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Solution Properties of Branched Macromolecules

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Book cover Branched Polymers II

Part of the book series: Advances in Polymer Science ((POLYMER,volume 143))

Abstract

Dilute and semi-dilute solution properties of several classes of branched macromolecules are outlined and discussed. The dilute solution properties are needed for a control of the chemical synthesis. The molecular parameters also determine the overlap concentration which is an essential quantity for description of the semi-dilute state. This state is represented by a multi-particle, highly entangled ensemble that exhibits certain similarities to the corresponding bulk systems. Because of the rich versatility in branching the present contribution made a selection and deals specifically with the two extremes of regularly branched polymers, on the one hand, and the randomly branched macromolecules on the other. Some properties of hyperbranched chains are included, whereas the many examples of slight deviations from regularity are mentioned only in passing. The treatment of the two extremes demonstrates the complexity to be expected in the general case of less organized but non-randomly branched systems. However, it also discloses certain common features.

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References

  1. Flory PJ ( 1953) Principles of polymer chemistry. Cornell University Press, Ithaca, NY

    Google Scholar 

  2. Flory PJ ( 1969) Statistical mechanics of chain molecules. Wiley, New York

    Google Scholar 

  3. Yamakawa H ( 1971) Modern theory of polymer solutions. Harper and Row, Evanstone, Il

    Google Scholar 

  4. De Gennes P-G (1979) Scaling concepts in polymer physics, Cornell University Press, Ithaca, NY

    Google Scholar 

  5. Doi M, Edwards SF ( 1986) The theory of polymer dynamics. Clarendon, Oxford

    Google Scholar 

  6. Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New York

    Google Scholar 

  7. Stauffer D (1985) Introduction to percolation theory. Taylor & Francis, London

    Google Scholar 

  8. Freire JJ (1998) Chapter in this book; there a list of relevant literature on the techniques of simulation

    Google Scholar 

  9. Burchard W (1988) Makromol Chem Macromol Symp 18:1

    CAS  Google Scholar 

  10. Burchard W (1993) TRIP 1:192

    CAS  Google Scholar 

  11. Burchard W (1966) Adv Colloid Interface Sci. 64:45

    Article  Google Scholar 

  12. Flory PJ (1941) J Am Chem Soc 63:3083

    Article  CAS  Google Scholar 

  13. Flory PJ (1946) Chem Revs 39:137

    Article  CAS  Google Scholar 

  14. Stockmayer WH (1943) J Chem Phys 11:45

    Article  CAS  Google Scholar 

  15. See textbooks on polymer science, eg [1] and more recent ones

    Google Scholar 

  16. Erlander SR, French D (1956) J Polymer Sci 20:7

    CAS  Google Scholar 

  17. Burchard W (1972) Macromolecules 5:604

    Article  CAS  Google Scholar 

  18. Daoud M, Cotton J-P, Farnoux B, Jannink G, Sarma G, Benoit H, Duplessix R, Picot C, de Gennes P-G (1975) Macromolecules 8:804

    Article  CAS  Google Scholar 

  19. Burchard W, Lang P, Schulz L ( 1992) Makomol Chem, Macromol Symp 58:21

    CAS  Google Scholar 

  20. Willner L, Jucknischke O, Richter D, Roovers J, Zhou L-L, Toporowski PM, Fetters LJ, Huang JS, Lin MY, Hadjichristidis N ( 1994) Macromolecules 27:3821

