Fractional shot noise of an SU(N) Kondo system

• Damian Krychowski and
• Stanisław Lipiński

Beilstein J. Nanotechnol. 2026, 17, 515–540, doi:10.3762/bjnano.17.34

• dot (N-QD) in the Kondo regime. Using the Kotliar–Ruckentein slave boson approach (SBMFA) for an N-level Anderson model, we define effectively noninteracting quasiparticles of the SU(N) Kondo system (N = 2, 3, 4, 5, 6). Kondo resonance transmission coefficients determine linear noise describing

• of the Kondo effect is to incorporate higher-rank SU(N) symmetries (N > 2). The first SU(N) generalizations of the Anderson model appeared in the literature on heavy fermion systems [36], where large N expansion proved to be a powerful approximation for describing magnetic atoms with orbital

• degeneracy. From the perspective of potential applications, it is important that the Anderson SU(N) model can be realized in a controlled way in various nanoscopic structures [37][38] and in correlated cold atomic gases [39][40]. A proposal of the SU(6) Kondo effect for a QD structure can be found in [39

Probing internal continua and atomic ultrafast charge transfer within size-controlled nanoparticles by post-collision interaction in core-hole clock spectroscopy

• Johannes Lütgert,
• Erika Giangrisostomi,
• Nomi L. A. N. Sorgenfrei and
• Alexander Föhlisch

Beilstein J. Nanotechnol. 2026, 17, 505–514, doi:10.3762/bjnano.17.33

• the existence of a faster charge transfer channel toward the CdSe core, only accessible for the inner-most shell layers, rather than a quantum confinement effect. By extending the traditional PCI model from free-electron systems to bound-state continua, we further establish a framework for

•  1b, yields the charge transfer time in relation to the core-hole lifetime based on a model of independent exponential decay [20][21]. The validity and capabilities of the CHC approach was demonstrated across various sample systems, including gas adsorbates on surfaces [22][23], two-dimensional

• obtained by fitting the resonant Auger spectra at the individual photon energies (see Supporting Information File 1, section 5, for detailed fitting procedure and model). Multiplying this ratio with the core-hole lifetime yields the charge transfer times presented in Figure 2a. To provide for a better

Defects and defect-mediated engineering of two-dimensional materials: challenges and open questions

• Arkady V. Krasheninnikov,
• Matthias Batzill,
• Anouar-Akacha Delenda,
• Marija Drndić,
• Chris Ewels,
• Katharina J. Franke,
• Mahdi Ghorbani-Asl,
• Alexander Holleitner,
• Ado Jorio,
• Ute Kaiser,
• Daria Kieczka,
• Hannu-Pekka Komsa,
• Jani Kotakoski,
• Manuel Längle,
• David Lamprecht,
• Yun Liu,
• Steven G. Louie,
• Janina Maultzsch,
• Thomas Michely,
• Katherine Milton,
• Anna Niggas,
• Hanako Okuno,
• Joshua A. Robinson,
• Marika Schleberger,
• Bruno Schuler,
• Alexander Shluger,
• Kazu Suenaga,
• Kristian S. Thygesen,
• Richard A. Wilhelm,
• E. Harriet Åhlgren and
• Carla Bittencourt

Beilstein J. Nanotechnol. 2026, 17, 454–488, doi:10.3762/bjnano.17.31

• system can have zero-dimensional (0D) defects and one-dimensional (1D) defects. 0D are point defects, like a vacancy or a substitutional atom, while 1D are line defects, like crystallite borders or mirror twin boundaries. Based on the geometrical model introduced in [81], it was possible to determine a

• ), mostly for pristine materials, but in a few cases also with defects [94][115]. In order to model Raman scattering, a model for the response of the material to incoming light is required. In the non-resonant case, Raman scattering is allowed when the atomic displacement of a vibrational mode leads to a

• polarizations of the incoming and the scattered light. In addition, first-principles calculations of polarizability are significantly more demanding than those of total energy, which is especially true for large systems with defects; this can limit the collection of training data. Finally, training a model for

Nanocarrier-integrated multilayer films produced by 3D printing for improved skin adhesion and curcumin photostability

• Thayse Viana de Oliveira,
• Ana Paula Farias Leão,
• Júlia Leão,
• Cesar Liberato Petzhold and
• Ruy Carlos Ruver Beck

