depletion flocculation in colloidal dispersions

Advances in Colloid and Interface Science 68 (1996) 57-96

ADVANCES IN COLLOID AND INTERFACE SCIENCE ELSEVIER

Depletion flocculation in colloidal dispersions

Paul Jenkinsa, Martin Snowdenb** aZan Wark Research Institute, University of South Australia, Warrendi Road,

b The Levels SA 5095, Australia

School of Chemical and Life Sciences, University of Greenwich, Woolwich, London SE18 6PF, UK

Abstract

The destabilisation of colloidal dispersions by free, non-adsorbing polymer molecules in solution is reviewed from both a theoretical and experimental perspective. Consid- eration is given to the behaviour of polymers in solution in relation to the theories proposed by Flory, de Gennes, and Scheutjens and Fleer. Similarly, the theories describing the behaviour of polymers at or near an interface are also examined together with an outline of their relevance to the depletion interaction. The depletion interaction itself is also comprehensively reviewed, taking into account the different segment density profile theories currently in the scientific literature. Experimental reports of the depletion phenomena are described, covering both hard sphere and soft sphere systems, with both the direct and indirect evidence of the depletion interaction being considered.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2. Origins of depletion flocculation . . . . . . . . . . . 3. The behaviour of polymers in solution . . . . . . .

3.1: Introduction . . . . . . . . . . . . . . . . . . . 3.2. The Flory-Huggins theory . . . . . . . . . . 3.3 The scaling theoryofde Gennes . . . . . . . 3.4. Mean-field or scaling approach? . . . . . . . .

. . . . . . . 57

. . . . . . . . 58

. . . . . . . . 59

. . . . . . . . 61

. . . . . . . . 61

. . . . . . . . 62

. . . . . . . . 64

. . . . . . . . 65

* Corresponding author.

OOOl-8686/96/$32.00 0 1996 - Elsevier Science B.V. All rights reserved. PII: SOOOl-8686(96)00304-1

58 P. Jenkins, M. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96

4. The behaviour of polymers near interfaces .................... 65 4.1. The scaling theory of de Gennes ....................... 65 4.2. Scheutjens-Fleer (S.F.) theory ........................ 67

5. Theoretical models for the depletion interaction ................. 70 5.1. Volume exclusion theories .......................... 7 1 5.2. Segment density profile theories ....................... 73

5.2.1. The scaling theory of de Gennes ................... 73 5.2.2. The theory of Feigin and Napper ................... 74 5.2.3. The theory of Scheutjens-Fleer-Vincent (S.F.V.) .......... 75 5.2.4. The pragmatic theory of Vincent ................... 78

5.3. Extensions to ‘soft’ systems .......................... 79 6. Experimental studies of depletion flocculation .................. 82

6.1. Direct evidence for the depletion force .................... 82 6.2. Phase separation studies ........................... 86

7. Concluding remarks ................................ 93 8. Acknowledgements ................................. 93 9. References. .................................... .93

1. Introduction

When a polymer is introduced into a colloidal system, flocculation is very often observed 111. This particle instability may be due to one of two distinct mechanisms - bridging or depletion flocculation.

Central to the understanding of the type of instability induced by polymer addition is the concept of the surface interaction parameter xS. This quantity is a measure of the net interactions between the surface, the polymer and the solvent. A positive value of xS leads to adsorption of polymer upon the particle surface. At low polymer concentrations, where the adsorption of polymer is insufficient to yield full surface coverage, a polymer chain may adsorb onto two discrete particles, causing aggregation. This is called ‘bridging flocculation’.

However, should xS be less than a certain critical value xS, crit (usually a few tenths of a kT), then adsorption of polymer onto the particle surface is unfavourable. In such circumstances, the centre of mass of the polymer coil is displaced from the interface, leading to a ‘polymer- depleted’ zone. This phenomenon has been termed ‘depletion’ and may induce weak, reversible particle aggregation.

The purpose of this review is to detail the effects of non-adsorbing polymers on the dispersion stability of colloidal particles. Initially, as an introduction, the origins of depletion flocculation are briefly discussed. The behaviours of polymers in solution and at interfaces are then described with reference to pertinent mean-field and scaling approaches.

P. Jenkins, M. SnowdenlAdu. Colloid Interface Sci. 68 (1996) 57-96 59

Then, the many models developed to describe the depletion potential are outlined. Finally, previous experimental studies are summarised.

2. Origins of depletion flocculation

The first experimental observation of depletion flocculation was reported by Traube [2], whilst investigating the creaming of natural rubber in the presence of water soluble polymers. Subsequently, various authors [3,4] described the aggregation of red blood cells using hydro- philic polymers.

However, it was not until some thirty years after these initial observations that Asakura and Oosawa [5,6] presented the first quantitative analysis of this phenomenon. These authors argued for the existence of a ‘depletion zone’ next to a surface in contact with a solution of non-adsorbing polymer, in which the polymer segment concentration is lower than that of the bulk solution, leading to a similar segment density profile as that shown in Fig. 1. The driving force for the existence of the depletion zone is the conformational entropy restrictions suffered by the polymer coils that, for non-adsorbing moieties, are not compensated by an adsorption energy. A working estimate for the depletion layer thickness,

Fig. 1. The segment volume fraction profile for a polymer solution next to a non-adsorbing interface. The depletion layer thickness A may be defined through the equality of areas a1 and a2. & is the volume fraction of the polymer in the bulk solution.

60 P. Jenkins, 134. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96

h>D h=D hS2A

26 D

Fig. 2. The interstitial free polymer concentration between two flat plates illustrated as a function of surface separation h. D represents the distance above which the polymer concentration between the plates may attain the bulk solution value. The lower graph shows how Scheutjens-Fleer-Vincent theory 171 can be used to estimate the depletion layer thickness A (see Section 5.2.3).

’ Depletion Zones /

I

Above ‘pi :

Fig. 3. A schematic illustration of depletion flocculation. $I& is the critical volume fraction of polymer required for flocculation. At polymer volume fractions greater than c)$ overlap of depletion zones occurs.

P. Jenkins, M. Snowden IAdv. Colloid Interface Sci. 68 (1996) 57-96 61

denoted by A, may be obtained by Scheutjens-Fleer-Vincent [7] theory, as will be discussed in Section 5.2.3. Figure 2 illustrates this method alongside the schematic polymer concentration between two non-adsorbing, flat plates.

Polymer molecules are similarly excluded from the interstices between particles, when the separation of the particles is less than the diameter of the free polymer coil, leading to the formation of a region of pure solvent. Figure 3 illustrates this for polymer volume fractions above and below the critical volume fraction of polymer required to induce flocculation, @h .

At sufficiently low particle concentrations, the solution outside the depletion zone retains its bulk polymer concentration. As particles approach, the volume of the depletion zone is decreased and hence, solvent is displaced from the inter-particle region towards the bulk solution. The free energy of the system is therefore lowered since the solvent is transferred from a region of higher chemical potential (the overlap of the depletion zones) to one of lower chemical potential (the bulk polymer solution).

3. The behaviour of polymers in solution

3.1. Introduction

Discussion of the role played by polymers in colloidal systems, whether they are free (as in depletion) or adsorbed (as in steric stabilisation), will necessarily need a consideration of polymer solution ther- modynamics. The best known, and still one of the most widely used theories, is that described by Flory [8,91 and Huggins [lo-121. The Flory-Huggins theory belongs to the group of polymer solution thermodynamic theories known as ‘mean-field’ theories. The key premise is that, at high polymer concentrations, the solution possesses an essentially homogeneous segment density. ‘Mean-field’ implies that interactions are the same for all segments owing to their identical local environment. Unfortunately, this mean-field concept breaks down at low concentrations when no overlap of polymer chains is present. In such circumstances, as illustrated in Fig. 4, spatial fluctuations in segment density invalidate the mean-field approach. De Gennes [131 has developed a scaling theory which deals with this very situation.

