Molecular Statistical Theory of the Frank Elastic Constants of Liquid Crystals

Abstract

The Frank elastic constants K ₁, K ₂, K ₃ are calculated in the mean field approximation by assuming that the intermolecular force is the sum of hard rod repulsion (length L and width D) and Maier-Saupe's type attraction. The main conclusions are as follows. (1) the inequlaities K ₃? K ₁ > K ₂ necessarily hold. (2) All the K'_is are nearly proportional to the square of the orientational order parameter S. (3) In thermotropic system, K ₁/S ², K ₁/K and K ₂/K(K: mean elastic constant) increase slowly and K _3/K, K ₃/K ₁ decreases with L/D, whereas K/kT _c is nearly constant for L/D = 3 ～ 5. These tendecies agree fairly well with the observations on liquid crystals with almost rigid structure. (4) In the lyotorpic system in which L/D ? 1, K ₁/3 = K ₂ = cLk _T/3π D ² and k ₁/K ₃ = 4D/5cL are obtained (c: packing fraction). The estimate for PBLG solution nearly agrees with recent experimental data.     

Antisolvent crystallization of aqueous ammonium dihydrogen phosphate solutions by addition of acetone at different rates

Results of an investigation of in situ measurements of laser‐beam intensity I_s and I_m transmitted through aqueous ammonium dihydrogen phosphate (ADP) solutions saturated at 30 °C and water, respectively, and temperature T_s of solution and T_m of water during feeding of antisolvent acetone at different rates R_A, using an indigenously designed experimental setup, are presented and discussed. It was found that: (1) for the measurement of MSZW, defined as the maximum volume fraction of acetone content Δx_max in the solution, obtained from temperature measurements are more reliable than transmitted laser‐beam intensity measurements for solutions, (2) two minima ΔT_min1 and ΔT_min2 associated with endothermic reactions, separated by a maximum ΔT_max due to exothermic reactions appear in the plots of temperature difference ΔT = T_s−T_m against acetone feeding time t, and (3) in the ΔT(t) plots there are time intervals Δt of constant rates R_T of increase in ΔT of aqueous ADP solutions, and these values of R_T increase linearly with acetone feeding time rate R_A. The experimental data on the observed dependence of MSZW on antisolvent feeding rate R_A, the appearance of minima ΔT_min1 and ΔT_min2 and maximum ΔT_max and their dependence on R_A, and the relationship between R_T and R_A are discussed from consideration of processes of nucleation and growth of crystallites.     

Thermodynamic properties of strontium tungstate crystal growth from sodium tungstate melts

Energy (E), enthalpy (ΔH_a), entropy (ΔS_a) and free-energy (ΔG_a) of activation and the pre-exponential factor (k₀) of the ordinary Arrhenius equation k_Dl = k₀e–E/RT were estimated for diffusion-controlled crystal growth of SrWO₄ from Na₂WO₄ melts. E increased slightly with increased cooling rates (R_T). k₀ was parallel to k_Dl and increased with increasing R_T. ΔH_a, ΔS_a, and ΔG_a did not change with the changes in R_T and crystallization temperature (T₀). The distance (d₁₂), between a diffusing particle and its host necessary for a successful diffusion, was estimated. Such distances slightly increased with T₀ and R_T.     

Formation Volume of Vacancies in Elements and the Defect Formation Mechanism of Melting

The formation volume V_v of vacancies is given by V_v = (h_v/L) ΔV_f with h_v = 8L (formation enthalpy h_v of vacancies and heat L of fusion given in same units; ΔV_f = change of volume due to melting). If there are phase transitions within the solid, L and ΔV_f must be replaced by (L + Δ H_t) and by (ΔV_f + ΔV_t), respectively (Δ H_t and Δ V_t refer to the heat (s) of transition (s) and to the volume change(s) due to transition(s), resp.). The pressure dependence of the melting point is dT_m/dp = (T_mV_v)/h_v. Independent of the sign of V_v any increase of the vacancy concentration above the maximum concentration possible within the solid decreases the melting point thus resulting in the observed surface melting. The melting point is fixed by the characteristics of vacancy formation (h_v, S_v, V_v) and by the bulk modulus of the solid (S_v = formation entropy of vacancies).     

