# Critical and near critical phase behaviour and interplay between the thermodynamic Casimir and van der Waals forces in confined non-polar fluid medium with competing surface and substrate potentials

## Abstract

We study, using general scaling arguments and mean-field type calculations, the behavior of the critical Casimir force and its interplay with the van der Waals force acting between two parallel slabs separated at a distance from each other, confining some fluctuating fluid medium, say a non-polar one-component fluid or a binary liquid mixture. The surfaces of the slabs are coated by thin layers exerting strong preference to the liquid phase of the fluid, or one of the components of the mixture, modeled by strong adsorbing local surface potentials ensuring the so-called boundary conditions. The slabs, on the other hand, influence the fluid by long-range competing dispersion potentials, which represent irrelevant interactions in renormalization group sense. Under such conditions one usually expects attractive Casimir force governed by universal scaling function, pertinent to the extraordinary surface universality class of Ising type systems, to which the dispersion potentials provide only corrections to scaling. We demonstrate, however, that below a given threshold thickness of the system for a suitable set of slabs-fluid and fluid-fluid coupling parameters the competition between the effects due to the coatings and the slabs can result in sign change of the Casimir force acting between the surfaces confining the fluid when one changes the temperature , the chemical potential of the fluid , or . The last implies that by choosing specific materials for the slabs, coatings and the fluid for one can realize repulsive Casimir force with non-universal behavior which, upon increasing , gradually turns into an attractive one described by an universal scaling function, depending only on the relevant scaling fields related to the temperature and the excess chemical potential, for . We presented arguments and relevant data for specific substances in support of the experimental feasibility of the predicted behavior of the force. It can be of interest, e.g., for designing nano-devices and for governing behavior of objects, say colloidal particles, at small distances. We have formulated the corresponding criterion for determination of . The universality is regained for . We have also shown that for systems with the capillary condensation phase diagram suffers modifications which one does not observe in systems with purely short-ranged interactions.

###### pacs:

64.60.-i, 64.60.Fr, 75.40.-s## I Introduction

When a fluctuating field is confined by material bodies, effective forces arise on them. This is due to the fact that the bodies impose boundary conditions on the medium, depending on their geometry, mutual position and material properties, which leads to a modification of the allowed fluctuations in the medium. The last leads to a dependence of the ground state, or the thermodynamic potential of the system (say the free energy) on the geometry of the system and on the distances between its (macroscopic) components. In order to change these distances one has to apply a force that depends on the induced change of the allowed fluctuations. If the fluctuations are long-ranged the corresponding forces are also long-ranged. The existence of such long-ranged fluctuation mediated forces is called the Casimir effect and the corresponding forces – Casimir like forces Casimir (1948, 1953); Casimir and Polder (1948); Fisher and de Gennes (1978), after the Dutch physicist Hendrik Casimir who in 1948 predicted an attractive force between two parallel perfectly conducting metal plates Casimir (1948) separated by a finite gap in vacuum at zero temperature. In order the force to be long-ranged (i.e., to decay in a power-law and not in an exponential with the distance way) the interactions in the system have to be mediated by massless excitations – photons, Goldstone bosons, acoustic phonons, etc. Considered in this general form the Casimir effect is a subject of investigations in condensed matter physics, quantum electrodynamics, quantum chromodynamics and cosmology. The results are summarized in an impressive number of reviews Plunien et al. (1986); Mostepanenko and Trunov (1988); Levin and Micha (1993); Mostepanenko and Trunov (1997); Milonni (1994); Kardar and Golestanian (1999); Bordag et al. (2001); Milton (2001, 2004); Lamoreaux (2005); Klimchitskaya and Mostepanenko (2006); Genet et al. (2008); Bordag et al. (2009); Klimchitskaya et al. (2009); French et al. (2010); Sergey et al. (2011); Klimchitskaya et al. (2011); Rodriguez et al. (2011); Milton et al. (2012); Brevik (2012); Bordag (2012); Cugnon (2012); Dalvit et al. (2011); Krech (1994); Brankov et al. (2000); Krech (1999); Gambassi (2009); Toldin and Dietrich (2010); Gambassi and Dietrich (2011); Dean (2012).

