# Attractive interaction and
bridging transition between neutral colloidal particles

due to preferential adsorption
in a near-critical binary mixture

###### Abstract

We examine the solvent-mediated interaction between two neutral colloidal particles due to preferential adsorption in a near-critical binary mixture. We take into account the renormalization effect due to the critical fluctuations using the recent local functional theory J. Chem. Phys. 136, 114704 (2012). We calculate the free energy and the force between two colloidal particles as functions of the temperature , the composition far from the colloidal particles , and the colloid separation . The interaction is much enhanced when the component favored by the colloid surfaces is poor in the reservoir. For such off-critical compositions, we find a surface of a first-order bridging transition in the -- space in a universal, scaled form, across which a discontinuous change occurs between separated and bridged states. This surface starts from the bulk coexistence surface (CX) and ends at a bridging critical line . On approaching the critical line, the discontinuity vanishes and the derivatives of the force with respect to and both diverge. Furthermore, bridged states continuously change into separated states if (or ) is varied from a value on CX to value far from CX with kept smaller than .

###### pacs:

64.70.pv,68.35.Rh,05.70.Jk,64.75.Xc## I Introduction

Much attention has been paid to the physics of fluids in restricted geometries Evansreview ; Gelb . The microscopic interactions between the fluid molecules and the solid surface can greatly influence the phase transition behavior of the confined fluid Is . The liquid phase is usually favored by the walls in fluids undergoing gas-liquid phase separation, while one component is preferentially attracted to the walls in binary mixtures. In the film geometry, narrow regions may be filled with the phase favored by the confining walls or may hold some fraction of the disfavored phase. Between these two states, there can be a first-order phase transition, called capillary condensation Evansreview ; Gelb ; Butt , depending on the temperature , and the reservoir chemical potential for each given wall separation . This phenomenon occurs both in one-component fluids and binary mixtures.

As another aspect, adsorption-induced density or composition disturbances are known to produce an attractive interaction between solid objects Hansen ; Evans-Hop . In binary mixtures, it is amplified when the solvent far from these objects is poor in the component favored by the surfaces Evans-Hop . Such solvent-mediate interactions should play an important role in reversible aggregation of colloidal particles in near-critical binary mixtures at off-critical compositions Beysens ; Maher ; Bonn ; Guo . In such situations, strong preferenial adsorption was observed by light scattering Beysens . It is worth noting that the colloid-wall interaction in a near-critical fluid has been measured directly Nature2008 ; Nellen . We mention some theoretical papers, which treated the solvent-mediated colloid interaction in an early stage Slu ; Two ; Lowen ; Netz ; Kaler .

However, other interactions come into play in real systems. First, we should account for the van der Waals (dispersion) interaction, which sometimes gives rise to intriguing effects in wetting behavior Is ; Butt ; Bonnreview ; Russel . In this paper, we examine importance of the van der Waals interaction as compared to the adsorption-induced interaction. Second, in aqueous fluids, the colloid surface can be ionized and the counterions and added ions form an electric double layer, resulting in the screened Coulomb interaction Is ; Butt ; Russel . This repulsive interaction can be very strong close to the surface, but it decays exponentially with the Debye screening length . Third, in near-critical fluids, the ion distributions and the critical fluctuations become highly heterogeneous around the colloid surfaces Okamoto . As a result, the wetting layer formation and the surface ionization are strongly coupled, which much complicates the colloid interaction.

On approaching the solvent criticality, the adsorption-induced interaction becomes long-ranged and universal Okamoto ; Fisher ; Fisher-Yang ; Gamb ; Upton , where the wall-induced heterogeneities extend over mesoscopic length scales. In the film geometry, some universal scaling relations are well-known and considerable efforts have been made to calculate Gamb ; Upton or measure Law ; Nature2008 ; Gamb the so-called Casimir amplitudes (coefficients in universal relations) Casimir , In these papers, near-critical fluids at the critical composition have mostly been treated along the critical path . On the other hand, Maciołek et al Evans-Anna found strong enhancement of one of the amplitudes in two-dimensional Ising films under applied magnetic field. In accord with their finding, we have recently found growing of the amplitudes at off-critical compositions OkamotoCasimir , which is particularly marked near a first-order capillary condensation line in the - plane. We have also examined phase separation dynamics around the capillary condensation line Yabunaka .

