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Papers by MABROUK BENHAMOU

The optical absorption spectra of the gold nanorods embedded in a dielectric medium are simulated... more The optical absorption spectra of the gold nanorods embedded in a dielectric medium are simulated using finite element method. The study is achieved for various shapes and forms of the nanoparticles and for different values of the dielectric constant of the host medium. Also, we quantify the influence of the aspect ratio and the dielectric constant of the surrounding medium on both longitudinal and transversal resonance modes. It is found that the obtained results are in good agreement with a previous theoretical work that is based on the classical electrostatic predictions and assuming that the gold nanorods behave as ellipsoidal particles. Finally, a comparison of the predicted results with experimental data dealt with the same system is very satisfactory.

The physical system we consider here is an interpenetrating polymer network made of two chemicall... more The physical system we consider here is an interpenetrating polymer network made of two chemically incompatible cyclic polymers A and B, linked each to other by topological trapping. When the system is cooled down, below some critical temperature, the system undergoes a microphase separation. The latter results from a competition between the usual macrophase separation and the fact that the cyclic polymer chains are connected. Using an extended de Gennes theory, we first compute the static structure factor allowing the study of the static critical properties of such a transition. We show that this structure factor exhibits a maximum at some finite value of the wave-vector, qc , of which the inverse, î஼∗ =qc, measures the size of microdomains. Second, we complete the study by kinetics of this microphase separation when the temperature of the system is lowered from an initial value towards a final one very close to the spinodal point. This kinetics is investigated through the relaxati...

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Introduction à la Physique des Polymères Mabrouk Benhamou

Dans ce travail, nous considerons une solution semi-diluee de polymeres en etoile identiques, fai... more Dans ce travail, nous considerons une solution semi-diluee de polymeres en etoile identiques, faits de longues chaines flexibles, du meme degre de polymerisation N. Nous determinons d'abord la forme du potentiel effectif de paire entre polymeres en etoile. De tel potentiel resulte des forces de volume exclu entre monomeres. Nous montrons que le potentiel de paire est logarithmique, en dessous d'une certaine distance caracteristique connue, σ, dependant du nombre de chaines attachees, f; et de la fraction volumique en monomeres, Φ 0 . Au dela de σ, le potentiel decroit d'une maniere exponentielle. En second lieu, nous investiguons la structure et la thermodynamique de ces polymeres en etoile. Dans ce but, nous utilisons la methode de l'equation integrale avec l'approximation spherique moyenne hybridee. La resolution numerique de cette equation donne le facteur de structure des polymeres en etoile, pour diverses valeurs de la densite de particule. Enfin, les relati...

IOP Conference Series: Materials Science and Engineering, 2017

In this work, we propose a model to study the Statistical Mechanics of a confined bilayer-membran... more In this work, we propose a model to study the Statistical Mechanics of a confined bilayer-membrane that fluctuates between two interactive flat substrates. From the scaling laws point of view, the bilayer-membranes and strings are very similar. Therefore, it is sufficient to consider only the problem of a string. We assume that the bilayer-membrane (or string) interact with the substrate via a Double Morse potential that reproduces well the characteristics of the real interaction. We show that the Statistical Mechanic of the string can be adequately described by the Schrödinger equation approach that we solve exactly using the Bethe method. Finally, from the exact value of the energy of the ground state, we extract the expression of the free energy density as well as the specific heat.

Journal of Molecular Liquids, 2017

The purpose of this work is a quantitative determination of the effective interaction potential, ... more The purpose of this work is a quantitative determination of the effective interaction potential, upon distance, between inert nanoparticles or (parallel or crossed) nanopores which are inserted in a bilayer-membrane. The expected effective pair-potential is the sum of a repulsive part and an attractive one. The expression of this repulsive part that originates from the thermal

In this paper, we consider a fluid membrane in suspension in a liquid that is confined to two par... more In this paper, we consider a fluid membrane in suspension in a liquid that is confined to two parallel plates. We assume that the distance between the two plates is small enough, when it is compared to the mean roughness of the membrane. In these conditions, the membrane may be regarded as a closed (or open) fluctuating curve in the two-dimensional Euclidean space (parallel to the two plates). We first derive the general curvature equation describing the equilibrium geometrical form of the one-dimensional membrane. Second, we determine the expression of the force experienced by the membrane. Such a force may be caused by the liquid flow or by an eventual interaction between the membrane and the surface of capillaries. To do calculations, use is made of Dierential

