RELATIONSHIP OF THE GLASS TRANSITION TEMPERATURE TO THE VISCOSITYTEMPERATURE CHARACTERISTICS OF LUBRICANTS RICHARD S. STEARNS, IRL N. DULING, AND ROBERT H. JOHNSON Research and Development Division, Sun Oil Co., Marcus Hook, P a .
The Williams-landel-Ferry approach in a somewhat modifled form may be used to correlate the viscositytemperature properties of fluids in the lubricating oil range. The quantity To is used as a unifying parameter, where To is the temperature at which the free volume available for hole formation reaches zero. TOIT,, where T, i s the measured glass transition temperature, is a constant having a value of 0.77. This relationship allows, in principle, the experimentally determined glass transition temperature, To, to be used in the viscosity equations when dealing with ideal fluids-that is, fluids for which the free volume i s proportional to T - To over the temperature range of interest. An apparent glass transition temperature, T,', i s estimated from the viscosity at 100" and 21 0" F. and used to construct a viscosity index scale which relates the viscosity index of a fluid to that of the normal paraffins. This scale i s compatible with, and a logical extension of, the ASTM-VI scale.
HE Williams-Landel-Ferry (WLF) (53) equation involving Tthe glass transition temperature, Tu,as a unifying parameter has been used very successfully to describe the rheological properties of polymers. While the glass transition temperature has been experimentally determined for many polymers, this parameter has not generally been determined for lower molecular weight materials-particularly fluids in the lubricant range-nor has T , been used to describe or correlate the rheological properties of lubricants. This paper shows that the experimentally measured value of Tu correlates with the rate of change of viscosity with temperature for a large variety of liquids and that an apparent T , obtained from the ratio of the log of the viscosity a t two temperatures can be used as the basis of a viscosity index scale. Equations bearing at least a formal similarity to the WLF equation, such as the modified Arrhenius equation of Cohen and Turnbull (70, 20, 29-33), Doolittle's equation ( I d ) , and the equation of Macedo and Litovitz (27), also express the transport behavior of fluids as a function of temperature with considerable success. The applications of these equations to a wide variety of materials over temperature ranges where the viscosity is changing by as much as 10 decades is well documented (70, 72, 23, 25-28, 39). The underlying theoretical basis for the above equations is the expansion of free volume ( V , of Figure 1) as temperature increases and the assumption that free volume available for redistribution into holes ( V , of Figure 1) disappears a t some temperature, To, which can be related to but is below the experimentally determined glass transition temperature, To. It is thus the movement of molecules into holes and the consequent migration of holes which account for the thermally activated transport phenomena such as viscosity (27,48). I t has generaHy been the practice to use values of absolute viscosity when dealing with viscosity-temperature relationships. Empirically we have found that the equation
306
l & E C P R O D U C T RESEARCH A N D D E V E L O P M E N T
ln
'I = In KV d
= In A
+ T -B To ~
(1)
which is equivalent to the modified Arrhenius equation but involves the kinematic viscosity, gives values of the parameters To and B, which have a high degree of internal consistency for fluids which are members of a repetitive series. O u r experience also indicates that the kinematic viscosity used in Equation l gives a better over-all fit to the viscosity-temperature data than use of the absolute viscosity, as demonstrated by Lewis (25) for the normal paraffins.
1
W
5
voc
p
vos
-I
VOL
vowl
76 der W A A L VOLUME
TEMPERATURE Figure 1. to liquid
Schematic concept of free volume as applied
Although Equation 1 must ultimately be regarded as empirical, it is nevertheless considered to be conceptually sound and to represent a limiting form of several more general expressions (26, 37, 42). For our purposes Equation 1 has the advantage that it involves only one adjustable parameter, To. I t also makes use of the kinematic viscosity, a parameter more frequently quoted in the literature than absolute viscosity, particularly when dealing with fluids of industrial importance. Although To is often determined from viscosity-temperature data far removed from To, in which case it may be only an empirical parameter, it does have theoretical significance as discussed by Gibbs (20). To may alternatively be described as the thermodynamic glass transition temperature (20), the temperature a t which the free volume available for hole formation reaches zero (29, 30, 32), or the temperature at which the excess configurational entropy reaches zero (20, 22). Unfortunately, To does not appear to be amenable to independent experimental measurement, because the barriers restricting rotation of molecular segments from one configuration to another become very high (the jump frequency becomes very low) as To is approached from above and perhaps reach infinity a t To. Experimentally, a glass transition temperature, T,, can be measured by several tecliniques involving such measurements as heat capacity, density, refractive index, etc., as a function of temperature. If the timle scale involved in these measurements is sufficiently long, the experimentally determined value of Tg approaches a lower limiting value, which appears to be independent of the modle of measurement. It is generally considered (20, 22, 23) that T , is a kinetically dependent pseudo-transition point taking place under isoviscous conditions or a t constant fractionall free volume, although there is certainly no agreement a t the present time concerning the precise theoretical interpretation. Gibbs (20) has suggested that T O I T ,is a constant, k . T o the extent that this is true, To can then be replaced by the term kT,, which a t least in principle eliminates the adjustable parameter from Equation 1. It can also be shown on the basis of the formalized volumetemperature relationships in Figure 1 (76, 37, 33) that
