Cylindrically confined assembly of diblock copolymer under oscillatory shear flow
Manipulating the self-assembly nanostructures with combined different control measures is emerging as a promising route for numerous applications to generate templates and scaffolds for nanostructured materials. Here, the two different control measures are a cylindrical confinement and an oscillat...
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| Zitieren: | Cylindrically confined assembly of diblock copolymer under oscillatory shear flow / Y.-Q. Guo, J.-X. Pan, J.-J. Zhang, M.-N. Sun, B.-F. Wang, H.-Sh. Wu // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33601: 1–12. — Бібліогр.: 73 назв. — англ. |
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irk-123456789-1562052019-06-19T01:25:51Z Cylindrically confined assembly of diblock copolymer under oscillatory shear flow Guo, Y.-Q. Pan, J.-X. Zhang, J.-J. Sun, M.-N. Wang, B.-F. Wu, H.-Sh. Manipulating the self-assembly nanostructures with combined different control measures is emerging as a promising route for numerous applications to generate templates and scaffolds for nanostructured materials. Here, the two different control measures are a cylindrical confinement and an oscillatory shear flow. We study the phase behavior of diblock copolymer confined in nanopore under oscillatory shear by considering different D/L₀ (D is the diameter of the cylindrical nanopore, L₀ is the domain spacing) and different shears via Cell Dynamics Simulation. Under different D/L₀ , in the system occurs different morphology evolution and phase transition with the changing of amplitude and frequency. Meanwhile, it forms a series of novel morphologies. For each D/L₀ , we construct a phase diagram of different forms and analyze the reason why the phase transition occurs. We find that although the morphologies are different under different D/L₀ , the reason of the phase transition with the changing of amplitude and frequency is roughly the same, all caused by the interplay of the field effect and confinement effect. These results can guide an experimentalist to an easy method of creating the ordered, defect-free nanostructured materials using a combination of the cylindrical confinement and oscillatory shear flow. Керування самоскупченими наноструктурами з допомогою поєднання рiзних заходiв керування виявляється багатообiцяючим шляхом для чисельних застосувань для генерування шаблонiв для наноструктурованих матерiалiв. У цiй статтi двома рiзними заходами керування є цилiндричне обмеження та осциляцiйний зсувний потiк. Дослiджується фазова поведiнка дiблочного кополiмера, обмеженого нанопорою пiд дiєю осциляцiйного зсуву з врахуванням рiзних D/L₀ (D — це дiаметр цилiндричної пори, L₀ — це розмiр домена) та рiзних зсувiв з допомогою симуляцiй комiркової динамiки. При рiзних D/L₀ , в системi вiдбувається еволюцiя рiзної морфологiї i фазовий перехiд зi змiною амплiтуди i частоти. Тим часом, утворюється ряд нових морфологiй. Для кожного D/L₀ , ми будуємо фазову дiаграму рiзних форм та аналiзуємо причини здiйснення фазового переходу. Встановлено, що, хоча морфологiї вiдрiзняються при рiзних D/L₀ , причина фазового переходу при змiнi амплiтуди i частоти є приблизно однаковою, а саме вона полягає в поєднаннi польового ефекту i ефекту просторового обмеження. Цi результати можуть дати експериментаторам простий метод створення впорядкованих, бездефектних наноструктурованих матерiалiв за рахунок поєднання цилiндричного обмеження та осциляцiйного зсувного потоку. 2016 Article Cylindrically confined assembly of diblock copolymer under oscillatory shear flow / Y.-Q. Guo, J.-X. Pan, J.-J. Zhang, M.-N. Sun, B.-F. Wang, H.-Sh. Wu // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33601: 1–12. — Бібліогр.: 73 назв. — англ. 1607-324X PACS: 64.75.Yz, 64.75.Va, 83.80.Uv, 83.10.Tv DOI:10.5488/CMP.19.33601 arXiv:1609.04704 http://dspace.nbuv.gov.ua/handle/123456789/156205 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 |
Manipulating the self-assembly nanostructures with combined different control measures is emerging as a
promising route for numerous applications to generate templates and scaffolds for nanostructured materials.
Here, the two different control measures are a cylindrical confinement and an oscillatory shear flow. We study
the phase behavior of diblock copolymer confined in nanopore under oscillatory shear by considering different
D/L₀ (D is the diameter of the cylindrical nanopore, L₀ is the domain spacing) and different shears via Cell
Dynamics Simulation. Under different D/L₀ , in the system occurs different morphology evolution and phase
transition with the changing of amplitude and frequency. Meanwhile, it forms a series of novel morphologies.
