T H E VARIATIOS O F THE VISCOSITY O F LIQUID W I T H TEJIPERXTURE BY E.
T. MADGE
The question of the variation of the viscosity of liquids with temperature although investigated as early as 1856 by Poiseuille' has always been approached with much diffidence, with the result that the formulas best representing the relation are empirical ones which hare little or no theoretical or physical interpretation. I t is probable that lack of attack on the question from a theoretical point of view is mainly due to our lack of knowledge of the true molecular state of a liquid, a subject which is only recently receiving close attention. It is not proposed t o consider the large number of empirical formulas that have been put forward except to state that of these it is probable that a logarithmic formula, rather than one involving merely polver terms, will have a greater theoretical significance. A number of empirical formulas break down as they consider the viscosity more as a function of the separation of the molecules, rather than with any thought as to the state of molecules themselves. The use of fluidity in place of viscosity sponsored chiefly by Bingham does not appear to lead us to any more definite picture of the viscosity process. Hence it, may he more profitable if we go hack t o the original ideas of Maxwell2 and proceeding therefrom in the light of recent knoivledge of the niolecular structure of liquids, try t o get a better conception of the viscosity nie chanism. In forming a physical conception of the viscosity of liquids Jlaxwell did not regard the phenomenon as an example of the diffusion of matter, but as the limiting case of an elastic solid when the material breaks down under stress. I t is seen of course at once, that if we consider the case of a liquid on the basis of the kinetic theory as similar to that of a gas we arc led to an erroneous result. For whereas the kinetic theory is successful in explaining the increase of the viscosity of a gas with temperature a similar treatment cannot also explain the decrease of the viscosity of a liquid xTith temperatures. In the case of an elastic body the shearing stress is proportional to the strain set up. If the body is viscous the stress no longer remains constant, but tends t o disappear at a rate depending on the stress and on the nature of the body. We are thus considering the viscous liquid as capable of supporting a certain amount of shear for a short time, then breaking down, and the shear recommencing. -As ?rIaxwell shewed, the quantity ET by which the rate of displacement must be multiplied to get the force, may be called the coefficient of viscosity.
1600
E. W. MADGE
It is the product of the modulus of elasticity E and a time T which may be called the time of relaxation of the elastic force (i.e. the time in which the stress has decreased to 'th of its original value.) e If the deformation is a simple shear the modulus called into play is the rigidity modulus. Thus we may write 7 = Er where E is the rigidity modulus governing shears applied to liquids and T is the relaxation time of the shearing force. Our problem is then what significance t'o attach to the rigidity of a liquid and to find what influence temperature has on this rigidity. We have also to interpret the relaxation time r and its relation to temperature If we can do this physically and mathematically we should have a pretty good picture of what is happening in a liquid. The problem has been treated theoretically by a number of investigators, such treatments really involving a relaxation time in one form or another. Some of the most successful were Graetz3 in 1888, Phillips' in 1921 and more reccntly Andrade5 in 1927. Phillips evolved a formula from a heterogeneous mixture of kinet,ic and quantum theory considerations, the main feature being that the interval between molecular collisions can be regarded as the half period in the sense of the quantum theory, and that the molecule can be regarded as moving with a constant velocity between consecutive collisions. The theory of Xndrade considered what amounted to a relaxation time. .indrade was among the first really to consider the fundamental viscosity process as a rolling of the molecules over one another without slip, and that the application of a shearing force to the liquid as a whole produced a molecular arrangement in the field similar to that. occurring in ferromagnetism and as we shall shew later having much in common with the orientation characteristics of the dipoles of dielectric liquids. Sbtoe had a somewhat similar idea in 1925, for he considered that the mutual action of the molecules played a much more important part in liquid viscosity bhan that played by momentum transfer. He developed this idea along the lines of Debye and Borns' early work, assuming the molecule to be a dipole of constant moment and considered the orientation of such a dipole in the direction of liquid flow by the couple exerted on it. By a lengthy discussion of the potential between two molecules and their thermal agitation he evolved a complicated formula for the viscosity of a liquid. SAto calculated the temperature dependence of viscosity for ethyl alcohol and methyl alcohol. The drawback to the formula however is that it is necessary to use the value of 7 at one definite temperature to calculate accurately the value of two constants of the equation, and thence deduce q a t all other temperatures. The formula therefore remains still somewhat of a n
VARIATION O F VISCOSITY WITH TEMPERATURE
1601
empirical one. Some deductions however made by Shto from this treatment are of great interest and will be mentioned in another connection later. As we shall see t,here is much experimental work which preceded this theory, and which may be quoted to support the ideas of hndrade and Shto. The former in his analysis makes a somewhat artificial assumption that the molecules would either be locked (normal condition) or free, and that they would be unlocked or freed by impact for a period T until locked again by another impact. Andrade however could not connect T with other properties of the liquid than that if the volume decreased T must decrease rapidly. The orientation theory of viscosity seems to be one of the most attractive ideas in fundamental viscosity theory that has been put forward. It is particularly attractive when we consider the step from liquids having long chain molecules to definite colloidal systems where the orientating particles are no longer of molecular dimensions, but can actually be seen Orientating giving similar effects to the orientation of the molecules themselves. The orientation theory is in no way contradictory to the early ideas of Maxwell but rather affords a better physical picture of his liquid supporting a shear for a short period of time, the period of shear support being comparable with the