Intermediate Frequency Regime Research Articles

Quasiperiodic disorder induced critical phases in a periodically driven dimerized p-wave Kitaev chain

The intricate relationship between topology and disorder in non-equilibrium quantum systems presents a captivating avenue for exploring localization phenomenon. Here, we look for a suitable platform that enables an in-depth investigation of the topic. To this end, we delve into the nuanced analysis of the topological and localization characteristics exhibited by a one-dimensional dimerized Kitaev chain under periodic driving and perform detailed analyses of the Floquet Majorana modes. Such a non-equilibrium scenario is made further interesting by including a spatially varying quasiperiodic potential with a temporally modulated amplitude. Apriori, the motivation is to explore an interplay between dimerization and a quasiperiodic disorder in a topological setting which is also known to demonstrate unique (re-entrant) localization properties. While the topological properties of the driven system confirm the presence of zero and π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} Majorana modes, the phase diagram obtained by constructing a pair of topological invariants (Z×Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z} \ imes \\mathbb {Z} $$\\end{document}), also referred to as the real space winding numbers, at different driving frequencies reveal intriguing features that are distinct from the static scenario. In particular, at either low or intermediate frequency regimes, the phase diagram concerning the zero mode involves two distinct phase transitions, one from a topologically trivial to a non-trivial phase, and another from a topological phase to an Anderson localized phase. On the other hand, the study of the Majorana π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} mode unveils the emergence of a unique topological phase, characterized by complete localization of both the bulk and the edge modes, which may be called as the Floquet topological Anderson phase. Moreover, different frequency regimes showcase distinct localization features which can be examined via the localization toolbox, namely, the inverse and the normalized participation ratios. Specifically, the low and high-frequency regimes demonstrate the existence of completely extended and localized phases, respectively. While at intermediate frequencies, we observe the critical (multifractal) phase of the model which is further investigated via a finite-size scaling analysis of the fractal dimension. Finally, to add depth into our study, we have performed a mean level spacing analyses and computed the Hausdorff dimension which yields specific characteristics inherent to the critical phase, offering profound insights into its underlying properties.

A Science Friction Story: Molecular Interactions in Semiflexible Polymer Networks

AbstractEstablished model theories, developed to capture the mechanical behavior of soft, complex materials composed of semiflexible polymers, assume that entropic interactions between filaments are primarily responsible for determining the mechanical response. In recent studies, the generally accepted tube model has been challenged in terms of this basic assumption about filament–filament interactions, but also because of its predictions regarding the frequency dependence of the elastic modulus in the intermediate frequency regime. A central question is how molecular interactions and friction between network constituents influence the rheological response of isotropic entangled networks of semiflexible polymers. It has been previously shown that friction forces between aligned pairs of actin filaments are not negligible. Here, the influence of friction forces and attractive interactions on network rheology is systematically investigated by means of targeted surface modification. It is shown that these forces have a qualitative and quantitative influence on the viscoelastic properties of semiflexible polymer networks and contribute to their response to nonlinear deformations. By comparing two polymer model systems with respect to their surface compositions, a possible explanation is given about the origin of acting forces on a molecular level.

Scattering of phonons by edge dislocation and thermal conductivity of nanocrystalline silicon

The temperature dependent thermal conductivity of nanocrystalline silicon with the grain size of 500 nm, 144 nm, and 76 nm, was calculated based on the intrinsic phonon properties using a first-principles-based method and recent analytical theory of phonon-dislocation scattering. The results demonstrate, in the nanocrystalline silicon, that the low temperature (T) limit of thermal conductivity is directly proportional to T2 instead of the conventional Casimir limit of T3. Calculated frequency-dependent scattering rates also indicate regimes of dominant scattering mechanisms: frequency-independent boundary scattering dominates in the very low phonon frequency regime of less than 0.6 THz; frequency-dependent boundary scattering dominates in the intermediate phonon frequency regime between 0.6 and 6 THz; and Umklapp scattering dominates in the high phonon frequency regime beyond 6 THz. The thermal conductivity is linear proportional to nanograin size in the region of d<100nm and it shows independence at d>1μm. This work proves the importance of dislocation-phonon scattering to thermal conductivity when the size of nanograins is small or at low temperature.