For each D/L₀ , we construct a phase diagram of different forms and analyze the reason why the phase transition occurs. We find that although the morphologies are different under different D/L₀ , the reason of the
phase transition with the changing of amplitude and frequency is roughly the same, all caused by the interplay
of the field effect and confinement effect. These results can guide an experimentalist to an easy method of
creating the ordered, defect-free nanostructured materials using a combination of the cylindrical confinement
and oscillatory shear flow. |
| format |
Article |
| author |
Guo, Y.-Q. Pan, J.-X. Zhang, J.-J. Sun, M.-N. Wang, B.-F. Wu, H.-Sh. |
| spellingShingle |
Guo, Y.-Q. Pan, J.-X. Zhang, J.-J. Sun, M.-N. Wang, B.-F. Wu, H.-Sh. Cylindrically confined assembly of diblock copolymer under oscillatory shear flow Condensed Matter Physics |
| author_facet |
Guo, Y.-Q. Pan, J.-X. Zhang, J.-J. Sun, M.-N. Wang, B.-F. Wu, H.-Sh. |
| author_sort |
Guo, Y.-Q. |
| title |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| title_short |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| title_full |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| title_fullStr |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| title_full_unstemmed |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| title_sort |
cylindrically confined assembly of diblock copolymer under oscillatory shear flow |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156205 |
| citation_txt |
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow / Y.-Q. Guo, J.-X. Pan, J.-J. Zhang, M.-N. Sun, B.-F. Wang, H.-Sh. Wu // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33601: 1–12. — Бібліогр.: 73 назв. — англ. |
| series |
Condensed Matter Physics |
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AT guoyq cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow AT panjx cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow AT zhangjj cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow AT sunmn cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow AT wangbf cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow AT wuhsh cylindricallyconfinedassemblyofdiblockcopolymerunderoscillatoryshearflow |
| first_indexed |
2025-07-14T08:35:26Z |
| last_indexed |
2025-07-14T08:35:26Z |
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1837610698715168768 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 3, 33601: 1–12
DOI: 10.5488/CMP.19.33601
http://www.icmp.lviv.ua/journal
Cylindrically confined assembly of diblock copolymer
under oscillatory shear flow
Y.-Q. Guo, J.-X. Pan, J.-J. Zhang, M.-N. Sun, B.-F. Wang, H.-Sh. Wu
School of Chemistry and Materials Science, Shanxi Normal University, Linfen, 041004, China
Received November 30, 2015, in final form January 18, 2016
Manipulating the self-assembly nanostructures with combined different control measures is emerging as a
promising route for numerous applications to generate templates and scaffolds for nanostructured materials.
Here, the two different control measures are a cylindrical confinement and an oscillatory shear flow. We study
the phase behavior of diblock copolymer confined in nanopore under oscillatory shear by considering different
D/L0 (D is the diameter of the cylindrical nanopore, L0 is the domain spacing) and different shears via CellDynamics Simulation. Under different D/L0 , in the system occurs different morphology evolution and phasetransition with the changing of amplitude and frequency. Meanwhile, it forms a series of novel morphologies.
For each D/L0 , we construct a phase diagram of different forms and analyze the reason why the phase tran-sition occurs. We find that although the morphologies are different under different D/L0 , the reason of thephase transition with the changing of amplitude and frequency is roughly the same, all caused by the interplay
of the field effect and confinement effect. These results can guide an experimentalist to an easy method of
creating the ordered, defect-free nanostructured materials using a combination of the cylindrical confinement
and oscillatory shear flow.
Key words: self-assembly, block copolymer, cylindrical confinement, oscillatory shear flow
PACS: 64.75.Yz, 64.75.Va, 83.80.Uv, 83.10.Tv
1. Introduction
Block copolymer is a class of soft material capable of self-assembling to form ordered structures at
nanometer scales. These structures possess a potential for various applications to nanotechnologies such
as lithographic templates for nanowires, photonic crystals, high-density magnetic storagemedia, drug de-
livery and biomineralization. So, the self-assembly of block copolymer has attracted much interest as an
efficient and effectivemeans to create structures at nanometer scales. Obviously, how tomanipulate these
self-assembled nanostructures have become the key of the research, so understanding the controlling fac-
tors is an intriguing field of research that holds promise for further expanding the nanofabrication. In
general, the control measures mainly include a confined environment [1, 2], imposed external field [3, 4],
substrate induction [5, 6], doping [7, 8], and branched molecular architectures [9] etc. Thereinto, confine-
ment effect makes the polymer system form a series of novel morphologies that are not accessible in the
bulk, and the external effect makes the structures more ordered. They both provide an effective route to
manipulate the self-assembled nanostructures.
