Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results.

The Mpemba effect is a fingerprint of the anomalous relaxation phenomenon wherein an initially hotter system equilibrates faster than an initially colder system when both are quenched to the same low temperature. Experiments on a single colloidal particle trapped in a carefully shaped double well potential have demonstrated this effect recently [A. Kumar and J. Bechhoefer, Nature 584, 64 (2020)]. In a similar vein, here, we consider a piece-wise linear double well potential that allows us to demonstrate the Mpemba effect using an exact analysis based on the spectral decomposition of the corresponding Fokker-Planck equation. We elucidate the role of the metastable states in the energy landscape as well as the initial population statistics of the particles in showcasing the Mpemba effect. Crucially, our findings indicate that neither the metastability nor the asymmetry in the potential is a necessary or a sufficient condition for the Mpemba effect to be observed.

Isolation of two different Ni2Zn complexes with an unprecedented cocrystal formed by one of them and a "coordination positional isomer" of the other.

A new homometallic trinuclear Ni(II) complex [(NiL)2Ni(NCS)2] (1) and three heterometallic trinuclear Ni(II)-Zn(II)-Ni(II) complexes [(NiL)2Zn(NCS)2] (2), [(NiL)2Zn(NCS)2(CH3OH)2]·2CH3OH (3) and {[(NiL)2Zn(NCS)2(CH3OH)2]} {[(NiL)2Zn(NCS)2]} (4) have been synthesized by using the "complex as ligand" approach with the "metalloligand" [NiL] (H2L = N,N'-bis(salicylidene)-1,3-propanediamine) and thiocyanate in different ratios. All the complexes have been structurally and magnetically characterized. In the isomorphous complexes 1 and 2, the two terminal square planar Ni atoms and the central octahedral nickel atom (in 1) or zinc atom (in 2) are arranged in a bent structure where two cis κN-SCN(-) thiocyanate ions are coordinated to the central atom. The chemical composition of 3 is very similar to that of 2 but, in 3, the central Zn atom is tetrahedral and the κN-SCN(-) thiocyanate ions occupy an axial position of each terminal nickel atom (which now are octahedral with the sixth position occupied by a methanol molecule). Complex 4 consists of two closely related trinuclear units 4A and 4B. In 4A, the coordination environments of the metals are identical to those of 3 whereas 4B is a "coordination position isomer" of complex 2 with the central square pyramidal Zn and one of the terminal square pyramidal Ni atoms coordinated by two κN-SCN(-) thiocyanate ions. Complex 4 is a unique example of a cocrystal formed by two similar trinuclear units (4A and 4B) where 4A is identical to an existing complex (3) and 4B is a "coordination position isomer" of another existing complex (2).

Di-, tri-, and tetranuclear nickel(II) complexes with oximato bridges: magnetism and catecholase-like activity of two tetranuclear complexes possessing rhombic topology.

Oxime-based tridentate Schiff base ligands 3-[2-(diethylamino)ethylimino]butan-2-one oxime (HL(1)) and 3-[3-(dimethylamino)propylimino]butan-2-one oxime (HL(2)) produced the dinuclear complex [Ni2L(1)2](ClO4)2 (1) and trinuclear complex [Ni3(HL(2))3(µ3-O)](ClO4)4·CH3CN (2), respectively, upon reaction with Ni(ClO4)2·6H2O. However, in a slightly alkaline medium, both of the ligands underwent hydrolysis and resulted in tetranuclear complexes [{Ni(deen)(H2O)}2(µ3-OH)2{Ni2(moda)4}](ClO4)2·2CH3CN (3) and [{Ni(dmpn)(CH3CN)2}2(µ3-OH)2{Ni2(moda)4}](ClO4)2·CH3CN (4), where deen = 2-(diethylamino)ethylamine, dmpn = 3-(dimethylamino)-1-propylamine, and modaH = diacetyl monoxime. All four complexes have been structurally characterized. Complex 1 is a centrosymmetric dimer where the square planar nickel(II) atoms are joined solely by the oximato bridges. In complex 2, three square planar nickel atoms form a triangular core through a central oxido (µ3-O) and peripheral oximato bridges. Tetranuclear complexes 3 and 4 consist of four distorted octahedral nickel(II) ions held together in a rhombic chair arrangement by two central µ3-OH and four peripheral oximato bridges. Magnetic susceptibility measurements indicated that dinuclear 1 and trinuclear 2 exhibited diamagnetic behavior, while tetranuclear complexes 3 and 4 were found to have dominant antiferromagnetic intramolecular coupling with concomitant ferromagnetic interactions. Despite its singlet ground state, both 3 and 4 serve as useful examples of Kahn's model for competing spin interactions. High-frequency EPR studies were also attempted, but no signal was detected, likely due to the large energy gap between the ground and first excited state. Complexes 3 and 4 exhibited excellent catecholase-like activity in the aerial oxidation of 3,5-di-tert-butylcatechol to the corresponding o-quinone, whereas 1 and 2 did not show such catalytic activity. Kinetic data analyses of this oxidation reaction in acetonitrile revealed that the catalytic activity of 3 (kcat = 278.4 h(-1)) was slightly lower than that of 4 (kcat = 300.0 h(-1)). X-band EPR spectroscopy indicated that the reaction proceeded through the formation of iminoxyl-type radicals.

