Steric Stablization- colloid dispersion-grafted polymers

Theory of Steric Stabilization of Colloid Dispersions by Grafted Polymers

EKATERINA B. ZHULINA, OLEG V. BORISOV, AND VICTOR A. PRIAMITSYN

Institute of Macromolecular Compounds of the Academy of Sciences of the USSR, 199904 Leningrad, USSR

Received February 7, 1989; accepted June 23, 1989

The structure and stabilizing properties of layers of long polymer chains, grafted at one end onto the surface of colloid particles, are considered. A mean-field analytical theory describing the interaction between two colloid particles covered with densely grafted polymer chains is developed. An analytical expression for the interaction potential of colloid particles, depending on the degree of polymerization, the thermodynamic stiffness of stabilizing chains, the solvent quality and the density of grafting of the chains onto the surface, is obtained. The shape and the parameters of this potential are determined by the value of the universal parameter 3' introduced, which depends on the magnitude of interparticle attraction, the total amount of grafted polymer per unit area, and relative deviation from the 0-temperature. In the case of relatively weak interparticle attraction, 3" < 1, with solvencies better than or equal to thai of the 0-solvent, the potential curves display a single secondary minimum and thermodynamic stability of dispersion can be attained. In the case of strong interparticle attraction, 3" > 1, with good solvents, the potential curves display primary and secondary minima separated by a maximum and the stability of dispersion is kinetic in character. © 1990 Academic Press, Inc.

INTRODUCTION

One of the methods for ensuring the aggregative stability of colloid dispersions is their steric stabilization by a polymer bonded to the surface of dispersed particles (1, 2). This bonding is achieved by adsorption or by the grafting of macromolecules onto the surface. As a result, a protective polymer layer is formed. This layer prevents particle aggregation due to van der Waals attraction.

Many papers deal with the theoretical de- scription of steric stabilization of dispersions by polymers. They are reviewed in detail, for example, in Ref. (1). Analytical theories of stabilization by grafted polymers (3, 4) are based on a monomolecular approximation according to which the polymer layer is considered to consist of separate noninteracting coils. This situation corresponds to that of relatively widely spaced polymer chains on the surface of the particles resulting from virtual nonoverlapping of neighboring macromolecules. In the present paper the stabilizing properties of layers of polymer chains grafted

at one end onto the surface of the colloid particle are considered under conditions of considerable coil overlapping leading to strong in- terchain interactions in the layer. We derive an analytical expression for the effective potential of interaction between two colloid particles, the surfaces of which are modified by the grafted polymer, and analyze the effect of several factors--the temperature (solvent quality), the degree of polymerization, the thermodynamic stiffness, and the grafting density of the chains--on the shape and parameters of the potential, and thus the effect on the aggregative stability of the dispersion.

Model

Let us consider a dispersion consisting of rigid colloid particles (dispersed phase), the surface of which is modified by polymer chains grafted onto it at one end. These chains consist of N >> 1 units immersed in a solvent (dis- persing medium) at a temperature T. A chain part with a length equal to the chain thickness,

Journal of Colloid and Interface Science, Vol. 137, No. 2, July 1990

495 0021-9797/90 $3.00 Copyright © 1990 by Academic Press, Inc. All fights of reproduction in any form reserved.

496 ZHULINA, BORISOV, AND PRIAMITSYN

a, is chosen as the chain unit and the asym- metry parameter of the Kuhn segment p is A/ a >~ 1. (We call the polymers with p = 1 flexible-chain polymers and those with p > 1 semirigid polymers.) It is assumed that the surface area of a particle per polymer chain is

~ R 2, where R ( T ) is the characteristic size of an isolated chain in solution, and that the surface is inert with respect to the polymer (the adsorption of the stabilizing chains is absent). It is also assumed that the characteristic size of the particles p exceeds the thickness of the polymer layer H0 considerably, so that the stabilizing layer on the surface of the particle may be considered to be planar (Fig. 1 ).

The interaction of two colloid particles modified by grafting a polymer onto them is composed of the van der Waals attraction of colloid particles AVa and the interaction between the "opposite" layers (grafted onto the surfaces of the neighboring particles) of polymer chains A Vp:

located at a distance 2H from one another, the theory ( 1 ) gives the expression

A S AVa = 48 H 2' [2.1]

and for the energy of the van der Waals attraction between spheres of radius p with a distance between the centers 2(p + H), it gives the expression

A p AV~ - 24 H ' [2.2]

where A is the effective Hamaker constant in which the properties of the medium are taken into account. Typical values of A range from 10 -21 J for organic dispersions and latices to 10 -19 J for sols of heavy metals. Before the interaction between the opposite polymer layers is considered, the equilibrium structure and properties of an individual layer are discussed.

AV= AV~ + AVp. [11

The contribution of the forces of electrostatic repulsion between the particles is neglected.

For the energy of the van der Waals attraction between two planar surfaces with area S

PLANAR LAYERS OF GRAFTED POLYMER CHAINS

The conformations of macromolecules in layers of polymer chains grafted at one end onto a planar surface have been analyzed in

I l / l i l [ / / / [ / [ / / [ [ / [ / [ /

H I

FIG. 1. Planar layers of grafted polymer chains: individual layer and two interacting layers.


C O L L O I D S T A B I L I Z A T I O N BY P O L Y M E R S 497

a number of theoretical papers (5-9). The scaling approach carried out by Alexander (5) and de Gennes (6) has shown that under conditions of considerable chain overlapping (

R 2) in a good solvent the macromolecules in a layer are stretched normal to the surface in proportion to their contour length, H0

aN. The effect of thermodynamic stiffness p and the solvent strength on the characteristics of the planar layer has been investigated in Refs. (7-9).

