Multi-component mixture of dipolar hard spheres with surface adhesion
Solution of the mean spherical approximation for multi-component model of dipolar hard spheres with surface adhesion is obtained using factorization technique pioneered by Wertheim and Baxter. For the sake of illustration numerical calculations for the dielectric constant of the two-component ver...
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Protsykevich, I.A. 2017-06-12T19:15:13Z 2017-06-12T19:15:13Z 2003 Multi-component mixture of dipolar hard spheres with surface adhesion / I.A. Protsykevich // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 629-636. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 82.70.Dd, 61.20.-p, 61.20.Gy, 61.20.Ne, 61.20.Qg, 02.30.Rz DOI:10.5488/CMP.6.4.629 https://nasplib.isofts.kiev.ua/handle/123456789/120764 Solution of the mean spherical approximation for multi-component model of dipolar hard spheres with surface adhesion is obtained using factorization technique pioneered by Wertheim and Baxter. For the sake of illustration numerical calculations for the dielectric constant of the two-component version of the model are carried out. На основі методу факторизації, започаткованого Вертхаймом та Бакстером, отриманий аналітичний розв’язок середньосферичного наближення для багатокомпонентної моделі дипольних твердих сфер з поверхневою липкістю. З метою ілюстрації виконані чисельні розрахунки діелектричної сталої двокомпонентного варіанту моделі. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Multi-component mixture of dipolar hard spheres with surface adhesion Багатокомпонентна суміш дипольних твердих сфер з поверхневим прилипанням Article published earlier |
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Multi-component mixture of dipolar hard spheres with surface adhesion |
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Multi-component mixture of dipolar hard spheres with surface adhesion Protsykevich, I.A. |
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Multi-component mixture of dipolar hard spheres with surface adhesion |
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Multi-component mixture of dipolar hard spheres with surface adhesion |
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Multi-component mixture of dipolar hard spheres with surface adhesion |
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Multi-component mixture of dipolar hard spheres with surface adhesion |
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multi-component mixture of dipolar hard spheres with surface adhesion |
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Protsykevich, I.A. |
| author_facet |
Protsykevich, I.A. |
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2003 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
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Багатокомпонентна суміш дипольних твердих сфер з поверхневим прилипанням |
| description |
Solution of the mean spherical approximation for multi-component model of
dipolar hard spheres with surface adhesion is obtained using factorization
technique pioneered by Wertheim and Baxter. For the sake of illustration
numerical calculations for the dielectric constant of the two-component version
of the model are carried out.
На основі методу факторизації, започаткованого Вертхаймом та
Бакстером, отриманий аналітичний розв’язок середньосферичного наближення для багатокомпонентної моделі дипольних твердих
сфер з поверхневою липкістю. З метою ілюстрації виконані чисельні розрахунки діелектричної сталої двокомпонентного варіанту
моделі.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120764 |
| citation_txt |
Multi-component mixture of dipolar hard spheres with surface adhesion / I.A. Protsykevich // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 629-636. — Бібліогр.: 21 назв. — англ. |
| work_keys_str_mv |
AT protsykevichia multicomponentmixtureofdipolarhardsphereswithsurfaceadhesion AT protsykevichia bagatokomponentnasumíšdipolʹnihtverdihsferzpoverhnevimprilipannâm |
| first_indexed |
2025-11-27T07:04:00Z |
| last_indexed |
2025-11-27T07:04:00Z |
| _version_ |
1850802904459378688 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 629–636
Multi-component mixture of dipolar
hard spheres with surface adhesion∗
I.A.Protsykevich
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received September 8, 2003
Solution of the mean spherical approximation for multi-component model of
dipolar hard spheres with surface adhesion is obtained using factorization
technique pioneered by Wertheim and Baxter. For the sake of illustration
numerical calculations for the dielectric constant of the two-component ver-
sion of the model are carried out.
Key words: mean spherical approximation, dipolar hard spheres,
adhesion, dielectric constant
PACS: 82.70.Dd, 61.20.-p, 61.20.Gy, 61.20.Ne, 61.20.Qg, 02.30.Rz
1. Introduction
Mean spherical approximation (MSA) [1] occupies a special place among the
liquid state integral equation theories [2–4] due to the availability of the analytical
solution of the corresponding Ornstein-Zernike (OZ) equation for a number of simple
albeit nontrivial models (see, for example [2–14]).