When the fluctuating field is the electromagnetic one the effect is known as the quantum electro-dynamical (QED) Casimir effect. There the Casimir force is caused by zero-point and thermal fluctuations of the electromagnetic field. In first approximation, it depends only on the velocity of light , Planck’s constant , the temperature , and the separation distance between the bodies , i.e., this force to a great extend is universal. A more advanced theory, the so-called Lifshitz theory, reveals the dependence of the Casimir force on the material properties of the bodies Dzyaloshinskii et al. (1961); Lifshitz and Pitaevskii (1980); Lifshitz (1956) and geometry of their boundary surfaces Emig et al. (2007); Kenneth and Klich (2008).

Thirty years after Casimir’s prediction, M. E. Fisher and P. G. de Gennes suggested that the fluctuating medium, confined between the bodies can be a fluid, the fluctuating field being the field of its order parameter, in which the interactions in the system are mediated not by photons but by different type of massless excitations like critical fluctuations or Goldstone bosons (spin waves). The corresponding Casimir effect is known as the thermodynamic Casimir effect Fisher and de Gennes (1978). When the confined fluid approaches it’s critical point, the corresponding fluctuations are the critical fluctuations of the order parameter and then the effect is usually called critical Casimir effect. In first approximation, the thermodynamic Casimir effect depends only on the gross features of the system – its dimensionality and the symmetry of the ordered state (both defining the so-called bulk universality class of the system) and on the boundary conditions (determined by the surface universality classes). Therefore, to a great extend the thermodynamic Casimir force is also universal. So far the critical Casimir effect has enjoyed two general reviews Krech (1994); Brankov et al. (2000) and some concerning specific aspects of it Krech (1999); Gambassi (2009); Toldin and Dietrich (2010); Gambassi and Dietrich (2011); Dean (2012).

Currently the Casimir effect is an object of intensive studies both in its original formulation due to Casimir as well as in its thermodynamic manifestation. In the present article we will report theoretical results dealing with the critical Casimir effect. Let us note, that the critical Casimir effect has been already directly observed, utilizing light scattering measurements, in the interaction of a colloid spherical particle with a plate Hertlein et al. (2008) both of which are immersed in a binary liquid mixture. The effect has been also studied in He Garcia and Chan (1999),Ganshin et al. (2006), as well as in He–He mixtures Garcia and Chan (2002) in the context of forces that determine the properties of a film of a substance in the vicinity of its bulk critical point. In Ref. Fukuto et al. (2005) and Ref. Rafaïet al. (2007) one has performed measurements of the Casimir force in thin wetting films of binary liquid mixture. One the theoretical side, the effect has been studied via exact calculations in the two-dimensional Ising model Evans and Stecki (1994); Nowakowski and Napiórkowski (2008, 2009); Abraham and Maciołek (2010); Rudnick et al. (2010); Douglas and Maciołek (2013); Israelachvili (2011a); Wu et al. (2012); Drzewinski et al. (2011); Maciołek et al. (2013), the three dimensional spherical model Dantchev (1996, 1998); Dantchev and Grüneberg (2009); Dantchev et al. (2006); Chamati and Dantchev (2004); Diehl and Rutkevich (2014); Dantchev et al. (2014); Diehl et al. (2012, 2014), via conformal-theoretical methods Affleck (1986); Blöte et al. (1986); Burkhardt and Eisenriegler (1995); Eisenriegler and Ritschel (1995); Hanke et al. (1998); BEK2015 (); DSE2015 (), within mean-field type calculations on Ising type Krech (1997); Gambassi and Dietrich (2006); Dantchev et al. (2007a); Parry and Evans (1992); Vasilyev et al. (2011) and models Bergknoff et al. (2011), through renormalization-group studies via -expansion Krech and Dietrich (1992a, b); Diehl et al. (2006); Grüneberg and Diehl (2008); Schmidt and Diehl (2008); Diehl and Schmidt (2011); Dohm (2009, 2013) and via fixed dimension techniques Dohm (2009, 2011, 2013) of models, as well as via Monte-Carlo calculations Krech and Landau (1996); Dantchev and Krech (2004); Hucht (2007); Hasenbusch (2009, 2010, 2011, 2013, 2015); Vasilyev et al. (2013, 2007, 2009); Mohry et al. (2010); Hucht et al. (2011); Hasenbusch (2012); Vasilyev and Dietrich (2013). The fluctuation of importance in all of the above mentioned models are of thermal origin since all these models possess non-zero critical temperature. In some systems, however, certain quantum parameters govern the fluctuations near their critical point which is usually close to or at the zero temperature Sachdev and Keimer (2011); Sachdev (2008, 2000, 2011). In this particular case one speaks of a quantum critical Casimir effect Chamati et al. (2000); Brankov et al. (2000); Pálová et al. (2009).

