Critical behaviour of conﬁned systems

The critical phenomena and peculiarities of phase transitions in the conﬁned ﬂuid systems are investigated. A system with the geometry of a planeparallel layer is chosen in order to discuss the inﬂuence of the space limitations on the critical characteristics of ﬂuids. The main ideas of the Munster...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	Condensed Matter Physics
Datum:	2000
Hauptverfasser:	Chalyi, A.V., Chalyi, K.A., Chernenko, L.M., Vasilev, A.N.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2000
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/121021
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:	Critical behaviour of conﬁned systems / A.V. Chalyi, K.A. Chalyi, L.M. Chernenko, A.N. Vasilev // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 335-358. — Бібліогр.: 11 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-121021
record_format	dspace
spelling	Chalyi, A.V. Chalyi, K.A. Chernenko, L.M. Vasilev, A.N. 2017-06-13T13:09:31Z 2017-06-13T13:09:31Z 2000 Critical behaviour of conﬁned systems / A.V. Chalyi, K.A. Chalyi, L.M. Chernenko, A.N. Vasilev // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 335-358. — Бібліогр.: 11 назв. — англ. 1607-324X DOI:10.5488/CMP.3.2.335 PACS: 05.70.Fh, 05.70.Jk https://nasplib.isofts.kiev.ua/handle/123456789/121021 The critical phenomena and peculiarities of phase transitions in the conﬁned ﬂuid systems are investigated. A system with the geometry of a planeparallel layer is chosen in order to discuss the inﬂuence of the space limitations on the critical characteristics of ﬂuids. The main ideas of the Munster iteration procedure were used to ﬁnd the pair and the direct correlation functions. Such an important characteristic of the system as the correlation length was found and correspondent results were analyzed in the terms of the scaling theory. Special attention is paid to the calculation of the shifts of the critical parameters (critical temperature and density). The three-moment approximation is used to investigate anisotropic liquids. The system of the Ornstein-Zernike (OZ) integral equations is involved to investigate the correlative properties of the binary ﬂuid mixtures. It is shown for the ﬂuids with the isomorphic character of the interaction that the approximation may be used that makes the system similar to the OZ one- component liquid model. The asymptotic formulae for the pair correlation functions are found and the validity of the Munster method for the binary mixtures is considered. The peculiarities of critical light opalescence for the systems with the special geometry are considered. Досліджено критичні явища та особливості фазових переходів в обмежених рідких системах. Для з’ясування характеру впливу просторової обмеженості на критичні характеристики рідини обрана система з геометрією плоского паралельного прошарку. З метою знаходження парних та прямих кореляційних функцій було використано ідеї ітераційного методу Мюнстера. Отримано вираз для радіуса кореляції флуктуацій параметра порядку і відповідні результати проаналізовані в термінах гіпотези подібності. Окрему увагу приділено розрахунку зсуву критичних параметрів (температури та густини). Для дослідження анізотропних систем використано тримоментне наближення. Аналіз кореляційних властивостей бінарних рідких сумішей проводився з використанням системи інтегральних рівнянь Орнштейна-Церніке (ОЦ). Показано, що для рідин з ізоморфним характером міжмолекулярної взаємодії може бути використане наближення, яке спрощує задачу до моделі однокомпонентної рідини. Знайдено асимптотичні формули для парних кореляційних функцій і показано принципову можливість застосування методу Мюнстера для розгляду бінарних сумішей. Розглянуто особливості критичної опалесценції світла для систем зі спеціальною геометрією. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Critical behaviour of conﬁned systems Критична поведінка обмежених систем Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Critical behaviour of conﬁned systems
spellingShingle	Critical behaviour of conﬁned systems Chalyi, A.V. Chalyi, K.A. Chernenko, L.M. Vasilev, A.N.
title_short	Critical behaviour of conﬁned systems
title_full	Critical behaviour of conﬁned systems
title_fullStr	Critical behaviour of conﬁned systems
title_full_unstemmed	Critical behaviour of conﬁned systems
title_sort	critical behaviour of conﬁned systems
author	Chalyi, A.V. Chalyi, K.A. Chernenko, L.M. Vasilev, A.N.
author_facet	Chalyi, A.V. Chalyi, K.A. Chernenko, L.M. Vasilev, A.N.
publishDate	2000
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Критична поведінка обмежених систем
description	The critical phenomena and peculiarities of phase transitions in the conﬁned ﬂuid systems are investigated. A system with the geometry of a planeparallel layer is chosen in order to discuss the inﬂuence of the space limitations on the critical characteristics of ﬂuids. The main ideas of the Munster iteration procedure were used to ﬁnd the pair and the direct correlation functions. Such an important characteristic of the system as the correlation length was found and correspondent results were analyzed in the terms of the scaling theory. Special attention is paid to the calculation of the shifts of the critical parameters (critical temperature and density). The three-moment approximation is used to investigate anisotropic liquids. The system of the Ornstein-Zernike (OZ) integral equations is involved to investigate the correlative properties of the binary ﬂuid mixtures. It is shown for the ﬂuids with the isomorphic character of the interaction that the approximation may be used that makes the system similar to the OZ one- component liquid model. The asymptotic formulae for the pair correlation functions are found and the validity of the Munster method for the binary mixtures is considered. The peculiarities of critical light opalescence for the systems with the special geometry are considered. Досліджено критичні явища та особливості фазових переходів в обмежених рідких системах. Для з’ясування характеру впливу просторової обмеженості на критичні характеристики рідини обрана система з геометрією плоского паралельного прошарку. З метою знаходження парних та прямих кореляційних функцій було використано ідеї ітераційного методу Мюнстера. Отримано вираз для радіуса кореляції флуктуацій параметра порядку і відповідні результати проаналізовані в термінах гіпотези подібності. Окрему увагу приділено розрахунку зсуву критичних параметрів (температури та густини). Для дослідження анізотропних систем використано тримоментне наближення. Аналіз кореляційних властивостей бінарних рідких сумішей проводився з використанням системи інтегральних рівнянь Орнштейна-Церніке (ОЦ). Показано, що для рідин з ізоморфним характером міжмолекулярної взаємодії може бути використане наближення, яке спрощує задачу до моделі однокомпонентної рідини. Знайдено асимптотичні формули для парних кореляційних функцій і показано принципову можливість застосування методу Мюнстера для розгляду бінарних сумішей. Розглянуто особливості критичної опалесценції світла для систем зі спеціальною геометрією.