Photoinduced prethermal order parameter dynamics in the two-dimensional large- N Hubbard-Heisenberg model

We study the microscopic dynamics of competing ordered phases in a two-dimensional correlated electron model, which is driven with a pulsed electric field of finite duration. In order to go beyond a mean-field treatment of the electronic interactions we adopt a large-$N$ generalization of the Hubbard model and combine it with the semiclassical fermionic truncated Wigner approximation as a time evolution method. This allows us to calculate dephasing corrections to the mean-field dynamics and to obtain stationary states, which we interpret as prethermal order. We use this framework to simulate the light-induced transition between two competing phases (bond density wave and staggered flux) and find that the post-pulse stationary state order parameter values are not determined alone by the amount of absorbed energy but depend explicitly on the driving frequency and field direction. While the transition between the two prethermal phases takes place at similar total energies in the low- and high-frequency regimes, we identify an intermediate frequency regime for which it occurs with minimal heating.

Magnetohydrodynamic instabilities in a self-gravitating rotating cosmic plasma

The generation of magnetohydrodynamic (MHD) waves and their instabilities are studied in galactic gaseous rotating plasmas with the effects of the magnetic field, the self gravity, the diffusion-convection of cosmic rays as well as the gas and cosmic-ray pressures. The coupling of the Jeans, Alfvén and magnetosonic waves, and the conditions of damping or instability are studied in three different cases, namely when the propagation direction is perpendicular, parallel and oblique to the static magnetic field, and are shown to be significantly modified by the effects of the Coriolis force due to the rotation of cosmic fluids and the cosmic-ray diffusion. The coupled modes can be damped or anti-damped depending on the wave number is above or below the Jeans critical wave number that is reduced by the effects of the Coriolis force and the cosmic-ray pressure. It is found that the deviation of the axis of rotation from the direction of the static magnetic field gives rise to the coupling between the Alfvén wave and the classical Jeans mode which otherwise results into the modified slow and fast Alfvén waves as well as the modified classical Jeans modes. Furthermore, due to the effects of the cosmic rays diffusion, there appears a new wave mode (may be called the fast Jeans mode) in the intermediate frequency regimes of the slow and fast Alfvén waves, which seems to be dispersionless in the long-wavelength propagation and has a lower growth rate of instability in the high density regimes of galaxies. The dispersion properties and the instabilities of different kinds of MHD waves reported here can play pivotal roles in the formation of various galactic structures at different length scales.

Density of states below the first sound mode in 3D glasses.

Glasses feature universally low-frequency excess vibrational modes beyond Debye prediction, which could help rationalize, e.g., the glasses' unusual temperature dependence of thermal properties compared to crystalline solids. The way the density of states of these low-frequency excess modes D(ω) depends on the frequency ω has been debated for decades. Recent simulation studies of 3D glasses suggest that D(ω) scales universally with ω4 in a low-frequency regime below the first sound mode. However, no simulation study has ever probed as low frequencies as possible to test directly whether this quartic law could work all the way to extremely low frequencies. Here, we calculated D(ω) below the first sound mode in 3D glasses over a wide range of frequencies. We find D(ω) scales with ωβ with β < 4 at very low frequencies examined, while the ω4 law works only in a limited intermediate-frequency regime in some glasses. Moreover, our further analysis suggests our observation does not depend on glass models or glass stabilities examined. The ω4 law of D(ω) below the first sound mode is dominant in current simulation studies of 3D glasses, and our direct observation of the breakdown of the quartic law at very low frequencies thus leaves an open but important question that may attract more future numerical and theoretical studies.