The rapid progress in nanotechnology has resulted in the growth of interest in fluctuation-induced phenomena, which play a dominant role between neutral non-magnetic objects at short separation distances (below a micrometer). The van der Waals and Casimir forces, both known under the generic name dispersion forces, play a key role in Micro- and Nano-Electromechanical Systems (MEMS/NEMS) Chan et al. (2001a); Delrio et al. (2005); Boström et al. (2012) operating at such distances. Indeed, upon scaling down devices, the dispersion forces can induce some usually undesirable non-linear behaviors in such systems Chan et al. (2001b). Irreversible phenomena appear such as stiction (i.e., irreversible adhesion) or pull-in due to mechanical instabilities Buks and Roukes (2001a, b). Therefore the ability to modify the Casimir interaction can strongly influence the development of MEMS/NEMS. Several theorems seriously limit, however, the possible search of repulsive Casimir forces Kenneth and Klich (2006); Silveirinha (2010); Rahi et al. (2010). Currently, apart from some suggestions for achieving Casimir repulsion in systems out of equilibrium Antezza et al. (2008); Bimonte (2009); Bimonte et al. (2011); Kirkpatrick et al. (2013); Messina and Antezza (2011); Kirkpatrick et al. (2014); Krüger et al. (2011); Banishev et al. (2012); Chang et al. (2011); Klimchitskaya et al. (2011), the only experimentally well verified way to obtain repulsive Casimir force is to have interaction between two different materials characterized by dielectric permittivities and such that Lifshitz (1956); Dzyaloshinskii et al. (1961); Lifshitz and Pitaevskii (1980)

(1) |

along the imaginary frequency axis, with being the dielectric permittivity of the medium in between them. In Refs. Milling et al. (1996); Meurk et al. (1997); Lee and Sigmund (2001, 2002); Munday et al. (2009); Ishikawa et al. (2011) Casimir repulsion was indeed observed experimentally for the sphere-plate geometry.

In the current article we study the interplay between the critical Casimir force and the van der Waals one in a system composed out of two flat parallel slabs both immersed in a critical fluid. Let us note that both the critical Casimir and van der Waals forces are fluctuation induced ones but due to the fluctuations of different entities. For terminological clarity, let us also remind that in colloid sciences fluid mediated interactions between two surfaces or large particles are usually referred to as solvation forces Evans (1990a, b); Evans and Stecki (1994). Thus, we study here a particular case of such a force when the fluid is near its critical point. In our system we suppose that the slabs are coated by thin layers of some substances, confining either a non-polar one-component fluid or a non-polar binary liquid mixture. We suppose that the liquid phase of the one-component fluid or one of the components of the binary liquid mixture are strongly adsorbed by both coating layers, i.e., they ensure the so-called boundary conditions. The slabs, on the other hand, influence the fluid by long-range competing dispersion potentials. In the case of a simple fluid these potentials increase the adsorption of the fluid near one of the surfaces, leading to preference there of its liquid phase, and decrease it near the other one. In the case of a binary fluid mixture the substrates prefer one of the components near the top and the other one near the bottom of the system. We will demonstrate that this experimentally realizable competition between the effects due to the coatings and the slabs can result in interesting effects like sign change of the Casimir force, acting between the surfaces confining the fluid when one changes , or . The last facts can potentially be used in designing nanodevices and for governing the behavior of objects at small, below micrometer, distances.

