20:
676:
1604:
2053:
In addition to aggregation, particles may also grow in size by condensation, deposition or by accretion. Hassan and Hassan recently proposed a condensation-driven aggregation (CDA) model in which aggregating particles keep growing continuously between merging upon collision. The CDA model can be
2044:
are an attractive alternative. Through this method, to compute the coagulation rates for different coagulation events, the simulation entries are virtualized to be equally weighted. The accuracy of this transformation can be adjusted such that just those coagulation events are considered while
338:
1346:
1122:
2593:
365:
1357:
2199:
2878:
2993:
108:
1141:
932:
1943:
856:
2306:
3743:
Kryven, I.; Iedema, P. D. (2013). "Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding".
2719:
671:{\displaystyle {\frac {\partial n(x_{i},t)}{\partial t}}={\frac {1}{2}}\sum _{j=1}^{i-1}K(x_{i}-x_{j},x_{j})n(x_{i}-x_{j},t)n(x_{j},t)-\sum _{j=1}^{\infty }K(x_{i},x_{j})n(x_{i},t)n(x_{j},t).}
3063:
3273:
Danov, Krassimir D.; Ivanov, Ivan B.; Gurkov, Theodor D.; Borwankar, Rajendra P. (1994). "Kinetic Model for the
Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems".
3708:
Landgrebe, J. D.; Pratsinis, S. E. (1990). "A Discrete-Sectional Model for
Particulate Production by Gas-Phase Chemical Reaction and Aerosol Coagulation in the Free-Molecular Regime".
1599:{\displaystyle K={\frac {2}{3}}{\frac {k_{B}T}{\eta }}{\frac {(x_{1}x_{2})^{\gamma }}{W}}\left(x_{1}^{1/y_{1}}+x_{2}^{1/y_{2}}\right)\left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),}
2060:
1647:
2029:
case, however, when two or more properties (such as size, shape, composition, etc.) are introduced, one has to seek special approximation methods that suffer less from
90:
The distribution of particle size changes in time according to the interrelation of all particles of the system. Therefore, the
Smoluchowski coagulation equation is an
2742:
2645:
2238:
1758:
3128:
2750:
3093:
2298:
1738:
1678:
757:
730:
2004:
1975:
348:
1760:
is the exponent of the product kernel, usually considered a fitting parameter. For cloud, the kernel for coagulation of cloud particles are usually expressed as:
897:
2616:
2278:
2258:
1718:
1698:
3817:"A Monte Carlo method for the simulation of coagulation and nucleation based on weighted particles and the concepts of stochastic resolution and merging"
914:
However, in most practical applications the kernel takes on a significantly more complex form. For example, the free-molecular kernel which describes
333:{\displaystyle {\frac {\partial n(x,t)}{\partial t}}={\frac {1}{2}}\int _{0}^{x}K(x-y,y)n(x-y,t)n(y,t)\,dy-\int _{0}^{\infty }K(x,y)n(x,t)n(y,t)\,dy.}
2889:
2009:
Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to
1341:{\displaystyle K={\frac {2}{3}}{\frac {k_{B}T}{\eta }}\left(x_{1}^{1/y_{1}}+x_{2}^{1/y_{2}}\right)\left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),}
1117:{\displaystyle K={\sqrt {\frac {\pi k_{B}T}{2}}}\left({\frac {1}{m(x_{1})}}+{\frac {1}{m(x_{2})}}\right)^{1/2}\left(d(x_{1})+d(x_{2})\right)^{2}.}
3213:
Blatz, P. J.; Tobolsky, A. V. (1945). "Note on the
Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena".
1766:
3511:
Marchisio, D. L.; Fox, R. O. (2005). "Solution of
Population Balance Equa- tions Using the Direct Quadrature Method of Moments".
3434:
Kreer, Markus; Penrose, Oliver (1994). "Proof of dynamical scaling in
Smoluchowski's coagulation equation with constant kernel".
769:
3546:
Yu, M.; Lin, J.; Chan, T. (2008). "A New Moment Method for
Solving the Coagulation Equation for Particles in Brownian Motion".
2022:
3186:
Smoluchowski, Marian (1916). "Drei Vorträge über
Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen".
3394:"An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach"
2588:{\displaystyle {\Big }n(x,t)=-n(x,t)\int _{0}^{\infty }K(x,y)n(y,t)dy+{{1} \over {2}}\int _{0}^{x}dyK(y,x-y)n(y,t)n(x-y,t).}
23:
This diagram describes the aggregation kinetics of discrete particles according to the
Smoluchowski aggregation equation.