Estimation of activation parameters for diffusion-controlled crystallization of barium tungstate from sodium tungstate melts by differential thermal analysis

The proximity (d₁₂) between a diffusing species and its host crystal necessary for a successful diffusion for diffusion-controlled crystallization of barium tungstate from sodium tungstate melts in platinum crucibles was estimated. These distances increased with increased cooling rates (R_T) and crystallization temperatures (T₀). Energy (E), enthalpy (ΔH_a), entropy (ΔS_a) and free-energy (ΔG_a) of activation and the pre-exponential factor (k₀) were evaluated using an ordinary Arrhenius equation k_D1 = k₀e^−E/RT, where k_D1 was the diffusion rate-constant. These parameters were virtually unaffected by the changes in T₀ and R_T.     

Structure and magnetic properties of heterometallic coordination carboxylate polymers with cobalt and lithium atoms

Reactions between lithium pivalate and cobalt coordination polymers [Co₅(OH₂)(OH)(Piv)₉)L ¹)₄] _n (I) and [Co₂(OH₂)(Piv)₄(L ²)₂] _n (II), where Piv—is the pivalate anion, L ¹ is pyrazine, and L ² is pyrimidine, result in new heterometallic polymers {[Li₂Co₂(Piv)₆(L ¹)₂]₂ MeCN} _n (III), {[Li₂Co₂(Piv)₆(L ²)]_0.5 MeCN} _n (IV), and [Li₂Co₂(Piv)₆(L ²)₂] _n (V). The resulting compounds contain tetra-nuclear {Li₂Co₂(Piv)₆} fragments connected by neutral bridging ligands (pyrazine or pyrimidine) into layer structures. Crystal structures III–V are determined, and the magnetic properties of III and IV are studied.     

Investigations on the homogeneity of single crystals of materials with variable composition grown by chemical transport reactions

To investigate the problem of homogeneity in crystals grown by chemical transport reactions (CTR) solid solutions Fe_1-yCo_yS, Fe_1-yNi_yS, and Zn_1-yCo_yO were choosen as model substances. Relative concentration profiles of Co and Ni, respectively, were measured by electron-probe microanalysis and light absorption (in Zn_1-yCo_yO only). No significant deviations from homogeneity were found for Fe_1-yMe_yS, whereas in Zn_1-yCo_yO the Co contents increase in the growth direction. A qualitative explanation due to different diffusivities in the solids can be given.     

Dynamics of structural defects of LPE-grown Gax-In1−xP Layers

The structural properties of epitaxial Ga_xIn_1−xP, LPE-grown on 〈111〉-oriented GaAs-substrate, have been investigated. Two groups of samples have been distinguished with respect to the lattice parameters of the layers (a₁) and the substrate (a_s): group A with a₁ < a_s and Δa/a_s = (0.85 ÷ 2.98)%, and group B with a₁ > a_s and Δa/a_s = (0.11 ÷ 1.98)%. Group A is characterized by a structure, including orthogonal, and slanted at 60° towards the surface, linear dislocations, originating from mismatch ones. Group B is characterized by a segmented-surface structure. The optimum conditions for the growth of GaInP layers on GaAs substrates (initial temperature, saturation and composition of the melt, as well as cooling rate) are discussed on the basis of the data obtained.     

A rietveld study of the cation substitution between uvarovite and yttrium-aluminum synthetic garnets,obtained by sol-gel method

A series of synthetic garnets solid solutions is compositions between Y₃Al₂Al₃O₁₂ (Y-Al garnet) (YAG) and Ca₃Cr₂Si₃O₁₂ (uvarovite) was synthesized using the sol-gel method. The expected general formula is (Ca_xY_1-x)₃ (Cr_xAl_1-x)₂ (Si_xAl_1-x)₃O₁₂ where x = 0 to x = 1. The cation distribution in dodecahedral, octahedral and tetrahedral sites and bond distances in these synthetic garnets were determined using the Rietveld method. It shows the incomplete substitution in small sites. The smaller site the smaller substitution there is.     

