Flory approach for polymers in the stiff limit
|
|
- Andrea Harmon
- 6 years ago
- Views:
Transcription
1 Flory approach for polymers in the stiff limit H. Nakanishi To cite this version: H. Nakanishi. Flory approach for polymers in the stiff limit. Journal de Physique, 1987, 48 (6), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1987 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.
2 05.30 On We J. Physique 48 (1987) JUIN 1987, 979 Classification Physics Abstracts Flory approach for polymers in the stiff limit H. Nakanishi (~) Collège de France, Physique de la Matière Condensée (+), 11, place Marcelin-Berthelot, Paris Cedex 05, France (Reçu le 1 er juillet 1986, révisé le 30 janvier 1987, accepté le 24 f6vrier 1987) 2014 Résumé. étudie, dans l approximation de Flory, la conformation de chaînes semi-rigides, linéaires ou branchées dans la limite de forte rigidité. On montre que la dimension marginale dc associée aux effets de volume exclu est en général réduite dans cette limite. En particulier, pour une chaîne linéaire en bon solvant on trouve dc 2 en accord avec des résultats récents obtenus numériquement par Lee et Nakanishi Abstract. present a Flory approximation for the conformation of semi-flexible linear and branched polymers in the extremely stiff limit. We show that the upper marginal dimension dc for the excluded volume effect is in general reduced in the stiff limit. In particular, for a linear chain in a good solvent we find dc 2 in agreement with the recent Monte Carlo result of Lee and Nakanishi. 1. Introduction. The conformation of semi-flexible long chain polymers has been of considerable interest from theoretical [1-4] and experimental [3-4] points of view. Recently, Lee and Nakanishi [5] studied the crossover behaviour of such a chain from its extremely stiff limit using the biased self-avoiding walk model (BSAW) [6], and pointed out a dramatic difference in the effects of excluded volume in d 2 dimensions and d 3 dimensions. They concluded, based on their Monte Carlo simulations, that the excluded volume effect is irrelevant in the stiff limit in d 3. In this paper, we present an analysis based on Flory approximations [7-9] for the conformation of stiff linear and branched polymers. This analysis results from an extension of the form of excluded volume parameter used by Odijk [10] for polyelectrolytes and also follows the work of Schaefer, Joanny and Pincus [3] who, however, did not discuss the present limit of extreme stiffness. Although Flory approximations are known to rely on «accidental» cancellations of neglected terms in the free energy, their predictions have been surprisingly (t) Present address : Department of Physics, Purdue University, W. Lafayette, IN U.S.A. (+ ) Unité Associ6e au CNRS (UA N 792). accurate for quantities such as the Flory exponent v and for the value of the upper marginal dimension dc for simple polymeric systems [9]. In fact, the use of Flory approximations for the evaluation of dc is much like the Ginzburg criterion in critical phenomena and is quite reasonable from the theoretical standpoint also. The BSAW model is simply that of an isolated self-avoiding random walk on a lattice with a variable parameter characterizing the stiffness. The statistical weights of BSAW are described by the probability p of taking a gauche step (a bend in the walk) and the probability 1- p of a trans step (continuing straight), at each stage of Monte Carlo simulation. Taking the step length (monomer dimension) as a, the persistence length is proportional to lp alp. In d dimensions, the generalization of Odijk s excluded volume (per persistence length) is then This result will be used for the discussion of stiff chains in good solvents and an analogous one will be used for 0 solvents. While these results make sense physically only for d, 2 because the BSAW model itself makes sense only for integer dimensions of two or greater, it is possible to give a formal analysis for an arbitrary value of d > 0. Indeed, e.g. for the 0 case, such an analysis may have a valid basis for d>, 3/2 as will be seen below. Of course, direct Article published online by EDP Sciences and available at
3 980 physical conclusions are to be drawn only for integer values of d>, 2. The following analyses always assume the limit of Np > 1 where N is the number of steps in the chain (thus proportional to the molecular weight) so that there are a large number of «blobs» of linear size of the order of the persistence length. 2. Linear chain in good solvent. We first discuss the case of a linear chain in a good solvent. In this case, the Flory free energy written as can be where the elastic contribution f el is given approximately by using the leading term in the free, stiff chain result [1] for Np > 1, as This is consistent with the expected form of the third virial coefficient [3, 11, 12]. From this, we obtain To see the validity of (5), we substitute it into the general expressions (7) for x and y. Requiring the results to satisfy XF >> 1 and YF >> 1, we obtain 1 :5: d :5: 4, which then leads to the stated result. On the other hand, if p is fixed and we ask at what size N * the chain conformation crosses over from Gaussian to the excluded volume type (5), then we must substitute the leading Gaussian form R oc (Nip a )112 into (7). In this case, For the Gaussian form to be valid for N N *, we must have XG - 1 and YG -- 1 at N *, which gives where R - (R2) 1/2 is the root-mean-square end-toend distance. The repulsive interaction contribution fint is given using (1) as to the leading order in monomer concentration. According to the usual Flory approximation, one simply minimizes the free energy (2)-(4), which will yield For d 3, this expression was already given by Schaefer et al. [3]. Aside from its inherently approximate nature, the intended validity of such an expression is limited from above by the upper marginal dimension de and from below by the neglect of the higher order terms in fi.t. In our problem, it is further constrained because of the assumption (1). Thus, in this problem, for the usual «Flory» limit of fixed p and large N, these constraints would restrict the validity of (5) to 2, d, 4 (with possible logarithmic corrections at either end). It is useful for later discussions to go through this argument in some detail. Let x be the first term in fint (righthand side of (4)) and y be the ratio of two successive terms in fi,,t. The value of y can be obtained from the following assumed form of the expansion of /;nt in terms of the concentration of «rods», c Np/Rd : where This results in N* - 1 /p for d 2 and N* ~ 11p3 for d 3, the latter of which was already given by Schaefer et al. [3]. We may, however, be interested in the «stiff» limit of N -> 00, p with Np Const. (> 1 ). This interest stems from the objective to write a scaling form [6] for (R2) about the singular point at p N The most obvious and natural variable is Np (the number of «rod-like» segments of the persistence length), but the arguments leading to equation (9) suggest a second important variable N/N *. Therefore, we propose a «scaling» relation, Unfortunately, equation (10) suggests that (R2) is in general dependent on N and p separately and that there is no reduction in the number of variables as true scaling relations would require. However, for d 2, the two variables in f (a, f3 ) are identical and (R2) does have a simple scaling form. For d 3, we have a -+ 0 in the stiff limit, and once again (R2) will have a simple (now Gaussian) scaling form. We show in figure 1 qualitative diagrams illustrating the crossover between Gaussian and excluded volume behaviours for two and three dimensions. We can also see that the upper marginal dimension de for the stiff limit is reduced to 2 by rewriting the Gaussian values xg and YG in equation (8) as The requirement for the consistency of the Gaussian approximation, XG 1 and YG -- 1 for fixed Np, then yields de 2 in agreement with the Monte Carlo result of reference [5].