Beilstein J. Nanotechnol. 2026, 17, 440–453, doi:10.3762/bjnano.17.30

• inhibiting MRSA growth in vivo in a murine wound model [21]. More recently, Terzopoulou et al. developed collagen patches containing chitosan nanoparticles loaded with curcumin, which showed efficacy against psoriasis in in vitro assays [22]. Similarly, Olmos-Juste et al. employed 3D printing to produce

• tangent (tan δ), were measured using an oscillatory frequency sweep over an angular frequency range of 1–600 rad/s at a constant strain of 0.5%. Flow behaviour was assessed using a shear rate ramp from 0.1 to 1.0 s−1 and subsequently from 1.0 to 0.1 s−1, in order to determine the best-fit flow model for

• generated model and, using a syringe fitted with a nozzle with a diameter of 0.41 mm, deposited the material layer by layer at room temperature. The printing parameters were set to a rectilinear infill pattern with 100% infill density and an extrusion speed of 6 mm/s. After printing, the films were dried at

First-principles study on elastic properties of Cu, (Cu_1−x,Ni_x)₃Sn and interfacial mechanical properties of (Cu_1−x,Ni_x)₃Sn/Cu in the lead-free solder joint

• Guomin Hua

Beilstein J. Nanotechnol. 2026, 17, 428–439, doi:10.3762/bjnano.17.29

• ductility in terms of elastic moduli. Subsequently, the orientation-dependent Young’s moduli of Cu and (Cu1−xNix)3Sn were calculated. Finally, tensile modulus, ultimate tensile stress, work of adhesion, and interfacial toughness of (Cu1−xNix)3Sn/Cu were calculated based the interface model with the

• efficiency without losing the accuracy by reducing the model size of the alloy systems. Considering the phase stability of (CuxNi1−x)3Sn [22], the content of Ni was set within the range from 0 to 30 atom %. As far as the calculations of the structure optimizations and elastic properties are concerned, a

Nanoinformatics: spanning scales, systems and solutions

• Iseult Lynch,
• Diego S. T. Martinez,
• Kunal Roy and
• Georgia Melagraki

Beilstein J. Nanotechnol. 2026, 17, 423–427, doi:10.3762/bjnano.17.28

• are challenging to experimentally measure. Using computational descriptors commonly used in nano-QSAR models, such as the potential energy of surface atoms and the water–NP surface energy, the model predicts that Pt NPs are more toxic than Au NPs, based on their surface properties, which drive

• reactivity [3]. Amini et al. combined atomistic molecular dynamics, a coarse-grained model of protein adsorption, and kinetic Monte Carlo simulations to predict the protein corona composition formed on aluminium surfaces with different crystal faces, (i.e., (100), (110), and (111)) from a simplified model of

• milk consisting of the six most abundant milk proteins found in natural cow milk and lactose, the most abundant sugar found in dairy products, based on their corresponding interaction strengths. The resulting freely accessible multiscale computational model enables predictions of the binding strength

Biomimetic nanoparticles in cancer photodynamic therapy: a review of targeted delivery systems and therapeutic outcomes

• Valentina I. Gorbacheva,
• Alexey S. Grabovoy,
• Polina S. Marukhina,
• Anastasiia O. Syrocheva and
• Ekaterina P. Kolesova

Beilstein J. Nanotechnol. 2026, 17, 396–422, doi:10.3762/bjnano.17.27

• genetically engineered HEK-293 cell line expressing synthetic SpyCatcher proteins was used to coat NPs, enabling targeted binding to both tumor cells and viral particles. Treatment with these nanoparticles in a mouse model of ovarian cancer suppressed tumor growth and improved survival outcomes [113]. For the

• demonstrated that the biomimetic membrane coating reduced macrophage-mediated clearance of the nanoparticles to a half, while maintaining efficient uptake by HeLa cells and preserving phototoxicity upon NIR irradiation. In vivo experiments in a CT-26 tumor mouse model showed that membrane-coated nanoparticles

Eco-efficient materials for agricultural crops based on a mineral rich in MOR- and HEU-type zeolites

• Esperanza Yamile de la Nuez-Pantoja,
• Inocente Rodríguez-Iznaga,
• Gerardo Rodríguez-Fuentes,
• Vitalii Petranovskii,
• Ariel Martínez García,
• José Juan Calvino Gámez and
• Daniel Goma Jiménez

