Powder Technology. 11 (1975) 41—44
`© 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-
`hesive, nun-'mtaemcting particulate systems are
`primarily mixed by this process. Cohesive, in-
`teracting particulate systems may also be mix-
`ed to a high degree of homogeneity. Many of
`the requirements of this type of mixing are
`different from those required for randomiza-
`tion. The process of mixing cohesive, interac-
`ting particulate systems follows a “disorder to
`order” 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
`random 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-
`proach to practical mixing problems.
`
`INTRODUCTION
`
`Powder mixing has been the subject of nu-
`merous investigations over the past decade
`[1,2]. For simplicity, most of the systems ex—
`amined have consisted of comparafively coarse,
`free-flowing particles and have led to the con-
`cept of randomization or “shuffling" of the
`particles as the mixing process. The randomi-
`zation may be brought about by a variety of
`mechanisms, including difl'usion and convec-
`tion, according to the variety of mixer employ-
`ed. Randomization requires equally sized and
`weighted particles, with little or no surface ef-
`
`fects, showing no cohesion or interparticle in—
`teraction, to achieve the best results- Undoubt-
`edly, this is an important process in powder
`technology today and has served a useful pur-
`pose in enabling mixing theories to be quanti-
`fied. However, it cannot be applied to all prac-
`tical mixing situations, especially for 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~
`teracting fine particles. There is no theoretical
`reason why fine particles cannot be mixed by a
`randomizing prowess. However, cohesive proper—
`ties and other surface phenomena usually devel-
`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 owned 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-
`face tension, frictional, electrostatic or any
`other form of adhesion. It results in an ordered
`
`mixing arrangement of the particles, which is
`best shown diagrammatically in Fig. 1(a) for an
`equal mixture of black and white particles, al-
`though ordered mixtures are more likely to oc~
`cur where there are a few large black particles
`and many small white particles. Figure 1(b)
`shows the same mixture in a randomized
`
`pattern.
`Indications that ordered mixing may Occur
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`(o)
`
`h
`
`i
`
`(m
`
`Fig. 1.(a}. Ordered mixture of equal proportions of
`black and white particles.
`
`Fig. 1(b). Randomized mixture of equal proportions of
`black and white particles.
`
`are available in the literature from mixing obser-
`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-
`ticulate system.
`Studies of homogeneity of powder mixtures
`have frequently been based on the heteroge-
`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 logarithnaic relationship [13] as would
`be required by the first-order kinetics sugges-
`ted for ordered mixing. It is the purpose of
`this paper to consider the possibilities of or-
`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-
`ticle = «Dz. The area occupied by each small
`particle adhering to a larger one will be the
`projected area 1rd2/4. Therefore the number
`of small particles adhering to each larger one
`in a monolayer = 4D2 1112.
`Allowing for the fact that the small partic-
`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 = tli‘IDzld2
`
`(1)
`
`A more exact solution [7] for small partic~
`les close-packed in an hexagonal arrangement
`is given by
`
`n=21r (D+d)2f
`
`«3d2
`
`(2)
`
`If the mixture contains 1% by weight of par—
`ticles of diameter, d , 5 X 10— 4 cm in partic-
`les of diameter, D, 5 X 10"2 cm, then the num-
`ber of small, Nd , and large ND, particles in a
`l-g sample is given by (the density for each is
`taken as 1.2 g era-'3):
`
`d3
`Nd7r T; 1.2 -=_o.01; Nd - 1.27 x 108
`
`D3
`ND 7r? 1-2 = 0-99;ND = 1-26 x 104
`Thus the number of small balls adhering to
`each large one is given by n = Nd/ND = 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-
`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-
`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 is pro-
`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-
`sidered to consist of many components. Alter—
`natively, an equilibrium situation could be es-
`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-
`_ 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-
`rence at all available sites and the excess fines
`
`may attempt to mix with this material in a ran-
`dom manner. Thus, a binary mixture could
`theoretically exist, but its composition would
`be different from that considered by the ingre-
`dient concentrations present. In the above ex-
`ample, if 10% fines had been used the concen-
`tration of coarse particles would be 90%. The
`coarse particles would come to equilibrium
`with 0.9% of fine material giving concentra-
`tions of 90.9 and 9.1 adhered units and excess
`
`fines respectively.
`
`Homogeneity
`Buslik [12] defined homogeneity, H, as the
`reciprocal of (WI), that weight of sample ne-
`cessary to give a standard deviation of 1%:
`
`H: 1/W1
`or using a logarithmic scale, where H.- is an in-
`dex of homogeneity,
`I
`.= —— log W1
`
`(3)
`
`For a randomized mixture, the standard de-
`viation, a, is given by
`
`02 = G(100 — G)w/W
`
`(4)
`
`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-
`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
`
`W1 = G(100—G)w
`Substituting in eqn. (3),
`
`Hi = — log [G(100 — G)w]
`
`(5)
`
`(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-
`mum sample size as a single hydrogen molecule,
`thus giving the limiting value which Ii. can at-
`tain as 23.5. Sodium chloride was treated simi-
`
`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 H,, which lies between 22.0 and 22.2.
`In a completely ordered system, the compo-
`nents are mixed so that they may be considered
`as a single material. Thus, the evaluation of H,
`
`43
`
`for such a system would follow exactly the
`same arguments as for sodium chloride. The
`sample size would necessarily have to be smal-
`ler than one complete ordered unit and the
`homogeneity would lie between the respective
`values for the weights of a large (W0) and a
`small (Wd) particle adhering to it.
`a
`3
`WD = 1.2«%— and Wd = 1.21r%
`
`giving 4.11 < H".< 10.11.
`Where only 0.5% of fines has been used, HI.
`still lies within the above limits providing an
`equilibrium situation can be attained. Where
`an excess of fines is used (Le. 10%), then the
`problem is complicated since both ordered mix-
`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-
`ed in large particles at a sample weight of 1 g,
`two mathematical treatments are possible.
`Firstly, the samples could consist of clumps
`of weight equivalent to a single particle, and
`being randomly distributed then H,- = — log
`G(100 — G)W =— log (99 X 1 X 1) = — 2.00.
`01', alternatively, if the individual samples
`were considered as pharmaceutical tablets re—
`quiring the range of contents to be i 15% of
`the mean, then
`
`30 = 0.15 X 1, and o = 0.05.
`
`If this system is randomized, substitution of
`eqn. (4) in eqn. (6) gives
`
`i=—log02W
`
`(7)
`
`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-
`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-
`ing will be proportional to the concentration
`of unmixed fine particles. Such a mixing rate
`is also applicable to random mixing [13] , and
`thus both mixing phenomena could not be se-
`parated by a simple examination of the kine-
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`44
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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-
`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-
`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
`homogeneity. One example is the incorpora-
`tion 0 6 tablet lubricants into tablet granules.
`Applications already exist and ordered mixing
`provides a means of understanding how such
`processes are possible. Undoubtedly, other ap-
`plications can be considered as soon as the
`mechanisms of ordered mixing are fully recog-
`nized and understood.
`
`REFERENCES
`
`1 5.8. Wiedenbaurn, Mixing of solids, Advan.
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`
`F.H.H. Valentin, Mxing of powders, pastes and
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`(Oct) (1967) 69 - 71.
`D.N. Travers and TLC. White, The mixing of rnicro-
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`2615.
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`.P.M.C. Lacey, Development in the theory of par-
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`M.D. Ashton and F.H.H. Valentin, The mixing of
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`K. Stange, Die Mischgiite einer Zufallsmischung
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`D. Buslik, A proposed universal homogeneity and
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