Solvent structure contributions to the small-angle scattering from colloidal dispersions

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Solvent structure contributions to the small-angle scattering from colloidal dispersions

M.J. Grimson, P. Richmond

To cite this version:

M.J. Grimson, P. Richmond. Solvent structure contributions to the small-angle scattering from col- loidal dispersions. Journal de Physique, 1985, 46 (3), pp.447-452. �10.1051/jphys:01985004603044700�.

�jpa-00209983�

Solvent structure contributions to the small-angle scattering

from colloidal dispersions

M. J. Grimson and P. Richmond

AFRC Food Research Institute, Colney Lane, Norwich NR4 7UA, U.K.

(Reçu ^{le ler octobre} 1984, accepte le 19 novembre 1984 )

Résumé.

Supposant ^{une forme} analytique approximative pour la force ^entre deux particules colloïdales, ^on analyse, dans le cadre d’une théorie de perturbation ^{fondée sur} l’approximation ^de phase aléatoire, l’influence de la structure du solvant sur la diffusion centrale de dispersions colloïdales. On observe _que la contribution à la force entre deux particules colloïdales due A la structure du solvant exerce en général ^une influence minime mais non

négligeable ^{sur la} compressibilité osmotique ^{et sur} la forme du facteur de structure statique ^{de la} dispersion ^aux petits ^{vecteurs d’ondes.}

Abstract

By postulating ^an approximate analytic form for the force between two colloidal particles arising

from the structure of the intervening solvent, the solvent structure contributions to the small angle scattering ^from

colloidal dispersions ^are analysed ^{within a} perturbation theory ^{based on} the random phase approximation. ^The

force between two colloidal particles ^{due to} the solvent structure is seen to make in general ^{a small but} non-negligible

contribution to both the osmotic compressibility and the small wavevector form of the static structure factor of the dispersion.

Classification Physics ^Abstracts

82.70D

1. Introduction.

Recently ^Grimson [1, 2] ^has presented ^an analysis

of the small angle scattering from colloidal dispersions

based on a perturbation theory ^{of the} liquid ^state using the random phase approximation (RPA). ^The pairwise interaction potentials ^{between the colloidal} particles ^were of the form familiar from the DLVO

theory of colloid stability [3] ^{and both hard} sphere

and one-component plasma ^{reference fluids were used}

in the perturbation theory to account for the funda- mental change ^{in the} predominant interaction bet-

ween the macroparticles ⁱⁿ going ^from systems ^of uncharged particles ^to dispersions ^of highly charged particles. ^{Two notable} points ^{arose from the} analysis.

Firstly, ⁱⁿ dispersions ^of highly charged polystyrene

latex particles, interpretation ^{of the} experimental scattering ^{data in terms of an} interaction potential

derived from Debye-Hfckel screening predicted ^ma- croparticle ^surface potentials ^{an order of} magnitude

lower than those deduced from other experiments.

Secondly, ^{when the} theory ^was applied ^{to micro-} emulsions, the value of the Hamaker constant cal- culated from light scattering measurements of the osmotic compressibility ^was larger ^{than the conven-}

tionally expected ^value (a point ^noted by other authors

[4, 5] in similar studies).

An explanation for these discrepancies ^{could lie}

in the quantitative accuracy of the RPA, ^{since the RPA} is only asymptotically ^{exact and} may be expected

to be of only qualitative ^or semi-quantitative ^value.

In a related perturbation theory ^{treatment of} phase equilibria in colloidal dispersions Victor and Hansen

[6, 7] used both the mean density approximation (MDA) ^{and the} RPA and found differences in the location of the spinodal ^{lines as} predicted by ^the

two approaches. Since the MDA is a higher ^order approximation than the RPA it might ^be expected

to be quantitatively superior ^{to the RPA. However}

while the MDA is more accurate than the RPA in its

prediction of the osmotic compressibility it is known to possess deficiences in its treatment of the small wavevector, q, behaviour of the static structure factor

S(q) [8]. ^A comparative study ^{of the} small q ^{form of} S(q) ^{for a} model fluid has shown that all of the theore- tical approaches ^{used to date are in} qualitative

agreement ^and only relatively ^small quantitative

differences exist between them [8]. Since it is not

possible identify ^the most accurate theoretical

approach ^{to the} prediction ^{of the small} -angle ^scat-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004603044700

448

tering ^at present, ^the computational simplicity ^and

remarkable _accuracy of the RPA makes it a valuable

approach ^{to be used in} further studies.