Mpemba effect for a Brownian particle trapped in a single well potential.

The Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piecewise linear single well potential that is devoid of any other metastable minima, we show the existence of the Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects.

Mpemba effect in driven granular gases: Role of distance measures.

The Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures.

Lattice models for ballistic aggregation: Cluster-shape-dependent exponents.

We study ballistic aggregation on a two-dimensional square lattice, where particles move ballistically in between momentum and mass conserving coalescing collisions. Three models are studied based on the shapes of the aggregates: In the first the aggregates remain point particles, in the second they retain the fractal shape at the time of collision, and in the third they assume a spherical shape. The exponents describing the power-law temporal decay of number of particles and energy as well as dependence of velocity correlations on mass are determined using large-scale Monte Carlo simulations. It is shown that the exponents are universal only for the point-particle model. In the other two cases, the exponents are dependent on the initial number density and correlations vanish at high number densities. The fractal dimension for the second model is close to 1.49.

Introducing N-Sulfinylamines into Visible-Light-Induced Carbene Chemistry for the Synthesis of Diverse Amides and α-Iminoesters.

A rare example of visible-light-mediated diverse reactivity of N-sulfinylamines with different types of carbene precursors has been disclosed. Acylsilanes and aryldiazoacetates have been utilized as nucleophilic and electrophilic carbene precursors into the N═S═O linchpin, to achieve valuable amides and α-iminoesters, respectively. Interestingly, diazocarbonyls can also participate in the amidation reaction with N-sulfinylamines via in situ generated ketenes. This operationally simple modular method offers a mild, transition-metal-free, and coupling-reagent-free protocol to fabricate structurally diverse amides and a promptly accessible technique to achieve α-iminoesters, where visible light remains as a key promoter.

Insertion of a hydroxido bridge into a diphenoxido dinuclear copper(II) complex: drastic change of the magnetic property from strong antiferromagnetic to ferromagnetic and enhancement in the catecholase activity.

A diphenoxido-bridged dinuclear copper(II) complex, [Cu(2)L(2)(ClO(4))(2)] (1), has been synthesized using a tridentate reduced Schiff base ligand, 2-[[2-(diethylamino)ethylamino]methyl]phenol (HL). The addition of triethylamine to the methanolic solution of this complex produced a novel triple bridged (double phenoxido and single hydroxido) dinuclear copper(II) complex, [Cu(2)L(2)(OH)]ClO(4) (2). Both complexes 1 and 2 were characterized by X-ray structural analyses, variable-temperature magnetic susceptibility measurements, and spectroscopic methods. In 1, the two phenoxido bridges are equatorial-equatorial and the species shows strong antiferromagnetic coupling with J = -615.6(6.1) cm(-1). The inclusion of the equatorial-equatorial hydroxido bridge in 2 changes the Cu···Cu distance from 3.018 Å (avg.) to 2.798 Å (avg.), the positions of the phenoxido bridges to axial-equatorial, and the magnetic coupling to ferromagnetic with J = 50.1(1.4) cm(-1). Using 3,5-di-tert-butylcatechol as the substrate, the catecholase activity of the complexes has been studied in a methanol solution; compound 2 shows higher catecholase activity (k(cat) = 233.4 h(-1)) than compound 1 (k(cat) = 93.6 h(-1)). Both complexes generate identical species in solution, and they are interconvertible simply by changing the pH of their solutions. The higher catecholase activity of 2 seems to be due to the presence of the OH group, which increases the pH of its solution.

Synthesis, crystal structures, magnetic properties and catecholase activity of double phenoxido-bridged penta-coordinated dinuclear nickel(II) complexes derived from reduced Schiff-base ligands: mechanistic inference of catecholase activity.

Three double phenoxido-bridged dinuclear nickel(II) complexes, namely [Ni(2)(L(1))(2)(NCS)(2)] (1), [Ni(2)(L(2))(2)(NCS)(2)] (2), and [Ni(2)(L(3))(2)(NCS)(2)] (3) have been synthesized using the reduced tridentate Schiff-base ligands 2-[1-(3-methylamino-propylamino)-ethyl]-phenol (HL(1)), 2-[1-(2-dimethylamino-ethylamino)-ethyl]-phenol (HL(2)), and 2-[1-(3-dimethylamino-propylamino)-ethyl]-phenol (HL(3)), respectively. The coordination compounds have been characterized by X-ray structural analyses, magnetic-susceptibility measurements, and various spectroscopic methods. In all complexes, the nickel(II) ions are penta-coordinated in a square-pyramidal environment, which is severely distorted in the case of 1 (Addison parameter τ = 0.47) and 3 (τ = 0.29), while it is almost perfect for 2 (τ = 0.03). This arrangement leads to relatively strong antiferromagnetic interactions between the Ni(II) (S = 1) metal centers as mediated by double phenoxido bridges (with J values of -23.32 (1), -35.45 (2), and -34.02 (3) cm(3) K mol(-1), in the convention H = -2JS(1)S(2)). The catalytic activity of these Ni compounds has been investigated for the aerial oxidation of 3,5-di-tert-butylcatechol. Kinetic data analysis following Michaelis-Menten treatment reveals that the catecholase activity of the complexes is influenced by the flexibility of the ligand and also by the geometry around the metal ion. Electrospray ionization mass spectroscopy (ESI-MS) studies (in the positive mode) have been performed for all the coordination compounds in the presence of 3,5-DTBC to characterize potential complex-substrate intermediates. The mass-spectrometry data, corroborated by electron paramagnetic resonance (EPR) measurements, suggest that the metal centers are involved in the catecholase activity exhibited by the complexes.

