# Hydrodynamics in bridging and aggregation of
two colloidal particles

in a near-critical binary mixture

###### Abstract

We investigate bridging and aggregation of two colloidal particles in a near-critical binary mixture when the fluid far from the particles is outside the coexistence (CX) curve and is rich in the component disfavored by the colloid surfaces. In such situations, the adsorption-induced interaction is enhanced, leading to bridging and aggregation of the particles. We realize bridging firstly by changing the temperature with a fixed interparticle separation and secondly by letting the two particles aggregate. The interparticle attractive force dramatically increases upon bridging. The dynamics is governed by hydrodynamic flow around the colloid surfaces. In aggregation, the adsorption layers move with the particles and squeezing occurs at narrow separation. These results suggest relevance of bridging in the reversible colloid aggregation observed so far. We use the local functional theory J. Chem. Phys. 136, 114704 (2012) to take into account the renormalization effect and the simulation method Phys. Rev. Lett. 85, 1338 (2000) to calculate the hydrodynamic flow around the colloidal particles.

## I Introduction

Colloidal suspensions exhibit various intriguing effects in a mixture solvent. For example, interplay between wetting and phase separation has been studied extensively in experiments Tanakareview (); Tanaka1 (); Mah (); Amis () and simulations Balzas (); La (); Araki (). Much attention has also been paid to a solvent-mediated attractive interaction among solid surfaces due to adsorption-induced concentration disturbancesFisher (); Krech (); Two (); Ga (); Gam (); Evans-Hop (), which is enhanced with increasing the correlation length near the criticality. In particular, reversible colloid aggregation has been observed at off-critical compositions outside the coexistence curve (CX) Be1 (); Be2 (); Be3 (); Be4 (); Bey (); Be5 (); Maher (); Guo (); Bonn (), where the solvent far from the particles is rich in the component disfavored by the colloid surfaces (near the disfavored branch of CX). It occurs while the particle radius still much exceeds . Remarkably, the face-to-face separation distance is of order and is much longer than in these aggregates Be1 (); Be2 (); Be3 (); Be4 (); Be5 (); Bey (). As a result, the aggregates fragment and the particles redisperse upon a reduction of adsorption caused by a small temperature change. Guo et al.Guo () furthermore observed liquid, fcc crystal, and glass phases of the aggregated particles also for . We mention a number of theoretical papers on this near-critical aggregation Slu (); Lowen (); Netz (); Kaler (); Peter (); Die (); Mohry (); Edison (). In non-critical solvents, on the other hand, the particles often stick and form fractal aggregates due to the attractive van der Waals interaction Russel ().

The adsorption-induced interaction decays exponentially as for if the solvent is at the critical composition without phase separation Fisher (); Two (); Gam (), as was measured directly Nature2008 (). Some authors Bonn (); Die () used this exponential expression to explain the observed aggregation. In their approach, close contacts of the particles in the range are needed for aggregation, since the factor is nearly zero for . In contrast, the van der Waals two-body potentialRussel (); Ross () is long-ranged for and grows as for , where is the Hamaker constant. In recent theoriesPeter (); Mohry (), the colloidal particles were assumed to form one-component systems interacting via an attractive solvent-mediated interaction potential () and a repulsive electrostatic potential, where the particle separation is smaller than in aggregation. Very recently, Edison et al. Edison () has examined phase behavior of a three-component model on a 2D lattice to obtain aggregation of the third component.

As a related problem, much attention has also been paid to the phase behavior of fluids between closely separated walls Evansreview (); Gelb (); Binder (), where narrow regions may be filled with the phase favored by the walls or may hold some fraction of the disfavored phase. As a result, there can be a first-order phase transition between these two states, called capillary condensation, when the fluid in the reservoir is in the disfavored phase. For fluids between parallel plates, there appears a first-order capillary condensation line outside (bulk) CX in the plane of the temperature and the reservoir chemical potential (which corresponds to a magnetic field for magnetic systems) Evansreview (); Gelb (); Binder (); Fisher (); Nakanishi (); Evans-Marconi (). Analogously, between large particles (or between a large particle and a wall), a bridging domain of the favored phase can appear, when they are surrounded by the disfavored phase Gam (); Yeomans (); Bauer (); Higashi (); Vino (); Seville (); Butt (); bubble (). This effect is relevant in the physics of wet granular matterGra (). Recently, we calculated phase diagrams of capillary condensation Oka-c () and bridging Oka-b (). We stress that the solvent-mediated interaction can be much enhanced near these transitions even in pretransitional states (before local phase separation) Oka-c (); Oka-b (); Two (); Anna (); Evans-Hop (). We also examined the dynamics of capillary condensation Yabu ().

