409:
84:
404:{\displaystyle \mathbf {B} =\mathbf {P_{\pm }} \left({\begin{matrix}2\\0\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\1\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\1\\1\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\2\\1\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}3\\0\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}3\\1\\0\end{matrix}}\right)}
614:. In addition to the bond length constraint, polymers should not be allowed to cross. This is done most efficiently by the use of a secondary lattice which is twice as fine as the original lattice. The secondary lattice tracks the midpoints of the bonds in the system, and forbids the overlap of bond midpoints. This effectively leads to disallowing polymers from crossing each other.
573:
As in the case of the
Carmesin-Kremer BFM, the Shaffer BFM is also constructed on a simple-cubic lattice. However, the lattice points, or vertices of each cube are the sites that can be occupied by a monomer. Each lattice point can be occupied by one monomer only. Successive monomers along a polymer
40:
systems. There are two versions of the BFM used: The earlier version was first introduced by I. Carmesin and Kurt Kremer in 1988, and the later version by J. Scott
Shaffer in 1994. Conversion between models is possible.
939:
560:
1149:
689:
866:
458:
612:
574:
backbone are connected by bond vectors. The allowed bond vectors must be one of: (a) A cube edge (b) A face diagonal or (c) A solid diagonal. The resulting bond lengths are
70:, which is taken from a set of typically 108 allowed vectors. There are different definitions for this vector set. One example for a bond vector set is made up from the six
482:
767:
798:
487:
The combination of bond vector set and monomer shape in this model ensures that polymer chains cannot cross each other, without explicit test of the local
62:
on a regular cubic lattice with each cube occupying eight lattice positions. Each lattice position can only be occupied by one monomer in order to model
958:
Carmesin, I.; Kremer, Kurt (1988). "The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions".
888:
500:
633:
1044:
Subramanian, Gopinath; Shanbhag, Sachin (2008). "On the relationship between two popular lattice models for polymer melts".
622:
In both versions of the BFM, a single attempt to move one monomer consists of the following steps which are standard for
770:
806:
1095:
Deutsch, H. P.; Binder, K. (1991). "Interdiffusion and self-diffusion in polymer mixtures: A Monte Carlo study".
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967:
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8:
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for example due to an electric field or an adsorbing force to the walls. In this case a
1001:
Shaffer, J. Scott (1994). "Effects of chain topology on polymer dynamics: Bulk melts".
623:
1120:
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The conditions to perform a move can be subdivided into mandatory and optional ones.
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The move does not lead to bonds that are not contained in the bond vector set.
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The move does not lead to bonds that are not contained in the bond vector set.
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from the interval [0, 1). If the
Metropolis rate is smaller than
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The lattice site to which the chosen monomer is going to be moved is empty.
71:
934:{\displaystyle \#MCS={\frac {\#{\text{ attempts}}}{\#{\text{ monomers}}}}}
1145:
555:{\displaystyle \mathbf {\Delta B} =\mathbf {P_{\pm }} \left(1,0,0\right)}
75:
979:
494:
The basic movement of a monomer cube takes place along the lattice axes
33:
1065:
1116:
1022:
78:
and sign variation of the three vector components in each direction:
882:
The number of Monte Carlo steps of the total system is defined as:
488:
55:
1153:
684:{\displaystyle \Delta \mathbf {B} \in \mathbf {P} _{\pm }(1,0,0)}
37:
704:
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so that each of the possible bond vectors can be realized.
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The move does not lead to overlapping of bond midpoints.
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705:Mandatory conditions for Carmesin–Kremer BFM
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879:the move is rejected, otherwise it is accepted.