The most probable origin ^{of the} discrepancies

noted above lies in the use of the DLVO form of the interaction potential ^between particles ^{of the dis-} persion. Indeed, ^{Ober and} Taupin [5] suggested ^that

the overestimation of Hamaker constant in the inter-

pretation ^{of the} light scattering studies of micro- emulsions arose from fitting the measured osmotic

compressibility by ^a model fluid of hard particles ^with long-ranged ^van der Waals interactions and neglecting

another attractive interaction between the macro-

particles. They indicated that this additional inter- action could be due to solvation forces between the colloidal particles that arise from the structuring ^of

solvent particles ^{between two} surfaces. The existence of solvation forces was predicted by theory [9] ^and

simulation [10, 11] ^{and the} oscillatory ^{nature of the}

forces was verified experimentally [12]. ^{In this} paper the role played by solvation forces in determining ^the small q ^form of S(q) ^is analysed ^{and used to} reinterpret

the anomalously large ^{value of the Hamaker constant}

inferred from light scattering studies of microemul- sions.

In the following ^{section the} relevant form of the

perturbation theory of the small angle scattering

based on the RPA developed ^earlier [1] is sketched.

In a section 3 a tractable analytic ^form for the solvation force is used to illustrate the effects of solvent struc- ture on the small q behaviour of S(q). ^The importance

of the solvent structure contributions is estimated in sections 4 and 5 by comparing the solvation force contributions to the small angle scattering ^{with those}

from the electrostatic repulsive ^{and van der Waals}

attractive forces between colloid particles ^familiar

from DLVO theory. ^The paper is concluded with a

discussion of the results.

2. Small angle scattering.

It is assumed that the potential energy of ^{a mono-}

disperse system ^{of colloidal} particles ^{with a hard core}

of diameter _{D may} be constructed from the sum of

pairwise interaction potentials 4J(r) ^which may be

split up ^as

with

The RPA consists of writing ^the direct correlation function of the dispersion ^as

where fl

(kB T)-1 ^and co(r) ^{is the} direct correlation function of the reference fluid i.e. a fluid at the same

temperature ^{T and bulk} density p ^as the true fluid,

but with a pair potential 4>o(r). The Fourier transform of c(r) ^{is defined} by

and is related to S(q) by

Thus in the RPA

It is known that c(r)

p4J(r) ^{as r}

^{oo and so}

the large ^r behaviour of c(r) will be well approximated by equation (2.4). The short- and intermediate _range behaviour of c(r) ^{is less} accurately ^described by ^the

RPA and this will lead to poor estimates of S(q) ^for

q > qp, ^where qp ^{is the} wavevector of the principal peak ⁱⁿ S(q). ^{So for the} perturbation theory ^{to be valid}

we require ^that S(q) ~ SO(q) for q > _qp. ^{However this}

is generally ^{correct as the} high q ^{form of} S(q) only provides information of the size and « steepness » ^of the « hard » core of interaction potential [13] ^{which is}

contained in the reference fluid

The RPA is expected ^to be reliable for q _qp

and at small q ^{it is} possible ^{to write}

which gives

In this _{paper the} effects of different contributions ^to the perturbation potential 01(r) ^on S(q

0) ^{and a}

are studied In each system ^to be considered in the

following the reference hard sphere fluid is assumed to have potential ^of equation (2.2). The direct corre-

lation function for this hard sphere fluid is known

analytically in the Percus-Yevick approximation ^and

it is _easy to show that for small q

with

and

where il = ’r pD’ ^{is the packing fraction [1]. For}

simplicity ^this approximate form for the reference

fluid is used throughout.