The scaling scheme of analysis used in Refs. (5-9) makes it possible to obtain only the asymptotic power dependences of the system characteristics (weaker dependences and nu- merical coefficients have been omitted). Thus, in these papers the result that there is no power dependence of unit density in a layer on the distance from the surface has been obtained, and hence it has been concluded that as a whole the planar layer is homogeneous. How- ever, as has recently been shown in Ref. (10) (see also Refs ( 11, 12) a planar layer of grafted chains is macroscopically inhomogeneous. This result has also been confirmed by cal- culations ( 13 ).

In order to calculate the potential of interaction between two polymer layers it is necessary to determine their structure in detail. Therefore we begin from a brief summary of the analytical theory which permits the de- scription of the structure and characteristics of the polymer layer located in a slit of width H ~< H0 and the structural rearrangement in a layer resulting from a change in the solvent strength (temperature). A detailed consider- ation of these problems is given in Ref. (10).

General Formalism

The equilibrium structure of the grafted polymer layer is determined by the minimization of its conformational free energy

z ~ F = z~/Tel -~- AFconc , [ 3 ]

which contains the contributions of elastic chain stretching in the layer, z~k/Wel, and volume

interactions, ZS~co,c. In the calculation per polymer chain we have

Z~co~c = -~ f[¢(x) ldx , [41

where f [ 4~] / a 3 is the density of free energy of volume interactions in a layer and 4~(x) is the average dimensionless concentration (volume fraction) of units at a height x from the grafting plane. In order to calculate the entropy of stretching, -~Fe l , the Gaussian approximation for the local stretching at the ends of a chain part consisting of ~n units by a distance 6x is used:

3kT (~x) 2 ~Fel - 2pa2 6n

The local chain stretching at the point x depends on the position x' of a free chain end. If the distribution function g(x') of free chain ends according to height ( f~ g(x')dx' = 1 ) is introduced and if we pass to the continuous limit, the elastic contribution (per chain) is given by

~ e l - 3kT foH fo x' 2pa2 g(x')dx' E(x, x')dx,

[51

where the function E(x, x') = dx/dn char- acterizes the local stretching of the chain at point x under the condition that its free end be at point x'. The physical meaning of the function E(x, x') is relatively simple: it determines the value of the elastic force Y = 3kTE(x, x')/pa 2 that acts in an extended chain at a height x.

The volume fraction of units q~(x) is ex- pressed by the functions E(x, x') and g(x') as

a3fx" g(x') dx'. [6] 4 ~ ( x ) = - - - - E(x, x')

The minimization of free energy AF determined by Eqs. [ 2 ] - [ 5 ] as the functional of two unknown functions, E(x, x') and g(x'), taking into account the normalization conditions



fo ~' dx E(x ,x '~ - N [71

~-~ ¢(x)dx = N, [81

makes it possible to obtain the expression for the function of local chain stretching,

E(x, x') = ~N ((x') 2 - x 2 ) 1 / 2 , [91

and the equation describing the density profile of units in a layer,

2a2p (X + #[c~]/(kT))

3

= - I r 2 X 2 / ( 4 N 2 ) , [10 ]

where ~[flq = 6f[¢]/Ma is the chemical potential and X = X(H) is the indefinite Lagrangian multiplier, the value of which is determined from the normalization condition [ 81- Atten- tion should be directed to the universal character of the function E(x, x'): it is independent of both the form of density of the free energy of volume interactions f [ ¢] and the slit width H. The explicit form ofE(x , x') is determined only by the Gaussian character of the local chain stretching. It follows from the form of E(x, x') that the greatest stretching of any chain of the layer occurs near the surface (x = 0). As the distance from the surface increases local chain stretching decreases, be- coming equal to zero at the free end.

The functional forms of the end distribution in a layer, g(x'), and the density profile ~(x) are determined by the explicit form of dependence of the chemical potential ~ = u[¢] on concentration 0(x). Under the conditions of rather widely spaced grafting ( a / a 2 ~ 1 ), it is possible to use the virial expansion for the density of free energy of volume interactions,

f [ ¢ ( x ) ] = kT[vck2(x) + w4~3(x)], [11]

where v ~ ( T - 0 ) / T and w = const(T) are the second and the third dimensionless virial coefficients of interaction between units (16, 17).

As is known, the thermodynamic quality of the solvent is determined by the sign and the value of the second virial coefficient v of unit interaction. The above expression for v ~ ( T - 0 ) /T refers to the polymer-solvent system, in which increasing temperature leads to im- provement of the solvent quality (the system displays an upper critical solution temperature). This particular situation will be analyzed in detail throughout this paper. It should be noted, however, that the theory developed can be easily generalized for the case of a system with a lower critical solution temperature if the temperature ranges T > 0 and T < 0 are discussed in terms of solvencies better or worse than that of the 0-solvent (v > 0 or v < 0, respectively).

When Eq. [ 10] is applied, the density profile of units in a layer is given by

V ¢(x ) = 7 w [(1 + ~2(X - x2)) 1/2

× s g n ( v ) - 1], [12]

where K 2 = 9rc2w/(8v2a2N2p) and X = -8a2N2p~/(3~r2).

In order to calculate g(x'), Eq. [6] is used as the integral equation with respect to g(x'). Its solution, with the application of Eqs. [9 ] and [ 12 ], is given by

g(x')

_ x'v [(1 + ~2 (X- H2))l /2sgn(v)- 1 3wNa3 I (H 2 - - Xt2) 1/2

+ ~ + arcsin ~- +- K--- ~-- x-- 2 .

[131

Equations [ 9 ], [ 12 ], and [ 13 ] make it possible to obtain the equilibrium value of the conformational free energy AF(H) of the layer in a slit of a fixed width H <~ Ho,

Journal of Colloid and Interface Science, VoL 137, No. 2, July 1990

COLLOID STABILIZATION BY POLYMERS 499

A F ( H ) - m/9N~w2 (K2X- 1) kT [ zv

Hv3cr [1 1 + 27a3w-------- ~ - ~ ( 1 + K2()k -- H2)) 3/2

3 1t X sgn(v) + ~ K2X - K2H 2

= m. A F ( H ) , [14]

where m = S~ ~r is the total number of chains in a layer and AF is the free energy per chain.

