1
|
Toledano Ó, Gálvez Ó. Energetics and structure of Langmuir monolayers of palmitic acid: a DFT study. Phys Chem Chem Phys 2019; 21:11203-11213. [PMID: 31099366 DOI: 10.1039/c9cp01563a] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
Langmuir monolayers are monomolecular wide films composed of amphiphilic molecules with a bi-dimensional structure typically formed at the air-water interface. They have been studied for many years because these monolayers have important applications in many research fields. Their phase diagrams present several condensed phases whose atomic structure is not yet completely known. We present a novel density functional study on palmitic acid dimers and monolayers. Our results reveal that dihydrogen contacts established among alkyl chains play a leading role in the final structure, regarding both dimers and for the arrangement of molecules in the monolayer. In addition, our calculations show that tilted phases at approx. 30° can be formed without significant loss of structure stability, a result that is in agreement with the experimental findings. Different structures for the high pressure phases, S and CS, are proposed here for the first time, being in good agreement with the diffraction data available. Furthermore, linear compressibility values also in accordance with previous experimental studies are presented for several structures.
Collapse
Affiliation(s)
- Óscar Toledano
- Dpto. Física Interdisciplinar, Facultad de Ciencias, Universidad Nacional de Educación a Distancia (UNED), Senda del Rey, 9, 28040 Madrid, Spain.
| | - Óscar Gálvez
- Dpto. Física Interdisciplinar, Facultad de Ciencias, Universidad Nacional de Educación a Distancia (UNED), Senda del Rey, 9, 28040 Madrid, Spain.
| |
Collapse
|
2
|
Kartashynska E, Vysotsky Y, Fainerman V, Vollhardt D, Miller R. Quantum-chemical analysis of condensed monolayer phases of N-alkanoyl-substituted alanine at the air/water interface. Colloids Surf A Physicochem Eng Asp 2018. [DOI: 10.1016/j.colsurfa.2018.03.027] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
|
3
|
Iakovlev A, Bedrov D, Müller M. Alkyl-Based Surfactants at a Liquid Mercury Surface: Computer Simulation of Structure, Self-Assembly, and Phase Behavior. J Phys Chem Lett 2016; 7:1546-1553. [PMID: 27045619 DOI: 10.1021/acs.jpclett.6b00494] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Self-assembled organic films on liquid metals feature a very rich phase behavior, which qualitatively differs from the one on crystalline metals. In contrast to conventional crystalline supports, self-assembled alkylthiol monolayers on liquid metals possess a considerably higher degree of molecular order, thus enabling much more robust metal-molecule-semiconductor couplings for organic electronics applications. Yet, compared to crystalline substrates, the self-assembly of organic surfactants on liquid metals has been studied to a much lesser extent. In this Letter we report the first of its kind molecular simulation investigation of alkyl-based surfactants on a liquid mercury surface. The focus of our investigation is the surfactant conformations as a function of surface coverage and surfactant type. First, we consider normal alkanes because these systems set the basis for simulations of all other organic surfactants on liquid mercury. Subsequently, we proceed with the discussion of alkylthiols that are the most frequently used surfactants in the surface science of hybrid organometallic interfaces. Our results indicate a layering transition of normal alkanes as well as alkylthiols from an essentially bare substrate to a completely filled monolayer of laying molecules. As the surface coverage increases further, we observe a partial wetting of the laying monolayer by the bulk phase of alkanes. In the case of alkylthiols, we clearly see the coexistence of molecules in laying-down and standing-up conformations, in which the sulfur headgroups of the thiols are chemically bound to mercury. In the standing-up phase, the headgroups form an oblique lattice. For the first time we were able to explicitly characterize the molecular-scale structure and transitions between phases of alkyl-based surfactants and to demonstrate how the presence of a thiol headgroup qualitatively changes the phase equilibrium and structure in these systems. The observed phenomena are consistent with available direct and indirect experimental evidence.
