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Shear-flow-induced microstructural distortion near the critical point
PHYSICA ELSEVIER
Physica A 235 (1997) 87-104
Shear-flow-induced microstructural distortion near the critical point Jan K.G. Dhont*, Henk Verduin van 't Hoff Laboratory, Utrecht University. Padualaan 8, 3584 CH Utrecht, Netherlands
Abstract Shear-flow-induced microstructural distortion of suspensions close to the critical point is described in detail on the basis of the Smoluchowski equation, where hydrodynamic interaction is neglected. The solution of the Smoluchowski equation exhibits scaling behaviour, predicting that the combined shear rate and correlation length dependence is characterized by a single dimensionless number ~ ~ ~ 4 (with ~ the shear rate and ~ the correlation length of the unsheared suspension). From the scaling relations for deformation of the structure factor, a scaling form for the turbidity is derived. Experiments concerning these phenomena are presented, some of which are preliminary. Two types of systems are used, both of which exhibit a gas-liquid critical point: stearyl silica in benzene and stearyl silica in cyclohexane with polydimethylsiloxane. Although there is an overall agreement with the theory, there remains a discrepancy in some of the quantitative comparisons, probably due to the neglect of hydrodynamic interaction and the approximate nature of the closure relation that is used for the three-particle correlation function.
1. Introduction In the vicinity o f the spinodal, large-scale microstructures exist, resulting in an upswing o f the static structure factor at small wave vectors. Such large-scale microstructures are severely affected by relatively small shear rates. In this paper we quantify the effects o f shear on these large-scale microstructures on the basis o f the Smoluchowski equation. A gas-liquid critical point may occur in colloidal systems where the interaction potential is (partly) attractive. Two examples o f such systems are silica cores coated with stearyl alcohol chains dissolved in benzene and the same particles dissolved in cyclohexane with free polymer added. Benzene is a poor solvent for stearyl alcohol so that the alcohol chain brushes on the surfaces o f two colloidal particles rather * Corresponding author. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved P H S0378-4371 (96)00330-5
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dissolve in each other than in the solvent. This results in a very short-ranged attractive interaction, super-imposed onto the hard-core repulsion [1]. For the latter system, the colloidal particles attract each other due to depletion forces: as two colloidal particles approach each other, polymer is expelled from the gap between these particles (for geometric and entropic reasons), resulting in an excess osmotic polymer pressure that drives the two colloidal particles towards each other [2-5]. Without polymer, stearyl coated silica particles behave as hard spheres, since cyclohexane is a good solvent for stearyl alcohol, contrary to benzene [6]. For low enough temperatures and large enough polymer concentrations, respectively, the two systems exhibit a critical point (phase diagrams of the systems used here can be found in Refs. [7] and [8], respectively). Experiments close to the critical point for both colloidal systems will be discussed in this paper. In Section 2, the Smoluchowski equation approach to describe critical phenomena in the mean field region is outlined. In particular, the Ornstein-Zernike static structure factor is rederived from the Smoluchowski equation. The Smoluchowski equation approach allows for a straightforward inclusion of shearing motion, contrary to the classic Ornstein-Zernike theory. The profound effects of shear flow on the static structure factor are discussed in Section 3. Section 4 deals with the shear rate dependence of the turbidity. Experimental results on the two above-mentioned colloidal systems are compared with the theoretical predictions in Section 5. A summary and some conclusions are given in Section 6.
2. The Smoluchowski equation approach to critical phenomena The Smoluchowski equation is an equation of motion for the probability density function P(rl ..... r x , t ) of the position coordinates {rj}, j = 1..... N of all N colloidal particles in the system. The stationary version of that equation for an unsheared suspension reads 0 = [t [Vr, a/,(rl ..... rN)] P(rl ..... rN) + Vr, P(rl ..... rN),
(1)
where Vr, is the gradient operator with respect to the position coordinate rl, and tb is the total potential energy of the assembly of N Brownian particles. In order to find an expression for the static structure factor, we must first recast this equation into an equation for the pair-correlation function g(r,r'). Since by definition, f dr3 • • - / " drN P(r, r', r3 ..... rN ) = PI (r)Pi (r')y(r, r ' ) ,
(2)
where the one-particle probability density function Pl(r) is equal to 1/V for a homogeneous system with volume V, such an equation of motion can be obtained from the Smoluchowski equation by integration. Assuming pair wise additive direct interactions, that is, assuming that the total potential energy q~ can be written as a sum of pair
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potentials, N
• (rl . . . . . r s ) =
V(rij),
Z i,j
(3)
l i
with rij ----Iri - rj I, the integration with respect to r3 . . . . . rN is easily performed for identical Brownian particles to obtain (with r = rl - r 2 ) 0 = Vrg(r) + where /3 =
N/V
fig(r)[XTrV(r) -
Find(r)] ,
(4)
is the number density o f colloidal particles, and (with r ~ = rl - r 3 )
Find(r) = --/3
fdr'
[Vr, V(r')]
ga(r,r')
g(r)
(5)
is the indirect force o f particle 2 on particle l, which is the contribution to the total force that is mediated via intervening particles. Similar to the definition o f the paircorrelation function (2), the three-particle correlation function g3(r,r ~) = g3(rl - r 2 , rl - r 3 ) is defined as
g3(rl-r2,rl-ra)=V3 fdr4...fdrNP(rl,r2,r3,r4
.... ,rN).
(6)
For the translationally invariant system under consideration here, the three-particle correlation function depends on the position coordinates rl, r2 and r3 only via their differences r = rl - r2 and r t = rl - r3. We wish to solve Eq. (4) for large distances r =] rl - r2 ], that is, for distances much larger than the range Rv o f the pair-interaction potential. On approach o f the critical point the indirect force becomes very long ranged. The range o f these indirect interactions is quantified by the correlation length ~, for which an explicit formula will be given later. This long range of the indirect force induces microstructural order over large distances, giving rise to a sharp increase of the static structure factor at small wave vectors. It is this "critical wave-vector range" in which we are interested here. In order to obtain a closed equation for the pair-correlation function, the threepanicle correlation function must be expressed in terms of pair-correlation functions. The most simple ansatz is the Kirkwood superposition approximation, which assumes that correlations are pair wise independent, g3(r,r t) =
g(r) g(r')g([ r
- r' [).
(7)
What is neglected here is the influence of a third particle on the correlation between two other particles. For the particular situation we are interested in here, the superposition approximation can be improved to some extent, by accounting for the effect that the presence o f the distant panicle 2 has on the correlation between the neighbouring particles 1 and 3. The crucial point here is, that in the integral (5) that defines the indirect force, the distance r ~ = rl - r3 is always smaller than the range o f the pairinteraction potential Rv, since for r r > Rv the pair-force ~7 r, V(r~) is zero. On the other
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J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
t
t[ II
tl
Rv
', ]
.....
11
i
® r>>R v Fig. 1. A sketch of an arrangement of particles 1, 2 and 3 for which a closure relation is needed. The dashed line is the total-correlation function h ~ g 1 relative to the position coordinate of particle 2. hand, the distance between particles 1 and 2 is much larger than Rv: these are the large distances for which we are seeking a solution of Eq. (4). A closure relation is therefore needed only for special configurations where particles 1 and 3 are close together, while particles 1 and 2 are far apart. Such a configuration is sketched in Fig. 1. The effect of the presence of the distant particle 2 is to enhance the number density around the neighbouring particles 1 and 3 to 15g(R), where R is the distance between particle 2 and the particles 1 and 3. The most obvious choice for R is the distance between particle 2 and the point in between particles 1 and 3, that is, R = I ½(rl + r 3 ) - r 2 I = l r - ½r' [. The effect of the distant particle on the correlation between the two neighbouring particles is accounted for by simply replacing g(r ~) in the superposition approximation (7) by the 1 t ]). same pair-correlation function at the enhanced density 15g(I r - ½F I) = 15+ 15h(I r - ~r We are interested here in the asymptotic solution of Eq. (4) for large distances r, where the total-correlation function h(] r - ½r~ ]) is small, since h(r) --+ 0 as r ---* ~ . The enhancement of the density around the two neighbouring particles can therefore be considered small, so that the pair-correlation function at the enhanced density may be Taylor expanded up to leading order,
g(rt)at the enhanced density
:
dg(rl)
g(rt) q- ~
15h(lr -
It
~ r I).
(8)
The correlation functions on the right-hand side are understood to relate to the number density t5 = N/V. Substitution of this result into the superposition approximation (7) yields an improved superposition approximation,
g3(r,r)'
--
g(r)g(Ir -
r'l)
g(r')+--~15h(lr
-½,'1
)
.