In this paper, we aim to investigate the interaction between two neutral colloidal particles due to preferential adsorption in a near-critical binary mixture. We shall see that the solvent-mediated interaction is much enhanced when the component favored by the colloid surfaces is poor in the reservoir, as in the case of the Casimir amplitudes. We also aim to examine the bridging transition between two colloidal particles Butt ; Butt1 , which is analogous to the capillary condensation transition in a film. That is, two large particles (or one large particle and a plate) are connected by the phase favored by the walls in bridged states, while they are disconnected by intrusion of the disfavored phase in separated states. Bridged states appear near the bulk coexistence curve as the separation distance is decreased. As previous papers on bridging, we mention numerical calculations of phenomenological models Yeomans ; Vino , density functional theories Bauer ; Evans-Hop , and a Monte Carlo study Higashi . We also note that a bubble bridging can occur between hydrophobic surfaces in water bubble , which is related to predrying of hydrophobic surfaces Teshi . Similarly, in the isotropic phase of liquid crystals, a nematic domain can appear between closely separated solid objects Zu ; Fukuda .

The organization of this paper is as follows. In Sec.II, we will summarize the results of the local functional theory of near-critical binary mixtures. In Sec.III, we will present a theory on the adsorption-induced interaction among colloidal particles together with some simulation results. In Sec.IV, we will numerically investigate the bridging transition near the bulk criticality.

## Ii Renormalized Ginzburg-Landau free energy

We consider near-critical binary mixtures using our local functional theory taking into account the renormalization effect near the bulk criticality, which is similar to the linear parametric model by Schofield et al. Sc69 ; Onukibook and the local functional model by Fisher et al. Upton ; Fisher-Yang . These authors treated near-critical fluids outside CX, while we define our model within CX. Furthermore, our model satisfies the two-scale-factor universalityOnukibook . The critical amplitude ratios from our model are in fair agreement with reliable estimates for Ising systems.

We assume an upper critical solution temperature at a given average pressure. The order parameter is proportional to , where is the composition and is its critical value. The physical quantities exhibit singular dependence on and the reduced temperature,

(2.1) |

Hereafter, ,, , , , and are the usual critical exponents for Ising-like systems Onukibook . At the critical composition with , the correlation length is written as , where is a microscopic length. The coexistence curve in the region is denoted by CX. We write in the coexisting two phases as with

(2.2) |

where is a constant.

We set up the singular bulk free energy , where the critical fluctuations with wave numbers larger than the inverse correlation length have been coarse-grained or renormalized. Including the square gradient term, is of the local functional form Fisher-Yang ; Upton ; OkamotoCasimir ,

(2.3) |

where the integral is within a cell. Outside CX (), the singular free energy density is written in the Ginzburg-Landau form,

(2.4) |

We do not write a constant term (),which is a singular contribution for . In this paper, is made dimensionless. Then, is dimensionless and in Eq.(2.2) is of order . In the mean field theory, , , and in are constants independent of and . In our renormalized functional theory, they depend on a nonnegative variable representing the distance from the criticality in the - plane. Outside CX, fractional powers of appear as Casimir-comment

(2.5) | |||||

(2.6) | |||||

(2.7) |

where is a universal number and is set equal to in our numerical analysis.

From , we have . We determine as a function of and by

(2.8) |

For , we simply have . For , we obtain , leading to the Fisher-Yang results Fisher-Yang : and . These authors introduced the local correlation length for .

In our scheme, and the susceptibility are related to the second derivative by

(2.9) |

For and , we find . On approaching CX (), we require to obtain and . The susceptibility on CX is determined by and is written as

(2.10) |

with . The correlation length on CX is written as .

We also need to determine inside CX ( and ) to discuss phase separation. Its simplest form is

(2.11) |

where is the free energy density on CX. Then, , , and are continuous across CX. We also set inside CX, which is the value of in Eq.(2.5) on CX. Here, we neglect the thermal fluctuations longer than . In our applications, the space regions inside CX are not wider than and the form in Eq.(2.11) is well justified. As an example, we may calculate the interface profile from Eqs.(2.3) and (2.11), where the surface tension is of the form Onukibook ,

(2.12) |

We shall see another example inside CX in Fig.1.

## Iii Colloidal particles in a near-critical fluid

We consider identical colloidal particles with common radius much larger than in a near-critical binary mixture. We seek equilibrium profiles of around these large particles. We assume far from them, where is proportional to the composition deviation far from the colloidal particles. In its calculation, we take the limit of strong preferential adsorption. This minimizes the grand potential , giving rise to attraction among the colloidal particles Slu ; Two ; Lowen ; Netz ; Okamoto . Typical reduced temperatures in this paper are from to in units of and are very small for large . Then the prewetting transition Bonnreview may be assumed to occur at lower temperatures. In fact, we realize thick adsorption layers in our numerical analysis.

