Some Complex Phenomena in Fluid and Solid Mechanics E-Book

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Beschreibung

This book is a tribute to Professor Abdelhak Ambari and brings together ten chapters written by colleagues who were fortunate enough to work with him.

The contributions presented in this book cover the research themes in which Abdelhak Ambari was interested, and to which he made valuable experimental and theoretical contributions. For example: rheology of complex fluids and polymers; hydrodynamic interactions; flows at low Reynolds numbers; characterization of porous media; hydrodynamic instabilities and solid mechanics; electrochemical metrology.

Some Complex Phenomena in Fluid and Solid Mechanics is aimed at a wide community of readers wishing to delve deeper into these scientific themes: since it is oriented toward the world of research, it will be a valuable tool for doctoral students and beyond. The book also provides undergraduate and graduate students with a good introduction to the techniques and approaches developed in fundamental and applied research in the fields of fluid mechanics, solid mechanics and instrumentation.

Leseprobe

Mathematical and MechanicalEngineering Set

coordinated byAbdelkhalak El Hami

Volume 13

Some Complex Phenomena in Fluid and Solid Mechanics

Edited by

Abdelghani Saouab

Stéphane Champmartin

Jaafar Khalid Naciri

First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-990-7

Preface

This preface aims to briefly outline the career path and portrait of the professor, researcher and exceptional human being Abdelhak Ambari.

Abdelhak Ambari (1950–2022) passed away at the age of 71. His life was shared between his native country, Morocco, and France. Five main periods of his life can be identified: (i) a Moroccan period of his youth, when he completed his secondary education and the preparatory classes for admission to grandes écoles; (ii) a second period of nine years in France, where he completed his engineering studies and his PhD; (iii) a third period (of five years) essential to his career, during which he brilliantly prepared his doctoral thesis; (iv) a fourth period of 12 years, during which Abdelhak Ambari chose to serve his native country Morocco as a research professor at EHTP (Hassania School of Public Works in Casablanca); (v) finally, the last period of 20 years, during which he was a Professor at ENSAM of Angers (École nationale supérieure des Arts et Métiers).

Abdelhak Ambari was born in Casablanca, Morocco, on September 10, 1950. From a very young age, he proved to be an outstanding student. Between 1970 and 1972, he followed the “Preparatory classes for admission to grandes écoles” in Advanced and Special Mathematics at one of the most prestigious high schools in Morocco at Casablanca, lycée Lyautey. He then joined ESTP (École Supérieure des Travaux Publics) in Paris, where he earned his engineering degree in 1975. At that time, given his passion for sciences and research, he decided to embrace a research professor career. He then enrolled in doctoral studies with one of the best universities in France: Pierre and Marie Curie University – Paris VI. In 1976, he earned a Diploma of Advanced Studies in Fluid Mechanics. He then continued by pursuing a PhD in Fluid Mechanics specializing in non-Newtonian fluids, and he defended his thesis on June 27, 1979. Given his passion for research, he continued his research activities and he earned the title of State Doctor in Physics on January 17, 1986. Throughout his studies, in order to finance his doctoral studies, he performed freelance activities and worked as a teaching assistant at ESPCI (École supérieure de physique et de chimie industrielles) in Paris and at EHTP in Casablanca.

Figure P.1.Ambari at l’ESPCI (early 1980s)

After earning his doctoral thesis, he decided to return to Morocco in order to promote and develop research in his native country. He held a Senior Lecturer position at EHTP and became a Professor in 1990. During this period, he initiated several actions that contributed to the boom of research in Morocco: the development of the hydraulics department, the implementation of the Diploma of Advanced Studies in Mechanics and Automation and the supervision of several doctoral thesis works. Abdelhak Ambari dedicated a lot of time and energy to the development and organization of research in Mechanics in Morocco: (i) he set up a research laboratory at EHTP with very limited resources; (ii) he initiated the creation of the school of Fluid Mechanics in Casablanca, which gave rise to the International Congress of Mechanics that is now at its 15th edition; (iii) he contributed to training a new generation of research professors and built up several bridges between Moroccan and French researchers.

