Effect of mesoscopic fluctuations on equation of state in cluster-forming systems

Equation of state for systems with particles self-assembling into aggregates is derived within a mesoscopic theory combining density functional and field-theoretic approaches. We focus on the effect of mesoscopic fluctuations in the disordered phase. The pressure - volume fraction isotherms are calc...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2012
Hauptverfasser:	Ciach, A., Patsahan, O.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2012
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/120267
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems / A. Ciach, O. Patsahan // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23604:1-16. — Бібліогр.: 27 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-120267
record_format	dspace
spelling	irk-123456789-1202672017-06-12T03:04:23Z Effect of mesoscopic fluctuations on equation of state in cluster-forming systems Ciach, A. Patsahan, O. Equation of state for systems with particles self-assembling into aggregates is derived within a mesoscopic theory combining density functional and field-theoretic approaches. We focus on the effect of mesoscopic fluctuations in the disordered phase. The pressure - volume fraction isotherms are calculated explicitly for two forms of the short-range attraction long-range repulsion potential. Mesoscopic fluctuations lead to an increased pressure in each case, except for very small volume fractions. When large clusters are formed, the mechanical instability of the system is present at much higher temperature than found in mean-field approximation. In this case phase separation competes with the formation of periodic phases (colloidal crystals). In the case of small clusters, no mechanical instability associated with separation into dilute and dense phases appears. Рiвняння стану для систем частинок, що самоскупчуються в агрегати, є отримане в рамках мезоскопiчної теорiї, що поєднує метод функцiоналу густини i теоретико-польовий пiдхiд. Ми дослiджуємо вплив мезоскопiчних флуктуацiй у невпорядкованiй фазi. Явно обчислено iзотерми ‘тиск – об’ємна частка’ для двох наборiв параметрiв потенцiалу короткосяжне притягання плюс далекосяжне вiдштовхування. В кожному випадку врахування мезоскопiчних флуктуацiй приводить до пiдвищення тиску, за винятком дуже малих об’ємних часток. Коли утворюються великi кластери, механiчна нестiйкiсть системи присутня при набагато вищих температурах, нiж це було отримано в наближеннi середнього поля. В цьому випадку фазове вiдокремлення конкурує iз формуванням перiодичних фаз (колоїдних кристалiв). У випадку малих кластерiв механiчна нестiйкiсть, пов’язана з вiдокремленням в розрiджену i густу фази, не виникає. 2012 Article Effect of mesoscopic fluctuations on equation of state in cluster-forming systems / A. Ciach, O. Patsahan // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23604:1-16. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.20.Gy, 64.10.+h, 64.60.De, 64.75.Yz DOI:10.5488/CMP.15.23604 arXiv:1201.0464 http://dspace.nbuv.gov.ua/handle/123456789/120267 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Equation of state for systems with particles self-assembling into aggregates is derived within a mesoscopic theory combining density functional and field-theoretic approaches. We focus on the effect of mesoscopic fluctuations in the disordered phase. The pressure - volume fraction isotherms are calculated explicitly for two forms of the short-range attraction long-range repulsion potential. Mesoscopic fluctuations lead to an increased pressure in each case, except for very small volume fractions. When large clusters are formed, the mechanical instability of the system is present at much higher temperature than found in mean-field approximation. In this case phase separation competes with the formation of periodic phases (colloidal crystals). In the case of small clusters, no mechanical instability associated with separation into dilute and dense phases appears.
format	Article
author	Ciach, A. Patsahan, O.
spellingShingle	Ciach, A. Patsahan, O. Effect of mesoscopic fluctuations on equation of state in cluster-forming systems Condensed Matter Physics
author_facet	Ciach, A. Patsahan, O.
author_sort	Ciach, A.
title	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems
title_short	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems
title_full	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems
title_fullStr	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems
title_full_unstemmed	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems
title_sort	effect of mesoscopic fluctuations on equation of state in cluster-forming systems
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/120267
citation_txt	Effect of mesoscopic fluctuations on equation of state in cluster-forming systems / A. Ciach, O. Patsahan // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23604:1-16. — Бібліогр.: 27 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT ciacha effectofmesoscopicfluctuationsonequationofstateinclusterformingsystems AT patsahano effectofmesoscopicfluctuationsonequationofstateinclusterformingsystems
first_indexed	2025-07-08T17:34:08Z
last_indexed	2025-07-08T17:34:08Z
_version_	1837101009863704576