During the past years, the effects of a confined environment have been extensively explored based
on a large number of experimental and theoretical studies. The internal branching is one of the con-
finements, i.e., one imposes some restrictions on the conformational freedom. For example, Ilnytskyi
et al. analyzed the peculiarities of the equilibrium morphologies observed for the star and comb diblock
polymers with equal molecular mass but with the differences in both intramolecular architecture and
composition fraction [9]. In addition, the geometrical confinement can be classified as one-, two-, and
three-dimensionally confined systems. The one-dimensionally confined system [10, 11] is the simplest
case, and its research is very mature. For a more complex three-dimensionally confined system, the stud-
ies are relatively few so far [12, 13]. The research area of two-dimensional confinement is developing
© Y.-Q. Guo, J.-X. Pan, J.-J. Zhang, M.-N. Sun, B.-F. Wang, H.-Sh. Wu, 2016 33601-1
http://dx.doi.org/10.5488/CMP.19.33601
http://www.icmp.lviv.ua/journal
Y.-Q. Guo et al.
rapidly. When placing the block copolymer in a cylindrical pore, a series of novel morphologies that
were not accessible in the bulk or in the one-dimensionally confined systems were observed at differ-
ent degrees of confinement D/L0 . Russell and co-workers discovered that the lamellae-, cylinder-, and
sphere-forming block copolymer of polystyrene-b-polybutadiene (PS-b-PBD) were confined in the pores
of anodized aluminumoxide (AAO) membranes experimentally [14–18]. They find that under cylindri-
cal confinement, block copolymer formed various kinds of novel structures that were not accessible in
the bulk, such as stacked disks, concentric cylinders, torus-like structure, core-shell structure, single row
and zigzag arrangement of spherical microdomains, single-, double-, triple-helical structures etc. Sun
et al. also observed the diameter-dependence of morphology confined within the ordered porous alu-
mina templates, but the block copolymer that they used was polystyrene-b-poly (methyl methacrylate)
(PS-b-PMMA) [19] and polyethylene-co-butylene-b-polyethylene oxide (PHB-b-PEO) experimentally [20].
For symmetric lamellae-forming diblock copolymer in cylindrical pores, many novel morphologies were
predicted based on dynamic density functional theory (DDFT) [21–23], Monte Carlo (MC) [24, 25], self-
consistent field theory (SCFT) [26], and simulated annealing technique (SAT) [27]. For the asymmetric
cylinder-forming diblock copolymer, the structures in the cylindrical pores severely deviated from the
bulk. The novel structures of spontaneous formation, such as stacked toroids, single helix, double helix
and perforated tubes, were observed by means of a SAT [28, 29] and SCFT [30–32]. For the asymmet-
ric sphere-forming diblock copolymer, the study of it was rare. Pinna et al. predicted tremendous rich
morphologies of sphere-forming diblock copolymer in cylindrical nanopores by using the cell dynam-
ics simulation (CDS) [33]. Recently, the phase behavior of sphere-forming triblock copolymer confined in
nanopores was investigated by Hao et al. by means of a DDFT [34]. They observed that typical structures
were different from the bulk morphologies, which were consistent with the available experiments.
As concerns the imposition of an external field, this is a versatile control measure to obtain the long-
range order and then to create microstructures with potential applications in biomaterials, optics, and
microelectronics. Earlier, during the study of block copolymer there were observed various alignments
in a lamellar block copolymer by using TEM and SAXS. Thereinto, polystyrene-polyisoprene is the most
representative. The parallel and perpendicular orientations were observed by K.I. Winey and S.S. Patel
et al. in 1993 [35] and 1995 [36], respectively; the transverse orientationwas found by V.K. Gupta et al. [37]
and Y.M. Zhang [38]. Then, B.S. Pinheiro and K.I. Winey observed the mixed parallel-perpendicular mor-
phologies in diblock copolymer systems at intermediate temperatures [39]. Afterwards, researchers the-
oretically predicted the phase transition of the lamellar [40–43], ring [44], hexagonal cylinders [45] di-
block copolymer subjected to shear flow further. These predictions were proved by researchers with
various numerical simulation methods, such as CDS [46–49], nonequilibrium molecular dynamics simu-
lation (NEMD) [50, 51], dissipative particle dynamics method (DPD) [52, 53], SCFT and lattice Boltzmann
(LB) method [54], Brownian dynamics (BD) [55], mean-field approach (MFA) [56], density functional the-
ory (DFT) [57, 58], MC [59] etc. For all these cases, shear flow plays an important role as a means for
aligning the microscopic domains.