3257:
3147:
2650:
3012:
899:
it could be mathematically proven that the solution of
Smoluchowski coagulation equations have asymptotically the
3891:
3478:
3869:
M. K. Hassan and M. Z. Hassan, “Emergence of fractal behavior in condensation-driven aggregation”, Phys. Rev. E
3896:
2300:
is the elapsed time. This reaction scheme can be described by the following generalized Smoluchowski equation
1132:
91:
36:
3816:
3006:
analysis reveals that the condensation-driven aggregation can be best described fractal of dimension
2194:{\displaystyle A_{x}(t)+A_{y}(t){\stackrel {v(x,t)}{\longrightarrow }}A_{(\alpha +1)(x+y)}(t+\tau ),}
3162:
2030:
1612:
870:
67:
3853:
M. K. Hassan and M. Z. Hassan, “Condensation-driven aggregation in one dimension”, Phys. Rev. E
3669:"Log-Normally Preserving Size Distribution for Brownian Coagulation in the Free-Molecule Regime"
2873:{\displaystyle v(x,t)={{\alpha x} \over {\tau (x)}}=\alpha x\int _{0}^{\infty }dyK(x,y)n(y,t).}
2034:
2026:
862:
760:
2724:
2621:
2207:
94:
of the particle-size distribution. In the case when the sizes of the coagulated particles are
1743:
3098:
3717:
3680:
3612:
3555:
3520:
3443:
3408:
3282:
3195:
3071:
2283:
1723:
1656:
735:
708:
1980:
1951:
8:
3157:
2006:
are the radius and fall speed of the cloud particles usually expressed using power law.
876:
95:
40:
28:
3721:
3684:
3616:
3559:
3524:
3447:
3412:
3286:
3199:
3130:
is always a conserved quantity which is responsible for fixing all the exponents of the
3770:
3752:
3581:
3459:
2601:
2263:
2243:
2041:
2010:
1703:
1683:
3653:
3369:
3352:
3329:
2037:
has been successfully applied to the coagulation equation in more than one dimension.
3836:
3729:
3585:
3374:
3333:
3298:
3253:
3230:
1650:
1128:
352:
3774:
3463:
19:
3828:
3797:
3762:
3725:
3688:
3649:
3620:
3571:
3563:
3532:
3528:
3493:
3451:
3416:
3364:
3325:
3290:
3222:
908:
904:
3766:
3393:
3316:
Thomas, D.N.; Judd, S.J.; Fawcett, N. (1999). "Flocculation modelling: a review".
3479:"Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems"
3131:
2999:
2883:
One can solve the generalized Smoluchowski equation for constant kernel to give
900:
702:
694:
682:
3788:
Kryven, I.; Iedema, P. D. (2014). "Topology Evolution in Polymer Modification".
3420:
62:
Simultaneous coagulation (or aggregation) is encountered in processes involving
923:
63:
48:
44:
3832:
3693:
3668:
3625:
3600:
3567:
2988:{\displaystyle n(x,t)\sim t^{-(2+2\alpha )}e^{-{{x} \over {t^{1+2\alpha }}}},}
3885:
3874:
3858:
3840:
3378:
3337:
3302:
3234:
2014:
51:
of particles as they coagulate (in this context "clumping together") to size
3801:
3497:
3294:
3152:
866:
79:
3167:
3226:
3455:
3135:
763:
to the equation exist when the kernel takes one of three simple forms:
3640:
Frenklach, M. (2002). "Method of Moments with Interpolative Closure".
3576:
3601:"Description of Aerosol Dynamics by the Quadrature Method of Moments"
915:
356:
1938:{\displaystyle K=\pi ^{2}|v(x_{1})-v(x_{2})|E_{coll}(x_{1},x_{2}),}
99:
75:
71:
3757:
3003:
2040:
When the accuracy of the solution is not of primary importance,
2017:
methods can be used when there is only one particle property (
903:
property. This self-similar behaviour is closely related to
851:{\displaystyle K=1,\quad K=x_{1}+x_{2},\quad K=x_{1}x_{2},}
3272:
919:
3101:
3074:
3015:
2892:
2753:
2727:
2653:
2624:
2604:
2309:
2286:
2266:
2246:
2210:
2063:
1983:
1954:
1769:
1746:
1726:
1706:
1686:
1659:
1615:
1360:
1144:
935:
879:
772:
738:
711:
368:
111:
3476:
2045:keeping the number of simulation entries constant.
355:sizes, then the discrete form of the equation is a
3707:
3122:
3087:
3057:
2987:
2872:
2736:
2713:
2639:
2610:
2587:
2292:
2272:
2252:
2232:
2193:
2048:
1998:
1969:
1937:
1752:
1732:
1712:
1692:
1672:
1641:
1598:
1340:
1116:
891:
850:
751:
724:
705:and describes the rate at which particles of size
670:
332:
16:Population balance equation in statistical physics
3357:Transactions of the American Mathematical Society
3315:
2714:{\displaystyle \int _{0}^{\infty }K(x,y)n(y,t)dy}
2618:grows due to condensation between collision time
2374:
2312:
3883:
3206:
2021:) of interest, the two principal ones being the
1127:Some coagulation kernels account for a specific
3179:
3134:. Such conservation law has also been found in
3058:{\displaystyle d_{f}={{1} \over {1+2\alpha }}.}
3814:
3666:
3266:
43:in a seminal 1916 publication, describing the
3667:Lee, K. W.; Chen, H.; Gieseke, J. A. (1984).
3309:
3212:
3787:
3742:
3510:
3477:Kryven, I.; Lazzari, S.; Storti, G. (2014).
