We present an approach for constructing thermodynamically consistent time-dependent models relevant to thin films of diblock copolymers in applied electric fields. The approach is based on the principles of linear irreversible thermodynamics, and, in this work, it is applied to study the effects of electric fields on thin films of incompressible diblock copolymers. Enforcement of local incompressibility constraint at all times leads to a local order parameter dependent transport coefficient in the model for the diblock copolymers. The dependence of the transport coefficient on the local order parameter is used to relate it with the diffusion constant of Rouse chains and leads to sensitivity of the model to initial conditions. In addition, transient behavior is found to be affected when compared with an ad hoc model assuming a constant transport coefficient. Numerical results such as electric field induced alignment of lamellae domains due to the field are found to be in qualitative agreement with experiments. This approach opens up a systematic way of developing kinetic models for simulating effects of electrolytes added to thin films containing diblock copolymers in the presence of applied electric fields.

1.
M.
Armand
and
J. M.
Tarascon
, “
Building better batteries
,”
Nature
451
,
652
657
(
2008
).
2.
T.
Ueki
and
M.
Watanabe
, “
Macromolecules in ionic liquids: Progress, challenges, and opportunities
,”
Macromolecules
41
(
11
),
3739
3749
(
2008
).
3.
D. T.
Hallinan
,Jr.
and
N. P.
Balsara
, “
Polymer electrolytes
,”
Annu. Rev. Mater. Res.
43
,
503
525
(
2013
).
4.
J. F.
Marko
and
Y.
Rabin
, “
Microphase separation of charged diblock copolymers: Melts and solutions
,”
Macromolecules
25
(
5
),
1503
1509
(
1992
).
5.
R.
Kumar
and
M.
Muthukumar
, “
Microphase separation in polyelectrolytic diblock copolymer melt: Weak segregation limit
,”
J. Chem. Phys.
126
(
21
),
214902
(
2007
).
6.
Z. G.
Wang
, “
Fluctuation in electrolyte solutions: The self energy
,”
Phys. Rev. E
81
(
2
),
021501
(
2010
).
7.
I.
Nakamura
,
N. P.
Balsara
, and
Z.-G.
Wang
, “
Thermodynamics of ion-containing polymer blends and block copolymers
,”
Phys. Rev. Lett.
107
(
19
),
198301
(
2011
).
8.
I.
Nakamura
,
A.-C.
Shi
, and
Z.-G.
Wang
, “
Ion solvation in liquid mixtures: Effects of solvent reorganization
,”
Phys. Rev. Lett.
109
(
25
),
257802
(
2012
).
9.
C. E.
Sing
,
J. W.
Zwanikken
, and
M.
Olvera de la Cruz
, “
Electrostatic control of block copolymer morphology
,”
Nat. Mater.
13
(
7
),
694
698
(
2014
).
10.
X.
Wang
,
M.
Goswami
,
R.
Kumar
,
B.
G. Sumpter
, and
J.
Mays
, “
Morphologies of block copolymers composed of charged and neutral blocks
,”
Soft Matter
8
(
11
),
3036
3052
(
2012
).
11.
A. A.
Teran
and
N. P.
Balsara
, “
Thermodynamics of block copolymers with and without salt
,”
J. Phys. Chemi. B
118
(
1
),
4
17
(
2014
).
12.
K.
Amundson
,
E.
Helfand
,
D. D.
Davis
,
X.
Quan
,
S. S.
Patel
, and
S. D.
Smith
, “
Effect of an electric field on block copolymer microstructure
,”
Macromolecules
24
(
24
),
6546
6548
(
1991
).
13.
Y.
Tsori
,
F.
Tournilhac
, and
L.
Leibler
, “
Orienting ion-containing block copolymers using Ac electric fields
,”
Macromolecules
36
(
15
),
5873
5877
(
2003
).
14.
C.-Y.
Lin
,
M.
Schick
, and
Andelman
D.
, “
Structural changes of diblock copolymer melts due to an external electric field: A self-consistent-field theory study
,”
Macromolecules
38
(
13
),
5766
5773
(
2005
).
15.