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/121021
citation_txt	Critical behaviour of conﬁned systems / A.V. Chalyi, K.A. Chalyi, L.M. Chernenko, A.N. Vasilev // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 335-358. — Бібліогр.: 11 назв. — англ.
work_keys_str_mv	AT chalyiav criticalbehaviourofconfinedsystems AT chalyika criticalbehaviourofconfinedsystems AT chernenkolm criticalbehaviourofconfinedsystems AT vasilevan criticalbehaviourofconfinedsystems AT chalyiav kritičnapovedínkaobmeženihsistem AT chalyika kritičnapovedínkaobmeženihsistem AT chernenkolm kritičnapovedínkaobmeženihsistem AT vasilevan kritičnapovedínkaobmeženihsistem
first_indexed	2025-11-25T06:43:14Z
last_indexed	2025-11-25T06:43:14Z
_version_	1850509672667152384
fulltext	Condensed Matter Physics, 2000, Vol. 3, No. 2(22), pp. 335–358 Critical behaviour of confined systems A.V.Chalyi 1 , K.A.Chalyi 2 , L.M.Chernenko 3 , A.N.Vasil’ev 4 1 Department of Physics, National Medical University, 13 Shevchenko Blvd., Kiev 01601, Ukraine 2 Dept. Bio. and Chem. Engineering, Gunma University, Kiryu 376-8515, Japan 3 Institute of Surface Chemistry of the National Academy of Sciences of Ukraine, 31 Nauki Avenue, Kiev 252028, Ukraine 4 Department of Physics, Kiev University, 6 Glushkova Avenue, Kiev 03022, Ukraine Received February 23, 2000 The critical phenomena and peculiarities of phase transitions in the con- fined fluid systems are investigated. A system with the geometry of a plane- parallel layer is chosen in order to discuss the influence of the space limita- tions on the critical characteristics of fluids. The main ideas of the Munster iteration procedure were used to find the pair and the direct correlation functions. Such an important characteristic of the system as the corre- lation length was found and correspondent results were analyzed in the terms of the scaling theory. Special attention is paid to the calculation of the shifts of the critical parameters (critical temperature and density). The three-moment approximation is used to investigate anisotropic liquids. The system of the Ornstein-Zernike (OZ) integral equations is involved to in- vestigate the correlative properties of the binary fluid mixtures. It is shown for the fluids with the isomorphic character of the interaction that the ap- proximation may be used that makes the system similar to the OZ one- component liquid model. The asymptotic formulae for the pair correlation functions are found and the validity of the Munster method for the binary mixtures is considered. The peculiarities of critical light opalescence for the systems with the special geometry are considered. Key words: critical phenomena, phase transitions, confined system, correlation function PACS: 05.70.Fh, 05.70.Jk 1. Introduction A number of achievements of the modern theory of critical phenomena and phase transitions are connected with the application of powerful methods of statistical c© A.V.Chalyi, K.A.Chalyi, L.M.Chernenko, A.N.Vasil’ev 335 A.V.Chalyi et al. physics, namely the method of collective variables by I.R.Yukhnovskii and his col- laborators [1]. The space limitation is one of the main factors determining the critical behaviour of the fluid systems undergoing phase transitions. In general case the whole set of the main results for the critical region may be received using only the pair correlation function of the order parameter fluctuations of the system. So it is crucial to get the information about the pair correlation function for the finite-size systems in order to use it hereafter. As it is well known, the pair correlation function of the density fluctuations may be expressed in the case of the space infinite one-component liquid system with a scaling formula [2]: G2(r) = A exp(−r/Rc)/r (1+η), (1.1) where η ≈ 0.034 is the critical exponent and Rc is the correlation length. Due to the long-ranged correlation at the critical point the correlation length demonstrates an anomalous growth. For many practical and theoretical purposes the OZ model of free fluctuation field [3] could be used. In the framework of this model it is possible to find the asymptotic formulae for the correlation functions and the corresponding correlation length and calculate the shifts of the critical parameters. Namely, for the abovementioned system the zeroth approximation for the pair correlation function is equivalent to the (1.1) when η = 0: G2(r) = A exp(−r/Rc)/r. (1.2) Unfortunately, this asymptotic expression has a singularity when its argument is approaching to zero. Therefore, it would be better to solve this problem using the Munster method. The main idea of this method is an application of the integral and differential OZ equations (or systems of OZ equations in the case of binary mixtures system) and corresponding iteration procedure in order to obtain new correct expressions for the correlation functions [4]. It is important to stress that even after the first step of iteration process the singularity [5] of the pair correlation function disappears. Let us consider the one-component liquid system. The OZ integral equation has the following form [3,6,7]: G2(r1, r2) = f(r1, r2) + ∫ ρ(r′)f(r1, r ′)G2(r ′, r2)dr ′. (1.3) As a rule the direct correlation function f(r1, r ′) differs significantly from zero only when the distance \|r1−r ′\| between the interacting particles is small. So in the case of an isotropic system when f(r1, r ′) = f(\|r1−r ′\|) and G2(r1, r ′) = G2(\|r1−r ′\|) one can transform the equation (1.3) using the Taylor expansion for the direct correlation function to the differential OZ equation [4]: (∇2 − κ2)G2(r) = −f(r) C2 . (1.4) 336 Critical behaviour of confined systems Here, the so-called two-moment approximation was used with two spatial moments of the direct correlation function C0 = ∫ ρ(r)f(r)dr, C2 = 1 6 ∫ ρ(r)f(r)r2dr, and κ2 = (1− C0)/C2, ρ(r) is the local density. When the system is considerably anisotropic one must take into account the first spatial moment of the direct correlation function. For this situation the differential OZ equation is 3 ∑ i,j=1 aij(r2) ∂2G2(r1, r2) ∂x1i∂x1j −−→ C 1 · −→∇1G2(r1, r2)− (1− C0)G2(r1, r2) = = −f(r1, r2), (1.5) where the following definitions were used: −→ C 1(r2) = ∫ ρ(r)f(r, r2)r dr, aij(r2) = aji(r2) = 1 2 ∫ ρ(x1, x2, x3)f(x1, x2, x3\|r2)xixj dx1dx2dx3 and C2 = 1 3 (a11 + a22 + a33). Lets assume that aii(r) ∼= C2 = const, −→ C 1(r) ≡ −→ C 1 and ρ(r) = 〈ρ〉 = const. Then we can get from (1.5) △G2 +G2xy a12 C2 +G2xz a13 C2 +G2yz a23 C2 −−→a −→∇G2 − κ2G2 = − f C2 , (1.6) where vector −→a = −→ C 1/C2. This equation may be simplified for the systems with a special symmetry. 