orientation time of the molecule. First of all it is proposed to put forward much available evidence in support of the theory arguing mainly from the similarity between viscous and electrical phenomena and then to predict certain effects that should be observed in connection with viscous flow of some liquids. It is well known that the motion of a perfect inviscid fluid in two dimensions is characterised by two functions both satisfying Laplace’s equation (a) the shear function expressing the lines of the flow of the fluid and the second function (b) the velocity potential expressing a system of curves cutting the shear lines orthogonally. The lines of flow are precisely analogous to lines of force in electric and magnetic fields. Surfaces of constant velocity potential correspond to electric or magnetic equipotential surface the velocity of the fluid (the gradient of the velocity potential) being the parallel of the electric or magnetic force (gradient of the corresponding potential). I t is bearing this analogy in mind that we propose to proceed. Raman and Krishnan’ did some experiments on the birefringence induced by flow in liquids and put forward a theory which amounted to an orientation theory to explain the results of many workers in the field, among whom Torlander and Walter* have given the most embracive results. As Raman and who examined nearly Krishnan point out the work particularly of 5’ and W., zoo liquids obliges us to conclude that the power to exhibit birefringence under mechanical flow in just as much a characteristic of pure liquids as for instance the power of exhibiting birefringence when placed in an electrostatic field. The action of the molecular orientation in producing birefringence is best shown by a simple experiment due to Zocher with larger particles.
E. W. MADGE
1602
ii beam of plane polarised light polarised for instance by a pile of plates passes through a cell containing a suspension of long-shaped particles in water such as magnesium carbonate or fine grained asbestos. If the light be observed through a Kicol>the field is uniformly dark or light. On rotation of the cell however, when the Nicol is in the extinction position the characteristic black cross is observed, the particles being orientated so that their long axes are along the lines of flow of the liquid. I t is something of this action that is pictured by Raman & Krishnan in their paper on birefringence in liquids, and it is to a similar source that we must look to I explain the viscosity phenomena in liquids. I I The dipole theory of molecules has been exI tremely successful in the hands more particularly I of Debye its author in explaining the dielectric behaviour of liquids. Debye shewed that liquids may be divided into two classes in view of their dielectric properties, non-polar and polar liquids. I n the first class the polarisation in an electric field and hence the dielectric constant is small, has a small temperature coefficient and may be esI plained siniply by electronic displacement. The molecule is considered solely as a polarisable systen1 in which the moment in an electric field is set up by the force causing a displacement of the charges. The second class of liquid is different. The molecule is represented by positive and negative charges separated by a finite distance. The polarisation is now represented by two terms-the displacement term a n d , in addit,ion t o this, a n orientation term--which is strongly dependent on temperature. Dipolar liquids whose polarisation include the term due to dipole orientation in the field, have large die1ect)ric constants and a large FIG.I temperature coefficient of dielectric constant. I n the second class the molecule has a permanent electric moment, and shews a polarisation not only by distortion but also by orientation for the orientation of smallest potential energy will be such that the electric moment points in the direction of the external field. We may expect that the amount of orientation created by the field will be larger the smaller the disturbance due to temperature motion. I t has recently been pointed out by Kyropulos8 how similar is the shape of the curve of viscosity and teniperature to that of D.E.K. and temperature particularly for polar liquids. Kyropulos has discussed qualitatively the importance of what he calls “flow orientation” in liquids with special reference
b A
I
VARIATION O F VISCOSITY WITH TEMPERATURE
1603
to lubrication. His papers deal mainly with oils and such substances where the molecules concerned are large and will cause large differences in the liquid behaviour on their orientat,ion. It is important at this point just t,o mention the similarity between dielectric constant, temperature behnTiour and viscosity temperature behaviour. Going a step further we must consider a further phenomenon wherein viscosit,y and dielectric constant are related, The phenomenon that is in mind is that, of the anomalous dispersion of dielectric constant'. It has been shexn theoretically by Debye and experimentally by Graffunder,'O Mizushima'2 and others that if the dielectric constant is measured at various temperatures for a constant frequency the dielectric constant passes u through a maximum. 20 Graffunder's results for glycerin shew this maximum very clearly and it is interesting to note that a curve of Bock's," determined for a much higher frequency for -700 -60 -80 o +.ZO +60 t the same liquid fits very well into the FIG.z graph. Kitchen and have made similar determinations for cast,or oil and give similar graphs. Owing to the highly viscous nature of these liquids the region of anomalous dispersion occurs in the longer wave portion of the spectrum. The physical explanation of the anomalous dispersion appears from Debye's theory to be simply this. For frequencies much higher than t'hat at which the maximum of dielectric constant occurs, the molecules cannot follow the changes in the field and the dielectric constant approaches its optical value. Since the dielectric constant a t constant frequency is a function of the temperature there must therefore be a temperature where the applied frequency coincides with the critical frequency, characteristic of the relaxation, and therefore in the neighbourhood of this frequency of the dielectric constant increases, since the dipoles which have been fixed a t low temperatures become liberated. This means that at the maximum of dielectric constant there is a kind of resonance between the field and the natural frequencies of the liquid. I n other words the relaxation time of the field bears a very simple relation to the relaxation time of the molecules of the liquid. Debye14 shews that the formula giving the critical frequency is Ymax
or
T
8n7a3 the relaxation time = zkT
2kT 87r7a3'
=-
a
=
effective radius of the molecule, assumed, spherical.
k
=
Boltzmann's constant.