Collapse
Affiliation(s)
- Anton Iakovlev
- Institut für Theoretische Physik, Georg-August-Universität Göttingen , Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Dmitry Bedrov
- Department of Materials Science & Engineering, University of Utah , 122 South Central Campus Drive, Salt Lake City, Utah 84112, United States
| | - Marcus Müller
- Institut für Theoretische Physik, Georg-August-Universität Göttingen , Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| |
Collapse
|
4
|
Steinrück HG, Will J, Magerl A, Ocko BM. Structure of n-Alkyltrichlorosilane Monolayers on Si(100)/SiO2. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2015; 31:11774-11780. [PMID: 26436472 DOI: 10.1021/acs.langmuir.5b03091] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The structure of n-alkyltrichlorosilane self-assembled monolayers (SAMs) of alkyl chain lengths n = 12, 14, 18, and 22 formed on the amorphous native oxide of silicon (100) has been investigated via angstrom-resolution surface X-ray scattering techniques, with particular focus on the proliferation of lateral order along the molecules' long axis. Grazing incidence diffraction shows that the monolayer is composed of hexagonally packed crystalline-like domains for n = 14, 18, and 22 with a lateral size of about 60 Å. However, Bragg rod analysis shows that ∼12 of the CH2 units are not included in the crystalline-like domains. We assign this, and the limited lateral crystallites' size, to strain induced by the size mismatch between the optimal chain-chain and headgroup-headgroup spacings. Analysis of X-ray reflectivity profiles for n = 12, 14, and 22 shows that the density profile used to successfully model n = 18 provides an excellent fit where the analysis-derived parameters provide complementary structural information to the grazing incidence results.
Collapse
Affiliation(s)
- H-G Steinrück
- Crystallography and Structural Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg , 91058 Erlangen, Germany
- Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory , Menlo Park, California 94025, United States
| | - J Will
- Crystallography and Structural Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg , 91058 Erlangen, Germany
| | - A Magerl
- Physics Department, Friedrich-Alexander-Universität Erlangen-Nürnberg , 91058 Erlangen, Germany
| | - B M Ocko
- Condensed Matter Physics & Materials Science Department, Brookhaven National Laboratory , Upton, New York 11973, United States
| |
Collapse
|
5
|
Theoretical description of 2D-cluster formation of nonionic surfactants at the air/water interface. Colloid Polym Sci 2015. [DOI: 10.1007/s00396-015-3630-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
|
6
|
Vysotsky YB, Belyaeva EA, Kartashynska ES, Fainerman VB, Smirnova NA. Quantum chemical approach in the description of the amphiphile clusterization at the air/liquid and liquid/liquid interfaces with phase nature accounting. I. Aliphatic normal alcohols at the air/water interface. J Phys Chem B 2015; 119:3281-96. [PMID: 25640463 DOI: 10.1021/jp512099x] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
A new model based on the quantum chemical approach is proposed to describe structural and thermodynamic parameters of clusterization for substituted alkanes at the air/liquid and liquid/liquid interfaces. The new model by the authors, unlike the previous one, proposes an explicit account of the liquid phase (phases) influence on the parameters of monomers, clusters and monolayers of substituted alkanes at the regarded interface. The calculations were carried out in the frameworks of the quantum chemical semiempirical PM3 method (Mopac 2012), using the COSMO procedure. The new model was tested in the calculations of the clusterization parameters of fatty alcohols under the standard conditions at the air/water interface. The enthalpy, Gibbs' energy and absolute entropy of formation for alcohol monomers alongside with clusterization parameters for the cluster series including the monolayer at air/water interface were calculated. In our calculations the sinkage of monomers, molecules in clusters and monolayers was varied from 1 up to 5 methylene groups. Thermodynamic parameters calculated using the proposed model for the alcohol monolayers are in a good agreement with the corresponding experimental data. However, the proposed model cannot define the most energetically preferable immersion of the monolayer molecules in the water phase.
