Chapter 27 Liquid bridges in granules

Chapter 27 Liquid bridges in granules

CHAPTER 27 Liquid Bridges in Granules Stefaan J.R. Simons Department of Chemical Engineering, University College London, Torrington Place, London WCI...

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CHAPTER 27

Liquid Bridges in Granules Stefaan J.R. Simons Department of Chemical Engineering, University College London, Torrington Place, London WCIE 7JE, United Kingdom Contents 1. Introduction 2. Adhesive forces arising from liquid bridges 2.1 The Young-Laplace equation 2.2 Numerical solution of the Young-Laplace equation 2.3 The rupture distance of a liquid bridge 2.4 The static liquid bridge force: capillary and surface tension effects 2.5 The viscous contribution to the general force expression 2.6 Rupture energy of a liquid bridge 3. Direct observation and measurement of liquid bridge behaviour 3.1. The micro-force balance 3.2. Particle wettability in relation to the geometry of a liquid bridge: approximated liquid bridge profiles 3.2.1. The toroidal and parabolic models 3.2.2. Comparison of the toroidal and parabolic approximations 4. Relating particle-binder interactions to granule behaviour 4.1. Compression of plastic agglomerates 4.2. Experimental validation of the hardness equation 4.3. An industrial case study: predicting pharmaceutical granulation performance from micro-scale measurements 4.3.1. Granulation of paracetamol 4.3.2. Binder selection criteria 4.3.3. Experimental procedure 4.3.4. Results and discussion 5. Conclusions References

1257 1259 1260 1262 1265 1266 1270 1271 1273 1274 1276 1278 1283 1292 1294 1298 1302 1303 1303 1305 1308 1312 1315

1. Introduction Many granulation processes rely on the addition of a binder (which is either sprayed, poured or melted onto a bed of dry particles) that will spread over the

*Corresponding author. E-mail: [email protected]

Granulation Edited by A.D. Salman, M.J. Hounslow and J. P.K. Seville ,i~ 2007 Elsevier B.V. All rights reserved

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S.J.R. Simons

particles on mixing to form liquid bridges between them. These bridges then create sufficient adhesion between the particles to enable them to stay together as nuclei agglomerates, which then go on to pick up more "wet" particles and grow and compact to form distinct granules. The growth mechanisms may also include breakage and deformation. In most cases, the liquid bridges go on to form solid bonds, either by drying, reacting or simply changing phase with temperature. Such processes can be referred to as wet or binder-induced granulation. Liquid bridges can be defined as lens-shaped liquid menisci (with the curved surface in contact with another fluid) supported by at least one solid surface [1]. The configuration assumed by a liquid bridge between two particles is termed "pendular". Tumbling drums, fluidised beds and high-shear mixers are devices commonly used to provide the appropriate conditions of agitation to the bed of powder, yet introducing a mechanical shear stress that may rupture bonds already formed within an agglomerate. The growth rate of granules, which depends on the equilibrium between bond formation and bond disruption, is therefore influenced by the type of apparatus used for granulation. It is clear that the modelling of pendular liquid bridges and their behaviour is of great importance to wet granulation processes. The prediction of growth kinetics, for instance, frequently includes parameters such as the strength of agglomerates [2,3], which in turn depends on the strength of individual liquid bridges [4]. In the published literature, work has concentrated on both the experimental investigation of the force developed by a liquid bridge [5,6] and on its modelling [7]. To describe the configuration assumed by a liquid bridge, some models assume that the liquid-to-solid contact angle remains either constant or zero throughout separation and that the liquid bridge shape can be described by either a toroidal approximation or a catenoid function [8]. In other approaches, the liquid bridge shape is not presupposed and results from the minimisation of the system free energy for a constant volume condition using the Young-Laplace relationship [7,9]. Very few models in the literature take into account the wetting hysteresis of the liquid on the solid surfaces [10,11]. In addition to bridge strength, several workers have also measured and theoretically predicted the pendular bridge rupture distance [7]. The rupture distance is a parameter which is required for discrete element modelling (DEM) of granules in the pendular or funicular state where bridges will break and reform as granules deform during collisions. The post-rupture liquid distribution on the particles is also important. Dabros and van de Ven [12] discussed the dispersion of droplets at the surface of similar sized spheres in agitation. When the size, shape and surface energy of the particles vary, the liquid distribution is expected to change. This is demonstrated experimentally through the occurrence of wetting segregation when powder mixtures of varying surface energy are co-granulated [13].

Liquid Bridges in Granules

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In many industries, granulation processes are often applied to mixtures of powders in which the component solids do not exhibit the same surface properties. Large discrepancies in surface energy create problems during granulation, as powders can be selectively wet at the expense of others [14]. In the pharmaceutical industry, most drugs have a low surface energy and therefore are poorly wet by common granulation liquids [15]. However, the behaviour of pendular liquid bridges when the liquid-to-solid contact angle is large has not been extensively covered in the literature. The majority of models dealing with pendular liquid bridges either assume zero [16], small [8] or fixed contact angles [9,17]. When the wetting hysteresis is important, the solid-liquid interfaces at the boundaries of the liquid bridge can remain constant with particle separation until a critical separation is reached, at which point the bridge liquid recedes from one particle surface and the corresponding solid-liquid interface reduces. The reduction of the solid-liquid interfacial area with particle separation is accentuated by low wetting hysteresis that drastically alters the shape of pendular liquid bridges, their rupture distance and post-rupture liquid volume distribution on the solid particles. In this chapter, the current theory on liquid bridges between pairs of particles will be presented, followed by a detailed review of the work carried out by the author and co-workers at University College London on the modelling of liquid bridges and the relationship between the micro- and macro-scale granule behaviour, developed from direct measurements and observations of liquid bridges between smooth spheres and between real pharmaceutical powders and binders.

2. ADHESIVE FORCES ARISING FROM LIQUID BRIDGES The formation of wet agglomerates is governed by the balance between the rupture energy of liquid bridges and the particle kinetic energy, and hence a knowledge of the liquid bridge adhesive force (also referred to as the strength of the liquid bridge), in relation to the type of liquid binder and the surface properties of the particulate, is required for a fundamental understanding of granule growth. Liquid bridge forces arise from both capillary and surface tension effects, which are static forces, and from a viscous component, which becomes more important during dynamic separation. The force that holds the particles together is ultimately related to the ability of a liquid binder to wet the particles to form effective bonds. Therefore, the study of the geometry assumed by a liquid bridge is essential in determining the adhesive force. Liquid bridges formed between two particles assume a lens-shaped profile, which can be described theoretically by the Young-Laplace equation. Being able to solve this equation for various conditions is important in the development of agglomeration models.

1260 2.1.

S.J.R. Simons The

Young-Laplace

equation

The configuration assumed by a liquid bridge, at rest, is implicitly defined by the Young-Laplace equation: 1 AP - 7 L ( 1 + ~21

(1t

Equation (1) relates the difference in hydrostatic pressure AP = P-Pext across the vapour-liquid or liquid-liquid interface (if the surrounding medium is a liquid) to the local radii of curvature rl and r2 (Fig 1) and to the interracial tension ~L between the liquid bridge and the external medium [18] P and Pext are the pressures inside the bridge and in the external medium, respectively The configuration of the liquid bridge can be readily obtained from equation (1) once the local radii of curvature are rewritten in analytical terms, which, by using a cylindrical coordinates system, results in [9]: AP 1 y" = 7L y(1 + y'2) 1/2 (1 + y,2)3/2

(2)

where y' an y" are the first and second derivatives of the liquid bridge profile, which is described by the function y(x). For relatively "large" liquid bridges, the geometry of the liquid bridge is influenced by the gravitational distortion. In this situation, the profile mean curvature is not uniform throughout the liquid bridge and equation (1) must therefore change accordingly. As detailed by Mazzone etal. [8], gravity has negligible effects on the bridge geometry when equation (3) is satisfied"

gL2Ap APR

(3)

where g is the gravitational acceleration, L is some characteristic length of the bridge and Ap the difference between the densities of the liquid binder and the external medium. "Small" liquid bridge volumes are defined as those meeting (3), as opposed to "large" volumes where gravity does have an effect on the geometry of the bridge.

r~

A(O,yA) /

G

(0,0)

,

~l q

/]

,

'~ ..

......~ ...........................................................................~

Fig. 1. Geometric parameters describing a liquid bridge between unequal spheres.

Liquid Bridges in Granules

1261

The solution of the Young-Laplace equation has been studied by Orr etal. [18] for liquid bridges formed between a sphere and a flat surface, in terms of elliptical integrals. He also classified possible liquid bridge geometries according to the meridional curvature, defined as -d/dx(sin CE/R), where s is the angle made by the normal to the meniscus with the axis of symmetry, as indicated in Fig~ 1. In Fig~ 2 the meridional curvature is negative in (a)-(c); it is negative in the top part and positive in the bottom branch of (d) and vice versa in (f); it is zero in (e) and positive for (g)-(i). Thick lines represent the inflection points~ Considering the mean curvature, it is negative in (a) (which leads to a pressure inside the

i

i (a) Nodoid segment

(d) Unduloid segment

(b) Catenoid segment

(e) Cylinder segment

(c) Unduloid segment

)

(f) Unduloid segment

C2 ) (g) Unduloid segment

(e) Zone of sphere

(i) Nodoid segment

Fig. 2. Profiles of axisymmetric menisci of uniform mean curvature (after Orr et al. [18]).

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S.J.R. Simons

meniscus lower than that of the surrounding medium), it is zero in (b) and positive in all the other situations. Equation (2) can be rearranged into a Bernoulli-type differential equation and, following classical integration, equation (4) is obtained, where B is an integral constant [9]: Y - APy 2 + B (1 + y'2) 1/2 27 L -

(4)

-

By substituting either of the two y'~ with cotg(e~+/~) into (4) (i refers to either point A or B in Fig. 1), the constant B can be evaluated, which, for point A, becomes AP e - YA sin(~A +/~A) ~ y2

(5)

If the coordinates of both points A and B are substituted into (4), the constant B can be eliminated" AP

YA sin(~A + / ~ A ) - YB sin(~g +/tg)

(6)

Equation (6) allows AP across the interface to be calculated once the interfacial tension 7L and the boundary values of the bridge profile, YA, YB, (0A+,~A), (~g +/~g), are known. However, equation (2) can only be solved analytically in a few cases, for instance, when the angle ~ +/~ = 90 ~ on both the equi-sized particles. Hence, in general, a numerical solution is required.

2.2. Numerical solution of the Young-Laplace equation A numerical method to solve the Young-Laplace equation is given in Ref. [17], where a symmetrical liquid bridge, between equi-sized particles of radius R, is considered, as indicated in Fig. 3.

Fig. 3. The Lian etal. physical model for a liquid bridge between equi-sized spheres (after Lian et al. [17]).

Liquid Bridges in Granules

1263

The solution of the bridge profile can be approximated through a truncated Taylor series to obtain the recurrence equation: 1

fi

Yi+I ~ YJ + YI(XJ+I - Xi) + ~ Yi (Xi+l - Xi) 2 + . . .

i - 0, 1,2, ...

(7)

where Y and X are dimensionless coordinates with respect to the particle radius R. Rearranging equation (4) and equation (2), the expressions for YI and YI' can be determined by considering that at Xc, Yc - sin/~ and Y'c - cotg (/~ + ~): ~

Yi

Y~ -

s i n / ~ sin(/~ + o) +

Yi =

H*(Y~-

y~+2H*(I+

sin2/~) '

9

- 1

(a)

(9)

where H ~ is the dimensionless mean curvature, H ~ = APR/27L. Owing to the configuration symmetry, Lian et aL [17] studied only the half profile, from the neck position Yo, to the contact point with the particle, Yc. Liquid bridge configurations were evaluated from contact through to rupture by considering a fixed contact angle ~ and a constant liquid bridge volume. The configurations assumed by the liquid bridge for different separation distances a, can be determined by specifying values for both H ~ and the half-filling angle/~. When the solutions of the Young-Laplace equation were analysed, two possible configurations resulted in agreement with equation (2) for the selected set of volume and contact angle (Fig. 4). The two possible solutions were also noted by Erie etal. [19] and De Bisschop etaL [9], who identified the stable configuration as the one that minimizes the free Helmotz energy E of the liquid bridge. In the analysis presented in Ref. [9], the rupture configuration was recognised as occurring when the two solutions became coincident. If we regard u(x) as the profile of the particle wetted by the liquid binder and Xc the abscissa of the binder to particle contact point (Fig. 3), the free Helmotz energy E can be written as [9]: E - 2~

[x xcl27LY(1

+

y'2)1/2

_

1

27 LCOS Ou(1 + u '2)1/2 dx

+ (mext - P ) Vbr (1 O)

J•

where the first term under the integral represents the energy stored within the liquid bridge surface, the second term is the energy contribution due to the surface of the particle wetted by the binder, -PVbr is the energetic contribution due to the "PV-work" and Pext Vbr is a constant (Vbr is the volume of the liquid bridge), whose physical meaning is the energy involved in displacing the suspending medium and replacing it with the liquid bridge.

1264

S.J.R. Simons ".,..% "% ",,,..

0.3-

(a)

(b)

"',,.. ~

"',.%. ".,. ".,., "%%

L.. O

:% \.

E

E

9

ffl

('-

\

Vb~* = 0.005

E~ %

0.1

._o E

Vbr

=

0.001

"N~ ,.,.,,o i

E

......"";

..... -

o.'1o

"

,

Vbr* = 0 . 0 0 5

Vb* = 0.001

-

0.20

(c) r* =

00

(/) E

e

(1)

S

1

~

%

=

Vbr* E

if) m

4,

E

2'

0

(d)

t, 0L 0 LL

E

E

0.;0

Dimensionless separation, 2a*

01

> L

9- -

,

0.~0

14

12.

O o-ffl r

,

0,0

'.00

0~0

Dimensionless separation, 2a* r'

Vbr* = 0.03

~~

I

..,,.--"..... i.Geel,.p,"*~

a

0.0 0.00

9

t~

~,~5Vbr* ~* =

= 0.03

~

0.001 %

5

Vbr* =

"~

0.03

.................... '

,.,'"

.,.........

/....,::::""-....I.............

.1

E D

0,

.2

0 (/) E:

~

~

J

'.

'

-

.......

r '

ooo o,o o~o o.~o Dimensionless separation, 2a*

.01

0,00

-

,

0.10

0.20

0.30

Dimensionless separation, 2a*

Fig. 4. Stable ( - - ) and unstable ( . . . . ) numerical solutions of the Young-Laplace equation at different dimensionless separation distances a*( = a/R) for a range of dimensionless liquid bridge volumes V~)r and zero contact angle in terms of (a) the dimensionless neck radius, (b) the half-filling angle fl, (c) the dimensionless mean curvature H*, and (d) the dimensionless total liquid bridge force F* (after Lian etal. [17]).

