INTERNATIONAL TEXT-BOOKS OF EXACT SCIENCE Editor : Professor E. N. da C. Andrade, D.Sc., Ph.D. THE VISCOSITY OF LIQUIDS THE VISCOSITY OF LIQUIDS BY EMIL HATSCHEK FELLOW OF THE INSTITUTE OF FHYSICS, LECTURER ON COLLOIDS AT THE SIR JOHN CASS TECHNICAL INSTITUTE LONDON G. BELL AND SONS, LTD. 1928 S3Z.S \\3(c5' Printed in Great Britain by Neill & Co., Ltd., Edinburgh. PREFACE The author has found in the teaching of Colloid Chemistry, more particularly in the laboratory, that the subject of Viscosity is one which seems to present difficulties to the majority of students. The more enterprising among them I have often asked him to ''recommend a book,'' but he has generally found it necessary to refer them to text-books or hand-books of physics for some points, and to original papers for others. In the present work the author has attempted to present, in a moderate compass and without excessive mathematical apparatus, as much of this scattered information as will give the reader a grasp of the fundamental principles and a general view of the subject. The task of selection from the enormous mass of material available has been difficult, but the author has had the advantage of being able to include some in¬ vestigations of great importance, like those of Bridgman, which have not so far found their way into any text-books. He has also thought it advisable to devote a chapter of some length to the viscosity of colloidal solutions, a subject of peculiar difficulty calling for a great deal of further research, which a summary of the present position may perhaps help to stimulate. The author has to thank the Director of the Shirley Institute of the British Cotton Industry Research Association for the loan of the blocks figs. 85 and 86 ; Messrs W. G. Pye & Co., v vi THE VISCOSITY OF LIQUIDS Cambridge, for the loan of the block fig. 24 ; and the Counci of the Physical Society of London for permission to repro duce fig. 23. His best thanks are also due to Mr R. H Humphry, M.Sc., for help in reading the proofs, and to M F. W. Clifford, Librarian of the Chemical Society, for mud valuable assistance. EMIL HATSCHEK. May 1928 CONTENTS CHAPTER I Fundamental Concepts and Historical Development PAGE Relative Motion in Liquids and Internal Friction or Viscosity . i Newton's Hypothesis and Treatment of the Problem of a Cylinder revolving in an Infinitely Extended Liquid ... 2 Definition, Dimensions, and Units of Viscosity Coefficient . . 5 Fluidity and Velocity Gradient ...... 5 History. Discrepancies between Hydrod^mamics of Ideal Liquid and Observation ........ 6 The Work of Coulomb ........ 7 Girard's Investigation on Flow of Water in Small Tubes . . 8 Navier's Hydrodynamic Equations for Viscous Liquids and Definition of Viscosity Coefficient ..... 9 Stokes's Solutions of Capillary and Concentric Cylinder Problems 9 Hägen's Investigation on Flow of Water through Capillaries . 10 The Work of Poiseuille . . . . . . . .10 Wiedemann's and Hagenbach's Deductions of Poiseuille's Formula and Calculation of Viscosity Coefficient . . . .11 Early Technical Viscosity Measurements . . . . .12 The Work of Osborne Reynolds . . . . . .13 Couette's Investigations with Concentric Cylinder System . . 13 Modern Developments. The Viscosity of Colloidal Solutions . 14 CHAPTER II Mathematical Theory of the Principal Methods of Determining the Coefficient of Viscosity Equation for Laminar Flow through Capillary. Poiseuille's Formula . . . . . . . . .17 Limits of Laminar Flow ; Reynolds Number and Criterion . 19 Kinetic Energy Correction and Couette's Correction ... 20 Divergent Views on Applicability of Corrections . . .22 The Work of Bond and Dorsey's Formulation of both Corrections 23 « • Vll viii THE VISCOSITY OF LIQUIDS PAGE Equations for Flow with Varying Head of Liquid . . .25 The Ostwald Viscometer ; Sources of Error ; Mean Head . . 26 Ubellohde's and Bingham's Pressure Viscometers ... 29 Theory of Concentric Cylinder Apparatus . . . . .30 Oscillating Sphere and Oscillating Disc Methods . . .32 Stokes's Formula and Falling Sphere Instruments • • • 33 Ladenburg's Corrections . . . . . . . -35 Bond's Equations for Viscous Spheres ..... 36 CHAPTER III The Design and Use of Viscometers Capillary Viscometers for Absolute Determinations Thorpe and Rodger's Instrument ...... Bingham and Jackson's Instrument . . . . . Capillary Instruments for Measuring Relative Viscosity Ostwald Viscometers. Applebey's and Washburn and Williams's Precision Instruments ....... Use of Viscometers, Thermostats and Regulators Special Viscometers : for Volatile Liquids . . . . For Opaque Liquids ....... For Fused Salts ........ Porous Cell Viscometer by Duclaux and Errera Concentric Cylinder Apparatus : Couette's Design Hatschek's Design ........ Searle's Design ........ Falling Sphere Viscometers : Gibson and Jacobs's Design . CHAPTER IV The Constancy of the Viscosity Coefficient Constancy assumed in Mathematical Treatment . . .59 Experiments at very Low Velocity Gradients : by Dufí . . 60 By Gurney ......... 61 By Griíñths ......... 62 CHAPTER V The Variation of Viscosity with Temperature Empirical Formulae for Variation of Viscosity with Temperature : by Poiseuille ........ By Meyer ......... By Graetz ......... By Slotte ......... 39 40 41 43 43 46 47 48 49 49 51 52 54 56 63 63 64 65 CONTENTS ix PAGE Thorpe and Rodger's Values for Viscosity of Organic Liquids at 20°, and Coefficients in Poiseuille's Interpolation Formula . 66 Coefficients of Expansion of Organic Liquids . . . .69 Batschinski's and Dorsey's Formulae ..... 70 Temperature Coefficient of Viscosity and Coefficient of Expansion 70 M^Leod's Formula connecting Viscosity and Free Space . . 70 Table and Discussion of Results ...... 72 Bingham's Empirical Fluidity Formulae ..... 74 Porter's Relation between Temperatures of Equal Viscosity of any Pair of Liquids ....... 74 Batschinski's Formula connecting Viscosity and Specific Volume . 76 Comparison of M^Leod's and Batschinski's Values for Free Space and Discussion of Discrepancies ..... 77 CHAPTER VI The Variation of Viscosity with Pressure The Work of Warburg and Sachs ...... 79 Cohen and Hauser's Work and the xAnomaly of Water . . 80 Faust's Investigation ........ 80 The Work of Bridgman, Description of Method .... 82 Table and Discussion of Results ...... 84 Failure of Batschinski's Formula tested by Bridgman's Results . 90 Application of M^Leod's Method to Bridgman's Data. . . 91 Free Space and Compressibility ...... 92 Viscosity not a Pure Function of Volume ..... 96 CHAPTER VII Viscosity and Constitution Early Work by Graham, Rellstab and Guerout . . . . The Work of Pribram and Handl and of Gartenmeister Relations between Viscosity and Molecular Weight The Work of Thorpe and Rodger, Temperatures of Comparison . " Molecular " Viscosity Constants ...... Bingham's Comparison at Temperatures of Equal Fluidity . Calculation of Association Factor ...... M^Leod's Treatment of Relation between Viscosity and Molecular Weight ......... Association Factors compared with Bingham's and other Authors' Dunstan's Relation between Viscosity and Molecular Volume Dunstan and Thole's Logarithmic Relation . . . . Logarithmic Increments for various Atoms and Groups Calculation of Viscosity from these Increments .... 98 99 99 TOO 101 102 103 104 103 108 108 109 no THE VISCOSITY OF LIQUIDS CHAPTER VIII The Viscosity of Solutions PAGE Non-electrolytes : Cane Sugar . . . . . .112 Temperature Coefficient and Relative Viscosity . . . .113 Arrhenius's Logarithmic Formula applied to Sugar Solutions . 115 " Negative Viscosity " of non-Electrolytic Solutions . . .116 Hydration and its Possible Effect . . . . . .116 Kendall and Monroe's Work on " Ideal " Solutions . . .117 Their Cube-root Formula . . . . . . • n? Electrolyte Solutions, General Characteristics , . . .119 Relative Viscosity at Different Temperatures . . . .122 Salts Producing " Negative Viscosity " . . . . .125 Merton's and Getman's Experimental Data . . . .126 Jones and Veazey's Hypothesis regarding " Negative Viscosity " . 127 Getman's Work on Potassium Iodide in Organic Solvents . . 128 Temperature Coefficients of Solutions of non-Dissociated Salts . 129 Taylor and Moore's Criticism of Jones and Veazey . . .129 Viscosity of Quaternary Ammonium-salt Solutions . . .130 Relation between Shrinkage in Solution and Viscosity . . 130 Grüneisen's Work on Anomaly of Electrolytes at Low Con¬ centrations , . . . . . . . .132 Applebey's Work on the Anomaly . . . . . -133 CHAPTER IX The Viscosity of Liquid Mixtures Difficulty of Problem : Ideal Law Unknown . . . . Kendall and Monroe's Review of Problem . . . . Empirical Formulae Tested ...... Table of Divergences ....... Data on Mixtures approximating Ideal . . . . Failure of all Formulae, including the Author's Cube-root Equation ......... Principal Types of Viscosity-ratio Curves of Mixtures Minima and Maxima displaced with Temperature Alcohol-water and Acetic Acid-water Mixtures .... Theories of Maxima, Opposing Views ..... Maxima produced by Chemical Combination .... Evidence from Vapour-pressure Curves ..... Position of Maximum and Composition of Assumed Compound Experimental Data : Tsakalatos ...... Kurnakow and Collaborators ...... Stranathan and Strong ....... 135 135 136 137 138 139 140 141 143 144 145 146 147 147 148 151 CONTENTS xi PAGE M^Leod's Treatment of Mixture Problem ; Contraction of Free Space . . . . . . . . . -152 Tests of Formula by Various Experimental Curves . «157 Contraction Calculated from Pressure Data . , .161 Mixtures of Components with widely different Viscosities . .162 Glycerin-water . . . . . . . . .162 Solutions of Gases in Liquids . . . . . . .163 CHAPTER X Viscosity and Conductivity Fundamental Difficulty : Impossibility of varying one Factor at a Time .......... 165 Early Experiments by Arrhenius and by Lüdeking on Solutions containing Gelatin . . . . . . .165 Investigation by Arrhenius with other non-Electrolytes . .166 Heber Green's Conductivity Formula . . , . .167 Views on Effect of Viscosity of Liquid as a Whole . .168 Temperature Coefficients of Viscosity and Conductivity . .168 Johnston's Formula for Ionic Mobility at Infinite Dilution . .169 Wien's Mathematical Treatment of Temperature Coefficients . 170 Kraus's Review of Problem . . . . . . .173 Viscosity Correction for Solutions with " Negative Viscosity " possible . . . . . . . . *175 Change of Conductivity and Viscosity with Pressure . .176 Rabinovich's Work at High Concentrations . . . .177 Anomalies caused by Viscosity Correction . . . .178 Viscosity and Conductivity of non-Aqueous Solutions . .178 Walden's Earlier Work ..... .178 Dutoit and Duperthuis's Criticism . . . . .178 Walden's Later Work . . . . . . .179 Recent Views of Viscosity Correction of Conductivity . .180 CHAPTER XI The Viscosity of Pitch-like Substances The Work of Barus, Röntgen, and Reiger . . . .182 Pochettino's Investigation by Three Different Methods . .183 Continuity of Log Viscosity-temperature Curve . . .185 Heydwilier's Data on Solid and Liquid Menthol . . .186 Trouton and Andrews's Investigations . . . .186 Elastic Effects , , . , . . . ,187 Xll THE VISCOSITY OF LIQUIDS CHAPTER XII The Viscosity of Colloidal Solutions The Anomaly of Colloidal Solutions .... Graham's Early Observations on Influence of Age Viscosity of Suspensoid Sols ..... Temperature Coefficients of Suspensoid and Emulsoid Sols The Concentration Function. Einstein's Formula Experimental Verification and Limits of Validity The Electro-viscous Effect ..... Hess's Formula ....... Arrhenius's Formula modified for Hydrated Colloids . Hatschek's Emulsoid Formula. .... Verification on Suspensions of Red Blood Corpuscles . Solvation Factors from Hatschek's Formula Evidence of Solvation from Swelling and Syneresis The Variation of Viscosity with Velocity Gradient, History Examples of Variation ...... Plasticity, Consistency, and " Yield Value " Grounds for retaining Term " Viscosity " . Empirical Formulae for Variable Viscosity, Parabolic . Herschel and Bulkley's ..... Bingham's Plasticity Formula .... Analytical Treatments of Problem .... Buckingham's Equation ..... Reiner and Riwlin's Equation for Elastic Sol in Cone. Cyl. App. Reiner and Riwlin's Equations for Particles of Variable Size Farrow, Lowe and Neale's Equation ..... Experimental Verification of the last Equation .... Maxwell's Theor^^ of Viscosity and its possible Application to Sols possessing Rigidity ....... Rigidity and Relaxation in Sols ...... Anomalous Turbulence in Sols. ...... PAGE 190 191 192 192 196 198 199 200 201 202 202 203 204 204 206 207 210 212 212 212 213 214 214 215 217 219 222 224 226 CHAPTER XIII Technical Viscometers Short Tube Viscometers, Standard Types ..... 230 Herschel's Formula and Conversion Tables . . . .231 MacMichael Viscometer . . . . . . . .233 Michell Viscometer ....... - 233 Index of Names ......... 235 Index of Subjects 238 THE VISCOSITY OF LIQUIDS CHAPTER I FUNDAMENTAL CONCEPTS AND HISTORICAL DEVELOPMENT If portions of a mass of liquid are caused to move relatively to one another, the motion gradually subsides unless sustained by external forces ; conversely, if a portion of a mass of liquid is kept moving, the motion gradually communicates itself to the rest of the liquid. These effects, which are matters of immediate observation, were ascribed by Newton to a ''de- fectus lubricitatis,'' i.e. a 'dack of slipperinessbetween the particles of the liquid, which may be fairly interpreted to mean a property resembling friction between solid surfaces, and Newton, in fact, uses the term ''attritus,'' i.e. friction, several times in the course of his deduction. The terms ''internal friction'' and "viscosity" have been used indifferently to describe this property of liquids; it is not clear when the latter word acquired a strict technical meaning, since etymo- logically (from viscum^ the mistletoe) it would seem to have been applied originally to liquids possessing the property to an abnormal degree. The corresponding terms in French are "frottement intérieur" and "viscosité," and in German "innere Reibung," "Viskosität," and, especially in recent literature, "Zähigkeit," which appears to have been used for the first time by Wiedemann^ in the strict technical sense. Newton was the first to formulate a hypothesis regarding the magnitude of the force required to overcome viscous resistance and to treat a case of motion in a viscous fluid. Newton's hypothesis is given in most text-books of physics, I 2 THE VISCOSITY OF LIQUIDS but the peculiar problem chosen and the unfamiliar method of treatment appear sufficiently interesting to justify quota¬ tion in extenso of the chapter in question [Principia, Lib. ii, Sect. ix). ''On the Circular Motion of Liquids Hypothesis That the resistance which arises from the lack of slipp crines s of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another. Proposition LI, Theorem XXXIX If a solid infinitely long cylinder in a uniform infinite liquid revolve round its axis with uniform motion, and the liquid he caused to revolve by the impulse thereof only, but every part of the liquid persevere uniformly in its motion ; I say that the periodic times of parts of the liquid are as their distances from the axis of the cylinder. Let AFL (ñg. i) be a cylinder revolving uniformly round the axis S, and let the liquid be divided by concentric circles BGM, CHN, DIO, EKP, etc., into innumerable concentric solid cylindrical shells of the same thickness. And as the liquid is homogeneous, the impressions made by contiguous shells on each other will [ex hypothesi) be as their mutual translations and as the contiguous surfaces in which the impressions are made. If the impression on any one shell is greater or smaller from the concave part than from the convex one, the stronger impression will prevail, and will accelerate or retard the motion of the shell, according as it is directed in the same or in the opposite sense as its own motion. If, therefore, any one shell is to persevere uniformly in its motion, the impressions from either part must be equal and opposite to one another.* Hence, since the impressions are as the contiguous surfaces and their relative translations, * For the error in this assumption, and consequently in the result, see Stokes, p. lo. HISTORICAL DEVELOPMENT 3 the translations will be inversely as the surfaces, i.e, inversely as the distances of the surfaces from the axis. But the difíerences of the angular motions round the axis are as these translations divided by the distances, or directly as the trans¬ lations and inversely as the distances ; that is, combining the ratios, inversely as the squares of the distances. Therefore, if at the several points of the infinite straight line SABGDEQ perpendiculars Ka, B&, Cc, E^, etc., be erected and made la 2nd ed., 1713.) inversely proportional to the squares of SA, SB, . . ., etc., and if a hyperbolic curve be imagined drawn through the ends of the perpendiculars, the sums of the differences, i.e. the total angular motions, will be as the corresponding sums of the lines Ka, Bö, etc., that is, if, to constitute a uniformly liquid medium, the number of circles is increased, and their width reduced, to infinity, as the hyperbolic areas AaQ, BöQ, CcQ, etc. And the times inversely proportional to these angular motions will also be inversely proportional to these areas. The periodic time of any one part D is therefore inversely as the area D¿Q, i.e. (from known quadratures of curves) directly as the distance SD. Q.E.D. Cowl. I. Hence the angular motions of parts of the liquid 4 THE VISCOSITY OF LIQUIDS are inversely as their distances from the axis of the cylinder and the absolute velocities are equal. Corol, 2. If a liquid be contained in an infinitely long cylindrical vessel and contain another interior cylinder; if both cylinders revolve about their common axis and the times of their revolutions be as their radii, and if every part of the liquid persevere in its motion, the periodic times of every part of the liquid will be as its distance from the axis of the cylinder. Corol. 3. If any common angular motion be added to, or withdrawn from, the cylinder and liquid moved in this manner, the motion of the parts against one another will not change, as the mutual friction of the parts of the liquid is not changed. For the translation of the particles against one another depends on the friction. Every part will persevere in that motion which is not more accelerated than retarded by the friction acting from either side in opposite directions. Corol. 4. Therefore, if the whole system of cylinders and liquid be deprived of all angular motion of the outer cylinder, there will be motion of the liquid in the cylinder at rest. Corol. 5. Hence, if the interior cylinder be rotated uni¬ formly, the liquid and the outer cylinder being at rest, the rotary movement will be communicated to the liquid and will gradually be propagated through the whole of it, nor will it cease to increase until the several parts of the liquid acquire the motion defined in the fourth corollary. Corol. 6. And because the liquid tends to propagate its motion yet further, the outer cylinder will also be carried round by it unless forcibly detained, and its motion will be accelerated until the periodic times of both cylinders become equal. If the outer cylinder be forcibly detained, it will tend to retard the motion of the liquid, and, unless the inner cylinder by some force impressed from outside maintain it, will gradu¬ ally cause it to cease. All which may be tried in deep and stagnant water." Newton's fundamental assumption amounts to the follow¬ ing : If two laminae having an area of contact A move with constant velocities v-^ and Vo, the force required to maintain FUNDAMENTAL CONCEPTS 5 the constant difference of velocity is (Vi -V2) F = yA (Zl-^2)' in which the z are measured in the direction perpendicular to the laminae. Since the velocity in the liquid changes con¬ tinuously, we may replace the differences by differentials and write : F-,Ag . . . . (I) in which, as experience has shown, 7^ is a characteristic constant for each liquid, which, at ordinary pressure, decreases with temperature, but in all simple liquids and true solutions is independent of dv/dz. By introducing the dimensions in equation (i), [F]=[MLT-2] [A] = [L^] [dv]=[LT-^] [dz]=[L], we find the dimensions of 7] to be r] is called the coefficient of viscosity, and its physical meaning becomes clear if the factors on the right hand of equation (i) are chosen =1. The viscosity coefficient is the force required per unit area to maintain unit gradient of velocity, or the force required per unit area to maintain unit difference of velocity between two parallel planes in the liquid unit distance apart. The coefficient is usually ex¬ pressed in dynes, centimetres and seconds, and the value T^r^i-ooo in these units is conveniently called a Poise (in honour of Poiseuille (see below)), and its hundredth part a Centipoise; the coefficient of viscosity of water at 20° is approximately a centipoise. The reciprocal of the viscosity is called the fluidity, (^, and is generally expressed as 1/(7^ in poises). The dimensions of the velocity gradient (which assumes considerable importance in anomalous liquids, such as the majority of colloidal solutions) are [T~^]. The only un¬ ambiguous way of expressing it is to state the change in 6 THE VISCOSITY OF LIQUIDS velocity per unit distance measured at right angles to the direction of the velocity: zb n cm. per sec./cm. Several authors ^ express the velocity gradient in radians per second/' one radian per second being taken as unit gradient. This is perhaps a convenient method of avoiding the some¬ what cumbersome expression given above, and unobjection¬ able as far as dimensions are concerned, but hardly conveys the idea of a gradient. Although Newton's hypothesis thus leads to an equation which, as will be shown later on, can be applied directly to important cases of motion in viscous liquids, the hydro¬ dynamics of the following century, as developed by Daniel Bernoulli, Euler and others, was exclusively concerned with ideal liquids, in which tangential forces between adjacent parts of the liquid were excluded. Even simple experiments could not fail to reveal very considerable discrepancies between the results of theory and the behaviour of real liquids, speaking of which Bernoulli ^ says in his Hydrodynamica : ''I attribute these enormous differences for the greatest part to the adhesion of the water to the sides of the tube, which adhesion can certainly exert an incredible effect in cases of this kind." The numerous and extensive investigations on the flow of liquids in pipes and open channels, carried out by the French engineers and physicists at the end of the eighteenth and the beginning of the nineteenth century, had produced a great mass of material which the hydrodynamics of ideal liquids failed to explain. Prony,^ who studied the subject both theoretically and experimentally, comments