3. Solvation forces.

The short-ranged solvation force between two walls

separated by ^a solvent arises from the structuring ^of

the solvent particles ^{near an} interface and the subse- quent rearrangement ^{of the solvent} particles ^{as the} separation ^{of the two walls is} varied. The solvation force for simple systems ^is typified by ^a decaying, oscillatory form, ^{but the} specific character of the force

depends ^{on both the nature} of the walls and type ^of

solvent and is usually only ^known numerically.

Since this study ^is principally concerned with the

qualitative effects of the solvation force on the small

angle scattering ^it is convenient to propose ^{an ana-}

lytically tractable form for the force which allows the solvent structure effects to be immediately appa- rent in the final result

A suitable analytic approximation ^{to the solvation}

force _{per unit} area between two walls of surface-to- surface separation h, fs(h), ^is [14]

wherefs ^{is a measure of the magnitude} of the solvation

force, a ^{is an inverse} length ^that characterizes the decay

of the force, ^b gives ^the period of oscillation of the force and 6 is a phase factor. This form for fs(h) qualitatively

retains all the principal features noted by theory,

simulation and experiment [9-12]. Previous studies have shown that b - 2 n/d ^{where d is the} diameter of the solvent particles ^{and that 1} ;$ a - I d ;$ ^10.

Since the decay length ^{of the} solvation force is

typically ^{much less than the diameter} of the colloidal

particle, a-’ D, ^the pair interaction potential

between two spherical ^colloidal particles ^{of centre-to-}

centre separation ^r

^{h +} D may be calculated from

equation (3.1) by ^the Derjaguin approximation [3]

to be

for

This expression ^for 0,(r) should be an accurate

representation of the solvation force interaction for h > d. Inaccuracies will exist for surface-to-surface

separations ^{h d where the nature of} the force is

critically dependent ^{on the «} hardness » of the surface and the solvent particles. ^{But since we} expect fs ^{to be}

very large [9, 10] ^this region ^{of the} potential ^should

be essentially inacessible and the colloid particle

can be treated as a hard core of diameter (D ⁺ 8) ^with

c - 0(d). As d ^{D we} may retain the interpretation

of D as the colloidal particle ^{diameter in the} following

with little error.

If we consider the case of a system of colloid particles

of hard sphere ^{diameter D} dispersed ^{in a solvent of}

hard particles of diameter d which give ^{rise to the} only « long-range » interaction between the colloid

particles. ^The perturbation potential could be written

as

with ps(r) given by equation (3.2). ^A small q expansion

of the Fourier transform of Pl(r) given by equations (3 . 2) ^and (3 . 3) ^{has the form}

where d ^{D and}

fo r 6 1.

To illustrate the role of the solvation force in the small angle scattering from the model dispersion consider, ^for simplicity, ^{the case when 6}

^{0 i.e. zero} phase ^{shift. It} immediately follows from equation (3. 5)

that

and

Since we physically expect a ^{b as the} decay length

of the solvation force is typically greater ^{than the}

period ^of oscillation, ⁱⁿ general ^{we would} expect F(a, b, 0) ^0. Thus, since from equations (2.6) ^and (2.11) ^{we know that} S0(q

0) > 0, it follows that

So we expect the presence of solvation forces between the colloidal particles ^{to raise the value of} S(q

0)

and thereby increase the osmotic compressibility xT of the dispersion ^as pxT

PS(q

0). ^Also using equations (2.8) ^and (2.10) ^{it is} possible ^{to write the}

coefficient of the q2 ^{term in the} small q expansion ^of c(q) ^{for the model} dispersion ^as

so that by ^the reasoning given ^{above we find that}

a ao. Thus the presence of solvation forces between the colloidal particles ^{tends to} reduce value of the coefficient a to compensate ^{for the} enhanced value of the osmotic compressibility.

