838:
419:
833:{\displaystyle \Delta P=\Delta P_{i}+\Delta P_{o}=2\gamma \left({\frac {1}{R_{i}}}+{\frac {1}{R_{i}+d}}\right)={\frac {2\gamma }{R_{i}}}\left(1+{\frac {R_{i}}{R_{i}+d}}\right)={\frac {2\gamma }{R_{i}}}\left(1+{\frac {R_{i}+d}{R_{i}+d}}-{\frac {d}{R_{i}+d}}\right)={\frac {4\gamma }{R_{i}}}\left(1-{\frac {1}{2}}{\frac {d}{R_{i}+d}}\right)\approx {\frac {4\gamma }{R_{i}}}+{\mathcal {O}}(d).}
20:
366:
For a gas bubble within a liquid, there is only one surface. For a gas bubble with a liquid wall, beyond which is again gas, there are two surfaces, each contributing to the total pressure difference. If the bubble is spherical and the outer radius differs from the inner radius by a small distance,
967:
A 1 mm bubble has negligible extra pressure. Yet when the diameter is ~3 ÎĽm, the bubble has an extra atmosphere inside than outside. When the bubble is only several hundred nanometers, the pressure inside can be several atmospheres. One should bear in mind that the surface tension in the
161:
23:
27:
26:
22:
21:
28:
968:
numerator can be much smaller in the presence of surfactants or contaminants. The same calculation can be done for small oil droplets in water, where even in the presence of surfactants and a fairly low interfacial tension
25:
61:
361:
411:
24:
926:
901:
986:
866:
262:
242:
319:
292:
218:
191:
44:
difference between the inside and the outside of a curved surface that forms the boundary between two fluid regions. The pressure difference is caused by the
264:) is the surface tension. Although signs for these values vary, sign convention usually dictates positive curvature when convex and negative when concave.
1091:
1075:
267:
The
Laplace pressure is commonly used to determine the pressure difference in spherical shapes such as bubbles or droplets. In this case,
156:{\displaystyle \Delta P\equiv P_{\text{inside}}-P_{\text{outside}}=\gamma \left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right),}
868:= 72 mN/m at 25 °C (298 K). The extra pressure inside the bubble is given here for three bubble sizes:
52:
327:
1132:
1065:
988:= 5–10 mN/m, the pressure inside 100 nm diameter droplets can reach several atmospheres.
1127:
1012:
1037:
1095:
1032:
Butt, Hans-JĂĽrgen; Graf, Karlheinz; Kappl, Michael (2006). "Physics and
Chemistry of Interfaces": 9.
370:
221:
1122:
1071:
908:
883:
971:
851:
247:
227:
997:
297:
270:
196:
169:
1050:
1002:
848:
A common example of use is finding the pressure inside an air bubble in pure water, where
45:
1007:
1116:
48:
of the interface between liquid and gas, or between two immiscible liquids.
16:
Pressure difference between the inside and the outside of a curved surface
41:
1064:
Gennes, Pierre-Gilles de; Francoise
Brochard-Wyart; David Quere (2004).
18:
32:
Experimental demonstration of
Laplace pressure with soap bubbles.
813:
974:
911:
886:
854:
422:
373:
330:
300:
273:
250:
230:
199:
172:
64:
980:
920:
895:
860:
832:
405:
355:
313:
286:
256:
236:
212:
185:
155:
356:{\displaystyle \Delta P=\gamma {\frac {2}{R}}}
8:
51:The Laplace pressure is determined from the
973:
910:
885:
853:
812:
811:
800:
786:
763:
753:
743:
724:
710:
687:
677:
659:
641:
634:
615:
601:
578:
567:
561:
542:
528:
505:
495:
484:
475:
455:
439:
421:
391:
378:
372:
343:
329:
305:
299:
278:
272:
249:
229:
204:
198:
177:
171:
137:
128:
117:
108:
91:
78:
63:
870:
1024:
1046:
1035:
7:
912:
887:
448:
432:
423:
331:
65:
14:
1067:Capillarity and Wetting Phenomena
824:
818:
1:
1092:"Antimicrobial Nanoemulsions"
406:{\displaystyle R_{o}=R_{i}+d}
222:principal radii of curvature
1149:
1070:. Springer. pp. 7–8.