In the present study we propose the method, which can be used to describe
the properties of the fluid of multicomponent dipolar hard spheres with surface
adhesion. It is based on the analytical solution of the corresponding version of the
MSA with the surface adhesion accounted for following Baxter [15]. In the relevant
publications of Blum and coworkers multi-component sticky hard-sphere ion-dipole
mixture with orientationally dependent stickiness for dipoles of the same size [16] and
one-component hard-sphere system with anisotropic adhesion of arbitrary symmetry
and electric multipoles [17] have been studied.
2. The model
We consider M -component adhesive hard-sphere fluid with the number density
of each species s ρs = Ns/V , hard-sphere diameter σs and dipolar moment ps.
∗This paper is dedicated to Professor Myroslav Holovko on the occasion of his 60th birthday.
c© I.A.Protsykevich 629
I.A.Protsykevich
In addition to hard-sphere and dipolar interaction there is the so-called “sticky”
interaction, which is characterized by the adhesion parameter Λst(Ω12), where Ω12 =
(Ω1, Ω2), Ω1 is a set of Euler angles that give the orientation of the molecule 1.
For the model at hand MSA consists of the following OZ equation and closure
conditions
hst(X12) = cst(X12) +
M
∑
u=1
ρu
∫
dX3hsu(X13)cut(X32), (1)
hst(X12) = −1 + Λst(Ω12)δ(r − σst), r12 < σst =
1
2
(σs + σt), (2)
cst(X12) = −βUst(X12) = −βpspt
{
s1s2
|r12|3
−
3s1r12r12s2r12
|r12|5
}
, r12 > σst, (3)
where hst(X12) = gst(X12) − 1 is the pair correlation function, gst(X12) is the pair
distribution function, cst(X12) is the direct correlation function, Ust(X12) is the elec-
trostatic pair potential, r12 is the distance between the particles, X12 = (X1, X2),
X1 = (~r1, Ω1), β = 1/kbT , ~s1 = ~r1/|~r1| is the unit vector and δ(r) is the Dirac
delta-function.
Solution of the present MSA problem is based on the Wiener-Hopf factorization
technique developed by Wertheim [18] and Baxter [19,20]. In order to consistently
account for the orientation dependencies, the technique developed by Blum and
co-workers [6–8] is utilized. According to this method all orientation dependent
functions are presented in orientational-invariant form
fst(X12) =
∑
mnl
fmnl
st (r12)Φ
mnl
00 (Ω1, Ω2, Ωr12
), (4)
where the linear symmetry of the dipoles is accounted for and Ωr12
is a set of Euler
angles, which defined the orientation of the vector ~r12. This vector connects the
centers of masses of the particles 1 and 2.
Φmnl
00 (Ω1, Ω2, Ωr) = ((2m + 1)(2n + 1))1/2
∑
µνλ
(
m n l
µ ν λ
)
× Dm
0µ(Ω1)D
n
0ν(Ω2)D
l
0λ(Ωr),
fmnl
st (r12) = (2l + 1) ((2m + 1)(2n + 1))1/2
∑
µνλ
(
m n l
µ ν λ
)
×
∫
dΩ1dΩ2dΩr12
D∗m
0µ (Ω1)D
∗n
0ν (Ω2)D
∗l
0λ(Ωr12
)fst(X12). (5)
Here a standard notation is used for the 3-j Wigner symbols and for generalized
spherical functions.
In terms of the rotational invariant expansion, the coefficients closure conditions
(2) and (3) are as follows:
h000
st (r) = −1 + Λ000
st δ(r − σ−
st), r < σst, (6)
630
Adhesive dipolar hard spheres
h110
st (r) = Λ110
st δ(r − σ−
st), r < σst, (7)
h112
st (r) = Λ112
st δ(r − σ−
st), r < σst, (8)
c000
st (r) = 0, r > σst, (9)
c110
st (r) = 0, r > σst, (10)
c112
st (r) = (10/3)
1
2 βpsptr
−3, r > σst. (11)
Relations (6)–(8) impose dipole symmetry on the adhesion parameter.