1604
E. W. MADGE
We may transpose this equation to read q = k T . T
4na3 kT of force and -has the dimensions ___ = 1LI.L-IT-2 4na3 (length)* which are also those of a rigidity modulus. kT . Thus this equation 11 = -T is of the Maxwell form. 11 = E.7. Here 4na3 therefore we have a method of actually determining the relaxation time characteristic of the viscosity which occurs most fundamentally in Maxwell’s original equations. As is seen from Graffunder’s curves, this relaxation time depends on temperature, decreasing with increasing temperature. Apart from this fundamental relationship that is disclosed, one or two other important predictions can be made. A similar prediction was made by Sbto but not from precisely the same point of view. It is agreed that if visocsity has such a large influence on the dielectric constant as has been shewn above in its influence on the rotational motion of the molecules; then the converse should hold i.e. the orientation of the molecules by a n electric field should influence the viscous flow particularly if the molecules are large. An attempt was made to demonstrate the effect by the author, but the fields available were not sufficiently large to give a positive result. A. Sellerio16however using more intense fields was able to shew a positive effect. He measured the variation of viscosity in an electric field by making use of a method of damped vibration obtaining the following results. % increase in Field Castor oil Carbon bisulphide Vaseline oil
viscosity
volt/cm.
2.4
23,000
5.3 6.0
12,000
16,000
The presence of this effect may be advanced as one more proof of our orientation theory of -.iscous flow. I n conclusion it is not irrelevant to remark that light may be thrown on the peculiar behaviour of many colloids in their viscosity relationships particularly a t very low rates of shear, by considering the mutual action of the colloidal particles more in the way described above than by other methods. As a n appendix it is thought of interest to mention a viscosity temperature relation that has been built up on the lines of Maxwell’s original considerations. We assume that the relaxation time approaches infinity as 7 approaches a definite temperature b which we may call the temperature of complete solidification in a way similar to that of Graetz. That is
VARIATION OF VISCOSITY WITH TEMPERATURE
w m m m * h O 0 0 m e a a m n m a i - - a L o * * m m 0 0 0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1605
1606
E. W. MADGE
Now as regards the rigidity modulus, some recent work by 1Iiss Tilleard as get unpublished has shewn that for metals a formula of the type E = xe-+ where N is a function of the atomic volume, is the ratio of the absolute temperature to the absolute temperature of fusion holds very well over a wide range of temperature. From the formula it is seen that at the melting point there still remains a definite rigidity. Thus besides the formula being one of the best known for rigidity and temperature we have some small justification in extrapolat,ingit to the liquid state. Thus we assume the variation of the rigidity modulus with teniperature for a liquid to be of the form E = deflT. Rigidity has a definite meaning with regard to colloids and there is no reason why a similar interpretation should not be applied to liquids. Hence combining the two formulas we are led to the formula
+
as an equation representing the variation of viscosity of a liquid lyith temperature. This equation was tested for a number of liquids and a comparison of some observed and calculated values is given in Tables 1-5. The formula has been found to hold for a nuinber of other liquids both normal and associating over a wide range of temperature. I n conclusion I should like t o thank Miss Tilleard for information on the temperature dependence of rigidity modulus and Dr. A. Ferguson for his continued interest.
Bibliography Poiseuille: MBm. Savants &rangers. 9, 433 (1846). Maxwell: Phil. hlag., (4) 35, 133 (1868). 3 Graets: Ann. Physik. Chem.. ( 3 )34. 2 j (1888). 4 Phillips: Proc. Kat. Acad. Sei., 7, I 72.177 11921). 5 Andrade: Engineering, 124, 461 ( 1 9 2 7 : . 8 %to: Sei. Reports Tohoku Imp. Univ.: 14, 403 ( 1 9 2 5 ) . 'Raman and Krishnan: Phil. Mag. Lpril (19281. * Vorlander and Walter: Z. physik. 'Chern., 118, I I I y z j ) . 9Kyropulos: Physik. Z., 29, 942-47 (19281;Z. tech. Physik, 10. 10 Graffunder: Ann. Physik, 70: 225 (19231. "Bock: 2. Physik, 31, j34 f1925'. l2 hlizushima: Physik. Z., 28, 418 ( ~ y z i l . 13 Kitchen and Muller: Phys. Rev., (2,; 32, 979 (1928). l1 Debye: "Handbuch der Radiologie, 4 , 643 (1924 '. 15 Sellerio: Nuovo Cimento, 11, 3 9 j (19161.
Bir?ni?igham, England.
2.
929).