In this paper, we demonstrate that the capillary force mediated by a bridging domain can give rise to the observed reversible aggregation. Let two colloidal particles be bridged by a columnar domain of the favored phase, where the column radius is of the order of the particle radius . Then, in terms of the surface tension , the interface free energy is of order and its derivative with respect to yields the capillary force of order between the particles Oka-b (); Seville (); Butt (). Such bridging can occur even for large of order , so it can be relevant to the reversible aggregation. We then need to investigate its dynamics to demonstrate its occurrence for near the disfavored branch of CX. It is a local phase separation process governed by the hydrodynamics, where squeezing also takes place as the particles approach. We note that the colloid hydrodynamics coupled with the concentration has been studied in various situations Tanakareview (); Araki (); Furukawa () using the model H equations Halperin (); Kawasaki-Ohta (); Onukibook ().

As a theoretical method, we use the renormalized local functional theory Oka-c (); Oka-b (); Fisher-Yang (); Up (); Upton () to account for the near-critical fluctuation effect. It is combined with the fluid-particle dynamics (FPD) simulation method Araki (); Tanakareview (); Tanaka2 (); Fujitani () to calculate the flow around the particles. In our case, the adsorption layers are thickened near the criticality Jasnow (); Liu (); Lawreview (), which strongly affect the flow in aggregation. We mention that Furukawa et al.Furukawa () used the FPD method to study two-particle aggregation at the critical composition. They assumed a small particle radius with in the mean-field theory, where the solvent-mediated force was appreciable even for because of relatively large . However, if , the force virtually vanishes at the critical composition.

The organization of this paper is as follows. In Sec.II, we will shortly explain the FPD method and the local functional theory applied to our problem. In Sec.III, we will explain the simulation method. In Sec.IV, we will present simulation results of bridging dynamics.

## Ii Theoretical Background

### ii.1 Critical behavior

We consider a near-critical binary mixture with an upper critical solution temperature for small reduced temperature at a given pressure, where is above the wetting temperature Bey (); Ross (). (For mixtures with a lower critical solution temperature such as 2,6-lutidine and water (LW), we should set .) The order parameter is a scaled concentration deviation slightly deviating from its critical value. For , a two-phase region appears in the - plane. The coexistence curve (CX) consists of the favored and disfavored branches as

(1) |

where is a constant. The critical exponentsOnukibook (); Vicari (); Aliu () are given by , , and . The correlation length depends on and and is given by for and and by for on CX, where is assumed to be common in the two cases and and are microscopic lengths. Here, is usually in the range for low molecular-weight binary mixtures. The ratio is a universal number in the renormalization group theoryOnukibook () and its reliable estimateAliu () is , so we suppose .

We express the free energy density in accord with the scaling relations in the vicinity of the critical point in the - plane including the region inside CX (, ). This is needed because changes from positive to negative around the colloidal particles in the presence of strong adsorption. To this end, we use a simple parametric form of devised in our previous paper Oka-c () (see the appendix).

At the starting point of our theory, the thermal fluctuations of and the velocity field with wave numbers larger than have already been coarse-grained or renormalized. Here, is the lower cut-off wave number of the renormalization effect, so the free energy density and the kinetic coefficients ( in eqn (6) and in eqn (9)) are renormalized ones depending on fractional powers of . Then, since our dynamic equations have renormalized coefficients, they can well describe nonequilibrium processes with spatial scales longer than even without the thermal noise terms Halperin (); Kawasaki-Ohta (); Onukibook (). We also neglect Brownian motions of the colloidal particles.

### ii.2 Strong adsorption near criticality

As the critical point is approached, the strong adsorption regime is eventually realized outside CX due to a nonvanishing surface field (see eqn (4) and (5)) Oka-c (); Oka-b (); Jasnow (); Fisher-Yang (); Lawreview (); Liu (). Here, let be positive near the colloid surfaces. If the distance from such a wall is shorter than the bulk correlation length in the case , behaves asJasnow (); Oka-c ()

(2) |

where is a constant and is a microscopic length. The surface value of at satisfies in the strong adsorption condition. Since , the preferential adsorption in the layer grows as per unit area, resulting in the well-known critical adsorption, where the integral in the region is negligible and the strong adsorption limit is well-defined.