The condition

O'aF(H)/ = 0, [151 /,,o

equivalent to that of the disappearance of pressure on the upper plane of the slit, com- monly solved with the normalization condition [ 81, determines the equilibrium height of the free layer and the corresponding value of the Lagrangian multiplier

rHg, v >/0

Xo= ~[H02- ]K-z, v < 0 . [16]

Here and below the subscript "0" refers to the characteristics of the free layer, which are considered in detail below.

Height Ho. The equation for the height of the free layer H0 provides a precise solution only at the 0-point (v = 0):

H ° = Ho (v = 0)

[ O- \ - 1 / 2 4(wp/2)1/4[--~) Na. [17] "K

Moreover, analyses of the temperature dependence of the layer height (10) show that the swelling coefficient of the layer with respect to its 0-dimensions, a = Ho(v)/Ho (v = 0), is a function of only one parameter,

v p , / 4 { ff ~1/2

/3- 3 .21 /4w3/4~] . [18]

The range/3 >> 1 corresponds to the conditions of a good solvent, that of [/31 ~< 1 corresponds to the 0-conditions, and that of/3 ~ - 1 corresponds to the conditions of a precipitant

(poor solvent). The dependence of the swelling coefficient a on/3 is given by

71"

2/3 2

Ol 2 7+(1+7 : j X arcsin(~-0/2 3 2 ) ~ / 2 ) , v>~0

/ 0/ /1 a2\

[19]

Figure 2 shows the dependence 0/(/3) plotted according to Eq. [ 19 ]. It is clear that a decrease in temperature (decrease in /3) leads to a monotonic decrease in the layer height H0. The high- (/3 >> 1) and low- (/3 ~ - 1 ) temperature asymptotics of 0/(/3) obtained from Eq. [19] are shown in Fig. 2 by broken lines. We have

(1_.,,./3)1/3, /3>> 1

0/(fl) = H61/31, f l ,~ - l . [201

These values make it possible to determine the size of the layer Ho under the conditions

7

2

/

J

/

-2 0 ~} 4 ~ l~J3

RG. 2. Swelling coefficient of a planar layer a vs parameter/3 = (vpl/4/3 • 2 l/4w3/~)( c~ / aZ) 1/2. Broken lines

show the high- and low-temperature asymptotics (Eq. [20]).


5 0 0 ZHULINA, BORISOV, AND PRIAMITSYN

of a good solvent (/3 >> 1 ) and a precipitant (/3 -1),

{ (8/Tr2)1/3(vp)l/3(ff/a2)-l/3Na ,

H0 = /3>>1

(2w/[v[)(cr/aZ)-lNa, fl .~ -1.

[211

Density profile. With the swelling coefficient a taken into account, the density profile of units in a free layer has the form

~0(z) 5o

( ( ),2) 4aft 1 + ~ ( 1 - - z 2) --1 , 71"

v > 0

4~1/31 + ( l - z 2) +1 ~r ~ '

V < 0 ,

[221

where the relative coordinate z = x/Ho and the average unit density in a layer 50 = Na3/ (crH0) were introduced. By using asymptotics [20], we easily obtain the asymptotic forms of the density profile in high- and low-temperature ranges and also at the 0-point,

~0(z)

{ ~ ( 1 - z 2 ) , fl>> 1 [23.1]

= (4/~r)Sg(1 -z2) 1/2, /3= 0 [23.2]

4 o , /3 ~ -1 , [23.31

where

5~ = 17r2/3(vp)-1/3(~r/a2)-2/3 [24.1]

(2/1/4 5g = \~Pl (a/a2)-l/2 [24.2]

5~ = [vl/(2w) [24.3]

are the asymptotic values of the average unit density in a layer.

Hence, the free planar layer of grafted chains is inhomogeneous as a whole at any degree of polymerization N >> 1. Figure 3 shows the de-

1.5

1.0

0.5

0 0.5 1.0

FIG. 3. Density profile in a free planar layer for the values of swelling coefficient a = 1.3, curve 1; 1.0, curve 2; 0.68, curve 3.

pendence of 4~0(z)/5 ° on az plotted from Eq. [22] in which Eqs. [17] and [21] are taken into account for the cases of a good solvent (~ = 1.3; curve 1), a 0-solvent (~ = 1; curve 2) and a poor solvent (a = 0.68; curve 3). Figure 3 shows that a decrease in temperature (solvent strength becomes inferior) causes not only a decrease in the layer height H0 but also the redistribution of unit density in the layer. Under the conditions of a good solvent (13 >> 1 ) the density profile is parabolic in shape (Eq. [ 23.1 ]). As the temperature decreases, the decrease in density on the layer periphery becomes more abrupt, and below the 0-point (/3 < 0) the unit density on the layer boundary (x = Ho) becomes abruptly equal to zero. The value of this jump, Aq5 = [vl/(2w), coincides with the equilibrium density of an isolated polymer globule ( 14, 18). In the limit fl ~ - 1 the density profile tends to change stepwise (Eq. [23.3]).

Distribution of free ends. The distribution function of free chain ends in a free layer is obtained from Eq. [ 13 ] with the application of condition [ 16 ] ;



go(z)

I l - z

v > O

2z, v = 0

+7 -77 ) 6a \

[251

Figure 4 shows the distribution of free chain ends for the values of a = 1.3 (good solvent, curve 1 ), a = 1 (0-solvent, curve 2), and = 0.68 (poor solvent, curve 3). The asymptotic forms of the distribution function of free chain ends are given by

3z(1 - z 2 ) 1/2, /~>~> 1

g o ( z ) : 2z, /3=0

z/(1 z2) 1/2, /3 < - 1 .