If a fluid system possesses a surface it breaks the spatial symmetry of the bulk system. The quantitative effects of the presence of a surface on the thermodynamic behavior of the system depends on the penetration depth of this symmetry breaking effect into the volume. There are two phenomena which increase the surface effects: long-range interactions and long-range correlations. They can act separately, or simultaneously, which leads to an interesting interplay of the effects due to any of them Dantchev et al. (2007b); Dantchev et al. (2009); Dantchev et al. (2007a); Maciołek et al. (2004); Barmatz et al. (2007). The penetration depth due to the correlations is set by the correlation length of the order parameter of the system; becomes large, and theoretically diverges, in the vicinity of the bulk critical point : , , and , , where and are the usual critical exponents. If the system is made finite, e.g., by the introduction of a second wall the behavior of the fluid is further enriched. When becomes comparable to the characteristic system size, say , the size dependence of thermodynamic functions enters through the ratio , i.e., takes a scaling form given by the finite-size scaling theory Barber (1983); Binder (1983); Cardy (1988); Privman (1990b); Parry and Evans (1990); Brankov et al. (2000) that incorporates, inter alia, shift of the critical point of the system Nakanishi and Fisher (1983a); Fisher and Nakanishi (1981); Binder et al. (2003); Nakanishi and Fisher (1983b); Evans and Marconi (1986). Below if the confining walls of the film geometry consist of the same material, one encounters the phenomenon of capillary condensation Evans (1990a); Evans and Marconi (1986); Okamoto and Onuki (2012); Dantchev et al. (2007a); Yabunaka et al. (2013) where the confinement of the fluid causes, e.g., the liquid vapor coexistence line to shift away from the coexistence line of the bulk fluid into the one-phase regime. We will demonstrate in the current article that in the envisaged realization of our system depending on the material properties of the slabs the phase diagram of the finite system might essentially differ from that one of the well studied case of a system with short-ranged type interactions and strongly adsorbing surfaces.

The article is arranged as follows. In Sec. II we recall and comment on the finite-size behaviour of systems with dispersion forces extending the known facts to the expected behavior of the Casimir and the net forces when they act between walls being semi-infinite slabs coated by some thin substances. By doing so, we especially pay attention to the conditions under which the effects stemming from these interactions are relevant. Section III presents the corresponding lattice gas models suitable for the investigation of fluid media with account of the long-ranged van der Waals interactions. Here we identify the main coupling parameters characterizing the interactions in the systems and in Sec. IV the equation for the equilibrium profile of the finite-size order parameter is obtained, which we later use to calculate the forces of interest. Section V presents the numerical results for the behaviour of the investigated forces followed by Sec. VI where the phase behaviour of the considered type fluid system is briefly discussed. The experimental feasibility of the predicted effects is discussed in Sec. VII. Here we also comment on the possible application of our findings in design of nanodevices and for governing the behavior of objects, say colloidal particles, at small distances. The article ends with a summary and discussion section – Sec. VIII. Important technical details concerning the Hamaker term for a van der Waals system of two different substances separated by a fluid medium are presented in Appendix A.

## Ii The thermodynamic Casimir force in a non-polar fluid film systems with dispersion forces

Let us consider some fluid medium confined between two parallel slabs of some materials and . Any of the slabs is coated by thin solid films of some other substances and , respectively (see Fig. 1). Let the slabs are situated at some distance from each other. We suppose that the thicknesses of the coating films are negligible. In the remainder of the text we are going to designate each slab and the thin solid film that coats it as a “wall”, and refer to any of its two components separately only when this is necessary.

If the fluid medium is in contact with a particle reservoir with a chemical potential , the grand canonical potential of this medium in excess to its bulk value depends on and, thus, one can define the effective force per cross sectional area and , due to the fluctuations of the medium and dispersion interactions in it as

(2) |

where the superscript designates the boundary conditions which the confining walls impose on the fluid medium (see above), is the excess grand canonical potential per unit area , is the total grand canonical potential, is the density of the bulk grand canonical potential, and not (a). Let us stress that, as pointed out in Ref. Dantchev et al. (2007a), one should keep in mind that the force , see Eq. (2), depends on how one defines the thickness of the film. This implies that a quantitative comparison between experimental data and theory is only possible if the data are accompanied by a precise definition of what is.