2. Plane-parallel layer system Let us consider the system with the geometry of a plane-parallel layer of the thickness 2h. Such systems have a significant practical importance because many real physical and biological systems are spatially finite-sized and their geometrical form may be taken as a plane-parallel [8]. 2.1. The correlation functions Taking the zeroth boundary conditions to satisfy the limiting transition to the infinite system one may find the direct and the pair correlation functions in the following form [8,9]: G2(ρ, z) = ∞ ∑ m=0 G2(m)(ρ) cos ( π(m+ 0.5)z/h ) (2.1) and f(ρ, z) = ∞ ∑ m=0 f(m)(ρ) cos ( π(m+ 0.5)z/h ) . (2.2) Using the zeroth approximation for the direct correlation function f (0)(r) = C0δ(r)/ 〈ρ〉 and taking into account that δ(z) = 1 h ∞ ∑ m=0 cos( π(2m+ 1)z 2h ), (2.3) 337 A.V.Chalyi et al. one has the following equations for the harmonics of correlation functions from the integral (1.3) and differential (1.4) OZ equations: G2(m)(ρ) = f(m)(ρ) + h〈ρ〉 ∫ f(m)(ρ1)G2(m)(\|−→ρ −−→ρ1 \|) d−→ρ1 (2.4) and (∇2 − κ2m)G2(m)(ρ) = −C0δ(ρ) 〈ρ〉hC2 , (2.5) where κ2m = κ2 + π2(2m+1)2/4h2. These equations may be used for the calculation of the correlation functions by the iteration procedure. So from the (2.5) one can find the asymptotic formula for the pair correlation function [5,8,9]: G (0) 2 (ρ, z) = C0 2πh〈ρ〉C2 ∞ ∑ m=0 K0(ρκm) cos ( π(2m+ 1)/2h ) . (2.6) Using then the integral OZ equation (2.4) it is possible to calculate the first approx- imation for the direct correlation function [5]: f (1)(ρ, z) = C0 2πh〈ρ〉C2 ∞ ∑ m=0 K0(ρ √ κ2m + C0/C2) cos ( π(2m+ 1)/2h ) . (2.7) Then one may find the first iteration for the pair correlation function [5] from equa- tion (2.5): G (1) 2 (ρ, z) = 1 2πh〈ρ〉C2 ∞ ∑ m=0 [K0(ρκm)−K0(ρκ̃m)] cos ( π(2m+ 1)/2h ) , (2.8) where κ̃m = √ κ2m + C0/C2. As it was stressed above the first iteration of the pair correlation function has no singularities and is more realistic than the asymptotic expression. 2.2. The correlation length It is natural to determine the correlation length as the normalized second spatial moment of the pair correlation function [8,9]: Rc = √ ∫ G2(r)r2dr ∫ G2(r)dr , (2.9) From (2.10) one can find, after integrating and leaving only the first harmonics in the (2.8), the following expression for the correlation length: R2 c = R2 xy +R2 z , (2.10) where R2 xy = 4 κ2 + π2 4h2 + 4 κ2 + π2 4h2 + C0 C2 (2.11) 338 Critical behaviour of confined systems and R2 z = ( 1− 8 π2 ) h2. (2.12) As it is seen from the (2.10)–(2.12) at the bulk critical point (κ = 0) the correlation length has a finite value. Thus the space limitation causes the shifts of the critical parameters [5,8]. 2.3. The shifts of the critical parameters According to the scaling-law theory, parameter κ may be presented in the fol- lowing form [2]: κ = τ νf1(△ρ/τβ) = △ρζf2(τ/△ρ1/β). (2.13) Here τ = (T−Tc)/Tc and△ρ = (ρ−ρc)/ρc are the deviations of the temperature and density from their critical values respectively, ν = 0.625, β = 0.325 and ζ = 1.923 are the critical exponents. The asymptotic behaviour of a scaling functions f1(u) and f2(u) are as follows: f1(u→ 0) = κ0 = const, f1(u → ∞) ∼ uζ (2.14) and f2(u→ 0) = κ0 = const, f2(u→ ∞) ∼ uν . (2.15) Thus using the expressions (2.10)–(2.12) for the correlation length and dependence of the parameter κ on the temperature and density one can find the shifts of the critical parameters with respect to its values at the bulk critical point. The new critical temperature is T ∗ c = Tc 1 + (κ0h∆)−1/ν (2.16) and the new critical density is ρ∗c = ρc 1 + (κ0h∆)−1/ζ , (2.17) where ∆ = √ 1 + 8/π2 + 16/(π2 + 4h2C0/C2). These results are in good agreement with the experimental data by Lutz [10]. 3. Anisotropic systems In many cases the real systems are essentially anisotropic and it leads to some considerable effects. Let us consider such systems. 339 A.V.Chalyi et al. 3.1. The spatially nonfinite system In the case of a spatially infinite system the asymptotic formula for the pair correlation function of the order parameter fluctuations [11] can be found from the equation (1.6): G (0) 2 (r) = C0 4π〈ρ〉C2 · exp(ar cos θ/2− r √ κ2 + a2/4) r = C0 4π〈ρ〉C2 · exp(az/2− r √ κ2 + a2/4) r . (3.1) Here the direction of the z-axis coincides with the direction of the vector −→a and θ is the angle between the vectors −→r and −→a . The following iterations may be found from the integral OZ equation (1.3). The first iteration for the direct correlation function is f (1)(r) = C0 4π〈ρ〉C2 · exp(ar cos θ/2− r √ κ2 + a2/4 + C0/C2) r = C0 4π〈ρ〉C2 · exp(az/2 − r √ κ2 + a2/4 + C0/C2) r (3.2) and the first iteration for the pair correlation function is G (1) 2 (r) = 1 4π〈ρ〉C2 · exp(ar cos θ/2− r √ κ2 + a2/4) r − 1 4π〈ρ〉C2 · exp(ar cos θ/2− r √ κ2 + a2/4 + C0/C2) r = 1 4π〈ρ〉C2 · exp(az/2 − r √ κ2 + a2/4) r − 1 4π〈ρ〉C2 · exp(az/2 − r √ κ2 + a2/4 + C0/C2) r . (3.3) The last formula presents the dependence of the correlative properties of the system on the angle θ and may be used to investigate the correlation length and shifts of the critical parameters [11]. 3.2. The correlation length Let’s find the correlation length of the spatially infinite anisotropic system. Tak- ing into account (2.13) and from (3.1) one has the following formula [11]: Rc(τ) = 1 √ κ2 + a2/4− a cos θ/2 = R0 √ τ 2ν + a2R2 0/4− aR0 cos θ/2 . (3.4) 340 Critical behaviour of confined systems Here R0 = κ−1 0 . It is clearly seen that critical temperature in this case depends on the direction and thus the anisotropy of the system causes the change in the critical parameters in comparison with the isotropic system. 3.3. The new critical parameters The new critical temperature T ∗ c may be found from the formula (3.4) for the correlation length: T ∗ c = Tc 1 + (R0a sin 2(θ/2))1/ν . (3.5) In the same way one may find the new critical density: ρ∗c = ρc 1 + (R0a sin 2(θ/2))β/ν . (3.6) The maximal shift of the critical parameters takes place when θ = π and there is no shift (in comparison with isotropic system) when θ = 0. 