(9)
It is essential to use this improved superposition appoximation to arrive at an expression for the correlation length that diverges on the spinodal. Simply using the original superposition approximation (7) leads to completely erroneous results, where a finite correlation length on the spinodal (c.q. the critical point) is found. This was first
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91
recognized by Fixman [9]. What is still neglected in this closure relation is the effect that particle 1 has on the correlation between particles 2 and 3, and of particle 3 on the correlation between particles 1 and 2. Substitution of the closure relation (9) into the expression (5) for the indirect force, and subsequent subsitution into the Smoluchowski equation (4) yields the following equation for the pair-correlation function: 0 = ~rg(r) q- fig(r)[~7rV(r ) A- p / dr t [Vr, V(rt)]
g(r')+~h(lr-
×g(lr-rtl)
2 l)
.
(10)
Since r' <<.Rv and r>>Rz, both correlation functions g(Ir- r' l) and h(lr- ½r' I) in the integrand can be Taylor expanded around r ~ = 0. These are smooth functions of r / for r >>Rv. The Taylor expansions read
g(lr-rll) =g(r)--r'.Vrg(r)+ h([ r -
½r'r': ~7r~rrg(r) - ~rrr 1 / / t"
~rrVrVrg(r)q-... ,
(11)
½r' l)
1 t_! = h(r) - ~rl, . '~,.h(r)+ . ~rr . :V,.Vrh(r) .
~8r'r'r':VrVrV,.h(r)+ .
. (12)
Furthermore, only linear terms in h(r) must be retained for the calculation of the asymptotic solution for large distances, since h(r) --~ 0 when r ~ c¢. Noting that g(r) =- h(r)+ 1, substitution of the Taylor expansions (11 )and (12) into Eq. (10) and keeping only linear terms in h(r) yields
0 = ~rh(r) + fl {h(r) + 1} [VrV(r)] / { l dg(r')} +fifl[~rh(r)]" dr' [V~,V(r')]r' g ( r ' ) + ~ f i - - ~ p --p[~
[VrVrVrh(r)]i/dr'
IVy, V(r')] r'r'r' { 1-~g(r')+ ~ - ~_dg(r') p--~p}.l (13)
The spherical angular integrations can be performed after subsitution of Vr' V(r') = ~' dV(rl)/dr ~, with f~ = r'/r ~ the unit vector along r ~, and using that
f d f' f' f' = ~rc 4 ,
J aft ?; Fir k r z = 4 n "
^1
^/
^1
(14)
[rij~kZ + 6ikrjl + 6iZ~jk] ,
(15)
where the integrations range over the unit spherical surface, J is the unit matrix and 6ij is the Kronecker delta. For r>>Rv, where VrV(r) = O, Eq. (13) reduces to 0
=
Vr
-;-=h(r)-flEV2h(r)
,
(16)
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where oo
II : fik~T -
r '3 d V ( r t )
(17)
o
dv(r,,{,q(r')+ 8l_dg(r' / PdT-J' "
oo
(18)
0
The quantities denoted here as II and E are shorthand notations for the expressions on the right-hand sides which are found from the Smoluchowski equation (13) after performing the spherical angular integrations. The expression on the right-hand side of Eq. (17) is precisely the osmotic pressure, which is denoted as II. The differential equation (16) is solved when (19)
k(r) = ~2V2rh(r),
where ~ JnTd
(20)
is the correlation length. The solution of this equation is the well-known OrnsteinZernike form h(r)
exp{-r/~}
= (ARv)
(21)
,
r
where A is a dimensionless constant. The correlation length measures the range of correlations and diverges on the spinodal where d I I / d f i = O. The integration constant A can be determined as follows. First Fourier transform the above expression to obtain an expression for the static structure factor at small wave vectors, ARv
S(k)
=
l +fih(k)
= 1 +4~fi~_2+k2.
From S ( k = O) = k B T / ( d H / d f i )
ARl ....
it now follows that
,,{Is l }
4 r t f i 32
[ dfi]
1
.
With the help of Eq. (20) the static structure factor is thus found to be equal to
L~2 + (k~)2 1 + (k ~)2
(fl~) s(k) =
Since fie ~ R 2 (see [10,11]) and k R v ~ 1, the second term in the numerator may be neglected. We thus finally arrive at the well-known Ornstein-Zernike expression for
J.K.G. Dhont, H. VerduinlPhysica A 235 (1997) 87-104
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the static structure factor [12]
S(k)-
1 1 fie ~ 2_{_k 2
(22)
'
We thus rederived the classic Ornstein-Zernike result, starting from the Smoluchowski equation (notice that Eqs. (4) and (5) are also valid for molecular systems). Contrary to the Ornstein-Zernike equation that leads to the expression (22) for the static structure factor, the Smoluchowski equation admits, rather straightforwardly, the inclusion of shearing motion.
3. The Ornstein-Zernike structure factor with shear flow
Let us now analyse the stationary Smoluchowski equation (neglecting the hydrodynamic interactions) for a sheared suspension, which reads N ~ P ( r , ..... rN 19) = 0 : ~ {Do(V2p+[3Vj • (P[Vj~b])) - V j . (F. riP) }, j=l
(23) where Do is the Stokes-Einstein diffusion coefficient, and F is the velocity gradient matrix, for which we take the simple form
F = 9
(o,o) 0 0 0 000
,
(24)
with 9 the shear rate. This corresponds to a shear flow in the x-direction with its gradient in the y-direction. Without shear flow, the Smoluchowski equation (23) reduces to Eq. (1) that was analysed in the previous section. Precisely as in the previous section, integration leads to
0 = 2DoVr" {Vrg(rl~))+[3g(rlg){VrV(r)-
Find(r]°))] } - - Vr.(F.r g(r 19)), (25)
where the indirect force is now a function of the shear rate through the shear rate dependence of the correlation functions, F~nd(rl'2) = - ~ f dr' [Vr' V(r')] g3(r,r'19) g(rtT)
(26)
The same closure relation (9) that was used for the unsheared system can be used here, except that the correlation functions are now shear rate dependent. Moreover, for the calculation of the asymptotic behaviour of the pair-correlation function at large distances, the same procedure as for the unsheared system can be employed here : use of the Taylor expansions (11) and (12) and linearization with respect to the
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total-correlation function yields 0 = 2O0Vr - { Vrh(r I~) +/3 (h(r I~) + 1} [Vr V(r)] +fS/~ [Vrh(r[~)]-
dr' [Vr, V(/)]r'
o(r'[~)+~5
dfi
[VrVrV h(rl#)]i/ar'
-~
ivy, V(r,)] r,r,r,{ ~g(r 1 i~)+~._~, 1 fidg~l~) }}
- V ~ - ( F . r h(r [~)).
(27)
The spherical angular integrations cannot be performed without knowledge of the anisotropic F-dependence of the pair-correlation function #(r'l ~). A crucial point here is that in the integrals, g(rq ~)) is always multiplied with the pair force ~Tr, V(/), which is zero for / > Rv. The shear rate dependence of the integrals is therefore related to the distortion of the pair-correlation function for short distances. The shear-induced distortion of the pair-correlation function for such short distances is much less pronounced than the distortion for larger distances, because the perturbing term (the last term in Eq. (27)) is larger for larger distances. Distortions for large distances are significant for shear rates where distortions for short distances are still insignificant. The order of magnitude of the combination F . r g(r]~) for small distances is ~rg÷, with g+ the contact value of the pair-correlation function. The order of magnitude of the first term between the curly brackets in Eq. (27) is dg/drl+, the slope of the paircorrelation function at contact of the hard cores. Since in equilibrium, without shear flow, the terms between the curly brackets in Eq. (27) cancel, the perturbing shear term is small in comparison to each of the terms between the curly brackets when ? r g+ ¢ COo [dg/drl+ ]. Hence,
Pe°~Rv ldln{g} [ ~ g(rl~)~oeq(r) dr I+
for r ~ R z ,
(28)
where geq(r) is the equilibrium pair-correlation function, without shear flow, and Pe ° is the bare Peclet number defined as
Pe ° - 7 R2".
(29)
2D0 The right-hand side in the inequality in (28) can be large for the systems with attractive pair-interaction potentials under consideration here, since at contact the numerical value of the pair-correlation function is large and the pair-correlation function decreases rapidly with increasing distance. Therefore, for not too large bare Peclet numbers, the shear rate dependence of the pair-correlation function in the integrals in Eq. (27) may be neglected. The spherical angular integrations can now be performed with the help of Eqs. (14) and (15) to obtain 0 = 2DoV 2 { // 7 P h(r]~))- B~2~72h(r]~)}-27r.(F-rh(rl~)),
(30)
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J.K.G. Dhont, H. VerduflllPhysica A 235 (1997) 87 104
with the pair-correlation fimction equal to the equilibrium pair-correlation function, that is, the pair-correlation function of the quiescent system, without shear flow. Both dII/dfi and E are thus the same where II and E are given in Eqs. (17) and (18),
quantities as encountered in the previous section where a quiescent system without shear flow is considered. The differential equation (30) is the generalization of the Smoluchowski equation (19) which now includes the effects of shear flow, Fourier transformation of the equation of motion (30) yields ~2kI
t" OS(klg) -- 2 D e f f ( k ) k 2 I S ( k ~3k2
I I)) -
seq(k))
(31)
kj is the jth component of k and where the wave-vector-dependent effective diffusion coefficient is equal to where
DefY(k) = Do~ [d~ +k2E] .
