Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions
The effects of polymer concentration and chain length on aggregation in associative polymer solutions, are studied using self-consistent e^ ld lattice model. Only two inhomogenous morphologies, i.e. microu^ ctuation homogenous (MFH) and micelle morphologies, are observed in the systems with differen...
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Інститут фізики конденсованих систем НАН України
2011
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nasplib_isofts_kiev_ua-123456789-1200382025-02-09T21:51:17Z Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions Вплив концентрацiї полiмера i довжини ланцюга на агрегацiю у фiзично асоцiйованих розчинах полiмерiв Han, X.-G. Zhang, X.-F. Ma, Y.-H. Zhang, C.-X. Guan, Y.-B. The effects of polymer concentration and chain length on aggregation in associative polymer solutions, are studied using self-consistent e^ ld lattice model. Only two inhomogenous morphologies, i.e. microu^ ctuation homogenous (MFH) and micelle morphologies, are observed in the systems with different chain lengths. The temperatures at which the above two inhomogenous morphologies r^ st appear, which are denoted by TMFH and Tm, respectively, are dependent on polymer concentration and chain length. The variation of the logarithm of critical MFH concentration with the logarithm of chain length full^ s a linear-t^ ting relationship with a slope equaling −1. Furthermore, the variation of the average volume fraction of stickers at the micellar core (AVFSM) with polymer concentration and chain length is focused in the system at Tm. It is founded by calculations that the above behavior of AVFSM, is explained in terms of intrachain and interchain associations. Використовуючи ґраткову модель самоузгодженого поля, вивчається вплив концентрацiї полiмера i довжини ланцюга на агрегацiю в асоцiативних полiмерних розчинах. У системах з рiзною довжиною ланцюга спостерiгається тiльки двi неоднорiднi морфологiї, а саме мiкрофлуктуацiйна однорiдна (MFH) i мiцелярна морфологiї. Температури, при яких вище згаданi двi морфологiї виникають впер-ше i позначаються як TMFH i Tm, вiдповiдно, є незалежними вiд концентрацiї полiмера i довжини ланцюга. Змiна логарифма критичної концентрацiї MFH зi змiною логарифма довжини ланцюга задовiльняє спiввiдношенню лiнiйного допасування з нахилом рiвним −1. Крiм того, змiна середньої об’ємної фракцiї стiкерiв при мiцелярному корi (AVFSM) з концентрацiєю полiмера i довжиною ланцюга сфокусована в системi при Tm. Знайдено шляхом розрахункiв, що вище згадана поведiнка AVFSM пояснюється в термiнах iнтраланцюгових та iнтерланцюгових асоцiацiй. This research is supported by the Innovation Fund of Inner Mongolia University of Science and Technology (Grant No. 2010NC065) the High Performance Computers of Inner Mongolia University of Science and Technology. 2011 Article Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions / X.-G. Han, X.-F. Zhang, Y.-H. Ma, C.-X. Zhang, Y.-B. Guan // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43601:1-11. — Бібліогр.: 41 назв. — англ. 1607-324X PACS: 61.25.Hp, 87.15.Nr, 82.60.Fa DOI:10.5488/CMP.14.43601 arXiv:1202.4595 https://nasplib.isofts.kiev.ua/handle/123456789/120038 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The effects of polymer concentration and chain length on aggregation in associative polymer solutions, are studied using self-consistent e^ ld lattice model. Only two inhomogenous morphologies, i.e. microu^ ctuation homogenous (MFH) and micelle morphologies, are observed in the systems with different chain lengths. The temperatures at which the above two inhomogenous morphologies r^ st appear, which are denoted by TMFH and Tm, respectively, are dependent on polymer concentration and chain length. The variation of the logarithm of critical MFH concentration with the logarithm of chain length full^ s a linear-t^ ting relationship with a slope equaling −1. Furthermore, the variation of the average volume fraction of stickers at the micellar core (AVFSM) with polymer concentration and chain length is focused in the system at Tm. It is founded by calculations that the above behavior of AVFSM, is explained in terms of intrachain and interchain associations. |
| format |
Article |
| author |
Han, X.-G. Zhang, X.-F. Ma, Y.-H. Zhang, C.-X. Guan, Y.-B. |
| spellingShingle |
Han, X.-G. Zhang, X.-F. Ma, Y.-H. Zhang, C.-X. Guan, Y.-B. Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions Condensed Matter Physics |
| author_facet |
Han, X.-G. Zhang, X.-F. Ma, Y.-H. Zhang, C.-X. Guan, Y.-B. |
| author_sort |
Han, X.-G. |
| title |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| title_short |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| title_full |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| title_fullStr |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| title_full_unstemmed |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| title_sort |
effects of polymer concentration and chain length on aggregation in physically associating polymer solutions |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2011 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120038 |
| citation_txt |
Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions / X.-G. Han, X.-F. Zhang, Y.-H. Ma, C.-X. Zhang, Y.-B. Guan // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43601:1-11. — Бібліогр.: 41 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-12-01T04:04:57Z |
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2025-12-01T04:04:57Z |
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1850277267956039680 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 4, 43601: 1–11
DOI: 10.5488/CMP.14.43601
http://www.icmp.lviv.ua/journal
Effects of polymer concentration and chain length
on aggregation in physically associating
polymer solutions
X.-G. Han1∗, X.-F. Zhang1, Y.-H. Ma1, C.-X. Zhang2, Y.-B. Guan2
1 Inner Mongolia Key Laboratory for Utilization of Bayan Obo Multi-Metallic Resources:
Elected State Key Laboratory, School of Mathematics, Physics and Biological engineering,
Inner Mongolia University of Science and Technology, 014010 Baotou, China
2 Department of Physics, Jilin University, 130021 Changchun, China
Received July 2, 2011, in final form September 19, 2011
The effects of polymer concentration and chain length on aggregation in associative polymer solutions, are
studied using self-consistent field lattice model. Only two inhomogenous morphologies, i.e. microfluctuation
homogenous (MFH) and micelle morphologies, are observed in the systems with different chain lengths. The
temperatures at which the above two inhomogenous morphologies first appear, which are denoted by TMFH
and Tm, respectively, are dependent on polymer concentration and chain length. The variation of the logarithm
of critical MFH concentration with the logarithm of chain length fulfils a linear-fitting relationship with a slope
equaling −1. Furthermore, the variation of the average volume fraction of stickers at the micellar core (AVFSM)
with polymer concentration and chain length is focused in the system at Tm. It is founded by calculations that
the above behavior of AVFSM, is explained in terms of intrachain and interchain associations.
Key words: concentration, chain length, aggregation, associative polymer
PACS: 61.25.Hp, 87.15.Nr, 82.60.Fa
1. Introduction
Physically, associating polymers are polymer chains containing a small fraction of attractive
groups along the backbones. The attractive groups, for example, solvophobic groups, tend to form
physical links which can play an important role in reversible junctions between different polymer
chains. The junctions can be broken and recombined frequently on experimental time scales. This
property of junctions makes associative polymer solutions behave reversibly when ambient condi-
tions, such as temperature and concentrations, change. This tunable characteristic of the system
produces extensive applications [1–4] that possess great potential as smart materials [5–7].
The attractive groups (also called sticker monomers) drive the self-assembly of the polymers,
leading to the formation of polymeric micelles in physically associating polymer solutions (PAPSs).
In telechelic [8, 9] and multiblock [10] associative polymers, flower micelles were observed. This
aggregation, as well as their ability to form bridges between micelles, profoundly affect their macro-
scopic properties, particularly their rheological behavior, which is required in many applications.
In telechelic associative polymers, the effects of architectural parameters of polymers have been
assessed, such as chain length, end-group length, and chemical composition [11–15]. In multi-
block associative polymers, the effect of chain architecture of polymer on the property of aggre-
gates [16, 17] was studied, which suggested that chain architecture can be an important factor in
controlling macroscopic properties of the systems. However, the above studies of multiblock PAPSs
were carried out in two dimensions and at low concentrations.
It is well known that self-consistent field theory (SCFT), as a mean-field theory, has been
applied to the study of a great deal of problems in polymeric systems [18–21]. Recently, SCFT has
∗E-mail: xghan0@163.com
c© X.-G. Han, X.-F. Zhang, Y.-H. Ma, C.-X. Zhang, Y.-B. Guan, 2011 43601-1
http://dx.doi.org/10.5488/CMP.14.43601
http://www.icmp.lviv.ua/journal
X.-G. Han et al.
been applied to the study of the properties of micelles in polymer solutions [22–24]. In the previous
paper [10], given a fixed chain length, we focused on the thermodynamic properties and structure
transitions in PAPSs. The microfluctuation homogenous (MFH) morphology, which corresponds to
the onset of gelation [10, 25], and micelle morphology were observed. The degrees of aggregations of
the above two morphologies are very different. The volume fraction of stickers in micelle morphology
is much bigger than that in MFH morphology. In this work, the property of the aggregation in
PAPAs is studied using self-consistent field lattice model. The temperatures at which MFH and
micelle morphologies first appear, which are denoted by TMFH and Tm, respectively, and the average
volume fraction of stickers at the micellar core, 〈φs(rco)〉, are calculated. Such calculations are
carried out for different chain lengths and polymer concentrations. It is found that TMFH, Tm and
〈φ(rco)〉 are dependent on chain length and polymer concentration, and the relationship of 〈φs(rco)〉
with chain length and polymer concentration is explained in terms of intrachain and interchain
associations.