Away from the critical temperature of the system it is customary to write the force acting between the plates of the fluid system in the form

(3) |

where one normally considers the case and omits the apparent dependence on the so-called retardation length Garcia and Chan (1999); Dantchev and Valchev (2012) . Here is the Hamaker term, whose dependence from the temperature and chemical potential is given by the so-called Hamaker constant Parsegian (2006); Israelachvili (2011a)

(4) |

The Hamaker constant, as it is clear from above, is a constant only in the sense that it is -independent. It depends on the temperature, chemical potential and on the material properties of the fluid medium and the walls. The factor in Eq. (4) is introduced there due to historical reason, according to which the interaction energy between two substrates at a finite separation in the case of standard van der Waals interaction (i.e., ), away from any phase transition region, is Parsegian (2006); Israelachvili (2011a)

(5) |

The Hamaker term takes into account the leading -dependent parts of the i) direct interaction between the slabs , ii) between each slab and the fluid medium – , and , as well as the iii) interactions between the portion of constituents of the fluid medium situated within the cavity bounded by the substrates – , i.e.,

(6) | |||||

Note that in Eqs. (3) – (6) both the slabs and the fluid medium are characterized by their bulk properties at the given temperature and chemical potentials.

Near the critical temperature of the bulk system Eq. (3) is no longer valid since the critical fluctuations of the order parameter lead to new contribution to the total force called thermodynamic (critical) Casimir force (see below). For such a system, following Ref. Dantchev et al. (2007a), near the bulk critical point Eq. (2) can be written in the form

(7) | |||||

In Eq. (7) is dimensionless, universal scaling function, and are the temperature and field relevant scaling variables, respectively, while , and are the irrelevant in renormalization group sense scaling variables associated with the interactions in the system. The freedom of choosing the precise definition of what is in systems with boundaries leads to the formal necessity to write as , with being a microscopic length. That will lead to further scaling corrections proportional to . Since in the current article we will keep in all calculated quantities only their leading dependence, we refrain from further refinement of the scaling Ansatz (7). By comparing Eq. (3) and Eq. (7) one immediately concludes that tends to zero away from the critical point, i.e., when at least one of the relevant scaling parameters and becomes large, i.e., when and/or .

As it is well known, the critical behavior of simple fluids and of binary liquid mixtures is described within the Ising universality class which determines the values of the critical exponents and ones the dimensionality of the system is fixed. When this universality class is characterized by critical exponents Hasenbusch (2010)

(8) |

In order to better reflect the actual properties of the non-polar fluids, instead of considering nearest-neighbour interactions we assume long-ranged pair ones between the fluid particles, decaying asymptotically for distances between each other, and substrate potentials acting on the fluid particles at a distance from each of the two slabs. We recall that when systems governed by such long-range interactions, usually termed subleading long-ranged interactions Dantchev and Rudnick (2001); Dantchev (2001), also belong to the Ising universality class characterized by short-ranged forces Pfeuty and Toulouse (1977). The last implies, among the others, that the critical exponents, e.g., do not depend on for such type of interactions. An important representative of such type of interactions are the non-retarded dispersion interactions with , which are one of the three types of van der Walls interactions. By varying the ratio between the strengths of the long-ranged – and the short-ranged – contributions one can quantitatively probe the importance of the long-ranged parts of the interactions within the fluid medium and study potential experiments in colloidal systems which allow for a dedicated tailoring of the form of the effective interactions between colloidal particles.