fulltext	Condensed Matter Physics, 2012, Vol. 15, No 2, 23604: 1–16 DOI: 10.5488/CMP.15.23604 http://www.icmp.lviv.ua/journal Effect of mesoscopic fluctuations on equation of state in cluster-forming systems A. Ciach1, O. Patsahan2 1 Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warszawa, Poland 2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received December 31, 2011, in final form March 6, 2012 Equation of state for systems with particles self-assembling into aggregates is derived within a mesoscopic theory combining density functional and field-theoretic approaches. We focus on the effect of mesoscopic fluc- tuations in the disordered phase. The pressure – volume fraction isotherms are calculated explicitly for two forms of the short-range attraction long-range repulsion potential. Mesoscopic fluctuations lead to an increased pressure in each case, except for very small volume fractions. When large clusters are formed, the mechanical instability of the system is present at much higher temperature than found in mean-field approximation. In this case phase separation competes with the formation of periodic phases (colloidal crystals). In the case of small clusters, no mechanical instability associated with separation into dilute and dense phases appears. Key words: clusters, self-assembly, equation of state, mesoscopic fluctuations PACS: 61.20.Gy, 64.10.+h, 64.60.De, 64.75.Yz 1. Introduction Recent experimental [1–4], theoretical [5–11] and simulation [8, 12–16] studies reveal that in many systems with competing interactions, clusters or aggregates of different sizes and shapes are formed. These objects in certain thermodynamic states can form ordered structures in space [5, 11, 12]. Notable examples include charged globular proteins in water [1, 3, 13, 17], and mixtures of small nonadsorbing polymers with charged colloids or micelles [1, 2, 18]. Interactions (or in fact effective interactions) in the latter systems can be described by themodel potential consisting of short-range attraction, resulting from solvophobic or depletion interactions, and long-range repulsion, resulting from screened electrostatic potential (SALR potential). Systems containing clusters or aggregates are inhomogeneous on the length scale associated with the average size of the aggregates and average distance between them. The corresponding length scale of the inhomogeneities is significantly larger than the size of the particles. Fluctuations on the mesoscopic length scale corresponding to displacements of the aggregates have an important impact on the grand potential, and thus on the equation of state (EOS). Derivation of an accurate EOS for inhomogeneous sys- tems is less trivial than in the case of homogeneous systems, since it is necessary to perform summation over different spatial distributions of the clusters and over all deformations of them. Contribution to the grand potential associated with mesoscopic fluctuations can be calculated in the field-theoretic approach [5, 11]. In principle, this contribution can be obtained in the perturbation expan- sion in terms of Feynman diagrams. In practice, an approximate result can be analytically obtained in the self-consistent Hartree approximation [5, 6, 19]. A formal expression for the fluctuation contribution to the grand potential has been derived in references [5, 6, 19]. However, its explicit formwith the chemical potential expressed in terms of temperature and density has not been determined yet. The EOS isotherms for various forms of the SALR potential were not analyzed, and the effect of the mesoscopic fluctuations on pressure remains an open question. © A. Ciach, O. Patsahan, 2012 23604-1 http://dx.doi.org/10.5488/CMP.15.23604 http://www.icmp.lviv.ua/journal A. Ciach, O. Patsahan It is important to note that various forms of the SALR potential are associated with different proper- ties of the systems. Depending on the ratios between the strengths and ranges of the attractive and re- pulsive parts of the potential, separation into uniform phases, formation of clusters of various sizes and shapes (globules, cylinders, slabs) in the so-called microsegregation, or isolated individual particles may occur. In certain conditions, the clusters can be periodically distributed in space in the periodic phases whose densities are smaller than the density of the liquid phase [5, 11, 12]. Possible types of the phase diagram for different SALR potentials are shown in figure 1 (see also references [5, 9, 10]). Properties T liquid gas µ T gas liquid p T gas liquid p µµ Figure 1. Types of possible phase diagrams for the SALR potentials, shown schematically. From the left to the right panel the role of the repulsion increases. ‘p’ indicates the stability region of the periodic phases of different symmetries. The disordered fluid close to the stability region of the periodic phases is inhomogeneous, i.e. clusters are formed, but their positions are correlated only at short distances. In this work we are interested in systems exhibiting the phase behaviour shown in the right (System 1) and the central (System 2) panel. of the disordered phase can be influenced by the periodic phases for thermodynamic states close to the stability of the latter. We expect that the disordered phase, although the long-range order is absent, is inhomogeneous on the mesoscopic length scale and resembles ‘molten periodic phases’. In this respect the inhomogeneous disordered phase is similar to microemulsion which can be interpreted as molten lyotropic liquid crystal. In this work we focus on the stable or metastable disordered inhomogeneous phase in which clusters are formed, but they do not form any ordered structure. We derive the EOS with the contribution from mesoscopic length-scale fluctuations included.We calculate the explicit form of the EOS for two represen- tative examples of the SALR potential within the self-consistent Hartree approximation. The first system corresponds to the formation of small clusters, and the gas-liquid separation is unstable for all tempera- tures in themean-field (MF) approximation (figure 1, right panel). In the second system, large clusters are formed. The gas-liquid separation is present in this system as a stable or a metastable transition for low temperatures (figure 1, central panel). We shall compare the effect of mesoscopic fluctuations in these two cases on the isotherms P (ζ), where ζ is the volume fraction of particles and P is pressure. In the next section we briefly summarize the mesoscopic approach. In section 3 the EOS is obtained by two methods. In section 3.1 we consider mesoscopic fluctuations about the average volume fraction, while in section 3.2 fluctuations about the most probable volume fraction are included. The formulas de- rived in section 3 are evaluated for the two versions of the SALR potential in section 4. We obtain a com- pletely different effect of the mesoscopic fluctuations in these two cases. The two approaches (sections 3.1 and 3.2) yield very close results provided that the relative fluctuation contribution to the average volume fraction is small. For larger fluctuation-induced shifts of the volume fraction, only qualitative agreement of the two methods is obtained. Short summary is presented in section 5. 