2. Theory
We consider a system of incompressible PAPSs, where nP polymers, each of which is composed
of Nst segments of sticker monomer type (attractive group) and Nns segments of nonsticky monomer
type, are distributed over a lattice. Each sticker monomer is a regularly placed apart l monomer
along the chain backbone. The degree of polymerization of chain is N = Nst +Nns. In addition to
polymer monomers, nh solvent molecules are placed on the vacant lattice sites. Sticker, nonsticky
monomers and solvent molecules are of the same size and each occupies one lattice site. The
total number of lattice sites is NL = nh + nPN . Nearest neighbor pairs of stickers have attractive
interaction −ǫ with ǫ > 0, which is the only non-bonded interaction in the present system. The
interaction energy is expressed as:
U = −
ǫ
2
∑
r
∑
r′
φ̂st(r)φ̂st(r
′), (2.1)
where
∑
r means the summation over all the lattice sites r and
∑
r′ means the summation over
the nearest neighbor sites of r. φ̂st(r) =
∑
j
∑
s∈st δr,rj,s is the volume fraction of stickers on site r,
where j and s are the indexes of chain and monomer of a polymer, respectively. s ∈ st means that
the sth monomer belongs to sticker monomer type. In this simulation, however, instead of directly
using the exact expression of the nearest neighbor interaction for stickers, we introduce a local
concentration approximation for the non-bonded interaction [10, 26].
∑
r′ φ̂st(r
′) in equation (2.1)
is replaced with z φ̂st(r), where z is the coordination number of the lattice used. Within this
approximation, the interaction energy is expressed as:
U
kBT
= −χ
∑
r
φ̂st(r)φ̂st(r), (2.2)
where χ is the Flory-Huggins interaction parameter in the solutions, which is equal to z
2kBT
ǫ. We
perform the SCFT calculations in the canonical ensemble, and the field-theoretic free energy F is
defined as follows:
F [ω+, ω−]
kBT
=
∑
r
{
1
4χ
ω2
−(r)− ω+(r)
}
− nP lnQP[ωst, ωns]− nh lnQh[ωh], (2.3)
where Qh is the partition function of a solvent molecule subject to the field ωh(r) = ω
+
(r), which
is defined as Qh = 1
nh
∑
r exp [− ωh(r)]. QP is the partition function of a noninteraction polymer
chain subject to the fields ωst(r) = ω
+
(r) − ω
−
(r) and ωns(r) = ω
+
(r), which act on sticker
and nonsticky segments, respectively. Following the scheme by Schentiens and Leermakers [27],
QP is expressed as QP = 1
NL
1
z
∑
rN
∑
αN
GαN(r,N |1), where rN and αN denote the position and
orientation of the Nth segment of the chain, respectively.
∑
rN
∑
αN
means the summation over
43601-2
Effects of polymer concentration and chain length on aggregation
all the possible positions and orientations of the Nth segment of the chain. Gαs(r, s|1) is the end
segment distribution function of the sth segment of the chain, which is evaluated from the following
recursive relation:
Gαs(r, s|1) = G(r, s)
∑
r′
s−1
∑
αs−1
λ
αs−αs−1
rs−r′
s−1
Gαs−1(r′, s− 1|1), (2.4)
where G(r, s) is the free segment weighting factor and is expressed as
G(r, s) =
{
exp[−ωst(rs
)], s ∈ st ;
exp[−ωns(rs
)], s ∈ ns .
The initial condition is Gα1(r, 1|1) = G(r, 1) for all the values of α1. In the above expression, the
values of λ
αs−αs−1
rs−r′s−1
depend on the chain model used. We assume that
λ
αs−αs−1
rs−r′s−1
=
{
0, αs = αs−1 ;
1/(z − 1), otherwise .