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49:
1150:Leibniz Institute of Polymer Research Dresden
1094:
746:If the move leads to an energetic difference
697:If all conditions are fulfilled, perform move
861:{\displaystyle p_{M}=e^{-\Delta U/k_{B}T}\,}
453:{\displaystyle 2,{\sqrt {5}},{\sqrt {6}},3}
994:
857:
607:{\displaystyle 1,{\sqrt {2}},{\sqrt {3}}}
1000:
66:. The monomers are connected by a bond
1162:
1156:) for simulating polymers with the BFM
741:
726:Mandatory conditions for Shaffer BFM
694:Check list of conditions (see below)
568:
710:Four lattice sites next to monomer
630:Select a monomer m and a direction
617:
13:
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36:the conformation and dynamics of
773:is applied: The Metropolis rate
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1097:The Journal of Chemical Physics
1046:The Journal of Chemical Physics
1003:The Journal of Chemical Physics
871:is compared to a random number
414:The resulting bond lengths are
678:
660:
1:
477:{\displaystyle {\sqrt {10}}}
7:
50:Carmesin and Kremer version
10:
1196:
944:
762:{\displaystyle \Delta U}
44:
26:bond fluctuation method
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22:bond fluctuation model
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793:{\displaystyle p_{M}}
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800:which is defined as
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771:Metropolis algorithm
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1109:1991JChPh..94.2294D
1058:2008JChPh.129n4904S
1015:1994JChPh.101.4205S
980:10.1021/ma00187a030
972:1988MaMol..21.2819C
742:Optional conditions
624:Monte Carlo methods
58:are represented by
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54:In this model the
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714:in the direction
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569:Shaffer's version
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1009:(5): 4205–4213.
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618:Monte Carlo step
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1170:Polymer physics
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64:excluded volume
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1175:Lattice models
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1136:External links
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1052:(14): 144904.
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960:Macromolecules
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30:lattice model
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74:below using
72:base vectors
53:
25:
21:
17:
15:
1146:Java applet
1103:(3): 2294.
76:permutation
1164:Categories
1144:– a
718:are empty.
34:simulating
1148:from the
1125:0021-9606
1074:0021-9606
1031:0021-9606
988:0024-9297
921:#
911:#
893:#
832:Δ
829:−
754:Δ
656:±
646:∈
638:Δ
522:±
506:Δ
363:±
350:∪
311:±
298:∪
259:±
246:∪
207:±
194:∪
155:±
142:∪
103:±
1082:19045165
691:randomly
489:topology
56:monomers
1154:Germany
1105:Bibcode
1054:Bibcode
1011:Bibcode
968:Bibcode
38:polymer
28:) is a
1123:
1080:
1072:
1029:
986:
354:
302:
250:
198:
146:
68:vector
945:Notes
60:cubes
45:Model
1142:JBFM
1121:ISSN
1078:PMID
1070:ISSN
1027:ISSN
984:ISSN
460:and
32:for
16:The
1113:doi
1062:doi
1050:129
1019:doi
1007:101
976:doi
24:or
18:BFM
1166::
1119:.
1111:.
1101:94
1099:.
1076:.
1068:.
1060:.
1048:.
1025:.
1017:.
1005:.
982:.
974:.
964:21
962:.
626::
491:.
484:.
470:10
1152:(
1127:.
1115::
1107::
1084:.
1064::
1056::
1033:.
1021::
1013::
990:.
978::
970::
905:=
902:S
899:C
896:M
877:r
873:r
853:T
848:B
844:k
839:/
835:U
825:e
821:=
816:M
812:p
786:M
782:p
757:U
716:d
712:m
679:)
676:0
673:,
670:0
667:,
664:1
661:(
651:P
642:B
600:3
595:,
590:2
585:,
582:1
549:)
545:0
542:,
539:0
536:,
533:1
529:(
518:P
513:=
509:B
448:3
445:,
440:6
435:,
430:5
425:,
422:2
398:)
391:0
384:1
377:3
370:(
359:P
346:)
339:0
332:0
325:3
318:(
307:P
294:)
287:1
280:2
273:2
266:(
255:P
242:)
235:1
228:1
221:2
214:(
203:P
190:)
183:0
176:1
169:2
162:(
151:P
138:)
131:0
124:0
117:2
110:(
99:P
94:=
90:B
20:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.