Experimental corroboration of theory for impedance response of solid electrolytes: Doped cubic garnet LLZO

• Theory is corroborated with experimentally recorded impedance for Al and Ta-doped LLZO. • Theory resolves impedance of solid electrolyte into contribution from grain and grain boundary. • Phenomenological time scales are characterized through impedance of SSE. • Al and Ta-substituted cubic garnet Li 7 La 3 Zr 2 O 12 are synthesized using solid-state method. We corroborate phenomenological theory for the electrochemical impedance response of solid state electrolyte (SSE) with experimental data for the garnet structured oxide LLZO synthesized using conventional solid-state method. Theoretical analysis of experimentally recorded impedance spectrum indicates three characteristic frequency regimes, viz. (i) high frequency regime, ion relaxation dynamics in grain (g) and space charge layer (SCL) in grain boundary (gb) controls the impedance response, (ii) intermediate frequency regime, depicts the ion transfer and reorganization across heterogeneous grain and grain boundary and (iii) low frequency regime, response is attributed to the capacitive behavior due to the formation of double layer at solid electrolyte/partially blocking metal interface. Theory identifies phenomenological time scales associated with different electrochemical processes occurring in grain, grain boundary and their interface. The material characterization of samples is carried out using powder X-ray diffraction, scanning electron microscopy and Raman spectroscopy. The characteristic phenomenological time scales are extracted from the impedance response of Ta-doped and Al-doped LLZO unraveling the influence of doping. Finally, theory provides an alternative approach as it has rigor of phenomenological ab initio methodology as well as captures the simplicity of equivalent circuit models.

Theory for influence of uncompensated solution resistance on EIS of diffusion limited adsorption at rough electrode

Theory is developed for the electrochemical impedance spectroscopy (EIS) of the diffusion-limited adsorption process coupled with reversible charge transfer at rough electrodes under the influence of ubiquitous uncompensated solution resistance. This study quantitatively relates the impedance response of rough electrode to its phenomenological components, viz., diffusion limited adsorption, reversible charge transfer and uncompensated solution resistance. The random roughness of electrode is expressed by the surface statistical property, i.e., power spectrum of roughness. The fractal nature of roughness is characterized in terms of fractal dimension, lower cut-off length and topothesy length. The high-frequency regime is controlled by the uncompensated solution resistance whereas the low-frequency regime is governed by the adsorption process. The magnitude of impedance as well as phase decreases with rise in adsorption isotherm (length) parameter. The intermediate frequency regime is controlled by the coupling of adsorption and uncompensated solution resistance with the diffusion process. The fractal roughness parameters has quantitative influence on the magnitude of impedance over whole frequency regime while the phase plot shows qualitative difference in the intermediate frequency regime. The governing length scales which controls the characteristic crossover frequencies are: diffusion length, adsorption-ohmic coupling length and topothesy length (or width of interface). The three emergent crossover frequencies are: (i) ohmic reduced inner crossover frequency (ii) adsorption roughness topothesy dependent pseudo-quasireversibility characteristic frequency (iii) outer crossover frequency. Synopsis: This study quantitatively relates the impedance response of rough electrode to its phenomenological components, viz., diffusion limited adsorption, reversible charge transfer and uncompensated solution resistance. The random roughness of electrode is expressed by the surface statistical property, i.e., power spectrum of roughness.

Static Structure Factor and Viscoelastic Properties of Dendrimer Grafted Nanoparticles in Solution.