4 Crossover Fig. 1. phase diagrams for stiff chains in (a) d 2 and (b) d 3 dimensions. The symbols 1 d, 2 d and 3 d refer to the effective dimensionality and RW and SAW refer to the absence or the presence of the excluded volume behaviour. 3. Linear chain in 0 solvent. For a linear chain in a 0 solvent, the leading tint has the form term in 4. Linear chain in poor solvent. For a linear chain in poor solvent, we may suppose that the sign of the leading interaction term is reversed, and thus the Flory approximation gives Note that an analog to the excluded volume f2 for this case is which replaces a2 d for a flexible chain. Thus analyses based on equation (12) cannot be valid for d 3/2 for small p ; for d > 3/2, however, such an analysis may still make sense formally as an analytic continuation in d into non-integral values in a sense somewhat similar to the so-called E-expansions [9]. For the usual Flory limit, an analysis analogous to that leading to equations (8) and (9) then shows that equation (13) should be limited in validity to 3/2 d 3 at best and that de 3 as usual for a 19 regime. For the «stiff» limit, however, the only place where (13) might apply is the lower bound of this range, d 3/2, similarly to the good solvent case. In the Flory limit of fixed p, the crossover size N * can be calculated similarly to equation (9) : For d 2, we have N*-p- 2 and thus, using equation (10), we see that a -+ 0 in f(a, (3 ) in the stiff limit. This leads to a Gaussian scaling form with vanishing excluded volume even in two dimensions. In fact, in the stiff limit de is reduced to 3/2. This form will result if, e.g., one supposes an impenetrable rod with a concentric sheath of attractive region as representing a stiff segment which likes to stick to other stiff segments. Normally, such a free energy is considered unphysical since it would be unbounded from below as R -> 0. To rectify this problem, one usually includes a positive higher order term in fint. An alternative remedy is simply to impose a hard-core excluded volume as an external constraint for R. The actual minimum thus depends in general on both this constraint and the higher order terms of fint- To see this, we first obtain the unconstrained minimum of f using the sum of equations (12) and (16) : and (3) for fel. Following de Gennes [13],,we write an expansion parameter, and the minimization with respect to this parameter yields For d 4, the righthand side of (19) is large and negative. Thus, we must have a 1 and conclude that
5 982 with no dependence on the stiffness p. However, since the hard-core excluded volume per persistence length is f2, assuming isotropy (i. e., as long as there is no transition to a nematic state), we must have Thus we surmise that the latter constraint actually determines the allowed minimum f and as long as the the polymer remains isotropic, equation (21) is appropriate Flory expression for a stiff linear chain in poor solvents (with Np > 1). In this case, higher order terms of lint have larger powers of p and thus the term (16) is leading in magnitude as required for consistency. 5. Branched polymer. Let us now consider an isolated branched polymer with fixed branching fugacity. To obtain the entropy term lei R/Ro in the Flory approximation, we estimate the Gaussian radius Ro in the following way. The number of monomers between two succes- to the sive branching units, n, is proportional reciprocal of the (fixed) branching fugacity A. Since the linear chain between such branching units has a linear size of 6 where for sufficiently large n (i.e., for large np), we have For fixed, small A, then, we have where N is essentially the nuniber of bifunctional monomers. In a good solvent, we also have as before, to the leading order assuming that the interactions arise from stiff segments of any branch. The minimization of F gives a Flory expression,, whose validity for fixed p and large N, n is restricted to 2 d 8 with d, 8 as usual. Since we are considering fixed A, we don t need to worry about the fact that the linear parts of the chain become Gaussian above d 4 [14]. The crossover size N * is obtained by considering The requirement is XG - 1 and YG > 1 at N *, which for all d 8. In particular, for d 3, N * p-6/5. For d 2, we have N * p - 2/3, which should be taken to suggest that the chain shows excluded volume behaviour whenever the conditions for the present approximations are met (i.e., Np > np > 1). The stiff limit for fixed A implies np - 0 and thus each linear segments must be considered to be a rigid rod of length na instead of (22). In this case the problem simply reduces to that of a branched polymer with only branching units each of size na. In particular, Flory expressions for such chains are well known [14] and we have d, 8 as for the Flory limit discussed above. More general situation of N, n and p being all variable parameters is left for a future study. In a 0 solvent, we take f el as in (24) but fint as in (12). Upon minimization, we get [14] which should be applicable only for 3/2 d 6 for the Flory limit with fixed small A. The crossover size is given by for all d 6 ; in particular, N * p - 2 for d 3 and N * p -1 for d 2. The situation is analogous to the linear chain in a good solvent in that in d 2 there is a confluence of the two variables Np and N/N *. However, the stiff limit with fixed A is again reduced to a branched polymer with only branching units and dc, remains to be 6. The results for poor solvents are similar to those for linear chains under the present conditions. 6. Concentrated solution of linear chains in good solvent. For a concentrated solution of polymers, we must in general consider screening effects even within the Flory approximation. Extremely polydispersed solutions of flexible, linear chains in good solvents have already been considered by Daoud and Family [15]. In this case, one considers a polydispersity of the form of the one-dimensional percolation problem so that the number of N-mers is proportional to qn -1 (1 _ q )2 where q is the independent probability that a bifunctional unit has reacted. Here, the screening coefficient NW 1 of the leading term of tint can be replaced by N - 1 when considering a typical chain of size N since all relevant averages (IV, NW, Nz) scale in the same way. (NW refers to the so-called weight average and NZ is the z-average.) Moreover, the screening coefficients of higher order terms in fint should also scale as N- 1 [16].
