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Solution Properties of Branched Macromolecules

  • Walther Burchard
Chapter
Part of the Advances in Polymer Science book series (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.

Keywords

Solution properties Regularly branched structures Randomly and hyperbranched polymers Shrinking factors Fractal dimensions Osmotic modulus of semi-dilute solutions Molar mass distributions SEC/MALLS/VISC chromatography 

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References

  1. 1.
    Flory PJ ( 1953) Principles of polymer chemistry. Cornell University Press, Ithaca, NYGoogle Scholar
  2. 2.
    Flory PJ ( 1969) Statistical mechanics of chain molecules. Wiley, New YorkGoogle Scholar
  3. 3.
    Yamakawa H ( 1971) Modern theory of polymer solutions. Harper and Row, Evanstone, IlGoogle Scholar
  4. 4.
    De Gennes P-G (1979) Scaling concepts in polymer physics, Cornell University Press, Ithaca, NYGoogle Scholar
  5. 5.
    Doi M, Edwards SF ( 1986) The theory of polymer dynamics. Clarendon, OxfordGoogle Scholar
  6. 6.
    Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New YorkGoogle Scholar
  7. 7.
    Stauffer D (1985) Introduction to percolation theory. Taylor & Francis, LondonGoogle Scholar
  8. 8.
    Freire JJ (1998) Chapter in this book; there a list of relevant literature on the techniques of simulationGoogle Scholar
  9. 9.
    Burchard W (1988) Makromol Chem Macromol Symp 18:1Google Scholar
  10. 10.
    Burchard W (1993) TRIP 1:192Google Scholar
  11. 11.
    Burchard W (1966) Adv Colloid Interface Sci. 64:45CrossRefGoogle Scholar
  12. 12.
    Flory PJ (1941) J Am Chem Soc 63:3083CrossRefGoogle Scholar
  13. 13.
    Flory PJ (1946) Chem Revs 39:137CrossRefGoogle Scholar
  14. 14.
    Stockmayer WH (1943) J Chem Phys 11:45CrossRefGoogle Scholar
  15. 15.
    See textbooks on polymer science, eg [1] and more recent onesGoogle Scholar
  16. 16.
    Erlander SR, French D (1956) J Polymer Sci 20:7Google Scholar
  17. 17.
    Burchard W (1972) Macromolecules 5:604CrossRefGoogle Scholar
  18. 18.
    Daoud M, Cotton J-P, Farnoux B, Jannink G, Sarma G, Benoit H, Duplessix R, Picot C, de Gennes P-G (1975) Macromolecules 8:804CrossRefGoogle Scholar
  19. 19.
    Burchard W, Lang P, Schulz L ( 1992) Makomol Chem, Macromol Symp 58:21Google Scholar
  20. 20.
    Willner L, Jucknischke O, Richter D, Roovers J, Zhou L-L, Toporowski PM, Fetters LJ, Huang JS, Lin MY, Hadjichristidis N ( 1994) Macromolecules 27:3821CrossRefGoogle Scholar
  21. 21.
    Allegra G, Colombe E, Ganazzoli F (1993) Macromolecules 26:330CrossRefGoogle Scholar
  22. 22.
    Benoit H, Joanny JF, Hadziioannou G, Hammouda B (1993) Macromolecules 26:5790CrossRefGoogle Scholar
  23. 23.
    Alessandrini JL, Carignano MA (1992) Macromolecules 25:1157CrossRefGoogle Scholar
  24. 24.
    Burchard W (1983) Adv Polym Sci 48:1CrossRefGoogle Scholar
  25. 25.
    Roovers J, Zhou L-L, Toporowski PM, van der Zwan M, Iatrou H, Hadjichristidis N (1993) Macromolecules 26:4324CrossRefGoogle Scholar
  26. 26.
    Bywater S (1979) Adv Polym Sci 30:89CrossRefGoogle Scholar
  27. 27.
    Cherayil BJ, Bawondi MG, Miyake A, Freed KF ( 1986) Macromolecules 19:2770CrossRefGoogle Scholar
  28. 28.