(32)
The equilibrium static structure factor s e q ( k ) is the Omstein-Zernike static structure factor (22) without shear flow. The solution of this differential equation reads
AS(k[,~) _= S(kt~ ) -
seq(k)
--
kl Pe k2
× [seq(v/k2÷x2÷k~)-seq(k)]exp{-kl@e(P(k)l~2=,-P(k))}, (33) where the functions Q and P are equal to
Q(k)=(kRv) 2 [l + (k~) 2] ,
(34)
k2
0
The + sign in the upper integration limit in Eq. (33) is to be used for positive (negative) values of kl Pe. The dressed Peclet number Pe that is introduced here is equal to
(fldII~-' Pe = k, --d-tip/ Pe°
-
")R2v 2Deff(k = O)
-
1 9~ 2 flE/R2v2D0
,
(36)
where the bare Peclet number Pe ° is defined in Eq. (29). The amount of distortion of long ranged correlations is measured by this dressed Peclet number, while the bare Peclet number measures the amount of distortion of short-ranged correlations. The numerical value of the dressed Peclet number is much larger than the bare Peclet number, since close to the spinodal fldII/dfi is small. This confirms the reasoning
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that led us to neglect the shear rate dependence of the integrals in the Smoluchowski equation (27). The shear-rate-dependent static structure factor must become equal to the equilibrium structure factor for vanishing shear rates. That this is indeed the case can be verified from the following representation of the delta distribution :
Let f(x) denote a .function in ~, with f'(x) =- d f(x)/dx > O,and l i m x ~ f(x) = cxD, then, 6(x-xo) = H(x-x°)limf'(X)exp{do e
f(x)--cf(x°)}'
where H(x) is the Heaviside unit step function. With ~: = Pe, f(x) = -4-P(k)M /kl ( + when kl > 0 and - when kl < 0), AS in Eq. (33) is easily seen to become equal to 0 for Pe --~ O. Notice that in Eq. (33) the Peclet number is always multiplied with ki, the component of the wave vector in the flow direction. Hence, S(klj)
I,,
= s~q(k)I,,=o
(37)
There is thus no distortion in directions perpendicular to the flow direction. This is true up to O(Pe°), since terms of this order were neglected when replacing g(r'lD in the integrals in Eq. (27) by ffeq(rt).
3.1. Scaling The expression (33) for the static structure factor looks quite complicated. It can be substantially simplified by scaling the wave vector to the correlation length (of the unsheared system). Let us therefore introduce the dimensionless wave vector, K = k ¢.
(38)
Define the relative static structure factor distortion tp as 7' - S(K [;0 - seq(K) Seq(K)-
(39)
1
Scaling the wave vectors in Eq. (33) to the correlation length, and substitution of the expression (22) for the equilibrium static structure factor, yields a relatively simple expression for the relative distortion, namely, ±vo
(K2-K~+X2)(K~-X2)exp
IP(KI2) = ~ l l Kz
2K, J '
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
97
where x
F( l )=faY
(x 2 -I 2 +
(1 q - K 2 - g 2 q- y2)
/(2
--(x-
x2)(K
- ,:2)(1
+ K 2 - x 2)
+½ (X3-K3)(l+ZK2-ZK2)+½(XS-KS),
(41)
and where 2 is a dimensionless number equal to
1 2 - flN/R~ Pe°
(~Tv) 4
(42)
Besides being a more simple expression, there is a fundamental feature about these new expressions, namely, that both the shear rate dependence and the distance from the spinodal (through the correlation length) are now entirely described in terms of the single dimensionless number 2. Identical numerical values of 2 give rise to the same relative distortion ~, considered as a function of the scaled wave vector K. A single numerical value of 2 relates to many different shear rates and temperatures. This scaling behaviour of the static structure factor has profound implications for the shear rate and temperature dependence of, for example, the turbidity, flow-induced dichroism (see [11]) and viscosity (see [13]). The transition from "weak" to "strong" shear flow occurs at 2 ~ l, 2 < 1 ~ weak shear flow, 2 > 1 =:> strong shear flow.
(43)
According to Eq. (42), 2 ~ ~ ~4, SO that on approaching of the critical point, smaller and smaller shear rates are sufficient to give rise to significant distortions. In other words, at a constant shear rate, distortions increase on approaching the critical point. This is a result of the unlimited increase of the correlation length ~. Using renormalization groups techniques, for molecular systems, Onuki et al. [ 14,15] arrived at a scaling involving the dimensionless group 2 ~ ~}~3 instead of 2 ~ ,)~4. The fourth power of the correlation length that we find here is probably a mean field exponent, while the third power holds beyond the mean field. The present theory is restricted to the mean field region due to the linearization of equations of motion with respect to the total-correlation function h. When fldlI/d~ is a very small number, which is the case very close to the critical point, the linear term in h in Eq. (30) (the first term between the curly brackets) is of no significance, and higher-order terms in h must be included. This linear term is dominant over the higher-order terms only when fldII/dfi is not very small, which is the case in the mean field region, but not beyond.
98
J.K.G. Dhont, H. VerduinlPhysica A 235 (1997) 8~104
3.2. Correlation lengths o f the sheared system Some numerical results for the structure factor are given m Fig. 2. As can be seen, the structure factor becomes extremely anisotropic, with a large distortion in all directions except perpendicular to the flow direction, where the distortion is o f O(Pe °), as discussed above. In directions where kj = 0, the correlation length ~0 o f the sheared system is a linear function o f Pe ° for not too large shear rates,
~o = ~ + ~_(I)Pe° + O((Pe°)2) ,
(44)
where ~(r) is a shear-rate-independent coefficient. Recall that { is the correlation length o f the unsheared equilibrium system. Whether the correlation length {0 increases or decreases on applying shear depends on the sign o f the numerical coefficient {(I), the calculation o f which requires a linear response analysis o f the Smoluchowski equation for short distances r 4 R z . In directions where kl ~ 0, a very small bare Peclet number gives rise to a large distortion close to the critical point since 2 is large also for very small shear rates. The correlation length o f the sheared suspension is now a non-analytic function o f Pe °.
KyC L
"
,1 \ -3 Fig. 2. The static structure factor as a function of KI and K2 with K3 0 (upper figures) and of Ki and K3 with K2 = 0 (lower figures), for )~ 10 and 100. The most left figure is the equilibrium Ornstein-Zernike static structure factor. A value of 1/100 is chosen for the quantity (Rv/()2(flE/R~,).
J.K.G. Dhont, 14. Verduin/Physica A 235 (1997) 87-104
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It is apparent from Fig. 2 that the sheared static structure factor decreases for very small wave vectors relative to the equilibrium static structure factor. Hence, no very long-ranged correlations are induced, and the correlation length always decreases due to shear flow, except may be in directions where kl = 0.
4. The temperature and shear rate dependence of the turbidity The most simple experimental quantity that measures changes in microstructural properties is the turbidity ~. The turbidity is a measure for the total scattered intensity of light, and is equal to -(1/L)ln{It/Io}, where Io is the incident intensity, and It is the intensity that is measured after the beam traversed the sample of thickness L. The difference between It and I0 is the result of scattering. Conservation of energy leads to an expression for the turbidity in terms of an integral of the scattered intensity over all scattering angles. When the scattered intensity at small angles is relatively large, this standard expression for the turbidity yields, upon substitution of Eq. (40) for the static structure factor (see [11]), C~
1
z ( ~ ) - zeq _ (koRv) 2 flE/R2v T(2),
(45)
with zeq the turbidity of the quiescent dispersion and where the turbidity scaling function T(2) is equal to
r(2)
=
de 0
dX
(46)
0
The relative distortion T t is the distortion (40) in forward direction, where O = 0 (q~ and O are the spherical angular coordinates of the wave vector, where the z-axis is in the direction of the laser beam). In Eq. (45), Cr is a constant that is related to the optical contrast of the colloidal particles and k0 = 27t/2, with 2 the wavelength of the light in the dispersion. For two experiments at two different shear rates and temperatures, such that the numerical value of 2 is equal for both experiments, the same turbidity change should be measured. In other words, when the shear rate dependence of the change of the turbidity at various temperatures is plotted as a function of 2, these data should collapse onto a single curve.