6 a 983 The stiff chain analog of the same problem is as follows. We have fe, as in (3), but f;nt is given to the leading order, by Assuming the screening coefficient of the k-th order term to be N k - 2IN k -[16], we would then have Thus the Flory expression for the radius is simply The N-dependence in (32) is the same as for the flexible case [15]. However, we also have For the Flory limit of fixed p, we have x oc y -1 and it is impossible to have x > 1 and y > 1 at the same time. This problem was first pointed out to us by Daoud [16] in the context of branched, flexible chains. This suggests that the expression (32) has no region of validity in the Flory limit. Since small p helps to achieve y > 1, one might suppose that the stiff limit would allow this for a range of d ; however, we note that, Thus, for the Flory limit, once again (36) appears to have no region of validity as pointed out by Daoud and Coniglio for the flexible case [16]. The stiff limit with constant branching fugacity does not help this situation because, as in 5, this limit reduces to branched polymers with only branching units. More general situation with A also being a variable may, however, resolve this problem ; this is left as a future study. We also have and thus N * p - 4/3 for d 3 and N * p- 1/2 for d 2 if the mean field percolation value of s 1/2 were used. The latter should be interpreted as suggesting that the typical chain with Np > np >> always of excluded volume type. 1 is 8. Summary. in the stiff limit, which would suggest the window to be 2/3 d 2. Since the excluded volume of the form (1) cannot be valid for d 2, this scheme cannot turn (32) into a consistent approximation [17]. The consideration of the Gaussian limit reveals that d,, 2 for both the Flory and stiff limits ; thus there is no reduction in d,. 7. Concentrated solution of branched polymers in good solvent. Lastly, it is possible to discuss a polydispersed melt of branched polymers as well. Again we treat the good solvent case only, and assume that the branching fugacity is fixed and small so that the size N essentially counts only the bifunctional monomers (which are stiffi. Since N z # NW in general, when we consider a typical polymer of size N - Nz, we must know the particular form of polydispersity for the problem at hand. Let us thus assume a particular relation NW oc Ns for our problem. Then, using (24) for tel and, to the leading order, we obtain a Flory expression for the stiff analog : In summary, by a Flory approximation applied to non flexible polymers we find that in general the upper marginal dimension dc for the excluded volume effect is reduced in the stiff limit of N... oo, p -> 0 with Np Const. > 1. For isolated linear chains, we find simple scaling forms for (R2) in d 2 and d 3 in both good and 0 solvents in this limit, with the scaling form for d 3 being that appropriate for Gaussian chains. For isolated branched polymers, with fixed, small branching probability ( p ), two relevant variables entering into (R2) coincide giving a simple scaling form for d 2 in 0 solvents (but not in good solvents). For both linear and branched polymers, the effect of stiffness seems stronger in a good solvent than in a 0 - solvent result somewhat surprising because the chains are more swollen in the former. In a highly polydispersed melt of linear polymers in a good solvent, for a typical chain, the marginal dimension dc does not seem to be reduced in the stiff limit. Acknowledgments. I am indebted to P. G. de Gennes for basic ideas that evolved into this work and to J. F. Joanny and M. Daoud for important comments correcting my misconceptions during the course of this work.
7 984 References [1] KRATKY, O. and POROD, G., Rec. Trav. Chim. Pays- Bas 68 (1949) [2] FLORY, P. J., Statistical Mechanics of Chain Molecules (Wiley, New York) [3] SCHAEFER, D. W., JOANNY, J. F. and PINCUS, P., Macromolecules 13 (1980) [4] AUSSERRE, D., HERVET, H. and RONDELEZ, F., J. Physique Lett. 46 (1985) L-929. [5] LEE, S. B. and NAKANISHI, H., Phys. Rev. B 33 (1986) [6] HALLEY, J. W., NAKANISHI, H. and SUNDARARA- JAN, R., Phys. Rev. B 31 (1985) 293. [7] FLORY, P. J., Principles of Polymer Chemistry, Chap. XII (Cornell University Press, Ithaca, New York) [8] FISHER, M. E., J. Phys. Soc. Japan 26 (1968) 44. [9] DE GENNES, P. G., Scaling Concepts in Polymer Physics, Chap. I (Cornell University Press, Ithaca, New York) [10] ODIJK, T. and HOUWAART, A. C., J. Polym. Sci. (Poly. Phys. Ed.) 16 (1978) 627. [11] STRALEY, J. P., Mol. Cryst. Liq. Cryst. 24 (1973) 7. [12] Writing fint/rd in terms of the monomer concentration 03A6 : fint/rd 03A6 (B203A6 + B303A ), and assuming that the series within the parentheses be of the form g(03a6 / 03A60) where 03A60 B-12 a-d is the close-packing concentration (independent of p), Ai+1/Ai Bi+1/pBi (03A60p)-1 we obtain ad/p. We thank J. F. JOANNY and M. DAOUD for illuminating discussions on this point. [13] DE GENNES, P. G., J. Physique Lett. 36 (1975) L-55 ; see also WILLIAMS, C., BROCHARD, F. and FRISCH, H. L., Ann. Rev. Phys. Chem. 32 (1981) 433. [14] The N-dependence in the expression for a good solvent was already given by J. ISAACSON and T. C. LUBENSKY, J. Physique Lett. 41 (1980) L- 469; for a 0398 solvent, by DAOUD, M. and JOANNY, J. F., J. Physique 42 (1981) [15] FAMILY, F. and DAOUD, M., Phys. Rev. B 29 (1984) 1506; see also DAOUD, M. and FAMILY, F., J. Physique 45 (1984) 151. [16] DAOUD, M. and CONIGLIO, A. (private communication). The general expression should be Nk-2z /Nk-1w for the term proportional to 03A6k so long as the polydispersity exponent 03C4 ~ 3. For the definition of this exponent, see, e.g., STAUF- FER, D., Phys. Rep. 54 (1979) 1. [17] We thank a referee for pointing out the constraint of d ~ 2 for the expression for 03A9 to make sense. In this case, this constraint actually eliminates a «window» of validity for a Flory expression.