on the lack of interest exhibited by the mathematical physicist : It is regrettable, and even astonishing, that the celebrated Euler, who in the course of his immense labours has often turned his attention to physico-mathematical problems and their practical applications, should not have endeavoured to treat the theory of liquids by taking into account the cohesion of the molecules and some kind of friction; even if he had introduced these resistances into the analysis in a purely hypothetical form, it would be interesting to know how he envisaged their effect; but I do not know of any paper by him where they are mentioned." HISTORICAL DEVELOPMENT 7 The investigations mentioned above had led to the general result, that the resistance to the flow of liquids in cylindrical tubes could be expressed as the sum of two terms, one containing the first power and the other the square of the velocity. The problem whether this second term becomes negligible when the motion is very slow, was attacked by Coulomb ^ in an entirely new way. Although he quotes Newton [Princ,, Lib. ii. Prop. 40) on the slow fall of spheres in air or water, he makes no reference to the hypothesis in Prop. 51. Cou¬ lomb's method consisted in causing a horizontal circular disc suspended from a wire to oscillate round its axis in air and in the liquid to be examined; the resistance of the latter was deduced from the logarithmic decrement of the amplitude. The resistance for slow motion was found to be simply pro¬ portional to the velocity, and, other things being equal, to the fourth power of the radius. Coulomb deduces from his experiments a ''resistance factor," viz. the force required per unit area to maintain unit velocity (which has a meaning for the particular apparatus used only). Coulomb found that the same relations as for water held for " a clarified lamp oil," the resistance factor of which turned out to be 17-5 times that of water. He made two further series of experiments, which, as he says, might have a bearing on future theories of the nature of liquids. The first series was undertaken for the purpose of ascertain¬ ing whether the nature of the surface in contact with the liquid affected the results. The disc was first covered with a thin layer of tallow, which did not cause any change in the logarithmic decrement; the greased disc was then covered with a thin coat of powdered stoneware, and the decrement still remained unaltered. The second series was directed to ascertaining whether pressure altered the frictional resistance. Coulomb points out that any difference in pressure due to varying the depth of liquid above the disc would be negligible compared with atmospheric pressure, and therefore carried out a number of determinations in a vacuum. No difference could be detected, whence Coulomb concludes that the frictional resistance is 8 THE VISCOSITY GE LIQUIDS • independent of pressure, and emphasises the difference between friction in liquids and that between solids. Coulomb quite clearly points out the advantages of his method, which allows small forces or moments to be measured with accuracy. Nevertheless Girard,® although he quotes Coulomb's paper, once more returned to the problem in its familiar form, and again attempted to determine the law of flow through cylindrical tubes. By a priori reasoning he arrived at the conclusion that with very slow flow in small tubes the resistance should be simply proportional to the velocity, and proceeded to test it with copper capillaries (drawn over steel wire) of 0-183 and 0-296 cm. diameter; they were made in 20 cm. lengths, and could be joined by screw couplings to make lengths up to 2200 cm. His results were best re¬ presented by the equation (Q = volume discharged in unit time, C a constant, íí=diameter and / ^length of tube, ^=pressure). Girard also followed Dubuat in studying the influence of temperature on flow; he found that the resistance decreased with rising temperature, and that the rate of decrease was greater at low temperature. Girard seeks the cause of the resistance purely in the con¬ ditions at the wall of the tube, and not in the body of the liquid. He assumes that a film of liquid adheres firmly to the wall of the tube, and gives rise to two kinds of resistance : one is the force with which all the molecules of the peri¬ meter adhere to the liquid layer which envelops and wets the wall." This resistance is simply proportional to the velocity, and would be the only one if the wall were absolutely smooth ; its asperities, however, are reproduced in the liquid film and give rise to turbulence, which produces a second resistance term containing the square of the velocity. This insistence on the surface condition of the wall is striking, in view of Coulomb's negative results. Up to this time (1813) there had been no attempt to find a general solution of the problem of motion in viscous liquids HISTORICAL DEVELOPMENT 9 corresponding to the general hydrodynamic equations for ideal liquids developed during the eighteenth century. Such equations were deduced for the first time by Navier ^ ten years later. It is not possible to say more of them here than that they contain, in addition to the density, which alone appears in the equations of the ideal liquid, a second character¬ istic constant, which is very clearly defined by Navier: " Let us assume that the velocities of the molecules of liquid contained in the same layer parallel to the plane (on which the liquid is assumed to rest) are equal among them¬ selves, and further that the velocities of all layers, as they are more and more distant from the plane, increase pro¬ gressively and uniformly in such a way that two layers whose distance from each other is equal to the unit of length have velocities whose difference is also equal to the unit of length. On this assumption the constant e (=our rj) represents in units of weight the resistance arising from the slipping over each other of any two layers, for an area equal to the unit of surface.'' The definition is exactly that of the coefficient of viscosity as it is deduced from Newton's hypothesis. Navier attempted to integrate the equations for the flow through a cylindrical tube, but obtained the incorrect solution that the volume discharged in unit time, other things being equal, was proportional to the cube of the radius—a solution which appeared to be confirmed by Girard's results. The equations found by Navier were again deduced by Poisson ^ from different assumptions, and finally by Stokes by a method differing from those of both his predecessors. Stokes integrated them for two cases of great importance: stationary flow through a cylindrical tube, and stationary motion between two coaxial cylinders. The cylinder revolving in an infinite liquid is a special case of the second problem, and Stokes remarks in reference to it: ''These cases of motion were considered by Newton {Princ., ii, 51). The hypothesis which I have made agrees in this case with his, but he arrives at the result that the velocity is constant, not that it varies inversely as the distance. This arises from his having taken, as the condition of there being no acceleration or retardation 10 THE VISCOSITY OF LIQUIDS of an annulus, that the force tending to turn it in one direction must be equal to that tending to turn it in the opposite direction, whereas the true condition is that the moment of the force tending to turn it in one direction must be equal to the moment of the force tending to turn it in the opposite direction. Of course, making this alteration, it is easy to arrive at the above result by Newton's reasoning." While the mathematical theory had taken its final shape, experimental investigation had likewise made important pro¬ gress. In 1839 Hägen studied the flow of water at differ¬ ent temperatures through brass tubes of the following dimensions :— The pressure was produced by a head of water, and the quantity discharged was determined by weighing. Hägen found that the volume discharged in unit time was pro¬ portional to the pressure, to a power of the radius and in¬ versely proportional to the length of the tube. The exponent, calculated by the method of least squares, was found to be 4*12, and Hägen suggested that the true value was 4. He also made observations on the departures from this law, which he ascribed to turbulence, and noticed that, other things being equal, turbulence set in more easily at lower viscosities, i.e. at higher temperatures. Hagen's work, although practically a complete anticipa¬ tion, has been overshadowed by that of Poiseuille,^^ a short abstract of which appeared in the Comptes Rendus of 1842, his first paper being printed in extenso in 1846. No doubt this is largely due to the extraordinary completeness and elegance of the investigation, which still deserves careful study, and the fortunate accident that Poiseuille approached the problem as a physician interested in the circulation of blood in capillary vessels, and not as a hydraulic engineer; he accordingly used (glass) capillaries of very much smaller bore than any of his predecessors, and had to deal with Length (cm.). 47*20 Radius (cm.). 0-127 0-207 0-294 108-70 104-40 HISTORICAL DEVELOPMENT 11 purely laminar flow. He used five capillaries with the follow¬ ing diameters:— A, 0-14 mm.; B, 0-113 mm.; C, 0-085 mm.; D, 0-044 mm.; E, 0-03 mm. Long series of measurements were carried out, in each of which one of the relevant factors only was varied at a time. The capillaries were joined to a bulb, and a constant volume defined by marks above and below the bulb forced through them by compressed air. The results were : 1. The quantity discharged in unit time is proportional to the pressure, provided the length of the tube exceeds a certain minimum, which increases with the radius. 2. The quantity discharged in unit time is inversely pro¬ portional to the length of the tube. 3. The quantity discharged in unit time is directly pro¬ portional to the fourth power of the radius. The quantity discharged in unit time is therefore PD^ Q=K-, where K is a constant characteristic of the liquid, the value of which increases with temperature. Poiseuille carefully in¬ vestigated the variation of K, and gave a formula containing two constants for calculating its value at any temperature. The meaning of the constant K became clear when Wiede¬ mann and Hagenbach,^^ independently of each other, de¬ duced mathematically the expression for the volume passed in unit time. Both make use of Newton's hypothesis without referring to him, or to Stokes's deduction of the velocity of flow. They obtained the equation ^ S-qL ' in which 7] is a constant characteristic of the liquid, which Wiedemann proposed to call the Zähigkeits-Koef&zient," i.e. the coefficient of viscosity. Hagenbach defines it as follows: ''We designate by the name Viscosity (Zähigkeit) 12 THE VISCOSITY OF LIQUIDS the force necessary for shifting a layer of liquid of unit area and of the thickness of one molecule past a second layer by the distance of two molecules in unit time/' The definition does not appear very useful as it stands, but as the thickness of a layer and the distance of two molecules in it are assumed equal, it amounts to postulating unit gradient, and therefore agrees with the accepted definition. Hagenbach calculated 7] from Poiseuille's data, and, taking the gramme and square metre as units, found for water at io°, \ ^=O-I335I- f Hagenbach mentions the interesting fact that practice had been in advance of theory, inasmuch as it had been found desirable in the arts to characterise certain liquids empirically by their viscosity. Dollfus is stated to have used an instrument called a viscosimeter " for testing the gum solu¬ tions used in cotton printing, while Schübler, in a paper on The Fatty Oils of Germany," included among their physical constants the fluidity ratio " (Flüssigkeits-Verhältnis). The instrument used by him was not much inferior to some still in use for technical purposes : it consisted of a glass cylinder about 10 cm. long and i-8 cm. diameter, with a tapering out¬ let of about 1-8 mm. diameter. A constant volume of liquid was charged into the instrument, and the time in seconds it took to empty read by a watch. The fluidity ratio " is the inverse ratio of the times of efflux, water being taken as standard liquid and its fluidity " as looo. Determinations were made at two temperatures—7-5° and 15"^ C. The fluidity " of, e.g., castor oil was found to be 2-6 at 7-5° C. and 4*9 at 15° C. The work of Wiedemann and Hagenbach gave the in¬ terpretation of Poiseuille's results, and made it possible to determine viscosity in absolute measure, but theory was still incomplete in one particular. There is nothing in the deduc¬ tion to indicate that the assumptions on which it rests cease to hold good at some limiting values of pressure or dimensions; both Hägen and Poiseuille, however, had found experimentally that the linear relation between discharge and pressure no longer held when the pressure was raised beyond, or the length HISTORICAL DEVELOPMENT 13 of the tube reduced below, a certain limit. This question was settled by the work of Osborne Reynolds, who showed that, with a given tube and liquid, there existed a critical velocity at which the flow changed abruptly from the laminar type, in which each particle moves with constant velocity parallel to the axis of the tube, to the turbulent type, in which the particles move in irregular paths. (Hägen had already clearly expressed the view that some such change in the nature of the flow occurred.) Other things being equal, the critical velocity is proportional, not to the viscosity, but to the viscosity divided by the density, a constant which has since received the name Kinematic Viscosity.'' During the thirty years following Hagenbach's work a considerable number of viscosity measurements by means of various capillary instruments were carried out ; several other methods were also^ developed, the mathematical theory of which is complicated and leads to approximate solutions only. In 1890 Couette took up the system of concentric cylinders, the velocity distribution for which had already been given by Stokes, who suggested that it might be studied experi¬ mentally by observing motes in the liquid." Couette calculated the moment exerted by the outer cylinder on the inner one, and constructed an apparatus in which this moment could be measured and the coeiflcient of viscosity deduced from it; it was found to agree with Poiseuille's values. Couette also found that beyond a certain velocity the linear relation between angular velocity and moment ceased to hold abruptly, and compared this change to that from laminar to turbulent flow in the capillary. He also examined whether a criterion analogous to that given by Reynolds for the tube could be postulated for the concentric cylinder system ; while he did not formulate it completely, he showed that, for the somewhat extreme case of wafer and air, the velocities at which turbulence set in were very approximately proportional to the kinematic viscosities. The capillary viscometer, however, has remained the most extensively used instrument, and, especially since Ostwald gave it the simple form called after him, has been employed in a very large number of investigations of varying accuracy. 14 THE VISCOSITY OF LIQUIDS Many of these were undertaken with the object, or in the hope, of finding relations between a physical property so easily measured as viscosity and the chemical constitution of pure liquids; an even greater number have attempted to connect quantitatively the viscosity of electrolyte solutions with their conductivity, or that of mixtures with the viscosi¬ ties and ratios of the components. All these investigations, more particularly those on liquid mixtures, have had to face the initial difficulty that there is so far no theory of the viscosity of simple liquids. It is not the purpose of this work to discuss such attempts at a theory as have been published beyond saying that, with one exception, they are all of earlier date than the very important work of Bridgman on the varia¬ tion of viscosity with pressure, and are quite irreconcilable with his results, to explain which will be the first object of any fresh attempt. The study of viscosity has received a new impetus from the development of colloid chemistry. Here, too, the tendency has been to find relations between the changes in viscosity and changes in other properties less easy to measure or, some¬ times, even to define. In this discipline an entirely new, and fundamental, difficulty has arisen, inasmuch as many of these colloidal systems exhibit the striking anomaly that the vis¬ cosity coefficient is not a constant, as defined on p. 5, but varies with the velocity gradient. This feature will be fully described in the chapter dealing with colloidal solutions. 1 G. Wiedemann, Pogg. Ann., 99, 221 (1856). 2 A. W. Duff, Phil. Mag. (6), 9, 685 (1905) ; L. E. Gurney, Phys. Rev., 26, 98 (1908). 2 Daniel Bernoulli, Hydrodynamica, sen de viribtcs et motibus fliiidofum commentavii, sect. 3, para. 27 (Argentor., 1738). ^ Prony, Recherches physico-mathématiques sur la théorie des eaux courantes (Paris, 1804). ® G. A. Coulomb, Mém. de VInst. National, 3, 261 (1798). ® Girard, Mém. de la classe des sciences math, et phys. de l'Institut, 14 (1813)- ' Dubuat, Principes d'hydraulique, nouv. ed. (Paris, 1786). ® Navier, Mém. de l'Acad. des Sciences, 6, 389 (1823). 9 S. D. Poisson, /. de VEc. Polyt., 13, 139 (1831). G. G. Stokes, Trans. Camb. Phil. Soc., 8, 287 (1845) ; Math, and Phys. Papers, 1, 75 (Cambridge, 1880). HISTORICAL DEVELOPMENT 15 G. Hagen, Ann. d. Phys., 46, 423 (1839) ; Abh. d. Berliner Akad. d. Wiss., Math. Abt., p. 17 (1854). 12 J. L. M. PoiSEUiLLE, Mém. Savants Étrangers, 9, 433 (1846). 13 G. Wiedemann, loe. cit. (i). 14 E. Hagenbach, Pogg. Ann., 109, 385 (i860). 13 Dollfus, Bnll. Sac. Ind. de Mtilhouse, No. 21, p. 14. 1® G. SciitiBLER, Erdmann's J. f. prakt. Chem., 2, 349 (1828). 1"^ Osborne Reynolds, Phil. Trans., A, 174, 935 (1883) ; 177, 171 (1886). 13 M. Couette, Ann. d. chim. et phys. (6), 21, 433 (1890) ; J. de Phys. (2), 9, 566 (1890). CHAPTER II ß 'dv dr MATHEMATICAL THEORY OF THE PRINCIPAL METHODS OF DETERMINING THE COEFFICIENT OF VISCOSITY The most widely used method for determining viscosity co¬ efficients is still in principle that of Poiseuille, though many modifications in experimental details have been made by successive observers. The liquid is forced through a capillary tube, and 7] is deduced from the volume dis¬ charged in unit time, the pressure, and the dimensions of the apparatus. The advantages of the method are: the com¬ parative simplicity and cheapness of the apparatus, the small quantity of liquid required for examination, the ease with which it can be maintained at constant temperature, and finally the circumstance that the mathematical theory can be de¬ veloped with perfect strictness and without approximations. Flow in a Capillary Tube. As related in the previous chapter, the laws governing the flow of a liquid through a tube of small diameter have been deduced from the Fig. 2.—Velocity and hydrodynamic equations by several physi- veiocity gradient in cists; the elementary treatment of the capillary tube. ' t n -, problem to be adopted here appears to have been first given by Hagenbach {loc. cit., p. ii). We con¬ sider a portion AB of a cylindrical tube having a circular cross-section of radius R (fig. 2). The distance AB=L, and 16 R B FLOW IN A CAPILLARY TUBE 17 a difference of pressure = P is maintained between A and B, which causes the liquid to flow through the tube. We assume the flow to be such that every particle of liquid moves parallel to the axis of the cylinder with a constant velocity v. For reasons of symmetry this velocity will be the same for all points lying on the same circle, so that we may consider the liquid composed of cylindrical laminae moving with velocities which are functions of their radii. The force exerted by the pressure P on a cylinder of radius r is Fp = 77f2P, while the r^stance^ round the surface of the cylinder, caused by the viscosity of the liquid, will, according to our funda¬ mental assumption, be given by the product : area x viscosity coefflcient x velocity gradient, i.e. dv Fv=2,TrL^-. If the motion of the cylinder is not to be accelerated, i.e. if V is to remain constant, the forces acting on the cylinder must be equal and opposite, Fp=—Fy, and therefore fP=—2L'n— . . • (l) dr The velocity gradient is therefore dv dr By integration we find : v = It now remains to determine the integration constant C, for which purpose it is necessary to make some assumption about the conditions at the boundary. The usual assumption is that the lamina in contact with the wall of the tube adheres to it ; in other words, that ^'=0 for 2 rr 2l^r¡ 4Lrj (2) (3) 18 THE VISCOSITY OF LIQUIDS The integration constant then becomes R^P G and the velocity 4L7J' . . . (4) 4LT? This is the equation of a parabola (fig. 2), the axis of which is the axis of v, while the axis of r is at the distance R2P/4L07 from the apex of the curve. Since v is the distance travelled in unit time, the particles of liquid which were in the plane AA at zero time will be on the surface of the paraboloid, the profile of which is given by equation (4) after unit time; in other words, the volume of this paraboloid is the volume of liquid Q which passes in unit time. The volume of this solid of revolution is rR Q=27r vrdr, '0 and by introducing the value of v, 277 , ttpr^ This expression is identical with that found empirically by Poiseuille (p. 11); the constant K in his formula accordingly has the value (provided P is expressed in the same units) K —77/1287^, whence ')7=77/i28K, by means of which expression the values of 77 can be calculated from Poiseuille's results, a calculation carried out apparently for the first time by Hagenbach [loc. cit., p. 12). As the flow is stationary, it follows that the volume Q¿ discharged in the time T is SLtj ■ All capillary viscometers are so designed that a constant volume is passed through the capillary and the time T of dis- FLOW IN A CAPILLARY TUBE 19 charge is measured. R, and L are therefore constants of the instrument, while at a given temperature is a constant for any normal liquid. It therefore follows that, if the instrument has such dimensions that Poiseuille's Law holds good, the product PT must be constant, and this relation can be used to test whether the instrument is, in fact, correctly proportioned. Conversely, once the instrument has been found correct, variation in the value of PT proves that the liquid does not behave normally (see Colloidal Solutions). Turbulent Flow. In the deduction of Poiseuille's Law laminar flow has been assumed, but it is impossible to say a priori that laminar flow will be maintained whatever the pressure, and therefore the mean velocity (?;==Q/77R^), may be. Poiseuille already found, with all his tubes, deviations from the law when the pressure, other things being equal, was increased beyond a certain limiting value. Osborne Reynolds ^ showed in an exhaustive experimental and mathematical in¬ vestigation that this deviation—which manifests itself as an apparent increase of r)—was due to a change from laminar to turbulent flow; with the latter the particles of liquid no longer move in straight lines, but in irregular paths which change with time. Reynolds found that, the more viscous the liquid, the less its tendency to form eddies. The constant which determines the type of flow is, however, not the viscosity coefflcient, but the viscosity coeflicient divided by the density, rj/p—an important constant which has since received the name kinematic viscosity.'