62 P. Jenkins, M. Snowden IAdv. Colloid Interface Sci. 68 (1996) 57-96

Fig. 4. The de Gennes [131 the correlation length 6.

representation of a semi-dilute polymer solution, showing

3.2. The Flory-Huggins theory

The Flory-Huggins theory [ 141 aims to predict the Gibbs free energy of mixing when a pure polymer is added to a pure solvent. The free entropy of mixing (a combinational term) and the enthalpy of mixing are calculated separately to yield the required energy term as shown in Eq. (11,

AGM = mM - TASK (1)

where the superscript M denotes mixing. A full description of Flory-Huggins theory, including derivation of

the expressions for the free entropy, enthalpy and energy of mixing is given by Flory [14]. Only a brief outline of the pertinent equations will be given here.

The Flory-Huggins theory is based on a lattice model. Polymer segments and solvent molecules are arranged in 3-dimensional space. Each lattice site is of an equal size, equivalent to the dimensions of a solvent molecule. Monodisperse polymer chains adopt an arrangement encompassing r consecutive lattice sites. An important observation is that r is usually less than the degree of polymerisation, i.e. each polymer segment may be formed from two or more ‘monomer’ units.

Given such a model, the configurational entropy may be calculated, using a statistical treatment, by consideration of the number of possible distinct arrangements of polymer chains that are obtainable. The entropy of mixing of nl solvent and n2 polymer chains (each consisting of r segments) may be expressed by the following simple equation,


ASM = - K(n, In +r + n2 In Q2) (2)

where K is the Boltzmann constant, @r and +2 are the volume fraction of solvent and polymer respectively.

The Flory-Huggins theory considers mixing as a pseudo-chemical reaction between polymer segment contacts (2-2) and solvent contacts (l-l), i.e.

l-l + 2-2+2 (1-2)

This allows the following relationship for the enthalpy of mixing to be derived.

~~ = nl43 x12 kT (3)

where,

z&2 x12 = kT (4)

z is the coordination number of the lattice. u12 denotes the internal energy of contact between a solvent (1) and polymer segments (2). X12kT is a measure of the difference in energy of a solvent molecule when immersed in a pure polymer compared with that in a pure solvent. The xl2 parameter is a central tenet of Flory-Huggins theory.

Combination of Eqs. (11, (2) and (3) yields the following equation for the free energy of mixing

AGM = kT In & + n,Q12

I’*’ Enthalpic

(5)

Equation (5) is the well-known Flory-Huggins expression for the free energy of mixing solvent with amorphous polymer and it is interesting to note that it contains no lattice parameters. Differentiation yields the familiar expression for the chemical potential of a solvent in a polymer solution.

* = o2 ( 1 1 - : + In (1 - $2) + X12 $g


where py is the standard chemical potential of the pure solvent. The second virial coefficient B,, is given by

B, = -g ( ‘82 - x12)

where vy is the solvent molar volume. When xl2 = l/2, it can be seen that B, becomes zero. This is the so-called ‘&condition’ at which the polymer solution will behave ideally and the polymer chain adopts its unpertur- bed dimensions. In a good solvent, however, the polymer segments are mutually repulsive and the chain expands beyond these ideal dimensions. This gives rise to an excluded volume effect.

3.3. The scaling theory of de Gennes

The approach of de Gennes [131 to polymer solution properties is more applicable to the case of semi-dilute solutions with large fluctuations in segment density. De Gennes suggested that such a solution may be regarded as a network with an average mesh size, 5, termed the ‘correlation length’. In de Gennes’ interpretation termed scaling theory, the polymer solution is considered to be composed of self-avoiding sub-chains, which form a close-packed array of ‘blobs’. Hence, the solution is regarded as inhomogeneous on the scale of the blob size (or correlation length) 6. The critical overlap volume fraction +;, for polymer coils in solution may be expressed by

q);= M2 $NA 1’2

(8)

where M, and p2 are the polymer molecular weight and number density respectively. NA is the Avogadro constant, rg is the radius of gyration of the polymer coil in dilute solution and b is a factor which depends upon the packing of the coils.

In dilute solutions, below OS, the blob has a size equivalent to the radius of gyration of the polymer coil. However, in semi-dilute solution, beyond +;, it obeys the scaling law given by Eq. (91, the blob size decreasing with increasing polymer volume fraction,

(9)

P. Jenkins, M. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96 65

Scaling theory is a very powerful tool, which has found application in many aspects of the behaviour of polymers in solution and at interfaces, including depletion [ 151. Its usefulness stems from the ease with which the laws may be tested experimentally, since attention is focused on exponents rather than difficult to determine pre-exponential factors.

3.4. Mean-field or scaling approach?

Much argument has been directed toward the validity of the two distinct approaches - scaling and mean-field. Fleer et al. [ 161 have argued that, in the case of polymer adsorption, the segment concentrations are high enough to warrant the use of a mean-field treatment. However, de Gennes [ 171 has proposed that a scaling approach is more appropriate due to large segment density variations within an adsorbed polymer layer. Schafer [18] has also discussed the applicability of scaling and mean- field theories. He concluded that the mean-field treatments are valid over a far wider range of polymer concentrations than are the scaling methods; the main exception being the case of dilute solutions. Signifi- cantly, de Gennes (1987) has suggested that a combination of the two approaches, to produce a unified treatment, may be viable in the future.

4. The behaviour of polymers near interfaces

The theories, previously outlined in Section 3, model polymer conformations as random walks in continuous space or on a lattice. However, near a surface, this description needs modification because the surface is impenetrable to polymer chains. This leads to a lower configurational entropy in the vicinity of the surface. For a case where the polymer segments have no affinity for the surface, a depletion zone will arise adjacent to the surface. The following section is intended to discuss the pertinent points of scaling and Scheutjens-Fleer theories as they are applied to polymer solutions at interfaces. In both cases depletion, or ‘negative adsorption’, will be emphasised.

4.1. The scaling theory of de Gennes

De Gennes 1191 has extended the scaling approach to describe the situation where a polymer solution is in contact with an impenetrable wall. He argued that the case where a polymer is repelled by the wall is analogous to the repulsion between polymer chains in semi-dilute solu-


Fig. 5. The segment density profile derived by de Gennes [ 191 for a polymer solution in contact with a repulsive wall.

tion, which leads to segment density variations of a characteristic size 5. This leads to the idea that a depletion zone, next to the wall, will have dimensions of the order of 6. At low polymer concentrations, where isolated coils are present, 5 may be equated with the dilute solution value for the radius of gyration r$ of the polymer coil. However, in the semi-dilute regime, the depletion layer thickness A is given by Eq. (10):

(10)

For a good solvent m = -3/4 and for a e-solvent m = -l/z. Moreover, the radius of gyration itself scales with a different index in the semi-dilute solution, as shown in Eq. (11):

(11)

Again the exponent n depends on the quality of the solvent. For a good solvent n = -l/s, whilst for a O-solvent n = 0.

The form of the concentration profile derived by de Gennes [19] for a polymer solution in contact with a repulsive wall is shown in Fig. 5.

Tbe polymer segment volume fraction as a function of distance from the wall may be described by the scaling relationship:

(12)

The parameter D has been linked by de Gennes to the ‘stickiness’ of the


polymer-surface interaction. In essence, D defines the distance from the surface beyond which short-range forces between polymer segments and surface sites no longer exist.

4.2. Scheutjens-Fleer (S.F.) theory

The Scheutjens-Fleer theory [20-221 is based upon a mean-field approach and evaluates the segment density profile of a homopolymer situated between two flat plates, as a function of separation.

