Phase transitions in aqueous suspensions of spherical colloid particles

Phase transitions in aqueous suspensions of spherical colloid particles

PHASE TRANSITIONS S. MARCELIA, 15 October i976 CHEMICAL PHYSICS LETTERS Volume 43, number 2 IN AQUEOUS SUSPENSIONS OF SPHERICAL COLLOID PARTICLES...

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PHASE TRANSITIONS S. MARCELIA,

15 October i976

CHEMICAL PHYSICS LETTERS

Volume 43, number 2

IN AQUEOUS SUSPENSIONS OF SPHERICAL

COLLOID PARTICLES

D.J. MITCHELL and B.W. NINHAM

Department of Applied Mathematics, Institute of Advanced Studies. Research School of Physical Sciences, The Australtin National University, Omberra, A.C.T. 2600, Australia Received 23 March 1976 Revised manuscript received 8 June 1976

Recent observations of order-disorder phase transitions in low density aqueous suspensions of electrostaticallystabilised latex spheres are explained using Gouy-Chapman theory and current theories of fmst-order phase transitions.

A problem which has excited a good deal of interest over the past few years has to do with the observations of a first-order solid-fluid-like phase transition in aqueous suspensions of electrostatically stabilised latex spheres. The system and those observations which have been made so far have been reported elsewhere [l-3]. For our purposes, the situation can be summed up as follows: A suspension of latex spheres, monodisperse with radii of the order of 103 A and a known number

of surface charge groups per particle * (x 104) in univalent salt concentrations mnging from 10-g-10-4M exhibits intense irridescent diffraction colours when

observed in white light. In a beaker containing such a suspension there is a phase boundary, not yet well studied and more or less sharp, which distinguishes the irridescent (ordered) phase in the lower part of the container from a milky white (disordered) phase in the upper part. Closer studies [2] reveal that the ordered system sits on a face-centred cubic lattice. Since the centre-to-centre distance of the spheres is of the order of the wavelength of visible light (= 5000 A), van der Waals forces are completely negligible, and the structure is determined solely by electrostatic forces (and ?f course gravity), a conclusion which had been reached much earlier by Langrnuir [4] in his pioneering studies on similar systems. In this paper we explore the nature of the phase transition using Gouy-Chapman theory, and exploiting earlier studies of fluid-solid phase transitions. * We are indebted to Professor R.H. OttewiU for this estimate.

Before launching into a detailed discussion of our particular system, it will be useful to recall some known results concerning first-order fluid-solid phase transitions in general_ For one-component systems with either hard core or soft core repulsions (l/&2, I/&i, I/+) such transitions have been studied extensively in the literature [5,6]. Also charged particles interacting via the repulsive Coulomb law l/r and immersed in a continuous uniform background of oppositeiy charged material which provides electroneutrality have been studied thoroughly [7,8] _ A revealing universality of the Monte Carlo data for systems interacting via these various potentials emerges if we appeal to Lindemann’s rule [9 ] _ This criterion for melting is intuitive and asserts that a solid will meIt when the root mean square displacement of particles from their lattice sites is a characteristic fraction of the lattice spacing (typically about 0.10). Hence if one determines the rms displacement from the harmonic modes of vibration of the lattice, one obtains a reasonable estimate of the density at which the iattice melts. A slightly cruder method is to consider the vibrations of one particle when alI its neighbours are fixed at their lattice sites. We illustrate the application of this rule for particles interacting via a potential u(r) = A /rn_ Consider the change in energy of a particle when it is displaced slightly from its lattice site on a fee lattice. For n large, this change in energy is due almost entirely to the nearest neighbours. Let the particle be displaced to x. y. z and let the nearest neighbours be at the point (b,O,O). The change in energy is, 353

&=A

15 October 1976

CHEMkAL PHYSICS LEPTIXS

Volume 43; nusnEter2 1 ffx - by&-y=-%- .+12

A SW--b"

-- 1 tr”

I 1.

.

tance t is (zq)h2/2 &, where zq is the charge of the particle, and $rR3 is the volume occupied per particle. The vdume per particfe for a fee I&t&e with nearest neighbour distance b is 2-lj2 63, whence b = (2si2~/3)‘/3R )

If we take the spherical average of this expression and multipiy by the number of nearest neighbours, viz. 12, we find for the displacement energy in this mean field harmonic approximation (21

(9

and

where I‘ = (zq)2/eRkT.

