Surface tension and disjoining pressure of free-standing smectic films above the bulk smectic-A-isotropic transition temperature A. V. Zakharov and Izabela Śliwa Citation: The Journal of Chemical Physics 140, 124705 (2014); doi: 10.1063/1.4869197 View online: http://dx.doi.org/10.1063/1.4869197 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Transition Helmholtz free energy, entropy, and heat capacity of free-standing smectic films in water: A mean-field treatment J. Chem. Phys. 141, 194706 (2014); 10.1063/1.4901926 A novel method for measuring the surface tension of free standing smectic films Rev. Sci. Instrum. 73, 114 (2002); 10.1063/1.1419223 Resonant x-ray scattering at the Se edge in liquid crystal free-standing films and devices Appl. Phys. Lett. 76, 1863 (2000); 10.1063/1.126193 Layering transition at the free surface of 12CB observed by scanning angle reflectometry J. Chem. Phys. 112, 946 (2000); 10.1063/1.480620 Fluctuations-induced anomalous heat capacity above the smectic-A to smectic-C transition J. Chem. Phys. 110, 2680 (1999); 10.1063/1.477990
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THE JOURNAL OF CHEMICAL PHYSICS 140, 124705 (2014)
Surface tension and disjoining pressure of free-standing smectic films above the bulk smectic-A-isotropic transition temperature 2,b) ´ A. V. Zakharov1,a) and Izabela Sliwa 1 Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia 2 Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna`n, Poland
(Received 13 January 2014; accepted 11 March 2014; published online 25 March 2014) We have carried out a numerical study of both the structural and thermodynamic properties of freestanding smectic films for the case of enhanced pair interaction in the bounding layers. Calculations, based upon the extended McMillan’s mean-field theory with anisotropic forces, show that the layerthinning transitions are characterized by abrupt drops to lower values, both for a disjoining pressure and a fluctuation-induced long-range interaction between the smectic film surfaces, and then continues to increase with a larger positive slope. Reasonable agreement between the theoretically predicted and the experimentally obtained data on the surface tension of the partially fluorinated 5-n-alkyl-2-(4-n-(perfluoroalkyl-metheleneoxy)phenyl) film has been obtained. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4869197] I. INTRODUCTION
One of the most interesting aspects of smectic liquid crystals (LCs) is that, under the appropriate conditions, they can be spread across an opening to form free-standing smectic films (FSSFs).1 In free-standing smectic-A (Sm-A) films, the layer normals are all aligned parallel to the film normal, creating a stack of a few smectic layers with two free bounding surfaces. Since there is no substrate, these films represent an excellent model of low-dimensional systems for the study of surface effects as the film thickness is reduced. The surface tension is clearly of fundamental importance to the stability of these films. Moreover, the presence of the surface tension is believed to be responsible not only for intriguing surface ordering phenomena exhibited by these films, but also for a disjoining pressure (DP) and a fluctuation-induced (FI) longrange interaction between the film surfaces. The FSSF/vapor interface appears to enhance the order of the surface layers so that they become ordered at temperatures well above the bulk Sm-A-Isotropic (AI) transition temperatures TAI (bulk).2 In particular, it was shown that the AI transition occurs through a series of layer-thinning transitions, causing the films to thin in a stepwise manner as the temperature is increased above the TAI (bulk),2, 3 whereas the film tension, at each thinning, abruptly jumps to a lower value and then continues to increase with a smaller slope.4, 5 The layer- thinning transition has been modeled as the successive melting and subsequent removal of interior layers of the film as the film temperature is increased in the range T > TAI (bulk).3, 5, 6 That transition has been attributed to the reduction of smectic fluctuations in the boundary layers. In such quasitwo-dimensional systems the melting originates in the intea) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. URL: www.ipme.ru.