The theory for the dynamics of multiscale branched polymeric structures is applied to understand the dendrimer-grafted nanoparticles in a dilute solution. The multiscale nature of dendrimer-grafted nanoparticles arises due to larger beads for the nanoparticles and the smaller beads for the polymeric structure connected through the harmonic springs. The multiscale generalized Gaussian structure approach allows us to study several viscoelastic properties: (i) storage and loss moduli and (ii) intrinsic viscosity. The influence of nanoparticles in the dendrimer structure is reflected in low and intermediate frequency regimes of the viscoelastic relaxation moduli. The increase in the size and the number fraction of nanoparticle shows an anomalous enhancement in the relaxation moduli. The increase in number fraction of nanoparticle in dendrimer-grafted nanoparticles decreases the transition frequency between solid- and liquid-like viscoelastic region. The intrinsic viscosity of dendrimer-grafted nanoparticles increases with increasing the size of nanoparticle. The inclusion of hydrodynamic interactions facilitates the dynamics of dendrimer-grafted nanoparticles. The Kratky plot of the static structure factor of all conformation of dendrimer-grafted nanoparticles is also analyzed as a function of number fraction and the size of the nanoparticles. At low wavenumbers, all conformations of dendrimer-grafted nanoparticles show a universal behavior. The compactness of dendrimer-grafted nanoparticles increases with the increase in number fraction and the size of the nanoparticles.

Higher order topological insulator via periodic driving

We theoretically investigate a periodically driven semimetal based on a square lattice. The possibility of engineering both Floquet Topological Insulator featuring Floquet edge states and Floquet higher order topological insulating phase, accommodating topological corner modes has been demonstrated starting from the semimetal phase, based on Floquet Hamiltonian picture. Topological phase transition takes place in the bulk quasi-energy spectrum with the variation of the drive amplitude where Chern number changes sign from $+1$ to $-1$. This can be attributed to broken time-reversal invariance ($\mathcal{T}$) due to circularly polarized light. When the discrete four-fold rotational symmetry ($\mathcal{C}_4$) is also broken by adding a Wilson mass term along with broken $\mathcal{T}$, higher order topological insulator (HOTI), hosting in-gap modes at all the corners, can be realized. The Floquet quadrupolar moment, calculated with the Floquet states, exhibits a quantized value of $ 0.5$ (modulo 1) identifying the HOTI phase. We also show the emergence of the {\it{dressed corner modes}} at quasi-energy $\omega/2$ (remnants of zero modes in the quasi-static high frequency limit), where $\omega$ is the driving frequency, in the intermediate frequency regime.

Photoprotected spin Hall effect on graphene with substrate induced Rashba spin–orbit coupling

We propose an experimental realization of the spin Hall effect in graphene by illuminating a graphene sheet on top of a substrate with circularly polarized monochromatic light. The substrate induces a controllable Rashba type spin–orbit coupling which breaks the spin-degeneracy of the Dirac cones but it is gapless. The circularly polarized light induces a gap in the spectrum and turns graphene into a Floquet topological insulator with spin dependent edge states. By analyzing the high and intermediate frequency regimes, we find that in both parameter limits, the spin-Chern number can be tuned by the effective coupling strength of the charge particles to the radiation field and determine the condition for the photoinduced topological phase transition.

Theory for impedance response of grain and grain boundary in solid state electrolyte

We develop a generic phenomenological theory for the electrochemical impedance response of solid state electrolytes (SSEs). The theory describes dynamics of various physical processes in grain (g) and grain boundary (gb). Leaking capacitor model for the dynamics of ions in grain accounts compositional heterogeneity, ion reorganization and transfer across the surface of the grain. Model for ion relaxation in the grain boundary is determined through generalized electric double layer (EDL) theory over heterogeneous surface (Singh and Kant, 2013). The theory accounts for ion transfer and reorganization in grain boundary has the following components: (i) ions distribution at energetically heterogeneous surface of grain; (ii) space charge layer (SCL) in the grain boundary. Predicted impedance response of SSE evinces three frequency regimes, viz. (i) high frequency regime, attributed to relaxation of ions in grain and grain boundary (SCL), (ii) intermediate frequency regime, related to ion reorganization and transfer across heterogeneous grain and grain boundary (iii) low frequency regime, ascribed to the SSE/metal interface. The model emphasizes that ions encounter energetic heterogeneity in the grain causing distribution of relaxation time with a characteristic exponent γg. Doped grain with aliovalent cations further enhances the structural/energetic heterogeneity and concentration of mobile ions therefore decreases the ion transfer resistance from grain to grain boundary. Doping also enhances the surface concentration of mobile ions at the grain boundary region resulting in enhanced capacitance and prolonged relaxation time of compact layer.