[26]

Hence, free chain ends are distributed throughout the thickness of the layer and the

character of their distribution is determined by the solvent strength (temperature).

Conformational free energy AFo. With the aid of Eqs. [9], [22], and [25] it is possible to calculate the value of free energy, &Fo, of a planar layer,

AFo = AF°(4/3a3/(3rr) + og 2 - - / 3 2 ) [27]

where

~XFg _ ~F0 (v = 0)

kT kT

= 3mN(w/2)~/2p-~/z(~/a2) -t [28]

is the free energy of the layer under the 0-conditions. Applying Eqs. [ 27 ] and [ 19 ] we obtain

ZXFo mkT

~ ~oTr2/3v2/3p-l/3(ff/a2)-2/3N, /3 >~ 1

[ - ( v 2 / 4 w ) N , /3 < -1.

[29]

It should be noted that because of the universal dependence ~ = ~(/3), the value of 2u%/AF~ is also a universal function of the parameter 13.

INTERACTION BETWEEN POLYMER LAYERS

3

2

0 0 .5 ~.o ~/H~

FIG. 4. Distribution function of free ends in a free planar layer for the values ofu = 1.3 ( 1 ); 1.0 (2); 0.68 (3).

Now we consider the interaction of two planar grafted polymer layers with one another in the process of their approach. It is assumed that during this approach the surface area of the particle per chain does not vary, cr = const(H), and the rate of rearrangement of layer structures is much greater than that of their approach, i.e., at any values of H ~ Ho the equilibrium has time to be established in the system.

The potential of interaction between two layers located at a distance of 2H from one another is determined as the difference between the conformational free energy of the system AF~(2H) and that of two free layers 2. ~V0,

&Vp(H) = AFz(2H) - 2. z3.Fo. [301



Interpenetration of Polymer Layers

For a correct calculation of dxF~(2H), it is necessary to solve the problem of chain conformations in the layers coming close to each other, i.e., to determine whether the polymer chains belonging to different layers intermingle during their approach or each layer becomes less extended, i.e., the layers are segregated. This problem was considered in Refs. ( 12, 15). As has been shown in these papers, under the condition H > R the polymer layers are virtually completely segregated. The reason for this is the considerable stretching of chains in the layers. The mixing of chains on the scale of the entire layer would lead to additional stretching of chains and an increase in the free energy of the system. It is clear, therefore, that the interpenetration of polymer layers should proceed on the scale of the unextended parts of the chains the ends of which at a fixed H are located near the middle of the clearance between the planes ( 15 ). If it is assumed that x ' = H in Eq. [ 9 ], the size of the unstretched terminal part of the chain consisting of v units will be determined from the condition

f2= a2pv= a2p -r E(x ,H)

~- aZpN(f/H) 1/2, [ 31 ]

which gives the characteristic size of the zone of interpenetration of the opposite polymer layers,

f "~ ( a 2 p N ) 2 / 3 H -1/3 ~ H . [32]

It is noteworthy that this result does not de- pend on the solvent strength because of the universality of the function E(x , x ') (Eq. [ 9 ] ). It is clear from Eq. [ 32 ] that the size of the interpenetration zone of opposite layers, ~', increases with decreasing H and with a loss in layer stretching ( H = N1/2ap 1/2) attains the size of the entire layer, f -~ H. Assuming in further discussion that H > R (and, hence, that f 4 H) we obtain AFz(2H) ~ 2. LXF(H), and the calculation of the potential of interaction between the layers

2~Vp(H) = 2 ( A F ( H ) - 2xFo) [331

reduces to the calculation of AF(H) (Eq. [141).

Potential of Interaction between Planar Polymer Layers

First, the form of A Vp at different temperatures T is considered. In the range of temperatures above the 0-point ( T > 0), the unit density in the free layer becomes equal to zero at the boundary of the layer x = Ho, Fig. 3, and the approach of the layers to each other ( H < Ho) leads to an increase in AVp directly from the moment of contact between the layers ( H = Ho). The physical reason for the increase in A Vp is the increase in the contribution of volume interactions, AFconc, with a simultaneous decrease in AFe~. A qualitatively different picture is observed below the 0-point (T < 0), where the unit density in the free layer undergoes a jump A~ = I v[/(2w) at the outer boundary of the layer, x = H0. This jump is equal to the equilibrium density of units in the polymer globule at a given temperature T (14). The existence of a jump in density at the layer boundary leads to an additional loss in free energy,

AFs ~ mfl2S" ~(Aq~)3aw_3, [34] kT

where S = m . a is the total area of the boundary and ~ ~ a/A4~ is the correlation radius of unit density in the polymer globule. With the disappearance of boundaries when the layers approach each other at a distance of H = H0, the losses in free energy related to these boundaries also disappear. This leads to the appearance of the potential well with a depth of approximately AFs at H = Ho. It is clear that at AFs >> kT the stabilizing properties of polymer layers disappear: the existence of the potential well of depth AF~ >> kT results in particle flocculation. The flocculation temperature T* will be evaluated from the condition AFs --- kT, which gives



6T* 0 - T* [ a2w 3/2\1/2 -- T ~ ~ T ) p - l / 4 . [35] T

For common flexible-chain polymers we have a = 5-8 A, w = 1, and dispersion loses stability virtually at once below the 0-point. (For o = 500 A we have 6T* /0 ~- 10-2). It should be noted that this conclusion has often been confirmed experimentally (see, e.g. Ref. ( 1 ) ).