For , depends on its value far from the wall. It decays as for at the critical composition, but it changes its sign from positive to negative for . In this paper, we set outside CX, where the thickness of the transition layer is about with enlarged composition disturbances around the particlesOka-c (); Oka-b (). In accord with this result, a light scattering experiment Bey () showed that the adsorption layers of colloidal particles in LW were much thicker near the disfavored branch of CX than near the favored one.

In the literature Krech (); Two (); Ga (); Gam (); Guo (); Bonn (); Nature2008 (); Peter (); Edison (); Die (); Mohry (), the adsorption-induced interaction has been called the critical Casimir interaction. However, the original (quantum) Casimir interaction stems from the ground-state fluctuations of the electromagnetic field between two mirrorsKardar (). In near-critical fluids, more analogous is the interaction arising from the thermal fluctuations of at zero-surface field , where the thermal average is homogeneous. Notice that the fluctuation-induced interactionKardar () is much weaker than the adsorption-induced one (with inhomogeneous )Fisher (); Fisher-Yang (). In fact, the Casimir amplitudes at (in the Neumann boundary condition) Krech1 () are much smaller than those under strong adsorption by one order of magnitude Krech (); Two (); Ga (); Gam (); Up (); Upton (); Oka-c ().

### ii.3 Fluid-particle dynamics (FPD) near criticality

In the FPD method Tanakareview (); Araki (); Furukawa (); Tanaka2 (); Fujitani (), the solid-liquid boundaries are treated as diffuse interfaces with thickness much shorter than the radius , which much simplifies the calculation of colloid motions. We introduce the colloid shape function ( here) with

(3) |

which tends to 1 in the colloid interior and to zero in the colloid exterior. We define even in the colloid interior and assume the total free energy of the form Araki (); Tanaka2 (); Furukawa ()

(4) |

where the integration is within the cell. In the first line, is the free energy density, is a weakly singular positive coefficient, and the term proportional to serves to fix at in the colloid interior. See the appendix for and . In the previous papersAraki (); Tanaka2 (); Furukawa (), the mean field form of has been used. In the second line, is the surface field assuming a large positive number Ross (); Oka-c (); Oka-b () and is an applied potential acting on the particles. In the limit , the second term tends to the usual surface integral and use of eqn (2) yields

(5) |

where is the surface value of . In the strong adsorption condition , we find from (see the appendix).

Assuming a homogeneous , we use the model H dynamics Halperin (); Kawasaki-Ohta (); Onukibook (). The order parameter is governed by

(6) |

where is the velocity field assumed to be incompressible (), is the kinetic coefficient, and is the (generalized) chemical potential. The particle velocities are given by the average interior velocity field,

(7) |

where . Neglecting the acceleration (the Stokes approximation), we determine from

(8) |

where , is a pressure ensuring , and is the viscous stress tensor of the form,

(9) |

with (). In the FPD method, the colloidal particles are treated as a highly viscous fluid. Then, is the viscosity in the liquid and is the viscosity in the colloid interior with . If we use eqn (6)-(9), in eqn (4) monotonically decreases in time () in nonequilibriumOnukibook ().

The kinetic coefficient in eqn (6) is strongly enhanced due to the convective motions of the critical fluctuations Halperin (); Kawasaki-Ohta (); Onukibook (). Because the solvent is close to CX in our situation, we set equal to its value on CX as

(10) |

where is the susceptibility on CX in eqn (A1) and is the diffusion constant in the Stokes form,

(11) |

with . For , this is much smaller than microscopic diffusion constants. In contrast, the viscosity in eqn (9) is only weakly singular and may be treated as a constant independent of and in near-critical fluids.

## Iii Simulation background

In the presence of colloidal particles with radius , we define characteristic values of and by Oka-b ()

(12) |

where the ratio is calculated to be 1.47 in our schemeOka-c (). We consider negative homogeneous with considerably large , so we treat the case . Hereafter, we measure time in our simulation in units of defined by

(13) |

where is the thermal relaxation time of the critical fluctuations Halperin (); Onukibook (). As the parameters in eqn (3) and (4), we set , , , and . The viscosity ratio is .