3.4. Plane-parallel layer Let us consider the anisotropic system with a geometry of a plane-parallel layer. In the situation when −→a = (0, 0, a) one may find, using the (1.3) and (1.6), the asymptotic expression and the first iterations for the direct and the pair correlation functions. So the asymptotic formula for the pair correlation function is G (0) 2 (r) = C0 exp( az 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 K0(ρ √ κ2m + a2/4) cos ( (m+ 0.5)zπ h ) . (3.7) The first approximation for the direct and pair correlation functions are as follows: f (1)(r) = C0 exp( az 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 K0(ρ √ κ̃2m + a2/4) cos ( (m+ 0.5)zπ h ) , (3.8) G (1) 2 (r) = exp(az 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 [ K0(ρ √ κ2m + a2/4)−K0(ρ √ κ̃2m + a2/4) ] × cos ( (m+ 0.5)zπ h ) . (3.9) In the case when vector −→a has no z-component one may accept without any restric- tion of the generality that −→a = (0, a, 0) and get, similar to the previous situation, the asymptotic formula for the pair correlation function G (0) 2 (r) = C0 exp( ay 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 K0(ρ √ κ2m + a2/4) cos ( (m+ 0.5)zπ h ) . (3.10) The first approximation for the direct correlation function is f (1)(r) = C0 exp( ay 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 K0(ρ √ κ̃2m + a2/4) cos ( (m+ 0.5)zπ h ) . (3.11) 341 A.V.Chalyi et al. And the first approximation for the pair correlation function G (1) 2 (r) = exp(ay 2 ) 2πhC2〈ρ〉 ∞ ∑ m=0 [ K0(ρ √ κ2m + a2/4)−K0(ρ √ κ̃2m + a2/4) ] × cos ( (m+ 0.5)zπ h ) . (3.12) When the vector −→a has an arbitrary direction the situation is much more com- plicated. In this case one may take −→a = (0, a sin θ, a cos θ). So the equation (1.6) transforms as follows: ∆G2 + s ∂2G2 ∂y∂z − a sin θ ∂G2 ∂y − a · cos θ · ∂G2 ∂z − κ2G2 = − f C2 , (3.13) where s = a23/ √ a22 · a33 ≈ a23/C2. Making the substitution x → x, y → (y − (sz)/2)/ √ 1− s2/4, z → z and using new variables x, u, z one has the follow equa- tions from (1.3) and (1.6) ∆G2 − a cos θ ∂G2 ∂z − [ 2a sin θ − as cos θ√ 4− s2 ] ∂G2 ∂u − κ2G2 = − f C2 (3.14) and G2(x, u, z) = f(x, u, z) + 〈ρ〉 √ 1− s2 4 × ∫ f(x′, u′, z′)G2(x− x′, u− u′, z − z′)dx′du′dz′. (3.15) The solution of this problem is as follows: 1) For the pair correlation function the asymptotic formula may be presented in such a way G (0) 2 (r) = C0 · exp [( az(2 cos θ − s · sin θ) + ay(2 sin θ − s · cos θ) ) /(4− s2) ] 2πhC2〈ρ〉 √ 1− s2/4 × ∞ ∑ m=0 K0 ( √ ( x2 + (2y − sz)2 4− s2 )( κ2m + a2(1− s · sin(2θ)/2) 4− s2 ) ) × cos ( (m+ 0.5)zπ h ) . (3.16) 2) The first approximation for the direct correlation function f (1)(r) = C0 · exp [( az(2 cos θ − s · sin θ) + ay(2 sin θ − s · cos θ) ) /(4− s2) ] 2πhC2〈ρ〉 √ 1− s2/4 × ∞ ∑ m=0 K0 ( √ ( x2 + (2y − sz)2 4− s2 )( κ̃2m + a2(1− s · sin(2θ)/2) 4− s2 ) ) × cos ( (m+ 0.5)zπ h ) . (3.17) 342 Critical behaviour of confined systems 3) The first approximation for the pair correlation function G (1) 2 (r) = exp [( az(2 cos θ − s · sin θ) + ay(2 sin θ − s · cos θ) ) /(4− s2) ] 2πhC2〈ρ〉 √ 1− s2/4 × ∞ ∑ m=0 [ K0 ( √ ( x2 + (2y − sz)2 4− s2 )( κ2m + a2(1− s · sin(2θ)/2) 4− s2 ) ) −K0 ( √ ( x2 + (2y − sz)2 4− s2 )( κ̃2m + a2(1− s · sin(2θ)/2) 4− s2 ) )] × cos ( (m+ 0.5)zπ h ) . (3.18) In the case when the anisotropy vector has its direction along the z-axis the corre- lation length may be found as follows: (Rc/h) 2 = 1 + 16 4κ2h2 + π2 − 8 a2h2 + π2 − 8ah tanh(ah/2) a2h2 + π2 + 16a2h2 (a2h2 + π2)2 . (3.19) In the limiting transition → 0 it gives the zeroth approximation of the correlation length of the isotropy system [8,9]. 4. The binary fluid mixture The integral OZ system for the binary mixture has the following form: Gij(r) = fij(r) + 2 ∑ k=1 〈ρk〉 ∫ Gik(r− r1)fkj(r1)dr1, (4.1) where fij(r) = fji(r) and Gij(r) = Gji(r) are the direct and the pair correlation functions of the components i and j respectively, 〈ρk〉 is the average density of the component k and i, j, k = 1, 2. It is also possible to find the asymptotic expressions for the correlation functions from the OZ differential system: 2 ∑ m=1 [ Bmj Bij (△− κ2mj)Gim(r) ] = −fij(r) Bij . (4.2) Here Aij=〈ρi〉 ∫ fij(r)dr, Bij=(〈ρi〉/6) ∫ fij(r)r 2dr and κ2ij=κ 2 ji = (δij −Aij)/Bij. 4.1. The zeroth approximation In order to receive the asymptotic formulae for the pair correlation functions let us take the Dirak delta-function as the zeroth approximation for the direct corre- lation functions. Taking in (4.2) fij(r) = Aijδ(r)/〈ρi〉 one may get the asymptotic formula for the pair correlation functions: 343 A.V.Chalyi et al. 1) For the spatially infinite system Gij(r) = Rij 4π△q2r [(S 2 ij − q21) exp(−q1r)− (S2 ij − q22) exp(−q2r)]. (4.3) 2) For the system with the geometry of a plane-parallel layer Gij(ρ, z) = ∞ ∑ m=0 Gij(m)(ρ) cos ( π(m+ 0.5)z/h ) (4.4) and Gij(m) = Rij 2πh△q2 [ (S2 ij − q21)K0 ( ρ √ q21 + π2(2m+ 1)2 4h2 ) − (S2 ij − q22)K0 ( ρ √ q22 + π2(2m+ 1)2 4h2 )] . (4.5) Here the following definitions are used: q21,2 = κ211 + κ222 − 2ξ2κ212 2(1− ξ2) [ 1± √ 1− 4 (κ211κ 2 22 − ξ2κ412)(1− ξ2) (κ211 + κ222 − 2ξ2κ212) 2 ] , (4.6) R12 = 1 1− ξ2 · A11/B11 − A22/B22 B11 − B22 · B12 〈ρ1〉 = 1 1− ξ2 · A11/B11 − A22/B22 B11 − B22 · B21 〈ρ2〉 , (4.7) R11 = 1 1− ξ2 · A11/B11 + ξ2κ212 〈ρ1〉 , (4.8) R22 = 1 1− ξ2 · A22/B22 + ξ2κ212 〈ρ2〉 , (4.9) S2 12 = A11 − A22 A11B22 − A22B11 , (4.10) S2 11 = (A11/B11)κ 2 22 + ξ2κ412 A11/B11 + ξ2κ212 , (4.11) S2 22 = (A22/B22)κ 2 11 + ξ2κ412 A22/B22 + ξ2κ212 (4.12) and △q2 = q22 − q21 , ξ 2 = B12B21/B11B22. At the critical point, the main singular part of the correlation length is deter- mined as follows: 1) For the infinite system R2 c = 1 q21 + 1 q22 = κ211 + κ222 − 2ξ2κ212 κ211κ 2 22 − ξ2κ412 . (4.13) 344 Critical behaviour of confined systems 2) For a plane-parallel layer R2 c = 1 q21 + π2 4h2 + 1 q22 + π2 4h2 . (4.14) The latter formula may be used to obtain the new critical temperature. It is obvious that in the case of a binary mixture system with the geometry of a plane- parallel layer the values of the critical parameters depend on the thickness of the layer as it was before for the one-component system. 