Note: Brownian motion of colloidal particles of arbitrary shape

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by  Bogdan Cichocki

Note: Brownian motion of colloidal particles of arbitrary shape
Bogdan Cichocki
Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
Maria L. Ekiel-Jez˙ewska∗ and Eligiusz Wajnryb
Institute of Fundamental Technological Research,
Polish Academy of Sciences, Pawin´skiego 5B, 02-106 Warsaw, Poland
(Dated: January 12, 2016)
6 Theanalyticalexpressionsforthetime-dependentcross-correlationsofthetranslationalandrota-
1 tionalBrowniandisplacementsofaparticlewitharbitraryshapearederived. Thereferencecenteris
0 arbitrary,andthereferenceframeissuchthattherotational-rotational diffusiontensorisdiagonal.
2

n
Ingeneral,thediffusionandmobilitymatricesofapar- averagesh...i oftheirproductswithrespecttotheparti-
a 0
J ticledependonthechoiceofthereferencecenterwhichis clepositionsandorientations,usingtheconditionalprob-
9 usedto describethe rotationalandtranslationalmotion. ability which satisfies the Smoluchowski equation [3, 4],
The requirementthatthe rotational-translationalmobil- with the diffusion matrix,

] itymatrixµrtissymmetricdefinesuniquelytheso-called
t
f centerofmobility. InRef.[1],thiscenterwasusedtoan- Dtt Dtr
o D = , (3)
s alyze the Brownian motion of a particle with arbitrary Drt Drr
. shape, and the corresponding analytical expressions for (cid:20) (cid:21)
t
a thetime-dependentcross-correlationsofthetranslational WeadopttheframeofreferenceinwhichDrr isdiagonal,
m
and rotational Brownian displacements were derived. i.e. Drr =D δ , with µ,ν =1,2,3.
- In this note, we derive the analogical analytical ex- µν µ µν
d When an arbitrary reference center is chosen, the ten-
pressions for the cross-correlations of the translational
n sor Drt contains non-vanishing antisymmetric part. In
o and rotational displacements of an arbitrary reference
this case to derive the rotational-translational correla-
c center. We denote the time-dependent position of this
1 [ dceicnutelarrausnRit(tv)e,catnodrswue(pin)(ttr)o,dpuc=e t1h,r2e,e3m, utotudaellsycrpibereptehne- tRuioe(nfp.)s[(1to])∆nweRitch(atn)thperifosocleelvoeawdliunaasgteimtdoiisdnipfitrhceaesteianonntae.ldoTginhiceaSlceowcr.arVyelIaaItsIioionnf
v particle orientation at time t [1, 2]. The translational 0
Eq.(79)fromRef.[1], butwithanothersecondrankten-
5
9 and rotational Brownian displacements are, s(cid:10)or A(p), i.e. (cid:11)
0 ∆R(t)=R(t)−R(0), (1)
2 1
.0 ∆u(t)= 1 3 u(p)(0)×u(p)(t). (2) A(p) =u(p)×Drt− 2u(p)V, (4)
1 2
0 Xp=1 where the components of vector V are given as
16 Theirdynamicalcross-correlationsareevaluatedasthe Vα= 3µ,ν=1ǫαµνDµrtν.
: The resulting off-diagonal Cartesian components are,
v P
i
X h∆u(t)∆R(t)i =− (Drt +Drt ) 1 + Dβ −Dγ +(Drt −Drt ) 1 + D−Dα e−f(−)t−e−fγ(1)t
ar 0,αβ (cid:20) αβ βα (cid:18)4 4∆ (cid:19) αβ βα (cid:18)6 2∆ (cid:19) (cid:21) f(−)−fγ(1)
− (Drt +Drt ) 1 − Dβ −Dγ +(Drt −Drt ) 1 − D−Dα e−f(+)t−e−fγ(1)t
αβ βα 4 4∆ αβ βα 6 2∆ f(+)−f(1)
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) γ
−Dαrtβ +Dβrtα e−fα(2)t−e−fβ(1)t + e−fα(1)t−e−fβ(1)t − Dαrtβ −Dβrtα e−fα(2)t−e−fβ(1)t − 1−e−fγ(1)t , (5)
4 " fα(2)−fβ(1) fα(1)−fβ(1) # 2 " fα(2)−fβ(1) 3fγ(1) #
1
withtheconventionthateverywhereinthisnote(α,β,γ) D = (D +D +D ), (8)
1 2 3
is a permutation of (1,2,3), and 3
∆= D2+D2+D2−D D −D D −D D . (9)
f(1) =3D−D , f(2) =3(D +D), α=1,2,3, (6) 1 2 3 1 2 1 3 2 3
α α α α
f(±) =6D±2∆, (7) q
2