M. W.
Matsen
, “
Stability of a block-copolymer lamella in a strong electric field
,”
Phys. Rev. Lett.
95
(
25
),
258302
(
2005
).
16.
Y.
Tsori
and
L.
Leibler
, “
Phase-separation in ion-containing mixtures in electric fields
,”
Proc. Natl. Acad. Sci.
104
(
18
),
7348
7350
(
2007
).
17.
A.
Panday
,
S.
Mullin
,
E. D.
Gomez
,
N.
Wanakule
,
V. L.
Chen
,
A.
Hexemer
,
J.
Pople
, and
N. P.
Balsara
, “
Effect of molecular weight and salt concentration on conductivity of block copolymer electrolytes
,”
Macromolecules
42
(
13
),
4632
4637
(
2009
).
18.
R.
Kumar
,
V.
Bocharova
,
E.
Strelcov
,
A.
Tselev
,
I. I.
Kravchenko
,
S.
Berdzinski
,
V.
Strehmel
,
O. S.
Ovchinnikova
,
J. A.
Minutolo
,
J. R.
Sangoro
 et al, “
Ion transport and softening in a polymerized ionic liquid
,”
Nanoscale
7
(
3
),
947
955
(
2015
).
19.
J. W.
Dugger
,
W.
Li
,
M.
Chen
,
T. E.
Long
,
R. J. L.
Welbourn
,
M. W. A.
Skoda
,
J. F.
Browning
,
R.
Kumar
,
B. S.
Lokitz
, and
B. S.
Lokitz
, “
Nanoscale resolution of electric-field induced motion in ionic diblock copolymer thin films
,”
ACS Appl. Mater. Interfaces
10
(
38
),
32678
32687
(
2018
).
20.
M. D.
Galluzzo
,
W. S.
Loo
,
E.
Schaible
,
C.
Zhu
, and
N. P.
Balsara
, “
Dynamic structure and phase behavior of a block copolymer electrolyte under dc polarization
,”
ACS Appl. Mater. Interfaces
12
(
51
),
57421
57430
(
2020
).
21.
L.
Onsager
, “
Reciprocal relations in irreversible processes. I
,”
Phys. Rev.
37
(
4
),
405
(
1931
).
22.
L.
Onsager
, “
Reciprocal relations in irreversible processes. II
,”
Phys. Rev.
38
(
12
),
2265
(
1931
).
23.
S. R.
De Groot
and
P.
Mazur
,
Non-equilibrium Thermodynamics
(
Courier Corporation
,
2013
).
24.
M.
Doi
, “
Onsager’s variational principle in soft matter
,”
J. Phys.: Condens. Matter
23
(
28
),
284118
(
2011
).
25.
R.
Kumar
,
J. P.
Mahalik
,
V.
Bocharova
,
E. W.
Stacy
,
C.
Gainaru
,
T.
Saito
,
M. P.
Gobet
,
S.
Greenbaum
,
B. G.
Sumpter
, and
A. P.
Sokolov
, “
A Rayleighian approach for modeling kinetics of ionic transport in polymeric media
,”
J. Chem. Phys.
146
(
6
),
064902
(
2017
).
26.
J.
Cummings
,
J. S.
Lowengrub
,
B. G.
Sumpter
,
S. M.
Wise
, and
R.
Kumar
, “
Modeling solvent evaporation during thin film formation in phase separating polymer mixtures
,”
Soft Matter
14
,
1833
1846
(
2018
).
27.
T.
Ohta
and
K.
Kawasaki
, “
Equilibrium morphology of block copolymer melts
,”
Macromolecules
19
(
10
),
2621
2632
(
1986
).
28.
L.
Leibler
, “
Theory of microphase separation in block copolymers
,”
Macromolecules
13
(
6
),
1602
1617
(
1980
).
29.
K. S.
Lyakhova
,
A. V.
Zvelindovsky
, and
G. J. A.
Sevink
, “
Kinetic pathways of order-to-order phase transitions in block copolymer films under an electric field
,”
Macromolecules
39
(
8
),
3024
3037
(
2006
).
30.
M.
