Gyan P. Johari'
National Research Council of Canada Ottawa, Ontario, Conada
Introduction to the Glassy State in the Undergraduate Currituhm
Several years ago, during a lecture on molecular glasses,
I was confronted with the realization that many of the fundamental concepts of the glassy state are unfamiliar to most physical chemists. We are all well aware of the great technical importance of the area, hut the topic of the glassy state also has considerable intrinsic scientific value, enough to generate publication of several scientific journals2 devoted to research exclusively in this area. The lack of familiarity, it appears to me, is simply because the topic has been ignored in the teaching of physical chemistry. There have been no reviews on it at the undergraduate level and it has not been treated in the undergraduate textbooks on chemistry. Many people, with some justification, might argue that a consideration of the glassy state belongs more to the realm of the physicist than to that of the chemist. But, inasmuch as the students are introduced to the basic concepts of the liquid and crystalline states in chemistry courses, the burden of teaching this topic should not he made to rest with the courses in physics and material sciences. The glasses show the characteristic properties of both the ~ ~ S t aand l s the liquids; they are systems in a thermodynamic non-equilibrium acquiring, at a very slow rate, thermodynamic equilibrium: they are an interesting rase ot metastabilitv: and. above all.. the". form adistmctlv different form of matter. I wish to outline in this article what I feel should he included in an introductory, phenomenological, and nonmathematical first exposure of the glassy state to the undergraduate. The topic can he easily covered in at most two lectures and the teacher can use his judgement as $0 the stage at which it should be introduced-preferably after the student has acquired an elementary knowledge of thermodynamics, liquids, and the crystalline state. Also, my purpose is not so much that of telling the teacher what should he taught about glasses hut rather that of giving him sufficient background to the topic to enable him to guide a student's efforts in the inquiries into this subject. An extensive coverage of this area can he found in several review articles3 and one m o n ~ g r a p h . ~ General Aspects
The term "glass" is commonly used to mean the optically transparent fusion product of inorganic materials which have been cooled to a rigid condition without crystallizine. This eenerallv means the ordinaw silicate dasses whi& are used for &aking windows and-labware. Literally thousands of glasses, each with its characteristic properties and chemical composition, have been made and thev do not necessarily contain inorganic materials. Examples of two familiar glasses made from cane sugar are lollipops and cotton candy; the former in t h i shape of a rigid block, and the latter as flexible fibers. Substances of quite diverse chemical composition have been obtained as glasses and it is becoming widely recognized that the property of glass-formation is not, strictly speaking, an atomic or molecular property hut rather one of a state of aggregation. Therefore, the word glass is a generic term and instead of speaking of "glass" we should speak of glasses, as we speak of crystals, liquids, metals, etc. We should concern ourselves with the description of glass as a class of matter.
r
-
Figure 1. Comparison of the radial distribution function of a glass with lhat of the gas, liquid, and crystallinestates.
Glasses are characterized by certain well-defined properties which are common to all. The X-ray and the electron diffraction studies show that they lack long range periodic order of the constituent atoms, ions, or molecules. That they resemble liquids and not the crystalline solids in their structure is illustrated in Figure 1 in which the radial distribution function of a glass is compared with that of the gas, liquid, and crystalline states. Unlike crystals, glasses do not have a sharp melting point and do not cleave in preferred directions. Glasses belong to the group of non-crystalline solids. Like crystalline solids, they show elasticity-a glass fiber can he bent almost double in the hand and when released springs back to its original shape-and like liquids, thev flow under a shear stress. as evident from the increased thickness of the glass at the bottom of windows in some vew old buildings. A lollipop cannot he permanently bent in one's hand b;t, when-lift for several days on a warm surface, flows and loses its shape. We see that the glassy form of matter combines the "short-time" rigidity of the crystalline state with the long-time fluidity of the liquid state. Glasses, like liquids, are isotropic, a property which is of immense value in their use for a variety of purposes. Unlike the crystalline solids, glasses are of rare occurIssued as NRCC No. 13408. 1 Present address: Glaciology Division, Environment Canada, 562 Booth Street, Ottawa, Ontario KlAOE7. ZFor example: Physics and Chemistry of Glasses, Journal o f Non-Crystalline Solids, Journal of the American Ceramics Soeiety, ete. 3Kauzrnan, W . , Chem. Reu., 43, 219 (1948); McKenzie, J. D. (Editor), "Modern Aspects of the Vitreous State," Vols. 1-3, But-
tenuorths, London. 4 Jones, G. O., "Glass." Methuen, London, 1956. Volume 51. Number 1. January 1974
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rence in nature. Obsidian, pitchstone, perlite, and pumice are the few naturally occurring glasses. These are associated with the volcanic rock and were formed on rapid cooling of viscous lava. Because of its quality to fracture to produce sharp edges, obsidian was used for making tools in the Monolithic period. Geologically ancient glasses are very rare and the glassy rocks known to us are of much younger age. I t is conceivable that the glassy materials were abundant in the ancient geological times hut nearly all of them have since become devitrified. The presence of glass on the lunar surface may indicate the occurrence of volcanic and meteoritic activity in recent times. Glass Formation
A glass is generally ohtained by undercooling a liquid below its freezing point and this has been considered as part of the definition of the glassy state, although as we shall see later it can also he obtained by compressing a liquid. The classical explanation for the formation of a glass is that when a liquid is cooled its fluidity decreases and, a t a certain temperature below the freezing point, it becomes nearly zero. Our liquid becomes "rigid." Let us take an example of a liquid and consider how its physical properties change on undercooling. Glucose, a familiar substance, is an example of a material which readily undercools to form a glass. It melts a t 141°C and once molten, it can be kept below this temperature for a sufficiently long time without crystallization. Its enthalpy, heat capacity, volume, and thermal expansivity are shown as a
Figure 2. The enthalpy, heat capacity. volume. and thermal expansivity of glucose as a function of temperature.