Hyperfine structure of EPR spectra of Cr5+ ions in irradiated sodium dichromate crystals

EPR spectra are studied of X-irradiated sodium dichromate crystals grown from an aqueous solution by evaporation at 31°C. Doublet lines of EPR-absorption are attributed to the Cr⁵⁺ ions in the CrO₄³⁻ and CrO₃⁻ radicals resulting from radiation decomposition of Cr₂O₇²⁻ and being in the lattice in two unequivalent positions. Hyperfine structure caused by interaction of an unpaired electron with Cr⁵³ nucleus were observed both at liquid nitrogen and room temperatures. For the line, caused by CrO₄³⁻ radical, g_y and A_y directions coincide and angles both between A_x, g_x and A_z, g_z make up ∼ 25°. The spectrum is described by usual spin Hamiltonian for S = 1/2 with following parameters (T = 77 K): for CrO: g_z = 1.984, g_y = 1.970, g_x = 1.961, |A_z| = 8.2 · 10⁻⁴ cm⁻¹, |A_y| = 13.7 × 10⁻⁴ cm⁻¹, |A_x| = 36.1 · 10⁻⁴ cm⁻¹; for CrO₃⁻: g_z = 1.915, g_y = 1.975, g_x = 1.985, |A_z| = 32.2 · 10⁻⁴ cm⁻¹.     

Cluster self-organization of inorganic crystal-forming systems: Templated nanocluster precursors and self-assembly of framework MT structures of A/B,Zr silicates (A = Na, K; B = Ca, Sr)

The basic concepts that are used to describe crystallization as a phenomenon of the hierarchical (cluster) self-organization of a chemical system are considered. The templation of theoretically possible nan-ocluster precursors composed of M octahedra and T tetrahedra by atoms of (A) alkaline and (B) alkaline earth metals is considered for the first time. A relationship between the A/B,M,T composition of templated nanocluster precursors with the composition of A/B,M silicates is established. The model that is developed is used to search for nanocluster precursors in framework MT structures of A/B,Zr silicates. Computer methods (TOPOS 4.0 program package) were used to perform complete 3D reconstruction of the self-assembly of all (four) structural types of A/B,Zr silicates (A = Na, K; B = Ca, Sr) with frameworks of the MT ₂O₇ type: nan-ocluster precursor S ₃⁰-primary chain S ₃¹-microlayer S ₃²-microframework S ₃³. The invariant type of mono-cyclic nanocluster precursor M ₂ T ₄ (with the point symmetries [`1]\bar 1 and 2), stabilized by one or two template cations (A and B), is determined. Bifurcations of the paths of evolution at the S ₃¹ level (structural branching point) are established for the self-assembly of the following frameworks: MT-1 in CaZrSi₂O₇ (gittinsite, C2), MT-2 in SrZrSi₂O₇ (P2₁/c); MT-3 in Na₂ZrSi₂O₇ (parakeldyshite,), K₂ZrSi₂O₇ (khibinskite, P2₁/b), and K₂ZrGe₂O₇ (C2/c); and MT-4 in Na₂ZrSi₂O₇ (H₂O)(C2/c), Na₃ScSi₂O₇ (Pbnm), and K₃ScSi₂O₇ (P6₃/mmc).     

Structure of antitumor agents 2-acetylpyridinethiosemicarbazone hemihydrate and 2-acetylpyridinethiosemicarbazone hydrochloride

The structures of two antitumor agents, 2-acetylpyridinethiosemicarbazone hemihydrate (1), C₁₈H₁₁N₄O_0.5S, colorless,M _r 203.3, monoclinic,P2₁ lc,a=16.713(3),b=9.460(2),c=12.642(2) ?, β=97.60(1)°,V=1981.2(6) ?³,Z=8,R=0.054,R _w =0.085 and 2-acetylpyridinethiosemicarbazone hydrochloride (2), C₁₈H₁₂N₄SCl, yellow,M _r =230.7, monoclinic,P2₁ ln,a=7.676(2),b=7.889(1),c=17.452(4), ?, β=100.96(2)°,V=1037.5(4) ?³ Z=4,R=0.041,R _w =0.076, have been determined. Both compounds exhibit an E configuration(S atomtrans to the N atom of the hydrazine group). Three hydrogen bonds link the two crystallographically independent molecules in a pairwise fashion in the hemihydrate. An intramolecular N−H...Cl bond lends extra conformational stability to the hydrochloride salt.     