This means that the chain is described as a random walk without the possibility of direct back-
folding. Although self-intersections of a chain are not permitted, the excluded volume effect is
sufficiently taken into account [28]. Another end segment distribution function Gαs(r, s|N) is eval-
uated from the following recursive relation:
Gαs(r, s|N) = G(r, s)
∑
r′
s+1
∑
αs+1
λ
αs+1−αs
r′
s+1
−rs
Gαs+1(r′, s+ 1|N), (2.5)
with the initial condition GαN(r,N |N) = G(r,N) for all the values of αN.
Using the expressions of the end segment distribution functions, the single-segment probability
distribution function P (1)(r, s) and the two-segment probability distribution function
P (2)(r1, s1; r2, s2) of the chain can be defined as follows:
P (1)(r, s) =
1
zNLQP
∑
r′s
∑
αs
Gαs(r′, s|1)Gαs(r′, s|N)
G(r′, s)
δr′s,r , (2.6)
which is the normalized probability that the monomer s of the chain is on the lattice site r;
P (2)(r1, s1; r2, s2) =
1
zNLQP
∑
r
′
s1
∑
αs1
∑
r
′
s2
∑
αs2
Gαs1 (r
′
, s1|1)δr′s1 ,r1
× G(r
′
, s1; r
′
, s2)G
αs2 (r
′
, s2|N)δr′s2 ,r2
(2.7)
and
G(r, s1; r, s2) =
∑
rs1+1
∑
αs1+1
.....
∑
rs2−1
∑
αs2−1
{
s2−1∏
s=s1+1
λ
αs−αs−1
rs−rs−1
G(r, s)
}
λ
αs2
−αs2−1
rs2−rs2−1
(for s2 > s1)
yield the probability that the monomers s1 and s2 of the chain are on the lattice sites r1 and r2,
respectively. It can be verified that
∑
r P
(1)(r, s) = 1, and
∑
r2
P (2)(r1, s1; r2, s2) = P (1)(r1, s1).
Equation (2.3) can be considered as the alternative form of the self-consistent field free energy
functional for incompressible polymer solutions [29]. When a local concentration approximation
for the non-bonded interaction is introduced, the SCFT description of lattice model for PAPSs
presented in this work is basically equivalent to that of the “Gaussian thread model” chain for the
similar polymer solutions [29]. The related details are presented in [10].
Minimization of the free energy function F with ω
−
(r) and ω
+
(r) leads to the following saddle
point equations:
ω
−
(r) = 2χφst(r), (2.8)
43601-3
X.-G. Han et al.
φst(r) + φns(r) + φh(r) = 1, (2.9)
where
φst(r) =
1
NL
1
z
nP
QP
∑
s∈st
∑
αs
Gαs(r, s|1)Gαs(r, s|N)
G(r, s)
(2.10)
and
φns(r) =
1
NL
1
z
nP
QP
∑
s∈ns
∑
αs
Gαs(r, s|1)Gαs(r, s|N)
G(r, s)
(2.11)
are the average numbers of sticker and nonsticky segments at r, respectively, and
φh(r) =
1
NL
nh
Q
h
exp [−ωh(r)] is the average numbers of solvent molecules at r.
The saddle point is calculated using the pseudo-dynamical evolution process presented by
Fredrickson et al. [30]:
ωnew
+
(r) = ωold
+
(r) + λ+(φst(r) + φns(r) + φh(r)− 1), (2.12)
ωnew
−
(r) = ωold
−
(r) + λ−(φst(r) −
ω
−
(r)
2χ
). (2.13)
The calculation is initiated from appropriately randomly-chosen fields ω
+
(r) and ω−(r), and
stopped when the change of free energy F between two successive iterations is reduced to the needed
precision. The resulting configuration is taken as a saddle point configuration. By comparing the
free energies of the saddle point configurations obtained from different initial fields, the relative
stability of the observed morphologies can be assessed.
3. Result and discussion
In our studies, the properties of associating polymers depend on three tunable molecular pa-
rameters: χ (The Flory-Huggins interaction parameter), N (Chain length) and l (The number of
nonsticky monomers between two neighboring stickers along the backbone, l equals 9 in this paper).
The calculations are performed in three-dimensional simple cubic lattice with periodic boundary
condition. The aggregation behavior is first sketched using the lattice with the size NL = 263.
Then, the obtained results are verified using larger size lattices. The results presented below are
obtained from the lattice with NL = 403. Three different morphologies, i.e., the homogenous,
micro-fluctuation homogenous and micelle morphologies, are observed in PAPSs. By comparing
the relative stability of the observed states, the phase diagram is constructed.