Let us assume for a LW mixture slightly outside CX, for which K, , and cp. Then, for nm, we obtain , , nm, cms, s, and ms. If we increase to m, becomes s.

We place two particles in the middle of a cylindrical cell with radius and height , so the system is in the region and . The particle centers are at , where is the surface-to-surface separation distance. This geometry is axisymmetric, so the time integration was performed in the 2D - plane, where the mesh size is and the time interval width is . The periodic boundary condition is imposed along the axis, while we set and on the side wall . Then, the total order parameter is conserved in time. As a result, the value of away from the particles, written as , exhibits a slight decrease of order after bridging. We also find that in the colloid interior remains smaller than for our choice of and , which assures the validity of our model.

Even in the axisymmetric geometry, integration of eqs (6) and (7) under eqn (8) is time-consuming. However, if we fix the particle positions, we may efficiently seek the equilibrium profiles of from the relaxation equationYabu (),

(14) |

where with being its value far from the particles. Here, is not coupled. At long times, the stationary solution satisfies .

## Iv Bridging and aggregation dynamics

### iv.1 Situations of bridging

In Fig.1, we illustrate how we performed our simulation outside CX. We initially set far from the particles. For this concentration, the reduced temperature on CX is given by from eqn (1), so we set . We plot the instability line from separated to bridged states (s b) and that from bridged to separated states (b s). To determine these two instability lines, we integrated the relaxation equation (14) at fixed particle positions for various and . We detected growth of small disturbances when we crossed the former (latter) line by gradually decreasing (increasing) from separated (bridged) states. Between these lines, both separated and bridged states remained stationary. Furthermore, between them, there is a first-order transition line, on which the free energy assumes the same value for these two statesOka-b ().

In our simulation, even if , the adsorption-induced force is significant from the beginning. Here, it is long-ranged, decaying as only for , due to the expanded adsorption layers close to the disfavored branch of CX (see Sec.IIA). See such examples in Fig.5 in our paper Oka-b (). Indeed, if the initial attractive force is equated to for case 1 in Fig.1, we have , while at the critical composition Die (); Oka-b ().

In real experiments, charges usually appear on the colloid surfaces in aqueous mixturesBe1 (); Be2 (); Be3 (); Be4 (); Maher (); Guo (); Bonn (); Nature2008 (). Supposing weak ionization, we neglect the effect of charges on the adsorption-induced interaction. However, we will assume a charge-induced repulsive interaction effective at short separation (in eqn (19) below )Russel (). We also neglect the van der Waals interactionRussel (); Ross () in our simulation. We note that Bonn et al.Bonn () observed colloid aggregation in refractive-index-matched systems, where the van der Waals interaction was suppressed.

### iv.2 Bridging after temperature quenching at fixed

In case 1 simulation, at , we lowered from () to () across the (s b) instability line, where outside CX. For , we realized the equilibrium at fixed particle positions as a stationary solution of eqn (14). The correlation length in the bulk is in the final state. Supposing optical tweezersYada (); Kimura (), we choose the potential in eqn (4) as

(15) |

where and are the coordinates of the particle centers and and are their initial values. The spring constant is set equal to , which depends on and is pNm for m. Then, the displacement is after bridging, where the separation is shortened from its initial length by . The time-dependent spring force is given by

(16) |

which is balanced with the force on the particles from the fluid in the Stokes approximationOka-b ().

In Fig.2, we give the initial and final profiles of in gradation. We can see thick adsorption layers covering the colloid surfaces due to large in eqn (4) and (5) and a bridging region with between the particles in the final state. In contrast, if the system would be quenched slightly inside CX, thick wetting layers with sharp interfaces would cover the whole particle surfacesAraki ().

In Fig.3, we display time evolution of vs on (a) the midplane () and (b) the plane containing the upper particle center () far from the other particle. In (a), increases between the particles. Its growth is rapid in the time range . For , it exhibits a flat profile in the middle region and an interface becomes well-defined. In (b), exhibits strong adsorption behavior, which only slightly depends on .