So by examining ^a typical ^{set of} parameter ^values

we have shown that the solvation force between colloid

particles ^{alters the} small q ^{form of} S(q) ^{in a fashion}

450

similar to that of an attractive interaction between the macroparticles. However the decaying oscillatory

form of 0. ,(r) ^{can be} also chosen such that it would appear ^{as an} effective repulsive interaction in the small angle scattering.

4. Charged ^colloidal particles.

Since the absolute magnitude of the solvation force

between colloid particles ^{is both} system dependent

and generally unknown, ^{in order to} gauge the relevance of solvent contributions in the analysis ^{of small} angle scattering data it is useful to compare the solvation force effects with those due to the electrostatic and

van der Waals components of the DLVO interaction

potential. First consider a dispersion ^of charged

colloid particles ^{in a} solvent of small neutral particles plus counterions where van der Waals interactions between the macroparticles ^are negligible. At the level of linear _response theory, ^the force between the colloid

particles may be written as the sum of electrostatic and solvation forces [15]. The solvation force inter- action can be assumed to be the same as in section 3.

The electrostatic contribution to the interaction

potential ^can be assumed to be given by ^the Debye-

Hfckel theory ^of screening which has the advantage

of simplicity ^and accuracy for low surface potentials

and strong screening.

In the thin double layer approximation (kD D > 1), Debye-Hfckel screening theory gives ^an electrostatic contribution to the interaction potential 0,,(r) ^{of the}

form [3]

where s is the dielectric constant of the system, .po

is the surface potential ^{of the} macroparticle and kD

is usual inverse Debye screening length. For r >> ^D

and we shall assume this form for 0,,(r) holds for all

r > D. This simplifies ^the analysis and should be accurate in the small q region.

Let the model charged particle dispersion ^{have a} perturbation potential

for kD D >> 1 we find

where

By comparing equations (3.4) ^and (4.4) ^{we see that}

the small q expansions of P I (q) ⁱⁿ this and the preced- ing ^{section are similar. For this} system ^of charged macroparticles the effects to PI (r) ^{on the small} angle scattering ^are contained in the behaviour of the coefficient C defined in equation (4.5).

Again ^{consider the case of 6}

^{0. If a} > J3b, ^we

find F > 0 and hence C > 0. By ^{the same} reasoning

used in section 3 this would imply ^that S(q = 0) S0(q=0) ^{and a} > a . But this is not the physical ^limit.

Typically a vi 3 b and ^{the sign of C depends on the}

relative magnitudes of the solvation and electrostatic forces. Define an electrostatic interaction strengthf,,

similar to the solvation force strength fs ^{defined in} equation (3.1) ^as

which allows equation (4. 5) ^{to be rewritten as}

Now analyse ^{two cases} (i) a

^{b. In this case}

and

Since we expect kD d ^{2 1C, for C 0 it requires} fs>fe

(ii) a

ko. ^Here

Typically b >> a ^{and so}

Again, ^{C 0} requires fs >> fe.

When C 0, ^{as before we find} S(q = 0) > So(q = 0)

and a ao and the « attractive » solvation force interaction is dominating ^the repulsive electrostatic

repulsion between the macroparticles ⁱⁿ regard ^{to the}

small angle scattering. ^{But from above we have seen}

that this requires fs >> fe ^{and so is likely to be the case} only ⁱⁿ dispersions ^of charged macroparticles ^with

small surface potentials. For many dispersion ^with fe > fs ^the opposite ^case would hold with C > 0. Then

S(q

0) So(q

0) ^{and a} > ao and the small

angle scattering ^{from the} dispersion would be domi- nated by the electrostatic interactions between the

macroparticles whose effective magnitude ^would only

be slightly diminished by ^the presence of attractive

solvation force contributions.

