General Synthetic Scheme Research Articles

General synthetic iterative scheme for non-equilibrium dense gas flows

The recently-developed general synthetic iterative scheme (GSIS), which is tailored for non-equilibrium dilute gas, has been extended to find the steady-state solutions of the non-equilibrium dense gas flows based on the Shakhov-Enskog model, resolving the problems of slow convergence and requirement of ultra-fine grids in near-continuum flows that exist in the conventional iterative scheme. The key ingredient of GSIS is the tight coupling of the mesoscopic kinetic equation and the macroscopic synthetic equations that are exactly derived from the kinetic equation. On the one hand, high-order terms computed from the velocity distribution function provide the higher-order constitutive relations describing the non-equilibrium effects for the macroscopic synthetic equations. On the other hand, the macroscopic quantities obtained from the macroscopic synthetic equations are used to guide the evolution of the velocity distribution function in the kinetic equation. The efficiency and accuracy of GSIS are demonstrated in several test cases, including the shock wave passing through a cylinder and the pressure-driven dense gas flows passing through parallel plates and porous media. The effects of denseness are analyzed in a wide range of gas rarefaction.

General synthetic iterative scheme for rarefied gas mixture flows

The numerical simulation of rarefied gas mixtures with disparate mass and concentration is a huge research challenge. Based on our recent kinetic modeling for monatomic gas mixture flows, this problem is tackled by the general synthetic iterative scheme (GSIS), where the mesoscopic kinetic and macroscopic synthetic equations are alternately solved by the finite-volume discrete velocity method. Three important features of GSIS are highlighted. First, the synthetic equations are precisely derived from the kinetic equation, naturally reducing to the Navier-Stokes equations in the continuum flow regime; in other flow regimes, the kinetic equation provides high-order closure of the constitutive relations to capture the rarefaction effects. Second, these synthetic equations, which can be solved quickly, help to adjust the kinetic system to relax rapidly toward the steady state. Furthermore, in such a two-way coupling, the constraint on the spatial cell size is relieved. Third, the linear Fourier stability analysis demonstrates that the error decay rate in GSIS is smaller than 0.5 for various combinations of mass, concentration and viscosity ratios, such that the error can be reduced by three orders of magnitude after 10 iterations. The efficiency and accuracy of GSIS are demonstrated through several challenging cases covering a wide range of mass ratio, species concentration, and flow speed.

Multiscale simulation of rarefied gas dynamics via direct intermittent GSIS-DSMC coupling

The general synthetic iterative scheme (GSIS) has proven its efficacy in modeling rarefied gas dynamics, where the steady-state solutions are obtained after dozens of iterations of the Boltzmann equation, with minimal numerical dissipation even using large spatial cells. In this paper, the fast convergence and asymptotic-preserving properties of the GSIS are harnessed to remove the limitations of the direct simulation Monte Carlo (DSMC) method. The GSIS, which leverages high-order constitutive relations derived from DSMC, is applied intermittently, which not only rapidly steers the DSMC towards steady state, but also eliminates the requirement that the cell size must be smaller than the molecular mean free path. Several numerical tests have been conducted to validate the accuracy and efficiency of this hybrid GSIS-DSMC approach.

Efficient parallel solver for rarefied gas flow using GSIS

Recently, the general synthetic iterative scheme (GSIS) has been proposed to find the steady-state solution of the Boltzmann equation in the whole range of gas rarefaction, where its fast-converging and asymptotic-preserving properties lead to the significant reduction of iteration numbers and spatial cells in the near-continuum flow regime. However, the efficiency and accuracy of GSIS have only been demonstrated in two-dimensional problems with small numbers of spatial cells and discrete velocities. Here, a large-scale parallel computing strategy is designed to extend the GSIS to three-dimensional flow problems, including the supersonic flows which are usually difficult to solve by the discrete velocity method. Since the GSIS involves the calculation of the mesoscopic kinetic equation which is defined in six-dimensional phase-space, and the macroscopic high-temperature Navier–Stokes–Fourier equations in three-dimensional physical space, the proper partition of the spatial and velocity spaces, and the allocation of CPU cores to the mesoscopic and macroscopic solvers, are the keys to improving the overall computational efficiency. These factors are systematically tested to achieve optimal performance, up to 100 billion spatial and velocity grids. For hypersonic flows around the Apollo reentry capsule, the X38-like vehicle, and the space station, our parallel solver can obtain the converged solution within one hour.