(7)

Monte Carlo studies [7] show that the Wigne~ lattice

(3) ar!d the idndemann ratio of this rms displac cFent the lattice spacing b is

to

f= (;z)1j2/b = /3kT/4n(n-l)u(b)]112

(41

_

Using the Monte Carlo data of ref. fsl we can evaluate this crude estimate of the tidemann f for various interactions of the form I/#, and see from table I that the ratios are virtually constant. The extreme example of the Wigner lattice is mathematically more complicated because of the long range nature of the Coulomb interaction. However to a good approximation each particle can be considered to be moving about in a Wigner-Seitz sphere of equal but opposite charge centred at the lattice site. The distribution of charge outside the sphere is sufficiently close to spherically symmetric distribution to be ignored. (A spherically symmetric d~t~but~on of charge would produce zero fieId inside the sphere.) By elementary electrostatics, the energy required to displace the particle from its lattice site by a disTattie 1

354

n

fmelt

4 6 9 12

osof! 0.101 0.102 O.r,06

mehs when the interaction parameter J? = 155k 10. At first sight this is an extraordinarily large value for r. [For a hard potential ~(b/r)~~ and lattice spacing b, the lattice undergoes a phase transition f5] when the interaction parameter o/kT = OSO’?.] However, in terms of the Undemann criterion the experimental value of r = 15.5 makes sense. The correspondingf values are

f me*t=0*079,

r= 165; f,,,=0*077,

f rnett= 0.075 f r=

145 _

r= 155; 031

The difference between the value f = 0.08 for the (soft) Wigner lattice and vahresfm 0.10 for potentials of the form l/r” reflects the long range nature of the Coulomb potential, A priori one expects the Wigner lattice to be more sensitive to small pe~urbations. The preceding considerations are useful, since they imply that whatever the form of any “‘effective” repulsive interaction ranging from very saft (unscreened Coulomb) or very hard, the simple Lindemann criterion is bound to provide melting curves which are at worst quaiitatively correct. We can now proceed to model the low density latex sphere problem. At the characteristic distances which concern us, it is clear that van der Waals attractive forces are negligible. The ordering of latex spheres has been described [I, 2,10-l 31 as a version of the U&wood-Alder hard sphere transition. This transition at high salt concentrations occurs when the density of the solid phase is near to that of a close packing ~gerne~t for which the volume fraction of latex is about 0.35. However, observations of the phase transition have been extended [I] to the volume fraetions of 0.03. It is seen sedately that such a system

CHEMICAL PIiYSICS LETTERS

Volume 43. number 2

is much closer to the Wigner lattice, so that many-body interactions cannot be ignored. Suppose first that the number of surface charges z. is fmed, and that the system is fully dialysed. In a regular array of such spheres, the maximum potential difference between the surface of a sphere and ~e.~idpo~t separating two spheres is clearly Qm, 9 zoq/ea. For a radius a $I:lOOO& temperature T = 20K, dielectric constant E x 80, qQimaxlkT M z&25. Hence if we can believe the Poisson-Boltzm~n equa~on, we can expect the counter-ion distribution to be practically uniform, provided z. < 125. This system can then be modelled in terms of the energetics of a Wigner lattice if we replace the sphere by a point charge of charge z = zoi+ra3n+(R)

,

(9)

where n,(R) is the (uniform) density of counter-ions defined by

n+(R) = Z& n(R3 -

a3)

(11)

nR3 n+(R) ,

(12)

where n.,.(R) is determined from the Poisson-Boltzma.nn equation. In the presence of added salt, the corresponding v&e of zeff will be seff=$rR3

[n+(R) -

?~_(R)l,

df@+2dql- -47~7

a
G

n_(r) = n_(R)e@@

.

