31:
1092:
The structure of a gel network can be conceptualised as a random graph. This analogy is exploited to calculate the gel point and gel fraction for monomer precursors with arbitrary types of functional groups. Random graphs can be used to derive analytical expressions for simple polymerisation
453:
approaches infinite size. The statistical approach assumes that (1) the reactivity of the functional groups of the same type is the same and independent of the molecular size and (2) there are no intramolecular reactions between the functional groups on the same molecule.
107:. Critical percolation theory was successfully applied to gelation in 1970s. A number of growth models (diffusion limited aggregation, cluster-cluster aggregation, kinetic gelation) were developed in the 1980s to describe the kinetic aspects of aggregation and gelation.
73:
between the chains, which lead to progressively larger polymers. As the linking continues, larger branched polymers are obtained and at a certain extent of the reaction, links between the polymer result in the formation of a single
86:
becomes very large. The onset of gelation, or gel point, is accompanied by a sudden increase in viscosity. This "infinite" sized polymer is called the gel or network, which does not dissolve in the solvent, but can swell in it.
765:
1025:
224:
358:
855:
103:. While the physical gels involve physical bonds, chemical gelation involves covalent bonds. The first quantitative theories of chemical gelation were formulated in the 1940s by
927:
885:
610:, respectively. The ratio of all A groups, both reacted and unreacted, that are part of branched units, to the total number of A groups in the mixture is defined as
451:
306:
276:
155:
628:
608:
581:
534:
407:
507:
481:
1047:
554:
380:
246:
115:
It is important to be able to predict the onset of gelation, since it is an irreversible process that dramatically changes the properties of the system.
1093:
mechanisms, such as step-growth polymerisation, or alternatively, they can be combined with a system of rate equations that are integrated numerically.
635:
1122:
Oliveira, J.T.; Reis, R.L. (2008). "Hydrogels from polysaccharide-based materials: Fundamentals and applications in regenerative medicine".
1052:
Note that the relation does not apply for reaction systems containing monofunctional reactants and/or both A and B type of branch units.
1279:
1254:
1139:
1029:
where r is the ratio of all A groups to all B groups. If more than one type of multifunctional branch unit is present and average
1087:
1427:
1303:
Stockmayer, Walter H. (February 1943). "Theory of
Molecular Size Distribution and Gel Formation in Branched-Chain Polymers".
1221:
418:
104:
1447:
1067:
1061:
423:
Flory and
Stockmayer used a statistical approach to derive an expression to predict the gel point by calculating when
937:
162:
1452:
1442:
313:
1338:
Flory, Paul J. (November 1941). "Molecular Size
Distribution in Three Dimensional Polymers. I. Gelation".
932:
This relation yields to an expression for the extent of reaction of A functional groups at the gel point
772:
1432:
1102:
308:
can be considered to be infinite, thus the critical extent of the reaction at the gel point is found as
1374:
1312:
1071:
890:
1398:
124:
860:
30:
1437:
1390:
1285:
1275:
1250:
1217:
1186:
1135:
46:
1382:
1347:
1320:
1242:
1209:
1176:
1168:
1127:
426:
281:
251:
130:
66:
613:
586:
559:
512:
385:
1075:
486:
460:
1378:
1316:
1181:
1156:
1032:
539:
365:
231:
1422:
1416:
1402:
96:
75:
1131:
1172:
100:
70:
38:
1394:
1289:
83:
79:
1190:
1049:
value is used for all monomer molecules with functionality greater than 2.
769:
The probability of obtaining the product of the reaction above is given by
1246:
760:{\displaystyle A-A+B-B+A_{f}\rightarrow A_{f-1}-(B-BA-A)_{n}B-BA-A_{f-1}}
1351:
1386:
1213:
62:
34:
17:
1324:
1157:"Hydrogel: Preparation, characterization, and applications: A review"
857:, since the probability that a B group reach with a branched unit is
1365:
Buffet, E.; Pulé, J. V. (1991-07-01). "Polymers and random graphs".
29:
887:
and the probability that a B group react with non-branched A is
556:
is the functionality. The extends of the functional groups are
99:. Gelation can occur either by physical linking or by chemical
1066:
Gelation of polymers can be described in the framework of the
278:
is the average functionality of reaction mixture. For the gel
58:
1270:
Rubinstein, Michael; Colby, Ralph H. (December 20, 2003).