General synthetic iterative scheme for polyatomic rarefied gas flows

Recently, the general synthetic iterative scheme (GSIS) has been proposed to solve the Boltzmann equation efficiently and accurately in the whole range of gas rarefaction. Here, the GSIS is extended and further improved to find the steady-state solutions in nonlinear polyatomic gas flows. The numerical method is developed within the unstructured finite volume framework, where the kinetic equation and synthetic equations are solved alternately. The synthetic equations, which are derived from the modified Rykov kinetic model, reduce to the two-temperature (translational and rotational) Navier–Stokes–Fourier equations in the continuum flow regime, and include high-order constitutive relations for rarefaction effects in other flow regimes. These high-order terms are obtained by solving the kinetic model with the discrete velocity method. When the synthetic equations are solved, their macroscopic quantities are used to guide the evolution of velocity distribution functions in the kinetic model. As a consequence of this two-way coupling, GSIS not only finds steady-state solutions quickly but also relieves the constraints on spatial cell size. Furthermore, the robustness of GSIS is improved by modifying the boundary macroscopic fluxes in the macroscopic implicit iteration. The efficiency and multiscale property of GSIS is demonstrated through several challenging test cases, including the one-dimensional normal shock wave with large internal-translational relaxation times, the two-dimensional hypersonic flow passing a cylinder, and the pump-driven flow in Europe’s demonstration fusion power plant.

General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows

General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows

Boosting the convergence of low-variance DSMC by GSIS

The low-variance direct simulation Monte Carlo (LVDSMC) is a powerful method to simulate low-speed rarefied gas flows. However, in the near-continuum flow regime, due to limitations on the time step and spatial cell size, it takes plenty of time to find the steady-state solution. Here we remove these deficiencies by coupling the LVDSMC with the general synthetic iterative scheme (GSIS) which permits the simulation at the hydrodynamic scale rather than the much smaller kinetic scale. As a proof of concept, we propose the stochastic-deterministic coupling method based on the Bhatnagar-Gross-Krook kinetic model. First, macroscopic synthetic equations are derived exactly from the kinetic equation, which not only contain the Navier-Stokes-Fourier constitutive relation, but also encompass the higher-order terms describing the rarefaction effects. Then, the high-order terms are extracted from LVDSMC and fed into synthetic equations to predict the macroscopic properties which are closer to the steady-state solution than LVDSMC. Finally, the state of simulation particles in LVDSMC is updated to reflect the change of macroscopic properties. As a result, the convergence to steady state is greatly accelerated, and the restrictions on cell size and the time step are removed. We conduct the Fourier stability analysis and simulate several canonical rarefied gas flows to demonstrate the advantages of LVDSMC-GSIS: when the Knudsen number is lower than 0.1, it can use the grid size about 10 times larger than that in traditional DSMC, and it can reduce the computational cost by two orders of magnitude in the flow regime.