The solution method proposed by Lian et aL [17] introduces a significant simplification by assuming a constant contact angle 8 throughout separation. In real cases, this situation does not apply because the contact angle varies according to the interaction exhibited with the particle. Effects of contact angle hysteresis will be detailed in Section 3.2. Another question arises as to whether the volume of the liquid bridge can be assumed constant during separation. Simons and Fairbrother [20] and more recently Pepin etal. [21] have shown that this volume, during separation, is either constant or varying according to the wettability of the powder. Between two perfectly wettable particles, the liquid binder tends to saturate the particles first, by forming a liquid reservoir around them, before being available to form liquid

Liquid Bridges in Granules

1265

bridges. Under such conditions, during separation, the volume of the binder can vary at the expense of the volume held in the reservoirs. To avoid the complexity of the solution of equation (2), simplified models exist for the description of the geometry assumed by a liquid bridge. These can then be used to derive parameters such as the rupture distance, post-rupture liquid distribution and apparent contact angles. Among these, the toroidal approximation, which approximates the liquid profile with an arc of circumference, has gained some popularity amongst researchers. We have developed a novel parabolic approximation. Both these methods will be discussed in Section 3.2.1 and their solutions compared with experimental data.

2.3. The rupture distance of a liquid bridge When the separation distance of a liquid bridge held between two particles is increased, the meniscus displaces until a certain critical bridge separation is attained, at which point the bridge becomes unstable and ruptures. The rupture of a liquid bridge is a very rapid process that involves some complex phenomena, which can only be studied using a relatively fast camera (i.e. 500 frames/s) [22]. Recorded sequences of pendant drops deformed under the effect of gravity show that near rupture the meniscus is similar to an umbilical cord, of finite length and very small radius, joining the two liquid masses which are about to separate. The phenomena involved in the disruption of a cylindrical meniscus were studied by Plateau [23] in terms of the increasing instability of the shape due to the formation of capillary waves generated by external disturbances. The capillary waves narrow the thin umbilical cord and eventually the bridge breaks. During rupture some satellite drops may form as a consequence of this process so that the volume of the bridge is not exactly conserved [22], while viscous dissipation also occurs due to the rapid process of liquid redistribution of the separated droplets. De Bisschop and Rigole [9] stated that as separation distance is increased the half-filling angle (/~ in Fig. 1) decreases continuously and rupture occurs when it reaches a minimum. It was subsequently observed by Mazzone et al. [8] that a minimum half-filling angle is indeed reached, but that stable bridges can exist on increasing the separation distance beyond this point with /~ actually increasing before rupture. As shown by Lian et al. [17] (see Fig. 4(b)), theory predicts this observed rise in the half-filling angle before the critical separation distance is reached. Following on from the solutions shown in Fig. 4, Lian etaL [17] derived a simple relationship between the rupture distance and the bridge volume. By plotting dimensionless volume V~r against the dimensionless rupture distance

1266 ama x --

S.J.R. Simons

amax/R, they proposed the following relationship" * x area

~ (1 + 0.50) ~~ *br

(11)

where 0 is the solid-liquid contact angle expressed in radians. It will be shown in Section 2.6 how this very useful equation can be used in the calculation of liquid bridge rupture energies.

2.4. The static liquid bridge force" capillary and surface tension effects The static force exerted by a liquid bridge is made up of two parts; that due to the surface (interfacial) tension and that due to the hydrostatic pressure within the bridge, determined from the Young-Laplace equation (1). To calculate the liquid bridge force formed between two particles, two different approaches are commonly used, which lead to slightly different values. In the first case, the force is determined by considerations at the neck of the bridge [24], as in equation (12), while the second method considers the interfaces between the particles and the liquid bridge [18], as in equation (13): Fn = 2~r17L -- ~r2Ap

(12)

Fb -- 2~R sin J~TL sin(/~ + 0) - ~R2sin2/~AP

(13)

with the symbols defined as before. In equation (12) the surface tension and the capillary pressure terms multiply the circumference and the area of the neck, respectively, while in equation (13) the circumference and the area of the contact between the particle and the liquid are used instead (see Fig. 1). In both equation (12) and equation (13), which are developed for spheres of identical size (for dissimilar particles R is replaced by the geometric average radius ,R; R - 2 R A R B / R A -4- RB), the effect of gravity is neglected. When "small" volumes of binder are administered between highly wettable particles in contact (contact angle, 0,-~0~ liquid bridges assume a nodoid configuration [18] with a negative mean curvature (q >0, r2<0 and I q l > I r l), as indicated in Fig. 5. The pressure deficiency (i.e. AP<0) across the liquid bridge leads to a higher adhesion force than the case where AP> 0, according to either equation (12) or equation (13). The magnitude of the liquid bridge force is difficult to compute exactly, even for simple geometries (sphere-to-plane or sphere-to-sphere), because the three-dimensional bridge forms an interface of constant curvature to satisfy equation (1). Fisher [24] developed the toroidal (circular) approximation, assuming that an arc of a circumference can approximate the exact nodoid configuration of a liquid

Liquid Bridges in Granules ~

1267 External

rticle Fig. 5. Nodoid configuration of a liquid bridge. bridge formed between perfectly wettable particles in contact (i.e. 0 = 0) (Fig. 5). At particle contact (i.e. a = 0): rl = R(sec/~ - 1)

(14)

r2 = R(1 + tan/~ - sec/~)

(15)

Substituting equation (14) and equation (15)into (12) then yields equation (16), which has been shown to compare favourably with the values obtained via the exact solution of the Young-Laplace equation (1) for the conditions mentioned above [8,17]: F -

2~RTL 1 + tan/~/2

(16)

Following substitution of equation (14) and equation (15) into equation (13), the torroidal approximation of the boundary method is F-

I2t - t + 1] 2~RTL [ i] + t2)2

(17)

where t = tan(fl/2). This yields values of F within a few percent of those given by equation (16) for half-filling angles between 10 and 40 ~ (coalescence limits for liquid bridges between spheres are 30 and 45 ~ for closed-packed and cubic arrangements, respectively [25] and below 10~ the contribution of the surface tension is negligible [26]). A parabolic approximation has been developed that results in a much simpler and more robust mathematical expression that can be used to evaluate the principal physical and geometric liquid bridge parameters (i.e. contact angle, curvature and strength of the liquid meniscus) [21,27,28]. The development of this approximation, its experimental validation and the comparison with the traditional toroidal method, will be detailed in Section 3.2. The configuration assumed by the liquid bridge in Fig. 5 is not a general case. In many granulation processes, particles exhibiting different surface energies are

1268

S.J.R. Simons

processed together and, as a result, some particles can be selectively wet at the expense of others [14]. In this scenario, during particle separation, the liquid binder can recede from those particles exhibiting lower surface energies which, as separation distance is increased, turns the profile of the liquid bridge from a nodoid geometry to one that is unduloid [28]. The geometry assumed by a liquid bridge in such conditions will be discussed in Section 3.2.2. Different workers [17,29,30] have investigated, theoretically, the effects that separation has on the strength of liquid bridges formed between perfectly wettable particles. The trend reported in Fig. 6 shows a decrease in the adhesion force (calculated using equation (16)) throughout separation as a consequence of the thinning of the bridge neck and the increase of the capillary pressure. It should be noted that, as the relative bridge volume q~ becomes smaller, the magnitude of the force becomes more sensitive to the variation in the interparticle distance. The trend shown in Fig. 6 is valid only for the case of quasi-static separation, constant volume and 0 = 0. It will be shown later how this trend differs under nonperfect wetting conditions, non-constant volumes and dynamic situations where viscous forces become dominant. Experimental data reported in [6,8,20] agree with the trend of Fig. 6, although the force appeared to reach a maximum at small but non-zero separation distances (Fig. 7), which is not predicted by theory. Mason and Clark [6] attributed this rise to an initial finite contact angle greater than zero that then reduces to zero as separation increases. The decrease in contact angle leads to a change in the profile curvature. In this situation, the 3.0

I

....

I .... V

2,5

2,0

..x-

.

.

.

.

.

.

.

1.S

1,0

0,5

0

0,05

0,10 a*

0,15

0,20

Fig. 6. Theoretical dimensionless adhesion force F*(= F/?R) of a liquid bridge between two equal spheres against the dimensionless separation a*(= a/R). The parameter, q~, is the volume of the liquid bridge divided by the volume of the spheres (after Schubert [30]; the author indicated the diameter of the spheres as R).

Liquid Bridges in Granules

1269

~9149 O ~ O

C

\ \ \ \\. ,,.

9

\\\

"C]

\k.

\

\

--.

o

\

0 U_

\

\

\

k

5

~.

,oo:2,,o-'.,L

~ ",., "\ ",,., ",.,.. "-.., k

\,,,'\"--.,.. .~..,.~..,,

9\._ %, ",....

....-,..,,"->.

,~,.

'l

~so

l 0

'

- ~,,o ~

.-7".1 -"'.-..,.

I00

ZOO

~

~

,~T

~

9

9

~ \n~

\

5O0

\2so "%

,\

.3~o

,

_

400

5OO

S e p a r a t i o n , x10 .3 cm

Fig. 7. Force/separation curves for oil bridges between two polythene sphere (radii 15 mm) suspended in water (after Mason and Clark [6]). 2.50

9 Measured force

2.00 t~

C

| Theoretical Force

O

z

1.50

2u 1.00 O L_ O I,I.

0.50

0.00 0

2

4

6

8

10

12

14

16

18

Separation (microns) Fig. 8. Force versus separation for a silicon oil liquid bridge holding two glass silanised ballotini of 23 l~m radii suspended in air (after Simons and Fairbrother [20]).

capillary pressure reaches a minimum, which leads to the initial increase in the liquid bridge force. Simons and Fairbrother [20] measured liquid bridge forces between particles in the micron size range (Fig. 8), using the micro-force balance described later in

1270

S.J.R. Simons

Section 3.1. Although the judgment of where the particle contact occurred was an arbitrary decision made by microscopic observation, the same trend as shown in Fig. 8 is clearly visible.

2.5. The viscous contribution to the general force expression During dynamic liquid bridge separations, the shear stress inside the liquid, caused by a velocity gradient in the direction orthogonal to that of separation, gives rise to an additional force, which depends on the viscosity of the liquid binder. The expression of the viscous force between two equi-sized spheres held by an infinite liquid bridge (the particles are submerged in the liquid)is given by equation (18), which is valid when the particle radius is large in comparison to the distance of closest approach (R,>a) [31,32]. In equation (18), ~/ is the viscosity of the liquid whilst the radius R is replaced by the geometric average radius,/~, for particles of dissimilar size. In this situation, at low Reynolds numbers (Re = vpR/~1, with v being the particle separation speed), the flow of liquid in the region between the surfaces may be described by the lubrication approximation [33], which assumes the flow to be similar to that between parallel plates where the velocity field is large in the direction orthogonal to that of separation and derivatives in the direction of separation are dominant. 3 1 da Fvis - ~ ~:~/R2 (18) adt Although for liquid bridges of finite volume such an analysis of the viscous contribution ignores the existence and influence of the bridge meniscus on the region of closest approach of the particles, the use of equation (18) is justified in the limit of small (< 10 -3) capillary numbers ( C a = Vfl/~L , the ratio between viscous and surface tension effects), small gap distances (a* ,-~ 0.01) and sufficient bridge volumes (V~r "~ 0.05) [34]. In fact, small capillary numbers imply that the viscosity does not affect the liquid bridge interface, while the restraints on the distance of closest approach and on the volume of the bridge justify a lubrication analysis for the viscous contribution. Under these circumstances, equation (18) can be added to either equation (12) or equation (13) for dynamic separations. By choosing equation (12), the more general expression of the liquid bridge force becomes 3 1 da F - 2~rlTL -- ~r2Ap + -2~tlR 2 adt

(19)

The force curves generated by equation (19), calculated using parabolic approximations to the bridge profile (Section 3.2.1), have been compared with experimental data obtained by the author and co-workers [35] and found to be accurate for spherical particles. A model for the hardness of wet granules developed from the basis of this expression is derived in Section 4.1 and compared with experimental results.

Liquid Bridges in Granules

1271

2.6. Rupture energy of a liquid bridge The rupture energy of a liquid bridge is usually calculated by integration of the force exerted by a meniscus throughout separation, from contact to rupture. Simplifications of the force expression are usually introduced, due to the difficulty in dealing with the general problem of the liquid bridge deformation. Two models, proposed by Simons et al. [26] and Pitois et al. [36], are discussed below. In both the models only the energy arising from capillary forces is evaluated. The model proposed by Simons etal. [26] is derived from the integration of the total liquid bridge force calculated using the toroidal approximation equation (16) and is written as: Xtan/~ F - ~TLR(1 + X tan/~ - X sec/~) X sec/~ - 1

(20)

where X = (1 + (a/2R)) and/~ is the half-filling angle, defined as in Fig. 1. In the integration of equation (20) through separation distance a,/~ was considered to be a constant. This approximation, due to the difficulty in being able to predict/~ for each value of a, seems reasonable for particles that exhibit a strong interaction towards the binder, where the solid-liquid interface stays almost constant (Section 3.2.2) [27]. Furthermore, it has been shown theoretically by Lian et al. [17] (see Fig. 4(b)) that the overall change in /~ is small for perfectly wetted spheres. The expression of the dimensionless rupture energy, IN* = W/~/LR2, calculated between any two configurations X m i n and Xmax is thus:

I xin

W* - 2re

(tan 2/~ cos/~ - tan ~) + X sin 2/~ + tan 2/? c o s 3/? In(X sec/~ - 1) Xmax

(21) in which X m i n = 1 and X m a x = 1 + amax/2, where amax is the rupture distance calculated using equation (11). A plot of W* against the dimensionless liquid bridge volume V~)r then leads to: W* = 3.6 ~/-V*br

(22)

Pitois et al. [36] used a cylindrical approximation to the bridge profile, leading to the following expression for the total force:

[ /

F - 2 ~ L R cos 0 1 -

1 4- ~a 2RJ

(23)

By using the approximation that ~ stays constant throughout separation, equation (23) can be integrated with respect to the separation distance a, to obtain the

1272

S.J.R. Simons

rupture energy, which in non-dimensional form reads as: W*-

2~a*cos0

1-

1+

(24)

,2 a;

If a~ - 0 and a~ -- (1 + 0.50)~~br (see equation (11)), equation (25)is obtained: W* = 2~cos0 [(1 + 0 . 5 0 ) ( 1 - C) ~~br + v/2V-~ ~r]

(25)

where C -- V/(1 + 2V~r//'C(1 4-- 0.50) 2, 0 is the contact angle expressed in radians, while W* and Vbr are defined as above. It can be seen that equation (22) and equation (25) depend only on global parameters, such as volume and contact angle. Rossetti et aL [37] have compared these two models with experimental data on liquid bridges obtained from the study of pairs of particles with similar and dissimilar surface energies, using the micromanipulation technique described in Section 3.1. They found that the model predictions were in reasonably close agreement in all cases, although equation (25) was slightly better than equation (22) when the assumption of perfect wetting was not valid, due to the inclusion of the contact angle. In dynamic situations, as experienced in a granulator, the rupture energy of liquid bridges plays a less significant role. Lian etal. [38] have studied, using computer simulations, the deformation behaviour of moist agglomerates formed in a gaseous system. The model is focused on the dissipation mechanisms of the kinetic energy upon reciprocal collision of two agglomerates, which is illustrated in Fig. 9. Dissipation of kinetic energy for the moist deformed agglomerates was not solely due

(a)

(b)

(c)

Fig. 9. Visualizations of computer simulated wet agglomerates for an interstitial fluid viscosity of 10 mPa s after impact at relative velocities of (a) 0.5 m/s, (b) 2.0 m/s and (c) 5.0 m/s (after Lian et al. [38]).