^ In a given tube the liquid with the lower kinematic viscosity becomes turbulent at lower velocity than that with the higher. Reynolds further showed that the conditions of flow for any tube and liquid could be characterised by a non-dimen¬ sional quantity, i.e. a pure number. The variables affecting the flow in a cylindrical tube can be combined to give such a non-dimensional expression as follows :— ) V in which V is the mean velocity and D the diameter of the tube. R is called the Reynolds number] conditions in different 20 THE VISCOSITY OF LIQUIDS tubes with different liquids will be the same when the Reynolds numbers are the same. More particularly there will be a change from laminar to turbulent ñow when R reaches a certain value, which experiment has shown to be 1400 to 2000 for capillaries. The critical velocity is therefore 2000rj This value of R is the Reynolds criterion for capillaries; the onset of turbulence is hastened by disturbances in the liquid before it enters the tube, so that the exact value of the criterion depends on the conformation of the orifice of the tube. By keeping the velocity well below the value corresponding to R^iqoo, it is generally possible to avoid turbulence. Even when the flow is laminar inside the tube at some distance from the ends, there are disturbances at the junction between it and the reservoir from which the liquid enters the tube, the exact nature of which need not be discussed. The effect they produce is that the formula—as Poiseuille found experimentally—does not hold strictly except for tubes of such length that the portions in which the flow is not laminar become negligible. A correction can be applied to the formula for these effects, which will be given below. Kinetic Energy Correction, It has further been assumed that the pressure is balanced by the frictional resistance, but this assumption does not quite correspond with conditions in the usual instruments, as was shown first by Hagenbach and again, in a very detailed study, by Couette.^ They point out that the formula (5) is strictly correct if a portion of a longer tube is considered, in which the liquid has reached a constant velocity; if, however, as is always done in viscosity measurements, L is taken as the whole length of the tube between two large reservoirs, and P as the difference of pressure between them, a portion only of this pressure serves to overcome viscous resistance, while the balance is used to impart velocity, i,e, kinetic energy, to the liquid. This por¬ tion can be calculated, and leads to a correction for the value FLOW IN A CAPILLARY TUBE 21 of rj calculated from equation (5), which is ttPR^ while the value corrected for kinetic energy is ttPR* mQp ''^SLQ SttL ' ' • ■ ^ ^ where p is the density of the liquid and m a numerical factor. Hagenbach found the value of m to he 1/ '^2 = 0793; Couette, whose calculation has been confirmed by Finkener and Wilberforce,^ finds w = i-oo; and avalué of w = i-i2 proposed by Boussinesq ^ has been used in a number of investigations. Couette suggested a second correction to allow for the non- laminar fiow at the ends of the capillary, which has been referred to above. This takes the form of a " fictitious lengthening " of the capillary, whereby equation (6) becomes »iQp 8Q(L+A) 8it(L+A) ■ ' ■ The value of A cannot be deduced theoretically, but must be found by experiment, and is generally of the order of a few diameters. The method by which Couette tested both correc¬ tions is generally applicable when L can be varied, and may therefore be described briefly. The results to be corrected were a set obtained by Poiseuille with two short tubes, A and B, of the same bore but different lengths. The values of 7] at 10° calculated from these by the uncorrected formula vary with the pressure. When the corrected formula (6) is applied the values of 7] become constant, i.e, no longer vary with the pressure, but the value found with A is different from that found with B, and both differ from the true value found with long tubes. If these differences are due to end effect, and if they can be corrected by adding an amount A to the length, A must have the same value for both tubes, since they have the same diameter and the shapes of the ends are the same. The equation La Lb 22 THE VISCOSITY OF LIQUIDS therefore holds, so that A can be calculated from a pair of observations : ^ EaLB(^A Vb) Vb^B ~VA^A Considerable divergence of opinion prevails regarding the conditions in which the Hagenbach-Couette correction is applicable. Grüneisen,® in a very exhaustive investigation on the range of validity of the Poiseuille formula, finds that it is not justified with instruments in which the capillary ends in large reservoirs, in which the liquid has a free surface of considerable area, since in such arrangements the gain or loss in velocity, and therefore in kinetic energy, is negligibly small. He tests the correction by applying it to a series of results obtained by Poiseuille with one of his longer tubes (R ==0-00567, L=2-357 cm.), in which the extreme values of the product PT do not differ from the mean value by more than o-i per cent. The kinetic energy correction should apply to this tube just as well as to the shorter tubes treated by Couette, since the conditions at the ends were exactly alike in both cases (see fig. 3). Grüneisen, however, shows that its application results in greatly increasing the discrepancies, so that the maximum deviation from the mean value becomes i-8 per cent. ; he considers the agreement obtained by Couette rather accidental. Graetz, in the chapter Friction in Liquids and Gases of Winkelmann's Handbuch der Physik, also expresses the view that the kinetic energy correction is not justified, at least in full, in the instrument used by Thorpe and Rodger (see fig. 9). These authors applied it to all their measurements, but found the A correction unnecessary. In accordance with Graetz's view, Thorpe and Rodger's results are recalculated and given without the kinetic energy correction in Landolt and Börn- stein's Tables, 4th edition; but in the 5th edition the present editor, Erk, adds a footnote (Ergänzungsband, p. 76), stating that later investigation, including work of his own, has shown the correction to be applicable, which should therefore be added to the figures in the tables. Bingham is emphatic in support of the correction, and FLOW IN A CAPILLARY LUBE 28 shows that it leads to concordant results. There is no diffi¬ culty in calculating the correction (6) when the tube, as in Bingham's improved Thorpe and Rodger instrument, has square ends, so that L is accurately known, while the A correction can be determined experimentally from two or more lengths of the same tube fixed in the same way into the same reservoirs. The matter, however, becomes more com¬ plicated in instruments in which the reservoirs are blown in one piece with the capillary, and the bore, as is recommended by most authors, expands gradually, so that L is indefinite. In such instruments it will generally be simplest to check the constancy of FT, and the way in which the correction can be found is indicated in a general review of the problem by Dorsey,^ based on data obtained by Bond.^ Dorsey trans¬ forms the equation (7) into where Q' is the volume discharged in time T, and v the average velocity {v' jT. The third term in brackets also has the dimensions of a length, so that the expression can be written: and the problem of finding the correction resolves itself into determining the sum in brackets, so that PT is constant for a given liquid, i.e. for a given kinematic viscosity. The value of A+Aj can be shown from Bond's investigation to depend on the average velocity v'. Bond determined the pressure required to drive equal volumes through two tubes of the same bore, but different lengths Li and Lg, discharging always into the same reservoirs, so that the end conditions were the same. The excess pressure P^ necessary to over¬ come the end effect is therefore the same for both lengths, and we can write: PT=^(L+A+A,). 7T7^ ^ I dlP2—\^<¿dp ¡ e> whence P = (L,P,-L,P,)/(L,-L,), 24 THE VISCOSITY OF LIQUIDS so that P^, or the additional length corresponding to it, can be deduced from a pair of determinations. Dorsey concludes from Bond's data for a range of velocities, that for Reynolds numbers {=2pvrlrj) doj M=277rL')7 . r—- . r=27rLr]r^—. dr dr Since r may be taken anywhere, this expression must be a constant for the apparatus, and equal to the moment main¬ taining rotation. By integration we find : ^ T , r =ATTï^y]œ +C. ^2 Since the outer cylinder rotates with the constant angular velocity O, while the inner one is prevented from rotating, we have the following conditions at the boundaries:— for f = R2, co = 0; for/=Ri, ch=o, from which the value of the integration constant is found : r_ ^ The moment is accordingly : R 2r> 2 m=4^L I = . . (II) in which K is an apparatus constant. M is usually measured by suspending the inner cylinder from a wire of known restoring moment N and determining the angle 9 by which it is deflected. We then have : NÖ-M-Kt^O; or, since N is also a constant for a given wire, i.e. for a given liquid at constant temperature the deflection is directly proportional to the angular velocity. If the de¬ flections 6 and 9' are found for two different liquids at the angular velocities O and íí', the relation between the vis¬ cosities is 9'Q 32 THE VISCOSITY OF LIQUIDS The linear relation between deflection and angular velocity holds up to a certain limiting value of the latter only, as was first shown by Couette. Beyond this limit turbulence sets in, and the deflection not only increases more rapidly than the angular velocity, but becomes rather irregular. From the experiments of Couette, and later ones by Mallock, a Reynolds number for the coaxial cylinder system can be deduced; the critical angular velocity is igoorj a {R,^-R,R,)p where the factor 1900 again has no dimensions. It is of interest, in view of some theoretical applications, to determine the velocity gradient in this arrangement. The angular velocity œ on any radius r is found as follows : since the moment is constant for any radius, we can write: ^ ^ The velocity gradient finally is 2Q (IC!) dr I-I^ In deducing equation (11) we have taken into account a length L of the inner cylinder. In practice, of course, both the outer and the inner cylinders have to be closed at the bottom, so that the moment tending to twist the inner cylinder is not due exclusively to the drag of the liquid between the cylindrical surfaces. If the formula is to apply, the effect of the end faces must be eliminated; the means adopted to achieve this end will be described in the next chapter. Oscillating Sphere and Disc. A number of other methods for determining r¡ need receive passing mention only, as they have not found extensive application; their mathe¬ matical theory is extremely complicated and generally in- THE OSCILLATING SPHERE AND DISC 83 volves simplifications, in consequence of which the results show rather considerable deviations from the most reliable data obtained by the transpiration method. Helmholtz and Piotrowski 21 caused a hollow metal sphere filled with liquid and suspended from a wire to perform torsional oscillations, and determined their period and logarithmic decrement ; from these figures and the constants of the apparatus r] can be deduced by very complicated calculations. The value found for the viscosity coeificient of water at 24-5° is over 20 per cent, higher than Poiseuille's figure for the same temperature. Meyer 22 used a horizontal circular disc suspended axially by a wire and oscillating in its own place in the liquid; the mathematical treatment is approximate only, as merely the cylinder of liquid, of which the disc is the cross-section, is taken into account, while the motion is propagated into the liquid beyond this boundary. The values of rj determined by Meyer, as well as by other investigators 23 using the same method, are higher than those found by the transpiration method ; a correction proposed by König 2^ reduces the dis¬ crepancies to about I per cent. A method used by Miitzel,25 the mathematical theory of which was also developed by Meyer, 26 somewhat resembles that used by Helmholtz and Piotrowski. A hollow cylinder filled with the liquid and suspended by a bifilar suspension is caused to oscillate round its axis änd the viscosity coeffi¬ cient deduced from the logarithmic decrement. In the investigations just mentioned, the object has been rather to verify experimentally the results of very difficult mathematical deductions than to devise convenient methods for the determination of viscosity coefficients. The solution of an equally or more difficult problem by Stokes 2? has, on the other hand, provided a comparatively simple method of measuring viscosity coefficients, which has found increasing application, especially for very viscous liquids. Stokes's Formula, Stokes shows that a sphere impelled by a constant force F in a viscous liquid eventually assumes a constant velocity V, and that the following linear relation holds :— F —GTTTjrY .... (14) 3 34 THE VISCOSITY OF LIQUIDS where r is the radius of the sphere. The most important special case, as far as the determination of viscosity coeffi¬ cients is concerned, is that in which the sphere falls (or rises) through the liquid owing to its weight (or buoyancy). The force then becomes Y=-TTr^g{p—p'), in which g is the gravity constant (==980 cm./sec.2) and p and p the densities of the sphere and the liquid respectively, and by introducing this value in (14) we obtain the equation known generally as Stokes's formula: .... (15) gr¡ This equation enables us to deduce the viscosity coefficient from the observed velocity of fall of a sphere of known mass and radius in a liquid of known density, provided a number of conditions assumed in deducing the formula are satisfied. These are : (1) The velocity is so small that higher powers of it may be neglected. (2) There is no slip between the liquid and the surface of the sphere. (3) The liquid is infinitely extended. As regards (i), Rayleigh has deduced a criterion by comparing the rates at which the terms included and those neglected in Stokes's deduction tend to decrease; according to him the velocity is small when R=^^=200 can be found as the sum of the values for the groups and atoms constituting a given compound, provided it is not associated. For associated liquids the temperatures calculated from con¬ stants derived from non-associated liquids are always con¬ siderably lower than the temperature actually observed, and in view of the additive character of these constants, Bingham considers that the degree of association can be found by simply dividing the observed by the calculated temperature. Table VII gives a number of observed and calculated values VISCOSITY AND CONSTITUTION 103 of the temperatures corresponding to <^ = 300, and the degree of association calculated from them as just described; the fourth column gives the degree of association for (¡>=200, that is, at lower temperatures. When the degree of association changes in the intervals contemplated, it is, as one would expect, higher at (j)~200, viz. at lower temperature. TABLE VII Association Factors at Fluidities 97=200 and 300 (Bingham) Substance. 95=300. 97=300. 97 = 200. Temp, obs. Temp, calc. Assoc. factor. Assoc. factor. Water ..... 358-5 162-7 2-20 2-31 Dimethyl ketone 289-5 207-5 1-23 1-23 Diethyl ketone 326-5 285-4 1-14 1-14 Methyl ethyl ketone 315-6 260-0 1-21 1-22 Methyl propyl ketone 330-0 285-4 1-17 i-r6 Acetic acid .... 407-9 236-3 1*73 1*77 Propionic acid 408-5 261-7 1*55 1*57 Butyric acid .... 413-7 287-1 1-48 1*51 Iso-butyric acid 413-7 278-9 1*49 1-51 Acetic anhydride 388-1 309-9 1-25 1-25 Benzene .... 348-1 303*7 1-14 1-17 Toluene .... 347-5 329-1 1-06 I-06 Ethyl benzene 362-1 354*5 1-02 1-02 o-Xylene .... 377*5 354*5 1-06 1-07 p-Xylene .... 357*4 354*5 I-00 I-00 Methyl alcohol 336-9 188-1 1-79 1-84 Ethyl alcohol (371-5) 213*5 (1-74) 1-83 Active amyl alcohol (408-7) 289-7 (1-41) 1*54 Inactive amyl alcohol (415-5) 289-7 (1-30) 1*55 Allyl alcohol .... 378-7 234-6 I-61 1-69 Methyl formate 297-5 236-3 1-26 (1-25) Ethyl formate 311-4 261-7 1-19 1-19 Propyl formate 333-9 287-1 1-16 1-17 Methyl acetate 306-0 261-7 1-17 1-17 Ethyl acetate 320-8 287-1 I-I2 1-17 Propyl acetate 343*0 312-5 1-09 1-12 Since this method of calculating association factors involves rather considerable assumptions, it is of interest to compare the values thus found with those given by other observers lOá THE VISCOSITY OF LIQUIDS using different methods. This will be done below, when M^Leod's treatment of the problem has been discussed. M^Leod's Theory. His paper,^ of which the following is a summary, begins by recalculating (as has been brieffy men¬ tioned on p. 73) the values of the free space at 0° from data for the small interval from o"^ to 10°, the object being to eliminate as much as possible the effect of any change in the degree of association. If is the volume occupied at 10° by the mass of liquid occupying i c.c. at 0°, the free space per c.c. at 10° will be (^0+^10 assuming the exponent A to be =1 {cf. p. 70), '^0 ^ 7]io ^0^10 Table VIII gives, in column 4, the values of Xq thus calculated, while column 5 shows those of v-^q. Column i gives the observed values of rj at 0°, and column 2 r¡ for each liquid with the free space adjusted to a fixed ratio to the total volume, viz. o*ioo.'' These values of 77 are calculated on the same assumption, that the viscosity varies inversely as the free space, and if this were strictly correct, the method would amount to comparing viscosities at temperatures of equal free space. M^Leod then plots these viscosities at free space ==o-i against the molecular weights (in the gaseous phase) ; a few typical points only are shown in fig. 35. They appear at first sight to be distributed quite irregularly, but it can be observed that the points for substances generally considered as non-associated fall approximately on a straight line pass¬ ing through the origin. Associated liquids have viscosities too high in proportion to their molecular weight to fall near this line—in other words, the molecular weight would have to be increased to bring them on it. M^Leod assumes as the normal line, r)=k'M, in the viscosity- molecular weight diagram that which passes through the point for octane, the degree of association of which is given by Ramsay and Shields as =0-93. M^Leod then calculates the factor a, by which the molecular weight has to be multi¬ plied to bring the points 7]-aM. on the octane line. VISCOSITY AND CONSTITUTION 105 TABLE VIII Viscosities Corrected for Free Space=o-ioo and Association Factors (M^Leod) Substance. ^0- no corr. for free space o-ioo. M. ^0- ^10- Assoc factor Pentane 0-283 0-342 72-1 0-1207 1-01497 0-96 Hexane 0-3965 0-388 86-1 0-0979 1-01285 0-92 Iso-hexane . 0-371 0-391 86-1 0-1054 1-01383 0-92 Heptane 0-519 0-448 lOO-I 0-0863 1-01223 0-9X Octane 0-703 0-523 1x4-1 0-0744 1-01203 0-93 Fthyl iodide . 0-719 0-653 156-0 0-0907 1-01155 0-85 Propyl iodide 0-938 0-735 170-0 0-0782 1-01146 0-88 Iso-propyl iodide . 0-8785 0-735 170-0 0-0836 1-01230 0-88 Acetone 0-394 0-447 58-0 0-1134 1-01383 x-56 Methyl ethyl ketone 0-5385 0-445 72-0 0-0827 I-0I220 x-25 Formic acid . 2-245 o-8oo 46-0 0-0356 1-00968 3*53 Acetic acid . 1-219 0-687 60-0 0-0564 I-OIO61 2-38 Propionic acid 1-519 0-855 74-0 0-0563 1-0x094 2-35 Butyric acid . 2-284 0-953 88-0 0-0417 x-OXO34 2-20 Iso-butyric acid 1-885^ 0-874 88-0 0-0463 x-0x000 2-00 Methyl alcohol 0-813 0-471 32-0 0-0580 x-OXX48 2-99 Fthyl alcohol 1-770 0-795 46-0 0-0450 X-0X05X 3-50 Propyl alcohol 3-882 0-902 60-0 0-0233 x-00797 3*95 Butyl alcohol 5-185 1-286 74-0 0-0248 X-00866 3*53 Iso-propyl alcohol . 4-564 1-150 60-0 0-0252 X-OX06X 3-89 Iso-butyl alcohol . 8-038 1-950 74-0 0-0243 x-OXX45 5*25 Fthyl acetate 0-578 0-474 88-0 0-0820 x-OX263 x-09 Propyl acetate 0-770 0-539 102-0 0-0700 x-OXX97 x-07 Fthyl ether . 0-286 0-353 74-1 0-1234 x *0x5x8 0-97 loo Fig. 35.—McLeod's graph illustrating association, i, octane ; 2, ethyl iodide ; 3, propyl iodide ; 4, chloroform ; 5, carbon tetrachloride ; 6, bromine ; 7, acetone ; 8, acetic acid ; 9, butyric acid ; 10, carbon di- sulphide ; 11, ethyl alcohol ; 12, butyl alcohol. 106 THE VISCOSITY OF LIQUIDS In Table IX the values thus calculated are shown in the first column, while the others give association factors found by different methods by (2) Ramsay and Shields {loc. cit.), (3) Bingham by the method described above, and (4) TraubeT^ TABLE IX Values of Association Factors by various Methods Substance. M«L. R. and S. B. T. Octane ..... 0*93 o*93 Ethyl iodide .... 0-85 I-OI Ethylene bromide . 1*21 • • 1-09 Bromine .... i-i8 • • 1*15 Ethylene chloride . 1-42 « • I-2I 1-46 Carbon tetrachloride 1-05 I-OI Carbon disulphide . 1-39 1-07 Acetone .... 1-56 1-26 1-23 1*53 Methyl propyl ketone I-I3 i-ii 1-17 1*43 Formic acid .... 3*53 3-61 • • I-80 Propionic acid 2-35 1*77 1*55 1-46 Butyric acid .... 2-20 1-58 1-48 Iso-butyric acid 2'00 1*45 Acetic anhydride 1-34 0-99 1-25 Propionic anhydride 1-22 é • i-ii Ethyl ether .... 0-97 0-99 Benzene .... 1*37 I-OI 1-14 1*05 Toluene .... I-08 • « I-06 Ethyl benzene 0-91 « « I-02 o-Xylene .... I-02 • • I-06 m-Xylene .... 0-92 • • i-oo Methyl alcohol 2-99 3*43 1-79 1*79 Ethyl alcohol. 3*50 2-74 1*74 1-67 Propyl alcohol 3*05 2-25 • • 1-66 Butyl alcohol. 3*53 1-94 Allyl alcohol .... 2-77 • • i-6i Iso-propyl alcohol . 