In terms of the rotational-invariants coefficients the set of the OZ equations (1)
will have the form similar to that presented in [7]
H̃mn
st,χ(k) − C̃mn
st,χ(k) =
M
∑
u=1
1
∑
l=0
(−1)χρuH̃
ml
su,χ(k)C̃ ln
ut,χ(k), (12)
where
F̃ mn
st,χ(k) = 2
∞
∫
0
dr cos(kr)Fmnl
st (r),
Fmn
st,χ(r) = 2π(−1)χ
∑
l
(
m n l
χ −χ 0
) ∞
∫
r
dt tPl
(
r
t
)
fmnl
st (t),
F̃ mn
st,χ(k) ≡ H̃mn
st,χ(k) or C̃mn
st,χ(k),
Fmn
st,χ(r) ≡ Jmn
st,χ(r) or S̃mn
st,χ(r),
fmn
st (r) ≡ hmn
st (r) or cmn
st (r) (13)
for the model at hand the set of equations (12) will be reduced to the set of three
independent sets of equations
H̃00
st,0(k) − C̃00
st,0(k) =
M
∑
u=1
ρuH̃
00
su,0(k)C̃00
ut,0(k),
H̃11
st,0(k) − C̃11
st,0(k) =
M
∑
u=1
ρuH̃
11
su,0(k)C̃11
ut,0(k),
H̃11
st,1(k) − C̃11
st,1(k) =
M
∑
u=1
ρuH̃
11
su,1(k)C̃11
ut,1(k). (14)
Closure conditions for the functions Jmn
st,χ(r) and Smn
st,χ(r) are as follows:
Smn
st,χ(r) = 0, mnχ = 000, 110, 111, r > σst,
Jmn
st,χ(r) = jmn,0
st,χ + jmn,2
st,χ r2, r < σst,
j00,0
st,0 = b00,0
st,0 − πσ2
st, j00,2
st,0 = π,
j11,0
st,0 = −
1
31/2
b110
st,0 −
1
301/2
b112
st,2, j11,2
st,0 =
(
3
10
)1/2
b112
st,2,
j11,0
st,1 = −
1
31/2
b110
st,0 +
1
1201/2
b112
st,2, j11,2
st,1 = −
(
3
40
)1/2
b112
st,2,
631
I.A.Protsykevich
bmnl
st,p = 2π(σst)
1−pΛmnl
st + 2π
∞
∫
σ+
st
drhmnl
xy (r)r1−p. (15)
3. General solution
All three sets of equations (14) are of the same form. Omitting the indices we
have
H̃st(k) − C̃st(k) =
M
∑
u=1
ρuH̃su(k)C̃ut(k), (16)
Jst(r) = j0
st + j2
str
2, r < σst, (17)
Sst(r) = 0, r > σst, (18)
where the function Jst(r) has a jump discontinuity at r = σst:
Jst(σ
+
st) − Jst(σ
−
st) = jstep
st , (19)
H̃st(k) = 2
∞
∫
0
dr cos(kr)Jst(r), (20)
C̃st(k) = 2
∞
∫
0
dr cos(kr)Sst(r). (21)
Using matrix notation we have
(
I + [ρ1/2] · [H̃(k)] · [ρ1/2]
)
×
(
I − [ρ1/2] · [C̃(k)] · [ρ1/2]
)
= I, (22)
where I is the unit matrix, the symbol “ · ” denotes matrix multiplication, square
brackets denote matrices of the order M , ρ
1/2
st = δstρ
1/2, δst is the Kronecker delta.
Using Wiener-Hopf factorization method we have
I − [ρ1/2] · [C̃(k)] · [ρ1/2] = [Q̃(k)] · [Q̃(−k)]T , (23)
[
I + [ρ1/2] · [H̃(k)] · [ρ1/2]
]
· [Q̃(k)] =
[
[Q̃(−k)]T
]−1
, (24)
where the upper index T denotes matrix transpose. Wertheim-Baxter factorization
correlation functions are of the following form
[Q̃(k)]st = δst − (ρsρt)
1/2
σst
∫
λts
drQst(r)e
ikr,
Qst(r) =
0, r < λts = (σt − σs)/2,
(
2π(ρsρt)
1/2
)−1 ∞
∫
−∞
dk
(
δst − Q̃st)k)
)
e−ikr, λts < r < σst ,
0, r > σst .
