Fluctuations in a fluid under a stationary heat flux. I. General theory

We present the basic formulas for a unified treatment of the correlation functions of the hydrodynamic variables in a fluid between two horizontal plates which is exposed to a stationary heat flux in the presence of a gravity field (Rayleigh-Bénard system). Our analysis is based on fluctuating hydrodynamics. In this paper (I) we show that in the nonequilibrium stationary state the hydrodynamic fluctuations evolve on slow and fast time scales that are widely separated. A time scale perturbation theory is used to diagonalize the hydrodynamic operator partially. This enables us to derive the eigenvalue equations for the nonequilibrium hydrodynamic modes. Therein we take into account the variation of the macroscopic quantities with position. The correlation functions are formally expressed in terms of the nonequilibrium modes. In paper II the slow hydrodynamic modes (viscous and viscoheat modes) will be determined explicitly for ideal heat-conducting plates with stick boundary conditions and used to compute the slow part of the correlation functions; in paper III the fast hydrodynamic modes (sound modes) will be explicitly determined for stick boundary conditions and used to compute the fast part of the correlation functions. In these papers we will also compute the shape and intensity of the lines measured in light scattering experiments.     

On the influence of the hydrodynamic interactions on the aggregation rate of magnetic spheres in a dilute suspension

Magnetostatic attraction may lead to formation of aggregates in stable colloidal magnetic suspensions and magneto-rheological suspensions. The aggregation problem of magnetic composites under differential sedimentation is a key problem in the control of the instability of non-Brownian suspensions. Against these attractive forces are the electrostatic repulsion and the hydrodynamic interactions acting as stabilizing effects to the suspension. This work concerns an investigation of the pairwise interaction of magnetic particles in a dilute sedimenting suspension. We focus attention on suspensions where the Péclet number is large (negligible Brownian motion) and where the Reynolds number (negligible inertia) is small. The suspension is composed of magnetic micro-spheres of different radius and density immersed in a Newtonian fluid moving under the action of gravity. The theoretical calculations are based on direct computations of the hydrodynamic and the magnetic interactions among the rigid spheres in the regime of low particle Reynolds number. From the limiting trajectory in which aggregation occurs, we calculate the collision efficiency, representing the dimensionless rate at which aggregates are formed. The numerical results show clear evidence that the hydrodynamic interactions are of fundamental relevance in the process of magnetic particle aggregation. We compare the stabilizing effects between electrostatic repulsion and hydrodynamic interactions.     

On the theory of rheological properties of magnetic suspensions

The paper deals with a theoretical study of influence of magnetic field on effective viscosity of suspension of non-Brownian magnetizable particles. It is supposed that experimentally observed magnetorheological effects are provided by chain-like aggregates, consisting of the particles. Unlike previous works on this subject, we take into account that the chains cannot be identical and estimate their size distribution. The following power law (η-η₀)/η₀∼Mn^-Δ, detected in many experiments, is obtained theoretically (η and η₀ are the suspension effective viscosity and the carrier liquid viscosity, respectively, Mn is the so-called Mason number, proportional to the shear rate and inversely proportional to the square of magnetic field). The calculated magnitude of the exponent Δ increases with the applied magnetic field from approximately 0.66 to 0.8-0.9 and slowly increases with the volume concentration ? of the particles. These results are in agreement with known experiments.     

The surface shear viscosity of a planar liquid-vapor interface I. Theory

In this paper we develop a theory for the calculation of the surface shear viscosity of a planar liquid-vapor interface. The theory is an extension of the generalized hydrodynamics formalism, originally developed for the calculation of linear transport coefficients in isotropic bulk fluids. We develop an expression for the surface shear viscosity in terms of the actual shear viscosity profile in the interfacial region. We derive an expression for this profile in terms of the first four moments of the autocorrelation function of the transverse parallel velocity (the component of velocity parallel to k, which is the projection of k on to the interface). Finally, we calculate these moments for a planar liquid-vapor interface.     

Inclusion of magnetic dipole-dipole and hydrodynamic interactions in implant-assisted magnetic drug targeting

Mathematical modelling of the implant-assisted magnetic drug targeting system of Avilés, Ebner and Ritter is performed. In order to model the agglomeration of particles known to occur in this system, the magnetic dipole-dipole and hydrodynamic interactions are included. Such interactions were calculated previously by Mikkelsen et al. under low magnetic fields (~0.05 T) in microfluidic systems. Here, a higher magnetic field (0.7 T) is considered and the effect of interactions on two nanoparticles with a seed implant is calculated. The calculations were performed with the open-source software OpenFOAM. Different initial positions are considered and the system performance is assessed in terms of capture cross section. Inclusion of both interactions was seen to alter the capture cross section of the system by up to 7% in absolute terms.     