In 1998, he decided to return to France for the position of Professor at ENSAM in Angers, and started once again from scratch, striving to build a new fluid mechanics research team. He was appointed the head of the Energy Department of ENSAM nationwide for several years, and he trained several PhD students and post-doctoral researchers. For 20 years, he played an essential role in emulating and leading the research in fluid mechanics at ENSAM by building collaborations, particularly with laboratories in Paris and Bordeaux. Being equally invested in research and education, he developed courses that contributed to the training of generations of student engineers in Angers. Combining the demanding nature of science and the pragmatic approach to industrial problems, he instilled an interest for fluid mechanics in many students, some of whom pursued academic careers in the field. He retired in 2019, leaving a lasting legacy and the memory of a profoundly human and altruistic person.

Throughout his entire career, Professor Abdelhak Ambari shared (with enormous generosity) his knowledge with his students, PhD students and postdoctoral researchers or with his colleagues. According to Professor Etienne Guyon (former Director of the Laboratory of Hydrodynamics and Physical Mechanics of ESPCI, former Director of Palais de la Découverte and the École normale supérieure in Paris), “Generosity is the word that comes to my mind when I think of Abdelhak, both with respect to human relations and to his spontaneous knowledge sharing”. Ambari was highly appreciated by his engineering students at EHTP in Casablanca and at ENSAM in Angers. He represented a father figure to them. A lot of his every day time was dedicated to exchanges and discussions with his research collaborators, and he often said that “he learned from his students as much as they learnt from them”. On the news of his passing, many of his students posted testimonials on the Internet, expressing gratitude and admiration. According to these testimonials, “(i) He was a highly competent, very generous and very modest professor; (ii) he was an outstanding professor; a great scientist, a methodical person who was listening to his students. (iii) An outstanding and kind professor who left no one indifferent”.

Professor Abdelhak Ambari was also an outstanding scientist. He had a very broad scientific culture and he worked on several research themes in Fluid Mechanics and in the Physics of Polymers. His PhD thesis, conducted under the supervision of Professor André Fortier, focused on the determination of rheological parameters of polymer solutions. His State Doctoral thesis, which focused on the study of the interaction between hydrodynamics and particle dynamics, was supervised by the distinguished Professor Pierre-Gilles de Gennes (Nobel Prize in Physics in 1991). For his outstanding work for the doctoral thesis, Abdelhak Ambari was awarded the Alain Brelot Prize by the French Society of Physics in 1987.

Figure P.2.A. Ambari receiving the Alain Brelot Prize of the French Society of Physics in July 1987.

Abdelhak Ambari had a lot of respect for his mentors. P.G. de Gennes had a remarkable influence on his work: he knew how to avoid the pitfalls of highly theoretical approaches and enjoyed formulating a physical problem in terms of simple concepts and images. Given his broad scientific culture, he perceived the analogies between various branches of physics. The most recent research themes of Abdelhak Ambari concerned several subjects: (i) low Reynolds number-flows in Newtonian and non-Newtonian fluids; (ii) transport phenomena in porous media; (iii) characterization of porous media; (iv) rheology of complex fluids; (v) capillary phenomena; and (vi) hydrodynamic interactions. Abdelhak Ambari was considered as one of the best specialists in the field of “Hydrodynamic Interactions”.

Many works of Abdelhak Ambari led to various national and international collaborations. Due to his empathic nature and open-mindedness, Abdelhak Ambari established contacts with bright scientific personalities, such as Professors Abdu Salam (Nobel Prize in Physics in 1979) whom he met in 1989 at the International Centre for Theoretical Physics at Trieste in Italy, Daniel D. Joseph of the University of Minnesota in the United States, J. Hinch whom he met at Cambridge, in the United Kingdom, and Professor Chhabra of the Indian Institute of Technology, whom he invited twice to ENSAM in Angers.

Figure P.3.During the defense of his doctoral thesis in 1986.