The diagonal Cartesian components read: Since 3 Dtt(±)=0, it is easy to see that W(t) is
α=1 αα
1/6 of the mean square Brownian displacement,
h∆u(t)∆R(t)i0,αα =−Dαrtα−8Dβrtβ e−fγ(2)tD−γe−fγ(1)t P W(t)=1h∆R(t)·∆R(t)i . (18)
6 0
−Dαrtα−Dγrtγ e−fβ(2)t−e−fβ(1)t + Dαrtα+Dβrtβ e−fγ(1)tt Moreover,it follows from our calculation that
8 D 2
β
+Dαrtα+2Dγrtγ e−fβ(1)tt. (10) W(t)=Dcmt+ 13 3 (D(Dβrtγ−+DDγrtβ)2)2 1−e−(Dβ+Dγ)t , (19)
β γ
αX=1 (cid:16) (cid:17)
To derive the translational-translational correlations,
where β,γ 6= α and D is the self-diffusion coefficient
the derivative (54) from Ref. [1] must be changed into cm
referring to the center of mobility,
d
dth∆R(t)∆R(t)i0 D = 1 TrDtt− 3 (Dβrtγ −Dγrtβ)2 . (20)
=2 Dtt(t) +hV(t)∆R(t)i +h∆R(t)V(t)i .(11) cm 3" Dβ +Dγ #
0 0 0 αX=1
The t(cid:10)wo last(cid:11)terms can be then calculated by decom- The importance of the center of mobility follows from
posing the vectorV, defined under Eq. (4), in the frame Eq. (19) – this is the only reference center for which the
u(p),p=1,2,3,andusingtheresultsfor u(p)(t)∆R(t) mean squaredisplacement is a linear function of time.[5]
0
obtained by the method described above. For completeness, we remind the rotational-rotational
(cid:10) (cid:11)
Theoff-diagonalCartesiancomponentswithα6=βread, correlations[1],whichdonotdependonthechoiceofthe
reference center. The off-diagonal elements vanish, and
1h∆R(t)∆R(t)i = 1−e−fγ(2)tDtt the diagonal elements, α=1,2,3, are given by
2 0,αβ f(2) αβ
1 γ h∆u(t)∆u(t)i = 1 − 3(D−Dα)+∆e−f(−)t
+ (Drt −Drt)(3Drt −Drt ) I(f(2),f(1);t) 0,αα 6 12∆
2 βγ γβ γα αγ γ α
3(D −D)+∆ 1 1
+1(Drt −Drt)(3Drt −Drt) I(f(2),f(1);t) − α12∆ e−f(+)t−4e−fα(2)t+4e−fα(1)t. (21)
2 αγ γα γβ βγ γ β
−(Drt −Drt )(Drt −Drt ) I(f(2),f(1);t), (12) In this work, the analytical expressions for the
αβ βα αα ββ γ γ
cross-correlations are more complex that their analogs
where in Ref. [1], but the advantage is that they apply to an
arbitrary reference center. Therefore, they can be easily
1 e−f1t e−f2t
I(f ,f ;t)= + − . (13) used to account for the experimental results, obtained
1 2 f f f (f −f ) f (f −f )
1 2 1 1 2 2 1 2 for any reference center. An example of the compari-
son of the theoretical expressions derived here with the
The diagonal Cartesian components are,
measurements(madeinRef.[2])canbefoundinRef.[6].
1 1−e−f(σ)t
2h∆R(t)∆R(t)i0,αα=W(t)+ f(σ) Dαttα(σ) M.L.E.-J. and E.W. were supported in part by
σ=± the Polish National Science Centre under Grant No.
X
3 2012/05/B/ST8/03010. M.L.E.-J. benefited from the scien-
+ P(σ)(α,β)I(f(σ),f(1);t), (14) tificactivities of theCOST Action MP1305.
β
σ=±β=1
X X
where
Dtt(±) = 1 ∓ 3Dα−D Dtt −1TrDtt ∗ Corresponding author: [email protected]
αα 2 4 ∆ αα 3 [1] B. Cichocki, M. L. Ekiel-Jez˙ewska and E. Wajnryb, J.
(cid:18) (cid:19)(cid:18) (cid:19)
D −D Chem. Phys. 142, 214902 (2015).
± β γ Dtt −Dtt , (15)
4∆ ββ γγ [2] D.J.Kraft,R.Wittkowski,B.tenHagen,K.V.Edmond,
D.J.PineandH.L¨owen,Phys.Rev.E88,050301(2013).
(cid:0) (cid:1)
and [3] N. van Kampen, Stochastic Processes in Physics and
Chemistry, 3rd Edition, North-Holland, 2007.
P(±)(α,β)=−(Drt −Drt)2 1 ∓ D−Dγ [4] R.B.JonesandP.N.Pusey,Annu.Rev.Phys.Chem.42,
αγ γα 3 ∆ 137 (1991).
(cid:18) (cid:19)
1 D −D [5] B. Cichocki, M. L. Ekiel-Jez˙ewska and E. Wajnryb, J.
+ (Drt )2−(Drt)2 ∓ α β , for α6=β, (16) Chem. Phys. 136, 071102 (2012).
αγ γα 2 2∆
(cid:18) (cid:19) [6] B. Cichocki, M. L. Ekiel-Jez˙ewska and E. Wajnryb,tobe
P((cid:2)±)(α,α)=−P(±(cid:3))(β,α)−P(±)(γ,α), (17) published.