Diffusion of a chain : concentration effects
Diffusion of a chain : concentration effects M. Daoud, G. Jannink To cite this version: M. Daoud, G. Jannink. Diffusion of a chain : concentration effects. Journal de Physique Lettres, 1980, 41 (9), pp.217220.
More informationDepletion layers in polymer solutions : influence of the chain persistence length
Depletion layers in polymer solutions : influence of the chain persistence length D. Ausserré, H. Hervet, F. Rondelez To cite this version: D. Ausserré, H. Hervet, F. Rondelez. Depletion layers in polymer
More informationConformation of confined macromolecular chains : cross-over between slit and capillary
Conformation of confined macromolecular chains : crossover between slit and capillary L. Turban To cite this version: L. Turban. Conformation of confined macromolecular chains : crossover between slit
More informationLight scattering by cooperative diffusion in semi-dilute polymer solutions
Light scattering by cooperative diffusion in semidilute polymer solutions M. Adam, M. Delsanti, G. Jannink To cite this version: M. Adam, M. Delsanti, G. Jannink. Light scattering by cooperative diffusion
More informationAdsorption of chain molecules with a polar head a scaling description
Adsorption of chain molecules with a polar head a scaling description S. Alexander To cite this version: S. Alexander. Adsorption of chain molecules with a polar head a scaling description. Journal de
More informationOn the nonrelativistic binding energy for positive ions
On the nonrelativistic binding energy for positive ions G.I. Plindov, I.K. Dmitrieva To cite this version: G.I. Plindov, I.K. Dmitrieva. On the nonrelativistic binding energy for positive ions. Journal
More informationIrregular wavy flow due to viscous stratification
Irregular wavy flow due to viscous stratification T. Shlang, G.I. Sivashinsky, A.J. Babchin, A.L. Frenkel To cite this version: T. Shlang, G.I. Sivashinsky, A.J. Babchin, A.L. Frenkel. Irregular wavy flow
More informationAccurate critical exponents from the ϵ-expansion
Accurate critical exponents from the ϵ-expansion J.C. Le Guillou, J. Zinn-Justin To cite this version: J.C. Le Guillou, J. Zinn-Justin. Accurate critical exponents from the ϵ-expansion. Journal de Physique
More informationCapillary rise between closely spaced plates : effect of Van der Waals forces
Capillary rise between closely spaced plates : effect of Van der Waals forces B. Legait, P.G. De Gennes To cite this version: B. Legait, P.G. De Gennes. Capillary rise between closely spaced plates : effect
More informationPolymer excluded volume exponent v : An experimental verification of the n vector model for n = 0
Polymer excluded volume exponent v : An experimental verification of the n vector model for n 0 J.P. Cotton To cite this version: J.P. Cotton. Polymer excluded volume exponent v : An experimental verification
More informationTemperature-concentration diagram of polymer solutions
Temperatureconcentration diagram of polymer solutions M. Daoud, G. Jannink To cite this version: M. Daoud, G. Jannink. Temperatureconcentration diagram of polymer solutions. Journal de Physique, 1976,
More informationDilution and polydispersity in branched polymers
Des J. Physique Lett. 45 (1984) L199 L203 ler MARS 1984, L199 Classification Physics Abstracts 36.20E 05.40 82.70G Dilution and polydispersity in branched polymers M. Daoud (*), F. Family (**) and G. Jannink
More informationTransfer matrix in one dimensional problems
Transfer matrix in one dimensional problems M. E. Mora, R. Pérez, Ch. B. Sommers To cite this version: M. E. Mora, R. Pérez, Ch. B. Sommers. Transfer matrix in one dimensional problems. Journal de Physique,
More informationA new simple recursive algorithm for finding prime numbers using Rosser s theorem
A new simple recursive algorithm for finding prime numbers using Rosser s theorem Rédoane Daoudi To cite this version: Rédoane Daoudi. A new simple recursive algorithm for finding prime numbers using Rosser
More informationA NON - CONVENTIONAL TYPE OF PERMANENT MAGNET BEARING
A NON - CONVENTIONAL TYPE OF PERMANENT MAGNET BEARING J.-P. Yonnet To cite this version: J.-P. Yonnet. A NON - CONVENTIONAL TYPE OF PERMANENT MAG- NET BEARING. Journal de Physique Colloques, 1985, 46 (C6),
More informationGENERALIZED OPTICAL BISTABILITY AND CHAOS IN A LASER WITH A SATURABLE ABSORBER
GENERALIZED OPTICAL BISTABILITY AND CHAOS IN A LASER WITH A SATURABLE ABSORBER E. Arimondo, F. De Tomasi, B. Zambon, F. Papoff, D. Hennequin To cite this version: E. Arimondo, F. De Tomasi, B. Zambon,
More informationAnisotropy dynamics of CuMn spin glass through torque measurements
Anisotropy dynamics of CuMn spin glass through torque measurements J.B. Pastora, T.W. Adair, D.P. Love To cite this version: J.B. Pastora, T.W. Adair, D.P. Love. Anisotropy dynamics of CuMn spin glass
More informationQuantum efficiency and metastable lifetime measurements in ruby ( Cr 3+ : Al2O3) via lock-in rate-window photothermal radiometry
Quantum efficiency and metastable lifetime measurements in ruby ( Cr 3+ : Al2O3) via lock-in rate-window photothermal radiometry A. Mandelis, Z. Chen, R. Bleiss To cite this version: A. Mandelis, Z. Chen,
More informationLandau-de Gennes free energy expansion for nematic polymers
Landau-de Gennes free energy expansion for nematic polymers V.V. Rusakov, M.I. Shliomis To cite this version: V.V. Rusakov, M.I. Shliomis. Landau-de Gennes free energy expansion for nematic polymers. Journal
More informationCase report on the article Water nanoelectrolysis: A simple model, Journal of Applied Physics (2017) 122,
Case report on the article Water nanoelectrolysis: A simple model, Journal of Applied Physics (2017) 122, 244902 Juan Olives, Zoubida Hammadi, Roger Morin, Laurent Lapena To cite this version: Juan Olives,
More informationMirage detection for electrochromic materials characterization. Application to iridium oxide films
Mirage detection for electrochromic materials characterization. Application to iridium oxide films V. Plichon, M. Petit To cite this version: V. Plichon, M. Petit. Mirage detection for electrochromic materials
More informationPossible long-range step interaction in 4He due to step oscillation
Possible longrange step interaction in 4He due to step oscillation M. Uwaha To cite this version: M. Uwaha. Possible longrange step interaction in 4He due to step oscillation. Journal de Physique, 1990,
More informationInterfacial interaction in PP/EPDM polymer blend studied by positron annihilation
Interfacial interaction in PP/EPDM polymer blend studied by positron annihilation Chen Wang, S. Wang, W. Zheng, Z. Qi To cite this version: Chen Wang, S. Wang, W. Zheng, Z. Qi. Interfacial interaction
More informationApplication of an aerodynamic code to marine propellers
Application of an aerodynamic code to marine propellers M. Schaffar, J. Haertig To cite this version: M. Schaffar, J. Haertig. Application of an aerodynamic code to marine propellers. Journal de Physique
More informationAN INTERNAL FRICTION PEAK DUE TO HYDROGEN-DISLOCATION INTERACTION IN NICKEL
AN INTERNAL FRICTION PEAK DUE TO HYDROGEN-DISLOCATION INTERACTION IN NICKEL K. Tanaka, T. Atsumi, M. Yamada To cite this version: K. Tanaka, T. Atsumi, M. Yamada. AN INTERNAL FRICTION PEAK DUE TO HYDROGEN-
More informationMethylation-associated PHOX2B gene silencing is a rare event in human neuroblastoma.