457:
Consider the polymerization of bifunctional molecules
1035:
940:
893:
863:
775:
638:
616:
589:
562:
542:
515:
489:
463:
429:
388:
368:
316:
284:
254:
234:
165:
133:
78:. At that point in the reaction, which is defined as
1124:Natural-Based Polymers for Biomedical Applications
1041:
1019:
921:
879:
849:
759:
622:
602:
575:
548:
528:
501:
475:
445:
401:
374:
352:
300:
270:
240:
218:
149:
1020:{\displaystyle p_{c}={\frac {1}{\{r\}^{1/2}}}}
219:{\displaystyle DP_{n}={\frac {2}{2-p.f_{av}}}}
1239:Introduction to Polymer Science and Chemistry
111:Quantitative approaches to determine gelation
8:
997:
960:
630:. This will lead to the following reaction
27:Formation of a gel from a mass of polymers
1180:
1034:
1004:
1000:
954:
945:
939:
898:
892:
868:
862:
838:
828:
818:
793:
780:
774:
745:
720:
680:
667:
637:
615:
594:
588:
567:
561:
541:
520:
514:
488:
462:
437:
428:
393:
387:
367:
353:{\displaystyle p_{c}={\frac {2}{f_{av}}}}
339:
330:
321:
315:
292:
283:
259:
253:
233:
204:
182:
173:
164:
141:
132:
1340:Journal of the American Chemical Society
127:number-average degree of polymerization
1114:
7:
1074:, which answers the question when a
1274:. Oxford: Oxford University Press.
850:{\displaystyle p_{A}^{n}p_{B}\rho }
248:is the extent of the reaction and
25:
82:, the system loses fluidity and
1305:The Journal of Chemical Physics
1088:Random graph theory of gelation
1367:Journal of Statistical Physics
993:
990:
978:
966:
922:{\displaystyle p_{B}(1-\rho )}
916:
904:
825:
811:
799:
786:
717:
695:
673:
119:Average functionality approach
1:
1206:Principles of Polymerization
1161:Journal of Advanced Research
1132:10.1533/9781845694814.4.485
1469:
1173:10.1016/j.jare.2013.07.006
1085:
1059:
880:{\displaystyle p_{B}\rho }
416:
413:Flory Stockmayer approach
37:prior (no gel) and after
382:is greater or equal to
95:Gelation is promoted by
57:) is the formation of a
1155:Ahmed, Enas M. (2015).
419:Flory–Stockmayer theory
1237:Chanda, Manas (2006).
1204:Odian, George (2004).
1043:
1021:
923:
881:
851:
761:
624:
604:
577:
550:
530:
503:
477:
447:
446:{\displaystyle DP_{n}}
403:
376:
354:
302:
301:{\displaystyle DP_{n}}
272:
271:{\displaystyle f_{av}}
242:
220:
151:
150:{\displaystyle DP_{n}}
42:
1428:Drug delivery devices
1247:10.1201/9781420007329
1103:Mechanics of gelation
1044:
1022:
924:
882:
852:
762:
625:
623:{\displaystyle \rho }
605:
603:{\displaystyle p_{B}}
578:
576:{\displaystyle p_{A}}
551:
531:
529:{\displaystyle A_{f}}
504:
478:
448:
404:
402:{\displaystyle p_{c}}
377:
355:
303:
273:
243:
221:
152:
33:
1126:. pp. 485–514.
1033:
938:
891:
861:
773:
636:
614:
587:
560:
540:
513:
509:and multifunctional
487:
461:
427:
386:
366:
314:
282:
252:
232:
163:
131:
105:Flory and Stockmayer
76:macroscopic molecule
1379:1991JSP....64...87B
1352:10.1021/ja01856a061
1317:1943JChPh..11...45S
502:{\displaystyle B-B}
476:{\displaystyle A-A}
409:, gelation occurs.
61:from a system with
1448:Physical chemistry
1387:10.1007/BF01057869
1214:10.1002/047147875x
1039:
1017:
919:
877:
847:
757:
620:
600:
573:
546:
526:
499:
473:
443:
399:
372:
350:
298:
268:
238:
216:
147:
125:Carothers equation
43:
1453:Polymer chemistry
1443:Organic chemistry
1346:(11): 3083–3090.
