mohammed chahid - Academia.edu (original) (raw)
Papers by mohammed chahid
Physica A: Statistical Mechanics and its Applications, 2008
ABSTRACT In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A ... more ABSTRACT In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A and B. Under a variation of a suitable parameter, such as temperature or difference of lengths of hydrophobic chains, these systems undergo a phase separation from a homogeneous liquid-phase to two distinct liquid-phases. Two physical situations can be distinguished: (1) The amphiphiles A and B prefer to jump from a monolayer to the other (flip-flop transition), (2) the mixture phase separates on each monolayer, and there is no jump from one sheet towards the second one (lateral transition). To investigate the associated critical phase behavior, we first introduce a field theory, constructed with two order parameters (or fields) φ and ψ, which are nothing else but the composition fluctuations relative to the monolayers. Beside the usual terms proportional to φ2, ψ2, φ4 and ψ4, the free energy contains an extra one, −Cφψ, which describes the lowest order coupling between the two monolayers. The coupling constant C is positive for the lateral phase separation, and negative for the vertical one. We show that its sign results from a competition between the chemical segregation of amphiphiles and the curvature asymmetry. With the help of this free energy, we first identify the liquid-phases, and show the existence of a critical point, Tc, of which the location depends naturally on the value of the coupling constant C. In particular, for those bilayer biomembranes or surfactants made of amphiphiles of the same chemical nature but with different lengths, and at fixed temperature, we show the existence of a critical line in the (Δc0,Δl)-plane, along which the bilayer undergoes a phase separation. Here, Δc0 and Δl account for the curvature gap and the length difference, respectively. Second, we determine the behavior of the composition fluctuations, φ and ψ, and the total one, Φ=φ+ψ, upon temperature, T, and chemical potential difference, Δμ, in the critical region. Third, we determine the critical behavior of the partial compressibilities, κφφ, κψψ and κφψ, and the overall one, κtot=κφφ+κψψ+2κφψ. Finally, we remark that the flip-flop phase separation shows some analogy with the classical para-ferrimagnetic transition of coupled paramagnetic materials of Curie–Weiss type.
International Journal of Academic Research, 2013
Physica A: Statistical Mechanics and its Applications, 2010
Physical Review Letters, 2010
Thin, mass-limited targets composed of V=Cu=Al layers with diameters ranging from 50 to 300 m hav... more Thin, mass-limited targets composed of V=Cu=Al layers with diameters ranging from 50 to 300 m have been isochorically heated by a 300 fs laser pulse delivering up to 10 J at 2 Â 10 19 W=cm 2 irradiance. Detailed spectral analysis of the Cu x-ray emission indicates that the highest temperatures, of the order of 100 eV, have been reached when irradiating the smallest targets with a high-contrast, frequency-doubled pulse despite a reduced laser energy. Collisional particle-in-cell simulations confirm the detrimental influence of the preformed plasma on the bulk target heating.
Physica A: Statistical Mechanics and its Applications, 2002
ABSTRACT The purpose of this work is the investigation of critical dynamic properties of two stro... more ABSTRACT 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-field approximation, and in order to get a correct critical dynamic behavior, use is made of the renormalization-group (RG) techniques applied to a field 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) ϕ and ψ, which are scalar fields associated with these sublattices. This free energy involves, beside quadratic and quartic terms in both fields ϕ and ψ, a lowest-order coupling, −C0ϕψ, where C0 is the coupling constant measuring the interaction between the two sublattices. Within the framework of mean-field theory, we first compute exactly the partial dynamic structure factors, when the temperature is changed from an initial value Ti to a final one Tf 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 ξ 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 effective action allowing to compute the free propagators in terms of wave vector and frequency ω. Third, through a supersymmetric formulation of this effective 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 τ and the thermal correlation length ξ, i.e., τ∼ξz, with the exponent z=(4−η)/(2ν+1), where ν and η are the usual critical exponents of Ising-like magnetic systems. At two dimensions, we find the exact value . At three dimensions, and using the best values for exponents ν and η, we find z=1.7562±0.0027.