Ce cours est destiné à l'étude du comportement asymptotique en temps des solutions d'une classe d... more Ce cours est destiné à l'étude du comportement asymptotique en temps des solutions d'une classe d'équations de réaction-diffusion, par la théorie du Groupe de Renormalisation. Cette théorie a été appliquée avec grand succès, d'abord en Physique des Particules, ensuite en Phénomènes Critiques. Ce n'est pas un hasard de s'intéresser à ce genre d'équations, car celles-ci gèrent beaucoup de phénomènes naturels, en relation avec la physique, la mécanique, la mécanique des fluides, la thermique, les milieux poreux, les fronts de diffusion, la chimie, la biologie, la botanique et même l'économie. Chapitre 1 Equations de réaction-diffusion. Le but de ce chapitre est de montrer comment on obtient les équations de réactiondiffusion, qui apparaissent dans plusieurs domaines des sciences de la nature.

Macromolecules, 1994

The static scattering function is calculated for a cross-linked blend in the presence of a low mo... more The static scattering function is calculated for a cross-linked blend in the presence of a low molecular weight solvent. The idea suggested by de Gennes to treat analogously the effects of cross-links in a network and those of electric charges in a dielectric medium is generalized to include the concept of screening. It is argued that this generalization could help to resolve the discrepancy between the model and the neutron scattering data in the low q region. Other phenomena are also examined such as the enhancement of compatibility due to cross-links and the kinetics of the microphase separation transition.

International Journal of Academic Research, 2013

Physica A: Statistical Mechanics and its Applications, 2010

We consider a bilayer membrane surrounded by small impurities, assumed to be attractive or repuls... more We consider a bilayer membrane surrounded by small impurities, assumed to be attractive or repulsive. The purpose is a quantitative study of the effects of these impurities on the statistical properties of the supported membrane. Using the replica trick combined with a variational method, we compute the membrane mean-roughness and the height correlation function for almost-flat membranes, as functions of the primitive elastic constants of the membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. As results, the attractive impurities increase the shape fluctuations due to the membrane undulations, while repulsive ones suppress these fluctuations. Second, we compute the equilibrium diameter of (spherical) vesicles surrounded by small random particles starting from the curvature equation. Third, the study is extended to a lamellar phase composed of two parallel fluid membranes, which are separated by a finite distance. This lamellar phase undergoes an unbinding transition. We demonstrate that the attractive impurities increase the unbinding critical temperature, while repulsive ones decrease this temperature. Finally, we say that the presence of small impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition, even the temperature and polarizability of the aqueous medium are fixed, in lamellar phases formed by parallel lipid bilayers.

Physica Scripta, 2011

We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fl... more We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fluid membrane with a change in some suitable parameter (temperature). Our purpose is to quantify the significant effects of the solvent quality and of the polydispersity of adsorbed loops formed by grafted polymer chains on the segregation phenomenon. To this end, we elaborate on a theoretical model that allows us to derive the expression for the mixing free energy. From this, we extract the phase diagram shape in the composition–temperature plane. Our main conclusion is that the polymer chain condensation is very sensitive to the solvent quality and to the polydispersity of loops of adsorbed chains.

Physica A: Statistical Mechanics and its Applications, 2002

The purpose of this work is the investigation of critical dynamic properties of two strongly coup... more The purpose of this work is the investigation of critical dynamic properties of two strongly coupled paramagnetic sublattices exhibiting a paramagnetic-ferrimagnetic transition. To go beyond the mean-ÿeld approximation, and in order to get a correct critical dynamic behavior, use is made of the renormalization-group (RG) techniques applied to a ÿeld model describing such a transition. The model is of Landau-Ginzburg type, whose free energy is a functional of two kinds of order parameters (local magnetizations) ' and , which are scalar ÿelds associated with these sublattices. This free energy involves, beside quadratic and quartic terms in both ÿelds ' and , a lowest-order coupling, −C0' , where C0 is the coupling constant measuring the interaction between the two sublattices. Within the framework of mean-ÿeld theory, we ÿrst compute exactly the partial dynamic structure factors, when the temperature is changed from an initial value Ti to a ÿnal one T f very close to the critical temperature Tc. The main conclusion is that, physics is entirely controlled by three kinds of lengths, which are the wavelength q −1 , the static thermal correlation length and an extra length Lt measuring the size of ordered domains at time t. Second, from the Langevin equations (with a Gaussian white noise), we derive an e ective action allowing to compute the free propagators in terms of wave vector q and frequency !. Third, through a supersymmetric formulation of this e ective action and using the RG-techniques, we obtain all critical dynamic properties of the system. In particular, we derive a relationship between the relaxation time and the thermal correlation length , i.e., ∼ z , with the exponent z = (4 − Á)=(2 + 1), where and Á are the usual critical exponents of Ising-like magnetic systems. At two dimensions, we ÿnd the exact value z = 5 4. At three dimensions, and using the best values for exponents and Á, we ÿnd z = 1:7562 ± 0:0027.