As we all know, the polymer system can form all kinds of novel structures that are different from the
bulk morphologies under cylindrical confinement and form ordered structures under shear flow. So, we
wonder what the phase behavior of polymer system is under both these two control measures, and it
should be interesting when these two control measures affect the system. Previously, Pinna et al. [60]
combined the two control measures of the one-dimension confinement and steady shear flow. They
demonstrated the shear alignment and the shear-induced transitions in sphere-forming diblock copoly-
mer single layer and bilayer films by cell dynamics simulation that was observed experimentally by Hong
et al. [61], and for the first time presented a nontrivial alignment mechanism of a single layer of spher-
ical domains in shear. On this basis, we dedicate ourselves to investigate the phase behavior of diblock
copolymer within cylindrical pores under oscillatory shear flow by means of the CDS of time dependent
Ginzbrug-Landau (TDGL) theory proposed by Oono and co-workers [62–66]. It will further provide some
guiding function for experimentalists. The other parts of this paper are organized as follows: section 2
is devoted to the description of the model and simulation method; section 3 is the numerical results and
discussions; and finally, section 4 gives a brief conclusion in this work.
33601-2
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow
2. Models and simulation methods
Our simulations are performed in a neutral cylindrical nanopore with a diameter D and a length Lz .
We employ a TDGL approach in the form of the CDS, which is a very fast computational technique. For AB
diblock copolymer, the structure can be described by an order parameterφ(r, t ) =φA−φB+(1−2 f ), where
φA and φB are the local volume fractions of A and Bmonomers, respectively. The fraction of A monomers
in a diblock copolymer chain is denoted by f , f = NA/(NA+NB). The kinetics andmorphological evolution
are described, in the spirit of linear irreversible thermodynamics, by the TDGL equation for a diffusive
field coupled with an external velocity field, which can be written as [67]
∂φ(r, t )
∂t
+v ·∇∇∇φ(r, t ) = M∇2
{
δF [φ(r, t )]
δφ(r, t )
}
, (2.1)
whereM is a phenomenological mobility constant and is set to 1. r is an external velocity field.
For simplicity, we set the shear rate as
ν(r, t ) = (
0, 0, γωy cos(ωt )
)
, (2.2)
where γ is amplitude, andω is frequency.We set z-axis (the direction of the arrows) is the shear direction.
Here, we use a two-order-parameter model in [68, 69]. The long-range part FL and the short-range
part FS are given by
FL = α
2
Ï
drdr′G(r,r′)[φ(r)−φ0][φ(r)−φ0], (2.3)
and
FS =
Ï
dr
{
f (φ)+ D
2
[∇∇∇φ(r)
]2
}
, (2.4)
respectively. The long-range part is relatively simple, in which G(r,r′) is the Green’s function defined by
the equation −∇2G(r,r′) = δ(r−r′), while α is a parameter that introduces a chain-length dependence to
the free-energy, φ0 is the spatial averages of φ, we should set φ0 = 0 in the case of symmetric copolymers.
As for the short-range part, D is a positive constant that plays the role of a diffusion coefficient, and
f (φ) =−A tanh(φ)+φ. We carry out computer simulations of themodel system by using the CDS proposed
by Oono and Puri [62–66]. In three-dimensional CDS, the system is discretized on a Lx ×Ly ×Lz cubic
lattice, and the order parameter for each cell is defined as φ(n, t ), where n = (nx ,ny ,nz ) is the lattice
position and nx , ny , and nz are integers between 1 and L. The Laplacian in CDS is approximated by
∇2φ(n) = 〈〈φ(n)〉〉−φ(n), (2.5)
where 〈〈φ(n)〉〉 represents the following summation of φ(n) for the nearest neighbors (n.), the next-
nearest neighbors (n.n.), and the next-next-nearest neighbors (n.n.n.) [66]
〈〈φ(n)〉〉 = 6
80
∑
n=n.
φ(r)+ 3
80
∑
n=n.n.
φ(r)+ 1
80
∑
n=n.n.n.
φ(r). (2.6)
In our simulations, we set ∆x,∆y,∆z are 1, and ∆t = 1.0. Then, equation (2.1) can be transformed into
the following difference equation:
φ(r, t +∆t ) =φ(r, t )− 1
2
γsin(ωt )y t
[
φ(x, y, z +1, t )−φ(x, y, z −1, t )
]+M
(〈〈Iφ〉〉− Iφ
)−αφ(r, t ) (2.7)
with
Iφ =−D
(〈〈φ〉〉−φ)− A tanh(φ)+φ. (2.8)
We have chosen z-axis as the flow direction, y -axis as the velocity gradient direction and x-axis as the
vorticity axis. In addition to Dirichlet boundary condition [33], a shear periodic boundary condition pro-
posed by Ohta et al. [70–72] has been applied to z direction along the pore. With the shear strain Υ, this
boundary condition is written as
φ(nx ,ny ,nz , t ) =φ(
nx +Nx L, ny +Ny L, nz +Ny L+Υ(t )Ny L
)
, (2.9)
where Nx , Ny , and Nz are arbitrary integers. All parameters in this paper are scaled, and all of them are
dimensionless [62].
33601-3
Y.-Q. Guo et al.