Methylation-associated PHOX2B gene silencing is a rare event in human neuroblastoma. Loïc De Pontual, Delphine Trochet, Franck Bourdeaut, Sophie Thomas, Heather Etchevers, Agnes Chompret, Véronique Minard,
More informationDispersion relation results for VCS at JLab
Dispersion relation results for VCS at JLab G. Laveissiere To cite this version: G. Laveissiere. Dispersion relation results for VCS at JLab. Compton Scattering from Low to High Momentum Transfer, Mar
More informationDetermination of absorption characteristic of materials on basis of sound intensity measurement
Determination of absorption characteristic of materials on basis of sound intensity measurement R. Prascevic, A. Milosevic, S. Cvetkovic To cite this version: R. Prascevic, A. Milosevic, S. Cvetkovic.
More informationRHEOLOGICAL INTERPRETATION OF RAYLEIGH DAMPING
RHEOLOGICAL INTERPRETATION OF RAYLEIGH DAMPING Jean-François Semblat To cite this version: Jean-François Semblat. RHEOLOGICAL INTERPRETATION OF RAYLEIGH DAMPING. Journal of Sound and Vibration, Elsevier,
More informationBehaviour of a Sm C disclination line s = + 1/2 with a partial twist under a magnetic field
Behaviour of a Sm C disclination line s = + 1/2 with a partial twist under a magnetic field H.P. Hinov To cite this version: H.P. Hinov. Behaviour of a Sm C disclination line s = + 1/2 with a partial twist
More informationInelastic scattering of 30 Mev polarized protons from 112Cd
Inelastic scattering of 30 Mev polarized protons from 112Cd R. De Swiniarski, G. Bagieu, DinhLien Pham, M. Massaad, J.Y. Grossiord, A. Guichard To cite this version: R. De Swiniarski, G. Bagieu, DinhLien
More informationLow-temperature state of Ising spin glass
Low-temperature state of Ising spin glass L.B. Ioffe, M.V. Feigel Man To cite this version: L.B. Ioffe, M.V. Feigel Man. Low-temperature state of Ising spin glass. Journal de Physique Lettres, 1983, 44
More informationBest linear unbiased prediction when error vector is correlated with other random vectors in the model
Best linear unbiased prediction when error vector is correlated with other random vectors in the model L.R. Schaeffer, C.R. Henderson To cite this version: L.R. Schaeffer, C.R. Henderson. Best linear unbiased
More informationHedgehog structure in nematic and magnetic systems
Hedgehog structure in nematic and magnetic systems N. Schopohl, T.J. Sluckin To cite this version: N. Schopohl, T.J. Sluckin. Hedgehog structure in nematic and magnetic systems. Journal de Physique, 1988,
More informationOn size, radius and minimum degree
On size, radius and minimum degree Simon Mukwembi To cite this version: Simon Mukwembi. On size, radius and minimum degree. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no.
More informationEFFECT OF THE ONE-DIMENSIONAL STRUCTURE ON THE ENERGY TRANSFER IN Li6Gd (BO3)3
EFFECT OF THE ONE-DIMENSIONAL STRUCTURE ON THE ENERGY TRANSFER IN Li6Gd (BO3)3 C. Garapon, B. Jacquier, Y. Salem, R. Moncorge To cite this version: C. Garapon, B. Jacquier, Y. Salem, R. Moncorge. EFFECT
More informationSound intensity as a function of sound insulation partition
Sound intensity as a function of sound insulation partition S. Cvetkovic, R. Prascevic To cite this version: S. Cvetkovic, R. Prascevic. Sound intensity as a function of sound insulation partition. Journal
More informationSmart Bolometer: Toward Monolithic Bolometer with Smart Functions
Smart Bolometer: Toward Monolithic Bolometer with Smart Functions Matthieu Denoual, Gilles Allègre, Patrick Attia, Olivier De Sagazan To cite this version: Matthieu Denoual, Gilles Allègre, Patrick Attia,
More informationImpulse response measurement of ultrasonic transducers
Impulse response measurement of ultrasonic transducers F. Kadlec To cite this version: F. Kadlec. Impulse response measurement of ultrasonic transducers. Journal de Physique IV Colloque, 1994, 04 (C5),
More informationConformation of branched polymers
Conformation of branched polymers M. Daoud, J.F. Joanny To cite this version: M. Daoud, J.F. Joanny. Conformation of branched polymers. Journal de Physique, 1981, 42 (10), pp.13591371. .