(25)
632
Adhesive dipolar hard spheres
After the inverse Fourier transform for the equations (23) and (24) we have
Sst(r) = −Qst(r) +
M
∑
u=1
ρu
{σsu;σtu+r}
∫
{λus;λut+r}
dtQsu(t)Qtu(t − r), (26)
Jst(r) = Qst(r) +
M
∑
u=1
ρu
σut
∫
λtu
dtJsu(|r − t|)Qut(t), (27)
where the upper and lower integration limits are represented by the smallest and
largest numbers in braces, respectively. Using the closure conditions (17) and taking
into account the relation (19) from the equation (27) at r < σst we have
Qst(r) = ast + (r − σst)bst + 1/2(r − σst)(r − λts)dst,
ast = jstep
st ,
bst =
(
[
I −
1
6
[ρtσ
3
t j
2
st]
]−1
·
[
σtj
2
st
]
)
st
,
dst =
2
[
0
0
−1
I −
1
6
[ρtσ
3
t j
2
st]
]−1
·
[
0
0
−1
j2
st
]
+
[
0
0
−1
bst
]
·
[
0
0
−1
ρsσsbst
]
− 2
[
1
6
−1
bst
]
·
[
1
6
−1
]−1
ρsast
st
. (28)
4. Wertheim-Baxter factorization correlation function
Using the results obtained in the previous section and taking into account the
rotationally-invariant indices m, n and χ Wertheim-Baxter factorization correlation
functions can be written as follows:
Qmn
st,χ(r) = amn
st,χ + (r − σst)b
mn
st,χ + 1/2(r − σst)(r − λts)d
mn
st,χ ,
a00
st,0 = 2πσstΛ
000
st ,
bst,0 =
π
∆
σt ,
dst,0 =
2π
∆
+
π2
∆2
ζ2σt −
2π
∆
M
∑
u=1
ρuσuaut,0 ,
a11
st,0 = −
2π
31/2
σstΛ
110
st +
4π
301/2
σstΛ
112
st ,
a11
st,1 = −
2π
31/2
σstΛ
110
st −
2π
301/2
σstΛ
112
st , (29)
where ∆ = 1 − π/6ζ3; ζp =
M
∑
u=1
ρu(σu)
p .
Coefficients of the Wertheim-Baxter factorization function b11
st,0 , d11
st,0 , b11
st,1 and
d11
st,1 follow from the solution of the set of equations
b11
st,0 =
1
1201/2
M
∑
u=1
ρuσ
3
ub
112
su,2b
11
ut,0 + 0.31/2σtb
112
st,2 ,
633
I.A.Protsykevich
d11
st,0 =
2
σt
b11
st,0 +
M
∑
u=1
ρuσub
11
su,0b
11
ut,0 − 2
M
∑
u=1
ρub
11
su,0a
11
ut,0 ,
b11
st,1 = −
1
4801/2
M
∑
u=1
ρuσ
3
ub
112
su,2b
11
ut,1 + 0.0751/2σtb
112
st,2 ,
d11
st,1 =
2
σt
b11
st,1 +
M
∑
u=1
ρuσub
11
su,1b
11
ut,1 − 2
M
∑
u=1
ρub
11
su,1a
11
ut,1 . (30)
This set of equations have to be supplemented by the additional equation for
the parameter b112
st,2. This equation can be obtained from the closure conditions for
c112
st (r) at r > σst
4π
3
βpspt = −C̃11
st,0(k = 0) + C̃11
st,1(k = 0). (31)
Introducing the quantities
Kmn,p
st =
σst
∫
λts
dr rpQmn
st (r), (32)
equation (31) can be written as follows
4π
3
βpspt = −K11,0
st,0 − K11,0
ts,0 +
M
∑
u=1
ρuK
11,0
su,0K
11,0
tu,0
+ K11,0
st,1 + K11,0
ts,1 −
M
∑
u=1
ρuK
11,0
su,1K
11,0
tu,1 , (33)
where
K11,0
st,χ = σsa
11
st,χ −
1
2
σ2
sb
11
st,χ −
1
3
σ3
sd
11
st,χ . (34)
Thus, we obtain a closed set of equations (30) and (33) for the coefficients of
the Wertheim-Baxter factorization correlation function and for the dipole-dipole
interaction parameter b112
st,2 .