4.2. The approximation of asymptotic behaviour Let us make the following substitution in the system (4.1): G ij=〈ρ〉/ √ 〈ρi〉〈ρj〉gij and fij = 〈ρ〉/ √ 〈ρi〉〈ρj〉Cij. The density 〈ρ〉 may be taken in any convenient way. The system of equations (4.1) transforms to the form as follows: gij(r) = Cij(r) + 〈ρ〉 2 ∑ k=1 ∫ gik(r− r1)Ckj(r1)dr1. (4.15) Taking into account the physical sense of the integrals in the (4.1) it is natural to accept that ∫ g12(r− r1)C11(r1)dr1 = ∫ g11(r− r1)C12(r1)dr1 (4.16) and ∫ g12(r− r1)C22(r1)dr1 = ∫ g22(r− r1)C12(r1)dr1. (4.17) Let us take into account the function V (r) = ∫ g12(r− r1)C22(r1)dr1 ∫ g11(r− r1)C12(r1)dr1 = 〈ρ2〉 ∫ G12(r− r1)f22(r1)dr1 〈ρ1〉 ∫ G11(r− r1)f12(r1)dr1 . (4.18) It is natural to suppose that the function V (r) weakly depends on the r e.i. V = const. In this case one has g22(r) = V g11(r), C22(r) = V C11(r) (4.19) and g12(r) = √ V g11(r), C12(r) = √ V C11(r). (4.20) Therefore, the system of the integral equations separates on the three equations similar to the OZ equation for one-component liquids: gij(r) = Cij(r) + 〈ρij〉 ∫ Cij(r1) · gij(r− r1)dr1, (4.21) where 〈ρij〉 = 〈ρ〉(1 + V )/V (i+j−2)/2. So one can find the pair and the direct corre- lation function for the binary mixture. Thus for the spatially infinite system: 345 A.V.Chalyi et al. 1) The asymptotic formula for the pair correlation function G (0) ij (r) = C0V (i+j−2)/2 4πC2(1 + V ) √ 〈ρi〉〈ρj〉 · exp(−κr) r . (4.22) 2) The first iteration for the direct correlation function f (1) ij (r) = C0V (i+j−2)/2 4πC2(1 + V ) √ 〈ρi〉〈ρj〉 · exp(−κ̃r) r . (4.23) 3) The first iteration for the pair correlation function G (1) ij (r) = V (i+j−2)/2 4πC2(1 + V ) √ 〈ρi〉〈ρj〉 · exp(−κr)− exp(−κ̃r) r . (4.24) Here C0 = 〈ρ1〉 ∫ f11(r)dr, C2 = (1/6)〈ρ1〉 ∫ f11(r)r 2dr, κ2 = [1 − (1 + V )C0]/(1 + V )C2 and κ̃2 = κ2 + C0/C2. In the case of a plane-parallel layer one has: 1) The zeroth iteration for the pair correlation function G (0) ij (r) = C0V (i+j−2)/2 2πhC2(1 + V ) √ 〈ρi〉〈ρj〉 ∞ ∑ m=0 K0(ρκm) cos ( πz(2m+ 1) 2h ) . (4.25) 2) The first iteration for the direct correlation function f (1) ij (r) = C0V (i+j−2)/2 2πhC2(1 + V ) √ 〈ρi〉〈ρj〉 ∞ ∑ m=0 K0(ρκ̃m) cos ( πz(2m+ 1) 2h ) . (4.26) 3) The first iteration for the pair correlation function G (1) ij (r) = V (i+j−2)/2 2πhC2(1 + V ) √ 〈ρi〉〈ρj〉 × ∞ ∑ m=0 [K0(ρκm)−K0(ρκ̃m)] cos ( πz(2m+ 1) 2h ) . (4.27) The shift of the critical temperature ∆Tc in this case in comparison with the critical temperature Tc of the one-component liquid is determined as ∆Tc = Tc 1 + (V R2 0C0/C2)−1/2ν . (4.28) The latter formula (4.28) may be used to find the parameter V from the correspond- ing experimental data. 346 Critical behaviour of confined systems 4.3. The diagonal form of the OZ system In the matrix form the integral OZ system may be presented as follows: ĝ(r) = Ĉ(r) + 〈ρ〉 ∫ Ĉ(r1)ĝ(r− r1)dr1. (4.29) Here the following definitions are used: ĝ(r) = ( g11(r) g12(r) g21(r) g22(r) ) and Ĉ(r) = ( C11(r) C12(r) C21(r) C22(r) ) . After the Fourier transforming one has ĝ(q) = Ĉ(q) + 〈ρ〉Ĉ(q)ĝ(q). (4.30) Let us consider the arbitrary matrix Ŝ(q) and Ĝ(q) = Ŝ(q)ĝ(q)Ŝ−1(q) F̂ (q) = Ŝ(q)Ĉ(q)Ŝ−1(q). Then for the new functions Ĝ(q) and F̂ (q) the equation (4.30) has got just the same form: Ĝ(q) = F̂ (q) + 〈ρ〉F̂ (q)Ĝ(q). (4.31) It makes possible to find the Ŝ in such a way that matrix Ĝ will have a diagonal form. For this reason let us calculate the eigenvalues of Ĝ from the equation: Det \|gij(q)− λ(q)δij \| = 0. (4.32) It gives λ1,2 = g11(q) + g22(q) 2 ± √ g212(q) + ( g11(q)− g22(q) 2 )2 . (4.33) The matrix Ŝ may be taken as follows: Ŝ(r) = 1 g12(q) ( [ √ 1 + s(q)2 − s(q)]−1 1 1 [s(q)− √ 1 + s(q)2]−1 ) , (4.34) where s(q) = g11(q)− g22(q) 2g12(q) = C11(q)− C22(q) 2C12(q) . (4.35) Thus one has G1,2(q) = g11(q) + g22(q) 2 ± g12(q) √ 1 + s(q)2 (4.36) and F1,2(q) = C11(q) + C22(q) 2 ± C12(q) √ 1 + s(q)2. (4.37) The system of integral equations (4.29) splits into two integral equations similar to the OZ equation for the one-component liquid system: G1,2(r) = F1,2(r) + 〈ρ〉 ∫ F1,2(r1)G1,2(r− r1)dr1. (4.38) The Munster scheme may be applied to the equation (4.38) in order to calculate the consequent iterations of the correlation functions because the initial iteration for the direct correlation function is commonly used as the delta-function. 347 A.V.Chalyi et al. 4.4. The Munster method for binary mixtures For the general case of binary systems, the Munster method is available for the investigation of the pair correlation functions. Let us consider the differential OZ system in the matrix form: △ĝ(r)− Ĉ−1 2 (Ê − Ĉ0)ĝ(r) = −Ĉ−1 2 Ĉ(r). (4.39) Here were used the definitions Ĉ0 = 〈ρ〉 ∫ Ĉ(r)dr, Ĉ0 = (1/6)〈ρ〉 ∫ Ĉ(r)r2dr and Ê is the unit matrix. It allows us to find the zeroth approximation for the matrix of the pair correlation functions using the initial iteration for the direct correlation functions from (4.39) Ĉ(0)(r) = (Ĉ0/〈ρ〉)δ(r). So, one has got △ĝ(r)− Ĉ−1 2 (Ê − Ĉ0)ĝ(r) = −Ĉ −1 2 Ĉ0 〈ρ〉 δ(r). (4.40) For the spatially infinite system the Fourier transformation of the zeroth approxi- mation for the matrix of the pair correlation functions ĝ(q) is ĝ(0)(q) = [q2Ê + Ĉ−1 2 (Ê − Ĉ0)] −1 Ĉ −1 2 Ĉ0 〈ρ〉 . (4.41) The first iteration for the direct correlation functions may be found using the equa- tion (4.30). It gives Ĉ(1)(q) = [q2Ê + Ĉ−1 2 (Ê − Ĉ0) + Ĉ−1 2 Ĉ0] −1 Ĉ −1 2 Ĉ0 〈ρ〉 . (4.42) It is important to stress that for conserving the initial symmetry of the integral system (4.29) the following equation has to take place: Ĉ2Ĉ0 = Ĉ0Ĉ2. (4.43) Then from the (4.39) one can get the first iteration for the pair correlation functions ĝ(1)(q) = [q2Ê + Ĉ−1 2 (Ê − Ĉ0)] −1 Ĉ −1 2 〈ρ〉 − [q2Ê + Ĉ−1 2 (Ê − Ĉ0) + Ĉ−1 2 Ĉ0] −1 Ĉ −1 2 〈ρ〉 . (4.44) Then, considering the system with the geometry of a plane-parallel layer, one can calculate the correlation functions in the following form: ĝ(ρ, z) = ∞ ∑ m=0 ĝ(m)(ρ) cos ( π(m+ 0.5)z/h ) (4.45) and Ĉ(ρ, z) = ∞ ∑ m=0 Ĉ(m)(ρ) cos ( π(m+ 0.5)z/h ) . (4.46) 348 Critical behaviour of confined systems The integral OZ equation (4.30) gives Ĝm(q) = f̂m(q) + h〈ρ〉f̂m(q)Ĝm(q). (4.47) And differential equation for the harmonics △ĝ(m)(ρ)− [ Ĉ−1 2 (Ê − Ĉ0) + π2(2m+ 1)2 4h2 Ê ] ĝ(m)(ρ) = −Ĉ −1 2 Ĉ0 h〈ρ〉 · δ(ρ). (4.48) Thus, using the analogy with the previous situation, one can find: 1) The zeroth approximation for the matrix of the pair correlation functions harmonics ĝ (0) (m)(q) = [ q 2Ê + π2(2m+ 1)2 4h2 Ê + Ĉ−1 2 (Ê − Ĉ0) ] −1 Ĉ−1 2 Ĉ0 h〈ρ〉 . (4.49) 2) The first approximation for the matrix of the direct correlation functions harmonics Ĉ (1) (m)(q) = [ q 2Ê + π2(2m+ 1)2 4h2 Ê + Ĉ−1 2 (Ê − Ĉ0) + Ĉ−1 2 Ĉ0 ] −1 Ĉ−1 2 Ĉ0 h〈ρ〉 . (4.50) 3) The first approximation for the matrix of the pair correlation functions har- monics ĝ (1) (m)(q) = [ q 2Ê + π2(2m+ 1)2 4h2 Ê + Ĉ−1 2 (Ê − Ĉ0) ] −1 Ĉ−1 2 h〈ρ〉 − [ q 2Ê + π2(2m+ 1)2 4h2 Ê + Ĉ−1 2 (Ê − Ĉ0) + Ĉ−1 2 Ĉ0 ] −1 Ĉ−1 2 h〈ρ〉 .(4.51) Comparing the equations (4.41), (4.42), (4.44) and (4.49), (4.50), (4.51) respec- tively, it is possible to make the following conclusion: the consequent iterations for the correlation functions could be expressed by the asymptotic formulae for the pair correlation functions. Just the same result was received for the one-component systems [11]. 