A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times

The phonon Boltzmann transport equation with dual relaxation times is often used to describe the heat conduction in semiconductor materials when the classical Fourier's law is no longer valid. For practical engineering designs, accurate and efficient numerical methods are highly demanded to solve the equation. At a large Knudsen number (i.e., the ratio of the phonon mean free path to a characteristic system length), steady-state solutions can be obtained via the conventional iterative scheme (CIS) within a few iterations. However, when the Knudsen number becomes small, i.e., when the phonon transport occurs in the diffusive or hydrodynamic regime, thousands of iterations are required to obtain converged results. In this work, a general synthetic iterative scheme (GSIS) is proposed to tackle the inefficiency of CIS. The key ingredient of the GSIS is that a set of macroscopic synthetic equations, which is exactly derived from the Boltzmann transport equation, is simultaneously solved with the kinetic equation to obtain the temperature and heat flux. During the iteration, the macroscopic quantities are used to evaluate the equilibria in the scattering terms of the kinetic equation, thus guiding the evolution of the phonon distribution function, while the distribution function, in turn, provides closures to the synthetic equations. The Fourier stability analysis is conducted to reveal the superiority of the GSIS over the CIS in terms of fast convergence in periodic systems. It is shown that the convergence rate of the GSIS can always be maintained under 0.2 so that only two iterations are required to reduce the iterative error by one order of magnitude. Numerical results in wall-bounded systems are presented to demonstrate further the efficiency of GSIS, where the CPU time is reduced by up to three orders of magnitude, especially in both the diffusive and hydrodynamic regimes where the Knudsen number is small.

3-Silaazetidine: An Unexplored yet Versatile Organosilane Species for Ring Expansion toward Silaazacycles.

Small-ring silacycles are important organosilane species in main-group chemistry and have found numerous applications in organic synthesis. 3-Silaazetidine, a unique small silacycle bearing silicon and nitrogen atoms, has not been adequately explored due to the lack of a general synthetic scheme and its sensitivity to air. Here, we describe that 3-silaazetidine can be easily prepared in situ from diverse air-stable precursors (RSO2NHCH2SiR12CH2Cl). 3-Silaazetidine shows excellent functional group tolerance in a palladium-catalyzed ring expansion reaction with terminal alkynes, giving 3-silatetrahydropyridines and diverse silaazacycle derivatives, which are promising ring frameworks for the discovery of Si-containing functional molecules.

Ligated aluminum cluster anions, LAln- (n = 1-14, L = N[Si(Me)3]2).

A wide range of low oxidation state aluminum-containing cluster anions, LAln- (n = 1-14, L = N[Si(Me)3]2), were produced via reactions between aluminum cluster anions and hexamethyldisilazane (HMDS). These clusters were identified by mass spectrometry, with a few of them (n = 4, 6, and 7) further characterized by a synergy of anion photoelectron spectroscopy and density functional theory (DFT) based calculations. As compared to a previously reported method which reacts anionic aluminum hydrides with ligands, the direct reactions between aluminum cluster anions and ligands promise a more general synthetic scheme for preparing low oxidation state, ligated aluminum clusters over a large size range. Computations revealed structures in which a methyl-group of the ligand migrated onto the surface of the metal cluster, thereby resulting in "two metal-atom" insertion between Si-CH3 bond.

General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows

The general synthetic iterative scheme (GSIS) is extended to find the steady-state solution of the nonlinear gas kinetic equation, resolving the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredient of GSIS is the tight coupling of gas kinetic and macroscopic synthetic equations, where the constitutive relations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated from the velocity distribution function; however, their constructions are simpler than our previous work (Su et al., 2020 [28]) for linearized gas kinetic equations. On the other hand, solutions of macroscopic synthetic equations are used to accelerate the evolution of gas kinetic equation at the next iteration step. A rigorous linear Fourier stability analysis of the present schemes in periodic system shows that the error decay rate of GSIS can be smaller than 0.5, which means that the deviation to steady-state solution can be reduced by 3 orders of magnitude in 10 iterations. Other important advantages of the GSIS are: (i) it does not rely on the specific form of Boltzmann collision operator, and (ii) it can be solved by sophisticated techniques in computational fluid dynamics, making it amenable to large scale engineering applications. In this paper, the efficiency and accuracy of GSIS are demonstrated by a number of canonical test cases in rarefied gas dynamics, covering different flow regimes.