### iii.1 Equilibrium relations

On the cell surface we assume for simplicity, but on the colloid surfaces we assume

(3.1) |

where is the normal unit vector from the interior to the exterior and is a large positive surface field arising from the short-range, fluid-surface interaction. In equilibrium, we minimize the grand potential, consisting of the bulk term and the surface term as

(3.2) |

Hereafter, is the space integral outside the colloidal particles and in the cell, while is the surface integral on the colloid surfaces. We define the grand potential density including the gradient contribution,

(3.3) |

where and is related to by

(3.4) |

In particular, close to the negative branch of CX. The is nonnegative in our case, tending to far from the colloidal particles. Minimization of yields Eq.(3.1) as the boundary condition and

(3.5) |

in the fluid region, where .

In equilibrium, is a function of the colloid centers (). In Appendix A, we will derive the following equilibrium relation,

(3.6) |

where . The integral is on the surface of the -th colloidal particle and is the normal unit vector from the colloid interior to the exterior. The is the stress tensor due to the order parameter deviations given by Onukibook

(3.7) |

This tensor satisfies the relation,

(3.8) |

which vanishes in equilibrium or under Eq.(3.5). Here, far from the colloidal particles with

(3.9) |

If we further use Eq.(3.5), we obtain a simpler expression,

(3.10) |

The expression (3.7) and the relation (3.8) are valid even in nonequilibrium and have in fact been used in dynamicsOnukibook ; Yabunaka . Note that the total stress tensor may be expressed as in binary mixtures, where is a large background pressure nearly uniform in the cell (with small variations arising from sounds and gravity).

### iii.2 Scaling and strong adsorption limit

We make Eq.(3.5) dimensionless by scaling the position by and by , where is is a characteristic order parameter around the colloidal particles of the form,

(3.11) |

Use of in Eq.(2.2) gives . By scaling and , we introduce two parameters,

(3.12) | |||

(3.13) |

The scaled correlation length is given by for on the critical path, for , and on CX. The CX curve is expressed as with from Eq.(2.2). In our calculations, we may use the scaled quantities only, where we need not specify the ratio . The scaling factors and . are needed when our theoretical results are compared with experimental data. For example, if , they are and , respectively.

We write the value of on the colloid surfaces as . For sufficiently large , the near-wall behaviors of and are expressed as OkamotoCasimir ; Rudnick ; Fisher-Yang

(3.14) | |||

(3.15) |

where is the distance from such a surface. We here assume that is shorter than the correlation length far from the surface. The length is of the order of the local correlation length near the surface () Fisher-Yang . In terms of in Eq.(2.2), we have

(3.16) |

where we assume so . In terms of , we also have . For , and become independent of or . From Eq.(3.1), we obtain the scaling relation,

(3.17) |

where . The strong adsorption condition is realized with increasing or on approaching the bulk criticality. In our numerical analysis, we assume to obtain . See Fig.1 for the near-wall behaviors of and in the strong adsorption.

The integral of in the near-wall layers with is proportional to and becomes negligible for large (since ), while that in the region grows as (critical adsorption) Bonnreview . It follows a well-defined preferential adsorption,

(3.18) |

which is independent of for large . On the other hand, the integral of in the layers with and the surface free energy in Eq.(3.2) ( are both proportional to and are large in magnitude. However, they are constants nearly independent of and and are irrelevant in the capillary condensation and the bridging transition (see discussions below Eq.(3.24)), which much simplifies our results.

In the strong adsorption regime, the profile of is highly nontrivial for negative , since changes from a large positive value near the surface to far from it. To illustrate this aspect, we here consider the simplest case of a single spherical particle curved , where is a function of the distance from the particle center. In this case, if approaches the CX value under the condition , the thickness of the adsorption layer increases logarithmically with increasing as curved

(3.19) |

It is also known that the contribution to from the transition region () is of order , where is the surface tension in Eq.(2.12).

In Fig.1, and are displayed around a single colloidal particle for , where on the positive and negative branches of CX.. For , the adsorption layer thickness is . For , it is thicker than by a few times and is of order in Eq.(2.19). Furthermore, for , exhibits a peak around with its area being about . However, the peak recedes and diminishes for smaller ( and ).

### iii.3 Two colloidal particles

As in Fig.2, we consider two colloidal particles with equal radius . In our numerical analysis, they are placed in the middle of a cylindrical cell with radius and height . The system is then in the region and . The particle centers are at with being the surface-to-surface separation distance. Hereafter, we set

(3.20) |

When the system lengths ( and ) much exceed , it is convenient to write as

(3.21) |

where is the value of for . That is, if is the grand potential for one isolated colloidal particle, we have . The dimensionless quantity is a universal function of , , and decaying to 0 for large . Note that it is independent of in the strong adsorption limit, as discussed in Subsec.IIIB. The adsorption-induced force between the two colloidal particles is given by

(3.22) |

The dimensionless functions and are related by

(3.23) |

We also have . In the derivative and the integral with respect to , and are fixed.