2. Short summary of the mesoscopic description We consider a local volume fraction of particles, i.e. the microscopic volume fraction averaged over mesoscopic regions, as an order parameter [6]. The corresponding mesoscopic volume fraction varies on a length scale larger than the size of the particles, and the characteristic size of inhomogeneities is the upper limit for the mesoscopic length scale. A particular form of the mesoscopic volume fraction can be considered as a constraint on the microscopic states. The corresponding mesostate is a subset of microstates compatible with the imposed constraint. The mesoscopic volume fraction (or the mesostate) 23604-2 Equation of state in cluster-forming systems was defined in references [5, 6]. For a one-component case we fix the mesoscopic length scale R Ê σ/2 and consider spheres SR (r) of radius R and centers at r that cover the whole volume V of the system. We define themesoscopic volume fraction at r by ζ(r) := 1 VS ∫ r′∈SR (r) ζ̂(r ′,M ), (2.1) where VS = 4πR3/3, and the microscopic volume fraction in the microstate M = {{ri }i=1,...,N } is defined by ζ̂(r,M ) := N ∑ i=1 θ (σ 2 −\|r−ri \| ) , (2.2) where θ(r ) is the Heaviside unit step function. The microscopic volume fraction is equal to 1 at points that are inside one of the hard spheres, and zero otherwise. Integrated over the system volume, it yields the volume occupied by the particles. The mesoscopic volume fraction at r is equal to the fraction of the volume of the sphere SR (r) that is occupied by the particles. Note that ζ(r) takes the same value if one particle is entirely included in this sphere, independently of the precise position of its centre. Thus, ζ(r) gives less precise information on the distribution of particles than ζ̂(r). In the disordered phase ζ(r) is independent of r and equals the fraction of the total volume that is occupied by the particles. The mesostate can be imagined as a fixed distribution of centers of clusters, with arbitrary distribution of particles within the clusters, and small modifications of their shapes. Probability of the mesostate ζ is given by [5, 6] p[ζ] = e−βΩco[ζ] Ξ , (2.3) where Ξ= ∫′ Dζe−βΩco[ζ]. (2.4) The functional integral ∫′ Dζ in (2.4) is over all mesostates, Ωco =U [ζ]−T S[ζ]−µN [ζ], (2.5) where U ,S, N are the internal energy, entropy and the number of molecules respectively in the system with the constraint of compatibility with the mesostate [ζ] imposed on the microscopic volume fractions. U is given by the well known expression U [ζ] = 1 2 ∫ r1 ∫ r2 Vco(r1 −r2)ζ(r1)ζ(r2), (2.6) where for spherically symmetric interactions Vco(r1 −r2) =V (r12)gco(r1 −r2), (2.7) r12 = \|r1−r2\|, v2V (r12) is the interaction potential, v =πσ3/6 is the volume of the particle, and gco(r1−r2) is the microscopic pair correlation function for the microscopic volume fraction in the system with the constraint of compatibility with the mesostate imposed on the microscopic states. The grand potential can be written in the form βΩ[ζ̄] =βΩco[ζ̄]− logΞfluc , (2.8) where Ξfluc = ∫ Dφe−βHfluc[ζ̄,φ], (2.9) Hfluc[ζ̄,φ] =Ωco[ζ̄+φ]−Ωco[ζ̄] = ∑ n=1 ∫ r1 . . . ∫ rn C co n [ζ̄] n! φ(r1) . . .φ(rn). (2.10) 23604-3 A. Ciach, O. Patsahan The average mesoscopic volume fraction, ζ̄, corresponds to the minimum ofΩ, andmust satisfy the equa- tion δβΩco[ζ̄] δζ̄(r) + 〈 δ(βHfluc) δζ̄(r) 〉 = 0, (2.11) where the averaging is over the fields φ with the probability ∝ exp(−βHfluc[ζ̄,φ]). Note that when C co n [ζ̄]= 0 for odd n, then the second term on the LHS in (2.11) vanishes, and the average volume fraction coincides with the most probable volume fraction ζ0 given by δβΩco[ζ] δζ(r) ∣ ∣ ∣ ∣ ζ=ζ0 = 0. (2.12) By contrast, when C co n [ζ̄], 0 for odd n, then ζ̄, ζ0. In order to evaluate the fluctuation contribution to Ω[ζ̄] we decompose Hfluc[ζ̄,φ] into two parts Hfluc[ζ̄,φ]=HG[ζ̄,φ]+∆H [ζ̄,φ], (2.13) where in the disordered phase HG[ζ̄,φ] = 1 2 ∫ k φ̃(k)C̃2(k, ζ̄)φ̃(−k), (2.14) and C̃2(k,ζ) is the Fourier transform of C2(r12,ζ) = δ2βΩ[ζ] δζ(r1)δζ(r2) . (2.15) The above function calculated for ζ= ζ̄ is related to the direct correlation function [20]. Assuming ∆H ≪HG, we obtain [5, 19] βΩ[ζ̄] ≈βΩco[ζ̄]− log ∫ Dφe−βHG +〈β∆H 〉G +O ( 〈β∆H 〉2 G ) , (2.16) where 〈. . .〉G denotes the averaging with the Gaussian Boltzmann factor ∝ e−βHG . Since P =−Ω/V , approximate EOS can be obtained from (2.16) calculated for ζ̄ satisfying (2.11), when Ωco[ζ̄] (see (2.5)) is known. The chemical potential in (2.16) should be expressed in terms of T and ζ̄; its form as a function of T and ζ̄ can be determined from equation (2.11). In order to evaluate the second term in equation (2.11), we need approximate forms of the correlation functions. In the lowest order approximation, it is necessary to determine 〈φ̃(k)φ̃(−k)〉 = G̃2(k, ζ̄) = 1/C̃2(k, ζ̄). (2.17) 3. Approximate results for the fluctuation contributions to the EOS, density and chemical potential In this sectionwe derive an explicit formof the EOS P (ζ̄,T ), the density shift,∆ζ= ζ̄−ζ0, and the chem- ical potential µ(ζ̄,T ) under the following assumptions: (i) local density approximation for S[ζ] and (ii) the lowest-order approximation for the second term in (2.11). In the local density approximation we have −T S[ζ] = ∫ r fh(ζ(r)), (3.1) where fh(ζ) is the free-energy density of the hard-sphere system with dimensionless density ρ∗ = 6ζ/π. We assume the Percus-Yevick approximation β fh(ζ) = ρ∗ ln(ρ∗)−ρ∗+ρ∗ [ 3ζ(2−ζ) 2(1−ζ)2 − ln(1−ζ) ] . (3.2) 23604-4 Equation of state in cluster-forming systems In the local density approximation C co n [ζ] are just functions of ζ in the disordered phase, and we can simplify the notation, introducing An(ζ) = dn[β fh(ζ)] dζn . (3.3) For n > 2 we have C co n [ζ] = An(ζ), (3.4) whereas for n = 2 C̃ co 2 (k,ζ) =βṼco(k)+ A2(ζ), (3.5) where in the disordered phase