    Article  CAS  Google Scholar 

  21. Allegra G, Colombe E, Ganazzoli F (1993) Macromolecules 26:330

    Article  CAS  Google Scholar 

  22. Benoit H, Joanny JF, Hadziioannou G, Hammouda B (1993) Macromolecules 26:5790

    Article  CAS  Google Scholar 

  23. Alessandrini JL, Carignano MA (1992) Macromolecules 25:1157

    Article  CAS  Google Scholar 

  24. Burchard W (1983) Adv Polym Sci 48:1

    Article  CAS  Google Scholar 

  25. Roovers J, Zhou L-L, Toporowski PM, van der Zwan M, Iatrou H, Hadjichristidis N (1993) Macromolecules 26:4324

    Article  CAS  Google Scholar 

  26. Bywater S (1979) Adv Polym Sci 30:89

    Article  CAS  Google Scholar 

  27. Cherayil BJ, Bawondi MG, Miyake A, Freed KF ( 1986) Macromolecules 19:2770

    Article  CAS  Google Scholar 

  28. Douglas JF, Roovers J, Freed KF (1990) Macromolecules 23:3168

    Article  Google Scholar 

  29. Daoud M, Cotton J-P (1982) J Physique (Paris) 43:531

    CAS  Google Scholar 

  30. Huber K, Burchard W, Bantle S, Fetters LJ (1987) Polymer 28:1997

    Article  CAS  Google Scholar 

  31. Richter D, Farago B, Huang JS, Fetters LJ, Even B ( 1989) Macromolecules 22:468

    Article  CAS  Google Scholar 

  32. Douglas JF, Freed KF (1984) Macromolecules 17:7461

    Google Scholar 

  33. Freed KF, Wang Q-S, Roovers J, Douglas JF (1988) Macromolecules 21:2219

    Article  CAS  Google Scholar 

  34. Horton JG, Squires GL, Boothoyd AT, Fetters LJ, Rennie AR, Glinka CJ, Robinson RA (1989) Macromolecules 22:681

    Article  CAS  Google Scholar 

  35. Grest GS, Fetters LJ, Huang JS, Richter D ( 1996) Adv Chem Phys 94:67

    Article  CAS  Google Scholar 

  36. Ganazzoli F, Allegra G (1990) Macromolecules 23:262

    Article  CAS  Google Scholar 

  37. Ganazzoli F, Fontelos MA, Allegra G (1993) Polymer 34:2615

    Article  CAS  Google Scholar 

  38. Burchard W, Kajiwara K, Nerger D (1982) J Polym Sci Phys Ed 20:157

    Article  CAS  Google Scholar 

  39. Voit BI( 1995) Acta Polym 46:87

    Google Scholar 

  40. Tomalia DA, Durst HD (1993) Top Curr Chem 165:194

    Google Scholar 

  41. Tomalia DA, Naylor A, Goddard W (1990) Angew Chem In Ed 29:138

    Article  Google Scholar 

  42. Tomalia DA, Hedstrand DM, Wilson LR (1990) In: Encyclopedia of polymer science and technology. Wiley, New York, pp 46–92