Abdelhak Ambari dedicated his doctoral thesis “to the memory of those who fought for the abolition of ignorance in developing countries”. He is now part of this family of noble people. This book is a tribute to him, and also to his family, his wife Annick and his son Mehdi, who contributed directly or indirectly to his work.

We would like to conclude with this beautiful testimonial:

The loss of Professor Abdelhak Ambari caused great sadness shared by the academic mechanics community in France, in Morocco and elsewhere. Abdelhak Ambari was the embodiment of professional generosity, quiet knowledge with immeasurable availability and great modesty. There is no greater loss for humanity than that of a human being with a well-rounded brain. People pass away, and loyal friends have a duty to continue their work.

Rest in peace, dear friend. You will be missed, and will forever have a place in our hearts.

This book has been written as a tribute to the memory of Professor Abdelhak Ambari. It is divided into 10 chapters authored by his French and Moroccan colleagues who had the chance to meet or collaborate with him.

Figure P.4.A. Ambari at Angers in 2020.

The contributions presented in this book focus on research themes that were of interest to Abdelhak Ambari during his career, and to which he brought his valuable experimental and theoretical contributions:

rheology of complex fluids and polymers;

hydrodynamic interactions;

low Reynolds number-flows;

characterization of porous media;

flows in porous media;

instabilities in hydrodynamics and in solid mechanics;

metrology by electrochemical methods.

We wish to thank all the authors for their contributions, and ISTE-Wiley for having kindly agreed to publish this book dedicated to the memory of our colleague and friend, Professor Abdelhak Ambari.

Abdelghani SAOUAB,

Stéphane CHAMPMARTIN,

Jaafar Khalid NACIRI.

August 2024

Note

Preface written by Abdelghani SAOUAB, Stéphane CHAMPMARTIN and Jaafar Khalid NACIRI.

1Kinetics of Microphase Separation in Interpenetrating Polymer Gels

The physical system considered here is an interpenetrating gel composed of two chemically different polymers, cross-linked in the high-temperature single-phase region. When the bath temperature is sufficiently lowered, below the usual critical temperature, the mixture tends to separate into two distinct phases; however, this does not occur, due to the permanent presence of cross-links. Below the spinodal temperature, a microphase separation takes place, accompanied by the appearance of microphase domains, contrary to the usual macrophase separation (in the absence of cross-links), where the phase domains are macroscopic.

This chapter summarizes the accomplishments in studying the kinetics of microphase separation in interpenetrating gels by a progressive reduction of the bath temperature. We particularly discuss the appearance of instabilities at the critical point. The chapter ends with a discussion on the effects of the presence of a good or a theta solvent, the initial composition fluctuations or charges in these kinetics.

1.1. Introduction

When the temperature is lowered, the polymer blends separately into two distinct phases, due to the chemical difference between the present species. This phase separation occurs even in the presence of a solvent. When the polymer chains are of high molar mass, the translational entropy is very small, and then, the chemical segregation between chemically different monomers is significantly amplified. In the temperature-composition phase diagram, a low or high critical separation temperature is then observed (de Gennes 1979a; Pincus 1981; Binder 1983) in certain cases. In the vicinity of this temperature, there are significant composition fluctuations, and there is a divergence in the correlation length that characterizes the spatial extent of these fluctuations.

Phase separation can be avoided by a chemical crosslinking of blend in the single-phase region to obtaining a gel termed interpenetrating gel. This gel can also be obtained by mixing, from the beginning, chemically different monomers and then initiating the polymerization process. In this case, the chemical cross-links are replaced by topological entanglements.