From the calculations in Appendix A, the normalized force is expressed in a convenient form,

(3.24) |

where the integral is on the plane with (the midplane between the two colloidal particles). From the geometrical symmetry, on this plane, we may set from Eq.(3.10). Also the integral may be replaced by , since depends only on . If , the midplane is far from the transition layers with thickness and becomes independent of or . In this paper, we thus calculate from Eq.(3.24).

Notice that we may use Eq.(3.15) on the midplane between the two colloidal particles for small mid , where we set with . In Eq.(3.24), the integral in the range then becomes

(3.25) |

To be precise, Eq.(3.24) yields as the coefficient in Eq.(3.25). In Appendix B, the Derjaguin approximation Is ; Butt ; Russel for small will yield

(3.26) |

with . The coefficient is somewhat larger than that from Eq.(3.24). This small- behavior stems from the de Gennes-Fisher theory for near-critical films Fisher ; Gamb ; Nature2008 . Furthermore, in Appendix B, we shall see that if exceeds the correlation length without bridging, and decay exponentially as

(3.27) |

where is determined by and from Eq.(2.9). These relations follow in separated states if the midpoint value of at is close to OkamotoCasimir .

### iii.4 Numerical results without bridging transition

In Figs.3-6, we present numerical results where there is no bridging transition. We aim to show that and are much more enhanced for than for .

In the left panel of Fig.3, we show curves of vs calculated from Eq.(3.24) and those from the Derjaguin approximation for and . They tend to a constant as as in Eqs.(3.25) and (3.26). Remarkably, for and , increases up to of order 10 to exhibit a peak as a function of , where the peak position is at from Eq.(B10). On the other hand, for and , exhibits only a rounded maximum of order 1 at from Eq.(B12). We recognize that the force is much enhanced for negative and the Derjaguin approximation nicely holds for . In the right panel of Fig.3, we present in gradation for in the plane, where is large in the region between the two colloidal particles.

In Fig.4, for , we show and vs at . We change as , 0.4, 0.2, and 0.008. For , the two collloidal particles are so separated such that resulting in a small . On the other curves of smaller , decreases from positive to negative with increasing and increases dramatically up to 219. The behavior of the latter curves are consistent with the theoretical expressions: and at , which follow from Eqs.(3.14) and (3.15) with as in Eq.(3.25).

Next, we plot and vs for six values of at in Fig.5 and for five values of with in Fig.6 on semi-logarithmic scales. In these examples, there is no bridging transition for any . For small , we have the behaviors in Eq.(3.26). For relatively large (), both and decay exponentially as . We confirm that the slopes of these curves are close to for , where is calculated from Eq.(2.9). We can again see that is well approximated by the Derjaguin approximation for .

### iii.5 Van der Waals interaction

So far, we have not explicitly accounted for the pairwise van der Waals interaction Is among constituent molecules, which was treated as one of the main elements causing colloid aggregation Beysens ; Petit . The resultant potential between two colloidal particles with equal radius is written as Butt ; Russel

(3.28) |

where is the center-to-center distance. The Hamaker constant is in many cases of order J, but it can change its sign Is ; Bonnreview and can be very small for some systems of colloids and binary mixtures Bonn . Without charges, the total potential is of the form,

(3.29) |

consisting of the adsorption-induced part and the van der Waals part. The former is very sensitive to and in the critical ranges, while the latter is insensitive to them. If we further include the charge effects, we should add an appropriate chrage-induced interaction in Eq.(3.29) Nature2008 ; Okamoto ; Bonn ; Beysens ; Gamb (see item (3) in Sec.V for more discussions).

The force from the van der Waals interaction reads

(3.30) |

As , we find This behavior is the same as that of in Eq.(3.26). So we compare the coefficients in front of the power of the two forces, and , to obtain the ratio,

(3.31) |

where the denominator is J for K. If is smaller than J, we have and the van der Waals interaction is weaker than the adsorption-induced interaction at least for small .

However, grows for with increasing as in Fig.3, so we need to examine the relative importance of the van der Waals interaction and the adsorption-induced interaction for larger . To this end, in Fig.7, we plot for four typical cases together with

(3.32) |

In Fig.7, while all the curves start from unity for , the normalized quantity increases up to a maximum about 10 for without bridging formation and can even be of order 100 close to a bridging transition with increasing . Thus, at an off-critical composition with , the adsorption-induced interaction can well dominate over the van der Waals interaction (even for J).