Figure 1 shows the phase diagram of the systems with different chain lengths N . In this study,
when N is changed, only MFH morphology and micelles are observed as inhomogeneous morpholo-
gies. The structural morphology of MFH morphology does not change, and micellar shape remains
spherelike (see figure 2). For N = 21, the χ value on micellar boundary (∼ 1/Tm) increases with
decreasing φ̄P. Approaching to critical micelle concentration (φ̄CMC = 0.04)1, micellar boundary
abruptly becomes steep. The χ value on MFH boundary (∼ 1/TMFH) also goes up with the decrease
in φ̄P, which resembles the behavior on the micellar boundary. The critical MFH concentration
(φ̄CFC = 0.37) is much higher than that of micellar morphology. Therefore, the MFH boundary
intersects with the micellar boundary at φ̄CFC. When N is increased, at fixed φ̄P, the χ value
on micellar boundary shifts slightly to larger value, and the χ value on MFH boundary decreases
markedly, which is different from that on micellar boundary. φ̄CFC also decreases with the increase
in N .
Figure 3 shows the variations of the logarithm of φ̄CFC with the logarithm of N . It is seen that
the straight line with a slope equaling −1 fits the results for all chain lengths considered in this
study. The appearance of MFH morphology is considered as the onset of gelation [10, 25]. Therefore,
1Critical micelle concentration (CMC) is generally considered as the minimum concentration of the micellar
appearance. In this study, the concentration which corresponds to the χ abrupt increase on micellar boundary is
regarded as CMC, which is denoted by φ̄CMC. When N = 21, φ̄CMC = 0.04. When N is increased φ̄CMC decreases.
For N = 101, the φ̄CMC is not observed till φ̄P = 0.015.
43601-4
Effects of polymer concentration and chain length on aggregation
0.00 0.15 0.30 0.45 0.60 0.75 0.90
0
2
4
6
8
P
Figure 1. (Color online) The phase diagram for the systems with different chain lengths N . The
boundaries of MFH and micelle morphologies are obtained. The red open and solid squares,
green open and solid triangles, blue open and solid diamonds, magenta open and solid pen-
tagons correspond to the boundaries of MFH and micelle morphologies for N = 21, 41, 81, 101,
respectively.
0 1 2
0.00
0.02
0.04
0.06
0.08
0.8
0.9
1.0
1.1
0.0
0.2
0.4
0.6
0.8
1.0
( )
st
r
r
ns
Figure 2. (Color online) The variations of the volume fractions of sticker and nonsticker compo-
nents of polymers in micelle morphology with r when φ̄P = 0.1 and χ = 5.5 in the systems with
N = 21 and 101. r is the distance from the sticker-rich core. The red solid and open squares,
blue solid and open triangles denote the volume fractions of stickers and nonstickers for N = 101
and 21, respectively.
critical MFH concentration φ̄CFC should correspond to the critical gelation concentration C∗, which
is analyzed in terms of the concept of chain overlap. The critical gelation concentration C∗, at which
quasi-ideal coils begin to overlap, the pervaded volume of one another is related to the chain length
as N = KC−2
∗ , where K is a constant [31–33]. The corresponding slope of the critical concentration
C∗ at quasi-ideal coil is −2. Similarly, at the excluded volume chain, the corresponding quantity
is −5/4. The slope of fitting straight line at associative polymer chain is the smallest of the above
three cases. It is demonstrated that the physically associating polymer chain in solution should be
elongated compared with the excluded volume chain and quasi-ideal coil.
In order to measure the degree of aggregation of stickers, an order-parameter-type variable [10]
is used in this work. It is expressed as:
Φ({φst(r)}) =
1
NL
∑
r
(φst(r)− φ̄st)
2 =
1
NL
∑
r
φ2
st(r) − φ̄2
st , (3.1)
43601-5
X.-G. Han et al.
1.20 1.35 1.50 1.65 1.80 1.95 2.10
-1.20
-1.05
-0.90
-0.75
-0.60
-0.45
-0.30
log
CFC
log N
Figure 3. The logarithm of critical MFH concentration φ̄CFC a function of the logarithm of chain
length N , which fulfils a linear-fitting relationship with the slope equaling -1.
0.00 0.75 1.50 2.25 3.00 3.75 4.50
0.000
0.015
0.030
0.045
0.060
0.075
Figure 4. (Color online) The variation of the order-parameter-type variable Φ with χ near the
HS-MFH and MFH-micelle transition points at φ̄P = 0.8 in the systems with different chain
length N . The red squares and blue triangles correspond to N = 21 and N = 81, respectively.