In Eq. (7), is the standard correction-to-scaling exponent for short-range systems, while and are the correction-to-scaling exponents due to the long-range parts of the interaction potentials between the constituents of the fluid medium and those of the confining walls. Further dependent contributions to the total forces such as next-to-leading order contributions to the Hamaker terms or higher order corrections to scaling are neglected because they are smaller than those captured in Eq. (7). The exponent , which appears in the expressions for and , is the standard one characterizing the decay of the bulk two-point correlation function at the critical temperature, is the (dimensionless) scaling field associated with the Wegner-type corrections, while and are dimensionless non-universal coupling constants: is proportional to the strength of the long-range part of the interaction potential between the particles of the fluid, whereas are proportional to the contrast between the potentials of the bounding slabs and those in the fluid medium (see below). For systems belonging to three-dimensional Ising universality class with “genuine” non-retarded van der Waals interaction one has , and Hasenbusch (2010). This leads to , , and Hasenbusch (2010). Within the mean-field theory with and one has, instead, and . One then has

(9) |

where all critical exponents take their mean-field values , , .

The peculiarities of the scaling theory for systems with dispersion forces are described in Refs. Dantchev and Rudnick (2001); Dantchev (2001); Dantchev et al. (2007b); Dantchev et al. (2009, 2006); Dantchev et al. (2007a). One obtains that despite these systems do belong to the Ising universality class with short-ranged forces, the finite-size quantities decay algebraically with towards their bulk values, and not in an exponential in way, when even for , where is the characteristic distance between the molecules of the fluid system. In this regime the dominant finite-size contributions to the free energy and to the force between the walls bounding the system stem from the long-ranged algebraically decaying parts of the interaction potentials. One can formulate a criterion clarifying when the long-ranged tails of the interactions can not be disregarded even in the critical region of the finite system. Neglecting the thickness of the coating layers, for the system under consideration the corresponding criterion states that the long range tails of the interactions can be disregarded only when Dantchev et al. (2007a); Dantchev et al. (2009, 2006)

(10) |

Using the scaling relations, the above can be formulated as a constraint on the thickness of the system under study. One obtains that when , where

(11) |

the effects of the long-ranged tails of the interaction can be neglected within the critical region of the finite system. For and with and from Eq. (8) for the three-dimensional Ising model one obtains

(12) |

We conclude that for moderate values of , i.e., when the behavior of the Casimir force can strongly depend on the details of the fluid-fluid [the amplitude depends on the details of the fluid-fluid interaction (see Eqs. (4.15) and (4.17) in Ref. [Dantchev et al., 2007a] and the text therein)] and substrate-fluid interactions which, as it turns out, can influence even the sign of the force. For large , i.e., for the behavior of the force shall approach the one of the short-ranged system. The last can be easily seen from Eq. (7) by simply expanding then the scaling function

(13) |

Here originates from the short-range interactions [see Eqs. (47)]. It is well known that , i.e., the force is attractive, under boundary conditions. provides the leading behaviour of the force near the bulk critical point (). There and represent only corrections to the leading dependance. The reader can refer for a detailed comment on that matter to Ref. Dantchev et al. (2007a) (see also Ref. Dantchev et al. (2006) for the properties of ). The validity of the proposed criterion as well as the statements made beneath it are well illustrated on Fig. 2.

The contribution of the dispersion forces to the total effective force can be distinguished from that of the critical Casimir force by their temperature dependence, because the leading temperature dependence of the former does not exhibit a singularity. Thus, one has

(14) |

where

(15) |

and

(16) |

One expects that near the bulk critical point

(17) |

where is a scaling function that for large enough (see below) approaches the scaling function of the short-ranged system . From Eqs. (7), (14) – (17) it follows that the scaling function of the critical Casimir force is proportional to the sum of and the singular part of the Hamaker term.

We will often compare the behavior of the system with subleading long-ranged interactions present with this one of a system with purely short-ranged interactions which will serve as a reference system. In such a purely short-range system one has . Then, at the bulk critical point the leading term of the thermodynamic Casimir force between the slabs bounding the fluid has the form

(18) |

where . Here is an universal dimensionless quantity, called Casimir amplitude, which depends on the bulk and surface universality classes (specified by the boundary conditions ). Since the Casimir force is proportional to the interaction between the walls can become rather strong in a system with high critical temperature such as, e.g., in classical binary liquid mixtures. Note that the sign of the force depends on the sign of the Casimir amplitude which, on its turn, depends on the boundary conditions . According to the usual convention negative sign corresponds to attraction, while positive sign means repulsion of the surfaces bounding the system.