3. Numerical results and discussion
In order to simulate the confined self-assembly of AB diblock copolymer under oscillatory shear flow,
we construct the cylindrical nanopore of diameter D in cubic lattice. The volume is V = Lx × Ly × Lz
with Lx = Ly = D +m and Lpore = Lz , wherem = 2 when D is odd andm = 3 when D is even. Polymers
cannot occupy the wall sites which are the lattice sites outside the cylinder with diameter D and the
wall is impenetrable. The extra m in Lx and Ly ensures that each site inside the cylindrical pore can
find its nearest neighbor, the next-nearest neighbor, and the next-next-nearest neighbor cells. These sites
are either inside the pore or in the wall. Thus, the polymers are confined in the cylindrical nanopore of
diameterD . In this paper, we investigate the phase behavior under oscillatory shear of diblock copolymer
in the cylindrical pore with different D/L0 , which L0 is the domain spacing, refer to [32].
Firstly, we investigate the effect of oscillatory shear flow in cylindrical nanopore of D/L0 = 1.01,
Lpore/L0 = 8.1. It is noted that the oscillatory shear is imposed on the disordered structures rather than
the final self-assembled structures. With the simulation on the continuous variation of shear amplitude
and shear frequency, we roughly construct a structure diagram in figure 1. In figure 1, symbol • repre-
sents single-helical structure that is abbreviated as SH, ? represents half-cylinder structure that is abbre-
viated as HC. The half-cylinder refers to A phase occupying half of the cylinder along the pipe axis.
In figure 1, the amplitude, γ, varies from 0 to 1.0, and the frequency, ω, varies from 0 to 0.1. We can
clearly see that in small amplitudes, such as γÉ 0.15, the field effect is too weak to have a sufficient capa-
bility to affect the morphology, therefore, no phase transition is observed at the given frequency, and the
single-helical structure is always preserved. Typical snapshots of this case are shown in figures 2 (a)–(d).
Moreover, in the small frequencies ω É 0.02, the structure is single-helical at all amplitudes. At ampli-
tudes 0.2 < γ < 0.35, with an increasing frequency, the single-helical structure transforms to the half-
cylinder, and then reverses to the single-helical structure. At the larger amplitude, when we increase the
frequency, the phase transition occursmore than once. In addition to the phase transition from the single-
helical structure to half-cylinder, and then to the single-helical structure, the phase morphology turns to
half-cylinder finally, as shown in figures 2 (e)–(h). Furthermore, we also observe that the morphology
turns directly to the half-cylinder from the single-helical structure with an increasing amplitude at the
frequency 0.09. At some frequencies, such as 0.07 ÉωÉ 0.08, while increasing the amplitude, the phase
transition occurs from single-helical structure to half-cylinder, then reverses to a single-helical structure,
Figure 1. Phase diagram of the diblock copolymer in nanopore D/L0 = 1.01 under oscillatory shear at
different amplitudes γ and frequencies ω. • represents single-helical structure which is abbreviated as
SH; ? represents half-cylinder structure which is abbreviated as HC.
33601-4
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow
Figure 2. Pattern evolution of diblock copolymer in nanopore of D/L0 = 1.01 under different oscillatory
shears: (a) γ= 0.01,ω= 0.00001, (b) γ= 0.01,ω= 0.001, (c) γ= 0.01,ω= 0.1, (d) γ= 0.01,ω= 1.0; (e) γ=
0.41,ω = 0.03, (f) γ = 0.41,ω = 0.04, (g) γ = 0.41,ω = 0.05, (h) γ = 0.41,ω = 0.09. Phase A is represented
by the black regions, phase B by the gray regions.
and then transforms to a half-cylinder structure. On the whole, the single-helical structure mainly con-
centrates on the region of the weak oscillatory shear flow, which can be seen in the bottom left-hand
corner in figure 1; the half-cylinder structure mainly concentrates on the region of the strong oscilla-
tory shear flow, which can be seen in the top right-hand corner in figure 1. Moreover, the critical shear
frequency of the phase transition becomes smaller and smaller with an increasing amplitude.