3-89 2-86 • • 1-66 Iso-butyl alcohol 5-25 1*95 In discussing his results, M^Leod says that his values of the association factor do not differ from those of other workers more than they differ amongst themselves,'' and the same remark applies to Bingham's figures. Comparison is difficult, as the determinations are all made at different temperatures VISCOSITY AND CONSTITUTION 107 by the various authors. M^Leod says that his values refer to 0°, but this is hardly correct, as he uses the values corrected to free space =o-i; Ramsay and Shields worked at 46°; Bingham's values are obtained at temperatures of equal fluidity =300, which differ widely, while Traube's data are for room temperature. M^Leod's flgures probably refer to lower temperatures than any of the others, and may there¬ fore be expected to be high. M^Leod summarises his views as follows: If, then, the association factors in column 6 can be considered justifiable, it follows that the viscosity of a liquid is a linear function of the molecular weight and varies inversely as the ' free space.' Thorpe and Rodger in comparing liquids at their boiling- points found constants for such groups as CHg, Br, I, OH, etc., depending on their chemical nature. It follows from the above statement that these groups have an influence which depends only on their molecular weight, and not on their contour, chemical nature, or position in the molecule. Viscosity in liquids is, in fact, as in gases, due to a transference of momentum, and does not depend on any property of the molecule except its weight. It is true the ' free space ' and the degree of association may be governed largely by the ' chemical affinity ' between the molecules, but ' internal friction ' in liquids is not itself a function of the surface of molecules, and is not therefore analogous to the friction between solid bodies." M^Leod is no doubt right in considering attempts to repre¬ sent the mechanism of viscosity as friction between molecular surfaces to be merely false analogies. His views on the respective effects of molecular weight and free space are supported by the fact {cf. p. 71) that isomers showing different viscosities also have different free spaces, the differences being in the right direction. On the other hand there are, as has been pointed out in Chapters V and VI, considerable diffi¬ culties in regard to the magnitude of the free space calculated by his method, and the further one that the exponent A is not actually =1, so that the relation between viscosity and free space is not as simple as assumed. Nevertheless, the whole treatment of the subject is of great interest, and should 108 THE VISCOSITY OF LIQUIDS be capable of being developed further, if the effect of tempera¬ ture at equal volume, and therefore equal free space, can be explained satisfactorily. The investigations so far described start from some a priori assumption, like the additive character of viscosity (Thorpe and Rodger), or fluidity (Bingham), or the function of free space (M^'Leod). One or two purely empirical relations between viscosity and molecular constants must now be mentioned. The first of these, put forward by Dunstan,^^ is that the ratio T^/molecular (volume) has low values not differing too much from the mean for non-associated liquids, while it is much higher for associated ones, and particularly those containing OH groups. A few examples will suffice. Substance (r] CD 0 M X Substance. X M 0 CO Benzene 65 Acetone . 43 Ethyl acetate 43 Water 493 Carbon disulphide 60 Ethyl alcohol . 189 Ethyl iodide 69 Methyl alcohol 138 Ethyl bromide 51 Glycol . • 2750 Toluene 53 Acetic acid 195 Chloroform 67 Lactic acid • 5410 Dunstan and Thole s Logarithmic Formula. A much more important relation found by Dunstan and Thole is the logarithmic one : log r¡—aM. +Ö, where M is the molecular weight, a is d. general constant, and & is a constant characteristic for any one homologous series. The relation fails, like others, for the lower terms, especially of the fatty acid series, but holds well for a number of series. Table X gives the viscosity (in poises) of a variety of these at 20°, and the values of the logarithmic increment for the CHg group; for convenience the logarithms of (7^x10^) are used : TABLE X Increment of log (17x10^) for CHg (Dunstan and Thole) Substance. Hexane Heptane Octane . Iso-hexane Iso-heptane Ethyl iodide Propyl iodide Iso-propyl iodide Iso-butvl iodide Methyl propyl ether Ethyl propyl ether . Propyl ether . Ethyl ether Propyl ether . Methyl propyl ether Methyl iso-butyl ether Ethyl iso-butyl ether Ethyl acetate Propyl acetate Methyl propionate . Methyl butyrate Acetone Methyl propyl ketone Methyl ethyl ketone Acetone Propyl alcohol Butyl alcohol . Iso-butyl alcohol Iso-amyl alcohol Methyl sulphide Ethyl sulphide r¡. 0-00320 0-00411 0-00538 0-00300 0-00379 0-00583 0-00737 0-00690 0-00870 0-002515 0-003175 0-00420 0-002345 0-00420 0-002515 0-003065 0-003785 0-00449 0-00581 0-00454 0-00575 0-003225 0-00501 0-00423 0-00501 0-0226 0-0295 0-0391 0-0509 0-0293 0-00455 A log [y] X lo^) for CHg. 0-109 0-117 0-102 ' 0-109 (paraffins). 0-102 o-ioi o-102 (alkyl iodides), o-ioi 0-126 O-III y0-108 (ethers). 0-092^ 0-112 0-103 -0-107 (esters). 'O-O96| 0-117^ 0-106 (ketones). O-ii6^ 0-115 -0-116 (alcohols). 0-09110-091 (sulphides). Mean value for 16 pairs = 0-107. 110 THE VISCOSITY OF LIQUIDS Dunstan and Thole then calculate the increments of log (t^xio^) for the chief groups at 20°. One example will be sufficient. The value for hydrogen is found by subtracting from log {rj X10^) for hexane, heptane, and octane respectively, 6, 7, and 8x0-107 (the average increment per CHg), and dividing the difference by 2 : From hexane . 0-931. From heptane . 0-932 (mean A log (07 xio^) for H =0-934). From octane . 0-938. The values for other atoms and groups are : OH (alcoholic) . 2-102 O (ethereal) . 0-098 CO (ketonic) . 0-407 C . . . —1-761 Double bond . 1-847 Iso-union . —0-030 The authors show that, by the use of these atom and group constants, it is possible to calculate the viscosity of even complicated compounds with fair accuracy. One example may be quoted: Iso-butyl ethyl ether, 7720—0-00376 poises (T. and R.), (CH3)2CH-CH2-0-C2H5 = 6CH2= 0-642 0 == 0-098 2H = 1-868 2-608 iso = —0-030 2-578 =log (0-00378 XIO^). 1 Thomas Graham, Ann. d. Chem. und Pharm., 123, 90 (1863). 2 L. Rellstab, Die Transpiration homologer Flüssigkeiten, Dissertation, Bonn, 1868. 3 A. Guerout, C.R., 81, 1025 (1875). 4 R. Pribram and A. Handl, Wien. Ber. (II), 78, 113 (1878) ; 80, 17 (1879) ; 84, 717 (1881). ^ R. Gartenmeister, Zeit, physik. Chem., 6, 524 (1890). ® T. E. Thorpe and J. W. Rodger, Phil. Trans., A, 185, 397 (1894) ; Proc. Roy. Soc., A, 60, 152 (1896). VISCOSITY AND CONSTITUTION 111 E. C. Bingham and J. P. Harrison, Zeit.physik. Chem., 66, i (1909) ; E. C. Bingham, Amer. Chem. 43, 302 (1910). ® Log. cit. (7). ® D. B. M^Leod, Trans. Faraday Soc., 21, 151 (1925)- W. Ramsay and J. Shields, Trans. Chem. Soc., 63, 1089 (1893). J. Traube, Ber., 30, 273 (1897). 12 A. E. Dunstan, Zeit, physik. Chem., 51, 738 (1909). Dunstan and Wilson, Trans. Chem. Soc., 91, 90 (1907) ; cf. also Dunstan and Thole, The Viscosity of Liquids, pp. 31-38 (1914). CHAPTER VIII THE VISCOSITY OF SOLUTIONS loo A. Non - Electrolytes. The investigations on solutions of non-electrolytes are not numerous ; the most extensively studied example is probably the solution of cane sugar (sucrose). Many of the earlier determinations, like those of Burkhardt,^ Rudorf,2 Grüneisen,^ and Heber Green,^ cover a small range of tem¬ perature only; Hosking,^ however, measured the viscosity of solutions containing up to 40 per cent, by weight, at tem¬ peratures between 0° and 90°, with Thorpe and Rodger's instru¬ ment. The latest data are those of Bingham and Jackson,^ who pro¬ posed this solution as a liquid suitable for calibrating viscometers. These are given in Table XI in centipoises, and the vis¬ cosities of the 40 and 60 per cent, solutions are plotted as ordinates against the temperatures as abscissae in fig. 36. 100' Fig. 36.—Viscosity-temperature curves of 40 and 60 per cent, cane sugar solu¬ tions (Bingham and Jackson). SOLUTIONS OF NON-ELECTROLYTES 113 TABLE XI Viscosity of Sucrose Solutions (Bingham and Jackson) Grm. of sucrose in loo grm. of solution. 0. 20. 40. 60. 0° 1*789 3-804 14*77 238 5 1-516 3-154 11*56 156 10 1-306 2*652 9-794 109*8 15 1*141 2*267 7*468 74-6 20 1*005 1*960 6*200 56-5 25 0*894 1*704 5-187 43-86 30 0*802 1*504 4*382 33-78 35 0*720 I-33I 3*762 26*52 40 0-653 1-193 3-249 21*28 45 0-596 1*070 2*847 17*18 50 0-550 0*970 2*497 14*01 55 0-507 0*884 2*219 11*67 60 0-470 0*808 1*982 9-83 65 0-436 0*742 1*778 8-34 70 0*406 0*685 1*608 7-15 75 0*379 0-635 1*462 6*20 80 0-356 0*590 1-334 5-40 85 0-334 0-550 1*221 4-73 90 0-315 • • 1*123 4-15 95 0*298 « • 1*037 3-72 100 0*282 • • 0*960 3-34 The great increase of viscosity with concentration at low temperature, and the rapid decrease in viscosity with tempera¬ ture at high concentration, are both strikingly illustrated by this graph. The temperature coefhcient, or, more precisely, the percentage decrease of viscosity per degree in the interval from i8° to 25°, Vis '^25.. a= X 25 18 VlS~^V25 has been calculated by Heber Green for the following con¬ centrations (C = mois./litre) :— i-o 1-2 1*4 C=o-o 0*1 0-2 0-4 0-6 0*8 a=2-390 2-454 2-520 2-637 2-778 2-945 0 II H 1*8 2-0 2-2 2-5 2-7 a =4'^^ 4.47 4-90 5-58 6-48 7-20 8 114 THE VISCOSITY OF LIQUIDS An important distinction between solutions of non-electro¬ lytes and of electrolytes shows itself in the relative viscosities at different temperatures, i.e. the ratios: 7] of solution at temperature t tjq of solvent at temperature í The values for the 60 per cent, sugar solution are as follows :— Temp.° o 10 20 30 50 70 90 100 V/Vo ■ i33'0 83-9 56-2 47-1 25-4 17-5 13-1 II-8 The relative viscosity, referred to water at the same temperature, decreases with rising temperature. We shall find later—as Ranken and Taylor seem to have pointed out for the first time—that the reverse holds good for electrolyte solutions. It is of interest to note that the volume change caused by dissolving sugar in water is not great. Green gives the follow¬ ing data for the molecular volume (i.e. volume of one grm.- moL): solid 215-6 c.c., at extreme dilution 210-0 c.c., and at high concentrations 220 c.c. The empirical formulae expressing the variation of viscosity with temperature do not appear to have been applied to solu¬ tions. The author has tested Batschinski's relation (p. 76) by Bingham and Jackson's data for the 60 per cent, solution; the free space at 0°, calculated from the volumes and vis¬ cosities at 0° and 50°, amounts to only 0-123 cent, of the total volume, and the viscosity at 25° calculated on this basis is about half the observed value. The problem of finding a connection between concentration and viscosity has received much more attention, though there has been one attempt only at rational treatment, by Einstein.^ This will be more conveniently discussed in the chapter on Colloidal Solutions; here it is sufficient to say that Einstein finds a linear increase in viscosity with concentration, a result which does not agree with observation except at very low concentrations. A number of empirical formulae have been proposed, the most useful of which is that of Arrhenius : ^ v/Vo = A", or log (7)/7)o) =c log A, SOLUTIONS OF NON-ELECTROLYTES 115 in which 7^0 is the viscosity of the solvent, rj that of the solu¬ tion with the concentration c, and A a constant. The concentration in this formula has been expressed in every possible way: volume of solute in volume of solution; weight of solute in volume of solution; molar ratio; mois, of solute in a fixed number of mois, of solvent, and mois, of solute in a fixed weight of solvent. As regards the sugar solution, Heber Green found that the formula did not hold for concentrations above o-2 mols./litre when expressed either as mols./litre or as volume of solute per unit volume of solution. He obtained the best agreement when using for c the value c=vlw, in which v is the volume of sugar and w that of water in unit volume of solution; vary¬ ing discrepancies arose according as the molecular volume of sugar or that of water was assumed to remain constant. Dunstan and Thole find very good agreement between Green's experimental values and those calculated from Arrhenius's formula when the concentration is expressed in mols./iooo grm. of water. Below are given the values of A found in this way, the constancy of which is satisfactory. Mols, of sucrose per 1000 grm. of water. V (centip.). A. 0 0-8953 0-4382 1-3083 2-38 0-9666 2-105 2-42 1-618 3-805 2-44 2-440 7-973 2-45 3-516 20-72 2-44 5-440 105-8 2-41 Applying the same method of calculation to Bingham and Jackson's data, the author finds fair agreement at 20° but considerable discrepancy at 0°. Table XI shows that the viscosity of sugar solutions at all concentrations and temperatures is higher than that of water, and this is the general rule with non-electrolytes. There are, however, instances of the phenomenon, which has been called 116 THE VISCOSITY OF LIQUIDS —not very felicitously—negative viscosity in electrolyte solutions, and held to be an exclusive characteristic of such, viz. lowering of the viscosity of the solvent by a solute. One example is p-nitrotoluene in ethyl alcohol, given by Wagner and Mühlenbein,whose data are given below. G is the concentration in mols./litre, p the density, and rj the relative viscosity, that of alcohol being taken as p-Nitrotoluene in Ethyl Alcohol c. p- V- I/I 0-83733 0-9989 1/2 0-81563 1-0039 1/4 0-80421 I-OOI8 The viscosity of the molar solution is smaller than that of the solvent—though the difference is probably not much greater than the experimental error—and it is obvious that the r¡-C curve has a maximum between C=i and G=1/2. In view of the next example to be quoted—urea in water— it is unfortunate that no determinations at other temperatures are available. Solutions of urea were investigated by Rudorf, who found the viscosity lower than that of water. Fawsitt,^^ working with specially pure material, was unable to confirm this result. Ranken and Taylor [loc. cit.) again investigated this solution, and found its viscosity at all concentrations higher than that of water at 25°; at 8°, however, the viscosity of a solution containing 0-03125 mols./litre was found to be 0-9985 [r¡ water ==i). It is worth pointing out that the density of the nitro- toluene solution increases continuously with concentration, as viscosities of liquid mixtures {q.v.) lower than that of either component are frequently ascribed to volume changes. As regards the sugar solution, conditions are no doubt complicated by the association of the solvent and by hydra¬ tion of the sugar molecules. Porter has deduced hydration numbers, i.e. the number of molecules of water associated with one molecule of sugar, on the assumption that a for¬ mula of the type P(î; —&)=RT represents osmotic pressure; SOLUTIONS OF NON-ELECTROLYTES 117 if h is calculated from the experimental data it is found to be greater than the volume occupied by the sugar, and the differ¬ ence is assumed to be due to water of hydration. In this way Porter finds that hydration decreases with increasing con¬ centration and with rising temperature ; at a concentration of go-i grm./litre the hydration number is 10-3 at 0° and 3-5 at 60°, while at a concentration of 256-7 grm./litre it is 5-5 at 0° and 3*3 at 60°. If either the mass or the volume of the hydrated molecule is the factor which determines the vis¬ cosity, the decrease of this volume or mass is in agreement with the fact which has been demonstrated above: that 7]l7¡Q decreases with rising temperature, or, in other words, the viscosity of a sugar solution decreases at a more rapid rate than that of water. Ideal Solutions, In view of these complicating factors it would be very desirable to have data on ideal solutions; unfortunately, the only determinations available were made at a single temperature. These are measurements on solu¬ tions of naphthalene and of diphenyl in benzene and in toluene, carried out at 25° by Kendall and Monroe. The freezing-point of the benzene solutions has been determined over the whole range of concentrations used and found in agreement with the ideal solution law ; a number of freezing- points have been determined for the toluene solutions and coincide with those for benzene, so that these solutions may also be assumed to be ideal ones. The investigation was undertaken with the object of testing the empirical formulae connecting viscosity with concentration: all of them, in¬ cluding Arrhenius's, failed to represent the results. Kendall and Monroe applied a relation which gives good results for mixtures of indifferent non-associated liquids : 7^1/3 7^^1/3 J — in which 7], 071, and 772 are respectively the viscosities of the mixture and of the two components, and a; is the molecular percentage of one of them. The formula was tested for solutions—where the viscosity of the solute, is of course unknown—by calculating 772 from the experimental data. The results are given in Table XIL 118 THE VISCOSITY OF LIQUIDS TABLE XII Viscosity of Ideal Solutions in Centipoises (Temp, 25°) (Kendall and Monroe) Per cent, of splute in solution. f] (obs.). rj2 (calc.). Weight. Volume. Molecular. Naphthalene in Benzene. o-o 0-0 o-o 0-6048 8-ii 7-3 5-10 0-6565 2-20 17-16 15-2 11-21 0-7261 2-30 22-97 20-6 15-38 0-7707 2-23 28-82 25-8 19-29 0-8263 2-27 34-10 30-8 23-98 0-8764 2-24 37-69 33-9 26-93 0-9178 2-27 M ean 2-25 Diphenyl in Benzene. 0-0 0-0 0-0 0-6051 18-08 15-8 10-06 0-7585 3-41 30-57 27-3 18-24 ■ 0-9014 3-40 53-03 48-9 36-38 1-298 3-51 M ean 3-44 Naphthalene in Toluene. 0-0 0-0 0-0 0-5526 5-73 4-8 4-19 0-5848 1-81 13-72 11-8 10-26 0-6394 1-80 20-12 17-4 15-33 0-6866 1-83 27-31 24-1 21-27 0-7470 1-86 M ean 1-825 Diphenyl in Toluene. 0-0 0-0 0-0 0-5520 21-38 18-6 13-98 07335 2-75 32-02 28-2 21-97 0-8587 2-81 38-97 34-8 27-61 0-9627 2-89 M ean 2-82 It will be seen that for any one system the constancy of r¡2 is satisfactory. The values of 7^2 for either solute, calculated from the data for benzene solutions, are, however, different from the corresponding values for toluene solutions, so that SOLUTIONS OF ELECTROLYTES 119 the physical meaning of 7^2 (viscosity of liquid naphthalene " at 25°; melting-point 80-5°) is quite obscure. The authors point out that the ratio, from benzene data)/(7y2 from toluene data), is practically identical for both solutes, viz. 1-23 for naphthalene and 1-22 for diphenyl. While this agreement is interesting, it is difficult to see any physical meaning in it, or to consider the cube-root formula as any¬ thing but a further interpolation formula. B. Electrolytes. Investigations on this subject have been extremely numerous since G. Wiedemann suggested a con¬ nection between the viscosity and conductivity of salt solu¬ tions and made determinations of the viscosity coefficient. References to some of the earlier papers are quoted at the end of this chapter. A fresh impetus was given to the study of the subject by the development of the electrolytic dissocia¬ tion theory; Arrhenius himself carried out numerous measurements with an early form of the Ostwald instrument, and represented the results by the experimental formula already mentioned. A few general characteristics of electro¬ lyte solutions may be summarised as follows :— All electrolytes, with the exception of some salts of potas¬ sium, rubidium, cœsium and ammonium, increase the viscosity of water (or other solvent). The change in viscosity is not great for concentrations of several mols./litre, as shown by the data in Tables XIII to XVI; thus the viscosity of a lithium nitrate solution containing 5*849 mois, in 1000 grm. of solution is only little more than three times that of water at 25*01°. A few highly soluble salts, however, give very viscous solutions at high concentrations, such as calcium chloride, and particularly zinc chloride (Table XVII). [Table XIII 120 THE VISCOSITY OF LIQUIDS TABLE XIII Relative Viscosities of Lithium Nitrate Solutions (Applebey) = viscosity of water at same temperature. G = mois, iñ I GOG grm. of solution. 0 0 M CO 0 25-01°. G. nho- G. Vha- c. nha- 0-0401 1-0032 0-00724 I-OOI24 0-0174 1-0026 0-0883 1-0058 0-0131 1-00200 0-0299 1-0040 0-1026 1-0076 0-0379 1-0047 0-0567 1-0067 0-2294 I-OI545 0-0784 1-00905 0-0825 1-0099 0-4179 1-0278 0-1446 I-OI545 0-I07I I-OI25 0-4818 1-0325 0-2653 1-0278 0-2333 1-0267 0-8577 i-o6i6 0-7034 1-0737 0-3238 I-0354 I-I34 1-0875 1-283 1-14395 0-3643 1-0405 1-572 I-I345 1-471 1-1699 0-5385 1-0597 2-099 1-2007 2-528 1-3498 0-8666 1-0980 2-508 1-2770 2-550 1-3579 0-9663 1-1112 3-120 1-4906 1-316 1-1567 3-279 1-5367 2-2719 1-3151 4-363 1-9346 3-8541 1-74075 4-578 2-0577 5-849 3-0255 TABLE XIV Relative Viscosities of Cesium Nitrate Solutions (Merton) = viscosity of water at same temperature. G = mois, in looo grm. of solution. 0° • loL 18°. 25°. G. ViVo- G. nhu- G. vho- ó 0-02513 0-9960 0-0292 0-9970 0-4529 0-9986 0-9988 0-05120 0-9901 0-0532 0-9932 1-0060 0-9660 0-9970 0-07609 0-9841 0-1609 0-9786 2-1415 0-9899 0-9926 0-09958 0-9793 0-2783 0-9625 3-1299 0-9844 0-9882 0-1984 0-9602 0-4196 0-9454 5-0677 0-9752 0-9811 0-2863 0-9941 0-5527 0-9309 6-5882 0-9681 0-9761 0-3965 0-9244 0-6423 O-92II 9-6569 0-9561 0-9671 12-3740 0-9470 0-9604 16-6355 0-9347 0-9518 SOLUTIONS OF ELECTROLYTES 121 TABLE XV Viscosity of Electrolyte Solutions in Centipoises (Herz and Martin) Solute Grm. in 100 c.c. NaCl 26*694. Na^SO^ ii-i20. KjSO, 10-962. KI 39*006. KI 12*020. Oxalic acid 8-552. Temperature 20° 1-835 1*410 1-135 0*888 0*951 1*193 30 1-471 1*124 0-934 » • • • 0*951 40 1*211 0*922 0*771 0-635 0*647 0*780 50 1*020 0*772 0*648 • • • • 0*654 ,, 60 0*877 0*659 0*571 0-493 0*481 0*561 70 0*767 0-573 0*501 • • • • 0*489 80 0*681 • • • • 0-398 0-374 0*430 90 0*634 • • 0*403 • • • • 0*382 TABLE XVI Viscosity of Ammonium Sulphate Solutions in Centipoises (Grunert) Concentration. 3-5 N. 1-75 N. 0*875 N. 0-4375 N. Temperature 20° 2-394 1-455 1*196 1*088 40 1*644 0*994 0*766 0*713 60 1*203 0*730 0-551 0*512 80 0-937 0*571 0*424 0-395 TABLE XVII Viscosity of Calcium Chloride and Zinc Chloride Solutions CaClg at 20°. ZnClg at 25°. Grm. per litre. 77 (centip.). Per cent, by weight. Grm.-equiv. per litre. Relative vis¬ cosity (77/770). 0 0*9974 74-87 22*8 153 204*2 1*704 68-43 19*3 33*7 321-3 2*542 61*72 16*1 12*7 415*9 3-817 56*22 13*1 6*91 473*4 4-946 49*55 11*3 4*51 504*2 6-143 44*20 9*57 3*41 542-5 7-603 36-75 7*40 2*50 575 9-733 28*71 5*38 2*00 22*17 3*92 1*67 18-36 3*15 1-546 12*22 2*51 1-372 6*14 0-95 1*202 3-08 0*46 1*108 122 THE VISCOSITY OF LIQUIDS Sodium hydroxide, which has been very carefully studied by Bousfield and Lowry,^^ also exhibits high viscosity at higher concentrations, as appears from the figures in Table XVIII, and the viscosity-concentration graph, fig. 37. Although the volume relations in solutions are obscure, the great contraction which occurs when sodium hydroxide is dissolved in water deserves mention. As the authors point out, 100 grm. of NaOH can be dissolved in i litre of water without an increase of volume, while 50 grm. produces a contraction of 3 c.c. TABLE XVIII Relative Viscosities and Fluidities of Sodium Hydroxide Solutions at 18° (Bousfield and Lowry) Concentration. V' (p. Per cent. Mols./litre. o-o 0-0 i-00 i-o 2-50 0-64 1-08 0-923 7-68 2-o8 1-43 0-700 14-28 4*15 2-25 0-444 20-14 6-13 3-84 0-260 25-0 7*94 6-69 0-145 30-2 10-03 11-81 0-0847 35-0 12-05 20-6 0-0485 40-0 14-29 32-3 0-0310 45-0 i6-6o 48-2 0-0207 50-6 19-37 74-7 0-0134 The viscosity of electrolyte solutions, like that of other liquids, decreases with rising temperature, but does so less rapidly than that of water, so that the relative viscosity, referred to water at the same temperature, increases with rising temperature. As has already been* stated, the change is in the opposite direction in solutions of non-electrolytes. Table XIX gives r]lr¡Q for a number of salt solutions; the figures show that the rule holds equally for salts which raise and for those which lower the viscosity of water. It is therefore possible SOLUTIONS OF ELECTROLYTES 123 that solutions of salts which, at ordinary temperature and concentration, increase the viscosity of water, may at a suitable concentration and sufficiently low temperature show an and at still lower temperature even < i; a possi¬ bility pointed out long ago by Arrhenius.