Adsorption of chain molecules in pores with crystalline walls: a density functional approach
A microscopic density functional theory of adsorption of chain molecules on the attractive solid surfaces of crystalline symmetry is developed. The proposed approach is based on the expansions of one-particle functions into two-dimensional Fourier series. Illustrative calculations for dimers, 8- a...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2009
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119764 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Adsorption of chain molecules in pores with crystalline walls: a density functional approach / K. Bucior, A. Patrykiejew, S. Sokołowski, O. Pizio // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 51-62. — Бібліогр.: 40 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119764 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1197642017-06-09T03:04:36Z Adsorption of chain molecules in pores with crystalline walls: a density functional approach Bucior, K. Patrykiejew, A. Sokołowski, S. Pizio, O. A microscopic density functional theory of adsorption of chain molecules on the attractive solid surfaces of crystalline symmetry is developed. The proposed approach is based on the expansions of one-particle functions into two-dimensional Fourier series. Illustrative calculations for dimers, 8- and 16-mers adsorbed on a (100) fcc crystal surface indicate the effect of the periodicity of the adsorbing potential on the structure of the adsorbed layer and thermodynamics of adsorption. Розвинуто мiкроскопiчну теорiю функцiоналу густини для вивчення адсорбцiї ланцюгових молекул на притягальних твердих поверхнях кристалiчної симетрiї. Запропонований пiдхiд ґрунтується на розкладах одночастинкових функцiй у двовимiрнi ряди Фур’є. Для iлюстрацiї проведено обчислення для димерiв, 8- i 16-мерiв, що адсорбованi на кристалiчнiй поверхнi (100) fcc, якi демонструють вплив перiодичностi адсорбуючого потенцiалу на структуру адсорбованого шару i термодинамiку адсорбцiї. 2009 Article Adsorption of chain molecules in pores with crystalline walls: a density functional approach / K. Bucior, A. Patrykiejew, S. Sokołowski, O. Pizio // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 51-62. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 68.08.-p, 68.43.Fg, 82.35.Gh, 68.43.-h DOI:10.5488/CMP.12.1.51 http://dspace.nbuv.gov.ua/handle/123456789/119764 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A microscopic density functional theory of adsorption of chain molecules on the attractive solid surfaces of
crystalline symmetry is developed. The proposed approach is based on the expansions of one-particle functions
into two-dimensional Fourier series. Illustrative calculations for dimers, 8- and 16-mers adsorbed on a
(100) fcc crystal surface indicate the effect of the periodicity of the adsorbing potential on the structure of the
adsorbed layer and thermodynamics of adsorption. |
| format |
Article |
| author |
Bucior, K. Patrykiejew, A. Sokołowski, S. Pizio, O. |
| spellingShingle |
Bucior, K. Patrykiejew, A. Sokołowski, S. Pizio, O. Adsorption of chain molecules in pores with crystalline walls: a density functional approach Condensed Matter Physics |
| author_facet |
Bucior, K. Patrykiejew, A. Sokołowski, S. Pizio, O. |
| author_sort |
Bucior, K. |
| title |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| title_short |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| title_full |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| title_fullStr |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| title_full_unstemmed |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| title_sort |
adsorption of chain molecules in pores with crystalline walls: a density functional approach |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119764 |
| citation_txt |
Adsorption of chain molecules in pores with crystalline walls: a density functional approach / K. Bucior, A. Patrykiejew, S. Sokołowski, O. Pizio // Condensed Matter Physics. — 2009. — Т. 12, № 1. — С. 51-62. — Бібліогр.: 40 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT buciork adsorptionofchainmoleculesinporeswithcrystallinewallsadensityfunctionalapproach AT patrykiejewa adsorptionofchainmoleculesinporeswithcrystallinewallsadensityfunctionalapproach AT sokołowskis adsorptionofchainmoleculesinporeswithcrystallinewallsadensityfunctionalapproach AT pizioo adsorptionofchainmoleculesinporeswithcrystallinewallsadensityfunctionalapproach |
| first_indexed |
2025-07-08T16:33:30Z |
| last_indexed |
2025-07-08T16:33:30Z |
| _version_ |
1837097195937988608 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 1, pp. 51–62
Adsorption of chain molecules in pores with crystalline
walls: a density functional approach
K.Bucior1, A.Patrykiejew1, S.Sokołowski1, O.Pizio2
1 Department for the Modelling of Physico-Chemical Processes, Maria Curie-Skłodowska University,
20-031 Lublin, Poland
2 Universidad Autonoma Metropolitana, Iztapalapa, México D.F., México
Received November 29, 2008, in final form February 7, 2009
A microscopic density functional theory of adsorption of chain molecules on the attractive solid surfaces of
crystalline symmetry is developed. The proposed approach is based on the expansions of one-particle func-
tions into two-dimensional Fourier series. Illustrative calculations for dimers, 8- and 16-mers adsorbed on a
(100) fcc crystal surface indicate the effect of the periodicity of the adsorbing potential on the structure of the
adsorbed layer and thermodynamics of adsorption.