    Google Scholar 

  43. Mekelburger H-B, Jaworek W, Vögtle F (1992) Angew ChemInt Ed 31:1571

    Article  Google Scholar 

  44. Newkome GR, Morrefield CN, Baker GR (12992) Aldichim Acta 25:31

    Google Scholar 

  45. Fréchet JMJ (1994) Science 263:1710

    Article  Google Scholar 

  46. Wooley KL, Hawker CJ, Fréchet JMJ (1991) J Am Chem Soc 113:4252

    Article  CAS  Google Scholar 

  47. Wooley KL, Hawkers CJ, Fréchet JMJ ( 1994) Angew Chem Int Ed 33:82

    Article  Google Scholar 

  48. Stockmayer WH (1944) J Chem Phys 12:125

    Article  CAS  Google Scholar 

  49. Zimm BH, Stockmayer WH (1949) J Chem Phys 17:1301

    Article  CAS  Google Scholar 

  50. Kajiwara K, Burchard W, Gordon M (1970) Brit Polymer J 2:110

    Article  CAS  Google Scholar 

  51. Burchard W (1972) Macromolecules 5:604

    Article  CAS  Google Scholar 

  52. Burchard W (1977) Macromolecules 10:919

    Article  CAS  Google Scholar 

  53. Gordon M (1962) Proc Roy Soc (London) 51:240

    Google Scholar 

  54. Gordon M, Scantlebury GR (1964) Trans Fard Soc 60:604

    Article  CAS  Google Scholar 

  55. Isaacson J, Lubensky TC (1980) J Physique (Paris) 41:L469

    Google Scholar 

  56. Boris D, Rubinstein M ( 1996) Macromolecules 29:7251

    Article  CAS  Google Scholar 

  57. Huglin MB ( 1972) Light scattering from polymer solutions. Academic Press, London

    Google Scholar 

  58. Glatter O, Kratky O (1982) Small angle X-ray scattering. Academic Press, London

    Google Scholar 

  59. Higgins JS, Benoit H ( 1994) Polymers and neutron scattering. Clarendon Press, Oxford

    Google Scholar 

  60. Berne BJ, Pecora R (1976) Dynamic light scattering. Wiley, New York

    Google Scholar 

  61. Siegert AJF (1943) MIT Rad Lab Report 465; see also [60]

    Google Scholar 

  62. Flammer J, Ricka J ( 1997) Apl Optics 36:508

    Article  Google Scholar 

  63. Burchard W, Schmidt M, Stockmayer W (1980) Macromolecules 13:1265

    Article  CAS  Google Scholar 

  64. Schmidt M, Stockmayer WH (1984) Macromolecules 17:509

    Article  CAS  Google Scholar 

  65. Burchard W, Schmidt M, Nerger D (1979) Polymer 20:582

    Article  Google Scholar 

  66. Einstein Q (1905) Ann Phys 17:549

    Article  CAS  Google Scholar 

  67. Einstein A (1906) Ann Phys 19:371

    Article  CAS  Google Scholar 

  68. Einstein A (1906) Ann Phys 19:289

    Article  CAS  Google Scholar 

  69. Einstein A (1911) Ann Phys 34:591

    Article  CAS  Google Scholar 

  70. Flory PJ, Fox TG ( 1951 ) J Am Chem Soc 73:1904

    Article  CAS  Google Scholar 

  71. Krigbaum WR, Carpenter DK (1955) J Phys Chem 59:1166

    Article  CAS  Google Scholar 

  72. McMillan WG, Mayer JE (1945) J Chem Phys 13:276

    Article  CAS  Google Scholar 

  73. Oseen CW (1927) Hydrodynamik. Akademie Verlag, Leipzig

    Google Scholar 

  74. McQuarrie DA ( 1973) Statistical mechanics. Harper and Row, New York

    Google Scholar 

