# Critical Casimir Interactions Between Spherical Particles in the Presence of the Bulk Ordering Fields.

###### Abstract

The spatial suppression of order parameter fluctuations in a critical media produces Critical Casimir forces acting on confining surfaces. This scenario is realized in a critical binary mixture near the demixing transition point that corresponds to the second order phase transition of the Ising universality class. Due to these critical interactions similar colloids, immersed in a critical binary mixture near the consolute point, exhibit attraction. The numerical method for computation of the interaction potential between two spherical particles using Monte Carlo simulations for the Ising model is proposed. This method is based on the integration of the local magnetization over the applied local magnetic field. For the stronger interaction the concentration of the component of the mixture that does not wet colloidal particles, should be larger, than the critical concentration. The strongest amplitude of the interactions is observed below the critical point.

###### pacs:

05.50.+q, 05.70.Jk, 05.10.Ln## I Introduction

In 1948 Hendrick Casimir predicted that in the vacuum between two parallel perfectly conducting plates an attractive force appears Casimir (). This force is caused by the suppression of the zero level quantum fluctuations of the electromagnetic field in the space between plates. This phenomena is known as the quantum Casimir effect.

In the vicinity of the second-order phase transition in the critical media long-range fluctuations of the order parameter arise. This phenomenon is observed, e.g., in the critical liquid binary mixture at the demixing point. Fisher and de Gennes predicted FdG (), that the confinement of these fluctuations produces effective forces acting on confining surfaces. The appearance of forces due to spatial suppression of fluctuations of an order parameter in a critical media is now known as the Critical Casimir (CC) effect Krech (); BDT (); Gambassi ().

The phenomenon of colloidal particle aggregation in the critical binary mixture was first reported in BE (). In the planar geometry the CC effect for critical binary mixtures measured experimentally via the influence on the thickness of the liquid wetting films Fukuto (). In this case the confining parallel surfaces are substrate-liquid and liquid-vapor interfaces. Later on, interaction forces between a colloidal particle and a flat substrate were measured directly nature (); PRE1 (); Nellen (). Critical depletion in colloidal suspensions was studied experimentally BCPP1 (); BCPP2 (). The colloidal aggregation in microgravity conditions, caused by CC interaction, was described in Ref MKG (). The controlled phase transition in colloidal suspension in the critical binary mixture was studied in NFHW (). In this article the interaction potential between colloidal particles was extracted from the pair correlation function. From the experimental point of view CC interactions provide the possibility of tuning an interaction between colloidal particles. By varying the temperature of the binary mixture in the vicinity of the consolute point it is possible to switch on interactions between colloids in controllable and reversible way.

The critical binary mixture consists of components A and B (with concentrations and , respectively) with the critical concentration and the critical temperature . The schematic phase diagram with the lower critical point (that corresponds to the water-lutidine mixture used in experiments BE (); nature (); PRE1 (); Nellen ()) is shown in Fig. 1(a). The state of such a system is characterized by the reduced temperature and chemical potentials , for the two components A and B with corresponding values , at criticality. It is convenient to represent chemical potentials as a combination of (which plays a role of the bulk ordering field) and (which describes the deviation of chemical potential for both components from the critical values). In the most general case, in the vicinity of the critical point the state of the binary liquid mixture is characterized by two scaling fields that are linear combinations of these three variables , , and (see Wilding () for detailed description).

A critical binary mixture belongs to the universality class of the Ising model which state is characterized by the reduced temperature and the bulk magnetic field . We consider the potential difference that is proportional to the bulk field and equal values of the reduced temperatures .

In accordance with the scaling theory Barber (); Privman () the CC interactions are characterized by the ratio of the linear size of the system and the bulk correlation length which is the function of the reduced temperature and the bulk field . For correct interpretation of experimental results we need information about CC interactions of colloids for the three-dimensional (3D) Ising universality class.

The CC force and its scaling function of the 3D Ising universality class for the film geometry and various boundary conditions were studied numerically without the bulk field EPL (); hucht (); PRE (); hasen1 (). Recently, Monte Carlo (MC) simulation results for the plane geometry with the bulk field were obtained Myhb (); Arxiv_exp_prot (). Results for the CC forces between a spherical particle and a plane for the 3D Ising universality class without the bulk field are published in Hsp (). The CC force between two colloidal particles for the Mean Field (MF) universality class was first studied in SHD () using the conformal transformation. Without the bulk field MF interactions between an elliptic particle and a wall were studied in KHD () and multi-particle interactions were studied in THD (). Recently, the results for CC force between two colloidal particles in the presence of the bulk ordering field for the MF universality class were published KM (). Results for CC force between two disks for the two-dimensional (2D) Ising model with the bulk field were obtained via the Derjuaguin approximation ZMD (). The alternative method that uses the distribution function of a mobile object position for the computation of the CC interaction was recently proposed HH (). In that article the CC interaction potential between a disk and a wall was computed for 2D geometry.