3433:
3185:
2054:understood by the following reaction scheme
681:There exists a unique solution for a chosen
907:which can be a characteristic feature of a
3875:https://doi.org/10.1103/PhysRevE.79.021406
3859:https://doi.org/10.1103/PhysRevE.77.061404
3815:Kotalczyk, G.; Kruis, F. E. (2017-07-01).
3736:
3427:
3247:
3756:
3692:
3639:
3624:
3575:
3545:
3368:
3344:
2042:stochastic particle (Monte Carlo) methods
320:
241:
3275:Journal of Colloid and Interface Science
18:
1740:is the continuous phase viscosity, and
3884:
3598:
3391:
3350:
3252:. John Wiley & Sons. p. 492.
3790:Macromolecular Theory and Simulations
3486:Macromolecular Theory and Simulations
688:
2598:Considering that a particle of size
873:kernels respectively. For the case
13:
2820:
2664:
2431:
2345:
2340:
2325:
2320:
2033:. Approximation based on Gaussian
578:
402:
372:
261:
138:
115:
14:
3908:
3370:10.1090/S0002-9947-1957-0087880-6
3215:The Journal of Physical Chemistry
1351:or Reaction-limited aggregation:
732:coagulate with particles of size
33:Smoluchowski coagulation equation
3821:Journal of Computational Physics
3250:Aerosols: Science and Technology
3863:
3847:
3808:
3781:
3701:
3660:
3633:
3592:
3539:
3504:
2049:Condensation-driven aggregation
2025:and sectional methods. In the
818:
785:
3533:10.1016/j.jaerosci.2004.07.009
3470:
3436:Journal of Statistical Physics
3401:Physica D: Nonlinear Phenomena
3385:
3241:
3148:Einstein–Smoluchowski relation
3117:
3105:
2937:
2922:
2908:
2896:
2864:
2852:
2846:
2834:
2795:
2789:
2769:
2757:
2702:
2690:
2684:
2672:
2634:
2628:
2579:
2561:
2555:
2543:
2537:
2519:
2469:
2457:
2451:
2439:
2418:
2406:
2394:
2382:
2369:
2357:
2240:denotes the aggregate of size
2227:
2221:
2185:
2173:
2168:
2156:
2153:
2141:
2129:
2117:
2109:
2102:
2096:
2080:
2074:
1993:
1987:
1964:
1958:
1929:
1903:
1880:
1876:
1863:
1854:
1841:
1834:
1824:
1820:
1807:
1798:
1785:
1779:
1720:is the Fuchs stability ratio,
1426:
1402:
1097:
1084:
1075:
1062:
1028:
1015:
997:
984:
701:, is known as the coagulation
662:
643:
637:
618:
612:
586:
556:
537:
531:
499:
493:
454:
397:
378:
351:, i.e. when particles join in
317:
305:
299:
287:
281:
269:
238:
226:
220:
202:
196:
178:
133:
121:
1:
3767:10.1016/j.polymer.2013.05.009
3654:10.1016/S0009-2509(02)00113-6
3353:"A scalar transport equation"
3330:10.1016/S0043-1354(98)00392-3
3173:
1133:diffusion-limited aggregation
347:is interpreted as a discrete
3730:10.1016/0021-9797(90)90445-T
92:integrodifferential equation
7:
3421:10.1016/j.physd.2006.07.024
3141:
1680:is the Boltzmann constant,
1642:{\displaystyle y_{1},y_{2}}
1131:of the clusters, as in the
98:, the equation involves an
85:
37:population balance equation
10:
3913:
3833:10.1016/j.jcp.2017.03.041
3694:10.1080/02786828408958993
3626:10.1080/02786829708965471
3568:10.1080/02786820802232972
3392:Wattis, J. A. D. (2006).
3248:Agranovski, Igor (2011).
3710:J. Colloid Interface Sci
2737:{\displaystyle \alpha x}
2640:{\displaystyle \tau (x)}
2233:{\displaystyle A_{x}(t)}
3163:Williams spray equation
2031:curse of dimensionality
1753:{\displaystyle \gamma }
3892:Differential equations
3802:10.1002/mats.201300121
3498:10.1002/mats.201300140
3351:Melzak, Z. A. (1957).
3295:10.1006/jcis.1994.1328
3124:
3123:{\displaystyle n(x,t)}
3089:
3059:
2989:
2874:
2738:
2715:
2641:
2612:
2589:
2294:
2274:
2254:
2234:
2195:
2035:radial basis functions
2000:
1971:
1939:
1754:
1734:
1714:
1694:
1674:
1643:
1600:
1342:
1118:
893:
852:
753:
726:
672:
582:
450:
334:
24:
3897:Statistical mechanics
3125:
3090:
3088:{\displaystyle d_{f}}
3060:
2990:
2875:
2739:
2716:
2642:
2613:
2590:
2295:
2293:{\displaystyle \tau }
2275:
2255:
2235:
2196:
2001:
1972:
1940:
1755:
1735:
1733:{\displaystyle \eta }
1715:
1695:
1675:
1673:{\displaystyle k_{B}}
1644:
1601:
1343:
1119:
894:
853:
754:
752:{\displaystyle x_{2}}
727:
725:{\displaystyle x_{1}}
673:
562:
424:
335:
22:
3673:Aerosol Sci. Technol
3605:Aerosol Sci. Technol
3548:Aerosol Sci. Technol
3194:: 557–571, 585–599.