5. Experimental The above theoretical predictions can be compared to experiments on suspensions close to their gas-liquid critical point. In the following two subsections, light scattering and turbidity measurements are discussed.
lO0
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
5.1. Stationary states Light scattering experiments under stationary shear are performed on a near critical system of stearyl silica (radius 13 nm) in cyclohexane with polydimethylsiloxane (radius of gyration 14nm) added. The silica volume fraction is about 0.19 and the volume fraction of polymer, as calculated from its radius of gyration, is about 1.2. The flowvorticity plane is probed, where the component of the scattering wave vector in the gradient direction is zero (k2=0). Fig. 3 shows a fit of an unsheared and a sheared structure factor with kl = 0, to the equilibrium Ornstein--Zernike structure factor (22), where the height of the top and the correlation length ~ are the fitting parameters. The fit is almost perfect for both cases, confirming both the Ornstein-Zernike behaviour of the quiescent suspension and the prediction (37) that the shear has a negligible effect in directions where kj = 0. Two-dimensional light scattering patterns are shown in Fig. 4 : the upper figures are experimental and the lower ones theoretical. The wave vectors shown here are rescaled with respect to the correlation length according to Eq. (38), where the correlation length is determined from fits as shown in Fig. 3. A quantitative comparison between experiment and theory can be made by fitting cross-sections of these scattering patterns at a constant value of/£3. The width of such sections as functions of Kl can be fitted with respect to 2, using our expressions for the static structure factor, where the heights of the curves are forced to coincide. Examples of such fits are given in Fig. 5. The solid lines are theoretical curves with values for 2 as indicated in the figures. As can be seen, these fits are quite satisfactory. Expression (42) for 2 predicts that a plot of )./~4 versus the shear rate is a straight line, independent of the concentration.
4
2 V
r~ 1 i
0
-2
i
-1
0 k
[106m
1
2
-']
Fig. 3. A fit of experimental light scattering data for an unsheared system (e) and a sheared system in the direction where kl = 0 ([]) to the equilibrium Ornstein-Zernike structure factor (22). The shear rate for the latter set of data is 4.48 s -~, corresponding to a value for 2 of 174 (see the upper middle scattering pattern in Fig.4).
J.K.G, Dhont, H. VerduinlPhysica A 235 (1997) 87-104
101
Fig. 4. Stationary structure factors in the flow-vorticity plane. Upper figures are experimental, and the lower ones calculated from Eqs. (39)-(41). The arrow in the upper middle figure indicates a cross-section at K3 = -2.0.
S [a.u.]
~.=0
-2.5
0~.0
~, = 30
~, = 345
I
-2.5
0.0
-2.5
0.0
2.5
K1
Fig. 5. Cross-sections of scattering patterns in the flow-vorticity plane at K3 = -2.0 as functions of KI. Solid lines are theoretical curves with a value for 2 as indicated.
This prediction is confirmed in Fig. 6, although it should be m e n t i o n e d that the two correlation lengths here are not v e r y different (~4 differs about 30% for the two p o l y m e r concentrations). N o t i c e that the slope o f this straight line is equal to ½DoflE, a l l o w i n g for the determination o f the C a h n - H i l l i a r d square gradient coefficient.
102
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
Z/~' [~-~] 100 80 60 40 20
'~ [s-l]
L
0
0
2
4
6
8
Fig. 6. A plot of 2/~4 versus the shear rate for two differentcorrelationlengths : 1.54pm ([2) and 1.45/~m (,).
These experimental results are certainly preliminary, and more detailed studies are desirable. 5.2. Turbidity measurements
The shear rate and correlation length dependence of the static structure factor determine, in an integrated form, that of the turbidity. Measurements are performed on a stearyl silica (radius 39 nm) in benzene close to the critical point at a volume fraction of 0.19 (for a more detailed account see [16]). Fig. 7(a) illustrates the enhanced effect of shear flow on approach of the critical point. Formally, this is due to the larger value of 2 for a given shear rate when the correlation length is larger. Physically, this means that a smaller shear rate is sufficient to affect larger structures. The scaling relation (45) predicts that when r ( ~ ) - req is plotted against ~ 4 , all the curves in Fig. 7(a) should collapse onto a single curve. This is indeed the case to within experimental errors. Moreover, the experimental points can be made to coincide with the theoretical scaling function T(2) in Eq. (46) by appropriately scaling numerical values of ~ 4 (see fig. 7(b)). However, the expected value for the scaling factor ~ 4 4 / 2 = 2Dofl~ (as obtained from an independent determination of/~Z from fits of scattered intensities to the Ornstein-Zemike structure factor, as discussed in the previous subsection) differs by an order of magnitude from the scaling factor used in fig. 7(b). This discrepancy is not due to uncertainties in the prefactor C~ in Eq. (45) for the turbidity: for the system that is used here, C~ is almost temperature independent, and its numerical value found by rescaling the turbidity axis to find the scaling function is in accordance with the numerical value found from an independent measurement of the optical contrast. The discrepancy is probably due to the neglect of hydrodynamic interaction and the approximate nature of the closure relation (9). Long-ranged hydrodynamic interactions
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104 0
I
I
il
103
I
I
T(
b-q
I G
u
20OC
i
-3000-
-4000-
-5000
I
40
81o
1801 120
0
500
1000
~500
X
Fig. 7. (a) Turbidity data at different temperatures as a function of the shear rate. The critical temperature of the suspension is 17.95°C. (b) The same data as in (a), but now plotted as T(2) ~ z(j) - zeq versus ), ~ j~_4. The solid line is the turbidity scaling function in Eq. (46) obtained by numerical integration.
probably do not account for this discrepancy: when these would have been important, a wave-vector-dependent contribution to the structure factor distortion as described in the previous subsection is expected, which does not seem to be the case. Short-ranged hydrodynamic interactions, however, do modify the mobility of particles and thereby "renormalizes" the Stokes-Einstein diffusion coefficient in Eqs. (42) and (29) for 2. It would be interesting to analyse the contribution of long ranged hydrodynamic interactions, that is, to include hydrodynamic interactions on the Oseen level The shear rate dependence of the turbidity in the vicinity of the critical point has been studied for binary fluid mixtures by Beysens et al. [17,18]. Their results indicate the same scaling as predicted here for colloidal systems (see also Ref. [11]).
6. Conclusions
The Smoluchowski equation approach to critical phenomena under shear reproduces all features that are seen experimentally. However, an order of magnitude discrepancy is found for the scaling factor to make experimental turbidity data coincide with the theoretically predicted scaling form. This is probably due to neglect of the hydrodynamic interaction and the approximations involved in the closure relation (9) for the three-particle correlation function. In view of the correctly predicted scaling behaviour and the shear rate and correlation length dependence of 2, this order of magnitude discrepancy is most likely a constant factor, independent of the shear rate and correlation length. The calculation of that factor requires consideration of hydrodynamic interactions or/and improvement over the closure relation used here.
104
J.K.G. Dhont, H. Verdum/Physica A 235 (1997) 8~104
T h e light s c a t t e r i n g e x p e r i m e n t s o n the s t e a d y state structure factors as d e s c r i b e d here are c e r t a i n l y p r e l i m i n a r y , a n d m o r e e x p e r i m e n t a t i o n w o u l d b e v a l u a b l e .
References [1] [2] [3] [4]
[5l
[6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
J.W. Jansen, C.G. de Kruif and A. Vrij, J. Colloid Interface Sci. 114 (1986) 471,481, 492, 501. S. Asakura and F. Oosawa, J. Chem. Phys. 22 (1954) 1255; J. Polym. Sci. 33 (1958) 183. A. Vrij, Pure Appl. Chem. 48 (1976) 471. B. Vincent, P.F. Luckham and F.A. Waite, J. Colloid Interface Sci. 73 (1980) 508; A.P, Gast, C.K. Hall and W.B. Russel, J. Colloid Interface Sci. 96 (1983) 251; M.M. Santore, W.B. Russel and R.K. Prud'homme, Macromolecules 22 (1989) 1317, H.N.W. Lekkerkerker, W.C.K. Poon, P.N. Pusey, A. Stroobants and P.B. Warren, Europhys. Lett. 20 (1992) 559. F.K.R. Li-ln-On, B. Vincent and F.A. Waite, Amer. Chem. Soc. Symp. Ser. 9 (1975) 165, C. Cowell, F.K.R. Li-ln-On and B. Vincent, J. Chem. Soc. Faraday Trans. 1 74 (1978) 337, C. Pathmamanoharan, H. de Hek and A. Vrij, Colloid Polym Sci. 259 (1981) 769. A.K. van Helden and A. Vrij, J. Colloid Interface Sci. 78 (1980) 312; M.M. Kops-Werkhoven and H.M. Fijnaut, J. Chem. Phys. 74 (1981) 1618; M.M. Kops-Werkhoven, C. Pathmamanoharan, A. Vrij and H.M. Fijnaut, J. Chem. Phys. 77 (1982) 5913. H. Verduin and J.K.G. Dhont, J. Colloid Interface Sci. 172 (1995) 425. N.A.M. Verhaegh, J.S. van Duijneveldt, J.K.G. Dhont and H.N.W. Lekkerkerker, Physica A 230 (1996) 409. M. Fixman, J. Chem. Phys. 33 (1960) 1357. M. Fixman, Adv. Chem. Phys. 6 (1964) 175. J.K.G. Dhont and H. Verduin, J. Chem. Phys. 101 (1994) 6193. L.S. Ornstein and F. Zernike, Proc. Sect. Sci. Med. Akad. Wet. 17 (1914) 793. J.K.G. Dhont, J. Chem. Phys. 103 (1995) 7072. A. Onuki and K. Kawasaki, Ann. Phys. 121 (1979) 456. A. Onuki, K. Yamazaki, K. Kawasaki, Ann. Phys. 131 (1981) 217. H. Verduin and J.K.G. Dhont, Phys. Rev. E 52 (1995) 1811. D. Beysens M. Gbadamassi, Phys. Rev. A 22 (1980) 2250. D. Beysens, in: Scattering Techniques Applied to Supramolecular Nonequilibrium Systems, eds. S. Chen, B. Chu and R. Nossal, NATO Advanced Study Institute, Series B: Physics Vol. 73 (Plenum Press, New York, 1981) p. 411.