Greybody Factors for Schwarzschild Black Holes: Path-Ordered Exponentials and Product Integrals

In earlier work concerning the sparsity of the Hawking flux, we found it necessary to re-examine what is known regarding the greybody factors of black holes, with a view to extending and expanding on some old results from the 1970s. Focusing specifically on Schwarzschild black holes, we have re-calculated and re-assessed the greybody factors using a path-ordered-exponential approach, a technique which has the virtue of providing a pedagogically useful semi-explicit formula for the relevant Bogoliubov coefficients. These path-ordered-exponentials, being based on a variant of the “transfer matrix” formalism, are closely related to so-called “product integrals”, leading to quite straightforward and direct numerical evaluation, while side-stepping any need for numerically solving the relevant ordinary differential equations. Furthermore, while considerable analytic information is already available regarding both the high-frequency and low-frequency asymptotics of these greybody factors, numerical approaches seem better adapted to finding suitable “global models” for these greybody factors in the intermediate frequency regime, where most of the Hawking flux is actually concentrated. Working in a more general context, these path-ordered-exponential techniques are also likely to be of interest for generic barrier-penetration problems.

Systematic bottom-up theory for the viscoelastic response of worm-like chain networks

Using the frequency-dependent force-extension response functions of single worm-like chains as the only input, the linear complex viscoelastic modulus of a polymeric network with given connectivity is derived from a systematic bottom-up theory via iterative coarse-graining. Choosing a cubic connectivity and accounting for random network orientation, we find excellent agreement with experimental data for actin networks under shear over the entire frequency range with the strength of the osmotic pressure that acts within the polymeric network as the only fitting parameter. In particular, we obtain a viscoelastic plateau regime at low frequencies and a crossover to an intermediate-frequency regime characterized by a power law behavior and an inhomogeneous shear deformation field.

Fast semi-analytical method for precise prediction of ion energy distribution functions and sheath electric field in multi-frequency capacitively coupled plasmas

We propose a fast semi-analytical method to predict ion energy distribution functions and sheath electric field in multi-frequency capacitively coupled plasmas, which are difficult to measure in commercial plasma reactors. In the intermediate frequency regime, the ion density within the sheath is strongly modulated by the low-frequency sheath electric field, making the time-independent ion density assumption employed in conventional models invalid. Our results are in a good agreement with experimental measurements and computer simulations. The application of this method will facilitate the understanding of ion–material interaction mechanisms and development of new-generation plasma etching devices.

Discrete multi-physics simulations of diffusive and convective mass transfer in boundary layers containing motile cilia in lungs

In this paper, the mass transfer coefficient (permeability) of boundary layers containing motile cilia is investigated by means of discrete multi-physics. The idea is to understand the main mechanisms of mass transport occurring in a ciliated-layer; one specific application being inhaled drugs in the respiratory epithelium. The effect of drug diffusivity, cilia beat frequency and cilia flexibility is studied. Our results show the existence of three mass transfer regimes. A low frequency regime, which we called shielding regime, where the presence of the cilia hinders mass transport; an intermediate frequency regime, which we have called diffusive regime, where diffusion is the controlling mechanism; and a high frequency regime, which we have called convective regime, where the degree of bending of the cilia seems to be the most important factor controlling mass transfer in the ciliated-layer. Since the flexibility of the cilia and the frequency of the beat changes with age and health conditions, the knowledge of these three regimes allows prediction of how mass transfer varies with these factors.