Beilstein J. Nanotechnol. 2026, 17, 381–395, doi:10.3762/bjnano.17.26

• model CHNS-932 was used, for which 6 mg of sample were used, dried before at 60 °C and pulverized in a mortar. For the determination of N content by Kjeldahl, samples were dissolved by hydrothermal treatments with HF, HClO4, and HCl, followed by a procedure similar to that described in [15]. In addition

Ferroelectric nanodot reservoir for neuromorphic computing

• Anna Razumnaya,
• Yuri Tikhonov,
• Dmitrii Naidenko,
• Léo Boron,
• Valerii Vinokur and
• Igor Lukyanchuk

Beilstein J. Nanotechnol. 2026, 17, 352–364, doi:10.3762/bjnano.17.24

• polarization dynamics across nanodots. The total electrostatic energy of the reservoir system becomes: Here, the system assumes the form of a frustrated Ising model with long-range antiferromagnetic interactions. The first term describes interactions between all pairs of sites via: The second term captures the

Calculation of the dynamic stiffness of a cantilever under torsional oscillation

• Keita Nishida,
• Yuuki Yasui and
• Yoshiaki Sugimoto

Beilstein J. Nanotechnol. 2026, 17, 303–308, doi:10.3762/bjnano.17.21

• underlying deformation mechanism. Here, we calculated the dynamic stiffness of cantilevers with tips in torsional oscillation using the equivalent spring-moment of inertia model. Then we found that the dynamic stiffness should be modified by 21–23% when the tip is considered. Dissipated energy is derived

• from the excitation amplitude through a proportionality coefficient that depends on the dynamic stiffness [21]. Thus, the present correction for the dynamic stiffness should be adopted for friction analyses with torsional oscillations in nc-AFM. Model We consider a cantilever with the dimensions length

• the deduced moment of inertia. The oscillator is modelled as an equivalent spring-moment of inertia model [22][24], and the static stiffness and the dynamic stiffness under torsional oscillation are calculated. The equation of motion of the oscillator under the torsional oscillation is given by where

Fast vortex dynamics and relaxation times in NbRe-based heterostructures

• Francesco De Chiara,
• Zahra Makhdoumi Kakhaki,
• Francesco Avitabile,
• Francesco Colangelo,
• Abhishek Kumar,
• Carmine Attanasio and
• Carla Cirillo

Beilstein J. Nanotechnol. 2026, 17, 292–302, doi:10.3762/bjnano.17.20

• superconducting coherence. This phenomenon, known as flux-flow instability (FFI), depends on several factors such as structural disorder and defects [3], pinning mechanisms [4][5], thermal effects [6], and sample geometry [7]. FFI is effectively described by the Larkin–Ovchinnikov (LO) model [8], which relates

• assumptions underlying the LO model remain valid and the deduced relaxation time can be regarded as an intrinsic property of the superconducting system [1]. Experimentally, biasing a superconducting strip with a current I, leads, in the flux flow regime, to the appearance of a jump to normal state at an

Advancing nanolithography: a comprehensive review of materials for local anodic oxidation with AFM

• Matteo Lorenzoni

Beilstein J. Nanotechnol. 2026, 17, 275–291, doi:10.3762/bjnano.17.19

• authors agree with an oxidation kinetic consistent with the Cabrera–Mott model [44][45]. As an electrochemical process, LAO is inherently current-dependent. Early studies showed that oxide growth evolves from an initial electronic tunneling regime to ionic transport as the film develops [46]. Dagata and

• thickness, and voltage, following the Cabrera–Mott model. This clean, precise approach highlights LAO’s potential for photonic and nanoelectronics applications, aligning with advances in LAO research. 2.5.3 Two-dimensional materials. 2D materials are at the forefront of LAO research, owing to their

Multilayered hyperbolic Au/TiO₂ nanostructures for enhancing the nonlinear response around the epsilon-near-zero point

• Fernando Arturo Araiza-Sixtos,
• Mauricio Gomez-Robles,
• Rafael Salas-Montiel and
• Raúl Rangel-Rojo

Beilstein J. Nanotechnol. 2026, 17, 251–261, doi:10.3762/bjnano.17.17

• followed by a postfocal maximum, albeit somehow deformed. This deformation is due to the complicated nonlinear absorption observed in the OA z-scan. In order to model the response, we used the approach found in [31], where the CA z-scan trace shows effects of both nonlinear absorption and refraction: where