Hence, since the range of the aggregative stability of dispersion is limited from below by the temperature T* ~< 0, the stabilizing properties of polymer layers at temperatures T >i 0 will be analyzed in detail. Applying Eqs. [14], [27], [28], and [33], we obtain

AV e = AF ° • qt(/3, q), [36]

where/3 is the combination of parameters determined by Eq. [ 18 ], q = H / H o is the relative compression of the polymer layers, and the function ~(/3, q) is given by

xIt(/3, q) = --2(/32 + a 2 + 4/3a3/(37r))

+ 2u2a2q 2 -- (8/37r)[2a/33q

-- 2/3(3U2/2 -- 1)(c~q) 3

+ ( U 2 - 1)3/2(aq)4], [37]

where u is the root of the transcendental equation

( a q ) -2 q- 4fl/(Traq)

= (2/Tr)((U 2 - 1)1/2 + u 2 a r c s i n l ) . [38]

When the compression of the polymer layers is strong, q ¢ 1, by developing Eqs. [36 ] - [38] for the small parameter q we obtain

A V v = ~ 12(o~q)2 +

= 2 m N ( v ~ + w4) 2)kr, [391

where ~ = a 3N/(~rH) is the mean unit density in the clearance between the planes. Hence, at q = H / H o ~ 1, the value of 2~Vp is completely determined by volume interactions between the polymer units in the clearance between the planes, and the entropy component

in A V v is not important. In the case of weak compression, 1 - q ~ 1, the behavior of 2xVp(q) is determined by both volume interactions and entropy component and depends drastically on the details of structure of free polymer layers.

Repuls ive Force

The derivative G v = - -OAVp/ (2OH) determines the repulsive force acting between the polymer layers when they approach each other at a distance 2H ~< 2H0. Applying Eqs. [36 ], [ 37 ], and also [ 17 ] and [ 28 ], we obtain

Go = ! 6 ( A F ~ / H O ) ( o t q ( u 2 _ 1 ) l / 2 _ / 3 ) 2 37r

X ( o ! q ( u 2 -- 1) 1/2 + i l l 2 ) . [40]

Figure 5 shows the dependence of normalized force Gp" H ~ / A F ° on the relative coordinate H / H ° = ~q at temperatures T = 0 (o~ = 1, curve 1) and T > 0 (o~ = 1.2, curve 2; and ~ = 2, curve 3). It is clear from Fig. 5 that an increase in the solvent strength (increase in temperature) leads to an increase in the stabilizing force G o at a fixed H value. Ap- plying Eqs. [ 17 ] and [ 28 ] for the height H ° and the free energy AF ° of the free polymer layer under the 0-conditions we find that at fixed H and T values the force G o increases with the thermodynamic chain stiffness p and the grafting density 1 / a.

It follows from Eqs. [38] and [40] that at weak compression of polymer layers (rq = 1 - q ~ 1 ) we have Gp ~ ( rq ) 3/2 under the 0- conditions (at/3 = 0) and Gp ~ (rq) 2 under the conditions of a good solvent (/3 >> 1 ). Hence, a detailed evaluation taking into account the conformational structure of the layers leads to a dependence of the repulsive force Gp(q) on deformation in the range of 6q ~ 1 that is weaker than the dependence obtained under the assumption of homogeneous structure of the layer (5-9) : Gp ~ 6q at any T. As mentioned above, in contrast to the scaling picture of the macroscopically homogeneous layer with a constant unit density and uniform

Journal of Colloid andlnterface Science, Vol. 137, No. 2, July 1990


15

10

o o15 ~ ~

3

2

1.0 1.5 2.0

FIG. 5. Normalized repulsion force between two polymer layers GpH°/2tF~ vs relative compression c~q = H/H~ for the values of c~ = 1.0 ( 1 ); 1.3 (2); 2.0 (3).

chain stretching, the planar layer of the grafted chains is inhomogeneous as a whole: the unit density decreases on the scale of the entire layer and the free chain ends are distributed throughout the layer thickness. An evaluation of this inhomogeneity results in an increase in the height of the free layer and a simultaneous decrease in layer resistance to compression at low compressions, ~q = (H0 - H ) / H o ~. 1.

In the range of strong compressions, q ~ 1, the layer structure approaches homogeneity. In this case layer elasticity is determined by volume interactions in the layers, and the dependence of force G o on deformation coincides with that obtained with the approximation for a homogeneous layer: Gp ~ q - 3 under the 0- conditions (/3 = 0) and Gp ~ q - 2 under the conditions of a good solvent (/3 >> 1 ).

INTERACTION POTENTIAL OF COLLOID PARTICLES WITH GRAFTED POLYMER

Planar Surfaces

According to Eqs. [1] and [36], the total interaction potential A V = A Vp + A V~ of two


planar-parallel surfaces modified by grafting polymer chains onto them is given by

A V = 25~F~.'~(/3, q) - U~/(aq) a, [41]

where -AU~) = - ( A S / 4 8 7 r ) ( H ° ) -2 is the attraction energy of two surfaces located at a distance of 2H~ from one another. In the range of strong compression of polymer layers, q ~ 1, the interaction potential of two surfaces is given by

A V ( q) : 2xF° { Tr/3/( aq)

+ (7r2/12 - ~U°o/2xF~)/(~q)Z}. [42]

The first term in Eq. [ 42 ] proportional to 1/q describes the contribution of pair interactions and the second term contains the contributions of ternary interactions of units in layers and that of van der Waals attraction of surfaces (both these contributions are proportional to 1/q2). In the range of maximum compressions corresponding to the conditions close to dense packing of units (~ --- 1 ), the interaction between layers should acquire the character of strong repulsion. This leads to the appearance of the following terms of higher

COLLOID STABILIZATION BY POLYMERS 5 0 5

order of magnitude in 1/q in Eq. [42] for V(q).