A parallelizable direct solution of integral equation methods for electromagnetic analysis

A parallelizable direct solution of integral equation methods is proposed for electromagnetic scattering analysis in low to intermediate frequency regime. There are mainly two parts of the proposed direct solution: forward decomposition and backward substitution. For the forward decomposition, the dense impedance matrix is decomposed of the product of several block diagonal matrices implicitly, which is shown to be O(Nlog2N) for both memory and CPU time cost. The final solutions are obtained with several matrix vector products (MVPs) in the part of backward substitution with O(Nlog2N) complexity as well. Both forward decomposition and backward substitution can be parallelized because of the group independence. Furthermore, an effective preconditioner with a reasonable selection criterion of the diagonal blocks region is proposed to accelerate the convergence of the iterative solver. The proposed solution is independent of the Green's function, and it is suitable for all the integral equation methods. Without loss of generality, the solution is proposed to solve the electric field integral equation (EFIE) in this work. Numerical tests demonstrate the effectiveness of the proposed solution for the electromagnetic analysis, especially for multiscale structures.

Transition to turbulence in pulsating pipe flow

Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $\unicode[STIX]{x1D6FC}$), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $\unicode[STIX]{x1D6FC}$. In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence (‘puffs’) analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero $\unicode[STIX]{x1D6FC}$ limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing $\unicode[STIX]{x1D6FC}$) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.

Interpretation of Nyquist Plot for Characterization of Electrode and Electrolyte Material Properties for Electrical Double Layer Capacitors

Electric double layer capacitors (EDLCs) store energy via ion adsorption in the electric double layer forming at the electrode/electrolyte interfaces. This charge storage mechanism is very fast and highly reversible resulting in large power density and long cycle life. Applications range from regenerative breaking for hybrid electric vehicles to the smart grid. Electrochemical impedance spectroscopy (EIS) has been used widely to determine the properties of the electrode and electrolyte materials for EDLCs. The impedance of an EDLC half-cell can be measured by applying a low-amplitude sinusoidal voltage to a steady-state potential and divide the imposed potential by the output sinusoidal current. Nyquist plots showing the imaginary part as a function of the real part of the impedance is informative to determine the properties including electrode and electrolyte resistance, charge transfer or polarization resistance at the electrode/electrolyte interface, and capacitance of the half-cell. However, the interpretation of Nyquist plots has been a “subject of controversy”, with no clear relations between the shape in high, intermediate, and low frequency regime of the plot and a particular property of the device. This study aims to derive guidelines for characterization of EDLC electrode and electrolyte materials from EIS measurements. To do so, Nyquist plots from EIC measurements were numerically reproduced for EDLC half-cells with planar electrodes in aqueous TEABF4 electrolyte. The continuum model used was based on the modified Poisson-Nernst-Planck model and accounted for (i) the Stern layer at the electrode/electrolyte interface, (ii) finite ion size, and (iii) binary and symmetric electrolyte. The effect of the electrode and electrolyte resistance, charge transfer or polarization resistance at the electrode/electrolyte interface, and the capacitances were separated numerically. This was done by varying the bulk ion concentration, electrolyte length, ion diffusion coefficient in the electrolyte, and/or the magnitude of steady-state potential. Therefore, the behaviour in each frequency regime in Nyquist plots that corresponded to a particular property was clarified.

Two-rate periodic protocol with dynamics driven through many cycles

We study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies $\omega_1$ and $\omega_2=r \omega_1$. Tuning of the ratio $r$ there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and XY models in $d=1$ and Kitaev model in $d=1$ and $2$ and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing to occur within some small/ intermediate frequency regimes in the $(\omega_1 ,r)$ plane (with $r\ne0$) when the ground state is evolved through single cycle of driving. However, evolved states soon become steady with long driving and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio $r$ in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow genuine non-area law of scaling and show a convergence (with a $r$ dependent pace) towards volume scaling behavior as the driving is continued for long time.