On the advanced integral equation theory description of dense Yukawa one-component plasma liquids

Different advanced bridge function closures are utilized to investigate the structural and thermodynamic properties of dense Yukawa one-component plasma liquids within the framework of integral equation theory. The isomorph-based empirically modified hypernetted-chain, the variational modified hypernetted-chain, the Rogers–Young, and the Ballone–Pastore–Galli–Gazzillo approaches are compared at the level of thermodynamic properties, radial distribution functions, and bridge functions. The comparison, based on accuracy and computational speed, concludes that the two modified hypernetted-chain approaches are superior and singles out the isomorph-based variant as the most promising alternative to computer simulations of structural properties of dense Yukawa liquids. The possibility of further improvement through artificial crossover to exact asymptotic limits is studied.     

Influence of surface tension on the conical meniscus of a magnetic fluid in the field of a current-carrying wire

We study the influence of surface tension on the shape of the conical meniscus built up by a magnetic fluid surrounding a current-carrying wire. Minimization of the total energy of the system leads to a singular second order boundary value problem for the function ζ(r)$\zeta (r)$ describing the axially symmetric shape of the free surface. An appropriate transformation regularizes the problem and allows a straightforward numerical solution. We also study the effects a superimposed second liquid, a nonlinear magnetization law of the magnetic fluid, and the influence of the diameter of the wire on the free surface profile.     

On the theory of the universal dielectric relaxation

Dielectric relaxation has been investigated within the framework of a modified mean field theory, in which the dielectric response of an arbitrary condensed matter system to the applied electric field is assumed to consist of two parts, a collective response and a slowly fluctuating response; the former corresponds to the cooperative response of the crystalline or noncrystalline structures composed of the atoms or molecules held together by normal chemical bonds and the latter represents the slow response of the strongly correlated high-temperature structure precursors or a partially ordered nematic phase. These two dielectric responses are not independent of each other but rather constitute a dynamic hierarchy, in which the slowly fluctuating response is constrained by the collective response. It then becomes clear that the dielectric relaxation of the system is actually a specific characteristic relaxation process modulated by the slow relaxation of the nematic phase and the relationship governing such a process can be defined as the universal dielectric relaxation law. Furthermore, we have shown that seemingly different relaxation relationships, such as the Debye relaxation law, the Cole-Cole equation, the Cole-Davidson equation, the Havriliak-Negami relaxation, the Kohlrausch-Williams-Watts function, Jonscher’s universal dielectric relaxation law, etc. are only the variants of this universal law under different circumstances.     

Quantum theory of light propagation I. General theory

We present a particular approach to quantum theory of light propagation in nonlinear medium using space and time dependent modal operators. Spatial and temporal evolutions of this space and time dependent modal operators are given by the Heisenberg-like equation involving the momentum operator and Heisenberg equation, respectively, which can be justified from point of view of quantum electrodynamics. This useful concept can be applied to an arbitrary nonlinear interaction.The author would like to express his sincere thanks to Professor J. Peina for advice, comments and stimulating discussions.This work was partially supported by the grant PV202/1994 of Czech Ministry of Education and by an internal grant of the Palacký University.     

Scaling laws and approximate expressions for the dynamic magnetic susceptibility of Brownian nanoparticles

We present simplified expressions for the out-of-phase component of the dynamic susceptibility χ″ of lognormal-sized magnetic nanoparticles under Brownian rotation. These expressions are based on transforming the general integral functions used for χ″ in the convolution of gaussian functions. χ″ can thus be expressed as a sum of gaussians with parameters directly related to those of the size distribution and to the saturation magnetization. The gaussian fit of χ″(ω) (where ω is the ac field frequency) is a simpler way to determine these structural and magnetic parameters as it avoids fitting χ″(ω) to an integral function. The expressions derived for χ″ suggest that χ″T data collapses in a ωη(T)/T scale (where T is the temperature and η the fluids viscosity), which is confirmed by numerical calculations. We also discuss the limits of validity of these approximations in real systems where both Néel and Brownian relaxation mechanisms coexist and we present further approximations for the relation of ω_χ with the average volume (being ω_χ the frequency at which χ″ is maximum). The ωη(T)/T scale can be used to qualitatively evaluate the dominance of the Brownian relaxation mechanism.     

Dynamics of a single peak of the Rosensweig instability in a magnetic fluid

To describe the dynamics of a single peak of the Rosensweig instability a model is proposed which approximates the peak by a half-ellipsoid atop a layer of magnetic fluid. The resulting nonlinear equation for the height of the peak leads to the correct subcritical character of the bifurcation for static induction. For a time-dependent induction the effects of inertia and damping are incorporated. The results of the model show qualitative agreement with the experimental findings, as in the appearance of period doubling, trebling, and higher multiples of the driving period. Furthermore, a quantitative agreement is also found for the parameter ranges of frequency and induction in which these phenomena occur.     