function of temperature in Figure 2. We observe that as the melt is cooled below the room temperature, its heat capacity decreases almost by a factor of two. The specific volume and the heat content show no analogous change, but they-show a slight discontinuity. There is no volume change or latent heat at this transition but the thermal expansivity decreases by a factor of four. Glucose stays optically transparent, and there is no change in the refractive index at this temperature, hut the temperature coefficient of the refractive index suddenly decreases. The transition occurs between 15-25°C. This temperature is known as the glass transition temperature and is denoted by T,. Below 15°C glucose is hard and brittle and can be easily {ractured by tapping. If kept pinched between the fingers, it softens and flows because the body heat raises its temperature. Its X-ray diffraction pattern is similar to that of the liquid. We note from Figure 2 that its thermal expansivity and heat capacity are much closer to that of the crystalline glucose, but its specific volume and heat content are almost continuous with those of the liquid and are nearly the same as those of the liquid. Glucose a t these temperatures below 15°C is in the glassy or uitreousstate. Liquids can also be ohtained in the glassy state by the application of pressure. Let us consider an example of this 24
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' 3 9 B . ' b "12 ' 1 16
""aI 20
PllibO,
Figure 3. Volume of selenium at 40'C asa function of Dresswe.
and see how the volume of a liquid changes with pressure. Figure 3 shows the volume of selenium at 40°C as a function of pressure up to 20 kbar. We see that near 11 kbar there is a discontinuity in the volume which is similar to that seen on cooling the glucose melt in Figure 2. The compressihility, obtained from the slope of the curve, decreases by about 40% a t 11 kbar, in very nearly the same way as the thermal expansivity. At higher pressures the comuressibilitv of the liauid selenium is very close to that of the crystalfine phase.'~he pressure at which there is a sudden decrease in the compressihility is known as the pressure of glass transition, P,, and selenium at a pressure above 11kbar is in the glassy state. There has been no measurement of the heat capacity of the liquid a t pressures high enough for the transition to be seen. We see from the foregoing examples that we can now have a phenomenological rather than a genetic definition of the glassy state: A glass is a form of matter which maintains the energy, the uolume, and the structure of liquid, but for which the changes in the energy and the uolume with temperature and pressure are similar in magnitude to those of the crystalline solid. The ability of a substance to form a glass does not depend upon any particular chemical or physical property. It is generally agreed that almost any substance, if cooled sufficiently fast, could be obtained in the glassy state; although in practice crystallization intervenes in many substances. Some of the typical substances which have been readily obtained as glasses are listed in the table. Thermodynamic Aspects
At the glass transition, the liquid and the glass differ in the second derivative of the free energy, G, with respect to temperature, T, and pressure, P, but not in the free enereies themselves. or in their first derivatives. In Fieure 2 the ;ohme of given by
is unchanged at the transition, hut the thermal expansivity The Glass Transition Temperature of Some Typical Substances Substances '"Pyrexg l a d '
sio* Na&Or.5H.O CaiNOsll-4HzO A1203 sulfur
Auai7Cca.~sSlo.ralloy Natural Rubber Polystyrene Sucmse mucone
Glass Transition Tern~(rattt, 'K
and the compressibility
undergo an abrupt change. Analogously, the enthalpy
does not change, but the heat capacity
changes a t the transition. These considerations indicate that the glass transition has more or less the characteristics specified for a second-order thermodynamic transition. Whether or not it is a thermodynamic transition is a question that we will discuss later in this article. Another thermodynamic property which changes a t the transition is the internal pressure, given by
which is generally obtained from
For several substances the internal pressure decreases sharply a t the transition. Glasses do not obey the third law of thermodynamics; the entropy of a glass does not vanish a t absolute zero. This residual entropy which is also called the frozen-in, or the configurational entropy, can be obtained from
where S~(g1as.s)and ST(crystal) are theintegralsof C,dln T for the glass and crystal, respectively. The enthalpy and free energy can similarly be evaluated. The residual entropy for most glasses is in the range 2.5-4 cal mole-' deg-l and the residual enthalpy in the range 0.6-5 kcal mole-'. The application of pressure on a liquid increases the glass transition temperature. This is illustrated in Figure 4 for molten poly(viny1 acetate). The glass transition temperature, T,, increases by about 13°C for 490 bar. The increase in T, with pressure is related to the changes in the thermal expansivity and the heat capacity by the equation5
where Au is the difference in the thermal expansivity of the liquid and AC, the difference between the heat capacity of the liquid and the glass.