Physica A: Statistical Mechanics and its Applications, 2007
ABSTRACT In this paper, we are interested in the critical behavior of the dynamic structure facto... more ABSTRACT In this paper, we are interested in the critical behavior of the dynamic structure factor of a crosslinked polymer blend made of two chemically incompatible polymer A and B, when it is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. Since the critical fluctuations occur over distances smaller than the mesh size (microdomains size), ξ*, the dynamic structure factor should be governed by a short-time behavior we want to determine. We demonstrate that the final time, t*, necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t*∼ξ*z, with z a dynamic critical exponent. The investigation of the dynamic structure factor is first achieved using a mean-field approach, based on an extended Van Hove theory, and second by a scaling argument. The Van Hove theory is valid as long as the fluctuations of composition can be underestimated. Within the framework of this model, we determine an exact form for the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0=6. Second, the study is extended to the case of crosslinked polymer blends of low-molecular-weight, where fluctuations of composition are strong enough near the spinodal temperature. Using a scaling argument, we prove that the scaling law for the dynamic structure factor is S(q,t)=qη-2f(qR(t)), with η the standard Ising critical exponent, f(x) a universal scaling function, and R(t)∼t1/z some time-characteristic length. The latter can be interpreted as the size of instabilities domains at time t , and it becomes of the order of ξ* at time t*. The growing process of instabilities is then stopped at the final time t*. We show that the dynamic exponent z is not trivial and has as three-dimensional value z≃5.969±0.001. In dimension 2, we find an exact value for this exponent that is z=234.
Journal of Magnetism and Magnetic Materials, 2001
ABSTRACT The purpose of the present work is a quantitative investigation of the biquadratic excha... more ABSTRACT The purpose of the present work is a quantitative investigation of the biquadratic exchange interaction effects on the paramagnetic–ferrimagnetic transition arising from two strongly coupled paramagnetic (1-spin) sublattices, of respective moments m and M. The free energy describing the physics of the system is of Landau type. In addition to the quadratic and quartic terms, in both m and M, this free energy involves two mixing interaction terms. The first is a lowest order coupling −CmM, where C0 is the new coupling constant. These two interactions enter in competition, and then, they induce drastic changes of the magnetic behavior of the material. The main change is that, the presence of this high order coupling tends to destroy the ferrimagnetic order of the system. We first show that the introduction of this biquadratic interaction does not affect the values of critical exponents. Also, we find that the compensation temperature (when it exists) and the compensation magnetic field are shifted to their lowest values, in comparison with the w=0 case. The Arrott-phase-diagram shape is also investigated quantitatively. We show the existence of three regimes depending on the values of w. When the latter is small, we find that the region of competition between the coupling C and the applied magnetic field H becomes more narrow under the effect of w (by competition, we mean the passage from the antiparallel state to the parallel one). While for higher values of w, this competition disappears completely, and then, the system loses its ferrimagnetic character. Kinetics of the phase transition is also examined, when the temperature is lowered from an initial value Ti to a final one Tf very close to the critical temperature Tc. As in the w=0 case, we find that kinetics is controlled by two kinds of relaxation times τ1 and τ2. The former is the relevant time, and is associated to long-wavelength fluctuations driving the system to undergo a phase transition. The second is a short time, which controls local dynamics. Near Tc, we show that, in particular, the longest relaxation time τ1 becomes less important in comparison with that relative to the w=0 case. Finally, we note that the existence of two relaxation times is consistent with the predictions of a recent experiment, which was concerned with the 1/2-spin compounds LixNi2−xO2, where the composition x is close to 1.
Journal of Magnetism and Magnetic Materials, 2000
The aim of this paper is the investigation of the critical properties of two strongly coupled par... more The aim of this paper is the investigation of the critical properties of two strongly coupled paramagnetic sublattices exhibiting a paramagnetic}ferrimagnetic transition, at some critical temperature ¹ greater than the room temperature. In order to take into account the strong #uctuations of the magnetization near the critical point, use is made of the renormalization-group (RG) techniques applied to an elaborated "eld model describing such a transition, which is of Landau}Ginzburg}Wilson type. The associated free energy or action is a functional of two kinds of order parameters (local magnetizations), which are scalar "elds and relative to these sublattices. It involves quadratic and quartic terms in both "elds, and a lowest-order coupling C , where C '0 stands for the coupling constant measuring the interaction between the two sublattices. We "rst show that the associated "eld theory is renormalizable at any order of the perturbation series in the coupling constants, up to a critical dimension d "4, and that, the corresponding counterterms have the same form as those relative to the usual -theory (C "0). The existence of the renormalization theory enables us to write the RG-equations satis"ed by the correlation functions. We solve these using the standard characteristics method, to get all critical properties of the system under investigation. We "rst determine the exact shape of the critical line in the (¹, C)-plane, along which the system undergoes a phase transition. Second, we determine the scaling laws of the correlation functions, with respect to relevant parameters of the problem, namely, the wave vector q, the (renormalized) coupling C and the temperature shift ¹!¹ . We "nd that these scaling laws are characterized by critical exponents, which are the same as those relative to Ising-like magnetic systems.
