Uniform Stationary Phase Research Articles

A channel-homogenized micro gas chromatography column for light alkane separation with uniform HKUST-1 stationary phase

BackgroundThe micro gas chromatography column (μGCC) is one of the key components of the miniaturized gas chromatography system. However, light alkanes are difficult to be separated by a micro gas chromatography column, especially for methane and ethane, because the length of μGCC is limited by the area of a silicon substrate. More importantly, the heterogeneous microchannel surface formed by silicon glass bonding causes uneven stationary phase coating and the forces between the untreated microchannel surfaces and the stationary phase materials are weak, which will prevent the improvement of separation performance. ResultsIn this paper, a micro gas chromatography column (μGCC) with uniform HKUST-1 stationary phase is reported. Significantly, an alumina film prepared by the atomic layer deposition (ALD) technique is used to homogenize the heterogeneous microchannels. The alumina is a hydrophilic material and the alumina made by the ALD technique is uniform. The forces between hydrophilic alumina film and HKUST-1 are strong, which can greatly improve the coating uniformity of the hydrophilic stationary phase HKUST-1. The test results show that the μGCC could baseline separate the light alkane mixtures (CH4, C2H6, C3H8, and C4H10) at the high testing temperature of 120 °C. The maximum resolution of the difficult-separated methane and ethane reached 19.2, which is 108 % higher than the μGCC using the same stationary phase without homogenizing the microchannel inner surface. SignificanceThe μGCC uses ALD alumina film to homogenize the microchannel inner surface; meanwhile, hydrophilic ALD alumina has a strong electrostatic attraction with the hydrophilic stationary phase HKUST-1. Homogeneous microchannel surface and strong electrostatic attraction are favorable to obtain uniform stationary phase which greatly improves the separation performance, resulting in a large resolution for methane and ethane. The μGCC has broad application prospects in light alkane separation.

Uniform Stationary Phase Estimate with Limited Smoothness

Uniform Stationary Phase Estimate with Limited Smoothness

An all-glass based micro gas chromatographic column for light hydrocarbon separation with HKUST-1 as stationary phase

BackgroundThe gas chromatography column is one of the key components of the gas chromatograph and typically be miniaturized using micro-electro-mechanical system (MEMS) technology. Due to the limited area of the Si wafer, the column length of micro gas chromatographic column (μGCC) is usually much smaller than that of commercial chromatographic columns. Therefore, it is always difficult to use μGCCs to separate small molecule gas components such as light hydrocarbons. More importantly, the heterogeneous microchannel surface formed by silicon glass bonding causes uneven stationary phase coating, further preventing the improvement of separation performance. ResultsIn this paper, a novel all-glass based μGCC with 2 m length for the separation of light hydrocarbons is proposed. The microchannels of the μGCC were directly prepared in the glass substrate by ultrafast laser assisted chemical etching (ULAE). The all-glass microchannels make the coating of the hydrophilic metal-organic frameworks (MOFs) stationary phase continuously because of the homogeneous material composition. Therefore, a widely used copper based hydrophilic MOFs HKUST-1 was used as stationary phase for coating and testing. The test results show that the μGCC which is an open tubular column can realize the baseline separation of light hydrocarbons at 100 °C. And the resolution of difficult separated compounds, methane and ethane, can reach 12.98, which is 201.86 % higher than the silica-based monolithic capillary column in the relevant research. The resolution of ethane and ethylene reaches 6.81 at 120 °C. SignificanceThe μGCC fabricated by ULAE method is composed of all-glass and has the uniform stationary phase coating because of the homogeneous microchannel surface which greatly improve the separation performance, resulting in a large resolution for methane and ethane. The all-glass μGCC has broad application prospects in light hydrocarbon separation.

A micro gas chromatography column fabricated by ultrafast laser-assisted chemical etching

A novel micro gas chromatography column (μGCC) with straight-sided elliptic microchannel cross sections was prepared by ultrafast laser-assisted chemical etching inside a glass substrate. The straight-sided elliptic cross sections resulted in a uniform distribution of the flow velocity field. This uniform distribution and the use of a microchannel with a consistent composition led to a uniform stationary phase coating. Upon the separation of a test mixture of C7–C11 alkanes, highly symmetrical chromatographic peaks were obtained. The tailing factors of the chromatographic peaks ranged from 0.95 to 1.05, which satisfies the relevant provisions of Pharmacopoeia of the People's Republic of China (ChP). The test results show that the improvement of the tailing factors is mainly due to the optimization of the cross-section structure rather than the optimization of the test conditions. Notably, due to the simple preparation processes and the low cost of the glass substrate, the total manufacturing cost of this novel μGCC is low.

