Br. Pofym. J. 1973, 5, 101-108

Formation of Films from Latices A Theoretical Treatment Geoffrey Mason

Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leics.

(Paper received 12 February 1973, amendedpaper accepted 26 March 1973)

Two published theoretical models for film formation from latices contain errors. These are corrected and new values of the criteria for film formation derived. In Brown’s model the error is quite small (less than an order of magnitude) but for Vanderhoff’s model the error completely invalidates the analysis. A new model is proposed in which the criterion for film formation is that the latex particles are liquid-a condition fulfilled above the polymer’s glass transition temperature.

1. Introduction

The formation of films from latices by coalescence of the dispersed phase is of con- siderable industrial importance and, as a consequence, the various possible mechan- isms and critical conditions have been investigated.’-* The analyses usually involve the pressure with which the latex particles are forced together and in some cases this has been erroneously calculated; it is the purpose of this work to show the correct solutions and, finally, to review the potential mechanisms and find the most plausible.

2. Brown’s mechanism

Brown2 examined a model for the system involving a lattice packing of elastic spheres being drawn together and compressed as interstitial water evaporated from the surface. He calculated the force between two deformed elastic spheres and also their mutual area of contact, thus obtaining the average pressure acting on the area of contact. Using the work of Haines,’ the negative liquid pressure caused by the menisci being just on the point of entering the system and drawing the spheres together was taken to be 1 I .3 a / R (see also ref. 8) where u is the surface tension of the air-water interface and R the radius of the spheres. Brown’sa proposal is that one of the conditions for film formation is that the capillary force, Fc, pulling the particles together must always be greater than the elastic force, FG, pushing the particles apart. These forces, Fc and FG, are calculated by taking the respective capillary and elastic pressures, Pc and PG, to act over some undefined area, A, and consequently Brown’s condition for film formation becomes

102 C. Mason

Syringe

Figure 1. An example of the capillary pressure and elastic deformation pressures acting over different areas. The capillary pressure on one particle acts nearly over the cross-sectional area of the tube. The

elastic deformation pressure acts only on the flattened region of the spheres.

However, the areas over which PC and PG act will not in general be equal: PC will act over some projected particle area and PG will act only over the area of contact between the spheres.

As an example let us consider two elastic spheres which fit, with a small clearance, into a tube filled with liquid as shown schematically in Figure 1. When the liquid is slowly removed from the tube by the syringe a negative pressure is developed in the liquid as menisci enter the small gap between the spheres and the tube, and this negative pressure draws the spheres together until the elastic force of repulsion between the spheres equals the force produced by the negative pressure. It is apparent that for equilibrium the capillary force must equal the elastic force; it is not true to say, as Brown’s2 analysis infers, that the negative pressure in the liquid equals the average repulsive pressure on the area between the compressed spheres.

Brown’s2 analysis can now be repeated correctly. The force exerted by the liquid on one latex particle is the sum of the capillary pressure and the surface tension forces. ReferencesS-l3 give a discussion of this point. If the meniscus is taken to be just entering the constriction in the pore space between three contacting spheres and we cut a section through this meniscus in a plane parallel to the sphere centres, the sum of the capillary pressure acting over the required area and the component of the surface tension force normal to the plane will be a constant. Fisher15 used this method to derive the force produced by a liquid drop between two contacting spheres. The force normal to the surface acting on a sphere at the surface will equal the capillary pressure multiplied by the average area of the surface occupied by the sphere. The surface ten- sion force serves to communicate the pressure acting over the constriction in the pore space to the spheres. If the capillary pressure is PC and the centre separation of the spheres is 2fR then the area over which the capillary pressure acts for one particle

Formation of Alms from latices 103

is the area of the regular hexagon, edge length 2fR/4/3. Thus the capillary force normal to the surface acting on one particle is 6fsRZPc /43 and this force acts through the three points of contact that our single particle makes with the layer beneath. The six surrounding contacts that our single particle makes are in the plane of the surface and carry none of the normal component of the capillary force. If the force due to capil- larity acting through a single point of contact is FC then this force is at an angle of cos+ 42/43 to h . c a p i h y fomeacting.norma1 to the surface and we have

3Fc 4 2 = - 6 f z R z P c 43 4

or, rearranging Fc = 4 2 f Z R z P c

Now BrownZ has given the repulsive force, FG, of two elastic particles with a Poisson's ratio of 0.5 as

16 3

FG fi - G wo (Lo R)'

where G is the shear modulus, and wo is the maximum deflection for each particle which will, in this case, be (1 - f ) R .

Consequently 16 - 3

FG * - 4 2 G R 2 ( 1 - f)"2

For the system to be continuously compressed as water is evaporated, Fc > FG or

16 - 3

.\r2 f ' Rz PC > - 4 2 G Rz ( 1 - f)"'

Giving

For compression to a solid mass containing no liquidfreduces to a limiting value of fm = 0.905 (see Brownz). If Pc is taken to be the value when the meniscus is just about to pass through the constriction between three mutually contacting spheres then, using HaineP incircle approximation, we obtain

U Pc = 12.9 - R

where u is the interfacial tension between water and air. The condition for film formation then becomes

which differs from Brown'sz value by about a factor of 2.