Naturally, when the temperature is reduced, the cross-linked polymer blend tends to separate in two phases, but the cross-links prevent the appearance of a macrophase separation, due to the elastic forces resulting from the presence of cross-links. As result of this competition, a microphase separation takes place (de Gennes 1979b; Binder and Frisch 1984; Battachy et al. 1991); this was theoretically studied for the first time by de Gennes (1979b) using a mean-field approach. The author has shown that the size of the microphase domains depends on the crosslinking rates and approximately corresponds to the average distance between two consecutive cross-links. A second important effect of cross-links is that it renders the polymer blend more compatible, and there is a deviation of the critical temperature that also depends on the cross-link density, defined as the number of cross-links per unit volume. Microphase separation in interpenetrating gels has been experimentally observed by Briber and Bauer (1988), who had used small-angle neutron scattering for the study of the structure of microphase domains in a cross-linked blend, which is composed of polystyrene and poly(vinyl methyl ether).

Let us consider an interpenetrating gel with high cross-link density. The typical distance scale of microphase separation is very small in comparison with the gyration-radius of uncross-linked polymer chains. Therefore, different kinetics are expected for non-cross-linked and cross-linked cases. The non-cross-linked polymer blends have been studied by de Gennes (1979a), Pincus (1981) and Binder (1983). They have shown that reptation of polymer chains has a significant contribution to the kinetics of phase separation. However, for cross-linked polymer blends, a small typical time scale is expected, since large-scale motions of cross-linked polymer chains are frozen by the presence of permanent cross-links. It can therefore be anticipated that, for the considered system, only Rouse motion is important. In this context, it had been shown that the dynamic structure factor of a polymer blend susceptible to phase separate can be decomposed in two modes (de Gennes 1981; Pincus 1981), namely a rapid mode, of low amplitude, and a low mode, of high amplitude. The first one corresponds to a local motion, while the second one is associated with a large-scale reptation of polymer chains. In a cross-linked system, above the freezing point, the reptation phenomenon is absent, because of constraints. This is why only a local motion of Rouse type is expected, which is essentially responsible for the microphase separation kinetics. Rouse time corresponds to a motion occurring very early in the case of a non-cross-linked blend, which was discussed by the above-mentioned authors.

Section 1.2 reviews the main results concerning the static study (or at thermodynamic equilibrium) of the microphase separation. Section 1.3 is dedicated to the kinetic study of this microphase separation. Finally, in section 1.4, we discuss the results dealt with the kinetic study and compare them to those related to non-cross-linked blends.

1.2. Static study of microphase separation

Let us consider a blend made of two chemically different polymers, A and B. N denotes the common polymerization-degree of polymer-chains initially in the single-phase region of the phase diagram (high-temperature phase). The monomer composition is denoted by Φ. Due to the symmetry, Φc = 1/2 is chosen as the critical monomer composition. Let us assume that the polymer chains are initially randomly distributed and there is no composition fluctuation. This is a high temperature approximation, which involves that the system is very far from its critical point. The blend is now strongly cross-linked, and by lowering the temperature, the interpenetrating gel is brought into the single-phase region. According to de Gennes (1979b), the free energy of such a system can be written, in a Landau approximation, as follows:

[1.1]

Here, χ denotes the Flory monomer–monomer interaction parameter, which is inversely proportional to temperature, χc = 2 / N is the critical value of the interaction parameter of the non-cross-linked mixture, is the local fluctuation of composition (or order parameter), defined by: , where is the composition of a species of the non-cross-linked blend, a is the common size of monomers, kB is the Boltzmann’s constant and T is the absolute temperature of the cross-linked blend. There, the gradient term takes into account the local fluctuations of composition (Joanny 1978) and the last term, which is discussed below, is the elastic contribution due to cross-links. Critical parameters, χc = 2 / N and Φc = 1 / 2, are determined in the Flory–Huggins (FH) approximation (Flory 1953). It is important to note that relation [1.1] can be considered as the sum of the expansion of FH free energy in the vicinity of the critical point and the elastic term. On the other hand, the resolution of equation [1.1] is equivalent to determining the effect of perturbations in the vicinity of the critical point. Previous studies on critical phenomena have shown that this effect can be either weak, causing a simple perturbation, or dramatic, leading to a change in the universality class of the problem. This will be the case in what follows.