3099:
3072:
3013:
2890:
2751:
2725:
2651:
2647:equal to inverse of
2622:
2602:
2307:
2284:
2264:
2244:
2208:
2061:
1999:{\displaystyle v(x)}
1981:
1970:{\displaystyle r(x)}
1952:
1767:
1744:
1724:
1704:
1700:is the temperature,
1684:
1657:
1613:
1358:
1142:
933:
877:
770:
736:
709:
366:
109:
96:continuous variables
3722:1990JCIS..139...63L
3685:1984AerST...3...53L
3617:1997AerST..27..255M
3599:McGraw, R. (1997).
3560:2008AerST..42..705Y
3525:2005JAerS..36...43M
3448:1994JSP....75..389K
3413:2006PhyD..222....1W
3287:1994JCIS..167....8D
3227:10.1021/j150440a004
3200:1916ZPhy...17..557S
3158:Smoluchowski factor
2824:
2668:
2509:
2435:
1587:
1551:
1508:
1475:
1329:
1293:
1250:
1217:
892:{\displaystyle K=1}
265:
174:
41:Marian Smoluchowski
29:statistical physics
3456:10.1007/BF02186868
3120:
3085:
3055:
2985:
2870:
2810:
2734:
2711:
2654:
2637:
2608:
2585:
2495:
2421:
2290:
2270:
2250:
2230:
2191:
1996:
1967:
1935:
1750:
1730:
1710:
1690:
1670:
1651:fractal dimensions
1639:
1596:
1555:
1519:
1479:
1446:
1338:
1297:
1261:
1221:
1188:
1114:
889:
848:
761:Analytic solutions
749:
722:
689:Coagulation kernel
668:
330:
251:
160:
25:
3751:(14): 3472–3484.
3648:(12): 2229–2239.
3050:
2978:
2799:
2611:{\displaystyle x}
2493:
2352:
2332:
2273:{\displaystyle t}
2253:{\displaystyle x}
2133:
2023:method of moments
2011:numerical methods
1713:{\displaystyle W}
1693:{\displaystyle T}
1653:of the clusters,
1439:
1397:
1375:
1181:
1159:
1129:fractal dimension
1032:
1001:
967:
966:
422:
409:
158:
145:
3904:
3877:
3867:
3861:
3851:
3845:
3844:
3812:
3806:
3805:
3785:
3779:
3778:
3760:
3740:
3734:
3733:
3705:
3699:
3698:
3696:
3664:
3658:
3657:
3637:
3631:
3630:
3628:
3596:
3590:
3589:
3579:
3543:
3537:
3536:
3508:
3502:
3501:
3483:
3474:
3468:
3467:
3431:
3425:
3424:
3398:
3389:
3383:
3382:
3372:
3348:
3342:
3341:
3324:(7): 1579–1592.
3313:
3307:
3306:
3270:
3264:
3263:
3245:
3239:
3238:
3210:
3204:
3203:
3183:
3129:
3127:
3126:
3121:
3094:
3092:
3091:
3086:
3084:
3083:
3064:
3062:
3061:
3056:
3051:
3049:
3035:
3030:
3025:
3024:
2994:
2992:
2991:
2986:
2981:
2980:
2979:
2977:
2976:
2975:
2956:
2951:
2941:
2940:
2879:
2877:
2876:
2871:
2823:
2818:
2800:
2798:
2784:
2776:
2743:
2741:
2740:
2735:
2720:
2718:
2717:
2712:
2667:
2662:
2646:
2644:
2643:
2638:
2617:
2615:
2614:
2609:
2594:
2592:
2591:
2586:
2508:
2503:
2494:
2492:
2487:
2482:
2434:
2429:
2378:
2377:
2353:
2351:
2343:
2338:
2333:
2331:
2323:
2318:
2316:
2315:
2299:
2297:
2296:
2291:
2279:
2277:
2276:
2271:
2259:
2257:
2256:
2251:
2239:
2237:
2236:
2231:
2220:
2219:
2200:
2198:
2197:
2192:
2172:
2171:
2135:
2134:
2132:
2112:
2107:
2095:
2094:
2073:
2072:
2005:
2003:
2002:
1997:
1976:
1974:
1973:
1968:
1944:
1942:
1941:
1936:
1928:
1927:
1915:
1914:
1902:
1901:
1883:
1875:
1874:
1853:
1852:
1837:
1832:
1831:
1819:
1818:
1797:
1796:
1759:
1757:
1756:
1751:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1679:
1677:
1676:
1671:
1669:
1668:
1648:
1646:
1645:
1640:
1638:
1637:
1625:
1624:
1605:
1603:
1602:
1597:
1592:
1588:
1586:
1585:
1584:
1575:
1563:
1550:
1549:
1548:
1539:
1527:
1513:
1509:
1507:
1506:
1505:
1496:
1487:
1474:
1473:
1472:
1463:
1454:
1440:
1435:
1434:
1433:
1424:
1423:
1414:
1413:
1400:
1398:
1393:
1389:
1388:
1378:
1376:
1368:
1347:
1345:
1344:
1339:
1334:
1330:
1328:
1327:
1326:
1317:
1305:
1292:
1291:
1290:
1281:
1269:
1255:
1251:
1249:
1248:
1247:
1238:
1229:
1216:
1215:
1214:
1205:
1196:
1182:
1177:
1173:
1172:
1162:
1160:
1152:
1123:
1121:
1120:
1115:
1110:
1109:
1104:
1100:
1096:
1095:
1074:
1073:
1052:
1051:
1047:
1038:
1034:
1033:
1031:
1027:
1026:
1007:
1002:
1000:
996:
995:
976:
968:
962:
958:
957:
944:
943:
909:phase transition
905:scale invariance
898:
896:
895:
890:
857:
855:
854:
849:
844:
843:
834:
833:
814:
813:
801:
800:
758:
756:
755:
750:
748:
747:
731:
729:
728:
723:
721:
720:
677:
675:
674:
669:
655:
654:
630:
629:
611:
610:
598:
597:
581:
576:
549:
548:
524:
523:
511:
510:
492:
491:
479:
478:
466:
465:
449:
438:
423:
415:
410:
408:
400:
390:
389:
370:
339:
337:
336:
331:
264:
259:
173:
168:
159:
151:
146:
144:
136:
113:
3912:
3911:
3907:
3906:
3905:
3903:
3902:
3901:
3882:
3881:
3880:
3873:021406 (2009),
3868:
3864:
3857:061404 (2008),
3852:
3848:
3813:
3809:
3786:
3782:
3741:
3737:
3706:
3702:
3665:
3661:
3638:
3634:
3597:
3593:
3544:
3540:
3509:
3505:
3481:
3475:
3471:
3432:
3428:
3396:
3390:
3386:
3349:
3345:
3314:
3310:
3271:
3267:
3260:
3246:
3242:
3211:
3207:
3184:
3180:
3176:
3144:
3132:dynamic scaling
3100:
3097:
3096:
3079:
3075:
3073:
3070:
3069:
3036:
3031:
3029:
3020:
3016:
3014:
3011:
3010:
3000:dynamic scaling
2998:which exhibits
2962:
2958:
2957:
2952:
2950:
2946:
2942:
2918:
2914:
2891:
2888:
2887:
2819:
2814:
2785:
2777:
2775:
2752:
2749:
2748:
2726:
2723:
2722:
2663:
2658:
2652:
2649:
2648:
2623:
2620:
2619:
2603:
2600:
2599:
2504:
2499:
2488:
2483:
2481:
2430:
2425:
2373:
2372:
2344:
2339:
2337:
2324:
2319:
2317:
2311:
2310:
2308:
2305:
2304:
2285:
2282:
2281:
2265:
2262:
2261:
2245:
2242:
2241:
2215:
2211:
2209:
2206:
2205:
2140:
2136:
2113:
2108:
2106:
2105:
2090:
2086:
2068:
2064:
2062:
2059:
2058:
2051:
1982:
1979:
1978:
1953:
1950:
1949:
1923:
1919:
1910:
1906:
1888:
1884:
1879:
1870:
1866:
1848:
1844:
1833:
1827:
1823:
1814:
1810:
1792:
1788:
1768:
1765:
1764:
1745:
1742:
1741:
1725:
1722:
1721:
1705:
1702:
1701:
1685:
1682:
1681:
1664:
1660:
1658:
1655:
1654:
1633:
1629:
1620:
1616:
1614:
1611:
1610:
1580:
1576:
1571:
1564:
1559:
1544:
1540:
1535:
1528:
1523:
1518:
1514:
1501:
1497:
1492:
1488:
1483:
1468:
1464:
1459:
1455:
1450:
1445:
1441:
1429:
1425:
1419:
1415:
1409:
1405:
1401:
1399:
1384:
1380:
1379:
1377:
1367:
1359:
1356:
1355:
1322:
1318:
1313:
1306:
1301:
1286:
1282:
1277:
1270:
1265:
1260:
1256:
1243:
1239:
1234:
1230:
1225:
1210:
1206:
1201:
1197:
1192:
1187:
1183:
1168:
1164:
1163:
1161:
1151:
1143:
1140:
1139:
1105:
1091:
1087:
1069:
1065:
1058:
1054:
1053:
1043:
1039:
1022:
1018:
1011:
1006:
991:
987:
980:
975:
974:
970:
969:
953:
949:
945:
942:
934:
931:
930:
901:dynamic scaling
878:
875:
874:
839:
835:
829:
825:
809:
805:
796:
792:
771:
768:
767:
743:
739:
737:
734:
733:
716:
712:
710:
707:
706:
691:
683:kernel function
650:
646:
625:
621:
606:
602:
593:
589:
577:
566:
544:
540:
519:
515:
506:
502:
487:
483:
474:
470:
461:
457:
439:
428:
414:
401:
385:
381:
371:
369:
367:
364:
363:
260:
255:
169:
164:
150:
137:
114:
112:
110:
107:
106:
88:
17:
12:
11:
5:
3910:
3900:
3899:
3894:
3879:
3878:
3862:
3846:
3807:
3780:
3735:
3700:
3659:
3642:Chem. Eng. Sci
3632:
3611:(2): 255–265.
