Adsorption from Binary Liquid Solutions into Mesoporous Silica: A Capacitance Isotherm on 5CB Nematogen/Methanol Mixtures
AAdsorption from Binary Liquid Solutions into Mesoporous Silica: ACapacitance Isotherm on 5CB Nematogen/Methanol Mixtures
Andriy V. Kityk a,b , Gennady Y. Gor c and Patrick Huber a,d,e a Hamburg University of Technology, Center for Integrated Multiscale Materials SystemsCIMMS, 21073 Hamburg, Germany; b Faculty of Electrical Engineering, Czestochowa University of Technology, 42-200Czestochowa, Poland; c Otto H. York Department Chemical and Materials Engineering, New Jersey Institute ofTechnology, University Heights, Newark, NJ 07102, USA; d Deutsches Elektronen-Synchrotron DESY, Center for X-Ray and Nano Science CXNS,22603 Hamburg, Germany; e Hamburg University, Centre for Hybrid Nanostructures CHyN,22607 Hamburg, Germany
ARTICLE HISTORY
Compiled February 16, 2021
ABSTRACT
We present a capacitance method to measure the adsorption of rod-like nematogens(4-cyano-4’-pentylbiphenyl, 5CB) from a binary liquid 5CB/methanol solution intoa monolithic mesoporous silica membrane traversed by tubular pores with radii of5.4 nm at room temperature. The resulting adsorption isotherm is reminiscent ofclassical type II isotherms of gas adsorption in mesoporous media. Its analysis by amodel for adsorption from binary solutions, as inspired by the Brunauer-Emmett-Teller (BET) approach for gas adsorption on solid surfaces, indicates that the firstadsorbed monolayer consists of flat-lying (homogeneously anchored) 5CB moleculesat the pore walls. An underestimation of the adsorbed 5CB amount by the adsorp-tion model compared to the measured isotherm for high 5CB concentrations hintstowards a capillary filling transition in the mesopores similar to capillary conden-sation, i.e. film-growth at the pore walls is replaced by filling of the pore centersby the liquid crystal. The experimental method and thermodynamic analysis pre-sented here can easily be adapted to other binary liquid solutions and thus allows acontrolled filling of mesoporous materials with non-volatile molecular systems.
KEYWORDS adsorption, binary solution, capacitance, isotherm, liquid crystal
1. Introduction
Spatial confinement of liquid crystals can alter their physico-chemical properties sub-stantially. Novel phase behaviour, complete suppression of phase transitions and inho-mogeneous structures and dynamics have been reported experimentally and validatedboth by analytical theory and computer simulation[1–14]. In particular the advent ofporous media with tailorable pore shapes and tuneable pore size from the macro-, viathe meso- to the microscale have resulted in an increased number of studies aimed at anunderstanding of liquid crystalline behaviour in these interface-dominated geometries[1, 11, 15–17]. a r X i v : . [ c ond - m a t . s o f t ] F e b oreover, mesoporous media with pores smaller than 50 nm exhibit structures sig-nificantly smaller than the wavelengths of visible light and can thus act as photonicmetamaterials. Their optical functionality is not determined by the properties of thebase materials, but rather by the precise pore shape, geometry, and orientation. Em-bedding molecular matter, most prominently liquid crystals in pore space providesadditional opportunities to control light-matter interactions at the single-pore, meta-atomic scale [18–21]. The resulting hybrid materials get their mechanical stability fromthe porous solid, whereas the liquid-crystalline pore filling adds an integrated func-tionality to the system. To exploit and predict the potential of liquid crystal-infusedsolids as functional nanomaterials, however, a detailed understanding of the physico-chemical changes of the confined compared to the unconfined, bulk liquid crystals isnecessary [17].Another challenge in the study of liquid crystals in porous media compared to othermolecular fillings is their often very low vapour pressure under ambient conditions.Therefore, the liquid crystals are typically filled via capillary action (spontaneousimbibition) into the porous solids [3, 22, 23]. This preparation scheme precludes apartial filling of pore space, e.g. the adsorption of a thin liquid crystalline film at thepore walls. Hence, almost solely completely with LC filled porous solids have beenexplored so far [11, 17].Here we present an experimental study, where we fill sequentially a mesoporousmonolithic silica membrane with pores 13 nm across by imbibition of binary mixturesof methanol and 5CB, 5CB x CH − x with distinct concentrations x of 5CB. We tryto infer the 5CB adsorption after each imbibition step. The resulting filling-fractionversus concentration isotherm is then analysed with a model for adsorption from binarysolutions.