Analysis shows that the shape of potential curves ~V(q) is profoundly affected by the sign of the parenthesis before the term 1 /(aq)2 in Eq. [ 42 ]. If

12AU ° A ( ~ ) 3 "Y-~r 2 A F ~ - 9 6 ~ r k ~ ~ < 1, [43]

then the potential of interaction between the colloid particles at any T >~ 0 ensures their repulsion at small distances: 2x V(q) increases monotonically with decreasing q. Condition [43 ] implies that the contribution of repulsive ternary interactions between units in the clearance between the surfaces is sufficient to suppress the van der Waals attraction and thus to ensure the dispersion stability with respect to "short-range" aggregation. The contribution of pair interactions at fl > 0 leads to additional stabilization (increase in repulsion) in the range q ~ 1 (Fig. 6).

The nonobservance of condition [43] implies that the repulsion between layers due to ternary interactions is not sufficient to overcome the van der Waals attraction of surfaces

at q ~ 1. Only when the density in the clearance comes close to the limiting value ~ --- 1 does the interaction between layers acquire the character of a repulsion that is stronger than the van der Waals attraction between the surfaces. Hence, if 3' > 1, the so-called "short- range" or primary minimum appears on the potential curves 2x V(q). Its position and depth are given by (in order of magnitude)

qmin ~ a - l ( w p ) - l / 4 ( a / a 2 ) -1/2 ~ 1 [44]

- 2XVmin - AS/a2(a/Na2) 2. [45]

The evaluation of Eq. [45] for the depth of the minimum is obtained from the condition A V m i n ~--- AVa(qmin ) , where AVa(qmin ) is t h e

energy of the van der Waals attraction of surfaces at a distance corresponding to dense packing of units in the clearance ~ = 1.

In the general case, /3 > 0, "y > 1, the potential curves AV(q) also exhibit a maximum. Its position and height are given by

7 r3 , - 1 qmax -- [46]

6 at3 ~2

AVmax = (36/Tr2)2xU °" [47] 3'('Y - 1) "

a

AF~°~I 6

aF~e~

8

6

#

° ~ / H , o 2

o o'.4 - ' - - ~ - - ~ - ~'.2 - ~ / .o °

FIG. 6. Normalized effective interaction potential between two planar surfaces modified by grafted polymer vs relative compression aq 0 = H/Ho for the values of 3" = 12/7r 2 (a) and 3' = 3-6/~r 2 (b). The values o f t are 0 ( 1 ); 0.2 (2); 0.4 (3); 0.6 (4); 0.8 (5); 1.0 (6).



This maximum appears as a result of the action of repulsive pair interactions between units. At q > qmax these interactions become sufficient to overcome the van der Waals attraction (weakened by repulsive ternary interactions).

Analysis of the shape of the potential A V(q) (Eqs. [ 3 6 ] - [ 3 8 ] ) shows that at any temperatures T > 0 and at relatively small q ~< 1 the potential curves exhibit the so-called "long- range" or secondary minimum. Its position and depth under the 0-condition are given by

q~nin = (1 - - 7 r Z ' y 2 / 3 / 1 6 ) l / 2 [48]

3 0 - - AV~nin ~ ~ A U o . [49]

When the solvent strength increases (3 increases), q ' in also increases, the depth of the "long-range" minimum decreases, and in the limit/3 >> 1 we have

- -AW~nin ~ A U ° / o L 2. [ 5 0 ]

Hence, in the case 7 < 1, the potential curves display only the long-range (secondary) minimum, the position and depth of which are given by Eqs. [48] and [50]. At 7 > 1, the short-range (primary) minimum (Eqs. [44] and [45 ]) and a maximum separating the two minima (Eqs. [46] and [47]) also appear on the curves AV(q) (Fig. 6).

Spherical Surfaces

Let us now consider the interaction potential AVS(H, P) of two spherical colloid particles of radius p > H0 modified by grafted polymer chains.

It is clear that in the case p >> Ho the polymer chains grafted onto the surface of the spherical particle do not "feel" the curvature of the surface and to a first approximation the characteristics of this layer (thickness, density profile, free end distribution) are described by Eqs. [ 17 ] - [ 26 ]. When the particle radius decreases to values of p ~< Ho, the structures of spherical and planar layers begin to differ greatly. Thus, the free chain ends in a spherical layer are markedly redistributed in the direction toward the periphery of the layer. The density profile

changes correspondingly: it acquires (in the main term) a power character in contrast to the nonpower dependences [22] and [23]. Local chain stretching also decreases with increasing distance from the surface of the sphere according to the power law, and the decrease in o at a fixed a leads to a decrease in the overall chain stretching and the layer thickness H0. These problems will be discussed in greater detail in subsequent papers; here, only the case P > H0 is considered.

Figure 7 shows two spherical particles with a grafted polymer located at a distance of 2H < 2H0 between the nearest points of their surfaces (the distance between the centers is 2 (p + H)) . In contrast to the planar case considered above, the number of polymer chains Af contained in the parts of spherical layers deformed upon their approach and contributing to AVSo(H, p) changes as the particles come closer to each other, and the deformations of these chains are different. The deformation of chains is at a maximum for chains with the ends located near the center of the contact surface and at a minimum for those with the ends located on the periphery of this surface. Neglecting chain redistribution from the region of interaction into the part of the layer which remains unperturbed, we obtain for the polymer part of the interaction potential of two spherical particles

K ° AV;(H, p) = APAg(o) )d f (O )

f f;o A#p(H')dH' [511 = ( H + p ) ( H ' + p ) 2 '

where df(O) = ( f / 2 ) s i n OdO is the number of chains for which the width of the clearance between the particles is H(O) = (H + p)/ cos 0 - p. The angle 0 is reckoned from the axis joining the centers of the interacting particles, the interaction region is limited by the surface of a cone with generator (p + H0) and cone angle 200 = 2 arccos((H + p)/(Ho + p)), and the change in 0 from 0 to 00 corresponds to the change in H(O) from the minimum value equal to H to the height of the