    Douglas JF, Roovers J, Freed KF (1990) Macromolecules 23:3168CrossRefGoogle Scholar
  29. 29.
    Daoud M, Cotton J-P (1982) J Physique (Paris) 43:531Google Scholar
  30. 30.
    Huber K, Burchard W, Bantle S, Fetters LJ (1987) Polymer 28:1997CrossRefGoogle Scholar
  31. 31.
    Richter D, Farago B, Huang JS, Fetters LJ, Even B ( 1989) Macromolecules 22:468CrossRefGoogle Scholar
  32. 32.
    Douglas JF, Freed KF (1984) Macromolecules 17:7461Google Scholar
  33. 33.
    Freed KF, Wang Q-S, Roovers J, Douglas JF (1988) Macromolecules 21:2219CrossRefGoogle Scholar
  34. 34.
    Horton JG, Squires GL, Boothoyd AT, Fetters LJ, Rennie AR, Glinka CJ, Robinson RA (1989) Macromolecules 22:681CrossRefGoogle Scholar
  35. 35.
    Grest GS, Fetters LJ, Huang JS, Richter D ( 1996) Adv Chem Phys 94:67CrossRefGoogle Scholar
  36. 36.
    Ganazzoli F, Allegra G (1990) Macromolecules 23:262CrossRefGoogle Scholar
  37. 37.
    Ganazzoli F, Fontelos MA, Allegra G (1993) Polymer 34:2615CrossRefGoogle Scholar
  38. 38.
    Burchard W, Kajiwara K, Nerger D (1982) J Polym Sci Phys Ed 20:157CrossRefGoogle Scholar
  39. 39.
    Voit BI( 1995) Acta Polym 46:87Google Scholar
  40. 40.
    Tomalia DA, Durst HD (1993) Top Curr Chem 165:194Google Scholar
  41. 41.
    Tomalia DA, Naylor A, Goddard W (1990) Angew Chem In Ed 29:138CrossRefGoogle Scholar
  42. 42.
    Tomalia DA, Hedstrand DM, Wilson LR (1990) In: Encyclopedia of polymer science and technology. Wiley, New York, pp 46–92Google Scholar
  43. 43.
    Mekelburger H-B, Jaworek W, Vögtle F (1992) Angew ChemInt Ed 31:1571CrossRefGoogle Scholar
  44. 44.
    Newkome GR, Morrefield CN, Baker GR (12992) Aldichim Acta 25:31Google Scholar
  45. 45.
    Fréchet JMJ (1994) Science 263:1710CrossRefGoogle Scholar
  46. 46.
    Wooley KL, Hawker CJ, Fréchet JMJ (1991) J Am Chem Soc 113:4252CrossRefGoogle Scholar
  47. 47.
    Wooley KL, Hawkers CJ, Fréchet JMJ ( 1994) Angew Chem Int Ed 33:82CrossRefGoogle Scholar
  48. 48.
    Stockmayer WH (1944) J Chem Phys 12:125CrossRefGoogle Scholar
  49. 49.
    Zimm BH, Stockmayer WH (1949) J Chem Phys 17:1301CrossRefGoogle Scholar
  50. 50.
    Kajiwara K, Burchard W, Gordon M (1970) Brit Polymer J 2:110CrossRefGoogle Scholar
  51. 51.
    Burchard W (1972) Macromolecules 5:604CrossRefGoogle Scholar
  52. 52.
    Burchard W (1977) Macromolecules 10:919CrossRefGoogle Scholar
  53. 53.
    Gordon M (1962) Proc Roy Soc (London) 51:240Google Scholar
  54. 54.
    Gordon M, Scantlebury GR (1964) Trans Fard Soc 60:604CrossRefGoogle Scholar
  55. 55.
    Isaacson J, Lubensky TC (1980) J Physique (Paris) 41:L469Google Scholar
  56. 56.
    Boris D, Rubinstein M ( 1996) Macromolecules 29:7251CrossRefGoogle Scholar
  57. 57.
    Huglin MB ( 1972) Light scattering from polymer solutions. Academic Press, LondonGoogle Scholar
  58. 58.
    Glatter O, Kratky O (1982) Small angle X-ray scattering. Academic Press, LondonGoogle Scholar
  59. 59.