Use of a reduced Schiff-Base ligand to prepare novel chloro-bridged chains of rare Cu(II) trinuclear complexes with mixed azido/oxo and chloro/oxo bridges.

Two mixed bridged one-dimensional (1D) polynuclear complexes, [Cu(3)L(2)(mu(1,1)-N(3))(2)(mu-Cl)Cl](n) (1) and {[Cu(3)L(2)(mu-Cl)(3)Cl].0.46CH(3)OH}(n) (2), have been synthesized using the tridentate reduced Schiff-base ligand HL (2-[(2-dimethylamino-ethylamino)-methyl]-phenol). The complexes have been characterized by X-ray structural analyses and variable-temperature magnetic susceptibility measurements. In both complexes the basic trinuclear angular units are joined together by weak chloro bridges to form a 1D chain. The trinuclear structure of 1 is composed of two terminal square planar [Cu(L)(mu(1,1)-N(3))] units connected by a central Cu(II) atom through bridging nitrogen atoms of end-on azido ligands and the phenoxo oxygen atom of the tridentate ligand. These four coordinating atoms along with a chloride ion form a distorted trigonal bipyramidal geometry around the central Cu(II). The structure of 2 is similar; the only difference being a Cl bridge replacing the mu(1,1)-N(3) bridge in the trinuclear unit. The magnetic properties of both trinuclear complexes can be very well reproduced with a simple linear symmetrical trimer model (H = -JS(i)S(i+1)) with only one intracluster exchange coupling (J) including a weak intertrimer interaction (j) reproduced with the molecular field approximation. This model provides very satisfactory fits for both complexes in the whole temperature range with the following parameters: g = 2.136(3), J = -93.9(3) cm(-1) and zj = -0.90(3) cm(-1) (z = 2) for 1 and g = 2.073(7), J = -44.9(4) cm(-1) and zJ' = -1.26(6) cm(-1) (z = 2) for 2.

Mpemba effect in driven granular Maxwell gases.

A Mpemba effect refers to the counterintuitive result that, when quenched to a low temperature, a system at higher temperature may equilibrate faster than one at intermediate temperatures. This effect has recently been demonstrated in driven granular gases, both for smooth as well as rough hard-sphere systems based on a perturbative analysis. In this paper, we consider the inelastic driven Maxwell gas, a simplified model for a granular gas, where the rate of collision is assumed to be independent of the relative velocity. Through an exact analysis, we determine the conditions under which the Mpemba effect is present in this model. For monodispersed gases, we show that the Mpemba effect is present only when the initial states are allowed to be nonstationary, while for bidispersed gases, it is present for some steady-state initial states. We also demonstrate the existence of the strong Mpemba effect for bidispersed Maxwell gas, wherein the system at higher temperature relaxes to a final steady state at an exponentially faster rate leading to smaller equilibration time.

Cis-trans isomerism in diphenoxido bridged dicopper complexes: role of crystallized water to stabilize the cis isomer, variation in magnetic properties and conversion of both into a trinuclear species.

The trans-[Cu(2)L(2)Cl(2)] (1), and cis-[Cu(2)L(2)Cl(2)]·H(2)O (2) isomers of a diphenoxido bridged Cu(2)O(2) core have been synthesized using a tridentate reduced Schiff base ligand 2-[(2-dimethylamino-ethylamino)-methyl]-phenol. The geometry around Cu(II) is intermediate between square pyramid and trigonal bipyramid (Addison parameter, τ = 0.463) in 1 but nearly square pyramidal (τ = 0.049) in 2. The chloride ions are coordinated to Cu(II) and are trans oriented in 1 but cis oriented in 2. Both isomers have been optimized using density functional theory (DFT) calculations and it is found that the trans isomer is 7.2 kcal mol(-1) more favorable than the cis isomer. However, the hydrogen bonding interaction of crystallized water molecule with chloride ions compensates for the energy difference and stabilizes the cis isomer. Both complexes have been converted to a very rare phenoxido-azido bridged trinuclear species, [Cu(3)L(2)(µ(1,1)-N(3))(2)(H(2)O)(2)(ClO(4))(2)] (3) which has also been characterized structurally. All the complexes are antiferromagnetically coupled but the magnitude of the coupling constants are significantly different (J = -156.60, -652.31, and -31.54 cm(-1) for 1, 2, 3 and respectively). Density functional theory (DFT) calculations have also been performed to gain further insight into the qualitative theoretical interpretation on the overall magnetic behavior of the complexes.