C̃ co 2 (k,ζ) is the Fourier transform of the function C co 2 (r12,ζ) = δ2βΩco[ζ] δζ(r1)δζ(r2) . (3.6) Equations (3.2) and (2.6) define the functional Ωco for a given form of Vco (equation (2.7)). From equation (2.3) it follows that the most probable fluctuations correspond to the wavenumbers k = kb for which Ṽco(k) assumes the minimum, and the inhomogeneities on the length scale 2π/kb are energetically favored when Ṽco(kb) < 0. In this work we focus on the effect of the self-assembly into aggregates. Therefore, we restrict our attention to Ṽco(k) which assumes the minimum for kb > 0, and Ṽco(kb) < 0 [5, 11]. Since the fluctuations with the wavenumber k ≈ kb are most probable, they yield the main fluctuation contribution to the grand potential (2.16). For such fluctuations we can make the approximation βṼco(k) ≈βṼco(kb)+βṼ (2) co (kb)(k −kb)2/2+ . . . (3.7) As the energy scale we choose the excess energy associated with the fluctuations having unit amplitude and the wavenumber kb, and introduce the notation β∗ = 1/T ∗ =β\|Ṽco(kb)\|, (3.8) v∗ 2 = Ṽ (2) co (kb) 2\|Ṽco(kb)\| (3.9) and v∗ 0 = Ṽco(0) Ṽco(kb) . (3.10) 3.1. Fluctuations around the average volume fraction In this subsection we consider fluctuations about the average value ζ̄. We shall first determine the chemical potential as a function of ζ̄ and T ∗ from (2.11). In order to calculate the second term in (2.11), we assume that relevant fluctuations are of small amplitudes, and truncate the expansion in (2.10) at the fourth-order term. Next we insert the derivative with respect to ζ̄ of the RHS of equation (2.10) truncated at the quadratic term in φ, and we obtain from (2.11) and (2.5) an approximate equation for the rescaled chemical potential, µ̄= 6µ/(πσ3), of the form βµ̄≈βµ̄MF(ζ̄)+ A3(ζ̄) 2 G (ζ̄), (3.11) where βµ̄MF(ζ̄) =βṼco(0)ζ̄+β f ′ h(ζ̄) (3.12) 23604-5 A. Ciach, O. Patsahan and the last term in (3.11) is the fluctuation contribution with G (ζ)= ∫ k G̃2(k,ζ) =G2(0,ζ). (3.13) The same expression can be obtained from 〈φ〉 = 0, when (2.13), (2.14) and the approximation e−βHfluc = e−βHG [ 1−β∆H +O ( ∆H 2 ) ] (3.14) are used. The approximation (3.11) is valid as long as the correction term is not larger than the MF result. When considering particular cases we shall verify if this is the case. The fluctuation contribution to the direct correlation function (2.15) is obtained by calculating the second derivative of the second term on the RHS of (2.8) with respect to ζ̄. In the consistent approximation we insert in the obtained expression the appropriate derivatives of Hfluc (equation (2.10)) with the expansion in φ truncated at the second order. The result is given by [5, 6, 19, 21] C̃2(k,ζ) ≈ C̃ co 2 (k,ζ)+ A4(ζ) 2 G (ζ). (3.15) Equations (3.15), (2.17) and (3.13) should be solved self-consistently. The fluctuation induced shift of the volume fraction, ∆ζ = ζ̄− ζ0, can be obtained from (2.11) by ex- panding the first term on the LHS about ζ0, C̃ co 2 (0,ζ0)∆ζ+ ∞ ∑ n=2 C co n+1[ζ0] n! ∆ζn =− 〈 δ(βHfluc) δζ̄(r) 〉 . (3.16) For small ∆ζ we can truncate the expansion in (3.16) at the first term. When the RHS in (3.16) is approxi- mated as in the calculation of µ̄ from (2.11), we obtain the result ∆ζ≈− A3(ζ0) 2C̃2(0,ζ0) G (ζ̄) ≈− A3(ζ0) 2C̃2(0,ζ0)+ A3(ζ0)G ′(ζ0) G (ζ0). (3.17) We used the approximations: A3(ζ̄) = A3(ζ0)+ A4(ζ0)∆ζ+O ( ∆ζ2 ) , G (ζ̄) = G (ζ0)+G ′(ζ)∆ζ+O ( ∆ζ2 ) and equation (3.15). For the potential given in (3.7) the approximate form of G is [5, 19, 21, 22] G (ζ) = 2a p T ∗ Z (ζ) , (3.18) where a = k2 b 4π √ v∗ 2 (3.19) and Z (ζ) = √ C̃2(kb,ζ). The above approximation is valid for C̃2(kb,ζ) ≪ β∗v∗ 2 k2 b [5, 19, 21, 22]. The equation (3.15) for k = kb takes the form Z (ζ)3 = Z (ζ)C̃ co 2 (kb,ζ)+ A4(ζ)a p T ∗, (3.20) and the explicit expression for Z is Z (ζ)= W (ζ) 6 + 2C̃ co 2 (kb,ζ) W (ζ) (3.21) with W (ζ) = { 108A4(ζ)a p T ∗+12 √ −12C̃ co 2 (kb,ζ)3 +81 [ A4(ζ)a p T ∗ ]2 }1/3 . (3.22) 23604-6 Equation of state in cluster-forming systems The fluctuation contribution in equation (2.16) for the approximations (3.15)–(3.20) was calculated in references [5, 19, 21, 22], and has the form βΩ[ζ̄]≈βΩco[ζ̄]+2a p T ∗Z (ζ̄)V − A4(ζ̄)G (ζ̄)2 8 V . (3.23) Taking into account (3.11) and (3.2), we obtain from (3.23) the explicit form of the EOS βP (ζ̄) =βP MF(ζ̄)+F (ζ̄,T ∗), (3.24) where βP MF(ζ̄) =− β∗v∗ 0 2 ζ̄2 + ζ̄ dβ fh(ζ̄) dζ̄ −β fh(ζ̄) =− β∗v∗ 0 2 ζ̄2 +ρ∗ ζ̄ 2 + ζ̄+1 ( 1− ζ̄ )3 (3.25) and F (ζ̄,T ∗) = a p T ∗A3(ζ̄)ζ̄ Z (ζ̄) −2a p T ∗Z (ζ̄)+ a2 A4(ζ̄)T ∗ 2Z (ζ̄)2 = a p T ∗ 2 [ 2A3(ζ̄)ζ̄− C̃ co 2 (kb, ζ̄) Z (ζ̄) −3Z (ζ̄) ] . (3.26) The second equality in (3.25) is valid for the PY approximation for fh. In order to obtain the last equality in (3.26), equation (3.20) was used. 3.2. Fluctuations around the most probable volume fraction In the previous subsection we considered fluctuations about the average value, which in general dif- fers from the most probable value of the volume fraction. In principle, it is possible to consider equations analogous to (2.8) and (2.10), but with ζ̄ replaced by ζ0. In this new approach equation (2.12) is satisfied, and thus the expansion in (2.10) starts with n = 2 (C co 1 [ζ0] = 0). On the other hand, 〈φ〉 = ζ̄−ζ0 , 0. The results obtained in the two approaches—with included fluctuations around the average value or around the most probable value— should be the same in the exact theory. However, when the fluctuation contri- bution is obtained in an approximate theory, the results may depend on the validity of the assumptions made in the two approaches. In this section we derive an alternative version of the EOS, based on the con- tribution from the fluctuations around the most probable value. From (2.12) we obtain for the chemical potential µ̄(ζ0) = µ̄MF(ζ0) (3.27) with µ̄MF given in equation (3.12). We consider (2.13) with ζ̄ replaced by ζ0, and the approximation (3.14). In the above, HG is given in equation (2.14) with C̃2(k,ζ0) = 1/G̃2(k,ζ0), where G̃2(k,ζ0) = 〈φ̃(k)φ̃(−k)〉, and ∆H = 1 2 ∫ r′ ∫ r′′ φ(r ′) [ C co 2 ( r ′−r ′′,ζ0 ) −C2 ( r ′−r ′′,ζ0 )] φ(r ′′)+ ∫ r A3(ζ0) 3! φ(r)3 + ∫ r A4(ζ0) 4! φ(r)4 + . . . (3.28) Note that the function G̃2(k,ζ0) defined