Pinna
and
AV
Zvelindovsky
, “
Large scale simulation of block copolymers with cell dynamics
,”
Eur. Phys. J. B
85
(
6
),
1
18
(
2012
).
31.
K.
Kawasaki
and
K.
Sekimoto
, “
Morphology dynamics of block copolymer systems
,”
Physica A
148
(
3
),
361
413
(
1988
).
32.
J. G. E. M.
Fraaije
,
B. A. C.
van Vlimmeren
,
N. M.
Maurits
,
M.
Postma
,
O. A.
Evers
,
C.
Hoffmann
,
P.
Altevogt
, and
G.
Goldbeck-Wood
, “
The dynamic mean-field density functional method and its application to the mesoscopic dynamics of quenched block copolymer melts
,”
J. Chem. Phys.
106
(
10
),
4260
4269
(
1997
).
33.
N. M.
Maurits
and
J. G. E. M.
Fraaije
, “
Mesoscopic dynamics of copolymer melts: From density dynamics to external potential dynamics using nonlocal kinetic coupling
,”
J. Chem. Phys.
107
(
15
),
5879
5889
(
1997
).
34.
J. G. E. M.
Fraaije
, “
Dynamic density functional theory for microphase separation kinetics of block copolymer melts
,”
J. Chem. Phys.
99
(
11
),
9202
9212
(
1993
).
35.
P. G.
de Gennes
, “
Dynamics of fluctuations and spinodal decomposition in polymer blends
,”
J. Chem. Phys.
72
(
9
),
4756
4763
(
1980
).
36.
K.
Binder
, “
Collective diffusion, nucleation, and spinodal decomposition in polymer mixtures
,”
J. Chem. Phys.
79
(
12
),
6387
6409
(
1983
).
37.
L.
Onsager
, “
Theories and problems of liquid diffusion
,”
Ann. N. Y. Acad. Sci.
46
(
5
),
241
265
(
1945
).
38.
S. R.
Ren
and
I. W.
Hamley
, “
Cell dynamics simulations of microphase separation in block copolymers
,”
Macromolecules
34
(
1
),
116
126
(
2001
).
39.
J. S. B.
Lev Davidovich Landau
,
M. J.
Kearsley
,
L. P.
Pitaevskii
,
E. M.
Lifshitz
, and
J. B.
Sykes
,
Electrodynamics of Continuous Media
(
Elsevier
,
2013
), Vol. 8.
40.
R.
Kumar
,
W.
Li
,
B. G.
Sumpter
, and
M.
Muthukumar
, “
Understanding the effects of dipolar interactions on the thermodynamics of diblock copolymer melts
,”
J. Chem. Phys.
151
(
5
),
054902
(
2019
).
41.
H.
Tang
, “
Rouse dynamics of block copolymers
,”
Macromolecules
29
(
7
),
2633
2640
(
1996
).
42.
G. H.
Fredrickson
and
H.
Orland
, “
Dynamics of polymers: A mean-field theory
,”
J. Chem. Phys.
140
(
8
),
084902
(
2014
).
43.
M. R.
Tonks
,
D.
Gaston
,
P. C.
Millett
,
D.
Andrs
, and
P.
Talbot
, “
An object-oriented finite element framework for multiphysics phase field simulations
,”
Comput. Mater. Sci.
51
(
1
),
20
29
(
2012
).
44.
H.
Kim
,
M. M. L.
Arras
,
J. P.
Mahalik
,
W.
Wang
,
D. M.
Yu
,
S.
Chernyy
,
M.
Goswami
,
R.
Kumar
,
B. G.
Sumpter
,
K.
Hong
,
G. S.
Smith
, and
T. P.
Russell
, “
Studies on the 3-lamellar morphology of miktoarm terpolymers
,”
Macromolecules
51
(
19
),
7491
7499
(
2018
).
45.
I. W.
Hamley
, “
Cell dynamics simulations of block copolymers
,”
Macromol. Theory Simul.
9
(
7
),
363
380
(
2000
).
46.
T. L.
Morkved
,
M.
Lu
,
A. M.
Urbas
,
E. E.
Ehrichs
,
H. M.
Jaeger
,
P.