The elastic contribution in equation [1.1] was introduced by de Gennes (1979b) by analogy with the polarization of a dielectric medium. The quantity C is the rigidity constant that was linked to the number, n, of monomers between two consecutive cross-links. According to de Gennes (1979b), its expression is given as:

[1.2]

In expression [1.1] of FH free energy, is the displacement vector between the centers of mass of two strands A and B (a strand is the segment of a polymer chain between two neighboring cross-links). The polarization in the problem of the dielectric medium is equivalent to the distance between the centers of mass of various strands in the problem of polymer, and the charge fluctuations corresponding to the composition fluctuations. This involves that the quantities and ϕ are not independent to each other, but linked by: , or in the reciprocal space, by:

[1.3]

We are therefore led to consider only the longitudinal modes, and the free energy can be written as follows:

[1.4]

Here, ϕq is the Fourier transform of the composition fluctuation:

[1.5]

The scattered intensity (or structure factor), S(q), in a light or neutron scattering, is given as:

[1.6]

It has a maximum for:

[1.7]

The value of the scattering intensity at this maximum diverges at spinodal for:

[1.8]

This deviation varies as the inverse of cross-links dose, n. Therefore, a divergence occurs for a non-zero value of the modulus of the transfer wave-vector, q, and this can be interpreted as a microphase separation. Then, there is a deviation of the critical temperature; the cross-linked system is more miscible than the non-cross-linked one.

The following section focuses on the study of kinetics of microphase in interpenetrating gels.

1.3. Kinetic study of microphase separation

As mentioned above, the main difference between demixing of a non-cross-linked polymer blend and a highly cross-linked polymer blend is that the distance scales being examined are highly different. Indeed, in the first case, the motions of polymer chains, at long distance, are controlled by reptation, while in the second case, there are local arrangements, over distances of the order of the size of microphase domains.

For a non-cross-linked polymer blend, de Gennes (1981) and Pincus (1981) have shown that the time dependence of the scattered intensity is the sum of two contributions. The first one is a rapid mode and it corresponds to a local reptation. The second having a high amplitude and being slower corresponds to a long distance reptation of the polymer chains and causes a disentanglement of species. In the present case, due to the permanent presence of cross-links, the latter should not be expected to bring their contribution, and longtime motions are then frozen. Local dynamics was not considered in the previous works, as it corresponds to kinetics during the first instants of the phase separation of blend. It is the only motion that is present here. It is the local Rouse dynamics that occur inside a tube formed by the other chains (de Gennes 1979a; Doi and Edwards 1986). This will appear in the expressions of Onsager coefficient that will be introduced below. Formally, we write equations similar to those related to a non-cross-linked blend (de Gennes 1980). More precisely, what we are looking for is the time evolution of the fluctuation at fixed wave-vector modulus, q. For this, the standard procedure that was thoroughly discussed by Binder (1983) is applied. This time evolution can be found by introducing a current, , for species A. The continuity equation is given as:

[1.9]

Here, t represents the time parameter. The current is linked to a chemical potential, , by the following relation:

[1.10]

being the Onsager coefficient that will be discussed below. Let us assume an exponential decrease of the fluctuation, as a function of the wave-vector, :

[1.11]

Here, τ(q) is the relaxation rate. Using relations [1.9] to [1.11] and keeping only the linear terms in the fluctuation of composition, the following result is found (Cahn 1968):

[1.12]

Equation [1.12] then provides the relaxation rates for the first instants, when the linearization procedure is valid. Positive relaxation rates correspond to stable cases when fluctuation decreases in time, while negative rates correspond to unstable modes. Such fluctuations increase in time (Binder and Heermann 1985). In order to determine the relaxation rate, we need to know Onsager transport coefficients, Λ(q), and therefore the nature of the strand motion. The shape of equation [1.12] indicates that the product of the last two terms is a diffusion-coefficient. As usual, this coefficient is the ratio of mobility, μ, to the correlation function, that is:

[1.13]