  75. Friedman HL (1985) A course in statistical mechanics. Prentice Hall, Englewood, NJ

    Google Scholar 

  76. Lang P, Burchard W, Wolfe MS, Spinelli HJ, Page L (1991) Macromolecules 24:1306

    Article  CAS  Google Scholar 

  77. Roovers J, Hajichristidis N, Fetters LJ (1983) Macromolecules 16:214

    Article  CAS  Google Scholar 

  78. Douglas JF, Freed KF (1984) Macromoleules 17:1854

    Article  CAS  Google Scholar 

  79. Schaefgen JR, Flory PJ ( 1948) J Am Chem Soc 70:2709

    Article  CAS  Google Scholar 

  80. Schulz GV (1939) Z Phys Chem B39:25

    Google Scholar 

  81. Zimm BH (1948) J Chem Phys 16:1099

    Article  CAS  Google Scholar 

  82. Feller W (1957) An introduction to probability theory and its applications, 3rd edn. Wiley, New York

    Google Scholar 

  83. Burchard W( 1974) Macromolecules 7:841

    Google Scholar 

  84. Benoit H (1953) J Polymer Sci 11:507

    Article  CAS  Google Scholar 

  85. Freire JJ, Pla J, Rey A, Prats R (1986) Macromolecules 19:452

    Article  CAS  Google Scholar 

  86. Freire JJ, Rey A, Garcia de la Torre J ( 1986) Macromolecules 19:457

    Article  CAS  Google Scholar 

  87. Rey A, Freire JJ, Garcia de la Torre J (1987) Macromolecules 20:342

    Article  CAS  Google Scholar 

  88. Kirkwood JG, Riseman J (1948) J Chem Phys 16:565

    Article  CAS  Google Scholar 

  89. Huber K, Burchard W, Fetters LJ (1984) Macromolecules 17:54

    Article  Google Scholar 

  90. Douglas JF, Freed KF (1984) Macromolecules 17:1854; 2344

    Article  CAS  Google Scholar 

  91. Casassa EF (1962) J Phys Chem 37:2176

    Article  CAS  Google Scholar 

  92. Trappe V, Bauer J, Weissmüller M, Burchard W ( 1997) Macromolecules 30:2365

    Article  CAS  Google Scholar 

  93. Trappe V, Burchard W (1994) Polymer Preprints (Am Chem Soc, Div Polymer Chem) 36:1

    Google Scholar 

  94. Weissmüller M, Burchard W (1995) Macromol Symp 93:301

    Google Scholar 

  95. Weissmüller M, Burchard W ( 1997) Acta Polym 48:571

    Article  Google Scholar 

  96. Burchard W, Kajiwara K, Gordon M, Kalal J, Kennedy JW (1973) Macromolecules 6:642

    Article  CAS  Google Scholar 

  97. Kajiwara K (1971) Polymer 12:57

    Article  CAS  Google Scholar 

  98. Marriman J, Hermans JJ (1961) J Phys Chem 65:385

    Article  Google Scholar 

  99. De Gennes P-G (1980) Compt Rend 291:17

    Google Scholar 

  100. Stauffer D, Coniglio A, Adam M (1982) Adv Polym Sci 44:103

    Article  CAS  Google Scholar 

  101. Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  102. Daoud M, Martin JE (1989) In: Avnir D (ed) The fractal approach to heterogeneous chemistry. Wiley, New York

    Google Scholar 

  103. Stanley HE (1972) Introduction to phase transition and critical phenomena. Clarendon Press, Oxford

    Google Scholar 

  104. Schaefer DW, Martin JE, Hurd AJ, Keefer KD (1985) In: Boccora N, Daoud M (eds) Physics of finely divided matter. Springer, Berlin Heidelberg New York