Physica A 235 (1997) 87-104
Shear-flow-induced microstructural distortion near the critical point Jan K.G. Dhont*, Henk Verduin van 't Hoff Laboratory, Utrecht University. Padualaan 8, 3584 CH Utrecht, Netherlands
Abstract Shear-flow-induced microstructural distortion of suspensions close to the critical point is described in detail on the basis of the Smoluchowski equation, where hydrodynamic interaction is neglected. The solution of the Smoluchowski equation exhibits scaling behaviour, predicting that the combined shear rate and correlation length dependence is characterized by a single dimensionless number ~ ~ ~ 4 (with ~ the shear rate and ~ the correlation length of the unsheared suspension). From the scaling relations for deformation of the structure factor, a scaling form for the turbidity is derived. Experiments concerning these phenomena are presented, some of which are preliminary. Two types of systems are used, both of which exhibit a gas-liquid critical point: stearyl silica in benzene and stearyl silica in cyclohexane with polydimethylsiloxane. Although there is an overall agreement with the theory, there remains a discrepancy in some of the quantitative comparisons, probably due to the neglect of hydrodynamic interaction and the approximate nature of the closure relation that is used for the three-particle correlation function.
1. Introduction In the vicinity o f the spinodal, large-scale microstructures exist, resulting in an upswing o f the static structure factor at small wave vectors. Such large-scale microstructures are severely affected by relatively small shear rates. In this paper we quantify the effects o f shear on these large-scale microstructures on the basis o f the Smoluchowski equation. A gas-liquid critical point may occur in colloidal systems where the interaction potential is (partly) attractive. Two examples o f such systems are silica cores coated with stearyl alcohol chains dissolved in benzene and the same particles dissolved in cyclohexane with free polymer added. Benzene is a poor solvent for stearyl alcohol so that the alcohol chain brushes on the surfaces o f two colloidal particles rather * Corresponding author. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved P H S0378-4371 (96)00330-5
88
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 8~104
dissolve in each other than in the solvent. This results in a very short-ranged attractive interaction, super-imposed onto the hard-core repulsion [1]. For the latter system, the colloidal particles attract each other due to depletion forces: as two colloidal particles approach each other, polymer is expelled from the gap between these particles (for geometric and entropic reasons), resulting in an excess osmotic polymer pressure that drives the two colloidal particles towards each other [2-5]. Without polymer, stearyl coated silica particles behave as hard spheres, since cyclohexane is a good solvent for stearyl alcohol, contrary to benzene [6]. For low enough temperatures and large enough polymer concentrations, respectively, the two systems exhibit a critical point (phase diagrams of the systems used here can be found in Refs. [7] and [8], respectively). Experiments close to the critical point for both colloidal systems will be discussed in this paper. In Section 2, the Smoluchowski equation approach to describe critical phenomena in the mean field region is outlined. In particular, the Ornstein-Zernike static structure factor is rederived from the Smoluchowski equation. The Smoluchowski equation approach allows for a straightforward inclusion of shearing motion, contrary to the classic Ornstein-Zernike theory. The profound effects of shear flow on the static structure factor are discussed in Section 3. Section 4 deals with the shear rate dependence of the turbidity. Experimental results on the two above-mentioned colloidal systems are compared with the theoretical predictions in Section 5. A summary and some conclusions are given in Section 6.
2. The Smoluchowski equation approach to critical phenomena The Smoluchowski equation is an equation of motion for the probability density function P(rl ..... r x , t ) of the position coordinates {rj}, j = 1..... N of all N colloidal particles in the system. The stationary version of that equation for an unsheared suspension reads 0 = [t [Vr, a/,(rl ..... rN)] P(rl ..... rN) + Vr, P(rl ..... rN),
(1)
where Vr, is the gradient operator with respect to the position coordinate rl, and tb is the total potential energy of the assembly of N Brownian particles. In order to find an expression for the static structure factor, we must first recast this equation into an equation for the pair-correlation function g(r,r'). Since by definition, f dr3 • • - / " drN P(r, r', r3 ..... rN ) = PI (r)Pi (r')y(r, r ' ) ,
(2)
where the one-particle probability density function Pl(r) is equal to 1/V for a homogeneous system with volume V, such an equation of motion can be obtained from the Smoluchowski equation by integration. Assuming pair wise additive direct interactions, that is, assuming that the total potential energy q~ can be written as a sum of pair
ZK.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
89
potentials, N
• (rl . . . . . r s ) =
V(rij),
Z i,j
(3)
l i
with rij ----Iri - rj I, the integration with respect to r3 . . . . . rN is easily performed for identical Brownian particles to obtain (with r = rl - r 2 ) 0 = Vrg(r) + where /3 =
N/V
fig(r)[XTrV(r) -
Find(r)] ,
(4)
is the number density o f colloidal particles, and (with r ~ = rl - r 3 )
Find(r) = --/3
fdr'
[Vr, V(r')]
ga(r,r')
g(r)
(5)
is the indirect force o f particle 2 on particle l, which is the contribution to the total force that is mediated via intervening particles. Similar to the definition o f the paircorrelation function (2), the three-particle correlation function g3(r,r ~) = g3(rl - r 2 , rl - r 3 ) is defined as
g3(rl-r2,rl-ra)=V3 fdr4...fdrNP(rl,r2,r3,r4
.... ,rN).
(6)
For the translationally invariant system under consideration here, the three-particle correlation function depends on the position coordinates rl, r2 and r3 only via their differences r = rl - r2 and r t = rl - r3. We wish to solve Eq. (4) for large distances r =] rl - r2 ], that is, for distances much larger than the range Rv o f the pair-interaction potential. On approach o f the critical point the indirect force becomes very long ranged. The range o f these indirect interactions is quantified by the correlation length ~, for which an explicit formula will be given later. This long range of the indirect force induces microstructural order over large distances, giving rise to a sharp increase of the static structure factor at small wave vectors. It is this "critical wave-vector range" in which we are interested here. In order to obtain a closed equation for the pair-correlation function, the threepanicle correlation function must be expressed in terms of pair-correlation functions. The most simple ansatz is the Kirkwood superposition approximation, which assumes that correlations are pair wise independent, g3(r,r t) =
g(r) g(r')g([ r
- r' [).
(7)
What is neglected here is the influence of a third particle on the correlation between two other particles. For the particular situation we are interested in here, the superposition approximation can be improved to some extent, by accounting for the effect that the presence o f the distant panicle 2 has on the correlation between the neighbouring particles 1 and 3. The crucial point here is, that in the integral (5) that defines the indirect force, the distance r ~ = rl - r3 is always smaller than the range o f the pairinteraction potential Rv, since for r r > Rv the pair-force ~7 r, V(r~) is zero. On the other
90
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
t
t[ II
tl
Rv
', ]
.....