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.
The European Physical Journal E, 2008
We consider a crosslinked polymer blend that may undergo a microphase separation. When the temper... more We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, zeta. First, by averaging over zeta, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q(-1), a time length scale R(t) approximately t(1/z), and the mesh size xi*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of xi*, the dynamics is stopped. The final time, t*, then scales as t* approximately xi*z, with the dynamic exponent z = 6-eta. Here, eta is the usual Ising critical exponent. Since the final size of microdomains xi* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.
The European Physical Journal E, 2009
We consider bilayer biomembranes or surfactants made of two chemically incompatible amphiphile mo... more We consider bilayer biomembranes or surfactants made of two chemically incompatible amphiphile molecules, which may laterally or transversely phase separate into macrodomains, upon variation of some suitable parameter (temperature, lateral pressure, etc.). The purpose is an extensive study of the dynamics of both lateral and transverse phase separations, when the bilayer is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. The critical dynamics are investigated through the partial dynamic structure factors of different species. Using a two-order parameter field theory, where the two fields are the composition fluctuations of one component in the leaflets of the bilayer, combined with an extended van Hove approach that is based on two coupled Langevin equations (with noise), we exactly compute these dynamic structure factors. We first find that the dynamics is governed by two time scales. The longest one, Tau, can be related to the thermal correlation length, Xi ~ Sigma|T - T(c)|(-1/2), by Tau ~ Xi(z), with the dynamic critical exponent z = 4, where Sigma is an atomic length scale, T the absolute temperature, and T(c) its critical value. The characteristic time Tau can be interpreted as the time required for the formation of the final macrophase domains. The second time scale is rather shorter, and can be viewed as the short time during which the unlike phospholipids execute local motion. Second, we demonstrate that the dynamic structure factors obey exact scaling laws, and depend on three lengths, namely the wavelength q(-1) (q is the wave vector modulus), the correlation length Xi, and a length scale R(t) ~ t(1/z) (z = 4) representing the size of macrophase domains at time t. Of course, the two lengths Xi and R(t) coincide at the final time Tau at which the bilayer reaches its final equilibrium state. Finally, the present work must be considered as a natural extension of our previously published one dealing with the study of lateral and transverse phase separations from a static point of view.
Physica A: Statistical Mechanics and its Applications, 2008
ABSTRACT In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A ... more ABSTRACT In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A and B. Under a variation of a suitable parameter, such as temperature or difference of lengths of hydrophobic chains, these systems undergo a phase separation from a homogeneous liquid-phase to two distinct liquid-phases. Two physical situations can be distinguished: (1) The amphiphiles A and B prefer to jump from a monolayer to the other (flip-flop transition), (2) the mixture phase separates on each monolayer, and there is no jump from one sheet towards the second one (lateral transition). To investigate the associated critical phase behavior, we first introduce a field theory, constructed with two order parameters (or fields) φ and ψ, which are nothing else but the composition fluctuations relative to the monolayers. Beside the usual terms proportional to φ2, ψ2, φ4 and ψ4, the free energy contains an extra one, −Cφψ, which describes the lowest order coupling between the two monolayers. The coupling constant C is positive for the lateral phase separation, and negative for the vertical one. We show that its sign results from a competition between the chemical segregation of amphiphiles and the curvature asymmetry. With the help of this free energy, we first identify the liquid-phases, and show the existence of a critical point, Tc, of which the location depends naturally on the value of the coupling constant C. In particular, for those bilayer biomembranes or surfactants made of amphiphiles of the same chemical nature but with different lengths, and at fixed temperature, we show the existence of a critical line in the (Δc0,Δl)-plane, along which the bilayer undergoes a phase separation. Here, Δc0 and Δl account for the curvature gap and the length difference, respectively. Second, we determine the behavior of the composition fluctuations, φ and ψ, and the total one, Φ=φ+ψ, upon temperature, T, and chemical potential difference, Δμ, in the critical region. Third, we determine the critical behavior of the partial compressibilities, κφφ, κψψ and κφψ, and the overall one, κtot=κφφ+κψψ+2κφψ. Finally, we remark that the flip-flop phase separation shows some analogy with the classical para-ferrimagnetic transition of coupled paramagnetic materials of Curie–Weiss type.