In the Scheutjens-Fleer theory, the polymer chains are described as walks upon a lattice, which is bordered by two flat plates (see Fig. 6). The space separating the plates is divided into M parallel lattice layers, numbered i = 1, 2, 3 ,..., M, each layer consisting of L lattice sites. The surfaces are bordered by layers 1 and M, segments in these layers are considered to be adsorbed.

Now, any lattice site in layer i, where 2 I i I (M - l), has a given number of neighbouring sites; a proportion of which, A,,,, are in the same layer and a further proportion, h,, are in adjacent layers, i.e. (i - 1) and (i + 1). For a simple cubic lattice, in which the lattice coordination number is 6, h, = 4/s and h, = l/6.

Any lattice site in the same layer is assumed to possess the same energy. Furthermore, the variations within each layer are ignored, allowing the averaged volume fraction of polymer segments in layer i, +!! , to represent the probability of a site in this layer being occupied by a polymer segment. This leads to Eq. (13) for the total energy, Gtotal, of the system,

M

G total = - kTn, xs + k’px12 c ni (46)

i=l

1 2 3 4 5 6 7 . . . . . . . . M

0 solvent molecule

??polymer segment

Fig. 6. The Scheutjens-Fleer [20-221 lattice model.

(13)

68 P. Jenkins, M. Snowden IAdv. Colloid InteTface Sci. 68 (1996) 57-96

where ni and nh are the number of solvent molecules and polymer segments in layer i, respectively. xl2 is the Flory-Huggins polymer-solvent interaction parameter. Silberberg 1231, followed by Roe 1241, adopted the concept of the surface interaction parameter xs, which may be related to the difference between the Flory-Huggins xls (solvent-surface) and x2s (polymer-surface) parameters. The site volume fraction (&, ) is an average over three layers as shown in Eq. (14).

(14)

Obviously, any chain may assume a large number of possible conformations. These ‘conformations’, as the Scheutjens-Fleer approach terms them, are defined by the layer numbers in which each of the successive chain segments are located. It must be realised that each conformation is thus a set of many different arrangements. Using the Bragg-Williams approximation of random mixing within each layer, each arrangement with a particular conformation must possess the same energy. As a result the energy of any system depends entirely on the chosen set of conformations. Differentiation of the partition function with respect to the number of chains possessing a particular conformation, allows evaluation of the equilibrium conformation distribution, i.e. the number of chains nd in a particular conformation d at equilibrium.

In the Scheutjens-Fleer theory, the conformation of any chain of r segments (and thus r - 1 bonds) is specified by rl segments in layer 1, r2 in layer 2, up to ri in layer i and q bonds parallel to the bounding surface. The probability that such a conformation c will be adopted by a chain is thus,

(15)

where PI, ~2, . . . . pi are weighting factors assigned to the energy of a segment, defined by xi2 and the layer number i. The measure of the probability of locating a free segment in layer i, denoted by pi, and derived from the partition function is

44 pi=beXp

$1 [-2x12(& - 441

Oi Pi=b exp

01 C -2xi2((&) - 6$] exp lx, - hi X121

(16)

(17)


where +i is the volume fraction of solvent in layer i or in bulk solution when i = b. Segment-solvent interactions are accounted for by the term exp[2X,,(($i ) - $i >I in both Eqs. (16) and 17). Equation (17) allows for adsorption of segments in the layers adjacent to the surface (i.e. i = 1, AI) by the utilisation of the factor exp@, - hl~r2).

The segment volume fraction $+ can be evaluated by application of these weighting factors to Eq. (18).

44 pi = r C ri(C) P(C) (18)

C

where & is the polymer volume fraction in bulk solution and hi is the number of segments of a particular conformation c situated in layer i. The random walk statistics and detailing of all possible chain conformations is achieved using the elegant matrix formalism of Di Marzio and Rubin [251.

The free energy of interaction between two plates Gti may thence be determined from the following expression:

G&W = WW - WI (19)

where 1((M> and $-> are the interfacial tensions at plate separations of M lattice layers and at infinite separation. @II is given by,

M 44

C In 7+x12 I (20) i=l 49

in which ‘yl is the interfacial tension for pure solvent, as is the area per site and 0,, is the adsorption excess (in equivalent monolayers) given by

M

cm i=l

The iterative method used to derive the polymer segment density profile has been described by Fleer et al. [161, but the salient steps are summarised by Fig. 7.

Fleer et al. 116,261 have presented calculations for segment density profiles for both adsorbing and non-adsorbing polymers in the presence of interfaces. Treatment of depletion in Scheutjens-Fleer theory may be achieved by equating xs to zero (or even making it negative). Figure 8


Estimation of 4’

1 Calculation of pi fro; equations 16 and 17 -

1 Calculation of P(c) from equation 15

1 Calculation of+’ from equation 18

J Repetition until convergence of $I i

Fig. 7. The Scheutjens-Fleer iterative procedure [ 161.

1

0.8

0.6

0.4

0.2

0 1 2 3 4 5 6 7 8 9 10

i

Fig. 8. Segment density profiles for non-adsorbing polymer chains between two flat plates at four different solution concentrations, calculated by the S.F.V. approach [26]. Polymer chain length, r = 1000, Flory-Huggins parameter, xIz = 0.44 and surface interaction parameter, Xa = 0.

features model profiles derived using the Scheutjens-Fleer approach [261 for chains of 1000 segments in a good solvent (xl2 = 0.44) inter- spersed between two non-adsorbing walls (x, = 0) in equilibrium with a bulk solution.

5. Theoretical models for the depletion interaction

Many authors have proposed models to describe the depletion attraction. However, these theories fall conveniently into two distinct classifica- tions: those which explain the effect due to volume exclusion and the others which consider the polymer segment density profile next to the surface.


5.1. Volume exclusion theories

As mentioned previously, Asakura and Oosawa were the first to recognise that an attractive force could act between particles suspended in a solution of macromolecules. Their original treatment [5] considered two flat plates immersed in solutions of spherical or cylindrical macromolecules. This was later [6] extended to embrace spherical particles.

The latter model considered hard, spherical particles of radius a and spherical polymer molecules, possessing a radius of gyration rg. Asakura and Oosawa allowed the macromolecules to freely interpenetrate with one another but to appear as hard spheres to the particles. This leads to exclusion of the centre of mass of the polymer spheres from a layer, of thickness rg, around the particles, thus creating a depletion zone.

Equation (22) gives an expression for the free energy Gdep required to bring two particles together from infinite separation to a surface-surface separation h, where h < 2rg

Gdep = Al-lAV c-22)

AV is effectively the extra volume available for occupation by the polymer molecules due to the overlapping of the depletion zones of neighbouring particles. It is given by

3r r3 4(a + rg) + 16(a + r.&3

Forh>2(a+rz):AV=O (23b)

w w Depletion Zones

Fig. 9. Definition of terms for the Asakura and Oosawa [5,6] model of depletion flocculation.

72 P. Jenkins, M. Snowo!enlAdv. Colloid Interface Sci. 68 (1996) 57-96

where r is the distance between particle centres and a is the particle radius (see Fig. 9).

Al-I represents the difference in osmotic pressures between the polymer-depleted (or pure solvent) region and the bulk polymer solution and for an ideal solution, is given by

Al-I=-pkT (24)

where p is the number density of polymer molecules in the bulk solution. Substitution of Eqs. (23) and (24) into Eq. (22) yields the following relationship for the attractive energy Gdep,

For r c 2a: Gdep = 0

For 2a c r c 2(a + rJ: Gdep = - f(a + rg)3 3r r3

4(a + rg) + 16(a + rg) 3 PkT 1 For 2(a + r,J c r: Gdep = 0 (25)

This result, obtained by Asakura and Oosawa, has subsequently been used by many other workers [27-311 to describe the depletion attraction in studies concerning the flocculation of a wide variety of experimental systems.