With the increase in χ, there are the appearances of two nonzero steps on the above two curves,
which denote the HS-MFH and MFH-micelle transitions, respectively.
where Φ is determined by the distribution of the volume fraction of stickers {φst(r)}, which is
a function of χ and φ̄P. For homogenous solutions (HS), Φ is equal to zero. Figure 4 shows the
variations of Φ in MFH and micelle morphologies with an increasing χ when φ̄P = 0.8 near HS-
MFH and MFH-micelle transition points, in the systems with N = 21 and N = 81. It is seen that
the degree of the aggregation of stickers increases with an increasing χ at fixed N . Compared with
N = 81, although the value of Φ at fixed χ above MFH-micelle transition point at N = 21 is larger,
the variations of Φ on the above two cases with χ are similar. However, near HS-MFH transition
point, the variation of Φ with χ at N = 21 is obviously different from that of N = 81. The value of
Φ at N = 21 increases more rapidly than that of N = 81 when χ is increased. The heat capacity
CV is proportional to the first derivative of Φ with respect to temperature [10, 34, 35]. When N is
increased, the maximum of the peak of CV decreases, and the corresponding half-width increases
near the HS-MFH transition. The shape of the peak of CV near MFH-micelle transition point does
not practically change with the increase in N (not shown). It is demonstrated that the change of
N has a greater effect on HS-MFH transition than that on MFH-micelle transition.
The aggregation number of micelles, which is used to account for some properties, for example,
43601-6
Effects of polymer concentration and chain length on aggregation
0.00 0.15 0.30 0.45 0.60 0.75 0.90
0.75
0.80
0.85
0.90
0.95
1.00
S co
r
P
Figure 5. (Color online) The average volume fractions of stickers at micellar core as a function of
φ̄P in micellar boundary systems with different chain lengths. The red squares, green triangles,
blue diamonds and magenta pentagons correspond to the boundaries of micelle morphology for
N = 21, 41, 81, 101, respectively.
the rheological behavior of associative polymers, is a major focus in many experimental [36–38]
and theoretic [39] studies. In this paper, the average volume fraction of stickers at micellar cores
〈φs(rco)〉 is similar to the average aggregation number of the micelle. Figure 5 shows the variations of
〈φs(rco)〉 on micellar boundary with φ̄P and N in the systems. Being given a fixed chain length, when
φ̄P is increased, 〈φs(rco)〉 decreases. When N is increased, at fixed φ̄P, the variation of 〈φs(rco)〉
with φ̄P, does not practically change for φ̄P 6 0.4, on the other hand, when 0.4 < φ̄P 6 0.9,
〈φs(rco)〉 increases. The decreasing tendency of 〈φs(rco)〉 with φ̄P becomes gentle with the increase
in N at high polymer concentrations. It is shown that the effect of the increase in N on 〈φs(rco)〉
is related to φ̄P.
In order to interpret the effects of N and φ̄P on the aggregation of stickers in micelle morphology,
we evaluate the probabilities that a polymer chain forms intrachain and interchain associations in
the system using an approach similar to the one presented in Refs. [10, 40, 41]. We suppose that
there are no other sticker aggregates in the micellar system except the micellar cores because the
volume fraction of stickers at micellar core approaches 1, and that of the rest of the system is less
than 10−1. A sticker in a particular chain can form an intrachain association, as well as interchain
association. Ignoring the probabilities that more than two stickers of a definite chain are attached to
a micellar core, the conditional probability that the sticker s1 concerns the intrachain association,
provided that the sticker s1 is at the micelle core rco, can be expressed as:
ploop(rco, s1) =
1
P (1)(rco, s1)
∑
s2∈st,s2 6=s1
P (2)(rco, s1; rco, s2), (3.2)
where
∑
s2∈st,s2 6=s1
means the summation over all the stickers of a polymer chain except the
s1th one, P (1)(rco, s1) and P (2)(rco, s1; rco, s2) are the single-segment and two-segment probability
distribution functions of a chain, respectively. Then, 1−ploop(rco, s1) is the conditional probability
that the sticker s1 is linked with those belonging to other chains when the sticker s1 is at rco, and
Plk(s1) =
∑
rco
P (1)(rco, s1) · ⌈1− ploop(rco, s1)⌉
is the probability that a sticker s1 of a chain is related to interchain association, where
∑
rco
means the summation over all the micellar cores of the system. The summation of Plk(s1) over
all the stickers in a chain, 〈n
lk
〉 =
∑
s1,s1∈st Plk(s1), can be viewed as the average sticker num-
ber from a particular polymer chain linked with other chains by sticker aggregates. The total
number of stickers participating in interchain association in the system is expressed as ninte =
43601-7
X.-G. Han et al.
nP〈nlk
〉. The total number of stickers belonging to intrachain association is defined as nintr =
(1/Nst)[
∑
s1,s1∈st{(1/Nm){
∑
rco
ploop(rco, s1)}}]·〈φs(rco)〉Nm, where Nm denotes the micellar num-
ber in the system.