    Google Scholar 

  105. Schaefer DW (1989) Science 243:1023

    Article  CAS  Google Scholar 

  106. Flory PJ (1936) J Am Chem Soc 58:1877

    Article  CAS  Google Scholar 

  107. Flory PF (1940) J Am Chem Soc 62:1561

    Article  CAS  Google Scholar 

  108. Zhou LL, Hadjichristidis N, Toporowski PM, Roovers J (1992) Rubber Chem Tech 65:303

    CAS  Google Scholar 

  109. Schossler F, Benoit H, Grubisic-Gallot Z, Strazielle C, Leibler L (1989) Macromolecules 22:400

    Article  Google Scholar 

  110. Weissmüller M, Burchard W ( 1997) Polymer Internat 44:380

    Article  Google Scholar 

  111. Adam M, Delsanti M, Munch JP, Durand D (1988) Phys Rev Lett 61:706

    Article  CAS  Google Scholar 

  112. Patton EV, Wesson JA, Rubinstein M, Wilson JC, Oppenheimer LE (1989) Macromole cules 22:1946

    Article  CAS  Google Scholar 

  113. Bauer J, Burchard W (1992) J Physique II (France) 2:1053

    Article  CAS  Google Scholar 

  114. Burchard W, Schulz L, Auersch A, Littke W (1990) Polymer Preprints 31(2):131

    CAS  Google Scholar 

  115. Trappe V, Richtering W, Burchard W ( 1992) J Physique II (France) 2:1453

    Article  CAS  Google Scholar 

  116. Colby RH, Rubinstein M, Gillmor JR, Mourey TH (1992) Macromolecules 25:7180

    Article  CAS  Google Scholar 

  117. De Gennes P-G (1977) J Physique (Paris) 38L:355

    Google Scholar 

  118. Colby R (1998) Personal communication, manuscript in preparation

    Google Scholar 

  119. Lusignan C P (1998) PhD Thesis. Pensylvania State University

    Google Scholar 

  120. Weissmüller M (1996) PhD Thesis. University of Freiburg, Germany

    Google Scholar 

  121. Daoud M, Family F, Jannink G (1984) J Physique Lett (Paris) 45:199

    CAS  Google Scholar 

  122. Martin JE, Adolf D, Wilxcoxon JP, Luck JM (1987) Phys Rev Lett 61:2620

    Article  Google Scholar 

  123. Bhatt M, Jamieson AM (1988) Macromolecules 21:3015

    Article  CAS  Google Scholar 

  124. Meyerhoff G, Appelt B (1979) Makromol Chem 12:968

    CAS  Google Scholar 

  125. Stockmayer WH, Fixman M (1953) Ann NY Acad Sci 57:334

    Article  CAS  Google Scholar 

  126. Kirkwood JG (1954) J Poly Sci 12:1

    Article  CAS  Google Scholar 

  127. Zimm BH (1956) J Chem Phys 24:269

    Article  CAS  Google Scholar 

  128. Zimm BH, Kilb RW (1959) J Polymer Sci 37:19

    Article  CAS  Google Scholar 

  129. Kurata M, Abe M, Imwama M, Matsushima M (1972) Polym J 3:729

    Article  CAS  Google Scholar 

  130. Roovers JEL, Bywater S (1972) Macromolecules 5:384

    Article  CAS  Google Scholar 

  131. Roovers JEL, Bywater S (1974) Macromolecules 7:443

    Article  CAS  Google Scholar 

  132. Berry GC (1971) J Polym Sci 9:687

    CAS  Google Scholar 

  133. Zilliox JG (1972) Makromol Chem 156:121

    Article  CAS  Google Scholar 

  134. Hadjichristidis N, Roovers JEL (1974) J Polymer Sci, Phys Ed 12:2521

    Article  CAS  Google Scholar 

  135. Roovers JEL, Toporowski PM, Martin J (1989) Macromolecules 22:1897

    Article  CAS  Google Scholar 

  136. Leukel J (1997) Ph D Thesis. University of Freiburg, Germany

    Google Scholar 

  137. Schmidt M, Nerger D, Burchard W (1979) Polymer 20:528

    Article  Google Scholar 

  138. Tanford C (1967) Physical chemistry of macromolecules. Wiley, New York, p 344

    Google Scholar 

  139. Akcasu AZ, Benmouna M (1978) Macromolecules 11:1187

    Article  Google Scholar 

  140. Schmidt M, Burchard W (1981) Macromolecules 14:370

    Article  CAS  Google Scholar 

  141. ter Meer H-U, Burchard W( 1980) Colloid Polym Sci 258:657

    Google Scholar 

  142. Douglas JF, Freed KF (1984) Macromolecules 17:2344

    Article  CAS  Google Scholar 

  143. Douglas JF (1996) Personal communication

    Google Scholar 

  144. Galinsky G, Burchard W (1996) Macromolecules 29:1498

    Article  CAS  Google Scholar 

  145. Bauer J, Burchard W (1993) Macromolecules 26:3103

    Article  CAS  Google Scholar 

  146. Akcasu AZ, Benmouna M (1978) Macromolecules 11:1193

    Article  CAS  Google Scholar 

  147. Huber K, Burchard W, Akcasu AZ ( 1985) Macromolecules 18:2743