11
i
® r>>R v Fig. 1. A sketch of an arrangement of particles 1, 2 and 3 for which a closure relation is needed. The dashed line is the total-correlation function h ~ g 1 relative to the position coordinate of particle 2. hand, the distance between particles 1 and 2 is much larger than Rv: these are the large distances for which we are seeking a solution of Eq. (4). A closure relation is therefore needed only for special configurations where particles 1 and 3 are close together, while particles 1 and 2 are far apart. Such a configuration is sketched in Fig. 1. The effect of the presence of the distant particle 2 is to enhance the number density around the neighbouring particles 1 and 3 to 15g(R), where R is the distance between particle 2 and the particles 1 and 3. The most obvious choice for R is the distance between particle 2 and the point in between particles 1 and 3, that is, R = I ½(rl + r 3 ) - r 2 I = l r - ½r' [. The effect of the distant particle on the correlation between the two neighbouring particles is accounted for by simply replacing g(r ~) in the superposition approximation (7) by the 1 t ]). same pair-correlation function at the enhanced density 15g(I r - ½F I) = 15+ 15h(I r - ~r We are interested here in the asymptotic solution of Eq. (4) for large distances r, where the total-correlation function h(] r - ½r~ ]) is small, since h(r) --+ 0 as r ---* ~ . The enhancement of the density around the two neighbouring particles can therefore be considered small, so that the pair-correlation function at the enhanced density may be Taylor expanded up to leading order,
g(rt)at the enhanced density
:
dg(rl)
g(rt) q- ~
15h(lr -
It
~ r I).
(8)
The correlation functions on the right-hand side are understood to relate to the number density t5 = N/V. Substitution of this result into the superposition approximation (7) yields an improved superposition approximation,
g3(r,r)'
--
g(r)g(Ir -
r'l)
g(r')+--~15h(lr
-½,'1
)
.
(9)
It is essential to use this improved superposition appoximation to arrive at an expression for the correlation length that diverges on the spinodal. Simply using the original superposition approximation (7) leads to completely erroneous results, where a finite correlation length on the spinodal (c.q. the critical point) is found. This was first
J.K.G. Dhont, H. V e r d u i n l P h y s i c a A 235 (1997) 8 7 - 1 0 4
91
recognized by Fixman [9]. What is still neglected in this closure relation is the effect that particle 1 has on the correlation between particles 2 and 3, and of particle 3 on the correlation between particles 1 and 2. Substitution of the closure relation (9) into the expression (5) for the indirect force, and subsequent subsitution into the Smoluchowski equation (4) yields the following equation for the pair-correlation function: 0 = ~rg(r) q- fig(r)[~7rV(r ) A- p / dr t [Vr, V(rt)]
g(r')+~h(lr-
×g(lr-rtl)
2 l)
.
(10)
Since r' <<.Rv and r>>Rz, both correlation functions g(Ir- r' l) and h(lr- ½r' I) in the integrand can be Taylor expanded around r ~ = 0. These are smooth functions of r / for r >>Rv. The Taylor expansions read
g(lr-rll) =g(r)--r'.Vrg(r)+ h([ r -
½r'r': ~7r~rrg(r) - ~rrr 1 / / t"
~rrVrVrg(r)q-... ,
(11)
½r' l)
1 t_! = h(r) - ~rl, . '~,.h(r)+ . ~rr . :V,.Vrh(r) .
~8r'r'r':VrVrV,.h(r)+ .
. (12)
Furthermore, only linear terms in h(r) must be retained for the calculation of the asymptotic solution for large distances, since h(r) --~ 0 when r ~ c¢. Noting that g(r) =- h(r)+ 1, substitution of the Taylor expansions (11 )and (12) into Eq. (10) and keeping only linear terms in h(r) yields
0 = ~rh(r) + fl {h(r) + 1} [VrV(r)] / { l dg(r')} +fifl[~rh(r)]" dr' [V~,V(r')]r' g ( r ' ) + ~ f i - - ~ p --p[~
[VrVrVrh(r)]i/dr'
IVy, V(r')] r'r'r' { 1-~g(r')+ ~ - ~_dg(r') p--~p}.l (13)
The spherical angular integrations can be performed after subsitution of Vr' V(r') = ~' dV(rl)/dr ~, with f~ = r'/r ~ the unit vector along r ~, and using that
f d f' f' f' = ~rc 4 ,
J aft ?; Fir k r z = 4 n "
^1
^/
^1
(14)
[rij~kZ + 6ikrjl + 6iZ~jk] ,
(15)
where the integrations range over the unit spherical surface, J is the unit matrix and 6ij is the Kronecker delta. For r>>Rv, where VrV(r) = O, Eq. (13) reduces to 0
=
Vr
-;-=h(r)-flEV2h(r)
,
(16)
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87 104
92
where oo
II : fik~T -
r '3 d V ( r t )
(17)
o
dv(r,,{,q(r')+ 8l_dg(r' / PdT-J' "
oo
(18)
0
The quantities denoted here as II and E are shorthand notations for the expressions on the right-hand sides which are found from the Smoluchowski equation (13) after performing the spherical angular integrations. The expression on the right-hand side of Eq. (17) is precisely the osmotic pressure, which is denoted as II. The differential equation (16) is solved when (19)
k(r) = ~2V2rh(r),
where ~ JnTd
(20)
is the correlation length. The solution of this equation is the well-known OrnsteinZernike form h(r)
exp{-r/~}
= (ARv)
(21)
,
r
where A is a dimensionless constant. The correlation length measures the range of correlations and diverges on the spinodal where d I I / d f i = O. The integration constant A can be determined as follows. First Fourier transform the above expression to obtain an expression for the static structure factor at small wave vectors, ARv
S(k)
=
l +fih(k)
= 1 +4~fi~_2+k2.
From S ( k = O) = k B T / ( d H / d f i )
ARl ....
it now follows that
,,{Is l }
4 r t f i 32
[ dfi]
1
.
With the help of Eq. (20) the static structure factor is thus found to be equal to
L~2 + (k~)2 1 + (k ~)2
(fl~) s(k) =
Since fie ~ R 2 (see [10,11]) and k R v ~ 1, the second term in the numerator may be neglected. We thus finally arrive at the well-known Ornstein-Zernike expression for
J.K.G. Dhont, H. VerduinlPhysica A 235 (1997) 87-104
93
the static structure factor [12]
S(k)-
1 1 fie ~ 2_{_k 2
(22)
'
We thus rederived the classic Ornstein-Zernike result, starting from the Smoluchowski equation (notice that Eqs. (4) and (5) are also valid for molecular systems). Contrary to the Ornstein-Zernike equation that leads to the expression (22) for the static structure factor, the Smoluchowski equation admits, rather straightforwardly, the inclusion of shearing motion.
3. The Ornstein-Zernike structure factor with shear flow
Let us now analyse the stationary Smoluchowski equation (neglecting the hydrodynamic interactions) for a sheared suspension, which reads N ~ P ( r , ..... rN 19) = 0 : ~ {Do(V2p+[3Vj • (P[Vj~b])) - V j . (F. riP) }, j=l
(23) where Do is the Stokes-Einstein diffusion coefficient, and F is the velocity gradient matrix, for which we take the simple form
F = 9
(o,o) 0 0 0 000
,
(24)
with 9 the shear rate. This corresponds to a shear flow in the x-direction with its gradient in the y-direction. Without shear flow, the Smoluchowski equation (23) reduces to Eq. (1) that was analysed in the previous section. Precisely as in the previous section, integration leads to
0 = 2DoVr" {Vrg(rl~))+[3g(rlg){VrV(r)-
Find(r]°))] } - - Vr.(F.r g(r 19)), (25)
where the indirect force is now a function of the shear rate through the shear rate dependence of the correlation functions, F~nd(rl'2) = - ~ f dr' [Vr' V(r')] g3(r,r'19) g(rtT)
(26)
The same closure relation (9) that was used for the unsheared system can be used here, except that the correlation functions are now shear rate dependent. Moreover, for the calculation of the asymptotic behaviour of the pair-correlation function at large distances, the same procedure as for the unsheared system can be employed here : use of the Taylor expansions (11) and (12) and linearization with respect to the
J.K.G. Dhont, H. Verduin/PhysicaA 235 (1997)87-104
94
total-correlation function yields 0 = 2O0Vr - { Vrh(r I~) +/3 (h(r I~) + 1} [Vr V(r)] +fS/~ [Vrh(r[~)]-
dr' [Vr, V(/)]r'
o(r'[~)+~5
dfi
[VrVrV h(rl#)]i/ar'
-~
ivy, V(r,)] r,r,r,{ ~g(r 1 i~)+~._~, 1 fidg~l~) }}
- V ~ - ( F . r h(r [~)).
(27)
The spherical angular integrations cannot be performed without knowledge of the anisotropic F-dependence of the pair-correlation function #(r'l ~). A crucial point here is that in the integrals, g(rq ~)) is always multiplied with the pair force ~Tr, V(/), which is zero for / > Rv. The shear rate dependence of the integrals is therefore related to the distortion of the pair-correlation function for short distances. The shear-induced distortion of the pair-correlation function for such short distances is much less pronounced than the distortion for larger distances, because the perturbing term (the last term in Eq. (27)) is larger for larger distances. Distortions for large distances are significant for shear rates where distortions for short distances are still insignificant. The order of magnitude of the combination F . r g(r]~) for small distances is ~rg÷, with g+ the contact value of the pair-correlation function. The order of magnitude of the first term between the curly brackets in Eq. (27) is dg/drl+, the slope of the paircorrelation function at contact of the hard cores. Since in equilibrium, without shear flow, the terms between the curly brackets in Eq. (27) cancel, the perturbing shear term is small in comparison to each of the terms between the curly brackets when ? r g+ ¢ COo [dg/drl+ ]. Hence,
Pe°~Rv ldln{g} [ ~ g(rl~)~oeq(r) dr I+
for r ~ R z ,
(28)
where geq(r) is the equilibrium pair-correlation function, without shear flow, and Pe ° is the bare Peclet number defined as
Pe ° - 7 R2".