The experimental and theoretical evidences accumulated till nowadays support the statement that the Casimir force is attractive when the boundary conditions on both plates are the same, or similar, and is repulsive when they essentially differ from each other, e.g., when in the case of a one-component fluid one of the surfaces adsorbs the liquid phase of the fluid while the other prefers the vapor phase. It is instructive to go back to the explicit physical units in Eq. (18) for the physically most relevant case of . One has

(19) |

where C (293.15 K). Sine as discussed above, for most systems and boundary conditions , when the thermodynamic Casimir force, for some space separation , will be of the same order of magnitude as the quantum one

(20) |

and they both shall be significant and consequently measurable at or below the micrometer length scale.

We turn now to description of the model and the procedure under which our results have been obtained.

## Iii The model

We are going to utilize the same type of model already used in Refs. Dantchev et al. (2007a, b) but amended to take into account the specific features of the system considered in the current article. Among them is the role of the two competing substrate potentials. Here, in order to introduce the notations needed further, we briefly recall the basic expressions of that model paying a bit more attention only to difference of the current model with that one studied in Refs. Dantchev et al. (2007b, a).

We consider a lattice-gas model of a fluid confined between two planar walls, separated at a distance from each other, with grand canonical potential given by

(21) |

where is a simple cubic lattice in the region occupied by the fluid medium – and is an external potential that reflects the interactions between the confining walls and the constituents of the fluid, given by

(22) | |||||

where , with

(23) |

This type of functional can be viewed as a modification of the model utilized by Fisher and Nakanishi Nakanishi and Fisher (1982, 1983a) in their mean-field investigation of systems governed by short-range forces.

In Eq. (21) the terms in curly brackets multiplied by correspond to the entropy contributions to the total energy, is the non-local coupling (interaction potential) between the constituents of the confined medium and is the chemical potential. Here and in the remainder of this paper, all length scales are taken in units of the lattice constant (for concrete values see Table. 1), so that the particle number density becomes simply a number density which varies in the range .

The variation of Eq. (21) with respect to leads to the equation of state for the equilibrium density

(24) | |||||

The advantage of this type of equation is that it lends itself to numerical solution by iterative procedures. For a given geometry and external walls-fluid potential its solution determines the equilibrium order parameter profile in the system. Inserting this profile into Eq. (21), renders the grand canonical potential of the considered system.

Denoting, as in Refs. Dantchev et al. (2007b, a) and , where , the equation of state Eq. (24) can be rewritten in the standard form

(25) | |||||

The bulk properties of the model are well known (see, e.g. Domb (1996); Baxter (1982) and the references therein). We recall that the order parameter of the system has a critical value which corresponds to so that . The bulk critical point of the model is given by with the sum running over the whole lattice. Within the mean-field approximation the critical exponents for the order parameter and the compressibility are and , respectively. The effective surface potential in Eq. (25) is given by

(26) |

where , and contributions of the order of , , etc., have been neglected,

(27) |

are (- and -independent) constants,

(28) |

is a proper lattice version of as the interaction energy between the fluid particles, and

(29) |

is the one between a fluid particle and a substrate particle, is the discrete delta function and is the Heaviside step function with the convention ; in Eq. (29) are the number densities of the coated slabs in units of (for concrete values see Table. 2). Note that the effective potentials result from the difference between the relative strength of the substrate-fluid interactions for substrates with density and that of the fluid-fluid interactions for a fluid with density . In Eq. (26) the restriction holds because we consider the layers closest to the substrate to be completely occupied by the liquid phase of the fluid (which implies that we consider the strong adsorption limit), i.e., , which is achieved by taking the limits ; thus, the actual values of will play no role. In order to preserve the monotonic behavior of as a function of the distance between the particles, in Eq. (28) we have to require that , i.e., , where

(30) |

From Eqs. (25) – (29) one can identify the dimensionless coupling constants , appearing in Eq. (7)

(31) |

Here , i.e., corresponds to walls “preferring” the liquid phase of the fluid, while , or mirrors the one with affinity to its gas phase. The marginal case which corresponds to will be commented further in the text.