5. Critical parameters and spatial insufficiency Let us consider a one-component liquid, which is filled in a cylindrical sample of the radius a that extends infinitely along the z-axes, i.e. 0 < x, y 6 a, −∞ < z <∞. Reduced geometry of the system leads to the change of critical characteristics while the cylinder radius became smaller. Thus the declination of the critical temperature T ∗ c , density ρ ∗ c and viscosity η∗ for finite-size near-critical liquid system takes place. The geometric factor set the characteristic of spatial limitation and defined as K = a/Rco, where Rco is the amplitude of the correlation length. The new values of critical parameters may be defined at the point of the correlation length maximum. 349 A.V.Chalyi et al. 100 1000 299.8 299.85 299.9 299.95 300 300.05 Tc (K) K Figure 1. Dependence of the new critical temperature T ∗ c (K) on geometrical factor K. Here, the assumed critical temperature for bulk system Tc = 300 K, critical exponent ν = 0.5. To determine such an important characteristic of a liquid as viscosity the scaling formula for the correlation length of finite-size cylindrical system has to be used. Correlation length Rc in a spatially limited system of cylindrical geometry appears dependent not only on thermodynamic variables (temperature, density, etc.), but also on the geometrical factor K, having the sense of molecular layers number, which is possible to arrange along the radius of the cylinder. The longitudinal component of the correlation length (Rc)z along the cylinder axes allows us to determine the new critical temperature T ∗ c (K). For a sufficiently close vicinity of critical isochore where ∆ρ≪ τβ , ρ ≈ ρc and T > Tc one can get the following formula for the critical temperature of liquid in the small volume with a cylindrical geometry: T ∗ c (K) = Tc[1 + (ψ1/K)1/ν ]−1. (5.1) From formula (5.1), it naturally follows that under transferring to the spatially unlimited system when the radius of the cylinder aspires to infinity a→ ∞ accom- panied by geometrical factor K → ∞, a new critical temperature T ∗ c (K) becomes equal to the bulk critical temperature Tc, i.e. there is no shift. Figure 1 illustrates the dependence of a new critical temperature T ∗ c (K) on the geometric factor K for a cylindrical sample with a homogeneous (zeroth) boundary condition of the first type with the mean-field value of exponent ν = 0.5 and the assumed bulk criti- cal temperature Tc = 300 K. The analysis of formula (5.1) shows that the shift of critical temperature for the cylindrical sample T ∗ c (K) in comparison with the critical temperature Tc of volumetric phase may be very considerable. For exam- ple, in the case of geometric factor K = 100 difference of critical temperature is 350 Critical behaviour of confined systems 100 1000 1 104292 294 296 298 300 302 ρc (K) K Figure 2. Dependence of the new critical density ρ∗c(K) on geometrical factor K. Here, the assumed critical density for bulk system ρc = 300 kg/m3, critical exponent δ = 3. ∆Tc = Tc − T ∗ c (K) = 0.173 K. It corresponds to the shift of critical point on ∆τ = 6.4 · 10−5 lower than bulk location. Using just the same way as for the temperature, one can consider the changing of a new critical density ρ∗c(K) in a spatially limited system of cylindrical geometry contrary to the value of critical density ρc for an unlimited volumetric phase. Then in the vicinity of critical isotherm, where ∆ρ ≫ τ β , for single-component liquid, the new value of critical density becomes equal to: ρ∗c(K) = ρc[1 + (ψ1/K)2/(δ−1)]−1. (5.2) From formula (5.2) follows that under transferring to the spatially unlimited sys- tem (K → ∞), ρ∗c(K) → ρc the shift of density is absent. Figure 2 illustrates the dependence of the new critical density ρ∗c(K) on the geometric factor K under a mean-field value of the critical exponent δ = 3 and ρc = 300 kg/m3. The analysis of formula (5.2) shows that the shift of the critical density for a cylindrical sample might be very significant. So, in the case of geometric factor K = 1000 and β = 0.5 (2/(δ − 1) = β/ν) the shift of the critical density is ρc − ρ∗c(K) = 0.72 kg/m3. It means that the shift of the critical point on ∆ρ = 2 · 10−3 lower than the bulk location. Using the above results, one could investigate the change of viscosity η. Combining the scaling relation for viscosity in the absence of shear η = ηB(Q0Rc) χ (5.3) and formula for the correlation length, the equation for viscosity η ∗ of a spatially- 351 A.V.Chalyi et al. 100 1000 1 1041.25 1.3 1.35 1.4 1.45 η∗(K) /(ηΒ A) K Figure 3. Dependence of the dimentionless viscosity η∗(K)/(ηBA) on geometrical factor K. Here, the assumed critical exponent ν = 0.5 and χ = 0.06, temperature declination τ = 10−5. limited liquid cylindrical system in near-critical state may be received η∗ ∼ ηBAKχ(K2τ 2ν + µ2 1) −χ/2, (5.4) where ηB is a background viscosity, χ = 0.06 is the critical exponent and A = (QoRco) χ is the system-dependent constant. As it is seen from (5.4), the viscosity in spatially limited system of cylindrical geometry depends not only on thermodynamic variable but also on the geometric factor K. In comparison with the spatially infinite system, for which according to the formula η/ηB ∼ τ−φ, (5.5) viscosity increases at the critical temperature point (T → Tc, τ → 0) up to infinity (φ = 0.41 is the critical exponent), for the cylindrical sample of radius a when τ → 0 the maximum value of viscosity is realized: η∗(τ = 0) ∼ 0.95aχηBQχ 0 . (5.6) Limiting transition to a case of a spatially unlimited system at K → ∞ has to be realized. From the formula (5.4) it could be seen, that this transition takes place, i.e. η ∼ τ−χ for K → ∞. Figure 3 demonstrates the dependence of viscosity η∗ on the geometric factor K at temperature deviation τ = 10−5. It is possible to make a conclusion that viscosity became smaller when reducing the size of the system. Figure 4 illustrates temperature dependence of η∗ at K = 300. Calculations show 352 Critical behaviour of confined systems η∗(K) /(ηΒ A) 2 10 5 4 10 5 6 10 5 8 10 5 0 1.3 1.4 1.5 1.6 1.7 .6.42 10 5 τ Figure 4. Temperature dependence of the dimentionless viscosity η∗(K)/(ηBA). Here assumed critical exponent ν = 0.5 and χ = 0.06, geometrical factorK = 300. that the anomalous growth of viscosity appears not at the bulk critical temperature T = Tc (τ = 0) but, as it was to be expected, at the new critical temperature T ∗ c (K), which is the same as determined from formula (5.1). This fact gives an opportunity to define a new critical temperature in one of the two ways – from maximum of correlation length or from maximum of viscosity. 