(29)
2D0 The right-hand side in the inequality in (28) can be large for the systems with attractive pair-interaction potentials under consideration here, since at contact the numerical value of the pair-correlation function is large and the pair-correlation function decreases rapidly with increasing distance. Therefore, for not too large bare Peclet numbers, the shear rate dependence of the pair-correlation function in the integrals in Eq. (27) may be neglected. The spherical angular integrations can now be performed with the help of Eqs. (14) and (15) to obtain 0 = 2DoV 2 { // 7 P h(r]~))- B~2~72h(r]~)}-27r.(F-rh(rl~)),
(30)
95
J.K.G. Dhont, H. VerduflllPhysica A 235 (1997) 87 104
with the pair-correlation fimction equal to the equilibrium pair-correlation function, that is, the pair-correlation function of the quiescent system, without shear flow. Both dII/dfi and E are thus the same where II and E are given in Eqs. (17) and (18),
quantities as encountered in the previous section where a quiescent system without shear flow is considered. The differential equation (30) is the generalization of the Smoluchowski equation (19) which now includes the effects of shear flow, Fourier transformation of the equation of motion (30) yields ~2kI
t" OS(klg) -- 2 D e f f ( k ) k 2 I S ( k ~3k2
I I)) -
seq(k))
(31)
kj is the jth component of k and where the wave-vector-dependent effective diffusion coefficient is equal to where
DefY(k) = Do~ [d~ +k2E] .
(32)
The equilibrium static structure factor s e q ( k ) is the Omstein-Zernike static structure factor (22) without shear flow. The solution of this differential equation reads
AS(k[,~) _= S(kt~ ) -
seq(k)
--
kl Pe k2
× [seq(v/k2÷x2÷k~)-seq(k)]exp{-kl@e(P(k)l~2=,-P(k))}, (33) where the functions Q and P are equal to
Q(k)=(kRv) 2 [l + (k~) 2] ,
(34)
k2
0
The + sign in the upper integration limit in Eq. (33) is to be used for positive (negative) values of kl Pe. The dressed Peclet number Pe that is introduced here is equal to
(fldII~-' Pe = k, --d-tip/ Pe°
-
")R2v 2Deff(k = O)
-
1 9~ 2 flE/R2v2D0
,
(36)
where the bare Peclet number Pe ° is defined in Eq. (29). The amount of distortion of long ranged correlations is measured by this dressed Peclet number, while the bare Peclet number measures the amount of distortion of short-ranged correlations. The numerical value of the dressed Peclet number is much larger than the bare Peclet number, since close to the spinodal fldII/dfi is small. This confirms the reasoning
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87 104
96
that led us to neglect the shear rate dependence of the integrals in the Smoluchowski equation (27). The shear-rate-dependent static structure factor must become equal to the equilibrium structure factor for vanishing shear rates. That this is indeed the case can be verified from the following representation of the delta distribution :
Let f(x) denote a .function in ~, with f'(x) =- d f(x)/dx > O,and l i m x ~ f(x) = cxD, then, 6(x-xo) = H(x-x°)limf'(X)exp{do e
f(x)--cf(x°)}'
where H(x) is the Heaviside unit step function. With ~: = Pe, f(x) = -4-P(k)M /kl ( + when kl > 0 and - when kl < 0), AS in Eq. (33) is easily seen to become equal to 0 for Pe --~ O. Notice that in Eq. (33) the Peclet number is always multiplied with ki, the component of the wave vector in the flow direction. Hence, S(klj)
I,,
= s~q(k)I,,=o
(37)
There is thus no distortion in directions perpendicular to the flow direction. This is true up to O(Pe°), since terms of this order were neglected when replacing g(r'lD in the integrals in Eq. (27) by ffeq(rt).
3.1. Scaling The expression (33) for the static structure factor looks quite complicated. It can be substantially simplified by scaling the wave vector to the correlation length (of the unsheared system). Let us therefore introduce the dimensionless wave vector, K = k ¢.
(38)
Define the relative static structure factor distortion tp as 7' - S(K [;0 - seq(K) Seq(K)-
(39)
1
Scaling the wave vectors in Eq. (33) to the correlation length, and substitution of the expression (22) for the equilibrium static structure factor, yields a relatively simple expression for the relative distortion, namely, ±vo
(K2-K~+X2)(K~-X2)exp
IP(KI2) = ~ l l Kz
2K, J '
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
97
where x
F( l )=faY
(x 2 -I 2 +
(1 q - K 2 - g 2 q- y2)
/(2
--(x-
x2)(K
- ,:2)(1
+ K 2 - x 2)
+½ (X3-K3)(l+ZK2-ZK2)+½(XS-KS),
(41)
and where 2 is a dimensionless number equal to
1 2 - flN/R~ Pe°
(~Tv) 4
(42)
Besides being a more simple expression, there is a fundamental feature about these new expressions, namely, that both the shear rate dependence and the distance from the spinodal (through the correlation length) are now entirely described in terms of the single dimensionless number 2. Identical numerical values of 2 give rise to the same relative distortion ~, considered as a function of the scaled wave vector K. A single numerical value of 2 relates to many different shear rates and temperatures. This scaling behaviour of the static structure factor has profound implications for the shear rate and temperature dependence of, for example, the turbidity, flow-induced dichroism (see [11]) and viscosity (see [13]). The transition from "weak" to "strong" shear flow occurs at 2 ~ l, 2 < 1 ~ weak shear flow, 2 > 1 =:> strong shear flow.
(43)
According to Eq. (42), 2 ~ ~ ~4, SO that on approaching of the critical point, smaller and smaller shear rates are sufficient to give rise to significant distortions. In other words, at a constant shear rate, distortions increase on approaching the critical point. This is a result of the unlimited increase of the correlation length ~. Using renormalization groups techniques, for molecular systems, Onuki et al. [ 14,15] arrived at a scaling involving the dimensionless group 2 ~ ~}~3 instead of 2 ~ ,)~4. The fourth power of the correlation length that we find here is probably a mean field exponent, while the third power holds beyond the mean field. The present theory is restricted to the mean field region due to the linearization of equations of motion with respect to the total-correlation function h. When fldlI/d~ is a very small number, which is the case very close to the critical point, the linear term in h in Eq. (30) (the first term between the curly brackets) is of no significance, and higher-order terms in h must be included. This linear term is dominant over the higher-order terms only when fldII/dfi is not very small, which is the case in the mean field region, but not beyond.
98
J.K.G. Dhont, H. VerduinlPhysica A 235 (1997) 8~104
3.2. Correlation lengths o f the sheared system Some numerical results for the structure factor are given m Fig. 2. As can be seen, the structure factor becomes extremely anisotropic, with a large distortion in all directions except perpendicular to the flow direction, where the distortion is o f O(Pe °), as discussed above. In directions where kj = 0, the correlation length ~0 o f the sheared system is a linear function o f Pe ° for not too large shear rates,
~o = ~ + ~_(I)Pe° + O((Pe°)2) ,
(44)
where ~(r) is a shear-rate-independent coefficient. Recall that { is the correlation length o f the unsheared equilibrium system. Whether the correlation length {0 increases or decreases on applying shear depends on the sign o f the numerical coefficient {(I), the calculation o f which requires a linear response analysis o f the Smoluchowski equation for short distances r 4 R z . In directions where kl ~ 0, a very small bare Peclet number gives rise to a large distortion close to the critical point since 2 is large also for very small shear rates. The correlation length o f the sheared suspension is now a non-analytic function o f Pe °.
KyC L
"
,1 \ -3 Fig. 2. The static structure factor as a function of KI and K2 with K3 0 (upper figures) and of Ki and K3 with K2 = 0 (lower figures), for )~ 10 and 100. The most left figure is the equilibrium Ornstein-Zernike static structure factor. A value of 1/100 is chosen for the quantity (Rv/()2(flE/R~,).
J.K.G. Dhont, 14. Verduin/Physica A 235 (1997) 87-104
99
It is apparent from Fig. 2 that the sheared static structure factor decreases for very small wave vectors relative to the equilibrium static structure factor. Hence, no very long-ranged correlations are induced, and the correlation length always decreases due to shear flow, except may be in directions where kl = 0.