1325:10.1063/1.1723803
1068:Erdős–Rényi model
1062:Erdős–Rényi model
1056:Erdős–Rényi model
1042:{\displaystyle f}
1015:
549:{\displaystyle f}
375:{\displaystyle p}
348:
241:{\displaystyle p}
214:
123:According to the
67:Branched polymers
47:polymer chemistry
16:(Redirected from
1460:
1407:
1406:
1362:
1356:
1355:
1335:
1329:
1328:
1300:
1294:
1293:
1267:
1261:
1260:
1234:
1228:
1227:
1201:
1195:
1194:
1184:
1152:
1146:
1145:
1119:
1048:
1046:
1045:
1040:
1026:
1024:
1023:
1018:
1016:
1014:
1013:
1012:
1008:
955:
950:
949:
928:
926:
925:
920:
903:
902:
886:
884:
883:
878:
873:
872:
856:
854:
853:
848:
843:
842:
833:
832:
823:
822:
798:
797:
785:
784:
766:
764:
763:
758:
756:
755:
725:
724:
691:
690:
672:
671:
629:
627:
626:
621:
609:
607:
606:
601:
599:
598:
582:
580:
579:
574:
572:
571:
555:
553:
552:
547:
535:
533:
532:
527:
525:
524:
508:
506:
505:
500:
482:
480:
479:
474:
452:
450:
449:
444:
442:
441:
408:
406:
405:
400:
398:
397:
381:
379:
378:
373:
359:
357:
356:
351:
349:
347:
346:
331:
326:
325:
307:
305:
304:
299:
297:
296:
277:
275:
274:
269:
267:
266:
247:
245:
244:
239:
225:
223:
222:
217:
215:
213:
212:
211:
183:
178:
177:
156:
154:
153:
148:
146:
145:
21:
1468:
1467:
1463:
1462:
1461:
1459:
1458:
1457:
1413:
1412:
1411:
1410:
1364:
1363:
1359:
1337:
1336:
1332:
1302:
1301:
1297:
1282:
1272:Polymer physics
1269:
1268:
1264:
1257:
1236:
1235:
1231:
1224:
1203:
1202:
1198:
1154:
1153:
1149:
1142:
1121:
1120:
1116:
1111:
1099:
1090:
1084:
1076:giant component
1072:Lushnikov model
1064:
1058:
1031:
1030:
996:
959:
941:
936:
935:
894:
889:
888:
864:
859:
858:
834:
824:
814:
789:
776:
771:
770:
741:
716:
676:
663:
634:
633:
612:
611:
590:
585:
584:
563:
558:
557:
538:
537:
516:
511:
510:
485:
484:
459:
458:
433:
425:
424:
421:
415:
389:
384:
383:
364:
363:
335:
317:
312:
311:
288:
280:
279:
255:
250:
249:
230:
229:
200:
187:
169:
161:
160:
137:
129:
128:
121:
113:
93:
28:
23:
22:
15:
12:
11:
5:
1466:
1464:
1456:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1415:
1414:
1409:
1408:
1357:
1330:
1295:
1281:978-0198520597
1280:
1262:
1255:
1229:
1222:
1196:
1167:(2): 105–121.
1147:
1140:
1113:
1112:
1110:
1107:
1106:
1105:
1098:
1095:
1086:Main article:
1083:
1080:
1060:Main article:
1057:
1054:
1038:
1011:
1007:
1003:
999:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
958:
953:
948:
944:
918:
915:
912:
909:
906:
901:
897:
876:
871:
867:
846:
841:
837:
831:
827:
821:
817:
813:
810:
807:
804:
801:
796:
792:
788:
783:
779:
754:
751:
748:
744:
740:
737:
734:
731:
728:
723:
719:
715:
712:
709:
706:
703:
700:
697:
694:
689:
686:
683:
679:
675:
670:
666:
662:
659:
656:
653:
650:
647:
644:
641:
619:
597:
593:
570:
566:
545:
523:
519:
498:
495:
492:
472:
469:
466:
440:
436:
432:
417:Main article:
414:
411:
396:
392:
371:
345:
342:
338:
334:
329:
324:
320:
295:
291:
287:
265:
262:
258:
237:
210:
207:
203:
199:
196:
193:
190:
186:
181:
176:
172:
168:
144:
140:
136:
120:
117:
112:
109:
97:gelling agents
92:
89:
55:gel transition
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1465:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1420:
1418:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1373:(1): 87–110.