Collapse
Affiliation(s)
- Yuri B Vysotsky
- Donetsk National Technical University , 58 Artema Street, 83000 Donetsk, Ukraine
| | | | | | | | | |
Collapse
|
7
|
Steinrück HG, Magerl A, Deutsch M, Ocko BM. Pseudorotational epitaxy of self-assembled octadecyltrichlorosilane monolayers on sapphire (0001). PHYSICAL REVIEW LETTERS 2014; 113:156101. [PMID: 25375723 DOI: 10.1103/physrevlett.113.156101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2014] [Indexed: 06/04/2023]
Abstract
The structure of octadecyltrichlorosilane self-assembled monolayers (SAMs) on sapphire (0001) was studied by Å-resolution surface-specific x-ray scattering methods. The monolayer was found to consist of three sublayers where the outermost layer corresponds to vertically oriented, closely packed alkyl tails. Laterally, the monolayer is hexagonally packed and exhibits pseudorotational epitaxy to the sapphire, manifested by a broad scattering peak at zero relative azimuthal rotation, with long powderlike tails. The lattice mismatch of ∼ 1%-3% to the sapphire's and the different length scale introduced by the lateral Si-O-Si bonding prohibit positional epitaxy. However, the substrate induces an intriguing increase in the crystalline coherence length of the SAM's powderlike crystallites when rotationally aligned with the sapphire's lattice. The increase correlates well with the rotational dependence of the separation of corresponding substrate-monolayer lattice sites.
Collapse
Affiliation(s)
- H-G Steinrück
- Crystallography and Structural Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - A Magerl
- Crystallography and Structural Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - M Deutsch
- Physics Department and Institute of Nanotechnology, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - B M Ocko
- Condensed Matter Physics and Materials Science Department, Brookhaven National Lab, Upton, New York 11973, USA
| |
Collapse
|
8
|
Yefet S, Sloutskin E, Tamam L, Sapir Z, Deutsch M, Ocko BM. Surfactant-induced phases in water-supported alkane monolayers: II. Structure. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2014; 30:8010-8019. [PMID: 24918630 DOI: 10.1021/la501589t] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The structure of the Langmuir-Gibbs films of normal alkanes C(n) of length n = 12-21 formed at the surface of aqueous solutions of C(m)TAB surfactants, m = 14, 16, and 18, was studied by surface-specific synchrotron X-ray methods. At high temperatures, a laterally disordered monolayer of mixed alkane molecules and surface-adsorbed surfactant tails is found, having thicknesses well below those of the alkanes' and surfactant tails' extended length. The mixed monolayer undergoes a freezing transition at a temperature T(s)(n,m), which forms, for n ≤ m + 1, a crystalline monolayer of mixed alkane molecules and surfactant tails. For n ≥ m + 2, a bilayer forms, consisting of an upper pure-alkane, crystalline monolayer and a lower liquidlike monolayer. The crystalline monolayer in both cases consists of hexagonally packed extended, surface-normal-aligned chains. The hexagonal lattice constant is found to decrease with increasing n. The films' structure is discussed in conjunction with their thermodynamic properties presented in an accompanying paper.
Collapse
Affiliation(s)
- Shai Yefet
- Physics Department and Institute of Nanotechnology, Bar-Ilan University , Ramat-Gan 52900, Israel
| | | | | | | | | | | |
Collapse
|
9
|
Calvente JJ, Molero M, Andreu R, López-Pérez G, Luque AM. Diffusional Surface Voltammetry as a Probe of Adsorption Energetics. Anal Chem 2011; 84:1034-41. [DOI: 10.1021/ac202564w] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Juan José Calvente
- Departamento
de Química Física, Facultad
de Química, Universidad de Sevilla, 41012 Sevilla, Spain
| | - Miguel Molero
- Departamento
de Química Física, Facultad
de Química, Universidad de Sevilla, 41012 Sevilla, Spain
| | - Rafael Andreu
- Departamento
de Química Física, Facultad
de Química, Universidad de Sevilla, 41012 Sevilla, Spain
| | - Germán López-Pérez
- Departamento
de Química Física, Facultad
de Química, Universidad de Sevilla, 41012 Sevilla, Spain
| | - Antonio M. Luque
- Departamento
de Química Física, Facultad
de Química, Universidad de Sevilla, 41012 Sevilla, Spain
| |
Collapse
|
10
|
|
11
|
Tamam L, Menahem T, Mastai Y, Sloutskin E, Yefet S, Deutsch M. Langmuir films of chiral molecules on mercury. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2009; 25:5111-5119. [PMID: 19256463 DOI: 10.1021/la804109h] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Homo- and heterochiral Langmuir films of a chiral derivative of stearic acid are studied in situ on the surface of liquid mercury as a function of surface coverage by surface tensiometry and surface-specific synchrotron X-ray diffraction and reflectivity. A transition from a phase of surface-parallel molecules to a phase of standing-up molecules is found. The former shows no surface-parallel long-range order. The standing-up phase of both homochiral and heterochiral compositions exhibit long-range order. However, the former has an oblique unit cell with parallel molecular planes, and the later has a centered rectangular unit cell with a herringbone molecular packing. For both cases, the standing-up molecules are tilted by 44 degrees from the surface normal and pack at a density of 19.5 A(2)/molecule in the plane normal to the molecular long axis. Important differences are found, and discussed, between this behavior and that of a Langmuir film of the nonchiral stearic acid on mercury.