A major disadvantage of the Asakura and Oosawa treatment is the inability to forecast dispersion restabilisation at high bulk polymer concentrations. This is a consequence of the theory predicting that the depletion interaction increases continuously with increasing polymer concentration. To address this, Walz and Sharma [32] have recently extended the Asakura and Oosawa model by the incorporation of a second order virial expansion of the single particle distribution function, in order to account for the effect of higher polymer concentrations. Just like the Asakura and Oosawa model, this extension gives rise to a depletion attraction at small particle-particle separations. However, at larger separation distances a repulsive maximum in the depletion energy is seen and this gives rise to a kinetic depletion ‘stabilisation’. This will be discussed in more detail in Sections 5.2.2 and 5.2.3.

Mao et al. [331 have extended the Asakura and Oosawa model in a similar manner to Walz and Sharma [321 but with the use of a third order virial expansion. Their treatment predicts the well-known attrac-


tive force at small particle-particle separations, a repulsive barrier at larger distances and beyond that a secondary minimum. However, these authors state that the repulsive barrier and the secondary minimum are too weak for kinetic stabilisation (due to the barrier) or flocculation into the secondary minimum to be significant, except in the case of extreme particle to polymer size ratios.

5.2. Segment density profile theories

The theory of Asakura and Oosawa and the extensions of it described by other workers, all treat the behaviour of polymer molecules, with respect to the colloidal particles, as ‘hard-sphere’ like. The polymer molecules are assumed to be completely excluded from the inter-particle region once the particle separation is less than the hard-sphere diameter of the polymer coil. However, polymer molecules are clearly not hard spheres but, in fact, are relatively flexible chains. Hence, they will never be able to totally vacate the region between the particles and there will always be residual polymer segment density in the depletion zone.

Many workers have presented theories which endeavour to take this chain flexibility into account. Effectively, these theories consider the consequences that incomplete depletion of the segmental concentration between particles, has on the colloidal interactions. Napper [l] has termed such treatments ‘Theories of Partial Depletion’, owing to the presence of residual polymer segments in the depletion zone, even at particle separations which give rise to depletion flocculation. This con- trasts to the Asakura and Oosawa approach, where no polymer exists between the particles at flocculation and has been designated as a ‘Theory of Full Depletion’ by Napper.

5.2.1. The scaling theory of de Gennes De Gennes [19] has derived an expression for the free energy of

depletion Gdep, using the concentration profile from Fig. 5. Joanny et al. [E] extended this treatment to spheres of radius a separated by a distance h, using the Derjaguin approximation. The relationship they obtained (for h cc a) is given by Eq. (26).

G dep (26)

It can be seen that this approach, like that of Asakura and Oosawa [5,61,


predicts an increasing depletion interaction with increasing polymer concentration. It is possible to extend this model to encompass restabilisation, by allowing the correlation length 5 (and thus A) to vary with free polymer concentration, in accordance with accepted scaling laws, i.e.

(27)

5.2.2. The theory of Feigin and Napper Feigin and Napper 134,351 proposed an approximate theory of depletion

flocculation and restabilisation. They calculated the segment density profile of macromolecules using rotational-isomeric-state Monte-Carlo procedures. The depletion layer thickness was found to be of the order of the radius of gyration of the polymer chain. The volume fraction of polymer, between the two parallel plates, was established as a function of separation and was found to drop to near zero, when the plate separation was reduced to values below approximately twice the radius of gyration of the polymer coil.

Combination of the evaluated segment density profile with the Flory- Huggins expression (Eq. (5)) for the free energy of mixing of polymer and solvent molecules, enabled calculation of the free energy of mixing as a function of plate separation. This analysis was then further extended, to consider spheres of radius a, using the Derjaguin approximation.

Figure 10 illustrates the free energy curves derived using the approach of Feigin and Napper. These interaction curves display a mini-

Fig. 10. The depletion energy, as calculated by Feigin and Napper [35], for two flat plates separated by a distance h, in a non-adsorbing poly(ethylene oxide) (M,, = 20 000) solution. (A) I$ = 0.07; (B) @ = 0.04; (0 I$ = 0.02.

P. Jenkins, M. Snowden IAdu. Colloid Interface Sci. 68 (1996) 57-96 75

mum at small particle separations, which is in agreement with other theories. However, at greater separations, a maximum is observed, which is due to the expulsion of polymer chains to the bulk solution. The minimum and maximum are both small at low polymer concentrations and hence flocculation does not ensue. However, as the polymer concentration rises, the minimum becomes sufficiently deep for flocculation of the particles to occur. Finally, at higher still polymer concentrations, the maximum becomes high enough to prevent flocculation. This depletion restabilisation (or more correctly ‘stabilisation’ in this treatment) is a kinetic effect, since the thermodynamic equilibrium position would be flocculation into the deep minimum. This agrees with the predictions made by the Walz and Sharma [32] extension of the Asakura and Oosawa model.

5.2.3. The theory of Scheutjens-Fleer-Vincent (S.F.V.) With an approach based upon the Scheutjens-Fleer theory [20-221

of polymers at interfaces, as described in Section 4.2, Scheutjens, Fleer and Vincent [7] have modelled the depletion interaction by setting xs = 0. They h ave succeeded in obtaining the attraction energy per unit area between two plates, immersed in a non-adsorbing polymer solution, as a function of plate separation.

Figure 11 illustrates an example of the results obtained for two different solution concentrations. The plots are linear over a wide range of plate separations, which correspond to the region where no polymer exists between the plates. In this region, the attraction is solely propor-

Fig. 11. Numerical calculations, based on the S.F.V. theory [7,16], of the normalised depletion energy per unit area as a function of separation (M/l) for two parallel plates (I = lattice element size). r = 1000, xIz = 0.5; (A) 4: = 0.1; (B) 41: = 0.2.


tional to the number of solvent molecules transferred from the solvent- rich region to the polymer-rich region. Extrapolation of this linear region to Gd,,Z2/kT = 0, allows an estimation of the depletion layer thickness A, as at this intercept M - 26.

Another interesting feature of these depletion energy curves is that they do indeed exhibit maxima at high polymer volume fractions. However, unlike the energy barriers found by Feigin and Napper, the ones predicted by the S.F.V. treatment are negligible in size. Thus, S.F.V. theory argues that these maxima have very little consequence for the interparticle interaction. Hence, the S.F.V. approach does not predict depletion restabilisation to have a kinetic origin.

For two spherical particles of radius a, separated by a surface-surface distance of h and immersed in a polymer solution, where p1 and ~1 are the chemical potential of the solvent in the bulk polymer solution and the pure solvent respectively, then the depletion attraction energy Gdep,is given by,

where VT is the molar volume of the pure solvent and A is the depletion layer thickness.

The first bracketed term on the right-hand-side of the equation (28) is equal to the osmotic pressure of the bulk polymer solution and is given by the Flory-Huggins expression

where xl2 is the Flory-Huggins polymer-solvent interaction parameter and r represents the number of effective segments per polymer chain.

When the two particles are in contact, i.e. h = 0, then Eq. (28) yields the following expression for the particle contact energy Gdep, o.

(30)

S.F.V. theory predicts that the depletion layer thickness A is a function of polymer concentration. Consider the case where the polymer volume


20 -

A/l r-‘(100 15 -

Fig. 12. Computed plots of the reduced depletion layer thickness Al1 as a function of bulk polymer volume fraction (I$ ), established by the Scheutjens-Fleer-Vincent theory [161 for various polymer chain lengths, r. (4) indicates the critical coil overlapvolume fraction of the polymer, 4;.

fraction increases to above that of chain overlap (($1. Here, the solution osmotic pressure increases and this enables narrower inter-particle gaps to be entered by the polymer coils. Figure 12 illustrates plots of A against @i derived by the S.F.V. approach, with an arrow indicating the critical coil overlap concentration +; for each chain length. In fact, Scheutjens, Fleer and Vincent have derived the analytical relationship shown in Eq. (31) which is valid for approximate conditions of &solvency and polymer volume fractions below 0.6:

(31)

where each polymer chain consists of r ‘effective’ segments each of length 1.