Liquid Bridges in Granules

1273

to the viscous resistance and breakage of the interstitial liquid bridges, but also due to rearrangement (plastic deformation) of the particle structure, which involves friction dissipation according to the theories of Johnson [39]. By setting the viscosity of the binder at 10 mPa s and the collision velocities in a range between 0.5 and 5 m/s, the viscous force was found to account for the dissipation of about 60% of the initial kinetic energy. Energy dissipated by friction was also very significant (--~30%), whilst the energy dissipated as a result of rupturing the internal liquid bridges was only a small proportion, at around 5%. It appears that there are limiting conditions to when either surface tension or viscosity dominate the energy dissipation, while there is sufficient evidence that friction plays an important role [21]. These limits depend not only on the values of these parameters, but also on the volume of liquid, since this governs, to a certain extent (Section 3.2.1) the bridge curvature and, hence, the capillary forces.

3. DIRECT OBSERVATION AND MEASUREMENT OF LIQUID BRIDGE BEHAVIOUR Micro-mechanistic approaches to determine granule properties and granulation performance have gained favour over the recent years, since many believe that it is the interfacial properties that are the governing parameters. The challenge is to relate what is observed at the solid-solid, solid-liquid and solid-vessel interfaces to multi-particle granules that often have unknown structures and compositions, particularly in relation to binder distribution, and that are experiencing complex shear conditions. Nevertheless, progress is being made in the fundamental understanding of such effects as granule strength [21,40], deformation [38,41] and attrition [42]. At UCL, micromanipulation techniques have been developed by the author and co-workers over the past decade that have had significant success in elucidating liquid bridge behaviour under a range of conditions, in both gaseous [20] and liquid media [37], with simulant (spherical) and real (irregular) particles [21,35], with good and poor wetting [27,28] and at room and high temperatures [43]. Among the major investigations that have taken place, the most important have been those involving particles of different surface energies, which have resulted in a new, parabolic approximation for bridge profiles [27,28] and a predictive model for granule strength [35]. Recently, a case study has been conducted on behalf of an international pharmaceutical company, to establish whether the micromanipulation approach can be used to select the optimal drug/excipient/binder system for successful granulation. These latter three studies will be detailed in the following sections, demonstrating the usefulness of micro-scale data to the prediction of macro-scale granule behaviour. First, however, the micromanipulation device, known as a micro-force balance (MFB), will be described.

1274

S.J.R. Simons

3.1. The micro-force balance The MFB takes the form of a specially adapted microscope stage, coupled, via a digital camera, to an image analysis and video recording system. A schematic of the complete experimental apparatus is shown in Fig. 10. The MFB itself is shown in more detail schematically in Fig. 11. The procedure for forming, observing and taking measurements of liquid bridge behaviour is as follows. Initially, a rigid micropipette, with a particle attached to one end, is clamped onto micromanipulator B, with the particle being placed under the objective lens

i .....................~'[ .....~.......

1..................~......Image ..........analyser 1

Video recorder

i

;amera

........~

Olympus BX60 Microscope

i

............

. . . . . . . . . . . .

!

Computer .......i......

.........................

Stage

Fig. 10. Schematic of the MFB equipment layout. Camera

Microscope BX60 Follower movement

B ~ reflective f o i l ~

Driven movement

Microman. A flexible micropipette

~

Microman. C feedingmicropipette

@

~--. RO incorporating PEC + LVDT

Microman.B rigid micropipette

Driven remotelyby PEC + LVDT

Fig. 11. Schematic of the experimental set-up of the MFB in a gaseous medium.

Liquid Bridges in Granules

1275

of the microscope. This pipette is held static throughout each experiment. Fine adjustment in all three dimensions is achieved using the individual plane micrometers of the micromanipulator. The second particle, attached to a pre-calibrated (for its spring constant) flexible micropipette, is then placed under the objective in contact with the first particle. Again, fine adjustment can be made using the micrometers of the micromanipulator. This micromanipulator A also incorporates a 30 l~m expansion piezo-electric crystal (PEC), which allows the pipette to be driven remotely. Since piezo-electric crystals exhibit non-linear expansion and hysteresis with respect to applied voltage, a linear variable differential transducer (LVDT) is fitted to monitor the PEC's expansion. To form the bridge, binder liquid is fed through a third micropipette onto the particles. The feeding micropipette is pre-loaded with the binder before being mounted on micromanipulator C. Once a drop of liquid binder is formed on the particle attached to the rigid pipette, the two particles are first brought together to form the bridge and then separated until the rupture of the bridge occurs. This is achieved by either applying a signal causing the PEC to expand or by acting manually on the micromanipulator. At this point, the flexible pipette is driven away and the force of the liquid bridge causes the flexible pipette to bend, with the bend being proportional to the force. Under electronic control, separation can take place at different speeds in the range 0.5-10 l~m/s. On the 90 ~ bend nearest to the pipette tip a small piece of aluminium foil is fixed (see Fig. 11). Owing to the separation movement imposed, the bend in the flexible pipette deflects proportionally to the strength exerted by the bridge. The deflection of the pipette is calculated as the difference in displacement between the base of the pipette and the centre of the bend, whose displacement is acquired by an optical follower with a resolution in the order of 75 nm. To control the optical follower, a reflecto-optic (RO) sensor is used to linearly detect the position of the edge of the reflective foil in its field. The RO sensor works by transmitting a beam of light and measuring how much is reflected back. The output from the sensor, when focused on the edge of the reflective foil, reads a constant voltage. Movement onto the reflective foil causes an increase in the output voltage and movement away, a decrease. To keep the sensor focused on the edge of the foil, control electronics are used to drive a second 15 #m expansion PEC. This expansion is measured by a LVDT. The flexible pipette is pre-calibrated to determine its spring constant (usually between 0.05 and 0.5 l~N/pm) and the total force exerted is thus calculated as illustrated in Fig. 12, which shows the steps to formation and separation of a liquid bridge. In Fig. 12(a), the flexible pipette (whose spring constant is ks), is approached by the rigid pipette onto which a liquid droplet has been previously administered. At a certain close distance between the two particles, the flexible pipette "jumps" towards the other pipette to form the liquid bridge, with "e" being

1276

S.J.R. Simons (a)

ei v

(b)

f~

Fref

ks e

=

v

follower movement Xf

(c)

8 driven movement X d

Fig. 12. Schematic showing the method used to calculate the strength of a liquid bridge during separation: (a) particles separated, (b)liquid bridge formation, and (c)liquid bridge separation. the deflection with respect to the undisturbed configuration (Fig. 12(b)). Figure 12(c) shows the separation sequence. When the thick base of the flexible pipette is driven away (distance Xd), the centre of the bend follows but is "retarded" by the strength of the bridge and, in general, the distance Xf is different from X~. The force of the bridge can eventually be calculated as: Fbr = ks(Xd -- Xf) -4- kse

(26)

The separation of particles and monitoring of the LVDTs is computer controlled via an analogue-to-digital interface. A complete description of the device and the computer code can be found in Ref. [44].

3.2. Particle wettability in relation to the geometry of a liquid bridge: approximated liquid bridge profiles Theoretical and experimental studies of liquid bridge forces and geometries have traditionally been carried out between pairs of similar and highly wettable particles, while the situation where the particles have different surface energies has generally been neglected. When different particles are formulated together, which is not unusual during the production of pharmaceutical and agricultural products, surface energy differences can cause preferential agglomeration of some species to occur, due to the fact that some particles are selectively wetted at the expense of others [14]. Particle wettability directly affects the geometry of a liquid bridge and the consequences are also reflected in other properties, such as the force of adhesion, the rupture energy and the post-rupture liquid distribution. The MFB described in the previous section has been used to observe and measure these

Liquid Bridges in Granules

1277

phenomena in a series of experiments involving glass spheres treated to exhibit different surface effects [27,28]. The experiments were carried out using clean glass ballotini in the size range 40-130 pm radius. Glycerol liquid bridges were formed between pairs of glass ballotini, either silanised (using a 2% solution of dimethyldichlorosilane in octamethylcyclotetrasiloxane) or kept in their natural state and the resulting geometries investigated during liquid bridge separation and rupture. The viscosity and surface tension of the glycerol were measured as 1630 mPa s and 63 mN/m, respectively, at 20 ~ The micromanipulation technique was used to observe the liquid bridge formation and rupture behaviour and to measure the liquid bridge geometry directly. A glycerol droplet was fed onto the surface of one of the particles using the feeding micropipette, which was then withdrawn. The two particles were brought into contact to form a liquid bridge and then axially separated with a constant speed of ~1 pm/s. The separation process was recorded with the camera and stills from the video were used for further image analysis. Glycerol exhibits good wettability towards untreated glass and moderate wettability with respect to silanised glass. During separation, the liquid binder can easily recede from particles exhibiting lower surface energies (poor wettability), which, as separation distance is increased, turns the profile of the liquid bridge from a nodoid geometry to one that is unduloid. This is the case for a liquid bridge formed between untreated and silanised glass ballotini (see Fig. 13). On the contrary, between two untreated glass particles a nodoid geometry is observed

.,.

.... ~,..

"

.... ~ ..~,.

d

Fig. 13. Evolution of the shape of a glycerol bridge between two glass spheres with increasing separation distance. The particle A is untreated whereas the particle B is silanised. Glass spheres of 119 (left) and 123 (right) l~m radii.

1278

S.J.R. Simons

Fig. 14. Evolution and rupture of a pendular glycerol bridge displaying fixed solid-liquid interfaces. Glass spheres of 125 (left) and 111 (right) l~m radii. Both particles are untreated glass ballotini. throughout separation, which results from a pinning of the three-phase contact line, leading to a reduction in the contact angle (large hysteresis) whilst the solid-liquid interface is almost constant, as illustrated in Fig. 14. A parabolic model was developed using physical data obtained from the micromanipulation experiments to approximate the various geometries [27,28]. This model will be detailed below and compared with the torroidal model introduced in Section 2.4.

3.2.1. The toroidal and parabolic models The toroidal approximation can be split into two categories, namely, whether the meniscus of the liquid bridge assumes a convex or a concave profile. This is not the case for the parabolic approximation, where a single equation can be used to describe both curvatures. Usually in the literature, the meniscus is considered to be concave [9,17,24] while in certain real cases, depending on the volume administered and the binder-to-particle wettability, a convex profile can result.

3.2.1.1. The concave toroidal model A schematic of the toroidal approximation for a concave geometry is shown in Fig. 15. The reference axes were chosen to simplify the expression of the liquid bridge profile. Two unequally sized spheres of radius RA and RB are separated by a distance a. The liquid bridge has a constant radius of curvature r2 in the plane of the page and r~ (evaluated at the narrowest point of the meniscus) in a plane perpendicular to the page. The x-axis is the axis of symmetry and the origin is taken as the point where the bridge is at its narrowest. The liquid bridge contacts each sphere at the ordinates YA and YB, with a half filling angle of j~A and j~B, respectively, and forms the contact angles (9A and ~B on each sphere. Equating the vertical (y) components of the bridge geometry gives: RA sin/~A + r2 sin(eA +/~A) = rl + r2

(27)

RB sin I~B -t- r2 sin(~g + ,8B) = rl + r2

(28)

Liquid Bridges in Granules

1279 /\ Y __(0, q+r 2)

/

--~

\

. . . .i... . Fig. 15. Schematic of the toroidal approximation for a concave profile.