Multiscale simulation of molecular gas flows by the general synthetic iterative scheme

The in-depth knowledge of rarefied gas dynamics is crucial to address challenges in a wide range of engineering problems, where gas flows are usually multiscale, i.e., covering a wide range of Knudsen numbers. As the traditional Navier–Stokes equations fail, gas kinetic equations are required to model the flows. So far, very few numerical methods are designed to efficiently solve the multiscale gas dynamics and reveal the role of internal degrees of freedom of gas molecules. In this work, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the gas kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state. In the macroscopic synthetic equations, while the momentum equation is the same as that used in the GSIS for monatomic gas, two energy equations are introduced here for polyatomic gases: one is for the translational energy and the other for the internal energy; these equations are derived exactly from the gas kinetic equations hence no approximation is made in final solutions. The Fourier stability analysis is performed to show that the GSIS permits fast convergence to steady-state solutions in the entire flow regime; meanwhile the asymptotic analysis shows that the GSIS recovers the Navier–Stokes equations when the Knudsen number is small, even on the spatial grid with cell size much larger than the molecular mean free path. With all these unique features, several challenging numerical examples are given to show that the proposed GSIS is a promising tool to simulate multiscale molecular gas flows and investigate the effects of internal degrees of freedom.

GSIS: An efficient and accurate numerical method to obtain the apparent gas permeability of porous media

The apparent gas permeability (AGP) of a porous medium is an important parameter to predict production of unconventional gas. The Klinkenberg correlation, which states that the ratio of the AGP to the intrinsic permeability is approximately a linear function of reciprocal mean gas pressure, is one of the most popular estimations to quantify AGP. However, due to the difficulty in defining the characteristic flow length in complex porous media where the rarefied gas flow is multiscale, the slope in the Klinkenberg correlation varies significantly for different geometries such that a universal expression seems impossible. In this paper, by solving the gas kinetic equation using the general synthetic iterative scheme (GSIS), we compute the AGP in porous media that are represented by Sierpinski fractals and pore body/throat systems. With the abilities of fast convergence to steady-state solution and asymptotic preserving of Navier-Stokes limit, it is shown that GSIS is a promising tool to simulate low-speed rarefied gas flow through complex multiscale geometries. A new definition of the characteristic flow length is proposed as a function of porosity, tortuosity and intrinsic permeability of porous media, which enables to find a unique slope in the Klinkenberg correlation for all the considered geometries. This research also shows that the lattice Boltzmann method using simple wall scaling for the effective shear viscosity is not able to predict the AGP of porous media.

Simplified Procedure for General Synthesis of Monosubstituted Piperazines-From a Batch Reaction Vessel to a Flow (Microwave) Reactor.

We reported a novel simplified synthetic procedure for the preparation of monosubstituted piperazine derivatives which can now be easily prepared in a one-pot-one-step way from a protonated piperazine with no need of introduction of a protecting group. Reactions, proceeding either at room or higher temperatures in common solvents, involve heterogeneous catalysis by metal ions supported on commercial polymeric resins. A general synthetic scheme was successfully applied to afford a wide range of monosubstituted piperazines. Furthermore, we picked up a set of piperazine derivatives and studied the possibilities of microwave acceleration of given synthetic reactions to make them even more efficient. Our research proceeded from a simple batch technique to the construction of a flow microwave reactor prototype and resulted in promising findings which are summarized and discussed in the article.

Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iterative scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the “converged” solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the number of iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iterative scheme (GSIS) to find the steady-state solutions of rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserve the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain the Newton's law for stress and the Fourier's law for heat conduction explicitly. For this reason, like the implicit UGKS, the constraint that the spatial cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at cell interfaces. What's more, as the GSIS does not rely on the specific collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are expected to accelerate the slow convergence in the simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.

Fast Convergence and Asymptotic Preserving of the General Synthetic Iterative Scheme

Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number $K$, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier--Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with $K$, while the second one implies that the GSIS asymptotically preserves the Navier--Stokes limit when $K$ approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.

Reductive Amination in the Synthesis of Pharmaceuticals.