S.F.V. theory explains depletion restabilisation through the rapid decrease of A at high polymer volume fractions (i.e. Q2 > @f >. The depletion attraction increases almost linearly at low polymer volume fractions. However, at higher polymer volume fractions, the decrease in the A2 term of Eq. (30) dominates over the developing chemical potential difference between the pure solvent and bulk polymer solution. This causes a maximum in the magnitude of the depletion attraction, as in Fig, 13, after which the interaction decreases with further increases in polymer volume fraction. Above a certain polymer volume fraction, +$ , the attraction may again become too weak to initiate flocculation. Scheutjens, Fleer and Vincent have proposed that since this stability is

78 P. Jenkins. M. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96

C

-10

s -20 .

: -30 u

-40

Fig. 13. The variation of the depletion interaction energy with bulk polymer volume fraction (I$“, ), as predicted by the S.F.V. theory 171.

not a consequence of a repulsive barrier, as in the treatment of Feigin and Napper [35], but merely of too weak an attraction, the term depletion restabilisation would be more intuitive as a description of this phenomenon. The restabilisation predicted by S.F.V. theory is thus a thermodynamic effect.

5.2.4. The pragmatic theory of Vincent Vincent [361 has described a pragmatic approach to treat depletion

layers at interfaces. In this treatment the compressional force experi- enced by a polymer coil near a wall is equated with the osmotic pressure of the bulk polymer solution. Using the equation derived by Evans et al. [37] for the elastic compression interaction energy for polymer chains between two hard walls, Vincent derived the following simple quadratic equation (32) to describe the depletion layer thickness A.

A=Sl+b2-b rg

(32)

where rg is the radius of gyration of the polymer chain and b is given by

(33)

where I is the polymer segment length, r is the effective number of segments in the polymer chain, & b the bulk polymer volume fraction


12

A/l 9

x = 0.5

Fig. 14. Computed plots of the depletion layer thickness A as a function of bulk polymer volume fraction (& 1, established by the pragmatic theory of Vincent [36] for various r values. (k) indicates the critical coil overlap volume fraction of the polymer, $.

and (ul - @)/IzTrepresents the osmotic pressure of the bulk solution (see Eq. (29)). Substitution for the osmotic pressure term in Eq. (33) yields the following equation for b.

(34)

where xl2 is the Flory-Huggins polymer-solvent interaction parameter. Clearly for small values of b, i.e. when (I! + 0, then A + rg as expected. Figure 14 illustrates plots of A against 4: derived by the pragmatic approach of Vincent, with an arrow indicating the critical coil overlap concentration 9; for each chain length. The general form of the plots are very similar to those shown in Fig. 12, obtained using the S.F.V. model. Typically, the S.F.V. and pragmatic models give good qualitative agreement. However, in order to make more quantitative comparisons, Fleer et al. [26] have suggested a procedure that can be used to convert the lattice dimensions used in S.F.V. to actual distances,

5.3. Extensions to ‘soft’ systems

The models of depletion flocculation outlined have all been developed to cope with the situation when a flexible polymer is adjacent to a hard wall. However, many colloidal particles possess surfaces covered with

80 P. Jenkins, h4. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96

irreversibly adsorbed chains. Such surfaces are termed ‘soft) and intro- duce extra complexity to the depletion interaction. In particular, the layer of adsorbed chains may be partially interpenetrated by the free chains in the bulk. This serves to reduce the depletion interaction.

All of the attempts to model this problem have been modifications of the theories which consider hard surfaces. Initially, Feigin and Napper [381 attempted to extend their rotational-isomeric-state Monte-Carlo approach (Section 5.2.2) to consider the consequences of pre-adsorbed chains, by assuming a uniform distribution of the adsorbed chains with average volume fraction e . They postulated that if the bulk volume fraction $$ was below $$ then no overlap of the adsorbed and free chains occurred. At low values of $i , the pre-adsorbed layer thus acts as a hard wall. Upon increasing e to values greater than @$ , interpenetration of chains was allowed until 6 and $8 became equal. In this case, only if the dimensions of the free chains exceeded those of the pre-adsorbed chains would depletion be observed.

The model of Feigin and Napper [381 is unrealistic in as much that some mixing of pre-adsorbed and free chains will always occur. The Scheutjens-Fleer theory was extended to consider terminally attached chains by Cosgrove et al. 1391. Subsequently van Lent et al. 1401 applied this extended model to the situation in which a wall covered with terminally attached chains was in the presence of free polymer. Figure 15 shows an example of the results obtained. Clearly, significant interpenetration of grafted and free chains is seen, in contradiction to the assumption made by Feigin and Napper in their treatment.

Distance

Fig. 15. Volume fraction profiles for ‘soft’ systems, as calculated by van Lent et al. [40] using an extension of S.F. theory. Note that some interpenetration of free and grafted chains is predicted.


free

-fJ --Q*

-A- Distance

Fig. 16. Volume fraction profiles for ‘soft systems’, as predicted by the pragmatic theory of Vincent 1421. Profile A shows the volume fraction profile of the free chains, whilst profile B illustrates the volume fraction profile of the grafted chains, assuming that no compression of the grafted chains by the free chains takes place. Profile C shows the altered profile that is obtained when compression, by an extent q, of the grafted chains is allowed for. As seen in S.F. theory, some interpenetration, to an extentp, of free and grafted layers is predicted. The effective depletion layer thickness 24, = 26 -p.

The pragmatic theory of Vincent was extended by Vincent et al. [41] and further by Jones and Vincent [421 to calculate the extent of interpenetration of terminally-grafted polymer and free polymer chains. The polymers they considered were of the same or chemically-different natures. The depth of interpenetration was obtained by equating the local pressure produced by the mixing of the chains with the osmotic pressure of the bulk polymer solution. Figure 16 shows the volume fraction profiles obtained by this approach. Small-angle neutron scattering experiments have provided some evidence that the terminally-attached chains may be compressed to some extent by the free chains [43]. In their later work, Jones and Vincent [421 accounted for this effect.

The S.F. and pragmatic approaches outlined in this section give good agreement with experimental data (see, for example, reference [42]). Both treatments have their own advantages and disadvantages. The pragmatic theory provides analytical solutions which allow values for the depletion layer thickness, degree of chain interpenetration and compression to be calculated. However, it does require assumptions to be made that describe the volume fraction profile. S.F. approach does not suffer this drawback, as it implicitly calculates the volume fraction profile. The major weakness of SF. theory is that it only applies to interactions between flat plates.

82 P. Jenkins, M. SnowdenlAdv. Colloid Znterface Sci. 68 (1996) 57-96

6. Experimental studies of depletion flocculation

6.1. Direct evidence for the depletion force

In dilute polymer solutions, the depletion layer thickness has been equated with the radius of gyration of the free polymer coils [5,6]. The first ‘direct’ evidence, for the existence of a polymer depletion layer adjacent to an interface of approximately these dimensions, was provided by Allain et al. [44] who used the evanescent wave technique to study a solution of free polystyrene (PS) in ethyl acetate next to a glass prism. The polystyrene, which possessed a molecular weight of around 100 000, contained a trace amount of 9-methacryloyloxymethylan- thracene. These fluorescently-labelled polymer segments are excited by the evanescent wave. Since the intensity of this wave falls exponentially with distance, it is only sensitive to polymer segments near the glass wall. A value of 4.5 + 0.5 nm was determined for the correlation length, < (which according to scaling theory represents the depletion layer thickness). This compared to a calculated value of 11 nm, for the radius of gyration of a polystyrene chain of this molecular weight. Ground- breaking though this result was, a couple of issues were not fully addressed during the experiment. Notably, light scattered by the prism surface was disregarded. Perhaps more importantly, since the minimum penetration distance of the evanescent wave was too great in comparison to the polymer chain dimensions, the chance of obtaining an accurate segment density profile was minimal.