^^ This case has Fig. 37.—Viscosity-concentration curve of sodium hydroxide solution at 18° (Bousfield and Lowry). not been realised experimentally. Although data are avail¬ able for the viscosity of concentrated (7*252 mois, in 100 mois, of water) calcium chloride solutions down to a tem¬ perature of — 49-16°, they cannot be utilised, as the expression ->7/7^0 has no meaning below the freezing-point of the solvent. The point, which is of considerable interest, could be decided by investigating the viscosity of solutions in a dissociating solvent of low freezing-point like alcohol. [Table XIX 124 THE VISCOSITY OF LIQUIDS TABLE XIX r}lr¡q of Electrolyte Solutions Electrolyte Grm. in 100 c.c. of sol. NaCl. 26-694. K2SO4. 10-962. KI. 39-006. (NH, 23-10, \ ^0 II-55- Temperature 20° 1-826 1-130 0-883 * 2-382 1-447 30 1-833 1-164 40 1-856 i-i8o 0-970 2-517 1-522 50 1-854 1-163 ,, 60 1-866 1-214 1-049 2-538 1-553 70 1-883 1-248 ,, 80 1-912 « • 1-263 2-632 1-603 90 2-012 1-279 From Herz and Martin's From Grunert's data. data. Electrolyte, cone. i mol./litre. NaCl. KCl. RbCl. NH4CI. Temperature 5° 1-065 ♦ 0-9307 0-9182 0-9637 10 1-076 0-9464 0-9326 0-9464 15 1-069 0-9588 0-9474 0-9553 20 1-084 0-9751 0-9632 0-9682 25 1-092 0-9866 0-9787 0-9765 30 1-108 0-9975 0-9850 0-9900 ^ ^ From Simon's data.^^ The converse, of course, holds for salts which, over a certain range of concentration and température^ lower the viscosity of water: at a suitable concentration and sufficiently high temperature r¡lr¡Q should become ==i, and at still higher tem¬ perature >1. There is experimental evidence that this range is reached, e.g.^ with potassium iodide, the concentration- viscosity curves for which have been determined at four tem¬ peratures, 10°, i8°, 30°, and 50°, by Getman (fig. 38). The SOLUTIONS OF ELECTROLYTES 125 dotted lines show the viscosity of water at the same tempera¬ tures and intersect the viscosity-concentration curves, except that at 10°; in other words, at every temperature there is a concentration at which the viscosity becomes equal to that of water. At 10° this has not been reached, although the solubility would have permitted it. The same behaviour is shown by ammonium salts, the vis¬ cosity of which has also been measured by Get- man at one tempera- ture, 25° (ñg. 39); with the chloride and the nitrate a concentration is reached at which the viscosity becomes equal to that of water, and there is no doubt that the same thing would happen with the bromide and iodide at higher temperature. The salts producing negative viscosity are 100 thus seen to conform to Fig. 38.—Viscosity of aqueous solutions of what appears to be the potassium iodide (Getman). Absc. grm. m TOO 0.0. of solution. general rule for electro¬ lytes, viz. their r]/r]Q decreases with falling temperature, and their peculiarity is merely that this ratio becomes smaller than unity at easily attainable concentrations and temperatures. They have, however, been the subject of many investigations and theoretical speculations, which may be conveniently dis¬ cussed at this place. Negative " Viscosity. The fact that some salts of potas¬ sium, rubidium, caesium and ammonium lower the viscosity 126 THE VISCOSITY OF LIQUIDS of water was discovered by Wagner.A number of these salts were again investigated by H. C. Jones and his col¬ laborators, as well as by Getman, while solutions of caesium Fig. 39.—Viscosity of aqueous solutions of ammonium salts (Getman). Absc. grm.-equivalents per litre. the temperature interval from 0° to 25° by Merton.^^ Get- man's data on potassium iodide and various ammonium salts have already been referred to (figs. 38 and 39). Merton's figures are given in Table XIV (p. 120) ; the concentrations are expressed as grm.-mols. in 1000 grm. of solution, and the viscosities as relative viscosities, referred to water at the same temperatures. The last two columns show 18° and 25° for the same concentrations, so that they can be com¬ pared directly ; rj/rjQis higher throughout at the higher tempera- SOLUTIONS OF ELECTROLYTES 127 ture. At the highest concentration and temperature the relative viscosity is still X2, the first term will be increased by a less amount than the second is reduced. Therefore a mixture of a liquid of low viscosity with one of high viscosity should give a sagged curve, which agrees with experience; in fact, mixing two such liquids would correspond to mixing equal parts of one and the same liquid at different temperatures, when the resulting viscosity is not the mean of the two values, but lies below it. If Xi and X2 are nearly equal, Ax will be small, and the curve very slightly sagged; if there is a considerable con¬ traction on mixing. Ax will be negative in both terms, so that a maximum results; while in the opposite case, i.e, expansion on mixing, the viscosity curve will have a minimum. M'^Leod states that he has attempted to apply the equation given above to various mixtures exhibiting decided maxima, but that the results, while reproducing the general trend of the experimental curves, were always lower than the observed values. He assumes a probable explanation of the discrepancy to be that the more viscous component does not gain an amount of free space proportional to its volume— as assumed in the formula—but somewhat less. Since there is so far no way of determining how the free space is divided between the two components, some further 156 THE VISCOSITY OF LIQUIDS simplifications become necessary. The linear law is still assumed to hold for mixtures of components with initial viscosities of the same order and without volume change on mixing; in addition, the free space is assumed to be the same in both components, and an arbitrary average value of it is chosen, viz. o-ioo. A special difficulty arises when water is the one component, as the free space of water has not been determined accurately. With these assumptions the formula becomes : where G is the contraction per unit volume. As the assumed free space of o*ioo will be too small for some and too great for other substances, the values of Ai and Ag will not be those found for the pure substances, but will have to be calculated from two sets of data for the mixture G is of course deduced from the density, so that the formula contains two arbitrary constants, the free space being assumed. M^'Leod has tested this formula on a number of the mixtures studied by Dunstan, Thole and Hunt,^® Thorpe and Rodger,^^ and Bramley.^2 The method of calculation is illustrated by Table XXIII. Column 4 gives the volume calculated on the assumption that there is no change on mixing, and column 5 the difference between the calculated and the observed volume of I grm. of mixture, divided by the calculated volume, i.e. the contraction per unit of original volume. The data for the two ratios marked by asterisks have been used for calculating the exponents, and lead to the values: 0*100 \ O-IOO—C Ai (pyridine) =4-363; Aa (water) =5-324. THE VISCOSITY OF LIQUID MIXTURES 157 TABLE XXIII Viscosity of Pyridine-water Mixture Calculated from Contraction (M^^Leod) Per cent, pyridine. Density of mixture. Volume of I grm. of mixture. Calcu¬ lated volume. C/Y. rj obs. 7] calc. O'OO 0-997x7 1-00290 X-00290 0-00 0-89X 15-33 1-00119 0-99882 X-00580 0-00694 x-246 x-290 25-55 1-00187 0-99818 X-007X9 0-00895 x-4027 1-434 30-99 1-00242 0-99764 X-00882 0-OXX08 l'ôgiô x-6oo *40-46 1-00359 0-99650 X-0X066 0-0X40X x-8630 1-863 50-03 1-00365 0-99642 X-0X250 0-0x588 2-05x5 2-040 61-46 1-00282 0-99725 X-OX472 0-0x722 2-2x55 2-157 64-99 1-00295 0-997x2 1-0x538 0-0x799 2-2438 2-237 *75-01 1-0027 0-99972 X-OX736 0-0x734 2-1151 2-XX5 79-80 0-99769 1*00235 X-OX826 0-0x563 X-920X X-9O8 87-96 0-99101 X-00925 X-OX987 0-OX04X 1-4424 1-434 94-96 0-98353 X-0X683 X-02XX6 0-00424 X-080X x-067 100-00 0-97832 X-022X7 X-022X7 0-00 0-8775 In fig. 60 the observed values of rj are plotted against the percentage composition in full line and crosses, and the calculated values in dotted line and circles. The agreement is remarkably good. Fig. 61 shows the graph, plotted in the same manner, for the mixture chloroform-ether, where the agreement is again very good. This curve, at the temperature used, has no maximum, nor does the density pass through one. The agreement is less satisfactory for the mixtures ethyl alcohol- water (fig. 62) and acetic acid-water (fig. 63). Regarding the latter, M^Leod points out that the discrepancies between the values determined experimentally by different observers are of nearly the same order as those between the observed and calculated values, M^Leod summarises his results as follows: the principal cause of the great increase in viscosity in the mixtures dis¬ cussed is the contraction which has taken place on mixing, and this contraction is analogous to that produced by cooling. 158 THE VISCOSITY OF LIQUIDS In support of this view he puts forward the following con¬ siderations : the density of a mixture of 90 per cent, of alcohol and 10 per cent, of water at 10° is 0-82654, and ■>7=2-o6 cp. Fig. 60.—Viscosity of mixtures (M^Leod). Pyridine--water at 25°. Absc. volume per cent, of first component. 0*81430, and the actual density is reduced to this figure when the temperature is about 24*1°. At this temperature the vis¬ cosity of the 90 per cent, mixture is about 1*47, while the viscosity calculated from the linear formula would be 1*44 at 10°. This indicates that the viscosity becomes practically normal when the abnormal density has been overcome by heating. M^Leod has applied the procedure just described to some of the mixtures of organic liquids measured by Bramley; THE VISCOSITY OF LIQUID MIXTURES 159 i,e, he has compared the viscosities at the temperatures required to counteract the contraction with the viscosities calculated for the lower temperature from the linear mixture formula. The agreement is moderate only, but tends to Fig, 6i.—Viscosity of mixtures (M^Leod). Ethyl ether-chloroform at o°. Absc. volume per cent, of first çomponent. support the view that the contraction is, at any rate, a very important factor in producing the high viscosity. The high values of the exponents A, which differ consider¬ ably from those for the pure substances (and therefore de¬ tract a good deal from the rational character of M'^Leod's formula) he attributes to association, which enhances the effect of contraction. In spite of the many difficulties, which M'^Leod himself recognises, his method of attacking the problem must be 160 THE VISCOSITY OF LIQUIDS considered the most promising one for the present, and until some theory of the viscosity of pure liquids is available— in the absence of which it is perhaps unreasonable to hope for a solution of the mixture problem. It seems, however. Fig. 62.—Viscosity of mixtures (M®Leod). Ethyl alcohol-water at 10°. Absc. volume per cent, of first component. worth trying whether the method could be improved in one particular: by assuming that the contraction on mixing is equivalent, not to a contraction by cooling, but to a reduc¬ tion of volume by pressure. Unfortunately, most mixtures have been studied at temperatures lower than 30°, for which Bridgman's data are available. The only example on which a rough test is possible is the alcohol-water mixture, the data for the mixture containing 50-94 per cent, of alcohol by volume at 25° being: r] of mixture at 30° =2-02, rj of alcohol THE VISCOSITY OF LIQUID MIXTURES 161 at 30° =1*003, r¡ of water at 30° =0*801, all in centipoises. The actual volume of i grm. of the mixture at 30° is 1*0904 c.c., while the calculated volume is 1*13056, so that the con- Fig. 63.—Viscosity of mixtures (M^Leod). Acetic acid-water at 15°. Absc. volume per cent, of first component. assume the compressibility of the mixture to be the mean of that of the components, it is (from Amagat's data for water and alcohol) about 69x10"® per atmosphere, so that the contraction would correspond to a compression of about 500 atmospheres. Bridgman's figures for the viscosities at this pressure and 30° are: 07 of water =0*896,77 of alcohol =1-5348. The linear mixture formula, with volume ratios, accordingly gives the viscosity of the mixture as =1*235, whereas the actual value, as given above, is 2*02. II 162 THE VISCOSITY OF LIQUIDS The calculation is approximate only, but the discrepancy is so great that it is obviously not possible to treat the con¬ traction caused by mixing as equivalent to compression by external pressure. This method of treatment, of course, also assumes the validity of the linear formula for the compressed mixture, an assumption which, as has been emphasised suffi¬ ciently, is not likely to be correct. It only remains to refer briefly to two types of mixtures: viz. mixtures of two components whose viscosities are very far apart, and solutions of gases in liquids, which--as far as they have been investigated—behave exactly like liquid mixtures. Mixtures of Components of widely Different Viscosities. The most characteristic of the former is the mixture glycerin- water, the latest data for which, by E. Müller,are given in Table XXIV. The values spread too far to be represented in a graph of reasonable scale, but comparison of the figures at 30° shows the extremely rapid drop in viscosity with increasing water content, the first 17 per cent, of which reduces the viscosity to less than 1/14 of the value for 99-19 glycerin. This large effect of small admixtures of water, and the difficulty of determining them, makes glycerin un¬ suitable as a standard liquid for calibrating viscometers for viscous liquids. TABLE XXTV Viscosity of Glycerin-water Mixtures in Centipoises (E. Müller) Tempera¬ ture. Weight per cent, of glycerin. 99-14. 81-98. 61-44.^ 39-3I- 20-29. 0. 17° « • 100-7 3-89 18 1393-0 • • • • 2-03 20 • • 72-6 12*27 « • 1-90 1-029 30 570-8 40-3 8-57 2-84 1-58 0-817 40 267-5 25-5 5-75 2*11 1-19 0-672 50 175-2 ^5*1 4-19 1-69 0-97 0-550 60 124-1 12-2 3*23 1*37 0-86 0-455 70 53*3 8-61 2-56 i-i6 0*72 0-403 80 32-8 5-68 2-04 I-OI 0-64 0-351 90 17-9 5-01 1-69 0-88 0-55 0-317 THE VISCOSITY OF LIQUID MIXTURES 163 The viscosity of solutions of gases in liquids has hardly been studied, but Lewis has investigated solutions of sulphur dioxide in carbon tetrachloride, methyl alcohol, ben¬ zene, acetone and ether. Solutions saturated at atmospheric pressure were examined in a viscometer of the Washburn A Fig, 64.—Lewis's visco¬ meter for solutions of gases in liquids. 0-20 Fig. 64a.—Viscosity of solutions of sulphur dioxide in organic solvents (Lewis). i, Carbon tetrachloride ; 2, methyl alcohol ; 3, benzene ; 4, acetone ; 5, ethyl ether. type; solutions containing up to 100 per cent, of SOg in the modified Ostwald instrument shown in fig. 64. The visco¬ meter was placed in a freezing mixture and filled through A with the solvent and with liquid sulphur dioxide, the con¬ tents being determined by weighing. It was then sealed off at A, placed in the thermostat at 25°, and the small bulb filled by inverting the instrument. The results are plotted in fig. 64A, and show that the dissolved gas behaves like a liquid; the curves are of the familiar types, except perhaps that for methyl alcohol, which shows a slight inflexion. The viscosity 164 THE VISCOSITY OF LIQUIDS of liquid sulphur dioxide at 25° was found to be 0-2550 centi- poise, conforming to the general rule that liquids with high vapour pressures have low viscosities. 1 J. L. M. PoiSEUiLLE, Ann, Chim. Phys. (3), 7, 50 (1843). 2 Thomas Graham, Phil. Trans., A, 167, 373 (1861). 3 J. Kendall and K. P. Monroe, J. Amer. Chem. Soc., 39, 1802 {igij). ^ E. C. Bingham, Amer. Chem. 35,195 {1906). 5 Ch. Lees, Phil. Mag. (6), 1, 128 (1901). ® K. Drucker and Kassel, Zeit, physik. Chem., 16, 367 (1911). ' S. Arrhenius, ihid., 1, 285 (1887). ® J. Wagner, ihid., 46, 867 (1903). ® F. Baker, Trans. Chem. Soc., 101, 1416 (1912). O. Faust, Zeit, physik. Chem., 97 (1912). A. Zawidzki, ihid., 39, 129 (1909). 12 A. F. Dunstan and F. B. Thole, J. Chem. Soc., 95, 1556 (1909)- F. H. Getman, J. Chim. phys., 4, 386 (1906). F. C. Bingham and R. F. Jackson, Scient. Pap. Bur. Stand., No. 298 (1917)- Dunstan and Thole, The Viscosity of Liquids, p. 44 (London, 1917)- 1® G. Senter, Physical Chemistry, p. 306 (1908). 17 F. W. Washburn, Tech. Quarterly, 21, 399 (1908) ; cf. also Trans, Chem. Soc., 95, 1556 (1909). 1® Loc. cit. (11). 10 N. A. Yajnik, M. D. Bhalla, R. C. Talwar and M. A. Soofi, Zeit, physik. Chem., 118, 305 (1925). 20 A. Findlay, ihid., 69, 203 (1909). 21 R. B. Denison, Trans. Faraday Soc., 8, 20 (1912). 22 D. F. Tsakalatos, Bull. Soc. chim. (4), 3, 234 (1908). 23 r. Kremann, Wien. Ber., 113, 878 (1904). 24 N. Kurnakow and S. Shemtchushni, Zeit, physik. Chem^., 83, 481 (1913). 23 N. S. Kurnakow, Zeit, anorg. Chem., 135, 81 (1924). 23 R. Kremann, loc. cit. (23). 27 J. D. Stranathan and J. Strong, J. Phys. Chem., 31, 1420 (1927)- 23 D. B. M^Leod, Trans. Faraday Soc., 19, 17 (1923). 23 Dunstan, Thole and Hunt, J. Chem. Soc., 91, 1728 (1907)- 2 3 Dunstan, Thole and Hunt, ihid. 21 T. F. Thorpe and J. W. Rodger, Trans. Chem. Soc., 11, 360 (1897). 22 A. Bramley, J. Chem. Soc., 109, 434 (1916). 22 F. Müller, Wien. Ber., 133, 133 (1925). 24 J, r. Lewis, J. Amer. Chem. Soc., 47, 626 (1925)- CHAPTER X VISCOSITY AND CONDUCTIVITY The earliest investigations on the viscosity of electrolyte solutions were undertaken by G. Wiedemann ^ with the object of finding a connection between this property and conductivity, and the subject has received a very large amount of attention since then, and more particularly since the theory of elec¬ trolytic dissociation began to develop. The assumption that the velocity of ions moving in a viscous medium must be a function of the viscosity is obvious, but equally so is the difficulty of finding a procedure for varying one, and only one, factor at a time. The viscosity of an electrolyte solution can be varied (a) by var3dng the temperature, (&) by varying the electrolyte concentration, {c) by the addition of a non- electrolyte, and {d) by varying the pressure, (b) must cause a change in conductivity even if there were none in viscosity, but the converse—that the conductivity is not altered directly—cannot a priori be said of any of the three other methods. Change in pressure appears, perhaps, most likely to alter viscosity without directly affecting conductivity much, but owing to experimental difficulties the investigations of this kind are scanty. Effect of non-Electrolytes, The variation in viscosity and conductivity caused by non-electrolytes and that produced by change of temperature are both easy to study, and both have been the subjects of very numerous investigations. As regards the former, it is interesting that at an early stage attempts were made to obtain an extreme viscosity range by using gelatin as the non-electrolyte—a course adopted by Arrhenius,^ and soon after by Lüdeking.^ The complications likely to 165 166 THE VISCOSITY OF LIQUIDS be caused by its anomalous viscous behaviour and its effect on the salt concentration were, of course, unsuspected at the time. Arrhenius used 4-2 per cent, of gelatin in zinc sulphate and sodium chloride solution, which set to a jelly at 24°; the conductivity curve between 15° and 32° showed no dis¬ continuity whatever at the setting-point. The conductivity was only about 20 per cent, lower than that of salt solutions of the same concentration without gelatin, and the tempera¬ ture coefficient of conductivity was practically identical for solutions with and without gelatin. This last result is par¬ ticularly striking, as the temperature coefficient of viscosity of gelatin sols near the setting-point is many times that of water. The investigation just mentioned was made in 1885. An explanation which Arrhenius gave about thirty years later will be discussed below. In 1889 Lüdeking again measured the viscosity and conductivity of zinc sulphate solutions con¬ taining up to 50 per cent, of gelatin. He found, like Arrhenius, no discontinuity in the temperature-conductivity curves at the setting-point, even with the highest gelatin concentra¬ tions. The effect of a given gelatin concentration, however, varied with the salt concentration, or, in other words, the temperature coefficient was no longer independent of the gelatin content, as Arrhenius had found for low gelatin concentrations. A few years earlier E. Wiedemann ^ had measured the viscosity and conductivity of zinc sulphate solutions in mix¬ tures of water and glycerin; as the latter is a dissociating solvent {cf, Getman, p. 128), complications are introduced. He found no proportionality between viscosity and resistance : at the same salt concentration a glycerin-water mixture con¬ taining 90 per cent, of the former had a relative viscosity =68*7 and a relative resistance = i2-i, the viscosity and resistance of the solution in water alone being taken as unity. Arrhenius ^ compared the effect of different non-electro¬ lytes in low concentrations, but found that the conductivities at equal viscosities were not the same. A i per cent, solution of cane sugar had a relative viscosity = 1-046, while 2-2 per VISCOSITY AND CONDUCTIVITY 167 cent, of methyl alcohol was required to give the same vis¬ cosity; the conductivity of a given KCl solution was lowered 3*0 per cent, by the sugar and 3-85 per cent, by the alcohol. Arrhenius concludes from this and similar evidence that the reduction of conductivity cannot be a function of the viscosity only. Heber Green,^ following the example of Martin and Masson,^ used sugar as non-electrolyte in an extensive investigation, with lithium chloride as electrolyte. He found that there was no simple proportionality between conductivity and fluidity; when the latter was varied by varying the sugar concentration, the concentration of LiCl remaining constant, the following relation was found to hold approximately:— The difference between the extreme values of K amounts to about 8 per cent, of the maximum. Some values of n, deduced by Green from his own and several other observers' results, are given below. n. Authors. ^ j- Martin and Masson. 070/ 070 W. H. Green, i-o Massoulier.® ^ ^ I Fawsitt.