Physica A: Statistical Mechanics and its Applications, 2007

... In the presence of a good solvent, the equilibrium profile depends, in addition, on the overa... more ... In the presence of a good solvent, the equilibrium profile depends, in addition, on the overall ... The gradient term accounts for the interfacial energy between A-rich and B-rich phases ... In equality (1), the vector P denotes the average displacement between the centres of masses of ...

Macromolecular Theory and Simulations, 1995

Cross‐linked mixtures of polystyrene and poly(vinyl methyl ether) exhibit a non‐vanishing zero‐an... more Cross‐linked mixtures of polystyrene and poly(vinyl methyl ether) exhibit a non‐vanishing zero‐angle intensity in small‐angle neutron scattering experiments. A possible explanation is that fluctuations in composition in the mixture may be frozen by the presence of cross‐links. Assuming this, we introduce a screening length κ by the condition that the scattered intensity should not be changed by cross‐linking. We find κ2 ∼ C/(χ − χi), where C is an elastic constant, χ and χi, respectively, the inverse temperature and that where cross‐linking is performed. When the temperature is varied, we find three regimes. In the first one, the scattered intensity is monotonously decreasing. In the second one, it has a finite maximum. In the last one, the maximum eventually diverges.

Journal of Molecular Liquids, 2003

ABSTRACT We consider colloidal particles clothed each by f end-grafted flexible polymer chains, a... more ABSTRACT We consider colloidal particles clothed each by f end-grafted flexible polymer chains, and immersed in a good solvent. We assume that these are small enough to be viewed as star-polymers. We suppose that the lengths of the f grafted chains is distributed at random, with a distribution P (n) (the probability to find a grafted chain having n monomers). The presence of a good solvent induces an effective force between particles. The aim is a quantitative study of the polydispersity effects on the interaction force. To make explicit calculations of the force, F (h), which depends on the interparticle-distance h, we choose a fractal distribution, i.e., P (n) similar to n(-alpha), if n < N, and P (n) = 0, if n > N. Here, 1 less than or equal to alpha < 2 is the polydispersity exponent and N is the polymerization degree of the longest grafted chain. We first show that the computation of the force depends. on wether N < N-c similar to f(1/(alpha-1)) or N > Nc. For N < N, the size of the polydisperse grafted layer is found to be af(1/5N(4-alpha)/5), while for N > N-c, the size of the clothed particle is dominated by that of the longest grafted chain, i.e., aN(3/5) To obtain the force expression, we use some method developed by Witten and Pincus for monodisperse star-polymers. For N < N, we show that the force decreases with distance h as: F (h) /k(B)T similar to A(f)h(-)omega, with the exponent omega = (13alpha - 7) /2(4 - alpha) and the f-dependent universal amplitude A(f) similar to f(9/2(4-alpha)). This force must be compared to that calculated by the authors for monodisperse star-polymers, i.e., F0(h) /k(B)T similar to A(f)(0)h(-1), with the amplitude A(f)(0) similar to f(3/2). The main conclusion is that the polydispersity drastically affects the force expression. Also, we write a formal expression of the force for any distribution. For N > N., however, we show the existence of two distance-ranges. For small distances, that is for h < 2R'(c) similar or equal to 2af(3/5(alpha-1)), the force expression is similar to that relative to the case with N < Nc. For high distances, that is for 2R'(c) < h < 2R' similar or equal to aN(3/5), the force decays as h(-1), independently on the polydispersity nature.