Key words: density functional, adsorption, chains, crystals
PACS: 68.08.-p, 68.43.Fg, 82.35.Gh, 68.43.-h
1. Introduction
Physical adsorption of nonspherical molecules on well defined crystal surfaces has been a fi-
eld of vivid activity for several years [1–11]. These studies indicated that periodic modulation of
an adsorbing potential may cause strong orientational effects in adsorbed films formed by linear
molecules on the surfaces of square and rectangular lattice symmetry [12–16]. Compared to the
films composed of spherical adsorbates that already exhibit very complex phase behavior of ad-
sorbed layers [17], the films formed by nonspherical molecules show many new and interesting
features [18,19]. In a majority of hitherto published studies, the adsorbate particles were composed
of a small number of atoms (or segments), most frequently two or three. Moreover, theoretical
approaches were usually based on the application of computer simulation techniques, whereas
analytical methods were rather seldom used [17–19].
Recently, several versions of a density functional approach for nonuniform molecular systems
were proposed. Among them the theories, which extend Wertheim’s thermodynamic perturbation
theory [20] to the case of nonuniform chain fluids [21,22], seem to be particularly convenient for the
purpose of studying the structure and various thermodynamic properties of nonuniform systems
composed of tangentially bonded chains [23–26].
The majority of previous studies of adsorption of chain particles using the density functional
approaches were restricted to the models with structureless adsorbing surfaces. The aim of this
work is to extend the density functional theory of Yu and Wu [21,22,25,26] to the case of crystalline
surfaces. Our extension is performed by using the ideas from the earlier studies of adsorption of
simple [27] and associating [28,29] fluids on crystalline surfaces. The present work concentrates on
the methodological aspects of the theory. Our principal objective is to develop equations for the
density profiles appropriate to the description of the fluid of chain molecules in contact with the
surfaces that exhibit finite corrugation of the external field. Theoretical deliberations are enhanced
by numerical results for simple model systems. Thermodynamic properties including the phase
behavior of more complex inhomogeneous fluids in contact with crystalline walls will become the
subject of future works.
c© K.Bucior, A.Patrykiejew, S.Sokołowski, O.Pizio 51
K.Bucior et al.
2. Theory
We consider a fluid of chain molecules composed of the same segments in contact with a single
wall of a regular crystal, characterized by the unit lattice vectors a1 and a2. Each fluid molecule is
built of M hard spheres of diameter σ each, which are tangentially jointed by a bonding potential
acting between the adjacent segments. Similarly to the approach of Wu et. al. [21,22,25,26] we
assume that the bonding potential between the two nearest-neighbor segments, vb(r), is of infinitely
short range, i.e.,
exp[−βvb(r)] = (1/(4πσ2))δ(r − σ). (1)
The potential (1) forbids any configuration that would lead to an overlap between two segments.
The total bonding potential, Vb(r1, r2, . . . , rM ) is a sum of the bonding potentials vb between the
segments, Vb(r1, r2, . . . , rM ) =
∑M−1
j=1
vb(|rj+1 − rj |).
Each segment of the chain particle interacts with a single pore wall via the potential v ′(r). This
potential results from the summation of Lennard-Jones (12,6) potentials
ugs(r) = 4εgs
[
(σgs/r)12 − (σgs/r)6
]
(2)
over all solid atoms. In the above r is the distance between a segment and an atom of the pore
wall and εgs and σgs are the energy and the size parameters. Following Steele [30] we apply a
two-dimensional Fourier representation of the potential v′(r),
v′(r) =
∑
qj>0
v′qj
(z) exp[−iqj · t], (3)
where z is the coordinate perpendicular to the surface, qj = 2πb1j1 + b2j2 are the reciprocal
lattice vectors, j1, j2 = 0,±1,±2, . . . and t = (x, y) is a two-dimensional vector parallel to the
surface. The analytical expressions for the Fourier coefficients vqj
(z) for crystal lattices of different
symmetry can be found in [30]. Briefly, the coefficients v′
qj
(z) depend only on the length of the
vectors qj and, therefore, one can define the functions fj(x, y) =
∑′
exp[−iqj · t], where the prime
means that the summation runs over all reciprocal vectors of a given length |qj |. The functions
fj(x, y) depend on the parameter σgs of the potential (2), whereas the coefficients v|qj |(z) depend
on σgs and εgs [30].