here differs from the correlation function, because in this case 〈φ(r)〉, 0. Taking into account that for HG of the form (2.13) there holds ∫ Dφe−βHGφ2n+1 = 0, we obtain the expression 〈φ(r)〉 ≈ − ∫ Dφ 1 3! ∫ r′ ∫ r′′ ∫ r′′′ A3(ζ0)φ ( r ′)φ ( r ′′)φ ( r ′′′)φ(r)e−βHG Ξfluc . (3.29) Finally, the lowest-order result is ζ̄= ζ0 +〈φ〉 ≈ ζ0 − A3(ζ0) 2C̃2(0,ζ0) G (ζ0). (3.30) 23604-7 A. Ciach, O. Patsahan At the same level of approximation C̃2(k,ζ0) is given in equation (3.15), except that all quantities are calculated at ζ0 which satisfies (2.12) rather than (2.11). This can be verified by a direct calculation of 〈φ(r1)φ(r2)〉 with the help of (3.14) and (3.28), in an approximation analogous to (3.29) (see refer- ence [21]). The above shift of the volume fraction differs from (3.17), because instead of G (ζ̄), there appears G (ζ0). The dependence of µ̄ on the average volume fraction is given in equations (3.27) and (3.30), with eliminated ζ0. In order to evaluate the EOS, we consider an equation analogous to (3.23), with Ωco[ζ0] calculated at its minimum ζ= ζ0. The EOS takes the form βP (ζ0) =βP MF(ζ0)+F0(ζ0,T ∗), (3.31) where βP MF is defined in (3.25), ζ0 satisfies (2.12), and F0(ζ0,T ∗) =−2a p T ∗Z (ζ0)+ a2 A4(ζ0)T ∗ 2Z (ζ0)2 . (3.32) The dependence of P on the average volume fraction ζ̄ is given by parametric equations (3.31) with (3.32) and (3.30). The approximate theory developed in this section is valid for small ∆ζ since we assumed that the relevant fluctuations are small and truncated the expansion in (2.10) at the term ∝ φ4. Moreover, to evaluate ∆ζ we neglected the terms of the order O ( ∆ζ2 ) . We may expect that if we obtain large ∆ζ and large discrepancies between the results obtained by the two methods, then the approximate theory is not sufficiently accurate. 3.3. Comparison between the two methods Let us focus on the chemical potential, and compare the two expressions, equations (3.11) and (3.27) where ζ̄ and ζ0 satisfy equations (2.11) and (2.12), respectively. We expand the RHS in equation (3.11) about ζ0, µ̄ ≈ βṼco(0)ζ0 +β f ′ h(ζ0)+∆ζ [ βṼco(0)+ A2(ζ0)+ A4(ζ0) 2 G (ζ0) + A3(ζ0) 2 G ′(ζ0) ] + A3(ζ0) 2 G (ζ0)+O ( ∆ζ2 ) . (3.33) From (3.17), (3.15) and (3.27) we obtain an equality of the two expressions for the chemical potential to the linear order in ∆ζ, when ∆ζ is given in (3.17). Similarly, to compare the two expressions for the EOS, equations (3.24) and (3.31), we expand the RHS of equation (3.24) about ζ0 to the linear order in ∆ζ. Taking into account (3.17) and (3.20), we arrive at equation (3.31), up to the terms proportional to A5. The latter are disregarded in an approximation consistent with the φ4 theory for the fluctuation contribution considered in this work. For relatively large ∆ζ, when the terms beyond the linear order become important, discrepancies between the results obtained by the two methods should be expected. 4. Explicit results for two model potentials In this subsection we shall compare the expressions for the chemical potential and for the pressure obtained by the two approaches for two systems showing a qualitatively different behavior. We shall evaluate the EOS (3.24) for the representative model SALR potential, Vco(r ) = [ − A1 r e−z1r + A2 r e−z2r ] θ(r −1), (4.1) 23604-8 Equation of state in cluster-forming systems where zi is the inverse range in σ−1 units. The function θ(r −1) is a very crude approximation for the pair distribution function. In Fourier representation, the above SALR potential takes the form Ṽco(k) = 4π [ A2e−z2 z2 2 +k2 ( z2 sin k k +cos k ) − A1e−z1 z2 1 +k2 ( z1 sin k k +cos k ) ] . (4.2) We choose two sets of parameters, considered in reference [11] in the context of most probable inho- mogeneous structures System 1 : A1 = 1, A2 = 0.05, z1 = 3, z2 = 0.5; System 2 : A1 = 1, A2 = 0.2, z1 = 1, z2 = 0.5. (4.3) The relevant parameters, kb, v∗ 2 and v∗ 0 , (see (3.9) and (3.10)) take the values: v∗ 2 ≈ 3.02 and System 1 : kb ≈ 1.79, v∗ 0 ≈−30.145; System 2 : kb ≈ 0.6089, v∗ 0 ≈ 0.035. (4.4) The two potentials in Fourier representation are shown in figure 2. In the first system small clusters are formed, since 2π/kb is small. Moreover, Ṽco(0) > 0, and the clusters repel each other. The gas-liquid separation is entirely suppressed due to the very short range of the attractive part of the potential. In the second system large clusters are formed, and Ṽco(0) < 0 (the clusters attract each other). Therefore, in MF, the metastable separation into disordered low- and high density phases occurs at low tempera- ture. Simulation results show that when large clusters are formed, gas-liquid separation occurs for low temperatures, and periodic phases are stable at higher temperatures [9]. Figure 2. The potential Ṽco(k) for System 1 (solid line) and for System 2 (dashed line). Ṽco(k) is in dimensionless units, k is in units σ−1, where σ is the particle diameter. Figure 3. The universal structural line (solid) in re- duced units (see (3.8)) and themetastable spinodal line of the separation into dilute and dense phases (dashed) for System 2. Note that the temperature scale is different from the corresponding scale in the theory based on ρ∗ [11], and the scaling factor is (6/π)2. We are interested mainly in the part of the phase diagram where the homogeneous structure is less probable than periodic distribution of particles in space, i.e. when Ωco does not assume a minimum for ζ0 = const (see (2.3)). We stress that the most probable structure differs from the average structure due to mesoscopic fluctuations. Cluster formation is associated with the excess volume fraction followed by a depleted volume fraction in mesoscopic regions, and the most probable mesoscopic structure associ- ated with cluster formation is periodic. Displacements of the clusters (i.e., mesoscopic fluctuations) can 23604-9 A. Ciach, O. Patsahan destroy the long-range order, though. Indeed, when temperature is sufficiently high, the average volume fraction ζ(r) takes the constant value as a result of the averaging over cluster displacements, and the dis- ordered inhomogeneous structure with short-range correlations of the cluster