    Article  CAS  Google Scholar 

  148. Kratky O, Porod G, Kahovec L (1951) Z Elektrochem 55:53

    CAS  Google Scholar 

  149. http://ne43neuiucedu/n-scatter/n-lengths/listhtml

  150. Burchard W, Schmidt M (1980) Polymer 21:745

    Article  CAS  Google Scholar 

  151. Burchard W (1986) In: Semlyen JA (ed) Cyclic polymers. Esevier, London

    Google Scholar 

  152. Simha R, Zakin JL (1960) J Chem Phys 33:1791

    Article  CAS  Google Scholar 

  153. Utracki L, Simha R (1963) J Polym Sci A1:1089

    Google Scholar 

  154. Des Cloizeaux J (1975) J Physique (Paris) 36:281

    Google Scholar 

  155. Stockmayer WH, Casassa EF (1952) J ChemPhys 20:1560

    CAS  Google Scholar 

  156. Burchard W (1990) Makromol Chem, Macromol Symp 39:179

    CAS  Google Scholar 

  157. Carnahan NF, Starling KE (1969) J Chem Phys 51:635

    Article  CAS  Google Scholar 

  158. Ohta T, Oono Y (1983) Physics Lett 79:339

    Google Scholar 

  159. Cotter MA, Martire DC (1970) J Chem Phys 52:1909

    Article  CAS  Google Scholar 

  160. Delay M, Tardieu A (1983) Nature 30L:415

    Article  Google Scholar 

  161. Delay M, Gromiec A (1983) Biopolymers 22:1203

    Article  Google Scholar 

  162. Coviello T, Kajiwara K, Burchard W, Dentini M, Crescenzi V (1986) Macromolecules 19:2826

    Article  CAS  Google Scholar 

  163. Dentini M, Coviello T, Burchard W, Crescenzi V (1988) Macromolecules 21:3312

    Article  CAS  Google Scholar 

  164. Coviello T, Burchard W, Dentini M, Crescenzi V, Morris VJ ( 1995) J Polym Sci, Phys Ed 33:1833

    Article  CAS  Google Scholar 

  165. Trappe V ( 1994) PhD Thesis. University of Freiburg

    Google Scholar 

  166. Aberle T, Burchard W (1997) Starch 49:215

    Article  CAS  Google Scholar 

  167. Merkle G, Burchard W, Lutz P, Freed KF, Gao J (1993) Macromolecules 26:2736

    Article  CAS  Google Scholar 

  168. Sato T, Norisuye T, Fujita H ( 1987) J Polym Sci, Phys Ed 25:1

    Article  CAS  Google Scholar 

  169. Nakamura Y, Akasaka K, Katayama K, Norisuye T, Teramoto A ( 1992) Macromolecules 25:1134

    Article  CAS  Google Scholar 

  170. Okumoto M, Nakamura Y, Norisuye T, Teramoto A (1998) Macromolecules 31:1615

    Article  CAS  Google Scholar 

  171. Bawn CEH, Freeman RFJ, Kamaliddin AR (1950) Trans Farad Soc 46:862

    Article  CAS  Google Scholar 

  172. Roovers J, Toporowski PM, Douglas J (1995) Macromolecules 28:7064

    Article  CAS  Google Scholar 

  173. Trappe V, Burchard W, Steinmann B (1991) Macromolecules 24:4738

    Article  CAS  Google Scholar 

  174. Trappe V, Richtering W, Burchard W( 1992) J Phys II (Paris) 2:1453

    CAS  Google Scholar 

  175. Trappe V, Bauer J, Weissmüller M, Burchard W (1997) Macromolecules 30:2365

    Article  CAS  Google Scholar 

  176. Daoud M (1976) PhD Thesis. Paris CEN Saclay

    Google Scholar 

  177. Auersch A, Littke W, Lang P, Burchard W (1991) J Cryst Growth 110:201

    Article  CAS  Google Scholar 

  178. Burchard W, Lang P, Schulz L, Coviello T (1992) Makromol Chem, Macromol Sympos 58:21

    CAS  Google Scholar 

  179. Aberle T, Burchard W (1997) Comp Theor Polym Sci 7:215

    Article  CAS  Google Scholar 

  180. Hanselmann R, Burchard W, Ehrat M, Widmer HM ( 1996) Macromolecules 29:3277

    Article  CAS  Google Scholar 

  181. Andres-Köksal K (1996) PhD Thesis. University of Freiburg (manuscript in preparation)

    Google Scholar 

  182. Bantle S, Schmidt M, Burchard W (1982) Macromolecules 15:1604

    Article  CAS  Google Scholar 

  183. Burchard W, Dolega R, Stockmayer WH (1995) Collect Czech Chem Commun 60:1653

    Article  CAS  Google Scholar 

  184. Bareiss R, Jung B, Khokhov AR (1996) Macromol Theory Simul 5:57

    Article  Google Scholar 

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Burchard, W. (1999). Solution Properties of Branched Macromolecules. In: Roovers, J. (eds) Branched Polymers II. Advances in Polymer Science, vol 143. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49780-3_3

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