In terms of the functional defined by Eq. (21) takes the form

(32) |

where

(33) |

do not depend on and, therefore, is a regular background term carrying an dependence and thus showing up in the corresponding force.

The only quantity in Eq. (7), which still has to be identified for our model, is the value of the coupling constant . According to Ref. Dantchev et al. (2007a)

(34) |

where and are coefficients in the Fourier transform of the interaction [see Eq. (28)] (for more details see Ref. Dantchev et al. (2007a)). It turns out that depends on .

In accordance with Eq. (7), for the finite-size behavior of the excess grand canonical potential per unit area of a liquid film in the case when both confining surfaces strongly adsorb the liquid phase, one expects

(35) | |||||

where and are the non-singular parts of the surface tensions at the surfaces of the confining walls, while the singular parts are incorporated in the scaling function , which arguments are as those of the function in Eq. (7).

In Eq. (35) is the Hamaker term which explicit form is derived in Appendix A. The result is

(36) |

Note that this result is in full agreement with the Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory Lifshitz (1956); Dzyaloshinskii et al. (1961). It provides, however, an easy possibility to study the and dependence of by studying the corresponding dependencies of , say, near the critical point of the fluid system. In accord with the DLP theory it teaches us that the sign of the Hamaker term depends on the contrast of material properties of the two bounding substrates with respect to the substance which is in-between them. Coating the substrate surfaces of the system with some additional material does not change the leading-order dependence of the interactions between the substrates and, therefore, does not change the above property. Let us stress that when , i.e., when the bounding substances are made from same material, the Hamaker term will be negative, independent on the properties of the fluid in-between them, i.e.,

(37) |

which corresponds to attraction between the confining walls. Similar is the situation when , i.e., when the fluid separating the substrates is replaced by vacuum. Then, again, irrespective of the material properties of the bounding substances, since . Further details on the behavior of are given in Appendix A.

In the next section IV we are going to present results for based on the model described in the current section.

## Iv Finite-size behaviour of the model in a film geometry

Let us start by rewriting the equations presented in Sec. III in the form suitable for studying of a system in a film geometry. Because of the translational symmetry of the system along the bounding surfaces, the quantities of interest depend only on the spacial coordinate along which the system is finite. Thus, one can write: , where , i.e., the local order parameter profile is given by , with . Hence the equation Eq. (25) for the equilibrium profile becomes

(38) |

where , is defined in Eq. (26), , , with . The functions , and are determined in Eqs. (C10), (C11) and (C12) of Ref. Dantchev et al. (2007a), respectively.

When the equilibrium finite-size order parameter tends to its bulk value . The corresponding equation for , performing the limit in Eq. (IV), reads

(39) |

wherefrom one immediately identifies the coordinates of bulk critical point

(40) |

and . Note that the position of the bulk critical point depends on , i.e., on the presence and the strength of the long-ranged tails in the fluid-fluid interactions.

For a fluid confined to a film geometry the natural quantity to consider is the excess grand canonical potential per unit area : . Using the result of Ref. Dantchev et al. (2009) [see Eq. (3.14) there], as well as the identity , one can write in the form

(41) | |||||

where

(42) | |||||

As it is clear from Eq. (21), the model presented above does not account for the direct wall-wall interaction, but only for the walls-fluid and fluid-fluid ones. Thus, in order to obtain the complete net force acting between the surfaces bounding the fluid one has to add to the force calculated from Eq. (41) via Eq. (2) the one due to the direct wall-wall interaction. Then the resulting net force will be the total force between the plates bounding the fluid. With the use of Eqs. (2), (61) and (41) one can write in the form

(43) | |||||

where the last term represents the direct wall-wall interaction. On the other hand, if one subtracts from the potential its regular part , i.e., if we consider the quantity

(44) |

then, in accord with Eqs. (14) – (16), the dependence of via Eq. (2) provides the singular part of the total force, i.e., the critical Casimir force – . Explicitly, one has

(45) | |||||