In Fig.4(a), we plot the spring force in eqn (16). It increases rather slowly for , but abruptly for up to the capillary forceOka-b (); Seville (); Butt (),

(17) |

where is the surface tension in eqn (A3). In Sec.I, the capillary force has already been discussed. See the appendix of our previous paper for its more systematic derivationOka-b (). This sharp increase of is simultaneous with the interface formation (see Fig.3(a)). Remarkably, becomes of order before the bridging transition (). As noted at the beginning of this section, the adsorption-induced force is long-ranged in pretransitional states close to the bridging transition.

We are interested in the fluid velocity outside the particles. In Fig.4(b), we thus show the average velocity amplitude in the fluid region defined by

(18) |

where tends to 1 outside the particles. We can see that has a small peak at , a minimum at , and a large sharp peak at . In the initial stage , the adsorption layers of the two particles weakly merge without domain formation, causing a weak convective flow. This initial motion decays for . However, an unstable mode emerges giving rise to an increase in from the center line . It is slowly growing and not noticeable for , but it grows abruptly in the late stage with outward expansion of an interface from the center line. This final acceleration of the growth is obviously due to the hydrodynamic transport.

In Fig.5, we display the profiles of the velocity field in (a) and the concentration flux in (b) at . Here, the vectors and are expressed as and , where and . At , is large in Fig.4(b). The maximum velocity is of order . We can see that (nearly) vanishes on the colloid surface, while is significant within the adsorption layer with . The flow is then directed outward on the midplane due to the incompressibility condition . This squeezing flow is blocked at the side wall at , but the flow in our case is appreciable only for a short time interval and the boundary disturbance should little affect the bridging dynamics. For LW, the typical velocity is estimated as cms for nm.

In Fig.6, we give the -components of the convective flux and the diffusive flux, and , separately, on the midplane at in (a) and 4 in (b). The velocity amplitude is one order of magnitude larger in (b) than in (a). Squeezing can be seen in the region in (b). We recognize that is mostly larger than by one order of magnitude, which demonstrates relevance of the hydrodynamic flow in bridging.

In their simulation, Araki and Tanaka Araki () pinned two particles in a binary mixture and quenched their system inside CX () to realize formation of wetting layers around the particles. They found that the spring force increased above the capillary force transiently in an initial stage for small separation distances. This effect has not yet been understood.

### iv.3 Bridging between aggregating two particles

In cases 2 and 3, we let the two particles approach from the initial separation for , where we set and . Then, we have . For , the particle positions were fixed and the fluid was in equilibrium, which was attained as a stationary solution of eqn (14). For , the motions are caused by the adsorption-mediated attractive interaction. Notice that the initial state is located slightly below the bs instability line in Fig.1. This is because the initial particle motions become extremely slow above the bs line in the present case.

The potential in eqn (4) is repulsive, preventing the particles from touching each other. We assume that it depends on exponentially as

(19) |

where in case 2 and in case 3 with . As a result, we have the final separation distance is in case 2 and in case 3. The above potential can arise for charged colloidal particles when the van der Waals interaction is negligibleBonn (). For weakly charged colloid particles, the screened Coulomb interaction in the Derjaguin, Landau, Verway, and Overbeek (DLVO) theory Russel () is of the form,

(20) |

where is the charge on a colloid particles, is the dielectric constant, and is the Debye wave number. If and , assumes the form in eqn (19) with

(21) |

where is the surface charge density and is the Bjerrum length. For exampleBonn (); Die (), we may set nm and nm to obtain with in nm, so the adopted values of in this paper can be realized in experiments.

In Fig.7, we give the profiles of in the initial and final states in cases 2 and 3. where the particles are bridged in the final state. The adsorption layers outside the bridged region () are nearly stationary as in Fig.2. In the final state in case 3, the adsorption layers of the two particles are not well separated. In Fig.8, we show on the midplane vs at several times. In case 2, becomes fairly flat with a well-defined interface in the middle for . In case 3, it depends on considerably even at the center for . The between the particles increases abruptly in the time range in case 2 and in case 3.

In Fig.9, we show the separation distance and the average velocity amplitude in eqn (18). Here, the maximum of is at in case 2 and is at in case 3. The final approach in case 3 is very fast, where the fluid between the particles is rapidly squeezed out. In Fig.10, we display (a) , (b) , and (c) at , where is the particle velocity with being the unit vector along the axis. The maximum velocity is of order , which is ten times larger than in Fig.5(a). Here, for LW, we have cms for nm. In (b), the adsorption layer moves with the particle and squeezing takes place between the particles. In (c), the convective transport within the adsorption layer is crucial. In Fig.11, we compare and as in Fig.6, which demonstrates that the convective transport dominates over the diffusive transport.