Later studies by Auserre et al. [45,46] sought to address these problems. Using the polysaccharide xanthan and fluorescently labelling it with amino-containing chromophores, these authors successfully employed the evanescent wave technique to solutions of xanthan in contact with quartz. A chain of xanthan (molecular weight 1800 0001, if modelled as a semi-flexible helix, is calculated to have a rg of approximately 160 nm. At low xanthan concentrations, the evanescent wave method measured the depletion layer thickness to be around 170 nm, in good agreement with the theoretical treatment of Asakura and Oosawa. However, on increasing the xanthan concentration above the critical concentration for chain overlap the depletion layer thickness was dra- matically reduced. This finding is in agreement with the Scheutjens- Fleer-Vincent treatment outlined previously.

More recently, Cosgrove et al. [471 have used nuclear magnetic resonance spectroscopy to observe the depletion of sodium polystyrene


sulphonate (NaPSS), a polyelectrolyte, from the surface of silica particles in aqueous solution. This method works by monitoring the spin-spin relaxation time T2 of the polymer. Depletion of polymer from the surface gives rise to an increase in bulk polymer concentration and hence a change in the polymer mobility and T2. If the volume fractions of particle and polymer are known accurately then the depletion layer thickness may be calculated from the change in T2. A was seen to decrease with increasing PSS concentration again indicating that polymer chains do not behave as hard spheres. Furthermore, A was found to increase with increasing PSS molecular weight. The thickness determined was found to be of the order of the radius of gyration of the polymer coil for only the highest molecular weight PSS samples used. For lower molecular weight PSS samples, A was closer to the extended polymer chain length (‘rigid rod’).

The surface-force apparatus, described in a review by Luckham and de Costello [48] has also been applied to the investigation of depletion. Luckham and Klein 149,501 have used this method, in particular the development of Israelachvili 151,521, to probe the depletion force in a number of systems. In their first study [491, they employed mica as the substrate, the two plates being immersed in toluene, to which non-adsorbing free polystyrene was added. No depletion attraction was observed between the surfaces, a result attributed to the fact that the calculated theoretical depletion attraction for this system was some 2-3 orders of magnitude smaller than the detection limit of the apparatus.

In a subsequent study, Luckham and Klein 1501 precoated the mica surfaces with an adsorbed layer of ‘Triton X-405’, a non-ionic surfactant, and then immersed them in water. Free poly(ethylene oxide) (PEO) homopolymer was then added to the system. Again, no depletion interaction was detected. This time, the authors suggested that the ‘Triton X-405’ was largely replaced on the mica surface by the higher molecular weight PEO from the bulk solution.

The surface-force apparatus has also been employed by Marra and Hair [53] with mica surfaces immersed in aqueous solutions of the polyelectrolyte, sodium poly(styrene sulphonate) (NaPSS). As both the mica surface and polyelectrolyte are negatively charged, the interaction is dominated by electrical double-layer repulsion. However, the existence of depletion of the polyelectrolyte chains near the surfaces was inferred, owing to the unexpected range of the double-layer forces.

Milling and Biggs 1541 have investigated the depletion interaction using the relatively new technique of Atomic Force Microscopy @FM).

84 P. Jenkins, M. SnowohlAdv. Colloid Interface Sci. 68 (1996) 57-96

0.2

0.1

T O 2 z. E IL -0.1

-0.2

-0.4 1 0 2 4 6 8 -0.3 1’01

0 10 20 30 40 50 Separation distance (nm)

Fig. 17. The depletion interaction between a stearylated silica sphere (a = 3.8 pm) and a stearylated silica plate mediated by a cyclohexane solution of PDMS (IN, = 119 500, M,JMw = 2.3,& = 0.064), measured using Atomic Force Microscopy. Reproduced from Ref. [541.

They mounted a silica sphere, coated with a grafted layer of crs chains (Si$-g-n&), onto the AFM cantilever and brought it towards a stearylated, polished silica surface in the presence of a cyclohexane solution of polydimethylsiloxane (PDMS), in order to establish the force against distance profile. The result, which may be seen in Fig. 17, is consistent with theories describing the depletion interaction, Furthermore, the authors found that the calculated value of A was of the order of the radius of gyration of the PDMS used for the polymer concentrations investigated. It is interesting to note that the maximum predicted by the treatment of Feigin and Napper was not observed and as such supports the S.F.V. treatment of depletion forces. However, it may be that this maximum is only appreciable at higher free polymer concentrations than used in the AFM study.

Another way of measuring A has been developed thanks to the knowledge that particles possessing unexpectedly high electrophoretic mobilities have been found in solutions of some non-adsorbing polymers [55,56]. Baumler and Donath [57] have explained this effect by the existence of a depletion layer with a lower viscosity than that of the bulk solution. With the assumption of an exponential viscosity profile in the


depletion layer, analytical expressions for the electrophoretic mobility as a function of depletion layer thickness has been derived for ‘smooth’ and ‘hairy’ particles ]58-611. Reasonable agreement between the theory and experiments was found for the measured mobility of human eryth- rocytes in aqueous solutions of poly(ethylene glycol) and dextran [62]. The thickness of the depletion layer was found to be approximately the same as the radius of gyration of the polymer chain used.

Krabi and Donath [62] also reported electrophoretic measurements of liposomes in high molecular weight dextran solutions, over a wide range of ionic strengths. The agreement between theory and experiment they obtained was very poor. This highlighted some of the limitations of the initial theories which assume arbitrary viscosity profiles in the depletion layer and also pay no attention to the arrangement of the polymer chains at the interface. The theory was subsequently extended by Donath et al. [63] with the use of a position-dependent viscosity profile based on the Navier-Stokes equation. The revised theory gave good agreement with experimental findings, relating to a system comprising liposomes in the presence of aqueous solutions of non-adsorbing dextran. The molecular weights of the two dextrans used were 110 000 and 500 000, which correspond to radius of gyrations of 9 nm and 19 nm, respectively. The value measured for A was 1.7 nm and appeared to be independent of the molecular weight of the dextran used. The authors accounted for the small value of the depletion layer thickness by considering that a polymer molecule is present in both the depletion layer and the bulk solution simultaneously and, therefore, is subject to bulk hydrodynamics. The independent nature of the depletion layer thickness with respect to the dextran molecular weight was explained by the fact that the two dextran samples were polydisperse and hence, there was a significant degree of overlap between their respective molecular weight distributions.

A further extension to the measurement of depletion layer thickness using electrophoretic mobility has been described by Krabi et al. [64]. In essence, these authors consider ‘hairy’ particles, i.e. those possessing an outer layer of grafted chains. This introduces extra complexity which involves consideration of the electrophoretic mobility of the ‘hairs’ and whether or not electroosmotic flow can occur within this grafted layer.

Small-angle neutron scattering has been singularly successful in measuring polymer volume fraction profiles near walls in adsorbing systems. In the case of depletion the variation in polymer segment density as a function of distance from the interface is very small. This

86 P. Jenkins, M. Snowden lAdu. Colloid Interface Sci. 68 (1996) 57-96

makes the collection of extremely accurate scattering data of tanta- mount importance; a condition which can only be met with the use of extremely long experimental times. For this reason no reports exist in the literature of the use of small-angle neutron scattering to determine depletion layer thicknesses.