Surface Displacement Field of a Coated Elastic Half-Space Under the Influence of a Moving Distributional Load

An analysis of the distributed moving load along the surface of a coated half space is presented. The formulation of the problem depends on the hyperbolic-elliptic asymptotic model developed earlier by the authors. The integral solution of the longitudinal and transverse displacements along the surface for the sub and super-Rayleigh cases are obtained by using the uniform stationary phase method. Numerical comparisons of the exact and asymptotic solutions of the longitudinal displacement are illustrated for the certain cross-sections of the profile.

The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space

This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.

Transmutation of non-local boundary conditions in ocean acoustics

We employ a transmutation technique to construct a new non-local boundary condition for the paraxial approximation in ocean acoustics. The transmutation operator introduced by De Santo and Polyanskii, when applied to the Helmholtz equation governing the acoustic pressure in the water column, leads to the so-called parabolic equation of Fock and Tappert. This transmutation operator acting on the N-D map at the water–bottom interface yields an intermediate non-local boundary condition for the parabolic equation which eliminates the backscattering terms in the Schwartz kernel of the N-D map. The kernel of the intermediate condition is approximated by a uniform stationary phase formula taking account of the possible coalescence of the brach points of the integrand with the stationary points of the phase, and it leads to a non-local boundary condition of Volterra-type for the parabolic equation. This condition is quite different than similar conditions derived by other approximations, in that the kernel of the Volterra operator is smooth, the smoothing effect coming from the fact that the horizontal range coordinate is scaled with the relative refraction index between the water column and the bottom.

The statistical theory of linear capillary chromatography with uniform stationary phase

Based on the random walk model and probability theory, general relations between the moments of column residence time and the moments of step sojourn time and step displacement are established. And starting from the mass-balances principle of solute molecules in the mobile and stationary phases, the moments of step sojourn time and step displacement are derived and expressed in terms of the basic parameters. Substituting the step moments into the general relations, the moments of column residence time are then obtained. The expression of retention time is completely identical to the well-known, the expressions of second moment or HETP unite and generalize various expressions of stochastic theory and mass balance theory, and the third and forth moments are given in more exact form.

Modeling the quasicompressional wave field of a rectangular transducer in a transversely isotropic solid.

A fast model for simulating the transient quasicompressional wave field of a rectangular ultrasonic transducer directly coupled to a transversely isotropic elastic half-space of general orientation is developed. The so-called two-tier asymptotic approach and the uniform stationary phase method are used to derive the high-frequency asymptotics of time-harmonic displacements. Then, transient fields are modeled by means of harmonic synthesis. In geometrical regions, the formulas involve elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral), and are applicable in the radiating near field. The asymptotics elucidate the physics in terms of various arrivals and give explicit dependence of the radiated waves upon model parameters. The asymptotic code is tested against a direct numerical solution. It is at least a thousand times faster but describes accurately both arrival times and amplitudes of various pulses radiated by the transducer.

Diffraction by polyhedral angle: Field in vicinity of singular ray

If the plane acoustic wave hits a polyhedral angle, the scattered field consists of an incident wave, of its reflections from the angle’s facets, of primary edge waves excited at angle’s edges, and of a spherical wave excited at the angle’s vertex. It is supposed for simplicity that the geometry of the problem is such that there is no secondary and subsequent edge waves. There are two types of light–shadow boundaries: first for incident and reflected waves and second for edge waves. As is well known, the field in penumbras is expressed via a Fresnel integral. The incident wave’s ray passing through the vertex and its reflections from facets are singular rays belonging to the light–shadow boundaries of both types. It will be shown that the field in the vicinity of a singular ray is expressed via a Fresnel intergal and via a generalized Fresnel integral G(p,q) which was introduced in [P. S. Clemmov and T. B. A. Senior, Proc. Cambridge Philos. Soc. 9, 570–572 (1953); see also V. A. Borovikov, Uniform Stationary Phase Method, IEE Electromagnetic Series 40, London, 1994]. This result is valid for an elastic incident wave as well.