0.00 0.15 0.30 0.45 0.60 0.75 0.90
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
int r
fint e
f
P
Figure 6. (Color online) The variations of the average fractions of intrachain and interchain
associations at a micellar core with φ̄P in the same systems presented in figure 5, denoted by
fintr and finte, respectively. The red open and solid squares, green open and solid triangles, blue
open and solid diamonds, magenta open and solid pentagons correspond to fintr and finte for
N = 21, 41, 81, 101, respectively.
Figure 6 shows the variations of the average fractions of intrachain and interchain associations
within a micelle, denoted by fintr = nintr/Nm and finte = ninte/Nm, respectively, with φ̄P and N in
the systems identical with those presented in figure 5. Being given a fixed chain length, when φ̄P
is increased, fintr does decrease markedly first, and then the decreasing tendency of fintr becomes
gentle when φ̄P exceeds some value. Except for the case of N = 21, finte at fixed N increases
markedly first, and then increases gently with the increase in φ̄P, which is contrary to fintr. At
N = 21, finte increases to φ̄P = 0.6, then decreases slightly with the increase in φ̄P. When N is
increased, fintr at fixed φ̄P increases, and corresponding finte decreases. When φ̄P is increased the
differences of fintr and finte in various N become small. In other words, the effect of chain length
on fintr and finte at low concentrations is more pronounced than that at high concentrations.
The relative magnitude of fintr and finte at fixed N is dependent on φ̄P. Except for N = 21,
when φ̄P is low, finte at fixed N is smaller than corresponding fintr. For N = 101, fintr equals 1,
and corresponding finte equals =0 at φ̄P = 0.015 and χ = 6.0. The micelle is absolutely aggregated
by intrachain association. When φ̄P exceeds some value, which is related to N , finte at fixed N is
larger than corresponding fintr. When N = 21, finte is always larger than fintr in the considered
range of φ̄P.
In light of the variations of 〈φs(rco)〉 with φ̄P and N in the systems, the effect of chain length
on aggregation of stickers is more pronounced at high concentrations. However, seen from respec-
tive value of fintr and finte at fixed φ̄P, the increase of chain length is more favorable to the
change of intrachain and interchain associations at low concentrations, which is different from
the above conclusions drawn in light of 〈φs(rco)〉. It is shown that [see figure 6], being given a
fixed chain length, although fintr is smaller than the corresponding finte at high concentrations,
the decrease of 〈φs(rco)〉 with the increase in φ̄P is similar to that of fintr, instead of finte, in
the considered range of φ̄P. When N is increased, fintr at fixed φ̄P goes up. It is notable that
the effect of chain length on fintr at low concentrations is much more pronounced than that at
high concentrations. However, the increase of fintr at fixed φ̄P does not result in the dramati-
cal change of 〈φs(rco)〉 at low concentrations. At high concentrations (φ̄P > 0.4), on the con-
trary, 〈φs(rco)〉 with a big chain length has a larger value. It is reasonable that 〈φs(rco)〉 is also
affected by finte. When φ̄P is low, finte at fixed φ̄P with a long chain length is much smaller
than that with a short chain. Therefore, the increase of fintr at fixed φ̄P is canceled by the de-
43601-8
Effects of polymer concentration and chain length on aggregation
crease of corresponding finte, thus increasing in N . With the increase in φ̄P, finte at fixed φ̄P
goes up markedly. The difference of finte among different N becomes small. The contribution of
fintr to 〈φs(rco)〉 resulting from the increase in N begins to appear. It is demonstrated that the
effects of chain length and polymer concentration on 〈φs(rco)〉 do interact each other. The in-
crease of chain length is favorable to intrachain association, and retains interchain association,
which is contrary to the effect of the increase in φ̄P. When φ̄P is increased, the effect of chain
length on intrachain and interchain associations is weakened. In other words, the polymer con-
centration and chain length simultaneously control the formations of intrachain and interchain
associations.