More informationEaster bracelets for years
Easter bracelets for 5700000 years Denis Roegel To cite this version: Denis Roegel. Easter bracelets for 5700000 years. [Research Report] 2014. HAL Id: hal-01009457 https://hal.inria.fr/hal-01009457
More informationATOMIC STRUCTURE OF INTERFACES IN GaAs/Ga1-xAlxAs SUPERLATTICES
ATOMIC STRUCTURE OF INTERFACES IN GaAs/Ga1-xAlxAs SUPERLATTICES J. Laval, C. Delamarre, A. Dubon, G. Schiffmacher, G. Teste de Sagey, B. Guenais, A. Regreny To cite this version: J. Laval, C. Delamarre,
More informationHIGH RESOLUTION ION KINETIC ENERGY ANALYSIS OF FIELD EMITTED IONS
HIGH RESOLUTION ION KINETIC ENERGY ANALYSIS OF FIELD EMITTED IONS J. Liu, T. Tsong To cite this version: J. Liu, T. Tsong. HIGH RESOLUTION ION KINETIC ENERGY ANALYSIS OF FIELD EMITTED IONS. Journal de
More informationCharacteristic numbers of polymers in dilute solution : a universal description of the cross-over effects
Characteristic numbers of polymers in dilute solution : a universal description of the crossover effects A. Dondos To cite this version: A. Dondos. Characteristic numbers of polymers in dilute solution
More informationFormation of microdomains in a quenched disordered heteropolymer
Formation of microdomains in a quenched disordered heteropolymer E.I. Shakhnovich, A.M. Gutin To cite this version: E.I. Shakhnovich, A.M. Gutin. Formation of microdomains in a quenched disordered heteropolymer.
More informationThe fractal nature of a diffusion front and the relation to percolation
The fractal nature of a diffusion front and the relation to percolation Bernard Sapoval, Michel Rosso, JeanFrançois Gouyet To cite this version: Bernard Sapoval, Michel Rosso, JeanFrançois Gouyet. The
More informationTHEORETICAL ANALYSIS OF THE TWO-TERMINAL MOS CAPACITOR ON SOI SUBSTRATE
THEORETICAL ANALYSIS OF THE TWO-TERMINAL MOS CAPACITOR ON SOI SUBSTRATE P. Paelinck, D. Flandre, A. Terao, F. Van de Wiele To cite this version: P. Paelinck, D. Flandre, A. Terao, F. Van de Wiele. THEORETICAL
More informationLAWS OF CRYSTAL-FIELD DISORDERNESS OF Ln3+ IONS IN INSULATING LASER CRYSTALS
LAWS OF CRYSTAL-FIELD DISORDERNESS OF Ln3+ IONS IN INSULATING LASER CRYSTALS A. Kaminskii To cite this version: A. Kaminskii. LAWS OF CRYSTAL-FIELD DISORDERNESS OF Ln3+ IONS IN INSULAT- ING LASER CRYSTALS.
More informationFORMAL TREATMENT OF RADIATION FIELD FLUCTUATIONS IN VACUUM
FORMAL TREATMENT OF RADIATION FIELD FLUCTUATIONS IN VACUUM Frederic Schuller, Renaud Savalle, Michael Neumann-Spallart To cite this version: Frederic Schuller, Renaud Savalle, Michael Neumann-Spallart.
More informationSolving the neutron slowing down equation
Solving the neutron slowing down equation Bertrand Mercier, Jinghan Peng To cite this version: Bertrand Mercier, Jinghan Peng. Solving the neutron slowing down equation. 2014. HAL Id: hal-01081772
More informationThe FLRW cosmological model revisited: relation of the local time with th e local curvature and consequences on the Heisenberg uncertainty principle
The FLRW cosmological model revisited: relation of the local time with th e local curvature and consequences on the Heisenberg uncertainty principle Nathalie Olivi-Tran, Paul M Gauthier To cite this version:
More informationStickelberger s congruences for absolute norms of relative discriminants
Stickelberger s congruences for absolute norms of relative discriminants Georges Gras To cite this version: Georges Gras. Stickelberger s congruences for absolute norms of relative discriminants. Journal
More information0.9 ev POTENTIAL BARRIER SCHOTTKY DIODE ON ev GAP GaxIn1-xASSi:H
0.9 ev POTENTIAL BARRIER SCHOTTKY DIODE ON 0.75-0.5 ev GAP GaxIn1-xASSi:H A. Deneuville, F. Valentin, S. Belkouch To cite this version: A. Deneuville, F. Valentin, S. Belkouch. 0.9 ev POTENTIAL BARRIER
More informationL institution sportive : rêve et illusion
L institution sportive : rêve et illusion Hafsi Bedhioufi, Sida Ayachi, Imen Ben Amar To cite this version: Hafsi Bedhioufi, Sida Ayachi, Imen Ben Amar. L institution sportive : rêve et illusion. Revue
More informationPasserelle entre les arts : la sculpture sonore
Passerelle entre les arts : la sculpture sonore Anaïs Rolez To cite this version: Anaïs Rolez. Passerelle entre les arts : la sculpture sonore. Article destiné à l origine à la Revue de l Institut National
More informationCr3+, Nd3+ multisites, pairs and energy transfer processes in laser crystal YAlO3
Cr3+, Nd3+ multisites, pairs and energy transfer processes in laser crystal YAlO3 J. Mares, G. Boulon, A. Brenier, L. Lou, S. Rotman, H. Luria To cite this version: J. Mares, G. Boulon, A. Brenier, L.
More informationRENORMALISATION ON THE PENROSE LATTICE
RENORMALISATION ON THE PENROSE LATTICE C. Godreche, Henri Orland To cite this version: C. Godreche, Henri Orland. RENORMALISATION ON THE PENROSE LATTICE. Journal de Physique Colloques, 1986, 47 (C3), pp.c3-197-c3-203.
More informationEvolution of the cooperation and consequences of a decrease in plant diversity on the root symbiont diversity
Evolution of the cooperation and consequences of a decrease in plant diversity on the root symbiont diversity Marie Duhamel To cite this version: Marie Duhamel. Evolution of the cooperation and consequences
More informationComment on: Sadi Carnot on Carnot s theorem.