International Journal of Academic Research, 2013
Physica A: Statistical Mechanics and its Applications, 2010
Physical Review Letters, 2010
Thin, mass-limited targets composed of V=Cu=Al layers with diameters ranging from 50 to 300 m hav... more Thin, mass-limited targets composed of V=Cu=Al layers with diameters ranging from 50 to 300 m have been isochorically heated by a 300 fs laser pulse delivering up to 10 J at 2 Â 10 19 W=cm 2 irradiance. Detailed spectral analysis of the Cu x-ray emission indicates that the highest temperatures, of the order of 100 eV, have been reached when irradiating the smallest targets with a high-contrast, frequency-doubled pulse despite a reduced laser energy. Collisional particle-in-cell simulations confirm the detrimental influence of the preformed plasma on the bulk target heating.
Physica A: Statistical Mechanics and its Applications, 2002
ABSTRACT The purpose of this work is the investigation of critical dynamic properties of two stro... more ABSTRACT 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-field approximation, and in order to get a correct critical dynamic behavior, use is made of the renormalization-group (RG) techniques applied to a field 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) ϕ and ψ, which are scalar fields associated with these sublattices. This free energy involves, beside quadratic and quartic terms in both fields ϕ and ψ, a lowest-order coupling, −C0ϕψ, where C0 is the coupling constant measuring the interaction between the two sublattices. Within the framework of mean-field theory, we first compute exactly the partial dynamic structure factors, when the temperature is changed from an initial value Ti to a final one Tf 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 ξ 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 effective action allowing to compute the free propagators in terms of wave vector and frequency ω. Third, through a supersymmetric formulation of this effective 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 τ and the thermal correlation length ξ, i.e., τ∼ξz, with the exponent z=(4−η)/(2ν+1), where ν and η are the usual critical exponents of Ising-like magnetic systems. At two dimensions, we find the exact value . At three dimensions, and using the best values for exponents ν and η, we find z=1.7562±0.0027.
Physica A: Statistical Mechanics and its Applications, 2007
ABSTRACT In this paper, we are interested in the critical behavior of the dynamic structure facto... more ABSTRACT In this paper, we are interested in the critical behavior of the dynamic structure factor of a crosslinked polymer blend made of two chemically incompatible polymer A and B, when it is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. Since the critical fluctuations occur over distances smaller than the mesh size (microdomains size), ξ*, the dynamic structure factor should be governed by a short-time behavior we want to determine. We demonstrate that the final time, t*, necessary to the appearance of microdomains alternatively rich in A and B-polymers, scales as t*∼ξ*z, with z a dynamic critical exponent. The investigation of the dynamic structure factor is first achieved using a mean-field approach, based on an extended Van Hove theory, and second by a scaling argument. The Van Hove theory is valid as long as the fluctuations of composition can be underestimated. Within the framework of this model, we determine an exact form for the dynamic structure factor, and in particular, we find that the corresponding dynamic exponent is z0=6. Second, the study is extended to the case of crosslinked polymer blends of low-molecular-weight, where fluctuations of composition are strong enough near the spinodal temperature. Using a scaling argument, we prove that the scaling law for the dynamic structure factor is S(q,t)=qη-2f(qR(t)), with η the standard Ising critical exponent, f(x) a universal scaling function, and R(t)∼t1/z some time-characteristic length. The latter can be interpreted as the size of instabilities domains at time t , and it becomes of the order of ξ* at time t*. The growing process of instabilities is then stopped at the final time t*. We show that the dynamic exponent z is not trivial and has as three-dimensional value z≃5.969±0.001. In dimension 2, we find an exact value for this exponent that is z=234.
Journal of Magnetism and Magnetic Materials, 2001
ABSTRACT The purpose of the present work is a quantitative investigation of the biquadratic excha... more ABSTRACT The purpose of the present work is a quantitative investigation of the biquadratic exchange interaction effects on the paramagnetic–ferrimagnetic transition arising from two strongly coupled paramagnetic (1-spin) sublattices, of respective moments m and M. The free energy describing the physics of the system is of Landau type. In addition to the quadratic and quartic terms, in both m and M, this free energy involves two mixing interaction terms. The first is a lowest order coupling −CmM, where C0 is the new coupling constant. These two interactions enter in competition, and then, they induce drastic changes of the magnetic behavior of the material. The main change is that, the presence of this high order coupling tends to destroy the ferrimagnetic order of the system. We first show that the introduction of this biquadratic interaction does not affect the values of critical exponents. Also, we find that the compensation temperature (when it exists) and the compensation magnetic field are shifted to their lowest values, in comparison with the w=0 case. The Arrott-phase-diagram shape is also investigated quantitatively. We show the existence of three regimes depending on the values of w. When the latter is small, we find that the region of competition between the coupling C and the applied magnetic field H becomes more narrow under the effect of w (by competition, we mean the passage from the antiparallel state to the parallel one). While for higher values of w, this competition disappears completely, and then, the system loses its ferrimagnetic character. Kinetics of the phase transition is also examined, when the temperature is lowered from an initial value Ti to a final one Tf very close to the critical temperature Tc. As in the w=0 case, we find that kinetics is controlled by two kinds of relaxation times τ1 and τ2. The former is the relevant time, and is associated to long-wavelength fluctuations driving the system to undergo a phase transition. The second is a short time, which controls local dynamics. Near Tc, we show that, in particular, the longest relaxation time τ1 becomes less important in comparison with that relative to the w=0 case. Finally, we note that the existence of two relaxation times is consistent with the predictions of a recent experiment, which was concerned with the 1/2-spin compounds LixNi2−xO2, where the composition x is close to 1.