4. Conclusion and summary
Using the self-consistent field lattice model, the effects of polymer concentration φ̄P and chain
length N on aggregation in physically associating polymer solutions are studied. When N is
changed, only two inhomogenous aggregates, i.e., the MFH and micelle morphologies, are ob-
served in PAPSs. When φ̄P is decreased, being given a fixed N , the χ values on MFH and micellar
boundaries increase. At fixed φ̄P, the increase in N remarkably decreases the χ value on MFH
boundary, but slightly increases the χ value on micellar boundary. The logarithm of critical MFH
concentration as a function of the logarithm of N fulfils a fitting straight line with a slope equal-
ing -1, which demonstrates that the associating polymer chain in solution should be elongated
compared with an excluded volume chain. Furthermore, on micellar boundary, the average volume
fraction of stickers at a micellar core, 〈φs(rco)〉, which is similar to the average aggregation number,
decreases at fixed N when φ̄P is increased. With the increase in N , on the other hand, 〈φs(rco)〉,
at fixed φ̄P, does not practically change when φ̄P 6 0.4. When φ̄P > 0.4, on the other hand, the
increase in N causes the increase of 〈φs(rco)〉 at fixed φ̄P. There is found a decreasing tendency
of variation of 〈φs(rco)〉 at fixed N , with φ̄P is determined by intrachain association, instead of
interchain association, and the magnitude of 〈φs(rco)〉 with different N is affected by intrachain
and interchain associations. At fixed φ̄P, when the difference of the contribution of interchain as-
sociation to 〈φs(rco)〉 between different N goes down to some extent, the corresponding effect of
intrachain association on 〈φs(rco)〉 begins to appear.
The architectural parameters of polymer are important to the properties of PAPSs. In this
paper, only the effect of chain length of the polymer is investigated. When the other parameters,
for example, the nonsticky monomer number between two neighboring stickers l, are changed, the
aggregation behavior is different. When l is increased, the χ values on MFH and micellar boundaries
rise. The increase of l has a different effect on the specific heat peaks for HS-MFH and MFH-micelle
transitions. Systemical studies are presented in our subsequent work. Our calculations are within
the mean field framework, which may also be taken as a starting point to the further study, i.e., the
direct calculation of the partition function of the system using complex Langevin simulations [29].
Acknowledgements
This research is supported by the Innovation Fund of Inner Mongolia University of Science and
Technology (Grant No. 2010NC065) the High Performance Computers of Inner Mongolia University
of Science and Technology.
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Effects of polymer concentration and chain length on aggregation
Вплив концентрацiї полiмера i довжини ланцюга
на агрегацiю у фiзично асоцiйованих
розчинах полiмерiв
К.Ґ. Ган1, К.-Ф. Жанг1, Й.-Г. Ма1, С.-К. Жан2, Й.-Б. Ґуан2
1 Школа математики, фiзики i бiотехнологiй, Унiверситет науки i технологiй Внутрiшньої Монголiї,
014010 Баоту, Китай
2 Фiзичний факультет, Унiверситет Джiлiн, 130021 Чанґчун, Китай
Використовуючи ґраткову модель самоузгодженого поля, вивчається вплив концентрацiї полiмера i
довжини ланцюга на агрегацiю в асоцiативних полiмерних розчинах. У системах з рiзною довжиною
ланцюга спостерiгається тiльки двi неоднорiднi морфологiї, а саме мiкрофлуктуацiйна однорiдна
(MFH) i мiцелярна морфологiї. Температури, при яких вище згаданi двi морфологiї виникають впер-
ше i позначаються як TMFH i Tm, вiдповiдно, є незалежними вiд концентрацiї полiмера i довжини
ланцюга. Змiна логарифма критичної концентрацiї MFH зi змiною логарифма довжини ланцюга за-
довiльняє спiввiдношенню лiнiйного допасування з нахилом рiвним −1. Крiм того, змiна середньої
об’ємної фракцiї стiкерiв при мiцелярному корi (AVFSM) з концентрацiєю полiмера i довжиною лан-
цюга сфокусована в системi при Tm. Знайдено шляхом розрахункiв, що вище згадана поведiнка
AVFSM пояснюється в термiнах iнтраланцюгових та iнтерланцюгових асоцiацiй.
Ключовi слова: концентрацiя, довжина ланцюга, агрегацiя, асоцiативний полiмер
43601-11
Introduction
Theory
Result and discussion
Conclusion and summary
|