positions is found [22, 23]. On the other hand, for low temperatures, the ordered periodic structures are stable [22, 23]. The phase- space region where the inhomogeneous phases (with either short- or long-range order) are stable is en- closed by the structural line [5, 23] given by T ∗ = 1/A2(ζ) and shown in figure 3. The structural line is also referred to as λ-line in literature [9, 10, 24–26]. Note that in the reduced units (see (3.8)) the structural line is universal. Figure 4. ∆ζ(ζ0) for System 1. Dotted, solid, dash-dotted, and dash lines correspond to T∗ = 0.02,0.01,0.007 and 0.0008, respectively. equa- tions (3.30) and (3.16) are not distinguishable on the plot. Figure 5. ∆ζ(ζ0) for System 2 from equation (3.30). Dotted, solid, dash-dotted and dash lines cor- respond to T∗ = 0.02,0.01,0.007,0.0008, respec- tively. Figure 6. Comparison of the two expressions, equa- tions (3.30) and (3.16), for ∆ζ(ζ0) for System 2. The lines with smaller \|∆ζ\| correspond to T∗ = 0.02; the solid line represents equation (3.16), and the dotted line represents equation (3.30). The remaining lines represent equation (3.16) (solid) and equation (3.30) (dashed) for T∗ = 0.01. We first compare the change of the average volume fraction induced by mesoscopic fluctu- ations. The shift ∆ζ calculated from (3.30) and (3.16) in System 1 is shown in figure 4. The shift is small for a relevant range of temperatures, and both formulas yield practically the same result — they are indistinguishable on the plot. The shift in- creases for a decreasing temperature. In System 2, the fluctuation contribution to the volume frac- tion is much larger than in System 1 (figure 5). As expected, when ∆ζ is not very small, ∆ζ/ζ0 ≈ 0.2, then the two approaches yield somewhat different results, as shown in figure 6 for System 2. In the next step, we study the chemical poten- tial. The fluctuation contribution in System 1 is small, except at very small volume fractions (fig- ure 7), whereas in System 2 it is substantial, and increases for decreasing temperature, as shown in figures 8 and 9. The two approaches yield sim- ilar results for small ∆ζ, whereas when ∆ζ/ζ0 > 0.25, significant discrepancy between the two ap- proaches is obtained. We can conclude that on the quantitative level the approximate theory is over- 23604-10 Equation of state in cluster-forming systems Figure 7. Left panel: the relative difference, (µ̄− µ̄MF)/µ̄MF, between µ̄ calculated from equation (3.11), and the MF approximation µ̄MF (3.12) for System 1 at T∗ = 0.007 and ζÊ 0.01. Right panel: µ̄− µ̄MF for System 1 (solid line) and System 2 (dashed line) for 0.0005 É ζÉ 0.1. simplified for the range of T and ζ̄ for which there are significant discrepancies between the two ap- proaches. Note that since µ̄− µ̄MF is large for volume fractions ζ < 0.03, for very small volume fractions our results are oversimplified. Finally, we present the isotherms obtained from (3.24) and (3.31) for the two systems in figures 10–14. In System 1 the pressure is much higher than found in MF, and for all temperatures it monotonously increases with ζ̄, as shown in figure 10. The increased pressure associated with mesoscopic fluctuations may result from the repulsion between the clusters, because in this case Ṽco(0) > 0. Figure 8. Chemical potential in MF (dotted line), and with the fluctuation contribution included according to equation (3.11) (solid) and equa- tion (3.27) with (3.30) (dashed line) for T∗ = 0.015 in System 2. The volume fraction is dimensionless, and the chemical potential is in πkT /6 units. Figure 9. Chemical potential in MF (dotted line), and with the fluctuation contribution included according to equation (3.11) (solid line), equa- tion (3.27) with equation (3.30) (dashed line), and equation (3.27) with equation (3.16) (dashed- dotted line) for T∗ = 0.007 in System 2. The vol- ume fraction is dimensionless, and the chemical potential is in πkT /6 units. 23604-11 A. Ciach, O. Patsahan Figure 10. The EOS isotherms for System 1 for T∗ = 0.02. Dashed line is the MF re- sult (equation (3.25)) and solid line represent equation (3.24), and equation (3.31) with equa- tion (3.30), indistinguishable on the plot. Figure 11. The EOS isotherms (3.24) (solid line) and (3.31) with equation (3.30) (dash-dotted line), and the MF approximation (3.25) (dashed line) for System 2 for T∗ = 0.015. Since in System 2 Ṽco(0) < 0, a mechanical instability develops at the MF spinodal line, with the metastable MF critical point T ∗ c ≈ 0.0009. What is really interesting is that such instability appears at much higher temperatures due to mesoscopic fluctuations. This is in strong contrast to the fluids with purely attractive interactions, where density fluctuations decrease the critical temperature with respect to the mean-field estimate. The present case with dominant fluctuations associated with mesoscopic wavelengths bears some resemblance to the restricted primitive model (RPM) of ionic systems. There is no gas-liquid instability in the RPM at the MF level of a mesoscopic theory analogous to the one con- sidered here, but when the short-wavelength charge-density fluctuations are included, such instability appears [24]. One could imagine that the mesoscopic fluctuations, i.e., displacements of the clusters from their most probable locations lead to their coalescence when Ṽco(0) < 0, and thus support the phase sep- aration. Note that properties of System 2 are completely different from the previously studied System 1. Figure 12. The EOS isotherms (3.24) (solid line) and (3.31) with equation (3.30) (dash-dotted line), and the MF approximation (3.25) (dashed line) for System 2 for T∗ = 0.0093. Left and right panels show large and small range of volume fractions, respectively. 