As seen infigure 2mentioned before, it shows themorphology evolutionwith an increasing frequency
at two typical amplitudes, γ= 0.01 and γ= 0.41. Phase A is represented by black regions, phase B by gray
regions. In figures 2 (a)–(d), no phase transition occurs during the frequency and in figures 2 (e)–(h), the
phase transition occurs more than once. These results are mainly due to the interplay of two factors: the
field effect caused by the oscillatory shear flow and the confinement effect produced by the confinement
boundary. Meanwhile, the field effect is the combined effect of both amplitude and frequency. For the
single-helical structure in the bottom left-hand corner, it is the same as the structure that is confined
in the cylinder with no shear of the same diameter, which is in agreement with the previous studies by
Morita [73] and Pinna [33]. They found that the helical structurewas usually observed at incommensurate
conditions. In our simulation, the length and diameter are slightly incommensurable with the domain
spacing; the amplitude and frequency are relatively low, the field effect is weaker while the confinement
effect is dominant. Thus, similarly, the single-helical structure is due to the extensional forces when the
length, diameter and the lamellae spacing are not commensurate. For the half-cylinder structure in the
top right-hand corner, the amplitude and frequency are higher. Relatively speaking, the field effect plays
an important role, so we get the half-cylinder structure along the direction of oscillatory shear flow. For
the repeated phase transition with an increasing frequency at a certain amplitude in figures 2 (e)–(h), we
consider that the coupling of amplitude and frequency plays a certain role. Comparedwith the field effect,
the confinement effect has an absolute superiority when the frequency is weaker. It is easier to form a
single-helical structure [figure 2 (e)]. With increasing the frequency, the movement of segments A and
B strengthens along the oscillatory shear direction, the coarse graining process is also intensified along
the field direction and then the ordered half-cylinder structure is formed [figure 2 (f)]. With a further
increase of the frequency, the movement is faster, more and more A and B segments accumulate, they
push each other and then it formes a single-helical structure again [figure 2 (g)], and the nature of this
33601-5
Y.-Q. Guo et al.
Figure 3. Simulation snapshots of the diblock copolymer in nanoporeD/L0 = 0.76 under oscillatory shear
at different amplitudes γ and frequencies ω. Phase A is represented by the black regions, phase B by the
gray regions.
structure is different from the helical under a weaker shear. When the frequency is high enough, the field
effect plays a leading role, it forms the ordered half-cylinder structure along the pipe axis [figure 2 (h)].
Then, we concentrate on the phase behavior under oscillatory shear flow in cylindrical nanopore
of D/L0 = 0.76 and Lpore/L0 = 8.1. The simulation snapshots under oscillatory shear flow at different
amplitudes γ and frequenciesω are shown in figure 3. As we can see in figure 3, we control the frequency
between 0 and 1, and consider the amplitude of various stages. Overall, it preferentially adopts a piled-
pancakes structure when the shear is low; while it adopts a half-cylinder structure that is parallel to the
pipe axis when the shear is high. When γ= 0.005 and γ= 0.01, the domain structure remains the piled-
pancakes in the entire frequency range. However, the number of pancakes decreases with an increase of
frequency when the amplitude γ is 0.01, at the same time, the layer spacing becomes wider. ∗ represents
the morphology and has no change, and it is used to distinguish these two different cases. When γ =
0.05 and γ = 0.1, with an increasing frequency, the piled-pancakes structure evolves into half-cylinder
structure due to the aggravation of the coarse graining process along the pipe axis. At the further increase
of frequency, at the coupling of amplitude and frequency, the system transforms to the piled-pancakes
structure and the number of pancakes decreases. Undoubtedly, then it turns to the half-cylinder structure
when the shear is strong. It is certain that the critical shear frequency of phase transition and frequency
range of each structure are different from each other. This phenomenon also exists in γ= 0.5 and γ= 1.0.
The higher the amplitude is, the lower the critical shear frequency is. The frequency range in the same
structure is smaller when the amplitude is higher, just like the piled-pancaked in 0 <ω< 0.0014, γ= 0.5
and 0 < ω É 0.0006, γ = 1.0; and the half-cylinder in 0.0014 É ω < 0.052, γ = 0.5 and 0.0006 < ω < 0.04,
γ = 1.0. Also, the number of pancakes that formed again are less than before, as well, the layer spacing
is wider than before in γ = 0.5. However, it forms a mixed structure between piled-pancakes and half-
cylinder just in a small range (i.e., 0.04 É ω < 0.06) under γ = 1.0. This is chiefly because at γ = 1.0, the
vibration amplitude along the pipe axis is large, it is difficult to stack and easy to stretch along the field
direction. Certainly, in the end of the frequency range, the system turns to a stable half-cylinder structure
of paralleling to the pipe axis.
Similar to the reason of the phenomenon in nanopore of D/L0 = 1.01, it is the result of the combina-
tion of the confinement effect and field effect. At a smaller amplitude, no phase transitionmainly depends
on the confinement effect. However, the phase transition at a larger amplitude is mainly decided by the
size of the frequency. The slower the frequency is, the weaker the field effect is. Thus, it is easier to form
piled-pancakes structure, as well as the structure with no shear, which is in agreement with the study
33601-6
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow
by Pinna [33], as a result of the competition between the effect of the neutral surface and the effect of
incommensurability. When the shear is weaker, the surface effect prevails at a small diameter, while
above a certain diameter, the bulk effect takes over. The first phase transition to the half-cylinder with
an increase of the frequency is due to the intensified graining process along the pipe axis caused by the
strong oscillation of A and B segments. In case of a further increase of the frequency, these segments
would be more piled up in z-axis and jostle each other to form the piled-pancakes structure due to the
too fast vibration. The last phase transition is caused by a strong field effect, and the half-cylinder struc-
ture is obtained so that A phase and B phase are completely separated. Furthermore, the decrease of the
critical frequency with an increase of amplitude also indicates that the phase transition occurs only if the
amplitude and frequency intercouple with each other.