b) Email: [email protected]
0021-9606/2014/140(12)/124705/5
rior of the film and penetrates toward the surface. As a result, the interior layers are squeezed by the bounding layers. In that case, the change F = FN−1 − FN of the total free energy of the smectic film can be obtained from the mean-field model3, 5, 6 for any N-layer FSSF at any temperature T from the range of its existence and is equal to a work which must be performed on the film unit surface area to decrease its thickness by one layer. This work is associated with an additional pressure (disjoining pressure) P = −F/L, acting on the film layers from the bounding free surface. Here, the FSSF composed of N discrete layers with a thickness L = Nd, whereas d is the molecular length. When the value F is positive, the disjoining pressure P prevents the thinning of FSSF, and the film layers are subjected to a stretching force. On the other hand, when F < 0, then the disjoining pressure promotes a thinning of the smectic film, and its layers are subjected to a compressive force. Another interesting phenomenon in FSSFs is the occurrence of the long-range FI interaction FI (L) between the FSSF surfaces.7–9 This FI interaction, due to the smectic layer displacement fluctuations, can be expected to play an important role in the layering and rupturing processes. In previous work,7 the surface effect was examined in smectic film of thickness L and was found that FI is given by F I (L) = −
kB T Li2 (x), 16π L
(1)
xn where kB is the Boltzmann constant, Li2 (x) = ∞ n=1 n2 , −α 2 x = ( +α ) , is the surface tension of the smectic film/vapor √ √ interface, = K/B and α = KB are the characteristic length scale and the characteristic surface tension of the film, respectively. In that model, the smectic elastic properties are assumed to be spatially uniform and characterized by the bending elastic constant K and the compressibility of the smectic film B, which are equal to those at the bulk Sm-A phase. This assumption probably is physically justified
140, 124705-1
© Author(s) 2014
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124705-2
´ A. V. Zakharov and I. Sliwa
J. Chem. Phys. 140, 124705 (2014)
for smectic films well below the TAI (bulk), when the smectic order is well developed in the whole volume of the film. However, when the Sm-A-I transition occurs through a sequence of layer-thinning transitions, as the temperature is increased above the TAI (bulk), the surface tension at the smectic/vapor interface acts to promote smectic order at the surface resulting in the surface layers ordering at a higher temperature than the interior. Therefore, the orientational q and translational σ order parameters (OPs) for smectic layers demonstrate strong ordering in the vicinity of the bounding surfaces with decreases rapidly with distance from those surfaces and this nonuniformity of the film should be taken into account when the DP and the long- range FI interaction are computed. It is well known that in macroscopic bulk smectic phase, the elastic constants K and B are proportional to q2 and σ 2 ,10 respectively. Although the smectic film layers have a microscopic thickness comparable to the molecular length d, their length and width are macroscopic. Hence, each film layer can be considered as a macroscopic ensemble, containing of Ni molecules, with total number of molecules M = N i=1 Ni in the whole volume of the smectic film. Both orientational qi and translational σ i OPs for the ith smectic layer can be introduced (for details see Ref. 3), and, as a result, the relations between the elastic constants Ki and Bi and qi2 and σi2 , respectively, should be similar to analogous relations for macroscopic bulk Sm-A phase. Therefore, the values of the elastic constants Ki and Bi for interior smectic film layers should be smaller than in the vicinity of the bounding surfaces and this nonuniformity of the elastic constants as well as the values of the surface tension , at given number of film layers N and temperature T, should be accounted for when the long-range FI (L) interaction is computed. II. MODEL AND CALCULATIONS