921:{\displaystyle \Delta P}
896:{\displaystyle \Delta P}
981:{\displaystyle \gamma }
861:{\displaystyle \gamma }
257:{\displaystyle \sigma }
237:{\displaystyle \gamma }
1045:Cite journal requires
1013:Two balloon experiment
982:
922:
897:
862:
834:
407:
357:
315:
288:
258:
238:
214:
187:
157:
53:Young–Laplace equation
33:
983:
923:
898:
863:
835:
408:
358:
316:
314:{\displaystyle R_{2}}
289:
287:{\displaystyle R_{1}}
259:
239:
215:
213:{\displaystyle R_{2}}
188:
186:{\displaystyle R_{1}}
158:
31:
972:
909:
884:
852:
420:
371:
328:
298:
271:
248:
228:
197:
170:
62:
978:
918:
893:
874:Bubble diameter (2
858:
830:
403:
353:
311:
284:
254:
234:
210:
183:
153:
34:
1133:Bubbles (physics)
1077:978-0-387-00592-8
965:
964:
806:
776:
751:
730:
700:
672:
621:
591:
548:
518:
490:
351:
244:(also denoted as
143:
123:
94:
81:
29:
1140:
1107:
1106:
1104:
1103:
1094:. Archived from
1088:
1082:
1081:
1061:
1055:
1054:
1048:
1043:
1041:
1033:
1029:
998:Ostwald ripening
987:
985:
984:
979:
927:
925:
924:
919:
902:
900:
899:
894:
871:
867:
865:
864:
859:
839:
837:
836:
831:
817:
816:
807:
805:
804:
795:
787:
782:
778:
777:
775:
768:
767:
754:
752:
744:
731:
729:
728:
719:
711:
706:
702:
701:
699:
692:
691:
678:
673:
671:
664:
663:
653:
646:
645:
635:
622:
620:
619:
610:
602:
597:
593:
592:
590:
583:
582:
572:
571:
562:
549:
547:
546:
537:
529:
524:
520:
519:
517:
510:
509:
496:
491:
489:
488:
476:
460:
459:
444:
443:
412:
410:
409:
404:
396:
395:
383:
382:
362:
360:
359:
354:
352:
344:
320:
318:
317:
312:
310:
309:
293:
291:
290:
285:
283:
282:
263:
261:
260:
255:
243:
241:
240:
235:
219:
217:
216:
211:
209:
208:
192:
190:
189:
184:
182:
181:
162:
160:
159:
154:
149:
145:
144:
142:
141:
129:
124:
122:
121:
109:
96:
95:
92:
83:
82:
79:
38:Laplace pressure
30:
1148:
1147:
1143:
1142:
1141:
1139:
1138:
1137:
1113:
1112:
1111:
1110:
1101:
1099:
1090:
1089:
1085:
1078:
1063:
1062:
1058:
1044:
1034:
1031:
1030:
1026:
1021:
1003:Kelvin equation
994:
970:
969:
907:
906:
882:
881:
850:
849:
846:
796:
788:
759:
758:
736:
732:
720:
712:
683:
682:
655:
654:
637:
636:
627:
623:
611:
603:
574:
573:
563:
554:
550:
538:
530:
501:
500:
480:
474:
470:
451:
435:
418:
417:
387:
374:
369:
368:
326:
325:
301:
296:
295:
274:
269:
268:
246:
245:
226:
225:
200:
195:
194:
173:
168:
167:
133:
113:
107:
103:
87:
74:
60:
59:
46:surface tension
19:
17:
12:
11:
5:
1146:
1144:
1136:
1135:
1130:
1128:Fluid dynamics
1125:
1115:
1114:
1109:
1108:
1083:
1076:
1056:
1047:|journal=
1023:
1022:
1020:
1017:
1016:
1015:
1010:
1008:Laplace number
1005:
1000:
993:
990:
977:
963:
962:
959:
956:
952:
951:
948:
945:
941:
940:
937:
934:
930:
929:
917:
914:
904:
892:
889:
879:
857:
845:
842:
841:
840:
829:
826:
823:
820:
815:
810:
803:
799:
794:
791:
785:
781:
774:
771:
766:
762:
757:
750:
747:
742:
739:
735:
727:
723:
718:
715:
709:
705:
698:
695:
690:
686:
681:
676:
670:
667:
662:
658:
652:
649:
644:
640:
633:
630:
626:
618:
614:
609:
606:
600:
596:
589:
586:
581:
577:
570:
566:
560:
557:
553:
545:
541:
536:
533:
527:
523:
516:
513:
508:
504:
499:
494:
487:
483:
479:
473:
469:
466:
463:
458:
454:
450:
447:
442:
438:
434:
431:
428:
425:
402:
399:
394:
390:
386:
381:
377:
364:
363:
350:
347:
342:
339:
336:
333:
308:
304:
281:
277:
253:
233:
207:
203:
180:
176:
164:
163:
152:
148:
140:
136:
132:
127:
120:
116:
112:
106:
102:
99:
90:
86:
77:
73:
70:
67:
15:
13:
10:
9:
6:
4:
3:
2:
1145:
1134:
1131:
1129:
1126:
1124:
1121:
1120:
1118:
1098:on 2010-04-17
1097:
1093:
1087:
1084:
1079:
1073:
1069:
1068:
1060:
1057:
1052:
1039:
1028:
1025:
1018:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
995:
991:
989:
975:
960:
957:
954:
953:
949:
946:
943:
942:
938:
935:
932:
931:
915:
905:
890:
880:
877:
873:
872:
869:
855:
843:
827:
821:
808:
801:
797:
792:
789:
783:
779:
772:
769:
764:
760:
755:
748:
745:
740:
737:
733:
725:
721:
716:
713:
707:
703:
696:
693:
688:
684:
679:
674:
668:
665:
660:
656:
650:
647:
642:
638:
631:
628:
624:
616:
612:
607:
604:
598:
594:
587:
584:
579:
575:
568:
564:
558:
555:
551:
543:
539:
534:
531:
525:
521:
514:
511:
506:
502:
497:
492:
485:
481:
477:
471:
467:
464:
461:
456:
452:
445:
440:
436:
429:
426:
416:
415:
414:
400:
397:
392:
388:
384:
379:
375:
348:
345:
340:
337:
334:
324:
323:
322:
306:
302:
279:
275:
265:
251:
231:
223:
205:
201:
178:
174:
150:
146:
138:
134:
130:
125:
118:
114:
110:
104:
100:
97:
88:
84:
75:
71:
68:
58:
57:
56:
54:
49:
47:
43:
39:
1100:. Retrieved
1096:the original
1086:
1066:
1059:
1038:cite journal
1027:
966:
875:
847:
365:
266:
165:
50:
37:
35:
1117:Categories
1102:2010-04-03
1019:References
413:, we find
976:γ
913:Δ
888:Δ
856:γ
793:γ
784:≈
741:−
717:γ
675:−
608:γ
535:γ
468:γ
449:Δ
433:Δ
424:Δ
341:γ
332:Δ
252:σ
232:γ
101:γ
85:−
72:≡
66:Δ
55:given as
1123:Pressure
992:See also
939:0.00284
844:Examples
220:are the
42:pressure
958:960000
878:) (ÎĽm)
93:outside
40:is the
1074:
961:9.474
950:0.947
947:96000
928:(atm)
166:where
80:inside
933:1000
903:(Pa)
1072:ISBN
1051:help
955:0.3
944:3.0
936:288
224:and
193:and
36:The
1119::
1042::
1040:}}
1036:{{
321::
294:=
1105:.
1080:.
1053:)
1049:(
916:P
891:P
876:r
828:.
825:)
822:d
819:(
814:O
809:+
802:i
798:R
790:4
780:)
773:d
770:+
765:i
761:R
756:d
749:2
746:1
738:1
734:(
726:i
722:R
714:4
708:=
704:)
697:d
694:+
689:i
685:R
680:d
669:d
666:+
661:i
657:R
651:d
648:+
643:i
639:R
632:+
629:1
625:(
617:i
613:R
605:2
599:=
595:)
588:d
585:+
580:i
576:R
569:i
565:R
559:+
556:1
552:(
544:i
540:R
532:2
526:=
522:)
515:d
512:+
507:i
503:R
498:1
493:+
486:i
482:R
478:1
472:(
465:2
462:=
457:o
453:P
446:+
441:i
437:P
430:=
427:P
401:d
398:+
393:i
389:R
385:=
380:o
376:R
349:R
346:2
338:=
335:P
307:2
303:R
280:1
276:R
206:2
202:R
179:1
175:R
151:,
147:)
139:2
135:R
131:1
126:+
119:1
115:R
111:1
105:(
98:=
89:P
76:P
69:P
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