Figure 4. The order-parameter profiles of φ along the pipe axis under different oscillatory shears. (a) γ=
0.1, from top to bottom, the frequencies are ω =0.0001, 0.04, 0.15, 0.25, 1.0, respectively. (b) The dotted
line represents γ= 0.5, ω= 0.052; the dash-dotted line represents γ= 1.0, ω= 0.04.
In figure 4, the abovementioned phase transition can be clearly seen in a different way. It displays the
order-parameter profiles ofφ along the pipe axis under different oscillatory shears. As seen in figure 4 (a),
γ= 0.1, from (a1) to (a5), the frequencies areω=0.0001, 0.04, 0.15, 0.25, 1.0, respectively. φ> 0 represents
A-rich domain, φ < 0 represents B-rich domain. This indicates that if the profile of φ along z-axis has
convexities and concavities, the convexity represents A phase, while the concavity represents B phase.
From figures 4 (a1) to (a2), the number of convexities decreases from 5 to 4 with an increase of frequency
from 0.0001 to 0.04 which corresponds to the domain morphologies in figure 3. An approximate straight
line in figure 4 (a3) represents φ≈ 0, it means that A and B phases are uniformly distributed in the z-axis.
Thus, the approximate straight line corresponds to the half-cylinder structure in 0.15 Éω< 0.23, γ= 0.1
as shown in figure 3. Then, the profile of φ along the z-axis changes from a straight line [figure 4 (a3)] to a
curved line [figure 4 (a4)], and then to a perfect straight line [figure 4 (a5)] with an increase of frequency
from 0.15 to 1.0. It means that the half-cylinder structure caused by the field effect is more perfect than
that caused by the intensity coarse graining process. In figure 4 (b), the dotted line represents the order-
parameter profile of γ = 0.5, ω = 0.052, and the dash-dotted line represents the order-parameter profile
of γ= 1.0,ω= 0.04. As we can see, they are completely consistent with the structures that an arrow points
to in figure 3.
For a relatively large nanopore with D/L0 = 4.05 and Lpore/L0 = 8.1, the phase diagrams of top
view under different oscillatory shears are shown in figure 5. The characteristic morphologies are found
33601-7
Y.-Q. Guo et al.
Figure 5. The top view phase diagrams of diblock copolymer in nanopore of D/L0 = 4.05 under different
oscillatory shears. Phase A is represented by the black regions, phase B by the gray regions.
mainly in the small frequency range of 0.004 ÉωÉ 0.008 in 0.6 É γÉ 1.0. In general, the most interesting
thing is the big response caused by a small effect, figure 5 is just this kind of situation. As we have seen,
when γ = 1.0, phase transition occurs repeatedly in a very small frequency range. The transformation
occurs from the initial concentric ring structure to the parallel lamellar structure, then the concentric
ring comes into being once again, and then transforms to the parallel lamellae with an increase of fre-
quency. Thereinto, there exists an intermediate state in the process of turning to the parallel lamellae for
the first time, it is the “Swiss roll” and “horseshoe” structures in ω= 0.0041 and ω= 0.0044, respectively.
At intermediate amplitude, γ = 0.8, since the amplitude is not enough large, the phase transition only
Figure 6. Pattern evolution of diblock copolymer in nanopore of D/L0 = 4.05 under different oscillatory
shears: γ= 1.0, (a) ω= 0.004, (b) ω= 0.0041, (c) ω= 0.0044, (d) ω= 0.005, (e) ω= 0.006, (f) ω= 0.008.
33601-8
Cylindrically confined assembly of diblock copolymer under oscillatory shear flow
occurs from the concentric ring to the parallel lamellae. Certainly, the intermediate state (i.e., the “Swiss
roll” and “horseshoe” structures) also exist. However, when the amplitude decreases to 0.6, we can find
that all the structures are a concentric ring in the given frequency range. Due to the amplitude being rel-
atively small, even if the frequency increases to 0.008, the field effect is still weaker than the confinement
effect.
As seen in figure 6, it shows the morphologies of full view that corresponds to γ = 1.0 in figure 5.
