`©Elsevier Sequoia S.A., Lausanne- Printed in The Netherlands
`
`Ordered Mixing: A New Concept in Powder Mixing Practice
`
`J.A. HERSEY
`
`Victorian College of Pharmacy. 381 Royal Parade. Parkville. Victoria 3052 (Australia)
`
`(Received May 16, 1974)
`
`SUMMARY
`
`A randomization concept of powder mixing
`has largely been explored in the past. Non-co(cid:173)
`hesive, non-interacting particulate systems are
`primaiily mixed by this process. Cohesive, in(cid:173)
`teracting particulate systems may also be mix(cid:173)
`ed to a high degree of homogeneity. Many of
`the requirements of this type of mixing are
`different from those required for randomiza(cid:173)
`tion. The process of mixing cohesive, interac(cid:173)
`ting particulate systems follows a "disorder to
`ordeJ:" concept and has been termed ordered
`mixing.
`Ordered mixing probably occurs widely in
`actual systems. Although the requirements for
`ordered mixing are different from those for
`J:andom mixing, the rate of mixing follows the
`same laws. Ordered mixtures are frequently
`more homogenous than random mixtures
`and, in certain cases, may offer a better ap(cid:173)
`proach to practical mixing problems.
`
`INTRODUCTION
`
`Powder mixing has been the subject of nu(cid:173)
`merous investigations over the past decade
`[1,2}. For simplicity, most of the systems ex(cid:173)
`amined have consisted of comparatively coru:se,
`free-flowing particles and have led to the con(cid:173)
`cept of J:andomization or "shufflipg" of the
`particles as the mixing process. The randomi(cid:173)
`zation may be brought about by a variety of
`mechanisms, including diffusion and convec(cid:173)
`tion, according to the variety of mixer employ(cid:173)
`ed. Randomization requires equally sized and
`weighted particles, with little or no surface ef-
`
`fects, showing no cohesion or interparticle in(cid:173)
`teraction, to achieve the best results. Undoubt(cid:173)
`edly, this is an important process in powder
`technology today and has served a useful pur(cid:173)
`pose in enabling mixing theories to be quanti(cid:173)
`fied. However, it cannot be applied to all prac(cid:173)
`tical mixing situations, especially :!'or cohesive
`or interacting particulate systems and may,
`therefore, not be unique amongst explanations
`of powder mixing phenomena.
`A concept of ordered mixing may be useful
`in explaining powder mixing of cohesive or in(cid:173)
`teracting fine particles. There is no theoretical
`reason why fine particles cannot be mixed by a
`randomizing process. However, cohesive proper(cid:173)
`ties and other surface phenomena usually devel(cid:173)
`op with increasing fineness and these will tend
`to order rather than to randomize the mixing
`operation. Where there are large differences in
`particle size, fine and coarse particles would
`tend to segregate at a faster rate than they
`would mix together, unless some interacting
`forces were utilized to mix the system. Ordered
`mixing may be considered to be different from
`random mixing since it does not require equally
`sized or weighted particles; it requires particle
`interaction, i.e. adsorption, chemisorption, sur(cid:173)
`face tension, frictional, electrostatic or any
`other fonn of adhesion. It results in an ordered
`mixing arrangement of the particles, which is
`best shown diagrammatically in Fig. l(a) for an
`equal mixture of black and white particles, al(cid:173)
`though ordered mixtures are more likely to oc(cid:173)
`cur where there are a few large black particles
`and m!my small white particles. Figure l(b)
`shows the same mixture in a randomized
`pattern.
`Indications that ordered mixing may occur
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`Col
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`Fig. L(a). Ordered mixture of equal proportions of
`black and white particles.
`Fig. l(b). Randomized mixture of equal proportions of
`black and white particles.
`
`are available in the literature from mixing obser(cid:173)
`vations [3,4] and from angle of repose date [5-
`7], which show that adhesion of fine particles
`considerably affects the properties of the par(cid:173)
`ticulate system.
`Studies of homogeneity of powder mL'!:tures
`have frequently been based on the heteroge(cid:173)
`neity of the completely randomized mixture
`[8 -10]. Such a basis would be unsatisfactory
`for ordered mixtures. However, other concepts
`of homogeneity, such as those using as a basis
`the required degree of homogeneity [11] or of a
`standard degree of heterogeneity [12], should
`provide a useful solution for the examination
`of homogeneity of ordered mixtures.
`The rate of mixing is generally described
`by a logarithmic relationship [13] as would
`be required by the first-order kinetics sugges(cid:173)
`ted for ordered mixing. It is the purpose of
`this paper to consider the possibilities of or(cid:173)
`dered mixing and to examine if such a concept
`would be useful in explaining homogeneity
`and rate of mixing in certain applications and,
`as such, provide a useful addition to the theory
`and practice of powder mixing operations.