Figure 4. Glass transition temperature as a function of pressure for molten poly(viny1 acetate).
I TEUPERArLTURE
-
1
Figure 5. Thermal expansivify versus temperature of a liquid.
Nature of the Glass Transition We are prepared to ask now, is glass transition a second-order transition in a truly thermodynamic sense? Our measurements of thermal expansivity and heat capacity suggest that this is possibly so. In seeking an answer to this, let us first examine what effect the rate of cooling has on the temperature of transition. We see in Figure 5 that the abrupt change in the a and C, occurs at a somewhat higher temperature along the curve 2 when the liquid is cooled at a faster rate, but occurs a t a lower temperature along the curve 3 when cooled a t a slower rate. A hasty experimenter observes the transition a t a higher temperature than a more patient experimenter. Since T, for a given material is governed by the experimental time scale, it cannot be assigned a unique value. A thermodynamic transition arises from a structural change in the system, or from the discreteness of the quantum energy levels; it is an equilibrium phenomenon and its temperature is independent of the thermal history of the system. (The freezing point of a liquid does not change with the rate of cooling!) The glass transition does not have a thermodynamic origin. Why then is there a sudden decrease in the thermal expansivity and the heat capacity? What are the molecular degrees of freedom which cease to contribute to these thermodynamic quantities at the glass transition temperature? The answer is provided by experiments which directly, or indirectly, measure the rate of molecular motion in liquids. The viscosity of a liquid reaches a value of 1013 P at the transition temperature. The molecular jump rate estimated from the viscosity is of the order of s-', or about 25 min for a single molecular jump. Direct measurements of the rates of structural change in the liquid also give us nearly the same value. We are aware that a large proportion of heat capacity of a liquid is contributed by the random changes in the arrangement of molecules. If the molecules in a liquid did not have time to jump, its heat capacity would decrease. This is precisely what happens at the transition; as a result of the slowness of molecular motion, the liquid is unable to change its structure in the duration of the experiment. Consequently, there is an abrupt decrease in the a and C, as we see in Figures 2 and 6. We already know that the time of a molecular jump increases with decreasing temperature, so, if our rate of cooling is such that we do not wait for more than 5 min during measurements, the transition will be seen at a somewhat higher temperature. If more time were allowed, the transition would be observed at a lower temperature. The glass transition is kinetic in origin. We realize now that the abrupt decrease in the a and JSee for example, Goldstein, M., J. Phys Chem., 77, 667 (1973).
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FREE ENERG
Figure 6. Volume, enthalpy, and entropy as a function of temperature.