104 G. Mason

Further assumptions have been made in this analysis. Tae first is that there is a thin film of liquid between the contacting spheres. If there is no water in this region the capillary forces on the particles will be reduced. Secondly, the elastic deformation equation only applies to small displacements and assumes that the areas of contact are circles which, with the large deformations involved in film formation, cannot be the case. Thirdly, it has been assumed that the capillary pressure remains constant as the latex particles move together, whereas, in fact, the constriction in the void space in the plane of three contacting spheres decreases in size as the particles move together.

It is possible to use a simple approximation to estimate the capillary pressure as the particles move together.

Let us assume that the curvature of the meniscus just passing through the constric- tion in the void space equals the curvature of the sphere that just passes through the constriction and that the latex particles simply overlap as they are brought closer together. The radius of the sphere just passing through the constriction becomes ( 2 J 4 - 1) R and consequently the capillary pressure is given by

If this is introduced into expression (1) we obtain the relation for film formation

The right hand side is infinite when f = 1 and also when f = 2 / 3 2 and clearly some- where between these values it passes through a minimum. In fact this minimum occurs when f = 0.942, giving the condition for film formation as

G < 266 R

which differs from Brown’s2 criterion by a factor of over 7.

3. Mechanism of Vanderhoff et al.

Brown’s2 criterion for film formation involves the radius of the latex particles but Vanderhoff et aL5 interpreted the available experimental evidence as indicating that film formation was independent of particle size and advanced an alternative theory of coalescence which depends on the forces developed by a pendular ring of liquid between two spheres. They also confuse force and pressure as did Brown2 but, in their case, it makes more difference. It is well established that the force of attraction between two touching spheres wetted by a drop of liquid is approximately constant at 2 r R a independent of the volume of l i q ~ i d . ’ ~ J ~ Consequently the pressure acting on the stab- ilising layer holding the spheres apart, being only a function of the capillary attraction force and the sphere elasticity, will be effectively constant over a wide range of liquid volumes. As Vanderhoff’~~ results show, the negative pressure in the liquid does rise

Formation of films from latices 105

to very high values as the liquid volume decreases but it acts over an ever decreasing area. The pressure on the stabilising layer remains constant and does not rise to the degree that Vanderhoff6 indicates.

4. Mechanism involving liquid droplets

Let us consider the special case of a water based emulsion of liquid polymer which is drying to produce a film. The situation is now completely different from the theories of Brown2 and Vanderhoff et aL6 as the system is defined only by capillary forces. Initially, when the spheres come into contact, the stabilising layer may be so weak that a film is formed immediately. Otherwise, because the curvature of the flattened region is zero the maximum pressure exerted by the liquid sphere on the stabilising layer can only equal the capillary pressure in the sphere. This maximum pressure will be 2u2/R where 0, is the interfacial tension between the polymer and the water and R is the sphere radius. If the film continues to dry the forces in the system can be analysed following Brown.2 If the sphere centre separation is 2fR, the capillary force acting on one point of contact is, as before,

Fc = JZ f 2 R2 Pc

The repulsive force between the two spheres will equal their internal pressure multiplied by the area of contact

2% FR = -7rRe(1 - fz) R

For continuous compression as water is evaporated we require

or 27r u2(1 - f2) Pc > -- - - d 2 R f 2

Now Pc, the capillary pressure, has its minimum effect when f = 1 ; in this event Haines’ approximation1* gives the maximum capillary pressure.

U Pc = 12.9 - R

For the minimum value of PC to give continual compression during drying:

u 27r u,(l - f 2 ) 12.9 - > - - - R 1 / 2 R f 2

This inequality is independent of the radius of the particles and clearly it holds when f = 1 as the r.h.s. tends to zero. When f = 0.9, the maximum compression, let us say, the inequality becomes

12.40 > u2

G. Mason

Figure 2. Diagrammatic representation of an edge in a highly compressed latex of liquid particles. The free surface of the latex particles has a radius of curvature r .

Usually the air-water interfacial tension, u, will be greater than the polymer-water interfacial tension, o,, so that remembering the factor 12.4, we can be confident that the inequality will be satisfied. Thus, as water is evaporated, the system will be con- tinuously compressed. The pressure on the stabilising layer will not change during this compression.

The approximations made for the initial stages of compression do not apply when there is scarcely any liquid left in the system. A better model would be to consider a packing of rhombic dodecahedra with small triangular channels filled with water along the dodecahedron edges and with a thin film of water and stabilising layer between the faces of the dodecahedra. One such channel is shown diagramatically in Figure 2. In this case the capillary pressure PC can have a maximum value of 12.9ulr where r is the radius of curvature of the dodecahedra at the edges. The pressure inside the particles relative to the pressure in the water is u2/r which consequently equals the pressure on the stabilising layer. As more water is evaporated the radius r will decrease and the pressure on the stabilising layer will increase until, eventually, this layer breaks down and coalescence results. So liquid latex particles with a stabilising layer always form a film-a conclusion independent of the particle size. Let us call this Mode I.