^ i-o / Heber Green makes some observations on the whole problem which deserve to be quoted: Arrhenius's and Lüdeking's work " (the reference is to solutions containing gelatin (see above)) indicates that in some cases at least the two fluidities (ionic and physical) are not identical; indeed in such cases they may be compared, on the one hand, to the passage of water through the interstices of a fine sponge, and, on the other, to the sponge itself being forced to flow through the tube in such a way that no circumferential slipping takes place. However, we find that several authors are inclined to assume a direct proportionality, and so quote the values Electrolyte. HCl KCl LiCl CUSO4 NaOH KCl Non-electrolyte. Sucrose ) > Glycerol Carbamide } > 168 THE VISCOSITY OF LIQUIDS A/^ or as 'ionisation coefficients/ Thus Bousñeld and Lowry/^ after pointing out that the viscosity of a 50 per cent, sodium hydroxide solution is approximately seventy times that of water, say : ' This increase of viscosity must produce a large efíect on the ionic mobility; the influence of this factor may be to some extent eliminated by dividing the molecular conductivity by the fluidity, and this ratio we have called the intrinsic conductivity " of the solution.' " In discussing his own results, Heber Green adds: " If we regard each ion as an independent particle threading its way between the molecules of which the solution is composed, then mobility, at any instant, will not depend on the fluidity of the solution, but on the possibility of the ion finding a pore-space through which to pass the section of liquid imme¬ diately in front of it, and this, if the dimensions of the ion compared with the pore-spaces are negligible, will be pro¬ portional to the 2/3 power of the total free space per unit volume of solution. If the ionic sizes be larger or the inter¬ stitial spaces smaller, then a higher and varying power will be required to express the relation between them." Although this attempt at a quantitative formulation is somewhat rash, it is hardly possible to disagree with the general view that a system of solvent molecules and solute molecules and ions, to which molecules-^of larger, but still comparable, size—of an indifferent substance are added, cannot be looked upon as a continuous medium of which merely the viscosity has been altered by the addition. In this connection the explanation by Arrhenius of his early experiments on the conductivity of salt solutions, given over thirty years later, is of interest. After pointing out {cf. above) that there is no discontinuity in the conductivity-temperature curve at the setting-point, and that the temperature coefficient in zinc sulphate and sodium chloride solution is practically the same with or without gelatin, he says: " This last circumstance indicates that the colloidal particles of gelatin do not alter their volume in the temperature interval mentioned above (15° to 32°), and that therefore the setting is not a consequence of increased hydration at falling temperature. One may VISCOSITY AND CONDUCTIVITY 169 picture this thus: that the gelatin particles form a kind of lattice with very large interstices, and that at the setting temperature they join to form a rigid system, without the interstices being thereby reduced/' Arrhenius therefore assumes that both in the gelatin sol and the gel the ions move through the interstices between the gelatin particles, the only difference between the two being that in the sol the gelatin particles are in motion, whereas in the gel they are joined together. It is difficult to see why ions in a sugar solution should not travel in some similar manner, though what Green calls the pore-spaces " will be very different in size and number, according as the non- electrolyte is gelatin, with a molecular weight " of the order lo^, or cane sugar with a molecular weight of 342. Attempts to co-ordinate the changes in viscosity and con¬ ductivity caused by variation of temperature have also been numerous since Grotrian found the temperature coefficients of both to be approximately the same. Further investigation has shown that the relation is not so simple, but Johnston deduced a parabolic formula for ionic mobility from conduc¬ tivity measurements by Noyes and his collaborators. This is analogous to Heber Green's formula given above; is the ionic mobility at infinite dilution: where K and n are constants. A number of values of n are given below. Ion. K. Na. Nn4. Ag. JBa. iCa. CI. n 0-887 0-97 0-891 0-949 0-986 1-008 0-88 /X^ at 0°. 40.4 26 40-2 32-9 33 30 41-1 Ion. NO3. Acetate. ISO4. | Oxalate. n 0-807 i-oo8 0-944 0-931 11^ at 0°. 40-4 20-3 42-3 39 The exponent is the greater the smaller the mobility of the ion; or, in other words, the more mobile ions show the greatest deviation from simple proportionality between fluidity and mobility. Both the calcium and the acetate ion have n>i) as it is highly improbable that their speed should increase more 170 THE VISCOSITY OF LIQUIDS rapidly than the fluidity, there must be a small error, unless the dimensions of these ions are assumed to alter. The results may be summarised as follows: for large ions the change in mobility and in fluidity are proportional to each other, but for smaller ones the speed changes relatively less than the fluidity of the solvent, and the change is the smaller the larger the original speed of the ion. Temperature Coefficients of Conductivity and Viscosity. The relation between conductivity and fluidity has recently been treated in a new way by M. Wien,^^ who was induced to take up the problem by some observations made by him and J. Malsch. They found that solutions heated by current of very short duration show an initial temperature coeffi¬ cient of conductivity which, for various electrolytes in the same solvent, were practically equal and agreed approxi¬ mately with the temperature coefficient of the viscosity of the medium. Only when the current was applied for longer periods the value of the temperature coefficient approached that of the permanent temperature coefficient, as observed when the liquid is kept continually at the higher temperature. This behaviour suggests the assumption that the effect of temperature change is a combination of two factors: an effect on the viscosity of the solvents which takes place in an imperceptibly short time, and a second effect, specific for the ion, which takes time to manifest itself. Wien finds that relations become clearest when the tem¬ perature coefficients are considered. He defines the tempera¬ ture coefficient a of fluidity p and the temperature coefficient ß of conductivity A in the region of temperature t as follows:— The temperature coefficient a of the fluidity of water and aqueous solutions can be represented by the equation CL—-T 1 A Pi dt A^ dt (I) I (2) a +bt By introducing this value for a and integrating, we obtain = ^)]'' • • • (3) VISCOSITY AND CONDUCTIVITY 171 where ^ = and q^bja. The equation has the same form as one of Slotte's formulae, and gives good agreement with Hosking's experimental dataT® The curve for ß, the temperature coefficient of conductivity, as a function of temperature, generally runs parallel to the a-t curve, so that in the interval from o° to ioo° the relation ß = a+y holds good. With a single exception y is negative, and for neutral salts approximately —o-2 to —0-3, for bases —0*5 to —0*7, and for acids —o-8 to —1-6. y is a quantity characteristic of the ion. In the majority of solutions y is, within the limits of ex¬ perimental error, independent of the temperature, and varies little, if at all, with concentration. Larger variations occur¬ ring only in solutions of several acids and acid salts are probably caused by changes in the constitution of the ions. By writing • • •' ^4) and integrating, the expression for the conductivity is obtained : . . . (5) in which the factor [ represents the variation of fluidity of the solution with temperature, and the exponential factor depends on the nature of the ion. The negative sign of y indicates that the conductivity of electrolyte solutions does not increase as rapidly with the temperature as the fluidity. {Cf. the exponents found by Johnston above.) In comparing the curves for A and ^ it has to be considered that the integration constant, and therefore the origin of the curves, is arbitrary. One may therefore either assume that A for some reason is greater than would correspond to the fluidity, and that the differ¬ ence decreases with rising temperature; or that A is smaller than corresponds to and that the difference increases with rising temperature. In very dilute solutions, when the temperature coefficient of viscosity can, without sensible error, be taken to be the same as that of water, y can be calculated immediately. 172 THE VISCOSITY OF LIQUIDS For higher concentrations a deviates markedly from the value for water, and must be determined from direct viscosity measurements. Wien remarks on the lack of adequate data, and takes Hosking's observations on viscosity and conductivity as the most suitable example. Table XXV gives a and ß, calculated from the respective measurements, and y as their difference. For the lithium chloride solution the constancy of y is satisfactory; for the sodium chloride solution it is so at the higher temperatures, while at the lower ones y varies irregularly. TABLE XXV Factors a, ß, and y in Wien's Equations NaCi, 4N. LiCl, 2-g7N. t. a per cent. ß per cent y- t. a per cent. ß per cent. y- 0 10 20 30 40 50 60 70 80 go 2-72 2-32 2-02 1-79 i-6i 1-46 1-34 1-24 1-14 2-36 2-o6 1-87 i-6i 1*41 1-23 1-07 o-gg 0-87 Mean 0-36 0-26 0-15 o-i8 0-20 0-23 0-27 0-25 0-2 7 0-25 0 10 20 30 40 50 60 70 80 go 2*76 2-34 2-04 I-80 1-62 1-47 1-34 I -24 I-I5 2*54 2-13 1-85 1-62 1*43 1-27 I-I3 1-03 0-93 Mean 0-22 0-2I o-ig o-i8 o-ig 0-20 0-2I 0-2I 0-22 0-20 At low concentrations—up to N/io for one or two electro¬ lytes—the constancy of y is so marked that Wien feels no doubt of the relation a-\-ß = y representing a real property of electrolyte solutions. The physical meaning of y, an effect which takes time to develop, is not clear, and Wien points out that possible explanations involve difficulties. Both dissociation and hydration may conceivably take time to attain their full values, but to account for the observed facts dissociation would have to decrease and hydration to increase with rising temperature, neither of which assumptions is probable. VISCOSITY AND CONDUCTIVITY 173 An empirical formula identical with equation (5) was used by Bousñeld and Lowry/^ who multiplied Slotte's equation by a dissociation factor Kraus's Summary. The most complicated case is obviously (&), change of viscosity caused by change in concentration of the electrolyte itself. The problem, as Kraus points out in a summary, the substance of which follows here, arises as soon as the concentration approaches normal, and frequently at lower concentrations. It cannot be assumed a priori that the viscosity of the solution is the only factor which influences the velocity of the ions. The fundamental difficulty in attacking the problem is that the law connecting concentration and conductivity in aqueous solutions of strong electrolytes is not known for concentrations at which the viscosity of the solution is appreciably higher than that of the pure solvent {cf. p. 180). Recourse must therefore be had to empirical relations, one of which, found by Kraus and Bray,^^ holds for a large number of electrolytes: (ca)Vr(l-a)-D(ca)^+K . . (6) where D, n, and K are constants, c the concentration, and a the degree of ionisation, a=A/Ao, A being the conductivity at concentration c, and Aq the equivalent conductivity at infinite dilution. Kraus considers that the wide range over which this formula applies justifies its use as an extrapolation formula to con¬ centrations at which the viscosity change becomes appreciable, when the influence of this factor can be determined by com¬ paring the observed values with those calculated from the formula. The use of an empirical equation in this manner is open to considerable objections, as has been pointed out by Rabinovich.^^ Kraus, however, considers that the validity of the procedure can be checked by applying it to a system with negative viscosity,'' such as potassium iodide in water. As has been shown above, the viscosity of such a solution passes through a minimum, and eventually becomes equal to that of water; the point at which this happens thus becomes a point of reference at which the observed value should agree 174 THE VISCOSITY OF LIQUIDS with that calculated from the formula, without any viscosity correction. In fig. 65 the observed conductivities are marked by crosses and plotted as ordinates against the logarithms of the con¬ centrations as abscissae. The points of curve A are cal- culated from Kraus and Bray's equation, using the value 81-12 for Ao, and represent the conductivities provided there is no change in the visco¬ sity with concentra¬ tion. Kraus assumes that, since the visco¬ sity is lowered by potassium iodide, the ions will not encoun¬ ter molecules or ag¬ gregates greater than those present in water, and that the conductivities can be corrected by simply multiplying by the relative viscosity If these cor¬ rected conductivities are introduced into Fig. 65.—Conductivity of aqueous solutions of potassium iodide (Kraus). A, Calcu¬ lated from Kraus and Bray's equation ; B, corrected for fluidity ; C, fluidity. Ord. right, cp ; left, Aq. Absc. log con¬ centration. equation (6), the curve B is obtained, which fits the experimental points fairly closely. Finally, C is the fluidity curve of the solution. If the logarithms of equation (6) are taken, we find, intro¬ ducing the value of a, n log (cA) =log [c(A - Ao)] +log DAo" ^ . (7) i.e. a linear relation between the logarithms. If the con¬ ductivities corrected for viscosity are introduced into this VISCOSITY AND CONDUCTIVITY 175 equation, the log-log curve becomes a straight line (fig. 66) ; uncorrected values give the points marked by crosses. The maximum difference amounts to about 30 per cent. Kraus considers that these results prove the applicability of equation (6) to solutions of potassium iodide in water up to 5*66 N, and show that the speed of the ions over this range is a linear function of the fluidity. The lowering of viscosity produced by this salt at 0° is the largest for which both con¬ ductivity and viscosity data were available when he wrote (since when Rabinovich has found that caesium chloride in high concentrations pro- ^ duces still lower viscosi¬ ties), but the same me- i thod of calculation was applied to solutions of ammonium chloride and -, potassium chloride at 18° With good agreement. ' Kraus further exam¬ ines the solution of lith¬ ium chloride—which in¬ creases the viscosity at all concentrations—using Washburn and Maclnnes's data. Here the log-log curves defined above are not straight when either the uncorrected values for conductivity or the values corrected by multiplica¬ tion by 7]/7]o are introduced into the equation; a straight line is, however, obtained when the observed values are corrected on the assumption that the speed of the lithium ion varies directly as the fluidity, while the speed of the chlorine ion remains constant. This method of correction, however, holds only up to normal concentration, beyond which further complications arise. The grounds on which this discrimination between the two ions is based will be more conveniently stated when a summary of Kraus's views on the effects of pressure has been given. Fig. 66.—Log- (corrected conductivity) log concentration graph of potassium iodide solutions (Kraus). 176 THE VISCOSITY OF LIQUIDS i:«00 uoo ' 0*9 The only data available for conductivity of salt solutions at high pressures were those of Tammann for N/io sodium chloride at pressures up to 4000 kg./cm.^ at 0° and 20°; the viscosity of water has since been determined by Bridgman {loc. cit., p. 82) at 0°, 10*3°, and 30°. In fig. 67 the relative resistances R/Rq (full lines) at the two temperatures, and the relative viscosities of water (dotted lines) at 0° and 10-3°, are plotted as ordinates against the pressures as abscissae. The viscosity of water, as has already been de¬ scribed, at low tem¬ peratures shows a marked minimum at about 1000 kg./cm.^, which fiattens with rising temperature and disappears above 30°. The y]-p curves thus bear an obvious resemblance to the 7¡-c curves of solutions of salts which lower the viscosity of water, and Tammann takes the view that there is parallelism between the efíect of external pressure and the effect of solutes on the internal pressure. The same kind of change in the molecular constitution of the liquid takes place in both cases, and there is general agreement that the change produced by pressure consists in the breaking up of molecular aggregates. There is an evident close connection between the R/Rq-/) curves and the r¡-p curves, but Kraus makes no attempt to formulate it quantitatively, and further data on the viscosity change with pressure in salt solutions would be required before this could be done. 1000 2000 3.000 4000 Fig. 67.—Relative resistance (full) of sodium chloride solutions and relative viscosity (dotted) of water. Ord. left, R/Rq ; right, i¡jr]Q. Absc. kg./cm.^ (Kraus after Tam¬ mann) . VISCOSITY AND CONDUCTIVITY 177 If the viscosity at atmospheric pressure is varied by varying the concentration of NaCl, the conductivity change is much smaller than the viscosity change, while the change in con¬ ductivity produced by pressure is approximately proportional to the fluidity change. This shows that the effect of viscosity change on the mobility of the same ions differs according to the means by which the viscosity change is brought about. Kraus next discusses solutions of salts which raise the viscosity of water. They also must be assumed to cause the breaking up of molecular aggregates, but the resulting decrease in viscosity is overbalanced by an increase due to large ions and non-ionised molecules''; the sum of the effects being that the r¡-c curve rises more slowly at low concentrations than at higher ones. These considerations explain why the conductivity of solutions with negative " viscosity can be corrected by multiplying by rj/rjQ, while this correction is not applicable in solutions with increased viscosity. Kraus then considers the effect of difference in the size of the ions, and concludes that, whèn this is sufficiently great, the speed of the smaller ion will in the limit " not be affected by the change in fluidity, while the speed of the larger ion will vary in proportion to it. The transport number must then be affected, and Kraus finds ample experimental evidence that there is such a change, and that it is in the right direction, i.e. corresponds to a smaller relative speed of the more slowly moving ion. Thus with increasing concentration the transport number of the lithium ion in lithium chloride decreases, while in hydrochloric acid that of the chlorine ion decreases. These conditions account for the possibility, mentioned above, of correcting the conductivities within limits by applying a correction for the lithium ion only. The conclusion to be drawn from Kraus's survey is that at present there is no method of correcting conductivities for the fluidity change, except in solutions with negative viscosity, where introduction of the values, corrected in direct ratio to the fluidity change, into an empirical equation leads to good agreement with the experimental data. In a recent paper Rabinovich ^3 expresses doubts regarding the validity of con¬ clusions drawn from the application of an empirical formula, 12 178 THE VISCOSITY OF LIQUIDS and describes an investigation of highly concentrated salt solutions undertaken with the object of testing whether the ordinary correction leads to useful results. The salts used include caesium chloride, which lowers, and halogen salts of cadmium, which raise the viscosity of water. The general result is that if the ordinary correction is applied, i.e. if the observed conductivities are multiplied by r¡lr]Q, the con¬ ductivity-concentration curves become anomalous ; with increasing concentration the decrease in molecular con¬ ductivity becomes slower, reaches a minimum, and then grows again. The simple assumption that the speed of the ions is directly proportional to the fluidity therefore leads to an over-correction the assumption leaves out of account the decrease of ionic diameter caused by decreased hydration. A further discussion on the question whether over-correction alone is sufficient for the anomaly, or whether other causes contribute, is outside the object of this chapter. Electrolytes in Organic Solvents. The anomalous behaviour of water plays a considerable part in the discussions which have been outlined above, and it is reasonable to expect that the relation between viscosity and conductivity may be simpler in other solvents. Waiden has investigated such solutions of tetraethyl ammonium iodide, a strong binary electrolyte with large cation, in a great number of organic liquids. He finds that the conductivity at infinite dilution is directly proportional to the fluidity (varied by varying the temperature), so that (-^oo A<„ ) 25 = (•)?„ ) 0=const. For the electrolyte mentioned, N(C2n5)4l, the constant varies between 0-634 and 0-777, with a mean value of 0-700. Dutoit and Duperthuis tested this relation, using sodium iodide as electrolyte, and several alcohols, acetone and pyridine as solvents. They found that the product varied with temperature, the variation amounting to +47 per cent, in iso-amyl alcohol, and +25 per cent, in iso-butyl alcohol, when the temperature was raised from 0° to 80°. They also found different values for the product in different solvents, which did not tend to a single value at high tempera- VISCOSITY AND CONDUCTIVITY 179 ture, the extreme values at o° being in the ratio lOO : 154, and at 80° in the ratio 100 : 215. They conclude that Walden's rule is valid, and that approximately, for a single instance only. Waiden, in reply, criticises the use of sodium iodide as electrolyte, since there is evidence of decomposition, and points out that Dutoit and Duperthuis determined the limiting con¬ ductivity by measurements at extremely low concentrations, while he obtained it by Kohlrausch's extrapolation formula, the validity of which for solvents other than water he considers proved by the work of Philip and Courtman.