Similarly, equating the horizontal (x) components gives: a = al + a2 = RA(COS/~A -- 1 ) + r2 COS(0A + ,SA) + RB(COS/~B -- 1) + r2 COS(0B +/3 B)

(29)

The equation for the upper toroidal bridge profile is given by equation (30) for a concave bridge, which occurs when (/~A+ j~B -4- {)A + {)B) < 2/'C[ X2 -t- (y -- rl - r2) 2 = r 2

(30)

which can be rearranged to give: (31)

y(x) - rl + r2 - q r 2 - x 2

The volume of revolution of the meniscus, Vm,A, from the point x = - r 2 COS(/~A+ 0A) (contact with particle A) to the point at which x = 0, is given by: V m , A - - /1:

L

o

y2

dx

(32)

r2 COS(flA+0A)

Substituting equation (31)into equation (32), gives:

/o

(2r 2 + 2Qr2 + r 2 -

V m , A - - /~

x 2

-

2(Q + r2)

r 2 - x2)dx

(33)

r2 COS(/?A+0A)

Integration of equation (33)leads to: Vm,A - -

/'C [(2r 2

-r2(rl+

+ 2Qr2 + r 2 ) x - - x3 5--(rl

+

r2) arcsin ~1 ~ - r 2 COS(/~A+0A)

r2)xqr 2

x2

(34)

1280

S.J.R. Simons

and when the integration limits are substituted, equation (35) follows: ~r 3 = COS(0A --I--J~A) 2 + 2 rlr2 ( -

~22rl 2

COS3(~)A3-Jr- 0A)

r~)( ~ ) 1 - ~2 COS(0A+/~A) sin(0A +/~A) + ~ + 0A +/~A

(35)

A similar expression exists for the volume of revolution Vm,B, obtained from integration of equation (34) from x = 0 to x = r2 COS(/~B+ 0B), where the profile contacts particle B:

v., ~r 3 = cos(0B +/~B) (2+ + 2(rlr2) ) ~rl 2 ( -

COS3(0B3+ 0g)

r,)( ~ ) 1- ~ COS(0B+ fiB) sin(0B + fiB) + ~ + 0B + fiB

(36)

TO find the exact value of the volume of liquid in the bridge, the volumes of the spherical caps enclosed at each end of the profile need to be subtracted. For the two spheres A and B, characterised by index i, this expression is given by: =R 3

Vcap,i - ~ - ( 2 - 3 cos/~i + c~

(37)

The volume of the bridge is eventually calculated as: Vbr-

~ i=A,B

Vm,i- Vcap,i

(38)

Equations (37) and (38) are also valid for the convex model. The area Abr of the meniscus interface can be calculated by: Abr

m

2~

f + r 2 COS(/~B-~-0B)

y(X)v/1 +

J Jr-r2COS(,SA+0A)

y'2(x)dx

which, after the igtegration limits are imposed, becomes" Abr = (kl + -~-) (/-c - ~)A -- ~A -- eB -- ~B -- COS(eA -Jr-~A) -- COS(~)B -Jr-,~B)) 12,/

(39)

(40)

3.2.1.2. The convex toroidal model A convex shape forms when the two spheres are close together and/or when the liquid forms a relatively large contact angle with the two particles, as indicated in Fig. 16. The upper and lower liquid bridge profiles are, in general, approximated by two different arcs of circumference for which r2 and q represent the radii of curvature in the plane of the page and in a plane perpendicular to it, respectively. The geometrical condition for the concave shape to occur is (,~A 4"/~g "{" ~A 4" ~B) > 2=. Since, in general, the upper and lower profiles are not

Liquid Bridges in Granules

1281

Y

x

Fig. 16. Schematic of the toroidal approximation for a convex profile. described by the same circumference, the centres of the two circumferences are not necessarily lying on the x-axis, as indicated in Fig. 16 for the upper profile. The expressions of the volumes and area of revolution can be determined in the same way as for the concave case by using the following expression for the convex profile: y(x) - r l - r2 4- v / r 2

-

(41)

x 2

AS in the concave case, the volume of revolution, Vm,A, is evaluated by integrating equation (32) from the point x = - r 2 COS(/~A+ 0A) (contact with particle A), to the point at which x = 0. When (41)is substituted into equation (32), equation (42) results: Vm,A -- /~ (2r 2 + 2Qr2 + r 2 - x 2 4- 2(rl + r2) 2 _ x2)dx (42)

/0

r2 COS(/~A+0A)

The integration of equation (42) gives:

[

Vm,A -- /~ (2r 2 + 2Qr2 + r2)x - - - ~ + (rl + r2)x

2

_

X 2

x] ~

+r2(Q + r2) arcsin r2 9 -r2 cOS(/~A+eA)

(43)

and when the integration limits are substituted, equation (44) follows: ~r23 = COS(0A + fiA) 2 + 2 r2rl -

(r,)( ~ - 1

r2rl 2

COS3(0 A3 +/~A)

~ ) COS(0A+/~A) sin(eA +/~A) + ~ +/~A -- eA

(44)

A similar expression exists for the volume of revolution Vm,B, obtained from integration of equation (33) from x = 0 to x = r2 COS(fiB+ DB), where the profile

1282

S.J.R. Simons

contacts particle B:

V.B ~:r23 = COS(~B+/~B) ( 2 + 2 __+( r2rl ~22rl 2 -

~ - 1

COS3(tgB3 +/~B)

COS(eB+/?A) sin(eB + #B) + ~ +/?B -- eB

(45)

The area Abr of the meniscus interface can be calculated from equation (39) by using equation (41) and after the integration limits are imposed, equation (46) results: Abr =

(rl )

r 2 - 1 (~gA+/~A + (~B + fiB -- ~ -- COS(L~A+ ~A) -- COS(0B + fiB))

(46)

3.2.1.3. The parabolic model Figure 17 is the schematic of the parabolic bridge profile approximation. The solid-liquid interface is a spherical cap, which has a maximum height of h~, L is the length of the liquid bridge and Ymin the minimum liquid neck radius of the pendular bridge. The x-axis is the symmetry axis of the system and the origin is set at the intersection between the x-axis and the half-cord YA. The liquid-to-solid contact points are P and Q on the two spheres, with co-ordinates of (0, YA) and (L, YB), respectively. The heights of the spherical caps on the particles, hi, are related to yj (i = A,B) by:

yi =
(47)

y(x), the function of the upper half of the liquid bridge profile, is approximated by a second-order polynomial equation in the form of:

y(x) = c~2x2 + 0~lx + c~o

(48)

in which the values of the three unknown parameters 0~2, C(1 and 0~o, come from the solution of the system given by equation (49), the symbols being defined as

Y

y

Fig. 17. Schematic of the parabolic approximation.

Liquid Bridges in Granules

1283

above: y(0) = YA

Vm--7:~0 y2(X)dx

-

-

vbr Jr- Vcap~A Jr Vcap,B

(49)

y ( L ) - YB The area of the liquid bridge, Abr, is calculated using equation (50), which leads to equation (51) once equation (48)is replaced and integrated between the integration limits: Abr -- 2~

/i'

y(X)v/1 + y'2(x)dx

(50)

~(~,(1 + 16~2~o- 2o~2 V/1 + ~2 + (-1 + 16c~2~zo- 4~2) arcsin h(c~l)) Abr ---- --

320~2 2L~2 + ~1)(1 + 8L2~z2+ 16~z20~0+ 8L~z2~Zl- 2~2)

1 -Jr-(2L0~2 + 0~1

32~2

~((-1 + 16~2c~o- 4oc2)arcsin h(2Lc~2 + o~1)) +

32o~2

(51)

3.2.2. Comparison of the toroidal and parabolic approximations To approximate the liquid bridge profile using either the toroidal or the parabolic model, some geometric quantities must be given or calculated through image analysis from a sequence of liquid bridge separations (see, for example, Figs. 13 and 14). These quantities are the particle radii RA,B, the half-filling angles ,~A,B, the separation distance a and the volume of the bridge Vbr. The application of the toroidal method for both the convex and the concave profile requires the determination of the contact angles 0A and ~B and the profile radii of curvature rl and r2. These parameters can be calculated iteratively through the solution of equations (27)-(29) and (38). Depending on the volume of the liquid bridge, a transition from a convex profile to one that is concave can occur during liquid bridge separation (see Fig. 14). The toroidal approximation, therefore, involves two sets of equations to be solved for the same sequence of separation. When the liquid bridge profile changes its configuration from a convex to a concave curvature, it will assume a cylindrical configuration, which cannot be solved by the toroidal model since rl ~ oo. The parabolic approximation is defined by the three parameters c~2, ~ and ~o, which can be calculated by solving the system equation (49). It is useful to note that by varying ~2, ~ and ~o, the single equation (48) can be used to approximate both convex and concave menisci. Hence, the parabolic approximation results in a mathematically simpler and much more robust expression than that of the toroid.

1284

S.J.R. Simons

Since the toroidal and the parabolic approaches are not the solution of the Young-Laplace equation (1) they do not comply with the constraint on the total mean curvature being constant along the profile of the liquid bridge. However, calculated values of the principal geometric parameters (contact angle, rupture distance) have been shown to be in fairly close agreement with experimental data, as will be described in the next section. The curvature of the profile has a role in the evaluation of the liquid capillary pressure and ultimately on the force exerted by the liquid bridge. Mazzone etal. [8] parameterised the dimensionless force versus the normalised bridge volume on the separation distance and compared the toroidal approximation with the numerical solution of the Young-Laplace equation (1). They showed that when the particles are nearly in contact only a slight discrepancy is noted between the two methods throughout the volume range. However, as the separation distance reaches just 10% of the particle radius, the toroidal method significantly underestimates the force as obtained from the numerical solution. Models intended to describe the process of liquid bridge separation always assume the contact angle between the particle and the binder to be fixed and equal to zero throughout separation [9]. However, this assumption is reasonable only for perfectly wet particles and therefore is not a general case. Predicting the behaviour of real cases is complex due to the difficulty of modelling the phenomena at the three-phase contact line. As has been shown in Figs. 13 and 14, the interface can either be "pinned" or recede, depending on whether the wettability between the particle and the liquid binder is good or poor, respectively. This situation can be explained in terms of the contact angle hysteresis. The pinning of a solid-liquid interface can continue until the contact angle between the binder and the particle reaches the receding values. At this point, since no further reduction of the contact angle appears feasible, the solid-liquid interface reduces. When the difference between the advancing and receding contact angle is large, the three-phase contact line is pinned and a reduction of the solid-liquid interface is not likely to occur, unless the receding contact angle is reached. The two cases of liquid bridges formed between particles with either good or poor wettability will be discussed separately to better highlight their peculiarities. 3.2.2.1. Liquid bridges formed between particles with good wettability Some useful simplifications can be made to use the parabolic and the toroidal methods to model the configurations assumed by liquid bridges during separation of pairs of particles both exhibiting good wettability towards the binder. An example is shown in Fig. 18, where a glycerol liquid bridge is formed between untreated glass particles and the last recorded frames before bridge rupture and the postrupture liquid distribution are shown. Despite the fact that small reductions in the solid-liquid interface might occur even for particles of good wettability, it can be assumed that these interfaces stay fixed during separation. This condition translates to the fact that the spherical cap

Liquid Bridges in Granules

1285

Fig. 18. Last recorded liquid bridge configuration and post-rupture liquid distribution for the separation of a glycerol liquid bridge (Vbr= 1383 x 10 3 #m3) formed between two untreated glass particles of radii RA = 92 l~m (left) and R B - 91 l~m (right).

heights on the particles, h~, or equivalently the half-filling angles,/~i where i = A,B (Fig. 15) remain constant during separation while the three-phase contact line is pinned. When the geometry of the particles, the volume and the initial configuration of the bridge are given (or calculated by image analysis), it is possible to calculate the liquid bridge profiles at different separation distances by solving the system equation (49) for the parabolic approximation or that formed by equations (27)-(29) and (38) for the toroidal method in which the appropriate equations for the convex and the concave profile must be chosen. The solution of these two systems leads to the determination of the unknown values in equation (48) for the parabolic model and in equations (31) and (41) for toroidal concave and convex shapes, respectively. To assess the validity of the approximations made (e.g. type of profile and fixed interface) the theoretical profile can be compared with that obtained from experimental observations in terms of the apparent contact angles measured at the liquid contact with particles A and B. The apparent contact angle can be calculated from the approximated liquid bridge profiles by means of equation (52) for spheres A and B, respectively, in which Y'A a n d Y'B represent the abscissa of the point of contact: 0A -- ~ + tan-1 (y~,) _ sin-1

YA

0a -- ~ -- tan -1 (y~) - sin -1

fa

(52)

Figure 19 represents a typical situation of the contact angle values for a pair of untreated glass particles (good wettability) held by a glycerol liquid bridge, as shown in Fig. 18. Data are plotted against a*, the separation distance normalised with respect to the geometric average radius (R-2RARB/RA + RB). These graphs clearly show that the approximation of keeping the solid-liquid interfaces

1286

S.J.R. Simons 4, P and fixed hA T and fixed hA ,L experimental

80 %-.

60 50 ~

~ ~ ~

.i.I~\ ~

4' P and fixed hB ~ T and fixed hB A experimental

~) 60 ~40 ~ 40 t--

20 E o

~o

20

0 .... 0.00

(a)

I .... 0.50

I .... 1.00 a*

', . . . . 1.50

I 2.00

0 .... 0.00

(b)

I .... 0.50

', . . . . 1.00

I ~ 1.50

2.00

a*

Fig. 19. Experimental and calculated apparent contact angles for the experiment presented in Fig. 18. P refers to the parabolic and T to the toroidal model, both calculated with the approximation of fixed solid-liquid interfaces: (a) refers to particle A and (b) to particle B.

fixed is not adequate for the untreated glass-glycerol system. The differences found with respect to the parabolic and toroidal models are to be attributed to small reductions in the solid-liquid interface with consequent rearrangement of the threephase contact line. From Fig. 19 it can be seen that initially the contact angles drop quickly, which is well predicted by both profile approximations. However, after a normalised separation of approximately 0.5 for both spheres A (Fig. 19(a)) and B (Fig. 19(b)) the agreement between theory and experiment is lost. The predicted contact angles drop towards zero whereas the measured contact angles level off at a value of approximately 25 ~ which can be assumed to be the receding contact angle for glycerol on these spheres. This decrease and then levelling effect of the contact angles indicates that the three-phase contact line is initially pinned on the particles, but that, after a certain particle separation, the contact line begins to slip on the particle surfaces since the reduction of the solid-liquid area is more energetically favourable than further pinning of the contact angle. Despite the fact that the assumption of fixed interfaces seems to be invalid for the prediction of the contact angles of the liquid bridge, it is more applicable when used to predict the distance at which rupture occurs, as will be shown later. The model can also be applied to binders that show a nearly 0 ~ receding contact angle towards the particle. A second approach to modelling the profile of glycerol liquid bridges by using the toroidal and the parabolic approximations is to measure the geometric parameters of the interface at any stage during separation. In Fig. 20 this approach has been used to predict the contact angles on both particles. Both the parabolic and the toroidal approximations are able to give good agreement with the experimental results. However, since the model requires some a priori knowledge of the solid-liquid interface, it cannot be considered to be predictive, but rather a model used to approximate a known liquid bridge configuration. The advantages of this latter approach lie in the possibility of calculating the liquid bridge force

Liquid Bridges in Granules P and m e a s u r e d hA

80 ~X '-

1287 --~-i 50

,,....-~.........e x p e r i m e n t a l

"~

60

60

....i:::.........T a n d m e a s u r e d hA

P a n d m e a s u r e d hB

~5

....i::::i~........ ; T and m e a s u r e d hB

~\

.......-~--- e x p e r i m e n t a l

2 _r u~ 4 0 to E o o

9

.. ~ , ~

~__

2O o 0

. . . . 0.00

(a)

I 0.50

. . . .

I 1.00

a*

. . . .

I 1.50

. . . .