3591:
3554:(9): 705–713.
3538:
3513:J. Aerosol Sci
3503:
3469:
3442:(3): 389–407.
3426:
3384:
3363:(2): 547–560.
3343:
3318:Water Research
3308:
3265:
3259:978-3527632084
3258:
3240:
3205:
3177:
3175:
3172:
3171:
3170:
3165:
3160:
3155:
3150:
3143:
3140:
3119:
3116:
3113:
3110:
3107:
3104:
3082:
3078:
3066:
3065:
3054:
3048:
3045:
3042:
3039:
3034:
3028:
3023:
3019:
2996:
2995:
2984:
2974:
2971:
2968:
2965:
2961:
2955:
2949:
2945:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2917:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2881:
2880:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2822:
2817:
2813:
2809:
2806:
2803:
2797:
2794:
2791:
2788:
2783:
2780:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2733:
2730:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2666:
2661:
2657:
2636:
2633:
2630:
2627:
2607:
2596:
2595:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2507:
2502:
2498:
2491:
2486:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2433:
2428:
2424:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2376:
2371:
2368:
2365:
2362:
2359:
2356:
2350:
2347:
2342:
2336:
2330:
2327:
2322:
2314:
2289:
2269:
2249:
2229:
2226:
2223:
2218:
2214:
2202:
2201:
2190:
2187:
2184:
2181:
2178:
2175:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2139:
2131:
2128:
2125:
2122:
2119:
2116:
2111:
2104:
2101:
2098:
2093:
2089:
2085:
2082:
2079:
2076:
2071:
2067:
2050:
2047:
1995:
1992:
1989:
1986:
1966:
1963:
1960:
1957:
1946:
1945:
1934:
1931:
1926:
1922:
1918:
1913:
1909:
1905:
1900:
1897:
1894:
1891:
1887:
1882:
1878:
1873:
1869:
1865:
1862:
1859:
1856:
1851:
1847:
1843:
1840:
1836:
1830:
1826:
1822:
1817:
1813:
1809:
1806:
1803:
1800:
1795:
1791:
1787:
1784:
1781:
1778:
1775:
1772:
1749:
1729:
1709:
1689:
1667:
1663:
1636:
1632:
1628:
1623:
1619:
1607:
1606:
1595:
1591:
1583:
1579:
1574:
1570:
1567:
1562:
1558:
1554:
1547:
1543:
1538:
1534:
1531:
1526:
1522:
1517:
1512:
1504:
1500:
1495:
1491:
1486:
1482:
1478:
1471:
1467:
1462:
1458:
1453:
1449:
1444:
1438:
1432:
1428:
1422:
1418:
1412:
1408:
1404:
1396:
1392:
1387:
1383:
1374:
1371:
1366:
1363:
1349:
1348:
1337:
1333:
1325:
1321:
1316:
1312:
1309:
1304:
1300:
1296:
1289:
1285:
1280:
1276:
1273:
1268:
1264:
1259:
1254:
1246:
1242:
1237:
1233:
1228:
1224:
1220:
1213:
1209:
1204:
1200:
1195:
1191:
1186:
1180:
1176:
1171:
1167:
1158:
1155:
1150:
1147:
1125:
1124:
1113:
1108:
1103:
1099:
1094:
1090:
1086:
1083:
1080:
1077:
1072:
1068:
1064:
1061:
1057:
1050:
1046:
1042:
1037:
1030:
1025:
1021:
1017:
1014:
1010:
1005:
999:
994:
990:
986:
983:
979:
973:
965:
961:
956:
952:
948:
941:
938:
888:
885:
882:
871:multiplicative
859:
858:
847:
842:
838:
832:
828:
824:
821:
817:
812:
808:
804:
799:
795:
791:
788:
784:
781:
778:
775:
746:
742:
719:
715:
690:
687:
679:
678:
667:
664:
661:
658:
653:
649:
645:
642:
639:
636:
633:
628:
624:
620:
617:
614:
609:
605:
601:
596:
592:
588:
585:
580:
575:
572:
569:
565:
561:
558:
555:
552:
547:
543:
539:
536:
533:
530:
527:
522:
518:
514:
509:
505:
501:
498:
495:
490:
486:
482:
477:
473:
469:
464:
460:
456:
453:
448:
445:
442:
437:
434:
431:
427:
421:
418:
413:
407:
404:
399:
396:
393:
388:
384:
380:
377:
374:
341:
340:
329:
326:
323:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
263:
258:
254:
250:
247:
244:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
172:
167:
163:
157:
154:
149:
143:
140:
135:
132:
129:
126:
123:
120:
117:
87:
84:
64:polymerization
49:number density
45:time evolution
39:introduced by
15:
9:
6:
4:
3:
2:
3909:
3898:
3895:
3893:
3890:
3889:
3887:
3876:
3872:
3866:
3860:
3856:
3850:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3811:
3803:
3799:
3795:
3791:
3784:
3776:
3772:
3768:
3764:
3759:
3754:
3750:
3746:
3739:
3731:
3727:
3723:
3719:
3715:
3711:
3704:
3695:
3690:
3686:
3682:
3678:
3674:
3670:
3663:
3655:
3651:
3647:
3643:
3636:
3627:
3622:
3618:
3614:
3610:
3606:
3602:
3595:
3587:
3583:
3578:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3542:
3534:
3530:
3526:
3522:
3518:
3514:
3507:
3499:
3495:
3491:
3487:
3480:
3473:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3437:
3430:
3422:
3418:
3414:
3410:
3407:(1–2): 1–20.