• S is the aperture transmittance, I0 is the peak irradiance, and is the effective length. Setting S = 1, we are left with ΔΨ0, which is related only to nonlinear absorption effects, and we can then model the fit for the OA z-scan results. For the CA z-scan, we set S = 0.5 (from experiment), and we

• describe both effects. Substituting in Equation 6, we are now able to model the effect of both nonlinear absorption effects. Then, for each different sample, we must find a set of constants β+, β−, , and that can be used to fit every z-scan data set obtained at different I0 values. Figure 5 shows the fits

Comparative study on 3D morphologies of delignified, single tracheids and fibers of five wood species

• Helen Gorges,
• Felicitas von Usslar,
• Cordt Zollfrank,
• Silja Flenner,
• Imke Greving,
• Martin Müller,
• Clemens F. Schaber,
• Chuchu Li and
• Stanislav N. Gorb

Beilstein J. Nanotechnol. 2026, 17, 239–250, doi:10.3762/bjnano.17.16

• 6.0.1 software (FEI SAS, Lyon, France). The segmented data were further processed in Blender software (Blender Foundation, Amsterdam, Netherlands, https://www.blender.org). Additional interactive 3D models of two samples for each species are available online with links for each model in Supporting

Gold nanoparticle-decorated reduced graphene oxide as a highly effective catalyst for the selective α,β-dehydrogenation of N-alkyl-4-piperidones

• Brenda Flore Kenyim,
• Mihir Tzalis,
• Marilyn Kaul,
• Robert Oestreich,
• Aysenur Limon,
• Chancellin Pecheu Nkepdep and
• Christoph Janiak

Beilstein J. Nanotechnol. 2026, 17, 218–238, doi:10.3762/bjnano.17.15

• size, stabilized by SiW9, and supported on the three-carbon materials AC, rGO, and CB by the RD and DP methods were synthesized as depicted in Figure 1. To assess the influence of the preparation methods and the type of the carbon supports, the catalytic activity was evaluated using the model reaction

• -, and N-benzyl-4-piperidone to the respective N-methyl-2,3-dihydropyridin-4(1H)-ones as a model reaction (Scheme 1). This transformation is an example of the synthesis of β-N-substituted α,β-unsaturated ketones and requires catalysts with strong oxidizing capabilities, such as gold-based catalysts. The

Micro- and nanoscale effects in biological and bioinspired materials and surfaces

• Thies H. Büscher,
• Rhainer Guillermo Ferreira,
• Manuela Rebora and
• Stanislav N. Gorb

Beilstein J. Nanotechnol. 2026, 17, 214–217, doi:10.3762/bjnano.17.14

• ] provided insights into biological plant materials combining experiments with simulations that provide inspiration for biomimetic actuators. For animals, Jain et al. [20] applied a multiscale computational model to gain detailed insights into the molecular and mechanical behavior of gecko setae during

Structure-dependent thermochromism of PAZO thin films: theory and experiment

• Georgi Mateev,
• Dean Dimov,
• Nataliya Berberova-Buhova,
• Nikoleta Kircheva,
• Todor Dudev,
• Ludmila Nikolova,
• Elena Stoykova,
• Keehoon Hong,
• Dimana Nazarova,
• Silvia Angelova and
• Lian Nedelchev

Beilstein J. Nanotechnol. 2026, 17, 186–199, doi:10.3762/bjnano.17.12

• ) calculations were employed to model possible supramolecular architectures, allowing us to explore the correlation between thermally induced optical responses and the underlying molecular-level organization. Results and Discussion Experimental results Temperature dependence of absorption spectra Absorption

• change in the optical properties of the isolated monomers. The possible dimers formed by the E-forms of the simplified PAZO side-chain model (shown in Figure 6D) are also presented in Figure 8. The aggregation upon heating hypothesis can be confirmed by simulating the absorption spectra of monomers and

• ωB97XD/6-31G(d,p) optimized structures in solvent chloroform (used to mimic the polarity of the polymer layer). Simulated (using the vertical excitation energies and oscillator strengths) UV spectra are given in Figure 9. One strong monomer band at 383 nm is predicted. The spectra of the model dimers

Safe and sustainable by design with ML/AI: A transformative approach to advancing nanotechnology