^"^ The discrep¬ ancies largely disappear when Walden's values for the con¬ ductivities are used. Values for the conductivity of potassium iodide in various organic solvents, taken from the literature, also give a constant product. Finally, Waiden investigates tetramethyl ammonium iodide in ten organic solvents, and again finds his rule confirmed satisfactorily. In a more recent paper Waiden returns to the subject, and suggests that deviations from the rule may be due to changes in the ionic diameter, i,e. the dimensions of the solvent envelope, caused by the changes of temperature. He therefore examines electrolytes consisting on the one hand of simple ions, and on the other of highly complex cations or anions, and finally a salt the anion and cation of which are both complex. He finds that his rule holds excellently for tetra-amyl ammonium iodide, N(C5nii)4l, the cation of which contains 65 atoms, and for triamyl ammonium picrate, Cn2(N02)30H.N(C5nii)3, the anion of which consists of 18 and the cation of 50 atoms. For the picrate the rule even holds in aqueous solution. Waiden concludes generally that the rule holds the better the larger the ions in comparison with the molecules of the liquid; deviations therefore occur principally in highly associated liquids. At higher concentrations no simple relation between con¬ ductivity and viscosity in organic solvents has yet been found. It is hardly necessary to do more than quote the conclusion reached by Robertson and Aeree as the result of a very comprehensive investigation of various electrolytes in ethyl 180 THE VISCOSITY OF LIQUIDS alcohol. The following remarks refer to sodium iodide: No method is known at present for . . , correcting the conductivi¬ ties for viscosity, with certainty. . . . The relations of the viscosities and the conductivities of N/i, N/2 and N/4 solutions of sodium iodide ... at 25° and 35°, for example, are such that if we were to apply Noyes' method " {i.e, multiplying by 7^/070) we should arrive at the conclusion that the per cent, of ionisation is practically the same for the N/i, N/2 and N/4 solutions of each salt. Such a conclusion is, however, not in harmony with all of the known facts of physical chemistry, and certainly is not borne out by our other experimental results." King and Partington in a very recent paper reach the same conclusion, that correction by multiplying by the viscosity ratio leads to impossible results for solutions of sodium thiocyanate in ethyl alcohol. Strong Electrolytes, The recent developments in the theory of strong electrolytes, as described authoritatively in the general discussion organised by the Faraday Society {Trans,, vol. xxiii, 1927), do not appear yet to have led to any relation between the mobility of an ion in a solution of appreciable concentration (free from non-electrolytes) and the actual macroscopic viscosity of the solution " (Allmand, loo, cit,, p. 349). The reader interested in the subject is referred to this report in general, and in particular to the contributions by Onsager, p. 356 ; Remy, p. 384 ; Ulich, pp. 388 and 415 ; Porter, p. 413 ; etc. i" ^ G. Wiedemann, Pogg. Ann., 99, 221 (1856). 2 S. Arrhenius, Öfvers. d. Stockh. Akad., No. 6, p. 121 (1885). ^ Ch. Lüdeking, Wied. Ann., 37, 172 (1889). ^ E. Wiedemann, ihid., 20, 537 (1883). ^ S. Arrhenius, Zeit, physik. Chem., 9, 487 (1892). ® W. H. Green, J. Chem. Soc., 93, 2023 (1908). ^ C. J. Martin and O. Masson, ihid., 79, 707 (1901). ^ Massoulier, C.P., 130, 773 (1900). ® E. E. Fawsitt, Pyoc. Roy. Soc. Edin., 25, 51 (1903). W. R. Bousfield and T. M. Lowry, Phil. Trans., A, 204, 253 (1905). S. Arrhenius, Meddel. K. Vetenskapakad. Nobelinst., 3, No. 13 (1916). 12 O. Grotrian, Pogg. Ann., 157, 130 (1875) ; 160, 238 (1877). J. Johnston, J. Amer. Chem. Soc., 31, loio (1909). M. Wien, Ann. d. Phys., 77, 560 (1925). VISCOSITY AND CONDUCTIVITY 181 J. Malsch and M. Wien, Physik. Zeitsch., 25, 559 (1924). R. Hosking, Phil. Mag., 49, 274 (1900). Loc. cit. (10). Chas. A. Kraus, J. Amer. Cham. Soc., 36, 35 (1914). Kraus and Bray, Trans. Amer. Electrochem. Sac., 26, 143 (1912). 20 A. J. Rabinovich, Zeit, physik. Chem., 99, 338 (1921). 21 W. E. Washburn and D. A. MacInnes, J. Amer. Chem. Sac., 33, 1686 (1911). 22 G. Tammann, Wied. Ann., 69, 770 (1899). 2® Loc. cit. (20). 2^ P. Walden, Zeit, physik. Chem., 55, 246 {1906). 25 P. Dutoit and H. Duperthuis, J. Chim. phys., 6, 726 (1908). 26 P. Walden, Zeit, physik. Chem., 78, 269 (1911). 27 J. C. Philip and H. R. Courtman, J. Chem. Soc., 97, 1261 (1910). 2® P. Walden, Zeit, anorg. Chem., 113, 85 (1920). 2® H. C. Robertson and S. F. Agree, J. Phys. Chem., 19, 413 (1915). 3® F. F. King and J. R. Partington, Trans. Faraday Soc., 23, 531 (1927). CHAPTER XI THE VISCOSITY OF PITCH-LIKE SUBSTANCES A NUMBER of bodies possess what Röntgen/ one of their first investigators, called the interesting property of behaving like brittle solids when subject to large stresses of short duration, and flowing like very viscous liquids when exposed to small, continuous stresses. Among the best investigated substances of this kind are marine glue (a mixture of rubber and shellac, with probably other ingredients), mixtures of turpentine and rosin, and pitch. Barus ^ determined the viscosity coefficient of marine glue by measuring the flow through a capillary, one end of which was luted into an evacuated flask. The experiment extended over about seven months, and led to a value of 7^ =2x10^ poises. In a later series of experiments he studied the effect of pressures up to 2000 atmospheres on the viscosity—which increased with pressure—and on its temperature coefficient. At atmospheric pressure the decrease in viscosity caused by raising the temperature 1° can be counteracted by an addi¬ tional pressure of about 67 atmospheres; at 500 atmospheres pressure an additional pressure of about 256 atmospheres is required to counteract the same temperature effect. These results are in agreement with the temperature-pressure re¬ lations found by Bridgman {cf. p. 88) for all liquids (except water). Röntgen {loe. cit.) also studied the effect of pressure on the viscosity of marine glue by a method which gives qualitative results only, and does not permit calculation of the viscosity coefficient. The velocity of penetration of loaded brass rods into the substance was determined, and Röntgen found, in 182 VISCOSITY OP PITCH-LIKE SUBSTANCES 183 agreement with Barus, that it decreased considerably at a pressure of 500 atmospheres. Reiger ^ studied mixtures of rosin and turpentine, the viscosity of which can be varied within fairly wide limits by varying the ratio of the ingredients. The measurements were carried out by the transpiration method, tubes of 0-3 to I cm. radius and up to 14 cm. long being used. The pressure was generated by a column of mercury kept con¬ stant by a simple arrangement. The volumes discharged were, other things being equal, proportional to the pressure within fairly wide limits. A mixture which spread spontane¬ ously in a few days on a plate at 10° showed the following coefficients:— At 18-5° 7] ==3-01 X10^ At 8-6° Tj ==67-2 X10®. Another mixture, which did not spread appreciably in one month at 10°, was forced through the capillary under a pressure of I'43 X 10® dynes/cm.2 (about 1-5 atmospheres), and showed a coefficient 7; =1-32X10^. An interesting procedure was adopted to test further whether the flow was in accordance with Poiseuille's Law. The capillary was partly filled with the mixture, which, on cooling, left an approximately flat surface ; after pressure had been applied for some time the surface had bulged out. A cast of it was taken, cut in half, the profile projected on a screen, and the ordinates and abscissae measured by tele¬ scope. The profile was found to approximate closely to a parabola, which shows that the flow was in conformity with Poiseuille's Law. R. Ladenburg ^ examined the same mixture both by the capillary and the falling-sphere method, and obtained results in good agreement when the corrections described on p. 35 had been applied to the measurements made by the second method. The most complete investigation, as far as temperature range is concerned, and also the most recent, is one by Pochettino ^ on pitch. It is interesting for the further reason that three different methods were used for three temperature 184 THE VISCOSITY OF LIQUIDS intervals, the results of which fall accurately on one smooth curve. The first method could be used for the interval from g° to 50°. The pitch was contained between two vertical coaxial cylinders of radius Rj and Rg, the length of the tube of pitch being L (fig. 68). The inner cylinder was loaded to the total ¡ weight P, and the velocity v-^ with which it descended was measured. It is easy to see that the difierential equa¬ tion for the arrangement is P dv 27rr dr Integration with the obvious boundary conditions, for f^^Rg, v=0, and for f = Ri, v=Vi, gives: _ P . R2 27TLVi For the interval from 34° to 80°, which partially overlaps the first, the falling-sphere method was employed. The pitch was enclosed in a thin alu¬ minium tube, and a lead sphere falling through it was observed from time to Fig. 68.—Determination , - -1 r -xr n of viscosity by axial dis- time by means of X-rays. Finally, placement of concentric abovc 8o°, the capillary-tubc method cylinders (Pochettino). -, was used. The viscosity coeificients found at temperatures between and 99*9° are given in Table XXVI, and the common logarithms of the viscosity are plotted as ordinates against the temperatures as abscissae in fig. 69. The fact, pointed out already, that the values fall on one curve is specially interesting, as all three methods are based on the assumption that the flow is viscous, and that 17 is a constant independent of the velocity gradient, so that any deviation from this behaviour would be expected to show itself in view of the very different velocity gradients in the three methods used. R A'? 5 • • •• • •*»« • • • t * fr ' • • ■ • ••» V/. VISCOSITY OF PITCH-LIKE SUBSTANCES 185 TABLE XXVI Viscosity of Pitch (Pochettino) Temp. rj (poises). log r¡. Temp. t] (poises). log t]. 0 9-0 2*35 X 10^® 10-371 0 45*1 1-64 X 10® 5-215 13-3 5-02 X 10^ 9-701 50-1 4-91 X 10^ 4-691 15-1 2-57 X 10® 9-409 59-0 1-26 X 10^ 4-100 17-9 6-33 X 10® 8-801 65-1 5-77 X 10® 3-761 19-0 4-52 X 10® 8-655 70-2 3-10 X 10® 3-491 20'I 3-30 X 10® 8-518 75*4 1-65 X 10® 3-217 25-2 3-49 X 10' 7-543 80-3 9-23 X 10® 2-965 29-8 7-30 X 10® 6-863 86-2 4-89 X io2 2-689 34-8 1-75 X10® 6-243 92-8 2*30 X 10® 2-362 39-4 6-17 X 10® 5-790 99-9 1-19 X 10^ 2-075 Although the ratio of the viscosities at 9° and 92*8° is 10®, there is no discontinuity in the log r]4 curve. Pochettino 186 THE VISCOSITY OF LIQUIDS points out that the portions between 9° and 32°, and between 75° and 99*9°, are very nearly straight, and refers to an earlier investigation by Heydwiller,^ who found that this relation, log 7;const., held good for menthol in both the solid and liquid states. Unlike pitch, menthol is a definite sub¬ stance with a well-defined freezing-point (41-4°), but super¬ cooling down to 34*9° was found possible. There is an enormous discontinuity between the viscosity in the solid and liquid states in the temperature range where both are possible. Heydwiller's results, which make no claim to great accuracy, are: Solid menthol. Liquid menthol. r, at 34° =6-15 X 10" 4 at 34-9° =0-2505 ^ . 7] at 37*2° =2-41 X 10^® 7] at 37-8°=o-2036/-^ Pochettino suggests that the region between 75° and 99-9° may correspond to the liquid state,'' but raises the question what the state of the substance can be assumed to be in the interval between 9° and 32°. Even assuming that the log 7]-t relation is general, which one set of experiments is hardly sufficient to prove, the comparison is obviously impossible in the absence of any discontinuity. A point of much greater interest is the absence of breaks in the curve at the points where the method of measurement changes, which—as has been pointed out—seems to prove that the viscosity coefficient is independent of the velocity gradient. Pitch is probably a disperse system, and it is reasonable to expect such anomalies of viscosity as are common in disperse systems, more especially a variation in the apparent viscosity with velocity gradient. That the behaviour of pitch differs from that of an ideally or purely viscous body is proved by some earlier experiments by Trouton and Andrews.^ These authors measured the vis¬ cosity of pitch by clamping cylinders of the material at both ends, one of which was kept fixed, while a constant torque T was applied to the other. The relative angular velocity œ with which two cross-sections (assumed to remain plane) unit distance apart are displaced against each other, as the cylinder is twisted, is a measure of the viscosity, VISCOSITY OF PITCH-LIKE SUBSTANCES 187 2T where R is the radius of the cylinder and r¡ is assumed to be independent of the velocity gradient. The formula was tested Fig. 70 —Effect of constant torque on pitch cylinder (Trouton and Andrews). Absc. time of application in minutes. Ord. twist in degrees. by using cylinders of different radius, when œ was found to be inversely proportional to the fourth power of the radius. Fig. 71.—Relaxation curve of twisted pitch cylinder (Trouton and Andrews). Absc. time in minutes. Ord. remaining torque. Elastic Effects, Tests at constant temperature reveal two departures from simple viscous deformation with constant rj, as shown by the two graphs, figs. 70 and 71. The first 188 THE VISCOSITY OF LIQUIDS of these shows the increase in twist (ordinates) with time (abscissae), when a constant torque is applied to the specimen: the twist during the initial period increases more rapidly than the time, a steady state being eventually reached. At the point marked with an arrow the total torque was removed, whereupon the cylinder did not remain stationary but turned backwards, i.e. energy in the form of elastic strain was stored in it. The rate of dissipation of this energy, or the relaxa¬ tion, can be measured by reducing the torque at intervals, so that no un¬ twisting takes place, and plotting remaining torque against time, which graph is shown in fig. 71. The authors point out that the relaxation curve is not, as might have been expected, logarithmic, but that the lower portion can be well represented by a rectangular hyperbola. Fig. 72 finally shows the relation between applied torque (abscissae) and velocity of twisting (ordinates) in degrees per minute. The velocity at first increases more slowly than the torque, but above a certain torque becomes proportional to it. The authors suggest as the cause of the anomalous behaviour that, during the first stage of deformation (while it increases more rapidly than the time of application of a constant torque), a store of elastic energy is gradually accumulated, which is preserved intact during the state of steady rotation, and is given out on removal of the stress to produce the return flow.'' The interest of this investigation, when correlated with present knowledge, lies in the fact that a large number of colloidal solutions have been proved to behave in a manner which is closely parallel to that just described. The anomaly is not bound up with a particular range of viscosity values; while Trouton found for pitch at 0° 07 ==5-1 X lo^^ poises, the apparent viscosity of many colloidal solutions exhibiting ing (ord.) of pitch cylinder as function of applied torque (absc.) (Trouton and Andrews). VISCOSITY OF PITCH-LIKE SUBSTANCES 189 anomalies is of the order of centipoises when determined by ordinary methods. 1 W. C. Röntgen, Wied. Ann., 45, 98 (1892). 2 C. Bakus, Phil. Mag. (5),,29, 337 (1890) ; Sillim J. (3), 45, 87 (1893). ^ R. Reiger, Drude's Ann., 19, 985 (1906). 4 R. Ladenburg, Ann. d. Phys., 22, 287 (1907). ^ A. PocHETTiNO, Nuovo Cimento, 8, 77 (1914). ® A. Heydwiller, Wied. Ann., 63, 56 (1897). F. T. Trouton and E. S. Andrews, Proc. Phys. Sac. Land., 19, 47 (1909) CHAPTER XII THE VISCOSITY OF COLLOIDAL SOLUTIONS Experience accumulated at an increasing rate during the last twenty years goes to show that the simple relations deduced in Chapter II for the flow through a capillary, and for the moment in the concentric cylinder apparatus, do not hold for a large number of colloidal solutions or even for coarser disperse systems, such as suspensions of microscopic particles. Other things being equal, the volume discharged in unit time through a capillary is not simply proportional to the pressure, but increases more rapidly than the pressure. Similarly, the moment in the concentric cylinder apparatus is not simply proportional to the angular velocity of the outer cylinder, but increases less rapidly than this velocity. This effect would be produced if the viscosity coefficient were not, as in normal liquids, a constant, but were to decrease with increasing pressure in the capillary, or with increasing angular velocity in the concentric cylinder apparatus. The variable obviously common to both methods of measurement is the velocity gradient, which increases with increasing pressure in the capillary and with increasing angular velocity in the cylinder apparatus ; so that the anomaly of disperse systems may be stated, af least formally, by saying that the viscosity coefficient is a function of the velocity gradient, of which so far nothing is known beyond the experimental fact that the coefficient always decreases with increasing velocity gradient. Whether a property of this kind is a viscosity, or what else it should be called, are questions which will be fully discussed later on; in the first instance it will be necessary to inquire what a large mass of experimental 190 VISCOSITY OF COLLOIDAL SOLUTIONS 191 material, determined and calculated by the methods applic¬ able to normal liquids, really means. Measurements have been made by determining the quan¬ tities Qi, Qg, . . . Qw discharged from a given capillary at the pressures P^, Pg, . . . P^, and from these data the viscosity coefficients r¡^, . . . have been calculated by Poiseuille's formula. Or the deflections 9^, ög, . . . 9^ at the angular velocities COi, CÜg, . . . CÚ^ have been measured, and a number of viscosity coeiflcients calculated from the formula rj = 9/Kco, Now the equations for the two forms of apparatus have been deduced on the assumption that is a constant. If, however, it is a function of dvjdr or rdœjdr, it is obvious that the differential equations must be different from those which hold for normal liquids. The method outlined above there¬ fore amounts to calculating a number of fictitious viscosity coefficients ; in other words, the rj of normal liquids which would produce the quantities Q (or the deflections 9) at the pressures P (or the angular velocities œ) employed, in an apparatus of the particular dimensions used. The large experimental material therefore has a qualitative significance only, a limitation to be remembered in considering it. It is also necessary to add that dilute sols of the suspensoid or lyophobe type, and emulsoid or lyophilic sols at temperatures of 40° and over, show no anomalies exceeding the probable experimental error. ^ Viscosity measurements on colloidal solutions were first carried out by Thomas Graham.^ When investigating silicic acid sols he discovered one of the fundamental properties of gelating sols, the continuous increase in viscosity with age. The ultimate pectisation of silicic acid is preceded by a gradual thickening in the liquid itself. The flow of liquid colloids through a capillary tube is always slow compared with the flow of crystalloid solutions, so that a liquid-transpira¬ tion tube may be employed as a colloidoscope. With a colloidal liquid alterable in viscosity, such as silicic acid, the increased resistance to passage through the colloidoscope is obvious from day to day. Just before gelatinising, silicic acid flows like an oil.'' Viscosity measurements—generally by means of the Ostwald 192 THE VISCOSITY OF LIQUIDS viscometer—subsequently became one of the most widely used methods of investigating colloidal solutions, and much of the material thus accumulated is difficult to interpret. In many of the investigations dealing, e.g., with the changes in viscosity caused by the addition of electrolytes, the results have had to be explained by ancillary or ad hoc hypotheses, which often do not admit of independent verification; the vast amount of work on protein sols is a very striking example of this kind. Many researches have been undertaken for technical ends, in the hope of connecting—though only empirically— an easily measured property like viscosity with technically important properties less easy to measure or even to define, like the nerve of india rubber, the strength of flour, or the adhesive properties of glue. Nevertheless, a few generalisations are possible which hold good independently of the particular chemical nature of the disperse phase. General Characteristics. The viscosity of stable suspensoid or lyophobic sols is a function of their concentration only and independent of their age or previous history. In many of these sols, however, the concentration is so low that the difference in viscosity between the sol and the dispersion medium is of the same order as the experimental error. Small changes in viscosity with age, such as have been noted in a few instances,^ may fairly confidently be ascribed to incipient coagulation or changes in the dispersion medium itself. In sols which admit of higher concentrations, but are of more doubtfully suspensoid character, such as glycogen ^ or sulphur ^ sols, the viscosity increases with concentration at a rate comparable with that of true solutions of non-electro¬ lytes, e.g. of cane sugar. The temperature coefficient of viscosity in suspensoid sols is—again within the limits of experimental error—merely that of the dispersion medium. A convenient way of examining the variation of the temperature coefficient is the following: if 7]i. and 7^/ are respectively the viscosities of the dispersion medium and of the sol at the temperature t, then 'r]t ht plotted against t will lie on a straight line parallel to the axis of t if the temperature coefficients of sol and dispersion medium are the same, Fig. 73 shows this graph for one VISCOSITY OF COLLOIDAL SOLUTIONS 193 of Oden's ® sulphur sols with a concentration of 7-68 per cent. The viscosity of the sol decreases somewhat less rapidly than that of water, so that in¬ creases with rising temperature. Both the sense and the magni¬ tude of the variation differ strikingly from that observed with emulsoid sols. m In the latter the temperature coefficient is always markedly greater than that of the disper- »o 20 30 40° sion medium. Fig. 74 shows Fig. 73.—Relative viscosity-tern- the values of for a 9-39 ™ per cent, casein (more correctly sodium caseinogenate) sol (Chick and Martin '). The shape Fig. 74.—Relative viscosity-temperature curve of casein sol (Chick and Martin). much more rapidly with rising temperature than that of water. The casein sol does not gelate at low temperature; the sols which do so have still higher temperature coefficients, especially in the vicinity of the gelation temperature. Another and even more marked characteristic of the 13 194 THE VISCOSITY OF LIQUIDS lyophilic sols is the rapid increase of viscosity with concentra¬ tion even at ordinary temperature, and in sols which do not of sol. yy,—Relative viscosity-concentration curves of various protein sols (Chick). A, sodium caseinogenate ; B, euglobulin (with NaCl) ; C, Pseudoglobulin ; D, serum albumin. Absc. per cent, by weight. set to gels. The high viscosity produced by small concentra¬ tions is particularly striking in sols with organic solvents. 