10 0

I 2.00

. . . . 0.00

(b)

.i 0.50

. . . ..

i. . . . .. 1.00

i 1.50

. . . .

i 2.00

a*

Fig. 20. Experimental and calculated apparent contact angles for the experiment presented in Fig. 18. P refers to the parabolic and T to the toroidal model, both calculated with parameters of solid-liquid interfaces measured during separation: (a) refers to particle A and (b) to particle B. from an approximated profile (either toroidal or parabolic) without the need to solve the Young-Laplace equation (1), although the error induced on the capillary pressure by the approximated profile increases with separation distance. One useful parameter that can be predicted using the toroidal and parabolic approximations is the separation distance at which the bridge ruptures. The simple criterion proposed by Lian etal. [17] (see Section 2.3) to evaluate the liquid bridge rupture distance can lead to ambiguous results when a large contact angle hysteresis exists during liquid bridge separation and a discretional judgment is required on deciding whether the advancing or the receding contact angle should be used in equation (11). A geometric rupture criterion has therefore been developed together with the parabolic approximation, which is based on the conservation of the liquid bridge area just before and after rupture [27,28]. This approximate rupture model has also been tested with the toroidal approximation, which gave the same order of accuracy in the prediction of the bridge rupture distance. The rupture criterion is iteratively applied by assuming a virtual rupture of the liquid bridge at each separation distance during the bridge elongation. The solid-liquid interfacial area is assumed to be held constant. If we consider the rupture of a liquid bridge of low volume (< 104 l~m3) between two spherical particles, the liquid binder tends to redistribute on the two particles by forming droplets that can be regarded as perfect caps of spheres (see Fig. 21 and the last frames in the sequences shown in of Figs. 14 and 18). To quantify the amount of binder left on each particle, it is assumed that, at each rupture, be it either virtual or real, the volume of the binder distributes between the two droplets in proportion to the amount on either side of the bridge at its thinnest point. For instance, if we consider a virtual rupture for the configuration shown in Fig. 15, the volume of liquid left on sphere A, VA, would be calculated according to equations (35)-(38). Figure 21 illustrates the post-rupture droplet shape: u~ is the liquid droplet radius, Ti the liquid droplet cap maximum height and ci the half-cord length, which

1288

S.J.R. Simons

~_

~ " i

~/

Ti

Fig. 21. Liquid droplet after rupture on a spherical particle L are all related by:

T2 + c2 ui =

2Tj

(53)

In the proximity of the effective rupture, the area of the liquid bridge Abr and the area of the droplets Adrop should be almost equal because the liquid bridge cannot adsorb any more energy from the surroundings to be turned into new liquid bridge surface. The area rupture criterion states that rupture occurs when equation (54) is satisfied, where Adrop is calculated by equation (55): Adrop -- Abr -- 0 (54) Adrop -- ~ 2~uiTi i=A,B

(55)

In Fig. 22, the evolution of Adrop and Abr calculated using both approximations is plotted versus normalised particle separation for the case of two untreated glass spheres, 101 pm RA and 100pm RB [27,28]. The liquid-vapour interfacial area the droplets would have if rupture occurred at a separation well before the observed rupture, is much superior to that of the liquid bridge, which indicates that the liquid bridge is perfectly stable. With elongation, the bridge liquid-vapour interfacial area increases, while the droplets' interfacial area levels off to a constant value. Variations in Adropwith a* occur when the solid-liquid interfaces vary and/or the position of the minimum liquid neck on the bridge profile changes. Even though the toroidal and the parabolic approximations have different mathematical expressions, their agreement is excellent in terms of the predicted interracial area.

3.2.2.2. Liquid bridges formed between particles with good and poor wettabilities The study of liquid bridges formed between particles of different wettabilities is summarised in Ref. [28]. From the batch of ballotini used in the work described in the previous subsection, a sample was soaked for 30 min in a silanising agent

Liquid Bridges in Granules 1.8E+05

1289

P Abr

1.6E+05

I ~ T Adrop t A T Abr

-

~'~ 1 . 4 E + 0 5 E observed rupture

~ 1.2E+05 1.0E+05

-

8.0E+04

' 0.0

I 0.4

'

I 0.8

'

I 1.2

'

I 1.6

'

I 2.0

'

I 2.4

a*

Fig. 22. Evolution of Adrop and A b r v e r s u s a* for the case of two untreated glass spheres, 101 pm RA and 100 l~m RB. The arrow indicates the observed rupture distance. P refers to the parabolic and T to the toroidal model, both calculated with the approximation of fixed solid-liquid interfaces. (2% solution of dimethyldichlorosilane in octamethylcyclotetrasiloxane) to modify the surface properties of the particles. The method of liquid bridge formation as well as the speed of separation (~1 pm/s) remained the same. The silanisation of the glass particles was intended to reduce the high wetting hysteresis shown by the glycerol on untreated glass. For pendular liquid bridges, a large or small wetting hysteresis drastically changes the shape adopted by the liquid bridge, as can be seen in Fig. 13. In this figure, a glycerol bridge has been created between a non-silanised glass sphere (A) and a silanised glass sphere (B). With increasing interparticle distance, the three-phase line is pinned on the non-silanised particle A, but reduces on particle B as the liquid recedes from the solid surface. To verify the applicability of the parabolic approximation to the case where one particle is silanised, the apparent contact angle of both particles, as calculated by equation (52), can be compared with the values measured experimentally. Because of the large interface reduction observed on the silanised particle, the parabolic model can only be applied using experimental values of the spherical cap heights, hi, which can be measured at any stage of the separation. The prediction of the contact angles for the experiment shown in Fig. 13 is very good, except for the last recorded configuration before rupture, where the parabolic model fails to account for the change in curvature of the liquid bridge in the proximity of particle B. Figure 23 shows that the contact angle remains more or less constant on the dewetting particle B, whereas the angle made by glycerol on particle A reduces with increasing particle separation.

1290

S.J.R. Simons 100 ,~

~

80

a

o

."o

_.e

60 D

C .i-, O tO O

40

~

r A theor.

D

A exp.

i•

20

r

D

D

O

O

- - z~ B t h e o r o B exp.

0 ' 0.500

'

'

I . . . . 1.000

I . . . . 1.500

I 2.000

. . . .

I 2.500

a * [-]

Fig. 23. Evolution of measured and calculated contact angles versus normalised separation distance for the case of the experiment shown in Fig. 13. A and B refer to the two particles, of which B is silanised.

160

/

140

[

~ .............~, ................ '~ .... ....... @ . , .................

.... /

.............,~...............L 234.6E 120

~

...... ~z~-

~

-~ ..............

+ ..........L 272.7E

100

L 301.4E -

E

-

e .... L 375.3E

:~

80 I

~ /

yB constant

:~

>,

60

t

L 272.7P

I

~

40

YA reducing

L 234.6P

;,:

L 301.4P

:,:

L 375.3P

20 0

' 0

I 50

'

I 100

'

I 150

'

I 200

'

I 250

'

I 300

'

I 350

'

I 400

x [~m]

Fig. 24. Evolution of the bridge profile between particle A, silanised, RA = 471~m, and particle B, unsilanised, R B - - 114 l~m. The legend indicates the liquid bridge length and the last shape for L = 375.3 l~m corresponds to the last bridge image before rupture.

Figure 24 shows the initial and the pre-rupture configurations of the liquid bridge between two particles during separation, measured experimentally and calculated using the parabolic approximation. Just before rupture for L>~375.3~m, the shape is unduloid. For this case, the third-order polynomial equation is unable to account for the more complex bridge shape. The liquid bridge shape shows a clear reduction of the solid-liquid interface on the silanised particle (A, RA = 471~m), on which the origin of the liquid bridge length is taken. On the contrary, the three-phase contact line on particle B

Liquid Bridges in Granules

1291

(untreated, RB = 114 pm) is pinned and the solid-liquid interface remains constant (the reverse of the experiment from which Fig. 13 was obtained). In Ref. [28] a fourth-order polynomial was used to fit the last configuration before rupture. However, this cannot be used as a predictive tool, not even for a fixed interface approximation, because it requires too many parameters that are not known a priori, as, for example, the values of the tangent of the liquid bridge at the point of contact with the particles. Nevertheless, the fourth-order polynomial model does allow the verification of a peculiar property of the dewetting phenomena, that is, that the increase of the liquid bridge area during separation is balanced by the reduction of the dewetting interface.

3.2.2.3. Post-rupture liquid distribution In the previous sections, it has been shown that the particle wettability significantly influences the geometry of the liquid bridge. As a consequence, the post rupture liquid distribution is also affected. The amount of liquid remaining on the particles after rupture will determine whether or not the formation of new liquid bridges is favoured or inhibited. Small amounts of liquid left on a particle will have less probability of forming new liquid bridges, which may result in segregation of particles exhibiting different wettabilities in a mixed formulation. On a theoretical basis, it seems reasonable to assume that after rupture the liquid binder distributes proportionally to the volumes of the two spherical particles. The validity of this assumption, however, is restricted to cases of similar and well-wetted particles and to liquid bridges perfectly symmetrical along the axis of separation. It can be seen in Fig. 13 that for particles of different wettabilities, the liquid distribution after rupture favours the particle exhibiting the strongest adhesion to the liquid, represented by the pinning of the three-phase contact line. Figure 25 illustrates the experimental binder volume distribution versus the solid fraction, as measured on particle A. The particle solid fraction is calculated as the ratio between the volume of particle (A) and the total volume of the two particles, Vs ( = VsA+ Vsg). VA represents the volume of liquid left on particle A after rupture. The experimental conditions are shown in Table 1. Figure 25 seems to show a relation between the solid fraction and the postrupture liquid distribution for liquid bridges formed between well-wetted particles (exps. A1-A7). Deviation from the theoretical trend (indicated by the line y = x) can be attributed to the dewetting of the solid-liquid interface, which, even when small, can influence the geometry of the liquid bridge and therefore the post rupture liquid distribution (e.g. see Fig. 18). A different situation is observed for liquid bridges formed between particles exhibiting good and poor wettability. In experiments A8 and A9 the liquid is almost completely redistributed on the untreated particle (particle A in exp. A8 and particle B in exp. A9) that exhibits higher wettability.

1292

S.J.R. Simons .

O exp. A1

0.8

X exp. A2 a exp. A3

Z 0.6

[] exp. A4 "

< 0.4

>

+

exp. A5 o exp. A6

0.2 0.0 ~ 0.0

+ exp. A7

I 0.2

'

I ' I o' 0.4 0.6 VsANs [-]

I 0.8

'

I 1.0

,5 exp. A8 o exp. A9

Fig. 25. Binder volume fraction versus particle solid fraction measured on particle A. In experiment A8 particle A is untreated whilst in experiment A9 particle A is silanised (see Table 1). Table 1. Experimental conditions between particles of untreated and silanised (marked with an asterisk) glass particles, attached by glycerol liquid bridges Experi-

Vbr x 10 3

ment

#m 3

RA (l~m) RB (#m) VsA/Vs VA/ Vbr (E) (P)

VA/ Vbr

(T)

A1 A2 A3 A4 A5 A6 A7 A8 A9

2175 622 10 3720 1383 1322 142 6850 10500

125 47 49 101 92 92 56 119 134"

No solution 0.102 0.705 0.500 0.495 0.501 0.477 Not applicable Not applicable

111 114 44 100 91 91 103 123" 103

0.588 0.065 0.582 0.511 0.512 0.506 0.132 0.454 0.648

0.709 0.044 0.499 0.710 0.659 0.593 0.365 0.980 0.010

VA/Vbr

0.532 0.160 0.532 0.506 0.479 0.509 0.364 Not applicable Not applicable

E, experimental" P, parabolic; T, torroidal.

For the experiments between untreated particles, the post-rupture liquid distribution has also been estimated using both the parabolic and toroidal models using the fixed interface approximation. The predictions of both models are presented in Fig. 26. Both models adequately predict the extent of liquid volume redistribution on the particles at rupture.

4. RELATING PARTICLE-BINDER INTERACTIONS TO GRANULE BEHAVIOUR Models that are used to describe wet granulation growth kinetics the agglomerate mechanical properties to determine, for instance, coalescence after inter-agglomerate collisions. These models elastic collisions between agglomerates with a layer of free liquid

usually rely on the success of either assume dissipating the

Liquid Bridges in Granules

1293

,~ 1 . 0 ~

<> P - fixed interface

.'. 0.8

E1 T- fixed interface

/ J

> >

<

0.6D

0.4 o

0.2 0.0

'

0.0

0.2

0.4 experimental

0.6

I

0.8

'

t

1.0

VA / Vbr [']

Fig. 26. Prediction of liquid volume distribution between the two particles using the toroidal T, and parabolic P model with the fixed interface approximation. kinetic energy of the impact [2], or pre-suppose the deformability of the agglomerate to build the growth kernel [3]. In the case of elastic collisions of moistened particles, viscous forces control coalescence [45]. Experimental work on wet agglomeration processes frequently shows that, initially, loose agglomerates are formed [45-47], which consolidate with agitation and increase in their moisture content. Models that are based on elastic collisions with a layer of free liquid would hold for the later stages of the granulation process [2]. The agglomerate hardness is clearly linked to its inner porosity but there is unfortunately no constancy of this factor [48]. Throughout wet granulation, agglomerates harden as they become less porous. The addition of liquid binder facilitates this porosity reduction as the binder can lubricate the interparticle contact points. However, when the mass is over-wet, further lubrication can also reduce the hardness of the agglomerates. Parallel to this, a number of simulations have proven the role of liquid viscosity, liquid surface tension and interparticle friction forces in the resistance to deformation of moist agglomerates [49] (see Section 2.6). Wet agglomerates mostly behave plastically until the yield strength is attained, where they rupture through crack propagation. The relative importance of the material properties and the agglomerate texture in the overall deformability is still controversial. Most models generally assimilate particles to spheres. There is a realistic probability that friction forces will increase the further the particle shape deviates from a sphere. Inside agglomerates, the shape of pendular liquid bridges is an important factor which determines the size and porosity of the agglomerate as well as its resistance to deformation. We have already seen how the volume of liquid of a pendular bridge is either constant or varying during separation according to the wettability of the powder (Section 3.2.2) [20,27,28]. When the bridging liquid poorly wets the powder, it is possible to obtain liquid bridges of fixed volume, as there is a clear three-phase contact line on both particles. In addition, when the wetting hysteresis

1294

S.J.R. Simons

of the particle surface is high, the apparent liquid-to-solid contact angle changes with interparticle distance as long as the three-phase line is pinned on the solid surface [27]. Conversely, when there is a reduced wetting hysteresis, the threephase line recedes and the bridge liquid dewets the particle with a constant apparent contact angle [28]. If the volume is fixed with a clear three-phase contact line, a certain range of bridge liquid volume can be observed for the same particles. The bridge volume can then not be determined from the properties of the materials, but varies according to the operating conditions of the wet granulation. On the other hand, if the bridge liquid perfectly wets the particles, a continuous liquid film forms on the particles and surrounding objects or particles, precluding the existence of a three-phase contact line. In this situation, there is a funicular saturation state of the agglomerate, regardless of the absolute saturation of the mass. The bridge liquid volume is not constant during particle separation, but there is a greater chance of relating this volume to the particle properties. On the particulate level, liquid bridges are responsible for the strength of a wet agglomerate, since they hold the particles together. On the wet agglomerate level, the hardness is related to three factors: the liquid binder surface tension and viscosity and the interparticle friction. A simple model has been developed [35], based on the powder and liquid binder properties, which shows that the forces due to interparticle friction are generally predominant in wet agglomerates made from non-spherical particles. This will be discussed in the following sections. Although mechanical interlocking is not predicted, this model yields accurate prediction of wet agglomerate hardness independently measured on wet masses of varying composition. This theoretical hardness could prove an interesting tool for wet granulation research and technology and represents where future research in this area should be focused, namely, on the use of micro-scale data to inform models across the length scales, from single liquid bridges to granule behaviour.