[tr,(r) - n_(r)],

r dr

dr2

R , (14)

with n+.(r) = n,(R) e-fl@,

(W

The boundary conditions are Ip(W=O,

UGa) at

%p/&=O,

For zo s 125, there are no sensible solutions as the spheres would be touching. For z. = 200, melting occurs for R = 2500 a. But for z. Z 125 the assumption of unifo~lty of counter-ion distribution has broken down. None the less, the main assumption necessary to use the criterion eq. (7) for melting is that the counter-ion drstribution is practically uniform over the range (R - fb) < r < R, and this will turn out to be so. Subject to that condition we can define an effective charge zeff which replaces io and still use eq. (7), where zeff is the adsorption excess of counter-ions, viz. Zeff = 2

of surface charge groups be determined self-consistently. We consider a fee array of charged spheres in saft water. To solve the non-linear Poisson-Roltzmann equation, we make the Wigner-Seitz approximation, i.e. replace the Wigner-Seitz cell by a sphere of the same volume. The equation is then to be solved with the boundary condition on the potential &#j& = 0 at r=R. Since r$is not defined up to a constant we may arbitrarily take $(R) = 0. Then 9 satisties the equation

t-=R,

W31

(10)

and R is the radius of the Wigner-Seitz sphere. By eq. (7) this system will then undergo a phase transition at r= 155 =&l+R&3]2.

15 October t976

(13)

where n+ refers to the sum of counter-ion and indifferent cation densities, and n_(R) means indifferent anion density. For our system, the Poisson-Boltzmann equation must also be solved subject to constraints imposed by the condition that the degree of dissociation

d+fdr=zq/ca2

at

r=a.

WC1

Here 4 is the unit charge, --zq the charge on the spheres, e the dielectric constant of water, a the radius of the spheres, and R the Wigner-Seitz radius. n,(r), n_(r) are the densities of positive (including counter-ions) and negative ions. These must satisfy the constraint [n,(r) - n_(r)] d3r = z ,

1

Wigner-Seitz cell

and f n_(r) d% =

a?11

total number of anions A . cell

0.8)

Eq. (17) is equivalent to the boundary condition eq. (16~). z is determined by the condition z

zo-z

-

z

2-i-A

n,(o)=K,

(19)

where z. is the total number of ion&able sites, +A) is the density of counter ions at the surface, and K is the dissociation constant. zeff is then given by eq. (13) and melting is determined by

zn+(~)/(z

I?= (zeffq)2/eRkT

= 155 .

The numerical routine is straightforward: eqs.(l4)~d(l5)~ou~~=~~,~=~~~,we~~e

Cw If we scale -

355

Volume 43, number 2 Table 2 r = 155;K= R (1000A)

yn+ (2/x)$

10-4-a

(a/RI3

,

1.6 1.7 1.8

2.44X 10-r 2.03x 10-r 1.71x10-’

2.0 2.4 3.0 5.0

1.25x 7.23x 3.70x 8.00x

10-l lo-* 10” 1O-3

2 (counter-

ions)

zeff

Salt

(moles/Iitre)

776 815 851

187 192 198

0.239x 10-a 0.176x 10r3 0.134x10-3

905 979 1047 1162

209 228 256 330

0.842x 0.407x 0.182x 0.355):

1241 1307 1375 1414 1435

390 467 551 611 656

0.143x lo=5 0.558X lO+j 0.237x 1O-6 0.143x10-e 0.965x lo-’

30.0 50.0

3.70X 1O-5 8.00x 10”

1522 1674

805 0144

0.339x lo-’ 0.846x 1O-8

3.98x lo-’

1844

1722

0

5573 5495 5573

198 238 356

with K= lo-* 1.71x10-’ 5.69x lo-* 5.12x10-3

0.278X 1O-3 0.549x 10-4 0.366X 1O-5

,-

---JOrdered

region

Disordered

)

I

0

I

10-s

02

Molar

I 0.4

06

08

-

r/R

Fig. 1. Charge profiles about a latex sphere. Vertical lines indicate the boundary of the sphere.