    Higgins JS, Benoit H ( 1994) Polymers and neutron scattering. Clarendon Press, OxfordGoogle Scholar
  60. 60.
    Berne BJ, Pecora R (1976) Dynamic light scattering. Wiley, New YorkGoogle Scholar
  61. 61.
    Siegert AJF (1943) MIT Rad Lab Report 465; see also [60]Google Scholar
  62. 62.
    Flammer J, Ricka J ( 1997) Apl Optics 36:508CrossRefGoogle Scholar
  63. 63.
    Burchard W, Schmidt M, Stockmayer W (1980) Macromolecules 13:1265CrossRefGoogle Scholar
  64. 64.
    Schmidt M, Stockmayer WH (1984) Macromolecules 17:509CrossRefGoogle Scholar
  65. 65.
    Burchard W, Schmidt M, Nerger D (1979) Polymer 20:582CrossRefGoogle Scholar
  66. 66.
    Einstein Q (1905) Ann Phys 17:549CrossRefGoogle Scholar
  67. 67.
    Einstein A (1906) Ann Phys 19:371CrossRefGoogle Scholar
  68. 68.
    Einstein A (1906) Ann Phys 19:289CrossRefGoogle Scholar
  69. 69.
    Einstein A (1911) Ann Phys 34:591CrossRefGoogle Scholar
  70. 70.
    Flory PJ, Fox TG ( 1951 ) J Am Chem Soc 73:1904CrossRefGoogle Scholar
  71. 71.
    Krigbaum WR, Carpenter DK (1955) J Phys Chem 59:1166CrossRefGoogle Scholar
  72. 72.
    McMillan WG, Mayer JE (1945) J Chem Phys 13:276CrossRefGoogle Scholar
  73. 73.
    Oseen CW (1927) Hydrodynamik. Akademie Verlag, LeipzigGoogle Scholar
  74. 74.
    McQuarrie DA ( 1973) Statistical mechanics. Harper and Row, New YorkGoogle Scholar
  75. 75.
    Friedman HL (1985) A course in statistical mechanics. Prentice Hall, Englewood, NJGoogle Scholar
  76. 76.
    Lang P, Burchard W, Wolfe MS, Spinelli HJ, Page L (1991) Macromolecules 24:1306CrossRefGoogle Scholar
  77. 77.
    Roovers J, Hajichristidis N, Fetters LJ (1983) Macromolecules 16:214CrossRefGoogle Scholar
  78. 78.
    Douglas JF, Freed KF (1984) Macromoleules 17:1854CrossRefGoogle Scholar
  79. 79.
    Schaefgen JR, Flory PJ ( 1948) J Am Chem Soc 70:2709CrossRefGoogle Scholar
  80. 80.
    Schulz GV (1939) Z Phys Chem B39:25Google Scholar
  81. 81.
    Zimm BH (1948) J Chem Phys 16:1099CrossRefGoogle Scholar
  82. 82.
    Feller W (1957) An introduction to probability theory and its applications, 3rd edn. Wiley, New YorkGoogle Scholar
  83. 83.
    Burchard W( 1974) Macromolecules 7:841Google Scholar
  84. 84.
    Benoit H (1953) J Polymer Sci 11:507CrossRefGoogle Scholar
  85. 85.
    Freire JJ, Pla J, Rey A, Prats R (1986) Macromolecules 19:452CrossRefGoogle Scholar
  86. 86.
    Freire JJ, Rey A, Garcia de la Torre J ( 1986) Macromolecules 19:457CrossRefGoogle Scholar
  87. 87.
    Rey A, Freire JJ, Garcia de la Torre J (1987) Macromolecules 20:342CrossRefGoogle Scholar
  88. 88.
    Kirkwood JG, Riseman J (1948) J Chem Phys 16:565CrossRefGoogle Scholar
  89. 89.
    Huber K, Burchard W, Fetters LJ (1984) Macromolecules 17:54CrossRefGoogle Scholar
  90. 90.
    Douglas JF, Freed KF (1984) Macromolecules 17:1854; 2344CrossRefGoogle Scholar
  91. 91.