Comparing it with relation [1.6], we find that the Onsager coefficient can be interpreted as the mobility of strands. In a highly cross-linked system, such as considered here, the motion of polymer-chains is local, over a distance d ≅ n1/2a, where a is the common size of monomers and n is the number of monomers between consecutive cross-links (polymerization-degree of strands). Notice that, for highly cross-linked polymer blends, with n < Ne, where Ne is the distance between consecutive entanglements, only Rouse motion is present. Onsager transport coefficient, Λ(q), is then proportional to Rouse mobility. It is emphasized that the case with n >> Ne is discussed in section 1.4 (see below). For high cross-linked blends, the number n(q) of monomers in a strand at distance q−1 is proportional to q−2. Within the framework of Rouse approximation, the mobility is as follows:

[1.14]

Then, we have:

[1.15]

Figure 1.1 shows the characteristic frequency, for various cases, Ω1(q) = τ−1 (q)/q4. Figure 1.1(a) describes the case in the absence of cross-links and for the first instants of dynamics. For χ < χc (or T > Tc), all frequencies are positive and therefore there is no instability. In the opposite case, T < Tc, however, we find that all modes with:

[1.16]

are stable, while the long wavelength modes with q < qc are unstable. Any fluctuation with such a wavelength increases in time. For a cross-linked polymer blend, for which the rigidity constant, C, is non-zero, the situation is quite different (Figure 1.1(b)). For χ < χc, the curves have a minimum that can be either positive or negative. For χ < χs, all frequencies are positive, and then, there is no instability. Let us recall that:

[1.17]

Figure 1.1.(a) Relaxation rate, as a function of the wave number, for a non-cross-linked blend. For χ > χc, the small q modes are unstable. (b) Relaxation rate for a cross-linked blend. For χc > χ > χs, the minimum is positive and then there is no instability. Above χs, the minimum becomes negative and the modes are unstable for a large q

For χ = χs, the minimum of the characteristic frequency becomes zero, and the latter corresponds to a microphase separation. For low temperatures, Ω1(q) has a negative part corresponding to the presence of instabilities in a finite interval of wave-vector, q. The most fast increasing instability corresponds to the minimum of the curve for:

[1.18]

The condition, χ ≅ χs, corresponds to the size, q*−1, of microdomains. As shown above, this can be written under the following scale form:

[1.19]

It is important to note that the fact that the short wavelength part, q, of the curves is canceled out is a consequence of the scattered intensity, S(q), which vanishes at q = 0. This is in perfect agreement with the results of the neutron scattering experiment realized by Briber and Bauer. Therefore, the previous considerations should not be expected to be valid in this region. This question will be reexamined in the next section. Finally, near typical size, q*−1, the characteristic frequency varies as follows:

[1.20]

with:

[1.21]

This characteristic frequency vanishes when χ is close to its spinodal value, χs. Therefore, the modes are slower than those of Rouse. It is important to note that according to relation [1.15], for χ = χc, the characteristic frequencies vary as follows:

[1.22]

Similar results have been previously found for a common polymer blend and have been interpreted as a weighted average of two characteristic rates, namely, local motions of Rouse type and long chain reptation.

The remaining part corresponds to a slow decrease related to the presence of spinodal decomposition.

1.4. Discussion

The previous sections discussed kinetics over short periods of time of microphase separation occurring in blends made of two chemically incompatible polymers, cross-linked in the single-phase region and brought to the demixing region by a progressive temperature decrease. According to our predictions, only a local motion of Rouse type is possible, due to the presence of permanent cross-links. This same motion occurs inside microphase domains. As a result, we found that the characteristic rate varies with the wave-vector as q6 at spinodal temperature. Furthermore, in the single-phase region, instabilities increase with a characteristic time, Ωq ≅ q4 (χs − χ), where χ is the segregation parameter, which varies essentially as the inverse of the absolute temperature, and χs is its spinodal value.