2. Experimental
A monolithic mesoporous silica SiO membrane of 280 µ m thickness is prepared bythermal oxidation of mesoporous silicon, pSi, at 1073 K for 12 h. The pSi membranesare synthesized by electrochemical anodic etching of highly p-doped (100)-orientedsilicon wafers employing a mixture of concentrated fluoric acid and ethanol (volumeratio 2:3) as electrolyte [24, 25]. The resulting pSiO membranes consist of an array ofparallel-aligned nanochannels of mean diameter D = 10 . ± ± . Thecapacitance C of the sample is measured by a Lock-in amplifier SR830 in internalreference mode at a frequency of 47 kHz, see the electric circuit in Fig. 1(c). In such asetup the measured voltage, U R at the resistor R , is given by U R = RU (cid:112) ( ωC ) − + R ≈ ωRCU (1)The simplification of Eq. 1 follows from the fact that here 1 / ( ωC ) (cid:29) R . The ratio ofthese values ranges from 20 to 50. Thus the correction to the measured C value dueto ignoring of R in the square root denominator is less than 0.001 C at maximum.2 igure 1. Experimental scheme for the measurement of an electrical capacitance isotherm from binary so-lutions. A liquid glass container with integrated sample holder for capacitance measurements on a monolithicmesoporous membrane in liquid environment. (a) The adsorption process takes place in the CH(1 − x )5CB( x )solution filled cell. (b) The capacitance C e is measured immediately after emptying of the liquid container. (c)Electrical circuit used in the capacitance measurements. In this case the capacitance C ∝ U R . Thus one can refer to the measured lock-in U R value as capacitance C presented in arbitrary units. The absolute capacitance in ourmeasurement is not important, only the relative change is of relevance. Moreover, thephase shift in the whole range of volume fractions x of the liquid mixture 5CB x CH − x embedded into nanoporous matrix changes roughly between 87 and 89 deg, i.e. byless than 2 deg only. This means that we are evidently far away from any dielectricrelaxation frequencies ( ωτ (cid:29)
3. Results and Discussion
In Fig. 1 a schematics of the experimental cell is shown. It consists of a syringe body.The mesoporous sample is fixed between two spring electrodes. A serious problem insuch experiments can be stray (parasitic) capacitances. For this reason our measure-ments have been done with and without the binary solution, see panel (a) and (b) ofFig. 1, respectively. The adsorption process takes place in the liquid-filled cell and canbe monitored in-situ by time-dependent capacitance measurements of the membrane C ( t ), see Fig. 2. This allows one to monitor for each distinct bulk concentration x theadsorption kinetics and in particular the reaching of an adsorption equilibrium in porespace.After equilibration the liquid cell is emptied and the capacitance C e is measured,see open symbols in Fig. 3. Thereby we have the same stray capacitance contribu-tion in each measuring point. The resulting equilibrium capacitance, C e vs x for thenanoporous membrane pSiO immersed into binary mixture 5CB x CH − x is shown inFig. 3. After measuring C e a new bulk solution 5CB x CH − x was inserted and the3 x=0.0132x=0.00818x=0.004 C [ a r b . un i t s ] time [s] x=0 Figure 2.