`
`THEORY
`
`Consider spheres of diameter D and diameter
`d, where D>>d. Surface area of a large par(cid:173)
`ticle = 1rD2
`• The area occupied by each small
`particle adhering to a larger one will be the
`projected area 7rd2/4. Therefore the number
`of small particles adhering to each larger one
`in a monolayer= 4D2Jd2.
`Allowing for the fact that the small partic(cid:173)
`les will not pack regularly or that there will be
`some areas of the larger particles devoid of bin-
`
`ding sites, only a fraction, f. of this number, n,
`will actually adhere in a monolayer:
`n = 4W2 /d2
`A more exact solution [7] for small partic(cid:173)
`les close-packed in an hexagonal arrangement
`is given by
`
`(1)
`
`(2)
`
`n = 27f (D + d)2 f
`v'3 d 2
`If the mixture contains 1% by weight Qf par(cid:173)
`ticles of diameter, d, 5 X lQ- 4 em in partic(cid:173)
`les of diameter, D, 5 X 10-2 em, then the num(cid:173)
`ber of small, Nd, and large Nv, particles in a
`1-g sample is given by (the density for each is
`taken as 1.2 g cm-3):
`d3
`Nd1r S 1.2 =.0.01; Nd = 1.27 X 108
`na
`Nv 1r61.2 = 0.99;ND = 1.26 X 104
`Thus the number of small balls adhering to
`each large one is given by n = NdfNv = 1.0
`X 104, and substituting this value in eqn. (1)
`gives f = 0.25.
`
`The equilibrium situation
`There are a definite number of small partic(cid:173)
`les that may adhere to a single large particle
`for any given system. Consider this to be the
`equilibrium situation. Thus, in the example
`given above there are 104 small particles assoc(cid:173)
`iated with each large particle. Each unit of the
`system is identical and it may be considered
`to consist of a single material.
`If a lower percentage of fine particles had
`been used then the total number of adherence
`sites in the larger particles would not have
`been filled. Under these conditions it hi pro(cid:173)
`bable that some large particles would have a
`larger number of small particles associated
`with them than others. In such a case each unit
`may be different and the system could be con(cid:173)
`sidered to consist of many components. Alter(cid:173)
`natively, an equilibrium situation could be es(cid:173)
`tablished by some sites on the larger particles
`being more active than others. These sites
`would be saturated more rapidly, resulting in a
`single component system in which the fraction
`f of fine particles adhering to large ones is re(cid:173)
`duced compared with the maximum possible
`value.
`If an excess of fines had been used, then the
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`equilibrium situation will be attained by adhe(cid:173)
`rence at all available sites and the excess lmes
`may attempt to mix with this material in a ran(cid:173)
`dom manner. Thus, a binary mixture could
`theoretically exist, but its composition would
`be different from that considered by the ingre(cid:173)
`dient concentrations present. In the above ex(cid:173)
`ample, if 10% fines had been used the concen(cid:173)
`tration of coarse particles would be 90%. The
`coarse particles would come to equilibrium
`with 0.9% of fine material giving concentra(cid:173)
`tions of 90.9 and 9.1 adhered units and excess
`fines respectively.
`
`Homogeneity
`Buslik [12] defined homogeneity, H, as the
`reciprocal of ( W1 ), that weight of sample ne(cid:173)
`cessary to give a standard deviation of 1%:
`H= 1/W1
`or using a logarithi:nic scale, where H, is an in(cid:173)
`dex of homogeneity,
`H,=-log w1
`(3)
`For a randomized mixture, the standard de(cid:173)
`viation, a, is given by
`(4)
`a 2 = G(100- G)w/W
`where G is the percentage of ingredient in the
`mixture, w is the effective mean particle weight
`and W is the sample weight. This equation ap(cid:173)
`plies where all particles are of the same weight;
`corrections must be applied in calculating the
`effective mean weight of multisized systems.
`If a = 1% as required by the definition of
`homogeneity, then
`= G(100- G}w
`W
`1
`Substituting in eqn. (3),
`H, =-log [G(100- G)w]
`(6)
`For a single component, the value of H, can
`be calculated from the minimum sample size
`that may be considered. For example, Buslik,
`in considering pure hydrogen gas, took the mini(cid:173)
`mum sample size as a single hydrogen molecule,
`thus giving the limiting value which H, can at(cid:173)
`tain as 23.5. Sodium chloride was treated simi(cid:173)
`larly, excepting that the minimum sample size
`could be either a sodium ion or a chloride ion
`giving rise to a discontinuity in the evaluation
`of If,, which lies between 22.0 and 22.2.
`In a completely ordered system, the compo(cid:173)
`nents are mixed so that they may be considered
`as a single material. Thus, the evaluation of H,
`
`(5)
`
`43
`
`for such a system would follow exactly the
`same arguments as for sodium chloride. The
`sample size would necessarily have to be smal(cid:173)
`ler than one complete ordered unit and the
`homogeneity would lie between the respective
`values for the weights of a large (Wn) and a
`small ( Wd} particle adhering to it.