6. Light scattering The study of light scattering, as it is well known, is a very fruitful method in molecular physics. Using the existing theoretical approaches and precision experi- mental data, it is possible to receive a reliable information about various equilibrium and kinetic properties of liquids in a broad interval of thermodynamic parameters modification including the critical region. The description of propagation of the electromagnetic waves (in particular – light) in near-critical liquids, is based on the well known conclusions, which are obtained within the framework of the main concepts of the critical light opalescence theory. To understand the specific features of the light scattering it is important to take into account the concept of spatially unlocal fluctuations in the proximity of the critical point that was suggested for the first time in the theory of OZ critical opalescence. The main point of this idea is the significance of the correlation length of fluctuations. From the standpoint of modern theories of phase transitions and critical phenomena, the OZ model actually corresponds to a model of free (noninteracting) 353 A.V.Chalyi et al. fluctuations and consequently cannot give an adequate account of interactions of the order parameters, which are strongly fluctuating in immediate proximity to the critical point. However, the modern experiments prove that for the light scattering in liquids that are close to the critical point the deviations of the data from predictions of the OZ theory are minor. In many experimental and theoretical studies of critical light opalescence, the approximation of single scattering is used. The traditional description of critical opalescence in an approximation of single scattering is based on the known bound between the integral light intensity I1 and the Fourier-image of conjugate correlation G2(q). Of special significance are the studies of critical light opalescence in individual liquids and liquid mixtures conducted under the natural conditions of the acting external gravitational field, which results in a sharp redistribution of physical prop- erties of a liquid depending on height. Strictly speaking, critical phenomena isomor- phic to phase transitions of the second type in such liquid systems in the presence of gravitational field happen in a rather narrow transitional stratum (inter-phase). The linear size of this transitional stratum appears to be about a correlation length. This important circumstance allows us to assert that the critical phenomena in liquids in conditions of the acting gravitational field (the so-called “gravitational effect”) happen, as a matter of fact, in spatially limited systems. Let us proceed immediately to the research of dispersing properties of a spatially limited liquid system near the point of phase transition and consider a spatially limited system, which has the form of a plane-parallel stratum (−∞ < x, y < ∞, −h 6 z 6 h). So, the description of light scattering near the critical point in the approximation of single scattering is based on the following expression: I1 ∼ Re ∫∫ G2(ρ, z) exp[−i(kxyρ+ kz)]ρ dρ dz, (6.1) where ρ = √ x2 + y2. One must take into account the main contribution inG2 for the plane-parallel stratum, which is set by the formula G2(ρ, z) = K0(ρ √ κ2 + π2/4h2)× cos(πz/2h), where κ = R−1 c = R−1 c0 τ ν is the inverse significance of a correlation length of spatially unlimited medium, R−1 c0 = κ0 is the amplitude of correlation length, K0 is the McDonald’s function, τ = (T − Tc)/Tc. Then executing an integration of expression (6.1) one receives I1(τ, θxy, θz) ∼ (π/h) cos(kzh) (κ2 + π2/4h2 + k2xy)(π 2/4h2 + k2z) , (6.2) where kxy = (4π/λ) sin(θxy/2), kz = (4π/λ) sin(θz/2) is the components of wave vector modification during scattering. By setting the components of the correlation length in a plane of the stratum (Rc)xy = 1/ √ κ2 + π2/4h2 and in a perpendicular direction, i.e. along the axes Oz, (Rc)z = 2h/π, it is possible to transform the formula (6.2) to the following form: I1(τ, θxy, θz) ∼ (4h) cos(kzh) (κ2 + π2/4h2)(1 + (Rc)2xyk 2 xy)(1 + (Rc)2zk 2 z) . (6.3) 354 Critical behaviour of confined systems Figure 5. Temperature dependence of the light scattering intensity at the small angles in the finite-size liquid system with the plane-parallel geometry. It is easy to see, that for h→ ∞ this result is in agreement with the formula which define intensity of scattering light in a spatially infinite system, which follows from main predictions of the OZ theory IOZ(k) = I(k → 0)/(1 +R2 ck 2). From the formula (6.2) follows, that for zero scattering angles and for the crit- ical temperature of volumetric (spatially unlimited) phase, i.e. for T = T c, when κ→ 0, the intensity of scattering has no singularity, and it appears proportional to a power of cube of the characteristic size of the system (to an element of dispersing volume) in the direction of space limitlessness (in the considered case – cube of a thickness of the stratum): I1(0, 0, 0) ∼ (16/π3)h3. From the formula (6.2) follows, that in the direction of very small scattering angles θxy ≈ 0 associated with the bounding surface. In the experiment, the anomalous growth of integral intensity of light scattering (critical opalescence) should be observed when the (R−2 c0 τ 2ν+π2/4h2) becomes equivalent to zero, i.e. not for the bulk critical temperature T = Tc. This fact gives an opportunity to define experimentally the new critical temperature for a spatially limited plane-parallel system from the maximum in temperature depen- dence of light intensity scattering. Figure 5 demonstrates the dependence of the light scattering intensity on temperature variable τ for the plane-parallel stratum with the thickness equal to 100Rc0, 150Rc0, 200Rc0, with a corresponding geometrical factor K = 2h/Rc0 is equal to 100, 150, 200 and for very small scattering angles θxy, θz ≈ 10−3. Using the similar way as above for a plane-parallel stratum we can now study another spatially limited system, which has got a geometry of cylinder. Taking into account the main contribution to the pair