Adsorption of 5CB from binary mixtures 5CB x CH − x in mesoporous pSiO as explored by elec-trical capacitance measurements. Marked by different colours are temporal intervals during which the 5CBadsorption into the mesoporous host is monitored by in-situ capacitance measurements upon immersion inbulk binary mixtures 5CB x CH − x with 5CB volume fraction x , as indicated in the figure. The gaps betweenthe shaded time intervals indicate the times, when the sample cell is empty, see Fig. 1(b). The capacitance C e is measured immediately after removing of the bulk solution, see open circles with distinct colours. After this,the cell is again filled with the bulk binary mixture of 5CB x CH − x with an increased volume fraction x . procedure repeated as a function of increasing x .To calculate the adsorption isotherm from the capacitance isotherm certain assump-tions have to be made. We anticipate that C e in the equilibrium state scales linearlywith the fractional volume content V ∗ ( x ) = V ( x ) /V of 5CB in the pore volume, V .The fractional volume content consists of the molecules located both in the adsorbed5CB layer at the pore walls plus molecules 5CB located in the binary mixture of thecore region of the pore filling. The second assumption is that the volume concentrationof the 5CB molecules inside the core region of the pore filling is the same as in the bulksolution surrounding the porous matrix, i.e. it is equal to x . Accordingly, the followingequations are valid: V ∗ ( x ) = C e ( x ) − C e ( x = 0) C e ( x = 1) − C e ( x = 0) , (2) V ( x ) = f V + (1 − f ) V x , (3)Thus for the fractional filling of adsorbed 5CB as a function of x , f ( x ) follows: f ( x ) = V ∗ ( x ) − x − x (4)The filling fraction f versus concentration x is shown in Fig. 4. An unknown parameterin such a calculation is the C e ( x = 1) value [see eq. 4], since at x = 1 the anisotropicnematic LC phase is expected inside the pores at T=296 K. Accordingly, our last data4 .0 0.2 0.4 0.6 0.8 1.01.21.41.61.82.02.2 C e [ a r b . un i t s ] x Figure 3.
Measured equilibrium capacitance, C e vs 5CB concentration x for a mesoporous membrane pSiO immersed into a binary mixture 5CB x CH − x . point corresponds indeed to x=0.99. The hypothetic value C e ( x = 1), which is quiteclose to C e ( x = 0 .
99) has been obtained (precisely adjusted) assuming that f ( x ) → x → q ∗ = q m b s x (1 − b l x )(1 − b l x + b s x ) (5)where q ∗ is the adsorbed amount, q m is the monolayer capacity, x is the solute concen-tration, b s and b l are the equilibrium constants for solute-adsorbent and solute-soluteinteractions respectively. This equation has been widely used for describing adsorptionof various solutes [29], e.g. in a recent work of the Findenegg group it was applied formodelling adsorption of proteins onto silica nanoparticles in aqueous media [30].We applied Eq. 5 to fit the 5CB experimental adsorption isotherm. Note that weexpressed the adsorbed amount q here in units of the complete filling, i.e. q = f = 1corresponds to a complete filling of the pores. Unlike the original BET equation, Eq. 5is a three-parametric dependence, thus we used the Trust Region Reflective algorithm,as implemented in SciPy [31] constraining the parameters as non-negative and usingvarious initial guesses. We also varied the interval of concentrations used for the fitting,and found that the solution for the monolayer capacity q m changes insignificantly forthe range of concentrations between 0.35 and 0.8. The solution for the interval [0, 0.75]gives b l = 0 . b s = 169 . q m = 0 . .0 0.2 0.4 0.6 0.8 1.05CB concentration x f illi n g f r a c t i o n f , q all experimental dataselected experimental datageneralized Brunauer-Emmett-Teller model Figure 4.