4.1. Compression of plastic agglomerates A perfectly plastic wet agglomerate is compressed between flat punches at a speed of v~. During compression, a certain number of structural modifications will occur. We assume for simplification that the agglomerate is formed of n touching particles at the interparticle contact of which liquid bridges can be found. The agglomerate liquid saturation is known and the liquid perfectly wets the solid particles. The shape of the agglomerate is assumed to be cylindrical with a radius of Rag and a length of M. We will assume in the following that neither the solid particles nor the liquid binder exhibit elastic deformation during the agglomerate compression. Experimental studies reported in the literature [46] have observed plastic deformation of moist agglomerates. From a global point of view, plastic agglomerates deform

Liquid Bridges in Granules

1295

against hard surfaces with a hardness of.Q, which is the ratio between the applied load, L', and the contact area, Ac, of the agglomerate with the hard surfaces between which it is compressed: L' s = -(56) Ac Experimentally, wet agglomerates can be submitted to deformation at variable speed with recording of the force necessary for deformation. The contact area Ac resulting from deformation increases with the absolute displacement. If neither the porosity of the assembly nor the co-ordination number of the particles is assumed to evolve initially during plastic deformation, the contact area of the agglomerate with the flat punches can be obtained from simple geometrical considerations. Consider a cylindrical agglomerate as described in Fig. 27. After an absolute compression of d, the agglomerate has flattened on both sides when in contact with the compression punches and resembles the schematic given in Fig. 28. From the agglomerate volume conservation, equation (57) can be derived:

Fag -

M~R2g -

Ac(2Rag - d)

+ 2M [R2gtan -' (, Rag ~ d ',~

\Rag-h(d)J-(Rag-h(d))

(Rag _ d ) J

(57)

where Fag is the volume of the agglomerate, M the test cylinder length, Rag the cylinder radius, Ac the contact area of the agglomerate with one flat punch and h(d) is the cylinder height which is not stressed after an absolute agglomerate compression of d. The number of particles n inside the agglomerate is given by equation (58), with ~ the average particle volume:

VagUS

n- ~

(58)

2Rag

--

Fig. 27. Cylindrical agglomerate.

~ ~ ~

M

1296

S.J.R. Simons 2Rag- d ~M ]

h(d)

Fig. 28. Compression of a cylindrical wet agglomerates. The co-ordination number c is given by equation (59) [51]: k c= ~ , 1 -~s

with k ~. 3 for a packing of spheres.

(59)

The volume of one liquid bridge Vbr is estimated by: Vbr - -

2VagUE 2~L~, -c n c ~s

(60)

where ~L and ~s are the liquid and solid volume fractions, respectively. The average interparticle distance H is given by equation (61), with c/, the average particle diameter:

H = 2 (3~4----~( Vag(ln-~ s ) + The average solid-solid contact agglomerate is estimated by:

area AAB

n

(61)

between two solid particles in the wet

2(Vag~sSps-~-d2) AAB -- C

~)_~_)

(2VSPs -- ~d2) ~s ~s --

(62)

C

where S is the true surface area of the powder and ps the powder true density. The solid volume fraction at the right-hand side of equation (62) estimates the effect of porosity and liquid content on the disruption of solid-solid interfacial --2 area. The term ~d in equation (62) implies that if parttcles were perfect spheres, the average solid-solid interparticle area would be taken as zero. Because the deformation of the mass is assumed plastic, the applied load L' corresponds to the yield strength of the mass at each moment. With increasing contact area, this force increases as the mass shows more resistance to deformation. If we consider that the mass hardness defined by equation (56) can be constant at the beginning of deformation, then the mass yield strength can be

Liquid Bridges in Granules

1297

calculated on an elemental surface which, in the case of a wet mass, can be the area occupied by a solid particle. One particle in the agglomerate assembly has c neighbours and NE, the number of interparticle contacts per unit area that are broken when one particle is moved, is given by: NE

-

(V) 2/3

(63)

In an agglomerate in which the liquid perfectly wets the solid particles, it can be considered that liquid covers the surfaces of the surrounding particles. In this case, AP in equation (19) will equal zero. In addition, the viscous force term needs to consider each particle as a whole and not just the interparticle contact. Hence, equation (18) can be re-expressed as: Fvisc -- 3~arlVi

~"

(64)

where the term ~/~L corrects the viscous force with the structural information of the wet agglomerate. The separation distance where rupture occurs can be predicted using the parabolic approximation, assuming that this occurs through the liquid's thinnest neck, Ymin (Fig. 17). For perfectly wetting liquids, liquid bridge volumes can be related to Ymin by an empirical relationship [21]: Vbr ~ 1.673y3in

(65)

The capillary force Fcap, developed by one liquid bridge is then given by: Fcap = 2~YminYLV

(66)

where ~LV is the surface tension of the vapour-liquid interface. The friction force of one interparticle contact is estimated by equation (67), developed from the expression for the work of adhesion of the liquid on the solid [21]: Aab Ffric -- ~Lv(COS 194- 1) H

(67)

where Aab is the interparticle contact area and H is the interparticle distance. Combining equations (56, 63-67), the elemental hardness Dca~ccan be derived as:

~

1[

2

+

+

(c))

2,3]

(68) The term 1/2 at the left-hand side of equation (68) arises from the fact that particles are randomly oriented inside the agglomerate. The elemental forces are equally distributed and only the cos~ fraction is measured, with ~ the angle

1298

S.J.R. Simons

made by each individual force vector the normal to the punch surface. If all possible 8 values are averaged, the calculated mass hardness can be given by equation (68).

4.2. Experimental validation of the hardness equation Equation (68) has been tested against data obtained from the crushing of cylindrical pellets made from a range of pharmaceutical powders and binders and from glass ballotini and silicon oils. The powders were sodocalcic glass beads of an average radius of 351~m (GB) [50]; a lactose DCL11 (L1), separated in three fractions (L1A), (L1B) and (L1C), with 90 and 180-1~m-mesh sieves; a lactose EFK sieved into coarse (L2) and fine (L3) fractions with a 100-1~m-mesh sieve; and a lactose 150 mesh (L4) [21]. Sugar beads, Suglets 30-35 (SB1) and Suglets 250-355 (SB2) were also used as a comparison to the glass ballotini. Finally, a crystalline drug powder (DP) magnesium stearate NF-VG-1-726 (MGST), which exhibits low interparticle friction. The liquid binders were silicon oils of increasing viscosity 96, 996 and 97920 mPa s, water, a 0.2% w/w sodium dodecyl sulphate aqueous solution and aqueous solutions of hydroxypropyl methylcellulose (HPMC) and polyvinylpyrrolidone (PVP) of increasing viscosity and varying surface tension. The hardness of pellets made from glass beads (GB) are reported in Ref. [50]. The hardness of pellets made from (L1), (L2), (L3), (DP) and (MGST) are reported in Ref. [21], where pellets of 16 mm diameter and 17mm height were compressed at 10 mm/s. In this work, pellets of 18 mm diameter on 17 mm length were made from (L4), (SB1) and (SB2) moistened with silicon oils of increasing viscosity. The pellets were carefully retrieved from the die on a pre-tared microscope slide, weighed and then deformed radial to the cylinder axis between the two flat punches of an TAXt2 | texture analyser (stable micro systems). The upper punch was lowered onto the pellets at speeds of between 0.1 and 10 mm/s. From the weight of the pellet and a knowledge of its composition, the pellet density and porosity could be calculated. Three pellets at least were characterised for each powder. The slope of the force versus contact area curve was taken as the mass hardness. Table 2 summarises the physical properties of the powders used in the compression study. The mass hardness is calculated from the slope of the graph of force versus contact area (Fig. 29) calculated from equations (56) and (57): The agglomerate standardised stress can be plotted versus capillary number to compare the results with that of Iveson etal. [50] (Fig. 30). The curve shown is the best-fit line to their data. In Fig. 30, the behaviour of agglomerates made from sugar beads is similar to that of agglomerates made from glass beads. Iveson etal. [50] stated that, at

Liquid Bridges in Granules

1299

Table 2. Physicochemical properties of the powders used in the experiments

d5o (pm) s(m2g -1) ps(g m1-1) Shape

L1A

L1B

L1C

L2

L3

DP

MGST

GB

L4

SB1

SB2

45 0.193 1.54 A

135 0.145 1.54 A

210 0.069 1.54 A

32 0.196 1.54 A

230 0.043 1.54 A

27.8 0.342 1.27 B

2.6 5.3 1.06 P

35 0.0697 2.457 S

57.51 0.189 1.54 A

545 0.0073 1.511 S

302.5 0.013 1.512 S

The shape descriptors are A, angular; B, beam; P, platelet; and S, spherical.

8.E-01 7E-01 ~ Z

6EOI

~

5.E-0I -

~

4.E-0I-

=

3.E-0I -

:~

2.E-01

-

1E01 0E+00

t

4-

I

5 E-05

0E+00

i

l E-04

2E04

i

2E-04

3E04

C o n t a c t area A c ( m 2)

Fig. 29. Measurement of mass hardness from experimental forces and calculated contact area.

- 1E+03

,t' oL1

l E+02

o_,,o

~ x~

x

~

Xx

x~

[] L2

~..~ 1.E+OI

,t

9

~

. &

99

. . . .

9"

~j~

9

J'~l

~--~AL3



o DP

=m

v

II

o

o,

-~ q~

o

1.E+O0

o

O

o MGST ISB • L4 9 GB

O

i

i

I

i

i

1 .E-10

1 .E-08

1 .E-06

1. E - 0 4

1 .E-02

Ca

V i

dq

2R~g

7LV

i

l .E+00

Fig. 30. Stress of wet agglomerates versus capillary number.

1E01 l E+02

1300

S.J.R. Simons

low capillary numbers Ca < 10 -4, the standardised stress is independent from the deformation speed, and friction is the predominant parameter in controlling the wet mass hardness. Above C a - 10 -4, the contribution of viscous forces to the deformation of the wet mass becomes predominant. Powders with non-spherical shapes show a different behaviour. Agglomerates made from lactose and drug powder exhibit higher stresses for equivalent capillary number, whereas agglomerates made from magnesium stearate exhibit lower stresses than that of agglomerates made from glass beads. In Fig. 31, the measured mass hardness is plotted versus theoretical mass hardness calculated using equation (68). From this figure, we can see that agglomerates made from L1, L2, L3, L4, SB1, SB2 and DP exhibit measured hardness that can be predicted from equation (68). The hardness of agglomerates made from glass beads are also well predicted below 105N m -2. Above this limit, which corresponds to capillary numbers that exceed 7 • 10 -2, the mass hardness is over estimated. The hardness of agglomerates made with magnesium stearate is over estimated by equation (68). The wet masses made from magnesium stearate have a low porosity and the liquid saturation of the pores created by the solid particles approaches 80%. It can be shown, therefore, that such agglomerates are actually in the funicular saturation state and that the particles cannot be considered as being in contact, but rather as being suspended in liquid with some air present. Hence, the model is not applicable. Further investigations on the measured mass hardness can be made by neglecting the capillary, friction and viscous elements of equation (68) in turn. For I.E+06

-

I.E+05

-

OL1 [] L 2 ix L 3

t"-,l

o DP 1.E+04 -

o MGST 9 SB

x L4 1.E+03

9 GB

I.E+02

1.E+02

1.E+03

I.E+04 f~calc ( N ' m - 2 )

Fig. 31. Measured versus calculated mass hardness.

1.E+05

I.E+06

Liquid Bridges in Granules

1301

1.E+06

1.E+05

9

/

$

OLI [] L2

E

o ~,

c-,l i

~

9

zX L3 oDP

o

Z

1.E+04

A

.

EL

o MGST 9 SB X L4 1.E+03

l .E+02 I.E+02

9 GB

1.E+03

1.E+04

1.E+05

1.E+06

~'~calc ( N ' m - 2 ) n e g l e c t i n g f r i c t i o n

Fig. 32. Measured versus theoretical hardness calculated with (68) but neglecting the elemental friction forces of the interparticle contacts. instance, by neglecting friction (Fig. 32) the measured hardness is underestimated for the lactose and drug powders. For the glass and sugar beads, capillary forces are dominant at the low deformation speeds used. The hardness for magnesium stearate is still overestimated, probably due to the reason given above and the difficulty in measuring particle surface area for such a powder (changing the powder surface area of magnesium stearate from 5.3 to 2.78 m2/g ensures a prediction of the measured mass hardness with an error of 17%). The simple model for the hardness of wet agglomerates given in equation (68) is valid only for particles which can be considered independent from one another during deformation and when the liquid bridge volume ensures that pendular liquid bridges can exist between touching particles. If particles have a shape that deviates significantly from the sphere or if they tend to aggregate, the model fails. Equation (68) shows that capillary, viscous and friction forces can be added to describe the yield strength of an agglomerate assembly. The contribution of viscous forces is well accounted for up to Ca = 1. The contribution of friction forces depends on the accuracy of physicochemical parameters of both the powder and liquid binder, such as the particle surface area, the powder density and the liquid surface tension and contact angle with the particle. For the wet masses made from magnesium stearate powder, the measured powder surface area and the high saturation level could be the origin of the overestimation of the mass hardness. In addition, the model has been developed based on crude assumptions about the particle shape and texture properties of the agglomerate.

1302

S.J.R. Simons

4.3. An industrial case study: predicting pharmaceutical granulation performance from micro-scale measurements The selection of an appropriate polymeric binder to be used to agglomerate drug with excipients is a critical issue for the development of high-shear wet granulation processes for pharmaceutical tablet systems. The aim of the study reported here, conducted on behalf of Merck Sharp & Dohme Ltd., was to determine the potential for successful granulation through measurement of the interactions of the polymer solutions with individual drug particles. Pharmaceutical powders frequently exhibit poor flow and compaction behaviour, making granulation necessary prior to tabletting. A granulation technique is selected to produce porous, free-flowing material that compacts at low pressures to form non-friable tablets. Although it is possible to produce binderless granules, it is usually desirable to incorporate a binding agent in the formulation to enhance granule and tablet strength. The ability for a binder to distribute between particles can be seen as the result of the competitive effect between the adhesion of the binder with the particle and the cohesion for itself. The more the binder is able to adhere to the drug (favoured by a high work of adhesion) compared to its tendency to self-associate (favoured by a low work of cohesion), the better the spreading and subsequent binding, which ultimately favours the mechanical properties of the agglomerate resulting from the formation of more uniformly distributed solid bridges during the drying phase. Pharmaceutical granules offer a further complexity in fully understanding the formation and breakage - that is, they are usually made up of mixtures of solid species (e.g. drug and excipient) that can exhibit very different interfacial behaviour when in contact with the liquid binder (which then is dried to a solid, often polymeric, bond). During pharmaceutical granulation, the objective is to produce granules that have, on average, a uniform (and repeatable) distribution of drug particles within the bulk carrier (excipient) solid. This can be difficult to achieve and both drug depletion and enrichment in granules can occur (Fig. 33).