The effective signal ^of the presence of solvation force interactions in dispersions ^of charged particles ^{lies not}

in studies of the osmotic compressibility ^{but in the} small q ^{form of} S(q). ^From equation (2.9) ^{we see that}

if a 0 then the gradient ^of S(q) ^at small q ^{would be} negative initially ^to produce ^a dip ^before rising ^{to the} principal peak ⁱⁿ S(q). By analogy ^to equation (3.6) ^we

have

and to make a 0 it would require ^{C 0 and from} equation (2.12)

5. Microemulsions.

Now consider the solvation force contribution to the small angle scattering ^{in relation to} contribution from the van der Waals interaction between colloidal

particles. ^In particular ^we pay attention to the analysis

of small angle scattering data from water-in-oil microemulsions in terms of a model system ^{of hard} spheres interacting ^via long range ^van der Waals interactions dispersed ^{in a} solvent of smaller hard

particles. ^In the absence of solvent contributions this model has been used previously ^to analyse scattering experiments ^on microemulsions with qualitative, ^{if not} quantitative, ^success [4, 5, 9, 16].

The van der Waals interaction between two spheres

of diameter D is given by

where A is the Hamaker constant which characterizes the strength ^{of the van} der Waals interaction. For r >> D

and we shall again ^{assume this} holds for all ^r > D [9].

The perturbation potential may then be taken ^to be

It should be noted that the Hamaker constant A refers to the bare interaction between the colloid particles

i.e. the interaction in vacuo. The summation of inter- actions 4>s(r) ⁺ W(r) then tends asymptotically ^{to the}

correct composite ^{form as r}

oo [17]. ^{We shall}

assume 4>s(r) has the form given ^{in section 3. So} equation (5.3) ^{limits the} subsequent analysis ^to

solvents whose dielectric constant is unity. ^{For solvents}

whose dielectric constant differs from unity ^the asymptotic ^{form of} 0. ,(r) ^must be such that the sum of interactions 0,,(r) ⁺ 0,(r) ^{has the form of the solvent}

screened van der Waals interaction as r

oo. Here

we find

The q3 ^{term in} equation (5.4) ^{arises from the} asympto- tic r-6 form of the pair potential [9]. ^{We are} primarily

interested in solvation force contributions to the osmotic compressibility ^{for the} interpretation ^of light scattering ^{data and the most} interesting property ^to calculate is

Following ^Cazabat [16] ^{it is} possible define the second osmotic virial coefficient /3v by ^the expansion

where II is the osmotic _pressure. Since

it follows that

and so for this model system

To estimate the effects of solvation forces on 13v ^{it is}

convenient to define a solvation force contribution to the second osmotic vivial coefficient, PVS , ^as

For simplicity ^we may ^assume 6

0 and b ~ 2 n/d.

If we introduce a decay length ^A for the solvation force

as a-1 = £4 ^then for £ » ^{1 we have}

Calculations for the solvation force between hard colloidal particles separated by ^a dense hard sphere

solvent [9] ^{indicate that}

So we might typically expect

452

So a solvation force interaction of the type ^introduced in section 3 provides ^a negative ^{but small} contribution to f3v. However the point ^to notice is that by systema- tically including ^{the solvation} force it is clear that the correct Hamaker constant to use when making comparisons ^with scattering ^{data is that} for colloid

particles interacting ^{in the} absence of solvent and this is

always greater than that when solvent is present.

6. Discussion.

In this paper it has been shown that it is possible ^to

estimate the contribution of solvent effects to the small angle scattering from colloidal dispersions. ^The

solvation force is capable ^of playing the role of either

an additional effective attractive or repulsive ^inter-

action between the macroparticles ⁱⁿ regard ^{to the} small q ^form of S(q) depending ^{on the} ratio of the decay length ^{of the force to} its oscillation period. Typically

it acts as an additional effective attractive interaction which tends to increase S(q

0) ^{and diminish a}

with respect ^{to the reference} system. ^{In a} dispersion

of charged ^colloidal particles the presence of solvation forces in the interaction can lead to a negative ^value

of ^a and hence make S(q

0) ^{a weak local maximum}

in S(q). ^This result may be of significance ^{for the} experimental detection of solvation forces in colloidal

dispersions.

