E. W. J. MARDLES AND I. E. PUDDINGTON 47 SOME EFFECTS ON THE FLOW OF CONCENTRATED SUSPENSIONS OF VARIATIONS IN PARTICLE SIZE AND SHAPE BY P. S. WILLIAMS Received 24th May, 1951 Classical colloids used in industry involve suspensions of particles with a wide range of size, shape and surface state. One of the most marked departures in behaviour from the laws of solution is observed in the viscosity or consistency of liquid paint. Flocculation of pigment, the strength of interparticle forces giving rise to thixotropy and false-body paints are but little understood, and the paper briefly reviews the present stage of knowledge on these matters. Studies have been made of the rheological properties of glass spheres, of pigment48 FLOW OF CONCENTRATED SUSPENSIONS size range, dispersed in stand-oil + paraffin, or glycerine.Such systems can be regarded as idealized paints in which some of the usually uncontrolled factors, e.g. surface condition and shape are known. The method of preparation and particle size analysis of these spheres is given. It is shown that the viscosity of a suspension of hard spherical particles, over the size range examined, is (a) Newtonian up to 50 yo by volume concentration, for particles > rp diam., (b) dependent on particle size, increasing as particle size decreases, and becoming non-Newtonian for particles > ~p diam. A co-axial cylinder viscometer, de- signed to enable a detailed study of stress-rate of shear-time relationships of materials exhibiting such viscosity anomalies as thixotropy, is described. From the physical point of view, paint, before application, is essentially a suspension of finely divided solid in a viscous liquid.We can at once justify the inclusion of paints in the classification of a colloid, when we consider the particle sizes of pigments. These sizes range from greater than 10 microns in diameter for the earth oxides to less than 0.01 microns for fine Prussian Blue and Carbon Blacks. It is only recently-since the application of the electron microscope to industrial problems-that the size and shapes of most pigments has been capable of resolution. The shapes of pigments may vary from nearly spherical to extremely acicular, while the size range may be very narrow or extremely wide. Illustrative of these variations are electron photomicrographs of Lemon Chrome, Prussian Blue, white lead, zinc oxide (Fig.I). It is well known that the f l ~ w properties of a suspension of small particles will depend to a first order on the viscosity of the suspending medium, and the concentration of the solid phase. Einstein 1 derived the relation where qs is the viscosity of the suspension, qo the viscosity of the sus- pending medium, v the volume concentration of the disperse phase, and k is a constant, equal to 2-5 for spheres. The conditions for the validity of this relationship are that the particles be uncharged, hard and non- interacting. Smoluchowski e has considered the case when particles are charged, and numerous investigators have proposed relationships covering the case when the concentration of the suspended phase increases so that there is interaction between the particles.In general, the viscosity of a concentrated suspension of small particles does not obey Newton’s law which states that for a liquid in laminar flow, the force per unit area required to maintain the flow, is proportional to the velocity gradient. On plotting rate of shear D against applied shear stress per unit area T a curve will be obtained and the ratio 7/0 varies with rate of shear. The differential d.r/dD may vary continuously with rate of shear, or may become constant beyond some limiting value of D. In many cases this ratio is not constant with respect to time of application of shear, but decreases, more or less slowly, to an equilibrium value. On leaving the suspension unsheared for some time, i t will be found that its viscosity will have regained its former value, only to de- crease again on being sheared.Freundlich 3 made extensive studies of such systems which he called thixotropic. Originally referring only to systems undergoing a reversible sol-gel-sol transformation, the term thixotropy now covers a wider range of phenomena, but all have the common feature of a decrease in viscosity with increasing rate of shear and a subsequent regain in viscosity when the material is left at rest. Several mechanisms have been postulated to account for the phe- nomenon of thixotropy 4, but as yet, little progress has been made in 7s = VO(I + kv), 1 Einstein, Ann. Physik, 1906, 19, 280. Smoluchowski, Kolloid-Z., 1916, 18, 194. Freundlich, Thixotropy (Hermann and Cie, Paris, 1935).Langmuir, J . Chern. Physics, 1938, 6, 873. Goodeve, Trans. Faraday SOC., 1939, 35, 342.P. S. WILLIAMS 49 deriving any quantitative theories, Recent theoretical work by Hamaker 6 and others present possible explanations for the thixotropic nature of various suspensions a t low concentrations. These authors derive curves of potential energy for spherical particles in suspension against distance of separation between them, for various ionic conditions. It