    Casassa EF (1962) J Phys Chem 37:2176CrossRefGoogle Scholar
  92. 92.
    Trappe V, Bauer J, Weissmüller M, Burchard W ( 1997) Macromolecules 30:2365CrossRefGoogle Scholar
  93. 93.
    Trappe V, Burchard W (1994) Polymer Preprints (Am Chem Soc, Div Polymer Chem) 36:1Google Scholar
  94. 94.
    Weissmüller M, Burchard W (1995) Macromol Symp 93:301Google Scholar
  95. 95.
    Weissmüller M, Burchard W ( 1997) Acta Polym 48:571CrossRefGoogle Scholar
  96. 96.
    Burchard W, Kajiwara K, Gordon M, Kalal J, Kennedy JW (1973) Macromolecules 6:642CrossRefGoogle Scholar
  97. 97.
    Kajiwara K (1971) Polymer 12:57CrossRefGoogle Scholar
  98. 98.
    Marriman J, Hermans JJ (1961) J Phys Chem 65:385CrossRefGoogle Scholar
  99. 99.
    De Gennes P-G (1980) Compt Rend 291:17Google Scholar
  100. 100.
    Stauffer D, Coniglio A, Adam M (1982) Adv Polym Sci 44:103CrossRefGoogle Scholar
  101. 101.
    Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San FranciscoGoogle Scholar
  102. 102.
    Daoud M, Martin JE (1989) In: Avnir D (ed) The fractal approach to heterogeneous chemistry. Wiley, New YorkGoogle Scholar
  103. 103.
    Stanley HE (1972) Introduction to phase transition and critical phenomena. Clarendon Press, OxfordGoogle Scholar
  104. 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 YorkGoogle Scholar
  105. 105.
    Schaefer DW (1989) Science 243:1023CrossRefGoogle Scholar
  106. 106.
    Flory PJ (1936) J Am Chem Soc 58:1877CrossRefGoogle Scholar
  107. 107.
    Flory PF (1940) J Am Chem Soc 62:1561CrossRefGoogle Scholar
  108. 108.
    Zhou LL, Hadjichristidis N, Toporowski PM, Roovers J (1992) Rubber Chem Tech 65:303Google Scholar
  109. 109.
    Schossler F, Benoit H, Grubisic-Gallot Z, Strazielle C, Leibler L (1989) Macromolecules 22:400CrossRefGoogle Scholar
  110. 110.
    Weissmüller M, Burchard W ( 1997) Polymer Internat 44:380CrossRefGoogle Scholar
  111. 111.
    Adam M, Delsanti M, Munch JP, Durand D (1988) Phys Rev Lett 61:706CrossRefGoogle Scholar
  112. 112.
    Patton EV, Wesson JA, Rubinstein M, Wilson JC, Oppenheimer LE (1989) Macromole cules 22:1946CrossRefGoogle Scholar
  113. 113.
    Bauer J, Burchard W (1992) J Physique II (France) 2:1053CrossRefGoogle Scholar
  114. 114.
    Burchard W, Schulz L, Auersch A, Littke W (1990) Polymer Preprints 31(2):131Google Scholar
  115. 115.
    Trappe V, Richtering W, Burchard W ( 1992) J Physique II (France) 2:1453CrossRefGoogle Scholar
  116. 116.
    Colby RH, Rubinstein M, Gillmor JR, Mourey TH (1992) Macromolecules 25:7180CrossRefGoogle Scholar
  117. 117.
    De Gennes P-G (1977) J Physique (Paris) 38L:355Google Scholar
  118. 118.
    Colby R (1998) Personal communication, manuscript in preparationGoogle Scholar
  119. 119.
    Lusignan C P (1998) PhD Thesis. Pensylvania State UniversityGoogle Scholar
  120. 120.
    Weissmüller M (1996) PhD Thesis. University of Freiburg, GermanyGoogle Scholar
  121. 121.
    Daoud M, Family F, Jannink G (1984) J Physique Lett (Paris) 45:199Google Scholar
  122. 122.
    Martin JE, Adolf D, Wilxcoxon JP, Luck JM (1987) Phys Rev Lett 61:2620CrossRefGoogle Scholar
  123. 123.