Collapse
Affiliation(s)
- L Tamam
- Physics Department, Bar-Ilan University, Ramat-Gan 52900, Israel
| | | | | | | | | | | |
Collapse
|
12
|
McGuiness CL, Blasini D, Masejewski JP, Uppili S, Cabarcos OM, Smilgies D, Allara DL. Molecular self-assembly at bare semiconductor surfaces: characterization of a homologous series of n-alkanethiolate monolayers on GaAs(001). ACS NANO 2007; 1:30-49. [PMID: 19203128 DOI: 10.1021/nn7000596] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Structural trends for a homologous series of n-alkanethiolate self-assembled monolayers (SAMs), C(n)H(2n+1)S- with 12 < or = n < or = 19, on GaAs(001), studied by a combination of grazing incidence X-ray diffraction and infrared spectroscopy, along with ancillary probes, show an overall decay in organization with decreasing n, with the largest changes occurring below n = 15-16. The long-chain monolayers form a mosaic structure with < or =10 nm domains of molecules organized in an incommensurate pseudo-hcp arrangement with nearest neighbor distances of 4.70 and 5.02 A, a 21.2 A(2) area per chain, two chains per subcell in a herringbone packing with a chain tilt angle of 14 degrees , and preferential domain alignment along the substrate [110]([110]) step edge direction. In contrast, for n < 14 no evidence of translational ordering is seen and the alkyl chains exhibit a loss of conformational ordering and coverage relative to the n > 16 cases. A 4'-methyl-biphenyl-4-thiolate companion SAM shows evidence for ordered structures but with lattice parameters close to those expected for a structure commensurate with the intrinsic GaAs(001) square lattice. These trends are explained on the basis of competitions between lattice, interfacial, and intermolecular forces controlling the nanoscale structures of the SAMs. Overall these results provide an important aspect of understanding the effects of SAM formation on surface properties such as electronic and chemical passivation.
Collapse
Affiliation(s)
- Christine L McGuiness
- Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park, Pennsylvania 16801-6300, USA
| | | | | | | | | | | | | |
Collapse
|
13
|
Tamam L, Kraack H, Sloutskin E, Ocko BM, Pershan PS, Ulman A, Deutsch M. Structure of Mercaptobiphenyl Monolayers on Mercury. J Phys Chem B 2005; 109:12534-43. [PMID: 16852550 DOI: 10.1021/jp050278j] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The molecular-scale structure and phase behavior of single-component Langmuir films of 4'-methyl-4-mercaptobiphenyl (MMB) and 4'-perfluoromethyl-4-mercaptobiphenyl (FMMB) on mercury were studied using surface tensiometry, grazing incidence X-ray diffraction, and X-ray reflectivity. At low coverages, a condensed but in-plane disordered single layer of surface-parallel molecules is found for both compounds. At high coverages, both compounds exhibit in-plane-ordered phases of standing-up molecules. For MMB, the biphenyl core dominates the structure, yielding a centered-rectangular unit cell with an area A(x) of 21.8 A(2)/molecule, with molecules tilted by approximately 14 degrees from the surface normal in the nearest-neighbor direction, and a coherence length xi of >1000 A for the crystalline domains. For FMMB, the perfluoromethyl group dominates the structure, yielding a hexagonal unit cell with untilted molecules, an area A(x) of 24.2 A(2)/molecule, and a much smaller xi of approximately 110 A. The structure is discussed in comparison with self-assembled monolayers of MMB on crystalline Au(111) and similar-length alkanethiolate SAMs on Au(111) and on mercury. The differences in the structure are discussed and traced to the differences in the substrate's surface structure, and in the molecular cross section and rigidity.