`da
`Da
`WD = 1.211"6 and Wd = 1.211"6
`
`giving 4.11 < H, < 10.11.
`Where only 0.5% of fines has been used, H,
`still lies within the above limits providing an
`equilibrium situation can be attained. Where
`an excess of fines is used (i.e. 10%), then the
`problem is complicated since both ordered mix(cid:173)
`ing will occur and randomized mixing can, in
`theory, also occur.
`If the same system of 1% by weight of fine
`particles is considered to be randomly dispers(cid:173)
`ed in large particles at a sample weight of 1 g,
`two mathematical treatments are possible.
`Firstly, the srunples could consist of clumps
`of weight equivalent to a single particle, and
`log
`being randomly distributed then H, = -
`G(100- G)W =-log (99 X 1 X 1) =- 2.00.
`Or, altematively, if the individual samples
`were considered as pharmaceutical tablets re(cid:173)
`quiring the range of contents to be± 15% of
`the mean, then
`3a = 0.15 X 1, and a = 0.05.
`If this system is randomized, substitution of
`eqn. (4) in eqn. (6) gives
`(7)
`H=-1oga 2 W
`'
`Since W = 1 g, then H, = +2.60; either index of
`homogeneity being considerably less than the
`homogeneity that could be obtained for the
`completely ordered system.
`
`RATE OF MIXING
`
`The rate of ordered mixing follows first(cid:173)
`order kinetics, since the rate of mixing will be
`proportional to the number of fine particles
`remaining to adhere onto the larger particles.
`For a given particulate system, the rate of mix(cid:173)
`ing will be proportional to the concentration
`of unmixed lme particles. Such a mixing rate
`is also applicable to random mixing [13], and
`thus both mixing phenomena could not be se(cid:173)
`parated by a simple examination of the kine-
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`tics of the mixing process. However, whereas
`coarse, equally sized and weighted particles
`randomize relatively quickly, a mixture of
`large and small particles would randomize very
`slowly and segregate very rapidly. This latter
`system, however, could mix extremely rapidly
`by an ordered process. Thus, examination of
`the actual rates of mixing might provide a use(cid:173)
`ful clue as to the type of mixing taking place.
`
`APPLICATION OF ORDERED MIXING
`
`The literature on powder mixing theory is
`unanimous in the requirement that only equal(cid:173)
`ly sized, equally dense particles can be mixed
`to fine-scale homogeneity. Such a situation is
`not true in practice, where small concentrations
`of fine powders are often incorporated into
`more coarse materials with a high degree of
`homow"neity. One example is the incorpora(cid:173)
`tion o {tablet lubricants into tablet granules.
`Applications already exist and ordered mixing
`provides a means of understanding how such
`processes are possible. Undoubtedly, other ap(cid:173)
`plications can be considered as soon as the
`mechanisms of ordered mixing are fully recog(cid:173)
`nized and understood.
`
`REFERENCES
`
`1 S.S_ Wiedenbaum,. Mixing of solids, Advan.
`Chem. Eng., 2 (1958) 238- 260.
`
`2 F.H.H. Valentin, Mixing of powders, pastes and
`non-Newtonian fluids, Chern. Process. Eng., 48
`(Oct.) (1967) 69- 71.
`3 D.N. Travers and R.C .. White, The mixing of JD.icro(cid:173)
`nized sodium bicarbonate with sucrose crystals,
`J. Pharm. PharmacoL, SuppL, 23 (1971) 2608-
`2618.
`4 J.A. Hersey, Avoiding powder mixing problems,
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`5 D.J. Craik, The flow properties of starch powders
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`73- 79.
`6 D.J. Craik.and B. F. Miller, The flow properties of
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`7 T.M. Jones and N. Pilpel, Some physical proper(cid:173)
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`8 .P.M.C. Lacey, Development in the theory of par(cid:173)
`ticle :mixing,. J. Appl. Chen1.. BiotechnoL, Lond .•
`4 (1954) 257- 268.
`9 M.D. Ashton and F.H.H. Valentin, The mixing of
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`Inst. Chern. Engrs., 44 (1966) T166- T188.
`10 K. Stange, Die Mischgiite einer Zufallsm.ischung
`als Grundlage zur Beurteilung von Mischversuchen,
`Chern. Ingr. -Tech., 26 (1954) 331-337.
`11 J.A. Hersey, The assessment of homogeneity in
`powder mixtures, J. Pharm. Pharmaco1., Suppl.,
`19 (1967) 1688-1768.
`12 D. Buslik, A proposed universal homogeneity and
`mixing index, Powder Technol., 7 (1973) 111 - 116.
`13 J.M. Coulson and N.K. Maitra, The mixing of solid
`particles, Ind. Chern. Mfr:, 26 (1950) 55 - 60.
`
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