Adsorption isotherm of 5CB in mesoporous silica. Plotted is the fractional 5CB filling adsorbedfrom binary mixtures 5CB x CH − x as inferred from electrical capacitance measurements (blue markers). Thedashed black line corresponds to a fit using the generalized BET model (Eq. 5). The solid markers were usedto fit the experimental data. f m = q m to the thickness of themonolayer h m using f m = R − ( R − h m ) R , (6)where R = D/ R = 5 . nm , Eq. 6 gives the monolayerthickness h m = 0 . nm . The 5CB molecule has a length of 1.9 nm and a diameterof 0.7 nm [32, 33]. Thus, presumably the 5CB molecules are lying flat on the sil-ica surfaces, similarly as it has been inferred for 8CB on planar quartz surfaces byBrewster-angle reflection ellipsometry and surface optical second harmonic generation[34].These conclusions with regard to the adsorption behaviour are consistent with opti-cal measurements [35] as well as dielectric spectroscopy studies upon sequential fillingof mesoporous silica with 7CB from binary 7CB/acetone solutions [26, 36–38]. In thoseexperiments also first a regime with monolayer formation at the pore walls and a sub-sequent formation of capillary bridges was inferred from distinct optical signatures(changes in the optical birefringence and light scattering) as well as distinct molecu-lar mobilities of these two molecular population in pore space. In particular, it wasfound that the first adsorbed layers at the pore walls are significantly slower in theirmolecular reorientation dynamics. The study presented here supports these functionaldifferences by a rather thermodynamic distinction between the two populations.
4. Conclusions
We presented an experimental study on the adsorption of 5CB from binary5CB/methanol solutions in mesoporous silica by capacitance measurements. The re-sulting adsorption isotherm is reminiscent of classical gas isotherms of type II indi-cating multilayer growth on pore walls with heterogeneous LC-wall interactions andcan be described by a generalized BET-isotherm, adapted for adsorption from binarysolutions. Deviations from the BET description at higher filling fractions, higher con-centrations, respectively, indicate a transition from LC-film growth at the pore wall tofilling of the pore centre reminiscent of the capillary condensation transition of gases inmesoporous media. The monolayer capacity derived from the sorption-isotherm analy-sis hints towards the formation of a layer with homogeneous LC anchoring at the porewalls, i.e. the first 5CB monolayer is lying flat on the silica surface.We hope that our study will stimulate complementary experimental and theoreticalstudies on adsorption of LC molecules from binary solutions. The technique presentedalong with the simple analysis schemes can be employed also for other molecularsystems with low vapour pressures, like discotic liquid crystals. Its applicability is alsoindependent of the mesogenic (liquid-crystalline) properties, necessitates however agood liquid solvent. Thus the experimental approach is very versatile and is suitablefor many molecular systems. From a more general perspective our techniques allowsone to prepare LC-mesoporous hybrid materials which could be of particular interestfor photonic [17, 39] and organic electronic applications [9, 17].7 . Acknowledgment
GG and PH dedicate this work to Professor Gerhard Findenegg (Technical UniversityBerlin), a pioneer in the field of molecular adsorption in porous materials and self-assembly in confinement. Funding by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) Projektnummer 192346071, SFB 986 “Tailor-MadeMulti-Scale Materials Systems” and the DFG Graduate School GRK 2462 ”Pro-cesses in natural and technical Particle-Fluid-Systems (PintPFS)” (Projektnummer390794421) is gratefully acknowledged. The presented results are part of a project thathas received funding from the European Union’s Horizon 2020 research and innovationprogramme under the Marie Sk(cid:32)lodowska-Curie grant agreement No. 778156. Supportfrom resources for science in the years 2018-2022 granted for the realization of interna-tional co-financed project Nr. W13/H2020/2018 (Dec. MNiSW 3871/H2020/2018/2)is also acknowledged.
References [1] G.P. Crawford and S. Zumer, editors.
Liquid crystals in complex geometries formed bypolymer and porous networks . Taylor and Francis; New York, U.S.A., 1996.[2] K. Binder, J. Horbach, R. Vink, and A. De Virgiliis. Confinement effects on phase behaviorof soft matter systems.
Soft Matter , 4(8):1555–1568, 2008.[3] Andriy V. Kityk, Matthias Wolff, Klaus Knorr, Denis Morineau, Ronan Lefort, andPatrick Huber. Continuous paranematic-to-nematic ordering transitions of liquid crystalsin tubular silica nanochannels.
Phys. Rev. Lett. , 101(18):187801, 2008.[4] A. V. Kityk and P. Huber. Thermotropic nematic and smectic order in silica glassnanochannels.