Journal of Magnetism and Magnetic Materials, 2000
The aim of this paper is the investigation of the critical properties of two strongly coupled par... more The aim of this paper is the investigation of the critical properties of two strongly coupled paramagnetic sublattices exhibiting a paramagnetic}ferrimagnetic transition, at some critical temperature ¹ greater than the room temperature. In order to take into account the strong #uctuations of the magnetization near the critical point, use is made of the renormalization-group (RG) techniques applied to an elaborated "eld model describing such a transition, which is of Landau}Ginzburg}Wilson type. The associated free energy or action is a functional of two kinds of order parameters (local magnetizations), which are scalar "elds and relative to these sublattices. It involves quadratic and quartic terms in both "elds, and a lowest-order coupling C , where C '0 stands for the coupling constant measuring the interaction between the two sublattices. We "rst show that the associated "eld theory is renormalizable at any order of the perturbation series in the coupling constants, up to a critical dimension d "4, and that, the corresponding counterterms have the same form as those relative to the usual -theory (C "0). The existence of the renormalization theory enables us to write the RG-equations satis"ed by the correlation functions. We solve these using the standard characteristics method, to get all critical properties of the system under investigation. We "rst determine the exact shape of the critical line in the (¹, C)-plane, along which the system undergoes a phase transition. Second, we determine the scaling laws of the correlation functions, with respect to relevant parameters of the problem, namely, the wave vector q, the (renormalized) coupling C and the temperature shift ¹!¹ . We "nd that these scaling laws are characterized by critical exponents, which are the same as those relative to Ising-like magnetic systems.
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.
The European Physical Journal E, 2008
We consider a crosslinked polymer blend that may undergo a microphase separation. When the temper... more We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, zeta. First, by averaging over zeta, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q(-1), a time length scale R(t) approximately t(1/z), and the mesh size xi*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of xi*, the dynamics is stopped. The final time, t*, then scales as t* approximately xi*z, with the dynamic exponent z = 6-eta. Here, eta is the usual Ising critical exponent. Since the final size of microdomains xi* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.
The European Physical Journal E, 2009
We consider bilayer biomembranes or surfactants made of two chemically incompatible amphiphile mo... more We consider bilayer biomembranes or surfactants made of two chemically incompatible amphiphile molecules, which may laterally or transversely phase separate into macrodomains, upon variation of some suitable parameter (temperature, lateral pressure, etc.). The purpose is an extensive study of the dynamics of both lateral and transverse phase separations, when the bilayer is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. The critical dynamics are investigated through the partial dynamic structure factors of different species. Using a two-order parameter field theory, where the two fields are the composition fluctuations of one component in the leaflets of the bilayer, combined with an extended van Hove approach that is based on two coupled Langevin equations (with noise), we exactly compute these dynamic structure factors. We first find that the dynamics is governed by two time scales. The longest one, Tau, can be related to the thermal correlation length, Xi ~ Sigma|T - T(c)|(-1/2), by Tau ~ Xi(z), with the dynamic critical exponent z = 4, where Sigma is an atomic length scale, T the absolute temperature, and T(c) its critical value. The characteristic time Tau can be interpreted as the time required for the formation of the final macrophase domains. The second time scale is rather shorter, and can be viewed as the short time during which the unlike phospholipids execute local motion. Second, we demonstrate that the dynamic structure factors obey exact scaling laws, and depend on three lengths, namely the wavelength q(-1) (q is the wave vector modulus), the correlation length Xi, and a length scale R(t) ~ t(1/z) (z = 4) representing the size of macrophase domains at time t. Of course, the two lengths Xi and R(t) coincide at the final time Tau at which the bilayer reaches its final equilibrium state. Finally, the present work must be considered as a natural extension of our previously published one dealing with the study of lateral and transverse phase separations from a static point of view.