• Georgia Melagraki

Beilstein J. Nanotechnol. 2026, 17, 176–185, doi:10.3762/bjnano.17.11

• networks, where standardized information on nanomaterial properties and toxicity profiles can be used to train increasingly robust ML models. Overall, the synergy between the SSbD concept, advanced ML/AI algorithms, and comprehensive regulatory directives fosters a future-oriented model of nanotechnology

• , inconsistently formatted, and insufficiently annotated for robust ML/AI model training [63][64]. Moreover, these datasets often arise from disparate sources – academic research labs, industrial R&D facilities, and public databases – each with its own protocols and measurement standards. Such heterogeneity

• annotations [65] and modelling-ready nanomaterials EHS and SSbD relevant databases including VINAS [66] and NanoPharos [67]. Another major hurdle is model interpretability, particularly for deep learning approaches that often function as “black boxes”. Despite their high predictive power, complex

Influence of surface characteristics on the in vitro stability and cell uptake of nanoliposomes for brain delivery

• Dushko Shalabalija,
• Ljubica Mihailova,
• Nikola Geskovski,
• Andreas Zimmer,
• Otmar Geiss,
• Sabrina Gioria,
• Diletta Scaccabarozzi and
• Marija Glavas Dodov

Beilstein J. Nanotechnol. 2026, 17, 139–158, doi:10.3762/bjnano.17.9

• interactions in complex organs like the brain, in vitro cell culture models, such as hCMEC/D3 and SH-SY5Y, are often used [9]. Therefore, the hCMEC/D3 model is useful for examining nanosystems, drug uptake, and transport across the BBB, with advantages like easy growth and mimicking basic BBB properties

• stability, toxicity, and therapeutic potential of nanodelivery systems, thus improving the translation of study results to real biological systems. In this direction, some of the prerequisites for establishing a relevant in vitro model for cell uptake studies include precise control over factors like

• set to 1.38 and 0.010, respectively. Water was set as the dispersant, the temperature was set to 25 °C, while the attenuation was set to 11. The 'general purpose (normal distribution)' was chosen as the analysis model. High-resolution automated electrophoresis of the adsorbed proteins onto

Capabilities of the 3D-MLSI software tool in superconducting neuron design

• Irina E. Tarasova,
• Nikita S. Shuravin,
• Liubov N. Karelina,
• Fedor A. Razorenov,
• Evgeny N. Zhardetsky,
• Aleksandr S. Ionin,
• Mikhail M. Khapaev and
• Vitaly V. Bol’ginov

Beilstein J. Nanotechnol. 2026, 17, 122–138, doi:10.3762/bjnano.17.8

• primary subject of this article is the detailed verification of 3D-MLSI software tool. The main task of this inductance extractor is an evaluation of two-terminal partial inductances [48] associated with equivalent scheme ones. The general mathematical model for all superconductor inductance calculations

Development and in vitro evaluation of liposomes and immunoliposomes containing 5-fluorouracil and R-phycoerythrin as a potential phototheranostic system for colorectal cancer

• Raissa Rodrigues Camelo,
• Vivianne Cortez Sombra Vandesmet,
• Octavio Vital Baccallini,
• José de Brito Vieira Neto,
• Thais da Silva Moreira,
• Luzia Kalyne Almeida Moreira Leal,
• Claudia Pessoa,
• Daniel Giuliano Cerri,
• Maria Vitória Lopes Badra Bentley,
• Josimar O. Eloy,
• Ivanildo José da Silva Júnior and
• Raquel Petrilli

Beilstein J. Nanotechnol. 2026, 17, 97–121, doi:10.3762/bjnano.17.7

• 7.4 containing 5-FU and/or R -PE, respectively, at final concentrations of 260 μg/mL and 1 mg/mL. Subsequently, the formulations were subjected to an ultrasonic bath for 15 min. To reduce the size of the vesicles, the sonication method with a probe ultrasound was applied (Qsonica Sonicator – Model

• previously described [24]. 2.3.4.3 Release profile of R-PE and 5-FU. The release profile of R-PE and 5-FU from liposomes and immunoliposomes was analyzed using the Korsmeyer–Peppas (Equation 3) model, commonly employed in drug release kinetics studies, using the DDSolver software [24]. The Korsmeyer–Peppas