A Theoretical Study of Molecular Ordering of Some Smectic Liquid Crystals

Abstract

A theoretical investigation of molecular ordering of smectic C liquuid crystals C₈H₁₇O-C₆H₄-C₆H ₄-COO-C₆H₄-OC₈H₁₇ { A }, C₉H₁₉O-C₆H₄-COO- C₆H₄-COO-C₆H₄-NO₂ { B } and its binary mixtures { A+B } has been carried out by the method based on the Rayleigth-Schrodinger perturbation theory. The mutual arrangement of two molecules corresponding to the minimum of the total interaction energy between them U_pair has been established. The curves of the dependence of U_pair and its various contributions on displacement of molecules from each other along the molecular long X-axis and angle θ between molecules' long axes has been also computed in the minimum point vicinity. The obtained results make it possible to determine the peculiarities of the structural organization of molecules, as well as to construct a model of the structure of A and B compounds in different phases taking into account the most probable packing of molecules.     

An optical microscopy study of the biaxial ‐ calamitic nematic lyotropic phase transition

Lyotropic nematic textures are investigated, using optical microscopy, near the reentrant isotropic (I_RE) ‐ discotic nematic (N_D^‐) ‐ biaxial nematic (N_B) ‐ calamitic nematic (N_C⁺) ‐ isotropic (I) phase transitions in a lyotropic mixture of potassium laurate, decanol and D₂O. The N_B, intermediate phase between the two uniaxial nematic ones, is characterized by optical birefringence measurements. In this way, using a polarizing microscope and a colour CCD digital camera, the N_B – N_C⁺ phase transition is identified as well as the domain of the N_B phase and N_D^‐ – N_B transition point in accordance with optical birefringence data (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structures of divalent transition metal salts of 4-aminonaphthalene-1-sulfonate: unexpected variations in layering pattern

Salts of 4-aminonaphthalene-1-sulfonate with divalent Mg, Mn, Co, and Ni cations have been crystallized and their structures determined by single crystal X-ray methods. The Mg and Mn salts are isostructural. Crystal data for hexa-aquamagnesium(II) 4-aminonaphthalene-1-sulfonate dihydrate, [Mg(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 2H ₂O: monoclinic, P2₁/c, Z = 2, a = 8.622(3), b = 7.043(3), c = 23.178(3) Å, =93.78(2)°, V = 1404.3(7) ų; hexa-aquamanganese(II) 4-aminonaphthalene-1-sulfonate dihydrate, [Mn(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 2H ₂O: monoclinic, P2₁/c, Z = 2, a = 8.652(3), b = 7.031(4), c = 23.402(2) Å, =93.09(2)°, V = 1421.5(9) ų. The structures are composed of alternating layers of octahedral metal–aqua complexes and sulfonate anions linked by an extensive network of hydrogen bonds. The extra water molecules of crystallization are located in the hexa-aquametal cation layers. The repeat unit along the c axis is a double layer. The Co and Ni compounds are isostructural with each other, but compared to the Mg and Mn compounds, have a strikingly different structure. Crystal data for hexa-aquacobalt(II) 4-aminonaphthalene-1-sulfonate trihydrate, [Co(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 3H ₂O: orthorhombic, Pbca, Z = 8, a = 8.518(1), b = 14.327(2), c = 45.367(6) Å, V = 5536(1) ų; hexa-aquanickel(II) 4-aminonaphthalene-1-sulfonate trihydrate, [Ni(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 3H ₂O: orthorhombic, Pbca, Z = 8, a = 8.4976(6), b = 14.288(1), c = 45.076(3) Å, V = 5472.9(7) ų. These structures also contain layers of octahedral hexa-aquametal complexes and additional water molecules of crystallization sandwiched by layers of sulfonate anions, however the stacking pattern is more complex with a quadruple layer repeat unit and two different types of anion layers.     