Collapse
Affiliation(s)
- Lilach Tamam
- Physics Department, Bar-Ilan University, Ramat-Gan 52900, Israel
| | | | | | | | | | | | | |
Collapse
|
14
|
Ocko BM, Kraack H, Pershan PS, Sloutskin E, Tamam L, Deutsch M. Crystalline phases of alkyl-thiol monolayers on liquid mercury. PHYSICAL REVIEW LETTERS 2005; 94:017802. [PMID: 15698133 DOI: 10.1103/physrevlett.94.017802] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2004] [Indexed: 05/24/2023]
Abstract
The structure of octadecanethiol monolyers on liquid Hg surfaces, measured with subangstrom resolution, evolves with increasing coverage from a laterally disordered phase of surface-parallel molecules to ordered rotator phases of surface-normal molecules. For the latter, an abrupt transition is found at 19 A(2)/molecule from a rectangular packing of molecules tilted by 27 degrees in the nearest-neighbor direction to a hexagonal unit cell of untilted molecules. The unit cell of the tilted phase is centered for the chains and noncentered for the headgroups. The thiol headgroups associate in pairs with a single Hg atom, and the bonds form long-range orientational order. The different order of thiols on Au(111) and on Hg highlights the subphase's role in determining the overlayer's structure.
Collapse
Affiliation(s)
- B M Ocko
- Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA.
| | | | | | | | | | | |
Collapse
|
15
|
Kraack H, Ocko BM, Pershan PS, Tamam L, Deutsch M. Temperature dependence of the structure of Langmuir films of normal-alkanes on liquid mercury. J Chem Phys 2004; 121:8003-9. [PMID: 15485263 DOI: 10.1063/1.1799993] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The temperature dependent phase behavior of Langmuir films of n-alkanes [CH3(CH2)(n-2)CH3, denote Cn] on mercury was studied for chain lengths 19< or =n< or =22 and temperatures 15< or =T< or =44 degrees C, using surface tensiometry and surface x-ray diffraction methods. In contrast with Langmuir films on water, where molecules invariably orient roughly surface normal, alkanes on mercury are always oriented surface parallel and show no long-range in-plane order at any surface pressure. A gas and several condensed phases of single, double, and triple layers of lying-down molecules are found, depending on n and T. At high coverages, the alkanes studied here show transitions from a triple to a double to a single layer with increasing temperature. The transition temperature from a double to a single layer is found to be approximately 5 degrees C, lower than the bulk rotator-to-liquid melting temperature, while the transition from a triple to a double layer is about as much below the double-to-single layer transition. Both monolayer and bulk transition temperatures show a linear increase with n with identical slopes of approximately 4.5 degrees C/CH2 within the range of n values addressed here. It is suggested that the film and bulk transitions are both driven by a common cause: the proliferation of gauche defects in the chain with increasing temperature.
Collapse
Affiliation(s)
- H Kraack
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | | | | | | | | |
Collapse
|
16
|
Kraack H, Ocko BM, Pershan PS, Sloutskin E, Tamam L, Deutsch M. Fatty acid Langmuir films on liquid mercury: X-ray and surface tension studies. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2004; 20:5375-85. [PMID: 15986676 DOI: 10.1021/la049977y] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
The structure and phase behavior of liquid-mercury-supported molecular films of fatty acids (CH3(CH2)n-2COOH, denoted CnOOH) were studied for molecular lengths 7 < or = n < or = 24, by surface tensiometry and X-ray methods. Two qualitatively different film structures were found, depending on coverage. For high coverage, the film consists of a monolayer of roughly surface-normal molecules, showing a pressure-dependent sequence of structures similar, though not identical, to that of the corresponding water-supported Langmuir films. At low coverage, phases consisting of surface-parallel molecules are found, not observed on the aqueous subphases employed to date. In this range, a two-dimensional (2D) gas followed by a single and, for 14 < or = n < or = 24, also by a double layer of surface-parallel molecules is found as coverage is increased. Depending on chain length, the flat-lying phases have a crystalline 2D-ordered, a smectic-like 1D-ordered, or a disordered in-plane structure consisting of molecular dimers. The structure and thermodynamics of the films are discussed.
Collapse
Affiliation(s)
- H Kraack
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | | | | | | | | | | |
Collapse
|