356

(21)

Thisis now integrated numerically starting at x = R/Q, y =y’= 0. For a given choice of n_(R), n,(R) is determined by satisfying the condition of ionic equilibrium. The number of counter ions which is a function of the ion density at the surface of the sphere has to be equal to the difference between total numbers of positive and negative ions. Subsequently n_(R) is varied to change the degree of screening until the salt concentration, where JI’= 155, is determined. Results are exhibited in table 2 for z = lo4 and two possible dissociation constants K= lo- Bs corresponding to carboxyl groups, and K= 10m2 corresponding to sulphate groups. The results for K= 10m4e8 correspond to the system most studied. The charge profile about a sphere is plotted in fig. 1, for various values of R/a at the phase transition. Note that over the important range of interest (R -j3)
lo4 104 lo4 10-5

2.92x lo-’ 1.00x 10” 3.6Ax104 2.04x 104 1.25x 10-4

1.8 2.6 5.8

= - [47rn+(R)je] fiq2a2 e-Y

+ [43r11- (R)/e]pq2a2e” .

7.0 10.0 14.0 17.0 20.0

136.0

15 October 1976

CHEMICAL PHYSICS LETTERS

1

salt

10-a

region

J 10-3

conceptration

Fig. 2. The phase diagram for the low density suspensions of latex spheres. The fulJ line is a theoretical result, and the circles are points within the coexistence region as determined by Hachisu and Kobayashi [ 11.

Volume 43, number 2

CHEMICALPHYSICS LETTERS

about 0.2, the ordering is better described as a KirkwoodAlder transition and consequently that region is not included in the figure. It should be noted [l ] that the experimental results show a large scatter due to the composition inhomogeneity, and the actual coexistence region is smaller than suggested by the experimental data. The quantum Wigner lattice solution of ref. ES] adopted in this work leads to an extremely narrow coexistence region, not shown in fig. 2. However, while the Wigner lattice result [8] for the width of the coexistence region is dependent on the quantum nature of the system, the limiting value of P is the same for both classical and quantum systems 171. The theoretical phase diagram shown in fig. 2 reflects only the limiting value r= 155. From fig. 2 we find that the trend in the behaviour of the experimental system [I 3, as well as the absolute values of the transition concentrations are correctly described by the proposed model. We conclude that the ordering of low density colloidal suspensions of spherical particles is governed solely by the repulsive electrostatic interaction and is well described by the concept of Wigner lattice.

References [ 1f S. Hachisu and Y. Kobayashi, f. ColIoid Interface Sci. 42 (1973) 342. [2] A. KOSC,M. Ozaki. K. Takano, Y. Kobayashi and S. Hachisu, 1. Colloid fntcrface Sci. 44 (1973) 330: [3]

(41 [S]

(61 (7I [8] 19 ]

[lo] Ill]

The authors are especially indebted to Professor RH. Ottewill who asked us to work on this problem, provided essential unpublished data on the system, and with whom and Drs. L.R. White and T.W. Healy -we had many useful discussions. We are grateful likewise to Professor Norman March who aIso drew our attention to recent work on the electron gas.

15 October 19’26

(121 [13]

S. Hachisu, Y. Kobaynshi and A. Kose, 3. CoIloid fnterface Sci. 46 (1974) 470. R.H. OttewiIl and G. Weise, private communi~tion, to be published; J.C. Brown, P.N. Pussey, J.W. Goodwin and R.H. OttewIlI, J. Phys. AS (1975) 664. I. Langmuir, J. Chem. Phys. 6 (1938) 873. W.G. Hoover, M. Ross, K.W. Johnson, D. Henderson, J.A. Barker and B.C. Brown, J. Chem. Phys. 52 (1970) 4931. W.G. Hoover, S.G. Gray and K.W. Johnson, J, Chem. Phys. 55 (1971) 1128. C.M. Care and N.H. March, Advan. Phys. 24 (1975) 101; N.H. March and M-P. Tosi, Phys. Letters SOA (1974) 224. J.P. Hansen, Phys. Rev. A8 (1973) 3096. F-A. Lindemann, Z. Phys. 11 (1910) 609; H-N-V. TemperIcy, Changes of state (Cleaver-Hufme Press, London, 1956). P.A. Hiltner and 1-M. Krieger, J. Phys. Chcm. 73 (1969) 2386. M, Wadnti and M. Toda, J. Phys. Sot. Japan 32 (1972) 1147. M. Wadati, Solid State 8 (1973) 51 I (in Japanese). W. vm Megen and 1. Snook, Chem. Phys. Letters 35 (1975) 399.

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