    Bhatt M, Jamieson AM (1988) Macromolecules 21:3015CrossRefGoogle Scholar
  124. 124.
    Meyerhoff G, Appelt B (1979) Makromol Chem 12:968Google Scholar
  125. 125.
    Stockmayer WH, Fixman M (1953) Ann NY Acad Sci 57:334CrossRefGoogle Scholar
  126. 126.
    Kirkwood JG (1954) J Poly Sci 12:1CrossRefGoogle Scholar
  127. 127.
    Zimm BH (1956) J Chem Phys 24:269CrossRefGoogle Scholar
  128. 128.
    Zimm BH, Kilb RW (1959) J Polymer Sci 37:19CrossRefGoogle Scholar
  129. 129.
    Kurata M, Abe M, Imwama M, Matsushima M (1972) Polym J 3:729CrossRefGoogle Scholar
  130. 130.
    Roovers JEL, Bywater S (1972) Macromolecules 5:384CrossRefGoogle Scholar
  131. 131.
    Roovers JEL, Bywater S (1974) Macromolecules 7:443CrossRefGoogle Scholar
  132. 132.
    Berry GC (1971) J Polym Sci 9:687Google Scholar
  133. 133.
    Zilliox JG (1972) Makromol Chem 156:121CrossRefGoogle Scholar
  134. 134.
    Hadjichristidis N, Roovers JEL (1974) J Polymer Sci, Phys Ed 12:2521CrossRefGoogle Scholar
  135. 135.
    Roovers JEL, Toporowski PM, Martin J (1989) Macromolecules 22:1897CrossRefGoogle Scholar
  136. 136.
    Leukel J (1997) Ph D Thesis. University of Freiburg, GermanyGoogle Scholar
  137. 137.
    Schmidt M, Nerger D, Burchard W (1979) Polymer 20:528CrossRefGoogle Scholar
  138. 138.
    Tanford C (1967) Physical chemistry of macromolecules. Wiley, New York, p 344Google Scholar
  139. 139.
    Akcasu AZ, Benmouna M (1978) Macromolecules 11:1187CrossRefGoogle Scholar
  140. 140.
    Schmidt M, Burchard W (1981) Macromolecules 14:370CrossRefGoogle Scholar
  141. 141.
    ter Meer H-U, Burchard W( 1980) Colloid Polym Sci 258:657Google Scholar
  142. 142.
    Douglas JF, Freed KF (1984) Macromolecules 17:2344CrossRefGoogle Scholar
  143. 143.
    Douglas JF (1996) Personal communicationGoogle Scholar
  144. 144.
    Galinsky G, Burchard W (1996) Macromolecules 29:1498CrossRefGoogle Scholar
  145. 145.
    Bauer J, Burchard W (1993) Macromolecules 26:3103CrossRefGoogle Scholar
  146. 146.
    Akcasu AZ, Benmouna M (1978) Macromolecules 11:1193CrossRefGoogle Scholar
  147. 147.
    Huber K, Burchard W, Akcasu AZ ( 1985) Macromolecules 18:2743CrossRefGoogle Scholar
  148. 148.
    Kratky O, Porod G, Kahovec L (1951) Z Elektrochem 55:53Google Scholar
  149. 149.
  150. 150.
    Burchard W, Schmidt M (1980) Polymer 21:745CrossRefGoogle Scholar
  151. 151.
    Burchard W (1986) In: Semlyen JA (ed) Cyclic polymers. Esevier, LondonGoogle Scholar
  152. 152.
    Simha R, Zakin JL (1960) J Chem Phys 33:1791CrossRefGoogle Scholar
  153. 153.
    Utracki L, Simha R (1963) J Polym Sci A1:1089Google Scholar
  154. 154.
    Des Cloizeaux J (1975) J Physique (Paris) 36:281Google Scholar
  155. 155.
    Stockmayer WH, Casassa EF (1952) J ChemPhys 20:1560Google Scholar
  156. 156.
    Burchard W (1990) Makromol Chem, Macromol Symp 39:179Google Scholar
  157. 157.