• is a semi-empirical model that encompasses release profiles of drugs through polymeric chains when the governing mechanism is a combination of Fickian and non-Fickian mechanisms: where Q is the amount of dose released at time t, k is the release rate constant, and n is the exponent. 2.3.5 Atomic

Microscopic study of the intermediate mixed state in intertype superconductors

• Vyacheslav D. Neverov,
• Alexander V. Kalashnikov,
• Andrey V. Krasavin and
• Alexei Vagov

Beilstein J. Nanotechnol. 2026, 17, 57–62, doi:10.3762/bjnano.17.5

• /bjnano.17.5 Abstract We present a comprehensive microscopic study of the intermediate mixed state in superconductors of the intertype (IT) regime separating types I and II. Using fully self-consistent Bogoliubov–de Gennes calculations for a lattice model, we analyze few-vortex configurations across the

• increases, shrinking to a single point at Tc. Based on these findings, we construct a phase diagram of the IT regime, which appears qualitatively consistent with that obtained earlier from perturbation theory for the conventional BCS model with a spherical Fermi surface. Results and Discussion Model and

• method The vortex configurations are analyzed within a microscopic lattice model of a superconductor described by the attractive Hubbard Hamiltonian: where () are the annihilation (creation) operator for an electron with spin σ at site i, is the electron number operator, tij = −t is the nearest

Subdigital integumentary microstructure in Cyrtodactylus (Squamata: Gekkota): do those lineages with incipiently expressed toepads exclusively exhibit adhesive setae?

• Philipp Ginal,
• Yannick Ecker,
• Timothy Higham,
• L. Lee Grismer,
• Benjamin Wipfler,
• Dennis Rödder,
• Anthony Russell and
• Jendrian Riedel

Beilstein J. Nanotechnol. 2026, 17, 38–56, doi:10.3762/bjnano.17.4

• The ER model revealed that the phylogeny of the sampled species is largely split into two groups (Figure 2). The first group [seta-group, i.e., a clade formed by the species groups pulchellus, intermedius, and brevipalmatus, as well as the species C. novaeguineae (novaeguineae group), C. louisiadensis

• (LMM) analysis, the ecotype model revealed highly significant effects between each ecotype (cave, crown, generalist, granite, intertidal, karst, terrestrial, and trunk) and apical diameter and effective bending stiffness, but only a significant effect between the crown ecotype and density (Supporting

• Information File 2). The structuretype-model revealed highly significant relationships between each derived microstructure type (prongs, spines, and setae) and apical diameter as well as effective bending stiffness. Only setae also had a highly significant relationship to density, showing lower densities than

Competitive helical bands and highly efficient diode effect in F/S/TI/S/F hybrid structures

• Tairzhan Karabassov,
• Irina V. Bobkova,
• Pavel M. Marychev,
• Vasiliy S. Stolyarov,
• Vyacheslav M. Silkin and
• Andrey S. Vasenko

Beilstein J. Nanotechnol. 2026, 17, 15–23, doi:10.3762/bjnano.17.2

• Hamiltonian in Equation 1 within the microscopic approach based on the quasiclassical Green’s functions in the diffusive limit, that is, when the coherence length ξ is much larger than the electron mean free path l. Such model can be described by the Usadel equations [79][80][81] Here D is the diffusion

• develop the model.

Quantitative estimation of nanoparticle/substrate adhesion by atomic force microscopy

• Aydan Çiçek,
• Markus Kratzer,
• Christian Teichert and
• Christian Mitterer

Beilstein J. Nanotechnol. 2026, 17, 1–14, doi:10.3762/bjnano.17.1

• observation, we refer to the approach of Weir and McGavin [8], who developed an analytic model for describing the coefficient of restitution of NPs rebounding from an ideally flat and rigid surface. Note that it delivers just a semi-quantitative trend and is not a rigorous description of the process. Their

• model provides a criterion to determine whether an NP escapes or is captured upon impact. This condition is expressed as: where F0 is the force acting to detach the NP from the surface, and 6πRγ is the adhesive force trying to hold the NP on the surface. R represents the radius of curvature defining the

• , and infinitely rigid, and its properties are therefore not considered in the model. The total velocity of a NP, vi, can be expressed as follows: where the first term corresponds to the initial NP velocity as it exits the NP source orifice, and the second term is the velocity added due to the substrate