Comment on: Sadi Carnot on Carnot s theorem. Jacques Arnaud, Laurent Chusseau, Fabrice Philippe To cite this version: Jacques Arnaud, Laurent Chusseau, Fabrice Philippe. Comment on: Sadi Carnot on Carnot
More informationStructural study of a rare earth-rich aluminoborosilicate glass containing various alkali and alkaline-earth modifier cations
Structural study of a rare earth-rich aluminoborosilicate glass containing various alkali and alkaline-earth modifier cations Arnaud Quintas, Daniel Caurant, Odile Majérus, Marion Lenoir, Jean-Luc Dussossoy,
More informationThe Core of a coalitional exchange economy
The Core of a coalitional exchange economy Elena L. Del Mercato To cite this version: Elena L. Del Mercato. The Core of a coalitional exchange economy. Cahiers de la Maison des Sciences Economiques 2006.47
More informationPolymer adsorption on colloidal particles
Polymer adsorption on colloidal particles P.A. Pincus, C.J. Sandroff, T.A. Witten To cite this version: P.A. Pincus, C.J. Sandroff, T.A. Witten. Polymer adsorption on colloidal particles. Journal de Physique,
More informationA new approach of the concept of prime number
A new approach of the concept of prime number Jamel Ghannouchi To cite this version: Jamel Ghannouchi. A new approach of the concept of prime number. 4 pages. 24. HAL Id: hal-3943 https://hal.archives-ouvertes.fr/hal-3943
More informationA MAGNETOSTATIC CALCULATION OF FRINGING FIELD FOR THE ROGOWSKI POLE BOUNDARY WITH FLOATING SNAKE
A MAGNETOSTATIC CALCULATION OF FRINGING FIELD FOR THE ROGOWSKI POLE BOUNDARY WITH FLOATING SNAKE Yan Chen, Fan Ming-Wu To cite this version: Yan Chen, Fan Ming-Wu. A MAGNETOSTATIC CALCULATION OF FRINGING
More informationNon-linear behaviour of gels under strong deformations
Nonlinear behaviour of gels under strong deformations S. Daoudi To cite this version: S. Daoudi. Nonlinear behaviour of gels under strong deformations. Journal de Physique, 1977, 38 (10), pp.13011305.
More informationThe dynamics of entangled polymers
dynamics of entangled polymers J.M. Deutsch To cite this version: J.M. Deutsch. dynamics of entangled polymers. Journal de Physique, 1987, 48 (1), pp.141150. .
More informationCompleteness of the Tree System for Propositional Classical Logic
Completeness of the Tree System for Propositional Classical Logic Shahid Rahman To cite this version: Shahid Rahman. Completeness of the Tree System for Propositional Classical Logic. Licence. France.
More informationParticle-in-cell simulations of high energy electron production by intense laser pulses in underdense plasmas
Particle-in-cell simulations of high energy electron production by intense laser pulses in underdense plasmas Susumu Kato, Eisuke Miura, Mitsumori Tanimoto, Masahiro Adachi, Kazuyoshi Koyama To cite this
More informationEarly detection of thermal contrast in pulsed stimulated thermography
Early detection of thermal contrast in pulsed stimulated thermography J.-C. Krapez, F. Lepoutre, D. Balageas To cite this version: J.-C. Krapez, F. Lepoutre, D. Balageas. Early detection of thermal contrast
More informationHook lengths and shifted parts of partitions
Hook lengths and shifted parts of partitions Guo-Niu Han To cite this version: Guo-Niu Han Hook lengths and shifted parts of partitions The Ramanujan Journal, 009, 9 p HAL Id: hal-00395690
More informationEntropies and fractal dimensions
Entropies and fractal dimensions Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Entropies and fractal dimensions. Philica, Philica, 2016. HAL Id: hal-01377975
More informationEFFECT OF TIP-SIZE ON STM IMAGES OF GRAPHITE
EFFECT OF TIP-SIZE ON STM IMAGES OF GRAPHITE C. Horie, H. Miyazaki To cite this version: C. Horie, H. Miyazaki. EFFECT OF TIP-SIZE ON STM IMAGES OF GRAPHITE. Journal de Physique Colloques, 1987, 48 (C6),
More informationEffect of thermal noise and initial conditions in the dynamics of a mean-field ferromagnet
Effect of thermal noise and initial conditions in the dynamics of a mean-field ferromagnet O. Golinelli, B. Derrida To cite this version: O. Golinelli, B. Derrida. Effect of thermal noise and initial conditions
More informationMean field theory for Heisenberg spin glasses
Mean field theory for Heisenberg spin glasses G. Toulouse, M. Gabay To cite this version: G. Toulouse, M. Gabay. Mean field theory for Heisenberg spin glasses. Journal de Physique Lettres, 1981, 42 (5),
More informationPhase and microphase separation in polymer blends
Phase and microphase separation in polymer blends M. Daoud To cite this version: M. Daoud. Phase and microphase separation in polymer blends. Journal de Physique IV Colloque, 1993, 03 (C1), pp.c1-211-c1-223.
More informationAnalysis of Boyer and Moore s MJRTY algorithm
Analysis of Boyer and Moore s MJRTY algorithm Laurent Alonso, Edward M. Reingold To cite this version: Laurent Alonso, Edward M. Reingold. Analysis of Boyer and Moore s MJRTY algorithm. Information Processing
More informationSoundness of the System of Semantic Trees for Classical Logic based on Fitting and Smullyan
Soundness of the System of Semantic Trees for Classical Logic based on Fitting and Smullyan Shahid Rahman To cite this version: Shahid Rahman. Soundness of the System of Semantic Trees for Classical Logic
More informationFull-order observers for linear systems with unknown inputs
Full-order observers for linear systems with unknown inputs Mohamed Darouach, Michel Zasadzinski, Shi Jie Xu To cite this version: Mohamed Darouach, Michel Zasadzinski, Shi Jie Xu. Full-order observers
More informationFinite element computation of leaky modes in straight and helical elastic waveguides
Finite element computation of leaky modes in straight and helical elastic waveguides Khac-Long Nguyen, Fabien Treyssede, Christophe Hazard, Anne-Sophie Bonnet-Ben Dhia To cite this version: Khac-Long Nguyen,
More informationThomas Lugand. To cite this version: HAL Id: tel
Contribution à la Modélisation et à l Optimisation de la Machine Asynchrone Double Alimentation pour des Applications Hydrauliques de Pompage Turbinage Thomas Lugand To cite this version: Thomas Lugand.