Appl. Phys. Lett. , 97(15):153124, October 2010.[5] Dirk M¨uter, Taegyu Shin, Bruno Dem´e, Peter Fratzl, Oskar Paris, and Gerhard H. Find-enegg. Surfactant Self-Assembly in Cylindrical Silica Nanopores.
The Journal of PhysicalChemistry Letters , 1(9):1442–1446, may 2010.[6] Christos Grigoriadis, Hatice Duran, Martin Steinhart, Michael Kappl, and GeorgeFloudas. Suppression of phase transitions in a confined rodlike liquid crystal.
ACS Nano ,5(11):9208–9215, 2011.[7] Takeaki Araki, Marco Buscaglia, Tommaso Bellini, and Hajime Tanaka. Memory andtopological frustration in nematic liquid crystals confined in porous materials.
Nat. Mat. ,10(4):303–309, April 2011.[8] X. J. Zhang, X. X. Liu, X. H. Zhang, Y. Tian, and Y. G. Meng. Ordering of the 7CB liquidcrystal induced by nanoscale confinement and boundary lubrication.
Liquid Crystals ,39(11):1305–1313, 2012.[9] H. Duran, B. Hartmann-Azanza, M. Steinhart, D. Gehrig, F. Laquai, X. L. Feng,K. Mullen, H. J. Butt, and G. Floudas. Arrays of aligned supramolecular wires bymacroscopic orientation of columnar discotic mesophases.
ACS Nano , 6(11):9359–9365,November 2012.[10] S. Yildiz, I. Koseoglu, and M. C. Cetinkaya. Temperature-dependent electro-optical andelastic properties of carbon nanotube doped polar smectogen octylcyanobiphenyl.
Journalof Molecular Liquids , 209:729–737, September 2015.[11] Patrick Huber. Soft matter in hard confinement: phase transition thermodynamics, struc-ture, texture, diffusion and flow in nanoporous media.
J. Phys.: Cond. Matt. , 27:103102,2015.[12] S. H. Ryu and D. K. Yoon. Liquid crystal phases in confined geometries.
Liquid Crystals ,43(13-15):1951–1972, October 2016.[13] C. F. Dietrich, P. Rudquist, K. Lorenz, and F. Giesselmann. Chiral structures from achiral icellar lyotropic liquid crystals under capillary confinement. Langmuir , 33(23):5852–5862, June 2017.[14] Kira E Klop, Roel PA Dullens, M Paul Lettinga, Sergei A Egorov, and Dirk GALAarts. Capillary nematisation of colloidal rods in confinement.
Molecular Physics , 116(21-22):2864–2871, 2018.[15] Jan P.F. Lagerwall and Giusy Scalia. A new era for liquid crystal research: Applicationsof liquid crystals in soft matter nano-, bio- and microtechnology.
Current Applied Physics ,12(6):1387–1412, 2012.[16] Sergej Schlotthauer, Robert A. Skutnik, Tillmann Stieger, and Martin Schoen. Defecttopologies in chiral liquid crystals confined to mesoscopic channels.
J. Chem. Phys. ,142(19):194704, May 2015.[17] Patrick Huber, Kathrin Sentker, Mark Busch, and Andriy V. Kity. Soft Matter underGeometrical Confinement. In Oleg Gang and Patrick Huber, editors,
Soft Matter AndBiomaterials On The Nanoscale: The Wspc Reference On Functional Nanomaterials-PartI (In 4 Volumes) , chapter Liquid Crystals Confined in Nanoporous Solids: From Funda-mentals to Functionalities. World Scientific Publishing, Singapore, 2020.[18] Mark Busch, Andriy V. Kityk, Wiktor Piecek, Tommy Hofmann, Dirk Wallacher, SylwiaCa(cid:32)lus, Przemys(cid:32)law Kula, Martin Steinhart, Manfred Eich, and Patrick Huber. A ferro-electric liquid crystal confined in cylindrical nanopores: Reversible smectic layer buckling,enhanced light rotation and extremely fast electro-optically active goldstone excitations.