where j , is the fractional free volume a t T,. f, is defined as (VL - V,)/VoL (5,43) and a,is the coefficient of expansion of the free volume which (can be approximated by (ffllqu,d f f s o l l d ) (43). aITg is fornially defined as the expansion volume ( 5 ) at T , and is equal to (VL- VOL)/VOL. The work of Ferry (76), Williams (52), and Simha (43) has shown that for a large variety of materials fo has a value of 0.025 and a,T, has a viilue of 0.113. If it is assumed that f,/a,T, is a constant for all materials independent of molecular weight and molecular structure, then k in Equation 2 should have a value in the neighborhood of 0.78. This would allow the viscosity of a fluid to be calculated a t any temperature if constants A and B and a11 experimental value of T , were available, providing Equation 1 is valid over the entire temperature range. T h e log of the ratio of viscosity a t two temperatures can be used to screen or rate fluids with respect to their viscositytemperature characteristics. For convenience and because it has gained some acceptance, the viscosity-temperature ratio (VTR) of Ramser (36) is used here as a means of describing the average change in viscosity with temperature of a fluid from measurements a t two temperatures. For the temperatures 100' and 210' F., VTR is defined as:
VTR =
-ST* Tz
- Ti
(-)
-!- dKV
100
T~
KV
dT
dT = 2.093 log KVzio - (3) KVioo
Making use of Equations 1 and 2, VTR may be formulated as:
VTR =
2.093 B
Tzio - Tg Tzlo - kTg
]
- Tloo - T g Ti00 - kTg
(4)
The constant B in Equations 1 and 4 is defined as the temperature-independent energy of activation. Cohen and Turnbull (70, 48) equate B to y/a,, where y is a factor relating the relative size of a molecule to the size of a hole necessary to receive it. The value of y is generally considered to lie somewhere between '/z and unity and to be relatively independent of molecular type. If now we replace B I T , in Equation 4 by y/a,T,, then on the assumption that y, k , and a j T pare constants, VTR should be dependent only on T,. T o test this conclusion, the glass transition temperatures of a large variety of fluids were measured by differential thermal analysis (DTA) techniques, and it was found that for these fluids the glass transition temperature, T,, could serve as a unifying parameter to characterize the viscosity-temperature properties. It was also found that a viscosity index (VI) could be based on an apparent glass transition temperature, To,, estimated from the viscosity of a fluid at two temperatures such as 100" and 210' F., which are the temperatures commonly used in determining the ASTM-VI. The T,-VI scale, which is independent of viscosity level for fluids which are members of a repetitive series, relates the viscosity index of a fluid to that of fluids in the normal paraffin series. Determination of ?',
Theoretically, all materials have a glass transition temperature (20, 49). Often the T , cannot be measured, however, because of the rapid rate of crystallization below the freezing point. For materials which can be supercooled without crystallization or for inherently amorphous materials, T , can be measured readily by several techniques, one of which is differential thermal analysis (DTA). The experimental technique for determining the glass transition temperature by DTA measurements has been adequately described ( 7 7,24,45). The apparatus used in this work was assembled from an F&M Model 240 temperature programmer, an L&N Catalog No. 9835-B amplifier, and a Moseley Model 1 X-Y recorder. A circular brass block l1/2 inches in diameter was drilled to hold a bayonet heater, a thermocouple, and small glass sample and reference tubes (5 x 150 mm.). The sample size was generally 60 mg. T h e samples were supercooled by immersion of the block in liquid nitrogen. The heating rate was generally 10' C. per minute. A typical thermogram is reproduced in Figure 2. T h e glass transition temperature has been taken as the intersection of the two extrapolated linear portions of the thermogram as indicated. T h e value of T o measured in this fashion agrees well with T , measured by other techniques (24, 45). T h e use of DTA, particularly for liquids, appears to be reliable, convenient, and the least time-consuming method of experimentally determining the value of T,. Where T o cannot be measured experimentally (as in the case of fluids which cannot be obtained in the amorphous state), the theory of corresponding states suggests that the VOL. 5
NO. 4
D E C E M B E R 1 9 6 6 307
is some doubt concerning the value of 0 in Equation 6 for the normal paraffins. A value of e as low as 0.46 can be obtained if the melting point of linear polyethylene is taken as 419’ K. (77) and the Tu for linear amorphous polyethylene is taken as 192’ K. The latter value comes from extrapolation of second-order transition data on ethylene-propylene copolymers (47). I n this work we have followed Beaman (3) and taken e to be 0.50. Table I lists calculated values of T , for the normal paraffins. The values of T ocalculated from the boiling point (40) and melting point (8) agree up to about 20 carbon atoms. The value of To estimated from the heat of vaporization (do), assuming C is unity, agrees with the other values up to about 10 carbon atoms. The average of To from the three different approximations over the carbon number range where each method appears applicable has been taken as representative of the glass transition temperature for any particular normal paraffin.
following relationships should hold at least for small, nonpolar molecules ( 4 4 9 ) .
To/T