Electrochromatography in microchips: reversed-phase separation of peptides and amino acids using photopatterned rigid polymer monoliths.

A microfabricated glass chip containing fluidic channels filled with polymer monolith has been developed for reversed-phase electrochromatography. Acrylate-based porous polymer monoliths were cast in the channels by photopolymerization to serve as a robust and uniform stationary phase. UV light-initiated polymerization allows for patterning of polymer stationary phase in the microchip, analogous to photolithography, using a mask and a UV lamp for optimal design of injection, separation, and detection manifolds. The monoliths are cast in situ in less than 10 min, are very reproducible with respect to separation characteristics, and allow easy manipulation of separation parameters such as charge, hydrophobicity, and pore size. Moreover, the solvent used to cast the polymer enables electroosmotic flow, allowing the separation channel to be conditioned without need for high-pressure pumps. The microchip was used for separation of bioactive peptides and amino acids labeled with a fluorogenic dye (naphthalene-2,3-dicarboxaldehyde) followed by laser-induced fluorescence detection using a Kr+ ion laser. The microchip-based separations were fast (six peptides in 45 s), efficient (up to 600,000 plates/m), and outperformed the capillary-based separations in both speed and efficiency. We have also developed a method for complete removal of polymer from the channels by thermal incineration to regenerate the glass chips.

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

The dynamic time–harmonic Green's tensor for a transversely isotropic homogeneous linearly elastic solid is studied. First, the representation of the Green's tensor as an integral over the unit sphere is obtained. It consists of three parts: quasi–longitudinal (P), shear–horizontal (SH) and quasi–shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far–field asymptotic approximation of P and SV parts is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.

A fast method for simulating the propagation of pulses radiated by a rectangular normal transducer into an elastic half-space

A new fast method for simulating the propagation of pulses radiated by a rectangular normal ultrasonic transducer which is directly coupled to an isotropic and homogeneous elastic half-space is proposed. First, the so-called two-tier approach introduced in Fradkin et al. [“The radiating near-field asymptotics of a time-harmonic circular normal ultrasonic transducer in an elastic half-space,” J. Acoust. Soc. Am. 104, 1178–1187 (1998)] and the uniform stationary phase method are used to obtain both nonuniform and uniform high-frequency asymptotics of the time-harmonic field. Then, the transient field is described by means of harmonic synthesis. The nonuniform asymptotics elucidate the physics and all the asymptotics give explicit dependence of the radiated waves on model parameters. The formulas are applicable in the radiating near field that is the near field with the evanescent wave zone excluded. The asymptotics involve in geometrical regions elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral). The code based on the uniform asymptotics has been tested in all regions against an exact numerical solution. It is at least 104 times faster but in many realistic cases the accuracy does not suffer. The trains of pulses generated by rectangular and circular transducers are compared.

A uniform semiclassical sudden approximation for rotationally inelastic scattering

The infinite-order-sudden (IOS) approximation is investigated in the semiclassical limit. A simplified IOS formula for rotationally inelastic differential cross sections is derived involving a uniform stationary phase approximation for two-dimensional oscillatory integrals with two stationary points. The semiclassical analysis provides a quantitative description of the rotational rainbow structure in the differential cross section. The numerical calculation of semiclassical IOS cross sections is extremely fast compared to numerically exact IOS methods, especially if high Δj transitions are involved. Rigid rotor results for He–Na2 collisions with Δj≲26 and for K–CO collisions with Δj≲70 show satisfactory agreement with quantal IOS calculations.

Semiclassical treatment of bound state systems. III. A uniform stationary phase integration for the multidimensional problem

The semiclassical bound state formalism is extended to the multidimensional problem by deriving an expression for the reduced density matrix which remains finite when the charge density is calculated. We derive a uniform stationary phase integration technique by mapping the phase of the semiclassical propagator onto a function which has the same behavior as the phase of the free particle propagator, enabling us to incorporate the singularity at t = 0 into the integration. The resulting rdm has the oscillatory behavior of a Bessel function of order d/2, where d is the dimensionality, rather than that of a simple trigonometric function.