5. Dielectric constant. Numerical calculations
The expression for the dielectric constant of the dipolar mixture ε was obtained
in [7,21]. For the model at hand we have
ε − 1
ε + 2
= Tr
(
[
[q1] · [q2]
−1 − I
]
·
[
[q1] · [q2]
−1 + 2I
]−1
)
, (35)
where
[q1] =
(
I − [ρ1/2] · [K11,0
st,0 ] · [ρ1/2]
)
×
(
I − [ρ1/2] · [K11,0
st,0 ] · [ρ1/2]
)T
,
[q2] =
(
I − [ρ1/2] · [K11,0
st,1 ] · [ρ1/2]
)
×
(
I − [ρ1/2] · [K11,0
st,1 ] · [ρ1/2]
)T
. (36)
634
Adhesive dipolar hard spheres
Table 1. Dielectric properties of the two-component dipolar hard-sphere fluid.
Hard spheres of species 1 has a diameter σ1 = 1 and hard-spheres size of species
2 is σ2 = 21/3σ1. The reduced total density is ρ∗ = ρ1σ
3
1 +ρ2σ
3
2 = 0.8. The dipole
moment of species 1 and 2 are such that when ρ2 = 0 (pure fluid 1) then y = 2.5
and when ρ1 = 0 (pure fluid 2) then y = 1.5, where y = 4π/9β
∑
s ρsp
2
s. Here
Λmnl
st is treated as a fitting parameter.
EMSA EMC ESDHS
X1 = ρ1/ρ y [6] [6] Λ110
st = 0, Λ112
11 = −0.14,
Λ112
12 = −0.18, Λ112
22 = −0.24
0.0 1.5 11.5 13.65 13.66
0.3 1.7 13.4 17.4 17.5
0.8 2.2 19.1 26.0 26.2
1.0 2.5 23.5 37.2 36.9
To calculate the dielectric constant of the system, the solution of the set of equa-
tions (30) and (33) has to be obtained. This is a set of highly nonlinear equations,
which can be solved only using the numerical methods, for example using the New-
ton’s method. To use this method it is important to have a sufficiently accurate
initial guess for the unknowns of the problem. In our study we start from the two-
component version of the model with hard spheres of the same size and dipolar
moment. For such a model, the dipole-dipole interaction parameter b112
st,2 is the same
for both species and can be obtained from the corresponding nonlinear equation.
All the rest of the unknown coefficients can be easily obtained from b112
st,2. Gradually
changing the ratio of the hard-sphere sizes and dipolar moments one can get solution
of the set of equations (30) and (33) for the two-component system with arbitrary
hard-sphere sizes and dipolar moments.
For the sake of illustration in table 1 we present the dependance of the dielectric
constant of the two-component mixture on the concentration of the first species
X1 = ρ1/ξ0 with the value of the adhesion constant Λmnl
st chosen so as to fit the
corresponding MC values. Good agreement between MC and theoretical predictions
in the whole range of concentrations with only one value for Λmnl
st shows that the
model at hand can be used to correlate the experimental data with the dielectric
properties of multi-component polar fluids.
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635
I.A.Protsykevich
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Багатокомпонентна суміш дипольних твердих сфер
з поверхневим прилипанням
І.А.Процикевич
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 8 вересня 2003 р.
На основі методу факторизації, започаткованого Вертхаймом та
Бакстером, отриманий аналітичний розв’язок середньосферично-
го наближення для багатокомпонентної моделі дипольних твердих
сфер з поверхневою липкістю. З метою ілюстрації виконані чисе-
льні розрахунки діелектричної сталої двокомпонентного варіанту
моделі.
Ключові слова: середньосферичне наближення, дипольні тверді
сфери, адгезія, діелектрична постійна
PACS: 82.70.Dd, 61.20.-p, 61.20.Gy, 61.20.Ne, 61.20.Qg, 02.30.Rz
636
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