23604-12 Equation of state in cluster-forming systems Figure 13. The EOS isotherms (3.24) (solid line) and (3.31) with equation (3.30) (dash-dotted line), and the MF approximation (3.25) (dashed line) for System 2 for T∗ = 0.007. Figure 14. The EOS isotherms (3.24) (solid line) and (3.31) with equation (3.30) (dash-dotted line), and the MF approximation (3.25) (dashed line) for System 2 for T∗ = 0.005. In this region of the phase diagram, the shift of the volume fraction is large. Therefore, on the quanti- tative level, the results are not sufficiently accurate. The inflection point on the P (ζ̄) isotherm appears at T ∗ c ≈ 0.0093 or T ∗ c = 0.007 according to (3.31) or (3.24) with (3.30), respectively. Both temperatures, how- ever, are much higher than in MF. Further studies are required to verify if the separation into disordered inhomogeneous phases, or periodic ordering of clusters occurs. If the ordered phases are formed, the still open question is for which part of the phase diagram such phases are globally stable. Finally, let us focus on the pressure for very small volume fractions. In the fluctuation correction to pressure (equation (3.26)) the first term comes from the fluctuation contribution to chemical poten- tial (see (3.11)). As shown in figure 7 (right panel), for very low volume fractions, our approximation is oversimplified, so the negative pressure is an artifact. For very low volume fractions, we should expect a perfect gas behavior, except that some fraction of particles should form clusters. Pressure should be pro- portional to the sum of the number densities of monomers and clusters. Since the number of clusters is smaller than the number of particles forming them, pressure should be smaller than in the corresponding perfect gas of isolated particles. Our theory agrees with this expectation (see figure 12, right panel). 5. Summary In this work, the effects ofmesoscopic fluctuations on the average volume fraction, chemical potential and pressure as functions of temperature and the average volume fraction were considered within the framework of the mesoscopic theory [5, 6]. We restricted our attention to a stable or metastable disor- dered phase. The fluctuation contribution to the quantities mentioned above was calculated in two ways. First, we considered the fluctuations about the average volume fraction, and derived equations (3.16), (3.11) and (3.24) for the volume fraction, chemical potential and pressure, respectively. In the second version, we considered the fluctuations about the most probable volume fraction, and obtained equa- tions (3.30), (3.27) and (3.31), with ζ0 satisfying (2.12). The chemical potential and the EOS as functions of the average volume fraction are given by parametric equations (3.30) and (3.27), and (3.31), respectively. Our expressions are derived under the assumption that the dominant fluctuations are of small ampli- tudes. Consistent with the above assumption, the two methods yield the same result to a linear order in the fluctuation contribution to the volume fraction ∆ζ. The fluctuation contributions to all three quantities were explicitly calculated for two versions of the SALR potential. In System 1 the zeroth moment of the effective interactions is positive and small clusters 23604-13 A. Ciach, O. Patsahan are formed. In System 2 the zeroth moment of the effective interactions is negative, and the clusters are large. We obtain nearly the same results independently of the method used when the fluctuation- induced shift of the volume fraction is very small (System 1). When ∆ζ/ζ0 ∼ 0.2 (low temperature in System 2), significant discrepancies between the twomethods appear for some part of the phase diagram. The largest discrepancies are present when both methods yield the results that strongly deviate from the MF predictions. The larger is the probability of finding inhomogeneous mesoscopic states compared to the homogeneous distribution of particles, the stronger are the discrepancies between the two methods. It is in this part of the phase diagram that the periodic order may appear. We conclude that the first method is superior to the second one because it is easier to implement. Exact results would be necessary to get a comparison between the accuracy achieved by these methods. We have found that mesoscopic fluctuations play a very important role and lead to a significant change of the chemical potential and pressure. The larger is the probability of finding the inhomo- geneities, i.e., the further away from the structural line on the low-T side of it (figure 2), the larger is the role of fluctuations. When small clusters are formed (System 1 in section 4), the fluctuation contri- bution to pressure increases monotonously with an increasing volume fraction. By contrast, for large clusters (System 2 in section 4) the fluctuation contribution to pressure is nonmonotonous; it is negligi- ble for small as well as for large volume fractions, whereas for intermediate volume fractions it is large and increases with a decreasing temperature (figures 10–13). Moreover, an inflection point at the pres- sure – volume fraction isotherm appears at the temperature and volume fraction both much larger than found in MF. Further studies are required to verify if the periodically ordered cluster phases are stable in System 2, or phase separation occurs due to the mechanical instability. Possible scenarios are: (i) phase separation at low T , and periodic structures at higher T , or (ii) the phase separation is only metastable, and finally (iii) the periodically ordered phases are only metastable. For System 1, the phase separation is not expected. In addition to the assumptions discussed earlier, we make an approximation concerning the form of the effective potential Vco. Note that in equation (4.1) we assumed that the pair distribution function vanishes for r < 1 in σ units. For the volume fraction, this is a poor approximation, and quantitative results for the structural line depend on the form of the pair distribution function (regularization of the potential [27]). In the future studies, the EOS for periodically ordered cluster phases should be determined in order to find the phase diagram. Our results indicate that despite the universal properties of the dependence of the most probable structures on the thermodynamic state, the effect of fluctuations on the average distribution of particles may depend on the form of the interaction potential, especially on the sign of the zeroth moment of the effective interactions. Acknowledgements A part of this work was realized within the International PhD Projects Programme of the Foundation for Polish Science, co-financed from European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation”. Partial support by the Ukrainian-Polish joint research project under the Agreement on Scientific Collaboration between the Polish Academy of Sciences and the National Academy of Sciences of Ukraine for years 2009–2011 is also gratefully acknowledged. 