It is rather surprising that the adsorption layer moves with the particle without large deformations in Fig.10(b), though there are no sharp interfaces (see Fig.3(b)). This aspect should be further examined, which should give rise to a decrease in the diffusion constant of suspended particles even due to critical adsorption.

## V Summary and remarks

We have numerically examined dynamics of bridging
and aggregation of two colloidal particles
in a near-critical binary mixture outside CX.
Particularly near the disfavored branch of CX,
the critical adsorption is strong and extended so that
the adsorption-induced interaction is
much enhanced.
As a result, the bridging transition readily
takes place among the colloidal particles when the face-to-face
separation is of the order the particle radius .
We have assumed that is
much longer than the correlation length .
We summarize our main results in the following.

In Sec.II, we have explained the critical behavior, the FPD method, and the model H equations in the Ginzburg-Landau scheme. In Sec.III, we have explained the simulation method and introduced characteristic order parameter , reduced temperature , and time .

In Sec.IV, to induce a bridging transition
in a separated state, we have crossed the instability line
in the - plane by temperature quenching (case 1)
and by two-particle aggregation (cases 2 and 3). In case 1, the spring force has been
calculated as in Fig.4(a), which increases
abruptly up to order upon
formation of a well-defined interface of a bridging domain.
In cases 2 and 3,
the separation
has been obtained as in Fig.9(a), where
the two particles initially approach
due to the adsorption-induced attraction
but eventually stop due to the
screened Coulomb interaction in eqn (19).
In case 2, the final distance
is large and a well-defined bridging domain appears.
In case 3, it is not large
such that the adsorption layers of the two particles overlap.
In the final stage in these cases,
the fluid between the particles is squeezed out,
leading to peaks in the average velocity
amplitude in Fig.9(b). In all these cases,
the convective transport of within the adsorption layers
is crucial.

Finally, we further mention
two experiments, which
suggest the presence of bridging
in aggregation.

(1)
Broide et al.Be5 () studied aggregation kinetics
after a temperature change at ,
where fluidlike aggregates
grew for with inter-particle
separations of order . They observed that
their average volume and average
length were related by
and the time-dependent scattered light intensity exhibited the Porod tailOnukibook ().
These behaviors are very different from those
of fractal aggregates induced
by the van der Waals interaction.

(2)
Guo et al.Guo () observed
a fcc crystal formed by the aggregated particles
with nm, where the lattice constant was nm
and the colloid volume fraction was
0.406. In their case, the corner-to-face distance of the lattice
was and the shortest
face-to-face distance of the particles was nm.
This separation distance
is much longer than ( nm).
We thus propose a bridging scenario of
crystal formation.

Acknowledgments. This work was supported by KAKENHI (No. 25610122) and Grants-in-Aid for Japan Society for Promotion of Science (JSPS) Fellows (Grants Nos. 241799 and 263111).

Appendix: Renormalized free energy density

To account for the statics
in near-critical fluids, we use the renormalized
local functional theoryFisher-Yang (); Up (); Upton (); Oka-c (); Oka-b ().
As in the linear parametric model by Schofield et al.Sch (),
we introduce a distance from the critical point in the -
plane, in terms of which the overall critical behavior can be
approximately described. We define by Oka-c ()

(A1) |

where is a constant. The coefficient in eqn (4) is set equal to , where is a constant and is the Fisher exponent.

The free energy density is determined such that the combination is a universal function of in accord with the two-scale factor universalityOnukibook (); Aliu (). Then, we obtain the correct exponent relations from the free energy , but the critical amplitude ratios Onukibook (); Aliu () are approximate depending on the choice of . In this paper, we use the choice in our previous workOka-c (). Then, we obtain , while it is 1.9 from numerical analysis of the Ising criticality Aliu ().

In the original Schofield modelSch (), the behavior of inside CX was not discussed. We assume inside CX (in the region and ) in the form,

(A2) |

where and on CX. Here, with . From eqn (4) and (A2) the surface tension is calculated as Oka-c ()

(A3) |

where is the correlation length on CX. See also Fig.3, where much exceeds near the colloid surfaces and tends to a value close to in the bulk. Thus, use of eqn (A2) is essential in our theory.

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