is shown that, as a result of attractive and repulsive forces acting on the particles, it is possible that the particles will fall into a weak potential trough at some distance from that of closest approach. Such an occurrence could account for observed thixotropic phenomena, The difficulties of investigating the flow behaviour of suspensions of fine particles are not entirely theoretical. The greatest difficulty is en- countered in collecting quantitative data on D - 7- t (time) relationships of concentrated suspensions of fine particles. It would seem that as a result of particle-particle, and particle-medium forces, a dispersed suspension may in time, arrange itself in a state of minimum energy.This may result in a flocculated strmture of the dis- persed phase in the suspending medium. The formation of this structure will be a t a certain rate when the system is unsheared, but may be a t a very different rate under shearing. Any experimental work must take into account the fact that the rheological properties of the material under investigation may be dependent on the previous history of that material.In order to obtain quantitative data relating to the rheological be- haviour of concentrated suspensions in general, and to derive theoretical relationships governing this behaviour, experimental determination of shear stress, rate of shear, time curves must be obtained. It must be emphasized that the rheological behaviour of concentrated suspensions of colloidal particles can seldom be deduced from results obtained on suspensions containing much larger particles. It is only when particles are of the order of I micron or less in diameter that the inter-particle forces, which play so great a part in determining rheological properties, become effective. The author is engaged in work on the effect of particle size and shape on the flow properties of concentrated suspensions of the kind encountered in the paint industry. This paper will present some results obtained on suspensions of glass and silica spheres of various diameters, which were chosen as representing ideal systems of a more simple nature than en- countered in practice.Experimental The method of production of glass and silica spheres is not new* but the method of collection differs from previous attempts, and results in the retention of the finest particles. Glass (Pyrex or other hard glass) is ground to a fine powder in a rubber-lined mill. This powder is then placed in a container connected in the oxygen supply of an oxy-hydrogen torch. The torch is fastened through the lid of a 5 gallon steel drum, and when the oxy- hydrogen supplies are correctly adjusted, the " bush " flame is thrust into the drum and the lid is fitted.This 5 gallon drum has a depth of I or z in. of water in the bottom and is surrounded by a water jacket. The glass powder reaching the flame zone melts ; surface tension forms i t into spheres and these are col- lected in the layer of water in the bottom of the drum. There are, therefore, no effluent gases to be led away which would normally carry off a high propor- tion of the smallest spheres. The spheres are centrifuged out of the water, cleaned, washed and dried. A gravity sedimentation method is used to obtain various sized spheres down to about I micron. The glass spheres are well dispersed by ball-milling in a small quantity of a solution of sodium hexametaphosphate.This suspension is added to a further quantity of water Verwey and Overbeck, Theory of the Stability of Lyophobic Collozds (Elsevier The spheres must now be fractionated. 0 Hamaker, Bet. trav. chim., 1936, 55, 1015 ; 1937, 56, 727 ; 1938, 57, 61. Publishing Co., New York, 1948). * Sollner, Ind. Eng. Chem. (Anal.), 1939, 1 1 , 48.50 FLOW OF CONCENTRATED SUSPENSIONS so that there results a suspension of glass spheres I yo by volume concentration, in 0.1 yo sodium hexametaphosphate solution. This suspension is contained in a 10-gallon drum 200 cm. high. A float resting on the surface of the suspension supports one limb of a syphon, and the syphon rate is adjusted so that the float falls a t the same rate as that of a par- ticular size of sphere.The syphoned suspension then contains all spheres less than this size, By repeating this process, for different syphoning rates, the spheres may be fractionated. The advantages of this method over collecting materid, which has settled after various periods of time, is that the spheres collected have never been at a higher concentration than that originally put in the drum, therefore there has been no deviation from Stokes' law daring the settling period. The use of an anti-flocculating agent is essentia.1 when suspending fine particles, or there will be a possibility of flocculation upsetting the size fractionation, The 10 gallons of dilute suspension are then put through a laboratory cofltinuous centri- fuge, and are collected, washed and dried.Pig. ;1 is electron photomicrographs of spheres produced and fractionated by the above dethod. Fig. 3 is of soheres The spheres are allowed to settle. produced from very &&silica pGwder, instead