Reductive amination plays a paramount role in pharmaceutical and medicinal chemistry owing to its synthetic merits and the ubiquitous presence of amines among biologically active compounds. It is one of the key approaches to C-N bond construction due to its operational easiness and a wide toolbox of protocols. Recent studies show that at least a quarter of C-N bond-forming reactions in the pharmaceutical industry are performed via reductive amination. This Review concisely compiles information on 71 medical substances that are synthesized by reductive amination. Compounds are grouped according to the principle of action, which includes drugs affecting the central nervous system, drugs affecting the cardiovascular system, anticancer drugs, antibiotics, antiviral and antifungal medicines, drugs affecting the urinary system, drugs affecting the respiratory system, antidiabetic medications, drugs affecting the gastrointestinal tract, and drugs regulating metabolic processes. A general synthetic scheme is provided for each compound, and the description is focused on reductive amination steps. The green chemistry metric of reaction mass efficiency was calculated for all reactions.

Investigation of Dopamine Analogues: Synthesis, Mechanistic Understanding, and Structure-Property Relationship.

Dopamine, perhaps the simplest molecule that covalently links catechol and amine, together with its derivatives, has shown impressive adhesive and coating properties with its polymers. However, the scope of the molecules is rather limited, and the polymerization mechanisms are still elusive. We designed a general synthetic scheme and successfully synthesized a series of dopamine analogues with different alkyl chain lengths between the catechol and amine. Taking these new dopamine analogues, together with the molecular systems that have separate catechol and alkyl amine, we show that having both catechol and amine in the molecular system, whether covalently linked via an alkyl chain or not, is sufficient to polymerize under a similar reaction condition to that of dopamine polymerization. However, the time-dependent UV-vis characterization of the individual polymerization indicates that the polymerization for individual molecular systems likely proceeds via different reaction intermediates, depending on the length of the alkyl chain and whether there is a covalent linkage. Interestingly, whereas the covalent linkage via an alkyl chain is not necessary for showing the adhesive property, it is required to achieve the impressive coating property. Our results offer new insights into the design and synthesis of dopamine analogues for future applications, as well as a further mechanistic understanding of the polymerization of these dopamine analogues.

Encoding Active Device Elements at Nanowire Tips.

Semiconductor nanowires and other one-dimensional materials are attractive for highly sensitive and spatially confined electrical and optical signal detection in biological and physical systems, although it has been difficult to localize active electronic or optoelectronic device function at one end of such one-dimensional structures. Here we report a new nanowire structure in which the material and dopant are modulated specifically at only one end of nanowires to encode an active two-terminal device element. We present a general bottom-up synthetic scheme for these tip-modulated nanowires and illustrate this with the synthesis of nanoscale p-n junctions. Electron microscopy imaging verifies the designed p-Si nanowire core with SiO2 insulating inner shell and n-Si outer shell with clean p-Si/n-Si tip junction. Electrical transport measurements with independent contacts to the p-Si core and n-Si shell exhibited a current rectification behavior through the tip and no detectable current through the SiO2 shell. Electrical measurements also exhibited an n-type response in conductance versus water-gate voltage with pulsed gate experiments yielding a temporal resolution of at least 0.1 ms and ∼90% device sensitivity localized to within 0.5 μm from the nanowire p-n tip. In addition, photocurrent experiments showed an open-circuit voltage of 0.75 V at illumination power of ∼28.1 μW, exhibited linear dependence of photocurrent with respect to incident illumination power with an estimated responsivity up to ∼0.22 A/W, and revealed localized photocurrent generation at the nanowire tip. The tip-modulated concept was further extended to a top-down/bottom-up hybrid approach that enabled large-scale production of vertical tip-modulated nanowires with a final synthetic yield of >75% with >4300 nanowires. Vertical tip-modulated nanowires were fabricated into >50 individually addressable nanowire device arrays showing diode-like current-voltage characteristics. These tip-modulated nanowire devices provide substantial opportunity in areas ranging from biological and chemical sensing to optoelectronic signal and nanoscale photodetection.

Naphthalimides for labeling and sensing applications

AbstractNaphthalimide has now become a class of most popular fluorophores for probe design, along with coumarin, fluorescein, rhodamine, BODIPY and cyanine. This account aims at the first-year graduate students as the primary audience and showcases the versatile design principles applicable to the naphthalimide fluorophore when designing a probe or label, with focused examples from the Qian research laboratory. We also provide a general synthetic scheme to naphthalimides of various substitution patterns.