FIG. 7. Schematic representation of two spherical particles modified by grafted polymer located at a distance of 2H < 2Ho between the surfaces.

unperturbed layer Ho. Here f = 4rcp2/a is the total number of chains on a particle and A I?p(H') is the interaction potential of planar layers at a distance of 2H' from one another (Eq. [36]) per chain

AIY'p(H') = AF ° . ~(fl, H'/Ho). [521

If the relative variables q = H/Ho and ~0 = p/H~ > 1 are introduced we obtain

AV;(q, w)

~q' AVp(q') = f--- (q + w/a) dq'. 1531

2 (q' + o~/a) 2

Without calculating the final expression in Eq. [53] let us analyze the behavior of AVe(q, co) at weak, q ~< 1, and strong, q ~ 1, compressions using the results obtained above for planar layers. Thus, at 6q = 1 - q ~ 1 Eq. [ 53 ] gives

f a AVe(q, o:) ~ ~ -~ .Al~p(q ) .6q [54]

and hence at low deformations the polymer part of the potential increases with 6q more slowly than in the case of planar surfaces: A V ~ ~ (tSq) 4 (instead o f ( 6 q ) 3) in the case of

a good solvent and AVe, ~ ( t~q) 7/2 (instead of (6q) 5/2) in the case of the 0-solvent.

At strong deformations, q ~ 1, we limit ourselves to the main terms in Eq. [53], ensuring the divergence of A Vp(q, ~o) at q --~ 0. Applying Eq. [39] we obtain at q ~ 1, w /a >>1,

f ~ V ~ ( q , o~) ~ - -

2w

) × AP~ 12aq 7r31n q + const(q) . [55]

As is shown by the analysis of the total potential AVS(q, ~o) for the interaction between spherical colloid particles sterically stabilized by the grafted polymer, the shape of the potential curves 2xVS(q, o)) is determined as in the case of planar surfaces by the value of the parameter 7 = A ( a / Na 2) 3(96rckTw) -1. Thus, in the case 7 < 1, ternary interactions between units ensure repulsion between the spherical particles when the distance between them becomes very small, q ~ 1. In this case the curves AVS(q, ~o) exhibit only the secondary minimum at q ~< 1, and the primary minimum at

Journal of ColloM and Interface Science, Vol. 137, No. 2, July 1990


q ~ 1 is absent. At 3" > 1, a deep primary minimum at q ~ 1 and also, at/3 > 0, a maximum separating the primary and secondary minima appear on the potential curves A VS(q, o~), just as in the case of planar surfaces. How- ever, the position of the maximum is displaced toward values of q that are lower than those in the case of planar surfaces (Eq. [ 46 ] ),

7r 3 " - 1 qmax 12 a/3 ' [56]

and the value of the maximum is

_ f & F ~ ) r / 3 1 n ( 1 2 c ~ / AVSmax 2w rr(3" - 1)el

= 1__22. AUO,,/3 In/ 12a/3_ / o0/~ >> 1, 7r3" \7r(3" - 1)e ] '

[571

where U~ 's is the energy of the van der Waals attraction between two spherical particles located at a distance of 2H ° between the nearest points of their surfaces and e is the base of natural logarithms. The value of this maximum increases as/3 In/3 and not as/32 (Eq. [47]). The position of the secondary minimum q' ia ~< 1 becomes slightly displaced toward lower values of q; its depth is given by

A w ' - . [581 --AVmi, = AU°'S/o~ 24

The total potential of interaction between spherical particles calculated from Eqs. [ 12 ] and [5 3] for various values of % o~, and/3 is shown in Fig. 8.

Stability of Sterically Stabilized Dispersions

1. At 3' < 1 and T I> 0, the curves of potential energy AV(q) have only one (secondary) minimum at q = q' in ~ 1. The relative depth of this minimum - A V ' i , / k T depends on the thickness of the polymer layer Ho, which in turn depends on the temperature (solvent strength). If - - A V ' i J k T ~< 1, then the dispersion is aggregatively stable. If - A V ' i n / k T > 1, then the particles coagulate at the secondary minimum, and long-range

aggregation occurs. In this case loose aggregates are formed in which the mean interparticle distance is ~ 2 q ' i n n o and the interpar- t ide space is filled with deformed polymer layers. It follows from Eqs. [50 ] and [58] that I AV'in [ / kT ~ lO-2 (A /kT) (p /Ho) x where x = 2 for planar surfaces and x = 1 for spherical particles. Hence, in order to avoid weak coagulation of organic dispersions and latices (for which A / ( k T ) ~ 1 ) it is sufficient to form a protective polymer layer on the particle surface. The thickness of this layer is comparable to the size of the particles (Ho ~< p), whereas for the steric stabilization ofsols of heavy metals ( A / ( k T ) ~ l0 2) the thickness of the protective polymer layer should markedly exceed the particle size (Ho > p). A decrease in temperature (inferior solvent strength) results in a decrease in the thickness of the polymer layer. This may lead to a loss in stabilizing properties of the layer, i.e., to weak coagulation. The temperature threshold of weak coagulation is determined by the condition I AV~nin I ~--- kT. Increases in the degree of polymerization and the grafting density and a decrease in particle size displace this threshold toward low temperatures. At I A V~nin l0

AUg ~ kO the layer retains its stabilizing properties up to the 0-point.