In the case of a slit-like pore of the width H , the potential of the segment-pore walls is the sum
of the potentials (3) exerted by two pore walls. In a general case the crystalline symmetry of the
walls may not be the same. Moreover, even for the same symmetry, there may be a shift between
crystal lattices forming two walls. In this work we consider the simplest case of the pore walls of the
same symmetry and assume that there is no phase shift between the segment-wall potentials, i.e.
v(r) =
∑
qj>0
[v′qj
(z) + v′qj
(H − z)] exp[−iqj · t]. (4)
The density functional theory of adsorption of chain molecules developed in [21,22,25,26] leads
to the following equation for the density profile of the segment j, j = 1, . . . , M
ρsj(rj) = exp(βµ)
∫
dr1dr2 . . . drMδ(r − rj) exp[−βVb(r1, r2, . . . , rM )]
M
∏
l=1
[g(rl)γ(rl)] , (5)
which, for identical segments and for the bonding potential given by equation (1), can also be
rewritten as
τsj(rj) = exp(βµ)γj(rj)G
j(rj)G
M+1−j(rj), (6)
where τsj(r) is one-particle cavity function, ρsj(r) = τsj(r)g(r), g(r) = exp[−βv(r)] and the
functions Gl(r) are given by the recurrence relation
Gl(r) = (1/(4πσ))
∫
dr′δ(|r − r′| − σ)Gl−1(r′)γl(r
′) for l > 2. (7)
52
Chains on crystalline walls
and G1(r) ≡ 1. In the above µ is the chemical potential of the fluid and γi(r) = exp[−βλi(r)],
λi(r) =
δFex
δρsi(r)
, (8)
with the excess free energy Fex =
∫
drΦex(r), Φex(r) = ΦHS(r) + ΦC(r) being the functional of
the free energy density due to hard-sphere interactions and chain connectivity and where ρs(r) =
∑M
j=1
ρsj(r) is the total segment density.
The free energy densities ΦHS and ΦC are given by
ΦHS = −n0 ln(1 − n3) +
n1n2 − n1 · n2
1 − n3
+ n3
2(1 − ξ2)3
n3 + (1 − n3)
2 ln(1 − n3)
36πn2
3(1 − n3)2
, (9)
and
ΦC =
1 − M
M
n0ζ ln
[
1
1 − n3
+
n2σζ
4(1 − n3)2
+
(n2σ)2ζ
72(1− n3)3
]
, (10)
where ξ(r) = |n2(r)|/n2(r), ζ = 1−n2 ·n2/(n2)
2, and where nα, α = 0, 1, 2, 3 and nα, α = 1, 2 are
scalar and vector averaged densities,
nα(r) =
∫
dr′ρs(r
′)wα(r − r′) for α = 0, 1, 2, 3,
nα(r) =
∫
dr′ρs(r
′)wα(r − r′) for α = 1, 2, (11)
with the weight functions wα and w given in [31]. Note that in equations (9) and (10) the spatial
dependence of all variables has been omitted for the sake of brevity. For detailed derivation of the
above equations and equation (6) see [21,22,25,26].
In order to rewrite equation (6) in a form suitable for the case of a crystalline surface we realize
that any one-particle function, χ(r) can be expanded into a two-dimensional Fourier series
χ(r) =
∑
qj
χqj
(z) exp[−iqj · t], (12)
where
χqj
(z) =
1
a1 × a2
∫∫
a1×a2
χ(r) exp[iqj · t]dt. (13)
The Fourier expansion coefficients χqj
(z) depend only on the length of the reciprocal lattice vector,
|qj |.
Using equations (6), (12) and (13), the expansion coefficients of the function τsj(r) can be
related to the expansion coefficients γi,q(z) and Gl
q(z) via
τsj,q(z) = exp(βµ)
∑
qa+qb+qc=q
γi.qa
(z)Gj
qb
(z)GM+1−j
qc
(z), (14)
and the recurrence relation (7) becomes
Gl
q(z) =
1
2σ
∑
qa+qb+qc=q
∫ z+σ
z−σ
dz′J0(|q|
√
σ2 − (z − z′)2)γi,qa
(z′)gqb
(z′)Gl−1
qc
(z′), (15)
where J0(u) is the zeroth-order Bessel function of the first kind. Note that for the model in question
the expansion coefficients γi,q(z′) are identical for all segments 1 6 i 6 M .
The last two equations comprise the solution for the segment density profile of a chain fluid
in contact with a crystalline surface. Their application, however, requires evaluating the Fourier
coefficients of the Boltzmann function g(r) and the function γ(r), according to equation (13). In
the case of the Boltzmann function the integration is instantaneous, whereas in order to calculate
53
K.Bucior et al.