When the solvation force interaction is included in a

microemulsion model of particles ^{with hard cores and}

van der Waals attractions, ^{the solvent structure effect}

is seen provide ^a small, ^but non-negligible, ^contribu-

tion to S (q

0) and the second osmotic virial coeffi- cient. The main observation is however that the

appropriate ^Hamaker constant to use in such theories is the bare constant for the interaction of colloid

particles ^{in the} absence of solvent. One source of error

in this analysis ^{lies in the use of a hard} sphere ^{fluid as}

the reference system. ^Evidence [18, 19, 20] suggests that microemulsion droplets ^are deformable wich would result in a soft sphere interaction between the

droplets. This would manifest itself in the high q ^form

of S(q) which would not be accurately ^fitted by ^a

hard sphere ^{model. For a more} quantitative ^deter-

mination of interaction potential parameters ^from

scattering experiments, ^extensive high q ^{data for} S (q)

is _necessary so that an accurate reference fluid can be chosen. The perturbation treatments of the type given

here will be able to give ^an indication of the form of the interaction potential by analysis of the small

angle scattering.

References

[1] GRIMSON, ^M. J., JCS Faraday ^{2 79} (1983) ^817.

[2] GRIMSON, ^M. J., ^{J. Chem.} Phys. ⁷⁹ (1984) ^5070.

[3] VERWEY, E. J. W. and OVERBEEK, ^J. Th., Theory of stability of lyophobic ^colloids (Elsevier, ^Amster- dam) ^1948.

[4] AGTEROF, ^{W. G.} M., ^VAN ZOMEREN, ^J. A. J. and VRIJ, A., Chem. Phys. ^{Lett. 43} (1976) ^363.

[5] OBER, ^{R. and} TAUPIN, C., ^J. Phys. ^{Chem. 84} (1980)

2418.

[6] VICTOR, ^{J. M. and} HANSEN, ^J. P., ^J. Physique ^{Lett. 45} (1984) ^L-307.

[7] VICTOR, J. M. and HANSEN, ^J. P., preprint.

[8] BOWLES, R. J. and SILBERT, M., ^J. Phys. ^{C 17} (1984)

207.

[9] GRIMSON, ^M. J., RICKAYZEN, ^{G. and} RICHMOND, P., Mol. Phys. ³⁹ (1980) ^61.

[10] SNOOK, I. K. and VAN MEGEN, W., Phys. ^{Lett. 74A} (1979) 332.

[11] LANE, ^{J. E. and} SPURLING, ^T. H., ^Chem. Phys. ^Lett.

67 (1979) ^107.

[12] HORN, R. G. and ISRAELACHVILI, J. N., Chem. Phys.

Lett. 71 (1980) ^192.

[13] JACOBS, ^{R. E.} and ANDERSEN, ^H. C., ^Chem. Phys. ¹⁰ (1975) 73.

[14] BORSTNIK, ^{B. and} PUMPERNIK, D., ^Mol. Phys. ⁴⁷ (1982) 363.

[15] GRIMSON, ^{M. J. and} RICKAYZEN, G., ^Chem. Phys. ^Lett.

86 (1982) ^71.

[16] CAZABAT, ^A. M., ^J. Physique ^{Lett. 44} (1983) ^L593.

[17] RICHMOND, P., ^In Microscopic Aspects of ^Adhesion

and Lubrication (J. ^M. Georges Editor, Elsevier)

1982.

[18] SAFRAN, ^S. A., ^J. Chem. Phys. ⁷⁸ (1983) ^2073.

[19] SORNETTE, ^{D. and} OSTROWSKY, N., ^J. Physique ⁴⁵ (1984) ^265.

[20] GLADWELL, N., GRIMSON, ^M. J., RAHALKAR, ^{R. R.}

and RICHMOND, P., JCS Faraday ² in press.