~,~Drug

Excipients

.

Binder .........

.............

c~ Dry power

Granulation

~ ~/"

binder Dry

Drying Wet Granule

Dry Granule

Fig. 33. Schematic representation of the action of binder during the processes of granulation and drying.

Liquid Bridges in Granules

1303

Most of the studies reported in the literature tend to focus on the effects on drug/ excipient distribution as a function of differences in primary particle size (see, for instance, Refs. [52,53]). Drug particles are usually very much smaller than excipient particles, which may be as large as 100 l~m and be comparable to the median final target granule size of approximately 200 pm. Hapgood etal. [54] conclude that coarse granules tend to always be drug-enriched since the drug particles are the finest component and preferentially granulate provided that the particles are capable of sustaining liquid bridges. This latter point is very important and is, in part, due to the wetting behaviour exhibited by the liquid binder on the particles (it can also be dependent on the amount of liquid required to saturate the excipient and, possibly, hydrate the binder before liquid is present on the particle surfaces). Hence, it would appear that a crucial step in optimising granulation performance would be to select the most appropriate drug/excipient/binder system to enhance the formation of liquid bridges between both solid species.

4.3.1. Granulation of paracetamol Paracetamol tablets can be produced via high-shear wet granulation of paracetamol crystals (the drug) with pharmaceutically inactive materials (excipients) including a polymeric binder. Typical binders are PVP, HPMC and hydroxypropyl cellulose (HPC), used in aqueous solutions of concentrations ranging between 0.25-7.0% Wtbinder/mlH20. The binder solution is sprayed onto the powder bed as it is being mixed. Usually a chopper blade, rotating at very high speeds (ca. 1000 rpm) is used to aid in the mixing process and to break-up any large agglomerates. The objective is to encourage a uniform mixture of the components. The resulting granulated material is then dried and fed to a tabletting machine that compresses the material in a die to produce uniformly sized and shaped tablets with the desired average content of drug.

4.3.2. Binder selection criteria During pharmaceutical granulation, the objective is to produce granules that have, on average, a uniform (and repeatable) distribution of drug particles within the bulk carrier (excipient) solid. This can be difficult to achieve and both drug depletion and enrichment in granules can occur [54]. One reason for this is the different surface properties of the solid species that can lead to different degrees of wetting with the binder liquid. A crucial step, therefore, is the choice of the most appropriate drug/ excipient/binder formulation to enhance the formation of liquid bridges between both solid species and, hence, that of granules. To minimise formulation development time, it is desirable to make an early decision on the type of binder for a drug, based on the binder's intrinsic ability to spread across the surface of the drug and adhere the drug into granules. In

1304

S.J.R. Simons

principle, the more the polymer spreads across the surface of the particles, the larger the surface area of contact within the granules and the greater the strength of adhesion. The ability for the binder to spread across the drug is determined by the spreading coefficient, SBD: SBD = Wago- WoB

(69)

The more the binder is able to adhere to the drug (favoured by a high work of adhesion, WaBD) compared to its tendency to self-associate (favoured by a low work of cohesion, Wcg), the better the spreading and subsequent binding. Two approaches to binder selection can be taken. In the first approach, the thermodynamics of the final, dry product are considered; in the second, spreading of the solution is considered. The result of drying is that the binder forms bonds between particles. If one assumes that the amount of bonding after drying is entirely determined by the thermodynamics of the dry materials, the binder can be selected on the basis of the dry polymer having a high spreading coefficient equation (69). The dry spreading coefficient can be predicted from the surface polarities of the dry binder and drug, usually derived from contact angle measurements of probe liquids such as water and diiodomethane. Examples of such predictions can be found in Ref. [55], where the spreading coefficient on paracetamol was predicted to increase in the order: Starch < PVP < Acacia < HPMC Measurements of the granule friability, tablet strength and capping index of paracetamol wet granulated with these binders were found to be in line with this ranking. Rowe [55] showed that selection between binder systems for a drug can be gauged simply from the surface polarity of the drug concerned. A disadvantage of this approach is that it does not consider the effect of the solvent. During granulation, the binder solutions form wet bridges between the particles, allowing wet granules to be formed. If it is assumed that the amount of bonding after drying is entirely determined by the contacts set up during wet granulation, the binder can be selected on the basis of it giving a high spreading coefficient of the liquid across the surface. The effect of the binder on the spreading coefficient is usually measured through the consequential decrease in the contact angle of the liquid on the material. Hence, the approach is usually to select the binder giving the smallest contact angle. This can be measured in many ways. The most common techniques- contact angle tensiometry and goniometry- involve powders or compacts, and suffer from many artefacts associated with the structure of the sample, e.g. solvent penetration between the particles. The most direct approach to measuring the relevant interactions between the liquid and solid is to measure the forces experienced between two drug particles separated by a liquid bridge using an MFB. This approach for studying drug materials is described below.

Liquid Bridges in Granules

1305

Fig. 34. An SEM image of a crystal of paracetamol. To improve the wet spreading coefficient, wetting agents are often added to binder solutions. It could be that a combination of the wet and dry spreading coefficients needs to be considered to optimise the binder distribution and subsequent granule formation. This was the focus of the work reported here.

4.3.3. Experimental procedure 4.3.3.1. Materials Needle-shaped crystals of paracetamol (Fig. 34), supplied by Sigma Aldrich, were adhered to the glass micropipettes using Loctite TM Super Glue GEL. PVP (Plasdone K-29/32) was obtained from ISP. HPMC (methocel, 6 cps grade) was obtained from Dow. Liquid binder was prepared with Analar water (BDH). Sodium lauryl sulphate (SLS) and sodium docusate (SD) wetting agents, in their solid state, were supplied by Merk Sharp and Dohme Ltd. The concentrations for the pure binders were 4% Wtb/mlH20 and for the mix of binder and wetting agent, 4% Wtb/mlH20 +0.5% Wtwa/mlH2o. The solutions were all prepared in distilled water. Table 3 gives the values of the liquid vapour surface tensions (TLV) measured using a Kruss 12 tensiometer. Table 3 shows that both wetting agents are surface active and are able to lower the surface tension of the pure binders, sodium docusate being the more active wetting agent.

4.3.3.2. Micromanipulation The MFB apparatus was used to manually elongate, along their axis, liquid bridges (of either HPMC or PVP solution) formed between paracetamol crystals

1306

S.J.R. Simons

Table 3. Surface tension ~LV of polymeric binders used

Solution

Surface tension (mN/m)

PVP 4% PVP 4% + SD 0.5% PVP 4% + SLS 0.5% HPMC 4% HPMC 4% + SD 0.5% HPMC 4% + SLS 0.5%

62.1 29.3 38.9 46.1 27.6 36.8

SLS, sodium lauryl sulphate; SD, sodium docusate.

FlexibleDosing [1 Binder

camera

Solution

/

videorecorder microscope~~~

"~O

li

1

manipulator

Fig. 35. Schematic of Method 1 used to measure the liquid bridge force between a reservoir of liquid binder and a single paracetamol crystal. The measurement of the deflection of the feeding pipette leads to the calculation of the liquid bridge force.

previously attached to the tips of the micropipettes, in a similar fashion as that described in Section 3.1. The movement of the flexible pre-calibrated micropipette was recorded and analysed to determine the maximum adhesion force exerted by the liquid bridges. Two methods were used to manipulate the particles and to obtain images of the separation sequence. Method 1 (Fig. 35)involved measurements on liquid bridges between a drug particle and a reservoir of solution binder held on the flexible micropipette; Method 2 (Fig. 36)involved measurements between two drug particles held by a liquid bridge. Adhesion forces were measured simultaneously. In each experiment, images were taken of two micropipettes, one of which was highly flexible in the direction of bridge separation with its tip in contact

Liquid Bridges in Granules

t

.........

1307

I

,-

camera / videorecorder

microscopel ' ~ ; l

I .....

.: ....

I

!

! t

Fig. 36. Schematic of Method 2 used to measure the liquid bridge force of a binder liquid bridge and two paracetamol crystals. The measurement of the deflection of the bent pipette leads to the calculation of the liquid bridge force. with the other side of the liquid bridge, either directly or through wetting a drug particle bonded to it. The second pipette was rigid with respect to the bridge forces, had a crystal always bonded to its tip and was moved through its micromanipulator to form and break the liquid bridge. The maximum force exerted by a liquid bridge, separated using either Method 1 or 2, was calculated from the maximum displacement of the flexible micropipette with respect to its initial, undisturbed position (see Fig. 12). This micropipette was previously calibrated by attaching known weights to determine its spring constant, as described in Section 3.1. Receding contact angles were measured through detailed analysis of the images of liquid bridge stretching. The baseline of the particle surface was taken before liquid contact and the tangents to the liquid profile were taken at the points of liquid contact. The contact angle was then measured from the angle between the baseline and tangent. Since this method does not account for the asperities and irregularities of the crystal surface, the values obtained are only indicative of the crystal-to-binder wetting behaviour. Other parameters, such as the reservoir volume, the volume of binder deposited onto a crystal and the geometry of the crystal were calculated directly from the images. The volume of the binder reservoir was calculated as the solid of revolution generated by a parabola (i.e. the approximation of the binder meniscus) around the axis of the feeding pipette, whilst the volume of binder left on the

1308

S.J.R. Simons

Fig. 37. Reservoir volume left on the feeding pipette before (left) and after contact with the paracetamol crystal. crystal was calculated as the difference of the reservoir volumes before and after the particle-binder contact, as illustrated in Fig. 37. Measurements of the maximum adhesive force and of the volume left on the crystal were carried out using feeding pipettes of different thickness. For the force measurements, very flexible pipettes (diameter of the thin end ~70 pm) were usedto increase the pipette deflection, while for the volume measurements thicker pipettes (diameter of the thin end -v130 pm) were employed. In the latter set of experiments, the amount of volume left on the crystal is the result of the balance of the binder adhesiveness between the pipette and the paracetamol. Since paracetamol exhibits high interaction with all the solutions tested, a thin pipette would favour the migration of all the binder towards the crystal, hindering any comparison between the different binders. To reduce any geometric influence, pipettes with similar tip diameters of ~130 l~m were employed.

4.3.4. Results and discussion 4.3.4.1. Residual film deposition In a previous study [56], a set of experiments was carried out to investigate the binder deposition on the crystal after contact with the binder solution. In that set of experiments, paracetamol crystals were engulfed in either HPMC 1% or PVP 1% solution, washed using a saturated paracetamol solution and then dried. Observations of the crystal engulfed in the HPMC 1% solution showed more dark patches than those observed when the crystal was contacted with the PVP 1% solution. This behaviour seemed to confirm higher adhesion with the crystal in favour of the HPMC solution. Experiments to investigate the residual film deposition were repeated in the present study using HPMC 4% and PVP 4% as binder solutions. In this set of experiments, the crystal was engulfed and dried to remove any side effects introduced by the crystal washing. Figures 38 and 39 show a comparison for the

Liquid Bridges in Granules Before contact

1309 After drying

Fig. 38. Observations of crystal engulfed in PVP 4% solution. Before contact (left) and after drying.

two solutions and illustrate the crystal faces before and after the engulfmentdrying process. The images do not show large differences in the crystal faces which indicates that the results obtained in the previous experimental study were affected by paracetamol deposition (from the saturated solution) and were not then due to binder transfer, as previously interpreted. This implies that the distribution of such binders during (high-shear) granulation would be poor without the addition of wetting agents. 4.3.4.2. Liquid bridge adhesiveness and volume deposition An extensive experimental programme was undertaken to measure the maximum liquid bridge force and the volume captured by a paracetamol crystal when put into contact and separated from a reservoir of binder. The binder solutions tested are those listed in Table 3.

1310

S.J.R. Simons Before contact

After drying

4

!iI

Fig. 39. Observations of crystal engulfed in HPMC 4% solution. Before contact (left) and after drying. Figure 40 shows the maximum force versus the volume of binder reservoir recorded during the liquid bridge deformation. The force was expected to decrease with the surface tension of the binder, although it is not solely dependent on that parameter. In fact, the PVP 4% + SLS 0.5% solution presents higher value of the adhesive force in comparison to the HPMC 4% solution, despite a lower liquid-vapour surface tension. The total force of the liquid bridge is the result of two effects: that due to the liquid-vapour surface tension and that due to the capillary pressure within the bridge, which depends on the geometry assumed by the liquid bridge during separation. The differences of the total force recorded using the method illustrated in Fig. 35 can be accounted for by differences in the capillary pressure within liquid bridges of different binders. Unfortunately, it is difficult to evaluate the variations of the capillary pressure for the experiments recorded. It is also observed that for the lower surface tension solutions, PVP 4% + SD 0.5%

Liquid Bridges in Granules

1311

40.00 . . ....... . .......... ,

35.00

............. .

.

.

.

.

.

.

.

.

.

!i]7!

15.00 10.00

~!ii!~ ..........~k~i 9

5.00

.

.

0.00 . . . . 0.0E+00

.

.

.

.

.

" ......... - "';~ ............

.............. <,~0 ~ ,i ~' .

.

L...~

"

.

~

.

.

.

I . . . . 5.0E-05

.

~

~s hpmc4% A pvp 4% + sis 0.5% & hpmc 4% + sis 0.5~ 0 pvp 4% + sd 0.5% ~ hpmc 4% + sd 0.5%

.

I . . . . 1.0E-04

I ' 1.5E-04

' I 2.0E-04

[ml]

Vfeed

Fig. 40. Maximum liquid bridge force, measured using method illustrated in Fig. 35, versus volume of binder formed on the feeding pipette [57]. ~:::~pvp 4 % :~i==h p m c 4 % a pvp 4 % + A hpmc 4% O pvp 4 % + hpmc 4%

1.0E-03 ,~

1.0E-04

O

sis 0 . 5 % + sis 0 . 5 % sd 0 . 5 % + sd 0 . 5 %

1.0E-05 > i: i:

&

1.0E-06 1.0E-07

. . . .

0.00

I

0.50

. . . .

I

. . . .

1.00 Vfeedl/3/P

I

1.50

. . . .

