of fatigue as given by thevarious authors have very different forms. The surprising fact is that all the previous investigators have used the same experimental data to verify their equations. This is because the test data a t presJanuary 1954
-{cox 24
--
STUART AND ANDERSON PONCELET N.W. 1AYLOR PRESTON A N D BAKER
I
1 0 -2
2
0
4
6
6
IO
LOG BREAKING T I M E I N SECONDS
Figure 1. Comparisonof Theories and Experimental Results for Fatigue of Borosilicate Glass Lines.
Theories
Circles. Experimental results
ent available do not extend to sufficiently long times to distinguish between the theories, which diverge as a function of time. A comparison of the theories and data is made in Figure 1. It would seem that further advances in theory will depend upon a number of experiments that remain to be done. For an excellent survey of the experimental data on fracture and fatigue, see (35). CORRELATION BETWEEN FRACTURE AND FLOW
Except for the Cox theory, all theories of fatigue have involved only two arbitrary constants in contrast to the four parameter equations developed for viscosity. All these theories assume that the network-modifying ions do not significantly affect fracture. If the network theory of the glassy state were correct, one would be led to expect some physical connection between the parameters in the theories of fracture and the parameters in the theories of viscosity. I n the first place, the primary network class of bonds should play a prominent role in the viscosity process, since according to the network theory the networkmodifying ions are attached weakly to the primary network and therefore the energy required to shear them is small compared t o that required to shear network-forming ions past each other. There should be, therefore, a simple relation between one of the parameters representing activation energy in the viscosity equation, and the parameter representing activation energy in the fracture process. Both Poncelet (27) and Cox (6) have postulated relationships between the fracture and flow processes in glassy materials. Poncelet has shown that a fracture surface propagates in a plane as a result of a transverse stress, and the chance that a bond is broken is unaffected by the state of affairs of neighboring bonds not in the fracture plane. He further shows that flow propagates as a result of a stress deviator tensor, and therefore a unit flow
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process requires the participation of all neighbors adjacent to a flow unit. Cox has formulated this idea in another way. He demonstrates that if the probability of the occurrence of one broken bond (as in fracture) is P, the probability that two network-forming ions will exchange places, requiring the simultaneous participation of x bonds (as in flow), is P*. As the probability of a broken bond is
P
ta e - f / k T
it immediately follows from Cox’s reasoning that b (flow) = xW (fracture) where W is the energy required to break a primary bond, b is the activation energy of the viscosity process due to the primary network, and x is the number of nearest neighbors to a network-forming ion. Now to test this relationship, available experimental data must be used. Unfortunately, Cox and Douglas failed to tabulate sufficient parameters in their theories of viscosity, so we must turn to the empirical model of Hodgdon and Stuart for values of the activation energy of viscosity. Stuart and Anderson have tabulated values for the activation energy found by their theory of fatigue. Using the previous relationship suggested by Cox and the equation between the activation energy and surface energy proposed by Stuart and Anderson, Equation 20, we obtain
where x is the number of nearest neighbors around a networkforming ion, Q is the activation energy of the fracture process, b is the effective activation energy of a primary bond in the viscosity process, and E is the surface energy per mole of surface particles. For a soda-lime glass, b = 135 kcal. per mole, Q = 26.4 kcal. per mole, and E = 300 ergs per sq. cm. = 11.1 kcal. per mole of surface particles yielding:
x
=
4.2
Le., predicting the necessary cooperation of about four neighbor ing bonds in the flow process. This recalls Taylor’s early idea (36)concerning the relationship between the activation energies of viscosity and electrical conductivity. It would seem that this avenue of approach might provide a reasonable basis for the correlation of the processes of fracture and flow. If such a conclusion is feasible, then the number of arbitrary parameters in the theories of viscosity can be reduced, since any information on the fracture precess applies equally well to the phenomenon of flow. SUMMARY