Journal of Magnetism and Magnetic Materials, 2000

The purpose of the present work is a quantitative study of the spin time relaxation within superw... more The purpose of the present work is a quantitative study of the spin time relaxation within superweak ferrimagnetic materials exhibiting a paramagnetic}ferrimagnetic transition, when the temperature is changed from an initial value ¹ to a "nal one ¹ very close to the critical temperature ¹. From a magnetic point of view, the material under investigation is considered to be made of two strongly coupled paramagnetic sublattices of respective moments and. Calculations are made within a Landau mean-"eld theory, whose free energy involves, in addition to quadratic and quartic terms in both moments and , a lowest-order coupling } C , where C(0 stands for the coupling constant measuring the interaction between the two sublattices. We "rst determine the time dependence of the shifts of the order parameters and from the equilibrium state. We "nd that this time dependence is completely controlled by two kinds of relaxation times and. The former is a long time and the second a short one, and they are associated, respectively, with long and local wavelength #uctuations. We "nd that, only the "rst relaxation time is relevant for physics, since it drives the system to undergo a phase transition. Spatial #uctuations are also taken into account. In this case, we "nd an explicit expression of the relaxation times, which are functions of temperature ¹, coupling constant C and wave vector q. We "nd that the critical mode is that given by the zero scattering-angle limit, i.e. q"0. Finally, we emphasize that the appearance of these two relaxation times is in good agreement with results reported in recent experimental work dealt with the Curie}Weiss paramagnet compound Li V Ni \V O , where the composition x is very close to 1.

Journal of Magnetism and Magnetic Materials, 1997

ABSTRACT We reconsider here the so-called super weak ferrimagnetism problem, arising from two str... more ABSTRACT We reconsider here the so-called super weak ferrimagnetism problem, arising from two strongly coupled paramagnetic sublattices, of Pauli or Curie—Weiss type, within a continuous Landau model. The parameters of the model are the absolute temperature T, the applied magnetic field H, the coupling C between the two sublattices, and two coupling constants b and B that traditionally multiply the quartic terms in magnetic moments relatively to the sublattices in the free energy. We first show that when the ratio λ = b/B < 1, the system can exhibit a compensation point. Second, we give the various curves of the magnetization versus T (H = 0), in the space of thermodynamic parameters T, C, b and B. Within the framework of this model, the classification of these curves is found to be in agreement with the classical one of Néel. We identify some temperature T1 above which the hysteresis phenomena is absent, and give its explicit dependence on the coupling C. We also investigate the shape of the Arrott diagram, within the parameter space, and we deduce various curves of the magnetization versus H (for fixed T and C). In the low temperature regime (ferrimagnetic phase) and for systems exhibiting a positive compensation point, we show the existence of some characteristic field H1, termed compensation magnetic field at which the overall magnetization vanishes. It is found that this latter is negative for H < H1 and becomes positive for strong fields H > H1. Our results are found to be in good qualitative agreement with recent experiments on lamellar Curie—Weiss paramagnetic compounds like LiNi1−xCoxO2 (0 ⩽ x < 1). This work must be considered as a natural extension of our earlier work which dealt with the critical behavior near the para—ferrimagnetic transition within these systems.

The Journal of Chemical Physics, 2003

We consider here a low-density assembly of spherical colloids immersed in a mixture of two incomp... more We consider here a low-density assembly of spherical colloids immersed in a mixture of two incompatible polymers A and B. We assume that, near the consolute point Tc of the host mixture, colloids adsorb preferentially A polymer. The preferential adsorption has as a consequence that particles aggregate in the nonpreferred B phase. We aim at the computation of the induced force F(r), responsible for this aggregation, as a function of the interparticle distance r. To achieve this, use is made of a field-theoretical approach based on ψ4 theory, where the field ψ is simply the composition fluctuation (order parameter). Combining this approach with the standard cumulants method, we first demonstrate that the effective pair potential is proportional to the two-point correlation function of the host mixture. Second, very close to the critical point, we find that the effective force is universal and decays with interparticle distance r>d0 according to: F(r)/kBTc=−(64π2/27)Nd02/r3, where N...

Fractals, 1999

The purpose of the present work is a systematic and analytical study of the scattering properties... more The purpose of the present work is a systematic and analytical study of the scattering properties of Gaussian D-manifolds (1≤D<2) and two-dimensional membranes (D=2). Here, D is the spectral (or internal) dimension of the polymeric complex. We first determine exactly their gyration radius and the higher order moments. Then, we construct a simple integral representation of the structure factor, in the case of D-manifolds (1≤D<2). By analytical continuation in dimension D, this representation enables us to get much information about those polymeric systems of non-integer internal dimension (ideal branched polymers, Gaussian polymeric fractals, etc.). For ideal linear polymers, with D=1, we recover the classical Debye function. An exact expression of the structure factor for Gaussian large two-dimensional membranes is also derived.