    Carnahan NF, Starling KE (1969) J Chem Phys 51:635CrossRefGoogle Scholar
  158. 158.
    Ohta T, Oono Y (1983) Physics Lett 79:339Google Scholar
  159. 159.
    Cotter MA, Martire DC (1970) J Chem Phys 52:1909CrossRefGoogle Scholar
  160. 160.
    Delay M, Tardieu A (1983) Nature 30L:415CrossRefGoogle Scholar
  161. 161.
    Delay M, Gromiec A (1983) Biopolymers 22:1203CrossRefGoogle Scholar
  162. 162.
    Coviello T, Kajiwara K, Burchard W, Dentini M, Crescenzi V (1986) Macromolecules 19:2826CrossRefGoogle Scholar
  163. 163.
    Dentini M, Coviello T, Burchard W, Crescenzi V (1988) Macromolecules 21:3312CrossRefGoogle Scholar
  164. 164.
    Coviello T, Burchard W, Dentini M, Crescenzi V, Morris VJ ( 1995) J Polym Sci, Phys Ed 33:1833CrossRefGoogle Scholar
  165. 165.
    Trappe V ( 1994) PhD Thesis. University of FreiburgGoogle Scholar
  166. 166.
    Aberle T, Burchard W (1997) Starch 49:215CrossRefGoogle Scholar
  167. 167.
    Merkle G, Burchard W, Lutz P, Freed KF, Gao J (1993) Macromolecules 26:2736CrossRefGoogle Scholar
  168. 168.
    Sato T, Norisuye T, Fujita H ( 1987) J Polym Sci, Phys Ed 25:1CrossRefGoogle Scholar
  169. 169.
    Nakamura Y, Akasaka K, Katayama K, Norisuye T, Teramoto A ( 1992) Macromolecules 25:1134CrossRefGoogle Scholar
  170. 170.
    Okumoto M, Nakamura Y, Norisuye T, Teramoto A (1998) Macromolecules 31:1615CrossRefGoogle Scholar
  171. 171.
    Bawn CEH, Freeman RFJ, Kamaliddin AR (1950) Trans Farad Soc 46:862CrossRefGoogle Scholar
  172. 172.
    Roovers J, Toporowski PM, Douglas J (1995) Macromolecules 28:7064CrossRefGoogle Scholar
  173. 173.
    Trappe V, Burchard W, Steinmann B (1991) Macromolecules 24:4738CrossRefGoogle Scholar
  174. 174.
    Trappe V, Richtering W, Burchard W( 1992) J Phys II (Paris) 2:1453Google Scholar
  175. 175.
    Trappe V, Bauer J, Weissmüller M, Burchard W (1997) Macromolecules 30:2365CrossRefGoogle Scholar
  176. 176.
    Daoud M (1976) PhD Thesis. Paris CEN SaclayGoogle Scholar
  177. 177.
    Auersch A, Littke W, Lang P, Burchard W (1991) J Cryst Growth 110:201CrossRefGoogle Scholar
  178. 178.
    Burchard W, Lang P, Schulz L, Coviello T (1992) Makromol Chem, Macromol Sympos 58:21Google Scholar
  179. 179.
    Aberle T, Burchard W (1997) Comp Theor Polym Sci 7:215CrossRefGoogle Scholar
  180. 180.
    Hanselmann R, Burchard W, Ehrat M, Widmer HM ( 1996) Macromolecules 29:3277CrossRefGoogle Scholar
  181. 181.
    Andres-Köksal K (1996) PhD Thesis. University of Freiburg (manuscript in preparation)Google Scholar
  182. 182.
    Bantle S, Schmidt M, Burchard W (1982) Macromolecules 15:1604CrossRefGoogle Scholar
  183. 183.
    Burchard W, Dolega R, Stockmayer WH (1995) Collect Czech Chem Commun 60:1653CrossRefGoogle Scholar
  184. 184.
    Bareiss R, Jung B, Khokhov AR (1996) Macromol Theory Simul 5:57CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Walther Burchard
    • 1
  1. 1.Insitute of Macromolecular ChemistryAlbert-Ludwigs-University of FreiburgFreiburgGermany

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