Conversely, Lee et al. 1641 have used neutron reflectometry to measure the depletion layer thickness of polystyrene at the air/toluene interface. Unfortunately, due to a lack of contrast between the depletion layer and the bulk solution at low polystyrene concentrations, reliable data were only collected for semi-dilute solutions. In this region, the behaviour of A was well modelled by scaling theory and exhibited a marked dependency on polymer concentration.

6.2. Phase separation studies

As described earlier (see Section 2) the first evidence for the existence of depletion zones next to surfaces was based on the fact that addition of non-adsorbing polymers to colloidal dispersions could invoke a weak and reversible flocculation. Direct evidence, such as that described in Section 6.1, has been limited and thus, much of the experimental work performed has focused on dispersion stability as a means to investigate the depletion force.

The reversible flocculation of colloidal dispersions leads to the formation of dilute and concentrated colloidal phases. The concentrated phase may be liquid-like or solid-like in appearance. Iridescent solid phases have been reported [30,66-691, although the solid phases have generally been found to be amorphous in nature. This iridescence has only been observed for aqueous charge stabilised dispersions, where flocculation occurs into a pseudo-secondary minimum formed by the superposition of electrical double layer and depletion forces. In general, iridescence is not observed in flocculated, non-aqueous, soft-sphere dispersions, where ‘rearrangement’ of particles is hindered, either by a strongly adhesive contact energy or entanglement of the stabilising polymer chains. Em- mett 1691 has studied a hard-sphere system comprising Sioz-g-&is dispersed in cyclohexane with free PDMS as the added polymer. He observed iridescence in sedimented samples prepared at free polymer volume fractions below @, but not for samples prepared at or above Qi . In contrast, Sperry [67] only found iridescence for large latex particles in the presence of free polymer at concentrations above $$ , illustrating the complex and very subtle behaviour associated with

P. Jenkins, 44. SnowdenlAdv. Colloid Interface Sci. 68 (1996) 57-96 87

sedimentation rates 1701. Establishment of a theoretical model for the solid phase is inhibited by the formation of such amorphous and hence, non-equilibrium solid phases.

Many authors have studied the effect of the free polymer molecular weight (M,) upon $f . In every case, it has been found that $$ decreases upon increasing M,. At higher free polymer concentrations $8 , restabilisation of dispersions has been observed [43,69,71-731. In non-aqueous systems, restabilisation has been seen in the hard-sphere system SiO,- g-nCi8/PDMS/cyclohexane 1691 and also for the soft-sphere systems SiOz-g-PSDWethylbenzene 1721 and SiOz-g-PWpolyvinylmethylether (PVME)/toluene [43]. Often, the location of @d has proved impossible due to the high viscosities of the non-aqueous polymer solutions. As described in Section 5, much critical discussion has focused on ‘restabilisation’ and, in particular, whether it has a thermodynamic or kinetic origin.

Several authors [28,74,751 have attempted to derive an empirical relationship of the form:

oh - ww (35)

Attempts to fit $d to such a universal exponent of iW, have met with failure, values of y lying between -0.25 1751 and -0.7 1281, whilst Napper [l] has derived a theoretical value of -0.45 + 0.05. In soft-sphere systems, interactions between the free polymer and the solvated polymer sheath further complicate matters.

For dispersions at @d , or just above, the weakness of depletion flocculation has been illustrated by the ease of redispersal upon dilution or by the application of shear. Further increase of +z, above $h , leads to a decrease of the volume fraction of colloidal particles in the dilute phase and an increase in the concentration in the particle rich phase as illustrated by Sperry 1671,

The effect of particle radius a reveals itself in two distinct ways. Firstly, by moderating the extent of the depletion energy (roughly proportional to particle radius); +d thus decreases with particle radius [27,71,76]. Secondly, by modifying the nature of the phase transition (gas/solid or gas/liquid) undergone by a dispersion beyond @d . The ratio of the depletion layer thickness to the particle radius, denoted by tiu, has been shown to alter the nature of the colloid-rich phase above +I . For systems of large particles and free polymers of relatively low molecular weights (i.e. where A << a) only phase equilibrium between


Fig. 18. Stearylated silica spheres (a = 81 nm) dispersed in toluene, flocculated by polystyrene homopolymers [77]. (A) Gas/solid transition (polystyrene M,,, = 15 800, i.e A << a). (B) Gas/liquid transition (polystyrene M, = 195 000, i.e A - a).

gaseous and solid states has been observed. However, if A _ a, dispersions may phase separate to form fluid-like, colloid rich phases, as shown in Fig. 18. Many authors [27,43,66,67,69,75-771 have reported this phenomenon, although in the studies of Sperry, the production of a liquid floe phase was found to be the general case not the exception. Sperry observed the formation of amorphous solid phases only at free polymer volume fractions well in excess of $4 .

Emmett and Vincent [78] have drawn an analogy between the phase behaviour of dispersions of weakly attracting colloidal particles and that of conventional molecular systems; the similarity arising from the common form of the pair potential, whose dominant feature in both cases is a shallow minimum, as shown in Fig. 19.

Lekkerkerker et al. [79] have described a theoretical treatment, based on simple statistical mechanics, of the phase behaviour of colloidal and non-adsorbing polymer mixtures. Superficially, their model predicts analogous phase diagrams to the approach adopted by Emmett and Vincent [78]; with gas/liquid coexistence seen for small particles flocculated by large polymers. However, a three phase coexistence region in the phase diagram, under certain conditions, is predicted. This coexistence region has yet to be observed experimentally.

P. Jenkins, M. SnowdenlAdv. Colloid Interfiie Sci. 68 (1996) 57-96 89

1

I 1-1 d

Fig. 19. The pair potential for weakly interacting (molecular and colloidal) is the effective range of the interaction.

particles. d

The presence of both bridging and depletion flocculation in the same system has been reported by Cawdery et al. [80,811, Jones and Vincent [42] and Snowden and coworkers [82]. Cawdery et al. [80,81] studied a system comprising PS latex, carrying a grafted layer of PEO chains (PS-g-PEO), in aqueous solutions of free poly(acrylic acid) (PAA) polymers. At low pH, and within a well-defined polymer concentration range, the particles are destabilised by a bridging mechanism. Above pH 4, no adsorption of PAA onto the particles occurs and the PS-g-PEO particles aggregate through the depletion interaction.

The system studied by Jones and Vincent [43], comprising SiO,-g- PS/poly(vinyl methyl ether)/toluene, exhibited bridging flocculation at low PVME concentrations, presumably due to adsorption of PVME into the toluene-swollen grafted layer around the silica particles (bridging flocculation being absent for higher grafted amounts and molecular weight of PS). Depletion flocculation was observed at higher free polymer concentrations.

Snowden et al. [821 studied aqueous silica dispersions in the presence of hydroxyethylcellulose (HE0 of&, _ 270 000. With no added electro- lyte, depletion flocculation was observed at high values of $$ . Suppress- ing the electrical double layer, with added NaCl, led to the observation


of bridging flocculation at considerably lower free polymer concentrations. This effect has been attributed to the depletion of polymer from a rigid layer of HEC adsorbed on the silica surface, with the depletion attraction beginning at a particle separation of ca. 4 rz. Smith and Williams [83] have made similar observations in a system comprising of PS latices in the presence of the same HEC polymer.

PS-g-PEO latices in aqueous solutions of free poly(acrylic acid) have also been investigated by Liang et al. 1841, who observed similar behaviour to that seen by Cawdery et al. 180,811. Liang et al. 1841 also presented rheological results, which indicate that above a critical volume fraction of free polymer, which corresponded to $4 , a rapid increase in the storage and loss moduli of the system was observed. Similar conclusions have been drawn by other workers [85,861 from rheological studies of concentrated PS-g-PEO latex particles flocculated by free poly(ethylene oxide). Hence, it seems that rheological measurements may be used as a method of detecting flocculation in concentrated colloidal dispersions.