the coefficients γsj,q(z) one must first calculate the coefficients λsj,q(z). To do that we consider
the integral
w̃α,q(z) =
∫
exp [iqj · t] wα(
√
z2 + t2)dt (16)
for the scalar weight functions, α = 0, 1, 2, 3, and analogous integrals for the x, y, z components of
the vector weight function. For scalar weight functions wα, α = 0, 1, 2, 3, and for z-components of
the vector functions wα,z, α = 1, 2, the last integral becomes
w̃α,q(z) = 2π
∫ σ/2
z
wα(R)J0(|q|
√
R2 + z2)RdR,
w̃α,z,q(z) = 2π
∫ σ/2
z
wα,z(R)J0(|q|
√
R2 + z2)RdR, (17)
whereas all in-plane vector components, w̃α,j,q, j = x, y are equal to zero. Therefore,
λsj,q(z) =
∑
α=0,1,2,3
∫ σ/2
−σ/2
(Φ′( z + z′))α,qw̃α,z,q(|z′|)dz′ +
∑
α=1,2
∫ σ/2
−σ/2
(Φ′( z + z′))α,qw̃α,q(|z′|)dz′,
(18)
where (Φ′(z))α,q is the q-th Fourier expansion coefficient of the partial derivative ∂Φex/∂nα. The
Fourier expansion coefficients of the scalar weighted densities are
nα,qj
(z) =
∫ σ/2
−σ/2
dzw̃α,qj
(|z′|)ρs,qj
(z + z′)dz′, (19)
and a quite analogous equation can be written for the z-components of the vector weighted densities,
while the x- and y-components of the vector local densities are equal to zero. Equations (17)–(19)
complete the description.
Let us summarize the proposed procedure. At the beginning of the calculations we evaluate the
Boltzmann function expansion coefficients, gq(z), from equation (13) and the functions w̃α,q(z);
they are calculated only once and stored. Next, we begin iterations from the assumed Fourier ex-
pansion coefficients τsj,q(z). At very low bulk densities we can apply the ideal gas approximation,
τsj,q(z) = δq0ρsb/M , where ρsb is the bulk total segment density. At the first step we evaluate the
expansion coefficients ρsj,q(z) =
∑
qa+qb=q τsj,qa
(z)gqb
(z) and then the weighted density coeffici-
ents from equation (19). The expansion (12) is used to calculate the weighed densities, and those,
in turn, to evaluate the values of ∂Φex/∂nα(r), and, consequently the coefficients (Φ′(z)))α,q by
integrating equation (13). Then, the coefficients λsj,q(z) are calculated from equation (19). These
coefficients yield the function λ(r), which is next used to calculate the coefficients γq(z), again
integrating exp[−βλ(r)] according to equation (13). Finally, the calculation of τsj,q(z) from equa-
tion (14) is preceded by the evaluation of the coefficients Gl
q(z) from equation (15). The calculations
are repeated until a required accuracy of the procedure is reached.
Previous numerical calculations for atomic adsorbates [27–29] have indicated that if the diame-
ter σ is equal or greater than the length of the lattice vectors a1 and a2, then the Fourier expansion
of the cavity function τsj(r) is very fast convergent and in many cases only the zeroth-order term,
τsj(0), suffices to obtain a quite accurate representation of the local density. In such cases one
needs to evaluate only very few coefficients Gl
q(z), λsj,q(z) and γq(z).
3. Results and discussion
Model calculations were carried out for chains adsorbed in pores with the walls built of (100)
fcc crystal surfaces, assuming the length of the unit lattice vector equal to σ, |a1| = |a2| = σ.
We have also assumed that σgs = σ. Similarly to the case of the potential expansion (3), all
Fourier expansion coefficients χq(z) depend only on the length of the vector, q = |q|. The Fourier
expansions can be simplified by introducing the functions fl(x, y) =
∑′
exp[−iq · t], where the
54
Chains on crystalline walls
summation runs over all reciprocal vectors of a given length, ql = |ql|. Of course, the functions
fl(x, y) are identical to those used in the expansions of the external potential (3) and they can be
found in [30].
1 1.5 2
z/σ
-20
-10
0
10
20
g l(z
)
l=0
l=1
l=2
l=3
l=4
l=5
Figure 1. Fourier expansion coefficients for the Boltzmann function. The values of all parameters
are given in the text.
In figure 1 we present the first six Fourier Boltzmann function coefficients gl(z) obtained for
kT/εgs = 0.25 and for H = 20σ. We see that the magnitude of g5(z) is quite small compared to
the coefficient g0(z). Therefore, the Fourier expansion of the Boltzmann function can be truncated
after l = 5 term.
In the case of chains built of up to 16 segments and for the pore considered above, we have also
tested the convergence of the Fourier expansion for one-particle background (or cavity) function
τ(r). We found that even for bulk total segment densities as high as ρsbσ
3 = 0.6 the truncation
of this expansion after the term τ1(z) was sufficient to obtain quite accurate results. Therefore,
all further calculations were carried out assuming that the Fourier expansions of the Boltzmann
function and of the cavity function can be truncated after l = 5 and l = 1 terms, respectively.
Figure 2. The change of the total segment densities of 8-mers along the diagonal of the unit
lattice cell, ρs(x, x, z) at the pore wall located at z = 0. Part a is for ρsbσ
3 = 0.1, whereas part
b – for ρsbσ
3 = 0.6. The pore width is H = 20σ and kT/εgs = 0.3.
55
K.Bucior et al.