20 195 where the viscosity at i per cent, may be sixty (rubber in benzene) to one thousand (blasting soluble nitrocellulose in acetone) times that of the solvent. Effect of Concentration in Various Types of Sols. The graphs, figs. 75 to 78, illustrate the variation of relative vis¬ cosity (that of the dispersion medium being taken as unity) with concentration in two lyophobe sols, sulphur and glycogen, and a number of Fig. 76.—Relative viscosity-concen- ^ - . tration curve of glycogen sol lyophilic Sols. The relative (Bottazzi and d'Errico). Absc. viscosities have all been calcu- grm. m 100 c.c. of sol. lated by Poiseuille's formula, and, since especially the lyophilic sols all exhibit the anomaly described at the be¬ ginning of this chapter, the data are subject to the quali¬ fications made there. Graham's observation that the viscosity of silicic acid sols increased steadily until gela¬ tion occurred has already been mentioned. This increase of viscosity is a characteristic pro¬ perty of all sols which set to jellies, but shows itself even when the concentrations are too low, or, in some instances. Fig. 78.—Viscosity - concentration curve of cellulose nitrate sol in acetone (Baker). Absc. grm. in 100 c.c. of sol. Ord. centipoises. 196 THE VISCOSITY OF LIQUIDS the temperatures too high, for the transformation to become complete. Fig. 79 shows relative viscosity of gelatin sols plotted against time,® while fig. 80 gives the same data for sols of cellulose acetate in benzyl alcohol.^ The Concentration Function. The first attempt to treat mathematically the viscosity of a disperse system is due to Einstein, who considered a suspension of rigid spheres in a viscous liquid. On the assumption that the aggregate volume of spheres was small compared with that of the liquid, and that the spheres were sufficiently far apart not to influence one another, he arrived at the following equation :— ij, = 7?(I+2-5.^) . . . (l) where is the viscosity of the suspension, 7} that of the dispersion medium, and ^ the aggregate volume of spheres in unit volume of suspension. In 1920 Einstein, in a short communication to the Kolloid-Zeitschrift, emphasised that this result was deduced strictly from the fundamental equations of hydrodynamics. The most striking feature of Einstein's formula is that the radius of the spheres does not appear in it, but only their aggregate volume, ix. the concentration; the viscosity grows in linear ratio with this concentration, provided the system conforms to the assumptions made in deducing the formula. Before discussing the various attempts to verify the equa¬ tion, it may be advisable to state here that, ten years after its publication, it was shown by Humphrey and Hatschek VISCOSITY OF COLLOIDAL SOLUTIONS 197 that the viscosity of suspensions of low concentrations (2 to 8 per cent.) was anomalous, i.e. varied with the velocity gradient. It is therefore again doubtful how the results obtained by using Poiseuille's formula can be interpreted. same at i8° ; C, lo per cent, sol at 23° ; D, the same at 23-5°. Absc. minutes. Ord. times of flow in seconds. A fundamental difficulty in verifying the formula for suspensoid sols is that it contains the volume of disperse phase, whereas, generally speaking, the weight only is known. Even if the density of the particles is the same as that of the material in bulk, as is probable for, e.g.^ gold sols, the particles do not consist of the pure substance. Silver sols prepared by 198 THE VISCOSITY OF LIQUIDS Carey Lea's method, which can be obtained in comparatively high concentrations, have particles containing notable amounts of impurities. It is also highly probable that in most sols the particles carry with them layers of liquid which increase their effective volume. Einstein himself applied his formula to a dilute cane-sugar solution, and found it necessary to assume hydration of the sugar molecules to obtain agreement with experimental data. The first experimental verification of Einstein's formula was undertaken by Bancelin,^^ who determined in the Ostwald viscometer the viscosity of suspensions of gamboge, made by Perrin's method, and containing globules of 0-3, i-o, 2-0, and 4-o/x radius. As required by the formula, the radius of the particles had no influence. The viscosity was a linear function of the concentration, with a proportionality factor of 2-9 instead of 2*5, as shown by the following table, which gives the results for particles o-3/x radius at 20° C. (water at 20° =1) :— q? (per cent.). Vs- 0-00 I'OOO 0-09 1-002 0-24 1-0069 0-33 1-0088 0-53 1-0128 0*66 1-0167 1-05 1-0276 2*11 1-0570 Within these limits of concentration the formula holds good as far as the linear increase in viscosity with concentra¬ tion is concerned. A more extensive range was investigated by Oden,^^ who determined the viscosity of sulphur sols prepared by Raffo's method, which can be obtained in high concentrations; it is, however, doubtful whether the sulphur particles can be considered as rigid spheres. Oden examined two sols, one with particles, whose diameter was estimated by the usual ultra-microscopic method at loo/x/x, while the other sol was amicroscopic with particles probably about lOfjufji diameter. Fig. 75 shows the results up to a concentra- VISCOSITY OF COLLOIDAL SOLUTIONS 199 tion of 50 per cent. The increase of at low concentrations is almost linear, but becomes markedly more rapid above about 10 per cent. The curve for the sol with smaller particles lies entirely above that for the coarser grained, the ratio of the ordinates being approximately constant. The viscosity is therefore not independent of the particle size, but a given aggregate volume of small particles produces a greater increase in viscosity than the same volume of large particles. It has been pointed out by Hatschek that the effect of the radius is easily explained if the particles are surrounded by adsorption layers which increase the effective volume. If the thickness of these layers is assumed to be the same for both sizes, the increase in effective volume will be proportional to the aggregate surface, and therefore greater for the sol with small particles. The thickness calculated from Oden's data is o-Sy/x/x, an entirely probable value which, however, cannot be checked in the absence of a third set of measure¬ ments. It must also be added that in sols of high con¬ centration the fundamental assumption made in deducing Einstein's equation, that the particles do not influence one another, is not likely to hold good. A further complication arises from the electric charges present on particles, at least in most aqueous sols and suspen¬ sions, which Einstein's deduction does not take into account. That such charges must increase the viscosity had been suggested already by Wo. Ostwald; the mathematical theory for a system of charged spheres in a liquid was devel¬ oped by Smoluchowski, who deduced the modified Einstein equation in which the new symbols mean : A the specific conductivity per c.c. of the liquid ; r the radius of the particles ; C the potential difference in the double layer ; D the dielectric constant of the liquid. It will be noticed that the radius of the particle appears in 200 THE VISCOSITY OF LIQUIDS this formula, and that the term in square brackets increases with decreasing radius ; i.e, the viscosity, other things being equal, increases in the same sense. The formula has been verified experimentally by Kruyt and his pupils on a number of sols. Hess has attempted to deduce synthetically an equation for the viscosity of a suspension. By assuming a simple arrangement of the particles in a liquid flowing through a capillary he arrives at the equation • • • • (3) in which the symbols have the same meaning as before, and a is a numerical factor >i. He calculated a from a series of measurements on suspensions of red-blood corpuscles— which, on account of their peculiar shape and easy deform- ability, are not very suitable for testing a formula to be applied to suspensions—and found that a varied from about 2-25 at low concentration (^=o-io) to about 1-15 at high concentration (9^—0-79). It is highly probable that the factor a also varies with the velocity gradient. The theoretical treatment of emulsoid or lyophilic sols presents even greater difficulties than that of suspensions, since some working hypothesis is necessary at the outset to account for the enormous increase in viscosity produced by small concentrations by weight of dispersed material. The assumptions usually made are of two kinds. One is that the molecules or particles of the disperse phase arrange themselves in long threads or ''ramifying aggregates'' (M^'Bain so that eventually there is a continuous network of disperse phase. No mathematical treatment of such systems has so far been attempted. The other assumption is that the particles of disperse phase are heavily hydrated (or solvated), so that they occupy a large multiple of the volume calculated from the weight concentration and density. A necessary corollary is that these solvated aggregates are, if not liquid, at least easily deformable, as otherwise the whole system could no longer behave like a liquid. Hydration has, for instance, been assumed by Arrhenius,^® VISCOSITY OF COLLOIDAL SOLUTIONS 201 who applied his empirical logarithmic formula to a series of viscosity determinations on protein sols carried out by Harriette Chick and collaborators.^^ The formula for this purpose was given the form : log^ = 0- . . (4) [loo— {n+T)p] in which p is the weight of dry substance dispersed in lOO grms. of sol, 6 a constant, and n the hydration factor, i.e. the number of grammes of water associated with i grm. of dry weight of disperse phase and thus withdrawn from the free dispersion medium. The values of 6 and n calculated from the experimental data show remarkable constancy, except in one or two in¬ stances at the lowest concentrations, and the formula ex¬ presses the viscosity of various protein sols with great accuracy. Arrhenius remarks on the fact that the hydration factor is practically constant, while in true solutions hydration always decreases with increasing concentration. As the reason of this difference he suggests that the vapour pressure of true solutions is lowered considerably with increasing concentra¬ tion of solute, while the lowering is imperceptible in colloidal solutions. The logarithmic formula fails when applied to sols of high viscosity in organic solvents, such as the cellulose nitrate sols investigated by Baker, or the cellulose acetate sols studied by Mardles.^^ The solvation factor becomes irregular and occasionally negative. Baker himself succeeded in representing his results with fair accuracy by a parabolic formula : r¡=rjo{l +acy, in which r]Q is the viscosity of the solvent, a and n are con¬ stants, and c the concentration in grm. per loo c.c. of sol. Hatschek has treated the problem of a dispersion of liquid, or elastically deformable, particles aggregating more than about one-half of the total volume by a geometrical method. When such a system is sheared the particles undergo deforma¬ tion, and, at low velocity gradients, recover their shape owing to interfacial tension or elasticity. When, however, a certain 202 THE VISCOSITY OF LIQUIDS velocity gradient is exceeded they are assumed to remain deformed throughout. During the first regime the viscosity of the system must vary with the velocity gradient; during the second it is constant, and given by Vs • • • (5) where the symbols have the same meaning as before, and ^ is assumed to be >0-5. The formula has hardly been tested for emulsions, which are systems conforming to the assumptions made in deducing it. It does not contain the viscosity coefficient of the dis¬ perse phase, which would be expected to play some part in determining the viscosity of the system. The formula, how¬ ever, represents fairly accurately the viscosity of suspensions of red-blood corpuscles—which are easily deformable—as was first shown by Trevan,^® whose data are given below. The first column gives the viscosity of the suspension (determined in an Ostwald viscometer), the second (/> determined in the usual way (centrifuging in graduated tubes), and the third cf) calculated from ^Jjh:z3}l ... (6) Vs Vs-

Q(N+3) 218 THE VISCOSITY OF LIQUIDS Since ^R/2/ is the shearing stress at the wall when stationary flow is established, it follows from (19) that the velocity gradient at the wall is äv Q{^+3) dr ¿77R® ■ ■ ■ ! It is possible thus to express the results of measurements in capillary tubes in terms of shearing stress and maximum velocity gradient, and to make them independent, not only of the dimensions, but of the type of instrument used. They can then be compared with the results of measurements in the concentric cylinder apparatus, and if N and r¡' are real constants of the liquid, they must satisfy the equations to be found for this apparatus. By introducing the expression (19) for the shearing stress into the differential equation of the concentric cylinders, we obtain : M , , ,27rRi2Lj ''Ra'N-R/^ ■ ■ in which all the symbols have the usual meaning. From simple static considerations it follows that the term in brackets is the shearing stress per unit area on the inner cylinder, and that therefore the velocity gradient at this surface is R 2N t> 2N • • • (^4) tv2 If the inner cylinder is suspended by a wire of torque per unit twist T, where ^ is the deflection of the inner cylinder. Measurements on the same sols (starch pastes containing formaldehyde, which prevents setting and retards the spon¬ taneous change of viscosity) were made both in capillaries and in a concentric cylinder apparatus, and the shearing stresses at the wall of the capillary or the surface of the inner cylinder calculated from the equations just given, as well as the velocity gradients at these surfaces. Approximate values of N were first obtained by plotting log P against log ijt (capillary), or log M against log i/O. VISCOSITY OF COLLOIDAL SOLUTIONS 219 Examination of the results, which it is not necessary to give in extenso, shows that neither N nor r¡' is a constant of the material at concentrations higher than i per cent. The values for I per cent, sols show reasonably good agreement. They are : From measurements in concentric cylinder apparatus— N=i-04, mean value of '^'=0-0373, maximum deviation 6 per cent, of mean value. From measurements in capillary viscometer— mean value of '^'==0*0353, maximum deviation 1-I per cent, of mean value. The 2 per cent, sol already shows much greater dis¬ crepancies : From measurements in concentric cylinder apparatus— N=i-46, mean value of 7^'=3-19, maximum deviation 16-3 per cent, of mean value. From measurements in capillary viscometer— N = I'4o, mean value of 77'=2*16, maximum deviation 2-8 per cent, of mean value. For higher concentrations the discrepancies between the values deduced with the two types of apparatus become very much greater. Farrow gives two graphs (figs. 85 and 86), in the first of which the velocity gradients are plotted against the shearing stresses, and in the second the logarithms of these quantities. The following gives the essential points of Farrow's own discussion of the results :— The fact that the experimental points for any one paste lie on a smooth curve shows that the method of expression used has succeeded in correlating the results of flow measure¬ ments made in viscometers of differing types and dimensions, over a total range of velocity gradient of 300,000-fold. It therefore seems possible to measure the flow of anomalous liquids in capillary instruments, and to express their condition at a given velocity gradient in terms of the constants of a simple equation (19) independently of the dimensions of the instrument. Over a range which is comparatively small compared with the total range of measurement, the log velocity gradient-log 220 THE VISCOSITY OF LIQUIDS stress points lie on a straight line. It is therefore probable that the assumption represented by equation (19) holds 2500 2000 g 1500 W Q O H 8 1000 w > 500 0 500 1000 ISOO 2000 STRESS Fig. 85.—Stress-velocity gradient graphs of starch pastes in different capillaries (Farrow, Lowe and Neale). approximately over a limited range, and that therefore the method of calculating velocity gradients at particular regions from total flow is a close approximation. VISCOSITY OF COLLOIDAL SOLUTIONS 221 Over a wide range the lines relating log stress and log H w h-l Q O H i—i O O tí > d o tí LO 2*0 LOO. STRESS Fig. 86.—Log stress-log velocity gradient graphs of starch pastes in capillaries and Couette (concentric cylinder) apparatus (Farrow, Lowe and Neale). velocity gradient, which should be straight if equation (19) is valid, appear to have a slight curvature, which is the 222 THE VISCOSITY OF LIQUIDS greater the more viscous the liquid. In other words, N, the slope of the log-log line, is constant over a small range only, and increases very slowly for the more viscous liquids as the velocity gradient decreases. On this account different values for N, and therefore for r¡\ are given by the same liquid in the capillary viscometer, where the gradient is high, and in the concentric cylinder apparatus, where it is low. It appears, therefore, that the original assumption is useful chiefly as a means of eliminating instrument dimensions and of obtaining values of pure shearing stress and velocity gradient in the liquid to a close approximation. It is, of course, not possible to calculate exact values of velocity gradient until an exact relation between it and shearing stress is found, and since such a relation must itself depend on viscometer measurements, the only method of progress is by successive approximations. This, however, is quite practic¬ able, since, particularly in the concentric cylinder apparatus, change of the stress-velocity gradient relation usually in¬ volves only a second order correction to the velocity gradient calculated. Farrow, Lowe and Neale add, what is indeed evident from the graphs, that they flnd no evidence of any yield value.'' A point deserving mention is that the graphs do not give any direct evidence of the inconstancy of 7^', which is con¬ siderable at the higher concentrations. Although the method developed by these authors is only, as they say, a first approximation, it is without doubt an important step towards the rational aim of co-ordinating the real variables, shearing stress and velocity gradient, instead of developing merely empirical relations for given instru¬ ments. If the direct determination of the velocity distri¬ bution, to which reference has been made, should prove impracticable, more complicated functions than that tested by Farrow will have to be tried in the same way. Maxwell's Theory of Viscosity. As has been mentioned before, many sols possess measurable rigidity, and this opens up one more theoretical possibility of arriving at a method for finding their true viscosity. A connection between elastic modulus and viscosity coeificient has been formulated by VISCOSITY OF COLLOIDAL SOLUTIONS 223 Maxwell as follows: a strain or deformation of some kind S is produced in a body (about the structure or state of aggregation of which no assumptions are made); a state of stress or elastic force F is thus excited, and the relation between stress and strain may be written: F=ES, where E is the modulus of elasticity for the particular kind of strain. In a body free from viscosity F will remain =ES, and dF dS , . Ht • • • • (25) If, however, the body is viscous, the stress F will not remain constant, but will tend to disappear at a rate depending on F and on the nature of the body. The simplest assumption is that this rate of disappearance is proportional to the stress, and the equation may then be written : If S is assumed to be constant, integration gives the equation F=ES^-^/^ .... (27) where T is an integration constant with the dimensions of a time. For t~T the stress F has decreased to rje of its initial value. When t = oo , the body has lost all internal stress. If S is variable, the most interesting case is that when dS/dt is constant, i.e. there is steady motion which continu¬ ally increases the displacement. Integration of equation (26) gives : F=ET^+Ce-'/T . . . (28) dt C is an integration constant. The equation shows that F tends to a constant value as the second term on the right tends to zero with increasing Maxwell concludes: ''The quantity ET, by which the rate of displacement must be multiplied to get the force, may be called the coefficient of viscosity. It is the product of the coefficient of elasticity E and a time T, which may be called ' the time of relaxation ' of the elastic force. In mobile fluids T is a very small fraction 224 THE VISCOSITY OF LIQUIDS of a second, and E is not easily determined experimentally. In viscous solids T may be several hours or days, and then E is easily measured. It is possible that in some bodies T may be a function of¥I' (My italics.—E. H.) This means that, for a body possessing both rigidity and viscosity, the simple relation should hold : 77=ET .... (29) E being the modulus of rigidity, and T the relaxation time,'' as defined above. It therefore would appear possible to find the real viscosity coefficient of a material possessing rigidity Ç' the particular kind of strain " in question being a shear) by determining the modulus and the time of relaxation; the first step being to verify whether the law of relaxation is that assumed by Maxwell. This can be done by maintaining a certain deforma¬ tion S constant and determining at intervals the stress F required to maintain it; if equation (27) holds, the F-t curve must be a logarithmic line. We have already described one investigation of the kind, that of Trouton and Andrews (p. 187), on pitch, in which the relaxation curve was found not to be logarithmic. Similar determinations of the relaxa¬ tion-time curve, as well as of the modulus, have been carried out with a number of colloidal solutions, and may be briefly described here. Rigidity in Colloidal Solutions. The first investigator to determine the modulus of rigidity and the relaxation of sols (a single gelatin sol was examined) was Schwedoff.^® A cylinder is suspended from a long wire coaxially in a cylindrical vessel containing the liquid to be investigated. The upper end of the wire is twisted through an angle if the liquid is simply viscous, the cylinder follows the wire with a decreasing velocity until it has moved through the angle p, i.e. until no torsion is left in the wire. If, however, the liquid also possesses rigidity, the cylinder does not follow, but rotates through an angle co{