In the present paper we propose the numerical method for the direct computation of CC interactions between particles for a 3D Ising model with the bulk ordering field. We present results for the interaction potential for two particles as a function of the bulk field at fixed temperatures and as functions of the temperature for fixed values of . The paper is organized as follows: in the second section we describe the numerical method. In the third section results of the MC simulation for the interaction energy between two particles are presented. The last section is the conclusion.

## Ii Method

We consider the Ising model on a simple cubic lattice with periodic boundary conditions, all distances are measured in lattice units. The system size is . In a site of the lattice the classical spin is located. The inverse temperature is . Our aim is to study the interaction between colloidal particles immersed in the critical binary mixture. Therefore we need the lattice representation of colloidal particles. The idea proposed by Martin Hasenbusch Hsp () is to draw a sphere of a certain radius around a selected spin. Then all spins within the sphere are considered to belong to the colloidal particle and fixed to be . In Fig. 1(b) we plot a cross section of a sphere of the radius , spins inside the sphere are denoted by filled squares. We consider the case of very strong positive surface fields for colloids. This choice corresponds to the symmetry-breaking Boundary Conditions (BCs) with completely ordered surface and usually denoted as BC (see Surf () for details). It means, that a neighbor spin , that is in a contact with a particle surface will be frozen ; such spins are denoted by filled circles. Let us denote as the set of all frozen spins in the system (spins in both colloidal particles and their neighbors, totally spins) and refer to this set as spins of colloidal particles. These spins are shown by filled symbols in Fig. 1(b). Fluctuating spins in the bulk are denoted by empty circles.

Let us denote as the free energy of an empty bulk system (see Fig. 1(c) top) with the standard Hamiltonian for a spin configuration

(1) |

where is the interaction constant, is the bulk magnetic field, the sum is taken over all neighbor spins, the sum over is taken over all spins of the spin configuration . The free energy of the system is expressed via the sum over all possible spin configurations as . The system with two colloidal particles of a radius at a distance (see Fig. 1(c) middle) is described by the same Hamiltonian Eq.(1). But all spins of colloidal particles and their neighbors should be frozen , , so the free energy is

(2) |

Here the product of the Dirac delta functions fixes the values of spins in colloidal particles to be . In this expression for a free energy we also count the interaction between frozen spins within particles. Let us consider the system with the Hamiltonian

(3) |

where the additional external local magnetic field is applied to spins of colloidal particles (see Fig. 1(c) bottom). The free energy of this system is given by the formula

(4) |

For zero additional field this free energy equals the free energy of the system without particles . We consider systems with certain bulk field at fixed inverse temperature . Therefore in this section we omit arguments of functions for the simplicity of notations. For a very strong additional field it has a limit , where is the total number of spins in the colloidal particle , because these spins became frozen by the local field . Let us introduce the variable . Then the magnetization of spins in colloids is expressed via the derivative of the free energy with respect to :

(5) |

Introducing the normalized (per total number of spins in particles) particle magnetization , we can express the free energy via an integral over the magnetization

(6) |

Selecting some big enough maximal value of the additional field we can express the free energy of the system with colloidal particles as

(7) |

The particle magnetization at zero additional field equals the bulk magnetization and it is equal to 1 at strong field . For this reason the result of the integration in Eq.(7) does not depend on the upper limit of the integration for big enough (we use the value ). In Fig. 1(d) we schematically plot the magnetization for the case of the negative bulk magnetic field . Graphically, the “insertion” free energy equals the area between lines and 1.

Our final aim is to compute the potential of the Casimir force between two quasi-spherical particles at the distance expressed in units . Up to a certain constant this potential may be expressed via the free energy . We select this constant equal to the value (with the sign “”) of the free energy at some maximal separation : . Therefore Graphically, in Fig. 1(d), this function equals the area between lines and with the minus sign. This method is optimized for the computation of the potential of the Casimir interaction . For the computation of the Casimir force between two particles it would be preferable to use the modification of the proposed method in which we interpolate between two configurations for distances and by varying the local field .