We begin with the presentation of the results illustrating the effect of the periodicity of the
adsorbing potential on the structure of adsorbed fluid. In figure 2 we show the changes of the total
segment local density of 8-mers with the distance from the pore wall, z, along the diagonal of the
unit lattice crystal cell, x = y. Again, the calculations were carried out for H = 20σ. Part a is for
the total segment bulk density ρsbσ
3 = 0.1 and part b – for ρsbσ
3 = 0.6. One observes that for
both bulk densities the periodicity of the adsorbing potential mainly affects the behavior of the
first fluid layer adjacent to the pore wall. Indeed, the in-plane changes of the density profile for
the second adsorbed layer are hardly seen on the scale of figure 2. Instantaneously, an increase of
the bulk density ρsb enhances the accumulation of segments located within the first adsorbed layer
over the so-called adsorbing site position, S, (x/σ = 1/2, y/σ = 1/2).
1 2
z/σ
0
0.05
0.1
0.15
ρ sj
(x
,y
,z
)σ
3
j=1
j=2
j=8
M=16
S
ρsbσ3=0.1
a
1 2
z/σ
0
0.1
0.2
0.3
ρ sj
(x
,y
,z
)σ
3
j=1
j=2
j=4
S
ρsbσ3=0.1
M=8
b
Figure 3. The density profiles of segments j over the S position of the unit lattice cell of the
pore wall at z = 0. Part a is for adsorbed 16-mers and part b – for adsorbed 8-mers. The bulk
density is ρsbσ
3 = 0.1. The pore width is H = 20σ and kT/εgs = 0.3.
More information on the structure of an adsorbed layer is provided by the inspection of densities
of consecutive segments. In figure 3a we show the changes of the profiles of the segments j = 1, 2
and 8 of 16-mers over the position S. The results presented in figure 3b were obtained for 8-mers
adsorbed over the same position of the crystal surface and for j = 1, 2 and 4. The bulk total segment
density is equal to ρsbσ
3 = 0.1 in both cases. It means that the bulk density of individual segments
in the system of 16-mers is two times lower than in the system involving 8-mers. Consequently,
the local densities of individual segments of 16-mers are lower than the local densities in the
system of 8-mers. For both chain lengths the local density of the first segment, j = 1, exhibits the
highest maximum. The first segment is preferentially adsorbed over the S position. Thus, the local
density of the adjacent segment, j = 2, over the same position should be less probable. In fact, the
maximum of the function ρs2(σ/2, σ/2, z) is the lowest in comparison with the maxima of all other
segments. For the adsorbed 8-mers the differences between the maxima for j = 2 and for j = 4 (the
middle segment) are small, whereas for the adsorbed 16-mers these differences are slightly more
pronounced. We also note that the second layer density profiles for all, but j = 1, segments are
almost identical. The second maximum of ρs1(σ/2, σ/2, z) is lower than all the remaining second
maxima.
A comparison of the total segment profiles of 16- and 8-mers over three positions of the crystal
lattice, i.e., over the S position, over the SP position (i.e. the saddle point position, x = 1/2, y = 0)
and over an atom of the crystal surface, A, (x = 0, y = 0) evaluated at three bulk densities,
ρsb = 0.1, 0.2 and 0.4 is shown in figure 4. Comparing the results shown in figure 3 with those
given in figure 4 one should have in mind that the total density profile is the sum of the local
densities of individual segments. The first maxima of the profiles of 16-mers are higher than the
56
Chains on crystalline walls
1 2
z/σ
0
10
20
30
ρ s(x
,y
,z
)/ρ
sb
ρsbσ3=0.1S
SP
A
a
M=16
M=16
M=16
M=8
M=8
M=8
1 2
z/σ
0
5
10
15
ρ s(x
,y
,z
)/ρ
sb
ρsbσ3=0.2
S
SP
b
A
1 2
z/σ
0
2
4
6
8
10
ρ s(x
,y
,z
)/ρ
sb
ρsbσ3=0.4
A
S
SP
c
Figure 4. A comparison of the total segment density profiles, ρs(x, y, z) of 16-mers (lines) and
8-mers (symbols) over the positions S, SP and A of the pore wall at z = 0 and at three bulk
densities, ρsbσ
3 = 0.1 (part a), 0.2 (part b) and 0.4 (part c). The pore width is H = 20σ and
kT/εgs = 0.3.
maxima of the profiles of 8-mers. This is particularly well seen at ρsb = 0.1 and for the profiles over
the S-position. At low bulk densities, the attractive fluid-solid interactions determine the adlayer
structure. If there is enough free space on the surface, all the chains can assume an in-plane
orientation and the effect of the promotion of some segments to higher layers is small. Moreover,
because of bonding, location of the j− 1 and j +1 segments on the wall enforces a similar location
of the segment j and this effect is more important for longer chains. With the bulk density increase
the differences between the total segment profiles of 16-mers and of 8-mers become smaller. This is
due to the fact that with the density increase, the hard-sphere and the segment-surface interaction
contributions to the free energy functional prevail and the contribution arising from the chain
connectivity plays a smaller role in determining the structure of the fluid adjacent to the wall.