## Iv Bridging transition between two colloidal particles

In this section, we study the bridging transition for between two colloidal particles in a near-critical binary mixture. In our case, assumes the profiles of in Fig.1 in separated states, where the adsorption layer has a thickness of order in Eq.(3.19). A bridging transition can then occur in a wide range of () under the condition . In the previous papers Vino ; Higashi ; Bauer ; Yeomans ; Evans-Hop , bridging between two spheres or between a sphere and a plate were studied numerically for small separation (say ) far from the criticality.

### iv.1 Phase diagrams

In Fig.8, we first show a phase diagram of the bridging transition in the -- space outside CX. We find a surface of a first-order bridging transition, bounded by CX and a bridging critical line. As functions of , the normalized separation may be written on the transition surface and on the critical line as

(4.1) |

respectively. Across this surface, discontinuities appear in and the adsorption in Eq.(3.18), which tend to vanish on approaching the critical line. The critical line tangentially ends on CX at . The maximum of at a transition is thus 2.6.

In Fig.9, phase diagrams in the - and - planes are presented, where is related to by Eq.(3.4) and scaled by . In these phase diagrams, a first-order bridging transition occurs at some in the region between CX and the bridging critical line, where the latter approaches CX tangentially. We also write cross-sectional bridging transition lines at fixed (equal to and ) on the bridging transition surface, each starting from CX and ending at a point on the critical line. In our case, these lines are nearly straight in the two phase diagrams in Fig.9. Previously, bridging transition lines at fixed separation were drawn Yeomans ; Bauer ; Gamb . For near-critical films, on the other hand, the capillary condensation line is detached from CX. As a result, it is considerably curved in the - plane OkamotoCasimir , but is nearly straight in the - plane Yabunaka .

The phase behavior at fixed separation is particularly intriguing. In the left panel of Fig.10, we show a phase diagram in the - plane, where we write the critical line and the transition line on CX. The latter is defined by

(4.2) |

where is the vallue of on the negative branch of CX. These two lines merge at on CX. Then, let us vary at fixed and . (i) If , separated states are realized without bridging for any . (ii) If ,we encounter the transition surface at a certain to find a discontinuous change. (iii) For , a bridging domain appears with a well-defined interface close to CX, but disconnection occurs continuously with incresaing the distance from CX. In this changeover, it is puzzling how the interface becomes ill-defined gradually (see Fig.17).

In the right panel of Fig.10, we plot the bridging radius vs at for , and , which correspond to the three marked points in the top panel of Fig.9. We determine from the condition at , where changes from positive to negative at with a bridging domain in the range . As a function of at each , is shortest at the transition and increases with decreasing . It is about for sufficiently small . Also it is smaller near the critical line. In fact, at the transition for .

The transition surface is determined from minimization of or maximization of from Eq.(3.21). In the left panels of Fig.11, we plot and vs for . The curve of from the Derjaguin approximation nicely agrees with that from Eq.(3.24) for . For this (, we find two stationary solutions satisfying Eqs.(3.1) and (3.5) in a window range ( for this example). Outside this range, one solution becomes unstable and the other one remains as a stable solution. In the bistable range, is larger on the equilibrium branch and smaller on the metastable one, so the transition is at the crosspoint of the two branches of .

In Fig.11, the slope of is very steep with bridging. It is at the transition, where . It is further amplified for smaller and is at . Here, for , a well-defined bridging domain exists and changes with a change of its surface area. In fact, use of the surface tension in Eq.(2.12) gives at . Thus, with a well-defined bridge, Eq.(3.24) yields the capillary force Butt ; Butt1 ,

(4.3) |

This relation is valid for . For smaller , the growth () in Eq.(3.25) becomes dominant. These features will be further examined in Figs.12, 13, and 16.

In the original units, the force with a well-defined bridge is of order , which increases as we move away from the bulk criticality. Also in Fig.14 below, we shall see that increases with lowering , where bridging occurs continuously. However, the exponential tail of the interaction in Eq.(3.27) in separated states () increases as the bulk criticality is approached, which was indeed observed experimentally Nature2008 .

In the right panels of Fig.11, we display in the plane of and in two bridged and one separated states at different . We can see that the bridging radius is larger in (a) (far below the bridging transition) than in (b) (close to it). The midplane between the two particles is filled with the phase outside the particles in (c) in a separated state.

In these phase diagrams the lowest value of is . With further loweing