Nanoscale , 9(48):19086–19099, December 2017.[19] Kathrin Sentker, Arne W. Zantop, Milena Lippmann, Tommy Hofmann, Oliver H. Seeck,Andriy V. Kityk, Arda Yildirim, A. Sch¨onhals, Marco G. Mazza, and Patrick Huber.Quantized Self-Assembly of Discotic Rings in a Liquid Crystal Confined in Nanopores.
Physical Review Letters , 120(6):067801, 2018.[20] Arda Yildirim, Kathrin Sentker, Glen Jacob Smales, Brian Richard Pauw, Patrick Huber,and Andreas Schoenhals. Collective orientational order and phase behavior of a discoticliquid crystal under nanoscale confinement.
Nanoscale Adv. , 1:1104–1116, 2019.[21] Kathrin Sentker, Arda Yildirim, Milena Lippmann, Arne Zantop, Florian Bertram,Tommy Hofmann, Oliver H Seeck, Andriy V. Kityk, Marco G. Mazza, Andreas Sch¨onhals,and Patrick Huber. Self-assembly of liquid crystals in nanoporous solids for adaptive pho-tonic metamaterials.
Nanoscale , 11(48):23304–23317, 2019.[22] S. Gruener and P. Huber. Imbibition in mesoporous silica: rheological concepts andexperiments on water and a liquid crystal.
J. Phys. : Cond. Matt. , 23(18):184109, May2011.[23] Andriy V Kityk, Mark Busch, Daniel Rau, Sylwia Calus, Carole V Cerclier, Ronan Lefort,Denis Morineau, Eric Grelet, Christina Krause, Andreas Sch¨onhals, et al. Thermotropicorientational order of discotic liquid crystals in nanochannels: an optical polarimetry studyand a landau–de gennes analysis.
Soft Matter , 10(25):4522–4534, 2014.[24] P. Kumar, T. Hofmann, K. Knorr, P. Huber, P. Scheib, and P. Lemmens. Tuning thepore wall morphology of mesoporous silicon from branchy to smooth tubular by chemicaltreatment.
Journal of Applied Physics , 103(2):024303, 2008.[25] M. J. Sailor.
Porous Silicon in Practice - Preparation, Characterization and Applications .Wiley-VCH, Weinheim, 2011.[26] Sylwia Ca(cid:32)lus, Andriy V. Kityk, Manfred Eich, and Patrick Huber. Inhomogeneous Re-laxation Dynamics and Phase Behaviour of a Liquid Crystal Confined in a NanoporousSolid.
Soft Matter , 11(16):3176–3187, 2015.[27] Matthias Thommes, Katsumi Kaneko, Alexander V. Neimark, James P. Olivier, FranciscoRodriguez-Reinoso, Jean Rouquerol, and Kenneth S. W. Sing. Physisorption of gases,with special reference to the evaluation of surface area and pore size distribution (iupactechnical report).
Pure and Applied Chemistry , 87(9-10):1051–1069, October 2015.[28] Fabrice Gritti and Georges Guiochon. New thermodynamically consistent competitiveadsorption isotherm in rplc.
Journal of Colloid and Interface Science , 264(1):43–59, 2003.[29] Amanollah Ebadi, Jafar S. Soltan Mohammadzadeh, and Anvar Khudiev. What is the orrect form of bet isotherm for modeling liquid phase adsorption? Adsorption , 15(1):65–73, 2009.[30] Jens Meissner, Albert Prause, Bhuvnesh Bharti, and Gerhard H. Findenegg. Characteri-zation of protein adsorption onto silica nanoparticles: influence of pH and ionic strength.