3406:
3402:
3395:
3388:
3380:
3376:
3371:
3366:
3362:
3358:
3354:
3347:
3339:
3335:
3331:
3327:
3323:
3319:
3312:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3269:
3261:
3255:
3251:
3244:
3236:
3232:
3228:
3224:
3220:
3216:
3209:
3201:
3197:
3193:
3190:(in German).
3189:
3182:
3178:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3139:
3137:
3133:
3114:
3111:
3108:
3102:
3095:th moment of
3080:
3076:
3052:
3046:
3043:
3040:
3037:
3032:
3026:
3021:
3017:
3009:
3008:
3007:
3005:
3001:
2982:
2972:
2969:
2966:
2963:
2959:
2953:
2947:
2943:
2934:
2931:
2928:
2925:
2919:
2915:
2911:
2905:
2902:
2899:
2893:
2886:
2885:
2884:
2867:
2861:
2858:
2855:
2849:
2843:
2840:
2837:
2831:
2828:
2825:
2815:
2811:
2807:
2804:
2801:
2792:
2786:
2781:
2778:
2772:
2766:
2763:
2760:
2754:
2747:
2746:
2745:
2731:
2728:
2721:by an amount
2708:
2705:
2699:
2696:
2693:
2687:
2681:
2678:
2675:
2669:
2659:
2655:
2631:
2625:
2605:
2582:
2576:
2573:
2570:
2567:
2564:
2558:
2552:
2549:
2546:
2540:
2534:
2531:
2528:
2525:
2522:
2516:
2513:
2510:
2505:
2500:
2496:
2489:
2484:
2478:
2475:
2472:
2466:
2463:
2460:
2454:
2448:
2445:
2442:
2436:
2426:
2422:
2415:
2412:
2409:
2403:
2400:
2397:
2391:
2388:
2385:
2379:
2366:
2363:
2360:
2354:
2348:
2334:
2328:
2303:
2302:
2301:
2287:
2267:
2247:
2224:
2216:
2212:
2188:
2182:
2179:
2176:
2165:
2162:
2159:
2150:
2147:
2144:
2137:
2126:
2123:
2120:
2114:
2099:
2091:
2087:
2083:
2077:
2069:
2065:
2057:
2056:
2055:
2046:
2043:
2038:
2036:
2032:
2028:
2027:multi-variate
2024:
2020:
2016:
2015:deterministic
2012:
2007:
1990:
1984:
1961:
1955:
1932:
1924:
1920:
1916:
1911:
1907:
1898:
1895:
1892:
1889:
1885:
1871:
1867:
1860:
1857:
1849:
1845:
1838:
1828:
1815:
1811:
1804:
1801:
1793:
1789:
1782:
1776:
1773:
1770:
1763:
1762:
1761:
1747:
1727:
1707:
1687:
1665:
1661:
1652:
1634:
1630:
1626:
1621:
1617:
1593:
1589:
1581:
1577:
1572:
1568:
1565:
1560:
1556:
1552:
1545:
1541:
1536:
1532:
1529:
1524:
1520:
1515:
1510:
1502:
1498:
1493:
1489:
1484:
1480:
1476:
1469:
1465:
1460:
1456:
1451:
1447:
1442:
1436:
1430:
1420:
1416:
1410:
1406:
1394:
1390:
1385:
1381:
1372:
1369:
1364:
1361:
1354:
1353:
1352:
1335:
1331:
1323:
1319:
1314:
1310:
1307:
1302:
1298:
1294:
1287:
1283:
1278:
1274:
1271:
1266:
1262:
1257:
1252:
1244:
1240:
1235:
1231:
1226:
1222:
1218:
1211:
1207:
1202:
1198:
1193:
1189:
1184:
1178:
1174:
1169:
1165:
1156:
1153:
1148:
1145:
1138:
1137:
1136:
1134:
1130:
1111:
1106:
1101:
1092:
1088:
1081:
1078:
1070:
1066:
1059:
1055:
1048:
1044:
1040:
1035:
1023:
1019:
1012:
1008:
1003:
992:
988:
981:
977:
971:
963:
959:
954:
950:
946:
939:
936:
929:
928:
927:
925:
921:
917:
912:
910:
906:
902:
886:
883:
880:
872:
868:
864:
861:known as the
845:
840:
836:
830:
826:
822:
819:
815:
810:
806:
802:
797:
793:
789:
786:
782:
779:
776:
773:
766:
765:
764:
762:
744:
740:
717:
713:
704:
700:
696:
686:
684:
665:
659:
656:
651:
647:
640:
634:
631:
626:
622:
615:
607:
603:
599:
594:
590:
583:
573:
570:
567:
563:
559:
553:
550:
545:
541:
534:
528:
525:
520:
516:
512:
507:
503:
496:
488:
484:
480:
475:
471:
467:
462:
458:
451:
446:
443:
440:
435:
432:
429:
425:
419:
416:
411:
405:
394:
391:
386:
382:
375:
362:
361:
360:
358:
354:
350:
346:
327:
324:
321:
314:
311:
308:
302:
296:
293:
290:
284:
278:
275:
272:
266:
256:
252:
248:
245:
242:
235:
232:
229:
223:
217:
214:
211:
208:
205:
199:
193:
190:
187:
184:
181:
175:
170:
165:
161:
155:
152:
147:
141:
130:
127:
124:
118:
105:
104:
103:
101:
97:
93:
83:
81:
77:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
21:
3870:
3865:
3854:
3849:
3824:
3820:
3810:
3793:
3789:
3783:
3748:
3744:
3738:
3716:(1): 63–86.