For the present study, the approximation made is that there is only one temperature in the problem. More precisely, crosslinking of the random blend of species is produced at high temperature, far away from the critical point. It is important to note that the above-mentioned theory of microphase separation shows that the theoretical scattering intensity vanishes when the scattering angle is zero. However, Briber and Bauer have experimentally shown that this is not the case. This discrepancy between theory and experiment has been explained by a series of published works (Vilgis et al. 1993; Benmouna et al. 1994b; Bettachy et al. 1995). In fact, the origin of this discrepancy is the existence of initial composition fluctuations at the moment of blend crosslinking. These fluctuations exist even at very high temperature.

It is important to note that the above considerations can be extended to the case of weakly cross-linked polymer gels, and let us extend kinetics study to these gels. For long polymer chains, the number of monomers between consecutive cross-links, n, is larger than the number of entanglements, Ne, and in this case, kinetics results from a reptation inside tubes between crosslinking points, over long periods of time. In this case, Onsager coefficient, Λ(q), no longer corresponds to the local motion of Rouse type, but to a local reptation motion of strands inside tubes. According to Pincus (1981), such a motion implies:

[1.23]

and therefore:

[1.24]

There is a change in behavior between relations [1.15] and [1.24] for n ≅ Ne. According to relation [1.24], there is a fast instability for q ≅ q*, for which the characteristic frequency increases as follows:

[1.25]

It is important to note that when the rigidity constant of gel, C, tends to zero, the previous relation becomes quantitatively similar to that of Pincus relative to a non-cross-linked polymer blend undergoing a macrophase separation:

[1.26]

With the deviation from the critical temperature:

[1.27]

This chapter, whose content is largely based on a previously published work (Battachy et al. 1992), focused on interpenetrating gels in a molten state. However, the extension of the present study has been achieved for interpenetrating gels in the presence of a solvent (Vilgis et al. 1993; Benmouna et al. 1994b; Bettachy et al. 1995; Benhamou et al. 1997; Derouiche et al. 2005; Benhamou et al. 2010). In this case, crosslinking was achieved at high temperature, therefore, under good solvent conditions. The essential change is that kinetics is this time governed by a local motion of Zimm type (Derouiche et al. 2005), where strands are subjected to hydrodynamic interactions.

Finally, it should be recalled that this study has been extended to the case of interpenetrating gels formed by a crosslinking of polyelectrolyte-chains (Benmouna et al. 1994a; Boussaid et al. 2009), and it has been shown that the sign of charges along chains and the ionic concentration of co-ions and counter-ions induce radical changes in kinetics of the microphase separation.

1.5. References

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Battachy, A., Derouiche, A., Benhamou, M., Daoud, M. (1992). Kinetics of microphase separation in cross-linked polymer blends.

Macromolecules

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Benhamou, M., Derouiche, A., Bettachy, A. (1997). Microphase separation of crosslinked polymer blends in solution.

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Benhamou, M., El Fazni, A., Bettachy, A., Derouiche, A. (2010). Theory of microphase separation in crosslinked polymer blends immersed in theta-solvent.

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Benmouna, M., Vilgis, T.A., Benhamou, M., Babaoui, M., Daoud, M. (1994a). Stability analysis and scattering properties of charged crosslinked blends in solution.

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Benmouna, M., Vilgis, T.A., Daoud, M., Benhamou, M. (1994b). Scattering and phase behavior of cross-linked blends.

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Bettachy, A., Derouiche, A., Benhamou, M., Benmouna, M., Vilgis, T.A., Daoud, M. (1995). Scattered intensity by a cross-linked polymer blend.

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Binder, K.J. (1983). Collective diffusion, nucleation and spinodal decomposition in polymer mixtures.

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Binder, K.J. and Frisch, H.L. (1984). Phase stability of weakly crosslinked interpenetrating polymer networks.

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Binder, K. and Heermann, D.W. (1985). Growth of domains and scaling in the late stages of phase separation and diffusion-controlled ordering phenomena. In

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Boussaid, A., Benhamou, M., El Fazni, A. (2009). Theory of microphase separation in charged crosslinked polymer blends with arbitrary charge distribution.

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Note

Chapter written by Mabrouk BENHAMOU.