More recently, Seeburgh and Berg 1871 have studied PS latices stabilised by adsorbed poly(ethylene oxide)/poly(propylene oxide) ABA block copolymers, known as ‘Pluronics’. These latices can be considered to be electrosterically-stabilised. Addition of free PEO homopolymer to the aqueous latex dispersions caused depletion flocculation, which the authors studied as a function of free polymer molecular weight, salt concentration and adsorbed ‘Pluronic’ molecular weight. Their findings are in general agreement with the pragmatic model of Vincent. How- ever, in certain cases the stability of the dispersions were found to increase with the addition of free polymer. Seeburgh et al. have attributed this phenomenon, not to depletion restabilisation, but instead to changes in conformation of the free polymer chains, possibly due to complexation with the cations in solution, an effect described by Bailey and Koleske 1881.

In a series of papers, Vincent and coworkers [41,42,72,81,89] have studied the effect of grafted polymer molecular weight and surface coverage on depletion flocculation in the following systems: SiOz-g- PS/PS/(toluene or ethylbenzene) and SiO,-g-PMMNPMMA/l, 4-dioxan (PMMA is polymethylmethacrylate). In every case, it was found that @i decreased with increasing free polymer molecular weight but increased with increasing grafted polymer molecular weight (ME). In the earlier studies [41,42,721, the effect of surface coverage was not fully investigated. The later, more detailed studies 181,891 presented two sets


of experimental findings. Firstly, a direct comparison of the soft-sphere system SiO,-g-PS was made with the hard-sphere system SiOs-g-&is (effectively zero coverage) in toluene solutions of PS and PDMS. It was found that the soft-sphere particles were significantly more stable (i.e. @& was higher) than the equivalent hard-sphere ones. Secondly, a study of the SiO,-g-PMMA/PMMA/l, 4-dioxan system revealed that the stability of the dispersions to depletion flocculation goes through a maximum with increasing surface coverage (r). The position of most stable coverage was found to depend upon ME.

The effect of solvency upon QJ has been explored for both hard-sphere [27,69,90] and soft-sphere systems [41]. For hard-sphere systems, $1 should increase as x is increased. The work of de Hek and Pathmama- noharan [27,90], using SiO,-g-&is particles in PS solutions, confirms this; on changing solvents from toluene (x = 0.44) to cyclohexane (&solvent for PS at 345”C), $$ was found to increase. Emmett [69], using similar particles dispersed in PDMS solutions, also found the same trend; in cyclohexane (x = 0.4) the dispersions were seen to be less stable than in bromocyclohexane (x = 0.5 at 30°C).

An interesting twist with regards to solvency has been investigated by Snowden and Vincent [91]. These workers studied polyW-isopropy- lacrylamide), termed poly(NIPAM), latices in the presence of aqueous solutions of the polyelectrolyte NaPSS. Poly(NIPAM) particles have the unusual property of contracting in size as the temperature is raised. For example, the ones used by Snowden and Vincent had a diameter of 460 nm at 25°C but upon raising the temperature to 40°C they had shrunk to 230 nm. This decrease in diameter is a consequence of the increase in the x parameter for the poly(NIPAM)/water system. In essence, this drives the poly(NIPAM) to maximise the number of polymer-polymer contacts, which it can achieve by collapsing in size. At 25”C, the poly(NIPAM) particles can be thought of as spongy microgels which are swollen with solvent. However, at 4O”C, the particles shrink and thus the solvent is driven out from the particle core. At this higher temperature the particles more closely resemble a hard-sphere in their nature. Addition of NaPSS to the poly(NIPAM) particles at 25°C produced no flocculation even up to polymer concentrations of 0.8%. However, flocculation was observed at polymer concentrations above 0.6% for the same system at 40°C. Furthermore, upon cooling these samples back to 25°C the dispersion redispersed fully. This stability/instability behaviour could be repeated over a number ofheating and cooling cycles. The authors ascribed the temperature induced instability of the poly(NIPAM) in the


presence of NaPSS to the increasing hard-sphere nature of the microgel particles at high temperatures. This finding is in direct agreement with the work of Milling et al. [89], who found that the concentration of polymer required for flocculation increased with enhanced ‘softness’ of the particle.

Most studies of the depletion interaction have considered systems containing a single, monodisperse polymer. Jenkins and Vincent [92] have considered depletion flocculation in non-aqueous SiO,-g-nC, dispersions, containing binary mixtures of non-adsorbing polymers. The binary mixtures used were either similar polymers of different molecular weights, or mixtures of chemically different polymers. The authors found that the order of mixing of the components crucially affected the minimum value of the total concentration of polymer required to floccu- late the dispersion. If the particles were added to a pre-mixed solution of the polymers, then the experimental results indicated that the larger polymer dominated the depletion interaction. Intuitively, it would be expected that the smaller polymers should dominate, in the sense that they ought to partially ‘fill-up’ any depletion layer formed by the large polymer. Theoretical modelling based on the S.F.V. treatment also predicted that the smaller polymer would dominate the depletion interaction under equilibrium conditions. Non-equilibrium effects, i.e. an inhomogeneous distribution of the large polymer caused by the presence of ‘pseudo-floes’, were used to explain the discordant nature of the findings.

Depletion flocculation has also been observed in systems where surfactants are used in place of polymers as the flocculant species. Ma [93] has described the depletion flocculation of PS latices by the micellar form of the non-ionic surfactant ‘Triton X-100’. More recently, Bibette and coworkers [94,95] have used the depletion interaction in systems comprising emulsions and surfactants (present in the form of micelles), as a method for effecting size partitioning of the emulsion.

Dickinson et al. [96] have investigated the creaming of an oil-in-water emulsion, formed using two surfactants ‘Tween 20’ (non-ionic) and sodium dodecyl sulfate (anionic), in the presence of dextran. These authors found that only very low concentrations of the polysaccharide dextran were needed to substantially increase the rate of creaming of the emulsion. Following osmotic pressure measurements, which exhibited a reduction in the osmotic pressure of the emulsion in the presence of dextran, these authors postulated that the enhanced creaming rate was caused by depletion flocculation of the emulsion droplets.

P. Jenkins, M. SnowdenlAdu. Colloid Interface Sci. 68 (1996) 57-96 93

7. Concluding remarks

As evidenced by the work outlined in this paper, the study of depletion flocculation has proved a fascinating area for perusal, by both experimen- talist and theoretician alike. Current theories, which all assume that equilibrium is attained within the depleting solution, adequately describe results for experimental systems, that comprise of monodisperse particles and polymers. In monodisperse systems, the equilibrium situation is almost certainly reached. One of the great advantages of the depletion attraction, in this instance, is that it allows independent variation of the range and magnitude of the particle interaction, through careful selec- tion of the free polymer molecular weight (and hence A) and polymer concentration respectively. It is impossible to achieve this with other interaction forces, such as the van der Waals or electrostatic potentials.

However, evidence is beginning to emerge to indicate that more complex systems, e.g. those containing mixtures of more than one type of polymer or particle, is dominated by non-equilibrium effects under certain conditions. Since most industrial formulations consist of a myr- iad of components, the understanding of such non-equilibrium systems is a challenging area for investigation that needs further exploration.

The direct measurement of the depletion force is also an aspect of depletion that has been somewhat neglected, due mainly to the lack of suitable investigative methods and instruments. With the advent of new tools, such as the colloid probe adaptation of the AFM, it is likely that the depletion force will be explored in much greater depth in the coming years.

8. Acknowledgements

It is a pleasure for the authors to thank Professor Brian Vincent for his helpful comments during the preparation of this manuscript.

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