Effect of buffer layers on the dislocation structure of heteroepitaxial systems

Transmission electron microscopy (TEM) as well as X-ray topography (XRT) and X-ray diffractometry have been used for investigation of the structure of the LPE heteroepitaxial system In_0.05Ga_0.95As-In_yGa_1−yAs_1−xP_x-GaAs(111) A. A critical value of the lattice misfit has been shown to exist at the metallurgical boundary ((Δa/a)* ≈ 10⁻³) which results in the change of the film nucleation and growth mechanism as well as the change of misfit dislocations (MDs) generation mechanism. With (Δa/a)₀ > (Δa/a)* the nucleation and growth mechanism is mixed: island growth at the first stages of growth and layer-by-layer growth at large thicknesses. MDs are created in an “island film” developing a non-ordered dislocation network. The density of threading dislocations (N_d) is ∼ 10⁸ cm⁻². With (Δa/a)₀ < (Δa/a)* there is layer-by-layer mechanism of film's nucleation and growth from the very first stages of crystallization. MDs are injected into continuous layer along the inclined slip planes {111}, thus forming a regular three-dimensional grid of MDs. N_d is less than 10⁶ cm⁻² in the case. A model of dislocation structure formation in heterolayers has been proposed. Within the frame of this model the two critical values of phosphorus concentration in the quaternary melt have been quantitatively determined. These are corresponding to the change of MD generation mechanism. The expected values of N_d for (Δa/a)₀ > (Δa/a)* and (Δa/a)₀ < (Δa/a)* have been theoretically determined.     

Complexes formed by the reactions of fluorinated and non-fluorinated organonitriles with [Zn(SO2)2][AsF6]2: A structural study

[Zn(SO₂)₂][AsF₆]₂ (1) has been prepared from zinc metal and AsF₅ in liquid sulfur dioxide, and crystallized from sulfur dioxide giving crystals of [Zn(SO₂)₄(AsF₆)₂] (1a). The compound 1 forms the homoleptic complexes [Zn(NCR)₆](AsF₆)₂ (R: CH₃ (2a), C₆H₅ (3a), C₆F₅ (3b)) with the corresponding nitriles. In these complexes, the metal center is octahedrally surrounded by six ligands (average Zn–N 2.13 ?). Reaction of 1 with the weaker ligand F₃CCN results in dicoordination at the metal center complemented by a two-dimensional polymeric network (2b) containing bridging AsF₆ ⁻ anions. [Zn(SO₂)₄(AsF₆)₂] (1a): monoclinic, P2₁/c, a=8.5176(8) ?, b=13.5787(14) ?, c=14.7661(15) ?, β=99.296(2)°; [Zn(NCCH₃)₆][AsF₆]₂ (2a): rhombohedral, R-3, a=11.2225(8) ?, b=11.2225(8) ?, c=16.9393(14) ?; [Zn (NCCF₃)₂][μ-FAsF₅)₂( (2b): monoclinic, P2₁/c, a=11.226(3) ?, b=6.6147(19) ?, c=10.460(3) ?, β=104.028(5)°; [Zn(NCC₆H₅)₆][AsF₆]₂ (3a): monoclinic, P2₁/n, a=13.7968(10) ?, b=19.8861(15) ?, c=16.1692(12) ?, β=91.563(2)°; [Zn(NCC₆F₅)₆][AsF₆]₂·2SO₂ (3b): monoclinic, P2₁/n, a=10.8448(9) ?, b=154456(13) ?, c=17.5206(15) ?, β=104.158(1)°.     

Structures of transition and alkaline earth metal salts of 5-aminonaphthalene-2-sulfonate and 6-aminonaphthalene-1, 3-disulfonate: Some unusual coordination behaviors