Mode 11. Obviously there are plastic materials with either a low shear modulus, or low elastic limit, for which the same analysis will hold. This is a modified form of Brown’s2 postulates in that he did not consider the elastic limit.

Mode 111. There will be other materials with a higher shear modulus and higher elastic limit which rupture the stabilising layer first and then, partially or completely,

Formation of films from latices 107

flow, driven by the polymer-water interfacial tension, in the manner suggested by Vanderhoff et aL5

Mode IV. Finally, there will be even more rigid materials with the highest shear modulus and elastic limit which break the stabilising layer almost immediately but do not flow under the influence of the polymer/water interfacial tension.

If we grade the elastic limit and the shear modulus qualitatively into low, medium, and high values according to the breakdown pressures of the stabilising layer, then we can examine these four modes in more detail.

In Mode I a film will always be formed irrespective of droplet size. Mode I1 will be the mechanism with a low elastic limit and any modulus, or with

a medium or even high elastic limit and a low modulus. As this mechanism depends on the relative magnitudes of the capillary and the elastic/plastic forces it will be a function of particle size.

Mode 111 will be the mechanism with a medium elastic limit and medium to high modulus and will be virtually independent of particle size until the rupture of the stabilising layer. The subsequent flow will be a function of particle size. It is illuminat- ing to estimate the forces and pressures involved in a system of this kind. The force of attraction between two spheres, with a drop of water forming a bridge at the point of contact, is constant at 27rRo. Suppose this force produces a circular area of flatten- ing of the spheres with a radius R/10, then the average on this area is

Now consider two spheres just starting to fuse with a radius of curvature at the point of contact of R/100, then the pressure difference across the interface will be

100 R R

2 a * - = 2 0 0 0 2 .

It is possible that 1 10

-a,= - - a

in which case the air-water interfacial tension produces a larger pressure than the polymer-water interfacial tension. This is the criterion for Mode I1 so we conclude that Mode 111 presupposes an extraordinary combination of physical properties.

Finally, Mode IV is the mechanism of a system with a high elastic limit and medium to high modulus: the stabilising layer is easily broken but the polymer will not flow.

5. Conclusions

The properties of the polymer are affected by time and temperature. However, it could be said that the polymer will behave as a viscous liquid above the glass transi- tion temperature and as an elastic solid below that temperature. As a viscous liquid, the polymer meets all requirements for film formation, independent of droplet size; but, as an elastic solid, the polymer fails to meet the film-formation requirements

108 G. Mason

except in special cases. It is therefore suggested that the physical nature of the polymer as indicated in particular by the glass transition temperature is the most important factor in film formation-not the stabilising layer and not the particle size.

Appendix

Nomenclature the air-water interfacial tension. the polymer-water interfacial tension. latex sphere radius. the force at a region of contact of two latex particles due to capillarity. the force at a region of contact of two latex particles due to their elastic deformation. the negative pressure produced by capillarity in the liquid filling the inter- stices between the latex particles. the average pressure acting on the mutual area of contact between two com- pressed spheres. an unspecified area. the fractional compression of a latex particle. polymer shear modulus. maximum deflection by compression of a particle. the minimum value off, the fractional compression. the force at a region of contact between two liquid particles due to their internal pressure. the radi.us of curvature of highly compressed liquid latex particles.

References

1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

Dillon, R. E.; Matheson, L. A.; Bradford, E. B. J. Colloid Sci. 1951,6, 108. Brown, G. L. J . Polym. Sci. 1956, 22,423. Protzman, T. F.; Brown, G . L. J . appl. Polym. Sci. 1960,4,81. Voyutski'l, S . S.; Starkh, B. W. ColloidJ. USSR 1952, 14, 314. Vanderhoff, J. W.; Tarkowski, H. L.; Jenkins, M. C.; Bradford, E. B. J. Mucromolec. Chem. 1966, 1, 361. Vanderhoff, J. W. Br. Polym. J . 1970, 2, 161. Haines, W. 9. J . agric. Sci. 1930, 20, 97. Hackett, F. E.; Stretton, J. S. J. agric. Sci. 1928, 18, 671. GJespie, T.; Settineri, W. J. J. Colloid Interface Sci. 1967, 24, 199. Gillespie, T.; Rose, G . D. J . Colloidlnrerfuce Sci. 1968, 26, 246. Cross, N. L.; Picknett, R. G. J . Colloid Interface Sci. 1968, 26, 247. Princen, H. M. J . Colloid Interface Sci. 1968, 26. 249. Derjaguin, B. J . Colloid Interface Sci. 1968,26, 253. Haines, W. B. J . Agric. Sci. 1927, 17, 264. Fisher, R. A. J . Agric. Sci. 1926, 16, 492. Mason, G.; Clark, W. C. Chem. Engng Sci. 1965,20,859.