2. Let us consider the case 3" >~ 1, which can be observed under conditions of strong interparticle attraction. In this case the A V(q) curves display a short-range minimum and at /3 >~ 3" - 1, a potential barrier separating the long- and short-range minima. The relative depth of the short-range minimum -AVmin/ k T is large and virtually independent of the parameter/3 ~ ( T - 0), whereas the height of the potential barrier increases at/3 >> 1 as /32 (in the case of planes) or as/3 In/3 (in the case of spheres). At/3 >> 1 the barrier height

Vmax >> k T so that the particles cannot sur- mount it and after collision either drift apart (at - A V ' i n ~< kT, the case of aggregative stability) or aggregate at the secondary minimum (--AV'mi, > kT, weak coagulation). In both cases the system is in the metastable state. A decrease in temperature (decrease in/3) leads



2 ̧

0

- 4 ¸

- 6

0.8 1.2

b 2" ~ " / ' ] ' $

6

4

-6

6.4 o'.8 1'.2

c '~ ca.~,V ¢

8

6

4

2

0 • ~ 1

-2[ - 4

FIG. 8. N o r m a l i z e d effective poten t ia l o f in terac t ion between two spherical part icles modif ied by grafted

po lymer vs relat ive compress ion c~q ~ H / H I for the values o f t = 12/7r 2, w = 10 ( a ) ; 3, = 12/7r 2, o~ = 2

( b ) ; and 3" = 3.6/7r2, w = 2 (c ) . Figures at the curves cor respond to the values o f f l = 0 (1 ) ; 0.2 (2 ) ; 0.4

( 3 ) ; 0.6 (4 ) ; 0.8 ( 5 ) ; 1.0 (6 ) ; 1.2 (7 ) ; 1.4 (8 ) ; 1.6 (9 ) .

to both a decrease in the relative height of the barrier AVmax/kT and an increase in the relative depth of the secondary minimum. At I Again I T=O ~ &U~ ~ kT, the dispersion retains aggregative stability down to very low values off l (¢~ ~> [(~, - 1)kT/&U°l 1/2 in the case of the planar surface and/3 >~ kT/AUg "s

in that of the spherical surface. At lower temperatures, AVmax ~< kT, slow coagulation starts at the primary minimum and near the 0-point (at/3 = "r - 1 ) the potential barrier disappears, which leads to fast coagulation (any collision of the particles results in their coalescence). In the case AU ° > kT, long-range aggregation

Journal of Colloid and Interface Science, V ol. 137, No, 2, July 1990


proceeds at temperatures much higher than the 0-point. A further decrease in temperature (decrease in/3) at T > 0 is accompanied by a slight decrease in the interparticle distance in the aggregates. Near the 0-point (at/3 _~ ~ - 1 ) at which the maximum on the AV(q) curves disappears, the aggregates undergo rearrangement (increase in density) caused by their transition from the metastable secondary minimum to the primary minimum corresponding to the thermodynamic equilibrium in the system. When the particles are aggre- gated at the short-range minimum, the stabilizing layers undergo considerable deformation (qmin "~ 1 ).

DISCUSSION

As is shown by the results of the preceding sections, the shape and the parameters of the potential AV(q) are determined not only by the particle size and the chemical nature of dispersion (via the values S(o) and A) but also by the characteristics of the stabilizing polymer layer: the surface concentration of units Na2/ ~r and the height of the free layer Ho. Let us estimate the value of the parameter 3" deter- mining the behavior of AV(q) at strong compressions q ~ 1. Under conditions of considerable overlapping of the stabilizing chains ~/Na 2 ~ 1, ensuring the stretching of chains in the layer (Ho > ap1/2Nl/2), we obtain for the typical (1) values ofA/ (kT) = 10°-10 2 and w - 1 (19), the value of 3" < 1.

Hence, in contrast to the theoretical pre- dictions (3), the stabilizing layer of overlapping polymer chains ensures the aggregative stability of dispersion at short distances q ~ 1 over the entire temperature range T >i 0. It should be noted that the indispensable condition of stabilization at the 0-point (inequality [43]) has a basis that is fundamentally different from that of the corresponding condition in Ref. [3]. Thus, according to Ref. [3], it is necessary, for stabilization of dispersion at the 0-point, that the entropy losses from the compression of the Gaussian coil in the clear-

ance between the planes (2xFel - Na2/H 2) exceed the van der Waals attraction A Va(H). As shown in the present paper, the compression of the polymer layer leads mainly to an increase in the free energy of volume interactions (ternary contacts at the 0-point), which results in the condition of stabilization [43]: 3' < 1.

As mentioned in the preceding section, under the condition 3" < 1 the aggregative stability of dispersion is determined by the depth of the secondary minimum - AV'in. It follows from Eqs. [49] and [50] that at fixed S and A we have - -AV' in ~ H~ 2 and the increase in stability is achieved by an increase in Ho. Since the height of the free layer Ho is proportional to the degree of polymerization N and increases with increasing chain stiffness and grafting density a2/a, the use of long semirigid (p > 1 ) chains densely grafted onto the particle surface would lead to a considerable increase in the aggregative stability of the dispersions. The attainment of densely grafted layers is, however, a complex experimental problem. Analysis of experimental data (19) shows that the degree of overlapping of polymer chains in stabilizing layers formed by di- block copolymers with an "adhesive" block is low. An increase in this degree might be attained by increasing the energy of bonding to the surface of the "adhesive" block, e.g., as a result of increasing the adsorption energy of "adhesive" units.

Note added in proof When this paper was being prepared for publication, we received a preprint of a paper by S. T. Milner, T. A. Witten, and M. E. Cares, who considered the structure of a planar polymer layer and obtained results for density profile and chain end distribution coinciding with our results (Eqs. [23.1 ] and [26.1]). Since that time several papers (20-22) by these authors dealing with the properties of grafted polymer layers have appeared in lit- erature.

REFERENCES

1. Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions," Academic Press, London, 1983.

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Journal of Colloid and Interface Science, Vol. 137, No. 2, July t990

COLLOID STABILIZATION BY POLYMERS 51 1

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4. Dolan, A. K., and Edwards, S. F., Proc. R. Soc. London Ser. A 337, 509 (1974) and 343, 427 (1975).

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Steric Stablization- colloid dispersion-grafted polymers

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