Salts of 5-aminonaphthalene-2-sulfonate with divalent Mg, Mn, Co, and Ni cations have been crystallized and their structures determined by single crystal X-ray methods. The Mg, Mn, and Co salts are isostructural. Crystal data for hexaaquamagnesium(II) 5-aminonaphthalene-2-sulfonate hexahydrate, [Mg(H₂O)₆](H₂NC₁₀H₆SO₃)₂·6H₂O: monoclinic, P2₁/c, Z=2, a=14.1329(18) ?, b=8.5789(11) ?, c=12.4880(17) ?, β=93.374(3)°, V=1511.5(3) ?³; hexaaquamanganese(II) 5-aminonaphthalene-2-sulfonate hexahydrate, [Mn(H₂O)₆](H₂NC₁₀H₆SO₃)₂·6H₂O: monoclinic, P2₁/c, Z=2, a=14.249(3) ?, b=8.5940(17) ?, c=12.505(3) ?, β=93.30(3)°, V=1528.8(6) ?³; hexaaquacobalt(II) 5-aminonaphthalene-2-sulfonate hexahydrate, [Co(H₂O)₆](H₂NC₁₀H₆SO₃)₂·6H₂O: monoclinic, P2₁/c, Z=2, a=14.1406(18) ?, b=8.5674(11) ?, c=12.4960(16) ?, β=93.297(2)°, V=1511.4(3) ?³. The structures are composed of alternating layers of octahedral metal–aqua complexes and sulfonate anions linked by hydrogen bonds. The three water molecules of crystallization are associated with the hexaaquametal cations and sulfonate O atoms. The repeat unit along the a-axis is a single layer. The Ni salt [crystal data for tetraaquabis(5-aminonaphthalene-2-sulfonato-N)nickel(II) dihydrate, [Ni(H₂O)₄(H₂NC₁₀H₆SO₃)₂]·2H₂O: triclinic, , Z=1, a=6.8524(10) ?, b=8.3094(12) ?, c=11.4832(17) ?, α=69.003(2)°, β=76.570(3)°, γ=83.952(2)°, V=593.56(15) ?³] has a very different structure with direct coordination of the amine N atom to Ni, a modified stacking pattern, and fewer free water molecules. Ag and Ni salts of an amine-substituted naphthalenedisulfonate have also been characterized. Crystal data for diaqua(6-ammonionaphthalene-1,3-disulfonato-O)silver(I) hydrate,[Ag(H₂O)₂(H₃NC₁₀H₅(SO₃)₂)]·0.42H₂O: monoclinic, P2₁/c, Z=2, a=9.099(3) ?, b=21.406(6) ?, c=7.629(2) ?, β=110.178(4)°, V=1394.9(7) ?³; tetraaquabis (6-ammonionaphthalene-1,3-disulfonato-O)nickel(II)tetrahydrate, [Ni(H₂O)₄(H₂NC₁₀H₅ (SO₃)₂)₂]·4H₂O: triclinic,, Z=1, a=6.7971(17) ?, b=10.661(3) ?, c=11.165(3) ?, α=68.308(4)°, β=88.292(5)°, γ=84.896(5)°, V=748.8(3) ?³. The silver salt contains six coordinate metal centers each of which bonds to four sulfonate O atoms from three different anions and two water molecules. The nickel salt contains octahedral cations with four water molecules and two trans sulfonate O atoms. Both structures are layered, but differently from each other and from those of the monofunctional naphthalenesulfonate salts.Unusual coordination of a sulfonate group to a transition metal and coordination of an amine group to nickel but not to cobalt or manganese have been observed in a series of metal aminonaphthalenesulfonate salts     

Differential Thermal Analysis Studies on the Crystallization of Strontium Tungstate from Sodium Tungstate Melts Thermodynamic and Kinetic Parameters

From the estimated diffusion rate-constants (k_D1) of strontium tungstate crystal growth from sodium tungstate melts in platinum crucibles, energy (E), enthalpy (ΔH_a), entropy (ΔS_a) and free-energy (ΔG_a) of activation and the pre-exponential factor (k₀) for the process were estimated using an ordinary Arrhenius equation k_Dl = k₀e^−E/RT. These thermodynamic parameters were virtually unaffected by the changes in crystallization temperatures (T₀) and cooling rates (R)_T. The distance (r₁₂), between a diffusing species and its host necesary for a successful diffusion, increased with T₀ and T_T but there was no direct correlation.     

DSC characterisation of various phase transitions in mesomorphic benzylidene anilines

To analyse the order of the phase transitions in a number of mesomorphic benzylidene anilines, the DSC (Differential Scanning Calorimetry) method proposed by NAVARD and HAUDIN is used. This method involves the measurement of the ratio (N) of the DSC peak heights H₂/H₁ for a given transition, performed with two different heating rates (T₂/T₁). N is a function of the ratio r₂/r₁(= R) and the magnitude of N provides a simple way to characterize the order of a phase transition.