23604-14 Equation of state in cluster-forming systems References 1. Stradner A., Sedgwick H., Cardinaux F., Poon W.C.K., Egelhaaf S.U., Schurtenberger P., Nature, 2004, 432, 492; doi:10.1038/nature03109. 2. Campbell A.I., Anderson V.J., van Duijneveldt J.S., Bartlett P., Phys. Rev. Lett., 2005, 94, 208301; doi:10.1103/PhysRevLett.94.208301. 3. Porcar L., Falus P., Chen W.-R., Faraone A., Fratini E., Hong K., Baglioni P., Liu Y., J. Phys. Chem. Lett., 2010, 1, 126; doi:10.1021/jz900127c. 4. Stiakakis E., Petekidis G., Vlassopoulos D., Likos C.N., Iatrou H., Hadjichristidis N., Roovers J., Europhys. Lett., 2005, 72, 664; doi:10.1209/epl/i2005-10283-y. 5. Ciach A., Phys. Rev. E, 2008, 78, 061505; doi:10.1103/PhysRevE.78.061505. 6. Ciach A., Mol. Phys., 2011, 109, 1101; doi:10.1080/00268976.2011.638329. 7. Sear R.P., Gelbart W.M., J. Chem. Phys., 1999, 110, 4582; doi:10.1063/1.478338. 8. Archer A.J., Wilding N.B., Phys. Rev. E, 2007, 76, 031501; doi:10.1103/PhysRevE.76.031501. 9. Archer A.J., Pini D., Evans R., Reatto L., J. Chem. Phys., 2007, 126, 014104; doi:10.1063/1.2405355. 10. Archer A.J., Phys. Rev. E, 2008, 78, 031402; doi:10.1103/PhysRevE.78.031402. 11. Ciach A., Góźdź W.T., Condens. Matter Phys., 2010, 13, 23603; doi:10.5488/CMP.13.23603. 12. De Candia A., Del Gado E., Fierro A., Sator N., Tarzia M., Coniglio A., Phys. Rev. E, 2006, 74, 010403(R); doi:10.1103/PhysRevE.74.010403. 13. Kowalczyk P., Ciach A., Gauden P.A., Terzyk A.P., J. Colloid Interface Sci., 2011, 363, 579; doi:10.1016/j.jcis.2011.07.043. 14. Sciortino F., Mossa S., Zaccarelli E., Tartaglia P., Phys. Rev. Lett., 2004, 93, 055701; doi:10.1103/PhysRevLett.93.055701. 15. Sciortino F., Tartaglia P., Zaccarelli E., J. Phys. Chem. B, 2005. 109, 21942; doi:10.1021/jp052683g. 16. Toledano J., Sciortino F., Zaccarelli E., Soft Matter, 2009, 5, 2390; doi:10.1039/b818169a. 17. Shukla A., PNAS, 2008, 105, 5075; doi:10.1073/pnas.0711928105. 18. Zhang T.H., Groenewold J., Kegel W.K., Phys. Chem. Chem. Phys., 2009, 11, 10827; doi:10.1039/b917254h. 19. Ciach A., Góźdź W.T., Stell G., J. Phys.: Condens. Matter, 2006, 18, 1629; doi:10.1088/0953-8984/18/5/016. 20. Evans R., Adv. Phys., 1979, 28, 143l doi:10.1080/00018737900101365. 21. Patsahan O., Ciach A., J. Phys.: Condens. Matter, 2007, 19, 236203; doi:10.1088/0953-8984/19/23/236203. 22. Brazovskii S.A., Sov. Phys. JETP, 1975, 41, 85. 23. Ciach A., Patsahan O., Phys. Rev. E, 2006, 74, 021508; doi:10.1103/PhysRevE.74.021508. 24. Ciach A., Stell G., J. Mol. Liq., 2000, 87, 255; doi:10.1016/S0167-7322(00)00125-2. 25. Ciach A, Góźdź W.T., Evans R, J. Chem. Phys., 2003, 118, 3702; doi:10.1063/1.1539046. 26. Stell G., In: New Approaches to Problems in Liquid-State Theory, edited by C. Caccamo, J.-P. Hansen, and G. Stell, Kluwer Academic Publishers, Dordrecht, 1999. 27. Patsahan O.V., Mryglod I.M., Condens. Matter Phys., 2004, 7, 755. 23604-15 http://dx.doi.org/10.1038/nature03109 http://dx.doi.org/10.1103/PhysRevLett.94.208301 http://dx.doi.org/10.1021/jz900127c http://dx.doi.org/10.1209/epl/i2005-10283-y http://dx.doi.org/10.1103/PhysRevE.78.061505 http://dx.doi.org/10.1080/00268976.2011.638329 http://dx.doi.org/10.1063/1.478338 http://dx.doi.org/10.1103/PhysRevE.76.031501 http://dx.doi.org/10.1063/1.2405355 http://dx.doi.org/10.1103/PhysRevE.78.031402 http://dx.doi.org/10.5488/CMP.13.23603 http://dx.doi.org/10.1103/PhysRevE.74.010403 http://dx.doi.org/10.1016/j.jcis.2011.07.043 http://dx.doi.org/10.1103/PhysRevLett.93.055701 http://dx.doi.org/10.1021/jp052683g http://dx.doi.org/10.1039/b818169a http://dx.doi.org/10.1073/pnas.0711928105 http://dx.doi.org/10.1039/b917254h http://dx.doi.org/10.1088/0953-8984/18/5/016 http://dx.doi.org/10.1080/00018737900101365 http://dx.doi.org/10.1088/0953-8984/19/23/236203 http://dx.doi.org/10.1103/PhysRevE.74.021508 http://dx.doi.org/10.1016/S0167-7322(00)00125-2 http://dx.doi.org/10.1063/1.1539046 A. Ciach, O. Patsahan Вплив мезоскопiчних флуктуацiй на рiвняння стану кластероутворювальних систем А. Цях1, О. Пацаган2 1 Iнститут фiзичної хiмiї, Польська академiя наук, 01-224 Варшава, Польща 2 Iнститут фiзики конденсованих систем Нацiональної академiї наук України, вул. Свєнцiцького, 1, 79011 Львiв, Україна Рiвняння стану для систем частинок, що самоскупчуються в агрегати, є отримане в рамках мезоскопi- чної теорiї, що поєднує метод функцiоналу густини i теоретико-польовий пiдхiд. Ми дослiджуємо вплив мезоскопiчних флуктуацiй у невпорядкованiй фазi. Явно обчислено iзотерми ‘тиск – об’ємна частка’ для двох наборiв параметрiв потенцiалу короткосяжне притягання плюс далекосяжне вiдштовхування. В ко- жному випадку врахування мезоскопiчних флуктуацiй приводить до пiдвищення тиску, за винятком дуже малих об’ємних часток. Коли утворюються великi кластери, механiчна нестiйкiсть системи присутня при набагато вищих температурах, нiж це було отримано в наближеннi середнього поля. В цьому випадку фазове вiдокремлення конкурує iз формуванням перiодичних фаз (колоїдних кристалiв). У випадку малих кластерiв механiчна нестiйкiсть, пов’язана з вiдокремленням в розрiджену i густу фази, не виникає. Ключовi слова: кластери, самоскупчення, рiвняння стану, мезоскопiчнi флуктуацiї 23604-16 Introduction Short summary of the mesoscopic description Approximate results for the fluctuation contributions to the EOS, density and chemical potential Fluctuations around the average volume fraction Fluctuations around the most probable volume fraction Comparison between the two methods Explicit results for two model potentials Summary

Effect of mesoscopic fluctuations on equation of state in cluster-forming systems

Institution

Ähnliche Einträge