C . which indicates the class transition is an artifact due torour inability to do experiments slowly enough'to allow the molecules in a liouid to chanee their confipurations. The slowness of ihe molecuiar motions' relative to the time available to us for our experiments creates thermodynamic nanequilibrium in a glass, and freezes-in, as illustrated in Figure 6, a certain amount of entropy, enthalpy, and volume. The Tg is the lowest practical limit of temperature a t which a liquid can exist in thermodynamic equilibrium. The molecular motions, however slow, occur in the glassy state and the ol and C , slowly increase and the frozen-in, or configurational, entropy, enthalpy, and volume decrease with time. Substances in the glassy state are redeeming their heat capacity and losing their entropy, enthalpy, and volume a t an extremely slow, almost imperceptible, rate. A glass is constantly tending to thermodynamic equilibrium. Let us now consider an academically interesting situation. Suppose we had enough time on our hands, so that we could wait for the thermodynamic equilibrium to be attained in the liquid and thus could follow the entropy along line FGH in Figure 6 . We see that our liquid is losing its entropy with decreasing temperature a t a much faster rate than the crystal. At a certain temperature at H,the entropy of the liquid and the crystal would be the same. And now if we could cool it still further, the liquid would have an entropy lower than the crystalline phase. Since the entropy is generally associated with disorder, it is inconceivable that a situation would ever arise in which the liquid would have a lower entropy than the ordered crystal. K a u ~ m a n n ,who ~ first pointed out this paradox, also considered, and rejected, the possibility that a decrease in the slope of the line G H may occur such that the liquid cannot approach the entropy of the crystal. Certain theories of the glassy state propose that the line FGH is continuous and a second-order transition in a true thermodynamic sense must occur a t H; otherwise our concept of entropy would be at fault. It tuns out, frustratingly, that we cannot in our lifetime do the experiment which would allow us to follow the line GH, because the time required for the attainment of thermodynamic equilibrium even 20 or 30°C below Tg is of the order of several hundred years. It is only the limitations on time available to us that prevents us from resolving this situation. See how nature, by exceeding far beyond a human life span the time molecules need to reorient, saves our concept of entropy from an embarrassing situation! Metastability, Non-Crystalline Solids and Glasses The glassy state is one example of the widespread condition in which matter can exist in a metastable state. Substances in the metastable state are by definition in a hieher free enerev than in the stable state: and this con,~ that there must cept, as introduckd by O ~ t w a l drequires exist between the metastable and the stable states of matter, intermediate states of higher energy which resist the transformation of the metastable into the stable state. 26
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Journal of Chemical Education
TEMPERATURE
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Figure 7. Free energy as a function oltemperature.
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Figure 8. A schematic illustration of the fraezing-in of the free energy excess required to farm metastable, noncrystalline states by sudden changes of athermadynamic intensivevariable.
The classic example given by him is of a marble placed in a bowl held at some distance above the table. Here the height of the rim of the bowl represents the energy barrier for the formation of the intermediate states. The metastability in a glass is introduced by two factors. One, as we already know, is the excess free energy of the undercooled or "superpressed" liquid over the crystalline solid as shown in Figure 7, and the second, which is of a more limited kind, arises from the energy barriers which resist changes in the position of the molecules. It is the frozen-in free energy of a glass. The free energy excess over the crystalline arrangement necessarily required to form metastable non-crystalline states can be frozen into the system by sudden changes of a thermodynamic intensive variable such as temperature or pressure. as in a -elass.. or as chemical potential, as in desiccated 'gels. In some amorphized solids it can be "pumped in" by the process involved. This is schematically iilustrated in ~ i ~ u8.i e We see that the periodic order of a crystalline solid can be destroyed to produce a non-crystalline solid by, (a) the high shearing stresses produced during grinding, (b) neutron or a-particle irradiation, and ( c ) oxidation at low temperature. Quartz has been amorphized by the first two methods. Certain liquids change their composition when BOstwald,W., "Lehrbueh der Allgemeinen Chemie," W. Engelmann, Leipzig, 1910, Vol. II/1., p. 514. 'Roy, R., J. NonCryst. Solids, 3,33 (1970).
the temperature is changed, and produce non-crystalline solids. Desiccated gels are produced via a reaction of this type. Vapors when slowly deposited on a cold substrate also produce non-crystalline solids. A large number of thin films in the semiconductor technology are produced by this method. Silica has been obtained in all the six categories of the non-crystalline state shown in Figure 8. A substance of any composition may exist in several non-crystalline forms, each having a different structure and set of physical proper tie^.^ An analogous situation that exists in the crystalline state is known as polymorphism (the occurrence of carbon as diamond and graphite). It is interesting to inquire, does polymorphism also exist in the non-crystalline state? Non-crystalline solids lack periodicity, so any difference in their atomic or molecular arrangement between the various categories cannot he detected by methods, such as X-ray and electron diffraction, presently available to us. The final answer to this could be found only when techniques for the structur-
al analysis of aperiodic lattices become available. Until then we are perhaps well-advised to maintain a t least a generic distinction between the various non-crystalline solids. The glass inherits its structure from the liquid; i t follows that a molecular theoiy of glass trgnsitions would evolve from considerations of the liquid state. Since there is as yet no quantitatively satisfactory theory of liquids valid over the complete range of their existence, we do not have a theory of glass transition. Glasses do present a somewhat simplified situation of a liquid in that they lack the complications which arise from the rotatory diffusion of molecules, and we may perhaps profit more by calling upon the glassy state to help formulate a molecular theory of liquids, than by attempting the reverse. One hopes that a widei interest in this topic, created by encouraging students to inquire into this area, would generate new concepts for the treatment of the molecular dynamic theories for the aperiodically arranged particles.
Volume57, Number I , January 1974
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