Mansky
, and
T. P.
Russell
, “
Local control of microdomain orientation in diblock copolymer thin films with electric fields
,”
Science
273
(
5277
),
931
933
(
1996
).
47.
T.
Thurn-Albrecht
,
J.
DeRouchey
,
T. P.
Russell
, and
R.
Kolb.
, “
Pathways toward electric field induced alignment of block copolymers
,”
Macromolecules
35
(
21
),
8106
8110
(
2002
).
48.
Y.
Tsori
and
U.
Steiner
,
Polymers, Liquids and Colloids in Electric Fields Interfacial Instabilities, Orientation and Phase Transitions
, (
World Scientific
,
2008
).
49.
Y.
Tsori
, “
Colloquium: Phase transitions in polymers and liquids in electric fields
,”
Rev. Mod. Phys.
81
,
1471
1494
(
2009
).
50.
A.
Böker
,
H.
Elbs
,
H.
Hänsel
,
A.
Knoll
,
S.
Ludwigs
,
H.
Zettl
,
V.
Urban
,
V.
Abetz
,
A. H. E.
Müller
, and
G.
Krausch
, “
Microscopic mechanisms of electric-field-induced alignment of block copolymer microdomains
,”
Phys. Rev. Lett.
89
,
135502
(
2002
).
51.
A.
Böker
,
A.
Knoll
,
H.
Elbs
,
V.
Abetz
,
A. H. E.
Müller
,
G.
Krausch
, and
G.
Krausch
, “
Large scale domain alignment of a block copolymer from solution using electric fields
,”
Macromolecules
35
(
4
),
1319
1325
(
2002
).
52.
A.
Böker
,
H.
Elbs
,
H.
Hänsel
,
A.
Knoll
,
S.
Ludwigs
,
H.
Zettl
,
A. V.
Zvelindovsky
,
G. J. A.
Sevink
,
V.
Urban
,
V.
Abetz
,
A. H. E.
Müller
, and
G.
Krausch
, “
Electric field induced alignment of concentrated block copolymer solutions
,”
Macromolecules
36
(
21
),
8078
8087
(
2003
).
53.
H.
Chen
and
I.
Nakamura
, “
Effects of the dielectric response of single-component liquids and liquid mixtures on electrochemical properties between charged plates
,”
J. Phys. Chem. C
119
(
44
),
24714
24723
(
2015
).
54.
B.
Zheng
,
X.
Man
,
Z.-C.
Ou-Yang
,
M.
Schick
, and
D.
Andelman
, “
Orienting thin films of lamellar block copolymer: The combined effect of mobile ions and electric field
,”
Macromolecules
51
(
19
),
7881
7892
(
2018
).
55.
P. J.
Scott
,
G. A.
Spiering
,
Y.
Wang
,
Z. D.
Seibers
,
R. B.
Moore
,
R.
Kumar
,
B. S.
Lokitz
, and
T. E.
Long
, “
Phosphonium-based polyzwitterions: Influence of ionic structure and association on mechanical properties
,”
Macromolecules
53
(
24
),
11009
11018
(
2020
).
56.
S.
Ham
,
C.
Shin
,
E.
Kim
,
D. Y.
Ryu
,
U.
Jeong
,
T. P.
Russell
, and
C. J.
Hawker
, “
Microdomain orientation of PS-b-PMMA by controlled interfacial interactions
,”
Macromolecules
41
(
17
),
6431
6437
(
2008
).
57.
X.
Man
,
J.
Tang
,
P.
Zhou
,
D.
Yan
, and
D.
Andelman
, “
Lamellar diblock copolymers on rough substrates: Self-consistent field theory studies
,”
Macromolecules
48
(
20
),
7689
7697
(
2015
).
58.
W.
Li
,
J.-M. Y.
Carrillo
,
J.
Katsaras
,
B. G.
Sumpter
,
R.
Ashkar
, and
R.
Kumar
, “
The influence of curvature on domain distribution in binary mixture membranes
,”
Soft Matter
15
(
33
),
6642
6649
(
2019
).

Supplementary Material

You do not currently have access to this content.