Obviously, with the bulk density increase we observe the formation of a multilayer structure at
the wall and at ρsb = 0.2, the second local density maximum is developing. However, the shape
and the location of the second maximum is almost independent of the position over the crystal
lattice.
57
K.Bucior et al.
Figure 5 compares the segment densities of the selected individual segments, i.e. the first (j = 1)
and of the middle (j = 4 for 8-mers and j = 8 for 16-mers) segments over the S, SP and A positions
of the crystal lattice. The first segments are preferentially adsorbed at the S-position and an increase
of the bulk density amplifies this tendency. Although it is not quite well seen on the figure scale, we
can state that the middle segment, compared to the first segment “prefers” the A-position. Over
the S and SP positions this tendency is opposite.
1 2
z/σ
0
0.05
0.1
0.15
ρ sj
(x
,y
,z
)σ
3
SP
A
M=16
ρsbσ3=0.1
a
j=1
j=8
j=1
j=8
j=1
j=8
S
1 2
z/σ
0
0.1
0.2
0.3
ρ sj
(x
,y
,z
)σ
3
S
SP
A
ρsbσ3=0.1
M=8
b
j=1
j=1
j=1
j=4
j=4
j=4
1 2 3
z/σ
0
0.2
0.4
ρ sj
(x
,y
,z
)σ
3
M=16
ρsbσ3=0.6
c S
SP
A
j=1
j=1
j=1
j=8
j=8
j=8
1 2 3
z/σ
0
0.5
1
ρ sj
(x
,y
,z
)σ
3
S
SP
A
M=8
ρsbσ3=0.6
d
j=1
j=1
j=1
j=4
j=4
j=4
Figure 5. A comparison of the density profiles of the first (lines) and of the middle (symbols)
segments of 16-mers (pars a and c) and of 8-mers over the positions S, SP and A of the pore
wall at z = 0. Parts a and b are for ρsbσ
3 = 0.1, whereas parts c and d – for ρsbσ
3 = 0.6. The
pore width is H = 20σ and kT/εgs = 0.3.
The numerical calculations performed in this work clearly demonstrate the effect of a periodic
variations of the adsorbing potential on the structure of the adsorbed layer. We now consider how
the energy periodicity of the pore walls effects the adsorption of a fluid in pores and the solvation
force. The amount of the confined fluid per unit surface area is given by
Γ =
1
a1 × a2
∫∫
a1×a2
ρs(r)dr =
∫ H
0
ρs0(z)dz. (20)
We recall that ρs0(z) is the zeroth-order expansion coefficient of the total segment density profile.
The calculation of the solvation force is carried out differentiating the grand potential Ω = F −
58
Chains on crystalline walls
∫
dr[µ − v(r)]ρs(r)
Fs = −
∂Ω
∂H
− p, (21)
where p is the pressure of a bulk fluid at the total segment density ρsb. It can be easily proven [21]
that the grand thermodynamic potential Ω is conveniently evaluated from
Ω =
M
∑
i=1
∫
drλ(r)ρsi(r) +
∫
drΦex(r) + kTΓ. (22)
The solvation force is an experimentally accessible quantity in measurements using surface force
apparatus [32]. Several groups investigated the dependence of solvation force on the specific prop-
erties of confining solid substrates, namely the shape of the surface, its symmetry, some aspects of
energy and geometrical heterogeneity, cf. [33–35].
1e-06 0.0001 0.01
ρsbσ3
0.0001
0.001
0.01
0.1
1
Γσ
2
M=2, kT/εgs=0.25
a
1e-06 0.0001 0.01 1
ρsbσ3
0.01
0.1
1
Γσ
3
M=8, kT/εgs=0.25
b
Figure 6. A comparison of the adsorption isotherms of dimers (part a) and 8-mers (part b). Solid
lines are the isotherms for pores with crystalline walls, dashed lines – for pores with the walls
interacting via the potential v0(z), whereas symbols have been obtained for the Boltzmann-
averaged potential from equation (23). The pore width is H = 20σ.
In figure 6 we show the adsorption isotherms obtained for dimers (part a) and for 8-mers
(part b). Solid lines denote the isotherms calculated for crystalline pore walls, assuming that
kT/εgs = 0.25. Dashed lines were evaluated for energy homogeneous surfaces. In this case the
adsorbing potential was equal to the zeroth order term in the Fourier expansion (4), v(z) =
v′0(z) + v′0(H − z). The isotherms for the latter model are lower than those evaluated assuming
periodic modulation of the adsorbing potential. The differences between the isotherms for uniform
and for crystalline pore walls increase with an increase of the chain length. The latter effect is
understandable, because the total fluid-pore potential energy increases with an increase of the
number of segments of a chain molecule. However, if we define the reference adsorbing potential
by
vref(z) = −kT 〈exp[−v(r)/kT ]〉q=0 , (23)
then the isotherms obtained assuming that the adsorbing potential is given by vref(z) become close
to those evaluated for crystalline pore walls. The “Boltzmann-averaged” potential (23) guarantees
that in the limit of low bulk densities, the isotherms for crystalline and uniform walls are iden-
tical. Instantaneously, our calculations indicate that approximation of the adsorbing potential by
equation (23) is reasonably accurate for dimers and longer chains, cf. figures (6a) and (6b). In fact,
the effects of energy heterogeneity of an adsorbing surface are less important for bigger adsorbate
59
K.Bucior et al.