In spite of the complicated nature of the glassy state, and the gross simplification required by the necessity of obtaining workable equations, it would seem that the thatistical approach has been moat fruitful in the interpretation of some macroscopic phenomena in terms of structure. Undoubtedly, much remains to be done, and as more structural features are incorporated into the theory, we may expect a better understanding of the physical properties of glass. NOMENCLATURE
arbitrary energy level current voltage resistivity P = - activation energy for resistivity process $I = Cox configurational energy 6 = internal friction w = frequency r = relaxation time coefficient of viscosity r l = b = activation energy of viscosity process D = Douglas configurational energy w = Douglas parameter for fractional number of bonds in highest energy state n / N = Cox’sparameter for relative number of bonds in highest energy state
fi
v
160
= =
=
x
=
b’
=
number of cooperating bonds in the flow process second activation energy of viscosity process used by Hodgdon and Stuart ur = rate of crack propagation v = transverse wave velocity Q = activation energy of the fracture process u = stress s = Joffe function for net attractive force between atoms Y = lattice frequency W = activation energy for the rupture of a primary network bond Y = Young’s modulus a = proportionality constant between stress and work done by that stress N = molar number of bonds = stress concentration factor due to the presence of a c sharp crack E = surface energy P = probability of a statistical event e, R, T , t, h, k all have their usual meaning A , A’, A “ , B , C, G, K , K1, Kz are constants a = crack depth y = mathematical dummy variable LITERATURE CITED (1)
Anderson, 0. L., and Stuart, D. A., paper presented a t annual meeting of American Ceramic Society, Pittsburgh, Pa., April
27, 1952. (2) Andrade, E. N. daC., “Viscosity and Plasticity,” 1st ed., pp. 1420, New York, Chemical Publishing Co., 1951. (3) Cole, H., J . SOC.Glass TechnoE., 31, 141-3 (1947). (4) Cox, 5.M., Jr., Ibid., 32, 127-46 (1948). ( 5 ) Ibid., pp. 340-65. (6) Ibid., pp. 350-3. (7) Douglas, R. W., Ibid., 31, 77 (1947). (8) Ibid., pp. 78-85. (9) Ibid., 33, 140-62 (1949). (10) Eyring, H., J . Chem. Phys., 4, 283 (1936). (11) Fitfigerald,J. V., J . Am. Ceram. SOC.,34, 314-19 (1951). (12) Frenkel, J., “Kinetic Theory of Liquids,” 1st ed., pp. 190-5, London Oxford Clarendon Press, 1946. (13) Glasstone, S., Laidler, K. J., and Eyring, H., “Theory of Rate Processes,” 1st ed., New York, McGraw-Hill Book Co., 1941. (14) Ibid., p. 517. (15) Gurney, C., Proc. Phys. Soc., 59, Pt. 2, 169 (1947); Phil. Mag., 39, 71 (1948). (16) Hodgdon, F., and Stuart, D. A., spring meeting, American Ceramic Society, 1949; J . A p p l . Phys., 21, 1160-70 (1950). (17) Holland, A. J., and Turner, W. E. S., J. SOC.Glass Technol.. 21, 392 (1937). (18) Joffe, A. F., “The Physics of Crystals,” 1st ed., pp. 12-20, Xew York, McGraw-Hill Book Co., 1928. (19) Johnson, J. R., Bristow, R. H., and Blau, H. H., J . Am. Ceram. SOC.,34, 314-19 (1951). (20) Jones, G. O., Repts. Progr. Phys., 12, 133 (1948). (21) Korn, G. A., J . Acoust. SOC. Amer., 21, 547 (1949). (22) Maurer, R. J., J . Chem. Phus., 9, 579-84 (1941). (23) iMott, M., and Gurney, R., “Electronic Processes in Ionic Crystals,” 1st ed., London, Oxferd University Press, 1940. (24) Poncelet, E. F., Colloid Chem., 6, 77-88 (1945). (25) Poncelet, E. F., “Fracturing of Metals,” 1st ed., pp. 201-27, Cleveland, Ohio, American Society for Metala, 1948. (26) Poncelet, E. F., Metals Technol., Tech. Publ. 1684 (1944). (27) Poncelet, E. F., Verres et refractaires, 3, 149-60, 289-99 (1949); 4. ., 158-71 - - - . - (1~m-1) \----,. (28) Ibid., 5, 69-80 (1951). (29) Poole, H. H., Phil. Mag., 42, 448 (1921). (30) Seddon. E., J . SOC.Glass Techol.. 23, 36 (1939). (31) Shand, E. B . , “An Experimental Study of Fracture,” paper presented at fall meeting of American Ceramic Society, 1952, to
be published.
?!.,
(32) Stevels,J. “Progress in the Theory of the Physical Properties of Glass, pp. 5-16, New York, Elsevier Publishing Co., 1948. (33) Ibid., pp. 56-60. (34) Stuart, D. A., and Anderson, 0. L., J . Am. Ceram. Soc., 36, 27 (19.53). . (35) Ibid., 36, 416-24 (1953). (36) Taylor, N. W., Ibid., 22, 1-8 (1939). (37) Taylor, N. W., J . A p p l . Phys., 18,943 (1947). (38) Rotger, R. von, Glass Tech. Ber., 19, 192-200 (1941). (39) Warren, B. E., J . AppZ. Phys., 8, 645 (1937); Chem. Rev., 26, 237 (1940). (40) Zachariasen, W. H., J . Am, Chem. SOC.,54, 3841-52 (1932). \----,
RECEIVED for review April 16,1953.
INDUSTRIAL AND ENGINEERING CHEMISTRY -
ACCEPTED September 30, 1953.
Vol. 46, No. 1