of Pyrex glass. As can be seen from these photo- FLOW micrographs, the size range and distribution of these spheres is very wide and any attempt a t connecting rheological behaviour with size de- mands some method for particle size analysis to within closer limits of accuracy than are readily obtainable by counting from photomicrographs. A method described by Loukom- sky and O'Brien has been adapted to deal with a wide range of particle L,Qu,o F,LM sizes. The method depends on obtaining a sediment " spectrum " along the length of a Sharples laboratory centrifuge bowl (Fig.4). The heavier particles are deposited NER in the first few centimetres after entry into the bowl, while the finer particles are deposited a t the top of RCMOVABLE the bowl. From an analysis of the weight of the deposit in each centi- meter interval, a particle size dis- tribution may be obtained. The merits of such a method are that -h(o'lOR cDupl1~6 ,ltWO EXIT --FEED NOZZLE SUSPENSION A' the limits of accuracy are due to experimental errors, and not due to FEED 4w oF ROiAiiON C r c A -Ch?rnlnc 1 o h n r Q t n r x r oz=ntrifiina c t o t i c t i r i l nrvnrc nf , - n i r n t ; m n C n n q n x x r * 1". 4. "Ilu.IyIu.3 'U""'C"L"'J u u I I c I I & u ~ u d * U * I U * I " W * b L I " I i . "I u"Uuc1116 C"" lu.. particles. Fig.5 shows particle size - .-. bowl. analyses on glass and silica spheres. The experimental data on the D-r-t relationships of materials described in this paper, were obtained using a co-axial cylinder viscometer designed and constructed at I.C.I. Paints Division. This viscometer (to be described in detail elsewhere) consists of a rotating outer cylinder, with a rigidly held co-axial inner cylinder. The material to be studied is placed in the annular gap between these two cylinders. The rotational speed of the outer cylinder can be varied over a wide range by means of an infinitely variable gear box. The torque on the inner cylinder is measured by a strain gauge torque unit, the output from the strain gauges being amplified and used to drive a D.C. recording milliammeter.By altering the gain of the amplifier, a full-scale deflection on the recwder can be obtained over a stress range of 5 to 50,000 dynes per sq. cm. Plate I is a photograph of the apparatus. The main features of interest in this viscometer are that the shearing stress may be recorded continuously during an experiment, the rate of shear is very nearly uniform throughout the sample and that both the sensitivity of the stress recording and the rate of shear may be altered without introducing any discontinuity in the experiment. Loukomsky and O'Brien, Proc. A.S.T.M., 1946, 1437.FIG. I .--Electron photomicrograph of typical pigments. FIG. 2.-Electron photomicro- FIG. 3.-Electron photomicro- graph of glass spheres, of dia- meter 4p and under. [To face $age 50.graph of silica spheres.PLATE I.-Co-axial cylinder viscometer and recording equipment.P. S. WILLIAMS 5 1 Results and Discussion Experiments were carried out in order to determine the effect of vari- ation of particle size on the viscosity of suspensions of fine particles at various concentrations. Shear stress, rate of shear curves are shown in Fig. 6 for suspensions of glass spheres in the size range 2-10 microns diameter, a t concentrations ranging from 26 % to 50 yo by volume, in a mixture of zinc iodide, glycerine and water, of viscosity 0.63 poises. i \ FIG. 5.-Particle size analyses of (u) silica spheres, (b) glass spheres. FIG. 6.--0-7 curves of glass spheres, Z - I O ~ diam. in ZnI, + glycerine + water mixture of r) = 0.63 poise. ( a ) 5 0 yo ; ( b ) 46.4 Yo ; (c) 40.4 "/b ; (d) 35-6 yo.This suspending medium was chosen since easy variation of density and viscosity may be obtained. Work reported by Vand 10 on the flow be- haviour of suspensions of glass spheres up to 50 Yo by volume concentra- tion in a similar medium, states that the suspensions exhibit a time thixo- tropy in the reverse sense to that which might be expected. On stirring the suspensions, an increase in viscosity was observed for suspensions of concentration above 40 yo by volume. The viscosity after stirring was greater than normal, but decreased exponentially to its normal value, lo Vand, J . Physic. Chem., 1948, 52, 277.52 FLOW OF CONCENTRATED SUSPENSIONS Over the range of rates of shear used in the present experiments (up to 50 sec.-l) the behaviour of the suspensions was Newtonian] with no evidence of variation of shear stress with time of application of the shear.Table I gives values of 7,. (experimental) and q,. calculated from Vand's equation, Agreement between the two values is not obtained. qr = I + 2 . 5 ~ + 7'349ca* qsp = kV* If, in Einstein's equation] the volume concentration V is replaced by the value V / ( I - S V ) , where S is the volume the sediment of a suspension would occupy, whose par- ticles themselves occupy unit volume, then on plotting V / T ~ ~ against V, a straight line would be obtained. Values of Y and V/qep obtained in the present experiment are given in Table I and pIotted in Fig. 7. The - 4 5 Volume uoncentration V FIG. '/.