4. The temperature and shear rate dependence of the turbidity The most simple experimental quantity that measures changes in microstructural properties is the turbidity ~. The turbidity is a measure for the total scattered intensity of light, and is equal to -(1/L)ln{It/Io}, where Io is the incident intensity, and It is the intensity that is measured after the beam traversed the sample of thickness L. The difference between It and I0 is the result of scattering. Conservation of energy leads to an expression for the turbidity in terms of an integral of the scattered intensity over all scattering angles. When the scattered intensity at small angles is relatively large, this standard expression for the turbidity yields, upon substitution of Eq. (40) for the static structure factor (see [11]), C~
1
z ( ~ ) - zeq _ (koRv) 2 flE/R2v T(2),
(45)
with zeq the turbidity of the quiescent dispersion and where the turbidity scaling function T(2) is equal to
r(2)
=
de 0
dX
(46)
0
The relative distortion T t is the distortion (40) in forward direction, where O = 0 (q~ and O are the spherical angular coordinates of the wave vector, where the z-axis is in the direction of the laser beam). In Eq. (45), Cr is a constant that is related to the optical contrast of the colloidal particles and k0 = 27t/2, with 2 the wavelength of the light in the dispersion. For two experiments at two different shear rates and temperatures, such that the numerical value of 2 is equal for both experiments, the same turbidity change should be measured. In other words, when the shear rate dependence of the change of the turbidity at various temperatures is plotted as a function of 2, these data should collapse onto a single curve.
5. Experimental The above theoretical predictions can be compared to experiments on suspensions close to their gas-liquid critical point. In the following two subsections, light scattering and turbidity measurements are discussed.
lO0
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
5.1. Stationary states Light scattering experiments under stationary shear are performed on a near critical system of stearyl silica (radius 13 nm) in cyclohexane with polydimethylsiloxane (radius of gyration 14nm) added. The silica volume fraction is about 0.19 and the volume fraction of polymer, as calculated from its radius of gyration, is about 1.2. The flowvorticity plane is probed, where the component of the scattering wave vector in the gradient direction is zero (k2=0). Fig. 3 shows a fit of an unsheared and a sheared structure factor with kl = 0, to the equilibrium Ornstein--Zernike structure factor (22), where the height of the top and the correlation length ~ are the fitting parameters. The fit is almost perfect for both cases, confirming both the Ornstein-Zernike behaviour of the quiescent suspension and the prediction (37) that the shear has a negligible effect in directions where kj = 0. Two-dimensional light scattering patterns are shown in Fig. 4 : the upper figures are experimental and the lower ones theoretical. The wave vectors shown here are rescaled with respect to the correlation length according to Eq. (38), where the correlation length is determined from fits as shown in Fig. 3. A quantitative comparison between experiment and theory can be made by fitting cross-sections of these scattering patterns at a constant value of/£3. The width of such sections as functions of Kl can be fitted with respect to 2, using our expressions for the static structure factor, where the heights of the curves are forced to coincide. Examples of such fits are given in Fig. 5. The solid lines are theoretical curves with values for 2 as indicated in the figures. As can be seen, these fits are quite satisfactory. Expression (42) for 2 predicts that a plot of )./~4 versus the shear rate is a straight line, independent of the concentration.
4
2 V
r~ 1 i
0
-2
i
-1
0 k
[106m
1
2
-']
Fig. 3. A fit of experimental light scattering data for an unsheared system (e) and a sheared system in the direction where kl = 0 ([]) to the equilibrium Ornstein-Zernike structure factor (22). The shear rate for the latter set of data is 4.48 s -~, corresponding to a value for 2 of 174 (see the upper middle scattering pattern in Fig.4).
J.K.G, Dhont, H. VerduinlPhysica A 235 (1997) 87-104
101
Fig. 4. Stationary structure factors in the flow-vorticity plane. Upper figures are experimental, and the lower ones calculated from Eqs. (39)-(41). The arrow in the upper middle figure indicates a cross-section at K3 = -2.0.
S [a.u.]
~.=0
-2.5
0~.0
~, = 30
~, = 345
I
-2.5
0.0
-2.5
0.0
2.5
K1
Fig. 5. Cross-sections of scattering patterns in the flow-vorticity plane at K3 = -2.0 as functions of KI. Solid lines are theoretical curves with a value for 2 as indicated.
This prediction is confirmed in Fig. 6, although it should be m e n t i o n e d that the two correlation lengths here are not v e r y different (~4 differs about 30% for the two p o l y m e r concentrations). N o t i c e that the slope o f this straight line is equal to ½DoflE, a l l o w i n g for the determination o f the C a h n - H i l l i a r d square gradient coefficient.
102
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104
Z/~' [~-~] 100 80 60 40 20
'~ [s-l]
L
0
0
2
4
6
8
Fig. 6. A plot of 2/~4 versus the shear rate for two differentcorrelationlengths : 1.54pm ([2) and 1.45/~m (,).
These experimental results are certainly preliminary, and more detailed studies are desirable. 5.2. Turbidity measurements
The shear rate and correlation length dependence of the static structure factor determine, in an integrated form, that of the turbidity. Measurements are performed on a stearyl silica (radius 39 nm) in benzene close to the critical point at a volume fraction of 0.19 (for a more detailed account see [16]). Fig. 7(a) illustrates the enhanced effect of shear flow on approach of the critical point. Formally, this is due to the larger value of 2 for a given shear rate when the correlation length is larger. Physically, this means that a smaller shear rate is sufficient to affect larger structures. The scaling relation (45) predicts that when r ( ~ ) - req is plotted against ~ 4 , all the curves in Fig. 7(a) should collapse onto a single curve. This is indeed the case to within experimental errors. Moreover, the experimental points can be made to coincide with the theoretical scaling function T(2) in Eq. (46) by appropriately scaling numerical values of ~ 4 (see fig. 7(b)). However, the expected value for the scaling factor ~ 4 4 / 2 = 2Dofl~ (as obtained from an independent determination of/~Z from fits of scattered intensities to the Ornstein-Zemike structure factor, as discussed in the previous subsection) differs by an order of magnitude from the scaling factor used in fig. 7(b). This discrepancy is not due to uncertainties in the prefactor C~ in Eq. (45) for the turbidity: for the system that is used here, C~ is almost temperature independent, and its numerical value found by rescaling the turbidity axis to find the scaling function is in accordance with the numerical value found from an independent measurement of the optical contrast. The discrepancy is probably due to the neglect of hydrodynamic interaction and the approximate nature of the closure relation (9). Long-ranged hydrodynamic interactions
J.K.G. Dhont, H. Verduin/Physica A 235 (1997) 87-104 0
I
I
il
103
I
I
T(
b-q
I G
u
20OC
i
-3000-
-4000-
-5000
I
40
81o
1801 120
0
500
1000
~500
X
Fig. 7. (a) Turbidity data at different temperatures as a function of the shear rate. The critical temperature of the suspension is 17.95°C. (b) The same data as in (a), but now plotted as T(2) ~ z(j) - zeq versus ), ~ j~_4. The solid line is the turbidity scaling function in Eq. (46) obtained by numerical integration.
probably do not account for this discrepancy: when these would have been important, a wave-vector-dependent contribution to the structure factor distortion as described in the previous subsection is expected, which does not seem to be the case. Short-ranged hydrodynamic interactions, however, do modify the mobility of particles and thereby "renormalizes" the Stokes-Einstein diffusion coefficient in Eqs. (42) and (29) for 2. It would be interesting to analyse the contribution of long ranged hydrodynamic interactions, that is, to include hydrodynamic interactions on the Oseen level The shear rate dependence of the turbidity in the vicinity of the critical point has been studied for binary fluid mixtures by Beysens et al. [17,18]. Their results indicate the same scaling as predicted here for colloidal systems (see also Ref. [11]).
6. Conclusions
The Smoluchowski equation approach to critical phenomena under shear reproduces all features that are seen experimentally. However, an order of magnitude discrepancy is found for the scaling factor to make experimental turbidity data coincide with the theoretically predicted scaling form. This is probably due to neglect of the hydrodynamic interaction and the approximations involved in the closure relation (9) for the three-particle correlation function. In view of the correctly predicted scaling behaviour and the shear rate and correlation length dependence of 2, this order of magnitude discrepancy is most likely a constant factor, independent of the shear rate and correlation length. The calculation of that factor requires consideration of hydrodynamic interactions or/and improvement over the closure relation used here.
104
J.K.G. Dhont, H. Verdum/Physica A 235 (1997) 8~104
T h e light s c a t t e r i n g e x p e r i m e n t s o n the s t e a d y state structure factors as d e s c r i b e d here are c e r t a i n l y p r e l i m i n a r y , a n d m o r e e x p e r i m e n t a t i o n w o u l d b e v a l u a b l e .
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