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CU-Boulder PHYS 7450 - Effective interactions in soft condensed matter physics

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EFFECTIVE INTERACTIONS IN SOFTCONDENSED MATTER PHYSICSChristos N. LIKOSInstitut f uK r Theoretische Physik II. Heinrich-Heine-UniversitaK tDuK sseldorf, UniversitaK tsstra}e1,D-40225 DuK sseldorf, GermanyAMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYOC.N. Likos / Physics Reports 348 (2001) 267} 439 267* Tel.: #49-211-811-3699; fax: #49-211-811-2262.E-mail address: [email protected] (C. N. Likos).Physics Reports 348 (2001) 267}439E!ective interactions in soft condensed matter physicsChristos N. Likos*Institut fu( r Theoretische Physik II, Heinrich-Heine-Universita( tDu( sseldorf, Universita( tsstra}e1,D-40225 Du( sseldorf, GermanyReceived September 2000; editor: M.L. KleinContents1. Introduction 2702. Soft matter 2722.1. What is soft matter 2722.2. Stabilization of colloidal suspensions 2762.3. The e!ective Hamiltonian 2822.4. Structure and thermodynamics ofone-component systems 2892.5. The importance of the volume terms 2942.6. Concluding remarks 2983. Polymer chains 2993.1. Conformations of single chains 2993.2. The e!ects of the solvent 3043.3. Polymer solutions 3073.4. E!ective interactions betweenpolymer chains 3143.5. The Gaussian core model 3213.6. The peculiarities of bounded inter-actions 3413.7. The volume terms for polymersolutions 3423.8. Concluding remarks 3434. Depletion interactions 3444.1. Hard spheres 3444.2. The depletion mechanism 3464.3. The depletion potential:quantitative results 3484.4. Tuning the attractions 3534.5. Solid-to-solid isostructuraltransitions 3574.6. Concluding remarks 3675. Star polymers 3685.1. General considerations 3685.2. Length scales of a star 3705.3. Conformations of a single star 3725.4. Scattering methods and the formfactor 3775.5. Concentration e!ects 3785.6. E!ective interactions betweenstar polymers 3815.7. Anomalous structure factor 3895.8. The phase diagram of star poly-mer solutions 3945.9. The three-body forces andthe volume terms 3975.10. Polydisperse stars 3995.11. Concluding remarks 4026. Current work 4026.1. Star polymer}colloidal mixtures 4030370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 0 - 1 573(00)00141-16.2. Polyelectrolyte stars 4076.3. Amphoteric polyelectrolytes 4086.4. Columnar assemblies of DNAmolecules 4117. Summary and conclusions 411Acknowledgements 412Note added in proof 413Appendix A. Elastic constants from thetheory of the harmonic solid 413Appendix B. The penetrable spheres model 415Appendix C. Clustering or reentrantmelting? An exact result 419Appendix D. Integral equation theories formulticomponent mixtures 426Appendix E. Star polymer solutions in theneighborhood of the  point 427References 430AbstractIn this work, we present a review of recently achieved progress in the "eld of soft condensed matter physics,and in particular on the study of the static properties of solutions or suspensions of colloidal particles. Thelatter are macromolecular entities with typical sizes ranging from 1 nm to 1 m and their suspensiontypically contain, in addition to the solvent, smaller components such as salt ions or free polymer chains. Thetheoretical tool introduced is the e!ective Hamiltonian which formally results by a canonical trace over thesmaller degrees of freedom for a "xed, `frozena con"guration of the large ones. After presenting the formalde"nitions of this e!ective Hamiltonian, we proceed with the applications to some common soft mattersystems having a variable softness and ranging from free polymer chains to hard colloidal particles. We beginfrom the extreme case of nondiverging e!ective interactions between ultrasoft polymer chains and derive anexact criterion to determine the topology of the phase diagrams of such systems. We use star polymers witha variable arm number f as a hybrid system in order to interpolate between these two extremes. By derivingan e!ective interaction between stars we can monitor the change in the phase behavior of a system as thesteepness of the repulsion between its constituent particles increases. We also review recent results on thenature and the e!ects of short-range attractions on the phase diagrams of spherical, nonoverlapping colloidalparticles.  2001 Elsevier Science B.V. All rights reserved.PACS: 82.70.Dd; 61.25.Hq; 61.20.!p; 64.70.DvKeywords: Soft matter; Colloids; Polymers; Liquid structureC.N. Likos / Physics Reports 348 (2001) 267}439 2691. IntroductionA common state of soft matter systems in room temperature and pressure is the liquid one anda great deal of this work deals with liquids. Even when solids are examined, the liquid state ofmatter is somehow always present, as in the modern approaches solids are considered as spatiallyinhomogeneous (periodically modulated) #uids. Hence, a theoretical understanding the structureand properties of #uids is of great importance in soft matter physics. Among the three states ofmatter, the liquid state was the last to be quantitatively analyzed and understood by the tools ofmodern physics. The nature of the gaseous phase has already been examined since the 19th centuryand the progress in understanding the physics of crystalline solids has experienced an explosionsince the middle of the 20th century, facilitated by the advances in quantum mechanics and theapplications of symmetry principles arising from lattice periodicity. On the contrary, the liquidphase is a peculiar state of matter, whereupon the system is on the one hand translationallyinvariant, in close similarity to the gases, but on the other hand the typically high densitiesencountered in this phase give rise to nontrivial short- and sometimes long-range correlationsbetween the constituent particles which are very di$cult to handle analytically. To paraphraseVictor Weisskopf [1]: if the world's most brilliant theoretical physicists were gathered in a desertisland, equipped with the laws of physics but otherwise isolated from the external world and wereasked to guess the forms of matter, then they would pretty soon come up with predicting theexistence of atoms, molecules and a bit later of the ideal gases. After some time they would realizethat atoms can also self-organize in periodic arrangements, thus discovering the crystalline solids.But they would probably never come up with the liquids } although in their island they would besurrounded by a large mass of the most abundant liquid in our planet!What is it that makes liquids so di$cult to study? For one thing, the theories of


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