-Plot of V against V/qaD for glass spheres, 2 - 1 0 ~ &am.in ZnI, + glycerine + water mixture of r) = 0.63 poise. value of S obtained from the graph is 1-9, in close agreement with values obtained by Robinson l1 using spheres between 10 and 30 p diam. TABLE I Volume Concentration V 50.0 46-6 40'4 35'6 31.6 26.0 rlf (Obs.) 50.0 25.0 9'3 6.0 4'7 3'3 tlf (Calc.) 4'1 3'7 3'2 2.8 2'5 2.15 V h p 0'0 I 0'02 0.05 0.07 0.09 0'12 The void space for ideal 6-fold packing is 0.4764 cm.-a per cm.-*, so that the value of S for such a sediment would be 1-91. There is obviously very close agreement between this value of S for 6-fold packing, and the value obtained experimentally. For a concentrated suspension showing Newtonian flow properties] the most likely spatial arrangement of the particles would be close to that of the 6-fold arrangement of a packed sediment.l1 Robinson, J . Physic. Chem., 1949, 53, 104.P. S. WILLIAMS 53 In order to test the above theory, namely, that on plotting V against V / T ~ ~ , a straight line should be obtained, with intercept on the V axis equal to r/S, an attempt was made to repeat the initial experiment, but using spheres of silica with diameters I to 0.1 p and 0.025 p. It was found very difficult to disperse these smaller spheres at concentrations approaching 40 yo by volume, or over. Fig. 8 and g show D-r-t curves for a suspension of 44 yo by volume silica spheres in a glycerine + water mixture of viscosity 0.35 poise. This suspension was observed to set to a gel after a period of about 24 hr. at rest. It was therefore decided to investigate the flow properties of suspensions of these spheres at a con- centration of about 10 yo by volume, in a linseed stand oil+paraffin mixture.This medium was chosen because of the excellent dispersing power of stand-oil, while the viscosiv of the mixture may be varied by alteration of the paraffin concentration. FIG. 8.-D-r curve for 44 yo silica spheres < ~p diam. in glycerine + watei mixture of 7 = 0.35 poise. FIG. g.--D-~-t curves for 44 yo silica spheres < I,U &am. in glycerine + water mixture of 7) = 0.35 poise. (a) immediately after stirring ; ( b ) after 12 hr. a t rest ; (c) after 24 hr. at rest. Fig. 10 and 11 show the 0--7 and D-T-~ relationships for these suspensions. The suspensions containing spheres down to O - I ~ diam.are Newtonian, but the finest spheres, of diameter 0-025p are thixotropic in flow behaviour, giving suspensions with a yield value even a t low con- centrations. The following conclusions may be drawn from the above results. The viscosity of a suspension of hard spherical particles over the size range examined is : (I) Newtonian up to 50% by volume concentration, for particles greater than about ~p diam.54 FLOW OF CONCENTRATED SUSPENSIONS (2) Dependent on particle size, increasing as particle size decreases, and becoming non-Newtonian for particles less than about ~p diam . (3) For suspensions exhibiting Newtonian flow behaviour, the vis- cosity against concentration relationship may be represented by the equation V / q s p = I l k - SV/k. This equation does not apply to thixotropic suspensions.The results presented in this paper indicate the complexity of the flow behaviour of concentrated suspensions of spherical particles in the colloidal size range. FIG. ~o.--D-r curves for (a) silica spheres, 0-025p diam. ; < ~p diam. ; (b). silica spheres (c) glass spheres 2-10p diam., at 11 % concentration in stand-oil + paraffin mixture of 9 = 11 poise. FIG. II.-D--~ curves for silica spheres, o'oz5p diam. in stand-oil + paraffin mixture. (a) 8-65 %, qmix 7-4 poisf ; (b) 8-17 %. qmix 5.3 poise ; (c) 7.7 %, Tmix 3'4 poise ; (a) 7'3 %, Vmix 2.6 poise ; (e) 6'7 %, qmix, 1-35 poise ; ( f ) 6.1 %, Attempts have been made to measure quantitative thixotropic char- acteristics of a system. Green and Weltman,f* and Moonep19 derive various coefficients, which are correlated to the observed flow behaviour of thixotropic systems. These authors have attempted to fit mathe- matical equations to the results they have obtained. In the present l2 Green and Weltman, I n d . Eng. Chem. (Anal.), 1943, 15, 201 ; 1946, 18, 167. l3 Mooney, J . Colloid Sci., 1946, I , 195. qmix 0.6 poise.P. S. WILLIAMS 55 author’s opinion, the factors to be considered in formulating any theory attempting to explain the rheological properties of concentrated sus- pensions of fine particles should consider (a) the force between the particles of a flocculate, tending to hold (b) the stability of a flocculate to the forces imposed upon it during It is not until a sound theory on these two important factors governing the flow behaviour of concentrated suspensions of colloidal particles has been derived, that a close understanding of this behaviour will be made. On the other hand, the gaining of quantitative data on these suspensions may well help the theoretician on his way. Mr. N. D. P. Smith, of I.C.I. Paints Division, Slough. Imperial Chemical Industries Limihd, them together, and shear. The author wishes to acknowledge with.thanks the helpful advice of Paints Division Research Laboratories.