correlation function of a system with a cylindrical geometry at zeroth boundary condition, given by the formula G2(ρ, z) = D1J0(µ1ρ/a) exp [ − √ κ20τ 2ν + µ2 1/a 2\|z\| ] (6.4) 355 A.V.Chalyi et al. Figure 6. Temperature dependence of the light scattering intensity at the small angles in log-log scale for the finite-size liquid system with a cylindrical geometry. and executing an integration, one can get I1(τ, kxy, kz) ∼ ∼ J1(µ1) √ a2κ20τ 2ν + µ2 1 1 (1 + (a2/2)[(1− 4/µ2 1)k 2 xy + (a2κ20τ 2ν + µ2 1)k 2 z ]) , (6.5) where J0 – cylindrical function of zero order, µ1 = 2.4048 is the first naught of the zero order cylindrical function. From this formula follows in particular, that the intensity of the light scattering remains final even for τ = 0, i.e. for the critical temperature of the volumetric phase. Figure 6 demonstrates the dependence of the light scattering intensity on the temperature variable τ with geometrical factor K = a/Rc0 is equal to 100, 300, 1000. When reaching the new critical temperature T ∗ c (a) of the liquid with a cylindrical form, the expression which defines component intensities in the direction of z-axes has a the following form: (I1)z ∼ [ Re ∫ exp(−ikzz) exp(− √ κ20τ ∗2ν + µ2 1/a 2z) dz ] [κ0τ ∗ν/(κ20τ ∗2ν + k2z)]. (6.6) From this follows, that at kz → 0 and τ ∗ → 0 the intensity of scattered light increases unlimitedly. This conclusion, as well as in the case of the plane-parallel stratum, could be explained for the direction of limitlessness (along z-axes for the cylinder or in the planes which are parallel to the boundary surface for the stratum) the correlation of the density fluctuations ceases decreasing exponentially at the new critical temperature T ∗ c and, as a corollary, only in these directions in the experiment a divergence of a correlation length and critical opalescence should be observed. In the above considered examples, the limited systems have the final sizes in fact only in one (for the stratum) or two (for the cylinder) directions. The availability of lim- itlessness even in one direction is the strongly requested condition for realization 356 Critical behaviour of confined systems of a defining indication of the critical condition – the divergence of the correlation length. However, in the system having the form of a sphere, i.e. a comprehensively limited system, the mentioned condition cannot be satisfied. The correlation length under such a geometry remains constant and could reach its maximum value, equal to the sphere’s radius s. Therefore in such systems the critical state (in its usual understanding) is absent. Accordingly, the critical state’s most characteristic mani- festations – critical opalescence of light – should not be observed. 7. Conclusions As it is shown, the Munster iteration procedure may be applied to the wide class of the systems: spatially infinite and finite-sized, one-component liquids and binary mixtures, isotropic systems and the systems having anisotropy. It allows us to cal- culate the nonsingular expressions for the pair correlation functions of the order parameter fluctuations, the correlation length and the shifts of the critical parame- ters. The abovementioned systems have a great practical application and, therefore, theoretical results received in the paper and especially the results of the critical light scattering might be used for the discussion of the corresponding experimental data. References 1. Yukhnovskii I.R. Phase Transitions of the Second Order. The Collective Variables Method. Kiev, Nauk. Dumka, 1985 (in Russian). 2. Patashinskii A.Z., Pokrovskii V.L. Fluctuation Theory of Phase Transition. Oxford, Pergamon Press, 1979. 3. Ornstein L.S., Zernike F. // Proc. Roy. Acad. Amsterdam, 1914, No. 17, p. 793–806. 4. Munster A. The theory of fluctuations. – In: Proceedings of the Enrico Fermi Inter- national School of Physics. New York, 1959. 5. Vasil’ev A.N. Chalyi A.V. // Ukr. Phys. J., 1998, No. 5, vol. 43, p. 572–576 (in Ukrainian). 6. Croxton C.A. Liquid State Physics. Cambridge University Press, 1974. 7. Stenley H. Introduction To Phase Transitions And Critical Phenomena. Clarendon Press, Oxford, 1971. 8. Chalyi A.V., Chernenko L.M. Phase transitions in finite-size systems and synaptic transmission. – In: Dynamical Phenomena at Interfaces, Surfaces and Membranes. New York, 1993, p. 457–464. 9. Chalyi A.V. // J. Mol. Liquids, 1993, vol. 58, p. 179–195. 10. Lutz H., Gunton J.D. et al. // Sol. State Comm. 1974, No. 14, p. 1075–1078. 11. Vasil’ev A.N., Chalyi A.V. // Ukr. Phys. J., 2000, No. 1, vol. 45, p. 118–123 (in Ukrainian). 357 A.V.Chalyi et al. Критична поведінка обмежених систем О.В.Чалий 1 , К.О.Чалий 2 , Л.М.Черненко 3 , О.М.Васильєв 4 1 Кафедра фізики, Національний медичний університет, 01601 Київ, бульв. Шевченка, 13 2 Факультет біо. та хім. інженерії, Університет Ганма, Кіру 376-8515, Японія 3 Інститут хімії поверхні НАН України, 252028 Київ, просп. Науки, 31 4 Фізичний факультет, Київський національний університет ім. Т.Шевченка, 03022 Київ, просп. Глушкова, 6 Отримано 23 лютого 2000 р. Досліджено критичні явища та особливості фазових переходів в об- межених рідких системах. Для з’ясування характеру впливу просто- рової обмеженості на критичні характеристики рідини обрана си- стема з геометрією плоского паралельного прошарку. З метою зна- ходження парних та прямих кореляційних функцій було використа- но ідеї ітераційного методу Мюнстера. Отримано вираз для радіу- са кореляції флуктуацій параметра порядку і відповідні результати проаналізовані в термінах гіпотези подібності. Окрему увагу приділе- но розрахунку зсуву критичних параметрів (температури та густи- ни). Для дослідження анізотропних систем використано тримомент- не наближення. Аналіз кореляційних властивостей бінарних рідких сумішей проводився з використанням системи інтегральних рівнянь Орнштейна-Церніке (ОЦ). Показано, що для рідин з ізоморфним ха- рактером міжмолекулярної взаємодії може бути використане на- ближення, яке спрощує задачу до моделі однокомпонентної рідини. Знайдено асимптотичні формули для парних кореляційних функцій і показано принципову можливість застосування методу Мюнстера для розгляду бінарних сумішей. Розглянуто особливості критичної опалесценції світла для систем зі спеціальною геометрією. Ключові слова: критичні явища, фазові переходи, обмежена система, кореляційна функція PACS: 05.70.Fh, 05.70.Jk 358

Critical behaviour of conﬁned systems

Institution

Ähnliche Einträge