With the aim to calculate the set of quantities such as the Helmholtz free energy and the surface tension of the smectic film/vapor interface, as well as the elastic constants Ki and Bi during the successive layer-thinning process we will use the extended McMillan’s mean-field approach with anisotropic forces.3 With that in mind, a free-standing Sm-A film composed of N discrete smectic layers with a thickness of the order of the molecular length d and with the total number of particles M will be considered. The molecules within each layer are assumed to interact only with molecules of the same layer and those of the two neighboring ones. In the framework of that approach, the effective anisotropic potentials i (i = 1, . . . , N) within the ith Sm-A layer can be introduced3 V0 W0 2π z1 1 (z1 , θ1 ) = − q1 + q2 + α cos 3 V0 d W0 σ1 + σ2 × P2 (cos θ1 ), V0 ⎤ ⎡ i+1 i+1 V0 ⎣ 2π zi ⎦ 1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Fri, 21 Nov 2014 18:59:02
THE JOURNAL OF CHEMICAL PHYSICS 140, 124705 (2014)
Surface tension and disjoining pressure of free-standing smectic films above the bulk smectic-A-isotropic transition temperature 2,b) ´ A. V. Zakharov1,a) and Izabela Sliwa 1 Saint Petersburg Institute for Machine Sciences, The Russian Academy of Sciences, Saint Petersburg 199178, Russia 2 Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna`n, Poland
(Received 13 January 2014; accepted 11 March 2014; published online 25 March 2014) We have carried out a numerical study of both the structural and thermodynamic properties of freestanding smectic films for the case of enhanced pair interaction in the bounding layers. Calculations, based upon the extended McMillan’s mean-field theory with anisotropic forces, show that the layerthinning transitions are characterized by abrupt drops to lower values, both for a disjoining pressure and a fluctuation-induced long-range interaction between the smectic film surfaces, and then continues to increase with a larger positive slope. Reasonable agreement between the theoretically predicted and the experimentally obtained data on the surface tension of the partially fluorinated 5-n-alkyl-2-(4-n-(perfluoroalkyl-metheleneoxy)phenyl) film has been obtained. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4869197] I. INTRODUCTION
One of the most interesting aspects of smectic liquid crystals (LCs) is that, under the appropriate conditions, they can be spread across an opening to form free-standing smectic films (FSSFs).1 In free-standing smectic-A (Sm-A) films, the layer normals are all aligned parallel to the film normal, creating a stack of a few smectic layers with two free bounding surfaces. Since there is no substrate, these films represent an excellent model of low-dimensional systems for the study of surface effects as the film thickness is reduced. The surface tension is clearly of fundamental importance to the stability of these films. Moreover, the presence of the surface tension is believed to be responsible not only for intriguing surface ordering phenomena exhibited by these films, but also for a disjoining pressure (DP) and a fluctuation-induced (FI) longrange interaction between the film surfaces. The FSSF/vapor interface appears to enhance the order of the surface layers so that they become ordered at temperatures well above the bulk Sm-A-Isotropic (AI) transition temperatures TAI (bulk).2 In particular, it was shown that the AI transition occurs through a series of layer-thinning transitions, causing the films to thin in a stepwise manner as the temperature is increased above the TAI (bulk),2, 3 whereas the film tension, at each thinning, abruptly jumps to a lower value and then continues to increase with a smaller slope.4, 5 The layer- thinning transition has been modeled as the successive melting and subsequent removal of interior layers of the film as the film temperature is increased in the range T > TAI (bulk).3, 5, 6 That transition has been attributed to the reduction of smectic fluctuations in the boundary layers. In such quasitwo-dimensional systems the melting originates in the intea) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. URL: www.ipme.ru.