molecules, i.e. longer chains in our case. We also note that the adsorption is higher for longer
molecules, especially at low bulk densities. For very low bulk densities this is quite obvious. Within
this region of densities, the attraction between molecules and the wall plays a predominant role.
Thus, higher attraction for longer chains results in higher adsorption. This behavior was already
observed in our previous works [36–38].
2 4
H/σ
0
1
2
F s/A
kT
M=2, ρsbσ3=0.001
v(r)
vref(z)
v0(z)
kT/εgs=0.25
a
2 3 4 5
H/σ
0
5
10
15
F s/A
kT
kT/εgs=0.25
vref(z)
v{r)
M=2, ρsbσ3=0.5
b
2 3 4 5
H/σ
0
5
10
15
F s/A
kT
ρsbσ3=0.1
ρsbσ3=0.5
ρsbσ3=0.5
c
M=8, kT/εgs=0.25
v(r)
vref(z)
v(r)
Figure 7. The plots of the solvation force per unit surface area A. Parts a and b are for dimers,
part c – for octamers. All the parameters are given in figures. The labels v0(z), vref (z) and v(r)
refer to external field potential models.
Figures 7a–7c present the plots of the solvation force. The calculations were carried out for
dimers (parts a and b) and octamers (part b) confined between crystalline walls (the curves ab-
breviated by the symbol v((r))) and between uniform walls interacting via the potential vref(z),
equation (23). In the case of dimers at low bulk density, we have also included the plot of the
solvation force obtained assuming that the segment-pore walls interaction potential is given by the
first term in the Fourier expansion (4), cf. figure 7a. Part a was evaluated at low bulk density, i.e.
low amount of confined fluid, whereas parts b and c were calculated for intermediate and high bulk
fluid densities.
At small wall-to-wall distances, the solvation force is strongly repulsive. Obviously, with an
increase of the bulk density, the plots of the solvation force exhibit more oscillations. When the bulk
60
Chains on crystalline walls
density increases (i.e. the amount of confined chains increases) the differences between solvation
force calculated for the crystalline walls and for the reference potential vref(z) decrease. Thus, in
several cases the results for a crystalline solid can be well approximated by those evaluated for
an uniform surface, exhorting the potential vref(z). The possibility of using the reference potential
vref(z) significantly simplifies the calculations. One should remember, however, that the potential
vref(z) is temperature-dependent.
Our work has concentrated on methodological aspects and its aim was to develop a suitable
numerical method for studying adsorption of chain particles on crystalline surfaces. Of course,
the approach presented here can be further used to investigate the phase transitions in the films
containing chain particles adsorbed on crystalline surfaces or in pores with crystalline walls.
We would also like to emphasize that the model of fluid-solid interaction considered in this
work can be treated as an intermediate one between the model of a flat (energy homogeneous) wall
and the model introduced by Tripathi and Chapman [39], that described adsorption of associating
fluids on the surfaces activated with polar sites to which fluid molecules can bond or associate.
This model was then generalized to the case of adsorption of chain molecules by Tscheliessnig et al.
[40], that considered the case when associative surface sites were capable of bonding the “heads”
of chain particles. In the limit of a complete association, the latter model seems to be suitable for
the description of the formation of “brushes”.
Acknowledgements
This project has been partially supported by the National University of Mexico under grant
IN–223808–2. O.P. is on sabbatical leave from Instituto de Quimica de la UNAM.
References
1. Jiang J.W., Sandler S.I., Fluid Phase Equilibr., 2005, 228, 189.
2. Piccolo L., Loffreda D., Aires F.J.C.S., Deranlot C., Jugnet Y., Sautet P., Bertolini J.C., Surface Sci.,
2004, 566, 995.