The three-dimensional morphology evolution with an increase of the frequency in figure 6 gives us a
better visual perception than that in figure 6. As for the reason why this phase transition occurs, we can
also interpret it as the interplay of the field effect caused by the oscillatory shear and the confinement
effect produced by the confinement boundary. The concentric ring structure is formed under this large
nanopore with no shear, and the same as the structure of the weaker shears, because the field effect is
almost neglected. When the shear is strong, the field effect prevails, then it forms the parallel lamellae
along the shear axis. The phase transition in the given frequency range is mainly induced by the ampli-
tude and frequency intercoupling with each other as before.
4. Conclusions
In the present work, we have combined two different control measures of the cylindrical confinement
and oscillatory shear flow to manipulate the self-assembly nanostructures and obtained novel morpholo-
gies. Specifically, we have predicted the phase behavior of diblock copolymer confined in nanopore under
oscillatory shear by considering the differentD/L0 and different shears. We have found that in nanopore
of D/L0 = 1.01, when the amplitude is small, there is no phase transition observed at a given frequency;
while at a certain amplitude, the system undergoes a phase transition from single-helical structure to
half-cylinder, and then reverses to a single-helical structure, transforms to half-cylinder structure with
an increase of shear frequency, finally. When we put the diblock copolymer in nanopore of D/L0 = 0.76,
the situation is similar, except that the structure of single-helical structure is replaced by piled-pancakes.
When the diblock copolymer is placed in a large nanopore ofD/L0 = 4.05, similarly, no phase transition is
observed at a smaller amplitude, and the phase transition occurs with an increase of the frequency under
a relatively bigger amplitude. It changes from a concentric ring to the parallel lamellar structure, then
reverses to a concentric ring, and finally, turns to a parallel lamellar structure. However, in the process of
phase transition, there exist two transient states, they are the “Swiss roll” and “horseshoe” structures. We
have discussed the transformation of three kinds of typical structures and constructed a phase diagram
of different forms with the changing amplitude and frequency. Although the morphologies at different
D/L0 are different, the reason for the phase transition with the change of amplitude and frequency is
roughly the same, which is the interplay of the field effect caused by the oscillatory shear and the con-
finement effect produced by the confinement boundary. These results can provide an easy method to
create the ordered, defect-free nanostructured materials for an experimentalist through the combined
control measures of the cylindrical confinement and oscillatory shear flow.
Acknowledgements
Project supported by the National Natural Science Foundation of China (Grant No. 21373131),
the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant
No. 20121404110004), and the Provincial Natural Science Foundation of Shanxi (Grant No. 2015011004),
the Research Foundation for Excellent Talents of Shanxi Provincial Department of Human Resources and
Social Security.
33601-9
Y.-Q. Guo et al.
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Цилiндрично обмежене скупчення дiблочного кополiмера
пiд дiєю осциляцiйного зсувного потоку
Ю.-К. Гуо,Ю.-К. Пен, Дж.-Дж. Жанг,М.-Н. Сун, В.-Ф. Вонг, Х.-Ш. Ву
Школа хiмiї та матерiалознавства,Шанхайський нормальний унiверситет, Лiнфен, 041004, Китай
Керування самоскупченими наноструктурами з допомогою поєднання рiзних заходiв керування виявля-
ється багатообiцяючим шляхом для чисельних застосувань для генерування шаблонiв для нанострукту-
рованих матерiалiв. У цiй статтi двома рiзними заходами керування є цилiндричне обмеження та осциля-
цiйний зсувний потiк. Дослiджується фазова поведiнка дiблочного кополiмера, обмеженого нанопорою
пiд дiєю осциляцiйного зсуву з врахуванням рiзних D/L0 (D — це дiаметр цилiндричної пори, L0 — це
розмiр домена) та рiзних зсувiв з допомогою симуляцiй комiркової динамiки. При рiзних D/L0 , в систе-мi вiдбувається еволюцiя рiзної морфологiї i фазовий перехiд зi змiною амплiтуди i частоти. Тим часом,
утворюється ряд нових морфологiй. Для кожного D/L0 , ми будуємо фазову дiаграму рiзних форм та ана-
лiзуємо причини здiйснення фазового переходу. Встановлено, що, хоча морфологiї вiдрiзняються при
рiзних D/L0 , причина фазового переходу при змiнi амплiтуди i частоти є приблизно однаковою, а са-
ме вона полягає в поєднаннi польового ефекту i ефекту просторового обмеження. Цi результати можуть
дати експериментаторам простий метод створення впорядкованих, бездефектних наноструктурованих
матерiалiв за рахунок поєднання цилiндричного обмеження та осциляцiйного зсувного потоку.
Ключовi слова: самоскупчення, блочний кополiмер, цилiндричне обмеження, осциляцiйний зсувний
потiк
33601-12
Introduction
Models and simulation methods
Numerical results and discussion
Conclusions
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