b) Email: [email protected]
0021-9606/2014/140(12)/124705/5
rior of the film and penetrates toward the surface. As a result, the interior layers are squeezed by the bounding layers. In that case, the change F = FN−1 − FN of the total free energy of the smectic film can be obtained from the mean-field model3, 5, 6 for any N-layer FSSF at any temperature T from the range of its existence and is equal to a work which must be performed on the film unit surface area to decrease its thickness by one layer. This work is associated with an additional pressure (disjoining pressure) P = −F/L, acting on the film layers from the bounding free surface. Here, the FSSF composed of N discrete layers with a thickness L = Nd, whereas d is the molecular length. When the value F is positive, the disjoining pressure P prevents the thinning of FSSF, and the film layers are subjected to a stretching force. On the other hand, when F < 0, then the disjoining pressure promotes a thinning of the smectic film, and its layers are subjected to a compressive force. Another interesting phenomenon in FSSFs is the occurrence of the long-range FI interaction FI (L) between the FSSF surfaces.7–9 This FI interaction, due to the smectic layer displacement fluctuations, can be expected to play an important role in the layering and rupturing processes. In previous work,7 the surface effect was examined in smectic film of thickness L and was found that FI is given by F I (L) = −
kB T Li2 (x), 16π L
(1)
xn where kB is the Boltzmann constant, Li2 (x) = ∞ n=1 n2 , −α 2 x = ( +α ) , is the surface tension of the smectic film/vapor √ √ interface, = K/B and α = KB are the characteristic length scale and the characteristic surface tension of the film, respectively. In that model, the smectic elastic properties are assumed to be spatially uniform and characterized by the bending elastic constant K and the compressibility of the smectic film B, which are equal to those at the bulk Sm-A phase. This assumption probably is physically justified
140, 124705-1
© Author(s) 2014
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Fri, 21 Nov 2014 18:59:02
124705-2
´ A. V. Zakharov and I. Sliwa
J. Chem. Phys. 140, 124705 (2014)
for smectic films well below the TAI (bulk), when the smectic order is well developed in the whole volume of the film. However, when the Sm-A-I transition occurs through a sequence of layer-thinning transitions, as the temperature is increased above the TAI (bulk), the surface tension at the smectic/vapor interface acts to promote smectic order at the surface resulting in the surface layers ordering at a higher temperature than the interior. Therefore, the orientational q and translational σ order parameters (OPs) for smectic layers demonstrate strong ordering in the vicinity of the bounding surfaces with decreases rapidly with distance from those surfaces and this nonuniformity of the film should be taken into account when the DP and the long- range FI interaction are computed. It is well known that in macroscopic bulk smectic phase, the elastic constants K and B are proportional to q2 and σ 2 ,10 respectively. Although the smectic film layers have a microscopic thickness comparable to the molecular length d, their length and width are macroscopic. Hence, each film layer can be considered as a macroscopic ensemble, containing of Ni molecules, with total number of molecules M = N i=1 Ni in the whole volume of the smectic film. Both orientational qi and translational σ i OPs for the ith smectic layer can be introduced (for details see Ref. 3), and, as a result, the relations between the elastic constants Ki and Bi and qi2 and σi2 , respectively, should be similar to analogous relations for macroscopic bulk Sm-A phase. Therefore, the values of the elastic constants Ki and Bi for interior smectic film layers should be smaller than in the vicinity of the bounding surfaces and this nonuniformity of the elastic constants as well as the values of the surface tension , at given number of film layers N and temperature T, should be accounted for when the long-range FI (L) interaction is computed. II. MODEL AND CALCULATIONS
With the aim to calculate the set of quantities such as the Helmholtz free energy and the surface tension of the smectic film/vapor interface, as well as the elastic constants Ki and Bi during the successive layer-thinning process we will use the extended McMillan’s mean-field approach with anisotropic forces.3 With that in mind, a free-standing Sm-A film composed of N discrete smectic layers with a thickness of the order of the molecular length d and with the total number of particles M will be considered. The molecules within each layer are assumed to interact only with molecules of the same layer and those of the two neighboring ones. In the framework of that approach, the effective anisotropic potentials i (i = 1, . . . , N) within the ith Sm-A layer can be introduced3 V0 W0 2π z1 1 (z1 , θ1 ) = − q1 + q2 + α cos 3 V0 d W0 σ1 + σ2 × P2 (cos θ1 ), V0 ⎤ ⎡ i+1 i+1 V0 ⎣ 2π zi ⎦ 1