## Iii Results

We perform numerical simulations for the system of the size . Two quasi-spherical particles of the radius are located at separation along the direction (see the cross section in Fig. 1(c)). For separation the particles are in the contact. The separation is the maximal possible interparticle separation in direction for this system. For accurate integration over the particle magnetization we use the histogram reweighting technique FS (); LB (). The probability distribution of the particle magnetization is proportional to the exponent . We compute this probability distribution for 16 values of the additional field 0,0.01,0.02,0.03,0.04, 0.05,0.07,0.1,0.16,0.23,0.4,0.5,0.7,1,1.5,2.5. The probability distribution for the value of the field may be expressed as

(8) |

where the normalization constant and the values of fields should be close enough to let the probability distributions intersect. In Fig. 2(a) we plot the probability as a function of for the set of reference points for , . In Fig. 2(b) we plot the magnetization as a function of for and various values of . For the curve for we denote by triangles values , for which the distribution in Fig. 2(a) is computed.

In accordance with the scaling concept the CC interactions between two similar colloidal particles of radius at distance at temperature , and the value of the bulk field are characterized by three variables: , and the bulk correlation length . Here is the reduced temperature ( is the inverse temperature). For the 3D Ising model the value of the critical inverse temperature is RZW (). In the general case the correlation length is an unknown function of the reduced temperature and of the bulk field . But for zero magnetic field the correlation length is and at the critical temperature the correlation length is where the value of the correlation length critical exponent is Hasnu (), PV () and critical amplitudes are EFS (), , and RZW ().

In the present paper we study two cases: the constant magnetic field and various temperatures and constant temperatures and various values of the magnetic field. In the first case we choose the scaling variable as an argument of the function because in the case of the variable for different values of we should perform computations for different temperatures (an alternative choice is the ratio , for this scaling variable the function is “stretched” in the vicinity of zero). The second reason for this choice is that it let us include the distance (when particles touch each other) into consideration. In the presence of the bulk ordering field, critical fluctuations on the system size scale should be suppressed, therefore in the present paper we do not study the influence of the system size.

In Figs. 3(a)-3(c) we plot the interaction potential as a function of the scaling variable for separations and values of the bulk field , respectively. In the case of zero bulk field Fig. 3(a), the attractive potential has a pronounced minimum in the vicinity of the critical point . For the negative value of the bulk field the amplitude of the attractive interaction increases several times. For big enough separations the width of the interaction potential well with respect to becomes very big. For shorter separations the minimum of the interaction disappears and the interaction within the investigated range has no minimum. The strongest interaction corresponds to the smallest value of . In Fig. 3(d) we plot the energy difference as a function of for separations with respect to maximal separation (the same maximal separation is used for the computation of the interaction potential ).

In Figs. 4(a)-4(c) we plot the CC interaction potential as a function of the scaling variable for various separations and temperatures (above , at , and below , respectively). In Fig. 4(d) we plot the magnetization profile as a function of coordinates for the value of the inverse temperature (the corresponding value of the scaling variable ) and the value of the magnetic field (the value of the scaling variable ) using the colormap. We observe, that for the interaction potential has a minimum as a function of . The depth of this minimum decreases with increasing separation . Above the minimum is smooth and is shifted for stronger negative values of . Below the minima become sharp and narrow, shifted to smaller (in the amplitude) values of the negative field . In Fig. 4(d) we observe for , (, ) the formation of the bridge of positive spins (which corresponds to component of the binary mixture) for small separations . For larger separation the bridge disappears. That correlates with the presence of an attractive potential in Fig. 4(c) for and and the absence of attraction for . It means, that the strong attraction for in Figs. 3(b) and 3(c) for and in Fig. 4(c) for is produced by the formation of the bridge of positive spins. This is confirmed by the energy difference in Fig. 3(d), which has the noticeable minimum for . It corresponds to the total decreasing of the area of the interface below due to the formation of the bridge. For the energy difference has no pronounced minimum, in this case the bridge is absent.

## Iv Conclusion

A numerical method for the computation of the potential of the CC interaction between particles immersed in the critical media is proposed. This method provides results for the 3D Ising universality class in the presence of non-zero bulk ordering field. The potential energy difference for two interparticle distances and has a simple graphical representation and is proportional to the area between graphs of the local magnetization for these two separations. We compute the interaction potential as a function of the temperature scaling variable for fixed values of the bulk ordering field and vice versa, as a function of the bulk field scaling variable for fixed temperatures. The strongest interaction for particles with boundary conditions (for colloidal particles with the surface that has a preference to component A) is observed for negative bulk fields (B-rich phase of the binary mixture) below the critical point (above the lower critical point in the phase diagram Fig. 1(a)). This aggregation region is shown in Fig. 1(a) (as observed in BE ()). For a small interparticle distances we observe the formation of the bridge of phase between particles that produces forces acting far away from criticality. As a result of the computation the interaction potential between two colloidal particles is provided that is convenient for comparison with experimental results nature (); NFHW (). The proposed method may be applied also to studying multi-particle interactions (which play a significant role in the critical aggregation in the vicinity of the critical point DVN ()) in a critical solvent.

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