More informationVibro-acoustic simulation of a car window
Vibro-acoustic simulation of a car window Christophe Barras To cite this version: Christophe Barras. Vibro-acoustic simulation of a car window. Société Française d Acoustique. Acoustics 12, Apr 12, Nantes,
More informationELASTIC WAVE PROPAGATION IN THREE-DIMENSIONAL PERIODIC COMPOSITE MATERIALS
ELASTIC WAVE PROPAGATION IN THREE-DIMENSIONAL PERIODIC COMPOSITE MATERIALS B. Auld, Y. Shui, Y. Wang To cite this version: B. Auld, Y. Shui, Y. Wang. ELASTIC WAVE PROPAGATION IN THREE-DIMENSIONAL PERI-
More informationELASTIC PROPERTIES OF THE ONE-DIMENSIONAL METAL Mo2 S3
ELASTIC PROPERTIES OF THE ONE-DIMENSIONAL METAL Mo2 S3 Alova, G. Mozurkewich To cite this version: Alova, G. Mozurkewich. ELASTIC PROPERTIES OF THE ONE-DIMENSIONAL METAL Mo2 S3. Journal de Physique Colloques,
More informationNEGATIVE ION IMAGING IN FIELD ION MICROSCOPY
NEGATIVE ION IMAGING IN FIELD ION MICROSCOPY R. Schmitz, L. Bütfering, F. Röllgen To cite this version: R. Schmitz, L. Bütfering, F. Röllgen. NEGATIVE ION IMAGING IN FIELD ION MICROSCOPY. Journal de Physique
More informationNumerical modification of atmospheric models to include the feedback of oceanic currents on air-sea fluxes in ocean-atmosphere coupled models
Numerical modification of atmospheric models to include the feedback of oceanic currents on air-sea fluxes in ocean-atmosphere coupled models Florian Lemarié To cite this version: Florian Lemarié. Numerical
More informationb-chromatic number of cacti
b-chromatic number of cacti Victor Campos, Claudia Linhares Sales, Frédéric Maffray, Ana Silva To cite this version: Victor Campos, Claudia Linhares Sales, Frédéric Maffray, Ana Silva. b-chromatic number
More informationCan we reduce health inequalities? An analysis of the English strategy ( )
Can we reduce health inequalities? An analysis of the English strategy (1997-2010) Johan P Mackenbach To cite this version: Johan P Mackenbach. Can we reduce health inequalities? An analysis of the English
More informationLaser induced fluorescence of MgO
Laser induced fluorescence of MgO G. Taïeb To cite this version: G. Taïeb. Laser induced fluorescence of MgO. Journal de Physique, 1981, 42 (4), pp.537540. .
More informationA Simple Proof of P versus NP
A Simple Proof of P versus NP Frank Vega To cite this version: Frank Vega. A Simple Proof of P versus NP. 2016. HAL Id: hal-01281254 https://hal.archives-ouvertes.fr/hal-01281254 Submitted
More informationThe generation of the Biot s slow wave at a fluid-porous solid interface. The influence of impedance mismatch
The generation of the Biot s slow wave at a fluid-porous solid interface. The influence of impedance mismatch T. Gómez-Alvarez Arenas, E. Riera Franco de Sarabia To cite this version: T. Gómez-Alvarez
More informationStress Dependency on the Ultrasonic Wave Velocity and Attenuation of Fe-C System
Stress Dependency on the Ultrasonic Wave Velocity and Attenuation of Fe-C System S. Takahashi, K. Takahashi To cite this version: S. Takahashi, K. Takahashi. Stress Dependency on the Ultrasonic Wave Velocity
More informationPrompt Photon Production in p-a Collisions at LHC and the Extraction of Gluon Shadowing
Prompt Photon Production in p-a Collisions at LHC and the Extraction of Gluon Shadowing F. Arleo, T. Gousset To cite this version: F. Arleo, T. Gousset. Prompt Photon Production in p-a Collisions at LHC
More informationOn infinite permutations
On infinite permutations Dmitri G. Fon-Der-Flaass, Anna E. Frid To cite this version: Dmitri G. Fon-Der-Flaass, Anna E. Frid. On infinite permutations. Stefan Felsner. 2005 European Conference on Combinatorics,
More informationA non-commutative algorithm for multiplying (7 7) matrices using 250 multiplications
A non-commutative algorithm for multiplying (7 7) matrices using 250 multiplications Alexandre Sedoglavic To cite this version: Alexandre Sedoglavic. A non-commutative algorithm for multiplying (7 7) matrices
More informationThe Mahler measure of trinomials of height 1
The Mahler measure of trinomials of height 1 Valérie Flammang To cite this version: Valérie Flammang. The Mahler measure of trinomials of height 1. Journal of the Australian Mathematical Society 14 9 pp.1-4.
More informationA SAXS STUDY OF SILICA AEROGELS
A SAXS STUDY OF SILICA AEROGELS A. Boukenter, D. Champagnon, J. Dumas, E. Duval, J. Quinson, J. Rousset, J. Serughetti, S. Etienne, C. Mai To cite this version: A. Boukenter, D. Champagnon, J. Dumas, E.
More informationEffect of the different parts of the interaction potential on the line profile in neutral atom-atom collisions
Effect of the different parts of the interaction potential on the line profile in neutral atomatom collisions N.F. Allard, Y.G. Biraud To cite this version: N.F. Allard, Y.G. Biraud. Effect of the different
More informationFrom Unstructured 3D Point Clouds to Structured Knowledge - A Semantics Approach
From Unstructured 3D Point Clouds to Structured Knowledge - A Semantics Approach Christophe Cruz, Helmi Ben Hmida, Frank Boochs, Christophe Nicolle To cite this version: Christophe Cruz, Helmi Ben Hmida,
More informationStatic and dynamic polystyrene dimensions in relation to the blob model and size-exclusion chromatography
Static and dynamic polystyrene dimensions in relation to the blob model and sizeexclusion chromatography Z. Gallot, G. Pouyet, Anne Schmitt To cite this version: Z. Gallot, G. Pouyet, Anne Schmitt. Static
More informationThe influence of the global atmospheric properties on the detection of UHECR by EUSO on board of the ISS
The influence of the global atmospheric properties on the detection of UHECR by EUSO on board of the ISS C. Berat, D. Lebrun, A. Stutz, E. Plagnol To cite this version: C. Berat, D. Lebrun, A. Stutz, E.
More information