Nonlinear neural networks. I. General theory

A neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through anonlinear operation. The original Hopfield model is linear, whereas, for instance, clipped synapses constitute a nonlinear model. Here a general theory is presented to obtain the statistical mechanics of a neural network with finitely many patterns and arbitrary (symmetric) nonlinearity. The problem is reduced to minimizing a free energy functional over all solutions of a fixed-point equation with synaptic kernelQ. The case of clipped synapses with bimodal and Gaussian probability distribution is analyzed in detail. To this end, a simple theory is developed that gives the spectrum ofQ and thereby all the solutions that bifurcate from the high-temperature phase.     

Molecular derivation of the hydrodynamic equations for a binary fluid

In this paper, the hydrodynamic equations and the associated transport coefficients are derived for a simple binary fluid from molecular considerations. This is a generalization of the methods of Felderhof and Oppenheim and of Selwyn to multicomponent systems. A linear response formalism is used to describe the relaxation of the binary system from an initial nonequilibrium state. Explicit molecular expressions are given for the transport coefficients in terms of time correlation functions of generalized current densities. These densities have the useful property of not containing a conserved part. The correlation functions are then related to a set of phenomenological coefficients in the theory of nonequilibrium thermodynamics. This explicit identification enables one to relate the correlation functions to experimentally measured transport coefficients.Supported by the National Science Foundation.     

On the mean-field theory of magnetic multilayers with bilinear and biquadratic Heisenberg exchange

A system consisting of several layers of magnetic ions interacting by both bilinear and biquadratic Heisenberg exchange is studied within the framework of the mean-field approximation. It is shown that for S = 1 there exist two types of ordering: ferromagnetic and ferroquadrupolar. The stability of phases as the function of temperature, biquadratic exchange and surface exchange is discussed analytically and numerically and it was shown that similar to bulk samples there appear first- and second-order transitions and a tricritical point may appear depending on system parameters.     

AC susceptibility of magnetic markers in suspension for liquid phase immunoassay

AC susceptibility of magnetic markers in solution was studied for biosensor applications. First, frequency dependence of the susceptibility was measured, and size distribution of the markers was estimated by analyzing the experimental result with the so-called singular value decomposition (SVD) method. The size distribution estimated with the magnetic measurement agreed with that obtained from conventional optical measurement. Next, susceptibility measurement was applied to the liquid-phase immunoassay without bound/free (B/F) separation. We performed the detection of biotin-coated polymer beads in suspension using avidin-coated magnetic markers. Changes of the susceptibility and the size distribution caused by the binding reaction were shown.     

Dynamic simulations of the dipolar driven demagnetization process of magnetic multi-core nanoparticles

In this work, we investigate dynamically the dipolar driven demagnetization process of magnetic multi-core particles by solving the Landau-Lifshitz equation for single-domain particles distributed on a three-dimensional sphere. We analyze the relaxation time in respect to different geometry and material parameters. Further we show that the demagnetization times differ from the behaviour of a single magnetic sphere in the case of low damping. To explain these dynamics nanoparticular systems of different dimensions are investigated. We show that deviations can be attributed to a confinement of the relaxation dynamics to a lower dimensional submanifold of the k-space.     

Enhancement of AC-losses of magnetic nanoparticles for heating applications

Aqueous ferrofluids of maghemite nanoparticles coated with carboxydextran were investigated with respect to their specific loss power (SLP) in dependence on frequency and field amplitude of magnetic AC-fields. In order to elucidate the effect of the size distribution on SLP fluid fractions with different mean particle core size were prepared by a magnetic separation procedure from the original ferrofluid. Structural characterisation by means of TEM and XRD as well as reconstruction of core size distributions from magnetisation curves reveals that the narrow size distributions of the fractions cover a range of mean core sizes from about 8 up to 20 nm. Spectra of the complex susceptibility were measured for a frequency range of 20 Hz to 1 MHz. From the imaginary part of the susceptibility the specific loss power is calculated in dependence on frequency. The results are compared with calorimetrical measurements performed in dependence on field amplitude up to 11 kA/m at 410 kHz. A very high specific loss power in the order of 400 W per gram maghemite was found at 410 kHz and 11 kA/m for the fluid fraction having the largest mean core diameter. A deviation from linear response behaviour is found for this sample showing a power law field dependence of the specific loss power SLPH^2.5. In addition to liquid suspensions measurements were performed with particles immobilised in mannitol or gel in order to elucidate the role of Brownian relaxation. The experimentally found dependence of SLP on the mean particle core diameter may be understood in the frame of the Debye dispersion model. Results are discussed with respect to applications of ferrofluids in RF-magnetic hyperthermia.     

Contribution to the nonlinear theory of sound and hydrodynamic turbulence of a compressible liquid

The interaction of sound with hydrodynamic turbulence has been studied in detail. The sound absorption decrement, the correlation time and length and the frequency diffusion coefficient for the acoustic wave packet are calculated. The spectral composition of the sound radiated by a unit, turbulent volume and the spectral energy density of sound in equilibrium with the turbulence are studied. The region of applicability of the kinetic equation for sound with a linear dispersion low is found.