3713:
3709:
3703:
3679:(1): 53–62.
3676:
3672:
3662:
3645:
3641:
3635:
3608:
3604:
3594:
3551:
3547:
3541:
3519:(1): 43–73.
3516:
3512:
3506:
3489:
3485:
3472:
3439:
3435:
3429:
3404:
3400:
3387:
3360:
3356:
3346:
3321:
3317:
3311:
3278:
3274:
3268:
3249:
3243:
3221:(2): 77–80.
3218:
3214:
3208:
3191:
3187:
3181:
3153:Flocculation
3067:
3002:. A simple
2997:
2882:
2597:
2203:
2052:
2039:
2018:
2008:
1947:
1608:
1350:
1126:
918:in a dilute
913:
860:
698:
692:
680:
344:
342:
89:
80:flocculation
76:emulsication
61:
56:
52:
32:
26:
3827:: 276–296.
3281:(1): 8–17.
3168:DLVO theory
2013:. Most of
68:coalescence
3886:Categories
3577:10397/9612
3492:(3): 170.
3174:References
3136:Cantor set
916:collisions
3841:0021-9991
3758:1305.1034
3586:120582575
3379:0002-9947
3338:0043-1354
3303:0021-9797
3235:0092-7325
3047:α
2973:α
2948:−
2935:α
2920:−
2912:∼
2821:∞
2812:∫
2805:α
2787:τ
2779:α
2729:α
2665:∞
2656:∫
2626:τ
2568:−
2532:−
2497:∫
2432:∞
2423:∫
2401:−
2346:∂
2341:∂
2326:∂
2321:∂
2288:τ
2183:τ
2145:α
2110:⟶
1858:−
1777:π
1748:γ
1728:η
1566:−
1530:−
1431:γ
1395:η
1308:−
1272:−
1179:η
947:π
579:∞
564:∑
560:−
513:−
468:−
444:−
426:∑
403:∂
373:∂
357:summation
262:∞
253:∫
249:−
209:−
185:−
162:∫
139:∂
116:∂
3796:: 7–14.
3775:96697123
3464:17392921
3188:Phys. Z.
3142:See also
2260:at time
926:system,
867:additive
863:constant
695:operator
353:discrete
100:integral
86:Equation
72:aerosols
55:at time
3745:Polymer
3718:Bibcode
3681:Bibcode
3613:Bibcode
3556:Bibcode
3521:Bibcode
3444:Bibcode
3409:Bibcode
3283:Bibcode
3196:Bibcode
3004:fractal
349:measure
47:of the
3839:
3773:
3584:
3462:
3377:
3336:
3301:
3256:
3233:
2204:where
1948:where
1609:where
869:, and
703:kernel
31:, the
3771:S2CID
3753:arXiv
3582:S2CID
3482:(PDF)
3460:S2CID
3397:(PDF)
3138:too.
2744:i.e.
924:phase
35:is a
3837:ISSN
3375:ISSN
3334:ISSN
3299:ISSN
3254:ISBN
3231:ISSN
3068:The
2280:and
1977:and
1649:are
693:The
3829:doi
3825:340
3798:doi
3763:doi
3726:doi
3714:139
3689:doi
3650:doi
3621:doi
3572:hdl
3564:doi
3529:doi
3494:doi
3452:doi
3417:doi
3405:222
3365:doi
3326:doi
3291:doi
3279:167
3223:doi
920:gas
759:.
343:If
70:of
27:In
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3855:77
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3792:.
3769:.
3761:.
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3724:.
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3644:.
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3607:.
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3403:.
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3355:.
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359::
345:dy
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3755::
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3720::
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1933:,
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