I

2.00

[-]

Fig. 41. Volume of binder left on the paracetamol crystal (Vleft) versus volume of liquid on the feeding pipette (Vfeed), parameterised towards the wet perimeter of the crystal (P)

[57].

(29.3 N/m) and HPMC 4% + SD 0.5% (27.6 mN/m), the values of the liquid bridge force are of comparable magnitude. Figure 41 shows the results of the experiments carried out to measure the amount of volume captured by the crystal after contact with the reservoir of binder and indicates that the solutions with lower liquid-vapour surface tension captured more volume of binder. The amount of liquid left on the surface of the crystal is a balance between the adhesion energy at the crystal-binder interface and the cohesion of the binder for itself. Whichever of the two is larger will determine whether the binder remains at the solid-liquid interface or recedes from it. The lower liquid-vapour surface tension promoted by the addition of the wetting agents favours the deposition of liquid on the crystal because it reduces the cohesiveness of the liquid for itself

1312

S.J.R. Simons

(note: the solid-liquid surface tension would also be reduced). Hence, the spreading and distribution of the binder on the solid improves with decreasing surface tension. It should also be noted that HPMC 4% + SD 0.5% has a higher viscosity than PVP 4% + SD 0.5%, which would infer slower rates of wetting for the former compared with the latter. The kinetic effects on the wetting/dewetting process due to viscosity were not studied here. The volume left on the crystal is an important parameter in the understanding of the manner in which the liquid distributes among particles during the particle-binder mixing. The higher the volume left on a crystal the better the binder distribution among crystals, which ultimately favours a homogeneous growth of the granules. In this scenario the binder is more evenly distributed within the bulk mass, forming a more homogeneous mixture. Since the mechanical strength of agglomerates is the result of the adhesive force of solid bridges formed during the drying phase, the distribution of binder just before drying is a fundamental parameter to achieve uniform and improved mechanical properties. In this analysis, it is assumed that all the binders are able to form solid bridges of adhesive strength higher than that measured for liquid bridges. The wet granulation phase should promote a homogeneous binder distribution, which is a prerequisite to the final mechanical properties of the agglomerate, obtained after granule drying. Therefore, a binder should be chosen for its spreading ability rather than for the adhesive force of single liquid bridges. From this experimental work, PVP 4% +0.5% SD seems to be the optimal binder for agglomeration of paracetamol. Another binder solution to take into consideration is the HPMC 4% + 0.5% SD, which has even better spreading ability but higher viscosity. The high viscosity though could retard the redistribution of liquid available on the surface of the crystal during particle mixing [45]. While having good binder distribution in a granulator promotes bridge formation and, hence, granule growth, the strength of the resultant granules depends on the dry binder bonds. Hence binder selection should be based on both the dry and wet binder surface energies.

5. CONCLUSIONS This chapter has detailed the current theoretical and experimental treatment of liquid binder bridges developed between particles in granules. The importance of these bridges in developing sufficient adhesion forces to keep particles together and in governing granule growth, deformation and fracture behaviour has been demonstrated. Novel, micro-scale approaches to the study of these interactions have also been described, in particular, the micromanipulation work carried out by the authors at UCL. The influence of both liquid and solid properties

Liquid Bridges in Granules

1313

in determining the nature of the interactions has been elucidated using such techniques and is now being applied to the prediction of both granule properties and granulation performance - relating micro-scale observations to meso- and macro-scale behaviour.

Nomenclature

a*, a~' a, ai C Ci

d e

g h(d) hi k

ks ri U Ui

~4 F/ X

Y, Yi Y', Y'i y" Ymin

Abr, Abr,i Ac

AAB Adrop B

C

Ca F

Fb

Dimensionless separation distance with respect particle radius, generic or calculated between two configurations (i = 1,2) Liquid bridge separation distance, generic or between particle and liquid bridge neck (i = 1,2) (m) Co-ordination number of particle assembly Droplet cord length on the/th (i = A,B) particle (m) Absolute agglomerate deformation (m) Average particle diameter (m) Deflection of flexible micropipette (m) Gravity acceleration (9.81 m/s 2) Non-deformed agglomerate cap height (m) Height of cap of sphere for the/th (i = A,B) particle (m) Packing constant Flexible micropipette spring constant (N/m) Liquid bridge radii of curvature (i = 1,2) (m) Analytical description of the profile assumed by a particle contacting a liquid bridge, u = u(x) (m) Liquid droplet radius on the/th (i = A, B) particle (m) Relative particle separation velocity (m/s) Average particle volume (m 3) Horizontal axis (m) Liquid bridge ordinate, generic or calculated at point A, B ( i - A , B ) (m) First derivative of y, generic or calculated at point A, B (i = A,B) Second derivative of y (m -1) Minimum bridge neck ordinate (m) Area of the liquid bridge profile, generic or of configuration i (i = 1,2) (m 2) Contact area of an agglomerate with a flat punch (m 2) Solid-solid contact area (m 2) Liquid droplet area (m 2) Integration coefficient, B - YA sin(~]A + ~A)(AP/2?L)YZA(m) Dimensionless parameter, c = V/(1+ 2~/br)1/3/~(1-Jr-~/2) 2 Capillary number Force between particles (N) Liquid bridge force calculated at the particle-meniscus boundary (N)

1314

Fcap Ffric

S .J.R. Simons

Capillary force (N) Friction force (N) Liquid bridge force calculated at the neck of the meniscus (N) Fn Total viscous force in a liquid bridge (N) Fvis R Geometric average radius, R = 2RARB/RA + Rg(m) Dimensionless mean curvature, H * = APR/27 H* Average particle separation distance in an agglomerate (m) H Applied load (N) L Characteristic length of liquid bridge (m) L M Length of test cylinder (m) Number of interparticle contacts per unit area (1/m 2) NE Pressure, generic or for configuration i (i = 1,2) (Pa) P,P~ External medium pressure (Pa) Pext Particle radius, generic or for particle/th (i = A,B) (m) R, Ri Agglomerate radius (m) Rag True surface area of a powder (m 2) S Spreading coefficient of binder on a drug particle (J) SBD Stokes number, St = 2mvo/3~rlR2 St Liquid droplet cap height on the/th (i = A,B) particle (m) Ti Volume of an agglomerate (m 3) Vag Volume of liquid bridge (m 3) Vbr Dimensionless liquid bridge volume with respect cubed particle raV~r dius Volume of cap of sphere for the/th (i = A,B) particle (m 3) Vcap,i Volumes of revolution of the liquid bridge meniscus (m 3) Vr. Vm,A, Vm,B Volumes of revolution of the liquid bridge meniscus, from particle A to liquid bridge neck and from neck to particle B, respectively (m 3) Rupture energy of liquid bridge (J) W Dimensionless rupture energy of liquid bridge, W* = W/TLR2 W* Wa Work of adhesion (J) Work of cohesion (J) Wc Dimensionless factor, X = (t + (a/2R)) X Dimensionless liquid bridge abscissa with respect particle radius, Xc evaluated at the contact with the particle Driven movement (m) xe Follower movement (m) xf Dimensionless liquid bridge abscissa with respect particle radius, xi evaluated at point i Xmin,max Dimensionless X calculated at minimum and maximum liquid bridge separation distance Dimensionless liquid bridge ordinate with respect to particle radius Yi, Yo, Yc First derivative of dimensionless liquid bridge ordinate evaluated at Y~i point i Second derivative of dimensionless liquid bridge ordinate evaluated yff i at point i

Liquid Bridges in Granules

1315

GREEK

7 7L

~LV

,/ 0, Oi Oi p ps

~s

Half-filling angle, generic or calculated at point A, B (i = A,B) (rad) Voidage fraction in an agglomerate Surface tension (N/m) Surface tension of liquid binder towards external medium (either gas or liquid) (N/m) Vapour-liquid surface tension (N/m) Dynamic viscosity (Pa s) Contact angle generic or calculated at point i (i = A,B) (rad) Intrinsic contact angle (rad) Mass density (kg/m 3) True powder mass density (kg/m 3) Liquid volume fraction in an agglomerate Solid volume fraction in an agglomerate Hardness of a material Angle between normal to liquid bridge profile and x-axis (rad)

REFERENCES

[1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14]

[15] [16] [17]

[18] [19]

[20] [21] [22] [23] [24]

[25] [26] [27]

J.F. Padday, Pure Appl. Chem. 48 (1976) 485. A.A. Adetayo, B.J. Ennis, AIChE J. 43 (1997) 927. N. Ouchiyama, T. Tanaka, Ind. Eng. Chem. Process Des. Dev. 21 (1982) 35. Z. Ning, R. Boerefijn, M. Ghadiri, C. Thornton, Adv. Powder Technol. 8 (1997) 15. J. Crassoux, E. Charlaix, H. Gayvallet, J.-L. Loubet, Langmuir 9 (1993) 1995. G. Mason, C.G. Clark, Chem. Eng. Sci. 20 (1965) 859. G. Lian, C. Thornton, M.J. Adams, Powders and Grains 93, C. Thornton (Ed.), A.A. Balkema, Rotterdam, 1993. D.N. Mazzone, G.I. Tardos, R. Pfeffer, Powder Technol. 51 (1987) 71. F.R.E. Bisschop, W.J.L. Rigole, J. Coll. Int. Sci. 88 (1982) 117. T.Y. Chen, J.A. Tsamopoulos, R.J. Good, J. Coll. Int. Sci. 151 (1992)49. E. Wolfram, J. Pinter, Acta Chim. Hung. 100 (1979) 433. T. Dabros, T.G.M. van de Ven, J. Coll. Int. Sci. 163 (1994) 28. M.J. Crooks, H.W. Schade, Powder Technol 19 (1978) 103. S.A. Schildecrout, J. Pharm. Sci. 36 (1984) 502. P.E. Luner, S.R. Babu, S.C. Mehta, Int. J. Pharm. 128 (1996) 29. Y. Pomeau, J. Coll. Int. Sci. 113 (1986) 5. G. Lian, C. Thornton, M.J. Adams, J. Coll. Int. Sci. 161 (1993) 138. F.M. Orr, L.E. Scriven, P. Rivas, J. Fluid Mech. 67 (1975) 723. M.A. Erie, D.C. Dyson, N.R. Morrow, AIChE J. 17 (1971) 115. S.J.R. Simons, R.J. Fairbrother, Powder Technol. 110 (2000)44. X. Pepin, S.J.R. Simons, S. Blanchon, D. Rossetti, G. Couarraze, Powder Technoi. 117 (2001) 123. J.F. Padday, J. Fluid Mech. 352 (1997) 177. J. Plateau, The Figures of Equilibrium of a Liquid Mass, Annual Report of the Smithsonian Institution, Washington, DC, 1864, p. 338. R.A. Fisher, J. Agric. Sci. 16 (1926) 492. R.W. Coughlin, B. Elbirli, L. Vergara-Edwards, J. Coll. Int. Sci. 87 (1982) 18. S.J.R. Simons, J.P.K. Seville, M.J. Adams, Chem. Eng. Sci. 49 (1994) 2331. X. Pepin, D. Rossetti, S.M. Iveson, S.J.R. Simons, J. Coll. Int. Sci. 232 (2000) 289.

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[28] X. Pepin, D. Rossetti, S.J.R. Simons, J. Coll. Int. Sci. 232 (2000) 298. [29] H. Schubert, Int. Chem. Eng. 21 (1981) 363. [30] H. Schubert, Kapillaritat in porosen feststoffsystemen, Springer, Berlin, Heidelberg, 1982. [31] O. Pitois, P. Moucheront, X. Chateau, J. Coll. Int. Sci. 231 (2000) 26. [32] A. Cameron, Basic Lubrication Theory, 2nd ed, Wiley, Chichester, 1976. [33] D.Y.C. Chan, R.G. Horn, J. Chem. Phys. 83 (1985) 5311. [34] X. Pepin, S. Blanchon, G. Couarraze, J. Pharm. Sci. 90 (2001) 332. [35] S.J.R. Simons, X. Pepin, D. Rossetti, Int. J. Min. Proc. 72 (2003) 463. [36] O. Pitois, P. Moucheront, X. Chateau, Eur. Phys. J. B 23 (2001) 79. [37] D. Rossetti, X. Pepin, S.J.R. Simons, J. Coll. Int. Sci. 261 (2003) 161. [38] G. Lian, C. Thornton, M.J. Adams, Chem. Eng. Sci. 53 (1998) 3381. [39] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [40] S.M. Iveson, J.A. Beathe, N.W. Page, Powder Technol. 127 (2002) 149. [41] J. Fu, A.D. Salman, M.J. Hounslow, Proc. World Congress of Particle Tech. 4, paper 113, July 21-25, Sydney, Australia, 2002. [42] A. Samimi, M. Ghadiri, R. Boerefin, R. Kohlus, Proc. 7th Int. Sym. Agglom., May 29-31, Albi, France, 2 (2001) 769. [43] P. Pagliai, S.J.R. Simons, D. Rhodes, Powder Technol. 148 (2004) 106. [44] R.J. Fairbrother, A microscopic investigation of particle-particle interactions in the presence of liquid binders in relation to the mechanisms of "wet" agglomeration processes, PhD Thesis, University of London, 1999. [45] B.J. Ennis, G. Tardos, R. Pfeffer, Powder Technol. 65 (1991) 257. [46] H.G. Kristensen, P. Holm, T. Schaefer, Powder Technol. 44 (1985) 227. [47] S.M. Iveson, J.D. Litster, B.J. Ennis, Powder Technol. 88 (1996) 15. [48] S.M. Iveson, J.D. Litster, Powder Technol. 99 (1998) 243. [49] C. Thronton, Kona 15 (1997) 81. [50] S.M. Iveson, N.W. Page, J.D. Litster, Proc. 7th Int. Symp. Agglom., May 29-31, Albi, France, 2 (2001) 541. [51] S.Y. Chan, N. Pilpel, D.C.H. Cheng, Powder Technol. 34 (1983) 173. [52] Y. Zhang, K.C. Johnson, Int. J. Pharma. 154 (1997) 179. [53] H. Vromans, H.G.M. Poels-Jansseen, H. Egermann, Pharm. Dev. Tech. 4 (1999) 297. [54] K. Hapgood, H.E. Hartman, C. Kaur, R. Plank, P. Harmon, J.A. Zega, Proc. World Congress of Particle Tech. 4, paper 706, July 21-25, Sydney, Australia, 2002. [55] R.C. Rowe, Int. J. Pharm. 58 (1990) 209. [56] S.J.R. Simons, D. Rossetti, P. Pagliai, R. Ward, S. Fitzpatrick, Powder Technol 140 (2004) 280. [57] S.J.R. Simons, D. Rossetti, P. Pagliai, R. Ward, S. Fitzpatrick, Chem. Eng. Sci. 60 (2005) 4055.