3. Engkvist O., Stone A.J., J. Chem. Phys., 2000, 112, 6827.
4. Curtarolo S., Stan G., Bojan, M.J., Cole M.W., Steele W.A., Phys. Rev. E, 2000, 61, 1670.
5. Minot C., Van Hove M.A., Biberian J.-P., Surf. Sci., 1996, 346, 283.
6. Marx D., Wiechert H., Adv. Chem. Phys., 1996, 95, 213.
7. Hansen F.Y., Bruch L.W., Taub H., Phys. Rev. B, 1995, 52, 8515.
8. Hansen F.Y. Bruch L.W., Phys. Rev. B, 1995, 51, 2515.
9. Marx D., Opitz O., Nielaba P., Binder K., Phys. Rev. Lett., 1993, 70, 2908.
10. Vakarin E., Duda Y., Holovko M., J. Chem. Phys., 1997, 107, 5569.
11. Holovko M., Vakarin E., Mol. Phys., 1996, 87, 1375.
12. Zeppenfeld, P., George J., Diercks V., Halmer R., David R., Comsa G., Marmier A., Ramseyer C.,
Girardet C., Phys. Rev. Lett., 1997, 78, 1504.
13. Hoang P.N.M., Picaud S., Girardet C., Meredith W., J. Chem. Phys., 1996, 105, 8453.
14. Vigiani A., Cardini G., Schettino V., J. Chem. Phys., 1997, 106, 5693.
15. Bruch L.W., Hansen F.Y., Phys. Rev. B, 1998, 57, 9285.
16. Trabelsi M., Coulomb J.P., Degenhardt D., Lauter H., Surf. Sci., 1997, 377–379, 38.
17. Mouritsen O.G., Berlinsky A.J., Phys. Rev. Lett., 1982, 48, 181.
18. Soko lowska Z., Soko lowski S., J. Colloid Interface Sci., 1988, 123, 73.
19. Lajtar L., Patrykiejew A. Soko lowski S., Surface Sci., 1988, 200, 288.
20. Wertheim M.S., J. Chem. Phys., 1987, 87, 7323.
21. Yu Y.-X., Wu J., J. Chem. Phys., 2002, 117, 2368.
22. Yu Y.-X., Wu J., J. Chem. Phys., 2003, 118, 3835.
23. Bryk P., Pizio O., Soko lowski S., J. Chem. Phys., 2005, 122, 194904.
24. Wu J., Li Z., Ann. Rev. Phys. Chem., 2007, 58, 85
25. Wu J., AIChE J., 2006, 52, 1169.
26. Li Z., Cao D., Wu J., J. Chem. Phys., 2005, 122, 224701.
27. Soko lowski S., Steele W.A., Langmuir, 1985, 1, 190.
28. Henderson D., Soko lowski S., Pizio O., J. Chem. Phys., 1995, 102, 9048.
61
K.Bucior et al.
29. Pizio O., Soko lowski S., Phys. Rev. E, 1996, 53, 820.
30. Steele W.A., Surface Sci., 1973, 36, 317.
31. Rosenfeld Y., Phys. Rev. Lett., 1989, 63, 980.
32. Israelachvili J.N., Adams G.E., J. Chem. Soc. Faraday Trans. I, 1978, 74, 975.
33. Ghatak C., Ayappa K.G., J. Chem. Phys., 2004, 120, 9703.
34. Henderson J.R., Molec. Simul., 2005, 31, 435.
35. Sprenger M., Schlesener F., Dietrich S., J. Chem. Phys., 2006, 124, 134703.
36. Bryk P., Soko lowski S., J. Chem. Phys., 2004, 121, 11314.
37. Bryk P., Pizio O., Soko lowski S., J. Chem. Phys., 2005, 122, 194904.
38. Bryk P., Bucior K., Soko lowski S., Żukociński G., J. Phys. Chem. B, 2005, 109, 2977.
39. Tripathi S., Chapman W.G., J. Chem. Phys., 2003, 119, 12611.
40. Tscheliessnig R., Billes W., Fischer J., Soko lowski S., Pizio O., J. Chem. Phys., 2006, 124, 164703.
Адсорбцiя ланцюгових молекул в порах з кристалiчними
стiнками: пiдхiд функцiоналу густини
К.Буцьор1, А.Патрикейев1, С.Соколовскi1, О.Пiзiо2
1 Факультет моделювання фiзико-хiмiчних процесiв, Унiверситет Марiї Кюрi-Склодовської, Люблiн,
Польща
2 Автономний унiверситет метрополiї, Iзтапалапа, Мехiко, Мексика
Отримано 29 листопада 2008 р., в остаточному виглядi – 7 лютого 2009 р.
Розвинуто мiкроскопiчну теорiю функцiоналу густини для вивчення адсорбцiї ланцюгових молекул
на притягальних твердих поверхнях кристалiчної симетрiї. Запропонований пiдхiд ґрунтується на
розкладах одночастинкових функцiй у двовимiрнi ряди Фур’є. Для iлюстрацiї проведено обчислен-
ня для димерiв, 8- i 16-мерiв, що адсорбованi на кристалiчнiй поверхнi (100) fcc, якi демонструють
вплив перiодичностi адсорбуючого потенцiалу на структуру адсорбованого шару i термодинамiку
адсорбцiї.
Ключовi слова: функцiонал густини, адсорбцiя, ланцюги, кристали
PACS: 68.08.-p, 68.43.Fg, 82.35.Gh, 68.43.-h
62
|