1372:
1368:
1361:
1358:
1353:
1349:
1345:
1341:
1334:
1331:
1326:
1322:
1318:
1314:
1310:
1306:
1299:
1296:
1291:
1287:
1283:
1277:
1273:
1266:
1263:
1258:
1256:9781420007329
1252:
1248:
1244:
1240:
1233:
1230:
1225:
1219:
1215:
1211:
1207:
1200:
1197:
1192:
1188:
1183:
1178:
1174:
1170:
1166:
1162:
1158:
1151:
1148:
1143:
1141:9781845692643
1137:
1133:
1129:
1125:
1118:
1115:
1108:
1104:
1101:
1100:
1096:
1094:
1089:
1081:
1079:
1077:
1073:
1069:
1063:
1055:
1053:
1050:
1036:
1027:
1009:
1005:
1001:
987:
984:
981:
975:
972:
969:
963:
956:
951:
946:
942:
933:
930:
913:
910:
907:
899:
895:
874:
869:
865:
844:
839:
835:
829:
819:
815:
808:
805:
802:
794:
790:
781:
777:
767:
752:
749:
746:
742:
738:
735:
732:
729:
726:
721:
713:
710:
707:
704:
701:
698:
692:
687:
684:
681:
677:
668:
664:
660:
657:
654:
651:
648:
645:
642:
639:
631:
617:
595:
591:
568:
564:
543:
521:
517:
496:
493:
490:
470:
467:
464:
455:
438:
434:
430:
420:
412:
410:
394:
390:
369:
360:
343:
340:
336:
332:
327:
322:
318:
309:
293:
289:
285:
263:
260:
256:
235:
226:
208:
205:
201:
197:
194:
191:
188:
184:
179:
174:
170:
166:
158:
142:
138:
134:
126:
118:
116:
110:
108:
106:
102:
98:
90:
88:
85:
81:
77:
72:
68:
64:
60:
56:
52:
48:
40:
36:
32:
19:
1433:Dosage forms
1370:
1366:
1360:
1343:
1339:
1333:
1311:(2): 45–55.
1308:
1304:
1298:
1271:
1265:
1238:
1232:
1205:
1199:
1164:
1160:
1150:
1123:
1117:
1091:
1082:Random graph
1065:
1051:
1028:
934:
931:
768:
632:
456:
422:
361:
310:
227:
159:
157:is given by
122:
114:
101:crosslinking
94:
54:
50:
44:
39:crosslinking
1417:Categories
1223:0471274003
1109:References
91:Background
1403:120859837
1395:1572-9613
985:−
976:ρ
914:ρ
911:−
875:ρ
845:ρ
809:ρ
806:−
750:−
739:−
730:−
711:−
702:−
693:−
685:−
674:→
655:−
643:−
618:ρ
494:−
468:−
192:−
84:viscosity
80:gel point
69:can form
1438:Colloids
1290:50339757
1191:25750745
1097:See also
1078:arises.
536:, where
63:polymers
51:gelation
35:Polymers
1375:Bibcode
1313:Bibcode
1182:4348459
1070:or the
18:Gelling
1401:
1393:
1288:
1278:
1253:
1220:
1189:
1179:
1138:
228:where
1399:S2CID
71:links
41:(gel)
1423:Gels
1391:ISSN
1286:OCLC
1276:ISBN
1251:ISBN
1218:ISBN
1187:PMID
1136:ISBN
583:and
362:If
1383:doi
1348:doi
1321:doi
1243:doi
1210:doi
1177:PMC
1169:doi
1128:doi
59:gel
45:In
1419::
1397:.
1389:.
1381:.
1371:64
1369:.
1344:63
1342:.
1319:.
1309:11
1307:.
1284:.
1249:.
1241:.
1216:.
1208:.
1185:.
1175:.
1163:.
1159:.
1134:.
929:.
483:,
65:.
49:,
1405:.
1385::
1377::
1354:.
1350::
1327:.
1323::
1315::
1292:.
1259:.
1245::
1226:.
1212::
1193:.
1171::
1165:6
1144:.
1130::
1037:f
1010:2
1006:/
1002:1
998:}
994:]
991:)
988:2
982:f
979:(
973:+
970:1
967:[
964:r
961:{
957:1
952:=
947:c
943:p
917:)
908:1
905:(
900:B
896:p
870:B
866:p
840:B
836:p
830:n
826:]
820:A
816:p
812:)
803:1
800:(
795:B
791:p
787:[
782:A
778:p
753:1
747:f
743:A
736:A
733:B
727:B
722:n
718:)
714:A
708:A
705:B
699:B
696:(
688:1
682:f
678:A
669:f
665:A
661:+
658:B
652:B
649:+
646:A
640:A
596:B
592:p
569:A
565:p
544:f
522:f
518:A
497:B
491:B
471:A
465:A
439:n
435:P
431:D
395:c
391:p
370:p
344:v
341:a
337:f
333:2
328:=
323:c
319:p
294:n
290:P
286:D
264:v
261:a
257:f
236:p
209:v
206:a
202:f
198:.
195:p
189:2
185:2
180:=
175:n
171:P
167:D
143:n
139:P
135:D
53:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