Colloid and Polymer Science , 293(11):3381–3391, nov 2015.[31] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, DavidCournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright,St´efan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, NikolayMayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C. J. Carey,˙Ilhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perk-told, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M.Archibald, Antˆonio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, Aditya Vijayku-mar, Alessandro Pietro Bardelli, Alex Rothberg, Andreas Hilboll, Andreas Kloeckner,Anthony Scopatz, Antony Lee, Ariel Rokem, C. Nathan Woods, Chad Fulton, CharlesMasson, Christian H¨aggstr¨om, Clark Fitzgerald, David A. Nicholson, David R. Hagen,Dmitrii V. Pasechnik, Emanuele Olivetti, Eric Martin, Eric Wieser, Fabrice Silva, FelixLenders, Florian Wilhelm, G. Young, Gavin A. Price, Gert-Ludwig Ingold, Gregory E.Allen, Gregory R. Lee, Herv´e Audren, Irvin Probst, J¨org P. Dietrich, Jacob Silterra,James T. Webber, Janko Slaviˇc, Joel Nothman, Johannes Buchner, Johannes Kulick,Johannes L. Sch¨onberger, Jos´e Vin´ıcius de Miranda Cardoso, Joscha Reimer, JosephHarrington, Juan Luis Cano Rodr´ıguez, Juan Nunez-Iglesias, Justin Kuczynski, KevinTritz, Martin Thoma, Matthew Newville, Matthias K¨ummerer, Maximilian Bolingbroke,Michael Tartre, Mikhail Pak, Nathaniel J. Smith, Nikolai Nowaczyk, Nikolay Shebanov,Oleksandr Pavlyk, Per A. Brodtkorb, Perry Lee, Robert T. McGibbon, Roman Feldbauer,Sam Lewis, Sam Tygier, Scott Sievert, Sebastiano Vigna, Stefan Peterson, Surhud More,Tadeusz Pudlik, Takuya Oshima, Thomas J. Pingel, Thomas P. Robitaille, Thomas Spura,Thouis R. Jones, Tim Cera, Tim Leslie, Tiziano Zito, Tom Krauss, Utkarsh Upadhyay,Yaroslav O. Halchenko, Yoshiki V´azquez-Baeza, and SciPy 1.0 Contributors. Scipy 1.0:fundamental algorithms for scientific computing in python.
Nature Methods , 17(3):261–272, 2020.[32] A. Leadbetter, R. Richardson, and C. Colling. The structure of a number of nematogens.
Journal de Physique Colloques , 36(C1):C1–37–C1–43, 1975.[33] Erin H. Lay, A. Kirakosian, J.-L. Lin, D. Y. Petrovykh, J. N. Crain, F. J. Himpsel,Rahul R. Shah, and Nicholas L. Abbott. Alignment of Liquid Crystals on Stepped andPassivated Silicon Templates Prepared in Ultrahigh Vacuum.
Langmuir , 16(16):6731–6738, aug 2000.[34] I. Drevenˇsek Olenik, K. Koˇcevar, I. Muˇseviˇc, and Th Rasing. Structure and polarity of8CB films evaporated onto solid substrates.
The European Physical Journal E , 11(2):169–175, jun 2003.[35] Patrick Huber, Mark Busch, Sylwia Calus, and Andriy V. Kityk. Thermotropic nematicorder upon nanocapillary filling.
Phys. Rev. E , 87(4):042502, April 2013.[36] G. P. Sinha and F. M. Aliev. Dielectric relaxation of nematic liquid crystal confined inporous matrices.
Mol. Cryst. Liq. Cryst. , 304:309–314, 1997.[37] G. P. Sinha and F. M. Aliev. Dielectric spectroscopy of liquid crystals in smectic, nematic,and isotropic phases confined in random porous media.
Phys. Rev. E , 58(2):2001–2010,August 1998.[38] Sylwia Ca(cid:32)lus, Lech Borowik, Andriy V. Kityk, Manfred Eich, Mark Busch, and PatrickHuber. Thermotropic interface and core relaxation dynamics of liquid crystals in silicananochannels: A dielectric spectroscopy study.
Physical Chemistry Chemical Physics ,17:22115–22124, 2015.[39] M. Spengler, R. Y. Dong, C. A. Michal, W. Y. Hamad, M. J. MacLachlan, and M. Giese.Hydrogen-bonded liquid crystals in confined spaces toward photonic hybrid materials.
Advanced Functional Materials , 28(26):1800207, June 2018., 28(26):1800207, June 2018.